Systems & Control Letters 61 (2012) 381–386

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Systems & Control Letters

journal homepage: www.elsevier.com/locate/sysconle

Static output feedback stabilization of interconnected systemsYoonsoo KimDepartment of Aerospace and System Engineering, and Research Center for Aircraft Parts Technology, Gyeongsang National University, Jinju 660-701, Republic of Korea

a r t i c l e i n f o

Article history:Received 12 June 2011Received in revised form19 December 2011Accepted 19 December 2011Available online 8 February 2012

Keywords:Interconnected systemsSpectral radiusSpectral normStabilizationVehicle formation

a b s t r a c t

This paper is concerned with the celebrated static output feedback control problem (SOFP) subject tolinear constraints on control input K , e.g. bounds on control magnitude, zeros in some elements ofK , etc. These constraints typically arise in the control of resource-limited systems interconnected toeach other, where the local control for each system makes use of its own and neighboring systems’outputs only, and its magnitude is bounded. This control problem can be approached by a spectral-normminimization technique (q-SNM), whose preliminary version was previously introduced with littlemathematical justification but shown to be promising for the regular SOFP without control constraints.This paper mathematically justifies q-SNM by showing its explicit relationship with spectral radius, andextends q-SNM to accommodate various linear control constraints. This paper also discusses the practicalapplication of q-SNM to vehicle formation control, which demonstrates the merit of q-SNM.

© 2011 Elsevier B.V. All rights reserved.

1. Problem statement and introduction

The celebrated static output feedback control problem (SOFP) isstated as follows: for a given linear system

x = Ax+ Bu; y = Cx, (1.1)

find a static gain matrix K such that u = Ky stabilizes the system,i.e. A + BKC has eigenvalues whose real parts are all strictlynegative. That is,α(A+BKC) < α0, whereα(·) denotes the spectralabscissa and α0 is a negative real number. This famous openproblem in control theorymay be complicated further by includinglinear constraints on the gainmatrix K = [kij] ∈ Rm×p (m and p arethe numbers of inputs and outputs, respectively): (1)−κ ≤ kij ≤ κfor all i, j and a constant κ > 0; and (2) k(i, j) = 0 for some i, j.Needless to say, the first constraint – each control gain has a limitin magnitude – is of practical interest due to actuator saturation,limited power resources, etc. The second constraint means thatthe jth output is not used for the design of the ith control input.This constraint arises in the situation in which the controller hasno direct access to a certain set of outputs or (even though it has)does not use the output information on purpose.

This situation associated with the second constraint is promi-nent in the control of interconnected systems, which is the maintopic of this paper. When an interconnected system (consisting ofseveral subsystems) is described as (1.1), the A, B and C matriceshave block structure, and the way of interconnection determinesthe zero structure of the K matrix. If the system is fully or simply

E-mail address: yoonsoo@gnu.ac.kr.

0167-6911/$ – see front matter© 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.sysconle.2011.12.007

connected (in the graph theoretical sense) but each subsystemusesits own output in control design, then K has a block diagonal struc-ture, i.e. K = diag(K (11), K (22), . . . , K (nn)), hence K (ii) is such thatu(i)= K (ii)y(i), where u(i), K (ii) and y(i) are quantities solely associ-

ated with the ith subsystem. In the literature, the control strategyin which such a block diagonal gain matrix is used is called decen-tralized control (see [1] for a recent overview on this topic). Tworecent works [2,3] dealt with this decentralized output feedbackcontrol problem. In particular, [2] allows for uncertain nonlineari-ties in the system. However, their control strategies are subject tosome technical assumptions that are necessary towarrant the exis-tence of their control laws. As a matter of fact, [2] assumes that thegain matrix must be related through a matrix equality to a config-urationmatrixΓ , which defines the interconnection between sub-systems, i.e. there must be a matrix F such that Γ = BFC . For [3],the system matrices A, B and C , as well as the matrices pertainingto a given performance objective, must be expressed in the formof I ⊗ Ma + P ⊗ Mb, where I is an identity matrix, Ma and Mb arereal matrices and P is a diagonalizable matrix. Although successfulfor a certain class of systems, these technical assumptions hamperapplication of the above strategies to general systems.

