Journal of System Design and Dynamics
Vol. 2, No. 2, 2008
Design of Passive Suspension System of Railway Vehicles via Control Theory* Hung Chi NGUYEN**, Akira SONE**, Daisuke IBA** and Arata MASUDA** **Department of Mechanical and System Engineering, Kyoto Institute of Technology Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan E-mail:
[email protected]
Abstract This study deals with the design of passive suspension system of railway vehicles. The proposed model has six-degree-of-freedom and can be designed via control theory. Since the classical fixed point theory is no longer applicable to the design of passive suspension system of railway vehicle, many methods have been developed to replace it. In this paper, By utilizing feedback control theories the problem is examined from the view of feedback control problem. Consequently, the “feedback gain” is a decentralized matrix composed of the suspension parameters to be optimized. Since minimizing H∞ norm of the system implies suppressing the peaks of the magnitude of frequency response of the system, parameters optimization of passive suspension systems become a H∞ static output feedback problems, and it is transformed to Bilinear-Matrix-Inequality (BMI) problem. One of the easiest methods to solve this BMI problem is alternative algorithm, which is derived from iterative schemes of alternation between analysis and synthesis via Linear Matrix Inequalities (LMIs). Finally, numerical simulations for the passive suspension system designed by control theory and fixed point theory will show the comparison of performance of each method. Key words: Passive Suspension, Alternative Algorithm, BMI Problem, Feedback Control, H∞ Optimization
1. Introduction
*Received 6 Nov., 2006 (No. 06-0193) [DOI: 10.1299/jsdd.2.518]
Nowadays, with the development of electronics and microprocessors, commercial railway vehicles with active suspensions or semi-active suspensions have become available. Although active suspensions and semi-active suspensions system can improve the ride comfort and track handling beyond that attainable by passive suspensions, the cost, weight, and the power requirement of active and semi-active suspensions remain prohibitive. Therefore, passive suspension systems remain dominant in the marketplace because they are simple, reliable, and inexpensive. Passive suspension system design was formerly used fixed-points theory (1),(2),(3). This design method is based on the existence of 3 fixed-points in frequency response curves of system. By choosing the optimal positions of these 3 points, designers are able to design the optimal parameters. But this method could not be applied for complex systems that have more than 2 degree-of-freedoms and moreover the results of design usually depend on the designer’s experiences. In the recent decades, there are numerous optimization methods that have been proposed to replace the classical method due to some problems. Some researchers have utilized the LQG optimal control theory for the design of passive mechanical systems (4),(5),(6),(7),(8) . L. Zuo, et al. in (4) and D. Iba, et al. in (3) utilized the H2 and H∞ norm
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optimization to design passive mechanical suspension (4),(5), MDOF tuned mass damper (3),(6),(7) and in vibration control of nuclear components (8) etc. This paper applies the H∞ norm optimization to design optimal parameters of springs and dampers of railway vehicles. The purpose of this optimization is: suppressing the peaks of the magnitude of frequency response curves of our system at resonance frequency. Because minimizing H∞ norm of the system implies suppressing the peaks of the magnitude of the frequency response of the system. Therefore, parameters optimization of passive suspension system becomes H∞ static output feedback problems. In other words, the passive suspension design is equivalent to design feedback gain of a controller with structured static output. This feedback gains is generated by the springs and damping elements which need to be designed. The design problem is transformed to the Bilinear Matrix Inequality (BMI) problem, which can be solved via the alternative minimization algorithm. The problem will be examined from the view of feedback control. Therefore, many difficult problems in passive suspension systems will become tractable in the framework of structural control. Applying feedback control theory in passive suspension design is one of solutions to avoid foregoing limitations of conventional method. Finally, we use numerical simulations to illustrate the body frequency response and body displacement of the two methods and compare the performance obtained by each.
