Damping force control device and method

[0002] The disclosure of Japanese Patent Application No. HEI 11-12865 filed on Apr. 20, 1999 including the specification, drawings and abstract is incorporated herein by reference in its entirety.

[0003] 1. Field of Invention

[0004] The invention relates to a damping force control device and a damping force control method for controlling damping forces of dampers disposed between a vehicle body and respective wheels of a vehicle.

[0005] 2. Description of Related Art

[0006] As a first related art of this kind, there is known a damping force control device which determines an actual damping coefficient based on the skyhook theory, that is, in accordance with a ratio between a vertical speed of the vehicle body and a vertical speed of the vehicle body relative to a wheel (e.g. Japanese Patent Application Laid-Open No. HEI 5-294122). This device is designed to decompose a vertical speed of the vehicle body at the location of each wheel into a roll movement speed, a pitch movement speed, a heave movement speed and a warp movement speed of the vehicle body. The pitch movement speed is multiplied by a pitch gain which changes in accordance with a differential value of a longitudinal acceleration. The roll movement speed is multiplied by a roll gain which changes in accordance with a differential value of a lateral acceleration. The heave and warp movement speeds are multiplied by constant gains respectively. The roll movement speed, the pitch movement speed, the heave movement speed and the warp movement speed that have thus been gain-adjusted are re-synthesized into a vertical speed of the vehicle body at the location of each of the wheels. The actual damping coefficient is determined according to a ratio between the re-synthesized vertical speed of the vehicle body at the location of each of the wheels and a speed of the vehicle body at the location of the wheel relative to the wheel. Thus the vehicle body at the location of each of the wheels is inhibited from vibrating vertically, and is effectively inhibited from making pitch or roll movements.

[0007] According to a second related art, an amount of vertical displacement of the vehicle body at the location of each of the wheels relative to the wheel is detected. Based on various equations of motion which consider pitch and roll movements of the vehicle body and the like, a vertical speed of the vehicle body at the location of each of the wheels is calculated by means of the aforementioned amount of relative displacement. By differentiating the aforementioned amount of relative displacement, a vertical speed of the vehicle body at the location of each of the wheels relative to the wheel is calculated. The actual damping coefficient is determined according to a ratio between the vertical speed and the relative speed for each of the wheels. In short, there is also known a damping force control device for controlling damping forces of the dampers based on the skyhook theory, by merely detecting the amount of relative displacement as mentioned above (Japanese Patent Application Laid-Open No. HEI 6-344743).

[0008] However, according to the aforementioned first and second related art, the algorithm for performing control to inhibit vertical vibrations of the vehicle body, namely, the algorithm based on the skyhook theory is directly corrected in accordance with pitch and roll movements of the vehicle body. Although the effect of inhibiting the vehicle body from vibrating in the pitch and roll directions can be accomplished through such correction, the basic performance of the damping force control for inhibiting vertical vibrations of the vehicle body is adversely affected.

[0009] The invention has been made with the aim of solving the above-stated problem. It is an object of the invention to provide a damping force control device and a damping force control method which, while ensuring the control performance for inhibiting vertical vibrations of a vehicle body, also bring about the effect of inhibiting the vehicle body from making pitch and/or roll movements.

[0010] In order to solve the aforementioned problem, a damping force control device according to a first aspect of the invention is provided with a controller that calculates, for each of the wheels, a first target damping force that inhibits vibrations of the vehicle body in a heave direction based on a single wheel model of the vehicle; calculates, for each of the wheels, a second target damping force that inhibits vibrations of the vehicle body in a pitch direction based on a model of front and rear wheels of the vehicle; determines an ultimate target damping force for each of the wheels based on the calculated first and second target damping forces; outputs a control signal corresponding to the determined ultimate target damping force to each of the dampers; and controls each damper such that a damping force exerted by each of the dampers is set to the determined ultimate target damping force.

[0011] In this case, for example, the controller calculates the first target damping force in accordance with a kinetic state quantity of the vehicle body in the lateral direction and the second target damping force in accordance with a kinetic state quantity of the vehicle body in pitch direction.

[0012] According to the first aspect of the invention, the controller calculates the first and second target damping forces, respectively. Through operations of the controller, control is performed such that a damping force exerted by each of the dampers is set to the determined ultimate target damping force for each of the wheels. In this case, the first target damping force is calculated to inhibit vibrations of the vehicle body in the heave direction, based on the single wheel model of the vehicle. The second target damping force is calculated, independently of the first target damping force, to inhibit vibrations of the vehicle body in the pitch direction, based on the model of front and rear wheels of the vehicle. Therefore, while the control performance intrinsic in the first target damping force that inhibits vertical vibrations of the vehicle body is ensured, a deficiency in damping force for pitch movements of the vehicle body is compensated for. Thus the vehicle body is effectively inhibited from vibrating vertically and also from making pitch movements. Consequently the vehicle achieves good riding comfort and high running stability.

[0013] In the aforementioned first aspect of the invention, the controller may be designed to determine the ultimate target damping force by selecting the greater one of the calculated first and second target damping forces for each of the wheels, and to output the selected control signal to each of the dampers and control each damper such that the damping force exerted by each of the dampers is set to the selected target damping force.

[0014] According to this aspect of the invention, control is performed such that the damping force of the damper for each of the wheels is set to the greater one of the first and second target damping forces. Thus, only if the magnitude of vibrations of the vehicle body in the pitch direction has increased to some extent, on the condition that the first target damping force is smaller than the second target damping force, the damping force of the damper at the location of each of the wheels is set to the second target damping force to inhibit vibrations of the vehicle body in the pitch direction, based on the model of front and rear wheels of the vehicle. Otherwise, the damping force of the damper at the location of each of the wheels is set to the first target damping force to inhibit vibrations of the vehicle body in the heave direction, based on the single wheel model of the vehicle. Accordingly, according to this aspect, while the control performance for inhibiting vertical vibrations of the vehicle body, which is intrinsic in the first target damping force calculated by the controller, is ensured more suitably, a deficiency in damping force for pitch movements of the vehicle body is compensated for. Thus the vehicle body is more effectively inhibited from vibrating vertically and also from making pitch movements. Consequently the vehicle achieves better riding comfort and higher running stability.

[0015] In a second aspect of the invention, the controller calculates, for each of the wheels, a second target damping force that inhibits vibrations of the vehicle body in a roll direction, based on a model of left and right wheels of the vehicle. In this case, for example, the controller calculates the second target damping force in accordance with a kinetic state quantity of the vehicle body in the roll direction.

[0016] In the second aspect of the invention, the second target damping force that inhibits vibrations of the vehicle body in the pitch direction in the first aspect of the invention is replaced by the second target damping force that inhibits vibrations of the vehicle body in the roll direction. Therefore, while the control performance for inhibiting vertical vibrations of the vehicle body, which is intrinsic in the first target damping force, is ensured, a deficiency in damping force for roll movements of the vehicle body is compensated for. Thus the vehicle body is effectively inhibited from vibrating vertically and also from making roll movements. Consequently the vehicle achieves good riding comfort and high running stability.

[0017] Furthermore, in a third aspect of the invention, the controller calculates a third target damping force. The controller determines an ultimate target damping force for each of the wheels, based on the calculated first, second and third target damping forces. Also in this case, the controller selects the greatest one of the calculated first, second and third target damping forces for each of the wheels, and outputs a control signal corresponding to the selected target damping force to each of the dampers and controls each damper such that the damping force exerted by each of the dampers is set to the selected target damping force.

[0018] In the third aspect of the invention, instead of the first and second target damping forces in the first and second aspects of the invention, the damping force exerted by the damper is so controlled as to be set to a target damping force that has been determined based on the independently calculated first through third target damping forces, for example, to the greatest one of the first, second and third target damping forces. In this case, the first target damping force is calculated to inhibit vibrations of the vehicle body in the heave direction, based on the single wheel model of the vehicle. The second target damping force is calculated to inhibit vibrations of the vehicle body in the pitch direction, based on the model of front and rear wheels of the vehicle. The third target damping force is calculated to inhibit vibrations of the vehicle body in the roll direction, based on the model of left and right wheels of the vehicle. These first through third target damping forces are calculated independently of one another. Hence, according to the third aspect of the invention, while the control performance for inhibiting vertical vibrations of the vehicle body, which is intrinsic in the first target damping force calculated by the controller, is ensured, deficiencies in damping force for pitch and roll movements of the vehicle body are compensated for. Therefore the vehicle body is effectively inhibited from vibrating vertically and also from making pitch or roll movements. Consequently the vehicle achieves good riding comfort and high running stability.

[0019] According to a damping force control method of a fourth aspect of the invention, a first target damping force that inhibits vibrations of the vehicle body in a heave direction is calculated for each of the wheels based on a single wheel model of the vehicle, and a second target damping force that inhibits vibrations of the vehicle body in a pitch direction is calculated for each of the wheels based on a model of front and rear wheels of the vehicle. An ultimate target damping force is then determined for each of the wheels, based on the calculated first and second target damping forces. A control signal corresponding to the determined ultimate target damping force is then outputted to each of the dampers, and each damper is controlled such that a damping force exerted by each of the dampers is set to the determined ultimate target damping force.

[0020] In a fifth aspect of the invention, instead of calculating the second target damping force that inhibits vibrations of the vehicle body in the pitch direction as in the damping force control method of the fourth aspect of the invention, a second target damping force that inhibits vibrations of the vehicle body in the roll direction is calculated for each of the wheels. An ultimate target damping force is determined for each of the wheels, based on the aforementioned first and second target damping forces calculated for each of the wheels.

[0021] Furthermore, according to a damping force control method of a sixth aspect of the invention, a first target damping force that inhibits vibrations of the vehicle body in a heave direction, a second target damping force that inhibits vibrations of the vehicle body in a pitch direction, and a third target damping force that inhibits vibrations of the vehicle body in a roll direction are calculated for each of the wheels. An ultimate target damping force is then determined for each of the wheels, based on the calculated first, second and third target damping forces.

[0022] Definitions of vibrations of the vehicle body in the heave, pitch and roll directions will be given hereinafter. It is assumed herein that X-axis represents the longitudinal direction of the vehicle body, that Y-axis represents the lateral direction of the vehicle body, and that Z-axis represents the vertical direction of the vehicle body. The vibrations of the vehicle body in the heave direction refer to proper vibrations of the vehicle body moving parallel to the Z-axis. The vibrations of the vehicle body in the pitch direction refer to proper vibrations of the vehicle body rotating around the Y-axis. The vibrations of the vehicle body in the roll direction refer to proper vibrations of the vehicle body rotating around the X-axis.

[0023] The foregoing and further objects, features and advantages of the invention will become apparent from the following description of preferred embodiments with reference to the accompanying drawings, wherein:

[0024] FIG. 1 is a schematic block diagram of a damping force control device for a vehicle;

[0025] FIG. 2 is a flowchart of a program executed by an electric control device shown in FIG. 1;

[0026] FIG. 3 is a detailed flowchart of a first damping force calculating routine shown in FIG. 2;

[0027] FIG. 4 is a model view of a single wheel of the vehicle;

[0028] FIG. 5 is a detailed flowchart of a second damping force calculating routine shown in FIG. 2;

[0029] FIG. 6 is a model view of front and rear wheels of the vehicle;

[0030] FIG. 7 is a detailed flowchart of a third damping force calculating routine shown FIG. 2;

[0031] FIG. 8 is a model view of left and right wheels of the vehicle;

[0032] FIG. 9 is a graph showing a data property in a table of relative speed against damping force;

[0033] FIG. 10 is a model view of another single wheel of the vehicle and relates to modifications of the invention;

[0034] FIG. 11 is a block diagram of a generalized model of a non-linear H∞ state feedback control system and relates to a first modification for calculating a first target damping force;

[0035] FIG. 12A is a graph showing a frequency weight for a sprung mass acceleration as an evaluated output;

[0036] FIG. 12B is a graph showing a frequency weight for a sprung mass speed as an evaluated output;

[0037] FIG. 12C is a graph showing a frequency weight for a speed of a sprung member relative to an unsprung member as an evaluated output;

[0038] FIG. 12D is a graph showing a frequency weight for a non-linear damping coefficient as a control input;

[0039] FIG. 13 is an image view showing operation and effect of the control based on the non-linear H∞ control theory;

[0040] FIG. 14A is a Lissajou's waveform diagram showing a characteristic of damping force against relative speed (F-V) in the damping force control according to the modification for calculating the first target damping force;

[0041] FIG. 14B is a Lissajou's waveform diagram showing a characteristic of damping force against relative speed (F-V) in the previously employed skyhook control;

[0042] FIG. 15 is a flowchart of a first damping force calculating routine according to the first modification for calculating the first target damping force;

[0043] FIG. 16 is a block diagram of a generalized model of a non-linear H∞ output feedback control system and relates to a second modification for calculating the first target damping force;

[0044] FIG. 17 is a flowchart of a first damping force calculating routine according to second and third modifications for calculating the first target damping force; and

[0045] FIG. 18 is a block diagram of a generalized model of a non-linear H∞ control system based on a Kalman filter and relates to the third modification for calculating the first target damping force.

[0046] Embodiments of the invention will be described hereinafter with reference to the drawings. FIG. 1 is a schematic block diagram showing a damping force control device for a vehicle according to the embodiments.

[0047] At locations of a front-left wheel FW1, a front-right wheel FW2, a rear-left wheel RW1 and a rear-right wheel RW2 of this vehicle, suspension systems 10A, 10B, 10C and 10D are disposed between a vehicle body BD (shown in FIGS. 4, 6 and 8) and the wheels FW1, FW2, RW1 and RW2 respectively. These suspension systems 10A, 10B, 10C and 10D, which are disposed between the vehicle body BD and the wheels FW1, FW2, RW1 and RW2 respectively, are provided with springs 11a, 11b, 11cand 11dand dampers 12a, 12b, 12cand 12drespectively. The springs 11a, 11b, 11cand 11delastically support the vehicle body BD against the wheels FW1, FW2, RW1 and RW2 respectively. The dampers 12a, 12b, 12cand 12dare disposed in parallel with the springs 11a, 11b, 11cand 11drespectively and damp vibrations of the vehicle body BD with respect to the wheels FW1, FW2, RW1 and RW2 respectively. Each pair of the springs and dampers 11aand 12a, 11band 12b, 11cand 12c, and 11dand 12dconstitutes a shock absorber. The dampers 12a, 12b, 12cand 12dare designed to be able to control a damping coefficient by changing an orifice opening degree. This vehicle is also provided with an electric control device 20 for variably setting damping coefficients (damping forces) of the respective dampers 12a, 12b, 12cand 12d. The electric control device 20 is composed of a microcomputer, its marginal circuits and the like, and repeatedly executes a program shown in FIG. 2 (including sub-routines in FIGS. 3, 5 and 7) at intervals of a predetermined period of time by means of a built-in timer, thus controlling orifice opening degrees of the respective dampers 12a, 12b, 12cand 12d. Sprung mass acceleration sensors 21athrough 21d, relative displacement amount sensors 22athrough 22d, a pitch angular velocity sensor 23 and a roll angular velocity sensor 24 are connected to the electric control device 20.