As [4] notes, decentralized control may not stabilize systemswith structurally-fixed unstable modes. Besides, there is no reasonfor not using the information available from other subsystems if noparticular implementation issues are present. In this regard, thispaper is concerned with static output feedback control design forarbitrarily interconnected systems (SOFAIS) in which the controlgain K is bounded and has the same structure as the configurationmatrix Γ , i.e. K (ij) is a zero matrix with appropriate dimensionif there is no direct communication from the ith subsystem tothe jth subsystem. Much work has been done in this direction.

382 Y. Kim / Systems & Control Letters 61 (2012) 381–386

Dynamic output feedback control design for a certain class ofinterconnected systems was discussed in [5], and the solutionidea therein is extended in [6] to cover arbitrarily interconnectedsystems via the use of linear matrix inequalities (LMIs). Note thatthese LMIs are only sufficient for the existence of their controllaws. In [7], a linear programming approachwas presented, but it isonly applicable to a class of interconnected systems whose closed-loop characteristic polynomials are such that their coefficientsare affinely dependent on the design parameters K . Using amatrix decomposition trick, [8] proposed a solution strategy (yetheuristic) for SOFAIS. One drawback of this heuristic strategy isthat it requires the existence of at least one subsystem whoseoutput must be available to all the other subsystems in thenetwork.

The aforementioned literature study clearly implies that thereare only a handful of sophisticated solution strategies for SOFAIS,and this motivates the present work. The present work is based onthe idea of directly minimizing the spectral radius of eθ(A+BKC), i.e.ρ(eθ(A+BKC))—the standard measure of stability for discrete linearsystems, where θ > 0 is a fixed real constant. This idea (calledq-SNM) was originally proposed in [9] to solve the regular SOFPwithout control constraints. Direct minimization of spectral radiusis known as a very difficult problem because, in general, ρ(·)

is neither convex nor locally Lipschitz [10]. However, it wasnumerically shown in [9] that q-SNM is indeed successful in findinga stabilizing gain for difficult benchmark problems, and alsogenerally performsbetter than apopular non-smooth optimizationtechnique (gradient sampling) discussed in [11].

Inspired by the success of q-SNM in handling SOFP, q-SNM’smathematical justification and its extension to SOFAIS areinvestigated in this paper. q-SNM is then compared with recentnumerical algorithms proposed in [12,13] to show its capabilityof solving a difficult benchmark problem even with additionalcontrol constraints. After this investigation, q-SNM is appliedto a vehicle formation control problem to further demonstrateits merit. In [14], vehicle formation stabilization was done viadecentralized output feedback control. In this present work,this stabilization problem is solved such that the local controlgain for each subsystem is calculated by making use of itsneighboring subsystems’ outputs (in addition to its own system’soutput), thereby yielding a smaller spectral abscissa than in thedecentralized control case. In summary, the main contributionsof this paper are as follows: (1) the clear relationship betweenρ(X) and q-SNM is given, i.e. the starting point of q-SNM is thewell-known mathematical relationship, ρ(X) = limq→∞ ∥Xq

∥1/q;

(2) the convergence of q-SNM is discussed; (3) an improvedq-SNM approach using the matrix exponential is proposed; and(4) this improved approach is applied to a practically interestingproblem.1

The paper is organized as follows. In Section 2, q-SNM’smathematical justification is given, then an improved q-SNMis presented and compared with other existing algorithms inthe literature. In Section 3, a practically motivated stabilizationproblem, vehicle formation control, is introduced and its previoussolutions are improved via the proposed approach in Section 2.After addressing several numerical issues in Section 4, concludingremarks follow in Section 5.

1 Another contribution is that the proposed method in this paper may be used tocontrol plants with uncertainties. Once a controller is designed so as to minimizethe spectral radius of the closed-loop system and so the closed-loop system mayhave a sufficient stability margin, some uncertainties in the original system can benaturally taken care of by the stability margin. See [15] for a more direct approachto static output feedback control of plants with uncertainties.

2. Method: q-SNM (the qth-order spectral normminimization)

2.1. Motivation for q-SNM

q-SNM is concernedwith finding amatrixwhose spectral radiusis minimized over a convex set C defined by linear constraints,i.e. q-SNM solves

P : argmin ρ(X) subject to X ∈ C. (2.2)

In view of the well-known relationship between ρ(X) and ∥X∥(spectral norm), ρ(X) = limq→∞ ∥Xq

∥1/q, the minimization prob-

lem P in (2.2) may be written as

argmin ∥Xq∥1/q subject to X ∈ C (2.3)

or

P (q): argmin ∥Xq

∥ subject to X ∈ C (2.4)

for a sufficiently large q. Note that (2.3) and (2.4) are equivalent. Infact, suppose X is a solution to (2.3), i.e. ∥Xq

∥1/q≤ ∥Xq

∥1/q for any

X ∈ C. Then, the equivalence of the two problems follows easily,since the inequality implies ∥Xq

∥ ≤ ∥Xq∥, and vice versa.

