Quadratic optimal control of stable abstract linear systems

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.

Rotational Motion Control by Feedback with Minimum L1-Norm

This paper deals with the linear quadratic regulation problems for the linear discrete-time systems with l input delays. The design of the optimal control law is transformed into solving one Diophantine equation and one spectral factorization with delays. A new and simple approach for the spectral factorization is proposed based on reorganized innovation analysis. The calculation of spectral factor comes down to solving l + 1 Riccati equations with the same dimension as the original systems.

Acausal models and balanced realizations of stationary processes

This paper deals with the problem of designing multirate-output contrlleers for sampled-dataH ∞-optimal control of linear continuous-time systems. Two formulations of the problem are studied. In the first, the intersample behavior of the disturbance and the controlled output signals is not considered, whereas in the second the continuous-time nature of these signals is taken into account. It is shown that, in both cases and unter appropriate conditions, it is plausible to reduce the repective initial problem to an associated discrete-timeH ∞-optimization problem for which a fictitious static state feedback controller is to be designed. This fact has a beneficial influence on the theoretical and numerical complexity of the problem, since only one algebraic Riccati equation is to be solved here, as compared to two algebraic Riccati equations needed in known techniques concerning theH ∞-optimization problem with dynamic measurement feedback.

Minimal square spectral factors

In this paper, we propose a method for estimating the algebraic Riccati equation (ARE) with respect to an unknown discrete-time system from the system state and input observation. The inverse optimal control (IOC) problem asks, “What objective function is optimized by a given control system?” The inverse linear quadratic regulator (ILQR) problem is an IOC problem that assumes a linear system and quadratic objective function. The ILQR problem can be solved by solving a linear matrix inequality that contains the ARE. However, the system model is required to obtain the ARE, and it is often unknown in fields in which the IOC problem occurs, for example, biological system analysis. Our method directly estimates the ARE from the observation data without identifying the system. This feature enables us to economize the observation data using prior information about the objective function. We provide a data condition that is sufficient for our method to estimate the ARE. We conducted a numerical experiment to demonstrate that our method can estimate the ARE with less data than system identification if the prior information is sufficient.

Optimal and robust control of a group of single-input linear systems are presented. The formulation of the optimal state feedback controller is developed for a group of linear systems derived from a nonlinear system under attractive domains. Choosing a particular linear system as the desired model, the rest of the systems are assumed to be parameter-uncertain. Selection of a suitable weighting matrix for the performance index is used as a key method to optimally control this particular system and robustly stabilize the rest of the linear systems, simultaneously. It is shown that the above problem can be solved if an algebraic Riccati equation (ARE) has a solution. The state weighting matrix of this ARE can be linearly formulated for all uncertain coefficients and factors. If the weighting matrix is selected to satisfy certain constraints, the solution of ARE is positive, and all closed loop systems are stable. Control of a group of linearized longitudinal motions of an aircraft is employed to illustrate the potential of the proposed method.

Undoubtedly one of the most important concepts in linear systems and control, both from a theoretical as well as from a practical point of view, is the algebraic Riccati equation. Since its introduction in control theory by Kaiman [16] the beginning of the sixties, the algebraic Riccati equation has known an impressive range of applications, such as linear quadratic optimal control, stability theory, stochastic filtering and stochastic control, stochastic realization theory, synthesis of linear passive networks, differential games and, most recently, H ∞ optimal control and robust stabilization. The purpose of the present paper is to give an expository survey of the main concepts, results and applications related to the algebraic Riccati equation.

New results in the Lyapunov-type algorithm for algebraic MCV Riccati equations

In this paper we solve two open problems in linear system theory: the computation of the inner-outer and spectral factorizations of a descriptor continuous-time system considered in the most general setting. Our method is based on descriptor state-space computations and relies on three techniques developed for rational matrices: compression of G to a surjective (full row rank) matrix, and dislocation of the unstable zeros/poles of G to stable locations, all achieved by left multiplication with all-pass factors. The proposed procedures are completely general being applicable for G polynomial/proper/improper, of arbitrary rank, with poles/zeros on the imaginary axis.

The problem of mixed H 2 /H ∞ control can be formulated as a two-player nonzero-sum differential game as done by Limebeer et al. in the 1990s. For linear systems the problem is characterised by two coupled algebraic Riccati equations. Solutions for such algebraic Riccati equations are not straight-forward to obtain, particularly for infinite-horizon problems. In this paper two algorithms for obtaining solutions for the coupled algebraic Riccati equations associated with the mixed H 2 /H ∞ control problem for scalar, linear systems is provided along with illustrative numerical examples.

The standard H/sub /spl infin// problem is solved for LTI systems with a single, pure input lag. The solution is based on state-space analysis, mixing a finite-dimensional and an abstract evolution model. Utilizing the relatively simple structure of these distributed systems, the associated operator Riccati equations are reduced to a combination of two algebraic Riccati equations and one differential Riccati equation over the delay interval. The results easily extend to finite time and time-varying problems where the algebraic Riccati equations are substituted by differential Riccati equations over the process time duration.

Optimal Guaranteed Cost Control of Discrete-time Uncertain Linear Systems

: Numerical issues related to the computational solution of the algebraic matrix Riccati equation were studied. The approach used the generalized eigenproblem formulation for the solution of general forms of algebraic Riccati equations arising in both continuous- and discrete-time applications. These general forms result from control and filtering problems for systems in generalized state space form. A Newton-type iterative refinement procedure for the generalized Riccati solution was derived. Balancing to improve numerical condition was studied. A Fortran package called RICPACK was developed. Numerical experiments with RICPACK were performed to investigate a number of proposed condition numbers. Experience with RICPACK to date indicates that it is the most powerful software yet developed to solve general classes of Riccati equations reliably for a few hundred or less problems. The special structure of models of physical systems given in linear second-order form was also examined. Exploiting that structure in solving associated Riccati equations was studied. Tests for controllability and observability were derived in terms of the original second-order-model matrices. Originator-supplied keywords include: Riccati equations, Generalized matrix Riccati equations, Discrete Riccati equation, Kalman filtering, Numerical condition, Balancing, Generalized eigneproblem, Numerical software, Second-order models.