Quadratic optimal control of stable well-posed 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.

     

On minimalization of time varying linear systems with well-posed boundary conditions

The notion of minimal realization of a pair of finite rank operators between Hilbert spaces is introduced. Constructions of such minimal realizations are made. A minimal realization of an integral operator with semi-separable kernel is constructed for the case that one of the realizations has analytic coefficients. A consequence for the characterization of minimal stability of an analytic minimal system is given.     

A characterization of constant-time linear control systems satisfying the Pontryagin maximum principle

Using the subdifferential, we extend the main characterization of Banach linear systems satisfying the Pontryagin maximum principle, given in our previous paper (Ref. 1), to the case whenF andX are locally convex spaces and the norm ofF is replaced by an arbitrary continuous convex functionalh onF.     

Partially stabilizing LQ-optimal control for stabilizable semigroup systems

The Linear-Quadratic optimal control problem with a partial stabilization constraint (LQPS) is considered for exponentially stabilizable infinite dimensional semigroup state-space systems with bounded sensing and control (having their transfer function with entries in the algebra . It is reported that the LQPS-optimal state-feedback operator is related to a nonnegative self-adjoint solution of an operator Riccati equation and it can be identified (1) by solving a spectral factorization problem delivering a bistable spectral factor with entries in the distributed proper-stable transfer function algebra _, and (2) by obtaining any constant solution of a diophantine equation over _. These theoretical results are applied to a simple model of heat diffusion, leading to an approximation procedure converging exponentially fast to the LQPS-optimal state feedback operator.     

q-Difference linear control systems

We propose q-difference versions of some basic results of complete controllability and observability of linear systems such as criterions of controllability and observability, controllability and observability canonical forms, the q-duality theorem, the interconnection between these concepts and that of polynomials without common zeros. As an appendix, is given a simple but computationally meaningful Maple Procedure for the controllability criterion for time constant systems.     

Weighted multiple interpolation and the control of perturbed semigroup systems

In this paper, the controllability and admissibility of perturbed semigroup systems are studied, using tools from the theory of interpolation and Carleson measures. In addition, there are new results on the perturbation of Carleson measures and on the weighted interpolation of functions and their derivatives in Hardy spaces, which are of interest in their own right.     

Resource-optimal control of linear systems

A numerical method for minimizing the resource consumption for linear dynamical systems is proposed. It is based on forming a finite-time control that steers the linear system from an arbitrary initial state to the desired terminal state in a given fixed time; this control gives an approximate solution of the problem. It is shown that the structure of the finite-time control makes it possible to determine the structure of the resource-optimal control. A method for determining an initial approximation is described, and an iterative algorithm for calculating the optimal control is proposed. A system of linear algebraic equations relating the deviations of the initial conditions in the adjoint system to the deviations of the phase coordinates from the prescribed terminal state at the terminal point in time is obtained. A computational algorithm is described. The radius of local convergence is found and the quadratic rate of convergence is established. It is proved that the computational procedure and the sequence of controls converge to the resource-optimal control.     

A characterization of well-posed minimum problems in a complete metric space

Given a functionalI:X R, defined on a complete metric space (X,d), we give a necessary and sufficient condition for the minimum problem forI onX to be well posed or well posed in the generalized sense.This work was performed under the auspices of the Italian Research Council, Grant No. 115-3083-0-5179.     

A control parametrization algorithm for optimal control problems involving linear systems and linear 1

Computational schemes based on control parametrization techniques are known to be very efficient for solving optimal control problems. However, the convergence result is only available for the case in which the dynamic system is linear and without the terminal equality and inequality constraints. This paper is to improve this convergence result by allowing the presence of the linear terminal inequality. For illustration, an example arising in the study of optimally one-sided heating of a metal slab in a furnace is considered.     

On attractors and stability for semigroup actions and control systems

The present paper is dedicated to the study of various aspects of attraction and stability for semigroup actions on topological spaces. The main purpose is to present the connections among the distinct notions of attractors and stable sets. The concept of Conley attractor is also investigated and related to the other notions of attractors. All the results are applied to the theory of control systems.     

A semigroup characterization of the multiparameter Wiener process

It is well known that the infinitesimal operator characterizes the Wiener process {W(t), t≥0}. In this paper we show that the family of operators plays the same role for the q-parameter Wiener process. Partially supported by CNPq Grant No. 200. 607/82.     

