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STATIC H∞ LOOP SHAPING CONTROL

E. Prempain ∗

∗ Department of Electrical Engineering & Electronics, University of LiverpoolBrownlow Hill, Liverpool, L69 3GJ, U.K., E-mail: [email protected]

Keywords: H∞ Loop Shaping; Static Output Feed-back; Linear Matrix Inequalities (LMIs).

Abstract

The aim of this paper is to introduce a novel method ofdesigning robust static output feedback controllers. Astatic output feedback version of the design procedureof Glover- McFarlane is presented and evaluated on nu-merical examples. Existence conditions for a static out-put loop shaping controller are given in terms of sim-ple linear sufficient conditions. In this framework, thedetermination of a static, sub-optimal, H∞ loop shap-ing controller reduces to an optimization problem overLinear Matrix Inequalities (LMIs). Two iterative algo-rithms are proposed to reduce the conservatism of theLMI conditions.

1 Introduction

The aim of this paper is to investigate the possibility ofexpanding H∞ optimization in static output feedbackdesign. H∞ optimization procedures developed overthe two last decades are now considered as standardtools for the control of multivariable systems. However,a drawback of modern linear control theory, includingH∞ control, is that the controller order is equal to thatof the augmented plant in the synthesis problem. Highorder controllers may lead to implementation issues. Inthis paper, the design of a stabilizing H∞ static outputfeedback gain is considered. The existence of a staticoutput feedback controller is known to be equivalent tothe existence of a positive definite matrix R satisfyingsimultaneously a linear matrix inequality and a Riccatitype matrix inequality. This problem cannot, in gen-eral, be cast as a Linear Matrix Inequality problem.The solution to this problem is still open and it hasbeen demonstrated that its constrained counterpart isindeed NP-hard [3]. It is not known if one can developa polynomial time algorithm for the unconstrained out-put feedback problem [3]. Fortunately, it is possible toderive sufficient Linear Matrix conditions from the nec-essary and sufficient conditions [7], [1], [6], [15]. Thebasic idea is to impose a certain structure on the Lya-punov matrix used in the synthesis procedure. Underthis assumption, the feedback synthesis reduces to an

LMI optimization problem.

This paper proposes alternate new LMI sufficient con-ditions for the determination of a stabilizing static-output feedback. The major advantage of using thesenew conditions is that they generalize easily to theH∞ design procedure of McFarlane and Glover [14].In this framework, two iterative algorithms are pro-posed to reduce the inherent conservatism of the LMIconditions of existence for a static output feedback con-troller.

The paper is organized as follows: The theoreticalbackground to the static output feedback problem isgiven in Section 2. The main results are presented inSection 3 where the notions of static control and staticrobust control in the context of coprime factors arepresented. Finally, the use of the algorithms is demon-strated on various numerical examples. Concluding re-marks end the paper.

The notation is standard:

Rm×n : real m× n matrices.In: n× n identity matrix.X > 0 (resp. ≥ 0): X is symmetric and positive defi-nite (resp. positive semi-definite).

2 Preliminaries

Let us consider a linear time invariant system describedby state-space equations

{x = Ax + Buy = Cx

(1)

where A ∈ Rn×n, u ∈ Rm is the control input, y ∈ Rp

is the measured output. The pairs (A,B) and (A,C)are assumed to be, respectively, stabilizable anddetectable, we assume that C and B are full rank.

Our aim is to compute a static output feedbacklaw u = Ky that ensures the stability of the closed-loop system Acl = A + BKC.

Lemma 1 (Finsler’s lemma) Let X be a given sym-metric matrix and let Z be a matrix such that

ξT Xξ < 0

Control 2004, University of Bath, UK, September 2004 ID-244

for all nonzero vector ξ such that Zξ = 0. Then thereexists a constant σ > 0 such that

X − σZT Z < 0.

Proof. See e.g. [4].

Lemma 2 The following statements are equivalent.

i) There exists a stabilizing static output feedbackgain.ii) There exists a positive-definite matrix R > 0 suchthat

NTB (AR + RAT )NB < 0,

NTC (R−1A + AT R−1)NC < 0,

where NB and NC denote bases of the null spaces ofBT and C, respectively.

Proof. See e.g. [11].