Clearly, P (1) is convex, has a global solution (minimum), andthus is solved easily. However, P (q) for q ≥ 2 is not convex andmay have numerous local solutions (minima). Two observationscan be made that shall motivate q-SNM: (1) Unlike ρ(Y ), ∥Y∥ isconvex in Y , and Y = Xq may be linearized around any fixedX = X0 for a finite q; and (2) P (q) comes closer to P as q getslarger. The first observation implies that a convex program can beemployed to find a local solution to (2.4). Note that, in general,even a local solution to P is difficult to obtain due to theundesirable properties of ρ(·), such as non-convex, non-locally-Lipschitz, non-differentiable, and so on. The second observationsuggests that one could consider solving P (q) initially with a smallvalue of q, and then with the values of q gradually being increased,if necessary. This second observation is particularly useful inreducing computational complexity, as smaller q involves lesscalculation when linearizing Xq. In fact, Taylor’s series expansionof Y = Xq around X0, i.e.

∥X − X0∥ ≤ δ (2.5)

with a sufficiently small δ > 0, yields the following linearization:

Y = Xq0 + (X − X0)X

q−10 + X0(X − X0)X

q−20 + · · ·

+ Xq−10 (X − X0), (2.6)

and the number of terms on the right-hand side is q+ 1.

2.2. Algorithm 1: q-SNM for general purposes

q-SNM first solves the convex program P (1) for X , and theniteratively solves the linearized version of non-convex P (q) (witha fixed q; initially q = 2) for X:

P(q)lin : argmin

X∥Y∥ subject to (2.5), (2.6) and X ∈ C. (2.7)

Once X is obtained, X is assigned to X0, and P(q)lin is solved again for

a new X . If the new X improves the previous X , i.e. ρ(X) < ρ(X0),then the same steps are repeated. Otherwise, q is increased toq + 1 and the aforementioned steps are repeated again. q-SNMis terminated with its output X∗ when q reaches a sufficientlylarge value qmax. Algorithm 1 in Table 1 shows an example pseudo-code for these steps. Note that, in Algorithm 1, some extra stepsassociatedwith the linearized region are included that do not affectthe convergence of this algorithm. The value of δ in (2.5), whichdefines a valid linearized region, is updated in a way to reflect thesize of X0. In the main loop of Algorithm 1, the value of δ is initially

Y. Kim / Systems & Control Letters 61 (2012) 381–386 383

Table 1Algorithm 1: q-SNM for general purposes.

Initial X∗: Solve P (1) for X = X∗ and q← 2.Main loop: Solve P

(q)lin for X = X (q) and δ← 0.1∥X∗∥.

If ρ(X (q)) < ρ(X∗), X∗ ← X (q), δ← 0.1∥X∗∥ and repeat the main loop;else if q > qmax , exit the main loop;

elseif δ < 10−5, q← q+ 1, δ← 0.1∥X∗∥ and repeat the main loop;else δ← 0.1δ and repeat the main loop.

Output: X∗

set to 10% of the size of X∗ (best solution so far), and is decreasedif no improvement on X∗ is made. If the decreased value of δ isless than a certain threshold (e.g. 10−5 in Algorithm 1), then q isincreased and δ is reset to 10% of the size of X∗.

The convergence of q-SNM is straightforward because (1) (2.7)is convex, thus convergent for any q; (2) q is strictly increased; and(3) q-SNM is terminatedwhen q reaches a finite value qmax. It is alsoeasy to see that Algorithm 1 indeed converges on a local solution tothe original spectral minimization problem (2.2).

Theorem 2.1. The output of Algorithm 1with a sufficiently large qmaxis arbitrarily close to a local solution to (2.2).