[0048] The sprung mass acceleration sensors 21athrough 21d, which are mounted to the vehicle body BD (the sprung member) at the locations of the wheels FW1, FW2, RW1 and RW2 respectively, detect vertical accelerations X_pb1″, X_pb2″, X_pb3″ and X_pb4″ of the vehicle body BD relative to absolute space at the locations of FW2, RW1 and RW2 respectively, and output detection signals indicative of the vertical accelerations X_pb1″, X_pb2″, X_pb3″ and X_pb4″ respectively. The vertical accelerations X_pb1″, X_pb2″, X_pb3″ and X_pb4″ assume positive values when the vehicle body BD is accelerated upwards and assume negative values when the vehicle body BD is accelerated downwards. The relative displacement amount sensors 22athrough 22d, which are mounted between the vehicle body BD (the sprung member) at the locations of the wheels FW1, FW2, RW1 and RW2 and the wheels FW1, FW2, RW1 and RW2 (unsprung members) respectively, detect amounts of vertical displacement (X_pw1−X_pb1), (X_pw2−X_pb2), (X_pw3−X_pb3) and (X_pw4−X_pb4) of the vehicle body BD at the locations of the wheels FW1, FW2, RW1 and RW2 relative to the wheels FW1, FW2, RW1 and RW2 respectively, and output detection signals indicative of the amounts of displacement (X_pw1−X_pb1), (X_pw2−X_pb2), (X_pw3−X_pb3) and (X_pw4−X_pb4) respectively. The amounts of relative displacement (X_pw1−X_pb1), (X_pw2−X_pb2), (X_pw3−X_pb3) and (X_pw4−X_pb4), which represent amounts of displacement with respect to a predetermined reference amount of displacement, assume positive values if they proceed in a decreasing direction (in which the dampers contract) and assume negative values if they proceed in an increasing direction (in which the dampers extend).

[0049] The pitch angular velocity sensor 23, which is composed of a rate sensor installed in the vicinity of the center of gravity of the vehicle body BD, detects a pitch angular velocity P_a′ of the vehicle body BD and outputs a detection signal indicative of the pitch angular velocity P_a′. The roll angular velocity sensor 24, which is composed of a rate sensor installed in the vicinity of the center of gravity of the vehicle body BD, detects a roll angular velocity R_a′ of the vehicle body BD and outputs a detection signal indicative of the roll angular velocity R_a′.

[0050] Next, it will be described how the thus-constructed vehicular damping force control device operates. After an ignition switch (not shown) has been turned on, the electric control device 20 starts repeatedly executing the program shown in FIG. 2 at intervals of a predetermined period of time. The execution of this program is started in step 100, and the first through third damping force calculating routines are executed in steps 102, 104 and 106 respectively.

[0051] The first damping force calculating routine, which is shown in detail in FIG. 3, is intended to calculate a first target damping force exerted by each of the dampers 12a, 12b, 12cand 12dfor inhibiting (vertical) vibrations of the vehicle body BD in the heave direction, based on a model of a single wheel of the vehicle.

[0052] Explanation of the first damping force calculating routine will be preceded by the description of how to calculate a first target damping force. FIG. 4 shows a model of a single wheel of the vehicle with one degree of freedom. Referring to FIG. 4, M_bdenotes a mass of the vehicle body BD. X_pbdenotes an amount of vertical displacement of the vehicle body BD (the sprung member) relative to a reference position in absolute space. X_pwdenotes an amount of vertical displacement of the wheel W (the unsprung member) relative to a reference position in absolute space. These amounts X_pband X_pwof displacement assume positive values when the vehicle body BD and the wheel W move upwards respectively. Ks denotes a spring constant of a spring 11, which comprehensively represents the springs 11a, 11b, 11cand 11d. Cs denotes a damping coefficient of a damper 12, which comprehensively represents the dampers 12a, 12b, 12cand 12d. Csk denotes a damping coefficient of a virtual damper 12A that has been sky-hooked according to the skyhook theory and is a preliminarily determined constant of an appropriate value.

[0053] First of all, a virtual model employing the virtual damper 12A instead of the damper 12 will be considered. Given that X_pb″ denotes a vertical acceleration of the vehicle body BD and that X_pb′ denotes a vertical speed of the vehicle body BD, an equation of motion of the vehicle body BD in the heave direction of the virtual model is expressed by a formula (1) shown below.

[0054] MbX_{pb″ =Ks (Xpx}−X_{pb)−CskXpb′}  (1)

[0055] Considering an actual model employing the existing damper 12, an equation of vertical motion of the actual model is expressed by a formula (2) shown below.

[0056] MX_{pb″ =Ks (Xp}−X_{pb) +Cs (Xpw}−X_pb))  (2)

[0057] A formula (3) shown below is derived from the aforementioned formulas (1) and (2). In accordance with a relation defined by the formula (3), an ideal damping force Fd applied to the vehicle body BD by the damper 12 (or the damper 12A) based on the skyhook theory is expressed by a formula (4) shown below.

[0058] Cs (X_pw′ - X_{pb′) +CskXpb′ =}  (3)

[0059] Fd=Cs (X_pw′ - X_pb′)=-CskX_pb  (4)

[0060] Next the first damping force calculating routine will be described. The execution of the first damping force calculating routine is started in step 150 in FIG. 3. In step 152, vertical accelerations X_{pb I″, Xpb2}″, X_pb3″ and X_pb4″ of the vehicle body BD at the locations of the wheels FW1, FW2, RW1 and RW2 are inputted from the sprung mass acceleration sensors 21athrough 21drespectively. Next in step 154, the inputted vertical accelerations X_pb1″, X_pb2″, X_pb3″ and X_pb4 are integrated, whereby vertical speeds X_pb1′, X_pb2′, X_pb3′, and X_pb4′ of the vehicle body BD at the locations of the wheels FW1, FW2, RW1 and RW2 are calculated respectively.

[0061] Next in step 156, the calculated vertical speeds X_pb1′, X_pb2′, X_pb3′ and X_pb4′ are multiplied by the skyhook coefficient C_skthat has appropriately been determined in advance through execution of a calculation expressed by a formula (5) shown below, whereby first target damping forces Fd1, Fd2, Fd3 and Fd4 exerted by the dampers 12a, 12b, 12cand 12drespectively are calculated.

[0062] Fdi=−C_skX_pbi′  (5)

[0063] In the aforementioned formula (5), “i” denotes a positive integer from 1 to 4. In this example, although the skyhook coefficient C_skis commonly used for the wheels FW1, FW2, RW1 and RW2, the front wheels FW1 and FW2 may employ a skyhook coefficient different from that of the rear wheels RW1 and RW2. In step 158, the execution of the first damping force calculating routine is terminated.

[0064] The second damping force calculating routine, which is shown in FIG. 5 in detail, is intended to calculate a second target damping force exerted by each of the dampers 12a, 12b, 12cand 12dfor inhibiting vibrations of the vehicle body BD in the pitch direction, based on a model of front and rear wheels of the vehicle.

[0065] Explanation of the second damping force calculating routine will be preceded by the description of how to calculate a second target damping force. FIG. 6 shows a model of front and rear wheels of the vehicle with two degrees of freedom. Referring to FIG. 6, X_bfdenotes an amount of vertical displacement of the vehicle body BD (the sprung member) at the locations of the front wheels FW1 and FW2 relative to a reference position. X_brdenotes an amount of vertical displacement of the vehicle body BD (the sprung member) at the locations of the rear wheels RW1 and RW2 relative to a reference position. X_wfdenotes an amount of vertical displacement of the front wheels FW1 and FW2 (the unsprung members) relative to a reference position. X_wrdenotes an amount of vertical displacement of the rear wheels RW1 and RW2 (the unsprung members) relative to a reference position. These amounts X_bf, X_wf, X_brand X_wrof displacement assume positive values when the vehicle body BD, the front wheels FW1 and FW2 and the rear wheels RW1 and RW2 move upwards respectively. K_sfdenotes a spring constant of a spring 11f, which comprehensively represents the springs 11aand 11bat the locations of the front wheels FW1 and FW2. K_srdenotes a spring constant of a spring 11r, which comprehensively represents the springs 11cand 11dat the locations of the rear wheels RW1 and RW2. C_sfdenotes a damping coefficient of a damper 12f, which comprehensively represents the dampers 12aand 12bat the locations of the front wheels FW1 and FW2. C_srdenotes a damping coefficient of a damper 12r, which comprehensively represents the dampers 12cand 12dat the locations of the rear wheels RW1 and RW2. Cp denotes a damping coefficient of a virtual damper 12pfor damping vibrations of the vehicle body BD in the pitch direction and is a preliminarily determined constant of an appropriate value.

[0066] First of all, a virtual model employing the virtual damper 12pinstead of the actual dampers 12fand 12rwill be considered. Given that X_pb″ denotes a vertical acceleration of the vehicle body BD and that P_a″ and P_a′ denote an angular acceleration and an angular velocity of the vehicle body BD in the pitch direction respectively, equations of motion of the vehicle body BD in the heave and pitch directions of the virtual model are expressed respectively by formulas (6) and (7) shown below. The angular acceleration P_a″ and the angular velocity P_a′ assume positive values in a rotational direction in which the vehicle body BD rises on the side of the front wheels FW1 and FW2 and falls on the side of the rear wheels RW1 and RW2.

[0067] M_bX_pb″=K_sf(X_wf−X_bf)+K_sr(X_wr−X_br)  (6)

[0068] I_pP_a″=−L_fK_sf(X_wf−X_bf)+L_rK_sr(X_wr−X_br)−C_pP_a′  (7)

[0069] In the aforementioned formula (7), I_pdenotes a moment of inertia of the vehicle body BD in the pitch direction, L_fa distance between a front axle and the center of gravity, and L_ra distance between a rear axle and the center of gravity.

[0070] Considering an actual model employing the existing dampers 12fand 12r, equations of motion of the actual model in the heave and pitch directions are expressed respectively by formulas (8) and (9) shown below.

[0071] M_bX_pb″=K_sf(X_wf−X_bf)+C_sf(X_wf′−X_bf′)+K_sr(X_wr−X_br)+C_sr(X_wr′−X_br′)  (8)

[0072] I_pP_a″=−L_f{K_sf(X_wf−X_bf)+C_sf(X_wf′−X_bf′)}+L_r{K_sr(X_wr−X_br)+C_sr(X_wr′−X_br′)}  (9)

[0073] Formulas (10) and (11) shown below are derived from the aforementioned formulas (6) through (9).

[0074] C_sf(X_wf′−X_bf′)+C_sr(X_wr′−X_br′)=0  (10)

[0075] L_fC_sf(X_wf′−X_bf′)−L_rC_sr(X_wr′−X_br′)+C_pP_a′=0  (11)

[0076] In accordance with a relation defined by these formulas (10) and (11), ideal damping forces PF_fand PF_rapplied to the vehicle body BD by the damper 12ffor the front wheels and by the damper 12rfor the rear wheels (or the damper 12p) for the purpose of damping vibrations of the vehicle body BD in the pitch direction are expressed respectively by formulas (12) and (13) shown below.

[0077] PF_f=C_sf(X_wf′−X_bf′)=C_pP_a′/(L_f+L_r)  (12)

[0078] PF_r=C_sr(X_wr′−X_br′)=−C_pP_a′/(L_f+L_r)  (13)

[0079] Next the second damping force calculating routine will be described. The execution of the second damping force calculating routine is started in step 160 in FIG. 5. In step 162, a pitch angular velocity P_a′ of the vehicle body BD is inputted from the pitch angular velocity sensor 23. Next in step 164, a target damping force PF_fof the damper 12ffor the front wheels and a target damping force PF_rof the damper 12rfor the rear wheels are calculated with a view to inhibiting vibrations of the vehicle body BD in the pitch direction, by making calculations according to after-mentioned formulas (14) and (15) respectively, wherein a damping coefficient C_pwhich is ideal for inhibiting vibrations of the vehicle body BD in the pitch direction and has appropriately been determined in advance, the inputted pitch angular velocity P_a′, the predetermined distance L_fbetween the front axle and the center of gravity, and the predetermined distance L_rbetween the rear axle and the center of gravity are used.

[0080] PF_f=C_pP_a′/(L_f+L_r)  (14)

[0081] PF_r=−C_pP_a′/(L_f+L_r)  (15)

[0082] Next in step 166, the calculated target damping force PF_fof the damper 12ffor the front wheels is set as second target damping forces PFd1 and PFd2 for the front-left and front-right wheels FW1 and FW2 respectively, and the calculated target damping force PF_rfor the damper 12rfor the rear wheels is set as second target damping forces PFd3 and PFd4 for the rear-right and rear-left wheels RW1 and RW2 respectively. In step 168, the execution of the second damping force calculating routine is terminated.

[0083] The third damping force calculating routine, which is shown in FIG. 7 in detail, is intended to calculate a third target damping force exerted by each of the dampers 12a, 12b, 12cand 12dfor inhibiting vibrations of the vehicle body BD in the roll direction, based on a model of left and right wheels of the vehicle.

[0084] Explanation of the third damping force calculating routine will be preceded by the description of how to calculate a third target damping force. FIG. 8 shows a model of left and right wheels of the vehicle with two degrees of freedom. Referring to FIG. 8, X_bmdenotes an amount of vertical displacement of the vehicle body BD (the sprung member) at the locations of the right wheels FW2 and RW2 relative to a reference position. X_bhdenotes an amount of vertical displacement of the vehicle body BD (the sprung member) at the locations of the left wheels FW1 and RW1 relative to a reference position. X_wmdenotes an amount of vertical displacement of the right wheels FW2 and RW2 (the unsprung members) relative to a reference position. X_whdenotes an amount of vertical displacement of the left wheels FW1 and RW1 (the unsprung members) relative to a reference position. These amounts X_bm, X_wm, X_bhand X_whof displacement assume positive values when the vehicle body BD, the right wheels FW2 and RW2 and the left wheels FW1 and RW1 move upwards respectively. K_sdenotes a spring constant of a spring 11m, which comprehensively represents the springs 11band 11dat the locations of the right wheels FW2 and RW2, and also denotes a spring constant of a spring 11h, which comprehensively represents the springs 11aand 11cat the locations of the left wheels FW1 and RW1. C_sdenotes a damping coefficient of a damper 12m, which comprehensively represents the dampers 12band 12dat the locations of the right wheels FW2 and RW2, and also denotes a damping coefficient of a damper 12h, which comprehensively represents the dampers 12aand 12cat the locations of the left wheels FW1 and RW1. C_rdenotes a damping coefficient of a virtual damper 12rfor damping vibrations of the vehicle body BD in the roll direction and is a preliminarily determined constant of an appropriate value.

[0085] First of all, a virtual model employing the virtual damper 12rinstead of the dampers 12mand 12hwill be considered. Given that X_pb″ denotes a vertical acceleration of the vehicle body BD and that R_a″ and R_a′ denote an angular acceleration and an angular velocity of the vehicle body BD in the roll direction respectively, equations of motion of the vehicle body BD in the heave and roll directions of the virtual model are expressed respectively by formulas (16) and (17) shown below. The angular acceleration R_a″ and the angular velocity R_a′ assume positive values in a rotational direction in which the vehicle body BD rises on the side of the left wheels FW1 and RW1 and falls on the side of the right wheels FW2 and RW2.

