Minimax Control Research Articles

Min-max optimal control of systems approximated by finite-dimensional models: Nonquadratic cost functional

An open-loop control problem with nonquadratic performance criterion for a dynamical system, described by an abstract linear equation of evolution and approximated by a finite-dimensional model, is solved. A min-max approach is taken: the value of the cost functional in theworst-case output error between the system and the model, under the assumption that the norm of this output error is estimated by the norm of the input, is minimized.

Min-Max Control of Norm Uncertain Systems: Post-Additive Functionals

The problem of the control of a process, which is described by an approximate model, is considered. The problem is reduced to the control of the model whose output is affected by an input dependent error constrained in norm, and a min-max approach is taken.Particular attention is deserved to the features of the penalty functional, since it is desired to syntetize the min-max control by mean of a feedback on the states of the model; for example a qua dratie functional has not this property. An L1 type functional is then proposed, which is not differentiable; by means of duality techniques, however, the minimization can be carried out. The min-max is obtained both by iterative procedures, both as a function of the actual value of the model state: this function is the solution of a set of a partial differential equation.The proposed methods are applied to a numerical example and compared.

An Algorithm for a Step by Step Minmax Control of Linear Systems with Power-Bounded Noise

For a linear discrete system subject to noise with a power-like bound, a step-by-step min-max control law is derived as an explicit function of the state in which a parameter appears which is easily determined by solving a scalar equation.Some interesting properties are shown of the corresponding trajectories of the state.

Min-Max Optimal Modeling and Control of Plants from Input-Output Data

The goal of this paper is to present an approach to the problem of the approximation of a complex plant which must to be controlled. Starting from input-output data of the actual system, the proposed procedure leads to determine an optimal reduced model, on which it is very easy to design a min-max control policy for the plant which guarantees a bound of the performance degradation.

On a class of differential games without saddle-point solutions

The minimax solution of a linear regulator problem is considered. A model representing a game situation in which the first player controls the dynamic system and selects a suitable, minimax control strategy, while the second player selects the aim of the game, is formulated. In general, the resulting differential game does not possess a saddle-point solution. Hence, the minimax solution for the player controlling the dynamic system is sought and obtained by modifying the performance criterion in such a way that (a) the minimax strategy remains unchanged and (b) the modified game possesses a saddle-point solution. The modification is achieved by introducing a regularization procedure which is a generalization of the method used in an earlier paper on the quadratic minimax problem. A numerical algorithm for determining the nonlinear minimax strategy in feedback form, in which Pagurek's result on open-loop and closed-loop sensitivity is used to nontrivially simplify the computational aspects of the problem, is presented and applied on a simple example.

Bayes and minimax controllers for a linear system with stochastic jump parameters

A procedure is given for determining an optimal zero-memory regulator for a linear stochastic plant with an incomplete probabilistic description. For situations in which the initial ambiguity is modeled probabilistically, the equations characterizing the appropriate Bayes control are derived. If no prior distribution is available, a minimax controller is sought and an algorithm for obtaining it is described.

Optimal control with minimax cost

The following paper discusses the optimal control of these systems, characterized by a set of n first-order state equations, in which performance is measured by Chebyshev-type functional over the state trajectory. The determination of control functions that minimize the maximum value of a given state function over the trajectory interval is shown to follow directly from the development of a differential minimax cost. The differential minimax cost allows the problem to be formulated as a coordinate minimization in the cost-augmented state space, and leads to the consideration of a set of suboptimal problems whose solutions are shown to converge to the required minimax control. The modifications required for the application of standard variational techniques to the reformulated problem are also discussed. The main result of this study is the demonstration of equivalence between the Chebyshev-type control problem and a more conventional Mayer-type formulation.

Conflicting and minimax controls

Conflicting and relaxed minimax controls for Weierstrass E condition or Pontryagin maximum principle

Certain minimax control problems with incomplete information: PMM Vol. 35, N≗6, 1971, pp. 952–961

We apply the minimax (the guarantee) approach to the optimal control problems for plants with incomplete information. We assume that the location of the phase vector is known to within a certain set to which it belongs; the system's position is updated by measurements during the motion. Using the dynamic programing method we investigate a way of contructing the control as a function of measurement results and of time. Similar problems were analyzed in [1–3].

Control Problems with Functional Restrictions

Unilateral and minimax control with functional restrictions, including pursuit and evasion games

Unilateral and Minimax Control Problems Defined by Integral Equations

Unilateral and Minimax Control Problems Defined by Integral Equations

Determination of Bounds on Performance Measure and Min-Max Controllers in Time-Lag Systems

The effect of bounded deterministic disturbances (external and/or internal) on a chosen performance index (PI) is investigated. Upper and lower bounds of the PI about a nominal value are determined by the application of the optimum control theory. A computational procedure is also proposed to determine an approximately optimal feedback controller for time-lag systems. Such a controller is designed here on the basis of min-max criteria. Numerical examples illustrate the proposed approach.

A minimax control problem for sampled linear systems

A linear differential system is subject to a bounded control and a bounded disturbance. The controller receives the value of the state at a finite number of fixed sampling times. The cost is a convex or concave function of the state at a fixed final time. Given any control law, there is a maximum cost over all perturbations, the guaranteed performance for this control law. It is desired to find the minimum of this number over all control laws. Fenchel's theory of conjugate convex functions is used to set up an algorithm dual to the dynamic programming approach. This algorithm deals with the support functions of reachable sets, more easily determined than other descriptions of these sets. A discrete minimax principle is generally incorrect for problems of this class.

Minimax Control of Linear Systems with Amplitude Bounded Disturbances

Optimal control problem formulated for linear constant-coefficient system subjected to class of amplitude bounded load disturbances

Minimax control of large launch boosters

Optimal control theory applied to piecewise constant approximation of large launch booster for first 84 seconds of flight

On Performance Losses in Some Adaptive Control Systems: I

In controlling dynamic systems with unknown parameters and/or systems operating in unknown environment, the systems suffer due to the unknowness of pertinent parameter values, compared with situations with perfect information where all pertinent information is available to control systems optimally. The paper defines the concept of loss of performance to represent the loss in performance of some adaptive control situations compared with perfect information situations and defines the optimal control problems as the one where the loss of performance is minimized. This concept is illustrated for a control system governed by a scalar linear differential equation with unknown gain. The minimax control policy is defined as the control policy which minimized the maximum possible loss in performance where no a priori knowledge on the unknown parameter is available. It also discusses the optimal estimation problem of the unknown parameter from the point of view of loss of performance.