The regulator problem for linear systems with constrained control: an LMI approach

^** Email: benzaouia{at}ucam.ac.ma A new methodology of the partial eigenstructure assignment bystate feedback via linear matrix inequality (LMI) is extendedto obtain a solution of the constrained regulator problem forlinear continuous-time and discrete-time systems by using theLMI formulation.     

Robust constrained linear regulator problem

^** Email: mesquine{at}ucam.ac.ma^*** Email: a.benlamkadem{at}ucam.ac.ma The robust constrained state and control regulator problem isconsidered. Necessary and sufficient conditions of positiveinvariance are established. A linear programming approach ispresented in order to construct, for an uncertain constrainedlinear system, a stabilizing linear state feedback control.The control law transfers asymptotically to the origin any initialstate belonging to a given set, while constraints on the stateand the control vectors are respected.     

On the Cauchy problem of degenerate hyperbolic equations

In this paper, we study a class of degenerate hyperbolic equations and prove the existence of smooth solutions for Cauchy problems. The existence result is based on a priori estimates of Sobolev norms of solutions. Such estimates illustrate a loss of derivatives because of the degeneracy.

     

Nonautonomous regulator problem in Hilbert spaces

In this paper, we study the quadratic optimal control problem on the half linett _o, for nonautonomous control processes in Hilbert spaces. We prove that the quadratic optimal control problem has a solution if, and only if, an associated Riccati equation has a positive solution fortt _o. The optimal control is given in feedback form. If a detectability assumption holds, then we prove that the optimal control is a stabilizing feedback control when the associated Riccati equation has a positive solution which is bounded fortt _o.This work was performed under the auspices of the National Research Council of Italy (CNR).     

A new equation for the linear regulator problem

In this article, a new equation is derived for the optimal feedback gain matrix characterizing the solution of the standard linear regulator problem. It will be seen that, in contrast to the usual algebraic Riccati equation which requires the solution ofn(n + 1)/2 quadratically nonlinear algebraic equations, the new equation requires the solution of onlynm such equations, wherem is the number of system input terminals andn is the dimension of the state vector of the system. Utilizing the new equation, results are presented for the inverse problem of linear control theory.     

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.     

New optimality condition of fixed-endpoint regulators for single-input linear digital systems and its application to fixed-endpoint LQ regulators

This note discusses the regulator problem with zero terminal state for a single-input, linear, time-invariant, discrete-time system. The fundamental structure of the regulator for a general additive performance index is obtained. The new optimality condition is expressed by the optimality condition of the induced free-endpoint regulator in the first phase and the deadbeat control in the second phase. Furthermore, a closed-loop expression for the quadratic performance index is obtained.     

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.     

Boundary-value technique to solve linear state regulator problems

In this paper, a method is proposed to solve free endpoint optimal control problems with linear state and quadratic cost. After translating the problem into a two-point boundary-value problem (TPBVP) that arises from the optimization of the Hamiltonian, two pointst ₁ andt ₂,t ₁<t ₂, belonging to the given interval of integration [t ₀,t _f], are chosen and conditions are derived at these points appropriately. Then, letting =(t–t ₀)/, =(t _f–t)/, the -scaled, the original, and the -scaled TPBVP's are solved on the intervals [t ₀,t ₁], [t ₁,t ₂], and [t ₂,t _f] respectively as boundary-value problems.The authors are deeply indebted to Dr. S. M. Roberts for his valuable comments and suggestions, which improved the clarity of the paper.     

Extensions of Lemke's algorithm for the linear complementarity problem

Lemke's algorithm for the linear complementarity problem fails when a desired pivot is not blocked. A projective transformation overcomes this difficulty. The transformation is performed computationally by adjoining a new row to a schema of the problem and pivoting on the element in this row and the unit constant column. Two new algorithms result; some conditions for their success are discussed.This research was partially supported by National Science Foundation, Grant GK-42092.     

On the minimal time function for distributed control systems in Banach spaces

In this paper, it is shown that the minimal time function is locally Lipschitz continuous for the control systemx=Ax+u in a Banach spadeE, under either of two conditions:A is linear and generates aC ₀-semigroup of bounded linear operators; orA is nonlinear, possibly multivalued, and dissipative. The main tool used for the nonlinear case is a result of Barbu concerning the null controllability of the system.     

