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Abstract

This article focuses on static output feedback $H_{\infty}$ control for active suspension system with input delay and parameter uncertainty. The parameter uncertainty of active suspension system model, input delay of actuator, input disturbance of road, and measurement output disturbance are simultaneously introduced in active suspension system. A kind of static output feedback $H_{\infty}$ controller is designed, which can consider the above factors, simultaneously. First of all, the mathematical model of quarter-vehicle active suspension system and the system state equation are established. Second, the static output feedback $H_{\infty}$ controller is designed by employing the Lyapunov–Krasovskii functional for system state equations without or with disturbance. The design problem of static output feedback $H_{\infty}$ controller for closed-loop system is transformed into the solving problem of linear matrix inequality. Finally, according to the designed controller and the specific vehicle parameters, the simulation model of quarter-vehicle active suspension system is established. And the simulations of three cases, for example, without disturbance, only with input disturbance, and with input disturbance and output disturbance, are exploited to demonstrate the feasibility and effectiveness of the proposed schemes.

Keywords

Active suspension system static output feedback input delay parameter uncertainty input disturbance output disturbance

Introduction

It is well known that suspension system is one of the important parts of vehicle chassis, which heavily affects the handling stability and ride performance which are at odds with each other.¹ Due to structural constraints, the passive suspension system can hardly improve the two properties at the same time. However, the active suspension system can synchronously give consideration to the two conflicting performances through adjusting the parameters of suspension system proactively according to the state of vehicle movement and the condition of road surface, rather than passive suspension whose parameters are fixed and designed by a compromise.² At present, active suspension system is faced with parameter uncertainty of system,³ input delay of actuator, input disturbance of uneven pavement, and measurement output disturbance of sensor,^4,5 which have great influences on the stability and control effect of active suspension system.^1,2,6

Therefore, in the recent decades, the control models and strategies of active suspension system are attracting more and more attention and exploration. A single-step method to acquire static output feedback controller for vehicle active suspension system was proposed based on finite frequency.⁷ A two-step computational approach was used for the design of static output feedback controller for vehicle suspension system.⁸ A kind of dynamic output feedback $H_{\infty}$ controller was studied with control delay and output constraints for vehicle suspension system.⁹ The dynamic output feedback $H_{\infty}$ control problem was addressed with time-varying input delay for active half-vehicle suspension system.¹⁰ According to the perturbation problem of system controller and parameter, the static output feedback for active suspension system was discussed based on the non-fragile $H_{2}$ /generalized $H_{2}$ optimal control.¹¹ The problem of non-fragile $H_{\infty}$ static output feedback control of vehicle active suspension with finite-frequency constraint was dealt with, and a new non-fragile $H_{\infty}$ finite-frequency control condition was established using generalized Kalman–Yakubovich–Popov lemma.¹² A sensor fault detection and isolation method was put forward for ordinary faults as vehicle active suspension system deadlocking, gain variation, and constant deviation.¹³ Aiming at the effects of actuator’s faults of semi-active suspension on the ride comfort of vehicle, an active fault-tolerant control strategy was proposed based on fault compensation.¹⁴ A model-free fractional-order sliding mode control based on an extended state observer for a quarter car active suspension systems was presented.¹⁵ An optimal fuzzy integrated control method was proposed and used in a half-vehicle model of active suspension system.¹⁶ A dynamic output feedback fuzzy control methodology for Takagi–Sugeno fuzzy system with time-varying input delay and output constraints was proposed and applied to active suspension system with different road conditions.¹⁷ A static disturbance compensator was proposed by employing linear matrix inequality (LMI), and the feedback controller and disturbance compensator were designed simultaneously for the disturbances in the linear time-invariant systems.¹⁸ A control method on optimal sliding mode for nonlinear active suspension system was put forward to obtain the better robustness and optimal suspension performance.¹⁹ A fuzzy control problem for active suspension system with uncertainty was investigated using dynamic sliding mode method, and the nonlinear active suspension system was described by adopting the Takagi–Sugeno fuzzy approach.²⁰ A robust non-fragile dynamic output feedback controller with fixed-order was developed for active suspension system of a quarter vehicle, in view of LMI and convex optimization.²¹ For an active hydraulically suspension system, a new approach to deal with the uncertain factors in the interval analysis method was proposed using the Chebyshev polynomial series.²² The quarter-vehicle model with asymmetrical damping was established, and the effects were analyzed under harmonic excitation.²³ The stability switches and bifurcation in nonlinear quarter-vehicle system were explored with small time-delayed feedback control.²⁴

However, most of the existing related literatures are concerned with one or two problems of parameter uncertainty of system, input delay of actuator, input disturbance of uneven road, and measurement output disturbance of sensor.²⁵ The design issues of active suspension controller are rarely involved with a comprehensive consideration of the above four factors, especially for static output feedback $H_{\infty}$ control. The main contributions of this article are as follows:

Based on $H_{\infty}$ control theory,²⁶ the Lyapunov–Krasovskii functional stability theory, and the LMI scheme,²⁷ a design methodology of static output feedback $H_{\infty}$ controller for active suspension system is proposed, which is related to the above four factors, simultaneously.

According to the designed controller and the specific vehicle parameters, the simulation model of quarter-vehicle active suspension system is established, and the simulations of three cases are exploited to demonstrate the feasibility and effectiveness of the proposed schemes.

This article will be organized as follows. In section “Problem formulation,” the typical structure and mathematical model of a quarter-vehicle active suspension system are provided, and the state-space equation of system is established. In section “Main results,” the static output feedback $H_{\infty}$ controller is designed by employing the Lyapunov–Krasovskii functional for system state equations without or with disturbance. In section “Numerical simulation,” the effectiveness of the designed controller is analyzed through numerical simulation based on a set of specific vehicle parameters. In section “Conclusion,” some conclusions and the future work are presented briefly.

