Active Suspension System via Linear Matrix Inequalities

We present an application of a new controller order reduction technique with stability and performance preservation based on linear matrix inequality optimization to an active suspension system. In this technique, the rank of the residue matrix of a proper rational approximation of a high-order Hoo controller subject to the H,,norm of a frequency-weighted error between the approximated controller and high-order H,, controller is minimized. However, since solving this matrix rank minimization problem is very difficult, the rank objective function is replaced with the nuclear-nonn that can be reduced to a semidefinite program, so that it can be solved efficiently Application to the active suspension system of the Automatic Laboratory of Grenoble provides a fourth-order controller. The experimental results show the control specifications are met to a large extent. Key Bnls.: H,0 control, order reduction, active suspension system, linear matrix inequalities 1. INTRODUCIONT An active suspension system is used for disturbance attenuation in a large frequency band and in the presence of the load variation. Active suspension systems are currently of great interest in both academia and industry. A literature survey on suspension systems shows that several models and controllers have been developed in attempts to enhance and improve the ride and handling qualities in today's vehicle (Kuo and Li, 1999; Sunwoo et al., 1991). Linear controllers are the main group of these controllers. In the linear control philosophy, it is assumed that the system's states exhibit only small variations around the equilibrium point, so that a linear approximation model can be used. Existing linear controllers range from PID to robust multivariable controllers (Thompson, 1976, 1989; Karnopp, 1983; Majeed, 1989; Ray, 1991). In Ray (1991), robustness analysis and synthesis methods based on stochastic stability robustness for a quarter-car model were presented, which can be applied to higher-order active suspension systems. However, such an approach requires large feedback gain and resonable phase must be selected. In Majeed (1989), the centralized/local optimal output feedback controller (CLOFC) was developed for active suspension systems. In Thompson (1976, 1989), optimal control theory was applied to the design of an active suspension system. The used performance index is based on ride quality, suspension deflection, and tire deflection. In Kamopp (1983), a combination Journal oa Vibration and Control, 10: 11811197, 2004 DOI: 10.1177/1077546304044617 © 2004 Sage Publications 1182 R. AMIRIFAR and N. SADATI of the H,, and LQR methods was used to improve the system performance when it is subject to external disturbances, e.g. road irregularities, and parameter uncertainties, e.g. vehicle weight as payload varies. Even though this method provided better performance, its application to a vehicle suspension system is difficult, since the H"" method often results in complex high-order controllers even if the design model is of resonable size (Landau et al., 1995). In order to reduce the order of high-order controllers, controller order reduction techniques can be used. There are a number ofapproaches for obtaining reduced-order controllers (Moore, 1981; Glover, 1984; Anderson and Liu, 1989; Anderson, 1993; Zhou and Doyle, 1998). What is very important is to remember that controller order reduction should aim to preserve the required closed-loop properties as far as possible (Cordons et al., 1999). Direct simplification of the controller using standard techniques, e.g. pole-zero cancellation within a certain radius, balanced reduction technique (Dullerud and Paganini, 2000), frequency-weighted Hankelnorm approximation (Glover, 1984; Latham and Anderson, 1985; Safonov et al., 1987), Schur decomposition method (Safonov and Chiang, 1988), without taking into account the closedloop behavior generally yields unsatisfactory results. The newer order reduction methods guarantee the preservation ofthe closed-loop stability and performance. An approach for controller order reduction based on the closed-loop identification was presented in Karimi et al. (2001) where the reduced-order controller is derived by the minimization of the output error between the closed-loop nominal system and the closed-loop system using the reduced-order controller, which result preserves the