This article was downloaded by: [DUT Library]On: 07 October 2014, At: 22:33Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number:1072954 Registered office: Mortimer House, 37-41 Mortimer Street,London W1T 3JH, UK

International Journal ofControlPublication details, including instructionsfor authors and subscription information:http://www.tandfonline.com/loi/tcon20

Nonlinear H infinitycontrol and relatedproblems of homogeneoussystemsYiguang Hong & Hongyi LiPublished online: 08 Nov 2010.

To cite this article: Yiguang Hong & Hongyi Li (1998) Nonlinear H infinitycontrol and related problems of homogeneous systems, International Journalof Control, 71:1, 79-92

To link to this article: http://dx.doi.org/10.1080/002071798221939

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy ofall the information (the “Content”) contained in the publicationson our platform. However, Taylor & Francis, our agents, and ourlicensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication arethe opinions and views of the authors, and are not the views of orendorsed by Taylor & Francis. The accuracy of the Content shouldnot be relied upon and should be independently verified with primarysources of information. Taylor and Francis shall not be liable for anylosses, actions, claims, proceedings, demands, costs, expenses,damages, and other liabilities whatsoever or howsoever causedarising directly or indirectly in connection with, in relation to orarising out of the use of the Content.

This article may be used for research, teaching, and private studypurposes. Any substantial or systematic reproduction, redistribution,reselling, loan, sub-licensing, systematic supply, or distribution inany form to anyone is expressly forbidden. Terms & Conditions ofaccess and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

DU

T L

ibra

ry]

at 2

2:33

07

Oct

ober

201

4

Nonlinear H ¥ control and related problems of homogeneous systems

YIGUANG HONG² and HONGYI LI²

In this paper, the problems of H¥ control and related L 2 gain of nonlinear homo-geneous systems are considered. First of all, L 2 gains of homogeneous stablesystems are estimated with a method in light of Hamilton± Jacobi inequalities.Secondly, by virtue of Hamilton± Jacobi± Isaacs (HJI) inequality, the nonlinearH¥ (induced L 2 gain) problem of a class of systems with homogeneous vector ® eldsis discussed, and global control law is given under an assumption similar to the oneof stabilizability of linear systems. Then, following the previous results, homoge-neous systems with higher degree perturbed terms are studied. Finally, the analysisof ® nite test of homogeneous systems with perturbed parameter is presented.

1. Introduction

The design and syntheses of nonlinear systems have aroused wide research inter-est in recent years. Nonlinear H¥ control is one of the important problems, con-cerning disturbance attenuation, model matching, and tracking. With the help ofdi� erential game theory and therefore, Hamilton± Jacobi± Isaacs (HJI) equations orinequalities (Basar et al. 1991, Isidori et al. 1995, van der Shaft 1992), the nonlinearH¥ control is widely discussed.

First of all, the formulation of nonlinear H¥ control is introduced brie¯ y. Con-sider an a� ne nonlinear control system

Çx = f (x) + g1 (x)u + g2 (x)w, x Î Rn, w Î Rp

z = h(x) + k(x)u, z Î Rl, u Î Rm } (1.1)

where f (0) = 0, h(0) = 0. Here w includes all external disturbance signals; u is thecontrol signal; z is the output variable; x is the state of the system. f (x) , g1 (x) , g2 (x) ,h(x) , k(x) are C1 smooth. In addition, kT (x)k(x) is of full rank, which is a basicassumption to assure that the (local) saddle solution of H¥ control problem existsuniquely (referred to Isidori et al. (1996)). Usually, we assume kT (x)h(x) = 0.

Nonlinear H¥ control via state feedback may be put in this way. Forw Î L p

2[0, T ) , T > 0 or T = ¥ , ® nd a positive constant (as small as possible) ¸*,and for ¸ > ¸*, a state feedback u = u(x) can be given such that

(1) if the initial condition x(0) = 0, we have,

òT

0zT z d¿ £ ¸

2 òT

0wTw d¿ (1.2)

(2) if w = 0, the closed loop system is asymptotically stable.

0020-7179/98$12.00 Ñ 1998 Taylor & Francis Ltd.

