Journal of System Design and Dynamics

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1 Fixed-Order Output Feedback Control and Anti-Windup Compensation for Active Suspension Systems Unggul WASIWITONO and Masami SAEKI Graduate School of Engineering, Hiroshima University Kagamiyama, Higashi-Hiroshima , JAPAN The Mechanical System Engineering, Hiroshima University Kagamiyama, Higashi-Hiroshima , JAPAN Abstract In this study, we apply an anti-windup scheme to the vehicle active suspension system. A fixed-order output feedback controller which does not explicitly take into account the actuator saturation constraint is first designed. Then, an anti-windup compensator is designed to handle the saturation constraint. Local control design technique based on the circle criterion and L 2 gain performance is used for the anti-windup compensator synthesis. A quarter car model is considered in this study and the effectiveness of the proposed approach is shown by a numerical example. Key words : Vehicle Suspension, Output-Feedback Control, Actuator Saturation, Anti- Windup. 1. Introduction Received 7 Oct., 2010 (No ) [DOI: /jsdd.5.264] Copyright c 2011 by JSME Vehicle suspension plays important roles in vehicle performance. The main function of the vehicle suspension is to improve ride and handling performance. The root mean square (rms) body acceleration is widely used to measure ride comfort while the rms tire deflection is a measure for handling performance (1). It is required that the rms body acceleration and the rms tire deflection should be as low as possible for the optimal performance of the vehicle suspension. However, these requirements are conflicting. The optimal body damping ratio regarding handling is approximately 0.40, while optimal comfort in contrast can be achieved with the body damping ratio around 0.16 (2). Various approaches have been proposed to improve the performance and to manage the trade-off between conflicting performance requirements of the vehicle active suspension system, such as adaptive control (3), nonlinear control (4), constrained H control (5), gain scheduling control (6) and multi-objective control (7). Recently, a number of important results on the H control of vehicle suspension have been reported in Refs. (8) (10). The most important objective for the vehicle suspension system is the improvement of the ride comfort (10) and a significant control input is often necessary to obtain better performance. However, in practice, the actuators which deliver the control signal are always subject to limits in their magnitude which is commonly known as actuator saturation. Most of the aforementioned vehicle active suspension control strategies deal with actuator saturation by keeping the control signals small, until the point where the actuator constraint is not met at all. The advantage of this approach is allowing one to use the unconstrained design methods and hence a linear analysis of the problem can be carried out. However, this is achieved at the cost of reducing achievable performance since we expect high performance to be associated with acting on or near constraints (11), (12). Moreover, most of the aforementioned control strategies for vehicle active suspension require that all the state variables can be measured or the implementation of an observer is needed. This often causes difficulties in implementation. 264

