A Behavioral Approach to the Control of Discrete Linear Repetitive Processes

This paper formulates the theory of linear discrete time repetitive processes in the setting of behavioral systems theory. A behavioral, latent variable model for repetitive processes is developed and for the physically defined inputs and outputs as manifest variables, a kernel representation of their behavior is determined. Conditions for external stability and controllability of the behavior are then obtained. A sufficient condition for stabilizability is also developed for the behavior and it is shown under a mild restriction that, whenever the repetitive system is stabilizable, a regular constant output feedback stabilizing controller exists. Next a notion of eigenvalues is defined for the repetitive process under an action of a closed loop controller. It is then shown how under controllability of the original repetitive process, an arbitrary assignment of eigenvalues for the closed loop response can be achieved by a constant gain output feedback controller under the above restriction. These results on the existence of constant gain output feedback controllers are among the most striking properties enjoyed by repetitive systems, discovered in this paper. Results of this paper utilize the behavioral model of the repetitive process which is an analogue of the 1D equivalent model of the dynamics studied in earlier work on repetitive processes

Reduced Order Controller Design for Robust Output Regulation

We study robust output regulation for parabolic partial differential equations and other infinite-dimensional linear systems with analytic semigroups. As our main results we show that robust output tracking and disturbance rejection for our class of systems can be achieved using a finite-dimensional controller and present algorithms for construction of two different internal model based robust controllers. The controller parameters are chosen based on a Galerkin approximation of the original PDE system and employ balanced truncation to reduce the orders of the controllers. In the second part of the paper we design controllers for robust output tracking and disturbance rejection for a 1D reaction-diffusion equation with boundary disturbances, a 2D diffusion-convection equation, and a 1D beam equation with Kelvin-Voigt damping.Comment: Revised version with minor improvements and corrections. 28 pages, 9 figures. Accepted for publication in the IEEE Transactions on Automatic Contro

Data-Driven Stabilizing and Robust Control of Discrete-Time Linear Systems with Error in Variables

This work presents a sum-of-squares (SOS) based framework to perform data-driven stabilization and robust control tasks on discrete-time linear systems where the full-state observations are corrupted by L-infinity bounded input, measurement, and process noise (error in variable setting). Certificates of state-feedback superstability or quadratic stability of all plants in a consistency set are provided by solving a feasibility program formed by polynomial nonnegativity constraints. Under mild compactness and data-collection assumptions, SOS tightenings in rising degree will converge to recover the true superstabilizing controller, with slight conservatism introduced for quadratic stabilizability. The performance of this SOS method is improved through the application of a theorem of alternatives while retaining tightness, in which the unknown noise variables are eliminated from the consistency set description. This SOS feasibility method is extended to provide worst-case-optimal robust controllers under H2 control costs. The consistency set description may be broadened to include cases where the data and process are affected by a combination of L-infinity bounded measurement, process, and input noise. Further generalizations include varying noise sets, non-uniform sampling, and switched systems stabilization.Comment: 27 pages, 1 figure, 9 table

Feedback control of spin systems

The feedback stabilization problem for ensembles of coupled spin 1/2 systems is discussed from a control theoretic perspective. The noninvasive nature of the bulk measurement allows for a fully unitary and deterministic closed loop. The Lyapunov-based feedback design presented does not require spins that are selectively addressable. With this method, it is possible to obtain control inputs also for difficult tasks, like suppressing undesired couplings in identical spin systems.Comment: 16 pages, 15 figure

Theory of nonlinear feedback under uncertainty

AbstractOur main purpose here is to demonstrate the potential of a new approach which is an important expansion of the feedback concept: we have chosen what seemed a natural way of tackling some traditional problems of the control theory and of comparing the results against those offered by conventional methods.The main problem considered is the output stabilization for uncertain plants. Using structural transformations, uncertain systems can change to the form convenient for output feedback design. Synthesis of observer-based control for asymptotical stabilization or uniform ultimate boundedness of the closed-loop system is provided.We consider the notions of asymptotic and exponential invariance of a control system implies its suboptimality.A method is described for stabilization of uncertain discrete-time plants of which only compact sets are known to which plants parameters and exogenous signals belong. New approaches for solving some central problems of mathematical control theory are considered for nonlinear dynamical systems. New criterious of local and global controllability and stabilizability are indicated and some synthesis procedures are suggested

LMI based Stability and Stabilization of Second-order Linear Repetitive Processes

This paper develops new results on the stability and control of a class of linear repetitive processes described by a second-order matrix discrete or differential equation. These are developed by transformation of the secondorder dynamics to those of an equivalent first-order descriptor state-space model, thus avoiding the need to invert a possibly ill-conditioned leading coefficient matrix in the original model

Error Correcting Codes for Distributed Control

The problem of stabilizing an unstable plant over a noisy communication link is an increasingly important one that arises in applications of networked control systems. Although the work of Schulman and Sahai over the past two decades, and their development of the notions of "tree codes"\phantom{} and "anytime capacity", provides the theoretical framework for studying such problems, there has been scant practical progress in this area because explicit constructions of tree codes with efficient encoding and decoding did not exist. To stabilize an unstable plant driven by bounded noise over a noisy channel one needs real-time encoding and real-time decoding and a reliability which increases exponentially with decoding delay, which is what tree codes guarantee. We prove that linear tree codes occur with high probability and, for erasure channels, give an explicit construction with an expected decoding complexity that is constant per time instant. We give novel sufficient conditions on the rate and reliability required of the tree codes to stabilize vector plants and argue that they are asymptotically tight. This work takes an important step towards controlling plants over noisy channels, and we demonstrate the efficacy of the method through several examples.Comment: 39 page

Analysis, estimation and control for perturbed and singular systems and for systems subject to discrete events.

Annual technical report for grant AFOSR-88-0032.Investigators: Alan S. Willsky, George C. Verghese.Includes bibliographical references (p. [10]-[15]).Research supported by the AFOSR. AFOSR-88-003