Construction of invariant and attractive sets for quantized-input linear systems

Picasso, Bruno; Gouaisbaut, Frédéric; Bicchi, Antonio

doi:10.1109/cdc.2002.1184607

Cited by 27 publications

(20 citation statements)

References 13 publications

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“…A simple modification of the above construction leads to a class of (Q ∆ , Q )-stabilizing feedbacks, as shown in the following proposition (see [16]). …”

Section: Uniform Quantized Feedbackmentioning

confidence: 99%

“…Sections 2 and 3 are devoted to a careful analysis of some examples of quantized feedback stabilization techniques for which we evaluate the various complexity and performance indices. We first consider a memoryless uniform quantizer like the one considered in [16]. Then we show that, by nesting scaled version of this quantizer, we obtain a quantized feedback map with performance similar to the logarithmic quantizer proposed in [5].…”

Section: Introductionmentioning

confidence: 99%

“…The stabilization problem by quantized feedback has been widely studied in the last few years: see [1,2,4,5,7,13,16,18,19] and the reference therein. Quantization can not be avoided in the digital control setting and it is indeed a natural way to insert into the control design complexity constraints of the controller and communication constraints of the channels which connect the controller and the plant.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Quantized Stabilization of Linear Systems: Complexity Versus Performance

Fagnani

Zampieri

2004

IEEE Trans. Automat. Contr.

100

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Quantized feedback control has been receiving much attention in the control community in the past few years. Quantization is indeed a natural way to take into consideration in the control design the complexity constraints of the controller as well as the communication constraints in the information exchange between the controller and the plant. In this paper we analyze the stabilization problem for discrete time linear systems with multidimensional state and one-dimensional input using quantized feedbacks with a memory structure, focusing on the trade off between complexity and performance. A quantized controller with memory is a dynamical system with a state space, a state updating map and an output map.The quantized controller complexity is modelled by means of three indices. The first index L coincides with the number of the controller states. The second index is the number M of the possible values that the state updating map of the controller can take at each time. The third index is the number N of the possible values that the output map of the controller can take at each time. The index N corresponds also to the number of the possible control values that the controller can choose at each time.In this paper the performance index is chosen to be the time T needed to shrink the state of the plant from a starting set to a target set. Finally, the contraction rate C, namely the ratio between the volumes of the starting and target sets, is introduced. We evaluate the relations between these parameters for various quantized stabilizers, with and without memory, and we make some comparisons. Then we prove a number of results showing the intrinsic limitations of the quantized control. In particular we show that, in order to obtain a control strategy which yields arbitrarily small values of T / ln C (requirement which can be interpreted as a weak form of the pole assignability property), we need to have that LN/ ln C is big enough.

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“…A simple modification of the above construction leads to a class of (Q ∆ , Q )-stabilizing feedbacks, as shown in the following proposition (see [16]). …”

Section: Uniform Quantized Feedbackmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Quantized Stabilization of Linear Systems: Complexity Versus Performance

Fagnani

Zampieri

2004

IEEE Trans. Automat. Contr.

100

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“…The stabilization by quantized feedback controllers has been widely investigated in the last few years (see [6,26,3,25,20,7,2,9,23] and the reference therein). There are two different situations in which quantization reveals to be a central feature in the control design.…”

Section: Introductionmentioning

confidence: 99%

A Symbolic Approach to Performance Analysis of Quantized Feedback Systems: The Scalar Case

Fagnani¹,

Zampieri²

2005

SIAM J. Control Optim.

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When dealing with the control of a large number of interacting systems, the fact that the flow of information has to be limited becomes an essential feature of the control design. The first consequence of the limited information flow constraint is that the signals that the controllers and the systems exchange have to be quantized. Though quantization has already been extensively considered in the control literature, its analysis from the point of view of the information flow demand has been considered only recently.Limiting the information flow between a plant and a controller will necessary lead to a performance degradation of the feedback loop, and we expect a trade-off between the achievable performance and the amount of information exchange allowed in the loop.Most of the success of modern digital communication theory in the last fifty years is due to the contributions of information theory, which proposed a symbolic based analysis of the communication channel performance. The same goal is much more difficult to be reached in digital control theory.The present paper proposes an attempt towards this direction. The main contribution of this paper is to provide a complete analysis of the trade-off between performance and information flow in the simple case of the stabilization of a scalar linear system by means of a memoryless quantized feedback map.

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“…1 They are generally computed in a forward manner by computing iteratively the union of the images of a reference point until a fixed-point is reached. It is impossible in general to obtain an explicit characterization of the mCI set, and several methods have been found for computing an overapproximation [13,16,17]. As far as we know, the problem of mCI synthesis for switched systems has not yet been studied.…”

Section: Introductionmentioning

confidence: 99%

Finite controlled invariants for sampled switched systems

Fribourg

Kühne

Soulat

2014

Form Methods Syst Des

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Abstract. We consider in this paper switched systems, a class of hybrid systems recently used with success in various domains such as automotive industry and power electonics. We propose a state-dependent control strategy which makes the trajectories of the analyzed system converge to finite cyclic sequences of points. Our method relies on a technique of decomposition of the state space into local regions where the control is uniform. We have implemented the procedure using zonotopes, and applied it successfully to several examples of the literature and industrial case studies in power electronics.

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Construction of invariant and attractive sets for quantized-input linear systems

Cited by 27 publications

References 13 publications

Quantized Stabilization of Linear Systems: Complexity Versus Performance

Quantized Stabilization of Linear Systems: Complexity Versus Performance

A Symbolic Approach to Performance Analysis of Quantized Feedback Systems: The Scalar Case

Finite controlled invariants for sampled switched systems

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