Sliding Mode Observation and Control for Semiactive Vehicle Suspensions Ravindra K. Dixit1 and Gregory D. Buckner2 1Emmeskay Inc., Plymouth, MI 48187, USA, [email protected] 2Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695, USA, [email protected] Keywords: robust, nonlinear, sliding mode observer, sliding mode controller, semiactive vehicle suspension, MR damper. Abstract: This paper investigates the application of robust, nonlinear observation and control strategies, namely sliding mode observation and control (SMOC), to semiactive vehicle suspensions using a model reference approach. The vehicle suspension models include realistic nonlinearities in the spring and magnetorheological (MR) damper elements, and the nonlinear reference models incorporate skyhook damping. Since full state measurement is difficult to achieve in practice, a sliding mode observer (SMO) that requires only suspension deflection as a measured input is developed. The performance and robustness of sliding mode control (SMC), SMO, and SMOC is demonstrated through comprehensive computer simulations and compared to popular alternatives. The results of these simulations reveal the benefits of sliding mode observation and control for improved ride quality, and should be directly transferable to commercial semiactive vehicle suspension implementations. 1 INTRODUCTION Active vehicle suspension systems were introduced in the early 1970’s to overcome the drawbacks of passive suspensions, namely the inherent tradeoff between ride quality and handling performance [1,2,5]. Despite their published benefits, these systems remain complex, bulky, and expensive and are not common options on production vehicles. Semiactive suspensions overcome many of these limitations, albeit with a reduction in achievable ride quality and handling performance [1,2], though some researchers have concluded that this reduction is quite small [12]. Semiactive suspensions can be considerably more cost effective, compact, and functionally simple as they require only a variable damper and a few sensors to achieve adequate performance. The recent introduction of commercial magnetorheological (MR) fluid dampers has enabled high- bandwidth, low-power control of suspension damping forces with very few mechanical parts [13,17]. These semiactive components contain suspensions of micron-sized, magnetizable particles in an oil-based fluid. In the presence of magnetic fields, these fluid particles become aligned with the field, dramatically increasing the fluid viscosity and effective damping. Recently, Carerra has introduced MagneShocks™ [13] and Delphi Automotive Systems has introduced Magneride™, both using MR fluid technology to enable semi-active damping adjustments at frequencies up to 1000 Hz. The performance of semiactive suspension systems relies heavily on real-time control strategies. Early research focused primarily on linear techniques, such as optimal control [15,18-20] and skyhook control [16,21-23]. However, vehicle suspensions contain dynamic nonlinearities associated with springs and dampers [4], sliding friction in joints [4,24], and suspension kinematics [25] which significantly affect ride quality and handling performance. Vehicle 2 suspensions are also subjected to parameter variations, like changes in damping and stiffness over extended time periods that adversely affect the robustness of many algorithms. For these reasons, recent semiactive control research has focused more on nonlinear control techniques. Gordon and Best [26] extended previous nonlinear optimal designs to semiactive systems and incorporated dynamic parameter optimization. Hedrick and Sohn [16] linearized the vehicle dynamics about an equilibrium point and applied skyhook control to control a semiactive MacPherson strut suspension. Henry and Zeid [27] derived a sub-optimal nonlinear control law, but did not consider spring and damper nonlinearities. Considerable research effort has concentrated on fuzzy logic, neural networks, and artificial intelligence techniques [14,28,29,30]. The primary advantages of these systems is that complete knowledge of the model dynamics may not be required, and hence nonlinearities may be incorporated easily. The primary drawbacks of such methods lie in the ad-hoc nature of their tuning and the inability to address robustness factors. Sliding Mode Control (SMC) is a highly effective nonlinear and robust control strategy. It is insensitive to unmodeled dynamics and parametric uncertainties and has been used effectively in fields like robotics, aerospace, and automotive systems [31]. Kim and Ro [32] developed SMC for a nonlinear active suspension system and compared it to a self-tuning controller. Results showed that SMC significantly improved robust tracking performance when vehicle parameters changed. Alleyne and Hedrick [33] used SMC for