3 Main results

3.1 Static Output Feedback Stabilization

Lemma 3 The following statements are equivalent.

i) There exists a stabilizing static output feedbackgain.ii) There exist a positive-definite matrix R > 0, amatrix L of compatible dimension and a positive realnumber γ such that

AR + RAT − γBBT < 0, (2)(A + LC)R + R(A + LC)T < 0. (3)

Proof. Using Finsler’s lemma the conditions of state-ment ii) of lemma 2 can be rewritten as ∃ R > 0, andpositive real numbers σ1, σ2 such that

AR + RAT − σ1BBT < 0, (4)AR + RAT − σ2RCT CR < 0. (5)

It is clear that if conditions (4) and (5) are satisfiedfor some positive real numbers σ1 and σ2, then theyare also satisfied if one replaces σ1 and σ2 withγ = max(σ1, σ2). Now, let L = −γRCT /2. With sucha L, condition (3) reduces to condition (5) and lemma3 is proven ¥

It is interesting to see that conditions of lemma3 are linear in R. This contrasts with the well-knownstatic output feedback conditions of lemma 2 whichinvolve R and R−1.

For a given matrix L, such that A + LC is sta-ble, the linear conditions in R and γ of lemma 3become sufficient conditions for the existence of astatic output feedback gain.

3.2 Static output feedback algorithm

The feasibility of the linear conditions of lemma 3 de-pends on the choice of the matrix L. Let us considerthe following lemma.

Lemma 4 Let L = −Y CT where Y ≥ 0 is the stabi-lizing solution to

AY + Y AT − αY CT CY (+BBT = 0. (6)

For any 0 < α < 2 the matrix (A + LC) is a stabilitymatrix.

Proof. Since (A,B) is stabilizable and (C, A) is de-tectable, it is well-known that the above Riccati equa-tion has a unique stabilizing positive semi-definite so-lution Y . First, note that α must be strictly positiveto guarantee the existence of a semi-definite positivesolution Y for (6). Now, (A+LC)Y +Y (A+LC)T :=Q = −BBT − (2−α)Y CT CY < 0. From [21], A + LCis stable if Y ≥ 0, Q < 0 and (Q, (A + LC)T ) is de-tectable. Hence, if the pair (A + LC, Q) is stabilizable,one can conclude on the stability of A + LC. Because,Q is a square and a full rank matrix, it is clear that thepair (A + LC,Q) is stabilizable and therefore A + LCis stable for any value of α in ]0, 2[ ¥

Remark: Lemma 4 guarantees the stability of A+LCfor 0 < α < 2. But, note that A + LC can be stableeven for α > 2.

To stabilize the triple (A,B,C) an iterative algorithmis proposed. The algorithm is based on the feasibil-ity conditions (2) and (3) which, as mentioned ear-lier, guarantee the existence of a static output feedbackcontroller. These conditions depend exclusively on thegain matrix L. The algorithm philosophy is to gener-ate an α-dependent set of matrices L on which the LMIconditions (2) and (3) are checked. In practice, it is in-teresting to extend the range of α beyond 2, providingthat A + LC is stable. The algorithm is described asfollows:

1. Set i = 1 and define the boolean variablefail. Setfail = true and define Λ = [α1, ..., αn] a row vec-tor of n logarithmically equally spaced points rep-resenting various values of α. n = 50, α1 = 0.01and αn = 100 suffice in practice.

2. while(fail)


3. Set α = Λ(i) in (6) and compute the correspondingY .

4. Let L = −Y CT and solve the LMI feasibility prob-lem (2) and (3) in the variables R > 0 and γ > 0.If the LMI conditions are feasible then stop (e.g.set fail=false), the triple (A,B,C) is static outputfeedback stabilizable.

5. i=i+1

6. endwhile

7. If the boolean variable fail is false (e.g. the triple(A, B,C) is static output feedback stabilizable),then solve the LMI problem in K: (A+BKC)R+R(A + BKC)T < 0.

3.3 Static H∞ Loop Shaping

Let Gs be a strictly proper plant of order n having astabilizable and detectable state-space realization:

Gs :=[

A BC 0

]

with A ∈ Rn×n,B ∈ Rn×m and C ∈ Rp×n.

A left coprime factorization of Gs = M−1N isgiven by (see [21])

[N , M

]=

[A + LC B L

C 0 I

].

Theorem 1 Let L = −Y CT where Y ≥ 0 is the sta-bilizing solution to

AY + Y AT − Y CT CY + BBT = 0. (7)

There exists a static controller K such that∥∥∥∥[

KI

](I + GsK)−1M−1

∥∥∥∥∞

< γ (8)

if γ > 1 and if there exists a positive definite matrix Rsolving the inequalities

R(A + LC)T + (A + LC)R < 0 (9)

AR + RAT − γBBT RCT −LCR −γIp Ip

−LT Ip −γIp

< 0. (10)

Proof. See [16].

The conditions of theorem 1 are sufficient for the exis-tence of a staticH∞ loop shaping controller. To reduce,the conservatism of these conditions, one can suggestto iterate on the matrix L. The algorithm philosophyis the same as in the pure stabilization case (section3.2), i.e. it is matter to generate an α-dependent set ofmatrices L on which the LMI conditions (9) and (10)will be checked.