Problem formulation

The typical structure and model of quarter-vehicle active suspension system with 2 degrees of freedom are shown in Figure 1, in which $m_{u}$ is the unsprung mass of suspension system and $m_{s}$ represents the sprung mass of suspension system; $z_{r}$ stands for the input displacement of road; $z_{u}$ and $z_{s}$ , respectively, denote the displacements of unsprung mass and sprung mass; $u (t - h (t))$ is the active control input of suspension system with a time-varying delay $h (t)$ ; $k_{s}$ and $c_{s}$ denote the stiffness and the damping of suspension system, respectively; $k_{t}$ and $c_{t}$ , respectively, stand for the stiffness and the damping of auto tire.

Figure 1.

Structure and model of active suspension system: (a) typical structure of active suspension system and (b) quarter-vehicle suspension model.

Based on Figure 1 and Newton’s second law, the dynamics differential equations of quarter-vehicle active suspension system can be established as follows

{\begin{matrix} m_{s} {\overset{\cdot\cdot}{z}}_{s} (t) = u (t - h (t)) - c_{s} ({\overset{\cdot}{z}}_{s} (t) - {\overset{\cdot}{z}}_{u} (t)) - k_{s} (z_{s} (t) - z_{u} (t)) \\ m_{u} {\overset{\cdot\cdot}{z}}_{u} (t) = - u (t - h (t)) + c_{s} ({\overset{\cdot}{z}}_{s} (t) - {\overset{\cdot}{z}}_{u} (t)) + k_{s} (z_{s} (t) - z_{u} (t)) \\ - c_{t} ({\overset{\cdot}{z}}_{u} (t) - {\overset{\cdot}{z}}_{r} (t)) - k_{t} (z_{u} (t) - z_{r} (t)) \end{matrix}

(1)

where $h (t)$ is a known time-varying delay and satisfies the inequalities below

0 \leq h (t) \leq h^{*} < \infty, \overset{\cdot}{h} (t) \leq ρ_{h} < 1

(2)

where $h^{*}$ and $ρ_{h}$ are two positive real numbers.

According to the structural characteristics and performance requirements of active suspension system, the suspension deflection $z_{s} (t) - z_{u} (t)$ , the tire deflection $z_{u} (t) - z_{r} (t)$ , the speed of sprung mass ${\overset{\cdot}{z}}_{s} (t)$ , and the speed of unsprung mass ${\overset{\cdot}{z}}_{u} (t)$ are defined as the state variables; the acceleration of sprung mass ${\overset{\cdot\cdot}{z}}_{s} (t)$ , the speed of sprung mass ${\overset{\cdot}{z}}_{s} (t)$ , the suspension deflection $z_{s} (t) - z_{u} (t)$ , and the tire deflection $z_{u} (t) - z_{r} (t)$ are chosen as the control output variables; and the speed of sprung mass ${\overset{\cdot}{z}}_{s} (t)$ and the suspension deflection $z_{s} (t) - z_{u} (t)$ are taken as the measured output variables. Therefore, the state vector $x (t) = [z_{s} (t) - z_{u} (t), z_{u} (t) - z_{r} (t), {\overset{\cdot}{z}}_{s} (t), {\overset{\cdot}{z}}_{u} (t)]^{T} \in R^{4}$ , the control output vector $z (t) = [{\overset{\cdot\cdot}{z}}_{s} (t), {\overset{\cdot}{z}}_{s} (t), z_{s} (t) - z_{u} (t), z_{u} (t) - z_{r} (t)]^{T} \in R^{4}$ , and the measured output vector $y (t) = [{\overset{\cdot}{z}}_{s} (t), z_{s} (t) - z_{u} (t)]^{T} \in R^{2}$ .

Considering the input delay of actuator, the parameter uncertainty of system, the input disturbance of road, and the output disturbance of measurement, according to equation (1), the quarter-vehicle active suspension system can be described by the following state-space equations

{\begin{matrix} \overset{\cdot}{x} (t) = (A + Δ A) x (t) + (B + Δ B) u (t - h (t)) + L_{1} w_{1} (t) \\ z (t) = C_{1} x (t) + Du (t - h (t)) \\ y (t) = C_{2} x (t) + L_{2} w_{2} (t) \end{matrix}

(3)

where $w_{1} (t) = {\overset{\cdot}{z}}_{r} (t)$ is the input disturbance of road, $w_{2} (t)$ is the measurement output disturbance which is produced by sensor, and

\begin{matrix} A = [\begin{matrix} 0 & 0 & 1 & - 1 \\ 0 & 0 & 0 & 1 \\ \frac{- k_{s}}{m_{s}} & 0 & \frac{- c_{s}}{m_{s}} & \frac{c_{s}}{m_{s}} \\ \frac{k_{s}}{m_{u}} & \frac{- k_{t}}{m_{u}} & \frac{c_{s}}{m_{u}} & \frac{- c_{s} - c_{t}}{m_{u}} \end{matrix}], B = [\begin{matrix} 0 \\ 0 \\ \frac{1}{m_{s}} \\ - \frac{1}{m_{u}} \end{matrix}], \\ L_{1} = [\begin{matrix} 0 \\ - 1 \\ 0 \\ \frac{c_{t}}{m_{u}} \end{matrix}], C_{1} = [\begin{matrix} \frac{- k_{s}}{m_{s}} & 0 & \frac{- c_{s}}{m_{s}} & \frac{c_{s}}{m_{s}} \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}] \\ D = [\begin{matrix} \frac{1}{m_{s}} \\ 0 \\ 0 \\ 0 \end{matrix}], C_{2} = [\begin{matrix} 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}], L_{2} = [\begin{matrix} β_{1} \\ β_{2} \end{matrix}] \end{matrix}

where $β_{1}$ and $β_{2}$ are the adjustable weights of measurement output disturbance $w_{2} (t)$ .

