Optimal stabilizing controllers for bilinear systems

Abstract

A new method for the design of stabilizing controllers for multivariable bilinear systems is presented. The design method is based on a Liapunov stability theorem and the solution of a Liapunov equation. The resulting non-linear control globally asymptotically stabilizes the closed-loop system and minimizes a generalized quadratic performance index. The effect of the weighting matrices on the closed-loop response is analysed. Simulation results demonstrate that the new design technique provides better control than linear feedback. Moreover, the linear feedback can destabilize the closed-loop system depending on the initial conditions, whereas the new control strategy results in global asymptotic stability. I. IntroductionThe theory of bilinear systems has undoubtedly reached a level of development that should permit its utilization in controlling physical processes. The papers by Bruni et a!. (1974) and Mohler and Kolodziej (1980) provide excellent reviews of the available results on the structural properties, controllability and observability of bilinear systems.While the structural properties of bilinear systems have been studied at length, the general regulator problem remains unsolved, and it is only in some special cases that studies on the stabilization of these systems have led to well-defined controller design methods. From a practical point of view there is a need for application-oriented controller design techniques for bilinear systems. In this perspective, Derese and Noldus (1980) proposed a controller design algorithm based on Liapunov stability theory. However, the controller arising from their technique does not use the 'bilinearity matrices' (matrices Ai in ( I ) below) which are the distinguishing feature of a bilinear system.In this paper, a brief overview of the available results on bilinear systems control is presented and a new controller design method is introduced. This method is based on LaSalle's stability theorem (LaSalle 1962) and yields a controller which encompasses the system bilinear characteristics, globally stabilizes the controlled system, and is shown to be optimal with respect to a performance criterion which is quadratic-inthe-state and quadratic-in-the-control.

Suboptimal control for the bilinear quadratic regulator problem: application to the activated sludge process

“…However, in this section, a Taylor series expansion around a fixed point in the state-space of the elements in the matrix in (8) is applied, which results in a transformation of the generalized power series into a smaller matrix dimension series expansion compared to (9). To see this, let a power-series expansion using the Kronecker product be defined as (10) where and the dimension of is . Remark 1: It is interesting to compare the power series expansion (10) with the sometimes used Kronecker product notation for the generalized power series expansion (11) Note that, eliminating the redundancy in the Kronecker product vectors by deleting the repeated entries, inevitably results in the reduced Kronecker product and, thus, the generalized power series expansion (9).…”

“…In most cases, it is not possible to formulate explicit expressions for the optimal feedback control law. Some of the obtained optimal control laws rely on a quadratic cost function, modified by incorporation of nonnegative state-dependent penalizing functions [7]- [9], [10]. An iterative method for the solution of the finite-time optimization problem is presented in [11].…”

“…In [8], both the finite-time and the infinite-time case are treated. The infinite-time case has also been investigated in [9] and [10]. The problem of finding stabilizing feedback controllers for bilinear systems has also been discussed by a number of authors.…”

“…Manuscript Some of the results in the literature are partly applicable on the control law developed here. For instance, stabilizing feedback control laws which are quadratic in the states have been treated in [9], [10], and [15]- [17]. In this brief, bilinear control systems with quadratic cost functions for the infinite-time case are considered.…”

Analytical prediction and experimental validation for longitudinal control of cable oscillations

“…The feedback bilinearization, resulting in equation (13), permits the design of the control term,ū c , such that the static equilibrium configuration is globally stable; following the work [13] a suitable expression forū c can be given, similar to equation (10).…”