Taming Internal Dynamics by Mismatched Ha-Optimized State Observer and

The feedback linearization control scheme is simple but quite effective for control of a single-input single-output (SISO) linear parameter-varying (LPV) plant. When the relative degree of the plant is less than its order, the feedback linearization control scheme generates the internal dynamics, which may or may not be stable. This paper presents the design methodology of a state observer that not only provides the feedback linearization controller with good estimation of state variables of the plant, but also stabilizes the overall closed-loop system. We propose to make the choice of system matrices of the state observer completely open in order to search for such an observer structure. Two application examples, including application to steering control for lateral motion of heavyduty vehicles, are presented to illustrate how the proposed mismatched observer stabilizes the closed-loop system. 1 I n t r o d u c t i o n Linear parameter-varying (LPV) systems are linear timevarying plants whose state-space matrices are fixed functions of a vector of known time-varying parameters, O(t). They can be generally described by state-space equations of the following form: x(t) = A(O(t))x(t) + B(O(t))u(t) y(t) = C(O(t))x(t) + D(O(t))u(t) (1) The simplest gain-scheduled controller design methodology for this class of plants is linear interpolation of locally designed linear controllers. This approach is effective only under the assumption that the scheduling parameter O(t) changes slowly [1]. Researchers in the H ~ robust control field have developed the linearly-interpolated controller that guarantees the stability and Hat norm performances of the closed-loop system for any t rajectory of the parameter O(t) [2]. Those linear interpolation methods essentially "approximate" a nonlinear plant by linear combination of a multiple of local linear models. On the other hand, the feedback linearization control (e.g. [5]) cancels all parameterdependent terms in the closed-loop input-output dynamics without using any linear approximation. It can be seen as a gain-scheduled state feedback controller in the sense that its state feedback gain matr ix is dependent on the scheduling parameters. When the relative degree of the plant is less than its order, the internal dynamics arises in the closed-loop system under the feedback linearization control, which may or may not be stable. This occurs when the zero dynamics of the system is unstable. This paper considers the stabilization of systems with unstable internal dynamics. Conventional methods for stabilization of the internal dynamics include employment of the "outer-loop" linear feedback controller (e.g. [3]). Employment of the "outer-loop" controller makes, however, the entire controller structure much more complicated. Furthermore, the closed-loop performance is likely dominated by the outer-loop controller, not by the inner-loop feedback linearization controller, which sometimes obscures the role of the feedback linearization controller. If the feedback linearization control law includes the state feedback terms and the state vector is not directly available, it is necessary to design a state observer. This paper presents the design methodology of a state observer that not only provides the feedback linearization controller with good estimation of state variables of the plant. but also stabilizes the overall closed-loop system. When the classical Luenberger-type state observer is used, the internal dynamics resulting from the state feedback control law is preserved in the observer state feedback system (the separation theorem for the closed-loop eigenvalues for observer state feedback control [4]). While it may be argued that the feedback linearization scheme should not be used if the zero dynamics is unstable, the use of mismatched observer avoids to make it internal dynamics. We make the choice of state matrices of the observer completely open such that we can search for the structure that works on stabilization of the closed-loop system, as well as state estimation. In the Luenberger observer, only the observer gain matr ix can be tuned to obtain the desired estimation error dynamics. The LMI-based H ~ optimization algorithm can be applied to optimize all of system matrices of the state observer such that it stabilizes the overall closed-loop system. By cooperating with the Hoc loop-shaping technique, the set of system matrices of the observer can be optimized such that it outputs the estimates of the plant state variables with the desired estimation error dynamics. The remainder of this paper is organized as follows. In the next section, the feedback linearization control scheme is outlined. The proposed design methodology of the state observer is presented in Section 3. Section 4 presents two application examples. In the first example, the feedback linearization controller and the proposed mismatched observer were applied to a second order system to illust ra te how the proposed mismatched observer stabilizes 0-7803-5519-9/00 $10.00 © 2000 AACC 7 1 5 the entire closed-loop system. The second example is the steering control problem for lateral control of heavy-duty vehicles (HDVs). Numerical simulations are conducted to show time-domain closed-loop performances of the designed controller and observer. 2 Feedback Linear izat ion Contro l Scheme The feedback linearization control scheme determines the control input such that it cancels all nonlinear terms in the closed-loop input-output dynamics. See e.g. [5] for its general formulation. Here, as an example, consider a SISO second-order dynamic system of the following form: ~(t) + A22(O)c~(t) + A21(O)q(t) = B(O)u(t) y(t) =C(O)q(t) (2) where q(t) is a vector of n state variables, u(t) is a scalar control input, y(t) is a scalar output and 0 is a vector of scheduling parameters. Suppose C(O)B(O) ~ 0 for any 0. Define the control law by 1 u(t) C(O)B(O) {C(0)A21 (O)q(t) + C(0)A22 (0)q(t) k l g ( t ) k2y( t ) k3 y(r)dr + v( t )} (3) where kl, k2 and k3 are constant. The closed-loop inputoutput dynamics becomes /: ~(t) + kly(t) + k2y(t) + k3 y(T)dT = v(t) (4) The integral term in the control law (3) is added to reject static disturbances. Notice that Equation (4) is completely independent of 0. Equation (2) can be rewritten in the standard state space representation by defining the state vector x = [q(t) T 0(t)W] v. This can be combined with the control law (3) to describe the closed-loop dynamics. Notice that there are only three tunable controller parameters, while the closed-loop system is (2n + 1)th order. Therefore, it is impossible to assign all closed-loop poles by the choice of the controller gains. The internal modes that cannot be controlled by the feedback linearization controller are called internal dynamics, which may or may not be stable. 3 M i s m a t c h e d and Hoe-Opt imized State Observer One way to deal with unstable internal dynamics is to employ an "outer-loop" linear controller [3]. The entire closed-loop configuration is shown in Figure 1. A linear time-invariant (LTI) Hat controller is used as an outerloop controller in order to stabilize the entire closed-loop system and enhance its robustness. In this paper, we propose to design the state observer such that it not only provides good estimates of the plant state variables, but also stabilizes the entire closed-loop system. The closed-loop configuration is shown in Figure 2. Advantages of the proposed approach are clear; first, the overall controller structure is much simpler. Furthermore, v