Design of Non-Fragile H1 Controller for Active Vehicle Suspensions HAIPING DU Control and Power Group, Department of Electrical and Electronic Engineering, Imperial College, Exhibition Road, London SW7 2BT (
[email protected])
JAMES LAM KAM YIM SZE Department of Mechanical Engineering, University of Hong Kong, Pokfulam Road, Hong Kong (Received 15 June 20041 accepted 8 September 2004)
Abstract: In this paper we present an approach to design the non-fragile H1 controller for active vehicle suspensions. A quarter-car model with active suspension system is considered in this paper. By suitably formulating the sprung mass acceleration, suspension deflection and tire deflection as the optimization object and considering a priori norm-bounded controller gain variations, the non-fragile state-feedback H1 controller can be obtained by solving a linear matrix inequality. The designed controller not only can achieve the optimal performance for active suspensions but also preserves the closed-loop stability in spite of the controller gain variations.
Key Words: Non-fragile, H1 control, linear matrix inequality, active suspension
1. INTRODUCTION One important function of a vehicle suspension system is to improve ride comfort through minimizing the vibration caused by driving on bumpy road surfaces. Other functions include maintaining vehicle safety and drive (road holding), minimizing road damage, and enhancing overall vehicle performance, etc. To meet these different demands, many types of suspension systems, ranging from passive, semi-active to active suspensions, are currently employed and studied. Active control of vehicle suspensions has received considerable attention since the late 1960s (Hrovat, 1997) due to its better performance than passive control. Various approaches have been proposed to improve the performance of active suspension designs, such as linear optimal control (Elmadany and Abduljabbar, 1999), fuzzy logic and neural network control (Al-Holou et al., 2002), adaptive control (Fialho and Balas, 2002), H1 control (Yamashita et al., 1994), nonlinear control (Karlsson et al., 2001), gain-scheduling control (Fialho and Balas, 2000) and preview control (Thompson and Pearce, 2001). Also, many approaches are presented to deal with the multi-objective requirement of vehicle suspensions (see Abdellari et al., 2000, and references therein). Apart from the performance of active suspensions to be considered when designing a controller, the issue of robustness is another important
Journal of Vibration and Control, 11: 225–243, 2005 1 22005 Sage Publications
DOI: 10.1177/1077546305049392
226 H. DU ET AL. factor to be addressed in active suspensions. This is because the presence of uncertainty in the system parameters and unmodeled dynamics in vehicle suspensions are unavoidable and generally decrease the control performance, if not inducing instability, of the system. Hence, robust control strategies have been applied to tackle these problems (Hayakawa et al., 1999). Recently, attempts to investigate the presence of the time delay in actuator dynamics for active suspensions have been made (Jalili and Esmailzadeh, 20011 Vahidi and Eskandarian, 2001). This represents a more realistic view in actuator modeling and may have a major effect on the control performance of the suspension system. Although the aforementioned control laws can improve the performance and robustness of active suspensions to some extent in theory, they invariably are designed under an implicit assumption that the controllers can be accurately realized, which is however impossible due to many physical limitations in controller implementation. Some examples are the effects of finite word length in any digital systems, round-off errors in numerical arithmetic, inherent imprecision in analog devices, aging of controller devices, etc. Consequently, even though controllers are robust with respect to system uncertainties, they may be very sensitive to their own uncertainties (implementation errors). Despite its importance, to the best of our knowledge, no more attempts have been made to consider the problems of controller gain variations for active suspensions, although the so-called fragility problem of controllers has been studied in the field of control theory recently (Keel and Bhattacharyya, 19971 Makila, 1998). During the last few years, efforts have been made to tackle the non-fragile controller design problem for linear systems (Dorato, 19981 Yang and Wang, 2003), including those with time delay (Liu et al., 2002). This also brings a new issue to controller synthesis for active vehicle suspensions such that the designed controller should be resilient or non-fragile with respect to its gain variations. In this paper we are concerned with the non-fragile H1 controller design problems for active vehicle suspensions. A quarter-car model is used to study the performance of a vehicle suspension system in terms of the bouncing motion, the tire deflection, and other performance features (Jalili and Esmailzadeh, 2001). Three main performance requirements for advanced vehicle suspensions (ride comfort, road holding, and suspension deflection) are considered by constructing an appropriate state-feedback H1 controller to provide a tradeoff between these requirements. Controllers subject to additive norm-bounded perturbations were studied previously to cope with the controller gain variations. By using a Lyapunov approach, a non-fragile static state-feedback H1 control law is obtained by solving a linear matrix inequality (LMI). A feasibility solution, if it exists, can be easily found using standard numerical algorithms. The rest of this paper is organized as follows. In Section 2 we present the description of suspension modeling of a quarter-car model. The formulation of an H1 control problem for a quarter-car model is given in Section 3. The main results of this paper are given in Section 4 in which a non-fragile static state-feedback H1 control law can be obtained based on the solvability of an LMI. In Section 5 we present the design results and discussion. The conclusions are given in Section 6. In our notation, 343 refers to either the Euclidean vector norm or the induced matrix 2norm. For a real symmetric matrix W , the notation of W 1 0 (W 2 0) is used to denote its positive (negative) definiteness. Also, I is used to denote the identity matrix of appropriate dimensions. To simplify the notation, 5 is used to represent a block matrix which is readily inferred by symmetry.
DESIGN OF NON-FRAGILE H1 CONTROLLER 227
123456 78 94 5 65 5 6 2
26 4628
2. MODELING OF QUARTER-CAR SUSPENSION A quarter-car model consists of one-fourth of the body mass, suspension components, and one wheel, as shown in Figure 1. This model has been used extensively in the literature and captures many essential characteristics of a real suspension system. The governing equations of motion for the sprung and unsprung masses of the quartercar model are given by m s z6 s 3t4 7 cs [8z s 3t4 9 z8 u 3t4] 7 ks [z s 3t4 9 z u 3t4] u3t4 m u z6 u 3t4 7 cs [8z u 3t4 9 z8 s 3t4] 7 ks [z u 3t4 9 z s 3t4] 7 kt [z u 3t4 9 zr 3t4] 9u3t45
(1) (2)
Here, m s is the sprung mass, which represents the car chassis1 m u is the unsprung mass, which represents the wheel assembly1 cs and ks are damping and stiffness of the uncontrolled suspension system, respectively1 kt serves to model the compressibility of the pneumatic tire1 z s 3t4 and z u 3t4 are the displacements of the sprung and unsprung masses, respectively1 zr 3t4 is the road displacement input1 u3t4 represents the active input of the suspension system. This input is normally generated by means of a hydraulic actuator placed between the two masses (see Lin and Kanellakopoulos 1997 for details). We assume that z s 3t4 and z u 3t4 are measured from their static equilibrium positions. After choosing a set of state variables as x1 3t4 z s 3t4 9 z u 3t46
x2 3t4 z u 3t4 9 zr 3t46
x3 3t4 z8 s 3t46
x4 3t4 z8 u 3t46
(3)
where x1 3t4 denotes the suspension deflection, x2 3t4 is the tire deflection, x 3 3t4 is the sprung mass speed, x4 3t4 denotes the unsprung mass speed, and defining
228 H. DU ET AL. x3t4
1
x1 3t4 x2 3t4 x3 3t4 x4 3t4
2T
6
73t4 z8r 3t46
(4)
where 73t4 represents disturbance caused by road roughness, equation (2) can be written in state-space description as x3t4 8 Ax3t4 7 Bu3t4 7 B7 73t45
(5)
Here 3
0 4 0 A 4 5 9ks 8m s ks 8m u
0 0 0 9kt 8m s
1 0 9cs 8m s cs 8m s
6 91 7 1 76 cs 8m s 8 9cs 8m s
3
6 0 4 7 0 7 B 4 5 18m s 8 6 918m u
3
6 0 4 91 7 7 B7 4 5 0 8 0
are constant matrices.
