Switching LPV control designs using multiple parameter-dependent Lyapunov functions
TL;DR: This paper studies the switching control of linear parameter-varying (LPV) systems using multiple parameter-dependent Lyapunov functions to improve performance and enhance control design flexibility.
About: This article is published in Automatica.The article was published on 2004-11-01. It has received 241 citations till now. The article focuses on the topics: Lyapunov function & Linear matrix inequality.
Citations
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TL;DR: This paper provides a survey of results in linear parameter-varying (LPV) control that have been validated by experiments and/or high-fidelity simulations.
Abstract: This paper provides a survey of results in linear parameter-varying (LPV) control that have been validated by experiments and/or high-fidelity simulations. The LPV controller synthesis techniques employed in the references of this survey are briefly reviewed and compared. The methods are classified into polytopic, linear fractional transformation, and gridding-based techniques and it is reviewed how in each of these approaches, synthesis can be carried out as a convex optimization problem via a finite number of linear matrix inequalities (LMIs) for both parameter-independent and parameter-dependent Lyapunov functions. The literature is categorized with regard to the application, the complexity induced by the controlled system’s dynamic and scheduling orders, as well as the synthesis method. Exemplary cases dealing with specific control design problems are presented in more detail to point control engineers to possible approaches that have been successfully applied. Furthermore, key publications in LPV control are related to application achievements on a timeline.
303 citations
TL;DR: A systematic switching LPV control design method is presented to determine if it is practical to use for flight control designs over a wide angle of attack region and parameter-dependent switching logics, hysteresis switching and switching with average dwell time, are examined.
Abstract: In flight control, the design objective and the aircraft dynamics may be different in low and high angle of attack regions. This paper presents a systematic switching LPV control design method to determine if it is practical to use for flight control designs over a wide angle of attack region. The approach is based on multiple parameter-dependent Lyapunov functions. A family of LPV controllers are designed, and each of them is suitable for a specific parameter subspace. The state of the controller is reset to guarantee the stability requirement of the Lyapunov function when the switching event occurs. Two parameter-dependent switching logics, hysteresis switching and switching with average dwell time, are examined. The proposed switching LPV control scheme is applied to an F-16 aircraft model with different design objectives and aircraft dynamics in low and high angle of attack regions. The nonlinear simulation results using both switching logics are compared.
270 citations
Cites background or methods or result from "Switching LPV control designs using..."
...In the previous research of switching LPV control [ 18 ], the state of the controller was preserved before and after switching, and this resulted in nonconvex matrix optimization problems....
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...Following the idea in [ 18 ], we partition the Lyapunov function matrices of the closed-loop system (3) according to the dimensions of the plant and controller states as...
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...Different from hysteresis switching, the synthesis condition of switching LPV control with average dwell time cannot be convexified under some conservative assumptions if the control state is preserved when switching [ 18 ]....
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...Recently, the authors have developed switching LPV control techniques under hysteresis and average dwell time switching logics [ 18 ]....
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...Proof: The proof is similar to that in [ 18 ]....
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TL;DR: A parameterized reduced-model is constructed and the corresponding existence conditions of such models are derived via strict LMI formulation such that the resulting model error system is exponentially stable and has a guaranteed l 2-l infin error performance.
Abstract: In this note, the model reduction problem for a class of discrete-time switched linear parameter varying systems under average dwell time switching is investigated. A parameterized reduced-model is constructed and the corresponding existence conditions of such models are derived via strict LMI formulation. The minimal average dwell time among all the subsystems and the desired reduced system are obtained such that the resulting model error system is exponentially stable and has a guaranteed l 2-l infin error performance. A numerical example is given to demonstrate the potential and effectiveness of the developed theoretical results.
202 citations
Cites background from "Switching LPV control designs using..."
...…Euclidean space and represents the set of nonnegative integers, the notation refers to the Euclidean vector norm. is the space of square summable infinite sequence and for , its norm is given by , is the space of all essentiallyboundedfunctionsandfor , its norm isgivenby . denotes…...
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TL;DR: In this article, the authors investigated the problem of exponential H∞ filter problem for a class of discrete-time polytopic uncertain switched linear systems with average dwell time switching and derived sufficient existence conditions for the desired filter in terms of a set of linear matrix inequalities.
Abstract: In this paper, the problem of exponential H∞ filter problem for a class of discrete-time polytopic uncertain switched linear systems with average dwell time switching is investigated. The exponential stability result of the general discrete-time switched systems using a discontinuous piecewise Lyapunov function approach is first explored. Then, a new µ-dependent approach is proposed, which means the analysis or synthesis of the underlying systems is dependent on the increase degree µ of the piecewise Lyapunov function at the switching instants. A mode-dependent full-order filter is designed such that the developed filter error system is robustly exponentially stable and achieves an exponential H∞ performance. Sufficient existence conditions for the desired filter are derived and formulated in terms of a set of linear matrix inequalities, and consequently the minimal average dwell time and the corresponding filter are obtained from such conditions for a given system decay degree. A numerical example is presented to demonstrate the potential and effectiveness of the developed theoretical results.
193 citations
TL;DR: A parameter-dependent multiple discontinuous Lyapunov function (PMDLF) approach is proposed to study the refined antidisturbance control problem of switched linear parameter-varying systems, and an example of an aero-engine control system is given to verify the availability of the acquired approaches.