Let $Δ A$ and $Δ B$ are the uncertainty parameter matrices with norm-bounded, then the uncertainty of system can be described as follows

[Δ A, Δ B] = HF (t) [E_{1}, E_{2}]

(4)

where $H$ , $E_{1}$ , and $E_{2}$ are the known real constant matrices with appropriate dimensions; $F (t)$ is the unknown time-varying matrix which satisfies $F^{T} (t) F (t) \leq I$ , where $I$ indicates the unit matrix with appropriate dimension.

Main results

In this section, the problem of static output feedback $H_{\infty}$ controller design will be solved for active suspension system with input delay and parameter uncertainty. Theorem 1 is aimed at the situation without disturbances, and Theorem 2 is aimed at the situation with input disturbance of road and measurement output disturbance of sensor.

Theorem 1

Consider the quarter-vehicle active suspension system (3) without disturbances, that is, $w_{1} (t) = 0$ and $w_{2} (t) = 0$ , if there exist positive-definite matrices $X, Z \in R^{n \times n}$ , arbitrary matrix $Y \in R^{m \times n}$ , and a scalar $α > 0$ , such that the following $LMI$ holds

[\begin{matrix} S & BY & {XE}_{1}^{T} \\ ★ & - (1 - ρ_{h}) Z & Y^{T} E_{2}^{T} \\ ★ & ★ & - α I \end{matrix}] < 0

(5)

where $S = X A^{T} + AX + Z + α H H^{T}$ , ★ denotes the matrix obtained by matrix symmetry, then the quarter-vehicle active suspension system (3) with the controller

u (t) = Ky (t), where K = Y X^{- 1} C_{2}^{†}

(6)

is asymptotically stable, that is, the control output also approaches to zero, where $C_{2}^{†}$ denotes the generalized inverse matrix of $C_{2}$ .

Proof

Define $\bar{A} = A + Δ A$ and $\bar{B} = B + Δ B$ , the system in equation (3) with $w_{1} (t) = 0$ and $w_{2} (t) = 0$ can be expressed as

{\begin{matrix} \overset{\cdot}{x} (t) = \bar{A} x (t) + \bar{B} u (t - h (t)) \\ z (t) = C_{1} x (t) + Du (t - h (t)) \\ y (t) = C_{2} x (t) \end{matrix}

(7)

According to equation (7) and $u (t) = Ky (t)$ , $\overset{\cdot}{x} (t)$ can be obtained

\overset{\cdot}{x} (t) = \bar{A} x (t) + \bar{B} K C_{2} x (t - h (t))

(8)

Considering the Lyapunov–Krasovskii function as follows

V (x (t)) = x (t)^{T} Px (t) + \int_{t - h (t)}^{t} x {(s)}^{T} Rx (s) ds

(9)

where $P$ and $R$ are symmetric positive-definite matrices.

From equations (8) and (9), we can obtain the derivative of $V (x (t))$ as

\begin{matrix} \overset{\cdot}{V} (x (t)) = & 2 x^{T} (t) P \overset{\cdot}{x} (t) + x^{T} (t) Rx (t) - (1 - \overset{\cdot}{h} (t)) x^{T} (t - h (t)) Rx (t - h (t)) \\ = & x^{T} (t) ({\bar{A}}^{T} P + P \bar{A} + R) x (t) - (1 - \overset{\cdot}{h} (t)) x^{T} (t - h (t)) Rx (t - h (t)) \\ + 2 x^{T} (t) P \bar{B} K C_{2} x (t - h (t)) \\ = & {[\begin{matrix} x (t) \\ x (t - h (t)) \end{matrix}]}^{T} [\begin{matrix} {\bar{A}}^{T} P + P \bar{A} + R & P \bar{B} K C_{2} \\ ★ & - (1 - \overset{\cdot}{h} (t)) R \end{matrix}] [\begin{matrix} x (t) \\ x (t - h (t)) \end{matrix}] \end{matrix}

(10)

The above formula (10) can be changed as

\overset{\cdot}{V} (x (t)) = {[\begin{matrix} x (t) \\ x (t - h (t)) \end{matrix}]}^{T} [\begin{matrix} L + Δ L \end{matrix}] [\begin{matrix} x (t) \\ x (t - h (t)) \end{matrix}]

(11)

where $L = [\begin{matrix} A^{T} P + PA + R & PBK C_{2} \\ ★ & - (1 - \overset{\cdot}{h} (t)) R \end{matrix}]$ , $Δ L = [\begin{matrix} Δ A^{T} P + P Δ A & P Δ BK C_{2} \\ ★ & 0 \end{matrix}]$ .

