Control design for large-scale Lur’e systems with arbitrary information structure constraints

Applied Mathematics and Computation 217 (2010) 1277–1286

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Control design for large-scale Lur’e systems with arbitrary information structure constraints A.I. Zecˇevic´ *, E. Cheng, D.D. Šiljak Department of Electrical Engineering, Santa Clara University, 500 El Camino Real, Santa Clara 95053, United States

a r t i c l e

i n f o

Keywords: Lur’e systems Information structure constraints Large-scale systems Linear matrix inequalities

a b s t r a c t In this paper, an LMI-based approach is proposed for the design of static output feedback for multi-nonlinear Lur’e–Postnikov systems. The resulting control laws ensure absolute stability and, at the same time, maximize the size of the nonlinear sectors. The proposed method is computationally efficient, and can accommodate feedback laws with arbitrary information structure constraints. The effectiveness of this approach is demonstrated on a large-scale example with 100 state variables. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction The Lur’e–Postnikov system [1] has traditionally been modeled as a feedback connection of a linear plant and a nonlinearity. While the linear plant is given precisely and is known to be asymptotically stable, the shape of the nonlinearity is assumed to be uncertain. The class of admissible nonlinear functions is quite general – they are only required to be continuous, and must lie within a sector delineated by two straight lines. In the 1950s, Lur’e formulated a finite system of quadratic equations (which he referred to as ‘resolving equations’) to examine the absolute stability of such models [2]. Since that time, the absolute stability problem has been the subject of extensive research (a complete account of the early developments in this field can be found in [3–5]). One of the most important results related to absolute stability has been the Popov criterion, which is a graphical construction that provides a straightforward way to maximize the nonlinear sector [6,7]. Popov subsequently extended his criterion to systems with multiple nonlinearities, but this did not lead to a suitable maximization of the size of the nonlinear sectors [8,9]. An alternative approach to absolute stability was proposed in Refs. [3,9,10], based on the concept of positive realness [11–14]. Conditions derived using the well-known positive real lemma (e.g., [9,14]) proved to be a powerful theoretical tool, which gave rise to a number of successful strategies for designing absolutely stabilizing controllers [15–18]. Although absolute stability has been an active research topic for more than four decades, the control of large-scale Lur’e– Postnikov systems has received little attention in the literature ([19–22] are the only relevant references that we could find on this topic). In Ref. [19], interconnections of absolutely stable Lur’e–Postnikov systems were studied, and a sufficient condition guaranteeing absolute connective stability for the entire system was derived using vector Lyapunov functions. References [20–22], on the other hand, are primarily concerned with the design of decentralized control laws, which are applied to stabilize special network structures whose nodes are described as Lur’e–Postnikov systems. The approach proposed in this paper is more general than the ones described above, and focuses on large-scale Lur’e systems with arbitrary information structure constraints. The problem of designing structurally constrained controllers is not a new one, and recent years have seen a number of innovative developments in this area (see, e.g., [23–35]). Many of these

* Corresponding author. E-mail address: [email protected] (A.I. Zecˇevic´). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.01.119

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methods extend beyond traditional decentralized control, and allow for various types of information exchange between the subsystems. One of the most general results of this kind was proposed in [35], where an algorithm was developed for designing gain matrices with arbitrary nonzero patterns. This method was based on linear matrix inequalities (e.g., [23,24]), which are known to be an effective tool for dealing with systems that contain multiple nonlinearities. Our objective in the following will be to extend this approach, and develop an LMI-based design algorithm for controlling large-scale multi-nonlinear Lur’e–Postnikov systems with arbitrary information structure constraints. The proposed method has a number of attractive features, three of which are singled out below: 1. 2. 3.

The resulting static output feedback laws establish absolute stability and, at the same time, maximize the size of nonlinear sectors. The proposed design algorithm does not require a stable linear plant. The method is computationally efficient, and is therefore suitable for large-scale applications. It can also accommodate gain matrices with arbitrary nonzero patterns.

