Improved optimisation approach to the robust H2/H1

control problem for linear systems

E.N. Goncalves, R.M. Palhares and R.H.C. Takahashi

Abstract: A strategy for robust H2/H1 state-feedback control synthesis, with regional poleplacement, applied to continuous- or discrete-time linear time-invariant uncertain systems ispresented. It is based on a multiobjective optimisation over the space of the controller parameters.In the case of systems with polytope-bounded uncertainty, the H2 and H1 norms, calculated in allpolytope vertices and in possible ‘worst case’ interior points are taken as the optimisationobjectives. An a posteriori exact cost norm computation based on a branch-and-bound algorithm isapplied for closed-loop performance assessment. The proposed strategy is applied to continuous-and discrete-time examples, including the design of a decentralised controller, and the results arecompared with LMI-based formulations.

1 Introduction

In the last decade considerable attention has been focusedon the formulation of the performance indices (or ‘cost’functions) and constraints that appear in controller synthesisproblems in terms of linear matrix inequalities (LMI) [1].A LMI characterisation of a controller synthesis problemmeans that the problem can be efficiently solved using oneof the commercially available LMI solvers. In the case ofsystems with polytopic uncertainties it is sufficient to checkthe performance indices and constraints in the polytopevertices instead of the whole polytope in a LMI formulation.This considerably simplifies the problem solution. Thestandard way of transforming performance indices andconstraints in a LMI formulation is the Lyapunov shapingparadigm that is based on the concept of quadratic stability.The main problem with this approach is that it requires aunique Lyapunov matrix variable for all performanceindices, constraints and polytope vertices. This results in aconservative solution for the case where the uncertainparameters are constant or only vary slowly [2].

In order to overcome this difficulty, the use of parameter-dependent Lyapunov functions that can lead to lessconservative results have been proposed, e.g. for the caseof discrete-time systems by [2] and for the case ofcontinuous-time systems by [4–6].

In this work we will propose a controller synthesisapproach that is based on a multiobjective optimisationproblem applied directly over the controller parameter

space. The proposed approach is an improved version of theapproach presented in [7]. In this case, the problem becomesnonconvex. At a first glance, this could be a majordrawback, since no optimisation algorithm exists that canguarantee systematic convergence for this class of problem.Fortunately, there are optimisation algorithms that oftenwork in practice for nonconvex and nondifferentiableproblems, as is the case here. This is the case of theclassical ellipsoid algorithm [8]. In this work we will applyan improved ellipsoid algorithm, called the cone-ellipsoidalalgorithm that has a strategy of matrix factorisation thatprovides a better numerical stability. We will attempt toverify using examples that with the proposed controllersynthesis procedure one can achieve better guaranteed coststhan with the LMI-based formulation under the sameconstraints. We will also attempt to verify that with theproposed procedure feasible solutions are achieved inconditions where the LMI-based formulation fails e.g. inallowing more restrictive regional pole placement.

The cost of dealing with a nonconvex problem formu-lation is the necessity of an a posteriori LMI-based (convex)analysis method to verify if the objectives are actuallyachieved or if it is necessary to improve the results. Thestandard LMI-based analysis formulations furnish onlyrough upper bounds on the performance indices and in orderto compute more precise ones, we will employ the standardLMI-based analysis formulations combined with a branch-and-bound algorithm. Using this approach means that theconservativeness is reduced by progressively partitioningthe uncertain parameter space and applying the analysisformulations to the generated subpolytopes. The partition-ing can be refined until the required accuracy is reached.

2 Problem formulation

Consider the linear time-invariant system represented by:

d½xðtÞ� ¼ AxðtÞ þ BuuðtÞ þ BwwðtÞz1ðtÞ ¼ Cz1xðtÞ þ Dzu1uðtÞ þ Dzw1wðtÞz2ðtÞ ¼ Cz2xðtÞ þ Dzu2uðtÞyðtÞ ¼ IxðtÞ

8>><>>: ð1Þ

with d½xðtÞ� � dxðtÞ=dt; t � t; for continuous-timesystems, and d½xðtÞ� � x½ðk þ 1ÞT�; t � kT ; for discrete-time systems, where T is the sample period.