According to equation (4), $Δ L$ can be expressed as

\begin{matrix} Δ L & = [\begin{matrix} E_{1}^{T} F^{T} (t) H^{T} P + PHF (t) E_{1} & PHF (t) E_{2} K C_{2} \\ ★ & 0 \end{matrix}] \\ = [\begin{matrix} PH \\ 0 \end{matrix}] F (t) {[\begin{matrix} E_{1}^{T} \\ C_{2}^{T} K^{T} E_{2}^{T} \end{matrix}]}^{T} + [\begin{matrix} E_{1}^{T} \\ C_{2}^{T} K^{T} E_{2}^{T} \end{matrix}] F^{T} (t) {[\begin{matrix} PH \\ 0 \end{matrix}]}^{T} \end{matrix}

(12)

It can be known from Lemma 1 that there exists a scalar $α > 0$ to make $Δ L$ satisfy the following matrix inequality

Δ L \leq α [\begin{matrix} PH \\ 0 \end{matrix}] {[\begin{matrix} PH \\ 0 \end{matrix}]}^{T} + α^{- 1} [\begin{matrix} E_{1}^{T} \\ C_{2}^{T} K^{T} E_{2}^{T} \end{matrix}] {[\begin{matrix} E_{1}^{T} \\ C_{2}^{T} K^{T} E_{2}^{T} \end{matrix}]}^{T}

(13)

According to equations (2), (11), and (13), it can be obtained that

L + Δ L \leq [\begin{matrix} A^{T} P + P A + R + α P H H^{T} P & P B K C_{2} \\ ★ & - (1 - ρ_{h}) R \end{matrix}] + α^{- 1} [\begin{matrix} E_{1}^{T} \\ C_{2}^{T} K^{T} E_{2}^{T} \end{matrix}] [\begin{matrix} E_{1} & E_{2} K C_{2} \end{matrix}]

(14)

Let $X = P^{- 1}$ , $Z = P^{- 1} R P^{- 1}$ , $Y = K C_{2} P^{- 1}$ , and equation (5) multiplied by diag ${P, P, I}$ on both sides, then the equivalent inequality of equation (5) can be obtained as follows

[\begin{matrix} A^{T} P + PA + R + α PH H^{T} P & PBK C_{2} & E_{1}^{T} \\ ★ & - (1 - ρ_{h}) R & C_{2}^{T} K^{T} E_{2}^{T} \\ ★ & ★ & - α I \end{matrix}] < 0

(15)

According to the Schur-complement theorem (Lemma 2), equation (15) is equivalent to the following inequality

[\begin{matrix} A^{T} P + P A + R + α P H H^{T} P & P B K C_{2} \\ ★ & - (1 - ρ_{h}) R \end{matrix}] + α^{- 1} [\begin{matrix} E_{1}^{T} \\ C_{2}^{T} K^{T} E_{2}^{T} \end{matrix}] [\begin{matrix} E_{1} & E_{2} K C_{2} \end{matrix}] < 0

(16)

Through equations (14) and (16), it is easy to get that

L + Δ L < 0

(17)

Therefore, $\overset{\cdot}{V} (x (t)) < 0$ according to equations (11) and (17), and the designed controller $u (t) = Y X^{- 1} C_{2}^{†} y (t)$ meets the design requirements. The proof is completed.

Theorem 2

Consider the quarter-vehicle active suspension system (3) with disturbances, if there exist positive-definite matrices $X, Z \in R^{n \times n}$ , arbitrary matrix $Y \in R^{m \times n}$ , and given scalars $α > 0$ , $γ > 0$ , such that the following $LMI$ holds

[\begin{matrix} S & BY & L_{1} & BK L_{2} & {XE}_{1}^{T} & {XC}_{1}^{T} \\ ★ & - (1 - ρ_{h}) Z & 0 & 0 & Y^{T} E_{2}^{T} & Y^{T} D^{T} \\ ★ & ★ & - γ I & 0 & 0 & 0 \\ ★ & ★ & ★ & - γ I & L_{2}^{T} K^{T} E_{2}^{T} & L_{2}^{T} K^{T} D^{T} \\ ★ & ★ & ★ & ★ & - α I & 0 \\ ★ & ★ & ★ & ★ & ★ & - γ I \end{matrix}] < 0

(18)

where $S = AX + X A^{T} + Z + α H H^{T}$ , ★ denotes the matrix obtained by matrix symmetry, then the quarter-vehicle active suspension system (3) with the controller

u (t) = Ky (t), where K = Y X^{- 1} C_{2}^{†}

(19)

is asymptotically stable for $H_{\infty}$ performance index $γ$ , where $C_{2}^{†}$ denotes the generalized inverse matrix of $C_{2}$ .

Proof

Similar to Theorem 1, define $\bar{A} = A + Δ A$ and $\bar{B} = B + Δ B$ , the system in equation (3) can be expressed as

{\begin{matrix} \overset{\cdot}{x} (t) = \bar{A} x (t) + \bar{B} u (t - h (t)) + L_{1} w_{1} (t) \\ z (t) = C_{1} x (t) + Du (t - h (t)) \\ y (t) = C_{2} x (t) + L_{2} w_{2} (t) \end{matrix}

(20)

According to equation (20) and $u (t) = Ky (t)$ , $\overset{\cdot}{x} (t)$ can be obtained

\dot{x} (t) = \bar{A} x (t) + \bar{B} K C_{2} x (t - h (t)) + \bar{B} K L_{2} w_{2} (t - h (t)) + L_{1} w_{1} (t)

(21)

Consider the Lyapunov–Krasovskii function as follows

V (x (t)) = x (t)^{T} Px (t) + \int_{t - h (t)}^{t} x {(s)}^{T} Rx (s) ds

(22)

where $P$ and $R$ are symmetric positive-definite matrices.

