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Fuzzy Sampled-Data Control for Uncertain Vehicle Suspension Systems Hongyi Li, Member, IEEE, Xingjian Jing, Member, IEEE, Hak-Keung Lam, Senior Member, IEEE, and Peng Shi, Senior Member, IEEE

Abstract—This paper investigates the problem of sampled-data H∞ control of uncertain active suspension systems via fuzzy control approach. Our work focuses on designing state-feedback and output-feedback sampled-data controllers to guarantee the resulting closed-loop dynamical systems to be asymptotically stable and satisfy H∞ disturbance attenuation level and suspension performance constraints. Using Takagi-Sugeno (T-S) fuzzy model control method, T-S fuzzy models are established for uncertain vehicle active suspension systems considering the desired suspension performances. Based on Lyapunov stability theory, the existence conditions of state-feedback and output-feedback sampled-data controllers are obtained by solving an optimization problem. Simulation results for active vehicle suspension systems with uncertainty are provided to demonstrate the effectiveness of the proposed method. Index Terms—Active suspension system, H∞ control, sampleddata control, T-S fuzzy model.

I. Introduction

D

URING THE past decades, vehicle suspension systems have been extensively studied since the systems play an important role in vehicle engineering and can be used to control vehicle attitude, reduce the influence of braking and vehicle roll during cornering maneuvers, and improve ride comfort and road handling capability [1]–[7]. Many researchers have investigated the problems by building the model and carrying out the control design to improve the

Manuscript received December 23, 2012; revised May 11, 2013; accepted August 9, 2013. This work was partially supported by the National Natural Science Foundation of China (61203002, 61333012), the Program for New Century Excellent Talents in University (NCET-13-0696), the Program for Liaoning Innovative Research Team in University (LT2013023), the Program for Liaoning Excellent Talents in University (LR2013053), a GRF project (ref. 517810) of H.K. RGC, internal research funds of Hong Kong Polytechnic Universities, the National Key Basic Research Program, China (2012CB215202) and the 111 Project (B12018). This paper was recommended by Associate Editor T. S. Li. H. Li is with the College of Engineering, Bohai University, Jinzhou 121013, China (e-mail: [email protected]). X. J. Jing is with the Department of Mechanical Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong (e-mail: [email protected]). H. K. Lam is with the Department of Informatics, King’s College London, London WC2R 2LC, U.K. (e-mail: [email protected]). P. Shi is with the School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide, SA 5005, Australia (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2013.2279534

suspension system performance and guarantee the stability [8], [9]. However, different performance requirements may contradict with each other, for example, minimization of suspension travel cannot be accomplished simultaneously with maximization of the ride comfort [2], [10], [11]. In the recent years, some researchers utilized control methods, such as fuzzy logic and neural network control [4], [12], linear optimal control [13], adaptive control [14] and H∞ control [15] to design active suspension systems and to improve the suspension performance. In [15], the authors pointed out that H∞ control method for active suspension systems is applicable to manage the trade-off between conflicting performance and thus to obtain a better compromise as H∞ control method can guarantee road holding and suspension stroke constraints within an acceptable level, and simultaneously improve ride comfort. In [2], [15], and [16], H∞ control design scheme with linear matrix inequalities (LMIs) approach has been utilized for the analysis and design active suspension systems resulting in better suspension performance. The researchers in [1], [2], and [16] pointed out that model uncertainty of suspension sprung and unsprung masses should be taken into account to improve the suspension performance in the active suspension control design. Since the uncertainties from vehicle sprung and unsprung masses vary in the interval, the polytopic parameter uncertainty type can be exploited to build the corresponding uncertainty model. More recently, the authors in [16] proposed a parameter-dependent controller design method for active suspension systems with uncertain sprung mass and the researchers in [2] designed a parameterindependent sampled-data H∞ controller to deal with the uncertainties and improve the suspension performances of a quarter vehicle suspension system. During the past years, the fuzzy logic control method has been extensively investigated in [17] and [18] since it is one of the most effective approaches to handle complex nonlinear systems and has been applied into various real systems. It has been shown that the Takagi-Sugeno (T-S) fuzzy model is an effective theoretical method and practical tool for representing complex nonlinear systems [19]–[21]. T-S fuzzy model based systems are described as a weighted sum of some simple linear subsystems, and thus are easily analyzable [22], [23]. Many stability analysis and controller synthesis results have been proposed for T-S fuzzy systems [24]–[40]. In [34], the authors investigated the generalized nonquadratic stability problem for continuous-time nonlinear models in the

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T-S form obtained by sector-nonlinearity approach. In [41], the authors proposed a novel fuzzy feedback linearization control strategy for some class of nonlinear control systems. The authors in [42] used the T-S fuzzy model to represent uncertain active suspension systems with input constraints and fuzzy H∞ controller design results were given. Following this result, the authors in [43] dealt with the problem of reliable fuzzy H∞ controller design for uncertain active suspension systems with actuator delay and fault. However, there are few results on output-feedback sampled-data controller for uncertain vehicle suspension systems. In practical control systems [44], computers are usually utilized as digital controllers to control continuous-time systems. A digital computer is applied to sample a continuous-time measurement signal to produce a discrete-time signal, and thus a discrete-time control input signal, which is further converted back into a continuous-time control input signal using a zeroorder hold. There exist both continuous-time and discretetime signals in the continuous-time framework within such control systems, which are named as sampled-data systems [45]. Due to the characteristic that the control signals are kept constant between any two consecutive sampling instants and only change at each sampling instant, it is more difficult to carry out control design for sampled-data systems. Recently, the researches in [46]–[49] discussed two main methods to develop stability analysis and control synthesis for sampleddata systems. The first one is to model a sampled data system as a discrete time system [48], in which a sampled data system with a delay is modeled as a discrete time system and a stability condition is derived. However, it should be mentioned that this method is very difficult to analyze or synthesize for complex systems. The second one is a delayed control input method by modeling the sampled-data system as a continuoustime system with a delayed control input, which was proposed in [49] and latter used in [50] and [51]. More recently, many sampled-data analysis and synthesis results have been reported for T-S fuzzy systems, for example, [52]–[56]. Among these results, the state-feedback control design method has been used in [53]–[57], and observer based control approach has been used in [52]. For the fuzzy systems, however, there are few sampled-data output-feedback controller design results. Therefore, this paper considers the problem of sampleddata H∞ control for uncertain active suspension systems via fuzzy control approach. Firstly, the vehicle dynamic system is built by taking into account vehicle sprung and unsprung mass variations, and suspension performances. Secondly, T-S fuzzy model method is adopted in this paper to represent the uncertain suspension systems. Then, the state-feedback and output-feedback sampled-data controllers are derived to guarantee that the closed-loop system is asymptotically stable, has H∞ disturbance attenuation level and satisfies the output constraints. Finally, simulation results are given to show the effectiveness of the proposed method. The main contributions of this paper can be summarized as below: 1) fuzzy based control model for uncertain active suspension system is developed for the control design; 2) when the state variables are measurable or known, a new state-feedback sampled-data control method is proposed for the fuzzy model to improve the

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Fig. 1.

