Fixed Order H Controller for Quarter Car Active Suspension System

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1 Fixed Order H Controller for Quarter Car Active Suspension System B. Erol, A. Delibaşı Abstract This paper presents an LMI based fixed-order controller design for quarter car active suspension system in order to ensure H performance and the stability in the predefined D-Region. The simulation results and dynamic response of the real system show the effectiveness of the proposed method over improvements on ride comfort. The proposed low-order controller is compared with full-order one in simulation and application phases. The advantages of using low-order controller contrary to high-order one are discussed. Keywords: fixed order control; quarter car active suspension system; regional pole assignment; H control I. INTRODUCTION In order to achieve the desired performance for any dynamic system such as disturbance rejection, maximum overshoot, settling time, etc. we can use the H control with the constraint on the location of the system s poles. Basic structure of H based control theory was formed by Zames [1], and by the developing over LMIs, this kind of convex structured problem is allowed to have solutions [2]. In the standard H control theory, the designed controller has the same degree as the system [3]. As a result of this designing constraint, the degree of the closed loop system is increased twofold. Usage of the high degree system is undesirable due to its fragile structure against any error happening in the system. Furthermore, the usage of the high degree controllers that obtained by using conventional methods, is not an easy work to accomplish in the embedded system. Although conventional methods seem to be feasible in theory especially for low order systems, it may not be easy in real life due to the necessity of the weighted filters located at the performance outputs, so this increases the degree of the system. As a matter of this fact, attraction to the fixed order H controller design has been increased recently. However, the fixed order controller design problem is defined in a non-convex cluster, and at present there is no algorithm to solve the problem globally, so that makes this approach a hard way. In [2] the performance criteria at issue are formulated in terms of LMIs and non-convex rank constraint is added for fixed order controller design. To deal with this non-convexity, usage of alternating projections method [4] and augmented Lagrangian method [5] give reasonable results. In addition to these approaches, for the solution of this problem nonlinear semi-definite programming can be another option [6]. Most of the previous studies on H controller designs are based on the state space representation. Contrary to the state space approaches, polynomial ones are more efficient in designing the controller due to the existence of the controller coefficients in the problem directly. In [7]-[9], they propose an inner LMI approximation technique to reduce the nonconvexity of the fixed order controller to convex cluster in terms of LMIs. These approaches are based on positivity of polynomials and Strictly Positive Realness (SPRness) of the closed loop transfer function. Since the problem presented in convex cluster, one can use interior point algorithm to find suboptimal solution. By using convex inner approximation and regional pole placement showed in [10] fixed order controller is designed for stability problem [7]. The H norm minimization is solved via the inner approximation technique by Henrion, but it includes some geometrical constraints [8]. Yang et al, extended this technique and these geometrical constraints are eliminated, so the stability in the predefined D-Region and H norm minimization are achieved with the fixed-order H controller design [9]. In a vehicle the frequency of the acceleration is important for human health and comfort. The effect of the vibration on humans is specified by the International Standard ISO , according to this report the most sensitive frequency range for humans is 4-8 Hz. In order to focus the acceleration over the specified frequency range a weighted filter is used on the performance output. In this study it is aimed to develop full order H controller with using conventional approach and fixed order H controller with using polynomial approach that minimizes the infinity norm of the transfer function between performance output and disturbance input, in addition to this, locates the closed loop poles in the predefined stable D-Region for the quarter car active suspension system which model is derived by using system identification methods and show efficiencies of both controllers in simulation and application. The quarter car active suspension system, which is constructed by a project supported by The Scientific and Technological Research Council of Turkey (TUBiTAK) under Grant 108E089, is used to demonstrate the effects of the proposed controller. Authors are with the Department of Control and Automation Engineering, Yıldız Technical University, Esenler, Istanbul, Turkey ( berol@yildiz.edu.tr, adelibas@yildiz.edu.tr).

