ℋ2 and ℋ∞ε-guaranteed cost computation of uncertain linear systems

H2 and H1 e-guaranteed cost computationof uncertain linear systems

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

Abstract: An approach to compute theH2- orH1-guaranteed costs with any prescribed accuracy ispresented. The proposed approach can be applied to uncertain state–space models of linear time-invariant systems, where the system matrices depend on uncertain parameters or vary in a polytopicdomain of the space of matrices. The developed approach is based on a new implementation of thebranch-and-bound algorithm with the upper-bound functions based on linear matrix inequality (LMI)characterisations. The branch operation is based on a new subdivision technique that can be appliedto any kind of polytope shape, not restricted to the hyper-rectangle case. When applied alone, theLMI-based analysis formulations can fail to compute the guaranteed costs, or they can present con-servative results. Examples are presented to illustrate that the proposed analysis approach overcomesthe problems faced with LMI-based formulations with reasonable computational time.

1 Introduction

In the last few decades, H2 and H1 control has frequentlybeen applied to guarantee upper bounds on controlledsignals for all admissible disturbances and model uncertain-ties. In this control strategy, it is fundamental to have accu-rate and efficient methods to compute theH2 andH1 normsof the system transfer functions. TheH2 norm of a preciselyknown system can be directly computed in terms of the con-trollability or observability Gramians. The H1 norm com-putation of precisely known systems cannot be directlyachieved, but there are several methods available for effi-cient approximated computation. In the case of uncertainsystems, it is necessary to compute the worst-case normfor all admissible models in the uncertain set, which is the‘cost’ to be minimised by the control synthesis approach.This problem is known to be NP-hard [Note 1] and thereis presently no way for determining such worst-casenorms accurately. There are strategies to compute the H2-and H1-guaranteed costs based on linear matrix inequality(LMI) formulations. The first LMI characterisations arebased on the concept of quadratic stability [1]. In thisconcept, the use of a single Lyapunov function for theentire uncertain domain can lead to conservative results.To overcome this conservatism, recent works have con-sidered parameter-dependent Lyapunov functions, slackmatrices and tuning parameters (e.g. [2–8] and referencestherein). Unfortunately, the values obtained with these

# The Institution of Engineering and Technology 2006

doi:10.1049/iet-cta:20050334

Paper first received 9th September 2005 and in revised form 3rd February 2006

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, Federal University ofMinas Gerais, Av. Antonio Carlos 6627, 31270-010, Belo Horizonte, MG,Brazil

R.C. Mesquita is with the Department of Electrical Engineering, FederalUniversity of Minas Gerais, Av. Antonio Carlos 6627, 31270-010, BeloHorizonte, MG, Brazil

E-mail: [email protected]

IET Control Theory Appl., Vol. 1, No. 1, January 2007

strategies are only upper bounds of the exact costs and theaccuracy of these approaches varies for each different case.

This paper presents a strategy for the computation of theworst-caseH2 andH1 norms with any prescribed accuracy,which is denoted as e-guaranteed cost, for uncertainsystems with a polytopic model or an affine parameter-dependent model. The proposed approach is based on abranch-and-bound (BnB) algorithm where the e-guaranteedcost is approximated, from below, by the norm computed inthe vertices of the polytopes, generated in the branch oper-ation and, from above, by its guaranteed costs computed bymeans of LMI-based formulations. The BnB algorithm hasalready been applied in previous works to reduce conserva-tism in robust stability and performance analysis of lineartime-invariant systems [9–13]. In previous works, theanalysis was aimed at linear-fractional models of uncer-tainty with uncertain parameters in given intervals. TheBnB procedures in the previous works were specificallybuilt for hyper-rectangle-shaped uncertainty sets, relyingon a kind of rectangle division algorithm. Notice that thisformer division procedure is very nice and easy, whenapplicable, because it always generates self-similar subsets.However, a much more involved problem occurs when theuncertainty set is not of a rectangular shape. For generic-shaped polytopes, the uncertainty set cannot be dividedinto self-similar subsets in most of the cases. The BnBalgorithm is more attractive nowadays than in the early1990s due to the continuous improvement of computerhardware and the development of new LMI-based guaran-teed cost formulations that can be efficiently incorporatedas upper-bound estimators. The major contribution of thiswork is not only to present an approach to compute theH2 and H1 costs with a prescribed accuracy, but it is alsoto present a much more efficient implementation of theBnB algorithm that can be applied to a larger class of uncer-tain systems than in previous works. To achieve these twogoals, the proposed implementation is based on up-to-dateLMI-based analysis formulations and a new polytope par-tition technique based on simplicial meshes. When

Note 1: NP-hard refers to the class of problems that can be described ascontaining the descision problems that are at least as difficult as any problemin NP. NP refers to the class of descision problems solvable in polynominaltime on a non-deterministic algorithm.

