A convex characterization of gain-scheduled H/sub /spl infin// controllers

doi:10.1109/9.384219

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A convex characterization of gain-scheduled H/sub /spl infin// controllers

Journal Article•DOI•

A convex characterization of gain-scheduled H/sub /spl infin// controllers

Pierre Apkarian¹, Pascal Gahinet²•Institutions (2)

Community emergency response team¹, French Institute for Research in Computer Science and Automation²

01 May 1995-IEEE Transactions on Automatic Control (IEEE)-Vol. 40, Iss: 5, pp 853-864

TL;DR: Extensions of H/sub /spl infin// synthesis techniques to allow for controller dependence on time-varying but measured parameters are discussed and simple heuristics are proposed to compute robust time-invariant controllers.

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Abstract: An important class of linear time-varying systems consists of plants where the state-space matrices are fixed functions of some time-varying physical parameters /spl theta/. Small gain techniques can be applied to such systems to derive robust time-invariant controllers. Yet, this approach is often overly conservative when the parameters /spl theta/ undergo large variations during system operation. In general, higher performance can be achieved by control laws that incorporate available measurements of /spl theta/ and therefore "adjust" to the current plant dynamics. This paper discusses extensions of H/sub /spl infin// synthesis techniques to allow for controller dependence on time-varying but measured parameters. When this dependence is linear fractional, the existence of such gain-scheduled H/sub /spl infin// controllers is fully characterized in terms of linear matrix inequalities. The underlying synthesis problem is therefore a convex program for which efficient optimization techniques are available. The formalism and derivation techniques developed here apply to both the continuous- and discrete-time problems. Existence conditions for robust time-invariant controllers are recovered as a special case, and extensions to gain-scheduling in the face of parametric uncertainty are discussed. In particular, simple heuristics are proposed to compute such controllers. >

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Journal Article•DOI•

Survey Research on gain scheduling

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Wilson J. Rugh¹, Jeff S. Shamma²•Institutions (2)

Johns Hopkins University¹, University of California, Los Angeles²

01 Oct 2000-Automatica

TL;DR: Current research on gain scheduling is clarifying customary practices as well as devising new approaches and methods for the design of nonlinear control systems.

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1,621 citations

Cites background from "A convex characterization of gain-s..."

...We should note that LPV approaches o!er another way to interpolate controllers (Packard, 1994; Apkarian & Gahinet, 1995 )....
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Journal Article•DOI•

Self-scheduled H ∞ control of linear parameter-varying systems: a design example

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Pierre Apkarian¹, Pascal Gahinet², Greg Becker³•Institutions (3)

Community emergency response team¹, French Institute for Research in Computer Science and Automation², University of California, Berkeley³

01 Sep 1995-Automatica

TL;DR: The methodology presented in this paper is applied to the gain scheduling of a missile autopilot and is to bypass most difficulties associated with more classical schemes such as gain-interpolation or gain-scheduling techniques.

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1,439 citations

Cites methods from "A convex characterization of gain-s..."

...A first technique for parameter-dependent controller synthesis is based on the small gain theorem and applicable to LPV plants with an LIT (linear fractional transformation) dependence on the parameter 0 (Packard, 1994; Apkarian and Gahinet, 1995)....
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...Once adequate matrices R and S have been computed, the Lyapunov matrix Xce common to all inequalities (29) and the vertex controllers Ri are obtained along the lines of Gahinet and Apkarian (1994) and Apkarian and Gahinet (1995)....
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Journal Article•DOI•

Parameterized linear matrix inequality techniques in fuzzy control system design

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Hoang Duong Tuan¹, Pierre Apkarian², Tatsuo Narikiyo³, Y. Yamamoto³•Institutions (3)

Toyota¹, Community emergency response team², Toyota Technological Institute³

01 Apr 2001-IEEE Transactions on Fuzzy Systems

TL;DR: This paper proposes different parameterized linear matrix inequality (PLMI) characterizations for fuzzy control systems and these characterizations are relaxed into pure LMI programs, which provides tractable and effective techniques for the design of suboptimal fuzzy control Systems.

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Abstract: This paper proposes different parameterized linear matrix inequality (PLMI) characterizations for fuzzy control systems. These PLMI characterizations are, in turn, relaxed into pure LMI programs, which provides tractable and effective techniques for the design of suboptimal fuzzy control systems. The advantages of the proposed methods over earlier ones are then discussed and illustrated through numerical examples and simulations.

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1,099 citations

Cites background or methods from "A convex characterization of gain-s..."