3. H1 SUSPENSION CONTROL Ride comfort, road holding ability, and suspension deflection are three main performance criteria in any suspension design. It is widely accepted that ride comfort is closely related to the vertical acceleration experienced by the car body. Consequently, to improve ride comfort amounts to keeping the transfer characteristics from road disturbance to car body (sprung mass) acceleration small over the frequency range of 0–65 rad s91 (Fialho and Balas, 2000). Due to the disturbances caused by road bumpiness, a firm uninterrupted contact of wheels to the road (good road holding) is important for vehicle handling and is essentially related to driving safety. To ensure good road holding, it is required that the transfer function from road disturbance to tire deflection z u 3t4 9 zr 3t4 be small. The structural features of the vehicle also constrain the amount of suspension deflection z s 3t4 9 z u 3t4 with a hard limit. Hitting the deflection limit not only results in rapid deterioration of the ride comfort, but at the same time increases the wear of the vehicle. Hence, it is also important to keep the transfer function from road disturbance to suspension deflection z s 3t49z u 3t4 small enough to prevent excessive suspension bottoming. In accordance with the aforementioned requirements, we formulate an H1 control problem to deal with the three different objectives for vehicle suspensions. It is standard in the H1 framework to use weighting functions to shape and compromise different performance objectives. In order to satisfy the performance requirement, we define the controlled output z3t4 as z6 s 3t4, z s 3t4 9 z u 3t4, and z u 3t4 9 zr 3t4, respectively, for the quarter-car model. We consider the case that all the state variables defined in equation (3) can be measured, which means that we can design a static state-feedback H1 controller. On the other hand, in order to compensate for the effect induced by the variations of the designed controller, a priori norm-bounded gain variation is considered in the design process. Therefore, considering a controller gain variation 9K , we formulate the following state-feedback non-fragile H1 control problem for vehicle suspensions. Given the system described by equations of the form
DESIGN OF NON-FRAGILE H1 CONTROLLER 229 x3t4 8
Ax3t4 7 Bu3t4 7 B7 73t4
z3t4 C x3t4 7 Du3t46
(6) (7)
where x3t4, 73t4, A, B, and B7 are defined as in equation (5), and 3
9ks 8m s
C 5 0
0 0
9cs 8m s 0 0
6 cs 8m s 0 8 0
(8)
where and are positive weighting scalars for the suspension deflection and tire deflection, respectively, 1 2T D 18m s 0 0 (9) find a state-feedback controller of the following form u3t4 3K 7 9K 4x3t46
(10)
where K is the state-feedback gain matrix to be designed, and 9K is an a priori normbounded gain variation of the form (Liu et al., 20021 Yang and Wang, 2003) 9K L F E
(11)
with L and E being known constant matrices of appropriate dimensions, and 3F3 1
(12)
such that the resulting closed-loop system is asymptotically stable and the H1 -norm of the closed-loop transfer function matrix from 7 to z is bounded by a constant 1 0 for any uncertainties F satisfying equation (12).
