Stefano Miani

Constrained stabilization of continuous-time linear systems

In this paper we consider the stabilization problem for linear continuous-time systems, under state and control constraints. We show that the largest domain of attraction to the origin can be arbitrarily closely approximated by a polyhedral domain of attraction associated to a certain (continuous) feedback stabilizing control and we show how to use existing numeric procedures for discrete-time systems to solve the continuous-time problem. We propose a new discontinuous stabilizing control law for scalar-input systems which has the advantage of being successfully applicable to systems with quantized control.

Constrained stabilization via smooth Lyapunov functions

Abstract We consider the problem of stabilizing a dynamic system by means of bounded controls. We show that the largest domain of attraction can be arbitrarily closely approximated by a “smooth” domain of attraction for which we provide an analytic expression. Such an expression allows for the determination of a (non-linear) control law in explicit form.

Robust performance with fixed and worst-case signals for uncertain time-varying systems

In this paper we focus our attention on the determination of upper bounds of the l∞ norm of the output of a linear discrete-time dynamic system driven by a step input, in the presence of both persistent unknown, but, l∞ bounded disturbances and memoryless time-varying model uncertainty. For the same type of systems we also analyze the transient behavior of the step response in terms of its overshoot. The problem is solved in a constructive way by determining appropriate invariant sets contained in a given convex region. Finally, we show how to extend these results to continuous-time systems.

A new class of universal Lyapunov functions for the control of uncertain linear systems

We analyze the problem of synthesizing a state feedback control for the class of uncertain continuous-time linear systems affected by time-varying memoryless parametric uncertainties. We consider as candidate Lyapunov functions the elements of the class /spl Sigma//sub p//sup z/ which is formed by special homogeneous positive definite functions. We show that this class is universal, in the sense that a Lyapunov function exists if and only if there exists a Lyapunov function in /spl Sigma//sub p//sup z/. We prove this result in a constructive way, showing that such Lyapunov function can always be obtained by smoothing a polyhedral function for which construction algorithms are available. We show how we can associate to the considered functions controllers in a global explicit form.

Enhancing Controller Performance for Robot Positioning in a Constrained Environment

The paper considers a novel technique for manipulator motion in a constrained environment due to the presence of obstacles. The basic problem is that of avoiding collisions of the manipulator with the obstacles. The main idea is to cover the free space (i.e. the points of the configurations space in which no collisions are possible) by a connected family of polyhedral sets which are controlled-invariant. Each of these polyhedral regions includes some crossing points to the confining regions. The tracking control is hierarchically structured. A high-level controller establishes a connected chain of regions to be crossed to reach the one in which the reference is included. A low-level control solves the problem of tracking, within a region, the crossing point to the next confining region and, eventually, tracking the reference whenever it is included in the current one. The scheme assures that the reference is asymptotically tracked and that the transient trajectory is completely included in the admissible configuration space. A connection graph associated with the cluster of regions, and the high-level control is achieved by solving a minimum-path problem. As far as the low-level control is concerned, we consider both speed-control and torque-control. We propose two types of controllers. The first type is based on a linear stabilizing feedback which is suitably adapted to achieve a local tracking controller. Such a controller is computed by the plane representation of the sets which is more natural and useful then the vertex representation considered in previous work. The second is a speed-saturated type of controller which considerably improves the performance of linear-based control laws. Both these controllers have a speed-control and torque-control version. Experimental results on a laboratory Cartesian robot are provided.

Constant signal tracking for state constrained dynamic systems

We deal with the possibility of tracking constant reference signals for a linear time-invariant dynamic system in the presence of state constraints. We resort to the theory of invariant sets due to their capability of well handling this kind of problems. We focus on the determination of suitable sets for the attainable steady state values and of suitable control laws which guarantee that every possible output steady state value belonging to this set can be reached from any initial state belonging to a proper set. Then, based on recent results on the possibility of associating to these explicit smooth control laws, we derive an explicit controller which allows the system to asymptotically track constant reference signals and guarantees that no constraint violation occurs. Finally, we report an example of the implementation of the proposed control law.

Optimal l∞ disturbance attenuation and global stabilization of linear systems with bounded control

This papers addresses the problem of globally minimizing the worst-case response to persistent l∞ bounded disturbances in linear systems with bounded control action. The main result of the paper shows that in the state-feedback case the best performance among all stabilizing controllers (possibly discontinuous, nonlinear time varying) is achieved by a memoryless, continuous, feedback control law. In the case of open-loop stable plants the proposed control law renders the system globally stable and provides the best possible l∞ attenuation in every neighbourhood of the origin. In the case of open-loop unstable plants this law optimizes performance in the region where a finite l∞ gain can be achieved. Copyright

Worst case l/sup /spl infin// to l/sup /spl infin// gain minimization: dynamic versus static state feedback

It has been recently shown that for the problem of optimal rejection of persistent disturbances using full-state feedback, static nonlinear controllers can recover the performance level achieved by any linear dynamic controller. In this paper we complement these results by showing that for this problem nonlinear dynamic time-varying finite dimensional, possibly discontinuous, compensators do not offer any advantage over memoryless time invariant nonlinear-compensators. Moreover, we show that the best possible (over the set of all stabilizing controllers) disturbance rejection can be achieved by using globally Lipschitz piecewise-linear controllers.

Step response bounds for time varying uncertain systems with bounded disturbances

In this paper we focus our attention on the determination of upper bounds of the l/sup /spl infin// norm of the output of a linear discrete-time dynamic system driven by a step input, in the presence of both persistent unknown-but-l/sup /spl infin// bounded disturbances and memoryless time-varying model uncertainty. For the same type of systems we also analyze the transient behavior of the step response in terms of its overshoot. The problem is solved in a constructive way by determining appropriate D-invariant sets contained in a given convex region. Finally, we show how to extend these results to continuous-time systems.

Control policies for multi-inventory systems with uncertain demand and setups

We consider multi-inventory systems in the presence of unknown but bounded demand and uncertain setups. The control input is assumed to be constant in its operating regime and incur setup whenever a variation of this regime is required. Both the setup times and setup configurations are unknown. We provide necessary and sufficient stabilizability conditions which turn out to be the same for the case in which there are no setups. We also face the problem of ultimately confining the state in an assigned constraint set and provide conditions on this set for the problem to be feasible. Furthermore, we consider the case in which the controls are quantized, as in the case of systems which work in a switching mode. Finally, we deal with the case in which multiple setups may happen during the planning horizon.