a nonlinear actuator in an active vehicle suspension system. Hedrick et al. [34] demonstrated the effectiveness of SMC for MR semiactive suspension systems using a model following approach. This paper investigates the application of robust, nonlinear observation and control strategies, namely sliding mode observation and control (SMOC), to semiactive vehicle suspensions using a 3 model reference approach. The vehicle suspension model includes realistic nonlinearities in the spring and MR damper elements, and the reference models are nonlinear versions of skyhook damping model. Since full state measurement is difficult to achieve in practice, a sliding mode observer (SMO) that requires only suspension deflection as a measured input is developed. The performance and robustness of sliding mode control (SMC), SMO, and SMOC is demonstrated through comprehensive computer simulations. The results presented quantify tracking errors between the plant and a skyhook reference model as a measure of performance, as skyhook models exhibit desirable ride quality and handling performance in most cases. 2 SYSTEM MODELS This section summarizes the linear and nonlinear models used to describe the vertical vehicle suspension dynamics. 2.1 Quarter-Vehicle Model The linear quarter-vehicle model (Figure 1) used routinely in the analysis and design of vehicle suspension systems [1,2,15,16,20] can be modified to include realistic nonlinearities associated with kinematics, bump-stops, stiction and hardening springs. Here x and x represent the sprung and unsprung mass absolute displacements, respectively, x s u 0 represents the road disturbance, k and k represent suspension and tire stiffness, respectively. The s t suspension damping coefficient is the manipulated variable of the control system, and is thus time varying b(t). 4 The suspension nonlinearities considered in this research include stiction, spring hardening and bump stops. Stiction is associated with Coulomb friction between the piston and cylinder, and occurs whenever the relative sliding velocity is zero and the static friction coefficient is significantly greater than the dynamic friction coefficient (Figure 2) [37,38]. This model incorporates the static friction force D (defined only when the suspension’s relative s velocity is zero) and the dynamic friction force C (for all other velocities): s  f v=0, f <D fst =  D sgna( f ) v=0, fa ≥Ds (1) s a a s  C sgn(v)+b v otherwise s 0 where f represents the stiction force and f is the force applied to the damper. st a Spring hardening and bump stops were incorporated using the polynomial stiffness model (2) formulated by Kim and Ro [32] (obtained by curve fitting measured data from the 1992 Hyundai Elantra front suspension) : ( ) ( ) f =k x −x +k x −x 2 s s1 s u s2 s u (2) ( ) +k x −x 3 s3 s u The nonlinear state equations for the semiactive, quarter-vehicle model are: m DxD =−f − f −b(t)(xD −xD ) s s s st s u (3) m xDD = f + f +b(t)(xD −xD )−k (x −x ) u u s st s u t u 0 5 2.2 MR damper dynamics Damping forces in MR components are controlled by manipulating flux-producing coil currents. This approach is represented in the block diagram of Figure 3. The desired damping coefficient b (t) determined by the suspension control algorithm must be related to a coil control current, d which depends on the nonlinear force-velocity characteristics of the MR damper. The damping relationships used in this research (Figure 4) are based on a generic composite of high performance MR dampers used for vibration isolation and automotive semiactive suspensions [23,40]. The stiction characteristics (Section 2.1) are lumped into this composite MR damper model. The steady-state damping coefficient b (t) is both bounded and proportional to the applied current i(t): ss ( ) ( ) b ≤b t ≤b for 0≤i t ≤i and 0 ss max max (4) ( ) ( ) b t =b +µi t ss 0 Here µ is a constant that linearly scales the damping resulting from the applied coil current i(t). The steady-state current required to achieve the desired damping coefficient b (t) is thus: d ( ) ( ) b t −b it = d 0 (5) µ Although the response times of commercial MR fluids are relatively fast, phase lag is associated with the inductance of the control coils. The dynamic response of a typical MR shock absorber can be modeled as first-order, linear, with a time constant τ of 1-10 ms [17,23,39]: ( ) b b s = ss (6) τs+1 6 The static friction force D is assumed equal to the dynamic friction force C [23,40] which is time- s s varying, bounded, and proportional to the applied current: ( ) C ≤C t ≤C smin s smax (7) ( ) ( ) C t =C +ψi t s smin 3 CONTROLLER DEVELOPMENT High-performance control of semiactive vehicle suspensions is complicated by nonlinearities and uncertainties in the system dynamics and by the need for accurate state information. For these reasons, practical implementations require