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Abstract-It is well known that unconstrained infinite-horizon optimal control may be used to construct a stabilizing controller for a nonlinear system. In this note, we show that similar stabilization results may be achieved using unconstrained finite horizon optimal control. The key idea is to approximate the tail of the infinite horizon cost-to-go using, as terminal cost, an appropriate control Lyapunov function. Roughly speaking, the terminal control Lyapunov function (CLF) should provide an (incremental) upper bound on the cost. In this fashion, important stability characteristics may be retained without the use of terminal constraints such as those employed by a number of other researchers. The absence of constraints allows a significant speedup in computation. Furthermore, it is shown that in order to guarantee stability, it suffices to satisfy an improvement property, thereby relaxing the requirement that truly optimal trajectories be found. We provide a complete analysis of the stability and region of attraction/operation properties of receding horizon control strategies that utilize finite horizon approximations in the proposed class. It is shown that the guaranteed region of operation contains that of the CLF controller and may be made as large as desired by increasing the optimization horizon (restricted, of course, to the infinite horizon domain). Moreover, it is easily seen that both CLF and infinite-horizon optimal control approaches are limiting cases of our receding horizon strategy.The key results are illustrated using a familiar example, the inverted pendulum, where significant improvements in guaranteed region of operation and cost are noted.Index Terms-Control Lyapunov functions (CLFs), model predictive control, nonlinear control design, optimal control, receding horizon control.

Abstract-We study the stability and region of attraction properties of a family of receding horizon schemes for nonlinear systems. Using Dini's theorem on the uniform convergence of functions, we show that there is always a finite horizon for which the corresponding receding horizon scheme is stabilizing without the use of a terminal cost or terminal constraints. After showing that optimal infinite horizon trajectories possess a uniform convergence property, we show that exponential stability may also be obtained with a sufficient horizon when an upper bound on the infinite horizon cost is used as terminal cost. Combining these important cases together with a sandwiching argument, we are able to conclude that exponential stability is obtained for unconstrained receding horizon schemes with a general nonnegative terminal cost for sufficiently long horizons. Region of attraction estimates are also included in each of the results.

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Model reduction for robust control: A Schur relative-error method," in Proc. Amer. Contr.Abstract-We study approximate input-output linearization of nonlinear systems which fail to have a well defined relative degree. For such systems, we provide a method for constructing approximate systems that are input-output linearizable. The analysis presented in this note is motivated through its application to a common undergraduate control laboratory experiment-the ball and beam system-where it is shown to be more effective for trajectory tracking than the standard Jacobian linearization. I. INTRODUCTIONThe conditions for feedback linearization of nonlinear systems are restrictive and it is of practical interest to investigate situations where these conditions fail but do so only slightly. Continuing the work of Krener [ 11, who gave conditions for approximate full state linearization of nonlinear multiinput systems, we discuss approximate input-output linearization of single-input single-output systems which fail to have relative degree. In contrast to extended linearization [2] and pseudolinearization [3], our technique does not approximate the system by a linear system or family of linear systems but rather by a single nonlinear system that is input-output linearizable.Our approach to the tracking problem differs from the recent work of Isidori and Byrnes [4] who provide (fragile) conditions under which one can exactly track the output of a finite-dimensional exosystem. In contrast, our goal is to provide approximate tracking of a large class of output signals that is valid under a wide class of nonlinear perturbations in the system model. In Section 11, we begin with an example drawn from undergraduate control laboratories, the ball and beam experiment, and show how the exact input-output linearization approach may fail. In Section U, we use the same example to motivate our approximate input-output linearization technique and provide a comparison with the standard Jacobian linearization. In Section IV, we define robust relative degree and present a method of approximate input-output linearization for SISO nonlinear systems.

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