Control system for resilient support mechanism such as vehicle suspension mechanism

[0002] 1. Field of the Invention

[0003] The present invention relates to a resilient support mechanism for resiliently supporting the component structure of a wheeled vehicle. More particularly, the present invention is concerned with a control system for a resilient support mechanism such as a suspension mechanism disposed between an unsprung mass and a sprung mass of the wheeled vehicle for controlling a damping force or a damping coefficient of a shock absorber or damper device assembled therein.

[0004] 2. Discussion of the Prior Art

[0005] In a conventional suspension mechanism of a wheeled vehicle, an amount of movement state of a sprung mass or an unsprung mass of the vehicle is detected to determine a target damping force or a target damping coefficient on a basis of the detected amount of movement state thereby to adjust the damping force or damping coefficient of a shock absorber or damper device assembled in the suspension mechanism to the target damping force or damping coefficient. Disclosed in Japanese Patent Laid-open Publication No. 10-119528 is a control system for the suspension mechanism in which the well-known sky nook theory is applied to determine a target damping coefficient based on acceleration of the sprung mass and relative velocity of the sprung mass to the unsprung mass in a vertical direction.

[0006] In the suspension mechanism described above, however, the damping force of the shock absorber or damper device is defined by the product of the relative velocity of the sprung mass to the unsprung mass and the damping coefficient, while the damping coefficient changes nonlinearly in accordance with the relative velocity of the sprung mass. For this reason, the design of the control system becomes very complicated. For example, it has been considered to estimate a plant indicative of a state space in the suspension mechanism for design of the control system. However, as the plant is bilinear, it is obliged to apply an approximate law to a range where a control input would not be realized in the resilient support mechanism such as a suspension mechanism, even if a linear control theory was applied to the bilinear system. For this reason, a control specification (a norm condition) given at a design stage may not be theoretically satisfied. As a result, the control input becomes discontinuous to cause a sense of incongruity in control of the suspension mechanism.

[0007] It is, therefore, a primary object of the present invention to provide a control system for a resilient support mechanism such as a vehicle suspension mechanism capable of enhancing riding comfort of the vehicle without causing any sense of incongruity in control of the suspension mechanism.

[0008] According the present invention, the object is accomplished by providing a control system for a resilient support mechanism including a damper or a shock absorber assembled therein for resiliently supporting thereon an object, wherein a control theory capable of providing a specification for design of the control system in a predetermined frequency range is applied to a nonlinear plant to calculate a target damping force and to adjust an actual damping force of the damper or shock absorber to the calculated target damping force. In a practical embodiment of the present invention, the resilient support mechanism is in the form of a resilient support mechanism in a wheeled vehicle for resiliently supporting a component member of the vehicle such as a vehicle body, a prime mover or the like. In the control system, either one of a nonlinear H_∞ state feedback control, a nonlinear H_∞ output feedback control or a nonlinear H_∞ control using a Kalman filter may be applied as the control theory.

[0009] With the control system, a norm condition given at a design stage is satisfied even in a resilient support mechanism in the form of a bilinear system such as a suspension mechanism of a wheeled vehicle so that a control input continuously changes to control a damper in the resilient support mechanism without causing any sense of incongruity in control of the support mechanism.

[0010] In the case that the present invention is applied to a suspension mechanism of a wheeled vehicle, it is preferable that the control system comprises detection means for detecting a state amount related to vertical movement of the sprung mass and unsprung mass members, memory means for memorizing a positive definite symmetric solution obtained on a basis of the control theory, and calculation means for calculating the target damping force based on the memorized positive definite symmetric solution and the detected state amount. Alternatively, the control system may comprises detection means for detecting a portion of a plurality of state amounts related to vertical movement of the sprung and unsprung mass members, estimation means for estimating the other portion of the state amounts by using an observer, memory means for memorizing a positive definite symmetric solution obtained on a basis of the control theory, and calculation means for calculating the target damping force based on the memorized positive definite symmetric solution, the detected state amount and the estimated state amount. In this case, it is desirable that a vertical displacement amount of a tire of a road wheel, a relative vertical displacement amount of the sprung mass member to the unsprung mass member, a vertical velocity of the unsprung mass member and a vertical velocity of the sprung mass member are detected as the state amounts.

[0011] With the above arrangement, a norm condition given at a design stage is satisfied even in such a bilinear control system described above so that a control input continuously changes to enhance running stability and riding comfort of the vehicle without causing any sense of incongruity in control of the suspension mechanism.

[0012] According to an aspect of the present invention, there is provided a control system for a resilient support mechanism such as a suspension mechanism of a wheeled vehicle wherein a damping coefficient of a damper assembled in the resilient support mechanism is divided into a linear portion and a nonlinear portion, and wherein the control system comprises first calculation means for calculating the nonlinear portion of the damping coefficient on a basis of a nonlinear H_∞ control theory and second calculation means for calculating a target damping coefficient based on synthesis of the linear portion of the damping coefficient and the calculated nonlinear portion, in the control system, a target damping force may be calculated by detecting or estimating a relative velocity of a sprung mass member to an unsprung mass member and multiplying the relative velocity by the calculated target damping coefficient.

[0013] According to another aspect of the present invention, there is provided a control system for a resilient support mechanism wherein a damping coefficient of a damper assembled in the support mechanism is divided into a linear portion and a nonlinear portion, and wherein each gain of the linear portion and nonlinear portion is defined in such a manner that a target damping force is determined in a range of an actual damping force produced by the damper. In the control system, it is preferable that the linear portion of the damping coefficient of the damper is defined to be located between a minimum damping force of the damper and a maximum damping force of the damper. With the control system, a control specification or a norm condition given at a design state is satisfied in such a manner that a control input continuously changes to control the support mechanism without causing any sensor of incongruity.

[0014] According to a further aspect of the present invention, there is provided a control system for a resilient support mechanism wherein a damper assembled in the support mechanism is constructed to be switched over at plural steps, wherein a damping coefficient of the damper is divided into a linear portion and a nonlinear portion, and wherein the linear portion of the damping coefficient is determined in such a manner that a damping force defined by the linear portion becomes approximately equal to a damping force produced by the damper at its either one of plural steps in a small range. In the control system, the linearity of the damping force relative to the relative velocity becomes strong in the small rang such that the nonlinear portion becomes approximately zero. With the control system, the changeover frequency of the damper is restrained to enhance the durability of the damper.

[0015] According an aspect of the present invention, there is provided a control system for a resilient support mechanism wherein a damping coefficient of a damper assembled in the support mechanism is divided into a linear portion and a nonlinear portion, wherein the nonlinear portion is applied as a control input to a generalized plant estimated to calculate a target damping force, and wherein the control input is applied with a predetermined frequency weight.

[0016] According to another aspect of the present invention, there is provided a control system for a suspension mechanism of a wheeled vehicle wherein a physical amount related to vertical movement of sprung and unsprung mass members is applied as an evaluation output of a generalized plant estimated to calculate a target damping force, and wherein the physical amount is applied with a predetermined frequency weight. In the control system, a vertical acceleration, a vertical velocity and a vertical displacement amount of the sprung mass member influential in resonance of the sprung mass member, a vertical velocity of the unsprung mass member, a relative velocity of the unsprung mass member to the sprung mass member, a tire displacement amount influential in resonance of the unsprung mass member can be utilized as the physical amount. Preferably, the vertical velocity of the sprung mass member influential in resonance of the sprung mass member and the relative velocity of the sprung mass member to the unsprung mass member influential in resonance of the unsprung mass member are utilized as that physical amount to restrain undesired influence to the vehicle in accordance with frequency ranges.

[0017] In the control system described above, it is preferable that the physical amount includes plural kinds of physical amounts, wherein each maximum range of frequency weights applied to the physical amounts is determined without causing any mutual interference. With such arrangement of the control system, it is able to independently restrain each undesired influence of the physical amounts to the suspension mechanism. In this case, it is preferable that the physical amounts includes at least two factors selected from a vertical velocity of the sprung mass member, a relative velocity of the sprung mass member to the unsprung mass member and a vertical acceleration of the sprung mass member to eliminate each resonance of the sprung and unsprung mass members and deterioration of riding comfort of the vehicle.

[0018] According to a further aspect of the present invention, there is provided a control system for a suspension mechanism including a damper disposed between an unsprung mass member and a sprung mass member of a wheeled vehicle, wherein a damping coefficient of the damper is divided into a linear portion and a nonlinear portion, wherein a control theory capable of providing a specification for design of the control system in a predetermined frequency range is applied to a nonlinear plant to calculate a target damping force and to adjust an actual damping force of the damper to the calculated target damping force, and wherein a travel speed of the vehicle or a mass of the sprung mass member is detected to control the linear portion of the damping coefficient in accordance with the travel speed of the vehicle or the mass of the sprung mass member. Since in the control system, the linear portion of the damping coefficient of the damper is varied in accordance with the travel speed of the vehicle or the mass of the sprung mass member, a sensory characteristic in a desired performance of the vehicle is enhanced.

[0019] In the control system, it is preferable that a change of the linear portion of the damping coefficient is prohibited when the calculated target damping force chances more that a predetermined value in response to change of the linear portion of the damping coefficient, and it is also preferable that a change of the frequency weight in the generalized plant is prohibited when the calculated target damping force changes more than a predetermined value due to change of the frequency weight. Such arrangement of the control system is useful to eliminate discontinuity of the target damping force and a second of incongruity in control of the damping force.

[0020] Other objects, features and advantages of the present invention will be more readily appreciated from the following detailed description of preferred embodiments thereof when taken together with the accompanying drawings, in which:

[0021] Other objects, features and advantages of the present invention will be more readily appreciated from the following detailed description of preferred embodiments thereof when taken together with the accompanying drawings, in which:

[0022] FIG. 1 is a schematic illustration of a quarter car model widely used to analyze the basic properties of suspension systems in wheeled vehicles;

[0023] FIG. 2 is a block diagram of a non-linear H_∞ state feedback control system in a first embodiment of the present invention;

[0024] FIGS. 3(A) to 3(C) each are a graph showing a gain in relation to a frequency region;

[0025] FIG. 3(D) is a graph showing a frequency weight in relation to a variable damping coefficient;

[0026] FIG. 4 is an image view showing functional effects based on a nonlinear H_∞ control theory;

[0027] FIG. 5(A) illustrates a Lissajous's figure showing a characteristic of a difference (F−V) between a damping force and a relative velocity under control of a damping force according to the present invention;

[0028] FIG. 5(B) is a Lissajous's figure showing a characteristic of a difference (F−V) between a damping force and a relative velocity under a conventional skyhook control;

[0029] FIG. 6 is a block diagram illustrating a damping force control system in a wheeled vehicle;

[0030] FIG. 7 is a flow chart of a control program executed by a microcomputer shown in FIG. 6;

[0031] FIG. 8 is a graph showing a data characteristic in a relative velocity-damping force table stored in the microcomputer of FIG. 6;

[0032] FIG. 9 is a block diagram of a generalized plant of a nonlinear H_∞ output feedback control system in a second embodiment of the present invention;

[0033] FIG. 10 is a flow chart of a control program executed by the microcomputer of FIG. 6 in the second embodiment;

[0034] FIG. 11 is a block diagram of a generalized plant of a nonlinear H_∞ output feedback control system of the Kalman filter base;

[0035] FIG. 12 is a flow chart of a modification of the control program executed by the microcomputer in a third embodiment of the present invention;

[0036] FIG. 13 is a flow chart of another modification of the control program executed by the microcomputer in a first modification of the foregoing embodiments;

[0037] FIG. 14 is a flow chart of a program for estimation of the mass of a sprung mass member in a second modification of the foregoing embodiments;

[0038] FIG. 15 is a graph showing a data characteristic in a relative velocity-damping coefficient table in a third modification of the foregoing embodiments; and

[0039] FIG. 16 is a flow chart of a fourth modification of the control program shown in FIGS. 7 and 10.