Quarter-car model.

suspension performances; and 3) when the state variables are unmeasurable or unknown, a novel dynamic output-feedback sampled-data controller is designed for the fuzzy model to improve the suspension performances. Notation: Rn denotes the n-dimensional Euclidean space. The superscript T denotes matrix transposition. ·∞ denotes the H∞ norm for matrices. The notation P > 0 (≥ 0) stands for a symmetric and positive definite (semi-definite) matrix. In symmetric block matrices or complex matrix expressions, an asterisk * is employed to represent a term that is readily induced by symmetry and diag{. . .} stands for a block-diagonal matrix. [A]s is used to denote A + AT for simplicity. Matrices, if the dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. The space of square-integrable vector functions over [0, ∞) is denoted by L2 [0, ∞), and for w = {w (t)} ∈ L2 [0, ∞), its ∞ |w (t)|2 dt. norm is denoted by w2 = t=0 II. Problem Formulation This paper considers the following two-degree-of-freedom quarter uncertain vehicle suspension system shown in Fig. 1. Since this model can capture many important characteristics of many complicated suspension models, considerable attention has been drawn to this model in the literature [2]. The dynamic governing equations of the suspension system are given as follows: ms z¨ s (t) + cs [˙zs (t) − z˙ u (t)] + ks [zs (t) − zu (t)] = u (t) , mu z¨ u (t) + cs [˙zu (t) − z˙ s (t)] + ks [zu (t) − zs (t)] +kt [zu (t) − zr (t)] + ct [˙zu (t) − z˙ r (t)] = −u (t) .

(1)

In Fig. 1, the sprung mass representing the car chassis is denoted by ms ; the unsprung mass representing mass of the wheel assembly is denoted by mu ; the active input of the suspension system is denoted by u (t); the displacements of the sprung and unsprung masses are denoted by zs and zu , respectively; the road displacement input is denoted by zr ; damping and stiffness of the suspension system are denoted by cs and ks respectively; and the compressibility and damping of the pneumatic tire are denoted by kt and ct , respectively. Due to a change in the number of passengers or the payload, vehicle load is often varying, which will accordingly lead the vehicle mass to a varying parameter. According the mechanical structure, it is assumed that the unsprung mass is a varying

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parameter. Therefore, the suspension system in (4) is an uncertain model that includes both the sprung mass ms and the unsprung mass mu . In (4), ms and mu denote ms (t) and mu (t), respectively. The parameter ms (t) and mu (t) are supposed to vary in a given range, that is, ms (t) ∈ [ms min , ms max ] and mu (t) ∈ [mu min , mu max ] . Three performances including ride comfort, suspension deflection, and road holding are considered in the control design. It is widely accepted that ride comfort can be generally quantified by the body acceleration in the vertical direction in the context of a quarter-vehicle model. Hence it is practical to choose body acceleration, z¨ s (t), as the first control output. One objective is to minimize the vertical acceleration, z¨ s (t), to secure vehicle travel comfort. The value of H∞ norm is defined as an upper bound of the root mean square gain. The goal is to minimize the H∞ norm of the transfer function from the disturbance w(t) to the control output z1 (t) = z¨ s (t) with an emphasis on ride comfort improvement. Meanwhile, the following performances will be considered. 1) The suspension deflection cannot exceed a maximum value constrained by mechanical structure |zs (t) − zu (t)| ≤ zmax

(2)

where zmax is the maximum suspension deflection. 2) The dynamic tire load has to be less than the static tire load in order to ensure a firm uninterrupted contact of the wheels on the road kt (zu (t) − zr (t)) < (ms (t) + mu (t)) g.

(3)

To satisfy the above-mentioned performance requirements, the controlled outputs are defined by z1 (t) = z¨ s (t), u (t) z2 (t) = zs (t)−z zmax

kt (zu (t)−zr (t)) (ms (t)+mu (t))g

T .

Denote x1 (t) = zs (t) − zu (t) as the suspension deflection, x2 (t) = zu (t) − zr (t) as the tire deflection, x3 (t)= z˙ s (t) as the sprung mass speed, x4 (t) = z˙ u (t) as the unsprung mass speed, and w(t) = z˙ r (t) as the disturbance input, respectively. Therefore, the model of the vehicle suspension system can be expressed into the following form: x˙ (t) = A (t) x (t) + B1 (t) w (t) + B (t) u (t) , z1 (t) = C1 (t) x(t) + D1 (t) u (t) , z2 (t) = C2 (t) x(t), y(t) = Cx(t) where

T x(t) = x1 (t) x2 (t) x3 (t) x4 (t) , ⎤ ⎡ 0 0 1 −1 ⎢ 0 0 0 1 ⎥ ⎥ A (t) = ⎢ ⎣ − mks(t) 0 − mcs(t) mcs(t) ⎦ , s s s cs +ct ks − mkut(t) mcus(t) − m mu (t) u (t) ⎡ ⎤ ⎤ ⎡ 0 0 ⎢ 0 ⎥ ⎢ −1 ⎥ ⎥ ⎥ ⎢ B (t) = ⎢ ⎣ m1(t) ⎦ , B1 (t) = ⎣ 0 ⎦ , s c t − mu1(t) mu (t)

(4)

3

ks cs cs 0− , C1 (t) = − ms (t) ms (t) ms (t) ⎡ 1 ⎤ 0 00 ⎢ ⎥ C2 (t) = ⎣ zmax ⎦, kt 0 00 (ms (t) + mu (t)) g 1 D1 (t) = , C= 1110 . ms (t) A general observation matrix C is adopted firstly for the dynamic output-feedback control design problem. Based on ms (t) ∈ [ms min , ms max ] and mu (t) ∈ [mu min , mu max ], it can be obtained that 1 1 1 1 =: m ˆ s , min =: m ˇ s, = = ms (t) ms min ms (t) ms max 1 1 1 1 max = =: m ˆ u , min = =: m ˇ u. mu (t) mu min mu (t) mu max max

The sector nonlinear method [20] is employed to represent 1 and mu1(t) by ms (t) 1 = M1 (ξ1 (t)) m ˆ s + M2 (ξ1 (t)) m ˇ s, ms (t) 1 = N1 (ξ2 (t)) m ˆ u + N2 (ξ2 (t)) m ˇu mu (t) where ξ1 (t) =

1 ms (t)

and ξ2 (t) =

M1 (ξ1 (t)) + M2 (ξ1 (t)) = 1,

1 mu (t)

are premise variables

N1 (ξ2 (t)) + N2 (ξ2 (t)) = 1.