[0086] M_bX_pb″=K_s(X_wm−X_bm)+K_s(X_wh−X_bh)  (16)

[0087] I_rR_a″=(K_s+K)(X_wh−X_bh−X_wm+X_bm)T/2−C_rR_a′  (17)

[0088] In the aforementioned formula (17), I_rdenotes a moment of inertia of the vehicle body BD in the roll direction, K a spring constant of the stabilizer, and T a tread of the vehicle.

[0089] Considering an actual model employing the existing dampers 12mand 12h, equations of motion of the actual model in the heave and roll directions are expressed respectively by formulas (18) and (19) shown below.

[0090] M_bX_pb″=K_s(X_wm−X_bm)+C_s(X_wm′−X_bm′)+K_s(X_wh−X_bh)+C_s(X_wh′−X_bh′)  (18)

[0091] I_rR_a″=((K_s+K)(X_wh−X_bh−X_wm+Xbm)+C_s(X_wh′−X_bh′−X_wm′ +X_bm′))T/2  (19)

[0092] Formulas (20) and (21) shown below are derived from the aforementioned formulas (16) through (19).

[0093] C_s(X_wh′−X_bh′+X_wm′−X_bm′)=0  (20)

[0094] C_s(X_wh′−X_bh′−X_wm′+X_bm′)T/2+C_rR_a′=0  (21)

[0095] In accordance with a relation defined by these formulas (20) and (21), ideal damping forces applied to the vehicle body BD by the damper 12mfor the right wheels and the damper 12hfor the left wheels (or the damper 12r) for the purpose of inhibiting vibrations of the vehicle body BD in the roll direction are expressed respectively by formulas (22) and (23) shown below.

[0096] RF_m=C_s(X_wm′−X_bm′)=C_rR_a′/T  (22)

[0097] RF_h=C_s(X_pwh′−X_pbh′)=−C_rR_a′/T  (23)

[0098] Next the third damping force calculating routine will be described. The execution of the third damping force calculating routine is started in step 170 in FIG. 7. In step 172, a roll angular velocity R_a′ of the vehicle body BD is inputted from the roll angular velocity sensor 24. Next in step 174, a target damping force RF_mof the damper 12mfor the right wheels and a target damping force RF_hof the damper 12hfor the left wheels are calculated with a view to inhibiting vibrations of the vehicle body BD in the roll direction, by making calculations according to after-mentioned formulas (24) and (25) respectively, wherein a damping coefficient C_rwhich is ideal for inhibiting vibrations of the vehicle body BD in the roll direction and has appropriately been determined in advance, the inputted roll angular velocity R_a′, and the predetermined tread T are used.

[0099] RF_m=C_rR_a′/T  (24)

[0100] RF_h=C_rR_a′/T  (25)

[0101] Next in step 176, the calculated target damping force RF_mof the damper 12_mfor the right wheels is set as third target damping forces RFd2 and RFd4 for the right wheels FW2 and RW2 respectively, and the calculated target damping force RF_hfor the damper 12hfor the left wheels is set as third target damping forces RFd1 and RFd3 for the left wheels FW1 and RW1 respectively. In step 178, the execution of the third damping force calculating routine is terminated.

[0102] Referring back to the description of the program shown in FIG. 2, after the first target damping forces Fd1, Fd2, Fd3 and Fd4, the second target damping forces PFd1, PFd2, PFd3 and PFd4, and the third target damping forces RFd1, RFd2, RFd3 and RFd4 have been calculated through the aforementioned processings of steps 102 through 106, one of the first through third target damping forces that has the greatest absolute value is selected for each of the wheels FW1, FW2, RW1 and RW2 in step 108. That is, respective absolute values |Fd1|, |PFd1|, |RFd1| of the first through third target damping forces Fd1, PFd1 and RFd1 for the front-left wheel FW1 are compared with one another. One of the first through third target damping forces Fd1, PFdI and RFd1, which corresponds to the greatest one of the absolute values |Fd1|, |PFd1| and |RFd1|, is set as a target damping force F1 for the damper 12a. Processings substantially identical to those for the front-left wheel FW1 are also carried out sequentially for the front-right wheel FW2, the rear-left wheel RW1 and the rear-right wheel RW2. Then the respective target damping forces Fi (“i” denotes an integer of 2 to 4) for the dampers 12b, 12cand 12dare sequentially determined.

[0103] Next in step 110, amounts (X_pw1−X_pb1), (X_pw2−X_pb2), (X_pw3−X_pb3) and (X_pw4−X_pb4) of relative displacement are inputted from the relative displacement amount sensors 22athrough 22drespectively. In step 112, the inputted amounts (X_pw1−X_pb1), (X_pw2−X_pb2), (X_pw3−X_pb3) and (X_pw4−X_pb4) of relative displacement are differentiated, whereby respective speeds (X_pw1′−X_pb1′), (X_pw2−X_pb2′), (X_pw3′−X_pb3′) and (X_pw4′−X_pb4′) of the vehicle body BD relative to the wheels FW1, FW2, RW1 and RW2 are calculated respectively.

[0104] After the aforementioned processing in step 112, with reference to a table of relative speed against damping force, respective orifice opening degrees OP1, OP2, OP3 and OP4 of the dampers 12a, 12b, 12cand 12d, which correspond to the target damping forces F1, F2, F3 and F4 determined by the aforementioned processing in step 108 and to the relative speeds (X_pw1′−X_pb1′), (X_pw2′−X_pb2′), (X_pw3′−X_pb3′) and (X_pw4′−X_pb4′) calculated through the aforementioned processing in step 112 respectively, are determined in step 114. The table of relative speed against damping force is preliminarily built into the microcomputer and stores data which represent, for each of the orifice opening degrees, a characteristic of change in damping force F of the dampers 12a, 12b, 12cand 12dwith respect to the relative speed (X_pw′−X_pb′). In determining each of the aforementioned orifice opening degrees OP1, OP2, OP3 and OP4, a curve which is located closest to a point determined by the damping force Fi and the relative speed (X_pwi′−X_pbi′) (“i” denotes an integer of 2 to 4) in the graph shown in FIG. 9 is retrieved. An orifice opening degree OP corresponding to the retrieved curve is selected for each pair of damping force and relative speed.

[0105] After the aforementioned processing in step 114, control signals indicative of the orifice opening degrees OP1, OP2, OP3 and OP4 determined in step 116 are outputted to the dampers 12a, 12b, 12cand 12drespectively. The orifice opening degrees of the dampers 12a, 12b, 12cand 12dare controlled to be set to the aforementioned orifice opening degrees OP1, OP2, OP3 and OP4 respectively. As a result, the dampers 12a, 12b, 12cand 12dgenerate the determined target damping forces F1, F2, F3 and F4 respectively.

[0106] As described hitherto, according to the aforementioned embodiment, through the processings of the first damping force calculating routine (FIG. 3), the first target damping forces Fd1, Fd2, Fd3 and Fd4 for inhibiting vibrations of the vehicle body in the heave direction are calculated for the respective wheels, based on the model of a single wheel of the vehicle using the skyhook theory. Through the processings of the second damping force calculating routine (FIG. 5), the second target damping forces PFd1, PFd2, PFd3 and PFd4 for inhibiting vibrations of the vehicle body in the pitch direction are calculated for the respective wheels, based on the model of front and rear wheels of the vehicle. Through the processings of the third damping force calculating routine (FIG. 7), the third target damping forces RFd1, RFd2, RFd3 and RFd4 for inhibiting vibrations of the vehicle body in the roll direction are calculated for the respective wheels, based on the model of left and right wheels of the vehicle. Through the processing of step 108 in FIG. 2, one of the aforementioned first through third target damping forces that has the greatest absolute value is selected as the target damping forces F1, F2, F3 and F4 for the locations of the wheels FW1, FW2, RW1 and RW2 respectively. The damping forces of the dampers 12a, 12b, 12cand 12dat the respective locations of the wheels FW1, FW2, RW1 and RW2 are controlled to be set to the aforementioned target damping forces F1, F2, F3 and F4 respectively.

[0107] As a result, only if vibrations of the vehicle body BD in the pitch or roll direction have become strong to a certain extent, on the condition that the first target damping forces Fd1, Fd2, Fd3 and Fd4 are smaller than the second target damping forces PFd1, PFd2, PFd3 and PFd4 or the third target damping forces RFd1, RFd2, RFd3 and RFd4 respectively, the damping forces of the dampers 12a, 12b, 12cand 12dat the locations of the respective wheels FW1, FW2, RW1 and RW2 are respectively set to the second target damping forces PFd1, PFd2, PFd3 and PFd4 or the third target damping forces RFd1, RFd2, RFd3 and RFd4 for inhibiting vibrations of the vehicle body in the pitch or roll direction, based on the model of front and rear wheels or left and right wheels. Otherwise, the damping forces of the dampers 12a, 12b, 12cand 12dare respectively set to the first target damping forces Fd1, Fd2, Fd3 and Fd4 for inhibiting vibrations of the vehicle body in the heave direction, based on the model of a single wheel of the vehicle. Thus, this embodiment compensates for a deficiency in damping force for pitch and roll movements of the vehicle body BD, while ensuring the control performance intrinsic in the first target damping forces Fd1, Fd2, Fd3 and Fd4 for inhibiting vertical vibrations of the vehicle body BD. Therefore, vertical vibrations of the vehicle body BD are effectively inhibited, and vibrations of the vehicle body BD resulting from pitch and roll movements thereof are also inhibited. Consequently the vehicle achieves good riding comfort and high running stability.

[0108] In the aforementioned embodiment, one of the first through third target damping forces that has the greatest absolute value is selected as the target damping forces F1, F2, F3 and F4 respectively for the locations of the wheels FW1, FW2, RW1 and RW2. However, in the case where the roll movements of the vehicle body BD are not very serious, calculation of the third target damping forces may be omitted, and either the first target damping forces Fd1, Fd2, Fd3 and Fd4 or the second target damping forces PFd1, PFd2, PFd3 and PFd4, which have the greater absolute values, may be selected as the target damping forces F1, F2, F3 and F4 respectively for the locations of the wheels FW1, FW2, RW1 and RW2. This also compensates for a deficiency in damping force for the pitch movements of the vehicle body BD, while ensuring the control performance intrinsic in the first target damping forces Fd1, Fd2, Fd3 and Fd4 for inhibiting vertical vibrations of the vehicle body BD. Therefore, vertical vibrations of the vehicle body BD are effectively inhibited, and vibrations of the vehicle body BD resulting from pitch movements thereof are also inhibited. Consequently the vehicle achieves good riding comfort and high running stability.

[0109] Also, in the case where the pitch movements of the vehicle body BD are not very serious, calculation of the second target damping forces may be omitted, and either the first target damping forces Fd1, Fd2, Fd3 and Fd4 or the third target damping forces RFd1, RFd2, RFd3 and RFd4, which have the greater absolute values, may be selected as the target damping forces F1, F2, F3 and F4 respectively for the locations of the wheels FW1, FW2, RW1 and RW2. This also compensates for a deficiency in damping force for the roll movements of the vehicle body BD, while ensuring the control performance intrinsic in the first target damping forces Fd1, Fd2, Fd3 and Fd4 for inhibiting vertical vibrations of the vehicle body BD. Therefore, vertical vibrations of the vehicle body BD are effectively inhibited, and vibrations of the vehicle body BD resulting from roll movements thereof are also inhibited. Consequently the vehicle achieves good riding comfort and high running stability.

[0110] In the aforementioned embodiments, one of the first through third target damping forces that has the greatest absolute value is set as an ultimate target damping force. However, an ultimate target damping force may be determined in the following manners by comparing the respective absolute values of the first through third target damping forces. That is, the first through third target damping forces may be summatively synthesized by increasing weights attributed to the first through third target damping forces in accordance with an increase in the respective absolute values. Alternatively, the two greatest ones of the first through third target damping forces may be selected and summatively synthesized. In conducting summative synthesis, a weight attributed to the greater one of the two selected target damping forces may be increased. In the aforementioned modification example, one of the first and second target damping forces that has the greater absolute value, or one of the first and third target damping forces that has the greater absolute value is determined as an ultimate target damping force. However, an ultimate target damping force may be determined based on the first and second target damping forces or the first and third target damping forces, by increasing a weight attributed to the greater one of the two target damping forces in conducting summative synthesis. This also serves to determine an ultimate target damping force based on the independently calculated first through third target damping forces, first and second target damping forces, or first and third target damping forces. Thus, a deficiency in damping force for pitch and roll movements of the vehicle body is compensated for, while the control performance intrinsic in the first target damping force for inhibiting vertical vibrations of the vehicle body is ensured. Therefore, vertical vibrations of the vehicle body are effectively inhibited, and vibrations of the vehicle body resulting from pitch and roll movements thereof are also inhibited. Consequently the vehicle achieves good riding comfort and high running stability.

[0111] Further, the aforementioned embodiment is designed to detect the vertical accelerations X_pb1″, X_pb2″, X_pb3″ and X_pb4″′ of the vehicle body BD as vertical kinetic state quantities of the vehicle body BD at the locations of the wheels FW1, FW2, RW1 and RW2 relative to absolute space, respectively. Apart from such a construction, the aforementioned embodiment may also be designed to detect the vertical speeds X_pb1′, X_pb2′, X_pb3′ and X_pb4′ of the vehicle body BD relative to the absolute space. Also, the aforementioned embodiment may be designed to calculate the vertical speeds X_pb1′, X_pb2′, X_pb3′ and X_pb4′ by detecting the amounts X_pb1, X_pb2, X_pb3and X_pb4of vertical displacement of the vehicle body BD at the locations of the wheels FW1, FW2, RW1 and RW2 relative to the absolute space, respectively. Also, as regards the amounts (X_pw1−X_pb1), (X_pw2−X_pb2), (X_pw3−X_pb3) and (X_pw4−X_pb4) of the vehicle body BD (the sprung member) at the locations of the wheels FW1, FW2, RW1 and RW2 relative to the respective wheels FW1, FW2, RW1 and RW2, the relative speeds (X_pw1′−X_pb1′), (X_pw2′−X_pb2′), (X_pw3′−X_pb3′) and (X_pw4′−X_pb4) may be detected, or the relative speeds (X_pw1′−X_pb1′), (X_pw2′−X_pb2′), (X_pw3′−X_pb3′) and (X_pw4′−X_pb4′) may be calculated by detecting the relative accelerations (X_pw1″−X_pb1″), (X_pw2″−X_pb2″), (X_pw3″−X_pb3″) and (X_pw4″−X_pb4″), respectively.

[0112] In the aforementioned embodiment, the pitch angular velocity sensor 23 includes a rate sensor provided to detect an angular velocity P_a′ of the vehicle body BD in the pitch direction. However, the angular velocity P_a′ may be detected from a difference in amount of vertical displacement between front and rear parts of the vehicle body BD. Instead of using the roll angular velocity sensor 24 composed of a rate sensor so as to detect an angular velocity R_a′ in the roll direction of the vehicle body BD, the angular velocity R_a′ may be detected from a difference in amount of vertical displacement between left and right parts of the vehicle body BD.

[0113] Furthermore, the sprung mass acceleration sensors 21athrough 21d, the relative displacement amount sensors 22athrough 22d, the pitch angular velocity sensor 23 and the roll angular velocity sensor 24 for directly detecting various kinetic state quantities are partially replaced, and the observer may be used to estimate and thus detect part of the various kinetic state quantities.