On the robustness of linear stabilizing feedback control for linear uncertain systems: Multi-input case

The robust stabilization of linear systems with constant uncertainties against structured perturbations using Lyapunov's theory is investigated. The only information needed on the uncertainties is the knowledge of their boundaries. The matching conditions of the uncertain systems are not required to be satisfied. It is first shown that, under some assumptions, the system can be transformed into a certain canonical controllable companion form. Then, under some additional assumptions, the existence of a linear controller which stabilizes the system based on Lyapunov's theory is shown.     

On the Pontryagin maximum principle for constant-time linear control systems in Banach spaces

We give some dual characterizations (i.e., in terms of certain suprema) of linear systems satisfying the Pontryagin maximum principle. We give several applications, among which a solution of a problem raised by Rolewicz.     

Riccati equations for generalized singular estimate control systems

We consider the Bolza problem associated with boundary/point control systems governed by strongly continuous semigroups. In continuation of our work in Lasiecka and Tuffaha [I. Lasiecka and A. Tuffaha, Riccati equations for the Bolza problem arising in boundary/point control problems governed by C ₀–semigroups satisfying a singular estimate, J. Optim. Theory Appl. 136 (2008), pp. 229–246; I. Lasiecka and A. Tuffaha, A Bolza optimal synthesis problem for singular estimate control systems, Control Cybernet 38(4B) (2009), pp. 1429–1460], we yet extend the theory to a more general class of control problems that are not analytic providing sharp blow-up rates for the regularity. Solvability of the associated Riccati equations and an optimal feedback synthesis are established. The presence of unbounded control actions, such as boundary/point controls, naturally lead to a singularity at the terminal point t?=?T of the optimal control and of the corresponding feedback operator as before. The class of control systems considered in this article is a generalization to the class usually referred to in the literature as ‘Singular Estimate Control Systems’. The prototype is still that of a PDE system consisting of coupled hyperbolic parabolic dynamics interacting on an interface with point/boundary control. The distinct feature of the class considered in this article is that the degree of unboundedness in the control is stronger than that allowed in the usual singular estimate control system configuration, giving rise to less regular optimal state trajectories.     

The Neumann problem for some degenerate elliptic equations

In the paper we study the equation L _u = f, where L is a degenerate elliptic operator, with Neumann boundary condition in a bounded open set μ. We prove existence and uniqueness of solutions in the space H(μ) for the Neumann problem.     

On the principal eigenvalue of degenerate quasilinear elliptic systems

We study the properties of the positive principal eigenvalue of a degenerate quasilinear elliptic system. We prove that this eigenvalue is simple, unique up to positive eigenfunctions and isolated. Under certain restrictions on the given data, the regularity of the corresponding eigenfunctions is established. The extension of the main result in the case of an unbounded domain is also discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On common linear copositive Lyapunov functions for pairs of stable positive linear systems

We study the common linear copositive Lyapunov functions of positive linear systems. Firstly, we present a theorem on pairs of second order positive linear systems, and give another proof of this theorem by means of properties of geometry. Based on the process of the proof, we extended the results to a finite number of second order positive linear systems. Then we extend this result to third order systems. Finally, for higher order systems, we give some results on common linear copositive Lyapunov functions.     

On the perturbation of the observability equation in linear control systems

The aim of this paper is to investigate stability and sensitivity of the observability variable in linear control systems, (LCS) for short. We first present two results of Hölder continuity in the abstract framework of the ordinary differential equation initial-value problem x′(t) = f(t,x(t)),x(t ₀) = x ₀. Afterwards, we apply our results to automatic systems, providing henceforth the sharpest bounds for the parametric input-output relation in LCS.     

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.     

On perturbation bounds of the linear complementarity problem

In this paper, new perturbation bounds for linear complementarity problems are presented based on the sign patterns of the solution of the equivalent modulus equations. The new bounds are shown to be the generalization of the existing ones.