2 The use of static-output feedback control for vehicle active suspension is proposed in Ref. (13) where genetic algorithm is utilized for the controller synthesis. In Ref. (14), the design of dynamic-output feedback controller for vehicle active suspension with the mixed H 2 /generalized H 2 based on LMI optimization (15) is presented. However, the controller s order is as high as the generalized plant. Recently, a fixed-order H control design using descent method in the controller gain space is presented in Ref. (16). This method treats the bilinear matrix inequality (BMI) that contains a Lyapunov matrix and the controller gain as variables directly. In addition, the effectiveness of the descent method compared with D-K iteration is shown in Ref. (17). In this study, we propose an alternative approach for the vehicle active suspension system. We perform some separation in the controller such that one part is devoted to achieve nominal performance and the other part is devoted to constraint handling. This is the approach taken in anti-windup compensation (18). Anti-windup has been studied extensively over the past decades (see e.g., the survey in Refs. (18) (20)). Originally, windup problems were encountered when using PI/PID controllers (19) where the integral states of the controller are wrongly updated. Later on, it is well known that windup not only occurs in integral controller, but whenever the controller contains badly damped or unstable modes. In addition, even in the absence of dynamic controller elements, saturation nonlinearities can also trigger undesired oscillations due to inappropriate plant state (21). Therefore, by considering the behavior of the controller and plant states has potential for the performance improvement of the anti-windup compensated system (22), (23). Furthermore, an attempt to use the anti-windup scheme for active vibration control is given in Refs. (24), (25) which shows the effectiveness of the anti-windup scheme for vibration isolation system. Because the improvement of the ride comfort can be seen as a vibration isolation problem, we aim to study the application of anti-windup scheme on the vehicle active suspension system. The anti-windup compensation involves a two-step design procedure. Namely, a controller is first synthesized for a nominal, usually linear system ignoring saturation. Then, the anti-windup compensator is designed such that, when saturation occurs, this compensator becomes active and leads to improved behavior during this period (26). Based on the two-step design procedure, in this study, a fixed-order output feedback controller which does not explicitly take into account the saturation constraints is first designed. The descent method (16) is used to solve the BMI problem of the fixed-order output feedback controller synthesis. Then, the anti-windup compensator is designed to handle the saturation constraints. Local control design technique which is based on the circle criterion and L 2 gain performance is used for the anti-windup compensator synthesis. The remainder of this paper is organized as follows. In Section 2, the quarter-car model is presented as well as the performance outputs need to be considered. The fixed-order output feedback controller design is presented in Section 3. In Section 4, the design of anti-windup compensator is presented. The effectiveness of the proposed anti-windup compensation for the vehicle active suspension system is shown by a design example in Section 5. Finally, some concluding remarks are given. Further, we use the following notation. M T, σ max (M) and M 1 are the transpose, the maximum singular value and the inverse matrix of M, respectively. G is the H norm of G (s). IfM = [ M ij ] is a symmetric matrix with block matrices Mij, the upper or lower block matrices are sometimes denoted by for space saving. 2. Problem Formulation The quarter car model shown in Fig. 1 is considered in this study. The quarter-car model is very often used for the vehicle suspension analysis and design, because of its simplicity yet capture many important characteristics of the full model. In Fig. 1, m s is the sprung mass, which represents the car chassis; m u is the unsprung mass, which represents mass of the wheel assembly; c s and k s are the damping and stiffness of the suspension system, respectively; k t and c t stand for the compressibility and damping of the pneumatic tire, respectively; z s 265

3 Fig. 1 Quarter-car model with an active suspension and z u are the displacements of the sprung and unsprung masses, respectively; z r is the road displacement input; and u is the active input of the suspension system. Define the following state variables: x 1 = z s z u, x 2 = z u z r, x 3 = ż s, x 4 = ż u (1) where x 1 denotes the suspension deflection, x 2 denotes the tire deflection, x 3 denotes the sprung mass velocity, and x 4 denotes the unsprung mass velocity. Further, define the disturbance input as w = ż r, and x p = [ ] T x 1 x 2 x 3 x 4, then, by applying Newton s second law of motion and using the static equilibrium position as the origin, the state-space form of the vehicle suspension system can be written as ẋ p = A p x p + B w w + B u u (2) where x p R n p is the suspension system state, w R n w is the disturbance input, u R n u is the control input, and A p = k s m s 0 c s c s, B w =, B u = 1 m s m s 0 k s m u k t c s m u m u c m s+c t c t s m u m u 1 m u Let us now focus on the controlled outputs. First of all, the rms body acceleration is a widely used measure to quantify the ride comfort. Hence, the sprung mass acceleration z s is chosen as the performance output, with z s = [ k s m s 0 c s m s c s m s ] xp + 1 m s u (3) According to ISO 2631, there are resonances at the natural frequencies of human body near 1 2 Hz in the horizontal direction and 4 8 Hz in the vertical direction (1). In these frequency ranges, the human body cannot endure vibrations. Therefore, we choose the following frequency-dependent performance weight of the vertical acceleration for ride comfort. W a (s) = s s s (4) s s s The second controlled output is the road holding ability. In order to ensure a firm uninterrupted contact of wheels to road, it s required that the transfer function from the road disturbance to tire deflection should be small. It is also important to keep the transfer function from the road disturbance to suspension deflection small to prevent excessive suspension bottoming, because there is a limitation in the suspension deflection. Furthermore, it is standard in the H framework to use weighting functions to shape and compromise the different performance objectives. Hence, we define the weighted performance outputs as follows z cl = W a (s) z s + α (z s z u ) + β (z u z r ) (5) 266