[0040] a. Problems in Design of a Model and a Control System:

[0041] First of all, a model of a vehicle suspension mechanism has been considered to represent a state space of the suspension mechanism. In FIG. 1, there is schematically illustrated a model of a suspension mechanism of a road wheel in a wheeled vehicle, wherein the character M_brepresents the mass of a sprung mass member 10 such a vehicle body, the character M_wrepresents the mass of an unsprung mass member 11 such as a lower arm or a road wheel, and the character K_trepresents a spring constant of a road wheel tire 12. The character K_srepresents a spring constant of a spring 13 disposed in the suspension mechanism between the sprung mass member 10 and unsprung mass member 11, the character C_srepresents a linear portion of a damping coefficient C of a shock absorber or damper device 14 in the suspension mechanism (hereinafter referred to a linear damping coefficient) and the character C_vrepresents a non-linear portion of the damping coefficient C (hereinafter referred to a non-linear damping coefficient). A total of the linear damping coefficient C_sand non-linear damping coefficient C_vis represented as an overall damping coefficient of the shock absorber 14. The reference numeral 15 indicates a road surface.

[0042] Assuming that each displacement amount of the spring mass member 10, unsprung mass member 11 and road surface 15 is respectively represented as x_pb, x_ow, x_pr, the following equations (1) and (2) are given.

[0043] M_bx_pb″=K_s(x_pw−x_pb)+C_s(x_pw′−x_pb′)+C_v(x_pw′−x_pb′)  (1)

[0044] 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′)  (2)

[0045] Provided that, the sign -′- represents once differentiation, and the sign -″- represents twice differentiation.

[0046] A control input in the suspension mechanism is a variable damping coefficient C_v. Assuming that the disturbance w₁from the road surface is a road surface velocity x_pr′ and that the variable damping coefficient C_vis applied as a control input to represent a state space of the suspension mechanism, the following equation is given.

[0047] x_p′=A_px_p+B_p1w₁+B_p2(x_p)u  (3)

[0048] where $\begin{array}{cc}{X}_p=\left[\begin{array}{c}{X}_{\mathrm{pr}}-X_{\mathrm{pw}}\\ X_{\mathrm{pw}}-X_{\mathrm{pb}}\\ X_{\mathrm{pw}}^{\prime }\\ X_{\mathrm{pb}}^{\prime }\end{array}\right]& \left(4\right)\\ A_p=\left[\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 1& -1\\ K_t/M_w& -K_s/M_w& -C_s/M_w& C_s/M_w\\ 0& K_s/M_b& C_s/M_b& -C_s/M_b\end{array}\right]& \left(5\right)\\ B_{\mathrm{p1}}=\left[\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right]& \left(6\right)\\ B_{\mathrm{p2}}\left(x_p\right)=\left[\begin{array}{c}0\\ 0\\ -\left(x_{\mathrm{pw}}^{\prime }-x_{\mathrm{pb}}^{\prime }\right)/M_w\\ \left(x_{\mathrm{pw}}^{\prime }-x_{\mathrm{pb}}^{\prime }\right)/M_b\end{array}\right]& \left(7\right)\end{array}$

[0049] In the present invention, the target for enhancement of the characteristic of the suspension mechanism is directed to simultaneously restrain the vertical velocity x_pb′ of the sprung mass (hereinafter referred to sprung mass velocity) greatly influential in vibration of the sprung mass member, the vertical acceleration x_pb″ of the sprung mass member 10 (hereinafter referred to the acceleration of the sprung mass member) greatly influential in riding comfort of the vehicle, and the vertical relative velocity x_pw′−x_pb′ of the unsprung mass member 11 to the sprung mass member 10 (hereinafter referred to the relative velocity x_pw′−x_pb′) greatly influential in vibration of the unsprung mass member 11. Thus, the sprung mass velocity x_pb′, acceleration x_pb″ and relative velocity x_pw′−x_pb′ each are used as an evaluation output z_pin the following description. As in the suspension mechanism, it is easy to detect the acceleration x_pb″ of the sprung mass member and the relative displacement amount x_pw−x_pbof the sprung mass member 10 to the unsprung mass member 11 (hereinafter referred to the relative displacement amount x_pw−x_pb), an observation output y_pis basically defined by the acceleration x_pb″ and relative displacement amount x_pw−x_pb. Assuming that an observation noise w₂is included in the observation output y_p, the state space of the suspension mechanism is represented as follows.

[0050] z_p=C_p1x_p+D_p12(x_p)u  (8)

[0051] y_p=C_p2x_p+D_p21w₂+D_p22(x_p)u  (9)

[0052] where $\begin{array}{cc}{z}_p=\left[\begin{array}{c}{x}_{\mathrm{pb}}^{″}\\ x_{\mathrm{pb}}^{\prime }\\ x_{\mathrm{pw}}^{\prime }-x_{\mathrm{pb}}^{\prime }\end{array}\right]& \left(10\right)\\ y_p=\left[\begin{array}{c}{x}_{\mathrm{pb}}\\ x_{\mathrm{pw}}-x_{\mathrm{pb}}\end{array}\right]& \left(11\right)\\ C_{\mathrm{p1}}=\left[\begin{array}{cccc}0& K_s/M_b& C_s/M_b& -C_s/M_b\\ 0& 0& 0& 1\\ 0& 0& 1& -1\end{array}\right]& \left(12\right)\\ D_{\mathrm{p12}}\left(x_p\right)=\left[\begin{array}{c}\left(x_{\mathrm{pw}}^{\prime }-x_{\mathrm{pb}}^{\prime }\right)/M_b\\ 0\\ 0\end{array}\right]& \left(13\right)\\ C_{\mathrm{p2}}=\left[\begin{array}{cccc}0& K_s/M_b& C_s/M_b& -C_s/M_b\\ 0& 1& 0& 0\end{array}\right]& \left(14\right)\\ D_{\mathrm{p21}}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]& \left(15\right)\\ D_{\mathrm{p22}}\left(x_p\right)=\left[\begin{array}{c}\left(x_{\mathrm{pw}}^{\prime }-x_{\mathrm{pb}}^{\prime }\right)/M_b\\ 0\end{array}\right]& \left(16\right)\end{array}$

[0053] However, the state space of the suspension mechanism is represented in the form of a bilinear system since the state amount x_pis included in the coefficient B_p2(x_p) as shown in the equation (3). As in the bilinear system the coefficient B_p1(o) becomes “0” even if the control input u is varied at the original point x=0, the control at the original point is impossible. For this reason, the design of the control system for the suspension mechanism may not be realized on a basis of the linear control theory. Accordingly, the present invention was made on a basis of a nonlinear H_∞ control theory to design a control system for restraint of the sprung mass velocity x_pb′, acceleration x_pb″ and relative velocity x_pw′−x_pb′. Hereinafter, various kinds of designs of the nonlinear H_∞ control system according to the present invention will be described.

[0054] b. First Embodiment:

[0055] b1. Design of a Nonlinear H_∞ State Feedback Control System.

[0056] For design of a nonlinear H_∞ state feedback control system, a generalized plant consisting of an evaluation output z_pand a control input u added with a frequency weight was estimated as shown in FIG. 2. In this case, the frequency weight is a dynamic weight which is given as a transfer function and changes in accordance with a frequency. With the frequency weight, the weight can be increased in a frequency range where it is desired to enhance control performance of the system and can be decreased in a frequency range where control performance of the system may be disregarded. In addition, after multiplied by frequency weights W_s(s), W_u(s), the evaluation output z_pand control input u were multiplied by functions a₁(x), a₂(x) of a state amount x as a nonlinear weight function. To obtain a solution based on a Riccati inequality, the nonlinear weights a₁(x), a₂(x) are defined as follows.

[0057] a₁(x)>0,a₂(x)>0  (17)

[0058] a₁(o)=a₂(o)=1  (18)

[0059] With the nonlinear weights, it was able to design a control system for positively restraining the L₂gain. The state space of the system is represented by the following equation.

[0060] x_p′=A_px_p+B_p1w₁+B_p2(x_p)u  (19)

[0061] Where the state space of frequency weights W_s(s) for the evaluation output z_pis expressed by the following equations.

[0062] x_w′=A_wx_w+B_wz_p  (20)

[0063] z_w=C_wx_w+D_wz_p  (21)

[0064] In the equations (20) and (21), the value x_wrepresents a state amount of the frequency weights W_s(s), the value z_wrepresents an output of the frequency weights W_s(s), and the characters A_w, B_w, C_w, D_wrepresent constant matrixes defined by the control specification. These constant matrixes A_w, B_w, C_w, D_weach are determined to reduce a gain in relation to the sprung mass acceleration x_bin a frequency range of about 3 to 8 Hz for enhancement of riding comfort of the vehicle as shown in FIG. 3(A) to reduce a gain in relation to the sprung mass velocity x_b′ in a frequency range of about 0.5 to 1.5 Hz for restraining resonance of the sprung mass member 10 as shown in FIG. 3(B) and to reduce a gain in relation to the relative velocity x_w′−x_b′ in a frequency range of about 10 to 14 Hz for avoiding resonance of the unsprung mass member 11 as shown in FIG. 3(C). Thus, each factor of the sprung mass acceleration x_b″, sprung mass velocity x_b′ and relative velocity x_w′−x_b′ for the evaluation output z_pis independently controlled in such a manner that the frequency ranges for reduction of the gains may not be overlapped.

[0065] The state space of frequency weights W_u(s) for the control input u is represented by the following equations.

[0066] x_u′=A_ux_u+B_uu  (22)

[0067] z_u=C_ux_u+D_uu  (23)

[0068] Where the value x_urepresents a state amount of the frequency weights W_u(s), the value z_urepresents an output of the frequency weight W_u(s), and the characters A_u, B_u, C_u, D_urepresents constant matrixes. These constant matrixes A_u, B_u, C_u, D_uare determined to restrain a gain in relation to the control input u in the high frequency range taking into account the frequency characteristics of an electric actuator for control of the damping coefficient as shown in FIG. 3(D). In this instance, the state space of the generalized plate in the nonlinear H_∞ state feedback control system is expressed as follows.