To develop the fuzzy model, the membership functions M1 (ξ1 (t)) , M2 (ξ1 (t)) , N1 (ξ2 (t)) and N2 (ξ2 (t)) can be calculated as 1 1 m ˆs− −m ˇs ms (t) ms (t) , M2 (ξ1 (t)) = , M1 (ξ1 (t)) = m ˆs−m ˇs m ˆs−m ˇs 1 1 m û− −m ˇu m (t) mu (t) N1 (ξ2 (t)) = u , N2 (ξ2 (t)) = . m û−m ˇu m û−m ˇu The membership functions are named as Heavy, Light, Heavy and Light as shown in Fig. 2. Then, the system with uncertainty in (4) is represented by the following fuzzy model: Model Rule i: IF ξ1 (t) is Mr (ξ1 (t)) and ξ2 (t) is Nr (ξ2 (t)) , THEN x˙ (t) = Ai x (t) + Bi u (t) + B1i w (t) , z1 (t) = C1i x (t) + D1i u (t) , z2 (t) = C2i x (t) where r = 1, 2 and i = 1, 2, 3, 4. As shown in Fig. 2, M1 (ξ1 (t)) denotes Heavy and M2 (ξ1 (t)) denotes Light; N1 (ξ2 (t)) denotes Heavy and N2 (ξ2 (t)) denotes Light. The matrices Ai , Bi , B1i , C1i , D1i and C2i can be obtained by replacing ms1(t) and mu1(t) with m ˆ s (or m ˇ s ) and m ˆ u (or m ˇ u ) in matrices A (t) , B (t) , B1 (t) , C1 (t) , D1 (t) and C2 (t) respectively.



The following sampled-data controller for the fuzzy system in (5) is given: Control Rule i: IF ξ1 (t) is Mr (ξ1 (t)) and ξ2 (t) is Nr (ξ2 (t)), THEN u(t) = Ki x(tk ), tk ≤ t < tk+1 , k = 0, 1, 2, . . . where i = 1, 2, 3, 4. In the 4 fuzzy rules, tk (k = 0, 1, 2, . . .) denotes the kth sampling instant, t0 ≥ 0, and lim tk = ∞. Hence, the k→∞ overall fuzzy control law is represented by

u(t) =

4

hj (ξ (t))Kj x(tk ), tk ≤ t < tk+1 , k = 0, 1, 2, . . . (7)

j=1

where Kj (j = 1, 2, 3, 4) are the local control gains, tk+1 is the next updating instant time of the ZOH after tk . As in [54], [57], the following assumption is made. Assumption 1: The sampling instants tk are assumed to satisfy 0 < tk+1 − tk hk ≤ h, k = 0, 1, . . . Fig. 2. (a) Membership functions M1 (ξ1 (t)) and M2 (ξ1 (t)) (b) Membership functions N1 (ξ2 (t)) and N2 (ξ2 (t)).

Fuzzy blending allows to infer the overall fuzzy model as follows: x˙ (t) = z1 (t) = z2 (t) =

4 i=1 4 i=1 4

(8)

where h is a positive scalar. Defining d (t) = t − tk , tk ≤ t < tk+1 , it can be obtained that u(t) =

4

hj (ξ (t))Kj x(t − d (t)), tk ≤ t < tk+1

(9)

j=1

hi (ξ (t)) [Ai x (t) + Bi u (t) + B1i w (t)] , hi (ξ (t)) [C1i x (t) + D1i u (t)] , hi (ξ (t)) C2i x (t) ,

(5)

i=1

where h1 (ξ (t)) = M1 (ξ1 (t)) × N1 (ξ2 (t)) , h2 (ξ (t)) = M1 (ξ1 (t)) × N2 (ξ2 (t)) , h3 (ξ (t)) = M2 (ξ1 (t)) × N1 (ξ2 (t)) , h4 (ξ (t)) = M2 (ξ1 (t)) × N2 (ξ2 (t)) . It should be noted that the fuzzy weighting functions hi (ξ (t)) 4 satisfy hi (ξ (t)) ≥ 0, i=1 hi (ξ (t)) = 1.

x˙ (t) =

III. State-feedback Controller Design

hi hj (Ai x(t) + Bi Kj x(t − d (t))

+B1i w(t)), 4 4 z1 (t) = hi hj C1i x(t) + D1i Kj x(t − d (t)) , i=1 j=1

z2 (t) = k = 0, 1, 2. . . .

4 4 i=1 j=1

In this section, we first consider the state-feedback controller design for the system in (5). Suppose that the updating signal successfully transmits signal from the sampler to the controller and to the ZOH at the instant tk . We assume that the sampling intervals are bounded by a constant h tk+1 − tk ≤ h,

where d (t) ≤ h and d˙ (t) = 1 for t = tk . Then, we know that d (t) is a fast time-varying delay. For brevity, we let hi =: hi (ξ (t)). Remark 1: In this paper, a sampled-data fuzzy controller with four fuzzy rules is designed based on the fuzzy model of the uncertain suspension systems. Parallel distributed compensation (PDC) approach and input delay method are used to present the state-feedback sampled-data for the system. In this paper, it is required that the T-S fuzzy model and fuzzy controller share the same premise membership functions and the same number of fuzzy rules. In future works, to enhance the design flexibility, we will consider that the fuzzy model and fuzzy controller use unmatched premise membership functions to develop more relax stability analysis and controller synthesis results. Applying the fuzzy controller (9) to system (5) yields the closed-loop system

(6)

Here h denotes the maximum time span between the time tk at which the state is sampled and the time tk+1 at which the next update arrives at the destination. The initial conditions of x(t) and u(t) are given as x(t) = ϕ (t) and u (t) = 0 for t ∈ [t0 − h, t0 ] , where ϕ (t) is a differentiable function.

4

hi C2i x(t).

(10)

i=1

Without loss of generality, it is assumed, w ∈ L2 [0, ∞), and w22 ≤ wmax < ∞. The objective in this paper is to design the controller such that the following requirements are satisfied: 1) the closed-loop system is asymptotically stable;


2) under zero initial condition, the closed-loop system guarantees that z1 2 < γ w2 for all nonzero w ∈ L2 [0, ∞), where γ > 0 is a prescribed scalar; 3) the following control output constraints are guaranteed: {z2 (t)}q ≤ 1, q = 1, 2 (11) where z2 (t)1 stands for the first row vector of z2 (t) and z2 (t)2 stands for the second row vector of z2 (t). For the control system in (10), we construct a proper Lyapunov functional and utilize the appropriate method to design H∞ controller via state-feedback sampled-data control approach. The following theorem can be established, which provides the condition under the H∞ performance. Theorem 1: Consider the closed-loop system in (10). For i, j = 1, 2, 3, 4, given scalars h > 0, q = 1, 2 and controller matrices Kj , if there exist symmetric matrices P > 0 and Q > 0, with appropriate dimensions such that the following LMIs hold: ii < 0, ij + ji < 0, √ −P ρ {C2i }Tq 0 Proof: For θ > 0 and Q −1 ¯ − P¯ ≥ 0 ¯ − P¯ Q ¯ θQ θQ ¯ − 2θ P. ¯ −1 P¯ ≤ θ 2 Q ¯ −P¯ Q ¯ − 2θ P¯ with −P¯ Q ¯ −1 P, ¯ In (15)–(16), by replacing θ 2 Q then, performing congruence transformation to (15)–(16) by diag{P, P, P, I, I, Q} with the change of matrix variables defined by P = P¯ −1 ,

