Performance enhancement stabilizing controllers design environment off-line controller design iterated and nested (S, Q) design direct adaptive-Q control indirect (S, Q) adaptive control adaptive-Q application to non-linear systems real-time implementation laboratory case studies.

High Performance Control

In this paper we first study the normalized right and left coprime factorizations of a nonlinear system from a state-space point of view. In order to do so, we introduce the notion of inner and co-inner nonlinear systems. Secondly we deal with balancing for unstable nonlinear systems. For linear systems there are several ways to approach this problem, and we generalize the ideas of balancing the normalized coprime factorization and of LQG balancing to the nonlinear case. LQG balancing can be seen as measuring the influence of a state component on a certain future and past energy function. We extend this interpretation to the nonlinear case. Furthermore we use the normalized coprime factorization to give another way of balancing the unstable nonlinear system. We give the relation between these two methods of balancing, which is similar to the relation in the linear case.

Normalized coprime factorizations and balancing for unstable nonlinear systems

In this paper a general approach is taken to yield a characterization of the class of stable plant controller pairs which is a generalization of the Youla parameterization for linear systems. This is based on the idea of representing the input-output pairs of the plant and controller as elements of the kernel of some related operator which is denoted the kernel representation of the system. It is demonstrated that in some sense the kernel representation is a generalization of the left coprime factorization of a general nonlinear system. Results giving one method of deriving a kernel representation for a nonlinear plant with a general state-space description are presented.

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The class of stabilizing nonlinear plant controller pairs

In this paper, we propose an iterative method for numerical solutions of unconstrained nonlinear optimal control problems, which is essentially based on the closed loop solutions of the linear quadratic optimal control problems. We also prove that accumulation points generated by the algorithm, if they exist, satisfy the weak necessary conditions for optimality. Moreover, we illustrate the numerical effectiveness of the present algorithm through five examples by simulation.

A Riccati-equation-based algorithm for nonlinear optimal control problems

In this paper, we consider the problem of generalizing elements of linear coprime factorization theory to a nonlinear context. The idea is to work with a suitably wide class of nonlinear systems to cover many practical situations, yet not cope with so broad a class as to disallow useful generalizations to the linear results. In particular, we work with nonlinear systems characterized in terms of (possibly time-varying) state-dependent matrices A(x), B(x), C(x), D(x) and an initial state x0. (This class clearly does contain the class of finite-dimensional linear (time-varying) systems.) We achieve first right coprime factorizations for idealized situations. To achieve stable left factorizations we specialize to the case where the matrices are output-dependent. Alternatively, we work with systems, perhaps augmented by a direct feedthrough term, where the input is reconstructible from the output. For nonlinear feedback control systems, with plant and controller having stable left factorizations, then under appropriate regularity-conditions earlier results have allowed the generation of the class of stabilizing controllers for a system in terms of an arbitrary stable system (parameter). Plant uncertainties, including unknown initial conditions are modelled by means of a Yula–Kucera-type parametrization approach developed for nonlinear systems. Certain robust stabilization results are also shown, and simulations demonstrate the regulation of nonlinear plants using the techniques developed. All the results are presented in such a way that specialization for the case of linear systems is immediate.

Coprime factorization over a class of nonlinear systems

Optimal control strategies for both non-linear and linear plants and indices are notoriously sensitive to modelling errors and external noise disturbances. In this paper a general framework to enhance robustness of an optimal control law is presented, with emphasis on the non-linear case. The framework allows a blending of off-line non-linear optimal control, on-line linear robust feedback control for regulation about the optimal trajectory and on-line adaptive techniques to enhance performance/robustness. The adaptive-Q techniques are those developed in previous work based on the Youla-Kucera parametrization for the class of all stabilizing two-degree-of-freedom controllers. Some general fundamental stability properties are developed which are new, at least for the non-linear plant and linear robust controller case. Also, performance enhancement results in the presence of unmodelled linear dynamics based on an averaging analysis are reviewed. A convergence analysis based on averaging theory appears possible in principle for any specific non-linear system but is beyond the scope of the present paper. Certain model reference adaptive control algorithms come out as special cases. A non-linear optimal control problem is studied to illustrate the efficacy of the techniques, and the possibility of further performance enhancement based on functional learning is noted.

Enhancing optimal controllers via techniques from robust and adaptive control

A general framework to enhance the robustness of an optimal control law is presented, with emphasis on the nonlinear case. The framework allows a blending of offline nonlinear optimal control, online linear robust feedback control for regulation about the optimal trajectory, and online adaptive techniques to enhance performance/robustness. Some general fundamental stability properties are developed for the nonlinear plant and linear robust controller case. Performance enhancement results in the presence of unmodeled linear dynamics based on an averaging analysis. A convergence analysis based on averaging theory appears possible in principle for any specific nonlinear system. Certain model-reference adaptive control algorithms come out as special cases. A nonlinear optimal control problem is studied to illustrate the efficacy of the techniques, and the possibility of further performance enhancement based on functional learning is noted. >

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The authors demonstrate via averaging theory an approach whereby any stable linear system can be approximated by a simple periodic-structure system. The control of continuous-time, linear, time-invariant plants via a periodic structure control scheme is proposed. It is established that for continuous-time minimal plants it is possible to design periodic-structure stabilizing first-order controllers which asymptotically approach the performance of an nth-order stabilizing time-invariant controller, such as an optimal (LQG) controller, in the limit as the switching rate increases. The proposed controllers suffer only a small loss of performance compared with the nth-order controller, are attractive from a computational point of view, and may be implemented in either discrete or continuous time. The question of achieving a low-order robust variable-structure controller for a high-order uncertain plant is addressed. Simulation results are shown which demonstrate the efficacy of the periodic structures proposed. >

Periodic structure controller design

A method of control is proposed whereby a high order plant is controlled at each time instant by one of a set of low order controllers, each designed to control one part of a partial fractions expansion of the plant. The method is motivated toward the control of flexible structures, such as large space structures, exhibiting large model order, model uncertainty, and decentralisation. Certain switching algorithms are presented, and simulation results demonstrate the efficacy of such methods. These results perhaps point one way forward for tackling difficult control problems.

L. Irlicht

Papers

Coprime factorization over a class of nonlinear systems

Enhancing optimal controllers via techniques from robust and adaptive control

Enhancing optimal controllers via techniques from robust and adaptive control

Periodic structure controller design

Switched controller design