Eduardo F. Costa

On the Observability and Detectability of Continuous-Time Markov Jump Linear Systems

The paper presents new observability and detectability concepts for continuous-time Markov jump linear systems with finite Markov state, which generalize previous corresponding concepts found in the literature. The concept can also assure that the solution of the coupled algebraic Riccati equation associated to the quadratic control problem is unique and stabilizing, making other concepts redundant. Tests for weak observability and detectability of a system are provided.

On the detectability and observability of discrete-time Markov jump linear systems

Abstract This paper presents a new detectability concept for discrete-time Markov jump linear systems with finite Markov state, which generalizes the MS-detectability concept found in the literature. The new sense of detectability can similarly assure that the solution of the coupled algebraic Riccati equation associated to the quadratic control problem is a stabilizing solution. In addition, the paper introduces a related observability concept that also generalizes previous concepts. A test for detectability based on a coupled matrix equation is derived from the definition, and a test for observability is presented, which can be performed in a finite number of steps. The results are illustrated by examples, including one that shows that a system may be detectable in the new sense but not in the MS sense.

Digital implementation of a magnetic suspension control system for laboratory experiments

In this paper, a digital control experiment using a magnetic suspension system is described. Although the results presented are for a quadratic optimal controller, the control system configuration used can cope with more advanced real-time control algorithms. Practical issues concerning parameter identification, computer simulation, discrete-time implementation, use of a digital signal processor based controller and filter design are addressed. Simulated and experimental responses are presented and analyzed. Finally, a detailed plan on reproducing the control experiment is included.

A New Approach to Detectability of Discrete-Time Infinite Markov Jump Linear Systems

This paper deals with detectability for the class of discrete-time Markov jump linear systems (MJLS) with the underlying Markov chain having countably infinite state space. The formulation here relates the convergence of the output with that of the state variables. Our approach introduces invariant subspaces for the autonomous system and exhibits the role that they play. This allows us to show that detectability can be written equivalently in term of two conditions: stability of the autonomous system in a certain invariant space and convergence of general state trajectories to this invariant space under convergence of input and output variables. This, in turn, provides the tools to show that detectability here generalizes uniform observability ideas as well as previous detectability notions for MJLS with finite state Markov chain, and allows us to solve the jump-linear-quadratic control problem. In addition, it is shown for the MJLS with finite Markov state that the second condition is redundant and that detectability retrieves previously well-known concepts in their respective scenarios.

On the design of guaranteed cost controllers for a class of uncertain linear systems

In this paper, we consider the guaranteed cost control problem (GCCP) for a class of uncertain linear systems with structured parametric uncertainty. We present an approach to the GCCP for systems with arbitrary rank uncertainty matrices based on a rank-one description of the uncertain linear system. A search for an adequate rank-one overbounding of the uncertainty matrices is included in an optimisation problem thus reducing the conservatism of previous methods based on rank-one description of the uncertain system. The paper extends previous results on the GCCP for a rank-one description of the uncertain system, an important description of uncertain systems. A feature of the proposed approach is that an upper bound on the guaranteed cost is minimized by solving an optimization problem with linear matrix inequalities. A numerical example is presented to illustrate the computational efficiency of the proposed approach. For comparison, we also include results for a polytopic description of the uncertain linear system.

Stabilizability and positiveness of solutions of the jump linear quadratic problem and the coupled algebraic Riccati equation

This note addresses the jump linear quadratic problem of Markov jump linear systems and the associated algebraic Riccati equation. Necessary and sufficient conditions for stability of the optimal control and positiveness of Riccati solutions are developed. We show that the concept of weak detectability is not only a sufficient condition for the finiteness of cost functional to imply stability of the associated trajectory, but also a necessary one. This, together with a characterization developed here for the kernel of the Riccati solution, allows us to show that the control solution stabilizes the system if and only if the system is weakly detectable, and that the Riccati solution is positive-definite if and only if the system is weakly observable. The connection between the algebraic Riccati equation and the control problem is made, as far as the minimal positive-semidefinite solution for the algebraic Riccati equation is identified with the optimal solution of the linear quadratic problem. Illustrative numerical examples and comparisons are included.

Quadratic costs and second moments of jump linear systems with general Markov chain

This paper presents an analytic, systematic approach to handle quadratic functionals associated with Markov jump linear systems with general jumping state. The Markov chain is finite state, but otherwise general, possibly reducible and periodic. We study how the second moment dynamics are affected by the additive noise and the asymptotic behaviour, either oscillatory or invariant, of the Markov chain. The paper comprises a series of evaluations that lead to a tight two-sided bound for quadratic cost functionals. A tight two-sided bound for the norm of the second moment of the system is also obtained. These bounds allow us to show that the long-run average cost is well defined for system that are stable in the mean square sense, in spite of the periodic behaviour of the chain and taking into consideration that it may not be unique, as it may depend on the initial distribution. We also address the important question of approximation of the long-run average cost via adherence of finite horizon costs.

Receding horizon control of Markov jump linear systems subject to noise and unobserved state chain

We study the solution of receding horizon control of discrete-time Markov jump linear systems subject to exogenous inputs (noise). The performance index is quadratic and the information available to the controller does not involve observations of Markov chain states. To solve this problem, a sequence of linear feedback gains that is independent of the Markov state is adopted. We propose an interactive method based on a variational procedure which attains the solution to the problem, and an illustrative example is presented.

On a Detectability Concept of Discrete-Time Infinite Markov Jump Linear Systems

Abstract This paper introduces a concept of detectability for discrete-time infinite Markov jump linear systems that relates the stochastic convergence of the output with the stochastic convergence of the state. It is shown that the new concept generalizes a known stochastic detectability concept and, in the finite dimension scenario, it is reduced to the weak detectability concept. It is also shown that the detectability concept proposed here retrieves the well-known property of linear deterministic systems that observability is stricter than detectability.

Technical Communique: Gain scheduled controllers for dynamic systems using sector nonlinearities

In this work we explore the use of gain scheduling for the control of nonlinear systems. The nonlinear system is represented locally by uncertain linear models using sector nonlinearities representation. The uncertain linear models are then used to design a family of robust controllers. We propose a gain scheduling procedure for the problem of guaranteed transition from an actual operating condition to a desired one by constructing a pre-specified path in the state space for the system operating points. As far as we know this problem has not been addressed before. The gain scheduling control procedure given is illustrated in the context of the regulator problem with state feedback.