Stochastic optimal linear feedback control systems using available measurements

The stochastic optimal control of linear systems with time-varying and partially observable parameters is synthesized under noisy measurements and a quadratic performance criterion. The structure of the regulator is given, and the optimal solution is reduced to a two-point boundary-value problem. Comments on the numerical solution by appropriate integration schemes is included.     

The galerkin method and feedback control of linear distributed parameter systems

In the development of feedback control theory for distributed parameter systems (DPS), i.e., systems described by partial differential equations, it is important to maintain the finite dimensionality of the controller even though the DPS is infinite dimensional. Since this dimension is directly related to the available on-line computer capacity, it must be finite (and not very large) in order to make the controller implementable from an engineering standpoint. In previous work, it has been our intention to investigate what can be accomplished by finite-dimensional control of infinite-dimensional systems; in particular, we have concentrated on controller design and closed-loop stability. The starting point for all of this is some means for producing a finite-dimensional approximation—a reduced-order model—of the actual DPS. When the “modes” of the DPS are known, the natural candidate for model reduction is projection onto the modal subspace spanned by a finite number of critical modes. Unfortunately, in real engineering systems, these modes are never known exactly and some other reasonable approximation must be used. In this paper, the model reduction is based on the well-known Galerkin procedure. We generate the Galerkin reduced-order model and develop a finite-dimensional controller from it; then we analyze the stability of this controller in closed loop with the actual DPS. Our results indicate conditions under which model reduction based on consistent Galerkin approximations will lead to stable finite-dimensional control.     

A convex optimal control problem involving a class of linear hyperbolic systems

In this paper, we consider a convex optimal control problem involving a class of linear hyperbolic partial differential systems. A computational algorithm which generates minimizing sequences of controls is devised. The convergence properties of the algorithm are investigated.     

On a class of optimal control problems with distributed and lumped parameters

The optimal control of moving sources governed by a parabolic equation and a system of ordinary differential equations with initial and boundary conditions is considered. For this problem, an existence and uniqueness theorem is proved, sufficient conditions for the Fréchet differentiability of the cost functional are established, an expression for its gradient is derived, and necessary optimality conditions in the form of pointwise and integral maximum principles are obtained.     

On curvature and feedback classification of two-dimensional optimal control systems

The goal of this paper is to extend the classical notion of Gaussian curvature of a two-dimensional Riemannian surface to two-dimensional optimal control systems with scalar input using Cartan’s moving frame method. This notion was already introduced by A. A. Agrachev and R. V. Gamkrelidze for more general control systems using a purely variational approach. Further, we will see that the “control” analogue of Gaussian curvature reflects similar intrinsic properties of the extremal flow. In particular, if the curvature is negative, arbitrarily long segments of extremals are locally optimal. Finally, we will define and characterize flat control systems. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 33, Suzdal Conference-2004, Part 1, 2005.     

On the robustness of linear stabilizing feedback control for linear uncertain systems

We investigate linear time-invariant scalar-input systems with constant uncertainties that are not required to satisfy matching conditions. In a previous paper, the existence of stabilizing discontinuous controllers is established under three assumptions for such systems. The first assumption requires controllability of the system for each uncertainty. The second is a condition on an uncertain Lyapunov equation, and the third is a boundedness condition related to the controllability matrix. In this paper, using the same assumptions, we show the existence of a linear stabilizing control. Our result is related to the high-gain theorem of classical control.     

On existence of optimal control governed by a class of the first-order linear dynamic systems on time scales

In this paper, by using the approximation of classical solution, we introduce the definition of solution and prove the existence and uniqueness of solutions of the first-order linear dynamic systems on time scales. Existence of Lagrange optimal control problem governed by the first-order linear dynamic systems on time scales is also presented. For illustration, some examples of optimal control problems on time scales are also discussed.     

Suboptimal linear feedback for a class of stochastic discrete-time systems

The well-known optimal linear feedback of the linear quadratic problem with jump Markov and independent disturbances is also a suboptimal feedback of the problem perturbed by a nonlinear element with a small parameter if the element is continuous. The assertion is shown and an asymptotic expansion is given.     

On the existence of an optimal feedback control for stochastic systems

We prove the existence of an optimal control for systems of stochastic differential equations without solving the Bellman dynamic programming equation. Instead, we use direct methods for solving extremal problems.     

On the optimal control of linear systems with linear performance criterion and constraints

We suggest an analytical-numerical method for solving a boundary value optimal control problem with state, integral, and control constraints. The embedding principle underlying the method is based on the general solution of a Fredholm integral equation of the first kind and its analytic representation; the method permits one to reduce the boundary value optimal control problem with constraints to an optimization problem with free right end of the trajectory.     