[0069] x′=Ax+B₁w₁+B₂(x)u  (24)

[0070] z₁=a₁(x)(C₁₁x+D₁₂₁(x)u)  (25)

[0071] Z₂=a₂(x)(C₁₂x+D₁₂₂u)  (26)

[0072] Where the factors x, A, B₁, B₂(x), C₁₁, D₁₂₁(x), C₁₂and D₁₂₂are represented by the following equations. $\begin{array}{cc}x=\left[\begin{array}{c}{x}_p\\ x_w\\ x_u\end{array}\right]& \left(27\right)\\ A=\left[\begin{array}{ccc}{A}_p& 0& 0\\ B_w{C}_{\mathrm{p1}}& A_w& 0\\ 0& 0& A_u\end{array}\right]& \left(28\right)\\ B_1=\left[\begin{array}{c}{B}_{\mathrm{p1}}\\ B_w{D}_{\mathrm{p11}}\\ 0\end{array}\right]& \left(29\right)\\ B_2\left(x\right)=\left[\begin{array}{c}{B}_{\mathrm{p2}}\left(x_p\right)\\ \begin{array}{c}{B}_w{D}_{\mathrm{p12}}\left(x_p\right)\\ B_u\end{array}\end{array}\right]& \left(30\right)\end{array}$  C₁₁=[D_wC_p1C_wo]  (31)

[0073] D₁₂₁(x)=[D_wD_p12(x_p)]  (32)

[0074] C₁₂=[ooC_u]  (33)

[0075] D₁₂₂=D_u  (34)

[0076] When the state space of the generalized plant represented by the foregoing equations (24) to (26) is substituted in expression in a condition defined by the following equation (35) to obtain a solution based on the Riccati inequality, the following equations (36) to (38) are given.

[0077] D_wD_p12(x)=0  (35)

[0078] x′=Ax+B₁w+B₂(x)u  (36)

[0079] z₁=a₁(x)C₁₁x  (37)

[0080] z₂=a₂(x)C₁₂x+a₂(x)D₁₂₂u  (38)

[0081] Since the factor A is a stable matrix indicative of the damping control system, a nonlinear H_∞ state feedback control law for the generalized plant was designed in such a manner as to satisfy the facts that a closed loop system is stable in internal exponent and that the gain L₂from the disturbance w from the road surface to the evaluation output z is less than a positive constant γ.

[0082] The nonlinear H_∞ state feedback control law can be obtained in the case that the following conditions are satisfied.

[0083] 1) In a condition where the positive constant is given under existence of D₁₂₂⁻¹, there exists a positive definite symmetric solution P which satisfies the following Riccati inequality (39).

[0084] 2) In the case that the nonlinear weights a₁(x), a₂(x) satisfy a condition for restriction defined by the following formula (40), one of control laws u=k (x) for making the closed loop internally stable and reducing the L₂gain less than γ is given by the following equation (41). $\begin{array}{cc}\mathrm{PA}+A^{T}P+\frac{{\mathrm{PB}}_1{B}_1^{T}P}{{\gamma }^2}+C_{11}^T{C}_{11}+C_{12}^T{C}_{12}<0& \left(39\right)\\ \left(1-{a_1\left(x\right)}^2\right)x^T{C}_{11}^T{C}_{11}x+\left[\frac{1}{{a_2\left(x\right)}^2}-1\right]x^T{\mathrm{PB}}_2\left(x\right)D_{122}^{-1}{D}_{122}^{-T}{B}_2^T\left(x\right)\mathrm{Px}\geqq 0& \left(40\right)\\ \begin{array}{c}u=k\left(x\right)\\ =-D_{122}^{-1}\left[\frac{1}{{a_2\left(x\right)}^2}{D}_{122}^{-1}{B}_2^T\left(x\right)P+C_{12}\right]x\end{array}& \left(41\right)\end{array}$

[0085] The nonlinear weights a₁(x), a₂(x) satisfying the condition for restriction defined by the equation (41) are expressed as follows. $\begin{array}{cc}{a}_1\left(x\right)=\sqrt{1+m_1\left(x\right)x^T{\mathrm{PB}}_2\left(x\right)D_{122}^{-1}{D}_{122}^{-T}{B}_2^T\left(x\right)\mathrm{Px}}& \left(42\right)\\ a_2\left(x\right)=\sqrt{\frac{1}{1+m_1\left(x\right)x^T{C}_{11}^T{C}_{11}x}}& \left(43\right)\end{array}$

[0086] In the equations (42), (43), m₁(x) is an appropriated positive definite function. As a result of calculation, the foregoing positive definite symmetric solution P could be obtained. Using the equation (43), the equation (41) is converted into the following equation. $\begin{array}{cc}\begin{array}{c}u=k\left(x\right)\\ =-D_{122}^{-1}\left(\left(1+m_1\left(x\right)x^T{C}_{11}^T{C}_{11}x\right)D_{122}^{-T}{B}_2^T\left(x\right)P+C_{12}\right)x\end{array}& \left(44\right)\end{array}$

[0087] For design of the control system based on the nonlinear H_∞ control theory, it is, in general, required to solve a partially differential inequality so called the Hamilton-Jacobi inequality. In the case that the nonlinear weights a₁(x), a₂(x) are applied with the condition for restriction defined by the equation (40), however, the design of the control system can be made by solution of the Riccati inequality in stead of the Hamilton-Jacobi inequality. Since the Riccati inequality can be solved in a simple manner by using a well-known software such as Matlab, the positive definite symmetric solution P can be found in a simple manner, and the control law u=k (x) can be derived.

[0088] In addition, the factor D₁₂₂does not appear in the Riccati inequality, and only the condition for restriction to the nonlinear weights and the control law are concerned with the Riccati inequality. This means that the control law using the factor D₁₂₂can be adjusted without resolution of the Riccati inequality. The adjustment of the control law means scaling to the control input. When the scaling of the control input is made ten times, the factor D₁₂₂becomes one tenth, the term B₂(x) of the equation (41) becomes one hundred times, and the term C₁₂of the equation (41) becomes ten times.

[0089] To confirm the role of the nonlinear weights, a generalized plant of a bilinear system was assumed to compare with the generalized plant using the nonlinear weights. In this instance, the nonlinear weights a₁(x), a₂(x) were defined as a₁(x)=1, a₂(x)=1, and the factors C₁₂, D₁₂₂were defined as C₁₂=o, D₁₂₂=I. Thus, the state space represented by the equation (35) to (38) are expressed as follows.

[0090] x′=Ax+B₁w+B₂(x)u  (45)

[0091] z₁=C₁₁x  (46)

[0092] z₂=u  (47)

[0093] Thus, the control law u=k (x) of the generalized plant is expressed as follows.

[0094] u=B₂^T(x)Px  (48)

[0095] Provided that, P is a positive definite symmetric solution which satisfies the following Riccati inequality. $\begin{array}{cc}\mathrm{PA}+A^{T}P+\frac{1}{{\gamma }^2}{\mathrm{PB}}_1{B}_1^{T}P+C_{11}^T{C}_{11}<0& \left(49\right)\end{array}$

[0096] On the other hand, a linear approximate system adjacent the original point of the generalized plant represented by the equations (45) to (47) is expressed as follows.

[0097] x′=Ax+B₁w  (50)

[0098] z₁=C₁₁x  (51)

[0099] z₂=u  (52)

[0100] The Riccati inequality (49) indicates that the closed loop of the generalized plant is internally stable and that the L₂gain is less than γ. That is to say, the L₂gain oI the bilinear system is determined by a value at the original point (x=o) shown in FIG. 4 since the bilinear system is defined as B₂(o)=o at the original point where the control input does effect to improve the L₂gain. In the case that the control input is defined as u=o, the generalized plant defined by the equations (45) to (47) coincides with the generalized plant defined by the equations (50) to (52). As a result, even in the case that the control input u to the generalized plant defined by the equations (45) to (47) is defined as u=o, the Riccati inequality indicates that the closed loop system is internally stable and that the L₂gain is less than γ. This means that even if the control input u becomes effective due to an increase of the state amount x, the L₂gain does not increase more than g_Oin a control system designed for the generalized plant defined by the equations (45) to (47), wherein the control output is defined by the following equations (53) and (54).

[0101] z₁=C₁₁x  (53)

[0102] z₂=u  (54)

[0103] In this instance, it is assumed that the control performance is enhanced by the control input or does not change as in the case or u=o. Accordingly, the control system can be designed on a basis of the following equations (55) and (56) so that the L₂gain of the plant is decreased by the nonlinear weights as shown by a characteristic line g₁in FIG. 4.

[0104] z₁=a₁(x)C₁₁x  (55)

[0105] z₂=a₂(x)u  (56)

[0106] In this embodiment, the damping coefficient C of damper 14 was divided into the linear damping coefficient C_sand the nonlinear variable damping coefficient C_vto design the control system wherein the variable damping coefficient is applied as the control input u. As shown in FIG. 5(A), the linear damping coefficient C was defined to be located approximately at a center between a minimum damping force characteristic line of the damper 14 (corresponding with a maximum opening degree of a damper orifice 14a) and a maximum damping force characteristic line (corresponding with a minimum opening degree of the damper orifice), and the gain of the control input u was adjusted in accordance with a frequency so that the damping coefficient C changes at both sides of the linear damping coefficient C_sand that the damping force defined by the damping coefficient C is placed between the minimum damping force characteristic line and the maximum damping force characteristic line. Thus, the variable damping coefficient C_vcan be determined in accordance with the design specification of the damper 14 in a simple manner. As a result, a desired control of the damping force can be realized in a range defined by actual function of the damper 14. For comparison with the damping force control in the present invention, illustrated in FIG. 5(B) is a Lissajous's figure in the case that the damping coefficient of the damper 14 is controlled on a basis of the conventional skyhook theory, wherein a desired control of the damping force may not be realized.

[0107] In the case that the damping force or damping coefficient of the damper 14 is switched over at plural steps, the linear damping coefficient C_sis determined in such a manner that he damping force defined by the linear damping coefficient C_sbecomes equal to a damping force produced by the damper 14 at either one of the plurality steps in a small range. In this kind of suspension mechanisms, a linearity relative to the relative velocity of the damping force is strong in a small range of the damping force where the variable damping coefficient C_vis calculated as “0”. For this reason, the damper 14 is maintained at either one of the plural steps without frequently being switched over. This is useful to enhance the durability of the damper 14.

[0108] b2. Practical Embodiment of the Present Invention in a Wheeled Vehicle.

[0109] Hereinafter, a practical embodiment in a wheeled vehicle, based on the nonlinear H_∞ state feedback control law will be described with reference to FIG. 6. Illustrated in FIG. 6 is a damping force control system which comprises a stepper motor for switching over an opening degree OP of an orifice 14aof the damper 14 at plural steps (N steps) and an electric actuator 21 such an ultrasonic motor. The stepper motor is designed to control the opening degree of the damper orifice 14ain such a manner that a damping force produced by the damper 14 at each step thereof becomes approximately equal to a damping force determined by a linear damping coefficient C_scalculated by processing of a control program shown by a flow chart in FIG. 7.