Abstract: In this paper, a parameter-dependent multiple discontinuous Lyapunov function (PMDLF) approach is proposed to study the $H_\infty$ refined antidisturbance control problem of switched linear parameter-varying systems. The $H_\infty$ refined antidisturbance means the disturbance appearing in the control channel can be accurately compensated by means of the estimation of the disturbance and the energy bounded external disturbance can be restrained. A key point is to set up a PMDLF framework that provides an effective tool for attenuating the energy bounded disturbances and rejecting the disturbances generated by the exosystem accurately. A parameter-driven and dwell time-dependent switching law is designed, and a solvability condition ensuring the $H_\infty$ refined antidisturbance performance is developed. Then, the $H_\infty$ refined antidisturbance switched parameter-dependent disturbance observers and the disturbance observer-based refined controllers are established to achieve required disturbance attenuation and rejection. Finally, an example of an aero-engine control system is given to verify the availability of the acquired approaches.
148 citations
Cites background or methods or result from "Switching LPV control designs using..."
...Compared with the literatures [1], [5], [12], [29], the positive definite matrix function Pk (t, ) is time and parameter-varying....
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...In [5], the average dwell time switching is designed to acquire the stability and L2 gain, and stability of all sub-systems are as a priori....
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...Afterwards, in [5], the LPV system is handled by using the hybrid control....
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...Here, we adopt the gridding method [5] to tackle this problem over the parameter space....
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...Inspired by [5], hybrid systems modeling methods are further developed to establish a switched LPV model, which exhibits the reliability and flexibility to cope with large-scale parameter variations [6], [7]....
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References
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Book•
24 Jun 2003
TL;DR: I. Stability under Arbitrary Switching, Systems not Stabilizable by Continuous Feedback, and Systems with Sensor or Actuator Constraints with Large Modeling Uncertainty.
Abstract: I. INTRODUCTION 1. Basic Concepts II . STABILITY OF SWITCHED SYSTEMS 2. Stability under Arbitrary Switching 3. Stability under Constrained Switching III. SWITCHING CONTROL 4. Systems not Stabilizable by Continuous Feedback 5. Systems with Sensor or Actuator Constraints 6. Systems with Large Modeling Uncertainty IV. SUPPLEMENTARY MATERIAL A. Stability B. Lie Algebras Notes and References Bibliography Index
5,844 citations
TL;DR: Bendixson's theorem is extended to the case of Lipschitz continuous vector fields, allowing limit cycle analysis of a class of "continuous switched" systems.
Abstract: We introduce some analysis tools for switched and hybrid systems. We first present work on stability analysis. We introduce multiple Lyapunov functions as a tool for analyzing Lyapunov stability and use iterated function systems theory as a tool for Lagrange stability. We also discuss the case where the switched systems are indexed by an arbitrary compact set. Finally, we extend Bendixson's theorem to the case of Lipschitz continuous vector fields, allowing limit cycle analysis of a class of "continuous switched" systems.
3,289 citations
07 Dec 1999
TL;DR: In this article, it was shown that scale-independent hysteresis can produce switching that is slow-on-the-average and therefore the results mentioned above can be used to study the stability of adaptive control systems.
Abstract: It is shown that switching among stable linear systems results in a stable system provided that switching is "slow-on-the-average". In particular, it is proved that exponential stability is achieved when the number of switches in any finite interval grows linearly with the length of the interval, and the growth rate is sufficiently small. Moreover, the exponential stability is uniform over all switchings with the above property. For switched systems with inputs this guarantees that several input-to-state induced norms are bounded uniformly over all slow-on-the-average switchings. These results extend to classes of nonlinear switched systems that satisfy suitable uniformity assumptions. In this paper it is also shown that, in a supervisory control context, scale-independent hysteresis can produce switching that is slow-on-the-average and therefore the results mentioned above can be used to study the stability of hysteresis-based adaptive control systems.
2,197 citations
01 Jan 1999
TL;DR: In this article, it was shown that switching among stable linear systems results in a stable system provided that switching is slow-on-the-average, i.e., the number of switches in any nite interval grows linearly with the length of the interval, and the growth rate is suciently small.
Abstract: It is shown that switching among stable linear systems results in a stable system provided that switching is \slow-on-the-average." In particular, it is proved that exponential stability is achieved when the number of switches in any nite interval grows linearly with the length of the interval, and the growth rate is suciently small. Moreover, the exponential stability is uniform over all switchings with the above property. For switched systems with inputs this guarantees that several input-to-state induced norms are bounded uniformly over all slow-on-the-average switchings. These results extend to classes of nonlinear switched systems that satisfy suitable uniformity assumptions. In this paper it is also shown that, in a supervisory control context, scale-independent hysteresis can produce switching that is slow-on-the-average and therefore the results mentioned above can be used to study the stability of hysteresis-based adaptive control systems.
2,028 citations
01 Jul 2000
TL;DR: In this paper, the authors introduce the concept of hybrid systems and some of the challenges associated with the stability of such systems, including the issues of guaranteeing stability of switched stable systems and finding conditions for the existence of switched controllers for stabilizing switched unstable systems.
Abstract: This paper introduces the concept of a hybrid system and some of the challenges associated with the stability of such systems, including the issues of guaranteeing stability of switched stable systems and finding conditions for the existence of switched controllers for stabilizing switched unstable systems. In this endeavour, this paper surveys the major results in the (Lyapunov) stability of finite-dimensional hybrid systems and then discusses the stronger, more specialized results of switched linear (stable and unstable) systems. A section detailing how some of the results can be formulated as linear matrix inequalities is given. Stability analyses on the regulation of the angle of attack of an aircraft and on the PI control of a vehicle with an automatic transmission are given. Other examples are included to illustrate various results in this paper.
1,647 citations