From equations (21) and (22), we can obtain the derivative of $V (x (t))$ as

\begin{matrix} \overset{\cdot}{V} (x (t)) = & 2 x^{T} (t) P \overset{\cdot}{x} (t) + x^{T} (t) Rx (t) - (1 - \overset{\cdot}{h} (t)) x^{T} (t - h (t)) Rx (t - h (t)) \\ = & x^{T} (t) (P \bar{A} + {\bar{A}}^{T} P + R) x (t) - (1 - \overset{\cdot}{h} (t)) x^{T} (t - h (t)) Rx (t - h (t)) \\ + 2 x^{T} (t) P \bar{B} K C_{2} x (t - h (t)) + 2 x^{T} (t) P [\bar{B} K L_{2} w_{2} (t - h (t)) + L_{1} w_{1} (t)] \\ = & {[\begin{matrix} x (t) \\ x (t - h (t)) \\ w_{1} (t) \\ w_{2} (t - h (t)) \end{matrix}]}^{T} Q_{1} [\begin{matrix} x (t) \\ x (t - h (t)) \\ w_{1} (t) \\ w_{2} (t - h (t)) \end{matrix}] \end{matrix}

(23)

where $Q_{1} = N_{1} + Δ N_{1}$ and

\begin{matrix} N_{1} = [\begin{matrix} PA + A^{T} P + R & PBK C_{2} & P L_{1} & PBK L_{2} \\ ★ & - (1 - \overset{\cdot}{h} (t)) R & 0 & 0 \\ ★ & ★ & 0 & 0 \\ ★ & ★ & ★ & 0 \end{matrix}] \\ Δ N_{1} = [\begin{matrix} P Δ A + Δ A^{T} P & P Δ BK C_{2} & 0 & P Δ BK L_{2} \\ ★ & 0 & 0 & 0 \\ ★ & ★ & 0 & 0 \\ ★ & ★ & ★ & 0 \end{matrix}] \end{matrix}

According to equation (4), $Δ L$ can be expressed as

\begin{matrix} Δ N_{1} = [\begin{matrix} PHF (t) E_{1} + E_{1}^{T} F^{T} (t) H^{T} P & PHF (t) E_{2} K C_{2} & 0 & PHF (t) E_{2} K L_{2} \\ ★ & 0 & 0 & 0 \\ ★ & ★ & 0 & 0 \\ ★ & ★ & ★ & 0 \end{matrix}] \\ = [\begin{matrix} PH \\ 0 \\ 0 \\ 0 \end{matrix}] F (t) {[\begin{matrix} E_{1}^{T} \\ C_{2}^{T} K^{T} E_{2}^{T} \\ 0 \\ L_{2}^{T} K^{T} E_{2}^{T} \end{matrix}]}^{T} + [\begin{matrix} E_{1}^{T} \\ C_{2}^{T} K^{T} E_{2}^{T} \\ 0 \\ L_{2}^{T} K^{T} E_{2}^{T} \end{matrix}] F^{T} (t) {[\begin{matrix} PH \\ 0 \\ 0 \\ 0 \end{matrix}]}^{T} \end{matrix}

(24)

It can be known from Lemma 1 that there exists a scalar $α > 0$ to make $Δ N_{1}$ satisfy the following matrix inequality

Δ N_{1} \leq α [\begin{matrix} PH \\ 0 \\ 0 \\ 0 \end{matrix}] {[\begin{matrix} PH \\ 0 \\ 0 \\ 0 \end{matrix}]}^{T} + α^{- 1} [\begin{matrix} E_{1}^{T} \\ C_{2}^{T} K^{T} E_{2}^{T} \\ 0 \\ L_{2}^{T} K^{T} E_{2}^{T} \end{matrix}] {[\begin{matrix} E_{1}^{T} \\ C_{2}^{T} K^{T} E_{2}^{T} \\ 0 \\ L_{2}^{T} K^{T} E_{2}^{T} \end{matrix}]}^{T}

(25)

Define $H_{\infty}$ performance index as follows²⁸

J_{T} = \int_{0}^{T} [γ^{- 1} z^{T} (t) z (t) - γ w_{1}^{T} (t) w_{1} (t) - γ w_{2}^{T} (t - h (t)) w_{2} (t - h (t))] dt

(26)

where $γ > 0$ .

For the arbitrary nonzero perturbations and $T > 0$ , according to equation (22) and the zero initial conditions, equation (26) can be changed as

\begin{matrix} J_{T} = \int_{0}^{T} [γ^{- 1} z^{T} (t) z (t) - γ w_{1}^{T} (t) w_{1} (t) - γ w_{2}^{T} (t - h (t)) w_{2} (t - h (t)) + \overset{\cdot}{V} (x (t), t) - \overset{\cdot}{V} (x (t), t)] dt \\ = \int_{0}^{T} [γ^{- 1} z^{T} (t) z (t) - γ w_{1}^{T} (t) w_{1} (t) - γ w_{2}^{T} (t - h (t)) w_{2} (t - h (t)) + \overset{\cdot}{V} (x (t), t)] dt - V (T) \\ \leq \int_{0}^{T} [γ^{- 1} z^{T} (t) z (t) - γ w_{1}^{T} (t) w_{1} (t) - γ w_{2}^{T} (t - h (t)) w_{2} (t - h (t)) + \overset{\cdot}{V} (x (t), t)] dt \end{matrix}

(27)

From equation (20), $z (t)$ can be expressed as

\begin{matrix} z (t) & = C_{1} x (t) + Du (t - h (t)) \\ = C_{1} x (t) + DK C_{2} x (t - h (t)) + DK L_{2} w_{2} (t - h (t)) \\ = [\begin{matrix} C_{1} & DK C_{2} & 0 & DK L_{2} \end{matrix}] [\begin{matrix} x (t) \\ x (t - h (t)) \\ w_{1} (t) \\ w_{2} (t - h (t)) \end{matrix}] \end{matrix}

(28)

Then

γ^{- 1} z^{T} (t) z (t) = {[\begin{matrix} x (t) \\ x (t - h (t)) \\ w_{1} (t) \\ w_{2} (t - h (t)) \end{matrix}]}^{T} Q_{2} [\begin{matrix} x (t) \\ x (t - h (t)) \\ w_{1} (t) \\ w_{2} (t - h (t)) \end{matrix}]

(29)

where $Q_{2} = γ^{- 1} [\begin{matrix} C_{1} & DK C_{2} & 0 & DK L_{2} \end{matrix}]^{T} [\begin{matrix} C_{1} & DK C_{2} & 0 & DK L_{2} \end{matrix}]$ .