4. NON-FRAGILE H1 CONTROLLER DESIGN To derive the main results, we need the following lemma. Lemma 1. (Xie et al., 1991) Given real matrices Y , M, and N of appropriate dimensions, then Y 7 M9N 7 N T 9T M T 2 0 for all 9 satisfying 393 1 if and only if there exists a constant 1 0 such that Y 7 M M T 7 91 N T N 2 05 System (7) with the static state-feedback control law u3t4 3K 7 9K 4x3t4 becomes
230 H. DU ET AL. x3t4 8
Ax3t4 7 B3K 7 9K 4x3t4 7 B7 73t4
z3t4 C x3t4 7 D3K 7 9K 4x3t45
(13)
In order to establish that system (13) is asymptotically stable with a disturbance attenuation 1 0, it is required that the associated Hamiltonian H 3x6 76 t4 V8 3x6 t4 7 z T 3t4z3t4 9 2 7 T 3t473t4 2 06
(14)
where V 3x6 t4 is a Lyapunov function given by V 3x6 t4 x T 3t4P x3t4
(15)
for some P 1 0. We take the derivative of V 3x6 t4 along the state trajectory of system (13)
9 V8 3x6 t4 x T 3t4 [A 7 B3K 7 9K 4]T P 7 P [A 7 B3K 7 9K 4] x3t4 727 T 3t4B7T P x3t46
(16)
then we obtain that 9
H 3x6 76 t4 x T 3t4 [A 7 B3K 7 9K 4]T P 7 P [A 7 B3K 7 9K 4] x3t4 727 T 3t4B7T P x3t4 7 z T 3t4z3t4 9 2 7 T 3t473t4
9
x T 3t4 [A 7 B3K 7 9K 4]T P 7 P [A 7 B3K 7 9K 4] x3t4 727 T 3t4B7T P x3t4 7 x T 3t4 [C 7 D3K 7 9K 4]T [C 7 D3K 7 9K 4] x3t4 9 2 7 T 3t473t4 T 3t4 3 [A 7 B3K 7 9K 4]T P 7 P [A 7 B3K 7 9K 4] 4 4 7 [C 7 D3K 7 9K 4]T [C 7 D3K 7 9K 4] 4 5 B7T P
(17) 6 P B7 7 7 7 3t4 8 9 2 I
2T 1 where 3t4 x T 3t4 7 T 3t4 . Therefore, the requirement that H 3x6 76 t4 2 0 in equation (14) for all 3t4 0 is implied by 3 4 4 4 5
[A 7 B3K 7 9K 4]T P 7 P [A 7 B3K 7 9K 4] 7 [C 7 D3K 7 9K 4]T [C 7 D3K 7 9K 4] B7T P
6 P B7 7 7 7 2 05 8 9 2 I
(18)
DESIGN OF NON-FRAGILE H1 CONTROLLER 231 Using the Schur complement, it is equivalent to 3
[A 7 B3K 7 9K 4]T P
P B7
4 4 7P [A 7 B3K 7 9K 4] 4 5 5
9 2 I 0
5
6 [C 7 D3K 7 9K 4]T 7 7 720 8 0 9I
(19)
and it is further equivalent to 3
3 A 7 B K 4T P 7 P3A 7 B K 4 P B7 4 5 9 2 I 5 5 0 3
6 PBL 1 7 5 0 8F E DL
6 3C 7 DK 4T 7 0 8 9I
3
6 P B L 2 1 0 0 7 5 0 8 F E DL
T
2 0 0
2 05
(20)
By using Lemma 1, the existence of inequality (20) for 1 0 is implied by 3
3A 7 B K 4T P 7 P3A 7 B K 4
4 4 4 4 4 5
7 P B L L T B T P 7 91 E T E
6 P B7
5
9 2 I
5
5
3C 7 DK 4T 7 P B L L T D T 7 7 7 7 2 05 7 0 8
(21)
9I 7 DL L T D T
Using the Schur complement, it is further equivalent to 3 4 4 4 4 4 4 4 5
3A 7 B K 4T P 7 P3A 7 B K 4
P B7
3C 7 DK 4T P B L
5
9 2 I
0
0
0
5
5
9I
DL
0
5
5
5
9 I
0
5
5
5
5
9 I
Pre- and post-multiplying equation (22) by respectively diag transpose, we obtain 3 4 4 4 4 4 4 4 5
X6
I6
ET
I6
X AT 7 AX 7 Y T B T 7 BY
B7
XC T 7 Y T D T
BL
X ET
5
9 2 I
0
0
0
5
5
9I
DL
0
5
5
5
9 I
0
5
5
5
5
9 I
6 7 7 7 7 7 2 05 7 7 8
I6
I
(22)
and its
6 7 7 7 7 720 7 7 8
(23)
232 H. DU ET AL. where X : P 91 and Y K X . This implies that equation (23) is satisfied, and hence H 3x6 76 t4 2 0 for all 3t4 0. Therefore, we can summarize the above development on the non-fragile H1 controller design procedure for a suspension system in the following theorem. Theorem 1. Consider suspension system (7), a non-fragile H1 state-feedback control gain can be found such that the closed-loop system is asymptotically stable with disturbance attenuation for all admissible controller gain uncertainties described by equations (11) and (12), if there exist matrices X 1 0, Y , and scalar 1 0, satisfying LMI (23). Moreover, a desired control gain is given by K Y X 91 .