4 where α>0isascalar weighting for the suspension deflection and β>0isascalar weighting for tire deflection. These weights are used to control the trade-off between the three control objectives. In practice, it is difficult to measure all the state variables. The most common sensors which can be found are Accelerometer that is use to measure the acceleration LVDT that is used to measure the suspension deflection However, when the vertical acceleration of sprung mass z s is used as measured output for feedback control, this condition leads to a BMI problem that is difficult to be cast into a LMI problem in the controller synthesis. On the other hand, the velocity sensor allowing the measurement of the sprung mass vertical velocity ż s is currently unavailable. Current velocity sensor (Piezo-Velocity Transducer) has a limitation on the measured amplitude range. To overcome the difficulty in obtaining the measured sprung mass velocity, one can obtain the velocity by the integration of the acceleration, as used in Ref. (27). Follow the same idea as in Ref. (27), in this study, we consider that the suspension deflection and the sprung mass velocity can be measured and define the following measured outputs y = x p = C y x p (6) which represents the sprung mass velocity and suspension deflection. Fig. 2 Linear closed-loop interconnection Augmenting the vehicle suspension system in Eq. (2) with the weighted performance outputs in Eq. (5), we get the state-space realization of system P s, as shown in Fig. 2, as follows ẋ s = A s x s + B sw w + B su u z cl = C s x s + D su u (7) y = C sy x s where x s R n s includes 3 state variables in the frequency-dependent weight W a (s). Our aim is to compute an output feedback controller K ẋ k = A k x k + B k y u = C k x k + D k y (8) that meets the various specifications mentioned above with A k R n k n k, (n k n s ). Moreover, actuator saturation is present almost everywhere in practical control systems. In this study, we use the anti-windup approach to deal with the actuator saturation problem. Hence, another problem that need to be addressed in this study is the design of anti-windup compensator that maintains closed-loop system stability and recovers as much as possible the performance loss when saturation occurs. 3. Output Feedback Controller Design The closed-loop system that is composed of the system in Eq. (7) and the output feedback controller in Eq. (8) can be written as (28) ẋ = A cl x + B cl w z cl = C cl x + D cl w (9) 267

5 where A = A cl C cl B cl D cl A s nk =, B 1 = C 1 = [ C s 0 ], C 2 = A + B 2 KC 2 B 1 C 1 + D 12 KC 2 0 B sw, B 2 = 0 0 I nk C sy 0, K = 0 B su I nk 0 A k C k, D 12 = [ 0 D su ] B k D k By using bounded real lemma the fixed-order H synthesis problem is first transformed into a matrix inequality condition as P (A + B 2 KC 2 ) + (A + B 2 KC 2 ) T P B T 1 P γi < 0 (10) C 1 + D 12 KC 2 0 γi The condition in Eq. (10) is a BMI, since there are product terms involving compensator parameter K and the Lyapunov parameter P. To solve this BMI problem, we use the descent method in the controller gain space proposed in Ref. (16). In the following section, a brief review of the optimization procedure is given Optimization procedure Let G (s) denote the closed-loop transfer function from w to z cl and consider the following level set in the gain space of K R (n k+n u ) (n k +n y). K (γ) = { K G (s) <γ} (11) The problem is to find a gain K that belongs to this level set. Namely, K K(γ) (12) Let consider the method of finding K that belongs to K (γ) for a given level γ = γ.in step 1, choose a starting point K 0. In step 2, obtain a sufficiently small γ, that satisfies Eq. (10) with the additional conditions P = P T > 0, and denoted it as γ 0. Then K 0 is an inner point of K(γ) and lies very close to the boundary, as shown in Fig. 3. In Step 3, obtain a feasible direction ΔK and find a new point K 1 that satisfies G (s, K 0 ) > G (s, K 1 ) (13) by applying a line search method (16) along the line defined by K = K 0 + α d ΔK, α d > 0 (14) Fig. 3 Level set K(γ 0 ) shrinks to K(γ 1 ) 268