INT. J. CONTROL, 1998, VOL. 71, NO. 1, 79± 92

Received 14 April 1997. Revised 25 November 1997.² Institute of Systems Science, Chinese Academy of Sciences, Beijing, 100080, China.

e-mail: yghong@iss03.iss.ac.cn

Dow

nloa

ded

by [

DU

T L

ibra

ry]

at 2

2:33

07

Oct

ober

201

4

To solve the problem, de® ne

H (V ,x,u,w) = ¶ V¶ x

( f + g1u + g2w) + zTz - ¸2wTw (1.3)

and its saddle solution

u*(V ,x) = - 12 (kTk)- 1gT

1¶ T V¶ x

, w*(V ,x) =1

2¸2 gT2¶ TV¶ x

(1.4)

Then inequality (1.2) can be satis® ed, if there is a nonnegative function V (x) suchthat the HJI inequality, H[V ,x,u*(V ,x),w*(V ,x)]£ 0, holds.

Unfortunately, HJI equations or inequalities are di� cult to analyse, even for theexistence of their smooth solutions, and therefore, in most cases, it is very di� cult topresent a concrete simple-structure control law for the nonlinear H¥ design. In fact,it is usually assumed beforehand that there exists (uniquely) a smooth solution of theinequality. In addition, for theoretic analysis, the study on viscosity solutions of theproblem is conducted (for example, Soravia 1996).

To provide practical H¥ control method for nonlinear systems, speci® c systemshave to be studied. For example, by backstepping algorithms, an H¥ design is givenfor the systems with the strict feedback forms or pure feedback forms (Pan et al.1996). In this paper, a research idea about nonlinear H¥ control is presented foranother particular class of nonlinear systems, i.e. homogeneous systems.

As is well known, systems with homogeneous properties have special character-istics and can therefore lead to better theoretic results than general a� ne nonlinearsystems. There are some good studies of homogeneous systems in di� erent aspects(for instance, Sepulchre et al. 1996, Hermes 1996).

Here we will focus on a special class of systems in the form of

Çx = f (x) + B1u + B2w, x Î Rn, w Î Rp

z = h(x) + Du, z Î Rl , u Î Rm } (1.5)

where f (x) = [f1 (x), . . . , fn (x)]T, h(x) = [h1 (x), . . . ,hl (x)]T, f is a homogenous vec-tor ® eld of degree k, h is a homogeneous function of degree k, B1, B2, D are constantmatrices with usual assumptions, B1 is of full column rank and DTD is of full rank.

The paper is organized as follows. Notice that nonlinear H¥ problem is studiedby virtue of L 2 gain from the viewpoint of game theory. Therefore, L 2 gain and itsestimation are considered ® rst. Then some of the H¥ control results of linear systemsare extended to the case of homogeneous nonlinear systems, whose properties arealso held mostly globally. Some examples and comments are given for illustration.During the discussion, a class of systems which can be approximated by homoge-neous systems (just like some systems approximated by linear systems) are alsoconsidered. Finally, the homogeneous systems with perturbed parameters aretaken into account, and from a robust viewpoint, the result of a ® nite test isobtained.

Without loss of generality, we assume DTD = sI, s > 0, hT (x)D = 0. In thepaper, we assume that k ³ 1 and can be expressed in the form of i /(2j + 1) ,i = 2, . . .; j = 0,1, . . . . Let i ´ i denote the Euclidean norm of vector spaces, andalso denote the norm for matrices like i Ai = s (A) , where s (A) is the maximumsingular value of the considered matrix A.

80 Y. Hong and H. L iD

ownl

oade

d by

[D

UT

Lib

rary

] at

22:

33 0

7 O

ctob

er 2

014

2. L2 gain of homogeneous systems

In this section, before dealing with H¥ control problem, analysis on L 2 gain ispresented for a class of homogeneous systems in the form of

Çx = f (x) + B2w

z = h(x) zi = hi (x), i = 1, . . . , l } (2.1)

where f (x) is a homogeneous vector ® eld of degree k, and h(x) is a homogeneousfunction of degree k. As is well known, L 2 gain and input± output stability of non-linear systems are deeply related to H¥ control and input-to-state stability, and all ofthese problems have been studied widely for many years (for example, Vidyasagar etal. 1982, Ryan 1995). Much work on input± output stability was done with the con-ditions like exponential stability.