q IEE, 2005

IEE Proceedings online no. 20045041

doi: 10.1049/ip-cta:20045041

E.N. Goncalves is with the Department of Electrical Engineering, FederalCenter of Technological Education of Minas Gerais, Av. Amazonas 7675-31510-470-Belo Horizonte, MG, Brazil

R.M. Palhares is with the Department of Electronics Engineering, FederalUniversity of Minas Gerais, Av. Antonio Carlos 6627-31270-010-BeloHorizonte, MG, Brazil

R.H.C. Takahashi is with the Department of Mathematics, FederalUniversity of Minas Gerais, Av. Antonio Carlos 6627-31270-010-BeloHorizonte, MG, Brazil

Paper received 14th May 2004

IEE Proc.-Control Theory Appl., Vol. 152, No. 2, March 2005 171

In (1), x 2 Rn is the vector of state variables, u 2 Rpu isthe vector of control inputs, w 2 Rpw is the vector ofexogenous inputs (such as disturbance signals, sensor noiseor reference signals), z1 2 Rmz1 is the vector of controlledvariables related to the H1 performance, z2 2 Rmz2 is thevector of controlled variables related to the H2 performance,and y 2 Rn is the vector of measured outputs. Matrices A,Bu;Bw;Cz1;Dzu1;Dzw1;Cz2 and Dzu2 in (1) are real matrices,and they vary within a fixed polytope of matrices, i.e.:

TðaÞ ¼A Bu Bw

Cz1 Dzu1 Dzw1

Cz2 Dzu2 0

264

375

8><>:¼

XN

i¼1

ai

Ai Bui Bwi

Cz1i Dzu1i Dzw1i

Cz2i Dzu2i 0

264

375a 2 O

9>=>; ð2Þ

with:

O a : ai 0XN

i¼1

ai ¼ 1

( )ð3Þ

where N is the number of polytope vertices and the vectora ¼ a1 . . . aN½ � parameterises the polytope.

The system is controlled by a state-feedback controller,K, such that the closed-loop system, with uðtÞ ¼ KxðtÞ;becomes:

d½xðtÞ� ¼ ½Aþ BuK�|fflfflfflfflfflffl{zfflfflfflfflfflffl}Acl

xðtÞ þ BwwðtÞ ð4Þ

z1ðtÞ ¼ ½Cz1 þ Dzu1K�xðtÞ þ Dzw1wðtÞ ð5Þ

z2ðtÞ ¼ ½Cz2 þ Dzu2K�xðtÞ ð6Þ

Let T1ða;KÞ denote the closed-loop transfer functionmatrix from w to z1 and T2ða;KÞ denote the closed-looptransfer function matrix from w to z2: Consider the vector ofcontrol objectives:

JðKÞ ¼maxa

T1ða;KÞk k1maxa

T2ða;KÞk k2

" #ð7Þ

with k � k1 and k � k2 respectively the H1 norm and the H2

norm of the argument. This vector carries the meaning of theworst case of H1 and H2 norms inside the polytope for agiven controller, K. Let G denote the set of all controllersthat satisfy the pole location constraints for the closed-loopsystem:

GfK : sðAclða;KÞÞ � D; 8a 2 Og ð8Þ

with D � C and sð�Þ the spectrum of the argument.The conceptual problem that is addressed here is stated

as:Multiobjective guaranteed cost H2=H1 problem: Find

controllers K� that belong to the Pareto-optimal set G�:

G�fK� 2G : 6 9K 2GjJðKÞ � JðK�Þ and JðKÞ 6¼ JðK�Þg ð9Þ

The vector comparison operators� and 6¼ are employed herewith the meaning usually adopted in vector optimisation [9]:x � y) xi � yi; for all i up to the dimension of the vectors xand y, and x 6¼ y) for at least one i xi 6¼ yi: The G� setdefined as in (9) may be interpreted as the result ofsimultaneous minimisation of the functionals in vector Jð�Þ.