201

necessary, a Delaunay triangulation is applied to decom-pose the polytope in a set of simplices. Further, simplicialmesh refinements are implemented with a simplex sub-division algorithm, developed specifically for this problem.One of the advantages of considering simplicial meshes isthe capability of dealing with polytopes of any shape, notrestricted only to the hyper-rectangle case. This featureallows the proposed analysis procedure to be applied toboth affine parameter-dependent and polytopic models,which is not possible with the works mentioned earlier.The choice of simplicial meshes also drastically improvesthe efficiency of the algorithm, considering that the boundfunctions are computed for simplices with dþ 1 vertices,instead hyper-rectangles with 2d vertices.

It should be noted that, at the present stage of knowledge,it is not expected that any LMI approach will reach theexact guaranteed cost for uncertain systems with arbitrarypolytopes and a given accuracy. There are however, cases,where the proposed procedure can compute the costswhen all LMI-based formulation fails. This means that aprocedure like the one as proposed here is likely to be ageneral framework for reaching such exact costs, incorpor-ating any LMI algorithm as the upper-bound generator. Theproposed performance analysis approach also has the capa-bility to find the uncertain domain coordinate of the worst-case norm. This is a useful feature in the implementationof the robust control synthesis procedure presented byGoncalves et al. [14, 15]. The proposed analysis approachhas already been applied to theH1-guaranteed cost compu-tation for uncertain time-delay systems [16].

2 Problem statement

Consider the linear time-invariant uncertain systemdescribed by

d½xðtÞ� ¼ AxðtÞ þ BwðtÞ

zðtÞ ¼ CxðtÞ þ DwðtÞð1Þ

where A [ Rn�n, B [ Rn�p, C [ Rq�n, D [ Rq�p andd[x(t)] W dx(t)/dt for continuous-time and d[x(t)] W x(tþ 1)for discrete-time systems.

Consider the system matrix

S WA B

C D

� �ð2Þ

This work is concerned with uncertain systems where thesystem matrix is not precisely known, but it belongs to aclosed convex polyhedral set: S [ PS.

In the case of polytopic models, the set PS is a polytope inthe system matrix space that is computed as

PS W Sð ~aÞ : Sð ~aÞ ¼XN

i¼1

aiSi; ~a [ Vp

( )ð3Þ

where

Si WAi Bi

C i Di

� �; i ¼ 1; . . . ;N ð4Þ

are the N polytope vertices, and the polytope coordinatevector a ¼ [a1

. . . aN]T [ Vp, with Vp defined as

Vp W ~a : ai � 0; 8i;XN

i¼1

ai ¼ 1

( )ð5Þ

In the case of affine parameter-dependent models, thesystem matrix is affine dependent on the uncertain

202

parameter vector p ¼ [p1 � � � pd]T

PS W Sð pÞ : Sð pÞ W S0 þ p1S1 þ � � � þ pdSd; p [ Va

� �ð6Þ

where p may belong to a hyper-rectangle delimited by theextreme values of the uncertain parameters, pi [ [ pi, �pi],i ¼ 1, . . . , d

Va W ½p1; p1� � ½ p2

; p2� � � � � � ½ pd; pd � ð7Þ

or any other closed convex polyhedral set, if there areadditional constraints.

The problem being considered is to compute the largestpossible Hq norm of the transfer matrix from w to z,Tzw(l) ¼ C(lI 2 A)21BþD, over all a [ V, with q [f2, 1g, l denoting s or z, a denoting a or p and V denotingVp or Va. The H2 (H1)-guaranteed cost, dgc (ggc), com-puted with LMI-based formulations are only an upperbound to the worst-case norm in the uncertain domain,without information about the gap between these values

maxa[VjjTzwjj2 � dgc

maxa[VjjTzwjj1 � ggc

ð8Þ

As the exact value is difficult to achieve, an approach tocompute the H2 (H1) e-guaranteed cost, dc (gc), which isdefined as the value that respects the following inequalities,is defined