[0110] The electric actuator 21 is activated under control of a microcomputer 20 through a driving circuit 22. The microcomputer 20 is connected to a tire displacement amount sensor 23, a stroke sensor 24, a sprung mass acceleration sensor 25 and an unsprung mass acceleration sensor 26. The tire displacement amount sensor 23 is arranged to detect a displacement amount x_pr−X_pwdefined by a relative displacement amount of road surface displacement x_prand unsprung mass displacement x_pw. For example, the displacement amount x_pr−x_pwis detected by an output of a strain sensor for detecting deformation of the tire and an output of a pressure sensor for detecting an air pressure of the tire. The stroke sensor 24 is disposed between the sprung mass member 10 and the unsprung mass member 11 to detect an amount of vertical relative displacement of the sprung mass member 10 to the unsprung mass member 11. The sprung mass acceleration sensor 25 is fixedly mounted on the sprung mass member 10 to detect vertical acceleration x_pb″ of the sprung mass member 10. The unsprung mass acceleration sensor 26 is fixedly mounted on the unsprung mass member 11 to detect vertical acceleration x_pw″ of the unsprung mass member 11.

[0111] The microcomputer 20 is provided to execute the control program of FIG. 7 at each lapse of a predetermined short time for applying a control signal indicative of a target opening degree OP of the damper orifice 14aat each step of the damper 14 to the driving circuit 22. The microcomputer 20 includes a relative velocity-damping table provided to memorize a data indicative of variation characteristics of a damping force F of the damper 14 in relation to a relative velocity x_pw′−x_pb′ at each step of the damper 14 as shown in FIG. 8.

[0112] Hereinafter, operation of the damping control system will be described. Assuming that an ignition switch (not shown) of the vehicle has been closed, the microcomputer 20 is activated to repeat execution of the control program of FIG. 7 at each lapse of the short time for control of the damping force of damper 14. The computer 20 starts at step 100 to execute the control program and inputs at step 102 detection signals respectively indicative of the tire displacement amount x_pr−x_pw, relative displacement amount x_pw−x_pb, sprung mass acceleration x_pb″ and unsprung mass acceleration x_pw″ from sensors 23, 24, 25 and 26. Subsequently, the computer 20 calculates at step 104 a sprung mass velocity x_pb′ and an unsprung mass velocity x_pw′ by time integration of the sprung mass acceleration x_pb″ and unsprung mass acceleration x_pw″ and calculates a relative velocity x_pw′−x_pb′ by time differentiation of the relative displacement amount x_pw−x_pb.

[0113] The following step 106, die computer 20 calculates B_p2(x_{p, Dp12}(x_p) and B₂(x) respectively on a basis of the following equations (57), (58) and (59). $\begin{array}{cc}{B}_{\mathrm{p2}}\left(x_p\right)=\left[\begin{array}{c}0\\ 0\\ -\left(x_{\mathrm{pw}}^{\prime }-x_{\mathrm{pb}}^{\prime }\right)/M_w\\ \left(x_{\mathrm{pw}}^{\prime }-x_{\mathrm{pb}}^{\prime }\right)/M_b\end{array}\right]& \left(57\right)\\ D_{\mathrm{p12}}\left(x_p\right)=\left[\begin{array}{c}\left(x_{\mathrm{pw}}^{\prime }-x_{\mathrm{pb}}^{\prime }\right)/M_b\\ 0\\ 0\end{array}\right]& \left(58\right)\\ B_2\left(x\right)=\left[\begin{array}{c}{B}_{\mathrm{p2}}\left(x_p\right)\\ B_w{D}_{\mathrm{p12}}\left(x_p\right)\\ B_u\end{array}\right]& \left(59\right)\end{array}$

[0114] In the equations (57) and (58), the characters M_w, M_brepresent the mass of the sprung mass member and the mass of the unsprung mass member of the vehicle, respectively. In the equation (59), the characters B_w, B_urepresent coefficient matrix related to frequency weights W_s(s), W_u(s) respectively defined by the foregoing equations (20) and (22), which coefficient matrix is preliminarily memorized in the computer 20.

[0115] After processing at step 106, the computer 20 calculates at step 108 a state variable x_wof the frequency weights on a basis of the following equation (60).

[0116] x_w′ =A_wx_w+B_wz_p  (60)

[0117] The equation (60) is the same as the foregoing equation (20), wherein the characters A_w, B_wrepresent coefficient matrixes respectively related to the frequency weight W_s(s) defined by the equation (20) and memorized in the computer 20.

[0118] Subsequently, the computer 20 calculates at step 110 a state variable x_uof the frequency weight, an extended state amount x_uand a control input u on a basis of the following equations (61), (62) and (63), which are the same as the foregoing equations (22), (27) and (44).

[0119] x_u′=A_ux_u+B_uu  (61)

[0120] $\begin{array}{cc}x=\left[\begin{array}{c}{x}_p\\ x_w\\ x_u\end{array}\right]& \left(62\right)\\ \begin{array}{c}u=k\left(x\right)\\ =-D_{122}^{-1}\left\{\left(1+m_1\left(x\right)x^T{C}_{11}^T{C}_{11}x\right)D_{122}^{-T}{B}_2^T\left(x\right)P+C_{12}\right\}x\end{array}& \left(63\right)\end{array}$

[0121] In the equation (61), the characters A_u, B_ueach represent a coefficient matrix related to the frequency weight W_u(s) defined by the foregoing equation (22), which coefficient matrix is memorized as a constant matrix in the computer 20. In the equation (63), the character D₁₂₂represents a coefficient matrix related to the frequency weight W_u(s) defined by the foregoing equation (34) and determined by the equation (23), which coefficient matrix is memorized as a constant matrix in the computer 20, and the factor m₁(x) represents an arbitrary positive constant function whose algorithm is memorized in the computer 20. The positive constant function m₁(x) may be defined as a positive constant such as “1.0”, and the factor C₁₁is defined by the foregoing equations (12) and (31). That is to say, the factor C₁₁is defined by the mass M_wof the unsprung mass member 11, the mass M_bof the sprung mass member 10, the constant K_sof the spring 13, the linear damping coefficient C_sof the damper 14, and the coefficient matrixes C_w, D_wrelated to the frequency weight W_s(s) defined by the equation (21), which factor C₁₁is preliminarily memorized in the computer 20. The factor B₂(x) is a matrix calculated at step 106 of the control program, the character P is a positive definite symmetric solution satisfying the equations (39) and (40), which matrix is preliminarily memorized as a constant matrix in the computer 20. The factor C₁₂represents a constant matrix defined by the equation (33) and including a coefficient matrix C_urelated to the frequency weight W_u(s), which constant matrix is memorized in the computer 20.

[0122] In calculation of the state variable X_u, extended state amount x and control input u at step 110, each value is applied with an initial value, and the equations (60) to (63) are repeatedly calculated to determine the respective values x_u, x, u. After processing at step 110, the computer 20 calculates at step 112 a target damping coefficient C of the damper 14 on a basis of the following equation (64).

[0123] C=C_s+C_v=C_s+u  (64)

[0124] At the following step 114, the computer 20 calculates a target damping force F on a basis of the following equation (65).

[0125] F=C(x_pw′−x_pb′)  (65)

[0126] where C is the target damping coefficient calculated at step 114, and x_pw′−x_pb′ is the relative velocity calculated by processing at step 104.

[0127] After processing at step 114, the computer 20 determines at step 116 the opening degree OP of the damper orifice 14awith reference to the relative velocity-damping force table shown in FIG. 8. In this determination, a nearest curve to a point defined by the damping force F and relative velocity x_pw′−x_pb′ in FIG. 8 is retrieved to determine the opening degree OP of damper orifice 14a. Thus, the computer 20 applies a control signal indicative of the determined opening degree OP of damper orifice 14ato the driving circuit 22. In turn, the electric actuator 21 is activated under control of the driving circuit 22 to adjust the damper orifice 14ato the determined opening degree OP. This causes the damper 14 to produce the calculated target damping force F thereby to restrain resonance of the sprung and unsprung mass members 10 and 11 and unpleasant feel to the passengers for enhancing riding, comfort of the vehicle.

[0128] c. Second Embodiment:

[0129] c1. Design of a Nonlinear H_∞ Output Feedback Control System:

[0130] For design of a nonlinear H_∞ output feedback control system, an observer is estimated which includes a portion of the state amount x_p(the tire displacement x_pr−x_pw, relative displacement amount x_pw−x_pb, unsprung mass velocity x_pw′ and sprung mass acceleration x_pb″). In this case, a generalized plant of the output feedback control system is estimated as shown in FIG. 9, wherein the frequency weight is added to the evaluation output z_pand the control input u. In the generalized plant, the evaluation output z_pis multiplied by a nonlinear weight function a₁(x, x{circumflex over ( )}) after multiplied by the frequency weight W_s(s), while the control input u is multiplied by a nonlinear weight function a₂(x, x{circumflex over ( )}) after multiplied by the frequency weight W_u(s). Each characteristic of the nonlinear weight functions a₁(x, x{circumflex over ( )}), a₂(x, x{circumflex over ( )}) is represented by the following inequalities (66), (67).

[0131] a₁(x,x{circumflex over ( )})>0,a₂(x,x{circumflex over ( )})>0  (66)

[0132] a₁(o,o)=a₂(o,o)=1  (67)

[0133] Where x{circumflex over ( )} represents a state amount including a portion of the estimation value.

[0134] Each state space of the control system, the frequency weight W_s(s) related to the evaluation output z_pand the frequency weight W_u(s) related to the control input u is expressed by the following equations (68) to (72) in the same manner as in the state feedback control system described above.

[0135] x_p′=Apx_p+B_p1w₁+B_p2(x_p)u  (68)

[0136] x_w′=A_wx_w+B_wz_p  (69)

[0137] z_w=C_wx_w+D_wz_p  (70)

[0138] x_u′=A_ux_u+B_uu  (71)

[0139] z_u=C_ux_u+D_uu  (72)

[0140] In addition, the state variable x_w, evaluation function z_w, constant matrix A_w, B_w, C_w, D_ware expressed in the same manner as in the state feedback control system described above. Provided that, the state space of the generalized plant in the nonlinear H_∞ output feedback control system is expressed by the following equations (73) to (76).

