Certainty Equivalence Approach Research Articles

Application of adaptive dual controllers to a DC-motor

A new dual pole-placement controller (APPC), and a dual linear quadratic Gaussian (LQG) controller with indirect adaptation are designed and applied to the speed control of a thyristor-driven DC motor. The suggested synthesis approach is based on bicriterial optimisation of two cost functions introduced to give the adaptive properties of caution and allow them to stimulate the parameter estimation. The controllers derived are computationally simple. The derived LQG controller is obtained using spectral factorisation and pole positioning, which corresponds to the optimum cost function of the LQG controller. The designed controllers are applied to the speed control of a thyristor-driven DC-motor. The described experimental application demonstrates the superiority of the designed controllers over the standard APPC and the LQG controller based on the certainty equivalence approach.

Optimal design of adaptive tracking controllers for non-linear systems

We pose and solve an ‘inverse optimal’ adaptive tracking problem for non-linear systems with unknown parameters. A controller is said to be inverse optimal when it minimizes a meaningful cost functional that incorporates integral penalty on the tracking error state and the control, as well as a terminal penalty on the parameter estimation error. The basis of our method is an adaptive tracking control Lyapunov function (atclf) the existence of which guarantees the solvability of the inverse optimal problem. The controllers designed in this paper are not of certainty-equivalence type. Even in the linear case they would not be a result of solving a Riccati equation for a given value of the parameter estimate. Our abandoning of the certainty-equivalence approach is motivated by the fact that, in general, this approach does not lead to optimality of the controller with respect to the overall plant-estimator system, even though both the estimator and the controller may be optimal as separate entities. Our controllers, instead, compensate for the effect of parameter adaptation transients in order to achieve optimality of the overall system. We combine inverse optimality with backstepping to design a new class of adaptive controllers for strict-feedback systems. These controllers solve a problem left open in the previous adaptive backstepping designs, i.e. obtaining transient performance bounds that include an estimate of control effort, which is the first such result in the adaptive control literature.

Uncertainty structures in adaptive and robust stabilization

We develop and synthesize results concerning the stabilization of parametrically uncertain linear systems via state feedback. Our main purpose is to bring together the previously disjoint areas of parametric robust control and adaptive control, and to examine and compare the uncertainty structures that can be accommodated by various approaches of adaptive stabilization and robust stabilization. Among adaptive approaches, we consider adaptive design based on the certainty equivalence approach and those based on Lyapunov design, which in turn include the cases of parameter-independent and parameter-dependent Lyapunov functions. Among robust approaches, we consider quadratic as well as non-quadratic stabilization. A hierarchy of uncertainty structures is revealed. As an example of the results established in this paper, we show that the broadest uncertainty structure stabilizable by adaptive control based on a parameter-independent Lyapunov function is the antisymmetric stepwise form, which is strictly larger than the so-called pure feedback form.

Comments on ‘Design of stochastic discrete-time linear optimal regulators’

A state-space one-stage optimal control problem discussed by Lam (1982) is reexamined. Contradictions found in Lam are resolved by the certainty equivalence approach.

Decisions under imperfect knowledge: The certainty equivalence theory as an alternative to the von Neumann-Morgenstern theory of uncertainty

This paper offers a modified version of the certainty equivalence (CE) theory of utility for uncertain prospects and a new set of axioms as its basis. It shows that the CE and the von Neumann-Morgenstern (NM) approaches to uncertainty are opposite in spirit: The CE approach represents a flight from the world of uncertainty to the rules of certainty while the NM approach represents a flight from the world of certainty to one of uncertainty. The two approaches differ even in their treatment of compound prospects and their actuarially identical simple counterparts.