2. Problem Formulations The design problem of the Body proposed six-degree-of-freedom m2 , J2 x3 x4 system has multiple masses m0, m1, l m2 and they are connected in series by springs k0, k1, k2 and dampers c0, c2 c2 k2 k2 Bogie c1, c2 as illustrated in Fig. 1. xtr1 and x2 xtr2 are displacements of rail tracks m1, J1 x1 excitations, l is standard distance of c1 k1 k1 c1 rail tracks, In this model, the motion Wheel x01 x02 of train body, bogie and train wheel c0 m0 , J0 c0 can be simultaneously translational k0 k0 x xtr1 tr2 and rotational in two-dimensional space x01, x02, x1, x2, x3, and x4 are Fig. 1. Six-DOF suspension system model of train translational motions at wheels, sides of bogie and sides of body respectively. J0, J1 and J2 are inertia moments of wheel-axle, bogie and body respectively. By examining the design problem from the view of feedback control, the springs feedback the relative displacements locally, the damping elements feedback the relative velocities locally, and the control forces are generated by springs and dampers which need to be designed (4). From this point, the equations of motion can be written as: MX + KX + CX = Eu + Fp f + Fv f (1) where: M, K, C are positive definite equivalent mass, stiffness and damping matrices respectively, u is control force, f and f are vectors of displacement excitations and velocity excitations as:
k0 0 0 K= 0 0 0
0 0 0 0 0
0 0 0 0 0
k0 0 0 0
c0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
c0 0 0 0 0 ; C = 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
k0 0 0 0 0 ; F = p 0 0 0 0 0 0
0
c0 0 0 c k0 0 0 0 0 ; Fv = ; 0 0 0 0 0 0 0 0 0
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l2 J m + 0 0 4 l2 l2 − J 0 + m0 4 l2 0 M = 0 0 0
l2 4
0
0
0
l2 4
0
0
0
− J 0 + m0 l2 J 0 + m0 l2
J 1 + m1
0
l2
l2 4
− J 1 + m1
l2 4
J 1 + m1
− J 1 + m1
0
l2
0
l2
l2
0
0
l2 4
0
l2 4
0 J 2 + m2
0
0
l2
l2 4
− J 2 + m2
0
0 0 0 ; 0 l2 − J 2 + m2 4 l2 l2 J 2 + m2 4 l2
l2
l2 4
−1 0 1 0 0 0 0 0 −1 0 1 0 0 0 0 0 0 −1 0 1 0 0 0 0 0 −1 0 1 0 0 0 0 1 0 −1 0 −1 0 1 0 1 0 −1 0 −1 0 1 0 E= ; 0 1 0 −1 0 −1 0 1 0 1 0 −1 0 −1 0 1 0 0 0 0 1 0 −1 0 0 0 0 0 1 0 −1 0 0 0 0 0 0 1 0 −1 0 0 0 0 0 1 0 −1 f = [ xtr1 Y = [ x01
xtr 2 ] ; u = K sY ; X = [ x01 T
x02
x1
x2
x1
x2
x3
x02
x1
x2
x4
x01
x02
x3 x1
x4 ] ; T
x2
x1
x2
x3
T x4 ] ;
By defining the state variable as: T −1 x s = X X − M Fv f The equation of motion can be written in state-space form as:
(2)
x s = Ax s + B1 f + B2 u
M Fv I 0 ; B1 = −1 ; B2 where A = −1 −1 −1 −M K −M C M ( Fp − CM Fv ) −1
(3)
0
; M E
=
−1
Based on the geometry of the train model, we write the vector of “measured” outputs, the relative displacements and velocities at the suspension connections as a linear combination of the states and inputs, that is: Y = C2 x s + D21 f (4) We can write the vertical displacement of train body (x3 and x4) as an output vector z, which can be expressed in the form: z = [ x3 where: C1 = GC2 ; D11 = GD21; G =
x4 ] = C1 x s + D11 f T
(5)
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
The forces generated by the suspension springs and dampers are determined from Y
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according to: u = K sY
where the “feedback gain” Ks is a decentralized matrix
(block-diagonal) composed of the suspension parameters to be optimized. In this problem Ks is shown as: Ks = diag ( k1
k1
k1
k1
k2
k2
k2
k2
c1
c1
c1
c1
c2
c2
c2
c2 )
(6)
Equations (3), (4) and (5) cast the design of the suspension system of train as a decentralized control problem, as indicated by the diagram (9) shown in Fig. 2. Based on this formulation, we use decentralized control techniques H∞ to directly optimize the stiffness and damping coefficients of springs and dampers to achieve performance (measured by z) under the disturbance of f. The goal of solving this problem is to determine the feedback law: u = K sY (7) The feedback gain Ks is a decentralized matrix composed of the parameters to be designed, and all parameters of springs and dampers which need to be designed are designable by determining Ks. D11
z
D21
f B1 u
C1
1 s
B2
xs
Y
C2
A Ks Fig. 2. Block diagram of feedback control system
3. H∞ Control Theory and H∞ Control Problem Based on BMI The H∞ control problem is defined as follows (10). Definition 1: Given a scalar γ > 0. The controller Ks is a “H∞ controller”, if two following conditions are met: • The closed loop system is asymptotically stable. •
Where
Gzf Gzf
∞
∞
