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In this paper a new approach to the robust asymptotic output tracking of an inverted pendulum is presented. The inverted pendulum is described by a third order nonlinear system. The proposed approach is as follows: First, we determine a static state feedback control law which solves the linear exact model matching (LEMM) problem resulting in an inputoutput (i/o) linearized closed -loop system. Second, in case where there exists a parameter uncertainty in the inverted pendulum model, we design a robust output tracking controller for the perturbed i/o linearized closed -loop system. The LEMM problem is solved using a new LEMM technique. This technique reduces the problem of finding the control law to that of solving a system of first order partial differential equations. Based on these equations, the general analytical solution for the feedback control law is derived. The robust design consists in determining the input signal of the perturbed i/o linearized closedloop system as a function of the state vector based on the construction of an appropriate Lyapunov function. LINTRODUCTIONThe contribution of this paper is two fold: A new technique to the linear exact model matching LEMM problem is presented, which subsequently is applied to an inverted pendulum controlled by a DC motor. Next, using this technique, a new approach to the robust controller design for the asymptotic output tracking of the inverted pendulum is established.The LEMM problem is part of a more general problem, known in the literature as the linearization problem. The problem of linearization has great theoretical and practical importance. The linearization problem may be distinguished into two major problems:. This problem consists of determining a state transformation and a control law (static or dynamic) which, when applied to a nonlinear system, modify a part of the input-state representation of the nonlinear system to the input-state representation of a linear system. I/O linearization problem [6] -[lo]. " 5 sproblem consists of determining a control law which, when applied to a nonlinear system, results in a dosed-loop system with linear i/o description. The closed-loop system may either allowed to be an arbitrary linear system [6] -[lo] or to have the Same i/o description with a prespecified linear model [6]. This last case constitutes the LEMM problem. ?he results reported on the LEMM problem are limited to the case of dynamic state feedback, yielding nonminimal controllers [6]. In this paper we present a new technique for the solution of the LEMM problem for a class of nonlinear systems via static state feedback. This technique reduces the problem of finding the desired control law to that of solving a system of first order partial differential equations, called LEMM design equations [ll] and [12]. Based on the LEMM design equations, the neceSSary and sufficient conditions for the problem to have a solution are derived. Furthermore, solving the LEMM design equations, the general form of the desired control law is characterized. The main a...

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The disturbance rejection with simultaneous input-output linearization and decoupling problem of nonsquare nonlinear systems via restricted state feedback is investigated in this paper. The problem is treated on the basis of an algebraic approach whose main feature is that it reduces the determination of the admissible state feedback control laws to the solution of an algebraic and a first order partial differential systems of equations. Verifiable necessary and sufficient conditions of algebraic nature based on these systems of equations are established for the solvability of the aforementioned problem. Moreover, an explicit expression for a special admissible restricted state feedback controller is analytically derived. [S0022-0434(00)02101-8]