performances of the nominal closed-loop system. Some progress was also achieved in Enns (1984), by introducing the weighting functions into the classical balanced reduction technique, in order to preserve the closed-loop stability In Anderson and Liu (1989), this approach is extended to find the weighting functions for the controller order reduction problem that can maintain the closed-loop performance. Linear matrix inequalities (LMIs) have emerged as a powerful formulation and design technique for a variety of linear control problems (Boyd et al., 1994). Since solving LMls is a convex optimization problem, such formulations offer a numerically tractable means of attacking problems that lack an analytical solution. In addition, a variety of efficient algorithms are now available to solve the generic LMI problems. Consequently, reducing a controller order reduction problem to an LMI problem can be considered as a practical solution to this problem. The novelty of this paper is the presentation of a control scheme for the active suspension system of the Automatic Laboratory of Grenoble (LAG; Karimi and Landau, 2002). This system has been the subject of the European Journal of Control benchmark on design and optimization ofrestricted-complexity controllers, 2002. Several control design methods have been presented and examined on the real system, including correlation based controller tuning (Miskovic et al., 2003), direct controller order reducton method (Constantinescu and Landau, 2003), genetic algorithms (Mauff and Duc, 2003), predictive control (Rossiter, 2003), crone control system design and closed-loop tuning (Lanusse et al., 2003), and multi-objective optimization (Cao and Yan, 2003). The stability of the system has been assured by all of the proposed controllers, but a very high performance low-order controller could not be achieved. In addition, a few of the proposed methods are straightforward to use. Our scheme is presented to remove the disadvantages of the available control schemes explained above. This scheme is done in the following three steps. A LOW-ORDER H,,, CONTROLLER DESIGN 1183 Figure 1. Closed-loop system. (i) Model order reduction: a low-order model of the system at low frequencies is obtained (is used in certain cases). (ii) Controller design: an Hos controller which meets the control specifications is designed. (iii) Controller order reduction: a new controller order reduction technique with stability and performance preservation via LMIs is used to reduce the order of the HO controller. In this technique, the rank ofthe residue matrix of a high-order controller subject to the error between the loop gain of the closed-loop nominal system and the loop gain of the closed-loop system using the reduced-order controller is minimized. However, since solving this matrix rank minimization problem is very difficult, similar to the approach of Fazel et al. (2001), the rank objective function is replaced with the nuclear-norm that can be reduced to a semidefinite program (SDP), so that it can be solved efficiently using the standard existing softwares (El Ghaoui, 1995; Mindenberge and Boyd, 1996; Wu and Boyd, 1996). The remainder of this paper is organized as follows. In Section 2, we show how the controller order reduction problem can be formulated as an LMI problem. The stability analysis of the closed-loop system using the reduced-order controller is discussed in Section 3. Section 4 is devoted to the active suspension system description. The application of the proposed approach to the active suspension system and the experimental results obtained by digital controller implementation are given in Section 5. Finally, in Section 6 we present our concluding remarks. 2. LMI FORMULATION OF CONTROLLER ORDER REDUCTION 2.1. Problem Statement Consider the nominal closed-loop system in Figure 1. G(s) is the nominal model of plant, K(s) is a high-order controller that achieves stability and performance, r(t) is the reference input, p(t) is the output disturbance, and y(t) is the system's output. Now, consider the following norm G(s) [K(s) K(-)] , (1) where K(s) is the reduced-order controller that must be determinated. This is equal to the following norms 1184 R. AMIRIFAR and N. SADATI |SP(s) [Sy (s) syr(s)]svp(s) , (2) |l,-Pl(s) [T(s) -T(s)]Sv,_(5 |W (3) where S,,p (s) is the output sensitivity function of the nominal closed-loop system, and $,p (s) is the output sensitivity