...Since in (4) is available online, system (4) also belongs to the more general class of gain-scheduling control systems intensively studied in control theory in the past decade (see, e.g., [12], [ 1 ], and [2])....
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...Only recently, however, this technique has received a systematic treatment within the framework and tools based on LMIs [12], [ 1 ], [2]....
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...our presented results based on a general theory of gain-scheduling control [ 1 ], [2], [12] have the following essential advantages over those of [14] and [15]: 1) The resulting optimization formulations are much simpler with much fewer variables involved....
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Journal Article•DOI•

Survey of Gain-Scheduling Analysis & Design

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Douglas J. Leith¹, William Leithead¹•Institutions (1)

University of Strathclyde¹

01 Jan 2000-International Journal of Control

TL;DR: The scope of this paper includes the main theoretical results and design procedures relating to continuous gain-scheduling (in the sense of decomposition of non-linear design into linear sub-problems) control with the aim of providing both a critical overview and a useful entry point into the relevant literature.

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Abstract: The gain-scheduling approach is perhaps one of the most popular non-linear control design approaches which has been widely and successfully applied in fields ranging from aerospace to process control. Despite the wide application of gainscheduling controllers and a diverse academic literature relating to gain-scheduling extending back nearly thirty years, there is a notable lack of a formal review of the literature. Moreover, whilst much of the classical gain-scheduling theory originates from the 1960s, there has recently been a considerable increase in interest in gain-scheduling in the literature with many new results obtained. An extended review of the gain-scheduling literature therefore seems both timely and appropriate. The scope of this paper includes the main theoretical results and design procedures relating to continuous gain-scheduling (in the sense of decomposition of non-linear design into linear sub-problems) control with the aim of providing both a critical overview and a useful entry point...

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933 citations

Journal Article•DOI•

Advanced gain-scheduling techniques for uncertain systems

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Pierre Apkarian¹, Richard Adams¹•Institutions (1)

Community emergency response team¹

01 Jan 1998-IEEE Transactions on Control Systems and Technology

TL;DR: Two alternative design techniques for constructing gain-scheduled controllers for uncertain linear parameter-varying systems are discussed and are amenable to linear matrix inequality problems via a gridding of the parameter space and a selection of basis functions.

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Abstract: This paper is concerned with the design of gain-scheduled controllers for uncertain linear parameter-varying systems. Two alternative design techniques for constructing such controllers are discussed. Both techniques are amenable to linear matrix inequality problems via a gridding of the parameter space and a selection of basis functions. These problems are then readily solvable using available tools in convex semidefinite programming. When used together, these techniques provide complementary advantages of reduced computational burden and ease of controller implementation. The problem of synthesis for robust performance is then addressed by a new scaling approach for gain-scheduled control. The validity of the theoretical results are demonstrated through a two-link flexible manipulator design example. This is a challenging problem that requires scheduling of the controller in the manipulator geometry and robustness in face of uncertainty in the high-frequency range.

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887 citations

Cites methods or result from "A convex characterization of gain-s..."

...In addition, these controllers have an LFT representation in terms of the nonlinear functions, i(:), and hence are computationally comparable to those of the LFT gain-scheduling approaches in [7, 8]....
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...Therefore, if the structure of the problem is ignored, the techniques in this paper or [7, 8] may o er no advantages over existing LFT gain-scheduling techniques....
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...Basic Technique with X( 2) and Y ( 2) 3:82 Projected Technique with X( 2) and Y ( 2) 3:82 LFT Technique in [8] 3:82 Table 2: Performance comparisons ( ) with ignored structure....
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...The corresponding levels are compared with the LFT gain-scheduling technique in [8], a technique that puts no bound on the parameter variation rates....
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...The Linear Fractional Transformation (LFT) gain-scheduling techniques in [7, 8, 9, 10] or the so-called quadratic gain-scheduled techniques in [11, 12] make use of a xed Lyapunov function, as opposed to one which depends on the scheduled variables, to characterize stability and performance....
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Journal Article•DOI•

A linear matrix inequality approach to H∞ control

[...]

Pascal Gahinet¹, Pierre Apkarian²•Institutions (2)

French Institute for Research in Computer Science and Automation¹, Community emergency response team²

01 Jan 1994-International Journal of Robust and Nonlinear Control

TL;DR: In this paper, the continuous and discrete-time H∞ control problems are solved via elementary manipulations on linear matrix inequalities (LMI), and two interesting new features emerge through this approach: solvability conditions valid for both regular and singular problems, and an LMI-based parametrization of all H ∞-suboptimal controllers, including reduced-order controllers.

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Abstract: The continuous- and discrete-time H∞ control problems are solved via elementary manipulations on linear matrix inequalities (LMI). Two interesting new features emerge through this approach: solvability conditions valid for both regular and singular problems, and an LMI-based parametrization of all H∞-suboptimal controllers, including reduced-order controllers. The solvability conditions involve Riccati inequalities rather than the usual indefinite Riccati equations. Alternatively, these conditions can be expressed as a system of three LMIs. Efficient convex optimization techniques are available to solve this system. Moreover, its solutions parametrize the set of H∞ controllers and bear important connections with the controller order and the closed-loop Lyapunov functions. Thanks to such connections, the LMI-based characterization of H∞ controllers opens new perspectives for the refinement of H∞ design. Applications to cancellation-free design and controller order reduction are discussed and illustrated by examples.