Then we know that, 1) the closed-loop system is asymptotically stable, and 2) under zero initial condition, the performance Tz1 w ∞ < γ is minimized subject to the output constraints in (11) with the disturbance energy under the bound wmax = (ρ − V (0))/γ 2 , where Tz1 w denotes the closed-loop transfer function from the road disturbance w(t) to the control output z1 (t). Proof: Proof of Theorem 1 is given in the Appendix. In what follows, a state-feedback sampled-data controller is derived for the system in (10) based on Theorem 1. Theorem 2: Consider the suspension system in (10). For i, j = 1, 2, 3, 4, given positive scalars h > 0 and θ > 0, if there ¯ > 0 with appropriate exist symmetric matrices P¯ > 0 and Q dimensions such that the following LMIs hold: ¯ ii < 0, ¯ ij + ¯ ji < 0, √ ¯ ¯ −P ρP {C2i }Tq 0 and q = 1, 2 and controller matrices Acj , Acdj , Bcj and Ccj , if there exist symmetric matrices P > 0 and Q > 0, with appropriate dimensions such that the following LMIs hold: ˜ ii < 0, ˜ ij + ˜ ji < 0, √ ¯ T −P ρ C 2ij q 0, θ > 0, and q = 1, 2, ˆ = Q1 Q2 > 0, if there exist symmetric matrices Q ∗ Q3 R > 0, S > 0, matrices Ai , Adi , Bi and Ci with appropriate dimensions such that the following LMIs hold: ⎡ ⎤ ˆ 1ii ˆ 2ii ˆ 3ii ⎣ ∗ −I 0 ⎦ < 0, (24) ˆ 4ii ∗ ∗ ⎡ ⎤ ˇ 1ij ˇ 2ij ˇ 3ij ˇ 4ij ˇ 5ij ⎢ ∗ −2I 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ ˇ 6ij hμ6ij 0 ⎥ < 0, i < j, (25) ⎢ ⎥ ⎣ ∗ ∗ ∗ ˇ 7ij 0 ⎦ ˇ 7ij ∗ ∗ ∗ ∗ ⎡ ⎤ −R −I R {C2i }Tq ⎣ ∗ −S {C2i }Tq ⎦ < 0, (26) ∗ ∗ −I RI >0 (27) I S where the parameters are listed in the next page. Then, there exists a dynamic controller such that the closedloop system in (20) is asymptotically stable. In this case, a desired output-feedback controller is given in the form of (19) with parameters as follows: Aci = N −1 (Ai − SBi Ci − SAi R) M−T , Acdi = N −1 (Adi − Bi Ci R) M−T , Bci = N −1 Bi , −T

Cci = Ci M

(28) (29) (30) (31)

where N and M are any nonsingular matrices satisfying MN T = I − RS.

(32)

Then a controller in the form of (19) exists, such that, 1) the closed-loop system is asymptotically stable, and 2) under zero initial condition, the performance Tz1 w ∞ < γ is minimized subject to output constraints (11) with the disturbance energy under the bound wmax = (ρ − V (0))/γ 2 .


7

⎤ ⎡ ⎤ ⎤ ⎡ ⎡ ⎤ ˆ λ2ii + Q ˆ 0 λ3ii λ1ii − Q λ4ii hλ5ii μ4ij ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ ˆ ˆ ∗ −2Q Q 0 ⎥ ⎢ ⎥, ˇ 2ij = ⎢ 0 ⎥ , ˆ 3ii = ⎢ hλ6ii ⎥ , ˆ 2ii = ⎢ 0 ⎥ , ⎦ ⎣ ⎣ ⎦ ⎣ ⎣ ⎦ ˆ 0 0 0 ⎦ 0 ∗ ∗ −Q 2 0 0 hλ7ii ∗ ∗ ∗ −γ I ⎤ ⎤ ⎡ ⎤ ⎡ ⎡ ⎡ ⎤ ˆ ˆ μ1ij s − 2Q μ2ij + 2Q 0 μ3ij hμ5ij μ6ij μ7ij ⎥ ⎢ ⎥ ⎢ ⎢ ⎢ ⎥ ˆ ˆ ∗ −4Q 2Q 0 ⎥ ⎥, ⎢ ˇ 4ij = ⎢ 0 ⎥ , ˇ 3ij = ⎢ hμ8ij ⎥ , ˇ 5ij = ⎢ μ9ij ⎥ , ⎣ ⎣ 0 ⎦ ⎣ 0 ⎦ ⎣ 0 ⎦ ˆ ∗ ∗ −2Q 0 ⎦ 0 hμ10ij 0 ∗ ∗ ∗ −2γ 2 I T T T T 0 0 B1i RC1i + Ci D1i [Ai R + Bi Ci ]s Ai + Ai , λ2ii = , λ3ii = , λ4ii = , T ∗ SB1i C1i Adi Bi Ci [SAi ]s CjT − CiT 0 RATi + CiT BiT ATi 0 ATdi RATi + CiT BiT ATi , λ = , λ = = , μ 6ii 7ii 7ij ATi ATi S 0 CiT BiT AT ATi S 0 0 i T T ˆ − 2θ R I , μ1ij = Ai R + Aj R + Bi Cj + Bj Ci s Ai + Aj + Ai + Aj , θ2 Q I S ∗ SAi + SAj s T T T T + CjT D1i + RC1j + CiT D1j RC1i 0 0 B1i + B1j , μ3ij = , μ4ij = , T T Adi + Adj Bj Ci + Bi Cj SB1i + SB1j C1i + C1j RATi + RATj + CjT BiT + CiT BjT ATi + ATj 0 0 , , μ = 6ij ATi + ATj ATi S + ATj S S Bi − Bj Bj − Bi T 0 ATdi + ATdj RI −I 0 0 R Ci − Cj 2 ˆ ˇ ˇ , μ = Q − 4θ , = 2θ , = , 9ij 6ij 7ij I S 0 −I 0 CiT BjT + CjT BiT 0 0 T T T T B1i + B1j B1i S + B1j S . ⎡

ˆ 1ii =

ˇ 1ij =

λ1ii = λ5ii = ˆ 4ii = μ2ij = μ5ij = μ8ij = μ10ij =

Note that the equality PP −1 = I leads to (32) holds. In fact, ¯ 44 in Theorem 2 that it can be seen from −R −I 0, therefore I − RS is nonsingular. This ensures that there always exist nonsingular matrices N and M such that (32) is satisfied. Setting R I I S

1 = . (34) , 2 = MT 0 0 NT

⎤ 0 T ⎢ S Bi − Bj ⎥ Cj − CiT ⎥ ⎢ ,

=⎣ ⎦,ϕ = 01×9 01×7 hS Bi − Bj ⎡ ⎤ ⎤ ⎡ 0 01×2 ⎢ Bj − Bi ⎥ ⎥ , χ = ⎣ R Ci − C j T ⎦ . σ=⎢ ⎣ ⎦ 0 1×7 01×7 h Bj − Bi ⎡

Then, we conclude from (34) that P 1 = 2 . It follows that:

T1 P 1 = T1 2 =

One can have

ϕT + ϕ T + σχT + χσ T ≤

T + ϕT ϕ + σσ T + χT χ

RI I S

which implies that the matrices 1 and 2 in (35) are square invertible. It is found that the matrix P can be constructed as P = 2 −1 1 , and we know P > 0. Due to the relation

which means ⎤ ˇ 2ij ˇ 3ij ˇ 1ij ⎣ ∗ −2I 0 ⎦ + ϕT + ϕ T + σχT + χσ T < 0. ˇ 6ij ∗ ∗

(35)

SAi R + SBi Ci + NAci M T + SAj R + SBj Cj +NAcj M T + S Bi − Bj Cj − Ci

⎡

= SAi R + SBi Cj + NAcj M T

(33)

For presenting the output-feedback controller, we partition P and its inverse as S N R M −1 P= = , P . NT Y MT T

+SAj R + SBj Ci + NAci M T and Bi Ci R+NAcdi M T + Bj Cj R+NAcdj M T + Bj −B i Ci − Cj R = Bi Cj R+NAcdj M T + Bj Ci R+NAcdi M T .



