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.

10aEmbedded Control10aRobotics1 aPicasso, B1 aBicchi, A. uhttps://www.centropiaggio.unipi.it/publications/hypercubes-are-minimal-controlled-invariants-discrete-time-linear-systems-quantized01484nas a2200157 4500008004100000245007500041210006700116300001400183490000700197520094100204653002101145653001301166100001501179700001501194856011701209 2007 eng d00aOn the Stabilization of Linear Systems Under Assigned I/O Quantization0 aStabilization of Linear Systems Under Assigned IO Quantization a1994-20000 v523 aThis 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.

10aEmbedded Control10aRobotics1 aPicasso, B1 aBicchi, A. uhttps://www.centropiaggio.unipi.it/publications/stabilization-linear-systems-under-assigned-io-quantization.html01395nas a2200157 4500008004100000245009300041210006900134260000900203300001200212520081100224653002101035653001301056100001501069700001501084856013801099 2006 eng d00aPractical stabilization of LTI SISO systems under assigned Input and Output quantization0 aPractical stabilization of LTI SISO systems under assigned Input cJuly a353-3583 aThis work is concerned with the practical stabilization of discrete–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.

10aEmbedded Control10aRobotics1 aPicasso, B1 aBicchi, A. uhttps://www.centropiaggio.unipi.it/publications/practical-stabilization-lti-siso-systems-under-assigned-input-and-output-quantization01559nas a2200157 4500008004100000245007800041210006900119300001400188490000700202520100400209653002101213653001301234100001501247700001501262856012401277 2005 eng d00aControl synthesis for practical stabilization of quantized linear systems0 aControl synthesis for practical stabilization of quantized linea a397–4100 v633 aIn this work we face the stability problem for quantized control systems (QCS). A discrete–time single–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–invariant sets are reviewed and a lower bound on the size of invariant sets is provided.

10aEmbedded Control10aRobotics1 aPicasso, B1 aBicchi, A. uhttps://www.centropiaggio.unipi.it/publications/control-synthesis-practical-stabilization-quantized-linear-systems.html01668nas a2200169 4500008004100000245009300041210006900134300001400203520104000217653002101257653001301278100001501291700001601306700001501322700001901337856014201356 2004 eng d00aControl of Distributed Embedded Systems in the Presence of Unknown–but–Bounded Noise0 aControl of Distributed Embedded Systems in the Presence of Unkno a1448-14533 aIn 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.

10aEmbedded Control10aRobotics1 aPicasso, B1 aPalopoli, L1 aBicchi, A.1 aJohansson, K H uhttps://www.centropiaggio.unipi.it/publications/control-distributed-embedded-systems-presence-unknown%E2%80%93%E2%80%93bounded-noise.html01298nas a2200157 4500008004100000245005500041210005400096520078700150653002100937653001300958100001500971700001900986700001601005700001501021856010401036 2004 eng d00aQuantised Control in Distributed Embedded Systems.0 aQuantised Control in Distributed Embedded Systems3 aTraditional 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).

10aEmbedded Control10aRobotics1 aBicchi, A.1 aJohansson, K H1 aPalopoli, L1 aPicasso, B uhttps://www.centropiaggio.unipi.it/publications/quantised-control-distributed-embedded-systems.html01704nas a2200145 4500008004100000245011100041210006900152260002500221520111100246653002101357653001301378100001501391700001501406856013701421 2004 eng d00aSome relations between ergodicity and minimality properties of invariant sets in quantized control systems0 aSome relations between ergodicity and minimality properties of i aMilanc21-22 October3 aLinear 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 'A'. 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.

10aEmbedded Control10aRobotics1 aPicasso, B1 aBicchi, A. uhttps://www.centropiaggio.unipi.it/publications/some-relations-between-ergodicity-and-minimality-properties-invariant-sets-quantized01752nas a2200205 4500008004100000245006600041210006500107260001300172300001200185520111000197653002101307653001301328100001501341700001601356700001601372700001501388700002001403700001201423856011101435 2003 eng d00aReceding-Horizon Control of LTI Systems with Quantized Inputs0 aRecedingHorizon Control of LTI Systems with Quantized Inputs bElsevier a259-2643 aThis 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.

10aEmbedded Control10aRobotics1 aPicasso, B1 aPancanti, S1 aBemporad, A1 aBicchi, A.1 aEngell, Gueguen1 aZaytoon uhttps://www.centropiaggio.unipi.it/publications/receding-horizon-control-lti-systems-quantized-inputs.html01423nas a2200193 4500008004100000245008600041210006900127260004100196300001200237490001400249520074400263653002101007653001301028100001501041700001501056700001401071700001301085856013101098 2003 eng d00aStabilization of LTI Systems with Quantized State-Quantized Input Static Feedback0 aStabilization of LTI Systems with Quantized StateQuantized Input aHeidelberg, GermanybSpringer-Verlag a405-4160 vLNCS 26233 aThis 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).

10aEmbedded Control10aRobotics1 aPicasso, B1 aBicchi, A.1 aPnueli, A1 aMaler, O uhttps://www.centropiaggio.unipi.it/publications/stabilization-lti-systems-quantized-state-quantized-input-static-feedback.html01664nas a2200169 4500008004100000245008500041210006900126260001300195300001200208520106100220653002101281653001301302100001501315700001801330700001501348856013101363 2002 eng d00aConstruction of invariant and attractive sets for quantized-input linear systems0 aConstruction of invariant and attractive sets for quantizedinput cDecember a824-8293 aIn 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.

10aEmbedded Control10aRobotics1 aPicasso, B1 aGouaisbaut, F1 aBicchi, A. uhttps://www.centropiaggio.unipi.it/publications/construction-invariant-and-attractive-sets-quantized-input-linear-systems.html