De® nition 1: System (2.1) is said to be L 2-stable or have a ® nite L 2 gain if thereexists a ® nite number ¸ > 0 such that when w Î L p

2[0, T ) (T > 0 or T = ¥ ), wehave

òT

0zT z d¿ £ ¸

2 òT

0wT wd¿ (2.2)

Set

H0 (V ,x,w) = ¶ V¶ x [f (x) + B2w]+ zTz - ¸

2wTw (2.3)

It is easy to see that the supreme value of w, from (2.3), is

w*(V ) =1

2 2 BT2¶ TV¶ x

(2.4)

sinceH(V ,x,w) £ H (V ,x,w*) - i w - w*i 2 £ H (V ,x,w*)

It is easy to see that, if there is a nonnegative function V to make H (V ,x,w*) £ 0hold, (2.2) will be satis® ed because

òT

0H d¿ = V[x(T )]- V[x(0)]+ ò

T

0zT z d¿ - ¸

2wTwd¿ £ 0 (2.5)

and V (0) = 0.Consider the free system corresponding to (2.1)

Çx = f (x), f (0) = 0, x Î Rn (2.6)

where f (x) is homogeneous of degree k and C1 smooth. Denote the solution of (2.6),corresponding to initial condition x(t0) = x0, by w (t,x0,t0) . Since (2.6) is time-invariant, it is easy to see that x(t) = w [t,x(t) ,t]= w [t,x(t0),t0].

Then a global result can be obtained in the following result.

Theorem 1: If (2.6) is asymptotically stable at its equilibrium x = 0, then it is L 2-stable.

Proof: At ® rst, take a Lyapunov candidate function as

V[x(t)]= ò¥

0i w (¿,x(t),0) i 2k d¿ = ò

¥

ti w (¿,x(t),t) i 2k d¿ (2.7)

According to Hahn (1976, Theorem 57.1), (2.7) is well de® ned. On account of

Nonlinear H¥ control of homogeneous systems 81D

ownl

oade

d by

[D

UT

Lib

rary

] at

22:

33 0

7 O

ctob

er 2

014

V (x0) = V (x(t) ) + òt

t0

i w (¿,x0,t0) i 2k d¿,it is simple to obtain

ÇV |(2.6) = ¶ V¶ x

f (x) = - i x(t) i 2k

and

V (¹x) = ¹2k ò

¥

0i w (¹k- 1

¿,x,0) i 2k d¿ = ¹2k- k+1V (x) , ¹ Î R

namely, V is a homogeneous function with degree k + 1.Therefore, if taking V0 = ¸V , then

H0[V0,x,w*(V0)]|(2.1) = ¸¶ V¶ x

f +14

¶ V¶ x

B2BT2¶ T V¶ x

+ hT h (2.8)

Notice that the right side of (2.8) is homogeneous of degree 2k, so we can only studyit on the unit sphere. Set

¸* = maxi xi =1

14

¶ V¶ x

B2BT2

¶ TV¶ x

+ hTh[ ] (2.9)

Then, when taking ¸ > ¸*, H0[V0,x,w*(V0)]£ 0. Thus, (2.2) can be obtainedeasily. h

Remark 1: In Ryan (1995), an elegant result was proposed on the input-to-statestability of another class of homogeneous systems in the form of Çx = f (x,w),where f is positively homogeneous with respect to x and w, i.e.f ( q x, q w) = q k f (x,w) , q ³ 0. The technique presented in that paper is also be-lieved, after slight changes, to be able to prove the above theorem.

The following result also shows the speciality of homogeneous properties.

Theorem 2: If (2.6) is asymptotically stable and there is a CK (K > k + 1) smoothnonnegative function V0 to make

H0[V ,x,w*(V )]= ¶ V¶ x

f +1

4¸2¶ V¶ x

B2BT2¶ TV¶ x

+ hTh £ 0 (2.10)

hold (globally) for (2.1) , then there is a homogeneous nonnegative function satisfying(2.10) .

Proof: Assume the lowest degree of V0 is q, namely, V0 (x) = Vq (x) + o( i xi q) ,where Vq is homogeneous of degree q. If q < k + 1, then it is obvious that theterm i BT

2 ( ¶ T V /¶ x) i , which is nonnegative, plays the main role on the left-handside of (2.10) locally near the origin of the space (x,w) and therefore, (2.10) can-not be satis® ed. Conversely, if q > k + 1, then hTh will play the main role locallyand (2.10) cannot hold either. Therefore, the lowest degree must be k + 1. TakeV = Vk+1, and the conclusion follows. h

In fact, the above result can be extended to a more general class of systems.Consider

Çx = f (x) + d f (x) + [B2 + g2 (x)]wz = h(x) + d h(x) } (2.11)

82 Y. Hong and H. L iD

ownl

oade

d by

[D

UT

Lib

rary

] at

22:

33 0

7 O

ctob

er 2

014

where f (x) , h(x) , B2 are de® ned as above, and d f (x) = o( i xi k) , d h(x) = o( i xi k) ,g2 (x) = o(1) . For convenience, (2.1) is called the homogeneous approximationsystem of (2.11).