[0114] Next, various modifications of the aforementioned first damping force calculating routine will be described. In these modifications, the aforementioned first target damping forces Fd1, Fd2, Fd3 and Fd4 are calculated based on a control theory which can handle a non-linear plant and provide a design specification in the form of a frequency range. Because the following description is premised on a single wheel model of the vehicle, attention will be focused on only one of the wheels FW1, FW2, RW1 and RW2, and the description will be made as to an embodiment for calculating only the first target damping force Fd which comprehensively represents the first target damping forces Fd1, Fd2, Fd3 and Fd4. As for the other wheels, the first target damping force can be calculated in the same manner. Prior to the description of the various modifications, the single wheel model of the vehicle according to the modifications will be described first.

[0115] a. Model

[0116] First of all, a model of a suspension system will be considered in an attempt to express the suspension system in a state space. FIG. 10 is a functional view of the suspension system for a single wheel of the vehicle. M_bdenotes a mass of the vehicle body (the sprung member) BD, M_wa mass of a wheel WH (more exactly, an unsprung member including a lower arm and the like), and K_ta spring constant of a tire TR. K_sdenotes a spring constant of the spring 11, C_s0a linear portion of a damping coefficient C_s(hereinafter referred to as a linear damping coefficient) of a damper provided in the suspension system, and C_va non-linear portion of the damping coefficient C_s(hereinafter referred to as a non-linear damping coefficient). The sum of the linear damping coefficient C_s0and the non-linear damping coefficient C_vis an overall damping coefficient of the damper 12 (C_s=C_s0+C_v). RD denotes a road surface. Given that X_pb, X_pwand X_prdenote amounts of displacement of the vehicle body BD, the wheel WH and the road surface RD respectively, equations (26) and (27) of motion are established as shown below.

[0117] M_bX_pb″=K_s(X_pw−X_pb)+C_s(X_pw′−X_pb′)+C_v(X_pw′−X_pb′)  (26)

[0118] M_wX_pw″=K_t(X_pr−X_pw)−K_s(X_pw−X_pb)−C_s(X_pw′−X_pb′)−C_v(X_pw′−X_pb′)  (27)

[0119] In the aforementioned formulas (26) and (27) and after-mentioned respective formulas, the sign (′) indicates single differentiation and the sign (″) indicates double differentiation.

[0120] The control input “u” in this suspension system is the non-linear damping coefficient C_v. Hence, if the suspension system is expressed in a state space with a road surface disturbance w₁and the non-linear damping coefficient C_vbeing used as a road surface speed X_pr′ and the control input “u” respectively, a formula (28) is established as shown below.

[0121] X_p′=A_pX_p+B_p1w₁+B_p2(X_p)u  (28)

[0122] In the aforementioned formula (28), X_p, A_p, B_p1and B_p2(X_p) are respectively expressed according to formulas (29) through (32) shown below. $\begin{array}{cc}{X}_p=\left[\begin{array}{c}\begin{array}{c}\begin{array}{c}{X}_{\mathrm{pr}}-X_{\mathrm{pw}}\\ X_{\mathrm{pw}}-X_{\mathrm{pb}}\end{array}\\ X_{\mathrm{pw}}^{\prime }\end{array}\\ X_{\mathrm{pb}}^{\prime }\end{array}\right]& \left(29\right)\\ A_p=\left[\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 1& -1\\ K_t/M_w& -K_s/M_w& -C_s/M_w& C_s/M_w\\ 0& K_s/M_b& C_s/M_b& -C_s/M_b\end{array}\right]& \left(30\right)\\ B_{\mathrm{p1}}=\left[\begin{array}{c}\begin{array}{c}\begin{array}{c}1\\ 0\end{array}\\ 0\end{array}\\ 0\end{array}\right]& \left(31\right)\\ B_{\mathrm{p2}}\left(X_p\right)=\left[\begin{array}{c}\begin{array}{c}\begin{array}{c}0\\ 0\end{array}\\ -\left(X_{\mathrm{pw}}^{\prime }-X_{\mathrm{pb}}^{\prime }\right)/M_w\end{array}\\ \left(X_{\mathrm{pw}}^{\prime }-X_{\mathrm{pb}}^{\prime }\right)/M_b\end{array}\right]& \left(32\right)\end{array}$

[0123] The suspension system of this modification simultaneously restricts a vertical speed X_pb′ of the vehicle body BD (hereinafter referred to as a sprung mass speed X_pb′) which greatly affects vibrations of the vehicle body BD (the sprung mass member), a vertical acceleration X_pb″ of the vehicle body BD (hereinafter referred to as a sprung mass acceleration X_pb″) which greatly affects riding comfort of the vehicle, and a vertical speed (X_pw′−X_pb′) of the wheel WH relative to the vehicle body BD (hereinafter referred to as a relative speed (X_pw′−Xpb′)) which greatly affects vibrations of the wheel WH. Accordingly, the sprung mass speed X_pb′, the sprung mass acceleration X_pb″ and the relative speed (X_pw′−X_pb′) are used as an evaluated output Z_p. In the suspension system, it is easy to detect the sprung mass acceleration X_pb″ and an amount (X_pw−X_pb) of displacement of the wheel WH relative to the vehicle body BD (hereinafter referred to simply as a relative displacement amount (X_pw−X_pb)). Therefore the sprung mass acceleration X_pb″ and the relative displacement amount (X_pw−X_pb) are basically used as an observed output yp. If the observed output y_pis expressed in a state space on the assumption that the observed output yp includes an observed noise w₂, formulas (33) and (34) are established as shown below.

[0124] Z_p=C_p1X_p+D_p12(X_p)u  (33)

[0125] y_p=C_p2X_p+D_p21w₂+D_p22(X_p)u  (34)

[0126] In the aforementioned formulas (33) and (34), Z_p, y_p, C_p1, D_p12(X_p), C_p2, D_p21and D_p22(X_p) are respectively expressed according to formulas (35) through (41) shown below. $\begin{array}{cc}{Z}_p=\left[\begin{array}{c}\begin{array}{c}{X}_{\mathrm{pb}}^{″}\\ X_{\mathrm{pb}}^{\prime }\end{array}\\ X_{\mathrm{pw}}^{\prime }-X_{\mathrm{pb}}^{\prime }\end{array}\right]& \left(35\right)\\ y_p=\left[\begin{array}{c}{X}_{\mathrm{pb}}^{″}\\ X_{\mathrm{pw}}-X_{\mathrm{pb}}\end{array}\right]& \left(36\right)\\ C_{\mathrm{p1}}=\left[\begin{array}{cccc}0& K_s/M_b& C_s/M_b& -C_s/M_b\\ 0& 0& 0& 1\\ 0& 0& 1& -1\end{array}\right]& \left(37\right)\\ D_{\mathrm{p12}}\left(X_p\right)=\left[\begin{array}{c}\begin{array}{c}\left(X_{\mathrm{pw}}^{\prime }-X_{\mathrm{pb}}^{\prime }\right)/M_b\\ 0\end{array}\\ 0\end{array}\right]& \left(38\right)\\ C_{\mathrm{p2}}=\left[\begin{array}{cccc}0& K_s/M_b& C_s/M_b& -C_s/M_b\\ 0& 1& 0& 0\end{array}\right]& \left(39\right)\\ D_{\mathrm{p21}}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]& \left(40\right)\\ D_{\mathrm{p22}}\left(X_p\right)=\left[\begin{array}{c}\left(X_{\mathrm{pw}}^{\prime }-X_{\mathrm{pb}}^{\prime }\right)/M_b\\ 0\end{array}\right]& \left(41\right)\end{array}$

[0127] However, since the coefficient B_p2(X_p) includes the state quantity X_pas indicated by the aforementioned formula (28), the state space expression of the aforementioned suspension system is a bilinear system. In the bilinear system, since B_p2(0)=0 at the origin X=0 despite a change in control input u, the execution of control is impossible in the vicinity of the origin. Therefore the control system for the aforementioned suspension system cannot be designed according to the linear control theory. Hence, an attempt is made to design the control system by means of the non-linear H∞ control theory for the purpose of gaining desired control performance. In other words, an attempt is made to design a control system for restricting the sprung mass speed X_pb′, the sprung mass acceleration X_pb″ and the relative speed (X_pw′−X_pb′).

[0128] Hereinafter, various examples of the non-linear H∞ control system according to the various modifications of the invention for calculating the first target damping force Fd, and concrete examples of calculation of the first target damping force Fd will be described.

[0129] b. First Modification

[0130] b1. Designing Example of Non-Linear H∞ State Feedback Control System

[0131] First of all, in order to make an attempt to design the non-linear H∞ state feedback control system, a generalized model of a state feedback control system as shown in FIG. 11 wherein a frequency weight is added to the evaluated output Z_pand the control input “u” is considered. In this case, the frequency weight means a dynamic weight that changes in accordance with a frequency and is given in the form of a transfer function. Use of the frequency weight makes it possible to increase a weight in a frequency range where the control performance is to be improved and to reduce a weight in a frequency range where the control performance is negligible. In addition, the evaluated output Z_pand the control input “u” is multiplied by frequency weights W_s(S) and W_u(S), respectively, and then multiplied, respectively, by functions a1(X) and a2(X) of the state quantity X. These functions are non-linear weight functions. In order to turn to Riccati's equation and find a solution, the non-linear weights a1(X) and a2(X) demonstrate characteristics defined by formulas (42) and (43) shown below.

[0132] a1(X)>0, a2(X)>0  (42)

[0133] a1(0)=a2(0)=1  (43)

[0134] These non-linear weights make it possible to design a control system for restricting an L2 gain more positively. A state space expression of this system is indicated by a formula (44) shown below.

[0135] X_p′=A_pX_p+B_p1w₁+B_p2(X_p)u  (44)

[0136] A state space expression of the weight W_s(S) by which the evaluated output Z_pis multiplied is indicated by formulas (45) and (46) shown below.

[0137] X_w′=A_wX_w+B_wZ_p  (45)

[0138] Z_w=C_wX_w+D_wZ_p  (46)

[0139] X_wdenotes a state quantity of the frequency weight W_s(S), and Z_wdenotes an output of the frequency weight W_s(S). A_w, B_w, C_wand D_ware constant matrices determined by a control specification. These constant matrices A_w, B_w, C_wand D_ware so determined as to reduce a gain for the sprung mass acceleration X_b″ in a frequency range of about 3 to 8 Hz (FIG. 12(A)) with a view to improving riding comfort of the passenger (eliminating a feeling of cragginess), reduce a gain for the sprung mass speed X_b′ in a frequency range of about 0.5 to 1.5 Hz (FIG. 12(B)) with a view to inhibiting resonance of the vehicle body BD, and reduce a gain for the relative speed (X_w′−X_b′)in a frequency range of about 10 to 14 Hz (FIG. 12(C)) with a view to avoiding resonance of the wheel WH. While the frequency ranges for reducing these respective gains are prevented from overlapping with one another and thus prevented from interfering with one another, respective elements constituting the evaluated output Z_p, namely, the sprung mass acceleration X_b″, the sprung mass speed X_b′ and the relative speed (X_w′−X_b′) are controlled independently.

[0140] A state space expression of the weight W_u(S) by which the control input “u” is multiplied is indicated by formulas (47) and (48) shown below.

[0141] X_u′=A_uX_u+B_uu  (47)

[0142] Z_u=C_uX_u+D_uu  (48)

[0143] X_udenotes a state quantity of the frequency weight W_u(S), and Z_udenotes an output of the frequency weight W_u(S). A_u, B_u, C_uand D_uare constant matrices determined by a control specification. In order to consider responding performance of an electric actuator for controlling the damping coefficient, the constant matrices A_u, B_u, C_uand D_uare determined such that a gain for the control input “u” is restricted in a high frequency range in accordance with a frequency characteristic of the actuator (FIG. 12(D)).

[0144] At this moment, a state space expression of a generalized model in the non-linear H∞ state feedback control system is indicated by formulas (49) through (51) shown below.

[0145] X′=AX+B₁w₁+B₂(X)u  (49)

[0146] Z₁=a1(X)(C₁₁X+D₁₂₁(X)u)  (50)

[0147] Z₂=a2(X)(C₁₂X+D₁₂₂u)  (51)

[0148] In the aforementioned formulas (49) through (51), X, A, B₁, B₂(X), C₁₁, D₁₂₁(X), C₁₂and D₁₂₂are expressed respectively by formulas (52) through (59) shown below. $\begin{array}{cc}X=\left[\begin{array}{c}\begin{array}{c}{X}_p\\ X_w\end{array}\\ X_u\end{array}\right]& \left(52\right)\\ A=\left[\begin{array}{ccc}{A}_p& 0& 0\\ B_w{C}_{\mathrm{p1}}& A_w& 0\\ 0& 0& A_u\end{array}\right]& \left(53\right)\\ B_1=\left[\begin{array}{c}\begin{array}{c}{B}_{\mathrm{p1}}\\ B_w{D}_{\mathrm{p11}}\end{array}\\ 0\end{array}\right]& \left(54\right)\\ B_2\left(X\right)=\left[\begin{array}{c}\begin{array}{c}{B}_{\mathrm{p2}}\left(X_p\right)\\ B_w{D}_{\mathrm{p12}}\left(X_p\right)\end{array}\\ B_u\end{array}\right]& \left(55\right)\end{array}$  C₁₁=[D_wVC_p1C_w0]  (56)

[0149] D_121(X)=[DwD_p12(X_p)]  (57)

[0150] C₁₂=[0 0 C_u]  (58)

[0151] D₁₂₂=D_u  (59)

[0152] Next, in order to find a solution based on Riccati's equation, if the state space expression of the generalized model expressed by the aforementioned formulas (49) through (51) is rewritten under the condition defined by a formula (60) shown below, formulas (60) through (63) are obtained as shown below.

[0153] D_wD_p12(X)=0  (60)

[0154] X′=AX+B₁w+B₂(X)u  (61)

[0155] Z₁=a1(X)C₁₁X  (62)

[0156] Z₂=a2(X)C₁₂X+a2(X)D₁₂₂u  (63)

[0157] Because A is a stable matrix indicative of a damping force control system, an attempt is made to design, for the aforementioned generalized model, a non-linear H∞ state feedback control rule u=k(X) which satisfies the condition (1) that the closed-loop system has a stable internal exponent and the condition (2) that the L2 gain from the road surface disturbance w₁to the evaluated output Z is equal to or smaller than a positive constant γ.

