Minimax Control Research Articles

State Feedback Minimax and Guaranteed Cost Controllers for Linear Systems with Integral Quadratic Constraint Uncertainty Descriptions

This paper considers the design of minimax and optimal guaranteed cost controllers for uncertain linear systems. The uncertainty is permitted to have a structured description and is bounded by an integral quadratic constraint. For known initial conditions on the system state, it is shown that minimax control can be realised by static full state feedback controllers. A new proof of minimax optimality is given using standard methods for linear systems. The design of such controllers requires the optimal solution of a single Riccati equation dependent upon a set of parameters whose dimension is equal to the number of uncertainty blocks. It is shown that this multivariable optimisation is convex and thus numerically tractable. No explicit criteria to determine the existence of a solution are known; however, if solutions exist, results pertaining to their location are given to assist numerical optimisation. The extension of the approach to arbitrary unknown initial conditions is considered but is shown, in general, not to be possible. A tractable approach, considering the initial condition to be a random variable whose occurrence satisfies a certain probability density function, is presented. The resulting optimisation problem is shown to be convex and thus numerically tractable.

Generalized Polynomial H ∞ Predictive Control Utilizing Minimax Prediction

This paper presents a predictor which minimizes the cost function of the prediction error in a minimax sense, yielding minimization of the peaks of the prediction error spectrum, rather than its integral on the unit circle. This predictor is then used for the derivation of a predictive minimax control law. This one degree of freedom control law minimizes the peaks of a generalized cost function in the frequency domain. The predictor and control algorithms are derived via embedding of the minimax problem within an LQ problem using polynomial methods.

Optimal Control of a Chaotic Map: Fixed Point Stabilization and Attractor Confinement

Optimal control is applied to a chaotic system. Reference is made to a well-known one-dimensional map. Firstly, attention is devoted to the stabilization of a fixed point. An optimal controller is obtained and compared with other controllers which are popular in the control of chaos. Secondly, allowance is made for uncertainty and emphasis is placed on the reduction rather than the suppression of chaos. The aim becomes that of confining a chaotic attractor within a prescribed region of the state space. A controller fulfilling this task is obtained as the solution of a min-max optimal control problem.

A local approach to solving the inverse minimax control problem for discrete-time systems

A local approach is used to solve the inverse problems of minimax control and of worst-case disturbance for linear discrete-time systems. It is shown that minimax control and worst-case disturbance with respect to a quadratic performance index are always those for some local criterion defined by a one-step increment of a state quadratic function along a system trajectory and current values of the control and the disturbance, while the inverse statement generally speaking is not true. Using a solution to the inverse problem of worst-case disturbance, which is of interest itself, the necessary and sufficient condition expressed in a frequency domain are derived under which a given local worst-case disturbance and a given local minimax control will be the worst-case disturbance and the minimax control respectively for some quadratic performance index with a suitable non-negative weight on the state. In addition, the set of all linear state feedbacks corresponding to minimax controls is revealed to be a subset of the set of all stable feedbacks corresponding to optimal controls in the absence of disturbances.

Discrete positive realness of the discrete min-max control law

It is shown that the discrete state dependent min-maxcontroller for an uncertain discrete system with matched uncertainties,is a Discrete Strictly Positive Real (d.s.p.r.) control law.That is to say that an output may always be defined such thatthe nominal open loop is d.s.p.r., and the min-max control lawmay be realized via that output.

H∞ optimization problem with sign-indefinite quadratic form

This paper presents the solution to min-max control problem arising when the matrix C 1 T C 1 of the cost function in the standard H ∞ control problem (Doyle et al., 1989) is replaced by an arbitrary matrix Q ≱ 0. This difference is proved to be sufficient for results obtained in (Doyle et al., 1989) not to cover such the case. Their derivations essentially base on the cost function being H ∞ norm and can not be adjusted to deal with sign-indefinite quadratic form. With some sort of strict frequency condition assumed, state space technique is fruitful to obtain the necessary and sufficient conditions of the solvability of the problem. The solution is given by two Riccati equations and has some difference when compared to that of (Doyle et al., 1989).

Optimal guaranteed control of linear systems under disturbances

This paper is devoted to the problem of the minimax control of a dynamical system with quadratic performance functional under external disturbances and geometric control constraints. The optimal guaranteed control strategy is obtained in explicit form.

Method of Moment Restrictions in Robust Control and Filtering

The minimax control and filtering problems are considered for the linear MIMO system acted on by a stochastic input. The input has a meaning of the disturbance in the control problem and it consists of the noise and signal in the filtering one. The performance index is the maximum of the system output variance over the class of inputs, which is specified by the moment restrictions. The problem is to minimize the performance index over the given class of regulators or filters. It includes the H2 and H∞ optimal control and filtering problems as the special cases. The solution obtained involves the solution of some auxiliary convex optimization problem and the solution of the standard model-matching problem. The approach is based on the theory of generalized Chebyshev-type inequalities.