The paper is organized as follows. In Section 2, we describe an LMI-based design approach that can produce gain matrices with an arbitrary nonzero structure. In Section 3, we consider certain improvements that can significantly lower the computational complexity of the algorithm. A large-scale example with 100 state variables is provided to demonstrate the effectiveness of this approach in controlling systems of high dimensionality. 2. Arbitrary information structure constraints Let us consider the multivariable Lur’e–Postnikov system

x_ ¼ Ax þ B/ /ðt; yÞ þ Bu u

ð1Þ

y ¼ Cx;

where x 2 Rn is the state of the system, y 2 Rq is the output, and u 2 Rm denotes the input vector. Matrices A; B/ ; Bu and C are constant, and we will assume that ðA; B/ ; CÞ is a minimal realization of the transfer function GðsÞ ¼ CðsI  AÞ1 B/ . The nonlinear function / : Rqþ1 ! Rq is piecewise continuous in t, locally Lipschitz in y, and belongs to the class UF of sector bounded functions defined as

UF ¼ f/ : ½/ðt; yÞ þ aFyT ½/ðt; yÞ  aFy 6 0; 8t P 0; 8y 2 Rq g;

ð2Þ

where a > 0 represents the sector bound parameter, and F is a q  q constant matrix. System (1) is said to be absolutely stable if its equilibrium x ¼ 0 is globally asymptotically stable for all / 2 UF (e.g., [36]). A standard design objective for Lur’e systems is to enlarge the set UF by maximizing parameter a. In the following, we propose to accomplish this by using output feedback laws of the form

u ¼ Ky;

ð3Þ

where K is a gain matrix with an arbitrary preassigned structure. To see how such a requirement can be incorporated into the control design, let us first consider a somewhat simpler problem, in which the system is given as

x_ ¼ Ax þ hðxÞ þ Bu

ð4Þ n

n

Following [24], we will assume that the nonlinearity h : R ! R satisfies a quadratic bound T

h ðxÞ hðxÞ 6 a2 xT HT Hx

ð5Þ

where H is a constant matrix of dimension n  n, and a is a scalar parameter that can be viewed as a measure of robustness with respect to uncertainties. Given an unconstrained state feedback law of the form

u ¼ Kx

ð6Þ

our immediate objective will be to maximize a by an appropriate choice of matrix K. The global asymptotic stability of the resulting closed-loop system

x_ ¼ ðA þ BKÞx þ hðxÞ

ð7Þ

can be established using a Lyapunov function

VðxÞ ¼ xT Px

ð8Þ

where P is a symmetric positive definite matrix (denoted P > 0). As is well known, a sufficient condition for stability is for the derivative of VðxÞ to be negative along the solutions of (7). Setting Y ¼ sP 1 (where s is a positive scalar), L ¼ KY, and c ¼ 1=a2 , the control design can now be formulated as the following LMI problem in c; jY ; jL ; Y and L [24,30].

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Problem 1. Minimize a1 c þ a2 jY þ a3 jL subject to

Y>0 2 AY þ YAT þ BL þ LT BT 6 4 I HY

I

YH

T

7 0 5<0

I

ð9Þ

3

ð10Þ

cI

0

c  1=a 2 < 0

ð11Þ

and

"

jL I

LT

L

I

#

 < 0;

Y

I

I

jY I

 > 0:

ð12Þ

Several comments need to be made regarding this design procedure. Remark 1. The coefficients a1 ; a2 and a3 represent positive weights which reflect the relative importance of optimization variables c; jY and jL . Typical values for these parameters are a1 ¼ a3 ¼ 0:01 and a2 ¼ 10. Remark 2. The obtained controller is linear, which makes its implementation straightforward and cost effective. Remark 3. The gain matrix is computed directly as K ¼ LY 1 , and its norm is implicitly constrained as kKk 6 kLkkY 1 k < pffiffiffiffiffiffi jL jY (by virtue of (12)). Remark 4. If the LMI optimization is feasible, the resulting gain matrix stabilizes the closed-loop system for all nonlinear. ities satisfying (5). Condition (11) additionally secures that a is greater than some desired value a The Lur’e problem that we will be interested in requires an output feedback law which must satisfy given structural constraints, while stabilizing the closed-loop system

x_ ¼ ðA þ Bu KCÞx þ B/ hðxÞ

ð13Þ

for all nonlinearities that conform to inequality (5). Note that in this case matrix H takes the special form H ¼ FC, due to the way in which set UF is defined. From an LMI perspective, the principal challenge in this context is to modify Problem 1 in a way that allows us to factorize the product LY 1 as

LY 1 ¼ KC:

ð14Þ

The following theorem provides a natural framework for incorporating this constraint into the optimization. Theorem 1. Let us assume that Problem 1 is feasible with matrices Y and L of the form