According to the proof of Theorem 1, if ¸ > ¸* then there is V0 such that

H0[V0,x,w*(V0)]|(2.11) = H0[V0,x,w*(V0)]|(2.1) + o( i xi 2k) £ 0

hold locally. Therefore, (2.11) is locally L 2-stable (when x and w are small enough).

3. Nonlinear H ¥ control of homogeneous systems

First of all, we give an assumption that is used in what follows:

Assumption 1: (Andreini et al. 1988): There is a linear subspace C and a scalarproduct k ,´ l , such that Rn = C % B1 Rm, and if u (x) denotes the projection on C off (x) along B1 Rm, then k x, u (x) l is positive de® nite in C .

About this assumption, we already know

Proposition 1 (Andreini et al. 1988): Assumption 1 holds if and only if there is apositive symmetric n ´ n matrix P such that

Ker BT1 P Ì {x Î Rn

: xTPf (x) < 0}Ä {0} (3.1)

Moreover, condition (3.1) is equivalent to the stabilizability in the linear case.

Remark 2: This assumption is more natural for homogeneous systems since iff Î C , then f ( q x) = q k f (x) is still in C . In fact, the reason why homogeneoussystems are easier to be analysed may be that homogeneity is similar to linearity.For example, many results in what follows are obtained because homogeneoussystems can be studied in unit sphere.

Remark 3: If the output of (1.5) is expressed as z = h(x) + Du, DTh /= 0, then wecan rede® ne ~u, ~f , ~h, ~D such that ~f (x) and ~h(x) are still homogeneous with thesame degree k and ~DT~h = 0. Moreover, (3.1) holds if and only ifKer BT

1 P Ì {x Î Rn : xTP~f (x) < 0}Ä {0}.

In this section, starting from Assumption 1, a simple feedback law is given for itsHJI inequality.

First of all, we discuss (1.5) in the case of B1 = B2, and the main result is thefollowing theorem.

Theorem 3: If system (1.5) satis® es Assumption 1 and ¸ > ¸* = ê êsÏ , the nonlinearH¥ control problem of (1.5) is solvable and its control law can be given in the formof

u(x) = - a 0

2s(xTPx) (k- 1) /2BT

1 Px, a 0 > 0

To get the main result, some lemmas should be proved ® rst. Let S still denote theunit sphere in Rn, and

C+ = {x Î Rn: xTPf (x) ³ 0}

C- = {x Î Rn: xTPf (x) < 0}

Nonlinear H¥ control of homogeneous systems 83D

ownl

oade

d by

[D

UT

Lib

rary

] at

22:

33 0

7 O

ctob

er 2

014

The following lemma shows the key point for the main result.

Lemma 1: If Assumption 1 holds, there are positive constant g 1, g 2 > 0 such thatfor any x Î S, at least one of the following inequalities holds:

xTPf (x) +1g 1

i h(x) i 2 £ 0, (3.2)

or

- i BT1 Pxi +

1g 2

i h(x) i £ 0 (3.3)

Proof: Note that Ker BT1 P and C+ are closed sets, while C- is open. According

to Assumption 1, Ker BT1 P Ì C- Ä {0}, so there is an open set G0, its closure G

and its complement set G0 (closed set), such that Ker BT1 P Ì G0 Ä {0} and

G Ì C- Ä {0}. Separate S = (S ´ G) Ä (S ´ G0) . Therefore, for any x Î S, thereare two cases: (1) x Î S ´ G and (2) x Î S ´ G0.

(1) Since G Ì C- Ä {0}and S ´ G are closed, xTPf (x) has a nonzero maximum- ·1 < 0 on S ´ G. Similarly, on S ´ G, hT (x)h(x) has its maximum ·2 ³ 0.Then take g 1·1 ³ ·2 to make (3.2) hold.