[0158] The aforementioned non-linear H∞ state feedback control rule u=k(X) can be obtained if the following conditions are established. That is, (1) if D₁₂₂⁻¹exists and a positive constant γ is given, a positive definite symmetric solution P satisfying the after-mentioned Riccati's equation (64) exists for the positive constant γ, and (2) if the non-linear weights a1(X) and a2(X) satisfy a restrictive condition expressed by a formula (65) shown below, one of the control rules u=k(X) for internally stabilizing the closed-loop system and making the L2 gain equal to or smaller than γ is given by a formula (66) shown below. $\begin{array}{cc}\mathrm{PA}+A^{T}P+\frac{{\mathrm{PB}}_1{B}_1^{T}P}{{\gamma }^2}+C_{11}^T{C}_{11}+C_{12}^T{C}_{12}\prec 0& \left(64\right)\\ \left(1-{\mathrm{a1}\left(X\right)}^2\right)X^T{C}_{11}^T{C}_{11}X+[\frac{1}{{\mathrm{a2}\left(X\right)}^2}-1]X^T{\mathrm{PB}}_2\left(X\right)D_{122}^{-1}{D}_{122}^{-T}{B}_2^T\left(X\right)\mathrm{PX}\ge 0& \left(65\right)\\ \begin{array}{c}u=k\left(X\right)\\ =-D_{122}^{-1}\left[\frac{1}{{\mathrm{a2}\left(X\right)}^2}{D}_{122}^{-1}{B}_2^T\left(X\right)P+C_{12}\right]X\end{array}& \left(66\right)\end{array}$

[0159] The non-linear weights a1(X) and a2(X) satisfying the restrictive condition of the formula (66) are exemplified respectively in formulas (67) and (68) shown below. $\begin{array}{cc}\mathrm{a1}\left(X\right)=\sqrt{1+\mathrm{m1}\left(X\right)X^T{\mathrm{PB}}_2\left(X\right)D_{122}^{-1}{D}_{122}^{-T}{B}_2^T\left(X\right)\mathrm{PX}}& \left(67\right)\\ \mathrm{a2}\left(X\right)=\sqrt{\frac{1}{1+\mathrm{m1}\left(X\right)X^T{C}_{11}^T{C}_{11}X}}& \left(68\right)\end{array}$

[0160] In the aforementioned formulas (67) and (68), m1(X) is an arbitrary positive definite function. As a result of calculations made by the computer, it has been made possible to find the positive definite symmetric solution P as described above. By using the aforementioned formula (68), the aforementioned formula (66) is converted into a formula (69) shown below. $\begin{array}{cc}\begin{array}{c}u=k\left(X\right)\\ =-D_{122}^{-1}\left(\left(1+\mathrm{m1}\left(X\right)X^T{C}_{11}^T{C}_{11}X\right)D_{122}^{-T}{B}_2^T\left(X\right)P+C_{12}\right)X\end{array}& \left(69\right)\end{array}$

[0161] This means that although a partial differential inequality called Hamilton-Jacobi's inequality must be solved in general in order to design a control system using the non-linear H∞ control theory, the control rule can be designed by solving Riccati's inequality instead of Hamilton-Jacobi's inequality, by imposing the restrictive condition of the aforementioned formula (65) on the non-linear weights a1(X) and a2(X) as described above. Riccati's inequality can be solved easily by using a known software, such as Matlab. Thus, this method makes it possible to easily find a positive definite symmetric solution P and to derive the control rule u=k(X).

[0162] The aforementioned D₁₂₂does not exist in Riccati's inequality and relates only to the restrictive condition imposed on the non-linear weights and the control rule. This means that the control rule using D₁₂₂can be adjusted to some extent without solving Riccati's inequality again. In other words, adjustment of the aforementioned control rule means the scaling of the control input “u”. If the scaling ratio is multiplied by 10, D₁₂₂is multiplied by {fraction (1/10)} and the terms of B₂(X) and C₁₂in the aforementioned formula (66) are multiplied by 100 and 10, respectively.

[0163] Next, in order to confirm a role played by the non-linear weights, a generalized model of a bilinear system that does not employ any non-linear weight will be considered and compared with the aforementioned generalized model that employs the non-linear weights. That is, the aforementioned non-linear weights a1(X) and a2(X) are defined as a1(X)=1 and a2(X)=1 respectively. Also, for the sake of simplification, it is determined that C₁₂=0 and that D₁₂₂=1, on the assumption that an orthogonal condition is satisfied. The state space expression indicated by the aforementioned formulas (61) through (63) is expressed according to formulas (70) through (72) shown below.

[0164] X′=AX+B₁W+B₂(X)u  (70)

[0165] Z₁=C₁₁X  (71)

[0166] Z₂=u  (72)

[0167] Thus, the control rule u=k(X) for the generalized model is expressed according to a formula (73) shown below.

[0168] u=B₂^T(X)PX  (73)

[0169] P is a positive definite symmetric solution that satisfies Riccati's inequality (74) shown below. $\begin{array}{cc}\mathrm{PA}+A^{T}P+\frac{1}{{\gamma }^2}{\mathrm{PB}}_1{B}_1^{T}P+C_{11}^T{C}_{11}\prec 0& \left(74\right)\end{array}$

[0170] A linear approximation system in the vicinity of the origin of the generalized model indicated by the aforementioned formulas (70) through (72) is expressed according to formulas (75) through (77) shown below.

[0171] X′=AX+B₁W  (75)

[0172] Z₁=C₁₁X  (76)

[0173] Z₂=u  (77)

[0174] Riccati's inequality in the aforementioned formula (74) means that the closed-loop system is internally stable for the generalized model and that the L2 gain is equal to or smaller than γ. That is, the L2 gain of the bilinear system is determined by a value at the origin (Z=0) shown in FIG. 13. This is because the bilinear system is B₂(0)=0 at the origin and therefore the control input “u” is so ineffective that the L2 gain cannot be improved in the vicinity of the origin. The generalized model with the control input u being equal to 0 (the formulas (70) through (72)) also coincides with the linearly approximated generalized model (the formulas (75) through (77)). Hence, Riccati's inequality in the formula (74) means that even in the case where the control input “u” is equal to 0 for the generalized model (the formulas (70) through (72)), the closed-loop system is internally stable and the L2 gain is equal to or smaller than γ. That is, even if the state quantity X has increased and the control input u has become effective, in the case where a control system is designed for the generalized model (the formulas (70) through (72)) with a control output expressed by formulas (78) and (79) shown below, it is merely guaranteed that the L2 gain does not become greater than g₀even after being multiplied by the control input “u”.

[0175] Z₁=C₁₁X  (78)

[0176] Z₂=u  (79)

[0177] That is, if the control output is expressed as in the aforementioned formulas (78) and (79), the control performance may improve through the use of the control input “u” but may not make any difference in comparison with the case of u=0. Thus, the non-linear weights a1(X) and a2(X) are applied to control outputs Z₁and Z₂respectively, whereby formulas (80) and (81) are established as shown below. This makes it possible to design a control system for making the L2 gain of the plant closer to the X axis which represents the origin level by means of the non-linear weights at locations remote from the origin, as indicated by a line g₁in FIG. 13.

[0178] Z₁=a1(X)C₁₁X  (80)

[0179] Z₂=a2(X)u  (81)

[0180] In this control, the damping coefficient C_sof the damper 12 is divided into the linear damping coefficient C_s0and the non-linear damping coefficient C_v, and the control system has been designed with the non-linear damping coefficient C_vbeing used as the control input “u”. As shown in FIG. 14A, the linear damping coefficient C_s0is approximately set to the vicinity of the center between a minimum damping force characteristic curve (corresponding to a maximum orifice opening degree) of the damper 12 and a maximum damping force characteristic curve (corresponding to a minimum orifice opening degree) of the damper 12, and the gain of the control input “u” is controlled in accordance with a frequency. The damping coefficient C_schanges on opposed sides of the linear damping coefficient C_s0, and the damping force obtained from the linear damping coefficient C_s0is designed to remain between the aforementioned minimum and maximum damping force characteristic curves. Accordingly, the non-linear damping coefficient C_vcan easily be determined in accordance with a design specification of the damper 12, and the damping force control can be performed within a range feasible with the actual damper 12. Consequently, the damping force control can be performed as desired. For comparison, FIG. 14B shows Lissajou's waveform diagram in the case where the damping coefficient of the damper 12 is controlled based on the skyhook theory. In this case, the control cannot be performed within the range feasible with the actual damper 12, and it is impossible to perform the control as desired.

[0181] Further, if the damping force (the damping coefficient) of the damper 12 is designed to be switched stepwise to one of a plurality of stages, the aforementioned linear damping coefficient C_s0is set such that the damping force determined by the linear damping coefficient C_s0becomes, within a range of small damping forces, substantially equal to a damping force generated by a predetermined one of the aforementioned stages of the damper 12. In the suspension system of this kind, in a range where the damping force is small, the linearity of the damping force with respect to the relative speed is strong. In other words, it is quite likely that the calculated non-linear damping coefficient C_vwill be equal to “0”. Thus, it is highly probable that the damper 12 be maintained at the predetermined one of the stages, and the frequency with which the damping coefficient is switched decreases. Therefore, high durability is rendered to the damper 12.

[0182] b2. Calculation Example of First Modification

[0183] Next, a calculation example of the first target damping force Fd using the aforementioned non-linear H∞ state feedback control rule will be described.

[0184] In this case, in addition to the sprung mass acceleration sensors 21a, 21b, 21cand 21d, the relative displacement amount sensors 22a, 22b, 22cand 22d, the pitch angular velocity sensor 23 and the roll angular velocity sensor 24, tire displacement amount sensors 25a, 25b, 25cand 25dand unsprung mass acceleration sensors 26a, 26b, 26cand 26d, which are provided for the wheels FW1, FW2, RW1 and RW2 respectively, are connected to the electric control device 20, as indicated by broken lines in FIG. 1. However, the following description will deal with an example of calculating the first target damping force Fd of a single wheel representative of the wheels FW1, FW2, RW1 and RW2. Therefore, in the following description, the tire displacement amount sensors 25a, 25b, 25cand 25dwill be referred to simply as a tire displacement amount sensor 25, and the unsprung mass acceleration sensors 26a, 26b, 26cand 26dwill be referred to simply as an unsprung mass acceleration sensor 26. Likewise, the sprung mass acceleration sensors 21a, 21b, 21cand 21dand the relative displacement amount sensors 22a, 22b, 22cand 22dwill be referred to simply as a sprung mass acceleration sensor 21 and a relative displacement amount sensor 22 respectively.

[0185] The tire displacement amount sensor 25 detects an amount (X_pr−X_pw) of displacement of the tire TR, which is a relative displacement amount between a road surface displacement X_prand an unsprung mass displacement X_pw. For example, the tire displacement amount (X_pr−X_pw) is detected based on outputs from a strain sensor for detecting a degree of deformation of the tire, a pressure sensor for detecting an air pressure of the tire and the like. The unsprung mass acceleration sensor 26 is fixed to the wheel WH and detects an unsprung mass acceleration X_pw″ indicative of a vertical acceleration of the wheel WH. The microcomputer in the electric control device 20 executes a first damping force calculating routine shown in FIG. 15 at intervals of a predetermined period of time by means of the built-in timer, and thereby calculates a first target damping force Fd.

[0186] The execution of the first damping force calculating routine is started in step 200. In step 202, detection signals indicative of the tire displacement amount (X_pr−X_pw), the relative displacement amount (X_pw−X_pb), the sprung mass acceleration X_pb″ and the unsprung mass acceleration X_pw″ are inputted from the tire displacement amount sensor 25, the relative displacement amount sensor 22, the sprung mass acceleration sensor 21 and the unsprung mass acceleration sensor 26 respectively. Then in step 204, the sprung mass acceleration X_pb″ and the unsprung mass acceleration X_pw″ are time-integrated, whereby a sprung mass speed X_pb′ and an unsprung mass speed X_pw′ are calculated. Also, the relative displacement amount (X_pw−X_pb) is time-differentiated, whereby a relative speed (X_pw′−X_pb′) is calculated.

[0187] Next in step 206, B_p2(X_p) and D_p12(X_p) are calculated according to after-mentioned formulas (82) and (83), which are identical to the aforementioned formulas (32) and (38) using the relative speed (X_pw′−X_pb′). B₂(X) is then calculated according to an after-mentioned formula (84), which is identical to the aforementioned formula (55) using B_p2(X_p) and D_p12(X_p). $\begin{array}{cc}{B}_{\mathrm{p2}}\left(X_p\right)=\left[\begin{array}{c}\begin{array}{c}\begin{array}{c}0\\ 0\end{array}\\ -\left(X_{\mathrm{pw}}^{\prime }-X_{\mathrm{pb}}^{\prime }\right)/M_w\end{array}\\ \left(X_{\mathrm{pw}}^{\prime }-X_{\mathrm{pb}}^{\prime }\right)/M_b\end{array}\right]& \left(82\right)\\ D_{\mathrm{p12}}\left(X_p\right)=\left[\begin{array}{c}\begin{array}{c}\left(X_{\mathrm{pw}}^{\prime }-X_{\mathrm{pb}}^{\prime }\right)/M_b\\ 0\end{array}\\ 0\end{array}\right]& \left(83\right)\\ B_2\left(X\right)=\left[\begin{array}{c}\begin{array}{c}{B}_{\mathrm{p2}}\left(X_p\right)\\ B_w{D}_{\mathrm{p12}}\left(X_p\right)\end{array}\\ B_u\end{array}\right]& \left(84\right)\end{array}$

[0188] In the aforementioned formulas (82) and (83), M_wand M_bdenote a mass of the wheel WH and a mass of the vehicle body BD, respectively. In the aforementioned formula (84), B_wand B_udenote coefficient matrices relating to the frequency weights W_s(S) and W_u(S) set in the aforementioned formulas (45) and (47) respectively. These coefficient matrices are constant matrices that are preliminarily stored in the microcomputer.

[0189] After the aforementioned processing of step 206, a state variable X_wof the frequency weights is calculated in step 208 according to an after-mentioned formula (85) identical to the aforementioned formula (45), using the evaluated output Z_p(the sprung mass speed X_pb′, the sprung mass acceleration X_pb″ and the relative speed (X_pw′−X_pb′)) which is a control target of this example that has been inputted in the processing is of step 202 or calculated in the processing of step 204 and is defined by the aforementioned formula (35).

[0190] X_w′=A_wX_w+B_wZ_p  (85)

[0191] In the aforementioned formula (85), A_wand B_wdenote coefficient matrices relating to the frequency weight W_s(S) set in the aforementioned formula (45). These matrices are constant matrices that are preliminarily stored in the microcomputer.

[0192] Next in step 210, a state variable X_uof the frequency weight relating to the control input “u”, an extended state quantity X and a control input “u” are calculated, using formulas (86) through (88) which are identical to the aforementioned formulas (47), (52) and (69) respectively.

[0193] X_u′=A_uX_u+B_uu  (86)

[0194] $\begin{array}{cc}X=\left[\begin{array}{c}\begin{array}{c}{X}_p\\ X_w\end{array}\\ X_u\end{array}\right]& \left(87\right)\\ \begin{array}{c}u=k\left(X\right)\\ =-D_{122}^{-1}\left(\left(1+\mathrm{m1}\left(X\right)X^T{C}_{11}^T{C}_{11}X\right)D_{122}^{-T}{B}_2^T\left(X\right)P+C_{12}\right)X\end{array}& \left(88\right)\end{array}$

[0195] In the aforementioned formula (86), A_uand B_uare coefficient matrices relating to the frequency weight W_u(S) set in the aforementioned formula (47). These matrices are constant matrices that are preliminarily stored in the microcomputer. D₁₂₂in the aforementioned formula (88), which is defined by the aforementioned formula (59), is a coefficient matrix relating to the frequency weight W_u(S) set in the aforementioned formula (48) and is a constant matrix preliminarily stored in the microcomputer. m1(X) is an arbitrary positive definite function, and an algorithm concerning the function is preliminarily stored in the microcomputer. The positive definite function m1(X) may be set to a positive constant, for example, “1.0”. C₁₁is defined by the aforementioned formulas (37) and (56). In other words, C₁₁is a constant matrix which is preliminarily stored in the microcomputer and which is defined by the mass M_wof the wheel, the mass M_bof the vehicle body BD, the spring constant K_sof the spring 11, the linear damping coefficient C_s0of the damper 12 and the coefficient matrices C_wand D_wrelating to the frequency weight W_s(S) set in the aforementioned formula (46). B₂(X) is a matrix calculated in the aforementioned step 206. P is a positive definite symmetric solution satisfying the aforementioned formulas (64) and (65) and is a constant matrix that is preliminarily stored in the microcomputer. C₁₂, which is defined by the aforementioned formula (58), is a constant matrix that includes the coefficient matrix C_urelating to the frequency weight W_u(S) set in the aforementioned formula (48) and is preliminarily stored in the microcomputer.