Proof. It is clear that there exists a sufficiently large finite qmaxsuch that a local solution to P (qmax) in (2.4) is arbitrarily close toa local solution to P in (2.2). Thus, if Algorithm 1 which minimizesP (q) using a typical gradient-followingmethod of solvingP

(q)lin was

initialized with q = qmax, its output would be arbitrarily close toa local solution to (2.2). Note that ρ(X (q)) < ρ(X∗) in Algorithm1 is the same as ∥(X (q))q∥ < ∥(X∗)q∥ for q = qmax. But even forAlgorithm 1 which begins with q = 2 as shown in Table 1, thealgorithm’s output X∗ must also be arbitrarily close to a localsolution to P because q is allowed to reach qmax in any case. Thisproves the claim. �

The advantages of Algorithm 1 with q gradually being increased,not q = qmax in the beginning, are twofold: (1) it helps X to quicklyapproach a local minimum X∗, as theminimization ofP (q) is easieror computationally faster for smaller q; and (2) for many practicalcases ρ(X) can be very close to ∥Xq

∥1/q even for a small q, and so

the algorithm allows to estimate a practical value (not just a largevalue) of qmax.

2.3. SOFAIS using improved q-SNM

If one was interested in static output feedback control for adiscrete linear system, i.e. find K such that u(k) = Ky(k) stabilizesx(k+1) = Ax(k)+Bu(k); y(k) = Cx(k) or x(k+1) = (A+BKC)x(k),then Algorithm 1 in Table 1 with A + BKC in place of X , could beapplied directly to find a K ∗ such that ρ(A+ BK ∗C) = ρ(X∗) < 1.In order for the continuous linear system in (1.1) still to be solvedin the q-SNM framework, the following equivalent discrete systemis considered: x(k + 1) = eθ(A+BKC)x(k) where e[·] is the matrixexponential and θ > 0 is a real number. Since α(A + BKC) < 0if and only if ρ(eθ(A+BKC)) < 1 for any positive θ , finding a gain Ksuch that α(A+ BKC) < 0 is essentially the same as finding a gainK such that ρ(eθ(A+BKC)) < 1. Hence, Algorithm 1 with eθ(A+BKC) inplace of X may be used to find such a stabilizing gain.2 However,eθ(A+BKC) is not linear in K and so needs to be linearized.

In [9], the first-order approximation of eθ(A+BKC), i.e. eθ(A+BKC)≈

I + θ(A + BKC), replaces X in Algorithm 1 to solve SOFP. In fact,this approach can also be used to solve SOFAIS. To see this, first

2 The proposed algorithm guarantees finding a local solution to P in (2.9), butmay not find a stabilizing gain K . Finding a stabilizing gain is guaranteed only ifevery local minimum of P is a stabilizing gain.

note that the two types of linear constraints on K = [kij]: (1)−κ ≤ kij ≤ κ for all i, j; and (2) k(i, j) = 0 for some i, j, can easilybe incorporated into the set C in P . That is, after the first-orderapproximation of eθ(A+BKC), one solves

P (q): argmin

K∥[I + θ(A+ BKC)]q∥ subject to K ∈ C. (2.8)

In this paper, however, eθ(A+BKC) is directly linearized withoutapproximation. To this end, now consider

P : argminK

ρ(eθ(A+BKC)) subject to K ∈ C (2.9)

Q(k): argmin

K∥[eθ(A+BKC)

]k∥ subject to K ∈ C (2.10)

for any positive θ and a sufficiently large positive k. Using aproperty of the matrix exponential, Q(k) can be rewrittenas argminK ∥eθk(A+BKC)

∥, subject to K ∈ C or Q : argminK∥eθ(A+BKC)

∥ subject to K ∈ C because θ could be chosen as θ/k.The definition of the matrix exponential X = eθM and Taylor’s

series expansion of thematrix exponential at X = X0 = eθM0 yields

X = X0 + θ1M + θ2[1MM0 +M01M]/2! + · · ·

+ θ q[1MMq−1

0 +M01Mq−2M0 + · · · +Mq−10 1M]/q! (2.11)

with a sufficiently large q, provided that

∥1M∥ = ∥M −M0∥ ≤ δ (2.12)

with a sufficiently small δ > 0. Then, a new, linearized version ofP in (2.9) may be proposed as follows:

Q(q)lin : argmin

K∥X∥ subject toM = A+ BKC,

(2.11), (2.12) and K ∈ C. (2.13)

Table 2 shows Algorithm 2, in which Q(q)lin is iteratively solved. Note

that P (1) in (2.8) is used to start Algorithm 2, because Q(1) is notconvex.