6 By iterating Steps 2 and 3 for the new point K = K 1 instead of K 0, we obtain new γ 0 and K 1 and denote them as γ 1 and K 2, respectively. In this way, we can construct a monotonically non-increasing sequence: γ 0 γ 1 γ 2 (15) and a sequence of subsets: K(γ 0 ) K(γ 1 ) K(γ 2 ) (16) If γ i becomes less than the given level γ for some i, a solution has been obtained LMI Sufficient condition In this Section, the LMI sufficient condition for the BMI condition in Eq. (10) is explained. Lemma 1 The BMI condition in Eq. (10) can be written in the form: MXQX T M T + MXN T + NX T M T + R < 0 (17) where X = M = P 0 0 K T I C T , Q =, R = 0 B 2 B T C1 T 0 γi 0 C 1 0 γi, N T = A B D T 12 Proof This can be easily shown by direct calculation. The decomposition of matrix Q is given by next lemma Lemma 2 Assume n s + n k n k + n u and let the rank of B 2 R (n s+n k ) (n k +n u ) be n k + n u. Then, an eigenvalue-eigenvector decomposition of the matrix Q becomes Q = [ ] Λ 0 0 U T 1 U 1 U 2 U 3 0 Λ 0 U2 T (18) U3 T where Λ=diag ( σ 1,σ 2,,σ nk +n u ) > 0 and σ i, i = 1,, n k + n u are nonzero singular values of the matrix B 2, and UU T = Ifor U = [U 1, U 2, U 3 ] with U 1 R n l (n k +n u ),U 2 R n l (n k +n u ), and U 3 R n l (n s (n k +n u )), where n l = (n s + 2n k + n u ). Proof See (16) From Lemma 2, Q = U 1 Λ 1 U T 1 + U 2Λ 2 U T 2 (19) where Λ 1 =Λ 2 =Λ> 0. Then, Eq. (17) is represented as MX ( U 1 Λ 1 U1 T + U ) 2Λ 2 U2 T X T M T + MXN T + NX T M T + R < 0 (20) by arranging this with respect to U 1 and setting Z = MXU 1 Λ (21) L = NU 1 Λ (22) 269

7 we represent Eq. (20) in the next form ZZ T ZL T LZ T MXU 2 Λ 2 U T 2 XT M T + R > 0 (23) where R = He { MX ( U 2 U2 T + U ) 3U } 3 T N T R (24) Theorem 1 For any Z q, if the next LMI with respect to X is satisfied H Z qzq T R U2 T XT M T Λ 1 2 ) } H := He {(Z q Λ 1 21 U T1 N X T M T then the BMI in Eq. (23) is satisfied Proof See (16) > 0 (25) Theorem 2 Suppose that Eq. (20) is satisfied for X = X a and set Z a = MX a U 1 Λ Then, Eq. (25) is also satisfied for X = X a and Z q = Z a. Proof See (16) Fig. 4 Level set K(γ 0 ), its convex approximation K c (γ 0 ), and a small sphere S where K in shown by star is an inner point of K(γ 0 ) S 3.3. Computing Feasible Direction In Step 3 of the optimization procedure by descent method, we need to obtain the feasible direction ΔK. The method of obtaining the feasible direction ΔK is explained as follows. In Fig. 4, the set K(γ 0 ) is shown by the region surrounded by the solid line and K 0 lies very close to the boundary. The arrow from K 0 to the inner point K in gives the feasible direction ΔK. To obtain the inner point K in, we derive a LMI condition that is sufficient for Eq. (10) with the additional conditions P = P T > 0 and that is also satisfied for K = K 0.Thesetof these gains that satisfy this sufficient condition is denoted by K c (γ 0 ), which is shown as the region surrounded by the dash line in Fig. 4. On the other hand, the sphere described by S = { K σ max (K K 0 ) <ɛ} (26) is shown by the disc in Fig. 4, and this condition can be represented by the next LMI condition ɛi K K 0 (K K 0 ) T < 0 (27) ɛi This condition tunes the approximation region around K 0. Setting the radius of ɛ sufficiently small will give a good feasible direction near K 0. By solving the feasibility problem of finding K and P that satisfy both the LMI sufficient condition and Eq. (27), we can obtain the inner point K in and then ΔK = K in K 0 is the feasible direction. 270