[0196] In calculating the state variable X_urelating to the frequency weight of the control input u, the extended state quantity X and the control input “u” in step 210, the respective values X_u, X and u are supplied with initial values, and calculations according to the aforementioned formulas (85) through (88) are repeatedly made until the respective values X_u, X and u converge. In this manner, the values X_u, X and u are determined.

[0197] After the aforementioned processing in step 210, since the control input “u” is equal to the non-linear damping coefficient C_v, an overall target damping coefficient C_sof the damper 12 is calculated in step 212 according to an after-mentioned formula (89) wherein the linear damping coefficient C_s0is added to the control input “u”. In step 216, the execution of the first damping force calculating routine is terminated.

[0198] C_s=C_s0+C_v=C_s0+u  (89)

[0199] Next in step 214, the first target damping force Fd is calculated according to an after-mentioned formula (90) wherein the calculated target damping coefficient C is multiplied by the relative speed (X_pw′−X_pb′) calculated in the aforementioned processing of step 204.

[0200] Fd=C_s(X_pw′−X_pb′)  (90)

[0201] c. Second Modification

[0202] c1. Designing Example of Non-Linear H∞ Output Feedback Control System

[0203] Next, the designing of the aforementioned non-linear H∞ state feedback control system will be further explained. That is, an estimated value will be obtained from an observer which includes part (e.g. the tire displacement amount (X_pr−X_pw) and the relative displacement amount (X_pw−X_pb) or the tire displacement amount (X_pr−X_pw), the relative displacement amount (X_pw−X_pb) and the unsprung mass speed X_pw′) of the state quantity X_p(the tire displacement amount (X_pr−X_pw), the relative displacement amount (X_pw−X_pb), the unsprung mass speed X_pw′ and the sprung mass acceleration X_pb″) in the control system, and the estimated value will be used in an attempt to design a non-linear H∞ output feedback control system. In this case, a generalized model of an output feedback control system as shown in FIG. 16 wherein frequency weights are added to the evaluated output Z_pand the control input “u” will be considered. In this case, the evaluated output Z_pis multiplied by a non-linear weight function a1(X, X{circumflex over ( )}) after being multiplied by the frequency weight W_s(S), and the control input “u” is multiplied by a non-linear weight function a2(X, X{circumflex over ( )}) after being multiplied by the frequency weight W_u(S). These non-linear weight functions a1(X, X{circumflex over ( )}) and a2(X, X{circumflex over ( )}) have characteristics indicated by formulas (91) and (92) shown below. This makes it possible to design a control system for restricting the L2 gain more positively. As described above, X{circumflex over ( )} denotes a state quantity partially including an estimated value.

[0204] a1(X, X{circumflex over ( )})>0, a2(X, X{circumflex over ( )})>0  (91)

[0205] a1(0, 0)=a2(0, 0)=1  (92)

[0206] A state space expression of this system, as well as a state space expression of the frequency weight W_s(S) by which the evaluated output Z_pis multiplied and a state space expression of the frequency weight W_u(S) by which the control input “u” is multiplied, is expressed according to formulas (93) through (97) shown below, as is the case with the aforementioned state feedback control system.

[0207] X_p′=A_pX_p+B_p1w₁+B_p2(X_p)u  (93)

[0208] X_w′=A_wX_w+B_wZ_p  (94)

[0209] Z_w=C_wX_w+D_wZ_p  (95)

[0210] X_u′=A_uX_u+B_uu  (96)

[0211] Z_u=C_uX_u+D_uu  (97)

[0212] The state variable X_w, the evaluated function Z_wand the constant matrices A_w, B_w, C_wand D_ware the same as in the case of the aforementioned state feedback control system.

[0213] However, a state space expression of the generalized model in this non-linear H∞ output feedback control system is indicated by formulas (98) through (101) shown below.

[0214] X′=AX+B₁w+B₂(X)u  (98)

[0215] Z₁=a1(X, X{circumflex over ( )})(C₁₁X+D₁₂₁(X)u)  (99)

[0216] Z₂=a2(X, X{circumflex over ( )})(C₁₂X+D₁₂₂u)  (100)

[0217] y=C₂X+D₂₁W+D₂₂(X)u  (101)

[0218] X, W, A, B₁, B₂(X), C₁₁, D₁₂₁(X), C₁₂, D₁₂₂, C₂, D₂₁and D₂₂(X) in the aforementioned formulas (98) through (101) are respectively expressed according to formulas (102) through (113) shown below. $\begin{array}{cc}X=\left[\begin{array}{c}\begin{array}{c}{X}_p\\ X_w\end{array}\\ X_u\end{array}\right]& \left(102\right)\\ W=\left[\begin{array}{c}{W}_1\\ W_2\end{array}\right]& \left(103\right)\\ A=\left[\begin{array}{ccc}{A}_p& 0& 0\\ B_w{C}_{\mathrm{p1}}& A_w& 0\\ 0& 0& A_u\end{array}\right]& \left(104\right)\\ B_1=\left[\begin{array}{cc}{B}_{\mathrm{p1}}& 0\\ B_w{D}_{\mathrm{p11}}& 0\\ 0& 0\end{array}\right]& \left(105\right)\\ B_2\left(X\right)=\left[\begin{array}{c}{B}_{\mathrm{p2}}\left(X_p\right)\\ B_w{D}_{\mathrm{p12}}\left(X_p\right)\\ \mathrm{Bu}\end{array}\right]& \left(106\right)\end{array}$  C₁₁=[D_wC_p1C_w0]  (107)

[0219] D₁₂₁(X)=[D_wD_p12(X_p)]  (108)

[0220] C₁₂=[0 0 C_u]  (109)

[0221] D₁₂₂=D_u  (110)

[0222] C₂=[C_p20 0]  (111)

[0223] D₂₁=[0 D_p21]  (112)

[0224] D₂₂(X)=D_p22(X_p)  (113)

[0225] Next, in order to find a solution based on Riccati's equation, the state space expression of the generalized model expressed by the aforementioned formulas (98) through (101) is converted under the condition prescribed in a formula (114) shown below, whereby formulas (115) through (118) are established as shown below.

[0226] D_wD_p12(X)=0  (114)

[0227] X′=AX+B₁w+B₂(X)u  (115)

[0228] Z₁=a1(X, X{circumflex over ( )})C₁₁X  (116)

[0229] Z₂=a2(X, X{circumflex over ( )})C₁₂X+a2(X, X{circumflex over ( )})D₁₂₂u  (117)

[0230] y=C₂X+D₂₁w+D₂₂(X)u  (118)

[0231] As is the case with the aforementioned state feedback control system, for the aforementioned generalized model, an attempt is made to design a non-linear H∞ output feedback control rule u=k(y) which satisfies the condition (1) that the closed-loop system has a stable internal exponent and the condition (2) that the L2 gain from w to Z is equal to or smaller than a positive constant γ. Furthermore, in the following description, the non-linear H∞ output feedback control will be classified into first through third types.

[0232] c1-1) Designing Example of Control System of First Type

[0233] The first type refers to a case where B₂(X) in the formula (106) and D₂₂(X) in the formula (113) are known functions, that is, a case where at least the relative speed (X_pw′−X_pb′) is observable and the observer gain L is a constant matrix.

[0234] The aforementioned non-linear H∞ output feedback control rule u=k(y) can be obtained if the following conditions are established. That is, (1) if D₁₂₂⁻¹exists, γ₁is a positive constant satisfying γ₁²I−D₂₁^TΘ^TΘD₁₂>0, γ₂>1, and positive definite symmetric matrices P, Q and a positive definite matrix Θ satisfying Riccati's inequality for designing an observer (observer gain) of an after-mentioned formula (119) and Riccati's inequality for designing a controller (control apparatus) of an after-mentioned formula (120) exist, and (2) if the non-linear weights a1(X, X{circumflex over ( )}) and a2(X, X{circumflex over ( )}) satisfy a restrictive condition expressed by formulas (121) and (122) shown below, one of the control rules according to an after-mentioned formula (123) is given by formulas (124) and (125) shown below.

[0235] AQ+QA^T+Q(C₁₁^TC₁₁+C₁₂^TC₁₂−C₂^TΘ^TΘC₂)Q+B₁(γ₁²I−D₂₁^TΘ^TΘD₂₁)⁻¹B₁^T<0  (119)

[0236] $\begin{array}{cc}\mathrm{PA}+A^{T}P+\frac{1}{{\gamma }_2^2}\mathrm{PL}{\Theta }^{-1}{\Theta }^{-T}{L}^{T}P+\frac{{\gamma }_2^2}{{\gamma }_2^2-1}\left(C_{11}^T{C}_{11}+C_{12}^T{C}_{12}\right)\prec 0& \left(120\right)\end{array}$  γ₂²−a1(X, X{circumflex over ( )})²>0, γ₂²−a2(X, X{circumflex over ( )})²>0  (121)

[0237] $\begin{array}{cc}\left[\frac{{\gamma }_2^2}{{\gamma }_2^2-1}-\frac{{\mathrm{a1}\left(X,X^{\bigwedge }\right)}^2{\gamma }_2^2}{{\gamma }_2^2-{\mathrm{a1}\left(X,X^{\bigwedge }\right)}^2}\right]X^{\bigwedge T}{C}_{11}^T{C}_{11}{X}^{\bigwedge }+\left[\frac{{\gamma }_2^2-{\mathrm{a2}\left(X,X^{\bigwedge }\right)}^2}{{\mathrm{a2}\left(X,X^{\bigwedge }\right)}^2{\gamma }_2^2}-\frac{{\gamma }_2^2-1}{{\gamma }_2^2}\right]\times X^{\bigwedge T}{\mathrm{PB}}_2\left(X\right)D_{122}^{-1}{D}_{122}^{-T}{B}_2^T\left(X\right){\mathrm{PX}}^{\bigwedge }\ge 0& \left(122\right)\\ {\Vert \left[\begin{array}{c}\mathrm{Z1}\\ \mathrm{Z2}\end{array}\right]\Vert }_2\le {\gamma }_1{\gamma }_2{\Vert W\Vert }_2& \left(123\right)\end{array}$  X′{circumflex over ( )}=(A+LC₂)X{circumflex over ( )}+(B₂(X)+LD₂₂(X))u−Ly  (124)

[0238] $\begin{array}{cc}u=-D_{122}^{-1}\times \left[\frac{{\gamma }_2^2-{\mathrm{a2}\left(X,X^{\bigwedge }\right)}^2}{{\mathrm{a2}\left(X,X^{\bigwedge }\right)}^2{\gamma }_2^2}{D}_{122}^{-T}{B}_2^T\left(X\right)P+C_{12}\right]X^{\bigwedge }& \left(125\right)\end{array}$

[0239] The observer gain L is expressed by a formula (126) shown below.

[0240] L=−QC₂^TΘ^TΘ  (126)

[0241] The sign “∥ ∥” denotes a Euclid-norm, and the sign “∥ ∥₂” denotes a norm in a square integrable function space L₂and is defined by an after-mentioned formula (127) for f(t)εL₂. $\begin{array}{cc}{\Vert f\left(t\right)\Vert }_2={\left({\int }_{\mathrm{t0}}^{\infty }{\Vert f\left(t\right)\Vert }^{2}dt\right)}^{1/2}& \left(127\right)\end{array}$

[0242] Θ is a positive definite matrix and Θ⁻¹exists. The use of Θ permits adjustment of the observer gain L. As is the case with the aforementioned state feedback control rule, the gain L of the controller can be adjusted using D₁₂₂. Furthermore, γ₁is an L2 gain of the observer and γ₂is an L2 gain of the controller. An L2 gain of the closed-loop system is determined as a product of γ₁and γ₂. Accordingly, the L2 gain of the system must be determined by suitably adjusting the observer and the controller.

[0243] The non-linear weights a1(X, X{circumflex over ( )}) and a2(X, X{circumflex over ( )}) satisfying the restrictive conditions of the aforementioned formulas (121) and (122) respectively are exemplified below.

[0244] a1(X, X{circumflex over ( )})=(γ₂²+(γ₂²−1)γ₂²εm₁(X, X{circumflex over ( )})X{circumflex over ( )}^T×

[0245] PB₂(X)D₁₂₂⁻¹D₁₂₂^−TB₂^T(X)PX{circumflex over ( )})^½/(γ₂²+

[0246] (γ₂²−1)m₁(X, X{circumflex over ( )})X{circumflex over ( )}^T×PB₂(X)D₁₂₂⁻¹D₁₂₂^−TB₂^T(X)PX{circumflex over ( )})^½  (128)

[0247] $\begin{array}{cc}\mathrm{a2}\left(X,X^{\bigwedge }\right)=\sqrt{\frac{1}{1+\mathrm{m1}\left(X,X^{\bigwedge }\right)X^{\bigwedge T}{C}_{11}^T{C}_{11}{X}^{\bigwedge }}}& \left(129\right)\end{array}$

[0248] In the aforementioned formulas (128) and (129), m1(X, X{circumflex over ( )}) is an arbitrary positive definite function, and ε is a positive constant satisfying inequalities ε<1 and εγ₂²>1. As a result of calculations made by the computer, it has been made possible to find the positive definite symmetric solution P as described above. Using the aforementioned formulas (128) and (129), the aforementioned formulas (124) and (125) are respectively converted into formulas (130) and (131) shown below.

[0249] X′{circumflex over ( )}=(A+LC₂)X{circumflex over ( )}+(B₂(X)+LD₂₂(X))u−Ly  (130)

[0250] $\begin{array}{cc}u=-D_{122}^{-1}\times [\frac{{\gamma }_2^2-1+{\gamma }_2^2{m}_1\left(X,X^{\bigwedge }\right)X^{\bigwedge T}{C}_{11}^T{C}_{11}{X}^{\bigwedge }}{{\gamma }_2^2}{X}^{\bigwedge }\times D_{122}^{-T}{B}_2^T(X)P+C_{12}]X^{\bigwedge }& \left(131\right)\end{array}$

[0251] Consequently, also in this case, a solution can easily be found by means of a known software in the same manner as in the case of the aforementioned state feedback control system. Therefore, this method makes it possible to easily find the positive definite symmetric solution P and to derive the estimated state quantity X′{circumflex over ( )} and the control rule u=k(y).

