Nicola Elia

Quantization of linear systems

In this paper, we show that the coarsest quantizer that quadratically stabilizes a single input linear discrete time invariant system is logarithmic, and can be computed by solving a special linear quadratic regulation problem. We provide a closed form for the optimal logarithmic base exclusively in terms of the unstable eigenvalues of the system. We show how to design quantized state-feedback in general, and quantized state estimators in the case where all the eigenvalues of the system are unstable. This leads to the design of output feedback controllers with quantized measurements and controls.

Controller Design via Infinite-Dimensional Linear Programming

This paper addresses the problem of synthesizing controllers to meet specifications that can be represented in terms of linear constraints. A duality result for the problem of minimizing the l1 norm of a closed loop map augmented with linear convex constraints is derived, and it is shown that under mild assumptions, there is no duality gap in the primal-dual programs. The utility of this result is shown through the solution of two problems: the no-overshoot problem, and minimizing the l1 norm of a system subjected to frequency domain constraints.

A quadratic programming approach for solving the l/sub 1/ multiblock problem

The authors present a new method to compute solutions to the general multiblock l/sub 1/ control problem. The method is based on solving a standard H/sub 2/ problem and a finite-dimensional semidefinite quadratic programming problem of appropriate dimension. The new method has most of the properties that separately characterize many existing approaches. In particular, as the dimension of the quadratic programming problem increases, this method provides converging upper and lower bounds on the optimal l/sub 1/ norm and, for well posed multiblock problems, ensures the convergence in norm of the suboptimal solutions to an optimal l/sub 1/ solution. The new method does not require the computation of the interpolation conditions, and it allows the direct computation of the suboptimal controller.

Minimization of the worst case peak-to-peak gain via dynamic programming: state feedback case

Considers the problem of designing a controller that minimizes the worst case peak-to-peak gain of a closed-loop system. In particular, we concentrate on the case where the controller has access to the state of a linear plant and it possibly knows the maximal disturbance input amplitude. We apply the principle of optimality and derive a dynamic programming formulation of the optimization problem. Under mild assumptions, we show that, at each step of the dynamic program, the cost to go has the form of a gauge function and can be recursively determined through simple transformations. We study both the finite horizon and the infinite horizon case under different information structures. The proposed approach allows us to encompass and improve earlier results based on viability theory. In particular, we present a computational scheme alternative to the standard bisection algorithm, or gamma iteration, that allows us to compute the exact value of the worst case peak-to-peak gain for any finite horizon. We show that the sequence of finite horizon optimal costs converges, as the length of the horizon goes to infinity, to the infinite horizon optimal cost. The sequence of such optimal costs converges from below to the optimal performance for the infinite horizon problem. We also show the existence of an optimal state feedback strategy that is globally exponentially stabilizing and derive suboptimal globally exponentially stabilizing strategies from the solutions of finite horizon problems.

Robust performance for fixed inputs

We address the problem of robust performance analysis when the exogenous input is assumed to be fixed and known. This differs from standard approaches in the literature which assume that the exogenous input is an unknown element in a class of norm bounded signals. When the performance is measured by the l/sub /spl infin// norm, and the nominal plant is perturbed by LTV perturbations of bounded l/sub /spl infin// induced norm, we propose upper and lower bounds for the measure of robust performance. Two upper bounds are derived. The first one can have direct application in robust performance synthesis problems. The second one provides a tighter bound. Both conditions are (usually) much less conservative than the condition resulting from assuming a worst case exogenous input. The necessary condition follows from the result of Khammash for robust steady state performance. For certain classes of input signals, these upper and the lower bounds coincide, providing a necessary and sufficient condition.<<ETX>>

Controller design with multiple objectives

This paper addresses the problem of synthesizing controllers that minimize the l/sub 1/ norm of the closed loop system subject to performance specifications expressed as linear constraints on the closed loop map. Convex constraints on the closed loop map, such as frequency point magnitude constraints, can be rewritten as an uncountable set of linear constraints. We use a previously derived duality result to approximate the convex constraint by a finite number of linear constraints and we derive bounds on the accuracy of the solutions of the approximate problems. The above mentioned duality result is also the basis for the analysis of the convergence properties of various computational methods.

Quantized Linear Systems

In this paper, we show that the coarsest quantizer that quadratically stabilizes a single input linear discrete time invariant system is logarithmic, and can be computed by solving a special LQR problem. We provide a close form for the optimal logarithmic base exclusively in terms of the unstable eigenvalues of the system. We show how to design quantized state-feedback in general, and quantized state estimators in the case where all the eigenvalues of the system are unstable. This leads to the design of output feedback controllers with quantized measurements and controls. The theory is then extended to sampling and quantization of continuous time linear systems sampled at constant time intervals. We show that there is a sampling time with an associated logarithmic quantization base that minimizes the density of both sampling and quantization and still ensures stability. The minimization of the density is related to the concept of minimal attention control recently introduced by Brockett. We show that the product of the optimal sampling time and the sum of unstable eigenvalues of the system is a constant independent of the system. Perhaps even more interestingly, the base of the optimal logarithmic quantizer is a constant independent from the system and the value of optimal sampling time. Finally, by relaxing the definition of quadratic stability, we show how to construct logarithmic quantizers with only finite number quantization levels and still achieve practical stability of the closed loop. This final result provides a way to practically implement the theory developed in this paper.

Verification of an automotive active leveler

We analyze an active leveler designed for automotive applications. The objective of the system is to maintain the height of the car body to a fixed value, despite changes in loads and driving conditions. The objective of the paper is to propose a verification method for checking that certain design specifications, or system performances are achieved. We are able to compute exact bounds on the maximum suspension deflection for the given model of the system and road disturbance. The motivation for this work comes from the disappointing results of Stanner et al. (1997) where the problem was approached by using HYTECH. The numerical and computational complexity problems reported in the above article have their common roots in the need to fit and approximate the actual model with a linear hybrid model.

A quadratic programming approach for solving the l/sub 1/ multi-block problem

We present a new method to compute solutions to the general multi-block l/sub 1/ control problem. The method is based on solving a standard H/sub 2/ problem and a finite-dimensional semidefinite quadratic programming problem of appropriate dimension. The new method has most of the properties that separately characterize many existing approaches, in particular, as the dimension of the quadratic programming problem increases, this method provides converging upper and lower bounds on the optimal l/sub 1/ norm and, for well posed multi-block problems, ensures the convergence in norm of the suboptimal solutions to an optimal l/sub 1/ solution. The new method does not require the computation of the interpolation conditions, and it allows the direct computation of the suboptimal controller.

Minimization of the worst-case peak to peak gain via dynamic programming: state feedback case

We consider the problem of designing a controller that minimizes the worst-case peak to peak gain of the closed loop system. We concentrate on the case where the controller has access to the state of a linear plant and it possibly knows the maximal disturbance input amplitude. We apply the principle of optimality and derive a dynamic programming formulation of the optimization problem. We show that, at each step of the dynamic program, the cost to go has the form of a gauge function and can be recursively determined through simple transformations. We study both the finite horizon and the infinite horizon cases. The proposed approach allows one to encompass and improve the recent results based on viability theory. The formulation presented allows one to consider, together with worst case inputs, fixed known inputs, and it can naturally incorporate actuator saturation constraints.