Y ¼ qY 0 þ QY Q Q T L ¼ LC U T

ð15Þ

where q; Y Q and LC are LMI variables (q > 0 plays the role of a scalar relaxation parameter). Suppose further that Y 0 ; Q and U are constant matrices that satisfy

U ¼ Y 0CT

ð16Þ

Q T C T ¼ 0:

ð17Þ

and

Then, the feedback law

u ¼ KCx

ð18Þ

with K ¼ q1 LC stabilizes system (4) for all nonlinearities that conform to bound (5). Proof. Conditions (15)–(17) ensure that

YC T ¼ qY 0 C T þ QY Q Q T C T ¼ qU

ð19Þ

and therefore

U T Y 1 ¼ q1 C

ð20Þ

as well. Since Problem 1 is assumed to be feasible under these circumstances, it will produce a gain matrix of the form

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K ¼ LY 1 ¼ LC U T Y 1 ¼ KC

ð21Þ

with K ¼ q LC . Such a feedback law maximizes the robustness parameter a, and guarantees that the system is stable for all nonlinearities that satisfy bound (5). h 1

Although Theorem 1 places no explicit constraints on the properties of matrix Y 0 , in practice it is important to select it in a way that is consistent with inequalities (9) and (10). Without such a preconditioning the LMI optimization described in the previous section could easily become infeasible, since variables q, Y Q and LC cannot always offset a poor choice of Y 0 . When A is a stable matrix the problem can be resolved rather easily, since Y 0 can be computed as the unique solution of the Lyapunov equation

AY 0 þ Y 0 AT ¼ I

ð22Þ

To see why such a choice is appropriate, it suffices to recall that Y 0 appears in inequalities (9) and (10) (since it is an additive component of matrix Y). The fact that Y 0 satisfies (22) and is positive definite is clearly conducive to the feasibility of Problem 1. The identification of an appropriate Y 0 becomes somewhat more complicated when A is an unstable matrix. The simplest way to resolve this problem would be to determine Y 0 as the solution of Lyapunov equation

ðA  bIÞY 0 þ Y T0 ðA  bIÞT ¼ I

ð23Þ

where b is chosen so that matrix A  bI is stable. One of the principal advantages of the method described above stems from the fact that it places no limitations on the structure of matrix K. To put this result in the proper perspective, we should first recall that the LMI procedure in Problem 1 allows us to assign an arbitrary nonzero pattern only to matrix L. This pattern is usually not preserved in the gain matrix, given that it is computed as K ¼ LY 1 . The exception, of course, is the case when Y is chosen to be a diagonal matrix, but this tends to be a restrictive requirement which often leads to infeasibility. For this reason, it is fair to say that optimization Problem 1 can realistically produce only certain special structures for K, such as block-diagonal or bordered block-diagonal (BBD). The design strategy associated with Theorem 1 removes this restriction, since K and LC now have identical nonzero patterns (note that q does not affect the matrix structure, since it is a scalar parameter). In view of the results outlined above, the LMI optimization associated with the Lur’e problem should be modified in the following manner. Problem 2. Minimize a1 c þ a2 jY þ a3 jL subject to

Y>0 2 AY þ YAT þ Bu L þ LT BTu 6 4 BT/ FCY

T

B/

YC F

I

0

0

cI

T

3 7 5<0

c  1=a 2 < 0

ð24Þ ð25Þ ð26Þ

and

"

jL I

LT

L

I

# < 0;



Y

I

I

jY I



> 0:

ð27Þ

with matrices Y and L that satisfy (15). The optimization procedure can be further enhanced by relaxing the two constraints described in (27). As noted in Remark 3, the purpose of these constraints is to ensure that the gain matrix is appropriately bounded. The following lemma shows how this can be accomplished with a single inequality. Lemma 2. Let

"

ql I LC

l > 0 be a given positive number. Then, inequality LTC qlI

# <0

ð28Þ

ensures that kKk < l. Proof. Let us first observe that matrix

" M¼

I ð1=qlÞLTC 0 I

#

is nonsingular by construction. If expression (28) is multiplied on the left by M T and on the right by M, we obtain