(2) Since Ker BT1 P Ì G0 Ä {0} and S ´ G0 are closed, i BT

1 Pxi has a nonzerominimum and hT (x)h(x) has a maximum on S ´ G0,which are denoted as

·3 = minx Î S ´ G0

i BT1 Pxi > 0, ·4 = max

xÎ S ´ G0

i h(x) i ³ 0 (3.4)

respectively Then take g 2·3 ³ ·4 to make (3.3) hold. h

H(V ,x,u,w) and u*,w* are de® ned as (1.3) and (1.4). Then we have

Lemma 2: Assumption 1 holds and ¸ > ¸* = ê êsÏ , then if a > 0 is large enough

V = a2

(xTPx) (k+1) /2 (3.5)

is a solution of the following HJI inequality

H (V ,x,u*,w*) = ¶ V¶ x

f - 14s

¶ V¶ x

B1BT1¶ TV¶ x

+1

4 2¶ V¶ x

B2BT2¶ T V¶ x

+ hTh £ 0 (3.6)

Proof: On account of homogeneity of (3.6), our discussion can be limited onsphere S. Set

b - = minxÎ S ´ G

(xTPx), b + = maxxÎ S ´ G0

(xTPx)

Note that

¶ V¶ x

= ¶¶ x

a2

(xTPx) (k+1) /2 =a (k + 1)

2(xTPx) (k- 1) /2xTP

Then (3.6) can be rewritten as

L a (x) = a (k + 1)2

(xTPx) (k- 1) /2xTPf - a 2 (k + 1)2

16s(xTPx)k- 1xTPB1B

T1 Px

+a 2 (k + 1)2

16¸2 (xTPx)k- 1xTPB2BT2 Px + hTh £ 0 (3.7)

84 Y. Hong and H. L iD

ownl

oade

d by

[D

UT

Lib

rary

] at

22:

33 0

7 O

ctob

er 2

014

Similar to the above lemma, we consider two cases:

(1) x Î S ´ G: take

a - =2g 1

(k + 1)B(k- 1) /2-,

and a > a - , then

L a (x) < L a - (x) £ g 1xTPf + i hi 2 £ 0

(2) x Î S ´ G0: take ·3, ·4 according to (3.4), and denote

· = maxxÎ S

xTPf (x), ~a = a (k + 1)2

(xTPx) (k- 1) /2

L a (x) £ ~a · -~a 2

41s- 1

¸2( ) ·23 + ·

24 = L (~a )

The next task is to solve the inequality

L (~a ) £ 0 (3.8)

Since ¸2 > s, only the positive root of (3.8) is meaningful. Therefore, we have

~a 1 =2· + 2{·

2 + [(1/s) - (1 /¸2)]·23·

24}1 /2

[(1 /s) - (1 /¸2)]·23

³ 0

and

a + =2~a 1

(k + 1) b (k- 1) /2+

If a > a + , we have

L a (x) < L a + (x) £ ~a 1· -~a 2

1

41s - 1

¸2( ) ·23 + ·

24 £ 0

Takea * = max ( a - , a + ) . (3.9)

and a > a *, (3.7) holds for any x Î S, and then, for any x Î Rn. h

In this way, a 0 in the controller of Theorem 3 can be taken to be a (k + 1) . Thusthe nonlinear H¥ control law can be expressed as

u(x) = - a (k + 1)2s

(xTPx) (k- 1) /2BT1 Px, a > a * > 0 (3.10)

Proof of Theorem 3: Take V as (3.5). By (3.6) and (3.7), we have

H (V ,x,u,w) £ H(V ,x,u,w) - H (V ,x,u*,w*) = i u - u*i 2 - ¸2 i w - w*i 2

H (V ,x,u*,w) £ H (V ,x,u*,w*) £ 0

Integrate two sides

òT

0H (V ,x,u*,w) d¿ = V (T ) - V (0) + ò

T

0(z- ¸

2wTw) d¿ £ 0

Since V (T ) > 0, V (0) = 0 (because x(0) = 0) , we have

Nonlinear H¥ control of homogeneous systems 85D

ownl

oade

d by

[D

UT

Lib

rary

] at

22:

33 0

7 O

ctob

er 2

014

òT

0(zTz - ¸

2wTw) d¿ £ 0

The next task is to study the stability of the closed loop system when w º 0. Takethe above V[x(t)]as the Liapunov function and its derivative is

ÇV (x) = ¶ V¶ x

f + B1 - 12 B1 T

¶ T V¶ x( )[ ]

= a (k + 1)2

(xTPx) (k- 1) /2xTPf - a 2 (k + 1)2

8s(xTPx) k- 1xTPB1B

T1 Px (3.11)