A characterization of a free elementary inverse semigroup

The purpose of this note is to give a characterization of a free inverse semigroup on a singleton set. It will be shown that if B is the bicyclic semigroup, then a certain inverse subsemigroup F of B × B is a free elementary inverse semigroup.     

Nonnegative control of finite-dimensional linear systems

We consider the controllability problem for finite-dimensional linear autonomous control systems with nonnegative controls. Despite the Kalman condition, the unilateral nonnegativity control constraint may cause a positive minimal controllability time. When this happens, we prove that, if the matrix of the system has a real eigenvalue, then there is a minimal time control in the space of Radon measures, which consists of a finite sum of Dirac impulses. When all eigenvalues are real, this control is unique and the number of impulses is less than half the dimension of the space. We also focus on the control system corresponding to a finite-difference spatial discretization of the one-dimensional heat equation with Dirichlet boundary controls, and we provide numerical simulations.     

The distinguishability of linear control systems

When observabilities of hybrid systems such as piecewise-affine hybrid systems are considered, the distinguishability of subsystems takes a very important role. However, properties and even definitions of such distinguishabilities have not been extensively studied in the literature. This paper will give a definition of the distinguishability of two linear control systems and present the necessary and sufficient condition of the distinguishability, giving an answer for the problem posed by Vidal et al.     

A characterization of the group congruences on a semigroup

In this paper we describe the group congruences on a semigroupS in terms of their kernels. In particular, we show that the least group congruence σ on a dense and uniteryE-semigroupS is defined by, for alla, b ∈ S, (a, b) ∈ σ iff (εe, f εE(S)) ea=bf Communicated by John M. Howie     

Constrained control for uncertain linear systems

The linear state feedback synthesis problem for uncertain linear systems with state and control constraints is considered. We assume that the uncertainties are present in both the state and input matrices and they are bounded. The main goal is to find a linear control law assuring that both state and input constraints are fulfilled at each time. The problem is solved by confining the state within a compact and convex positively invariant set contained in the allowable state region.It is shown that, if the controls, the state, and the uncertainties are subject to linear inequality constraints and if a candidate compact and convex polyhedral set is assigned, a feedback matrix assuring that this region is positively invariant for the closed-loop system is found as a solution of a set of linear inequalities for both continuous and discrete time design problems.These results are extended to the case in which additive disturbances are present. The relationship between positive invariance and system stability is investigated and conditions for the existence of positively invariant regions of the polyhedral type are given.The author is grateful to Drs. Vito Cerone and Roberto Tempo for their comments.     

Constraint controllability for linear control systems

Summary We investigate some controllability properties of linear control systems in finite or infinite dimensions.     

Quadratic control for linear periodic systems

We propose a new quadratic control problem for linear periodic systems which can be finite or infinite dimensional. We consider both deterministic and stochastic cases. It is a generalization of average cost criterion, which is usually considered for time-invariant systems. We give sufficient conditions for the existence of periodic solutions.Under stabilizability and detectability conditions we show that the optimal control is given by a periodic feedback which involves the periodic solution of a Riccati equation. The optimal closed-loop system has a unique periodic solution which is globally exponentially asymptotically stable. In the stochastic case we also consider the quadratic problem under partial observation. Under two sets of stabilizability and detectability conditions we obtain the separation principle. The filter equation is not periodic, but we show that it can be effectively replaced by a periodic filter. The theory is illustrated by simple examples.This work was done while this author was a visiting professor at the Scuola Normale Superiore, Pisa.     

On guaranteed stability of uncertain linear systems via linear control

This paper addresses the problem of stabilizing an uncertain linear system. The uncertaintyq(·) which enters the dynamics is nonstistical in nature. That is, noa priori statistics forq(·) are assumed; only boundsQ on the admissible variations ofq(·) are taken as given. The results given here applied to so-called matched systems differ from previous results in two ways. Firstly, the stabilizing control in this paper is linear; for this same class of problems, many of the existing results would require a nonlinear control. Furthermore, those results which do in fact yield linear controls are only valid when a certain matrix (q) (formed using the given data) is negative definite for allq Q. In contrast, the theory given here only requires compactness of the bounding setQ. Secondly, we show that the so-called matching conditions (used in earlier work) can be generalized so as to encompass a larger class of dynamical systems.This research was supported by the US Department of Energy under Contract No. ET-78-S-01-3390.