[0141] x′=Ax+B₁w+B₂(x)u  (73)

[0142] z₁=a₁(x,x{circumflex over ( )})(C₁₁x+D₁₂₁(x)u)  (74)

[0143] z₂=a₂(x,x{circumflex over ( )})(C₁₂x+D₁₂₂u)  (75)

[0144] y=C₂x+D₂₁w+D₂₂(x)u  (76)

[0145] Provided that, x, w, A, B₁, B₂(x), C₁₁, D₁₂₁(x), C₁₂, D₁₂₂, C₂, D₂₁, D₂₂(x) in the above equations (73) to (76) are expressed as follows. $\begin{array}{cc}x=\left[\begin{array}{c}{x}_p\\ x_w\\ x_u\end{array}\right]& \left(77\right)\\ w=\left[\begin{array}{c}{w}_1\\ w_2\end{array}\right]& \left(78\right)\\ A=\left[\begin{array}{ccc}{A}_p& 0& 0\\ B_w{C}_{\mathrm{p1}}& A_w& 0\\ 0& 0& A_u\end{array}\right]& \left(79\right)\\ B_1=\left[\begin{array}{cc}{B}_{\mathrm{p1}}& 0\\ B_w{D}_{\mathrm{p11}}& 0\\ 0& 0\end{array}\right]& \left(80\right)\\ B_2\left(x\right)=\left[\begin{array}{c}{B}_{\mathrm{p2}}\left(x_p\right)\\ \begin{array}{c}{B}_w{D}_{\mathrm{p12}}\left(x_p\right)\\ B_u\end{array}\end{array}\right]& \left(81\right)\end{array}$  C₁₁=[D_wC_p1C_wo]  (82)

[0146] D₁₂₁(x)=[D_wD_p12(x_p)]  (83)

[0147] C₁₂=[ooC_u]  (83)

[0148] D₁₂₂=D_u  (85)

[0149] C₂=[C_p2oo]  (86)

[0150] D₂₁=[oD_p21]  (87)

[0151] D₂₂(x)=D_p22(x_p)  (88)

[0152] To obtain a solution based on the Riccati inequality, the expression of the state space of the generalized plant represented by the foregoing equations (73) to (76) is substituted as follows.

[0153] D_wD_p12(x)=o  (89)

[0154] x′=Ax+B₁w+B₂(x)u  (90)

[0155] z₁=a₁(x,x{circumflex over ( )})C₁₁x  (91)

[0156] z₂=a₂(x,x{circumflex over ( )})C₁₂x+a₂(x,x{circumflex over ( )})D₁₂₂u  (92)

[0157] y=C₂x+D₂₁w+D₂₂(x)u  (93)

[0158] Similarly, in the state feedback control system, a nonlinear H_∞ output feedback control law u=k (y) is designed in such a manner to satisfy the facts that a closed loop system is stable in internal exponent and that the gain L₂from w to z is less than a positive constant γ. In this embodiment, three types of nonlinear H_∞ output feedback control systems will be described hereinafter.

[0159] c1-1) Design of a First Type Control System:

[0160] In the control system of this type, B₂(x) of the equation (81) and D₂₂(x) of the equation (88) each are a measurable known function, and an observer gain L is in the form of a constant matrix.

[0161] The foregoing nonlinear H_∞ output feedback control law u=k (y) is obtainable if the following conditions are satisfied.

[0162] 1) D₁₂₂⁻¹exists, γ₁is a positive constant defined as γ₁²I−D₂₁^TΘ^TΘD₁₂>0, γ₂is defied as γ₂>1, the following Riccati inequality (94) for design of an observer (an observer gain) and a positive definite symmetric matrix P, Q and a positive definite matrixΘ satisfying the following Riccati inequality (95) for design of a controller are given.

[0163] AQ+QA^T+Q(C₁₁^TC₁₁+C₁₂^TC₁₂^−C₂^TΘ^TΘC₂)Q +B₁(γ₁²I−D₂₁^TΘ^TΘD₂₁)⁻¹B₁^T<0  (94)

[0164] $\begin{array}{cc}\mathrm{PA}+A^{T}P+\frac{1}{{\gamma }_2^2}\mathrm{PL}{\Theta }^{-1}{\Theta }^{-T}{L}^{T}P+\frac{{\gamma }_2^2}{{\gamma }_2^2-1}\left(C_{11}^T{C}_{11}+C_{12}^T{C}_{12}\right)<o& \left(95\right)\\ {\gamma }_2^2-{a_1\left(x,x^{^}\right)}^2>0,{\gamma }_2^2-{a_2\left(x,x^{^}\right)}^2>0\left[\frac{{\gamma }_2^2}{{\gamma }_2^2-1}-\frac{{a_1\left(x,x^{^}\right)}^2{\gamma }_2^2}{{\gamma }_2^2-{a_1\left(x,x^{^}\right)}^2}\right]x^{^T}{C}_{11}^T{C}_{11}{x}^{^}+[\frac{{\gamma }_2^2-{a_2\left(x,x^{^}\right)}^2}{{a_2\left(x,x^{^}\right)}^2{\gamma }_2^2}-\frac{{\gamma }_2^2-1}{{\gamma }_2^2}]& \left(96\right)\end{array}$  xx{circumflex over ( )}^TPB₂(x)D₁₂₂⁻¹D₁₂₂^−TB₂^T(x)Px{circumflex over ( )}≧0  (97)

[0165] $\begin{array}{cc}{\Vert \left[\begin{array}{c}{z}_1\\ z_2\end{array}\right]\Vert }_2\leqq {\gamma }_1{\gamma }_2{\Vert W\Vert }_2& \left(98\right)\end{array}$

[0166] If the nonlinear weights a₁(x, x{circumflex over ( )}), a₂(x, x{circumflex over ( )}) satisfy the conditions defined by the inequalities (96) and (97), one of control laws defined by the inequality (98) is obtained by the following equations (99) and (100).

[0167] x′{circumflex over ( )}=(A+LC₂)X{circumflex over ( )}+(B₂(x)+LD₂₂(x))u−Lyu=−D₁₂₂⁻¹  (99)

[0168] $\begin{array}{cc}x\left[\frac{{\gamma }_2^2-{a_2\left(x,x^{^}\right)}^2}{{a_2\left(x,x^{^}\right)}^2{\gamma }_2^2}{D}_{122}^{-T}{B}_2^T\left(x\right)P+C_{12}\right]x^{^}& \left(100\right)\end{array}$

[0169] Provided that, the observer gain L is represented as follows.

[0170] L=−QC₂^TΘ^TΘ  (101)

[0171] The character “∥ ∥” represents a Euclidean norm, and “∥ ∥₂” represents a norm on a square integrable function space L₂which is defined in relation to f(t)εL₂. by the following equation (102). $\begin{array}{cc}{\Vert f\left(t\right)\Vert }_2={\left({\int }_{\mathrm{to}}^{\infty }{\Vert f\left(t\right)\Vert }^{2}dt\right)}^{1/2}& \left(102\right)\end{array}$

[0172] The factorΘ is a positive definite matrix under existence of Θ⁻¹, and the observer gain L can be adjusted by using the positive definite matrix Θ. Similarily, in the state feedback control law, the controller gain L can be adjusted by using D₁₂₂. In addition, γ₁is a L₂gain of the observer, and γ₂is a L₂gain of the controller. A gain L₂of the closed loop system is determined by product of γ₁and γ₂. Accordingly, the L₂gains of the control system must be determined by adjustment of the observer and controller.

[0173] Here, the nonlinear weights a₁(x, x{circumflex over ( )}), a₂(x, x{circumflex over ( )}) satisfying the formulae (96) and (97) are expressed as follows. $\begin{array}{cc}\begin{array}{c}{a}_1\left(x,x^{^}\right)=({\gamma }_2^2+\left({\gamma }_2^2-1\right){\gamma }_2^2ϵm_1\left(x,x^{^}\right)x^{^T}\times \\ {{\mathrm{PB}}_2\left(x\right)D_{122}^{-1}{D}_{122}^{-T}{B}_2^T\left(x\right){\mathrm{Px}}^{^})}^{1/2}/\\ ({\gamma }_2^2+\left({\gamma }_2^2-1\right)m_1\left(x,x^{^}\right)x^{^T}\times \\ {{\mathrm{PB}}_2\left(x\right)D_{122}^{-1}{D}_{122}^{-T}{B}_2^T\left(x\right){\mathrm{Px}}^{^})}^1\end{array}& \left(103\right)\end{array}$ $\begin{array}{cc}{a}_2\left(x,x^{^}\right)=\sqrt{\frac{1}{1+m_1\left(x,x^{^}\right)x^{^T}{C}_{11}^T{C}_{11}{x}^{^}}}& \left(104\right)\end{array}$

[0174] In the equations (103) and (104), m₁(x, x{circumflex over ( )}) is an appropriated positive definite function, ε is a positive constant defined as ε<1 and εγ₂²>1. Thus, the foregoing positive definite symmetric solution P is calculated by the computer 20. Using the equations (103) and (104), the equations (99) and (100) are converted into the following equations (105) and (106).

[0175] x′{circumflex over ( )}=(A+LC₂)x{circumflex over ( )}+(B₂(x)+LD₂₂(x)u−Lyu=−D₁₂₂⁻¹  (105)

[0176] $\begin{array}{cc}x\left[\frac{{\gamma }_2^2-1+{\gamma }_2^2{m}_1\left(x,x^{^}\right)x^{^T}{C}_{11}^T{C}_{11}{x}^{^}}{{\gamma }_2^2}{\mathrm{xD}}_{122}^{-T}{B}_2^T\left(x\right)P+C_{12}\right]x^{^}{x}^{^}& \left(106\right)\end{array}$

[0177] In this case, the positive definite symmetric solution can be obtained in a simple manner using a conventional software, similarly in the state feedback control system. With this method, an estimated state amount x′{circumflex over ( )} and control law u=k (y) can be also derived in a simple manner.

[0178] c1-2) Practical Embodiment of a Damping Force Control system in a Wheeled Vehicle on a Basis of the First Type Control Law.

[0179] In this embodiment, the damping force control system is constructed as shown in FIG. 6, without the provision of the tire displacement amount sensor 23 and unsprung mass acceleration sensor 26, and the microcomputer 20 is arranged to execute a control program shown by a flow chart in FIG. 10. The other construction and components of the control system are substantially the same as those in the damping force control system in the first embodiment.

[0180] In operation of the control system, the computer 20 starts at step 100 to repeat execution of the control program at each lapse of the predetermined short time and inputs at step 102adetection signals respectively indicative of a relative displacement amount x_pw−x_pbof the sprung mass to the unsprung mass and acceleration x_pb″ of the sprung mass from the stroke sensor 24 and sprung mass acceleration sensor 25. Thus, the computer 20 calculates at step 104aa relative velocity x_pw′−x_pb′ of the sprung mass to the unsprung mass and a vertical velocity x_pb′ of the sprung mass in the same manner as in the first embodiment.