maxa[VjjTzwjj2 � dc � ð1þ eÞmax

a[VjjTzwjj2

maxa[VjjTzwjj1 � gc � ð1þ eÞmax

a[VjjTzwjj1

ð9Þ

The proposed analysis approach is based on a BnB algor-ithm. The idea behind this algorithm is simple. The globalmaximum of the norm in the polytopic domain is approxi-mated by a lower- and an upper-bound function, andthe gap between them informs the calculus accuracy. Thebound functions tend to the global maximum when thepolytope is progressively decomposed into smaller sub-polytopes. In this work, the upper-bound function is theH2 (H1)-guaranteed cost, dgc (ggc), computed with LMI-based formulations, and the lower-bound function is theworst-case H2 (H1) norm, dw.c. (gw.c.), at the vertices ofthe polytope and subpolytopes. The algorithm stops whenthe e-guaranteed cost, dc (gc), achieves the prescribed rela-tive accuracy, e.

3 BnB algorithm

For the sake of completeness, the BnB algorithm isrevisited. Much of the following notation and terminologyis from Balakrishnan et al. [10]. The BnB algorithm canbe used to find the global maximum of a function f (a):Rd! R over a d-dimensional polytope Pinit. For a sub-

polytope P # Pinit, one can define

FmaxðPÞ W maxa[P

f ðaÞ ð10Þ


The BnB algorithm computes Fmax(Pinit) based on twofunctions, Flb(P) and Fub(P), defined over fP: P # Pinitg.These two functions must satisfy the following conditions

FlbðPÞ � FmaxðPÞ � FubðPÞ ð11Þ

8e . 0; 9d . 0 such that 8P # Pinit; sizeðPÞ � d

)½FubðPÞ �FlbðPÞ�

FlbðPÞ� e ð12Þ

The condition (11) establishes that the functions Flb(P) andFub(P) compute a lower and an upper bound on Fmax(P),respectively. The condition (12) establishes that, as themaximum distance between the vertices of P, denoted bysize(P), goes to zero, the difference between the upperand the lower bounds converges to zero.

The BnB algorithm applied in this work is as follows [10]:

Algorithm : Branch-and-Bound

k 0; L0 fPinitg;

L0 FlbðPinitÞ;

U0 FubðPinitÞ;

While ðUk � LkÞ=Lk . e do

P arg maxP[Lk

FubðPÞ;

subdivide P in P1; . . . ;Ps

Lkþ1 fLk � Pg< fP1; . . . ;Psg;

Lkþ1 maxP[Lkþ1

FlbðPÞ;

Ukþ1 maxP[Lkþ1

FubðPÞ;

eliminate all P [ Lkþ1 such as FubðPÞ , Lkþ1;

k k þ 1;

end

end

4 BnB algorithm applied to H2 and H1

e-guaranteed cost computation

To apply the BnB algorithm to the Hq e-guaranteed costcomputation, q [ f2, 1g, it is necessary to find out the func-tions Flb(P) and Fub(P) that satisfy the conditions (11) and(12). The lower-bound function can be defined as the worst-case Hq norm computed at the vertices of the polytope

FlbðPÞ ¼ max1�i�N

jjTzw;ijjq ð13Þ

where Tzw,i ¼ Ci(lI 2 Ai)BiþDi, with Di ¼ 0 for q ¼ 2 inthe case of continuous-time systems.

The upper-bound function Fub(P) can be defined as theguaranteed costs that are computed by means of any LMIoptimisation approach. It is obvious that these bound func-tions satisfy condition (11). As the LMI-based guaranteedcost formulations also compute the norm for preciselyknown systems, they also satisfy condition (12), consideringthat the polytope tends to a point because of successivesubdivisions. In this work, the upper-bound function isimplemented based on the work of Oliveira et al. [3–5].This implementation is based on the combination ofLemmas 1 and 2 in the case of H2-guaranteed costof continuous-time systems [4], on Lemma 1 in the caseof H1-guaranteed cost of continuous-time systems [5], onTheorem 3 in the case of H2-guaranteed cost of discrete-time systems [3] and on Theorem 4 in the case ofH1-guaranteed cost of discrete-time systems [3].


An additional feature of the proposed BnB implementationis the capability to identify stable systems that are not robust,which means that it is not possible to compute the guaranteedcost. The proposed implementation verifies if the system overa polytope (or subpolytope) vertex is stable before the normcomputation. If an unstable system is found, the algorithmstops and returns the corresponding coordinate.

