Franco Blanchini

Survey paper: Set invariance in control

The properties of positively invariant sets are involved in many different problems in control theory, such as constrained control, robustness analysis, synthesis and optimization. In this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems.

Nonquadratic Lyapunov functions for robust control

Abstract This paper deals with the problem of controlling a linear continuous-time system with structured time-varying parameter uncertainties and input disturbances with a Lyapunov-function approach. In contrast with most of the previous results in the literature, we do not confine our attention to the class of quadratic Lyapunov functions. Conversely, the basic motivation of this paper is to determine whether there exist other functions that can be conveniently used as candidate Lyapunov functions. This question has a positive answer: the proposed class is that of polyhedral norms or, more generally, of polyhedral Minkowski functionals. We show that the class of these functions is universal in the sense that if the problem of ultimately bounding the state in an assigned convex set via state feedback control can be solved via a Lyapunov function and a continuous state-feedback compensator then it can be solved via a polyhedral Lyapunov function and a (possibly different) continuous control. Moreover, we show that the control can be piecewise linear. A numerical technique for constructing the controller is presented for the case in which the uncertainty constraint sets are polyhedral.

Feedback control for linear time-invariant systems with state and control bounds in the presence of disturbances

The problem of the stabilizing linear control synthesis in the presence of state and input bounds for systems with additive unknown disturbances is considered. The only information required about the disturbances is a finite convex polyhedral bound. Discrete- and continuous-time systems are considered. The property of positive D-invariance of a region is introduced, and it is proved that a solution of the problem is achieved by the selection of a polyhedral set S and the computation of a feedback matrix K such that S is positively D-invariant for the closed-loop system. It is shown that if polyhedral sets are considered, the solution involves simple linear programming algorithms. However, the procedure suggested requires a great amount of computational work offline if the state-space dimension is large, because the feedback matrix K is obtained as a solution of a large set of linear inequalities. All of the vertices of S are required. >

A separation principle for linear switching systems and parametrization of all stabilizing controllers

In this paper, we investigate the problem of designing a switching compensator for a plant switching amongst a (finite) family of given configurations (Ai, Bi, Ci). We assume that switching is uncontrolled, namely governed by some arbitrary switching rule, and that the controller has the information of the current configuration i.As a first result, we provide necessary and sufficient conditions for the existence of a family of linear compensators, each applied to one of the plant configurations, such that the closed loop plant is stable under arbitrary switching. These conditions are based on a separation principle, precisely, the switching stabilizing control can be achieved by separately designing an observer and an estimated state (dynamic) compensator. These conditions are associated with (non-quadratic) Lyapunov functions. In the quadratic framework, similar conditions can be given in terms of LMIs which provide a switching controller which has the same order of the plant. As a second result, we furnish a characterization of all the stabilizing switching compensators for such switching plants. We show that, if the necessary and sufficient conditions are satisfied then, given any arbitrary family of compensators K i,(s), each one stabilizing the corresponding LTI plant (Ai, Bi, Ci) for fixed i, there exist suitable realizations for each of these compensators, which assure stability under arbitrary switching.

Persistent disturbance rejection via static-state feedback

In contrast with /spl Hscr//sub /spl infin// and /spl Hscr//sub 2/ control theories, the problem of persistent disturbance rejection (l/sup 1/ optimal control) leads to dynamic controllers, even when the states of the plant are available for feedback. Using viability theory, Shamma showed (1993), in a nonconstructive way, that in the state-feedback case the same performance achieved by any dynamic linear time-invariant controller can be achieved using memoryless nonlinear state feedback. In this paper we give an alternative, constructive proof of these results for discrete- and continuous-time systems. The main result of the paper shows that in both cases, the l/sup 1/ norm achieved by any stabilizing state-feedback linear dynamic controller can be also achieved using a memoryless variable structure controller. >