[0181] At the following step 106a, the computer 20 calculates B_p2(x_p), D_p12(xp) based on the following equations (107) and (108) and calculates B₂(x) based on the following equation (109). Additionally, the computer 20 calculates D₂₂(x) based the following equations (110) and (111). $\begin{array}{cc}{B}_{\mathrm{p2}}\left(x_p\right)=\left[\begin{array}{c}\begin{array}{c}\begin{array}{c}0\\ 0\end{array}\\ -\left(x_{\mathrm{pw}}^{\prime }-x_{\mathrm{pb}}^{\prime }\right)/M_w\end{array}\\ \left(x_{\mathrm{pw}}^{\prime }-x_{\mathrm{pb}}^{\prime }\right)/M_b\end{array}\right]& \left(107\right)\end{array}$ $\begin{array}{cc}{D}_{\mathrm{p12}}\left(x_p\right)=\left[\begin{array}{c}\begin{array}{c}\left(x_{\mathrm{pw}}^{\prime }-x_{\mathrm{pb}}^{\prime }\right)/M_b\\ 0\end{array}\\ 0\end{array}\right]& \left(108\right)\\ B_2\left(x\right)=\left[\begin{array}{c}\begin{array}{c}{B}_{\mathrm{p2}}\left(x_p\right)\\ B_w{D}_{\mathrm{p12}}\left(x_p\right)\end{array}\\ B_u\end{array}\right]& \left(109\right)\\ D_{\mathrm{p22}}\left(x_p\right)=\left[\begin{array}{c}\left(x_{\mathrm{pw}}^{\prime }-x_{\mathrm{pb}}^{\prime }\right)/M_b\\ 0\end{array}\right]& \left(110\right)\end{array}$  D₂₂(x)=D_p22(x_p)  (111)

[0182] In the equations (107) to (110), M_w, M_b, B_w, B_ueach represent a constant matrix of the same value as that in the first embodiment.

[0183] After processing at 106a, the computer 20 calculates at step 110aan estimated state amount x{circumflex over ( )} and a control input u based on the following equations (112) and 113) in the same manner as in the first embodiment.

[0184] x′{circumflex over ( )}=(A+LC₂)x{circumflex over ( )}+(B₂(x)+LD₂₂(x))u−Ly  (112)

[0185] $\begin{array}{cc}u=-D_{122}^{-1}x\left[\frac{{\gamma }_2^2-1+{\gamma }_2^2{m}_1\left(x,x^{^}\right)x^{^T}{C}_{11}^T{C}_{11}{x}^{^}}{{\gamma }_2^2}{\mathrm{xD}}_{122}^{-T}{B}_2^T\left(x\right)P+C_{12}\right]x^{^}{x}^{^}& \left(113\right)\end{array}$

[0186] In the equation (112), A is a constant matrix determined by the foregoing equation (79) which is memorized in the computer 20, and L is a constant matrix defined by the foregoing equation (101) which is memorized in the computer 20. The constant matrix L is a gain of an observer determined by a positive definite matrix Q, the constant matrix C₂determined by the foregoing equations (14), (86) and a positive definite matrix Θ. C₂is the foregoing constant matrix memorized in the computer 20, and B₂(x) and D₂₂(x) each are a matrix calculated at step 106a. The factor y represents the relative displacement amount x_pw−x_pbapplied by processing at step 102aand a sprung mass velocity x_pb′ calculated by processing at step 104a.

[0187] In the equation (113), D₁₂₂is defined by the equation (85) and is a coefficient matrix related to the frequency weight W_u(s) defined by the equation (23), which coefficient matrix is memorized in the computer 20. γ₂is a positive constant defined as γ₂>1, m₁(x, x{circumflex over ( )}) is an appropriated positive constant function whose algorithm is preliminarily memorized in the computer 20. The positive constant function m₁(x) may be set as “1.0”. C₁₁is defined by the foregoing equations (12) and (82). That is to say, C₁₁is defined by the mass M_wof the unsprung mass member, the mass M_bof the sprung mass member, the constant K_sof the spring 13, the linear damping coefficient C_sof damper 14 and the coefficient matrixes C_w, D_wrelated to the frequency weight W_s(s) determined by the foregoing equation (21), which C₁₁is preliminarily memorized as a constant matrix in the computer 20. B₂(x) is a matrix calculated by processing at step 106a. P is a positive definite symmetric solution satisfying the foregoing equations (94) and (95), which is memorized as a constant matrix in the computer 20. C₁₂is defined by the foregoing equation (84), which is memorized in the computer 20 as a constant matrix including a coefficient matrix C_urelated to the frequency weight W_u(s) determined by the foregoing equation (23).

[0188] After processing at step 110a, the computer 20 calculates at step 112 to 118 a target damping coefficient C of the damper 14 and a target damping force F in the same manner as in the first embodiment and determines the opening degree OP of damper orifice 14ato produce the target damping force F. Thus, the damper 14 is adjusted in a simple manner to produce the target damping force F without detection of the unsprung mass velocity x_pw′ and tire displacement amount x_pr−x_pw.

[0189] c2-1) Design for the Second Type Control System:

[0190] In the control system of this type, it is assumed that B₂(x) and D₂₂(x) in the foregoing equations (81) and (88) each are an unknown function. In a bilinear system of this kind, B₂(x), D₂₂(x) each are a primary Junction of x. When the generalized plant represented by the foregoing equations (90) to (93) is rewritten taking into account the above fact, the plant is represented by the following equations (114) to (117), wherein B₂₀, D₂₂₀and d₁₂₂each are in the form of a constant matrix.

[0191] x′=Ax+B₁w+B₂₀xu  (114)

[0192] z₁=a₁(x{circumflex over ( )})C₁₁x  (115)

[0193] z₂=a₂(x{circumflex over ( )})C₁₂x+a₂(x)d₁₂₂u  (116)

[0194] y=C₂x+D₂₁w+D₂₂₀xu  (117)

[0195] Assuming that the observer gain L is a constant matrix in the generalized plant as described above, a nonlinear H_∞ output feedback control law can be designed as follows.

[0196] 1) In the case that γ₁is a positive constant defined as γ₁²I−D₂₁^TΘ^TΘD₁₂>0, γ₂is a positive constant defined as γ₂>1 and a positive constants ε is defined as a ε₁²−u²>0, the following Riccati inequality (118) for design of the observer (the observer gain) is given, and a positive definite symmetric matrix P, Q and a positive definite matrix Θ satisfying the following Riccati inequality (11) for design of a controller are given. In addition, in the case that nonlinear weights a₁(x, x{circumflex over ( )}), a₂(x, x{circumflex over ( )}) satisfy the condition for restriction defined by the following equations (120) and (121), one of control laws defined by the following equation (122) is given by the following equations (123) and (124). $\begin{array}{cc}\mathrm{AQ}+{\mathrm{QA}}^T+Q(C_{11}^T{C}_{11}+C_{12}^T{C}_{12}-C_2^T{\Theta }^T\Theta C_2+{ϵ}^{2}I+{ϵ}_1^2{D}_{220}^T{\Theta }^T\left(I+\Theta D_{21}\times {\left({\gamma }_1^{2}I-D_{21}^T{\Theta }^T\Theta D_{21}\right)}^{-1}{D}_{21}^T{\Theta }^T\right)\Theta D_{220})Q+{B_1\left({\gamma }_1^{2}I-D_{21}^T{\Theta }^T\Theta D_{21}\right)}^{-1}{B}_1^T+\left({B_1\left({\gamma }_1^{2}I-D_{21}^T{\Theta }^T\Theta D_{21}\right)}^{-1}{D}_{21}^T{\Theta }^T\Theta D_{220}+B_{20}\right)\times \left(D_{220}^T{\Theta }^T\Theta {D_{21}\left({\gamma }_1^{2}I-D_{21}^T{\Theta }^T\Theta D_{21}\right)}^{-1}{B}_1^T+B_{20}^T\right)<0& \left(118\right)\end{array}$ $\begin{array}{cc}\mathrm{PA}+A^{T}P+\frac{1}{{\gamma }_2^2}\mathrm{PL}{\Theta }^{-1}{\Theta }^{-T}{L}^{T}P+\frac{{\gamma }_2^2}{{\gamma }_2^2-1}(C_{11}^T{C}_{11}+C_{12}^T{C}_{12)}<0& \left(119\right)\end{array}$  γ₂²−a₁(x,x{circumflex over ( )})²>0,γ₂²−a₂(x,x{circumflex over ( )})²>0  (120)

[0197] $\begin{array}{cc}\left[\frac{{\gamma }_2^2}{{\gamma }_2^2-1}-\frac{{a_1\left(x,x^{^}\right)}^2{\gamma }_2^2}{{\gamma }^2-{a_1\left(x,x^{^}\right)}^2}\right]x^{^T}{C}_{11}^T{C}_{11}{x}^{^}+[\frac{{\gamma }_2^2-{a_2\left(x,x^{^}\right)}^2}{{a_2\left(x,x^{^}\right)}^2{\gamma }_2^2}-\frac{{\gamma }_2^2-1}{{\gamma }_2^2}]\frac{1}{d_{122}^2}{\mathrm{xx}}^{^T}{\mathrm{PB}}_{20}{x}^{^}{x}^{^T}{B}_{20}^T{\mathrm{Px}}^{^}\geqq 0& \left(121\right)\end{array}$  ∥[z₁^Tz₂^T]^T∥₂≦γ₁γ₂∥w∥₂  (122)

[0198] x′{circumflex over ( )}=(A+L(L(u)C₂)x{circumflex over ( )}+(B₂₀+L(u)D₂₂₀)x{circumflex over ( )}u−L(u)y  (123)

[0199] $\begin{array}{cc}u=-\frac{1}{d_{122}}\left[\frac{{\gamma }_2^2-{a_2\left(x,x^{^}\right)}^2}{{a_2\left(x,x^{^}\right)}^2{\gamma }_2^2{d}_{122}}{x}^{^T}{B}_{20}^{T}P+C_{12}\right]x^{^}& \left(124\right)\end{array}$

[0200] Provided that, the observer L (u) is represented as follows.

[0201] L(u)=−-QC₂^TΘ^TΘ  (125)

[0202] The factor Θ is a constant definite matrix under existence of Θ⁻¹, and the observer can be adjusted by using the factor Θ. Similarly, in the state feedback control law, the gain L of the controller can be adjusted by using the factor d₁₂₂.

[0203] Here, the nonlinear weights a₁(x, x{circumflex over ( )}), a₂(x, x{circumflex over ( )}) satisfying the formulae (120) and (121) is expressed as follows. $\begin{array}{cc}\begin{array}{c}{a}_1\left(x,x^{^}\right)=({\gamma }_2^2{d}_{122}^2+\left({\gamma }_2^2-1\right){\gamma }_2^2ϵm_1\left(x,x^{^}\right)\times \\ x^{^T}{\mathrm{PB}}_{20}{x}^{^}{x}^{^T}{B}_{20}^T{\mathrm{Px}}^{^}{)}^{1/2}/\\ ({\gamma }_2^2{d}_{122}^2+\left({\gamma }_2^2-1\right)m_1\left(x,x^{^}\right)\times \\ x^{^T}{\mathrm{PB}}_{20}{x}^{^}{x}^{^T}{B}_{20}^T{\mathrm{Px}}^{^}{)}^{1/2}\end{array}& \left(126\right)\end{array}$ $\begin{array}{cc}{a}_2\left(x,x^{^}\right)=\sqrt{\frac{1}{1+m_1\left(x,x^{^}\right)x^{^T}{C}_{11}^T{C}_{11}{x}^{^}}}& \left(127\right)\end{array}$

[0204] In the equations (126) and (127), m₁(x, x{circumflex over ( )}) is an appropriated positive definite function, and ε is a positive constant defined as ε<1, εγ₂²>1. Thus, the positive definite symmetric solution P is calculated by the computer 20. Using the equations (126) and (127), the foregoing equations (123) and (124) are converted into the following equations (128) and (129).

