Computation of optimal control for linear systems with delay

A numerical algorithm is presented to calculate an optimal control recursively for linear multivariable systems with delay. The algorithm is based on the method of steepest descent in Hilbert space. The optimal control of a multivariable system with delay for a quadratic criterion function is given by the Riccati partial differential equations. These are simultaneous partial differential equations which are difficult to solve numerically. In most computational algorithms, errors are inevitable, a vast memory is required, and a lot of computational time is needed. This makes it impractical to use available computers for these algorithms. The algorithm presented here gives optimal control effectively without a large memory requirement. It also provides a practical computational method for obtaining the optimal control of multivariable systems with delay. Several numerical computations are performed to show how the effectiveness of the algorithm compares with other methods.     

A Riccati equation approach to the design of stabilizing controllers and observers for a class of uncertain linear systems

This paper presents a procedure for designing a full state observer and feedback control law which will stabilize a given uncertain linear system. The uncertain linear systems under consideration are described by state equations which depend on uncertain parameters. These uncertain parameters may be time varying. Their values, however, are constrained to lie within known compact bounding sets. The design procedure involves solving two algebraic Riccati equations. A feature of the design procedure presented is the fact that it reduces to the standard LQG design procedure if the system contains no uncertain parameters.     

Solution of optimal control problem of linear diffusion equations via Walsh functions

An attempt is made in this note to illustrate the use of Walsh functions in solving Riccati matrix equations arising in optimal control studies of linear diffusion equations with quadratic performance index.     

An optimal control approach to robust tracking of linear systems

In our early work, we show that one way to solve a robust control problem of an uncertain system is to translate the robust control problem into an optimal control problem. If the system is linear, then the optimal control problem becomes a linear quadratic regulator (LQR) problem, which can be solved by solving an algebraic Riccati equation. In this article, we extend the optimal control approach to robust tracking of linear systems. We assume that the control objective is not simply to drive the state to zero but rather to track a non-zero reference signal. We assume that the reference signal to be tracked is a polynomial function of time. We first investigated the tracking problem under the conditions that all state variables are available for feedback and show that the robust tracking problem can be solved by solving an algebraic Riccati equation. Because the state feedback is not always available in practice, we also investigated the output feedback. We show that if we place the poles of the observer sufficiently left of the imaginary axis, the robust tracking problem can be solved. As in the case of the state feedback, the observer and feedback can be obtained by solving two algebraic Riccati equations.     

Sequential decomposition of the matrix Riccati equation and its application to the linear quadratic regulator problem

The differential matrix Riccati equation for the multi-input-multi-output linear quadratic optimal regulator problem is considered. Two methods are presented, successive and parallel, that decompose this equation into a set of Riccati equations that correspond to optimal regulator problems of possibly reduced dimensions. The additivity of the solutions to the equations obtained by the successive sequential decomposition method (SDM) and the additivity of the inverses of the solutions to the equations obtained by the parallel SDM are established. Some duality relations between the successive and the parallel methods are presented via the use of the adjoint Riccati equation. The theory developed is extended to the algebraic matrix Riccati equation as a limiting case. The application of the SDMs in the infinite-time linear quadratic regulator problem is investigated. Special attention is paid to the partially ‘cheap’ problem where the cost of some of the regulator controls or of their combinations tends asymptotically to zero. Explicit expressions for the asymptotic optimal cost are derived and the behaviour of the asymptotic optimal root loci is investigated.     

Synthesis of linear quadratic control laws on basis of linear matrix inequalities

Classic problems of control law construction for a linear dynamic object that is optimal by the quadratic criterion in the determinate and stochastic cases are reduced, as is known, to solving nonlinear matrix Riccati equations. It is shown that the notion of H ₂-norm of a system transfer matrix makes it possible to formulate and solve the stated problems in terms of linear matrix inequalities.     

Optimal regulators for linear systems with no control cost

This article is concerned with the theory of optimal feedback regulator for the linear system =Ax +Bu with the cost functional given byJ(u) = 1/2(Mx(T),x(T)). Due to absence of the usual positive definite quadratic cost for controls, this is a nonstandard problem.Two sets of results are presented: one for bounded and one for unbounded controls. For bounded controls, the control law is given by solving a system of coupled nonlinear differential equations of the Riccati type; and for unbounded controls, the optimal control law is determined by solving a parameterized family of matrix Riccati differential equations.     