8 4. Anti-Windup Control In the previous section, the design of the fixed-order output feedback control is given. However, the actuator saturation is not considered in the design problem. Therefore, we add the anti-windup compensation as a constraint handling. In this section the design of the antiwindup compensator is given Performance index A general structure of the anti-windup compensator is shown in Fig. 5. In this paper, we consider the static anti-windup compensator, which is most desirable from practical point of view. Namely, the controller with the anti-windup compensator is described by ẋ k = A k x k + B k y +Λ u v u = C k x k + D k y (28) where the constant matrix Λ u R n k n u is the anti-windup compensator to be designed. Fig. 5 Block diagram of anti-windup compensator In Refs. (22), (23), the anti-windup compensator design considering the behavior of controller state is proposed. The same method is used here for the anti-windup compensator design. Fig. 6 shows the system configuration for the design. The difference between the states of the plants P(s) and P(s) is denoted by Δx p = x p x p, and that of the controllers K(s) and K(s) is denoted by Δx c = x c x c. These are introduced to estimate the differences of the states of the actual feedback system with the linear model. Fig. 6 System configuration for design The general goal of the anti-windup compensator is to guarantee the stability of the closed-loop system subject to the actuator saturation, and to recover as much as possible the 271

9 performance lost during saturation. The anti-windup compensator design often assumes the saturation as a sector-bounded nonlinearity and absolute stability conditions are applied for the stability analysis. Generally, the L 2 gain was picked as an appropriate induced system norm to measure the anti-windup compensator s performance (26), namely, the problem is to minimize δ u subject to where z w 2 <δ u w 2 (29) z w = W pδx p W c Δx c (30) The constant weights W p and W c are used for the tuning of the balance between the plant-state difference Δx p and the controller-state difference Δx c. In the next section, we will derive an LMI-based synthesis procedure Reduction to LMI problem By using the deadzone function v =Ψ(u) = u Φ(u) (31) the system in Fig. 6 can be represented as that of Fig. 7, with the system G w in this figure is described by ẋ w = A w x w + B ww w + (B v1 + B v2 Λ u ) v (32) u = C uw x w (33) z w = C zw x w + D zw w (34) where A w R n d n d with n d = 2(n p + n k ) A w = A lw 0, A lw = A p + B u D k C y B u C k 0 A lw B k C y A k B ww = [ B w ] T, Bv1 = [ B u 0 B u 0 ] T B v2 = [ I 0 I 0 ] T, Cuw = [ D k C y C k 0 0 ] z w = 0 0 W p 0, D zw = W z 0 Fig. 7 Nonlinear feedback system with sector condition Considering that Ψ( ) satisfies the sector condition (29) in the finite interval [ Ξ, Ξ] with Ξ=(1/ (1 κ i )) U i then the following inequality condition holds v T N u [v κu] 0 (35) where N u = diag[n 1, n 2,, n nu ] > 0 κ = diag[κ 1,κ 2,,κ nu ], 0 κ i 1 272