nonlinear controllers and observers that are robust to uncertainties and disturbances. In this section, a robust sliding mode controller is developed for the nonlinear, semiactive vehicle suspension system (3). Ideally, this controller will be robust to parameter variations (so-called structured uncertainties) and unmodeled dynamics (unstructured uncertainties). The controller is developed using a model reference approach that emulates the performance of the well-known skyhook damping model [21,23,32,34]. Advanced control strategies like model reference sliding mode control require knowledge of the absolute sprung mass velocity and displacement, states that are not readily measurable. This fact is frequently overlooked in simulations, but cannot be ignored in vehicle implementations. In this research, a robust observer is developed to provide state estimation using only measured suspension deflection as an input. 7 3.1 Model Reference Control Model reference control (MRC) is a strategy based on specifying the desired closed-loop performance through the selection of a stable reference model (Figure 5). In MRC, the plant output y is made to track a reference model output y using feedback control u. r Practical challenges to this approach are imposed by physical plant constraints [23,34], minimum phase plant requirements [47], and accurate knowledge of states and inputs that may not be measurable [32,44]. 3.1.1 Skyhook Reference Models An excellent reference model for semiactive vehicle suspension control can be derived from Karnopp's skyhook damping strategy [16,21-23]. An inertially grounded damper (the “skyhook” damper, b ) provides damping proportional to the absolute velocity of the sprung mass. A fourth- sky order realization of this reference model (which assumes road inputs are known) is shown in Figure 6. The nonlinear state equations for this reference model are: m DxD =−f −b (xD −xD )−b xD s sr s s sr ur sky sr (8) m DxD = f +b (xD −xD )−k (x −x ) u ur s s sr ur t ur 0 Here f and f are the nonlinear stiction and hardening spring relations defined in (1) and (2). st Because these suspension nonlinearities significantly affect the achievable performance of semiactive suspensions, and because plant “followability” is an inherent requirement of MRC, this nonlinear skyhook model is a more suitable reference model than linear alternatives. 8 3.2 Sliding mode control Sliding Mode Control (SMC) is a high-performance, robust control strategy for uncertain nonlinear systems. The control law consists of two components: a stabilizing equivalent control law u and eq a performance term u . Model reference SMC is based upon the formulation of a “sliding surface” p of tracking errors, such that perfect tracking is equivalent to remaining on this surface for all time. The performance term is discontinuous over the sliding surface, and governs the reachability and global asymptotic stability of the system. The reference tracking errors e and sliding surface s(e,t) are defined: e=(x −x) (xD −xD) m dn−1xr −dn−1x (9)  r r  dtn−1 dtn−1  s(e,t)=eD+λe where λ is a positive design scalar and x(t) and x(t) represent the states of the reference model and r the plant, respectively. A system remaining on such a sliding surface is said to be in “sliding mode” and has zero tracking error, as the unique solution to s(e,t)=0 is e(t)=0 [45]. This reduces a high-order tracking problem to a first-order regulator problem in s. For the specific case of tracking a skyhook reference model, the sliding surface can be defined: [ ( ) ( )] e= x t −x t sr s (10) s(e,t)=e(cid:1)+λe ( ) where x t is the sprung mass vertical displacement of the skyhook reference model. sr Constructing a control strategy so that the system is in sliding motion is accomplished by formulating a continuous feedback control law u and adding to it a discontinuous performance eq term u : p 9 u=u +u (11) eq p 3.2.1 Equivalent Control law The equivalent control law is derived from the nonlinear state equations (3) and the defined sliding surface (10). Neglecting the MR damper dynamics (6), the quarter-vehicle dynamics can be expressed: DxD =− fs + fst − b0 (xD −xD )− 1 (xD −xD )b(t) s m m s u m s u (12) s s s ( ) ( ) = f x +g x u where u=b(t) is the desired damping coefficient determined by the controller, and: f(x)=− fs + fst − b0 (xD −xD ) m m s u (13) s s g(x)=− 1 (xD −xD ) m s u s ( ) ( ) Since f x and g x are not known precisely (due to parametric uncertainties and unmodeled dynamics), SMC must assume that modeling uncertainties are bounded. The additive plant uncertainties can be bounded by a constant (or known function) F =F(x,xD) [45]: f(x)− ˆf(x)≤F (14) where ˆf(x)and gˆ(x) are uncertain models of f(x) and g(x). The multiplicative plant uncertainties can be bounded by known functions (or constants) g =g (x,xD) and min min g = g (x,xD) [45]: max max 10