Intelligent controllers for dynamic systems with state and input constraints and subjected to model uncertainties. - Leizer Schnitman (2001)
This thesis deals with the control of nonlinear systems governed by nonlinear state equations of the form =f(x)+g(x)u with state and input constraints, and subjected to model uncertainties. Classical linearization methods lead to adequate results when the model is accurate and the reference signals are well-conditioned and have low amplitude. However, this may not be the case for inputs such as large steps or when uncertainties are not to be neglected. The problem of bounds on control and state variables is tackled by using the reference governor approach which guarantees that the constraints are satisfied for a general class of input commands while stability is also assured. The main idea behind the reference governor is to manage the reference signal which is supplied to the inner loop controller in such a way that violation of the constraints is avoided. Model uncertainties are treated by using fuzzy logic adaptation features. Fuzzy blocks are used in order to approximate the nonlinear functions f(x) and g(x) that appear in the state equations. This thesis also describes some theoretical difficulties encountered in some of previous work in the field, especially when fuzzy or neural estimators are used for the estimation of the nonlinear functions. Despite the apparent success (because the tracking error seems to be tending to zero), some subtle details related to the proof of stability are pointed out. Basically, the difficulties may stem from the fact that the Lyapunov method is used assuming that the tracking error converges to zero. Moreover, an external control signal us which acts like a sliding mode control is used to guarantee stability and it assumes that the function estimates converge to their real values. Here, it is shown that making use of this previous approach the tracking error does not tend to zero when t. An explicit equation for computing the steady-state error is presented. Moreover, despite many propositions for the introduction of an external control signal us to force stability, it is verified that its use is not necessary to assure the tracking error stability if some free parameters are properly initialized. Also, it is noted that the tracking error is bounded and can be reduced through a convenient choice of design parameters. Finally it is shown that in the method presented in the previous work, the estimates of f(x) and g(x) are not required to be accurate to achieve stability. In order to solve the detected theoretical problems, this thesis proposes a new approach for the control of the considered class of nonlinear systems. It also uses fuzzy blocks as estimators for f(x) and g(x). However, the proposed structure and the control and adaptation laws, are then applied so that the tracking error converges to zero and the function estimates converge to their real values.In order to highlight the controllers properties, a nonlinear and open-loop unstable magnetic levitation system is used as an example.