Due to the nonsingular matrices M and N , the controller matrices Aci , Acdi , Bci , and Cci can be then obtained by solving equations (28)–(31). Then, performing congruence −1 −1 −1 transformation to (33) by diag −1 re1 , 1 , 1 , I, I, 1 spectively, the following inequality holds: ⎤ ⎡ ˜ 1ij + ˜ 1ji ˜ 2ji ˜ 2ij + ˘ 3ij ⎣ ∗ −2I 0 ⎦ 0 and Q−1 > 0, from (θQ − P) Q−1 (θQ − P) ≥ 0 we can conclude that −PQ−1 P ≤ θ 2 Q − 2θP. Then, we know that ⎡ ⎤ ˜ 1ij + ˜ 2ij + ˘ 3ij ˜ 1ji ˜ 2ji ⎣ ⎦ < 0. ∗ −2I 0 ∗ ∗ −2PQ−1 P

(37)

(38)

For (38), via performing congruence transformations by diag I, I, I, I, I, P −1 Q , we know that condition in (22) holds. Furthermore, performing congruence transformation to (24) by −1 −1 −1 diag −1 ,

,

, I, I,

and using the equality (37), 1 1 1 1 we have ⎡ ⎤ ˆ 1ii ˆ 2ii ˇ 3ii ⎣ ∗ −I ⎦ VL


Fig. 3. Body acceleration responses of the open- and closed-loop systems under different parameter θ.

9

Fig. 5. Tire deflection constraint responses of the open- and closed-loop systems under different parameter θ.

Fig. 6. Body acceleration responses of the open- and closed-loop systems under different sampling interval h. Fig. 4. Suspension deflection constraint responses of the open- and closedloop systems under different parameter θ.

where A and L are the height and the length of the bump, respectively. Assume A = 50 mm, L = 6 m and the vehicle forward velocity as V = 35 (km/h). We hope that the desired state-feedback sampled-data controllers in Tables II and III can be designed such that: 1) the sprung mass acceleration z1 (t) is as small as possible; 2) the suspension deflection is below the maximum allowable suspension stroke zmax = 0.1 m, which means z1 (t)2 < 1; and 3) the controlled output defined satisfy z2 (t)2 < 1. Fig. 3 shows the responses of body vertical accelerations, Fig. 4 plots the responses of the constrained suspension deflection, and Fig. 5 demonstrates the responses of the tire deflection constrains for the passive and closed-loop systems with different state-feedback sampleddata controllers for the same sampling interval h = 10 ms and the different chosen design parameters θ. In order to make the comparisons for suspension performance under different chosen design parameter θ more clear, we give a very short response time 1 s in the second figures in Figs. 3–5. It can be observed from Fig. 3 that the values of the vertical acceleration z¨ (t) in closed-loop system is much less than the the open-loop systems and an improved ride comfort has been achieved via the designed controller. In addition, it can be seen that, from Figs. 4 and 5, the suspension deflection constraint x1 (t)/zmax < 1 and the relation dynamic tire load constraint kt x2 (t)/(ms (t)+mu (t))g < 1 are guaranteed, which implies the road holding capability is ensured by the desired sampled-data controller. Furthermore, as shown in Table II, Figs. 3–5 demonstrate that we can achieve better suspension performances when the chosen design parameter θ is smaller. Moreover, we can see that the change of suspension performances of these figures are clear, when implies the flexibility of the chosen parameter θ. Then, Figs. 6–8 show

Fig. 7. Suspension deflection constraint responses of the open- and closedloop systems under different sampling interval h.

the responses of the responses of body vertical accelerations, the responses of the constrained suspension deflection, and responses of the tire deflection constrains, respectively, for the passive and closed-loop systems with different state-feedback sampled-data controllers for the different sampling interval h and the same chosen design parameter θ = 1. From Figs. 6–8, it can be seen that better suspension performances can be obtained when the sampling interval h is smaller, which justifies the computational results in Table III. By comparing some existing sampled-data control for fuzzy systems, it can be found that the proposed sampled-data state-feedback control results in [54]–[57] are not available for uncertain suspension systems since only stabilization problems for fuzzy systems were investigated in [54]–[57]. In addition, the performance constrains and performance control problems have not been considered in [54]–[57]. 2) Output-feedback sampled-data controller: Secondly, we consider the dynamic output-feedback sampled-data controller design for suspension systems in (20). For a given θ = 0.1 and the sampling interval h = 10 ms, by using Theorem 4, we can obtain the dynamic output-feedback controller (19) with the attenuation rate γmin = 123.1687. In the following part,



Fig. 8. Tire deflection constraint responses of the open- and closed-loop systems under different sampling interval h.

Fig. 10.

Half-vehicle model.

wheels respectively. ksf and ksr denote the front and rear tire stiffness. uf (t) and ur (t) are the front and rear actuator force inputs, respectively. The dynamic equations of half-vehicle suspension model can be established ms z¨ c (t) + ksf zsf (t) − zuf (t) + csf [˙zsf (t) −˙zuf (t)] + ksr [zsr (t) − zur (t)] + csr [˙zsr (t) − z˙ ur (t)]

Fig. 9. Body acceleration responses, suspension deflection constraint responses, tire deflection constraint responses and observed output y of the open- and closed-loop systems.

the effectiveness of the proposed output-feedback sampleddata controller can be verified. First, Fig. 9 illustrates that the values of the vertical acceleration in closed-loop system is much less than the the open-loop systems and an improved ride comfort has been achieved via the designed controller. In addition, Fig. 9 plots the output constraints can also be satisfied, which means that the suspension deflection constraint and the road holding performance have been guaranteed. B. Half-Car Model In this subsection, we continue to confirm the effectiveness of the proposed method for uncertain half-vehicle suspension systems. The half-vehicle model is shown in Fig. 10. zsf (t) and zsr (t) denote the front body displacement and the rear body displacement, respectively. l1 denotes the distance between the front axle and the center of mass, l2 denotes the distance between the rear axle and the center of mass, ϕ (t) denotes the pitch angle, and zc (t) denotes the displacement of the center of mass. ms , muf and mur denote the mass of the car body, the unsprung masses on the front and rear wheels, respectively. Iϕ denotes the pitch moment of inertia about the center of mass. zuf (t) and zur (t) are the front and rear unsprung mass displacements, respectively. zur (t) and zrr (t) denote the front and rear terrain height displacements, respectively. csf and csr are the stiffness of the passive elements of the front and rear

= uf (t) + ur (t), Iϕ ϕ¨ (t) − l1 ksf zsf (t) − zuf (t) −l1 csf z˙ sf (t) − z˙ uf (t) + l2 ksr [zsr (t) − zur (t)] +l2 csr [˙zsr (t) − z˙ ur (t)] = −l1 uf (t) + l2 ur (t), muf z¨ uf (t) − ksf zsf (t) − zuf (t) −csf z˙ sf (t) − z˙ uf (t) + ktf zuf (t) − zrf (t) = −uf (t), mur z¨ ur (t) − ksr [zsr (t) − zur (t)] −csr [˙zsr (t) − z˙ ur (t)] + ktr [zur (t) − zrr (t)] = −ur (t).