function of the closed-loop system using K(s). T(s)is the complementary sensitivity function of the nominal closed-loop system, and T(s) is the complementary sensitivity function of the closed-loop system using K(s), defined by SVp (s) 1 + G(s)K(s)' S p(s) 1 + G(s)k(s)' G(s)K(s) G_______ T(s) = 1 G(s)K(s)' T(s) 1 + G(s)k(s) Also, the input sensitivity functions are defined as K(s) _ k(s) Sup (s) -1 + G(s)K(s) 7 Sup (s) I + G(s)k(s) (6) Equations (2) and (3) show that by minimizing a weighted norm of K(s) -k(s) in equation (1), a frequency-weighted norm of Syp (s) -Sp (s) and T(s)T(s) will be minimized. Since the inverse of the output sensitivity function has appeared as the weighting function, the optimization will be performed well in frequencies where the output sensitivity function is small enough. On the other hand, the output sensitivity function is the transfer function from the output disturbance p(t) to the system's output y(t); therefore, the performance in disturbance rejection will be preserved well, as seen in experimental results. Now, we consider a certain family ofK(s) as follows K(s) = Rop + E i,(7) where Ri, pi C C are a set ofcomplex numbers with conjugate symmetry. pi are considered as fixed scalars and residues Ri are the variables that are used for reducing the order of controller. By defining the residue matrix R as R = diag(Ri), (8) the order of a minimal state-space realization ofK(s) can be given by deg[K(s)] = rank (R) . (9) A LOW-ORDER Ho, CONTROLLER DESIGN 1185 The goal is to determine the values of Ri that minimize deg[K(s)] subject to a set of preserving stability and performance constraints. We see that in order to preserve the performance in disturbance rejection k(s) should satisfy the following norm inequality G(s)[K(s)k(s)] < c, (10) where £ s 0 is a very small scalar. The above inequality can be approximated by G(iw9k) [K(i°k )k(iwk)] < E, k= 1, ..., (11) where N is a finite number of points in the frequency range of interest. Now, the controller order reduction problem can be expressed as min rank (R) subject to G(iOk)[K(jwk) K(ik)] < E k = 1, ..., (12) Rj = Ri, forp =P., where Ri are the optimization variables. Note that G( jOk )K( Iwk) is a linear function ofRi. 2.2. Definition 1 The nuclear-norm of a matrix, for example Rkxk, is defined as k tIRJr* = >Zi (R) , (13) i=1 where ai (R) denotes the ith singular value of R. In Fazel et al. (2001), it was shown that by solving the nuclear-norm minimization problem, a lower bound on the optimal value of the rank minimization problem can be obtained. Therefore, equation (12) can be approximated as a heuristic problem, given by min t subject to I[Rt * < t. G(ICork)[K( Iwk) k(iwok)] .0 (16) Xe CO, where X CmC , y yHe C mxnM Z = ZH e CnXnf, and t C R are the variables and Co is a set of LMI constraints. See Fazel et al. (2001) for the proof ofLemma 1. Using Lemma I and the Schur complement formula (Boyd et al., 1994), equation (14) can be expressed as min trace (Y) + trace (Z) subject to [RH ] >.0 [ EH(ik) E(JkE ) . > 07 k = 1,...,N, (17) R C CO, Rj = Ri, forpp Pi where ReC flXfn y yH C Cnxn Z= ZH C Cnxn are the LMI variables and E(ijCk) = G(jCk) [K(iwjk) k(i)]k (18) 2.4. Remark I Equation (12) is not equivalent to equation (14). In other words, the minimum value of rank (R) is not equal to the one of IJRII*. So, solving equation (14) will not give the minimum possible rank, although it will reduce the rank ofR in certain cases. A LOW-ORDER Hoc CONTROLLER DESIGN 1187 Figure 2. Closed-loop system using reduced-order controller. Figure 3. Rearrangement of the closed-loop system using reduced-order controller. 2.5. Remark 2 The poles ofK(s) cannot be changed by the optimization. This implies that the methodology may produce reduced-order controllers with poor performance, or be unable to reduce the order. So, the proposed approach is in some ways unnecessarily restrictive. 3. STABILITY ANAlYSIS The following theorem provides an upper bound for stability preservation. 3.1. Theorem I Assume that K(s) and K(s) have the same number of poles in Re(s) > 0 and no poles on the imaginary axis. If G(s) EK(s)k(s)j < flSVI (S)lK1 (19) then K(s) stabilizes G(s). 3.2. Proofof Theorem I Consider the closed-loop system using K(s) in Figure 2 and rearrangement of it in Figure 3. According to Figure 3 and using the small gain theorem (Zhou and Doyle, 1998), it can be concluded that if 1188 R. AMIRIFAR and N. SADATI