3.4 LMI solution and controllerreconstruction

1. Classical loop shaping: Select W1 and W2 to geta desired open loop shape. The nominal plant Gand the shaping functions W1 and W2 are com-bined to form the shaped plant Gs = W2GW1.Let (A,B, C) be a realization of Gs. Compute thefull-order McFarlane-Glover solution (for instance,by using the commands given in [2]). Tune W1 andW2 so that the closed-loop H∞ attenuation levelγopt is satisfactory, see e.g. [18] and [21] for a com-prehensive treatment on weighting function selec-tion. Next, these weights are used in the staticoutput version of the design problem.

2. Set i = 1 and define the boolean variablefail. Setfail = true and define Λ = [α1, ..., αn] a row vec-tor of n logarithmically equally spaced points rep-resenting various values of α. n = 50, α1 = 0.01and αn = 100 suffice in practice.

3. Solve (7) and compute the matrix R > 0 and thescalar variable γ solution to the LMI system (9)and (10). If these LMI conditions are feasible thenset fail = false (i.e. the triple (A,B, C) is staticoutput feedback stabilizable).

4. while(fail)

5. Set α = Λ(i) in (6) and compute the correspondingY .

6. Let L = −Y CT and solve the LMI feasibility prob-lem (9) and (10) in the variables R > 0 and γ > 0.If the LMI conditions are feasible then stop (i.e.set fail=false), the triple (A,B,C) is static out-put feedback stabilizable.

7. i=i+1

8. endwhile

9. if (NOT fail)

10. The controller K can be computed by making useof the Lyapunov matrix R and γ and the analyticformulae given in [10].

11. The final feedback controller KST is then con-structed using the static output feedback con-troller K with the shaping functions W1 and W2

such that KST = W1KW2. The order of KST isequal to sum of the orders of the weighting func-tions.

12. endif

The Glover McFarlane approach only makes sense fornormalized coprime factors of the plant (α = 1). Whenα 6= 1, the algorithm returns a γ which may not be


a good robustness/design indicator. In this case, theH∞ norm of the closed-loop must be re-evaluated withthe controller KST and the augmented plant used inthe classical Glover-McFarlane design procedure [14].

4 Examples

4.1 Static Stabilization Example

Let us consider the following simple example

Gs =−s + 1

(εs + 1)(s− 2).

In [20] it is shown that this plant is static output feed-back stabilizable for −0.5 < ε ≤ 0. Solving the condi-tions of section 3, with ε = .45, and α = 0.1, we get afeasible K = −1.995 .

4.2 Static Glover-McFarlane H∞ loop shapingcontroller

This example is adapted from [17]. The plant transferfunction is given by

G(s) =200

10s + 10.001s + 1

(0.005s + 1)2

the weights were chosen as

10−1

100

101

102

10−5

10−4

10−3

10−2

10−1

100

101

[rad/s]

Static Output FeedbackFull order Glover McFarlane

Figure 1: Singular value plot of the sensitivity func-tions. Static based output feedback controller and fullorder controller.

W1 =s + 2

s,

W2 = 1.

The synthesis procedure described in section 3 was im-plemented using the solver in the LMI toolbox [8]. In

this case, after one iteration (α = 1) the algorithm re-turns the following static controller

K = −0.502,

with γ = 5.42.

The controller of interest is KST = W1KW2, a1st order controller (the static-based H∞ loop shapingcontroller).

The full order loop shaping controller (KFO) wasdesigned with the same weighting functions W1

and W2. As expected, the best γ attenuation levelis achieved with the full order controller KFO forwhich γ = 2.31. The closed-loop sensitivity functions(I + GKST )−1 and (I + GKFO)−1 for the static-basedcontroller and the full order controller are given infigure 1. Clearly, the static and full order loop shapingcontrollers present similar responses.

4.3 Benchmark examples from COMPleib 1.0.

COMPleib consists of examples collected from the en-gineering literature and contains models from real-woldapplication as well as pure academic models [13]. Thecurrent version of COMPleib contains about 80 plantswhich are static output feedback stabilizable. The li-brary is useful for testing linear and non-linear semi-definite optimization solvers such as SeDuMi [19] andPENBMI [12].

The static version of the Glover and McFarlane de-sign procedure described in this paper has been testedon the Aircraft Models (AC), Helicopter Models (HE),Jet Engine models (JE), Reactor Models (REA) andDecentralized Interconnected Systems (DIS) which areavailable in COMPleib. This represents a collection ofabout 40 plants. For each plant, the static version ofthe Glover and McFarlane design procedure has beenapplied with W1 and W2 equal to the identity matrixas we were just interested in testing the ability of ouralgorithm to find a static feedback controller. For anyof these plants, the algorithm of section 3.3 successfullyfound a static output feedback controller.