For a test purpose, Algorithm 2 is applied to the followingbenchmark system found in [9]:

A =

1.38006 −0.2077 6.7150 −5.6760−0.5814 −4.2900 0.0000 0.67501.0670 4.2730 −6.6540 5.89300.0480 4.2730 1.3430 −2.1040

,

B =

0.0000 0.00005.6790 0.00001.1360 −3.14601.1360 0.0000

, C =1 0 1 −10 1 0 0

.

For this system subject to the control constraints that |K(i, j)| ≤ 1for all i, j and K(2, 2) = 0, Algorithm 2 with θ = 1 and α0= −0.15 returns K ∗ = [0.0611 − 0.8658; 1.0000 0.0000] atq = 2 in 3.53 s.3The corresponding eigenvalues of A + BK ∗C

3 A personal computer equipped with an Intel(R) Core(TM)2 CPU 2.00 GHz and3GBRAMwas used to produce all simulation results in this paper. Free optimizationsoftware, SeDuMi [16] and YALMIP [17], was used for solving P (1) and Q

(q)lin .

384 Y. Kim / Systems & Control Letters 61 (2012) 381–386

Table 2Algorithm 2: q-SNM using direct linearization of eθ(A+BKC) .

Initial X∗: Solve P (1) for K = K ∗ . If α(A+ BK ∗C) < α0 , go to Output; else q← 2.Main loop: Solve Q

(q)lin for K = K (q), δ← 0.1∥A+ BK ∗C∥, X (q)

← eθ(A+BK (q)C) , X∗ ← eθ(A+BK∗C) .If α(A+ BK (q)C) < α0 , K ∗ ← K (q) and go to Output.If ρ(X (q)) < ρ(X∗), K ∗ ← K (q), δ← 0.1∥X∗∥ and repeat the main loop;else if q > qmax , exit the main loop;

elseif δ < 10−5, q← q+ 1, δ← 0.1∥X∗∥ and repeat the main loop;else δ← 0.1δ and repeat the main loop.

Output: K ∗

are −0.1539 ± j0.7400,−9.5509 and −9.8720. Note that somenumerical algorithms available in the literature, e.g. [12], fail tofind a stabilizing controller for the above benchmark system, evenwith no control constraints imposed. Although HIFOO (a MATLABpackage for fixed-order controller design using a non-smoothoptimization technique in [13]) indeed returns a (large) stabilizinggain K = [−350.568.65; 639.1 − 125.8] in the case of no controlconstraints, it does not properly handle SOFP with controlconstraints yet.4 Amore thorough numerical comparison betweenq-SNM and a non-smooth optimization algorithm proposed in [11]can be found in [9].

3. Practical example: vehicle formation control

The vehicle formation control problem is to design a controllerfor each vehicle such that a set of dynamic vehicles at arbitrarylocations converges to a desired formation. Here the formation isdefined as a state in which each pair of vehicles in the set keeps afixed relative distance while moving at the same speed.

Following the notation and problem formulation in [14],assume that all N vehicles have the same dynamics: xi = Avehxi +Bvehui, i = 1, 2, . . . ,N, xi ∈ R2n where the entries of xi representn configuration variables for vehicle i and their derivatives, uirepresents the control inputs, and

Aveh = diag

0 1a121 a222

, . . . ,

0 1an21 an22

,

Bveh = In ⊗01

.

Here In is an identitymatrixwith dimension n and⊗ the Kroneckerproduct. Then, the vehicle formation control problem may beposed as an SOFP, i.e. find a matrix F such that

x = Ax+ BFL(x− h), (3.14)

where A = IN ⊗ Aveh, B = IN ⊗ Bveh, L = LG ⊗ I2n (LG is theLaplacianmatrix of a communication graphG—see [18] for details),and h is a constant vector associated with a desired formation.In [14], F is restricted to the specific form F = IN ⊗ Fveh forthe purpose of decentralized control. This restriction simplifies theanalysis and gives rise to algebraic conditions on the elements ofevery stabilizing gain Fveh.