3 Proposed design procedure

Consider the set of points initialised as the polytope vertices~OO:

~OOfa : ai ¼ 1; aj ¼ 0 8j 6¼ i; i ¼ 1; . . . ;Ng ð10Þ

Consider also the ‘worst-case’ points inside the polytope,given a controller K:

að2Þ ¼ argmaxa

T2ða;KÞk k2

að1Þ ¼ argmaxa

T1ða;KÞk k1 ð11Þ

Define now the set:

~GGfK : sðAclða;KÞÞ � D 8a 2 ~OOg ð12ÞWe will now apply the following auxiliary problem:

Auxiliary problem: Given a g>0; find the controller ~KK�

such that:

~KK� ¼ argmin

Kmaxa

T2ða;KÞk k2 ð13Þ

subject to

K 2 ~GGT1ða;KÞk k1 � g

�The proposed design procedure can be stated as:

Design procedureStep 1: Initialise

i 0,~OO0 set of polytope vertices,dc;0 sufficiently large.

Step 2: i iþ 1,~OOi ~OOi�1.

Step 3: Solve the auxiliary problem, finding Ki ~KK�.

Step 4: Compute the exact H2 cost, dc;i, and find að2Þ for Ki.

If að2Þ =2 ~OOi and ðdc;i�1 � dc;iÞ=dc;i�1> ed; then ~OOi ~OOi [ að2Þ.Step 5: Compute the exact H1 cost, gc, and find að1Þ for Ki.

If gc>g and að1Þ =2 ~OOi; then ~OOi ~OOi [ að1Þ.Step 6: Check the regional pole placement constraints, if it isfound a aðpÞ that violates these constraints then~OOi ~OOi [ aðpÞ.Step 7: If ~OOi 6¼ ~OOi�1 go to step 2. Otherwise stop.

These steps are further explained in the following Sections.The sufficiently small constant ed is the procedure stopcriterion that specifies the relative accuracy required for theH2 cost.

3.1 Solution of the auxiliary problem

The auxiliar scalar optimisation problem can be solved bythe ellipsoidal algorithm described by the recursiveequations [10, 11]:

xkþ1 ¼ xk � b1

Qkmk

ðmTk QkmkÞ

12

ð14Þ

Qkþ1 ¼ b2 Qk �b3ðQkmkÞðQkmkÞT

mTk Qkmk

� �ð15Þ

with

b1 ¼1

d þ 1b2 ¼

d2

d2 � 1b3 ¼ 2b1

IEE Proc.-Control Theory Appl., Vol. 152, No. 2, March 2005172

where x 2 Rd is the optimisation parameter vector with theentries of the controller gain matrix, and d is the dimensionof the parameter vector space Let f ðxÞ: Rd ! R be theobjective function and gðxÞ: Rd ! Rr be the constraintvector. In the conventional ellipsoidal method, the vector mk

is calculated as the gradient (or sub-gradient) of the mostviolated constraint when xk is not a feasible solution, or asthe gradient (or sub-gradient) of the objective function whenxk is a feasible solution. In this work, the vector mk will becalculated based on the cone ellipsoid algorithm proposed in[10]. In this method, when xk is not feasible, the vector mk

will be the normalised vector mk ¼ m=kmk calculated asthe sum of the gradients (or sub-gradients) of the activeconstraints:

m ¼Hf ðxÞ if gjðxÞ< 0; 8j ¼ 1; . . . ; rPr

j¼1

sjðxÞ if 9jjgjðxÞ 0

8<: ð16Þ

with

sjðxÞ ¼0 if gjðxÞ< 0

HgjðxÞ if gjðxÞ 0

�ð17Þ

where Hð:Þ means the function gradient (or sub-gradient)that are approximated by forward-differencing.

The optimisation algorithm stops whenðfmax � fminÞ=fmin � e, where fmax and fmin are the maximumand minimum objective function values in the last Neiterations and e is the relatively accuracy required.