It is easy to see that ÇV (x) is negative de® nite. h

Remark 4: That ¸2 > s [i.e. (1 /s) - (1 / 2) > 0], is the natural case for the H¥problem. In this case, take ¸* = ê êsÏ . From the above proofs, we know when®

¸*, a will tend to ¥ .Similar to the above section, a more general class of nonlinear systems

expressed by

Çx = f (x) + d f (x) + [B1 + g1 (x)]u + [B2 + g2 (x)]w, x Î Rn, u Î Rm

z = h(x) + d h(x) + [D + d (x)]u, z Î Rl } (3.12)

can be studied, where f (x) and h(x) are of homogeneity degree k, d f (x) = o( i xi k) ,d h(x) = o(||x||k), g1 (x) = o(1) , g2 (x) = o(1) , and d (x) = o(1) . In fact, a correspond-ing system in the form of (1.5) may be called the homogeneous approximationsystem of (3.12).

The following result exhibits the robustness of homogeneous system (1.5) withhigher degree perturbations, locally around the origin of (x,w) .

Theorem 4: Suppose Assumption 1 holds. Then, for some ¸ > ¸*, the nonlinear H¥control problem of system (3.12) has a solution locally.

Proof: According to Theorem 3, for some ¸ > ¸*, there is V = a xTPx such thatthe H¥ control problem of (1.5) has a solution by satisfying thatH (V ,x,u*,w*)|(1.5) is of homogeneity degree 2k and negative de® nite. Therefore.

H (V ,x,u*,w*)|(3.12) = H(V ,x,u*,w*)|(1.5) + o( i xi 2k) < 0

holds locally when x and w are small enough at the origin. Also when w º 0, (3.12) isasymptotically stable. h

Remark 5: van der Schaft (1992) showed the relationship between nonlinearsystems and their linear approximation systems about H¥ control.

Next, e� ort is made to extend the above result to the case when the matchingcondition holds, namely, there is a matrix X such that B2 = B1X. Then we have

Proposition 2: The following statements are equivalent:

(i) Ker BT2 É Ker BT

1 ; or equivalently, the matching condition holds;(ii) there is a constant g > 0, such that i BT

2 Qxi £ g i BT1 Qxi is satis® ed for any Q

with det (Q) /= 0 and any x Î Rn.

86 Y. Hong and H. L iD

ownl

oade

d by

[D

UT

Lib

rary

] at

22:

33 0

7 O

ctob

er 2

014

Proof: (ii) to (i) is obvious We only prove (i) to (ii). Because B1 is of full rank,BT

1 Q can be expressed as BT1 Q = Q1[I 0]Q2, where Q1 Î Rm ´ m and Q2 Î Rn n are

of full rank. Denote Q2x = [xT1 ,xT

2 ]T, then BT1 Qx = Q1x1 and

BT2 Qx + BT

2 QQ- 12

x1

x2[ ] = [~B1

~B2] x1

x2[ ]

From (i), it follows that ~B2 = 0. Then (ii) follows when taking g = i ~B1Q- 11 i . h

With this proposition, results similar to Theorem 3 and Theorem 4 can beobtained easily.

Finally, in the general case that B1 does not have the above relations with B2, thefollowing result can still be derived by virtue of the above analyses.

Theorem 5: If Assumption 1 holds, then there is ¸* selected large enough, such thatfor any ¸ > ¸*, the nonlinear H¥ control of system (1.5) can be solved.

This conclusion is consistent with linear H¥ theory when system (1.5) is linear.

4. Examples and discussions

In this secion, further discussions are given following the above theoretic results.

Example 1: Let us have a look at the linear case, which is homogeneous ofdegree 1. Consider a linear system in the form of

Çx = Ax + B1u + B2w

z =Cx

Du[ ], DTD = sI, D Î Rm ´ m

üïïýïïþ

(4.1)

Take

V = a2

xTPx, a > 0 (4.2)

Since (4.1) satis® es the saddle point condition, we have

u* = - a2s

BT1 Px, w* =

a2¸2 BT

2 Px (4.3)

Then (3.7) is (Algebraic Riccati Inequality)

a PA + a ATP - a 2

2sPB1B

T1 P +

a 2

2¸2 PB2BT2 P + CTC £ 0 (4.4)

which becomes the standard H¥ problem (see Green et al. 1995).Next, which kind of system satis® es Assumption 1 needs to be studied.