The polytope Pinit can be parameterised by the vectors aor p. In the case of polytopic models, it is more appropriateto consider the uncertain domain represented in the(N 2 1)-dimensional space

V W a ¼ ½a1 � � �aN�1�T : ai � 0; 8i;

XN�1

i¼1

ai � 1

( )

ð14Þ

The vector a can be easily computed from a, just includingthe last entry as aN ¼ 1 2

Pi¼1N21 ai. The set Pinit ¼ V is a

simplex. The simplex is so named because it represents thesimplest polytope in any given space with the lower numberof vertices. The simplex in the d-dimensional space is a poly-tope with dþ 1 vertices (generalisation to d-dimensionalspaces of the triangle in two-dimensional space and thetetrahedron in three-dimensional space). In the case ofaffine parameter-dependent models, it is more efficient towork in the uncertain parameter space parameterised by pthan in the system matrix space parameterised by a. For duncertain parameters, the first case corresponds to a hyper-rectangle in the d-dimensional space, while the second casecorresponds to a simplex in the (2d 2 1) -dimensional space.

The partition of Pinit in the BnB algorithm can be accom-plished by several different ways. It is interesting to dealwith a simplicial mesh because simplex is the simplest poly-tope with the lower number of vertices. This leads to lowercomputational times when computing the guaranteed costsand the norms over the polytope vertices. Another advan-tage came from the fact that any polytope can be exactlydecomposed in a set of simplices by means of an operationknown as triangulation. As already mentioned, the simplexis also the shape of the uncertain domain related topolytopic models. In the case of affine parameter-dependentsystems, a triangulation technique is necessary to decom-pose a polytope with general shape in a set of simplices.The Delaunay triangulation is a good choice for thisoperation. The Delaunay triangulation maximises theminimum angle between edges over all possible triangu-lations to result in a set of ‘well-shape’ simplices. Thereis a close relationship between the Delaunay triangulationof a point set and the convex hull of the ‘lifting transform-ation’ [17] of these points in one higher dimension. So,algorithms to compute the convex hull in (dþ 1) dimen-sions can be used to compute the Delaunay triangulationin d dimensions. This is done in the MATLABw functiondelaunayn(.) that is based on the Quickhull algorithm [18].

In order to refine the simplicial mesh, the simplices aresplit with a simplex edgewise subdivision technique basedon an abacus model of a simplex introduced inEdelsbrunner and Grayson [19]. This technique allows tosplit a d-simplex into 2d simplices. This is accomplishedby adding a new vertex at the middle of each simplexarrow (the main difficulty in this procedure is in keepingthe information about which vertices belong to a simplex).This technique has two advantages. First, all simplicesachieved with the subdivision have the same volume,which guarantees that the volume tends to zero with succes-sive subdivisions. Second, the number of congruence classesof simplices that result from successive refinements are

203


limited to d!/2, which is the optimum value for simplexsubdivision [20]. This feature means that this subdivisiontechnique avoids creating degenerated simplices, that is, sim-plices with too small angles between edges, that operate infavour of the BnB algorithm convergence. It is interestingto mention that the bisection strategy, applied in previousworks for the case of hyper-rectangles, does not guarantee alimited number of congruence classes for d . 2 whenapplied to simplex subdivision [20]. The combination of theDelaunay triangulation and the edgewise simplex subdivisionallows the proposed approach to be applied in the case ofuncertainty domain shape not constrained to the hyper-rectangle case. It must be noted that the previous BnBalgorithms in the control theory are not equipped to dealwith regions different from hyper-rectangles.

5 Edgewise subdivision of a simplex

Based on the conceptual idea presented by Edelsbrunnerand Grayson [19], the formalisation of a general algorithmto implement the edgewise subdivision of a d-dimensionalsimplex in kd simplices is proposed here. The same notationused by Edelsbrunner and Grayson [19] is adopted here.Consider a d-simplex s defined as a sequence of dþ 1points, P0, P1, . . . , Pd, that are affinely independent in Rd.Consider the notation

Px1x2...xk¼ðPx1þ Px2

þ � � � þ PxkÞ

kð15Þ

The edgewise subdivision of s in kd will be derived fromthe points P0, P1, . . . , Pd and new points Px1x2. . .xk

, asdefined in (15). The points that define each new simplexwill be obtained from a matrix M [ Nk�(dþ1), called thecolour scheme, whose entries are integer numbers in therange [0, d], called the colours, that represent the indicesof the points P0, P1, . . . , Pd. The ith column of M willdefine the point Px0,ix1,i. . .xk21,i

of the new simplex. In thedeveloped algorithm, the indices of the rows of M startwith ‘0’, instead of ‘1’, as presented by Edelsbrunner andGrayson [19].