In this paper, this restriction is relaxed so that (3.14) becomes

x = Ax+ BK(x− h), (3.15)

where K = FLw and Lw is a weighted Laplacian matrix with theweights undetermined.5The matrix F here is no longer restrictedto be IN ⊗ Fveh. As a result, K still has the same structure as FL

4 HIFOO indeed offers a way to limit the size of the gain matrix. However, for thisspecific benchmark system, it fails to find a stabilizing gain K such that |K(i, j)| ≤ 1for all i, j. Besides, HIFOO cannot handle a constraint such as K(2, 2) = 0 at all.5 Unlike L having off-diagonal entries of integer numbers only, Lw allows any real

numbers in its off-diagonal entries while satisfying Lw1N = 0 like L.

in the sense that K(1N ⊗ γ ) = 0 for any γ ∈ R2n, but belongsto a larger set of stabilizing gains than the FL in (3.14) does. Here1N denotes an all-one vector in RN . For instance, consider a five-vehicle formation stabilization problem inwhich vehicle 1 can talkto vehicles 2 and 5, vehicle 2 to vehicles 1 and 3, vehicle 3 tovehicles 2 and 4, vehicle 4 to vehicle 3 only, and vehicle 5 to vehicle1 only. Then, the K matrix looks like

K =

In ⊗ K11 In ⊗ K12 0 0 In ⊗ K15In ⊗ K21 In ⊗ K22 In ⊗ K23 0 0

0 In ⊗ K32 In ⊗ K33 In ⊗ K34 00 0 In ⊗ K43 In ⊗ K44 0

In ⊗ K51 0 0 0 In ⊗ K55

(3.16)

with Kij ∈ R1×2. Note that (3.16) can easily be constructed once thevehicle formation control problem with n = 1 is solved.

In [14], the key facts needed to show the convergence of xin (3.14) to formation h are that (1) when the vehicles are information, L(x − h) = 0, and thus (3.14) is reduced to x = Ax;and (2) the eigenvalues of A + BFL are those of Aveh + λBvehFveh,where λ is an eigenvalue of LG. Hence, for a connected graph Gwhose LG has only one zero eigenvalue, the convergence of x in(3.14) to formation h is guaranteed if Fveh is chosen such that all thereal parts of the eigenvalues of Aveh+ λBvehFveh for each nonzero λare strictly negative. In this present work, the convergence of x in(3.15) to formation h must be guaranteed. Using the definition offormation in [14], it is easy to see that K(x − h) = 0 if and only ifthe vehicles are in formation. Thus, all one needs to show is that theeigenvalues of A+BK include those of Aveh, and then choose K suchthat the real parts of the eigenvalues of A+ BK except for those ofAveh are all strictly negative. To show that the eigenvalues of A+BKinclude those of Aveh, suppose that λ and z are an eigenvalue and itsassociated eigenvector of Aveh, i.e. Avehz = λz. Then, the propertyof K , i.e. K(1N ⊗ γ ) = 0 for any γ ∈ R2n, yields that

(A+ BK)(1N ⊗ z) = ((IN ⊗ Aveh)+ (IN ⊗ Bveh)K)(1N ⊗ z)= 1N ⊗ (Avehz) = λ(1N ⊗ z).

Eigenvalues of Aveh may be removed from those of A + BK byusing an orthonormal matrix P whose columns are perpendicularto the space spanned by the eigenvectors of Aveh. Thus, for thevehicle formation control, the problem P in (2.9) can be changedto the following:

P : argminK

ρ(PT eθ(A+BK)P) subject to K ∈ C, (3.17)

whereC defines the zero structure of K , and eigenvalues of PT (A+BK)P are the same as those of A + BK , except for those of Aveh.Consider again the five-vehicle formation stabilization problem inwhich Aveh = [0 1; 0 0.1], Bveh = [0; 1] and K has the same zerostructure as in (3.16). For n = 1, applying Algorithm 2 with θ = 1and α0 = −0.35 in Table 2, to P in (3.17) with these data, yieldsK ∗ at q = 4 in 29.92 s, and the corresponding α(PT (A+BK ∗C)P) =−0.3566. The magnitude of each element of K ∗ is bounded by 1.

For a comparison, a gain matrix K ∗∗ is carefully chosenas directed in [14] to solve the same five-vehicle formationstabilization problem as above. The gain matrix K ∗∗ involves only

Y. Kim / Systems & Control Letters 61 (2012) 381–386 385

(a) Using K ∗∗ . (b) Using K ∗ .

Fig. 1. Five-vehicle formation stabilization: vehicle trajectory (left) and control input (right).