(41)

By defining the following scales: a1 =

1 l2 + 1, m s Iϕ

a2 =

1 l1 l2 − , ms Iϕ

a3 =

1 l2 + 2 m s Iϕ

and setting the following state variables: x1 (t) = zsf (t) − zuf (t) is the suspension deflection of the front car body, x2 (t) = zsr (t) − zur (t) is the suspension deflection of the rear car body, x3 (t) = zuf (t) − zrf (t) is the tire deflection of the front car body, x4 (t) = zur (t) − zrr (t) is the tire deflection of the rear car body, x5 (t) = z˙ sf (t) is the vertical velocity of the front car body, x6 (t) = z˙ sr (t) is the vertical velocity of the rear car body, x7 (t) = z˙ uf (t) is the vertical velocity of the front wheel, and x8 (t) = z˙ ur (t) is the vertical velocity of the rearwheel.In addition, the z˙ (t) disturbance input is defined as w (t) = rf . The dynamic z˙ rr (t) equations in (41) can be rewritten in the following form: x˙ (t) = A(t)x (t) + B(t)u (t) + B1 (t)w (t)

(42)


The second output vector z2 (t) consists of suspension performance constrains (43) and (44). Then, half-vehicle suspension system can be described as follows:

where x(t) = x1T (t) x2T (t) x3T (t) x4T (t) T xT (t) x6T (t) x7T (t) x8T (t) , 5 uf (t) z˙ rf (t) u (t) = , w (t) = , ur (t) z˙ rr (t) 04×4 a12 (t) A(t) = , a21 (t) a22 (t) T 0 0 0 0 a1 a2 − m1uf 0 B(t) = , 0 0 0 0 a2 a3 0 − m1ur T 0 0 −1 0 0 0 0 0 B1 (t) = , 0 0 0 −1 0 0 0 0 ⎡ ⎤ 1 0 −1 0 ⎢ 0 1 0 −1 ⎥ ⎥ a12 (t) = ⎢ ⎣0 0 1 0 ⎦, 00 0 1 ⎤ ⎡ −a1 ksf −a2 ksr 0 0 ⎢ −a2 ksf −a3 ksr 0 0 ⎥ ⎥ ⎢ ktf a21 (t) = ⎢ ksf , 0 − muf 0 ⎥ ⎦ ⎣ muf ksr 0 − mktrur 0 mur ⎤ ⎡ −a1 csf −a2 csr a1 csf a2 csr ⎢ −a2 csf −a3 csr a2 csf a3 csr ⎥ ⎥ c a22 (t) = ⎢ ⎣ mcsf 0 − msfuf 0 ⎦ uf csr 0 − mcsrur 0 mur

x˙ (t) = A(t)x(t) + B1 (t)w(t) + B(t)u(t), z1 (t) = C1 (t)x(t) + D1 (t)u(t), z2 (t) = C2 (t)x(t)

D1 (t) = ⎡

(43)

where zf max and zr max are the maximum suspension deflection hard limits, under any road disturbance input and vehicle running conditions. (3) Road holding: the dynamic tire loads should not exceed the static tire loads for both the front and rear wheels ksf zuf (t) − zrf (t) ≤ Ff ,

|ksr (zur (t) − zrr (t))| ≤ Fr (44) where Ff and Fr are static type loads that can be calculated by Fr (l1 + l2 ) = ms gl1 + mur g (l1 + l2 ) , Ff + Fr = ms + muf + mur g.

1 1 ms ms − Il1ϕ Il2ϕ 1 zf max

⎢ 0 ⎢ C2 (t) = ⎢ ⎣ 0 0

(2) Suspension defection: consider the following constraints: |zsr (t) − zur (t)| ≤ zr max

(47)

where A(t), B1 (t) and B(t) are defined in (42), and k − sf − ksr 0 0 C1 (t) = l1 kmsfs l2mkssr − Iϕ 0 0 Iϕ c csf csr csr − msfs − m ms ms s , l1 csf l c − l2Icϕsr − 1Iϕsf l2Icϕsr Iϕ

where t is used to characterize the parameter uncertainty from vehicle load variations, which will be described in detail subsequently. (1) For half-vehicle suspension system, the ride comfort performance is closely referred to the heave and the pitch accelerations. The heave and the pitch accelerations are included into the first control output vector z¨ c (t) z1 (t) = . ϕ¨ (t)

zsf (t) − zuf (t) ≤ zf max ,

11

(48)

, 0

1 zr max

0 0

0 0 0 0 0 0 ksf 0 0 Ff ksr 0 Fr 0

0 0 0 0

0 0 0 0

⎤ 0 0⎥ ⎥ ⎥. 0⎦ 0

It can be found that the suspension suspension system in (47) is an uncertain model, where the sprung mass ms , the front and rear wheels unsprung masses muf and mur vary in given ranges. The sprung mass ms , the front and rear wheels unsprung masses muf and mur are uncertainties, which vary in a given range, that is, ms ∈ [ms min , ms max ], muf ∈ muf min , muf max and mur ∈ [mur min , mur max ] . This means the uncertain mass ms is bounded by its minimum value ms min and its maximum value ms max . In addition, the uncertain mass muf is bounded by its minimum value muf min and its maximum value muf max , and mur is bounded by its minimum value mur min and its maximum value mur max . Next, we can obtain the values of m1s , m1uf and m1ur from ms ∈ [ms min , ms max ] , muf ∈ muf min , muf max and mur ∈ [mur min , mur max ] . Then, we have 1 1 1 =m ˆ s, =m ˇ s, =m ˆ uf , ms min ms max muf min 1 1 1 =m ˇ uf , =m ˆ ur , =m ˇ ur . muf max mur min mur max 1 , 1 ms muf

We can represent

and

1 mur

by

1 = M1 (ξ1 (t)) m ˆ s + M2 (ξ1 (t)) m ˇ s, ms 1 ˆ uf + N2 (ξ2 (t)) m ˇ uf , = N1 (ξ2 (t)) m muf 1 = O1 (ξ3 (t)) m ˆ ur + O2 (ξ3 (t)) m ˇ ur mur where ξ1 (t) =

1 , ms

ξ2 (t) =

1 mu f

and ξ3 (t) =

1 mur

(45)

M1 (ξ1 (t)) + M2 (ξ1 (t)) = 1, N1 (ξ2 (t)) + N2 (ξ2 (t)) = 1,

(46)

O1 (ξ3 (t)) + O2 (ξ3 (t)) = 1.