Robust rate control for integrated services packet networks

Research on congestion-control algorithms has traditionally focused more on performance than on robustness of the closed-loop system to changes in network conditions. As the performance of the control loop is strictly connected with the quality of service, these systems are natural candidates to be approached by the optimal control theory. Unfortunately, this approach may fail in the presence of transmission delay variations, which are unavoidable in telecommunication systems.In this paper, we first show the fragility of optimal controllers and demonstrate their instability when the control delay is not known exactly. Then we propose a robust control algorithm based on a classical proportional integral derivative scheme which does not suffer from this fragility phenomenon. Its stability versus the control delay variations, as well as versus sources that transmit less than their computed share, is studied with Nyquist analysis. The control algorithm is implemented within a simulator in the framework of the asynchronous transfer mode (ATM) ABR transfer capability. The final part of the paper shows some selected results assessing the performance of the control algorithm in a realistic network environment. ABR was chosen as an example, but the control studied here can be applied in any data network to obtain a robust and reliable congestion-control scheme.

Co-Positive Lyapunov Functions for the Stabilization of Positive Switched Systems

In this paper, exponential stabilizability of continuous-time positive switched systems is investigated. For two-dimensional systems, exponential stabilizability by means of a switching control law can be achieved if and only if there exists a Hurwitz convex combination of the (Metzler) system matrices. In the higher dimensional case, it is shown by means of an example that the existence of a Hurwitz convex combination is only sufficient for exponential stabilizability, and that such a combination can be found if and only if there exists a smooth, positively homogeneous and co-positive control Lyapunov function for the system. In the general case, exponential stabilizability ensures the existence of a concave, positively homogeneous and co-positive control Lyapunov function, but this is not always smooth. The results obtained in the first part of the paper are exploited to characterize exponential stabilizability of positive switched systems with delays, and to provide a description of all the “switched equilibrium points” of an affine positive switched system.

Stabilization of LPV Systems: State Feedback, State Estimation, and Duality

In this paper we consider the problem of stabilizing linear parameter varying (LPV) systems by means of gain scheduling control. This technique amounts to designing a controller which is able to update its parameters on-line according to the variations of the plant parameters. We first consider the state feedback case and show a design procedure based on the construction of a Lyapunov function for discrete-time LPV systems in which the parameter variations are affine and occur in the state matrix only. This procedure produces a nonlinear static controller. We show that, different from the robust stabilization case, we can always derive a linear controller, that is, nonlinear controllers cannot outperform linear ones for the gain scheduling problem. Then we show that this procedure has a dual version which leads to the construction of a linear gain scheduling observer. The two procedures may be combined to derive an observer-based linear gain scheduling compensator.

Stabilizability of switched linear systems does not imply the existence of convex Lyapunov functions

Counterexamples are given which show that a linear switched system (with controlled switching) that can be stabilized by means of a suitable switching law does not necessarily admit a convex Lyapunov function. Both continuous and discrete-time cases are considered. This fact contributes in focusing the difficulties encountered so far in the theory of stabilization of switched systems. In particular the result is in contrast with the case of uncontrolled switching in which it is known that if a system is stable under arbitrary switching then admits a polyhedral norm as a Lyapunov function

Constrained control for uncertain linear systems

The linear state feedback synthesis problem for uncertain linear systems with state and control constraints is considered. We assume that the uncertainties are present in both the state and input matrices and they are bounded. The main goal is to find a linear control law assuring that both state and input constraints are fulfilled at each time. The problem is solved by confining the state within a compact and convex positively invariant set contained in the allowable state region.It is shown that, if the controls, the state, and the uncertainties are subject to linear inequality constraints and if a candidate compact and convex polyhedral set is assigned, a feedback matrix assuring that this region is positively invariant for the closed-loop system is found as a solution of a set of linear inequalities for both continuous and discrete time design problems.These results are extended to the case in which additive disturbances are present. The relationship between positive invariance and system stability is investigated and conditions for the existence of positively invariant regions of the polyhedral type are given.