To improve the numerical stability of the squaresymmetric non-negative definite Q matrix computation,we apply the UDUT factorisation, similar to the algorithmemployed with the recursive least-squares parameterestimation. The Q matrix can be rewritten as:

Q ¼ UDUT ð18Þwhere U is an upper-triangular square matrix with unitarydiagonal elements and D is a diagonal matrix. Thefactorised matrix updating is given by:

Ukþ1 ¼ Uk�UU ð19Þ

Dkþ1 ¼ �DD ð20Þwhere �UU and �DD are obtained from the UDUT factorisation:

�UU �DD �UUT ¼ b2 Dk �

b3ðDkUkmkÞðDkUkmkÞT

mTk Qkmk

� �ð21Þ

3.2 Guaranteed cost computation andregional pole placement verification

The ‘exact’ H2 ðH1Þ cost computation is based on a branch-and-bound algorithm where the upper bound function is theH2 ðH1Þ guaranteed cost, dc ðgcÞ and the lower bound is theworst-case norm at the vertices of polytope, dw:c ðgw:cÞ [12].The difference between the upper and lower boundfunctions goes to zero when the polytope is subdividedinto smaller polytopes. Besides the application of thisalgorithm in the synthesis, it will be applied to compare thecontroller performances for the different synthesis strat-egies. The guaranteed cost computations are based on theformulations presented in [13] with exception of the H1guaranteed cost for continuous-time systems that it iscomputed by lemma 1 in [14].

The robust pole placement in LMI regions is verifiedby a branch-and-bound method with the conventional

LMI-based analysis formulation as presented in [15]. Inthis case two possibilities can occur: (i) the polytope ispartitioned until all subpolytopes result in feasible problemsindicating that the regional pole placement constraints aresatisfied in the whole polytope; and (ii) otherwise asubpolytope vertex that violates one of the regional poleplacement constraints, aðpÞ, is eventually found. This secondcase is processed in step 6.

4 Example 1

Consider the state-space description of a satellite systemconsisting of two rigid bodies (main module and sensormodule) connected by an elastic link [2]:

d

dtxðtÞ ¼

0 0 1 0

0 0 0 1

� kJ1

kJ1

� fJ1

fJ1

kJ2

� kJ2

fJ2

� fJ2

2664

3775xðtÞ

þ

0

01J1

0

2664

3775uðtÞ þ

0

01J1

0

2664

3775wðtÞ ð22Þ

z1ðtÞ ¼ 0 1 0 0½ �xðtÞ ð23Þ

z2ðtÞ ¼1 0 0 0

0 1 0 0

0 0 0 0

24

35xðtÞ þ

0

0

1

24

35uðtÞ ð24Þ

where x ¼ ½ y1 y2 y1 y2 �T ; y1 and y2 are the yawangles for the main body and the sensor module, u is thecontrol torque, and w is a torque disturbance on the mainbody. It is considered that J1 ¼ 1 and J2 ¼ 1: The constantuncertain parameters k and f are in the ranges:

0:09 � k � 0:4 and 0:0038 � f � 0:04

The control purpose is to design a robust state-feedbackcontroller that achieves a trade-off between the kT2k2 andkT1k1 norms, with kT1k1�1; and places the closed-loopeigenvalues into the intersection of the half-plane regionReðsðAclÞÞ� � 0:1, and the cone sector centred at the originwith inner angle 3p=4; p� jffsðAclÞj�3p=8, for all possiblevalues of the parameters k and f.

The solutions obtained by the proposed procedure, forg ¼ 0:1; 0:2; . . . ; 0:9; 1:0, will be compared with the sol-utions calculated with the function msfsyn provided by theMatlab LMI Control Toolbox [2]. For the proposedprocedure we use initial values of x0 ¼ ½1111� andQ0 ¼ 103I4, and the stop criteria e ¼ 0:01 and Ne ¼ 10.

Figure 1 shows the Pareto-like trade-off curves generatedby the two approaches where the kT2k2 and kT1k1guaranteed costs are computed with the branch-and-boundalgorithm with a 1% relative accuracy. It can be verified thatthe proposed approach has two advantages in relation to theLMI-based approach. The first advantage is that the proposedprocedure curve covers all the specified H1 norm rangewhile the msfsyn function curve achieves resultsonly between 0.04 to 0.33. The second advantage of theproposed procedure is that it achieves lower kT2k2

guaranteed costs, for the same g specification, than theLMI-based approach.