Simultaneously, $- γ w_{1}^{T} (t) w_{1} (t) - γ w_{2}^{T} (t - h (t)) w_{2} (t - h (t))$ can be expressed as

- γ w_{1}^{T} (t) w_{1} (t) - γ w_{2}^{T} (t - h (t)) w_{2} (t - h (t)) = {[\begin{matrix} x (t) \\ x (t - h (t)) \\ w_{1} (t) \\ w_{2} (t - h (t)) \end{matrix}]}^{T} Q_{3} [\begin{matrix} x (t) \\ x (t - h (t)) \\ w_{1} (t) \\ w_{2} (t - h (t)) \end{matrix}]

(30)

where $Q_{3} = [\begin{matrix} 0 & 0 & 0 & 0 \\ ★ & 0 & 0 & 0 \\ ★ & ★ & - γ I & 0 \\ ★ & ★ & ★ & - γ I \end{matrix}]$ .

Therefore, equation (27) can be expressed as

J_{T} \leq \int_{0}^{T} {[\begin{matrix} x (t) \\ x (t - h (t)) \\ w_{1} (t) \\ w_{2} (t - h (t)) \end{matrix}]}^{T} [\begin{matrix} Q_{1} + Q_{2} + Q_{3} \end{matrix}] [\begin{matrix} x (t) \\ x (t - h (t)) \\ w_{1} (t) \\ w_{2} (t - h (t)) \end{matrix}] dt

(31)

Let $X = P^{- 1}$ , $Z = P^{- 1} R P^{- 1}$ , $Y = K C_{2} P^{- 1}$ , and equation (18) multiplied by diag ${P, P, I, I, I, I}$ on both sides, then the equivalent inequality of equation (18) can be obtained as follows

[\begin{matrix} PA + A^{T} P + R + α PH H^{T} P & PBK C_{2} & P L_{1} & PBK L_{2} & E_{1}^{T} & C_{1}^{T} \\ ★ & - (1 - ρ_{h}) R & 0 & 0 & C_{2}^{T} K^{T} E_{2}^{T} & C_{2}^{T} K^{T} D^{T} \\ ★ & ★ & - γ I & 0 & 0 & 0 \\ ★ & ★ & ★ & - γ I & L_{2}^{T} K^{T} E_{2}^{T} & L_{2}^{T} K^{T} D^{T} \\ ★ & ★ & ★ & ★ & - α I & 0 \\ ★ & ★ & ★ & ★ & ★ & - γ I \end{matrix}] < 0

(32)

According to the Schur-complement theorem (Lemma 2), equation (32) is equivalent to the following inequality

S_{11} - S_{12} S_{22}^{- 1} S_{12}^{T} < 0

(33)

where

S_{11} = [\begin{matrix} PA + A^{T} P + R + α PH H^{T} P & PBK C_{2} & P L_{1} & PBK C_{2} \\ ★ & - (1 - ρ_{h}) R & 0 & 0 \\ ★ & ★ & - γ I & 0 \\ ★ & ★ & ★ & - γ I \end{matrix}]

S_{12} = [\begin{matrix} E_{1}^{T} & C_{1}^{T} \\ C_{2}^{T} K^{T} E_{2}^{T} & C_{2}^{T} K^{T} D^{T} \\ 0 & 0 \\ L_{2}^{T} K^{T} E_{2}^{T} & L_{2}^{T} K^{T} D^{T} \end{matrix}], S_{22} = [\begin{matrix} - α I & 0 \\ ★ & - γ I \end{matrix}]

Define $M_{1} = S_{11} - S_{12} S_{22}^{- 1} S_{12}^{T}$ , then

\begin{matrix} M_{1} = [\begin{matrix} PA + A^{T} P + R & PBK C_{2} & P L_{1} & PBK C_{2} \\ ★ & - (1 - ρ_{h}) R & 0 & 0 \\ ★ & ★ & 0 & 0 \\ ★ & ★ & ★ & 0 \end{matrix}] \\ + [\begin{matrix} 0 & 0 & 0 & 0 \\ ★ & 0 & 0 & 0 \\ ★ & ★ & - γ I & 0 \\ ★ & ★ & ★ & - γ I \end{matrix}] \\ + α [\begin{matrix} PH \\ 0 \\ 0 \\ 0 \end{matrix}] {[\begin{matrix} PH \\ 0 \\ 0 \\ 0 \end{matrix}]}^{T} + α^{- 1} [\begin{matrix} E_{1}^{T} \\ C_{2}^{T} K^{T} E_{2}^{T} \\ 0 \\ L_{2}^{T} K^{T} E_{2}^{T} \end{matrix}] {[\begin{matrix} E_{1}^{T} \\ C_{2}^{T} K^{T} E_{2}^{T} \\ 0 \\ L_{2}^{T} K^{T} E_{2}^{T} \end{matrix}]}^{T} \\ + γ^{- 1} [\begin{matrix} C_{1}^{T} \\ C_{2}^{T} K^{T} D^{T} \\ 0 \\ L_{2}^{T} K^{T} D^{T} \end{matrix}] {[\begin{matrix} C_{1}^{T} \\ C_{2}^{T} K^{T} D^{T} \\ 0 \\ L_{2}^{T} K^{T} D^{T} \end{matrix}]}^{T} \end{matrix}