[0205] x′{circumflex over ( )}=(A+L(u)C₂)x{circumflex over ( )}+(B₂₀+L(u)D₂₂₀)x{circumflex over ( )}u−L(u)y  (128)

[0206] $\begin{array}{cc}u=-\frac{1}{d_{122}}\left[\frac{{\gamma }_2^2-1+{\gamma }_2^2{m}_1\left(x,x^{^}\right)x^{^T}{C}_{11}^T{C}_{11}{x}^{^}}{d_{122}{\gamma }_2^2}x{B}_{20}^{T}P+C_{12}{x}^{^}\right]& \left(129\right)\end{array}$

[0207] In this case, the positive definite symmetric solution is obtained in a simple manner using a conventional software, similarly in the state feedback control system. With this method, an estimated state amount x′{circumflex over ( )} and the control law u=k (y) can be also derived in a simple manner.

[0208] c2-2) Practical Embodiment of a Damping Force Control System in a Wheeled Vehicle on a Basis of the Second Type Control Law:

[0209] In this embodiment, a damping force control system based the second type control law is constructed as shown in FIG. 6, without the provision of the stroke sensor 24, and the microcomputer 20 is arranged to execute the control program shown in FIG. 10 without calculating the relative displacement amount x_pw−x_pband relative velocity x_pw′−x_pb′ applied from the stroke sensor 24 and executing the processing at step 106a.

[0210] In operation of the control system, the computer 20 repeats execution of the control program at each lapse of the predetermined short time and inputs at step 102aa sprung mass acceleration x_pb″ to calculate a sprung mass Velocity x_pb′ at step 104a. At the reflowing step 110a, the computer 20 calculates an estimated state amount x′{circumflex over ( )} and a control input u on a basis of the following equations (130) and (131).

[0211] x′{circumflex over ( )}=(A+L(u)C₂)x{circumflex over ( )}+(B₂+L(u)D₂₂₀)x{circumflex over ( )}u−L(u)y  (130)

[0212] $\begin{array}{cc}u=-\frac{1}{d_{122}}\left[\frac{{\gamma }_2^2-1+{\gamma }_2^2{m}_1\left(x,x^{^}\right)x^{^T}{C}_{11}^T{C}_{11}{x}^{^}}{d_{122}{\gamma }_2^2}x{B}_{20}^{T}P+C_{12}{x}^{^}\right]& \left(131\right)\end{array}$

[0213] In the equations (130) and (131), A, L, C₂, γ₂, m₁(x, x{circumflex over ( )}), C₁₁, P, C₁₂are the same as those in the first type control system, and B₂₀, D₂₂₀, d₁₂₂each are an appropriate constant matrix preliminarly memorized in the computer 20. In this case, the value y represents a sprung mass velocity x_pb′ calculated by processing at step 104a.

[0214] After processing at step 110a, the computer 20 calculates a target damping coefficient C and a target damping force F of the damper 14 by processing at step 112 and 114. In this instance, the target damping force F is calculated on a basis of the estimated relative velocity x_pw′−x_pb′ calculated by processing at step 10a. Subsequently, the computer 20 determines at step 116 and 118 the opening degree OP of the damper orifice 14ato produce the target damping force F. Anus, the damper 14 is adjusted in a simple manner to produce the target damping force F without detection of the relative displacement amount x_pw−x_pb.

[0215] c3-1) Design of a Control System of the Third Type:

[0216] In the control system of this type, B₂(x) and D₂₂(x) in the foregoing equations (81) and (88) each are a unknown function, and an observer gain L is a function matrix as in the second type control system.

[0217] Assuming that the observe gain L is a function of the control input, a nonlinear H_∞ output feedback control law for the generalized plant represented by the equations (114) and (117) of the second type is designed under the following conditions.

[0218] 1) In the case that γ₁is a positive constant defined as γ₁₂I−D₂₁^TΘ^TΘD₁₂>0, γ₂is a positive constant defined as γ₂>I and a positive constant ε is defined as ε₁²−u²>0, the following Riccati inequality (132) for design of an observer (an observer gain) is given, and a positive definite symmetric matrix P, Q and a positive definite matrix Θ satisfying the following Riccati inequality (132) for design of a controller are given.

[0219] 2) In addition, in the case that nonlinear weights a₁(x, x{circumflex over ( )}), a₂(x, x{circumflex over ( )}) satisfy the condition for restriction defined by the following equations (134) and (135), one of control laws defined by the following equation (136) is given by the following equations (137) and (138).

[0220] AQ+QA^T+Q(C₁₁^TC₁₁+C₁₂^TC₁₂−ε²I)Q +B₁(γ₁²I−D₂₁^TΘ^TΘD₂₁)⁻¹B₁^T+B₂₀B₂₀^T<0  (132)

[0221] $\begin{array}{cc}\mathrm{PA}+A^{T}P+\frac{1}{{\gamma }_2^2}{\mathrm{PL}}_1{\Theta }^{-1}{\Theta }^{-T}{L}_1^{T}P+\frac{{ϵ}_1^2{d}_{122}^2}{{\gamma }_2^2}{\mathrm{PL}}_2{\Theta }^{-1}{\Theta }^{-T}{L}_2^{T}P+\frac{{\gamma }_2^2}{{\gamma }_2^2-1}\left(C_{11}^T{C}_{11}+C_{12}^T{C}_{12}\right)<o& \left(133\right)\end{array}$  γ₂²−a₁(x,x{circumflex over ( )})²>0,γ₂²−a₂(x,x{circumflex over ( )})²>0  (134)

[0222] $\begin{array}{cc}\left[\frac{{\gamma }_2^2-{a_2\left(x,x^{^}\right)}^2}{{a_2\left(x,x^{^}\right)}^2{\gamma }_2^2}-\frac{{\gamma }_2^2-1}{{\gamma }_2^2}\right]\frac{1}{d_{122}^2}{x}^{^T}Px\left(B_{20}+L_1{\Theta }^{-1}{\Theta }^{-T}{L}_2^{T}P\right)x^{^}+\left[\frac{{\gamma }_2^2}{{\gamma }_2^2-1}-\frac{{a_1\left(x,x^{^}\right)}^2{\gamma }_2^2}{{\gamma }_2^2-{a_1\left(x,x^{^}\right)}^2}\right]x^{^T}{C}_{11}^T{C}_{11}{x}^{^}\geqq 0& \left(135\right)\end{array}$  ∥[z₁^Tz₂^T]^T∥₂≦γ₁γ₂∥w∥₂  (136)

[0223] x′{circumflex over ( )}=(A+L₁C₂)x{circumflex over ( )}+(B₂₀+L₂D₂₂₀)x{circumflex over ( )}u−(L₁+L₂u)y  (137)

[0224] $\begin{array}{cc}u=-\frac{1}{d_{122}}\left[\frac{{\gamma }_2^2-{a_2\left(x,x^{^}\right)}^2}{{a_2\left(x,x^{^}\right)}^2{\gamma }_2^2{d}_{122}}{x}^{^T}x\left(B_{20}^{T}P+\frac{1}{{\gamma }^2}{\mathrm{PL}}_1{\Theta }^{-1}{\Theta }^{-T}{L}_2^{T}P\right)+C_{12}\right]x^{^}& \left(138\right)\end{array}$

[0225] Provided that, the observer gain L (u) is represented as follows.

[0226] L(u)=−QC₂^TΘ^TΘ−uQD₂₂₀^TΘ^TΘ=L₁+uL₂  (139)

[0227] In the equation (139), L₁, L₂are represented by the following equations (140) and (141).

[0228] L₁=−QC₂^TΘ^TΘ  (140)

[0229] L₂=−QD₂₂₀^TΘ^TΘ  (141)

[0230] The factor Θ is a constant definite matrix under existence of Θ⁻¹, and the observer can be adjusted by the factor Θ. Similarly, in the state feedback control law, the gain L of the controller can be adjusted by the factor d₁₂₂.

[0231] Here, the nonlinear weights a₁(x, x{circumflex over ( )}), a₂(x, x{circumflex over ( )}) satisfying the formulae (134) and (135) are expressed as follows. $\begin{array}{cc}\begin{array}{c}{a}_1\left(x,x^{^}\right)=({\gamma }_2^2{d}_{122}^2+\left({\gamma }_2^2-1\right){\gamma }_2^2ϵm_1\left(x,x^{^}\right)x^{^T}P\times \\ \left(B_{20}+L_1{\Theta }^{-1}{\Theta }^{-T}{L}_2^{T}P\right)x^{^}{x}^{^T}\times \\ {\left(B_{20}^T+{\mathrm{PL}}_2{\Theta }^{-1}{\Theta }^{-T}{L}_1^T\right){\mathrm{Px}}^{^})}^{1/2}/\\ ({\gamma }_2^2{d}_{122}^2+\left({\gamma }_2^2-1\right)m_1\left(x,x^{^}\right)x^{^T}P\times \\ \left(B_{20}+L_1{\Theta }^{-1}{\Theta }^{-T}{L}_2^{T}P\right)x^{^}{x}^{^T}\times \\ {\left(B_{20}^T+{\mathrm{PL}}_2{\Theta }^{-1}{\Theta }^{-T}{L}_1^T\right){\mathrm{Px}}^{^})}^{1/2}\end{array}& \left(142\right)\end{array}$ $\begin{array}{cc}{a}_2\left(x,x^{^}\right)=\sqrt{\frac{1}{1+m_1\left(x,x^{^}\right)x^{^T}{C}_{11}^T{C}_{11}{x}^{^}}}& \left(143\right)\end{array}$

[0232] In the equations (142) and (143), m1(x, x{circumflex over ( )}) is an appropriated positive definite function, and ε is a positive constant defined as ε<1, εγ₂²>1. Thus, the positive definite symmetric solution P is calculated by the computer 20. Using the equations (142) and (143), the foregoing equations (137) and (138) are converted into the following equations (144) and (145).

[0233] x′{circumflex over ( )}=(A+L₁C₂)x{circumflex over ( )}+(B₂x{circumflex over ( )}+L₂D₂₂x{circumflex over ( )})u−L(u)y  (144)

[0234] $\begin{array}{cc}u=-\frac{{\gamma }_2^2-1+{\gamma }_2^2{m}_1\left(x,x^{^}\right)x^{^T}{C}_{11}^T{C}_{11}{x}^{^}}{d_{122}^2{\gamma }_2^2}x\left(B_{20}^T+\frac{1}{{\gamma }_2^2}{\mathrm{PL}}_1{\Theta }^{-1}{\Theta }^{-T}{L}_2^T\right){\mathrm{Px}}^{^}-\frac{1}{d_{122}}{C}_{12}{x}^{^}& \left(145\right)\end{array}$

[0235] In this case, the positive definite symmetric solution P is obtained in a simple manner using a conventional software, similarly in the state feedback control system. With this method, an estimated state amount x′{circumflex over ( )} and the control law u=k (y) can be also derived in a simple manner.