Obviously, linear systems are quali® ed. What follows are some other examples,which shows the assumption is valid for many homogeneous systems.

For simplicity, set B = (0,Im)T, where Im is a unit matrix in Rm ´ m . Then thesystem in question can be described as

Çx1 = f 1 (x)

Çx2 = f 2 (x) + u, x2 Î Rm

Nonlinear H¥ control of homogeneous systems 87D

ownl

oade

d by

[D

UT

Lib

rary

] at

22:

33 0

7 O

ctob

er 2

014

If it satis® es Assumption 1, then C = {x: x2 = Fx1} where F is a matrix, andu (x) = (In- m ,FT)Tf 1. Then ® nding a P Î Rn ´ n is reduced to ® ndingQ Î R(n- m) ´ (n- m) and F Î Fm ´ (n- m) , since

k x, u (x) l = x1T

Q f 1 (x1,Fx1) < 0 (4.5)

P =P1 PT

2

P2 P3

éë

ùû, P1 = Q + FTP3F, P2 = - P3F, P3 > 0 (4.6)

The following examples represent four di� erent kinds of systems that satisfyAssumption 1.

Example 2: Consider a system with dominating diagonal terms like

Çx1 = - 4xk1 + 3x1x

k- 12 + xk- 2

1 x2x3

Çx2 = x31x

k- 32 - 4xk

2 + 5xk3

Çx3 = f3 (x1,x2,x3) + u

where k ³ 3 and B = (0,0,1)T. Take Q = I, F = 0, then with (4.6), we haveBTP = (0,0,P3) . Therefore, all the vectors in the form of ~x = (x1,x2,0)T form thespace Ker BTP. Now consider

~xTPf (~x) = - 4xk+11 + 3x2

1xk- 12 + x3

1xk- 22 - 4xk+1

2 ,and notice that

x21x

k- 12 £ 2

k + 1xk+1

1 +k - 1k + 1

xk+12 , x3

1xk- 22 £ 3

k + 1xk+1

1 +k - 2k + 1

xk+12

then ~xTPf (~x) < 0 if ~x /= 0, namely Assumption 1 holds.

Example 3 (Andreini et al. 1988): Consider a locally controllable system withm = n - 1 and homogeneity degree is k

Çx1 = f1 (x)

Çx2 = f2 (x) + u1

. . .

Çxn = fn (x) + un- 1

üïïïïïýïïïïïþ

(4.7)

The H¥ problem of (4.7) has a state feedback solution since the local controllabilityof (4.7) implies Assumption 1.

Then the task is to get F = (c1, . . . ,cn- 1)T with Q = 1 such that

x1Q f1 (x1,c1,x1, . . . ,cn- 1x1) < 0, x /= 0

Because of local controllability, f1 (0,x2, . . . ,xn) is not identical to 0. Therefore,f1 (1,x2, . . . ,xn) is a vector ® eld of degree k. Then it is easy to get an F to makef1 (1,c1, . . . ,cn- 1) < 0. Namely

x1Q f1 (x1,c1x1, . . . ,cn- 1x1) = xk+11 f1 (1,c1, . . . ,cn- 1) < 0, x /= 0

From (4.6), P can be constructed.

88 Y. Hong and H. L iD

ownl

oade

d by

[D

UT

Lib

rary

] at

22:

33 0

7 O

ctob

er 2

014

Example 4: Consider (1.5) with f (x) = f1 + ´´´+ f L , where fi, i = 1, . . . , L havethe same homogeneity degree. If there is a P such that all fi , i = 1, . . . , L satisfyAssumption 1, then f (x) does.

Example 5: Consider a system of degree 3

Çx1 = x21x2

Çx2 = x32 + u + w

z =x2

1x2 - x32

u[ ]

üïïïïïïýïïïïïïþ

where B1 = B2 = (0,1)T, f (x) = (x21x2,x3

2)T, h(x) = x2

1x2 - x32 and s = 1. Notice

that the system is not (locally) controllable. Take

P =2 11 2[ ]

If we let C- = {(x2,x2)T Î R2

: x2 (x2 + x1) < 0}, then Ker BTP Ì C- Ä {0}.If x2

1 + x22 = 1

|xTPf (x)| = |(2x21 - x1x2 + zx2

2) (x1 + x2)x2|³ 1

2(x1 - x2)2|(x1 + x2)x2|

³ (x1 - x2)2 1

[1 /(x1 + x2)2]+ (1/x2

2)

³ 13[(x1 - x2) (x1 + x2)x2]2 = 1

3 i h(x) i

Hence g 1 = 3.According to Theorem 3, for any ¸ > ¸* = 1,

u(x) = - 2a (2x21 + 3x1x2 + 2x2

2) (x1 + 2x2)

is a suitable control, if a is large enough. Here set ¸ = ê ê ê2Ï . To get a , the constants, ·,·4, ·3, b - , b + are needed at ® rst.

