Shaik Fiaz

Equivalence of rational representations of behaviors

This article deals with the equivalence of representations of behaviors of linear differential systems. In general, the behavior of a given linear differential system has many different representations. In this paper we restrict ourselves to kernel and image representations. Two kernel representations are called equivalent if they represent one and the same behavior. For kernel representations defined by polynomial matrices, necessary and sufficient conditions for equivalence are well known. In this paper, we deal with the equivalence of rational representations, i. e. kernel and image representations that are defined in terms of rational matrices. As the first main result of this paper, we will derive a new condition for the equivalence of rational kernel representations of possibly noncontrollable behaviors. Secondly we will derive conditions for the equivalence of rational representations of a given behavior in terms of the polynomial modules generated by the rows of the rational matrices. We will also establish conditions for the equivalence of rational image representations. Finally, we will derive conditions under which a given rational kernel representation is equivalent to a given rational image representation.

Tracking and regulation in the behavioral framework

Given a plant together with an exosystem generating the disturbances and the reference signals, the problem of asymptotic tracking and regulation is to find a controller such that the plant variable tracks the reference signal regardless of the disturbance acting on the system. If a controller achieves this design objective, we call it a regulator for the plant with respect to the given exosystem. In this paper, we formulate the asymptotic tracking and regulation problem in the behavioral framework, with control by interconnection.

Regular Implementability and Stabilization Using Controllers With Pre-Specified Input/Output Partition

This paper deals with the problems of regular implementability and stabilization of a given plant in the context of finite-dimensional linear differential system behaviors. In particular we solve the problems of regular implementability and stabilization using controllers in which a pre-specified subset of the plant control variables is free. We will also extend the results to the situation in which the set of plant control variables is partitioned into two complementary subsets. Variables from one subset should become controller inputs, while variables from the other should become controller outputs. In other words, we consider the problems of regular implementability and stabilization using controllers with a priori given input/output structure.

Realization and elimination in rational representations of behaviors

Abstract This article deals with the relationship between rational representations of linear differential systems and their state representations. In particular we study the relationship between rational representations on the one hand, and output nulling and driving variable representations on the other. In the input–output framework it is well-known that every controllable and observable realization of the transfer matrix of the system yields a minimal input/state/output representation. If a proper rational matrix is used for a rational kernel or image representation of the system, then the question arises under what conditions realizations of this rational matrix give rise to state representations. We will establish conditions under which realizations of the proper rational matrix appearing in a rational image representation yield minimal driving variable representations of the system. Likewise, we find conditions under which realization leads from rational kernel representations to minimal output nulling representations. We also study the converse problem, namely state elimination from driving variable and output nulling representations. We will establish closed form expressions for kernel and image representations of the external behaviors associated with these particular state representations.

Optimal Robust Stabilization and Dissipativity Synthesis by Behavioral Interconnection

Given a nominal plant, together with a fixed neighborhood of this plant, the problem of robust stabilization is to find a controller that stabilizes all plants in that neighborhood (in an appropriate sense). If a controller achieves this design objective, we say that it robustly stabilizes the nominal plant. In this paper we formulate the robust stabilization problem in a behavioral framework, with control as interconnection. We also formulate a relevant behavioral

The decentralized implementability problem

\mathcal{H}_{\infty}

The internal model principle: Asymptotic tracking and regulation in the behavioral framework

synthesis problem, which will be instrumental in solving the robust stabilization problem. We use both rational and polynomial representations for the behaviors under consideration. Necessary and sufficient conditions for the existence of robustly stabilizing controllers are obtained using the theory of dissipative systems. We will also find the optimal stability radius, i.e., the smallest upper bound on the radii of the neighborhoods for which there exists a robustly stabilizing controller. This smallest upper bound is expressed in terms of certain storage functions associated with nominal control system.

Small gain theorem and optimal robust stabilization in a behavioral framework

This paper deals with the problems of decentralized implementability and decentralized regular implementability in the context of finite-dimensional linear differential system behaviors. Given a plant behavior with a pre-specified partition of the system variable and a desired behavior, the problem of decentralized implementability is to find a controller which is decentralized with respect to the given partition and implements (regularly) the desired behavior with respect to the plant. In this paper we formulate these problems in the behavioral framework, with control as interconnection and we also provide necessary and sufficient conditions for the solvability of these problems.

49TH IEEE CONFERENCE ON DECISION AND CONTROL (CDC)

Given a plant, together with an exosystem generating the disturbances and the reference signals, the problem of asymptotic tracking and regulation is to find a controller such that the to-be-controlled plant variable tracks the reference signal regardless of the disturbance acting on the system. If a controller achieves this design objective, we call it a regulator for the plant with respect to the given exosystem. In this paper we formulate the asymptotic tracking and regulation problem in the behavioral framework, with control as interconnection. The problem formulation and its resolution are completely representation free, and specified only in terms of the plant and exosystem dynamics.

New results on the equivalence of rational representations of behaviors

Given a nominal plant, together with a fixed neighborhood of this plant, the problem of robust stabilization is to find a controller that stabilizes all plants in that neighborhood (in an appropriate sense). If a controller achieves this design objective, we say that it robustly stabilizes the nominal plant. In this paper we formulate the robust stabilization problem in a behavioral framework, with control as interconnection. We use both rational as well as polynomial representations for the behaviors under consideration. We obtain a behavioral version of the ‘small gain theorem’ and then obtain necessary and sufficient conditions for the existence of robustly stabilizing controllers using the theory of dissipative systems. We will also find the smallest upper bound on the radii of the neighborhoods for which there exists a robustly stabilizing controller. This smallest upper bound is expressed in terms of certain storage functions associated with the nominal control system.