5. DESIGN RESULTS AND PERFORMANCE EVALUATION Now we apply the proposed approach to design a non-fragile suspension controller based on the quarter-car model described in Section 3. The quarter-car model parameters have the following values (Tam and Wen, 1996): ms cs
50455 kg6
m u 62 kg6
400 Ns m91 6
ks 136 100 N m91 6
kt 2526 000 N m91 5
For later comparison, we design a state-feedback H1 controller to system (7) without fragility consideration. In other words, controller gain variations are not taken into account. This controller is referred to as a fragile controller here. The obtained controller gain matrix is K 103
1
9054047 9758240 9456599 050299
2
with 3K 3 951156 103 , and the achieved H1 -norm of the closed-loop transfer function matrix from 7 to z is 0 758325. The closed-loop frequency responses from disturbance to sprung mass acceleration, suspension deflection, and tire deflection are depicted in Figures 2(a), 3(a), and 4(a), respectively. It can be seen from these figures that the H1 controller gives a closed-loop which satisfies the three different performance criteria. Both the sprung mass acceleration and the suspension deflection are better than the uncontrolled suspension, especially in the range of sprung mass resonance, apart from a slight deterioration of the tire deflection between the sprung mass resonance and unsprung mass resonance. To test the fragility of the controller, we construct the minimum norm destabilizing perturbation on the controller gain by computing the real stability radius (Qiu et al., 1995). Such a controller perturbation is given by 9K
1
659148 9658961 152068 944356770
2
with 39K 3 44357861, which is 4587% of 3K 3. Then, we consider the controller gain variations defined by L [16 16 16 1]6 E
diag316 16 16 146 and 250 in equation (11). Using the proposed method in Section 4,
DESIGN OF NON-FRAGILE H1 CONTROLLER 233
123456 8 5 65 5
652 2 5 5 2 45 6 543
665 28
234 H. DU ET AL. we set 16 and 32 in equation (8), 12 in equation (23), and we obtain the following non-fragile controller constructed based on K Y X 91 where X 1 0 and Y satisfy equation (23): K 104
1
052275 158020 9054338 051181
2
with 3K 3 158711 104 . The achieved H1 -norm of the closed-loop transfer function matrix from 7 to z is 0 758470, which has no significant difference from the performance using the fragile controller. The closed-loop frequency responses from disturbance to sprung mass acceleration, suspension deflection, and tire deflection are depicted in Figures 2(b), 3(b), and 4(b), respectively. It can be seen from these figures that the performance of the sprung mass acceleration, suspension deflection, and tire deflection are better than the uncontrolled suspension system, except that the sprung mass acceleration deteriorates a little between the sprung mass resonance and unsprung mass resonance. However, the effect of suspension deflection and tire deflection in unsprung mass resonance is significant. To test the fragility of the controller, we also construct the worst perturbation to the controller gain according to (Qiu et al., 1995). The minimum norm destabilizing controller perturbation is given by 9K 103
1
050081 9050080 050208 9156406
2
with 39K 3 156403 103 5 Thus, the required destabilizing norm is much larger than previous one and is nearly 8577% of 3K 3, and hence it has better non-fragility characteristics. To further validate the results, we compare the two controllers with a road bump disturbance input. We generate 10 sets of randomly generated perturbations to the two controllers, respectively. The time-domain simulation results for the system with fragile controller and non-fragile controller are shown in Figures 5(a), 6(a), 7(a), and Figures 5(b), 6(b), 7(b), respectively. It is observed that the fragile controller will give a destabilizing effect under certain perturbations, while the non-fragile controller maintains its performance throughout. The frequency responses from disturbance to sprung mass acceleration, suspension deflection, and tire deflection for the non-fragile controller are shown in Figures 8, 9, and 10, respectively. Notice that the perturbed controllers not only maintained closed-loop stability, their performances are also acceptable. Different non-fragile controllers can be obtained by varying in L. In Table 1, a number of values of are used to obtain different non-fragile controllers with different corresponding H1 performance with 10 and 20 in equation (8), and 12 for all instances except 120 for 2000 and 1200 for 5000 in equation (23). It can be seen from these results that when increases, the obtained destabilizing norm 39K 3 increases with the increases of K . From these results, we can conclude that when gain variations are taken into consideration in the design, the approach presented in this paper gives controllers that are more non-fragile under the same level of performance.