2 II. PROBLEM FORMULATION Consider a continuous time LTI MIMO system: x = Ax + B u u + B w w y = C y x + D yu u + D yw w z = C z x + D zu u + D zw w (1) where AεR n n, B [B w D zu B u ]εr n m, C [ C z C y ] εr p n, D [ D zw D yw D ] εr p m are the system matrices and x yu R n, w, u, y, and z are the state vector, exogenous input, control input, measured output and performance output respectively. Assuming (A, B u ) is controllable and D yw = 0. The output feedback controller K has the form: B k K ( x K ) = ( A k ) ( x K ) (2) u C k D k u where x K R n k is the state vector and A k, B k, C k, D k are the system matrices of the controller. Figure 1 shows the output feedback control scheme, P is the generalized MIMO plant which is given in (1), and K is the controller. III. FIXED ORDER H CONTROLLER Since standard H controller design gives controller of the same degree as the system, in this section as shown in the Figure 2, it is aimed to find the fixed-order H Controller b K (s)/a K (s) for the system given in the equation (1) by using the convex inner approximation technique which is proposed in [7]-[9]. The core idea of this approach is based on the positivity of polynomials and Strictly Positive Realness (SPRness) of the closed loop transfer function. Figure 2 The Overall System Structure The D-Stability region, which is defined in the Equation (3), can be formed by a single matrix such as: D = {s C [ 1 s ] [ δ 11 δ 12 ] δ 22 δ 12 [ 1 ]} 0, (4) s here is a symmetric matrix, and has one positive and one negative eigenvalues. Standard choices for are; the left half plane, δ 11 = 0, δ 12 = 1, δ 22 = 0, the circle centered at origin with radius r, δ 11 = r 2, δ 12 = 0, δ 22 = 1 or the left side of the vertical strip placed at k, δ 11 = 2k, δ 12 = 1, δ 22 = 0. Figure 1 The Output feedback control scheme. In control theory not only the steady state response, but also the transient response of the system should be taken into account. By putting eigenvalues of the closed loop system in the predefined region, one can deal with this issue. Considering the dynamic response of the system, these regions can be chosen as a circle, conic, vertical strip or intersection of them. In the complex plane a region can be defined in the form of D = {s C δ 11 + δ 12 s + δ 12 s + δ 22 ss < 0} (3) Definition 2.1 (D-Stability): [10] The dynamic system given in (1) is D-Stable if all its poles lie in D-Region. In brief, it is aimed to find an LTI control law u = K(s)y for the system given in (1) such that: Closed loop poles lie in D-Region. The infinity norm of the transfer function between performance output and exogenous disturbance input should be smaller than a positive scalar valued γ, P CL (s) < γ. Let the one-dimensional boundary of the D-Stability region be; D = {s C δ 11 + δ 12 s + δ 12 s + δ 22 ss } = 0. (5) A polynomial is called D-Stable if its roots belong to D. Likewise, a rational function is D-SPR if and only if its real part is positive when evaluated along D. Let us consider three polynomials in monomial basis, c(s) = c 0 + c 1 s + + c n s n, n(s) = n 0 + n 1 s + + n n s n, d(s) = d 0 + d 1 s + + d n s n. c(s) is called central polynomial and it is the main design parameter. n(s) and d(s) are numerator and denominator of the closed loop transfer function, P CL (s) = n(s)/d(s). Coefficient vectors of these polynomials are c = [c 0 c 1 c n], n = [n 0 n 1 n n], d = [d 0 d 1 d n ].

3 Lemma 4.1: For a given D-Stable polynomial c(s), polynomial d(s) is also D-Stable if the rational function c(s)/d(s) is D-SPR. Since, the SPRness of a transfer function is closely related with the stability, this approach takes an important place in control theory. To ensure the SPRness of a rational function, G(s) = c(s)/d(s), two requirements should be satisfied [11], d(s) is Hurwitz, Re{G(jw)} > 0 for all real w. The first requirement means that the roots of d(s) should be in the left half plane. Let us focus on the second requirement; Re c(s) d(s) = 1 2 (c (s) d (s) + c(s) d(s) ) = 1 (s)d(s) + d (s)c(s) 2 (c d ) ε, ε > 0. (6) (s)d(s) From this equation, the positivity condition can be defined as follows: p(s) = d (s)c(s) + c (s)d(s) 2εd (s)d(s) 0 (7) for all s D, and with using coefficient vectors of polynomials, Hermitian matrix P(c) is P(c) = d c + c d 2εd d 0 (8) here 2εd d can be negligible, because ε can be any positive scalar, then we may set it as a very small valued number. In order to reduce the stability problem into LMIs, a projection matrix with dimensions (2n) (n + 1) should be used, S [ S ] =, S (9) [ 0 0 1] then the projected stability matrix can be defined as D(Q) = S T ( Q)S (10) where denotes Kronecker Product. By using the following lemma, one can find the output feedback controller parameters (b k, a k ) that stabilizes the closed loop transfer function, which is shown in Figure 2. Lemma 4.2: [7] Given a D-Stable polynomial c(s) of degree n, polynomial d(s) is also D-Stable if there exists a matrix Q of size n solving the LMI P(c) D(Q) = c T d + d T c S T ( Q)S 0 (11) Let us assume that the polynomial d(s) is affected by an additive norm-bounded uncertainty such as d δ (s) = d(s) + δn(s), δ γ 1 (12) where δ is a real-valued scalar of unstructured uncertainty, whose magnitude is smaller than γ 1. According to the small-gain theorem [3], robust stability of polynomial is equivalent to the H performance constraint n(s) γ (13) d(s) on the closed loop transfer function. Theorem 4.1: [9] For a given D-Stable polynomial c(s) and a positive scalar γ, the transfer function n(s)/d(s) is also D- Stable and ensures the H performance constraint if there exists a symmetric matrix Q = Q T and a scalar β such that [ ct d + d T c βc T c S T ( Q)S n T n βγ2] 0 (14) IV. APPLICATION TO QUARTER CAR ACTIVE SUSPENSION SYSTEM In this section we apply the proposed and the full-order techniques to the quarter car active suspension system which is shown in Figure 3. We show the effectiveness and applicability of the fixed order controller using this prototype. The system model of any dynamic system can be derived in two ways; physical laws based mathematical methods and system identification methods. In system identification there is no need any prior knowledge about system, the only thing to do is that firstly exciting the system with suitable input signals and observe the outputs, then using these inputs and outputs the system model can be derived. Here, Matlab System Identification Toolbox is used to derive the system model. The real time system is excited by 20 normal distributed random sine signals, then the response of this system is collected. By using these inputs and outputs, the different type system models are derived via Matlab System Identification Toolbox. After testing these models on the system, among models the more appropriate one is selected. The transfer functions between outputs and inputs of this model are P 11 (s) = 1174s s s P 12 (s) = 18.51s s s s s s 2747 P 21 (s) = s s s P 22 (s) = (15)