5 Conclusions

In this paper, a simple method for static outputfeedback control synthesis is presented. The methodhas been extended to the well-known McFarlane andGlover H∞ design method. The effectiveness of thisiterative synthesis method has been demonstratedon various numerical examples. Unlike previousalgorithms, the algorithm does not require (A,B, C)to be a minimum phase plant [20], [1], [9]. Theproposed algorithm is also much simpler than other


iterative algorithms [11], [5]. It worth mentioningthat the static version of the Glover-McFarlane designprocedure, presented in this paper, has been usedto design a low order controller for the BEll 205helicopter and this controller has been successfullyimplemented and flight tested [16].

Future work is necessary to clarify the relation-ship between the choice of the gain matrix L and thefeasibility of the LMI conditions. Future work alsoshould focus on a possible extension of the proposedmethod to robust gain-scheduling control.

References

[1] S. Akoachere, Edwards C., and Spurgeon S.K. AnLMI approach to the sliding mode static outputfeedback problem. Proc. European Control Conf.Karlsruhe, 1999.

[2] G. J. Balas, J. C. Doyle, K. Glover, A. Packard,and R. Smith. µ-analysis and synthesis toolbox,version 3. The MathWorks, Inc., January 2001.

[3] V. Blondel and J. N. Tsitsiklis. NP-hardness ofsome linear control design problems. SIAM Jour-nal on Control and Optimization, 35:2118–2127,1997.

[4] S. Boyd, L. el Ghaoui, E. Feron, and V. Balakr-ishnan. Linear Matrix Inequalities in System andControl Theory. SIAM. Studies in Applied Math-ematics, 1994.

[5] Y. Y Cao and Y. X. Sun. Static output feedbacksimultaneous stabilization: Ilmi approach. Int.Jour. of Cont, 70(5):803–814, 1998.

[6] C. A. R Crusius and A. Trofino. Sufficientlmi conditions for output feedback control prob-lems. IEEE Transactions on Automatic Control,44(5):1053–1057, May 1999.

[7] C. Edwards and S. K. Spurgeon. Sliding ModeControl: Theory and Applications. Taylor & Fran-cis, 1998.

[8] P. Gahinet, A. Nemirovski, A. J. Laub, andM. Chilali. LMI control toolbox. The Math Works,May 1995.

[9] G. Garcia, B. Pradin, S. Tarbouriech, and F. Zeng.Robust stanilization and guaranteed cost controlfor discrete-time linear systems by static outputfeedback. Automatica, 39:1635–1641, 2003.

[10] T. Iwasaki and R. E. Skelton. All controllers forthe general H∞ control problem: LMI existenceconditions and state space formulas. Automatica,30(8):1307–1317, 1994.

[11] T. Iwasaki, R. E. Skelton, and J. C. Geromel.Linear quadratic suboptimal control with staticoutput feedback. System and Control Letters,23(6):421–430, 1994.

[12] M. Kocvara and M. Stingl. PENNON-a codefor nonlinear and semidefinite programming. Op-timization Methods and Software, 8(3):317–333,2003.

[13] F. Leibfritz and W. Lipinski. Compleib 1.0 -user manual and quick reference. Tech.-Report.,University of Trier, http://www.mathematik.uni-trier.de:8080/ leibfritz/publications.htm 2004.

[14] D. McFarlane and K. Glover. A loop shap-ing design procedure using H∞ synthesis. IEEETransactions on Automatic Control, 37(6):759–769, 1992.

[15] E. Prempain and I. Postlethwaite. Static outputfeedback stabilization with H∞ performance for aclass of plants. Systems & Control Letters, 43:159–166, 2001.

[16] E. Prempain and I. Postlethwaite. Static H∞loop shaping control of a fly-by-wire helicopter.CDC04, December 2004.

[17] S. Skogestad and I. Postlethwaite. Multivariablefeedback control, analysis and design. Wiley, June1996.

[18] S. Skogestad and I. Postlethwaite. Multivariablefeedback control. analysis and design. Wiley, Jan-uary 1997.

[19] J. F. Sturm. SeDuMi-a Matlab∗ toolbox for op-timization over symmetric cones. Departmentof Econometrics, Tilburg University, The Nether-lands, October 2001.

[20] V. L. Syrmos, C. T. Abdallah, P. Dorato, andK. Grigoriadis. Static Output Feedback - A Sur-vey. Automatica, 33(2):125–137, 1997.

[21] K. Zhou, J. Doyle, and K. Glover. Robust andOptimal Control. Prentice-Hall International, Inc,1995.


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