Feedback Control Design of Constrained Parabolic Systems in Uncertainty Conditions

The paper presents recent results by the authors on minimax control design of parabolic systems in uncertainty conditions under hard control and state constraints. The design procedure involves multi-step approximations and takes into account monotonicity properties of the parabolic dynamics. The results obtained justify a suboptimal three-positional structure of feedback controllers in the Dirichlet and Neumann boundary conditions and provide calculations of their optimal parameters that ensure the required state performance and stability under any admissible perturbations.

Embedding adaptive JLQG into LQ martingale control with a completely observable stochastic control matrix

With jump linear quadratic Gaussian (JLQG) control, one refers to the control under a quadratic performance criterion of a linear Gaussian system, the coefficients of which are completely observable, while they are jumping according to a finite-state Markov process. With adaptive JLQG, one refers to the more complicated situation that the finite-state process is only partially observable. Although many practically applicable results have been developed, JLQG and adaptive JLQG control are lagging behind those for linear quadratic Gaussian (LQG) and adaptive LQG. The aim of this paper is to help improve the situation by introducing an exact transformation which embeds adaptive JLQG control into LQM (linear quadratic Martingale) control with a completely observable stochastic control matrix. By LQM control, the authors mean the control of a martingale driven linear system under a quadratic performance criterion. With the LQM transformation, the adaptive JLQG control can be studied within the framework of robust or minimax control without the need for the usual approach of averaging or approximating the adaptive JLQG dynamics. To show the effectiveness of the authors' transformation, it is used to characterize the open-loop-optimal feedback (OLOF) policy for adaptive JLQG control.

ON THE LYAPUNOV APPROACH TO ROBUST STABILIZATION OF UNCERTAIN NONLINEAR SYSTEMS

In the Lyapunov approach employed in this paper, known in the literature as Lyapunov control, or min-max control, robust, global uniform asymptotic stability is achieved by a discontinuous control law which ensures that the Lyapunov derivative is negative despite bounded uncertainty. For that, it is assumed that the uncertainties satisfy certain matching conditions, and that a Lyapunov function for the nominal plant is available. To obtain lower control magnitudes, this paper develops control laws which counter the uncertainties on a component-wise basis, rather than the usual normic one. Both the basic discontinuous control law, which is proved to provide robust global uniform asymptotic stability, and a continuous app roximation, which is proved to ensure global uniform ultimate boundedness, are derived. Application to model following is given. We adapt recent results on robust quadratic stabilization of nominally linear time-invariant plants subjected to nonlinear, bo unded and unmatched uncertain perturbations, to extend our results to this important class of systems; this is illustrated by two examples.

A Separation Theorem for Expected Value and Feared Value Discrete Time Control

We show how the use of a parallel between the ordinary (+, X) and the (max, +) algebras, Maslov measures that exploit this parallel, and more specifically their specialization to probabilities and the corresponding cost measures of Quadrat, offer a completely parallel treatment of stochastic and minimax control of disturbed nonlinear discrete time systems with partial information. This paper is based upon, and improves, the discrete time part of the earlier paper [9].

Robustness of minimax controllers to nonlinear perturbations

We study the robustness of minimax controllers, originally designed for nominal linear or nonlinear systems, to unknown static nonlinear perturbations in the state dynamics, measurement equation, and performance index. When the nominal system is linear, we consider both perfect state measurements and general imperfect state measurements; in the case of nominally nonlinear systems, we consider perfect state measurements only. Using a differential game theoretic approach, we show for the former class that, as the perturbation parameter (say, e>0) approaches zero, the optimal disturbance attenuation level for the overall system converges to the optimal disturbance attenuation level for the nominal system if the nonlinear structural uncertainties satisfy certain prescribed growth conditions. We also show that anH∞-controller, designed based on a chosen performance level for the nominal linear system, achieves the same performance level when the parameter e is smaller than a computable threshold, except for the finite-horizon imperfect state measurements case. For that case, we show that the design of the nominal controller must be based on a decreased confidence level of the initial data, and a controller thus designed again achieves a desired performance level in the face of nonlinear perturbations satisfying a computable norm bound. In the case of nominally nonlinear systems, and assuming that the nominal system is solvable, we obtain sufficient conditions such that the nominal controller achieves a desired performance in the face of perturbations satisfying computable norm bounds. In this way, we provide a characterization of the class of uncertainties that are tolerable for a controller designed based on the nominal system. The paper also presents two numerical examples; in one of these, the nominal system is linear; in the other one, it is nonlinear.