Adaptive feedback control for a class of chaotic systems

In this paper, the problem of control for a class of chaotic systems is considered. The nonlinear functions of chaotic systems are not necessarily to satisfy the Lipsichtz conditions, but bounded by a polynomial with the gains unknown. Employing adaptive method, the corresponding controller which renders the closed-loop system asymptotically stable is constructed. The designed controller is robust with respect to certain class of disturbances in the chaotic systems. Simulations on unified chaotic systems and Arneodo chaotic system are performed and the results verify the validity of the proposed techniques.     

Suboptimal feedback control of linear gyroscopic systems

A simplified model of reduced dimensionality is presented for a class of linear gyroscopic systems with quadratic performance indices. This model is based on the concept of weakly coupled subsystems and can be used in the synthesis of suboptimal controllers. Controllers based on this model compare favorably with both optimal and conventional controllers.This research was supported by NSF Grant No. GK-3273 and a grant from the Graduate School of the University of Minnesota.     

Suboptimal feedback control of linear periodic systems

A method is developed for the approximate design of an optimal state regulator for a linear periodically varying system with quadratic performance index. The periodic term is taken to be a perturbation to the system. By making use of a power-series expansion in a small parameter, associated with periodic terms, a set of matrix equations is derived for determining successively a feedback gain. Given periodic terms of a Fourier-series form, explicit solutions are obtained for those matrix equations. A sufficient condition for existence and periodicity of the solution is also shown. Further, the performance degradation resulting from a truncation of the power-series solution is investigated. The method may effectively be used in a computer-programmed computation.     

On the existence of the optimal control for linear uncontrollable systems

The optimal control of time-invariant, uncontrollable linear systems with respect to an integral quadratic performance index over an infinite time interval is considered. It is proved that the asymptotic stability of the uncontrollable part of the system is a necessary and sufficient condition for the existence of the solution.This research was supported by the CNR (Consiglio Nazionale delle Ricerche), Rome, Italy.     

On control and synchronization in chaotic and hyperchaotic systems via linear feedback control

This paper presents the control and synchronization of chaos by designing linear feedback controllers. The linear feedback control problem for nonlinear systems has been formulated under optimal control theory viewpoint. Asymptotic stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function which can clearly be seen to be the solution of the Hamilton–Jacobi–Bellman equation thus guaranteeing both stability and optimality. The formulated theorem expresses explicitly the form of minimized functional and gives the sufficient conditions that allow using the linear feedback control for nonlinear system. The numerical simulations were provided in order to show the effectiveness of this method for the control of the chaotic Rössler system and synchronization of the hyperchaotic Rössler system.     

On a class of nonsmooth optimal control problems

In the paper the Zangwill-Pietrzykowski penalties are used for the augmentation of state-space and terminal state inequality constraints in an optimal control problem with a nonsmooth cost of a special structure having a straightforward practical application. Directional derivatives of the augmented cost are derived. In such a way bundle methods for nonsmooth minimization may be utilized.     

On a class of inverse optimal control problems

Four distinct, though closely related, inverse optimal control problems are considered. Given a closed, convex setU in a real Hilbert spaceX and an elementu ₀ inU, it is desired to find all functionals of the form (u,Ru) such that (i)R is a self-adjoint positive operator and (u,Ru) is minimized over the setU at the pointu ₀, (ii)R is self-adjoint, positive definite and (u,Ru) is minimized overU atu ₀, (iv)R is self-adjoint, positive definite and (u,Ru) is uniquely minimized overU atu ₀. The interrelationships among the sets of solutions of these problems are pointed out. Necessary and sufficient conditions which explicitly characterize the solutions to each of these problems are derived. The question of existence of a solution (namely, Given a particular setU and a particular elementu ₀, under what conditions does there exist an operatorR having certain required properties?) is discussed. The results derived are illustrated by an example.     

Global synchronization criteria for a class of third-order non-autonomous chaotic systems via linear state error feedback control

This paper investigates the global synchronization of a class of third-order non-autonomous chaotic systems via the master–slave linear state error feedback control. A sufficient global synchronization criterion of linear matrix inequality (LMI) and several algebraic synchronization criteria for single-variable coupling are proven. These LMI and algebraic synchronization criteria are then applied to two classes of well-known third-order chaotic systems, the generalized Lorenz systems and the gyrostat systems, proving that the local synchronization criteria for the chaotic generalized Lorenz systems developed in the existing literature can actually be extended to describe global synchronization and obtaining some easily implemented synchronization criteria for the gyrostat systems.