10 The next theorem guarantees the L 2 gain condition in Eq. (29) for the nonlinearity that satisfies Eq. (35). Theorem 3 For a given κ, if there exist a positive-definite symmetric matrix Q R n d n d,a diagonal matrix T = diag[t 1, t 2,, t nu ] > 0, a matrix M R n k n u, and a scalar δ u > 0 that satisfy the next matrix inequality QA T w + A w Q B v1 T + B v2 M + QCuwκ T B ww QCzw T 2T 0 0 δ u I D T zw δ u I < 0 (36) then, the feedback system described by Eq. (34) with Λ u = MT 1 is asymptotically stable and the L 2 gain from w to z w is less than δ u when the condition u i (1/ (1 κ i )) U i, i = 1, 2,..., n u holds. Proof Consider a Lyapunov quadratic function V (x) = x T Px, P = P T > 0, P R n d n d (37) In order to show that the closed-loop system of Fig. 7 is asymptotically stable and the L 2 gain from w to z w is less than δ u, we may show that the Lyapunov function in Eq. (37) satisfies the next dissipation inequality. dv <δ u w T w 1 z T dt δ wz w (38) u By using the sector condition Eq. (35) and the S-procedure, we obtain dv + 1 z T dt δ wz w δ u w T w 2v T N u (v κu) < 0 (39) u Subtituting the derivation of the Lyapunov function and others variables, then applying the Schur complement, we obtain A T wp + PA w PB v1 + PB v2 Λ u + CuwκN T u PB ww Czw T 2N u 0 0 δ u I D T zw δ u I Applying a simple congruence transformation block-diag ( P 1, N 1 u define Q = P 1,T = N 1 u, and M =Λ u N 1 u we have Eq. (36) < 0 (40), I, I ) to Eq. (40) and This design problem is an LMI problem with respect to the variables Q, T, δ u, M, and therefore it can be solved easily by a numerical optimization method. Note that if A p is stable and the above design problem is satisfied for κ = I, the sector condition is satisfied for the deadzone function globally and the system is guaranteed to be globally asymptotically stable. Further, it is reported in Ref. (30) that the performance of the anti-windup compensated system can be significantly improved by reducing the value of the upper bound of the sector condition from 1. Therefore, for the anti-windup compensator synthesis we use 0 <κ<i, and then further analyze the stability of the closed-loop system augmented with the obtained anti-windup compensator using Popov criterion (22), which is less conservative than the circle criterion. 5. A Design Example In this section, we apply the output feedback control with anti-windup scheme for the vehicle active suspension on the quarter-car model described in Section 2. The parameters for the quarter-car model are taken from Refs. (5), (10), where the same parameters for the quarter-car model are used, except for the actuator saturation bound. For this study, we use 273

11 the actuator saturation bound u max = 1.5 kn (5) which is lower than that used in Ref. (10). However, for comparison, the proposed approach is compared with the passive suspension and the state feedback control reported in Ref. (10) as the controller proposed in Ref. (10) gives better ride comfort compared with that proposed in Ref. (5). First, we design the fixed-order output feedback controller. In this study, we want to synthesize a second order controller. For this purpose, we apply the descent method (16), with K 0 A k0 C k0 B k0 D k0 = (41) as the starting point, ɛ = 2e 7, and convergence parameter 1e 12. By setting α = 75 and β = 50 the sequence of γ i converged only after 8 iterations, as shown in Fig. 8, with the following dynamic output-feedback controller A k C k B k D k = (42) Fig. 8 Sequence of L 2 gain obtained by descent method Fig. 9 Frequency response of body vertical acceleration (thin solid line: passive suspension; thin dashed line: state feedback control (10) ; thick solid line: output feedback control) After the output feedback controller is obtained, the performance of the closed-loop suspension system with this controller is compared with the passive suspension system and that using the state feedback controller (10). Fig. 9 shows the frequency response from the road disturbance to the body acceleration. It can be seen that the output feedback controller yields 274

12 the least value of H norm over the frequency range 1 8 Hz compared with the passive system and the state feedback controller, which shows that an improvement in ride comfort has been achieved. Let us now consider the case of an isolated bump in a road surface. The corresponding disturbance input is given by (10) ( ) 2πv0 t w = A m sin L b if 0 t L b v 0 ; and w = 0 otherwise where A m = 0.5 m represents the amplitude of the bump, L b = 5 m is the length of the bump and v 0 is the vehicle velocity. Fig. 10 shows the bump response for the case of vehicle velocity 90 km/h that represents the disturbance with frequency 5 Hz. Clearly, the vehicle suspension system with the output feedback controller achieves better performance on ride comfort (lower peak) than that with the state feedback controller. From Fig. 10b, we note that the improvement in the ride comfort requires a larger actuator force. However, in this case, the actuator force is still below the saturation bound. Fig. 10 Bump response for vehicle velocity 90 km/h (thin solid line: passive suspension; thin dashed line: state feedback control (10) ; thick solid line: output feedback control) Figure 11 shows the bump response for the case of vehicle velocity 25 km/h that represents the disturbance with frequency 1.4 Hz. When there is no control input saturation the ride comfort performance of the output feedback controller (thin dash-dotted line) is better compared with the state feedback controller (thin dashed line). On the other hand, there is a Fig. 11 Bump response for vehicle velocity 25 km/h (thin dashed line: state feedback control (10) ; thin dash-dotted line: output feedback control without saturation; thick dashed line: output feedback control with saturation; thick solid line: output feedback control with anti-windup) 275