Fig. 2. Effect of θ on α (top) and computational time (bottom).

two negative parameters, f1 and f2, i.e. for n = 1 K ∗∗ = (IN ⊗[f1f2])(LG ⊗ I2). Both values of f1 and f2 are chosen as −1/2,so that the magnitude of each entry of K ∗∗ is less than 1, andα(PT (A + BK ∗∗)P) is minimized. As a result, one obtains α(PT

(A + BK ∗∗C)P) = −0.2 (> −0.3566). Fig. 1 shows the resultantvehicle trajectories and control inputs for 15 s when K ∗∗ and K ∗are used respectively. The vehicles are initially lined up in a rowat the start (marked with an ‘x’) and required to form a pentagonformation (marked with circles in Fig. 1(b)). As expected, thevehicles converge to the required formation much quicker whenK ∗ is used.

4. Numerical issues

The algorithms implementing q-SNM need three fixed param-eters, qmax, α0 and θ , to start up. For the purpose of minimizingα(A + BKC), relatively large values of qmax and α0 are desired.However, numerical experience suggests that solutions are nor-mally not improved after q reaches 10, or sometimes even a muchsmaller number. For this reason, in the previous practical example,qmax was chosen as 10 so that Algorithm 2 could be terminated atq = 10 in case the desired spectral abscissa α0 is not achieved.The values of |α0|were chosen large enough to claim that the pro-posed gains K ∗ improve the previously proposed stabilizing gainsK ∗∗. Clearly, larger values of qmax and |α0| typically require longercomputational time.

The value of θ needs more attention. In theory, any positivevalue of θ could be used for the spectral abscissa minimization.

However, as Algorithm 2 searches for a locally optimal solution,its performance is indeed dependent on the value of θ . Fig. 2shows the changes in the assumed spectral abscissa and therequired computational time as the value of θ varies for the vehiclestabilization problem. The figure suggests that large values of θare less likely to return small spectral abscissa and to requiremore computational power for the algorithm to converge thansmall values of θ . In fact, this phenomenon is expected becausethe algorithm’s output is likely to be more sensitive to a smallchange in the parameters, when a larger value of θ is used andso equations such as (2.11) involve larger numbers. The author’snumerical experience shows that a good choice of θ is less thanor equal to 1. Note that the main loop of Algorithm 2 solves asemi-definite program Q

(q)lin , and has a theoretical computational

complexity of O(n8) at each iteration (see [19] for a detail).

5. Conclusion

In this paper, the qth-order spectral-norm minimizationmethod (q-SNM) was formally introduced and implementedthrough an algorithm for solving the static output feedback controlproblem (SOFP) coupledwith linear control constraints. It was firstshown that the proposed algorithms successfully handle a difficultbenchmark problem coupledwith these control constraints, unlikethe existing methods in the literature. It was then applied tothe vehicle formation control and greatly improved the previoussolutions in the literature. It was shown that one of the parameters,θ , which needs to be fixed to start Algorithm 2, should ideally bechosen as a small number to reduce the algorithm’s sensitivitywith respect to parameter changes and its computational time. Itis believed that the algorithms proposed in this paper may greatlycontribute to solving challenging SOFAIS arising in various fields.

It is true that the current version of q-SNM is not scalable, i.e.its computational cost grows fast as the problem size does. Thus, aproper scalingmethodneeds to be sought to improve the computa-tional aspect of q-SNMwhile not compromising the solution qual-ity. Also, the solution quality may need to be evaluated by someother practical performance measure which reflects such as rel-ative distances between agents. For this purpose, one may startsearching for a control gain in the direction of improving the per-formance measure, in the neighborhood of the control gain forwhich the stability measure (spectral radius) was minimized.

Acknowledgments

This work was supported by the Priority Research CentersProgram through theNational Research Foundation of Korea (NRF),

386 Y. Kim / Systems & Control Letters 61 (2012) 381–386

funded by the Ministry of Education, Science and Technology(Grant No. 2011-0031383), and partially by the Degree andResearch Center for Aerospace Green Technology (DRC) of theKorea Aerospace Research Institute (KARI), funded by the KoreaResearch Council of Fundamental Science & Technology (KRCF).An earlier version of this paper was presented at the 11thInternational Conference on Control, Automation and Systems,KINTEX, Republic of Korea, October 26–29, 2011. The author thanksanonymous reviewers for their thoughtful comments on the initialdraft of this paper.

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