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erfect Regulation ith Cheap Contr ol or Uncertain Linear Systems Li Xie and Ian. R. Petersen Uni ersity of Ne South ales at the Australian Defence orce Academy Ian Petersen is currently on secondment to the Australian Research Council.

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Outline Introduction Perfect Re gulation with Norm Bounded Uncertainty The Main Result Illustrati Example Conclusions

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Intr oduction An important problem in control system design is concerned with the maximally achie able performance and fundamental performance limitations. or linear systems, the cheap control problem has attracted much attention. This problem consists of an optimal linear re gulator problem such that scalar weighting coef ˆcient on the control input in the quadratic cost functional tends to zero. If the optimal cost approaches zero as the weighting on the control input tends to zero, cheap control is said to pro vide perfect re gulation.

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Recently the problem of performance limitations in feedback design has recei ed great deal of interest from man researchers. This paper is concerned with the problem of perfect re gulation with cheap control for linear uncertain systems. The moti ation for studying this problem comes from desire to xtend kno wn performance limitation results for linear systems to linear uncertain systems. emplo Riccati equation approach to quadratic guaranteed cost control to in estig ate the limiting case of quadratic cost functional with cheap control for linear uncertain systems.

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Using the limiting beha vior of the minimal positi deˆnite stabilizing solution to the Riccati equation, the performance limit for the uncertain systems with norm-bounded uncertainty and inte gral quadratic constraint uncertainty respecti ely is deri ed ˆnd that perfect re gulation with cheap control can be achie ed if the uncertain system has particular special structure.

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Norm Bounded Uncertainty The class of uncertain systems under consideration is described by the follo wing state equation where is the state, is the control input, and is time-v arying matrix of uncertain parameters.

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is the set of all admissible uncertainties for the uncertain system and is deˆned as follo ws: Associated with this system is the cost functional Rx Gu dt where

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Deˆnition control la is said to deˆne quadratic guaranteed cost control with associated cost matrix for the uncertain system and cost functional if GK for all non-zero and for all If the control la is quadratic guaranteed cost control with cost matrix 0, then the corresponding alue of the cost functional satisˆes the bound for all admissible uncertainties

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Lemma (Petersen McF arlane) Suppose that there xists constant such that the Riccati equation has solution and consider the control la

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Then gi en an 0, there xists matrix such that and this control la is quadratic guaranteed cost control for the uncertain system with cost matrix and thus Con ersely gi en an quadratic guaranteed cost control with cost matrix there xists constant such that the abo Riccati equation has stabilizing solution where 10

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Ne xt, let where 0. Suppose that there xist and such that the abo Riccati equation has positi deˆnite solution Pr oposition Suppose that has full column rank. Then with and for an ˆx ed the unique minimal positi deˆnite stabilizing solution to the Riccati equation, tends to the minimal positi deˆnite stabilizing solution of the Riccati equation: as 11

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The Main Result From the abo Proposition, we see that when has full column rank, for gi en taking the limit does not ensure perfect re gulation for the uncertain system. achie perfect re gulation, we will assume that is singular Hence, there xist tw unitary matrices and such that we ha singular alue decomposition of Letting 11 12 we ha the follo wing equalities 11 11 12 12

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Theor em Consider the abo Riccati equation and suppose that does not ha full column rank. Then there xists unitary matrix such that and can be written as in abo e. If 12 has full ro rank, then for an gi en there xists constant such that there is the unique positi deˆnite stabilizing solution to the Riccati equation with Furthermore, lim 0. Con ersely let be the minimal positi deˆnite solution to the Riccati equation with gi en and where 0. If lim 0, then 12 has full ro rank. 13

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Illustrati Example Consider the uncertain system described by the state equation 16 121 where is ector subject to the bound Associated with this system is the cost functional dt 14

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Let 33. Notice that does not ha full column rank. Here we consider tw cases; (i) 121 and (ii) 121 1. or the ˆrst case, 12 does not ha full ro rank; for the second case, 12 has full ro rank. plot of the maximal eigen alue of 33 ersus for both cases is sho wn belo When 12 has full ro rank, from the ˆgure belo lim 33 and the maximal achie able performance is zero. 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 −1 The maximal eigenvalue of X( , 0.33) 12 does not have full row rank. 12 has full row rank. Figure 1: The maximal eigen alue of 33 ersus 15

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Conclusions ha deri ed the cheap control performance limit for the uncertain systems with both norm-bounded uncertainty Similar results hold for uncertain systems with inte gral quadratic constraint uncertainty Also the condition under which perfect re gulation can be achie ed has been gi en. It is noted that the uncertain systems under consideration require special structure to achie perfect re gulation. 16

R Petersen Uni ersity of Ne South ales at the Australian Defence orce Academy Ian Petersen is currently on secondment to the Australian Research Council brPage 2br Outline Introduction Perfect Re gulation with Norm Bounded Uncertainty The Main Resul ID: 22838

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Page 1

erfect Regulation ith Cheap Contr ol or Uncertain Linear Systems Li Xie and Ian. R. Petersen Uni ersity of Ne South ales at the Australian Defence orce Academy Ian Petersen is currently on secondment to the Australian Research Council.