4 In order to define a convex problem, the roots of the central polynomial are chosen in the D-region, around -10.6, so the central polynomial is c(s) = s s s s s s s s (18) The second order H controller is obtained, b k (s) a k (s) = 399s2 3356s s 2, (19) s Figure 3 The Quarter Car Active Suspension System Prototype. by solving the convex optimization problem: minimize γ subject to LMI problem (14) using YALMIP [14] via SeDuMi [15] solver. The second order H controller yields γ = In suspension system design the ride comfort, which indicates how much vibrations caused by road roughness affects passengers comfort, is the first objective to be accomplished. This performance issue is related with the vehicle vertical acceleration. The researches over frequency range of vibration which affects human body shows that the most effective frequency range for human body is 4-8 Hz [12]. In order to focus the acceleration over the specified frequency range a second order weighted filter, which is proposed in [13], 86.51s W(s) = s 2 (16) s is used on the performance output. As mentioned before, the main motivation of this work is to show that the fixed order H controller provides similar performance as full order one. The full order H controller is designed by using the theory given in [10]. In this study, the D-region is constructed by the intersection of two regions, one is the circle centered at the origin with radius 150, and the other is the left side of the vertical strip placed at 0.1. Our goal is to find the second order H controller, that places the closed-loop poles in the predefined region and ensure the H of the transfer function, P CL (s) < γ, between weighted performance output, the vehicle vertical acceleration, and disturbance input, the road roughness. The full order H controller, b k (s) a k (s) = 412.7s s s s s s s s s s s s (17) which ensures γ = Although the full order controller provides the optimal result, it can be hard to implement this on a real system because of its high degree. In this matter of fact, low-order controllers are preferred. Figure 4 Frequency responses of the open loop system (blue solid line), the closed loop system with full-order (red dashed line) and second-order controller (green dotted line). In order to show the effectiveness of the proposed controller, two different road types are used on the system, shown in following figures. The stochastic road profile is generated by normal distributed 20 random sine signals which have different amplitudes, frequencies and phases. Figure 5 Stochastic road profile

5 Figure 6 Bump type road profile. The vehicle body accelerations of the system for simulation and application phases are showed in Figure 7-8 and Figure In the simulation results, one can see that the full-order controller achieves the best performance in all applied controllers as expected. However, it may not so easy to observe the similar scenario in the application results. The effect of the proposed controller can be seen in the application phase, this controller supplies nearly the same improvement to the system as the full order controller. Since the conservatism of the proposed technique is dependent on the central polynomial, the designer can obtain a better dynamic response by changing the location of this polynomial. Figure 8 Simulation result of the open loop system (blue solid line), the closed loop system with full-order (red dashed line) and second-order controller (green dotted line) under the bump type road profile. Figure 9 Performance response of the passive system (blue solid line), the system with full-order (red dashed line) and second-order controller (green dotted line) under the stochastic road profile. Figure 7 Simulation result of the open loop system (blue solid line), the closed loop system with full-order (red dashed line) and second-order controller (green dotted line) under the stochastic road profile. Figure 10 Performance response of the passive system (blue solid line), the system with full-order (red dashed line) and second-order controller (green dotted line) under the bump type road profile.