Consider x(n)i,j the entry of the ith row and jth column of

the nth colour scheme, M(n), n ¼ 0, . . . , kd 2 1 and xm,m ¼ 0, . . . , d 2 1, the digits of n represented in the basek, that is, n ¼ xd21kd21

þ . . . þ x1kþ x0. Each colourscheme M(n), n ¼ 0, 1, . . . , kd 2 1, will be created row byrow starting with x(n)

0,0 ¼ 0, as presented in the followingalgorithm.

Algorithm: Colour schemes

for n ¼ 0; 1; . . . ; kd� 1 do

xd�1 . . . x0 conversion of n to base k;

colour 0;

for i ¼ 0; 1; . . . ; k � 1 do

xðnÞi;0 colour;

for j ¼ 1; . . . ; d do

if xd�j ¼ i then colour colour þ 1;

xðnÞi;j colour;

end

end

end

end

204

In the BnB algorithm, to compute the 1-guaranteed costs,only consider the edgewise subdivision with k ¼ 2. In thiscase, the edgewise subdivision will be accomplished byintroducing one new point at the middle of each edge ofthe simplex P. These new points will provide the conditionsto split P into 2d new simplices. The problem solved by thepresented algorithm is how to combine these points inhigher dimensions.

Consider the subdivision of a tetrahedron. Passing to thealgorithm the values d ¼ 3 and k ¼ 2, it returns 8 matrices,the colour schemes. Each matrix informs how to combinethe four original vertices to generate each subtetrahedron.For example, the following colour scheme

M ð6Þ ¼0 0 0 1

1 2 3 3

� �

indicates that the seventh subtetrahedron is defined by theset fP01, P02, P03, P13g, as shown in Fig. 1, where thepoint Px0,ix1,i

is computed by (15): Pij ¼ (Piþ Pj)/2.

6 Complexity of the proposed polytopesubdivision technique

The computational time required by the proposed approachdepends on the uncertain space dimension, d, the systemsize, n, and the number of polytope vertices, N. Thesystem size affects the guaranteed cost computations, withinfluence over the number of optimisation variables of theLMI-based formulations. In the case of LMI formulationsbased on parameter dependent Lyapunov functions, thecomputational time depends on the number of polytope ver-tices drastically. The number of vertices of the initial poly-tope related to affine parameter-dependent models affectsonly the first guaranteed cost computation because, afterthe decomposition of the polytope in a set of simplices bythe Delaunay triangulation, the number of vertices is fixedas dþ 1. The uncertain space dimension, d, defines howmany guaranteed cost computations are necessary foreach iteration because it defines how many partitions arederived with the subdivision technique. The number of sim-plices generated by the simplex edgewise subdivision tech-nique is equal to 2d in (Table 1). Both the Delaunaytriangulation and the simplex edgewise subdivision maylead to a prohibitive computational cost for d � 5 if theguaranteed cost computation is not fast enough, whichhappens in the case of higher-order systems combinedwith complex LMI formulations (a large number of decisionvariables). It is clear that the efficiency of the algorithmdepends on the appropriate choice of the LMI-based analy-sis formulation, considering the trade-off between conserva-tism and complexity.

P

P

P

P

P

P

1

2

3

12

13

23

P0

P03

P02

P01

Fig. 1 Subtetrahedron defined by M (6)

IET Contr

Table 1: Number of simplices generated by the subdivision technique

Dimension d 1 2 3 4 5 6 7

Delaunay triangulation of a hyper-cube — 2 6 24 103 648 3642

Simplex edgewise subdivision (2d) 2 4 8 16 32 64 128

7 Numerical illustrative examples

The following examples will be presented to illustrate theeffectiveness of the proposed approach to compute H2

and H1 costs in the case of both continuous and discrete-time systems. The LMI control toolbox for MATLABw

has been used in a Pentium IV 2.8 GHz, 1 GB RAM com-puter. The options [1024 500 109 10 1] are used with thefunction mincx(.).