Jump linear quadratic regulator with controlled jump rates

Deals with the class of continuous-time linear systems with Markovian jumps. We assume that jump rates are controlled. Our purpose is to study the jump linear quadratic (JLQ) regulator of the class of systems. The structure of the optimal controller is established. For a one-dimensional (1-D) system, an algorithm for solving the corresponding set of coupled Riccati equations of this optimal control problem is provided. Two numerical examples are given to show the usefulness of our results     

LMI optimization for nonstandard Riccati equations arising instochastic control

We consider coupled Riccati equations that arise in the optimal control of jump linear systems. We show how to reliably solve these equations using convex optimization over linear matrix inequalities (LMIs). The results extend to other nonstandard Riccati equations that arise, e.g., in the optimal control of linear systems subject to state-dependent multiplicative noise. Some nonstandard Riccati equations (such as those connected to linear systems subject to both state- and control-dependent multiplicative noise) are not amenable to the method. We show that we can still use LMI optimization to compute the optimal control law for the underlying control problem without solving the Riccati equation     

Differential Riccati equation‐based adaptive stabilization for switched linear systems

In this paper, a novel adaptive control design is proposed to stabilize a class of switched linear systems with parametric uncertainties and disturbances. In particular, a family of differential Riccati equations that holds for the finite switching intervals are established, of which the closed‐form solutions are exploited to develop the Lyapunov function for the closed‐loop system. This Lyapunov function is imposed to be decreasing at and between two arbitrary switching instants by solving a group of matrix inequalities, based on which the adaptive controllers for switched linear system with and without disturbances are proposed, respectively. In the nondisturbed setting, asymptotic stability of the adaptive switched linear system is achieved, while the system is bounded in a mean square sense in the disturbed setting. A numerical example of flight control for F4E fighter aircraft is used to illustrate the proposed adaptive control methodology.     

具有对称循环结构的大系统Riccati方程的求解

本文研究了具有对称循环结构的大系统的代数Ｒｉｃｃａｔｉ方程和Ｌｙａｐｕｎｏｖ矩阵方程的求解问题，结果表明，这类系统的代数Ｒｉｃｃａｔｉ方程和Ｌｙｐａｐｕｎｏｖ矩阵方程的求解问题可以简化为求解Ｎ／２＋１个独立的低阶方程，做为一个应用，这类系统的二次型最优控制问题和鲁棒二次型最优控制问题也可以简化。     

Some new results on algebraic Riccati equations arising in linear quadratic differential games and the stabilization of uncertain linear systems

This paper presents some new results on algebraic Riccati equations arising in linear quadratic differential games. The first result is a uniqueness result for solutions of the Riccati equations under consideration. The second result is concerned with Riccati equations which arise in differential games in which the weighting on the minimizing control is allowed to approach zero. It is shown that if a certain minimum phase condition is satisfied then the corresponding solution to the Riccati equation will also approach zero.     

Algorithms for solving a class of problems of <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript>-optimization of control systems

Problems of synthesis of feedbacks providing H _∞-norm-maximal suppression of disturbances, acting on a linear stationary SISO system with account of limitedness of control resources, are considered. Two methods for solving are proposed. The first one is based on the Nevanlinna-Pick interpolation, and the second one is constructed on the ground of guaranteeing singularities of controllers. Within the framework of these approaches, simple computation algorithms that do not require solving Riccati equations or linear matrix inequalities are generated. The application of the developed algorithms is illustrated by an example.     

SDP-based approximation of stabilising solutions for periodic matrix Riccati differential equations

Numerically finding stabilising feedback control laws for linear systems of periodic differential equations is a nontrivial task with no known reliable solutions. The most successful method requires solving matrix differential Riccati equations with periodic coefficients. All previously proposed techniques for solving such equations involve numerical integration of unstable differential equations and consequently fail whenever the period is too large or the coefficients vary too much. Here, a new method for numerical computation of stabilising solutions for matrix differential Riccati equations with periodic coefficients is proposed. Our approach does not involve numerical solution of any differential equations. The approximation for a stabilising solution is found in the form of a trigonometric polynomial, matrix coefficients of which are found solving a specially constructed finite-dimensional semidefinite programming (SDP) problem. This problem is obtained using maximality property of the stabilising solution of the Riccati equation for the associated Riccati inequality and sampling technique. Our previously published numerical comparisons with other methods shows that for a class of problems only this technique provides a working solution. Asymptotic convergence of the computed approximations to the stabilising solution is proved below under the assumption that certain combinations of the key parameters are sufficiently large. Although the rate of convergence is not analysed, it appeared to be exponential in our numerical studies.     