If we consider that the best controller is the first onewhere the relative reduction in kT2k2 is lower than therelative increase in kT1k1, then we would choose

IEE Proc.-Control Theory Appl., Vol. 152, No. 2, March 2005 173

the controller computed with g ¼ 0:2. With g ¼ 0:2, thecontroller computed with the function msfsyn is:

K ¼ �16:6769 2:9441 �6:7480 �59:6433½ �

this corresponds to the exact costs gc ¼ 0:07 and dc ¼ 2:26.For the same g, the controller computed with the proposedprocedure is:

K ¼ �6:3806 �0:2348 �3:6711 �17:2426½ �

which corresponds to exact costs of gc ¼ 0:20 and dc ¼ 1:76:This H2 guaranteed cost is 22% lower than that obtainedwith the LMI-based approach.. The CPU times (Pentium 42.2 GHz, 256 Mb) required in this case is 10.355 s in step 3(optimisation), 1.672 s in step 4 ðH2 exact cost compu-tation), 4.427 s in step 5 (H1 exact cost computation), and2.604 s in step 6 (pole placement validation).

As an example of the capability of the proposedprocedure to achieve feasible solutions we now consideran additional constraint: a disk region centred at the origin,jsðAclÞj � r. In this case we are interested in finding whichapproach achieves a feasible solution when decreasing thedisk radius with the other constraints fixed. For g ¼ 0:2,the LMI-based approach achieves a feasible solution forr 3:70 whereas the proposed approach achieves feasiblesolution for r 2:41, that is 35% smaller. With the functionmsfsyn, the resulting controller with r ¼ 3:70 is:

K ¼ �13:0114 �0:5045 �5:7376 �50:7299½ �

whereas with the proposed procedure the resulting con-troller with r ¼ 2:41 is:

K ¼ �6:0180 �1:0098 �3:9611 �16:2547½ �

which approximately corresponds to the same exact costsobtained before.

5 Example 2

Consider the example of a discrete-time reliable controlsystem first investigated in [16], where the matrices of thecontinuous-time model are:

A¼

�2 1 1 1

3 0 0 2

�1 0 �2 �3

�2 �1 2 �1

26664

37775 Bu ¼

0 0

b1 0

0 0

0 b2

26664

37775 Bw ¼

1 0 0

0 0 0

1 0 0

0 0 0

26664

37775

Cz2 ¼1 0 �1 0

0 0 0 0

0 0 0 0

264

375 Dzu2 ¼

0 0

1 0

0 1

264

375

The discrete-time system matrices will be computedassuming a zero-order hold with a sampling period equalto 0.1 s considering b1 ¼ 1 and b2 ¼ 1, as in [3].

In [3], the first design is a decentralised state-feedbackcontroller where the goal is to minimise the H2 norm of thediscrete closed-loop transfer-function T2 from w to z2: TheLMI-based strategies provide the following results [3]:

. Standard Lyapunov shape paradigm with block-diagonalLyapunov variable:

K ¼ �1:4891 �1:6600 0 0

0 0 �0:0651 �0:0063

� �

T2k k22¼ 0:1741

. Extended controller characterisation with a block-diag-onal multiplier:

K ¼ �1:4236 �1:2918 0 0

0 0 �0:1211 �0:0137

� �

T2k k22¼ 0:1579

Since this is not an uncertain system, in this case only thestep 3 of the proposed procedure will be required. Considerthe following state-feedback gain structure:

K ¼ x1 x2 0 0

0 0 x3 x4

� �With the initial conditions x ¼ 0 and Qð0Þ ¼ 0:5I4, and thestop criteria e ¼ 0:01 and Ne ¼ 10, the controller parameteroptimisation generates the following gain matrix with0.4310 s of CPU time (Pentium 4 2.2 GHz):

K ¼ �0:4539 �0:3076 0 0

0 0 �0:3818 �0:1964

� �

T2k k22 ¼ 0:0745

The proposed design approach results an improvement of53% over the extended controller parametrisation.

In [17] a structured controller synthesis approach isproposed that uses LMI and alternating projection method(which also results in a nonconvex formulation) thatachieves an equivalent result, but in a longer CPU time,34.44 s (Pentium 4 2.5 GHz):

K ¼ �0:4591 �0:3079 0 0

0 0 �0:4801 �0:2574

� �

T2k k22 ¼ 0:0747

Note that the method in [17] requires an initial stabilisingcontroller gain to start the optimisation algorithm.