ð29Þ

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ql I

0

0

ð1=qlÞLC LTC  ql I

<0

ð30Þ

which is equivalent to

LC LTC < q2 l2 I

ð31Þ

(since q > 0). Recalling that kKk2 ¼ kq LC k2 , we directly obtain kKk2 < l. 1

h

3. Applications to large-scale Lur’e systems In cases when the Lur’e system is large, it is usually convenient to represent it as an interconnection of N subsystems, which takes the form

x_ i ¼ Aii xi þ

N P j¼1

Aij xj þ Bui ui þ B/i /i ðt; yÞ

ð32Þ

yi ¼ C i xi ; In (32), xi 2 Rni denotes the subsystem states, while yi 2 Rqi and ui 2 Rmi represent the local outputs and inputs, respectively. We will additionally assume that every nonlinear function /i : Rqþ1 ! Rqi is piecewise-continuous in t and locally Lipschitz in y. Setting AD ¼ diag fA11 ; . . . ; ANN g; AC ¼ ðAij Þ; B/D ¼ diag fB/1 ; . . . ; B/N g; BuD ¼ diag fBu1 ; . . . ; BuN g; C D ¼ diag fC 1 ; . . . ; C N g, and / ¼ ½/T1 ; . . . ; /TN T , system (32) can be expressed in compact form as

x_ ¼ ðAD þ AC Þx þ BuD u þ B/D /ðt; yÞ y ¼ C D x:

ð33Þ

It can be shown that the aggregate nonlinear function /ðt; yÞ in (33) is piecewise-continuous in t and locally Lipschitz in y. We will further assume that /ðt; yÞ belongs to the class UF of sector bounded functions defined in (2) with a constant matrix F. In applying the proposed LMI procedure to large-scale Lur’e systems, computational complexity becomes a critical consideration. To see why this is so, it suffices to observe that an unconstrained symmetric matrix Y in Problem 2 introduces

gðYÞ ¼

nðn þ 1Þ 2

ð34Þ

LMI variables into the optimization. Since this number becomes prohibitively large as n increases beyond a certain limit, the structure of matrix Y proposed in (15) is clearly better suited for large-scale applications. If we elect to adopt such a strategy, gðYÞ reduces to

gðYÞ ¼

lðl þ 1Þ þ1 2

ð35Þ

where l is the dimension of matrix Y Q . The impact of such a reduction can be very significant when l  n. It is important to keep in mind, however, that lowering l below a certain threshold can decrease the size of the sector, and can ultimately lead to infeasibility. The following example illustrates this effect. Example 1. Let us consider a large-scale Lur’e–Postnikov system with multiple nonlinearities, in which matrices A; Bu ; B/ ; C, and F are sparse (their nonzero patterns are shown in Figs. 1–5). The structure of Bu ; B/ and C is block-diagonal, which ensures that the system can be expressed in the form shown in (33). Matrix A (whose dimension is 100  100) was chosen to be unstable, with 26 eigenvalues in the right half plane. Our objective in the following will be to design an output feedback law that guarantees absolute stability of the system, while maximizing the sector bound parameter a. In doing so, we will assume that the gain matrix can be decomposed as

K ¼ KD þ KC

ð36Þ

where K D corresponds to the decentralized part of the control. Matrix K C is associated with information exchanges between the subsystems, whose nature is defined by the availability of communication channels. In this particular example, matrix K is assumed to have the structure shown in Fig. 6. It turns out that Problem 2 becomes infeasible if purely decentralized control is used (i.e., if K ¼ K D ). As a result, the supplemental term K C plays an essential role in stabilizing this system. The results obtained with the gain matrix K ¼ K D þ K C are summarized in Table 1. The maximal sector bound was found to be a ¼ 0:1672, with a corresponding value of gðYÞ ¼ 1; 541. Note that in this case feasibility alone requires gðYÞ ¼ 1276 variables, which is obviously too large. Example 1 indicates that large-scale Lur’e systems require an alternative form for matrix Y, which can result in better convergence properties. To see how this issue can be resolved, let Q m be an n  n matrix with rank nq that satisfies

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0 10 20 30 40 50 60 70 80 90 100 0

10

20

30

40

50 60 nz = 467

70

80

90

100

40

45

50

Fig. 1. The nonzero pattern of matrix A.

0 10 20 30 40 50 60 70 80 90 100

0

5

10

15

20

25 30 nz = 160

35

Fig. 2. The nonzero pattern of matrix Bu .

Q Tm C T ¼ 0:

ð37Þ

Given expression (37), we now propose to look for matrix Y in the modified form

Y ¼ qY 0 þ Q m Y D Q Tm þ QY Q Q T

ð38Þ

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0 10 20 30 40 50 60 70 80 90 100

0

5

10

15

20

25 30 nz = 143

35

40

45

Fig. 3. The nonzero pattern of matrix B/ .