[0252] c1-2) Calculation Example of First Type

[0253] Next, a calculation example of the first target damping force Fd using the aforementioned control rule of the first type will be described. In this case, the tire displacement amount sensor 25 (the tire displacement amount sensors 25a, 25b, 25cand 25din FIG. 1) and the unsprung mass acceleration sensor 26 (the unsprung mass acceleration sensors 26a, 26b, 26cand 26din FIG. 1) are omitted, and the microcomputer executes a first damping force calculating routine shown in FIG. 17 instead of the first damping force calculating routine shown in FIG. 15. In other respects, this example is identical to the aforementioned first modification.

[0254] Also in this case, the execution of the first damping force calculating routine in FIG. 17 is started in step 200. In step 202a, detection signals indicative of a relative displacement amount (X_pw−X_pb) and a sprung mass acceleration X_pb″ are inputted from the relative displacement amount sensor 22 and the sprung mass acceleration sensor 21 respectively. In step 204a, as is the case with the aforementioned first modification example, a relative speed (X_pw′−X_pb′) and a sprung mass speed X_pb′ are calculated.

[0255] Next in step 206a, B_p2(X_p) and D_p12(X_p) are calculated according to after-mentioned formulas (132) and (133), which are identical to the aforementioned formulas (32) and (38) using the relative speed (X_pw′−X_pb′). B₂(X) is then calculated according to an after-mentioned formula (134), which is identical to the aforementioned formula (106) using B_p2(X_p) and D_p12(X_p). D₂₂(X) is calculated according to after-mentioned formulas (135) and (136), which are identical to the aforementioned formulas (41) and (113) using the relative speed (X_pw′−X_pb). $\begin{array}{cc}{B}_{\mathrm{p2}}\left(X_p\right)=\left[\begin{array}{c}\begin{array}{c}\begin{array}{c}0\\ 0\end{array}\\ -\left(X_{\mathrm{pw}}^{\prime }-X_{\mathrm{pb}}^{\prime }\right)/M_w\end{array}\\ \left(X_{\mathrm{pw}}^{\prime }-X_{\mathrm{pb}}^{\prime }\right)/M_b\end{array}\right]& \left(132\right)\\ D_{\mathrm{p12}}\left(X_p\right)=\left[\begin{array}{c}\begin{array}{c}\left(X_{\mathrm{pw}}^{\prime }-X_{\mathrm{pb}}^{\prime }\right)/M_b\\ 0\end{array}\\ 0\end{array}\right]& \left(133\right)\\ B_2\left(X\right)=\left[\begin{array}{c}\begin{array}{c}{B}_{\mathrm{p2}}\left(X_p\right)\\ B_w{D}_{\mathrm{p12}}\left(X_p\right)\end{array}\\ B_u\end{array}\right]& \left(134\right)\\ D_{\mathrm{p22}}\left(X_p\right)=\left[\begin{array}{c}\left(X_{\mathrm{pw}}^{\prime }-X_{\mathrm{pb}}^{\prime }\right)/M_b\\ 0\end{array}\right]& \left(135\right)\\ D_{22}\left(X\right)=D_{\mathrm{p22}}\left(X_p\right)& \left(136\right)\end{array}$

[0256] In the aforementioned formulas (132) through (135), M_w, M_b, B_wand B_udenote the same values or constant matrices as in the aforementioned first modification example.

[0257] After the aforementioned processing of step 206a, an estimated state quantity X{circumflex over ( )} and a control input u are calculated using formulas (137) and (138) which are identical to the aforementioned formulas (130) and (131), in the same manner as in the aforementioned first modification example.

[0258] X′{circumflex over ( )}=(A+LC₂)X{circumflex over ( )}+(B₂(X)+LD₂₂(X))u−Ly  (137)

[0259] $\begin{array}{cc}u=-D_{122}^{-1}\times [\frac{{\gamma }_2^2-1+{\gamma }_2^2{m}_1\left(X,X^{\bigwedge }\right)X^{\bigwedge T}{C}_{11}^T{C}_{11}{X}^{\bigwedge }}{{\gamma }_2^2}{X}^{\bigwedge }\times D_{122}^{-T}{B}_2^T\left(X\right)P+C_{12}]X^{\bigwedge }& \left(138\right)\end{array}$

[0260] In the aforementioned formula (137), A is a constant matrix that is preliminarily stored in the microcomputer and determined by the aforementioned formulas (104), (30) and (37). L is a constant matrix that is preliminarily stored in the microcomputer and defined by the aforementioned formula (126), and is a gain of the observer that is determined by the positive definite symmetric matrix Q, the constant matrix C₂determined by the aforementioned formulas (39) and (111), and the positive definite matrix Θ. C₂is also the aforementioned constant matrix that is preliminarily stored in the microcomputer. B₂(X) and D₂₂(X) are matrices calculated in the aforementioned step 206a. Further, y is an observed value which represents, in the first type, the relative displacement amount (X_pw−X_pb) inputted through the processing of the aforementioned step 202aand the sprung mass speed X_pb′ calculated through the processing of the aforementioned step 204a.

[0261] In the aforementioned formula (138), D₁₂₂is a coefficient matrix that is defined by the aforementioned formula (110) and relates to the frequency weight W_u(S) set in the aforementioned formula (48), and is a constant matrix that is preliminarily stored in the microcomputer. γ₂is a positive constant satisfying the aforementioned inequality γ₂>1. m1(X, X{circumflex over ( )}) is an arbitrary positive definite function, and an algorithm concerning the function is preliminarily stored in the microcomputer. The positive definite function m1(X) may be set to a positive constant, for example, “1.0”. C₁₁is defined by the aforementioned formulas (37) and (107). In other words, C₁₁is a constant matrix which is preliminarily stored in the microcomputer and which is defined by the mass M_wof the wheel WH, the mass M_bof the vehicle body BD, the spring constant K_sof the spring 11, the linear damping coefficient C_S0of the damper 12 and the coefficient matrices C_wand D_wrelating to the frequency weight W_s(S) set in the aforementioned formula (46). B₂(X) is a matrix calculated in the aforementioned step 206a. P is a positive definite symmetric solution satisfying the aforementioned formulas (119) and (120) and is a constant matrix that is preliminarily stored in the microcomputer. C₁₂, which is defined by the aforementioned formula (109), is a constant matrix that includes the coefficient matrix C_urelating to the frequency weight W_u(S) set in the aforementioned formula (48) and is preliminarily stored in the microcomputer.

[0262] After the processing of the aforementioned step 210a, through processings of the same steps 212 and 214 as in the first modification example, an overall target damping coefficient C_sof the damper 12 and a first target damping force Fd are calculated. In step 216, the execution of the first damping force calculating routine is terminated.

[0263] c2-1) Designing Example of Control System of Second Type

[0264] The second type refers to a case where B₂(X) in the aforementioned formula (106) and D₂₂(X) in the aforementioned formula (113) are unknown functions, that is, a case where the relative speed (X_pw′−X_pb′) is unknown and the observer gain L is a constant matrix.

[0265] In a bilinear system of this kind, B₂(X) and D₂₂(X) are linear functions of X. In consideration of this, the generalized model expressed by the aforementioned formulas (115) through (118) is rewritten, whereby formulas (139) through (142) are established as shown below. It is to be noted herein that B₂₀, D₂₂₀and d₁₂₂are constant matrices.

[0266] X′=AX+B₁w+B₂₀X_u  (139)

[0267] Z₁=a1(X{circumflex over ( )})C₁₁X  (140)

[0268] Z₂=a2(X{circumflex over ( )})C₁₂X+a2(X)d₁₂₂u  (141)

[0269] y=C₂X+D₂₁W+D₂₂₀X_u  (142)

[0270] An attempt will be made to design a non-linear H∞ output feedback control rule for this generalized model. In the case where the observer gain L is given as a constant matrix, an output feedback control rule can be designed according to the following theorem. That is (1) if, given that γ₁is a positive constant satisfying an inequality γ₁²I−D₂₁^TΘ^TΘD₁₂>0, that γ₂is a positive constant satisfying an inequality γ₂>1 and that a positive constant ε satisfying an inequality ε_I²−u²>0 exists, positive definite symmetric matrices P and Q and a positive definite matrix Θ which satisfy Riccati's inequality for designing the observer (observer gain) of an after-mentioned formula (143) and Riccati's inequality for designing the controller of an after-mentioned formula (144) exist, and (2) if the non-linear weights a1(X, X{circumflex over ( )}) and a2(X, X{circumflex over ( )}) satisfy restrictive conditions prescribed in formulas (145) and (146) shown below, one of the control rules according to an after-mentioned formula (147) is given by formulas (148) and (149) shown below.

[0271] AQ+QA^T+Q(C₁₁^TC₁₁+C₁₂^TC₁₂−C₂^TΘ^TΘC₂+

[0272] ε²I+ε₁²D₂₂₀^TΘ^T(I+ΘD₂₁×(γ₁²I−D₂₁^TΘ^TΘ

[0273] D₂₁)⁻¹D₂₁^TΘ^T)ΘD₂₂₀)Q+B₁(γ₁²I−D₂₁^TΘ^TΘD₂₁)⁻¹B₁^T+

[0274] (B₁(γ₁²I−D₂₁^TΘ^TΘD₂₁)⁻¹D₂₁^TΘ^TΘD₂₂₀+B₂₀)×

[0275] (D₂₂₀^TΘ^TΘD₂₁(γ₁²I−D₂₁^TΘ^TΘD₂₁)⁻¹B1^T+B₂₀^T)<0  (143)

[0276] $\begin{array}{cc}\mathrm{PA}+A^{T}P+\frac{1}{{\gamma }_2^2}\mathrm{PL}{\Theta }^{-1}{\Theta }^{-T}{L}^{T}P+\frac{{\gamma }_2^2}{{\gamma }_2^2-1}\left(C_{11}^T{C}_{11}+C_{12}^T{C}_{12}\right)\prec 0& \left(144\right)\end{array}$  γ₂²−a1(X, X{circumflex over ( )})²>0, γ₂²−a2(X, X{circumflex over ( )})²>0  (145)

[0277] $\begin{array}{cc}\left[\frac{{\gamma }_2^2}{{\gamma }_2^2-1}-\frac{{\left(\mathrm{a1}\left(X,X^{\bigwedge }\right)\right)}^2{\gamma }_2^2}{{\gamma }_2^2-{\left(\mathrm{a1}\left(X,X^{\bigwedge }\right)\right)}^2}\right]X^{\bigwedge T}{C}_{11}^T{C}_{11}{X}^{\bigwedge }+[\frac{{\gamma }_2^2-{\left(\mathrm{a2}\left(X,X^{\bigwedge }\right)\right)}^2}{{\left(\mathrm{a2}\left(X,X^{\bigwedge }\right)\right)}^2{\gamma }_2^2}-\frac{{\gamma }_2^2-1}{{\gamma }_2^2}]\frac{1}{d_{122}^2}\times X^{\bigwedge T}{\mathrm{PB}}_{20}{X}^{\bigwedge }{X}^{\bigwedge T}{B}_{20}^T{\mathrm{PX}}^{\bigwedge }\ge 0& \left(146\right)\end{array}$  ∥[Z₁^TZ₂^T]^T∥₂≦γ₁γ₂∥W∥₂  (147)

[0278] X′{circumflex over ( )}=(A+L(u)C₂)X{circumflex over ( )}+(B₂₀+L(u)D₂₂₀)X{circumflex over ( )}u−L(u)y  (148)

[0279] $\begin{array}{cc}u=-\frac{1}{d_{122}}\times \left[\frac{{\gamma }_2^2-{\mathrm{a2}\left(X,X^{\bigwedge }\right)}^2}{{\mathrm{a2}\left(X,X^{\bigwedge }\right)}^2{\gamma }_2^2{d}_{122}}{X}^{\bigwedge T}{B}_{20}^{T}P+C_{12}\right]X^{\bigwedge }& \left(149\right)\end{array}$

[0280] The observer gain L(u) is expressed by a formula (150) shown below.

[0281] L(u)=−QC₂^TΘ^TΘ  (150)

[0282] Θ is a positive definite matrix and Θ⁻¹exists. The use of Θ permits adjustment of the observer gain L(u). As is the case with the aforementioned state feedback control rule, the gain L of the controller can be adjusted using d₁₂₂.

[0283] The non-linear weights a1(X, X{circumflex over ( )}) and a2(X, X{circumflex over ( )}) satisfying the restrictive conditions of the aforementioned formulas (145) and (146) respectively are exemplified in formulas (151) and (152) shown below.

[0284] a1(X, X{circumflex over ( )})=(γ₂²d₁₂₂²+(γ₂²−1)γ₂²ε m1(X, X{circumflex over ( )})×

[0285] X{circumflex over ( )}^TPB₂₀X{circumflex over ( )}X{circumflex over ( )}^TB₂₀^TPX{circumflex over ( )})^½/(γ₂²d₁₂₂²+

[0286] (γ₂²−1)m1(X, X{circumflex over ( )})×X{circumflex over ( )}^TPB₂₀X{circumflex over ( )}X{circumflex over ( )}^TB₂₀^TPX{circumflex over ( )})^½  (151)

[0287] $\begin{array}{cc}\mathrm{a2}\left(X,X^{\bigwedge }\right)=\sqrt{\frac{1}{1+\mathrm{m1}\left(X,X^{\bigwedge }\right)X^{\bigwedge T}{C}_{11}^T{C}_{11}{X}^{\bigwedge }}}& \left(152\right)\end{array}$

[0288] In the aforementioned formulas (151) and (152), m1(X, X{circumflex over ( )}) is an arbitrary positive definite function, and ε is a positive constant satisfying inequalities ε<1 and εγ₂²>1. As a result of calculations made by the computer, it has been made possible to find the positive definite symmetric solution P as described above. Using the aforementioned formulas (151) and (152), the aforementioned formulas (148) and (149) are respectively converted into formulas (153) and (154) shown below.

[0289] X′{circumflex over ( )}=(A+L(u)C₂)X{circumflex over ( )}+(B₂₀+L(u)D₂₂₀)X{circumflex over ( )}u−L(u)y  (153)

[0290] $\begin{array}{cc}u=-\frac{1}{d_{122}}\times \left[\frac{{\gamma }_2^2-1+{\gamma }_2^2{m}_1\left(X,X^{\bigwedge }\right)X^{\bigwedge T}{C}_{11}^T{C}_{11}{X}^{\bigwedge }}{d_{122}{\gamma }_2^2}\times B_{20}^{T}P+C_{12}\right]X^{\bigwedge }& \left(154\right)\end{array}$

[0291] Consequently, also in this case, a solution can easily be found by means of a known software in the same manner as in the case of the aforementioned state feedback control system. Therefore, this method makes it possible to easily find the positive definite symmetric solution P and to derive the estimated state quantity X′{circumflex over ( )} and the control rule u=k(y).

[0292] c2-2) Calculation Example of Second Type

[0293] Next, a calculation example of the first target damping force Fd using the control rule of the second type will be described. In this case, the relative displacement amount sensor 22 of the first type shown in FIG. 6 (the relative displacement amount sensors 22a, 22b, 22cand 22dshown in FIG. 1) is omitted. The inputting of the relative displacement amount (X_pw−X_pb) from the relative displacement amount sensor 22 in step 202aof FIG. 17, the calculation of the relative speed (X_pw′−X_pb′) in step 204aof FIG. 17, and the arithmetic processing in step 206aare omitted. Then calculations are made according to the aforementioned control rule of the second type.