13 performance degradation for the output feedback control when control input saturation occurs (thick dashed line). Let apply the anti-windup scheme and synthesize the anti-windup compensator using Theorem 3. In this example, we obtain a good response by setting W p = 1e 3, W c = 1e 3 and κ = 0.8 with the following anti-windup compensator Λ u = [ ] T (43) It can be clearly seen from Fig. 11 that by applying the anti windup scheme, the improvement in ride comfort performance is obtained (thick solid line). Moreover, the suspension deflection for the case of saturated system (with and without anti-windup) is lower compared with that of unsaturated system, as shown in Fig. 12a. It s likely that the actuator force is limited by the saturation bound. In terms of road holding ability, better dynamic tire load is obtained by the anti-windup scheme compared with that without anti-windup, as shown in Fig. 12b. These show that the output feedback control with anti-windup realizes the active suspension performance very well. Fig. 12 Bump response for vehicle velocity 25 km/h (thin solid line: passive suspension; thin dashed line: state feedback control (10) ; thin dash-dotted line: output feedback control without saturation; thick dashed line: output feedback control with saturation; thick solid line: output feedback control with antiwindup) The above anti-windup compensator is obtained using κ = 0.8. This means that the asymptotic stability is guaranteed only for the magnitude of the controller output u to be less that U/ (1 κ). However, in this case the asymptotic stability is guaranteed for the magnitude of the controller output u < 7.5 kn that is sufficiently large. Moreover, it is well-known that in some cases the resulting stability condition by the circle criterion is extremely conservative. Therefore, the stability of the closed-loop system augmented with the obtained anti-windup compensator is further analyzed using Popov stability analysis in (22) and the asymptotic stability is guaranteed for the larger magnitude of the controller output, in this case for u < kn. 6. Conclusion The design method of the fixed-order output feedback control with anti-windup scheme for vehicle active suspension systems is presented. By using the descent method proposed in Ref. (16), the fixed-order output feedback controller is easily obtained and it is shown by simulation that the ride comfort in the frequency ranges 1 8 Hz is improved. Then, the antiwindup compensator is added to the linear controller to ensure that stability is maintained and to recover as much as possible the performance loss when saturation occurs. Simulation results have shown the potential benefit of the proposed vehicle active suspension system in achieving the best possible ride comfort. However, in this study, the sprung mass velocity is considered as an output by the integration of the measured acceleration. Because of the 276