Example 1: Consider the uncertain discrete-time lineartime-invariant system described by

xðt þ 1Þ ¼ AxðtÞ þ BwwðtÞ þ BuuðtÞ

zðtÞ ¼ CzxðtÞ þ DzuuðtÞ

yðtÞ ¼ CyxðtÞ

ð16Þ

where x(t) [ Rn, w(t) [ Rn, u(t) [ Rn21 and z(t) [ R2n21.For evaluating the efficiency of the proposed robust perform-ance analysis approach, an exhaustive numerical comparisonwith LMI-based analysis formulations is considered here.For each pair (n, N), with n [ [2, 4] and N [ [2, 4], 100robustly stable polytopic systems are generated as follows:(i) the N polytope vertices are generated with matricesAi [ Rn�n, Bw,i [ Rn�n and Bu,i [ Rn�n21, i ¼ 1, . . . , N,whose entries are real numbers uniformly distributed in theinterval [21, 1], with 1 � maxi r(Ai) � 1.5, where r(.)denotes the spectral radius. Cz and Dzu are fixed such thatz ¼ [x1

. . . xn u1. . . un21]T and Cy ¼ I; (ii) using the design

procedure presented by Goncalves et al. [14] a stabilisingstate-feedback controller that minimises the worst-casekTzw(z)k2 is computed.

The following H2-guaranteed cost formulations areanalysed: quadratic stability (QS), presented by Palhareset al. [1]; Theorem 3 (T.3), presented by de Oliveiraet al. [3]; Lemmas 1 and 5 (L.1&5), presented by deOliveira et al. [4]; Lemmas 3 and 6 (L.3&6), also presentedby de Oliveira [4] and Lemma 2 (L.2) presented by Xieet al. [7]. Table 2 presents the rate of success to computethe H2-guaranteed cost with the LMI-based approachesand the H2 e-guaranteed cost computed with the BnB for

ol Theory Appl., Vol. 1, No. 1, January 2007

e ¼ 0.01. Table 3 presents the average relative gapbetween each H2-guaranteed costs and the lowest one.The results in Table 2 show that all LMI-based formulationsfail to compute theH2-guaranteed cost in several cases. For(n, N ) ¼ (4, 4), their rate of success are all ,50%. Note thatthe proposed approach computes the H2 1-guaranteed costswith the prescribed accuracy for all 900 uncertain systems.In contrast with the proposed approach, Table 3 shows thatthe analysed LMI-based formulations alone do not provideaccurate H2-guaranteed costs. In this example, for all com-binations of the pair (n, N ), the BnB algorithm requiresmore computational time than the LMI-based formulationsalone, but they are not prohibitive. For (n, N ) ¼ (4, 4), theaverage computational time required by the BnB algorithmis 225.585 s.

To illustrate the branch operation, consider the systemthat demands more computational time (134.828 s) tocompute the H2 e-guaranteed cost for (n, N ) ¼ (3, 3).This system is given by

S ¼ a1

0:1043 0:3958 1:1169 �0:1311

�0:6602 0:4884 �1:4101 �1:0514

0:4593 0:3685 �0:6147 �1:7599

1:0000 0 0 0

0 1:0000 0 0

0 0 1:0000 0

0:0258 �1:6053 2:4398 0

�0:0906 �1:0613 2:4551 0

266666666666664

�1:4848 1:3283

�3:3302 �3:0469

0:5132 2:4032

0 0

0 0

0 0

0 0

0 0

377777777777775

Table 2: Rate of success (%) to compute H2-guaranteed cost (Example 1)

n N QS [1] T.3 [3] L.1&5 [4] L.3&6 [4] L.2 [7] BnB

2 2 67 90 100 100 100 100

3 36 85 99 99 99 100

4 29 78 98 96 96 100

3 2 45 87 94 97 98 100

3 4 57 81 83 89 100

4 0 34 63 68 65 100

4 2 23 78 88 98 99 100

3 0 37 57 75 74 100

4 0 4 29 34 38 100

205

206

Table 3: Average relative gap (%) of H2-guaranteed cost (Example 1)

