Hybrid stabilization of planar linear systems with one-dimensional outputs

Benassi, Carlo; Gavioli, Andrea

doi:10.1016/s0167-6911(02)00126-3

Cited by 12 publications

(15 citation statements)

References 7 publications

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“…During the preparation of this manuscript, it was pointed out that the result of Santarelli et al [12] is similar to the results of [18] and [19], with the exception that the framework of [12] employs continuous-time nonlinear feedback laws, while [18] and [19] employ hybrid feedback automata. The reader who is interested in exploring the differences between the switched feedback approach we take here and the hybrid feedback approach of [18] and [19] is encouraged to explore these two references.…”

Section: S(t) R (M +1) ∀T>0supporting

confidence: 52%

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Comparison between a switching controller and two LTI controllers for a class of LTI plants

Santarelli

Dahleh

2008

Intl J Robust & Nonlinear

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SUMMARYWe consider the design of three different control architectures for a step response tracking problem within a class of linear time-invariant (LTI) plants. Our goal is to motivate the use of a particular switching architecture that has been the subject of our prior work. We present the design of the particular switching architecture that we use and characterize its step response performance (measured in terms of the percentage overshoot and 1% settling time of the step response). We then compare the response of the switching controller with two other forms of LTI control in a servo configuration, one in which the order of the controller is constrained to be first order (which matches the order of the dynamics of the switching controller) and one in which the order of the controller is unconstrained. We will show that the switching architecture can outperform first-order LTI control, first in the context of a particular example. We shall then provide a weak generalization to extend this result to a more general class of plants. We shall also show that, while the LTI control of unconstrained order can outperform the switching architecture, the performance improvement is bounded (in a sense to be defined). Moreover, one method of designing close-to-optimal controllers will be discussed, which yields controllers of very high order.

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Section: S(t) R (M +1) ∀T>0supporting

confidence: 52%

“…The reader who is interested in exploring the differences between the switched feedback approach we take here and the hybrid feedback approach of [18] and [19] is encouraged to explore these two references.…”

Section: S(t) R (M +1) ∀T>0mentioning

confidence: 99%

Comparison between a switching controller and two LTI controllers for a class of LTI plants

Santarelli

Dahleh

2008

Intl J Robust & Nonlinear

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“…Given a plant P with binary control input u(t) ∈ U = {0, 1}, binary sensor output y(t) ∈ Y = {−1, 1} and upper bounded 4 performance output v(t) ∈ R. Consider a system M with inputs u(t) ∈ U and w(t) ∈ W = {0, 1}, and with outputs u(t) ∈ U,ỹ(t) ∈ Y andv(t) ∈ V, where V is a finite discrete subset of R. Suppose that M has the internal structure shown in Figure 3, whereM is a deterministic finite state machine. To ensure well-posedness, we require that there be no direct feedthrough from inputỹ to outputŷ inM : In other words,ŷ(t) is only allowed to be a function of the input u(t) and the state ofM at time t. Memoryless system φ is defined byỹ…”

Section: A a Notion Of Approximationmentioning

confidence: 99%

“…In contrast, the setting considered in this note where fixed, binary sensors are used leads to a more difficult state estimation (and thus controller synthesis) problem 3 . This case has been much less well studied, and has only begun to receive attention in the recent past [4], [20], [23]. Specifically, the use of finite state approximations in this context remains minimally explored.…”

Section: Introduction a Motivation And Overviewmentioning

confidence: 99%

Finite Approximations of Switched Homogeneous Systems for Controller Synthesis

Tarraf

Megretski

Dahleh

2011

IEEE Trans. Automat. Contr.

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We demonstrate the use of a new, control-oriented notion of finite state approximation for a particular class of hybrid systems. Specifically, we consider the problem of designing a stabilizing binary output feedback switching controller for a pair of unstable homogeneous second order systems. The constructive approach presented in this note, in addition to yielding an explicit construction of a deterministic finite state approximate model of the hybrid plant, allows us to efficiently establish a useable upper bound on the quality of approximation, and leads to a discrete optimization problem whose solution immediately provides a certifiably correct-by-design controller for the original system. The resulting controller consists of a finite state observer for the plant and a corresponding full state feedback switching control law. I. INTRODUCTION A. Motivation and OverviewFinite state approximations and abstractions of hybrid plants have been explored as a means for addressing instances in which the interaction between analog and discrete dynamics 1 leads to complex analysis and synthesis problems that cannot be adequately handled by traditional methods. Some of the early work in this area explored the use of 'qualitative' models, namely non-deterministic automata whose output behavior contains that of the original hybrid system [19]. In [21], an approach for deriving non-deterministic finite state approximations of systems with quantized outputs was proposed and used in conjunction with the supervisory control theory developed by Ramadge and Wonham [22] to design controllers meeting the desired specifications. Another line of research, inspired by formal verification techniques in computer science, explored the construction of deterministic discrete abstractions of hybrid systems [2]. Early work in this area focused on identifying classes of systems that admit finite bisimulation abstractions [11], [15]. Having recognized that these classes are fairly limited, more recent work has focused on finding discrete abstractions that are related to the original hybrid system by an approximate simulation or bisimulation relation [10], [25], [26]. The resulting controller design problem is then a two step procedure in which a finite state supervisory controller is first designed, and subsequently refined to yield a certified hybrid controller for the original plant [27].In this note, we demonstrate the use of a new 2 , alternative notion of approximation that, in addition to yielding a finite state deterministic approximate model of the original hybrid

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“…In [2,3], the following result is obtained for B and C being nonzero matrices of rank 1: system (1.1) is stabilizable by a linear hybrid feedback control (LHFC) if and only if for at least one α ∈ R, the matrix A + αBC does not have nonnegative real eigenvalues. This result gives a necessary and sufficient stabilization condition, and it is straightforward that making use of hybrid feedback controls provides a better stabilization criterion compared to any one we can obtain exploiting ordinary feedback controls.…”

Section: Introductionmentioning

confidence: 99%

Efficient criteria for the stabilization of planar linear systems by hybrid feedback controls

Litsyn

Myasnikova

Nepomnyashchikh

et al. 2004

Abstract and Applied Analysis

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We suggest some criteria for the stabilization of planar linear systems via linear hybrid feedback controls. The results are formulated in terms of the input matrices. For instance, this enables us to work out an algorithm which is directly suitable for a computer realization. At the same time, this algorithm helps to check easily if a given linear 2 × 2 system can be stabilized (a) by a linear ordinary feedback control or (b) by a linear hybrid feedback control.

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Hybrid stabilization of planar linear systems with one-dimensional outputs

Cited by 12 publications

References 7 publications

Comparison between a switching controller and two LTI controllers for a class of LTI plants

Comparison between a switching controller and two LTI controllers for a class of LTI plants

Finite Approximations of Switched Homogeneous Systems for Controller Synthesis

Efficient criteria for the stabilization of planar linear systems by hybrid feedback controls

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