6 TABLE I. INFINITY NORM COMPARISON popularity in industry contrary to applicable developments on complex controllers. The main conservatism of the proposed method is the choosing the central polynomial. The study, which can provide an algorithm to find a much more suitable central polynomial is seen to be an attractive research topic. Concerning the main purpose of the controllers as H norm minimization of the system located between the performance output and the disturbance input, the efficiency of the controller can be measured by the ratio of these signals two norms. From the Table I one can also observe that the full-order controller ensures lower bound of γ value in simulation results, but in application phase the results exceed this value (γ = 4.68), on the other hand the second order controller yields lower results than (γ = 17.29), in both phases. Unmodeled environmental noise sources and perturbations can arise during operation. We cannot observe the effects of these unwanted factors in the simulation phase. But in the real time application, a simple parameter variation may cause serious results, especially for higher order systems. The same scenario appears in our system as well. From the Table I it is clear that, although in the simulation the full-order controller supplies reasonable better results than the second-order controller, in the application the difference is reduced to an acceptable level. When working over real time vehicle applications, unlike laboratory circumstances, the factors which are caused by noises and disturbances will affect the dynamic response of the system. In this case the low-order controller will act as a less sensible structure. The simulation results of the second-order contrary to the full-order controller are more similar to the real time results. These mainly shows the fragile structure of the high-order controllers for real time application. V. CONCLUSION In this paper, fixed-order controller design is taken into account for quarter car active suspension system. In order to overcome the difficulties on the representation of the fixedorder controller design problem, which is non-convex, the proposed techniques define a convex body in the exact solution set by using LMI regions. Therefore, this technique provides a suboptimal solution for minimizing the infinity norm of the system. By using this technique, a second-order controller is designed, and the effect of this controller is compared with the full-order one and the passive system on two different road types. Although the performances of the controllers are drastically different then each other with respect to simulation results, they represent a similar performance in real time applications. The low order controller s application results are consistent with the simulated ones. Therefore, they have still preserved their REFERENCES [1] G. Zames, "Feedback and Optimal Sensitivity: Model Reference Transformations, Multiplicative Seminorms, and Approximate Inverses."Automatic Control, IEEE Transactions on , 1981 [2] Gahinet, P. ve Apkarian, P., A Linear Matrix Inequality Approach to H_ Control, Int. J. Robust and Nonlinear Control, 4: (1994) [3] Zhou, K., Doyle, J.C. ve Glover, K.,. Robust and Optimal Control, Upper Saddle River, Printice-Hall, New Jersey, (1996). [4] K. M. Grigoriadis and R. E. Skelton, Low order control designfor LMI problems using alternating projection methods,, Automatica, vol. 32, no. 8, pp , 1996 [5] B. Fares, P. Apkarian, and D. Noll, An augmented Lagrangian method for a class of LMI-constrained problems in robust control theory, International Journal of Control, vol. 74, no. 4, pp , [6] P. Apkarian, D. Noll, J. B. Thevenet, and H. D. Tuan, A spectral quadratic-sdp method with applications to fixed-order H2 and H1 synthesis, European Journal of Control, vol. 10, no. 6, pp , [7] Henrion, D., Sebek, M. ve Kucera, V.. Positive Polynomials and Robust Stabilization with Fixed-Order Controllers, IEEE Trans. Automatic Control, 48 (7): , (2003). [8] Henrion, D., LMI Optimization for Fixed-Order H_ Controller Design, In Proceedings of the IEEE Conference on Decision and Control, 5: , (2003). [9] Yang, F., Gani, M. ve Henrion, D., Fixed Order Robust H_ Controller Design with Regional Pole Assignment, IEEE Trans. Automatic Control, 52 (10): , (2007). [10] Chilali, Mahmoud, and Pascal Gahinet. "H Design with Pole Placement Constraints: an LMI Approach." Automatic Control, IEEE Transactions on 41.3: , (1996). [11] Henrion, Didier. "Linear Matrix Inequalities for Robust Strictly Positive Real Design." Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on 49.7: , (2002). [12] International Organization for Standardization, (1997) Mechanical Vibration and Shock Evaluation of Human Exposure to Whole Body Vibration part 1: general requirements, ISO :1997. [13] Zuo, L., and S. A. Nayfeh. (2003) "Low order continuous-time filters for approximation of the ISO human vibration sensitivity weightings." Journal of sound and vibration 265.2: [14] Löfberg, J., (2004). YALMIP: A Toolbox for Modeling and Optimization in Matlab, Proceedings of the IEEE Symposium on Computer-Aided Control System Design (CACSD), Taipei, Taiwan, [15] Sturm, J.F., (1999). Using SeDuMi 1.02, a MATLAB Toolbox for Optimization Over Symmetric Cones, Optimization Methods and Software, (3):

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