Rational representations and controller synthesis of L2 behaviors

This paper considers linear dynamical systems restricted to square integrable trajectories. Following the behavioral formalism, a number of relevant classes of linear and shift-invariant L2systems are defined. It is shown that rational functions, analytic in specific half-spaces of the complex plane, prove most useful for representing such systems. For various classes of L2 systems, this paper provides a complete characterization of system equivalence in terms of rational kernel representations of L2 systems. In addition, a complete solution is given for the problem when selected (non-manifest) variables of an L2 system can be completely eliminated from their behavior. This elimination theorem has considerable independent interest in general modeling problems. It is shown that the elimination result is key in the solution of the problem for realizing an arbitrary L2 system as the interconnection of a given L2 system and a to-be-synthesized L2 system. In the context of control, this problem amounts to characterizing the existence and parameterization of all controllers that, after interconnection with a given plant, constitute a desired controlled system

Regulation and robust stabilization: a behavioral approach

In this thesis we consider a number of control synthesis problems within the behavioral approach to systems and control. In particular, we consider the problem of regulation, the H! control problem, and the robust stabilization problem. We also study the problems of regular implementability and stabilization with constraints on the input/output structure of the admissible controllers. The systems in this thesis are assumed to be open dynamical systems governed by linear constant coefficient ordinary differential equations. The behavior of such system is the set of all solutions to the differential equations. Given a plant with its to-be-controlled variable and interconnection variable, control of the plant is nothing but restricting the behavior of the to-be-controlled plant variable to a desired subbehavior. This restriction is brought about by interconnecting the plant with a controller (that we design) through the plant interconnection variable. In the interconnected system the plant interconnection variable has to obey the laws of both the plant and the controller. The interconnected system is also called the controlled system, in which the controller is an embedded subsystem. The interconnection of the plant and the controller is said to be regular if the laws governing the interconnection variable are independent from the laws governing the plant. We call a specification regularly implementable if there exists a controller acting on the plant interconnection variable, such that, in the interconnected system, the behavior of the to-becontrolled variable coincides with the specification and the interconnection is regular. Within the framework of regular interconnection we solve the control problems listed in the first paragraph. Solvability conditions for these problems are independent of the particular representations of the plant and the desired behavior.

On strict passivity and its application to interpolation and Hl control

The authors introduce the L2-system and derive necessary and sufficient conditions for these systems to be strictly passive. Strictly passive L2-systems are characterized as having a representation in terms of a co-J-lossless matrix. A state space proof is developed and provides a Riccati equation characterization of a strictly passive L 2-system, as well as a formula for the co-J-lossless matrix representation. Applications to Nevanlinna-Pick interpolation and an H∞ filtering problem are considere

Distance between Behaviors and Rational Representations

In this paper we study notions of distance between behaviors of linear differential systems. We introduce four metrics on the space of all controllable behaviors which generalize existing metrics on the space of input-output systems represented by transfer matrices. Three of these are defined in terms of gaps between closed subspaces of the Hilbert space L2(R). In particular we generalize the “classical” gap metric. We express these metrics in terms of rational representations of behaviors. In order to do so, we establish a precise relation between rational representations of behaviors and multiplication operators on L2(R). We introduce a fourth behavioral metric as a generalization of the well-known ν-metric. As in the input-output framework, this definition is given in terms of rational representations. For this metric, however, we establish a representation-free, behavioral characterization as well. We make a comparison between the four metrics and compare the values they take and the topologies they induce. Finally, for all metrics we make a detailed study of necessary and sufficient conditions under which the distance between two behaviors is less than one. For this, both behavioral as well as state space conditions are derived in terms of driving variable representations of the behaviors

Supervisory Control Synthesis of Discrete-Event Systems using Coordination Scheme

Supervisory control of discrete-event systems with a global safety specification and with only local supervisors is a difficult problem. For global specifications the equivalent conditions for local control synthesis to equal global control synthesis may not be met. This paper formulates and solves a control synthesis problem for a generator with a global specification and with a combination of a coordinator and local controllers. Conditional controllability is proven to be an equivalent condition for the existence of such a coordinated controller. A procedure to compute the least restrictive solution is also provided in this paper and conditions are stated under which the result of our procedure coincides with the supremal controllable sublanguage.Comment: 29 pages, 11 figure