[0294] Also in this case, the execution of the first damping force calculating routine in FIG. 17 is started in step 200. A sprung mass acceleration X_pb″ is inputted in step 202a, and a sprung mass speed X_pb′ is calculated in step 204a. In step 210a, a control input u and an estimated state quantity X′{circumflex over ( )} including an estimation of the relative speed (X_pw′−X_pb′) are calculated using formulas (155) and (156) which are identical to the aforementioned formulas (153) and (154) respectively.

[0295] X′{circumflex over ( )}=(A+L(u)C₂)X{circumflex over ( )}+(B₂₀+L(u)D₂₂₀)X{circumflex over ( )}u−L(u)y  (155)

[0296] $\begin{array}{cc}u=-\frac{1}{d_{122}}\times \left[\frac{{\gamma }_2^2-1+{\gamma }_2^2{m}_1\left(X,X^{\bigwedge }\right)X^{\bigwedge T}{C}_{11}^T{C}_{11}{X}^{\bigwedge }}{d_{122}{\gamma }_2^2}\times B_{20}^{T}P+C_{12}\right]X^{\bigwedge }& \left(156\right)\end{array}$

[0297] In the aforementioned formulas (155) and (156), A, L, C₂, T2, m1(X, X{circumflex over ( )}), C₁₁, P and C₁₂are the same as in the case of the first type. B₂₀, D₂₂₀, d₁₂₂are the aforementioned appropriate matrices that are preliminarily stored in the microcomputer. In this case, y is an observed value which represents the sprung mass speed X_pb′ calculated through the aforementioned processing of step 204a.

[0298] After the aforementioned processing of step 210a, through the processings of steps 212 and 214 substantially identical to those of the first type, an overall damping force C_sof the damper 12 is calculated and a first target damping force Fd is calculated. In this case, when calculating the first target damping force Fd in step 214, the estimated relative speed (X_pw′{circumflex over ( )}−X_pb′{circumflex over ( )}) calculated in step 210ais utilized.

[0299] c3-1) Designing Example of Control System of Third Type

[0300] The third type also refers to a case where B₂(X) in the aforementioned formula (106) and D₂₂(X) in the aforementioned formula (113) are unknown functions, that is, a case where the relative speed (X_pw′−X_pb′) is unknown and the observer gain L is a function matrix.

[0301] Also in the third type, an attempt will be made to design a non-linear H∞ output feedback control rule for the generalized model expressed by the aforementioned formulas (139) through (142) of the second type. In the case where the observer gain L is given as a function of the control input u, an output feedback control rule can be designed according to the following theorem. That is (1) if, given that γ₁is a positive constant satisfying an inequality γ₁²I−D₂₁^TΘ^TΘD₁₂>0, that γ₂is a positive constant satisfying an inequality γ₂>1 and that a positive constant ε satisfying an inequality ε₁²−u²>0 exists, positive definite symmetric matrices P and Q and a positive definite matrix Θ which satisfy Riccati's inequality for designing the observer (observer gain) of an after-mentioned formula (157) and Riccati's inequality for designing the controller of an after-mentioned formula (158) exist, and (2) if the non-linear weights a1(X, X{circumflex over ( )}) and a2(X, X{circumflex over ( )}) satisfy restrictive conditions prescribed in formulas (159) and (160) shown below, one of the control rules according to an after-mentioned formula (161) is given by formulas (162) and (163) shown below.

[0302] AQ+QA^T+Q(C₁₁^TC₁₁+C₁₂^TC₁₂+ε²I)Q+B₁(γ₁²I−D₂₁^TΘ^TΘD₂₁)⁻¹B₁^T+B₂₀B₂₀^T<0  (157)

[0303] $\begin{array}{cc}\mathrm{PA}+A^{T}P+\frac{1}{{\gamma }_2^2}{\mathrm{PL}}_1{\Theta }^{-1}{\Theta }^{-T}{L}_1^{T}P+\frac{{ϵ}_1^2{d}_{122}^2}{{\gamma }_2^2}{\mathrm{PL}}_2{\Theta }^{-1}{\Theta }^{-T}{L}_2^{T}P+\frac{{\gamma }_2^2}{{\gamma }_2^2-1}\left(C_{11}^T{C}_{11}+C_{12}^T{C}_{12}\right)\prec 0& \left(158\right)\end{array}$  γ₂²−a1(X, X{circumflex over ( )})²>0, γ₂²−a2(X, X{circumflex over ( )})²>0  (159)

[0304] $\begin{array}{cc}\left[\frac{{\gamma }_2^2-{\mathrm{a2}\left(X,X^{\bigwedge }\right)}^2}{{\mathrm{a2}\left(X,X^{\bigwedge }\right)}^2{\gamma }_2^2}-\frac{{\gamma }_2^2-1}{{\gamma }_2^2}\right]\frac{1}{d_{122}^2}{X}^{\bigwedge T}P\times \left(B_{20}+L_1{\Theta }^{-1}{\Theta }^{-T}{L}_2^{T}P\right)X^{\bigwedge }\times X^{\bigwedge T}\left(B_{20}^T+{\mathrm{PL}}_2{\Theta }^{-1}{\Theta }^{-T}{L}_1^T\right){\mathrm{PX}}^{\bigwedge }+[\frac{{\gamma }_2^2}{{\gamma }_2^2-1}-\frac{{\mathrm{a1}\left(X,X^{\bigwedge }\right)}^2{\gamma }_2^2}{{\gamma }_2^2-{\mathrm{a1}\left(X,X^{\bigwedge }\right)}^2}]X^{\bigwedge T}{C}_{11}^T{C}_{11}{X}^{\bigwedge }\ge 0& \left(160\right)\end{array}$  ∥[Z₁^TZ₂^T]^T∥₂≦γ₁γ₂∥W∥₂  (161)

[0305] X′{circumflex over ( )}=(A+L₁C₂)X{circumflex over ( )}+(B₂₀+L₂D₂₂₀)X{circumflex over ( )}u−(L₁+L₂u)y  (162)

[0306] $\begin{array}{cc}u=-\frac{1}{d_{122}}\times [\frac{{\gamma }_2^2-{a_2\left(X,X^{\bigwedge }\right)}^2}{{a_2\left(X,X^{\bigwedge }\right)}^2{\gamma }_2^2{d}_{122}}{X}^{\bigwedge T}\times \left(B_{20}^{T}P+\frac{1}{{\gamma }^2}{\mathrm{PL}}_1{\Theta }^{-1}{\Theta }^{-T}{L}_2^{T}P\right)+C_{12}]X^{\bigwedge }& \left(163\right)\end{array}$

[0307] The observer L(u) is expressed by a formula (164) shown below.

[0308] L(u)=−QC₂^TΘ^TΘ−uQD₂₂₀^TΘ^TΘ=L₁+uL₂  (164)

[0309] L₁and L₂shown in the aforementioned formula (164) are respectively expressed by formulas (165) and (166) shown below.

[0310] L₁=−QC₂^TΘ^TΘ  (165)

[0311] L₂=−QD₂₂₀^TΘ^TΘ  (166)

[0312] Θ is a positive definite matrix and Θ⁻¹exists. The use of Θ permits adjustment of the observer gain L. As is the case with the aforementioned state feedback control rule, the gain L of the controller can be adjusted using d₁₂₂.

[0313] The non-linear weights a1(X, X{circumflex over ( )}) and a2(X, X{circumflex over ( )}) satisfying the restrictive conditions of the aforementioned formulas (159) and (160) respectively are exemplified respectively in formulas (167) and (168) shown below.

[0314] a1(X, X{circumflex over ( )})=(γ₂²d₁₂₂²+(γ₂²−1)γ₂²ε m1(X, X{circumflex over ( )})X{circumflex over ( )}^TP×

[0315] (B₂₀+L₁Θ⁻¹Θ^−TL₂^TP)X{circumflex over ( )}X{circumflex over ( )}^T×

[0316] (B₂₀^T+PL₂Θ⁻¹Θ^−TL₁^T)PX{circumflex over ( )})^½/(γ₂²d₁₂₂²+(γ₂²−1)m1(X, X{circumflex over ( )})X{circumflex over ( )}^TP×

[0317] (B₂₀+L₁Θ⁻¹Θ^−TL₂^TP)X{circumflex over ( )}X{circumflex over ( )}^T×(B₂₀^T+PL₂Θ⁻¹Θ^−TL₁^T)PX{circumflex over ( )})^½  (167)

[0318] $\begin{array}{cc}\mathrm{a2}\left(X,X^{\bigwedge }\right)=\sqrt{\frac{1}{1+\mathrm{m1}\left(X,X^{\bigwedge }\right)X^{\bigwedge T}{C}_{11}^T{C}_{11}{X}^{\bigwedge }}}& \left(168\right)\end{array}$

[0319] In the aforementioned formulas (167) and (168), m1(X, X{circumflex over ( )}) is an arbitrary positive definite function, and ε is a positive constant satisfying inequalities ε<1 and εγ₂²>1. As a result of calculations made by the computer, it has been made possible to find the positive definite symmetric solution P as described above. Using the aforementioned formulas (167) and (168), the aforementioned formulas (162) and (163) are respectively converted into formulas (169) and (170) shown below.

[0320] X′{circumflex over ( )}=(A+L₁C₂)X{circumflex over ( )}+(B₂X{circumflex over ( )}+L₂D₂₂X{circumflex over ( )})u−L(u)y  (169)

[0321] $\begin{array}{cc}u=-\frac{{\gamma }_2^2-1+{\gamma }_2^2{m}_1\left(X,X^{\bigwedge }\right)X^{\bigwedge T}{C}_{11}^T{C}_{11}{X}^{\bigwedge }}{d_{122}^2{\gamma }_2^2}\times \left(B_{20}^T+\frac{1}{{\gamma }_2^2}{\mathrm{PL}}_1{\Theta }^{-1}{\Theta }^{-T}{L}_2^T\right){\mathrm{PX}}^{\bigwedge }-\frac{1}{d_{122}}{C}_{12}{X}^{\bigwedge }& \left(170\right)\end{array}$

[0322] Consequently, also in this case, a solution can easily be found by means of a known software in the same manner as in the case of the aforementioned state feedback control system. Therefore, this method makes it possible to easily find the positive definite symmetric solution P and to derive the estimated state quantity X′{circumflex over ( )} and the control rule u=k(y).

[0323] c3-2) Calculation Example of Third Type

[0324] Next, a calculation example of the first target damping force Fd using the control rule of the third type will be described. In this case, the construction is the same as in the aforementioned case of the second type.

[0325] Also in this case, the execution of the first damping force calculating routine is started in step 200. After processings in steps 202aand 204awhich are substantially identical to those of the aforementioned first type, an estimated state quantity X′{circumflex over ( )} and a control input u are calculated using formulas (171) and (172) which are identical to the aforementioned formulas (169) and (170), in the same manner as in the aforementioned case of the second type.

[0326] X′{circumflex over ( )}=(A+L₁C₂)X{circumflex over ( )} ×(B₂X{circumflex over ( )}+L₂D₂₂X{circumflex over ( )})u−L(u)y  (171)

[0327] $\begin{array}{cc}u=-\frac{{\gamma }_2^2-1+{\gamma }_2^2{m}_1\left(X,X^{\bigwedge }\right)X^{\bigwedge T}{C}_{11}^T{C}_{11}{X}^{\bigwedge }}{d_{122}^2{\gamma }_2^2}\times \left(B_{20}^T+\frac{1}{{\gamma }_2^2}{\mathrm{PL}}_1{\Theta }^{-1}{\Theta }^{-T}{L}_2^T\right){\mathrm{PX}}^{\bigwedge }-\frac{1}{d_{122}}{C}_{12}{X}^{\bigwedge }& \left(172\right)\end{array}$

[0328] In the aforementioned formulas (171) and (172), A, C₂, B₂₀, D₂₂₀, γ₂, m1(X, X{circumflex over ( )}), C₁₁, d₁₂₂, P, C₁₂are the same as those in the aforementioned case of the second type. L, L₁and L2 are gains defined by the aforementioned formulas (164) through (166). Furthermore, also in this case, y is an observed value which represents the sprung mass speed X_pb′ calculated through the aforementioned processing of step 204a.

[0329] After the aforementioned processing in step 210a, through the processings of steps 212 and 214 substantially identical to those of the second type, an overall damping force C_sof the damper 12 is calculated and a first target damping force Fd is calculated. In step 216, the execution of the first damping force calculating routine is terminated.

[0330] d. Third Modification

[0331] d1. Designing Example of Non-Linear H∞ Control System of Kalman Filter Base

[0332] For the aforementioned model marked with a, an attempt will be made to design an output feedback system which employs a Kalman filter as an observer, on the condition that the bilinear terms B_p2(X_p) and D_p2(X_p) be known, namely, that the relative speed (X_pw′−X_pb′) be observable.

[0333] In the third modification, the same reference characters as in the aforementioned case of the second modification are used, and the coefficients and variables concerning the model are accompanied by a suffix p. A state space expression of the suspension system is indicated by formulas (173) and (174) shown below.

[0334] X_p′=A_pX_p+B_p1w₁+B_p2(X_p)u  (173)

[0335] y_p=C_pX_p+D_p1w₂+D_p2(X_p)u  (174)

[0336] If D_p1=I, the Kalman filter in the case of t→∞ is expressed according to a formula (175) shown below.

[0337] X₀′=A_pX₀+B_p2u+K(C_pX₀+D_p2(X_p)u−y)  (175)

[0338] X₀and X₀′ are estimated state quantities in the Kalman filter, and the filter gain K is expressed by a formula (176) shown below.

[0339] K=−ΣC_p^TW⁻¹  (176)

[0340] The estimated error covariance Σ is a positive definite symmetric solution of Riccati's equation (177) shown below.

[0341] A_pΣ+ΣA_p^T+B_p1VB_p1^T−ΣC_p^TW^−TC_pΣ=0  (177)

[0342] V is a covariance matrix of w₁, and W is a covariance matrix of w₂.

[0343] FIG. 18 shows a block diagram of a generalized model of this system. In this case, “a product obtained by multiplying the estimated state quantity X₀by the frequency weight W(S)”, which is an output from the observer, and “a product obtained by multiplying the control input u by the frequency W_u(S)” are used as evaluated outputs Z. In other words, the Kalman filter is used herein as a detector, and the control system is so designed as to reduce an output from the Kalman filter. The third modification is different from the aforementioned first and second modifications in this respect. However, if the state has been estimated successfully, it is considered that the performance equivalent to that of the first and second modifications will be obtained. A state space expression of the system indicated by the block diagram in FIG. 18 is expressed according to formulas (178) through (184) shown below.

[0344] X_p′=A_pX_p+B_p1w₁+B_p2(X_p)u  (178)

[0345] X₀′=A_pX₀+B_p2(X_p)u+L(C₂X₀+D_p2(X_p)u−y)  (179)

[0346] y=C_pX_p+D_p1w₂+D_p2(X_p)u  (180)

[0347] X_w′=A_wX_w+B_wC_SX₀  (181)

[0348] Z₁=a1(X_p, X₀, X_w, X_u)(C_wX_w+D_wC_SX₀)  (182)