TABLE IV Half-Car Model Parameters

The membership functions can be calculated as M1 (ξ1 (t)) =

1 ms

−m ˇs

,

M2 (ξ1 (t)) =

m ˆs−

1 ms

, m ˆs−m ˇs m ˆ uf − m1uf N1 (ξ2 (t)) = , N2 (ξ2 (t)) = , m ˆ uf − m ˇ uf m ˆ uf − m ˇ uf 1 −m ˇ ur m ˆ ur − m1ur m , O2 (ξ3 (t)) = O1 (ξ3 (t)) = ur . m ˆ ur − m ˇ ur m ˆ ur − m ˇ ur m ˆs−m ˇs 1 −m ˇ uf muf

Fig. 11.

Responses of the heave accelerations and the pitch acceleration.

Fig. 12.

Responses of the front and rear suspension deflection constraints.

Based on the fuzzy modeling method proposed for a quartervehicle suspension system, the following fuzzy model can be easily obtained: Model Rule i: IF ξ1 (t) is Mr (ξ1 (t)), ξ2 (t) is Nr (ξ2 (t)), and ξ3 (t) is Or (ξ3 (t)), THEN x˙ (t) = Ai x (t) + Bi u (t) + B11 w (t) z1 (t) = C1i x (t) + D1i u (t) z2 (t) = C2i x (t) where r = 1, 2, i = 1, 2, . . . , 8. Following the similar statefeedback and dynamic output-feedback sampled-data controllers design approach presented in the Sections III and IV, the state-feedback and dynamic output-feedback sampled-data controller design results can also be derived. To save the space of the paper, the simulation results for half-vehicle suspension system are provided as follows. Firstly, half-car model parameters are listed in Table IV. The sprung mass ms , the front and rear unsprung masses muf and mur are assumed that ms belongs to the range [621 kg, 759 kg], muf belongs to the range [39.6 kg, 40.4 kg] and mur belongs to the range [44.55 kg, 45.45 kg], respectively. The maximum allowable front and rear suspension strokes are assumed as zf max = 0.1 m and zr max = 0.1 m, respectively. By using the state-feedback control method results and above parameters. it can be found that the minimum guaranteed closed-loop system H∞ performance index can be computed as γmin = 12.5453 and admissible control gain matrices Kj (j = 1, 2, . . . , 8). To check the effectiveness of the design controller, we hope that the desired controller to satisfy: 1) the first control output z1 (t) including the heave acceleration z¨ c (t) and the pitch acceleration ϕ¨ (t) is as small as possible, and 2) the suspension deflection is below the maximum allowable suspension strokes zf max = 0.1 m and zr max = 0.1 m, which means that z2 (t)1 < 1 and z2 (t)1 < 1; 3) the controlled output defined in (47) satisfies z2 (t)3 < 1 and z2 (t)4 < 1. In order to evaluate the suspension characteristics with respect to ride comfort, vehicle handling, and working space of the suspension, the variability of the

road profiles is taken into account. In the context of active suspension performance, road disturbances can be generally assumed as shocks. Shocks are discrete events of relatively short duration and high intensity, caused by, for example, a pronounced bump or pothole on an smooth road surface. In this paper, this case of road profile is considered to reveal the transient response characteristic, which is given by A (1 − cos( 2πV t)), if 0 ≤ t ≤ VL L zrf (t) = 2 (49) 0, if t > VL where A and L are the height and the length of the bump, respectively. Assume A = 0.1 m, L = 2.5 m and the vehicle forward velocity as V = 20 km/h. In this section, we assume that the road condition zrr (t) for the rear wheel is the same as the front wheel but with a time delay of (l1 + l2 )/V . Fig. 11 to Fig. 13 plot responses of the heave and pitch accelerations, the front and rear suspension deflection constrains, the relation of dynamic front and rear tire deflection constrains of the


13

1.1003 3.8302 3.2124 4.1587 −0.5581 0.1646 0.0002 0.0098 , −0.2037 −0.9087 −0.3333 0.8088 0.0559 −0.6115 −0.0074 −0.0185 1.1018 3.8252 3.2245 4.1604 −0.5579 0.1648 0.0004 0.0097 103 × , −0.2146 −0.9154 −0.4306 0.7711 0.0575 −0.6110 −0.0077 −0.0183 1.0849 3.7988 3.2227 4.2763 −0.5534 0.1590 0.0002 0.0100 103 × , −0.1788 −0.8553 −0.3701 0.5758 0.0476 −0.6008 −0.0074 −0.0186 1.0861 3.7955 3.2348 4.2776 −0.5534 0.1591 0.0004 0.0100 103 × , −0.1899 −0.8670 −0.4700 0.5395 0.0498 −0.6002 −0.0077 −0.0184 −0.4479 1.9475 1.9010 2.7776 −0.6834 0.0133 0.0061 0.0158 103 × , −2.2589 −3.3496 −2.2660 −1.4856 −0.1268 −0.7972 −0.0001 −0.0107 −0.4557 1.9419 1.8078 2.7444 −0.6811 0.0141 0.0058 0.0161 103 × , −2.2603 −3.3493 −2.2621 −1.4857 −0.1273 −0.7975 −0.0002 −0.0107 −0.4224 1.9921 1.8706 2.5624 −0.6904 0.0236 0.0063 0.0156 103 × , −2.2811 −3.3858 −2.2384 −1.3373 −0.1230 −0.8052 −0.0002 −0.0108 −0.4343 1.9857 1.7766 2.5259 −0.6890 0.0239 0.0059 0.0159 103 × . −2.2805 −3.3852 −2.2339 −1.3363 −0.1232 −0.8052 −0.0002 −0.0109

K1 = 103 × K2 = K3 = K4 = K5 = K6 = K7 = K8 =

Fig. 13.

Responses of the dynamic front and rear tire stroke constraints.

open- and closed-loop systems. It can be observed that the required performances can be guaranteed by the state-feedback sampled controller. Due to page limitation, the simulation results on the dynamic output-feedback controller are omitted.

VI. Conclusion The problem of sampled-data H∞ control design has been investigated for uncertain active suspension systems in this paper. In this paper, both sprung and unsprung mass variations for building suspension systems and the suspension performance have been considered. The sector nonlinearity method has been used to construct the uncertain system by T-S fuzzy system. Both state-feedback and output-feedback sampleddata controllers have been designed such that the resulting closed-loop T-S fuzzy system is asymptotically stable with an H∞ performance, and simultaneously satisfies the constraint suspension performance. Finally, the simulation results for

the uncertain quarter-vehicle suspension model have been provided to validate the effectiveness of the proposed design method and these simulation results have clearly demonstrated that the designed sampled-data controllers can improve suspension performances. It has been shown that the proposed control design method in this paper is also available for uncertain half vehicle suspension systems. In future work, based on the piecewise and parameter dependent Lyapunov function methods, we will further improve the suspension performances via sampled-data control approach. In addition, in order to propose more general fuzzy control results, research work will be done without requiring that both the T-S fuzzy model and the fuzzy controller share the same number of rules and/or the same set of premise membership functions. Thus, it offers a greater design flexibility for the fuzzy controller and is possible to reduce the controller complexity by employing a smaller number of rules and simple membership functions. Furthermore, the fault-tolerant control design problem [59], [60] will be investigated for vehicle suspension systems.