DESIGN OF NON-FRAGILE H1 CONTROLLER 235
123456 8 5 65 5
652 2 5 5 2 45 6 462 66 28
236 H. DU ET AL.
123456 8 5 65 5
652 2 5 5 2 45 6 256 66 28
DESIGN OF NON-FRAGILE H1 CONTROLLER 237
123456 8 543
665 28
238 H. DU ET AL.
123456 8 462 66 28
DESIGN OF NON-FRAGILE H1 CONTROLLER 239
123456 !8 256 66 28
240 H. DU ET AL.
123456 "8 5 65 5
652 2 5 5 2 45 6 543
665 28
123456 #8 5 65 5
652 2 5 5 2 45 6 462 66 28
DESIGN OF NON-FRAGILE H1 CONTROLLER 241
123456 7$8 5 65 5
652 2 5 5 2 45 6 256 66 28
6 78 4 5% 6 22&23 3 2 5 2656 5 326 5658 0
K
9K
39K 3
50
7.8114
[2569226 94543106 103 93592636 051083]
[94538936 4537676 1549196 952057143]
520.7534
100
7.8128
[1599926 91599076 2573976 964352013]
643.2134
200
7.8375
103
[90512916 5501246 94599856 056693]
103
[0511196 90511106 0500926 9151137]
1.1249 103
500
7.8510
104
[0507576 6588526 90553376 052725]
103
[90503736 0503626 0509916 9352944]
3.2963 103
1000
7.8560
105
[90512246 3579166 90512916 050763]
103
[90505636 0505236 0568226 9859923]
9.0184 103
2000
7.8543
105
[90534536 6517966 90526176 051029]
104
[90501966 0501806 0511576 9152958]
1.3013 104
5000
8.4498
105
[90542476 7567036 90535746 051423]
104
[90503306 0503026 0516846 9157876]
1.7961 104
103
[91504626 95515466 95517826 052195]
242 H. DU ET AL.
6. CONCLUSIONS In this paper we address the non-fragile H1 controller design problem for active vehicle suspensions. Based on the solvability of an LMI, in this paper we present an approach to design a non-fragile static state-feedback H1 controller using a quarter-car model. The extension of the proposed approach to more complex vehicle models does not present any conceptual difficulty. The performance of the non-fragile controller is validated using numerical simulation. It is also demonstrated that the deterioration of the performance a controller designed without fragility consideration may be unacceptable in practice. Acknowledgments. The work was f inancially supported in part by RGC Grants HKU 7082/97E and 7103/01P.
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DESIGN OF NON-FRAGILE H1 CONTROLLER 243
Vahidi, A. and Eskandarian, A., 2001, “Predictive time-delay control of active suspensions,” Journal of Vibration and Control 7, 1195–1211. Xie, L., de Souza, C., and Fragoso, M., 1991, “H1 filtering for linear periodic systems with parameter uncertainty,” Systems and Control Letters 17(5), 343–350. Yamashita, M., Fujimori, K., Hayakawa, K., and Kimura, H., 1994, “Application of H1 control to active suspension systems,” Automatica 30(11), 1717–1720. Yang, G. and Wang, J., 2003, “Non-fragile H1 output feedback controller design for linear systems,” Journal of Dynamic Systems, Measurement, and Control 125(1), 117–123.