Fig. 1 Pareto-like trade-off curves obtained with the msfsynfunction and the proposed procedure for g ¼ 0:1; 0:2; . . . ; 0:9; 1:0

IEE Proc.-Control Theory Appl., Vol. 152, No. 2, March 2005174

6 Example 3

In this example, the control objective is to design a reliablecontrol for example 2 in three different scenarios [3]: (i) thenominal plant ðb1 ¼ 1; b2 ¼ 1Þ; (ii) outage of the firstactuator ðb1 ¼ 0; b2 ¼ 1Þ; and (iii) outage of the secondactuator ðb1 ¼ 1; b2 ¼ 0Þ. The system is modelled by apolytope of matrices with three vertices corresponding tothese scenarios.

Considering a (centralised) robust state-feedback control,the quadratic robust controller is [3]:

K ¼ �0:5473 �0:7060 �0:5289 �0:6190

�0:6590 �0:9840 �0:6073 �0:3613

� �

T2k k22 ¼ 0:67

The extended robust controller is [3]:

K ¼ �0:6521 �0:6783 �0:3365 �0:5451

�0:6294 �0:9690 �0:6510 �0:4170

� �

T2k k22 ¼ 0:60

The relation between the optimisation parameters and thestate-feedback gain in the proposed procedure is:

K ¼ x1 x2 x3 x4

x5 x6 x7 x8

� �With the initial conditions x ¼ 0 and Qð0Þ ¼ I8, and thestop criteria e ¼ 0:01 and Ne ¼ 10; the controller parameteroptimisation achieves the following controller after 42iterations and 4.3570 s of CPU time:

K ¼ �0:0597 �0:8743 0:1480 �0:3660

�0:8879 �0:0463 �0:5690 0:1281

� �Considering the three scenarios, the worst-case norm isachieved in the vertex corresponding to the outage of thefirst actuator:

T2k k22 ¼ 0:4323

The exact cost considering the whole polytope is:

T2k k22 ¼ 0:6332

As one can see from Fig. 2, the worst-case norm does notoccur at the polytope vertices and it is necessary to repeatsteps 3, 4 and 6 including interior points to the initial set.With ed ¼ 0:1; three external iterations of the proposedalgorithm are necessary which means that it is necessary to

include two additional interior points (see Fig. 3) obtainedby the guaranteed cost computation (branch-and-boundstrategy). After three iterations and 163.8460 s of total CPUtime we achieve the following controller:

K ¼ �0:2177 �0:9050 0:1076 �0:3246

�0:8988 0:0001 �0:5862 0:1267

� �Considering the three scenarios, the worst-case norm is:

T2k k22 ¼ 0:4539

The exact cost considering the whole polytope is:

T2k k22 ¼ 0:4874

which is 19% better than the LMI-based result.

7 Conclusions

A strategy for a robust multiobjective H2=H1 controldesign with regional pole placement for continuous- ordiscrete-time linear time-invariant systems with polytope-bounded uncertainty was presented. The proposed pro-cedure relies on an efficient optimisation algorithm appliedto the multiobjective problem of H2 or H1 normminimisation over the set of polytope vertices. The controlsynthesis, performed directly in the space of the controllerparameters, avoids the conservatism that appears in LMI-based approaches. The combination of a LMI-basedanalysis formulation and a branch-and-bound strategy wasemployed to validate the performance and the regional poleplacement of the closed-loop system in the whole polytope.

Considering the analysed examples, the proposed syn-thesis procedure presents better results than LMI-basedapproaches and can find feasible solutions where the LMI-based formulations are unfeasible. In this way, the proposedprocedure can be considered as an alternative strategy whenthe LMI-based approaches do not produce a result thatsatisfies the constraints. The main concern when using theproposed procedure is the CPU time required especially forthe steps 4, 5, and 6 of the algorithm when more than oneiteration is required.

8 Acknowledgment

This work was supported in part by Conselho Nacional deDesenvolvimento Cientıfico e Technologico–CNPq, Brazil.

Fig. 3 Normalised parameter space subdivision used by thebranch-and-bound algorithm in the third iteration

Fig. 2 kHzwðzÞk2 mesh in the first iteration

IEE Proc.-Control Theory Appl., Vol. 152, No. 2, March 2005 175

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