Appendix Proof: In order to develop a state-feedback sampled-data control design results, we consider the following LyapunovKrasovskii functional: V (t) = xT (t) Px (t) 0 t +h x˙ T (s) Q˙x (s) dsdα. −h

t+α

The derivative of V (t) along the solution of system (10) is expressed as V˙ (t) = 2xT (t) P x˙ (t) + x˙ T (t) h2 Q˙x (t) t −h x˙ T (s) Q˙x (s) ds. t−h



Using Jensen’s inequality [58], it can be seen that t −h x˙ T (s) Q˙x (s) ds t−h

≤ − [x (t) − x (t − d (t))]T Q [x (t) − x (t − d (t))] − [x (t − d (t)) − x (t − h)]T Q [x (t − d (t)) − x (t − h)] . (50) Since zT1 (t) z1 (t) 4 4 T C1i T C1i D1i Kj α (t) ≤ hi hj α (t) T KjT D1i

i=1

where θmax (·) represents maximal eigenvalue. From the above inequality, it leads to that the constraints in (11) are guaranteed, if 4 1 1 hi P − 2 {C2i }Tq {C2i }q P − 2 < I ρ· (54)

i=1 j=1

where

αT (t) = xT (t) xT (t − d (t)) .

i=1

which means 4 1 1 hi ρ · P − 2 {C2i }Tq {C2i }q P − 2 − I < 0

Furthermore V˙ (t) + zT1 (t) z1 (t) − γ 2 wT (t) w (t) 4 4 hi hj βT (t) (1ij + 2ij T2ij ≤

i=1

which can be guaranteed by the feasibility of (14). The proof is completed.

i=1 j=1

+3ij Q−1 T3ij )β (t) where

(51)

βT (t) = xT (t) xT (t − d (t)) xT (t − h) wT (t) . For (12)–(13), we obtain 4 4

hi hj ij

i=1 j=1

=

4 i=1

h2i ii +

3 4

hi hj ij + ji < 0.

(52)

i=1 j=i+1

Based on (52) and by using Schur complement to (51), one has V˙ (t) + zT1 (t)z1 (t) − γ 2 wT (t)w(t) < 0.

2 max {z2 (t)}q t>0 4 T T ≤ max hi x (t){C2i }q {C2i }q x(t) t>0 i=1 2 4 1 1 1 1 T − T − = max hi x (t)P 2 P 2 {C2i }q {C2i }q P 2 P 2 x(t) t>0 i=1 2 4 1 1 < ρ · θmax hi P − 2 {C2i }Tq {C2i }q P − 2 , q = 1, 2

(53)

On the other hand, w (t) = 0, it is easy to see that V˙ (t) < 0, then system in (10) is asymptotically stable. Next, we establish the H∞ performance of the system in (10) under zero initial conditions. Under zero initial conditions, we have V (0) = 0 and V (∞) ≥ 0. Integrating both sides of (53) yields z1 2 < γ w2 for all nonzero w ∈ L2 [0, ∞), and then the H∞ performance is established. In the following part of this proof, we will prove that the output constraints in (11) are guaranteed. From inequality (53), one can have V˙ (t) − γ 2 wT (t)w(t) < 0. Integrating both sides of the above inequality from zero to any t > 0, we obtain t V (t) − V (0) < γ 2 wT (τ)w(τ)dτ < γ 2 w22 . 0

It can be seen from the Lyapunov-Krasovskii functional that one has xT (t)Px(t) < ρ with ρ = γ 2 wmax + V (0). Similar to [2], the following inequality holds:

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Hongyi Li (M’13) received the B.S. and M.S. degrees in mathematics from Bohai University, Jinzhou, China, in 2006 and 2009, respectively, and the Ph.D degree in intelligent control from the University of Portsmouth, Portsmouth, U.K., in 2012. He is currently a Professor at the College of Engineering, Bohai University. He was a Research Associate with the Department of Mechanical Engineering, University of Hong Kong and Hong Kong Polytechnic University, Hong Kong, in 2010 and 2012, respectively. His current research interests include fuzzy control, robust control, and their applications. Dr. Li is an Associate Editor/Editorial Board member for several international journals, including Neurocomputing, Circuits, Systems, and Signal Processing, Shock and Vibration, and so on.


Xingjian Jing (M’13) received the B.S. degree from Zhejiang University, Hangzhou, China, in 1998, the M.S. degree from Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang, China, in 2001, and the Ph.D. degree in nonlinear systems and signal processing from the Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield, U.K., in 2008. He is currently an Assistant Professor with the Department of Mechanical Engineering, Hong Kong Polytechnic University (PolyU), Hong Kong. Prior to joining PolyU in 2009, he had been a Research Fellow with the Institute of Sound and Vibration Research, University of Southampton, Southampton, U.K., where his research focused on biomedical signal processing. His current research interests include nonlinear analysis and design in the frequency domain, system identification, signal processing and control of complex nonlinear systems, intelligent computing methods and their applications in nonlinear mechanical systems (sound and vibration control), nonlinear physiological systems (neural systems), and robotic systems. Dr Jing is an active reviewer for many known journals and conferences, and also serves on the editorial boards of several international journals.

Hak-Keung Lam (M’98–SM’10) received the B.Eng. (Hons.) and Ph.D. degrees from the Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hong Kong, in 1995 and 2000, respectively. From 2000 to 2005, he was a Post-Doctoral Fellow and a Research Fellow, respectively, with the Department of Electronic and Information Engineering, Hong Kong Polytechnic University. In 2005, he joined Kings College London, London, U.K., as a Lecturer and currently is a Senior Lecturer. He has co-edited two edited volumes, Control of Chaotic Nonlinear Circuits (World Scientific, 2009) and Computational Intelligence and Its Applications (World Scientific, 2012). He has co-authored the book Stability Analysis of FuzzyModel-Based Control Systems (Springer, 2011). His current research interests include intelligent control systems and computational intelligence. Dr Lam is an Associate Editor for the IEEE Transactions on Fuzzy Systems and International Journal of Fuzzy Systems.


Peng Shi (M’95–SM’98) received the B.Sc. degree in mathematics from Harbin Institute of Technology, Heilongjiang, China, the M.E. degree in systems engineering from Harbin Engineering University, Heilongjiang, China, the Ph.D. degree in electrical engineering from the University of Newcastle, New South Wales, Australia, and in mathematics from the University of South Australia, Adelaide, Australia, and the D.Sc. degree from the University of Glamorgan, Treforest, U.K. Currently, he is a Professor at the University of Adelaide, Adelaide, and Victoria University, Melbourne, Australia. Prior to this, he had been a Lecturer at the Heilongjiang University, Heilongjiang, China, a Post-Doctorate and Lecturer at the University of South Australia, a Senior Scientist in the Defence Science and Technology Organisation, Canberra, Australia, and a Professor at the University of Glamorgan, U.K. His current research interests include system and control theory, computational intelligence, and operational research. Dr Shi is a fellow of the Institution of Engineering and Technology, and the Institute of Mathematics and its Applications, U.K. He has also been in the editorial board of a number of journals, including Automatica, the IEEE Transactions on Automatic Control, the IEEE Transactions on Fuzzy Systems, the IEEE Transactions on Cybernetics, the IEEE Transactions on Circuits and Systems-I, and IEEE Access.