Quantized linear systems are a widely studied class of nonlinear dynamics resulting from the control of a linear system through finite inputs. The stabilization problem for these models shall be studied in terms of the so called practical stability notion that essentially consists in confining the trajectories into sufficiently small neighborhoods of the equilibrium (ultimate boundedness) . In this paper, we study the problem of describing the smallest sets into which any feedback can ultimately confine the state, for a given linear single-input system with an assigned finite set of admissible input values (quantization). We show that a controlled invariant set of minimal size is contained in a family of sets (namely, hypercubes in controller-canonical form), previously introduced in [14, 15] . A comparison is presented which quantifies the improvement in tightness of the proposed analysis technique with respect to classical results using quadratic Lyapunov functions.

}, keywords = {Embedded Control, Robotics}, author = {B. Picasso and A. Bicchi} } @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 {PB06, title = {Practical stabilization of LTI SISO systems under assigned Input and Output quantization}, year = {2006}, note = {ADHS{\textquoteright}06

}, month = {July}, pages = {353-358}, abstract = {This work is concerned with the practical stabilization of discrete{\textendash}time SISO linear systems under assigned quantization of the input and output spaces. A controller is designed which guarantees effective practical stability properties. Unlike most of the existing literature, quantization is supposed to be a problem datum rather than a degree of freedom in design. Moreover, in the framework of control under assigned quantization, results are concerned with state quantization only and do not include the quantized output feedback case considered here. While standard stability analysis techniques are based on Lyapunov theory and invariant ellipsoids, our study of the closed loop dynamics involves a particularly suitable family of sets, which are hypercubes in controller form coordinates.

}, keywords = {Embedded Control, Robotics}, author = {B. Picasso and A. Bicchi} } @article {PB04-rendi, title = {Control synthesis for practical stabilization of quantized linear systems}, journal = {Rend. Sem. Mat. Univ. Pol. Torino, Control Theory and Stabil., I}, volume = {63}, number = {4}, year = {2005}, pages = {397{\textendash}410}, abstract = {In this work we face the stability problem for quantized control systems (QCS). A discrete{\textendash}time single{\textendash}input linear model is considered and, motivated by technological applications, we assume that a uniform quantization of the control set is a priori fixed. As it is well known, for QCS only practical stability properties can be achieved, therefore we focus on the existence and construction of quantized controllers capable of steering a system to within invariant neighborhoods of the equilibrium. The main contribution of the paper consists in a theorem which provides a condition for the practical stabilization in a fixed number of steps: not only the result is interesting in itself, but also it enables to construct a family of stabilizing controllers by means of Model Predictive Control (MPC) techniques. In the last part of the paper some results on the characterization of controlled{\textendash}invariant sets are reviewed and a lower bound on the size of invariant sets is provided.

}, keywords = {Embedded Control, Robotics}, author = {B. Picasso and A. Bicchi} } @conference {PPBJ04, title = {Control of Distributed Embedded Systems in the Presence of Unknown{\textendash}but{\textendash}Bounded Noise}, booktitle = {Proc. IEEE Int. Conf. on Decision and Control}, year = {2004}, pages = {1448-1453}, abstract = {In this paper we consider the problem of controlling multiple scalar systems through a limited capacity shared channel. Each system is affected by process noise and can be controlled byactuators with values in a {\em fixed}inite set. The control objective is to bound the evolution of the systems in specified sets (controlled invariance). Our goal is to find an optimal allocation of the shared communication resource to the different control activities and to identify correct choices for the design parameters. The paper provides fundamental conceptual tools to attack the design problem in the formal framework of an optimization problem. Namely, we give a feasibility criterion to decide whether a set of design parameters conforms with a control specification (i.e., with the controlled invariance of a specified set for each system). Moreover, we offer the explicit computation of the minimum bit rate necessary for the controlled invariance of a set, which is of utmost importance for solving the optimization problem.

}, keywords = {Embedded Control, Robotics}, author = {B. Picasso and L. Palopoli and A. Bicchi and K. H. Johansson} } @conference {BJPP, title = {Quantised Control in Distributed Embedded Systems.}, booktitle = {Proc. 16th Int. Symp. on Mathematical Theory of Networks and Systems}, year = {2004}, abstract = {Traditional control design is based on ideal assumptions concerning the amount, type and accuracy of the information flow that can be circulated across the controller. Unfortunately, real implementation platforms exhibit non-idealities that may substantially invalidate such assumptions. As a result, the systems closed-loop performance can be severely affected and sometimes stability itself is jeopardised. These problems show up with particular strength when multiple feedback loops share a limited pool of computation and communication resources. In this case the designer is confronted with the challenging task of choosing at the same time the control law and the optimal allocation policy for the shared resources (control algortihm/system architecture co-design).

}, keywords = {Embedded Control, Robotics}, author = {A. Bicchi and K. H. Johansson and L. Palopoli and B. Picasso} } @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-ADHS, title = {Receding-Horizon Control of LTI Systems with Quantized Inputs}, booktitle = {Analysis and Design of Hybrid Systems 2003}, year = {2003}, pages = {259-264}, publisher = {Elsevier}, organization = {Elsevier}, abstract = {This paper deals with the stabilization problem for a particular class of hybrid systems, namely discrete-time linear systems subject to a uniform (a priori fixed) quantization of the control set. Results of our previous work on the subject provided a description of minimal (in a specific sense) invariant sets that could be rendered maximally attractive under any quantized feedback strategy. In this paper, we consider the design of stabilizing laws that optimize a given cost index on the state and input evolution on a finite, receding horizon. Application of Model Predictive Control techniques for the solution of similar hybrid control problems through Mixed Logical Dynamical reformulations can provide a stabilizing control law, provided that the feasibility hypotheses are met. In this paper, we discuss precisely what are the shortest horizon length and the minimal invariant terminal set for which it can be guaranteed a stabilizing MPC scheme. The final paper will provide an example and simulations of the application of the control scheme to a practical quantized control problem.

}, keywords = {Embedded Control, Robotics}, author = {B. Picasso and S. Pancanti and A. Bemporad and A. Bicchi}, editor = {Gueguen Engell and Zaytoon} } @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} } @conference {PGB02, title = {Construction of invariant and attractive sets for quantized-input linear systems}, booktitle = {Proc. IEEE Int. Conf. on Decision and Control}, year = {2002}, month = {December}, pages = {824-829}, abstract = {In this paper, the problem of the stabilization of a discrete-time linear system subject to a fixed and uniformly quantized control set is considered. It is well known that, working with quantized inputs, the states of the system (except for a negligible set of initial conditions) cannot reach asymptotically the equilibrium point. Our aim is then to find an invariant and attractive neighborhood of the equilibrium and provide with a controller which steers the system into it. We construct a continuous and increasing family of invariant sets including one which is, in a specific sense, minimal. The invariance and attractivity properties of such sets are revised in the finite control set case: we propose a family of controllers taking on a finite number of values and ensuring the system convergence to the minimal invariant set. Some consequences of our technique are underlined with particular regard to the usage of Model Predictive Control tools. In the last section an example which shows the effectiveness of our results is presented.

}, keywords = {Embedded Control, Robotics}, author = {B. Picasso and F. Gouaisbaut and A. Bicchi} }