Infinite dimensional singular discrete control systems and feedback law

Strict Dissipativity for LQ Problem with Left-invertible Underlying Semigroups

LetA, ℬ be evolution operators (possibly nonlinear) which act within a Banach spaceB andu(·) a measurable, real valued, control function. We study control systems of the form ∂υt/∂t=Aυt +u(t)ℬυt, υ0=ϕ ∈B. An observation of this system is defined to be a continuous linear mapg:B→ℝk. Our main result gives a computable sufficient condition to assure that fort > 0 and sufficiently small, the observation of the reference solution (which corresponds tou(t)≡0) at timet is interior to the set of observations of all solutions at timet. An example to illustrate the theory is the local controllability, via tension, of various observations of a vibrating string.

We consider the infinite dimensional linear control system described by the population dynamics model of Lotka-McKendrick with spatial diffusion. Considering control functions localized with respect to the spatial variable but active for all ages, we prove that the whole population can be steered to zero in any positive time. The main novelty we bring is that, unlike the existing results in the literature, we can also control the population of ages very close to 0. Another novelty brought in is the employed methodology: as far as we know, the present work is the first one remarking that the null controllability of the considered system can be obtained by using the Lebeau-Robbiano strategy, originally developed for the null-controllability of the heat equation.

In this work we study the structure of “approximate” solutions for a nonautonomous infinite dimensional discrete-time control system determined by a sequence of continuous functions \(v_i: X \times X \to R^1\), \(i=0,\pm 1,\pm 2,\dots \) where X is a metric space.

Optimal control of stochastic functional neutral differential equations with time lag in control

The problem of robust strictly passive analysis for a class of uncertain discrete singular time-delay systems is investigated in this paper. The discrete singular systems under consideration involve constant time-delay and norm-bounded uncertainties. Based on an integral inequality, a new sufficient condition is firstly obtained, which guarantees that the discrete singular time-delay systems are admissible and strictly passive. Meanwhile, the sufficient condition for robust strictly passive is also obtained in terms of linear matrix inequality (LMI). A numerical example is also given to demonstrate the applicability of the proposed method.

Weak convergence of random sets in Banach spaces

The Linear Quadratic Regulator (LQR), which is arguably the most classical problem in control theory, was recently related to kernel methods in [1] for finite dimensional systems. We show that this result extends to infinite dimensional systems, i.e. control of linear partial differential equations. The quadratic objective paired with the linear dynamics encode the relevant kernel, defining a Hilbert space of controlled trajectories, for which we obtain a concise formula based on the solution of the differential Riccati equation. This paves the way to applying representer theorems from kernel methods to solve infinite dimensional optimal control problems.

The internal properness and stability of singular decentralized control systems are discussed, and a sufficient and necessary condition that singular systems can be properly stabilized by normal decentralized compensators is presented, i.e. there do not exist any unstable finite fixed modes or impulsive fixed modes in the singular system.

By complementarily fusing the robust stabilizability condition, the orthogonal-functions approach (OFA) and the hybrid Taguchi-genetic algorithm (HTGA), an integrative method is proposed in this paper to design the robust-stable and quadratic-optimal static output feedback controller such that i) the linear singular control system with structured parameter uncertainties is regular, impulse-free and asymptotically stable and ii) a quadratic finite-horizon integral performance index for the linear nominal singular control system can be minimized. Based on some essential properties of matrix measures, a new sufficient condition is presented for ensuring that the linear singular system with structured and quadratically-coupled structured parameter uncertainties is regular, impulse free and asymptotically stable. By using the OFA and the robust stabilizability condition, the dynamic-optimization problem for the robust-stable and quadratic-optimal static output feedback control design of the linear uncertain singular system is transformed into a static-constrained-optimization problem represented by the algebraic equations with constraint of robust stabilizability condition; thus greatly simplifying the robust-stable and quadratic-optimal static output feedback control design problem of the linear uncertain singular system. Then, for the static-constrained-optimization problem, the HTGA is employed to find the robust-stable and quadratic-optimal static output feedback controller of the linear uncertain singular control system. One design example of the robust-stable and quadratic-optimal static output feedback controller for a mass-spring-damper mechanical system with structured parameter uncertainties is given to demonstrate the applicability of the proposed integrative approach.

