George Bitsoris

Unconstrained and constrained stabilisation of bilinear discrete-time systems using polyhedral Lyapunov functions

The constrained and unconstrained stabilisation problem of discrete-time bilinear systems is investigated. Using polyhedral Lyapunov functions, conditions for a polyhedral set to be both positively invariant and domain of attraction for systems with second-order polynomial nonlinearities are first established. Then, systematic methods for the determination of stabilising linear feedback for both constrained and unconstrained bilinear systems are presented. Attention is drawn to the case where no linear control law rendering the pre-specified desired domain of attraction positively invariant exists. For this case, an approach guaranteeing the existence of a possibly suboptimal solution is established.

Design techniques for the control of discrete-time systems subject to state and control constraints

The regulation problem of linear discrete-time systems under state and control constraints is investigated. In the first part of the paper, necessary and sufficient conditions for the existence of a solution to the constrained regulation problem are established. The constructive form of the proof of this result also provides a method for the derivation of a control law transferring to the origin any state belonging to a given set of initial states, while respecting the state and control constraints. In the second part of the paper, design techniques are proposed in which the determination of the control law is reduced to simple linear programming problems.

Invariant set computation for constrained uncertain discrete-time linear systems

In this article a novel approach to the determination of polytopic invariant sets for constrained discrete-time linear uncertain systems is presented. First, the problem of stabilizing a prespecified initial condition set in the presence of input and state constraints is addressed. Second, the problem of computing an estimate of the maximal positively invariant or controlled invariant set for this class of systems is investigated. An illustrative example, showing the effectiveness of the proposed methods, is presented.

Further results on the linear constrained regulation problem

The problem of constrained regulation of linear systems around an equilibrium situated in the interior of a domain of attraction has been extensively investigated. In many engineering problems however, like obstacle avoidance problems, the regulation around an equilibrium situated on the boundary of the domain of attraction is necessary. For this kind of problems, the classical methods cannot be applied and design control methods are missing. Using invariant set techniques, the present paper proposes design methods for guaranteeing convergence to an equilibrium situated on the boundary of the feasible region, all by respecting the state constraints. A collision avoidance numerical example is presented for illustrating the theoretical results of the paper.

Explicit robustness and fragility margins for linear discrete systems with piecewise affine control law

In this paper, we focus on the robustness and fragility problem for piecewise affine (PWA) control laws for discrete-time linear system dynamics in the presence of parametric uncertainty of the state space model. A generic geometrical approach will be used to obtain robustness/fragility margins with respect to the positive invariance properties. For PWA control laws defined over a bounded region in the state space, it is shown that these margins can be described in terms of polyhedral sets in parameter space. The methodology is further extended to the fragility problem with respect to the partition defining the controller. Finally, several computational aspects are presented regarding the transformation from the theoretical formulations to explicit representations (vertex/halfspace representation of polytopes) of these sets.

A fault detection scheme based on controlled invariant sets for multisensor systems

The present paper uses set theoretic methods for the design of a fault tolerant control scheme in the case of a multisensor application. The basic principle is the separation of invariant sets for the estimations of the state and tracking error under healthy and faulty functioning. The fault scenario is based on abrupt changes of the observation equations. The main contribution is the introduction of controlled invariant sets in the fault detection mechanism. The control action is chosen so that the closed loop invariance is assured for a candidate region which accounts for the bounds on the exogenous signals (additive disturbances, noise and reference/set-points).

Low Complexity Constraint Control Using Contractive Sets

This paper proposes a simple, non-iterative way of calculating low-complexity controlled contractive sets, and shows how such set can be used for controller design for systems with state and input constraints. The result is a low complexity controller, suitable for implementation in embedded control systems with modest computational power and available memory.

A Dual Comparison Principle for the Analysis and Design of Discrete-Time Dynamical Systems

Abstract In this article, two new comparison principles for studying positive invariance and stability of linear and nonlinear discrete-time systems are presented. First, a general comparison principle enabling one to determine necessary and sufficient conditions of positive invariance of sets described by their “border surfaces” is developed. Then, a dual comparison principle describing the dynamics of the “vertices” of these sets is established. Both results can be used for the stability analysis and the control design of nonlinear systems.

Explicit fragility margins for PWA control laws of discrete-time linear systems

The paper considers a given discrete-time linear dynamic in closed loop with a piecewise affine control law defined over a bounded polyhedral partition of the state space χ. The objective is to describe the largest error affecting the control law coefficients over each region in the above partition, for which the positive invariance of the set χ is guaranteed with respect to the closed loop dynamics. This subset describes in fact a robust invariance margin with respect to a subset of closed-loop parameters and subsequently represents the explicit fragility margin of the given piecewise affine controller. The fragility characterizes the impact of an error in the implementation of a given controller. It is shown that the largest subset of errors in the state feedback gains is represented by a polyhedral set explicitly constructed with operations over convex domains. A series of remarks on the structure of the numerical problems are discussed and an example is provided for illustration.

Ultimate boundedness and robust stabilization of bilinear discrete-time systems

In this paper nonlinear time-varying and bilinear discrete-time systems with additive bounded disturbances are considered. First, conditions guaranteeing uniform ultimate boundedness for time-varying nonlinear systems are established. Then, algebraic conditions ensuring the existence of polyhedral Lyapunov functions, uniform boundedness and positive invariance for closed-loop bilinear systems are obtained. Finally, these results are applied to various robust stabilization problems for bilinear systems subject to persistent additive disturbances. It is shown that these problems can be reduced to a single or a series of linear programming problems.