Stabilization of continuous-time jump linear systems

We investigate almost-sure and moment stabilization of continuous time jump linear systems with a finite-state Markov jump form process. We first clarify the concepts of /spl delta/-moment stabilizability, exponential /spl delta/-moment stabilizability, and stochastic /spl delta/-moment stabilizability. We then present results on the relationships among these concepts. Coupled Riccati equations that provide necessary and sufficient conditions for mean-square stabilization are given in detail, and an algorithm for solving the coupled Riccati equations is proposed. Moreover, we show that individual mode controllability implies almost-sure stabilizability, which is not true for other types of stabilizability. Finally, we present some testable sufficient conditions for /spl delta/-moment stabilizability and almost-sure stabilizability.     

Optimal control of discrete-time linear fractional-order systems with multiplicative noise

A finite horizon linear quadratic (LQ) optimal control problem is studied for a class of discrete-time linear fractional systems (LFSs) affected by multiplicative, independent random perturbations. Based on the dynamic programming technique, two methods are proposed for solving this problem. The first one seems to be new and uses a linear, expanded-state model of the LFS. The LQ optimal control problem reduces to a similar one for stochastic linear systems and the solution is obtained by solving Riccati equations. The second method appeals to the principle of optimality and provides an algorithm for the computation of the optimal control and cost by using directly the fractional system. As expected, in both cases, the optimal control is a linear function in the state and can be computed by a computer program. A numerical example and comparative simulations of the optimal trajectory prove the effectiveness of the two methods. Some other simulations are obtained for different values of the fractional order.     

多重延时线性系统最优控制的Walsh级数分析法

本文导出了Walsh级数的延时算子和超前算子,并利用这两个算子和Walsh级数的运算特性给出了一套多重延时线性系统最优控制的新的简便算法,直接从性能泛函着手避免了求解带多重延时的非线性Riccati方程,使最优控制的求解转换成了代数极值问题的求解,这是一种值得讨论的新算法。     

A new optimal multirate control of linear periodic andtime-invariant systems

The optimal multirate design of linear, continuous-time, periodic and time-invariant systems is considered. It is based on solving the continuous linear quadratic regulation (LQR) problem with the control being constrained to a certain piecewise constant feedback. Necessary and sufficient conditions for the asymptotic stability of the resulting closed-loop system are given. An explicit multirate feedback law that requires the solution of an algebraic discrete Riccati equation is presented. Such control is simple and can be easily implemented by digital computers. When applied to linear time-invariant systems, multirate optimal feedback optimal control provides a satisfactory response even if the state is sampled relatively slowly. Compared to the classical single-rate sampled-data feedback in which the state is always sampled at the same rate, the multirate system can provide a better response with a considerable reduction in the optimal cost. In general, the multirate scheme offers more flexibility in choosing the sampling rates     

The interior-point method for linear programming

A robust, reliable, and efficient implementation of the primal-dual interior-point method for linear programs, which is based on three well-established optimization algorithms, is presented. The authors discuss the theoretical foundation for interior-point methods which consists of three crucial building blocks: Newton's method for solving nonlinear equations, Joseph Lagrange's methods for optimization with equality constraints, and Fiacco and McCormick's barrier method for optimization with inequality constraints. The construction of the primal-dual interior-point method using these methods is described. An implementation of the primal-dual interior-point method, its performance, and a comparison to other interior-point methods are also presented     

Design of fixed-point smoother algorithm for linear discrete time-delay systems

This paper investigates the fixed-point smoothing problems for linear discrete-time systems with multiple time-delays in the observations.The linear discrete-time systems considered have l + 1 output channels.One is instantaneous observation and the others are delayed.The fixed-point smoothers involving recursive algorithm and non-recursive algorithm are designed by using innovation analysis theory without relying on the system augmentation approach.Also, it is further shown that the design of fixed-point smoother comes down to solving l + 1 Riccati equations with the same dimensions as the original systems.