Spectral factorization for multiple input delayed discrete-time systems with applications to control

We consider the nonstandard infinite horizon quadratic cost minimization problem for a stable abstract linear control system, and show that it can be reduced to a J-inner coprime factorization problem (or equivalently, to a canonical spectral factorization problem) in the (often finite-dimensional) control space. More precisely, we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution. If both the system and the spectral factor are regular together with their adjoints, then the feedback operator can be expressed in terms of the Riccati operator, and the Riccati operator is a self-adjoint solution of an algebraic Riccati equation. This Riccati equation is similar to the classical algebraic Riccati equation, but one of its coefficients differs from the expected one. We apply our main theorems to prove the first available versions of the positive real and bounded real lemmas for abstract linear systems. Similar results are true for unstable systems.

Acausal models and balanced realizations of stationary processes

We consider the infinite horizon quadratic cost minimization problem for a stable time-invariant well-posed linear system in the sense of Salamon and Weiss, and show that it can be reduced to a spectral factorization problem in the control space. More precisely, we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution. If both the system and the spectral factor are regular, then the feedback operator can be expressed in terms of the Riccati operator, and the Riccati operator is a positive self-adjoint solution of an algebraic Riccati equation. This Riccati equation is similar to the usual algebraic Riccati equation, but one of its coefficients varies depending on the subspace in which the equation is posed. Similar results are true for unstable systems, as we have proved elsewhere.

This correspondence investigates the continuity behavior of the spectral factorization mapping for trigonometric polynomials. It is clear that this factorization mapping is continuous on the space of all trigonometric polynomials of a fixed degree N which means that a small perturbation in the given spectrum yields always a bounded error in the spectral factor. The correspondence derives a lower bound on the continuity constant of the spectral factorization mapping which shows that the error in the spectral factor grows at least proportional with the logarithm of the degree N of the given spectrum.

The solution is given to a time-varying optimal state feedback problem with stochastic disturbances. The system is composed of a plant and disturbance model represented by polynomials in the delay operator, z/sup -1/, leading to a solution involving spectral factorisation of operator equations and Diophantine operator equations. The cost function is over infinite time and the assumption is made that the system is time-varying for T steps into the future from the current sample and time-invariant thereafter. For a time-invariant system over infinite time, the optimal controller is a constant state-feedback matrix gain. Thus, with the assumption of time-invariance from T to /spl infin/, the feedback gain may be calculated using constant system polynomials. The solution of the spectral factors and Diophantine equations may then be computed recursively, for a scalar plant, working from T steps ahead to the current time. The controller calculated for the current time is then applied to the system. If the input non-linearity of a plant is represented in time-varying form, the time-varying ideas may be used to produce a nonlinear controller for the system. The example in this paper is for a smooth saturation non-linearity represented by a tanh function. Simulation results are given and it is clear that performance gains over a time-invariant controller are possible.

Minimal square spectral factors

A stochastic linear quadratic (LQ) control problem is indefinite when the cost weighting matrices for the state and the control are allowed to be indefinite. Indefinite stochastic LQ theory has been extensively developed and has found interesting applications in finance. However, there remains an outstanding open problem, which is to identify an appropriate Riccati-type equation whose solvability is {\it equivalent} to the solvability of the indefinite stochastic LQ problem. This paper solves this open problem for LQ control in a finite time horizon. A new type of differential Riccati equation, called the generalized (differential) Riccati equation, is introduced, which involves algebraic equality/inequality constraints and a matrix pseudoinverse. It is then shown that the solvability of the generalized Riccati equation is not only sufficient, but also necessary, for the well-posedness of the indefinite LQ problem and the existence of optimal feedback/open-loop controls. Moreover, all of the optimal controls can be identified via the solution to the Riccati equation. An example is presented to illustrate the theory developed.

The sampled-data linear quadratic tracking problem has been transformed into the well known discrete-time linear quadratic tracking problem, just as the sampled-data linear quadratic regulator problem may be transformed into the well known discrete-time linear quadratic regulator problem.

ABSTRACTThis work proposes a new technique to characterize the suboptimality of stabilizing control laws for the infinite‐horizon linear quadratic regulation problem, wherein the norm of the state feedback gain matrix is conditioned to be lesser than a given bound. In contrast to existing techniques where the upper bound on the cost is pre‐specified, this work proposes to compute an upper bound associated with the suboptimal control law(s). To derive the norm‐bounded state feedback gain matrix, sufficient conditions of Krotov global optimal control framework are analyzed for the constrained linear quadratic problem, which typically renders the optimal control problem non‐convex. To address this issue, suitable convex optimization problems are formulated to compute the upper bound independent of initial conditions. The effectiveness of the proposed technique and its comparison with state‐of‐the‐art methods are demonstrated through numerical examples.

A Class of Linear Quadratic Gaussian Hybrid Optimal Control Problems with Realization–Independent Riccati Equations

Linear Quadratic Optimal Control of a Wave Equation with Boundary Damping and Pointwise Control Input

This paper studies the consensus problem for discrete-time multi-agent systems with state, input and different communication delays, where the input delays can be arbitrarily large yet exactly known. Moreover, the communication delays are different for different agents and can also be arbitrarily large. A nested predictor based protocol is established to solve the problem. It is shown that both the input and communication delays can be compensated completely. Furthermore, a linear matrix inequalities based approach is provided to design the state feedback gains. Finally, a numerical example is worked out to illustrate the effectiveness of the proposed approach.

A fundamental matrix based approach to design predictor-based control laws for discrete-time systems with state and input delays

A perturbation analysis is applied to the Riccati equation of the linear quadratic regulator problem. In the limit where the weighting on the control input is zero, it is shown that the asymptotic expansion of the Riccati equation is equivalent to the limiting Riccati equation obtained by using the Goh transformation in the singular optimal control problem. Although the Goh transformation leads to an elegant general form, the perturbation approach is more direct and insightful. Furthermore, the asymptotic behavior of the Riccati equation near the limit is captured by using the perturbation approach.

Control of time-delay systems represents one important class of control problems. A lot of work has been done in recent years. Much of the research work has been focused on the stability analysis and stabilization of time-delay systems based on the Lyapunov functional and linear matrix inequality (LMI) approaches. While the LMI approach does provide an e±cient tool, the results are mostly only su±cient and only numerical solutions are available. Our research aims to present analytical solutions by some algebraic approaches, which can provide an insightful understanding of control problem of time-delay systems. On the other hand, there has also been significant recent interest in networked multi-agent system (MAS) where unavoidably there exist delays in information acquisition as well as information exchange between agents in addition to possible input/state delays in each agent. This motivates the study of MASs with input and communication delays. The thesis is divided into two parts: Part I is focused on optimal control of single agent systems with multiple input delays; Part II is on distributed consensus control of MAS with communication and input delays. More specifically, in Part I, we shall study the linear quadratic regulation (LQR) and tracking for a single agent with linear dynamics. We first focus on LQR for linear discrete-time systems with multiple delays in a single input channel. By introducing a backward stochastic system, we establish a relationship between the LQR problem for the original system and optimal estimation for a backward stochastic system which can be solved by using projection approach. The explicit optimal controllers are developed for both finite time and infinite time horizon cases. A parallel result for linear continuous-time systems is also established, where the optimal feedback gain is given in terms of the solution of Riccati-type partial differential equations (PDEs).