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3,200 citations

Book•

Feedback Control Theory

[...]

John Doyle, Bruce A. Francis, Allen Tannenbaum

01 Jan 1992

2,581 citations

Journal Article•DOI•

Self-scheduled H ∞ control of linear parameter-varying systems: a design example

[...]

Pierre Apkarian¹, Pascal Gahinet², Greg Becker³•Institutions (3)

Community emergency response team¹, French Institute for Research in Computer Science and Automation², University of California, Berkeley³

01 Sep 1995-Automatica

...read moreread less

1,439 citations

Journal Article•DOI•

On the input-output stability of time-varying nonlinear feedback systems Part one: Conditions derived using concepts of loop gain, conicity, and positivity

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George Zames¹•Institutions (1)

Massachusetts Institute of Technology¹

01 Apr 1966-IEEE Transactions on Automatic Control

TL;DR: In this article, the authors outline a stability theory for input-output problems using functional methods and derive open loop conditions for the boundedness and continuity of feedback systems, without, at the beginning, placing restrictions on linearity or time invariance.

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Abstract: The object of this paper is to outline a stability theory for input-output problems using functional methods. More particularly, the aim is to derive open loop conditions for the boundedness and continuity of feedback systems, without, at the beginning, placing restrictions on linearity or time invariance. It will be recalled that, in the special case of a linear time invariant feedback system, stability can be assessed using Nyquist's criterion; roughly speaking, stability depends on the mounts by which signals are amplified and delayed in flowing around the loop. An attempt is made here to show that similar considerations govern the behavior of feedback systems in general-that stability of nonlinear time-varying feedback systems can often be assessed from certain gross features of input-output behavior, which are related to amplification and delay. This paper is divided into two parts: Part I contains general theorems, free of restrictions on linearity or time invariance; Part II, which will appear in a later issue, contains applications to a loop with one nonlinear element. There are three main results in Part I, which follow the introduction of concepts of gain, conicity, positivity, and strong positivity: THEOREM 1: If the open loop gain is less than one, then the closed loop is bounded. THEOREM 2: If the open loop can be factored into two, suitably proportioned, conic relations, then the closed loop is bounded. THEOREM 3: If the open loop can be factored into two positive relations, one of which is strongly positive and has finite gain, then the closed loop is bounded. Results analogous to Theorems I-3, but with boundedness replaced by continuity, are also obtained.

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1,309 citations

"A convex characterization of gain-s..." refers methods in this paper

...To apprehend this problem with Small Gain Theory,we must rst gather all parameter-dependent components into a single uncertainty block....
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...The synthesis of gain-scheduledH1 controllers relies on the Small Gain Theorem [35, 9]....
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...Su cient conditions for solvability are then provided by Small Gain Theory [35, 9]....
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...If there exists a scaling matrix L 2 L and an LTI controlstructure K( ) such that the nominal closed-loop system Fl(Pa( );K(( )) is internally stableand satis es L1=2 00 I Fl(Pa( ); K( )) L 1=2 00 I 1 < ; (2:15)then Fl(K( ); ) is a -suboptimal gain-scheduled H1 controller.7 Proof: The proof is a straightforward application of the Small Gain Theorem....
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...Speci cally, consider the set of positive de nite similarity scalingsassociated with the structure in (2.3):L = fL > 0 : L = L; 8 2 g Rr r with r = KXi=1 ri: (2:13)This set enjoys the following immediate properties:(P1) Ir 2 L (P2) L 2 L ) LT 2 L (P3) L 2 L ) L 1 2 L (P4) L1 2 L ; L2 2 L ) L1L2 = L1L2; 8 2 (P5) L is a convex subset of Rr r.Given L , the set of scalings commuting with the repeated structure is readily deducedas: L = L1 L2LT2 L3 > 0 : L1; L3 2 L and L2 = L2;8 2 : (2:14)From Small Gain Theory, a su cient condition for robust performance in the face of theuncertainty , or equivalently for the existence of gain-scheduled controllers, is asfollows....
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Proceedings Article•DOI•

The LMI control toolbox

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Pascal Gahinet, A. Nemirovskii, A.J. Laub, M. Chilali

14 Dec 1994

TL;DR: This paper describes a new MATLAB-based toolbox for control design via linear matrix inequality (LMI) techniques, and its contents and capabilities are presented.

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Abstract: This paper describes a new MATLAB-based toolbox for control design via linear matrix inequality (LMI) techniques. After a brief review of LMIs and of some of their applications to control, the toolbox contents and capabilities are presented. >

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1,218 citations