(34)

Through equations (23), (25), (29), (30), and (34), it is easy to find

[Q_{1} + Q_{2} + Q_{3}] \leq M_{1} < 0

(35)

Therefore, it can be known from equation (31): $J_{T} < 0$ . When $T$ tends to positive infinity, that is

\int_{0}^{+ \infty} z^{T} (t) z (t) d t < γ^{2} \int_{0}^{+ \infty} [w_{1}^{T} (t) w_{1} (t) + w_{2}^{T} (t - h (t)) w_{2} (t - h (t))] d t

(36)

The designed controller $u (t) = Y X^{- 1} C_{2}^{†} y (t)$ meets the design requirements. The proof is completed.

Numerical simulation

In this section, two numerical examples are presented to demonstrate the feasibility and effectiveness of the proposed schemes. The selected model parameters of quarter-vehicle active suspension system are shown as $m_{s} = 973 kg$ , $m_{u} = 114 kg$ , $k_{s} = 42, 720 N / m$ , $k_{t} = 101, 115 N / m$ , $c_{s} = 1095 Ns / m$ , and $c_{t} = 14.6 Ns / m$ .⁹

Simulation without disturbances

According to equation (7), the control block diagram of active suspension system without disturbances is expressed as Figure 2.

Figure 2.

Block diagram of control system without disturbances.

Based on MATLAB/Simulink software, the simulation model of quarter-vehicle active suspension system without disturbances is established as Figure 3.

Figure 3.

Simulation model of control system without disturbances.

The initial value of system simulation is set as $x (t) = [- 0.03; - 0.012675; 0; 0]$ . On the basis of equation (2), the function of input delay with time-varying is taken as $h (t) = 1 - 1 \times \exp (- 0.01 \times (t + 1)) \times \sin (t) \times (\cos (t))^{2}$ , and the function curve and derivative curve of input delay $h (t)$ are shown in Figure 4. According to the LMI equation (5), $K$ can be obtained as $100 \times [2.52070.9761]$ . And in accordance with equation (4), the uncertain parameters with time-varying of system are respectively expressed as

\begin{matrix} Δ A = 0.01 \times H \times [\begin{matrix} \sin (t) & 0 & 0 & 0 \\ 0 & \sin (t) & 0 & 0 \\ 0 & 0 & \sin (t) & 0 \\ 0 & 0 & 0 & \sin (t) \end{matrix}] \times E_{1} \\ Δ B = 0.01 \times H \times [\begin{matrix} \sin (t) & 0 & 0 & 0 \\ 0 & \sin (t) & 0 & 0 \\ 0 & 0 & \sin (t) & 0 \\ 0 & 0 & 0 & \sin (t) \end{matrix}] \times E_{2} \end{matrix}

(37)

where $H = 0.2 \times I_{4}$ , $E_{1} = I_{4}$ , and $E_{2} = 1 / 4 \times [1; 1; 1; 1]$ .

Figure 4.

Input delay signals of control system: (a) function of input delay and (b) derivative of input delay.

The simulation time is set to 20 s, then the control output signals of system without disturbances, for example, acceleration of sprung mass, speed of sprung mass, suspension deflection, and tire deflection, can be easily obtained as Figure 5, and the control input signal of active suspension system with input delay can also be easily gained as Figure 6.

Figure 5.

Control output signals of system without disturbances: (a) acceleration of sprung mass, (b) speed of sprung mass, (c) suspension deflection, and (d) tire deflection.

Figure 6.

Control input of active suspension system with input delay.

From Figure 5, it can be seen that the acceleration of sprung mass, speed of sprung mass, suspension deflection, and tire deflection of active suspension system with input delay and parameter uncertainty are asymptotically stable, and the designed static output feedback $H_{\infty}$ controller can meet the performance requirements. From Figure 6, the input delay of about 0.5 s can be seen when the actuator is started, and the control force and its change satisfy the design requirements.

Simulation with disturbances

According to equation (3), the control block diagram of active suspension system with the input disturbance of road and the output disturbance of measurement is expressed as Figure 7.

Figure 7.

Block diagram of control system with $w_{1} (t)$ and $w_{2} (t)$ .

Based on MATLAB/Simulink software, the simulation model of quarter-vehicle active suspension system with $w_{1} (t)$ and $w_{2} (t)$ is established as Figure 8. The following will be two simulation examples, for example, only with $w_{1} (t)$ and with $w_{1} (t)$ and $w_{2} (t)$ , to demonstrate the feasibility and effectiveness of the proposed schemes.

Figure 8.

Simulation model of control system with $w_{1} (t)$ and $w_{2} (t)$ .

Simulation with input disturbance of road

In order to exhibit the effectiveness of the proposed static output feedback $H_{\infty}$ controller to the input disturbance of road, an isolated bump on a smooth road surface is defined as follows⁹

\begin{matrix} z_{r} (t) = {\begin{matrix} \frac{A}{2} (1 - \cos (\frac{2 π V}{L}) t), & if & 0 \leq t \leq \frac{L}{V} \\ 0, & if & t > \frac{L}{V} \end{matrix} \end{matrix}

(38)

where $A$ and $L$ , respectively, indicate the height and length of the isolated bump and $V$ indicates the forward velocity of vehicle. Here, let $A = 1.1 m$ , $L = 5 m$ , and $V = 12.5 m / s$ , then the input disturbance of the isolated bump is shown as Figure 9.

