Nonlinear control via approximate input-output linearization: the ball and beam example

Abstract

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.

Orbital stabilization of a class of underactuated mechanical systems

“…In [7] tracking of a periodic reference was investigated with the result that a tracking controller based on the Jacobi linearization fails for large amplitudes. However, if the nonlinear feedback proposed in this contribution is used, a periodic regime at the desired frequency can be stabilized and much larger amplitudes than in the case of the linear tracking controller can be achieved, cf.…”

ZD, ZG and IOL Controllers and Comparisons for Nonlinear System Output Tracking with DBZ Problem Conquered in Different Relative‐Degree Cases

“…In light of the complexity and uncertainty of nonlinear systems, which widely exist in scientific and engineering fields, many efforts have been devoted to the solving of nonlinear system problems [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. As an important part of nonlinear system problems, the tracking control of nonlinear systems has attracted much attention in recent decades [6,7,9,10,[14][15][16][17].…”

“…Note that, in the rest of this paper, the argument t is omitted sometimes for the presentation convenience (e.g., by writing u(t) as u). For decades, the above tracking control problem has been investigated since it is frequently encountered in practical applications [6][7][8][9][10][11][12][13][14]. For solving this problem, a conventional method is input-output linearization (IOL) [9][10][11][12].…”

“…That is, if the nonlinear system encounters the division by zero (DBZ) problem (or say, singularity problem), the IOL controller cannot be implemented. The DBZ problem is a challenging subject in the area of tracking control [13,14,[19][20][21].…”

Fuzzy-modal control for the ball and beam system

“…The simulation of the "ball and beam" system is of great practical importance, because the transient processes here are similar to the dynamics of the aircraft during takeoff and landing, as well as when moving in the turbulent zone [1,2]. To solve this problem, various approaches can be used, including modal control and fuzzy logic.…”