0 5 10 15 20 25 30 35 40 45 0

10

20

30

40

50 60 nz = 133

70

80

90

100

Fig. 4. The nonzero pattern of matrix C.

where Y D represents an unknown diagonal (or block-diagonal) matrix of dimension n  n, and Q is a n  l matrix that satisfies (17). We should note in this context that the construction of Q is straightforward, and amounts to selecting l columns of matrix Q m . Since Y Q has dimension l  l, the number of LMI variables associated with matrix Y becomes

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0 5 10 15 20 25 30 35 40 45 0

5

10

15

20

25 nz = 65

30

35

40

45

35

40

45

Fig. 5. The nonzero pattern of matrix F.

0 5 10 15 20 25 30 35 40 45 50 0

5

10

15

20

25 nz = 222

30

Fig. 6. The nonzero pattern of matrix K.

gðYÞ ¼

N X ni ðni þ 1Þ lðl þ 1Þ þ þ 1: 2 2 i¼1

ð39Þ

where ni denotes the size of the ith diagonal block of Y D . In order to fully exploit this possibility, we should also observe that the standard procedure for computing matrix Y 0 is not very practical for large systems, since it entails the solution of a

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Table 1 Optimization results for Example 1. Dimension of Y Q

(g(Y))

LMI optimization

Maximized a

55  55 54  54 52  52 50  50 Lower than 50  50

1541 1486 1379 1276 Less than 1276

Feasible Feasible Feasible Feasible Infeasible

0.1672 0.1483 0.1335 0.0589 N/A

Dimension of Y Q

ðgðYÞÞ

LMI optimization

Maximized a

49  49 45  45 40  40 35  35 30  30 25  25 20  20 15  15 13  13 Lower than 13  13

1339 1149 934 744 579 439 324 234 205 Less than 205

Feasible Feasible Feasible Feasible Feasible Feasible Feasible Feasible Feasible Infeasible

0.1659 0.1658 0.1608 0.1559 0.1550 0.1449 0.1382 0.1289 0.1287 N/A

Table 2 Optimization results with a modified Y.

Table 3 Optimization results with the simplification. Dimension of Y Q

ðgðYÞÞ

LMI optimization

Maximized a

49  49 30  30 25  25 20  20 15  15 13  13 88 55 No Y Q

1339 579 439 324 234 205 150 129 114

Feasible Feasible Feasible Feasible Feasible Feasible Feasible Feasible Feasible

0.2700 0.2697 0.2697 0.2672 0.2665 0.2650 0.2645 0.2644 0.2643

Lyapunov equation (either (22) or (23), depending on the stability properties of matrix A). A computationally attractive alterðrÞ native would be to look for an approximate solution Y 0 of rank r  n, which can be obtained using Krylov space techniques (see e.g., [37–39]). To demonstrate the effectiveness of the proposed modifications, in Table 2 we show results obtained for the system in Example 1 in which matrix Y has the form described in (38). In these simulations, we chose a block-diagonal structure for matrix Y D , which introduces gðY D Þ ¼ 113 LMI variables. It is readily observed that feasibility is attained with a much smaller number of variables than was the case in Table 1. The value a ¼ 0:1287 that is produced by l ¼ 13 is actually quite close to a ¼ 0:1659 (which is the maximal value obtained by the optimization). With that in mind, it is reasonable to conclude that choosing l ¼ 13 represents an acceptable compromise between sector maximization and computational savings. The results of the optimization can be further enhanced by replacing the two inequalities in (27) with a single one. As shown in Lemma 2, constraints of this type ensure that kKk < l (where l is a preassigned positive number). To demonstrate the potential impact of such a simplification, we repeated the simulations shown in Table 2 using condition (28) instead of (27). In these experiments, parameter l was selected to match the value of kKk that is associated with the unmodified version of Problem 2. The results obtained in this manner are presented in Table 3. It is readily observed that the values of a are significantly larger, and that the number of variables in Y Q can be further reduced (the simulation is actually feasible even without Y Q ). 4. Conclusions In this paper, we developed a design strategy for stabilizing large Lur’e–Postnikov systems with multiple nonlinearities. The proposed approach, which was formulated in the framework of linear matrix inequalities, can accommodate controllers with arbitrary information structure constraints. The resulting output feedback laws guarantee absolute stability and, at the

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