Minimax control under a bound on the partial covariance sequence of the disturbance

A discrete-time linear SISO plant that is acted on by a stochastic disturbance is considered. It is assumed that a restriction on the partial covariance sequence of the disturbance is given. The cost function is a maximum of the closed-loop system output variance over the class of disturbances. The problem is to find the optimal linear stabilizing regulator. The transfer function of the optimal closed-loop system is constructed in explicit form. It is shown that the minimax regulator is intermediate between the H ∞ and H 2 optimal regulators. As a result of the problem solution, generalizations of the Caratheodory-Fejer and Nevanlinna-Pick theorems are obtained.

Minimax control of switching systems under sampling

We consider a general class of systems subject to two types of uncertainty: A continuous deterministic uncertainty that affects the system dynamics, and a discrete stochastic uncertainty that leads to jumps in the system structure at random times, with the latter described by a continuous-time finite state Markov chain. When only sampled values of the system state is available to the controller, along with perfect measurements on the state of the Markov chain, we obtain a characterization of minimax controllers, which involves the solutions of two finite sets of coupled PDEs, and a finite dimensional compensator. For the linear-quadratic case, a complete characterization is given in terms of coupled generalized Riccati equations, which also provides the solution to a particular H ∞ optimal control problem with randomly switching system structure and sampled state measurements.

An adaptive controller based on disturbance attenuation

This paper discusses the control of linear systems with uncertain parameters in the control coefficient matrix, under the influence of both process and measurement noise. A disturbance attenuation approach is used, and from this a multiplayer game problem is generated. First, the minimax formulation is presented, which represents an upper bound on the game cost criterion. Second, a dynamic programming approach is used to solve the game. It is necessary to significantly extend the method over earlier implementations, as the class of problems does not satisfy certain assumptions generally made. It is shown that for this class of problems, the controller determined from the dynamic programming approach is equivalent to the minimax controller. Therefore, the minimax controller is also a saddlepoint strategy for the differential game. Controller development appears to be much simpler from the dynamic programming standpoint. A simple scalar example is presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Minimax control of nonlinear evolution equations

In this paper we study the minimax control of systems governed by a nonlinear, time-dependent evolution inclusion of the subdifferential type. Using some continuity and lower semicontinuity results for the solution map and the cost functional, respectively, we are able to establish the existence of an optimal control. The abstract results are then applied to obstacle problems, semilinear systems with weakly varying coefficients (e.g. oscillating coefficients), and differential variational inequalities.

Design of Simple Adaptive Nonlinear Robust Controller.

In this paper, we propose an adaptive nonlinear robust controller which estimates the upper bound of uncertain elements. The structure of our controller is much simpler than those of other available controllers. After separating the original nonlinear dynamic system into linear and nonlinear parts, we first determine the virtual input to stabilize the linear part, so that we can obtain a unique solution to the algebraic Lyapunov equation. Next, we determine a min-max (or bang-bang) controller with an estimated upper bound for uncertainties that contain the original unknown nonlinear elements and the virtual input. The controller is applied to control a pantograph-type 2-link manipulator. The effectiveness of the proposed method is illustrated by some computer simulations. Specifically, we examine the relationships among the control perfarmance, the estimation rate of an upper bound, and the magnitude of a boundary layer in the computer simulations.

Minimax optimal control of uncertain systems with structured uncertainty

AbstractThis paper considers a minimax control problem for an uncertain system containing structured uncertainties. The uncertainties in this system are assumed to satisfy a certain integral quadratic constraint. For a given initial condition, the minimax optimal controller is constructed by solving a parameter‐dependent Riccati equation of the game type. This controller leads to a closed‐loop uncertain system which is absolutely stable.

Control of an aircraft landing in windshear

The problem of the feedback control of an aircraft landing in the presence of windshear is considered. The landing process is investigated up to the time when the runway threshold is reached. It is assumed that the bounds on the wind velocity deviations from some nominal values are known, while information about the windshear location and wind velocity distribution in the windshear zone is absent. The methods of differential game theory are employed for the control synthesis. The complete system of aircraft dynamic equations is linearized with respect to the nominal motion. The resulting linear system is decomposed into subsystems describing the vertical (longitudinal) mo- tion and lateral motion. For each subsystem, an auxiliary antagonistic differential game with fixed terminal time and convex payoff function depending on two components of the state vector is formulated. For the longitudinal motion, these components are the vertical deviation of the aircraft from the glide path and its time derivative; for the lateral motion, these components are the lateral deviation and its time deriva- tive. The first player (pilot) chooses the control variables so as to minimize the payoff function; the interest of the second player (nature) in choosing the wind disturbance is just opposite. The linear differential games are solved on a digital computer with the help of corresponding numerical methods. In particular, the opti- mal (minimax) strategy is obtained for the first player. The optimal control is specified by means of switch surfaces having a simple structure. The minimax control designed via the auxiliary differential