14 wide availability of the acceleration sensor, further research to study the possibility of using sprung mass acceleration as output in the fixed-order output feedback controller synthesis will be useful from practical point of view. In addition, it is also possible to prevent the suspension bottoming by modifying the actuator command before the stroke reaches the limitation, which is another future work needs to be considered. References ( 1 ) Hrovat D., Survey of advanced suspension developments and related optimal control applications, Automatica, vol. 30, no. 10 (1997), pp ( 2 ) Koch G., Fritsch O., Lohmann B., Potential of low bandwidth active suspension control with continuously variable damper, Control Engineering Practice, vol. 18, no. 11 (2010), pp ( 3 ) Fialho I., Balas G. J., Road adaptive active suspension design using linear parametervarying gain-scheduling, IEEE Transaction of Control Systems Technology, vol. 10, no. 1 (2002), pp ( 4 ) Karlsson N., Dahleh M., Hrovat D., Nonlinear H control of active suspensions, in Proceedings of the American Control Conference, (2001), pp ( 5 ) Chen H., Guo K., Constrained H control of active suspensions: An LMI approach, IEEE Transaction on Control Systems Technology, vol. 13, no. 3 (2005), pp ( 6 ) Fialho I., Balas G. J., Design of nonlinear controllers for active vehicle suspensions using parameter-varying control synthesis, Vehicle System, vol. 33, no. 5 (2000), pp ( 7 ) Gao H., Lam J., Wang C., Multi-objective control of vehicle active suspension systems via load-dependent controllers, Journal of Sound and Vibration, vol. 290, no. 3-5 (2006), pp ( 8 ) Turkay S., Akcay H., Aspects of achievable performance for quarter-car active suspensions, Journal of Sound and Vibration, vol. 311, no. 1-2 (2008), pp ( 9 ) Gao H., Sun W., Shi P., Robust sampled-data H control for vehicle active suspension systems, IEEE Transaction on Control Systems Technology, vol. 18, no. 1 (2010), pp (10) Sun W., Gao H., Kaynak O., Finite frequency H control for vehicle active suspension system, IEEE Transactions on Control System Technology, early access (2010). (11) Dona J. A., Goodwin G. C., Seron M. M., Anti-windup and model predictive control: reflections and connections, European Journal of Control, vol. 6, no. 5 (2000), pp (12) Goodwin G. C., Seron M. M., Dona J. A., Constrained control and estimation: An optimization approach, Springer, (2005), pp. 4. (13) Du H., Zhang N., Multi-objective static output feedback control design for vehicle suspension, Journal of System, vol. 2, no. 1 (2008), pp (14) Sun, P. Y., Chen H., Multi objective output feedback suspension control on a half-car model, in Proceedings of 2003 Conference on Control Applications, (2003), pp (15) Scherer C., Gahinet P., Chilalli M., Multi objective output feedback control via LMI optimization, IEEE Transactions on Automatic Control, vol. 42, no. 7 (1997), pp (16) Saeki M., Static output feedback design for H control by descent method, in proceedings of the 45th IEEE Conference on Decision & Control, (2006), pp (17) Wasiwitono U., Saeki M., Takamatsu S., Design of model-following anti-windup compensator that minimizes L 2 gain, in Proceedings of International Conference on Modelling, Identification and Control, (2010), pp (18) Tarbouriech, S., Turner, M., Anti-windup design: An overview of some recent advances and open problems, Control Theory & Applications, IET, vol. 3, No. 1 (2009), pp

15 (19) Kothare M. V., Campo P. J., Morari M., Nett C. N., A unified framework for the study of anti-windup designs, Automatica, vol. 30, no. 12 (1994), pp (20) Galeani S., Tarbouriech S., Turner M., Zaccarian L., A tutorial on modern anti-windup design, European Journal of Control, vol. 3, no. 4 (2009), pp (21) Hippe P., Wurmthaler C., Systematic closed-loop design in the presence of input saturations, Automatica, vol. 35, no. 4 (1999), pp (22) Wasiwitono U., Takamatsu S., Saeki M., Ochi K., Wada N., Dynamic anti-windup compensator design considering behavior of controller state, Journal of System, vol. 4, no. 4 (2010), pp (23) Takamatsu S., Wasiwitono U., Saeki M., Wada N., Anti-windup compensator design considering behavior of controller state, in proceeding of the 19th IEEE International Conference on Control Applications, (2010), pp (24) Teel A. R., Zaccarian L., Marcinkowski J. J., An anti-windup strategy for active vibration isolation systems, Control Engineering Practice, vol. 14, no. 1 (2006), pp (25) Tliba S., Varnier M., Dealing with actuator saturation for active vibration control of a flexible structure piezo-actuated, in proceeding of the 19th IEEE International Conference on Control Application, (2010), pp (26) Turner M. C., Postlethwaite I., A new perspective on static and low order anti-windup synthesis, International Journal of Control, vol. 77, No.1 (2004), pp (27) Chamseddine A., Noura H., Control and sensor fault tolerance of vehicle active suspension, IEEE Transaction on Control Systems Technology, vol. 16, No. 3 (2008), pp (28) Apkarian P., Noll D., Tuan H. D., Fixed-order H control design via a partially augmented Lagrangian method, International Journal of Robust and Nonlinear Control, vol. 13, no. 12 (2003), pp (29) Khalil H. K., Nonlinear systems, Prentice-Hall, Inc, (1996), pp (30) Wu F., Lu B., Anti-windup control design for exponentially unstable LTI systems with actuator saturation, System and Control Letters, vol. 52, no. 3-4 (2004), pp