The main objective of this article is to review the major progress that has been made on controllability of dynamical systems over the past number of years. Controllability is one of the fundamental concepts in the mathematical control theory. This is a qualitative property of dynamical control systems and is of particular importance in control theory. A systematic study of controllability was started at the beginning of sixties in the last century, when the theory of controllability based on the description in the form of state space for both time-invariant and time-varying linear control systems was worked out. Roughly speaking, controllability generally means, that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. It should be mentioned, that in the literature there are many different definitions of controllability, which strongly depend on a class of dynamical control systems and on the other hand on the form of admissible controls. Controllability problems for different types of dynamical systems require the application of numerous mathematical concepts and methods taken directly from differential geometry, functional analysis, topology, matrix analysis and theory of ordinary and partial differential equations and theory of difference equations. In the paper we use mainly state-space models of dynamical systems, which provide a robust and universal method for studying controllability of various classes of systems. Controllability plays an essential role in the development of modern mathematical control theory. There are various important relationships between controllability, stability and stabilizability of linear both finite-dimensional and infinite-dimensional control systems. Controllability is also strongly related to the theory of realization and so called minimal realization and canonical forms for linear time-invariant control systems such as the Kalmam canonical form, the Jordan canonical form or the Luenberger canonical form. It should be mentioned, that for many dynamical systems there exists a formal duality between the concepts of controllability and observability. Moreover, controllability is strongly connected with the minimum energy control problem for many classes of linear finite dimensional, infinite dimensional dynamical systems, and delayed systems both deterministic and stochastic. Finally, it is well known, that controllability concept has many important applications not only in control theory and systems theory, but also in such areas as industrial and chemical process control, reactor control, control of electric bulk power systems, aerospce engineering and recently in quantum systems theory.

This paper deals with the reduced order controller design for infinite dimensional port Hamiltonian systems (IDPHS). Firstly, a structure preserving and passive LQG control design equivalent to Control by Interconnection is proposed. Based on this LQG controller, a structure preserving reduction method is used to approximate both the closed loop IDPHS and the LQG controller. This closed loop reduction guarantees that the reduced order controller will ensure acceptable closed loop performances on the infinite dimensional system. The proposed methods is applied to the control of a vibro-acoustic system.

The concept of zeros at infinity is generalized to some particular cases of infinite dimensional control systems: those described with bounded operators and having a finite number of inputs and outputs. These zeros are characterized with the help of four equivalent descriptions. Two geometric characterizations are provided as well as a matricial one using some particular Toeplitz matrices. The last one is directly deduced from the Structure Algorithm. Finally, for systems having a finite number of disturbance inputs, we show that the Disturbance Decoupling Problem with Measurement of the Disturbance is solvable if and only if the orders of the zeros at infinity are the same for the system with and without the disturbance. Applications to invertibility tests or to ideal observability are also given.

The robust stability problem of linear discrete singular time-delay systems with structured (elemental) parameter uncertainties is investigated. Under the assumptions that the nominal discrete singular time-delay system is regular, impulse-free and asymptotically stable, some new sufficient conditions are proposed to preserve the assumed properties when structured (elemental) parameter uncertainties are added into the nominal discrete singular time-delay system. Further new sufficient conditions are also proposed for guaranteeing that the linear discrete singular time-delay system with both structured (elemental) and unstructured (norm-bounded) parameter uncertainties has the properties of regularity, impulse-immunity and asymptotic stability. Four examples are given to demonstrate the applicability of the proposed sufficient conditions.

The stabilization problem of singular linear large-scale control systems with input feedbacks is investigated by using Riccati matrix equation, generalized Lyapunov function method, system decomposition method, singular systems theory and matrix theory. Some sufficient conditions for determining the asymptotical stability and unstability of the corresponding singular closed-loop large-scale systems are given. At last, an illustrate example is given to show the application of main result.