n N QS [1] T.3 [3] L.1&5 [4] L.3&6 [4] L.2 [7] BnB

2 2 66.34 40.14 20.65 20.27 21.53 0.00

3 105.28 77.77 26.61 30.52 30.87 0.00

4 159.68 75.22 33.05 54.20 40.41 0.00

3 2 175.30 87.16 40.11 31.94 41.93 0.00

3 165.65 112.74 96.48 66.05 103.89 0.00

4 — 251.72 120.98 125.44 112.98 0.00

4 2 114.79 90.01 54.53 43.19 58.83 0.00

3 — 128.76 92.11 135.31 153.84 0.00

4 — 675.27 140.67 161.86 311.71 0.00

þa2

1:0082 �0:1766 �1:7410 �1:9883

0:9079 �0:2808 1:3338 �0:6546

0:2246 �0:2852 1:1973 �0:2919

1:0000 0 0 0

0 1:0000 0 0

0 0 1:0000 0

0:0258 �1:6053 2:4398 0

�0:0906 �1:0613 2:4551 0

266666666666664�1:4925 �1:0440

�1:2915 0:2366

�3:3380 2:7317

0 0

0 0

0 0

0 0

0 0

377777777777775

þa3

0:9842 0:2584 �1:0661 �0:8363

�0:3825 �0:2019 0:6192 2:2160

0:9410 0:1153 �1:3990 �1:7613

1:0000 0 0 0

0 1:0000 0 0

0 0 1:0000 0

0:0258 �1:6053 2:4398 0

�0:0906 �1:0613 2:4551 0

266666666666664�3:5992 0:2433

�2:4479 �3:6709

�2:1300 �1:2462

0 0

0 0

0 0

0 0

0 0

377777777777775

In this case, all LMI-based analysis formulations fail tocompute the H2-guaranteed cost. The number of necessaryrefinements on the polytope subdivision depends on the gra-dient of kTzwkq near the maximum points. This fact can beobserved in Figs. 2 and 3 that present the surface of the H2

norm for a [ V and the final polytope subdivision after 83iterations of the BnB algorithm. In this case, the maximumH2 norm is equal to 75.8267 for a ¼ [0.7813 0]T, witha3 ¼ 1 2 (a1þ a2) ¼ 0.2188, and the H2 e-guaranteedcost is computed as dc ¼ 76.5657.

Example 2: In this example, consider the H1-guaranteedcost computation for discrete-time systems. The same 900random polytopic systems analysed in the previousexample are considered again. The followingH1-guaranteed cost formulations are analysed: quadraticstability (QS), presented by Palhares et al. [1]; Theorem 4(T.4), presented by de Oliveira et al. [3]; Lemma 4 (L.4),

0 0.25 0.5 0.75 1 00.25

0.50.75

10

10

20

30

40

50

60

70

80

α2α1

H2 n

orm

Fig. 2 H2 norm for a [ V (Example 1)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

α1

α2

Fig. 3 Polytope subdivision in theH2 e-guaranteed cost compu-tation (example 1)


Lemma 5 (L.5) and Lemma 6 (L.6), presented by deOliveira [5]. Table 4 presents the rate of success to computethe H1-guaranteed cost with the LMI-based approaches andthe H1 e-guaranteed cost computed with the BnB approachfor e ¼ 0.01. Table 5 presents the average relative gapbetween each H1-guaranteed costs and the lowest one.Table 6 lists the normalised average computational times.The analysed LMI-based formulations alone achieve fewresults with accuracy ,1% but they also present low rate ofsuccess and low accuracy in some cases. Different from theH2 case, besides the 100% of rate of success and the betteraccuracy, Table 6 shows that the proposed approach also


requires less computational time than Lemmas 4 and 6 pre-sented by de Oliveira et al. [5] in all considered situations.In Table 6, the normalised average computational timesare the average time of each approach, considering the 100tests for each (n, N), divided by the maximum one. For(n, N) ¼ (4, 4), the proposed analysis approach leads to theaverage computational time as 36.55 s.

Example 3: In this example, consider a control surfaceservo for an underwater vehicle presented by Leibfritzand Lipinski [21]. It included three uncertain parameters

Table 4: Rate of success (%) to compute H1-guaranteed cost (Example 2)

n N QS [1] T.4 [3] L.4 [5] L.5 [5] L.6 [5] BnB

2 2 67 90 93 100 100 100

3 36 85 90 99 100 100

4 29 77 89 95 98 100

3 2 45 87 83 98 98 100

3 4 57 64 89 92 100

4 0 34 41 65 84 100

4 2 23 78 81 99 99 100

3 0 37 40 74 89 100

4 0 4 10 38 54 100

Table 5: Relative gap (%) of H1-guaranteed cost (Example 2)

n N QS [1] T.4 [3] L.4 [5] L.5 [5] L.6 [5] BnB

2 2 111.31 95.61 2.61 3.71 1.35 0.00

3 729.88 419.80 24.10 11.09 3.61 0.02

4 572.14 49.85 4.31 8.45 1.27 0.02

3 2 881.91 880.23 165.14 16.51 9.73 0.03

3 635.17 219.39 100.84 80.23 11.17 0.02

4 — 838.53 64.94 68.17 34.89 0.01

4 2 259.74 193.12 484.08 21.21 14.21 0.02

3 — 358.45 262.90 124.14 51.20 0.01

4 — 3468.84 177.20 489.90 25.12 0.00

Table 6: Normalised average computational time (Example 2)