Page 2

Outline Introduction Perfect Re gulation with Norm Bounded Uncertainty The Main Result Illustrati Example Conclusions

Page 3

Intr oduction An important problem in control system design is concerned with the maximally achie able performance and fundamental performance limitations. or linear systems, the cheap control problem has attracted much attention. This problem consists of an optimal linear re gulator problem such that scalar weighting coef ˆcient on the control input in the quadratic cost functional tends to zero. If the optimal cost approaches zero as the weighting on the control input tends to zero, cheap control is said to pro vide perfect re gulation.

Page 4

Recently the problem of performance limitations in feedback design has recei ed great deal of interest from man researchers. This paper is concerned with the problem of perfect re gulation with cheap control for linear uncertain systems. The moti ation for studying this problem comes from desire to xtend kno wn performance limitation results for linear systems to linear uncertain systems. emplo Riccati equation approach to quadratic guaranteed cost control to in estig ate the limiting case of quadratic cost functional with cheap control for linear uncertain systems.

Page 5

Using the limiting beha vior of the minimal positi deˆnite stabilizing solution to the Riccati equation, the performance limit for the uncertain systems with norm-bounded uncertainty and inte gral quadratic constraint uncertainty respecti ely is deri ed ˆnd that perfect re gulation with cheap control can be achie ed if the uncertain system has particular special structure.

Page 6

Norm Bounded Uncertainty The class of uncertain systems under consideration is described by the follo wing state equation where is the state, is the control input, and is time-v arying matrix of uncertain parameters.

Page 7

is the set of all admissible uncertainties for the uncertain system and is deˆned as follo ws: Associated with this system is the cost functional Rx Gu dt where

Page 8

Deˆnition control la is said to deˆne quadratic guaranteed cost control with associated cost matrix for the uncertain system and cost functional if GK for all non-zero and for all If the control la is quadratic guaranteed cost control with cost matrix 0, then the corresponding alue of the cost functional satisˆes the bound for all admissible uncertainties

Page 9

Lemma (Petersen McF arlane) Suppose that there xists constant such that the Riccati equation has solution and consider the control la

Page 10

Then gi en an 0, there xists matrix such that and this control la is quadratic guaranteed cost control for the uncertain system with cost matrix and thus Con ersely gi en an quadratic guaranteed cost control with cost matrix there xists constant such that the abo Riccati equation has stabilizing solution where 10

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Ne xt, let where 0. Suppose that there xist and such that the abo Riccati equation has positi deˆnite solution Pr oposition Suppose that has full column rank. Then with and for an ˆx ed the unique minimal positi deˆnite stabilizing solution to the Riccati equation, tends to the minimal positi deˆnite stabilizing solution of the Riccati equation: as 11

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The Main Result From the abo Proposition, we see that when has full column rank, for gi en taking the limit does not ensure perfect re gulation for the uncertain system. achie perfect re gulation, we will assume that is singular Hence, there xist tw unitary matrices and such that we ha singular alue decomposition of Letting 11 12 we ha the follo wing equalities 11 11 12 12

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Theor em Consider the abo Riccati equation and suppose that does not ha full column rank. Then there xists unitary matrix such that and can be written as in abo e. If 12 has full ro rank, then for an gi en there xists constant such that there is the unique positi deˆnite stabilizing solution to the Riccati equation with Furthermore, lim 0. Con ersely let be the minimal positi deˆnite solution to the Riccati equation with gi en and where 0. If lim 0, then 12 has full ro rank. 13

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Illustrati Example Consider the uncertain system described by the state equation 16 121 where is ector subject to the bound Associated with this system is the cost functional dt 14

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Let 33. Notice that does not ha full column rank. Here we consider tw cases; (i) 121 and (ii) 121 1. or the ˆrst case, 12 does not ha full ro rank; for the second case, 12 has full ro rank. plot of the maximal eigen alue of 33 ersus for both cases is sho wn belo When 12 has full ro rank, from the ˆgure belo lim 33 and the maximal achie able performance is zero. 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 −1 The maximal eigenvalue of X( , 0.33) 12 does not have full row rank. 12 has full row rank. Figure 1: The maximal eigen alue of 33 ersus 15

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Conclusions ha deri ed the cheap control performance limit for the uncertain systems with both norm-bounded uncertainty Similar results hold for uncertain systems with inte gral quadratic constraint uncertainty Also the condition under which perfect re gulation can be achie ed has been gi en. It is noted that the uncertain systems under consideration require special structure to achie perfect re gulation. 16

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