Figure 9.

Input disturbance of an isolated bump.

Let $w_{2} (t) = 0$ in Figure 8. The initial value of system simulation is set as $x (t) = [0; 0; 0; 0]$ . According to the LIM (18), $K$ can be obtained as $100 \times [1.53101.9742]$ . On the other hand, the related passive suspension simulation model can be obtained by removing the control input and input delay in Figure 8.

The simulation time is also set to 20 s, then the time response comparison of control output signals to the above bump disturbance between passive suspension and active suspension can be obtained as Figure 10, and the control input signal of active suspension system can be gained as Figure 11.

Figure 10.

Time response comparison to bump disturbance between passive suspension and active suspension: (a) acceleration of sprung mass, (b) speed of sprung mass, (c) suspension deflection, and (d) tire deflection.

Figure 11.

Control input of active suspension system with bump disturbance.

From Figure 10, it is obvious that the acceleration of sprung mass, speed of sprung mass, suspension deflection, and tire deflection of active suspension system attenuate faster than those of passive suspension system, which shows the designed static output feedback $H_{\infty}$ controller for active suspension system has better response performance to the isolated bump and the proposed scheme satisfy the design requirements. From Figure 11, the input delay of about 0.5 s can be seen when the actuator is started, and the control force and its change satisfy the design requirements.

Simulation with input disturbance and output disturbance

In order to exhibit the effectiveness of the proposed static output feedback $H_{\infty}$ controller to both the input disturbance of road and the output disturbance of measurement, the measurement output disturbance of sensor is taken as $w_{2} (t) = 0.1 \times \sin (t)$ , the corresponding gain matrix $L_{2} = [1; 0.1]$ , then the overall measurement output disturbance of sensor can be expressed as Figure 12.

Figure 12.

Output disturbances of measurement.

The time response comparison of control output signals of active suspension can be obtained as Figure 13, between only with input disturbance of road and both input disturbance and output disturbance. From Figure 13, it can be seen that the designed static output feedback $H_{\infty}$ controller has a good effect to suppress the measurement output disturbance, and the two contrast curves of acceleration of sprung mass, speed of sprung mass, suspension deflection, and tire deflection of active suspension system are basically same.

Figure 13.

Time response comparison to different disturbances of active suspension: (a) acceleration of sprung mass, (b) speed of sprung mass, (c) suspension deflection, and (d) tire deflection.

The comparison of control input signals of active suspension can be obtained as Figure 14, between only with input disturbance of road and both input disturbance and output disturbance. From Figure 14, it can be easily seen that the control input signals of the two cases are roughly consistent, which indicates that the designed static output feedback $H_{\infty}$ controller has good control ability to the input disturbance of uneven pavement and the measurement output disturbance of sensor.

Figure 14.

Control input comparison of active suspension system with different disturbances.

Conclusion

This article has investigated a class of problem about the static output feedback $H_{\infty}$ control for active suspension system which involves parameter uncertainty, input delay, input disturbance, and measurement output disturbance simultaneously. First, the basic concept, existing problems, and research status of active suspension system have been introduced, and the main contribution of this article has been given. Second, the mathematical model of quarter-vehicle active suspension system and the system state equation have been established. Third, a design methodology of static output feedback $H_{\infty}$ controller has been proposed, and the $H_{\infty}$ controller has been designed by employing the Lyapunov–Krasovskii functional. Finally, the simulation model of quarter-vehicle active suspension system has been established based on MATLAB/Simulink software, and the simulations of three cases, for example, without disturbance, only with input disturbance, and with input disturbance and output disturbance, have been exploited to demonstrate the feasibility and effectiveness of the proposed schemes. In the future work, the active suspension system with multiple control parameters and the implementation of proposed method will be the main objectives of next exploration.

Footnotes

Appendix 1

Handling Editor: Jan Torgersen

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was jointly supported by the National Natural Science Foundation of China (grant 11705122), the Research Foundation of Department of Education of Sichuan Province (grants 17ZA0271, 16ZA0263, 17ZB0194, and 18ZB0418), the Open Foundation of Enterprise Informatization and Internet of Things Key Laboratory of Sichuan Province (grants 2016WYJ03 and 2017WZY01), the Open Foundation of Artificial Intelligence Key Laboratory of Sichuan Province (grants 2016RYJ04, 2015RYJ03, 2016RYY01, and 2017RYJ01), the Open Foundation of Sichuan Provincial Key Lab of Process Equipment and Control (grant GK201612), the Open Foundation of Material Corrosion and Protection Key Laboratory of Sichuan province (grant 2017CL09), the Innovation Platform & Key Project of International Cooperation of General Institutes of Higher Education of Guangdong Province & Overseas (including Hong Kong, Macao, and Taiwan) (grant 2015KGJHZ025), Research & Innovation Foundation of General University Graduate of Jiangsu Province (grant CXZZ130659), and Natural Science Foundation of Sichuan University of Science & Engineering (grants 2018RCL17, 2018RCL18, 2017RCL10, 2017RCL52, and 2014RC11).

ORCID iD

Ping He

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Static output feedback H ∞ control for active suspension system with input delay and parameter uncertainty

Abstract

Keywords

Introduction

Problem formulation

Main results

Theorem 1

Proof

Theorem 2

Proof

Numerical simulation

Simulation without disturbances

Simulation with disturbances

Simulation with input disturbance of road

Simulation with input disturbance and output disturbance

Conclusion

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

ORCID iD

References