n N QS [1] T.4 [3] L.4 [5] L.5 [5] L.6 [5] BnB

2 2 0.14 0.15 0.48 0.23 1.00 0.42

3 0.04 0.04 0.42 0.06 1.00 0.18

4 0.01 0.01 0.30 0.02 1.00 0.11

3 2 0.11 0.10 0.53 0.19 1.00 0.36

3 0.03 0.03 0.42 0.06 1.00 0.22

4 0.01 0.01 0.34 0.03 1.00 0.18

4 2 0.11 0.10 0.59 0.18 1.00 0.38

3 0.01 0.03 0.41 0.07 1.00 0.19

4 ,0.01 0.01 0.22 0.03 1.00 0.20

207

in the original model

dx

dt¼

0 p1 0 0

�p1 �p2 �4100 0

p3 0 �p3 0

0 0 0 0

0 0 1600 �450

0 0 0 81

0 0 0 0

0 0 0 0

266666666666664

0 0 0 0

0 0 0 0

�700 0 0 0

1400 0 0 0

�110 0 0 0

0 �1 0 �900

0 0 0 110

0 12 �1:1 �22

377777777777775

x

þ

0 0

9900 0

0 0

0 0

0 0

0 0

0 0

0 99

266666666664

377777777775

wþ

0 0

4:6 99 000

0 0

0 0

0 0

0 0

0 0

0 0

266666666664

377777777775

u

z1 ¼ ½ 0 0 0 0 0 0 1 0 �xþ ½ 0 0 �w

þ ½ 1 0 �u

y ¼0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

� �x

with p1 [ [663, 1037], p2 [ [93.6, 146.4] and p3 [ [25.74,40.26], resulting in an eight-vertex polytope in the 3-dspace. It was designed as a fourth-order structureddynamic output feedback controller to minimise the worst-case kTz1wk1 of the 12th-order closed-loop system based onthe procedure presented by Goncalves et al. [15]

K ¼Ac Bc

Cc Dc

" #

with

Ac ¼

0 0 0 �9:7499

1 0 0 �20:2208

0 1 0 �13:0829

0 0 1 �5:5692

2664

3775

Bc ¼

0:9154 10:7133

0:8171 5:3653

�0:3397 �18:5070

�1:6809 �15:6363

2664

3775

Cc ¼0 0 0 1

0 0 0 1

� �

Dc ¼0:0004 �0:9924

�0:1557 �1:8732

� �

208

In the case of H1-guaranteed cost, the followingformulations have been considered: quadratic stability, pre-sented by Palhares et al. [1]; Lemma 1, presented by deOliveira [5]; Lemma 2 presented by de Oliveira et al. [5](or Theorem 2, presented by He et al. [8]) and Lemma 3presented by de Oliveira et al. [5]. All the LMI-basedformulations failed to compute the guaranteed H1 costfor this example. The proposed approach, implementedwith Lemma 1 [5], achieved a H1 e-guaranteed costequal to 0.9377 with accuracy e ¼ 0.1. In this case, threeiterations with 34 min and 9 s of computational time werenecessary. Notice that this example illustrates the useful-ness of the proposed approach to assess the performanceachieved with this controller.

8 Conclusions

A new approach to compute the H2- and H1-guaranteedcosts based on a BnB algorithm is presented in this paper.The upper-bound function in the BnB algorithm can bechosen from any available LMI-based guaranteed cost for-mulation. The proposed approach can be applied to per-formance analysis of uncertain linear time-invariantsystems represented by polytopic or affine parameter-dependent models. The appealing feature of the proposedapproach is the possibility of computing the H2- orH1-guaranteed costs with any prescribed accuracy, evenwhen all LMI-based guaranteed cost formulations fail.Exhaustive numeric tests illustrate that the proposedapproach has a better performance than recent LMI-basedformulations, considering both the criteria rate of successand accuracy. In some cases, it also requires less compu-tational time than the more complex LMI-based formu-lations. This work also presented an efficient and simplealgorithm to subdivide a d-simplex in kd simplices thatwas applied with the Delaunay triangulation to implementthe branch operation.

9 Acknowledgments

This work has been supported in part by the Brazilianagencies CNPq and FAPEMIG.

10 References

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ℋ2 and ℋ∞ε-guaranteed cost computation of uncertain linear systems

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