@article {PB07-TAC, title = {On the Stabilization of Linear Systems Under Assigned I/O Quantization}, journal = {IEEE Transactions on Automatic Control}, volume = {52}, number = {10}, year = {2007}, pages = {1994-2000}, abstract = {

This paper is concerned with the stabilization of discrete-time linear systems with quantization of the input and output spaces, i.e., when available values of inputs and outputs are discrete. Unlike most of the existing literature, we assume that how the input and output spaces are quantized is a datum of the problem, rather than a degree of freedom in design. Our focus is hence on the existence and synthesis of symbolic feedback controllers, mapping output words into the input alphabet, to steer a quantized I/O system to within small invariant neighborhoods of the equilibrium starting from large attraction basins. We provide a detailed analysis of the practical stabilizability of systems in terms of the size of hypercubes bounding the initial conditions, the state transient and the steady state evolution. We also provide an explicit construction of a practically stabilizing controller for the quantized I/O case.

}, keywords = {Embedded Control, Robotics}, author = {B. Picasso and A. Bicchi} } @conference {PB04mi, title = {Some relations between ergodicity and minimality properties of invariant sets in quantized control systems}, booktitle = {Bifurcations in nonsmooth and hybrid dynamical systems: analysis, control and applications}, series = {S.I.C.C. (Societ\à Italiana Caos e Complessit\à)}, year = {2004}, note = {

poster presentation

}, month = {21-22 October}, address = {Milan}, abstract = {

Linear dynamical systems controlled by quantized inputs exhibit phenomena which are typically non-linear, including chaotic behaviours. We consider discrete-time single-input models of the type x(k+1)=Ax(k)+bu(k). The construction of invariant sets for this class of hybrid systems is of utmost importance for the stabilization problem. We first review a technique to construct invariant sets when an arbitrary quantized input set is assigned. We hence study minimality properties for invariant sets when inputs take integer values. There is a relation between a so-called strong minimality property and ergodicity of the closed-loop dynamics, in particular, ergodicity implies strong minimality. A condition ensuring strong minimality is given in terms of the coefficients of the characteristic polynomial of the matrix {\textquoteright}A{\textquoteright}. Two examples are presented: the first one shows that strong minimality does not imply ergodicity. The second one shows that our condition for strong minimality is only sufficient: this is done by exhibition of an ergodic dynamics for which our condition is not satisfied.

}, keywords = {Embedded Control, Robotics}, author = {B. Picasso and A. Bicchi} } @inbook {PB03-HSCC, title = {Stabilization of LTI Systems with Quantized State-Quantized Input Static Feedback}, booktitle = {Hybrid Systems: Computation and Control}, series = {Lecture Notes in Computer Science}, volume = {LNCS 2623}, year = {2003}, pages = {405-416}, publisher = {Springer-Verlag}, organization = {Springer-Verlag}, address = {Heidelberg, Germany}, abstract = {

This paper is concerned with the stabilization problem for discrete-time linear systems subject to a uniform quantization of the control set and to a regular state quantization, both fixed a priori. As it is well known, for quantized systems only weak (practical) stability properties can be achieved. Therefore, we focus on the existence and construction of quantized controllers (mapping a quantized state into a quantized input set) capable of steering a system to within invariant neighbourhoods of the equilibrium. Such analysis is helpful because it allows to decide a priori whether adesired control objective can be achieved by using a \textit{given} technology (actuators, sensors, communication and computational means).

}, keywords = {Embedded Control, Robotics}, author = {B. Picasso and A. Bicchi}, editor = {A. Pnueli and O. Maler} }