[0236] c3-2) Practical Embodiment of a Damping Force Control System in a Wheeled Vehicle on a Basis of the Third Type Control Law:

[0237] In this embodiment, a damping force control system based on the third type control law is the same in construction as the control system based on the second type control law.

[0238] In operation of the control system, the computer 20 repeats execution of the control program of FIG. 10 at each lapse of the predetermined short time. After processing at 102aand 104a, the computer 20 calculates an estimated state amount x′{circumflex over ( )} and a control input u on a basis of the following equations (146) and (147).

[0239] x′{circumflex over ( )}=(A+L₁C₂)x{circumflex over ( )}+(B₂x{circumflex over ( )}+L₂D₂₂x{circumflex over ( )})u−L(u)y  (146)

[0240] $\begin{array}{cc}u=-\frac{{\gamma }_2^2-1+{\gamma }_2^2{m}_1\left(x,x^{^}\right)x^{^T}{C}_{11}^T{C}_{11}{x}^{^}}{d_{122}^2{\gamma }_2^2}x\left(B_{20}^T+\frac{1}{{\gamma }_2^2}{\mathrm{PL}}_1{\Theta }^{-1}{\Theta }^{-T}{L}_2^T\right){\mathrm{Px}}^{^}-\frac{1}{d_{122}}{C}_{12}{x}^{^}& \left(147\right)\end{array}$

[0241] In the equations (146) and (147), A, C₂, B20, D220, γ₂, m₁(x, x{circumflex over ( )}), C₁₁, d₁₂₂, P, C₁₂are the same as those in the second type control system, and L, L₁, L₂each are a gain defined by the foregoing equations (139) to (141). In addition, the value y represents the sprung mass velocity x_pb′ calculated by processing at step 104a.

[0242] After processing at step 110a, the computer 20 calculates an overall target damping coefficient C and a target damping force F of the damper 14 by processing at step 112 and 118. Thus, the computer 20 determines the opening degree OP of the damper orifice 14ato produce the target damping force F, and the damper 14 is adjusted in a simple manner to produce the target damping force F without detection of the relative displacement amount x_pw−x_pbas in the second type control system.

[0243] d. Third Embodiment:

[0244] d1. Design of a Nonlinear H_∞ Control System of the Kalman Filter Base:

[0245] An output feedback system with a Kalman filter used in an observer is designed in a condition where the bilinear factors B_p2(x_p), D_p2(x_p) are known. In this embodiment, the same reference characters as those in the second embodiment represent the same factors as those in the second embodiment, and the coefficients aid variables related to a plant are suffixed with “p”. The state space of the suspension mechanism is expressed by the following equations (148) and (149).

[0246] x_p′=A_px_p+B_p1w₁+B_p2(x_p)u  (148)

[0247] y_p=C_px_p+D_p1w₂+D_p2(x_p)u  (149)

[0248] In the case that D_p1is defined as d_p1=I, the Kalman filter in the case of t→∞ is represented by the following equation.

[0249] x_o′=A_px_o+B_p2u+K(C_px_o+D_p2(x_p)u−y)  (150)

[0250] Provided that, x_o, x_o′ each are an estimated state amount in the Kalman filter. The gain of the Kalman filter is defined by the following equation (151).

[0251] K=−ΣC_p^TW⁻¹  (151)

[0252] where the estimated error covariance Σ is a positive symmetric solution of the following Riccati equation (152).

[0253] A_pΣ+ΣA_p^T+B_p1VB_p1^T−ΣC_p^TW⁻¹C_pΣ=O  (152)

[0254] where V is a covariance matrix of w₁, and W is a covariance matrix of w₂.

[0255] Illustrated in FIG. 1 is a block diagram of a generalized plant of the system in which an estimated state amount x_omultiplied by a frequency weight Ws (s) and a control input u multiplied by a frequency weight W_u(s) are used as an evaluation output z. That is to say, the Kalman filter is used as a detector in such a manner that the output of the Kalman filter decreases in the control system. Although the control system is different from the first and second embodiments in this point, the same performance as that of the first and second embodiments is obtainable if the state estimation is made in an appropriate manner. The state space of the control system shown iin FIG. 11 is expressed as follows.

[0256] x_p′=Apx_p+B_p1w₁+B_p2(x_p)u  (153)

[0257] x_o′=A_px_o+B_p2(x_p)u+L(C₂x_o+D_p2(x_p)u−y)  (154)

[0258] y=C_px_p+Dp_p1w₂+D_p2(x_p)u  (155)

[0259] x_w′=A_wx_w+B_wC_sx_o  (156)

[0260] z₁=a₁(x_p,x_o,x_w,x_u)(C_wx_w+D_wC_sx_o)  (157)

[0261] x_u′=A_ux_u+B_uu  (158)

[0262] z₂=a₂(x_p,x_o,x_w,x_u)(C_ux_u+D_uu)  (159)

[0263] Provided that, x_pis a state amount of the system, the equation (153) is an expression of the state space of the system, x_ois an estimated state amount, the equation (154) is an expression of the state space of the observer, the value y is a measured output, and the value x_wis a state of the frequency weight. Evaluation outputs z₁, z₂are multiplied by a nonlinear weight designed later.

[0264] In the control system, a control law u=k (x_o) is designed for control of a state of the observer wherein a closed loop system is stable in internal exponent and an L₂gain from w to z is less than a positive constant γ. This control system is characterized by an input x_oto the frequency weight W_s(s) as represented by the following equation (160).

[0265] x_w′=A_wx_w+B_wC_sx_o

[0266] W_s(s):

[0267] z₁=C_wx_w+D_wC_sx_o  (160)

[0268] In the case that an error variable is defined by the following equation (161), an error system is represented by the following equations (162) and (163).

[0269] x_e=x_p−x_o  (161)

[0270] x_e′=(A_p+LC_p)x_e+B_p1w₁+LD_pw₂  (162)

[0271] y_e−y=C_px_o−D_p2(x_p)u=C_px_e+D_p1w₂  (163)

[0272] The equations (162) and (163) are converted into the following equations (164) and (165) by multiplying the factor y by a constant matrix (a scaling matrix) including a reverse matrix.

[0273] x_pe′=(A_p+LC_p)x_pe+B_p1w₁+LD_pw₂  (164)

[0274] y_e⁻=ΘC_px_e+ΘD_p1w₂  (165)

[0275] For the converted error system, an observer gain L is designed in such a manner that an L₂gain of from an external disturbance input w=[w₁^Tw₂^T] to y_e⁻ under existence of a positive constant γ₁becomes γ₁(∥y_e⁻∥₂≦γ₁∥w∥₂).

[0276] In case γ₁is a positive constant satisfying γ₁I−D_p1^TΘ^TΘD_p1>O, the gain L of ∥y_e⁻∥₂≦γ₁∥w∥₂is represented by the following equation (166).

[0277] L=−QC_p^TΘ^TΘ  (166)

[0278] Provided that, Q is a positive definite symmetric matrix satisfying the following Riccati equation (167). $\begin{array}{cc}{A}_{p}Q+{\mathrm{QA}}_p^T+\frac{1}{{\gamma }_1}{B}_{\mathrm{p1}}{B}_{\mathrm{p1}}^T-{\mathrm{QC}}_p^T{\Theta }^T\Theta C_{p}Q=o& \left(167\right)\end{array}$

[0279] In this case, it is to be noted that the Riccati equation (167) is an order of the plant which is smaller than that of the generalized plants in the first and second embodiments.

[0280] Subsequently, the equation (154) related to the plant is rewritten as follows.

[0281] x_o′=Ax_o+B₂(x_p)u+L(C₂x_o+D_p2(x_p)u−y =Ax_o+B₂(x_p)u+LΘ⁻¹y_e{tilde over ( )}  (168)

[0282] Using an observer represented by the equation (168), a controller is designed in such a manner that an L₂gain of from an observer error y_e{circumflex over ( )} to an evaluation output z under existence of a positive constant γ₂becomes less than γ₂(∥z∥₂∥y_e⁻∥₂). Here, in the case that the observer is used to provide a generalized plant in comply with a state variable x_w, x_u, the state space of the plant is represented by the following equations (169) to (171).

[0283] x_k′=Ax_k+B₂(x_p)u+L₁Θ⁻¹y_e⁻  (169)

[0284] z₁=a₁(x_p,x_k)C₁₁x_k  (170)

[0285] z₂=a₂(x_p,x_k)C₁₂x_k+a₂(x_p,x_k)D₁₂u  (171)

[0286] Provided that, each variable matrix and constant matrix in the equations (169) to (171) are represented by the following equations (172) to (179). $\begin{array}{cc}{x}_k=\left[\begin{array}{c}\begin{array}{c}{x}_0\\ x_w\end{array}\\ x_u\end{array}\right]& \left(172\right)\\ A=\left[\begin{array}{ccc}{A}_p& o& o\\ B_w{C}_s& A_w& o\\ o& o& A_u\end{array}\right]& \left(173\right)\\ B_2\left(x_p\right)=\left[\begin{array}{c}\begin{array}{c}{B}_{\mathrm{p2}}\left(x_p\right)\\ 0\end{array}\\ B_u\end{array}\right]& \left(174\right)\\ L_1=\left[\begin{array}{c}\begin{array}{c}L\\ 0\end{array}\\ 0\end{array}\right]& \left(175\right)\\ z=\left[\begin{array}{c}{z}_1\\ z_2\end{array}\right]& \left(176\right)\end{array}$  C₁₁=[D_wC_sC_wo]  (177)

[0287] C₁₂=[ooC_u]  (178)

[0288] D₁₂=D_u  (179)

[0289] The state amount x_kdefined above does not include a state amount x_p.

[0290] If there exists D₁₂⁻¹, a positive definite symmetric solution P of the following Riccati inequality (180) is obtainable. When the nonlinear weights a₁(x_p, x_k), a₂(x_p, x_k) satisfy the following equation (181), the controller is given by the following equation (182). $\begin{array}{cc}\mathrm{PA}+A^{T}P+\frac{\mathrm{PL}{\Theta }^{-1}{\Theta }^{-T}{L}^{T}P}{{\gamma }_2^2}+C_{11}^T{C}_{11}+C_{12}^T{C}_{12}<o& \left(180\right)\\ \left(1-{a_1\left(x_p,x_k\right)}^2\right)x_k^T{C}_{11}^T{C}_{11}{x}_k+[\frac{1}{{a_2\left(x_p,x_k\right)}^2}-1]\times x_k^T{\mathrm{PB}}_2\left(x_p\right)D_{12}^{-1}{D}_{12}^{-T}{B}_2^T\left(x_p\right)P{x}_k\geqq 0& \left(181\right)\\ u=-D_{12}^{-1}[\frac{1}{{a_2\left(x_p,x_k\right)}^2}{D}_{12}^{-1}{B}_2^T\left(x_p\right)P+C_{12}]x& \left(182\right)\end{array}$