· = maxxÎ S

x2 (x1 + x2) (2x21 - x1x2 + 2x2

2) £ 3 ê ê ê2Ï /2

·4 = maxxÎ S ´ G0

[(x1 - x2) (x1 + x2)x2]2 = ê ê ê6Ï /9

·3 = minx Î S ´ G0

(x1 + 2x2)2 = 1 /2

b - = minx C- (x2

1 + 3x1x2 + x22) = 1 /2, b + = min

xÎ S ´ C+(2x2

1 + 3x1x2 + x22) = 1 /2

a - =2 ´ 3

(3 + 1)12(3- 1) /2 = 3, a + = ~a 1 £ 68

Therefore, take a * = max ( a - , a + ) , so a > 68. Then the control law can be given by(3.10).

Nonlinear H¥ control of homogeneous systems 89D

ownl

oade

d by

[D

UT

Lib

rary

] at

22:

33 0

7 O

ctob

er 2

014

5. Finite test of H ¥ control of homogeneous systems with perturbed parameters

It is well known that, for linear systems with perturbed parameters, there aremany testing methods based on Khalitonov and Edge Theorem (Bartlett et al. 1988,Barmish 1994). The methods provide a way to use the boundary or vertices to checkthe stability on the whole parameter sets. In the nonlinear case, there seems noe� ective method to realize ® nite test due to the complexity of nonlinearity. In theH¥ study, there are some results in frequency domain to deal with linear H¥ controlof perturbed parameter systems, but no corresponding discussion is shown fornonlinear systems, maybe because the technique of frequency domain fails in thissituation.

In this section, also on account of the advantages of homogeneous properties, wegive a method to test the stability and even nonlinear (global) H¥ control of homo-geneous systems with perturbed parameters.

Consider a family of systems

Çx = fa (x) + B1u + B2w, a Î X

z = h(x) + Du, z Î Rl } (5.1)

where fa (x) = å Ni=1 ai f i (x) , a = (a1, . . . ,aN)T Î RN denotes parameter vector, and

f i is homogeneous vector ® eld with degree k. Set f0 (x) = f (x) and C-a = {x Î Rn

:

xTPfa (x) < 0}Ä {0}. Then we have:

Theorem 6:

(1) For any a1, a2 Î RN, if Ker BT1 P Ì C-

a1 ´ C-a2 , then Ker BT

1 P Ì C-a if

a = (1 - q )a1 + q a2, q Î [0,1].(2) X = {a Î RN

: Ker BT1 P Ì C-

a } is convex.(3) For any a Î X , the H¥ problem of (5.1) can be solved.

Proof:

(1) Because 1 ³ q ³ 0 and a = q a2 + (1 - q )a1, fa = q fa2 + (1 - q ) fa1 . It is easyto see that if " 0 /= x Î C-

a1 ´ C-a2 , then x Î C-

a .(2) In light of (1)

Ker BTP Ì {x Î Rn: xTPfa < 0}Ä {0}

Hence X is convex.(3) According to Theorem 3, the conclusion follows. h

In fact, for any given parameter set ~X , checking (3.1) on it is equivalent to

checking on its hull: hul (~X ) = {q a2 + (1 - q )a1: a1,a2 Î ~

X , q Î [0,1]}.

Theorem 7: Given convex set X . For any a Î ¶ X ( ¶ X is the boundary of X ) , if(5.1) satis® es (3.1) , then for any a Î X (3.1) holds. Moreover, let a 0 = maxaÎ X a (a) ,for any a Î X , the H¥ problem of (5.1) can be solved via

u(x) = - a 0 (k + 1)2s

(xTPx) (k- 1) /2BT1 Px (5.2)

Proof: On account of continuity of a (a) with respect to a and Theorems 3 and6, the conclusion can be obtained directly. h

If the convex set X is selected as polytopes with the form

90 Y. Hong and H. L iD

ownl

oade

d by

[D

UT

Lib

rary

] at

22:

33 0

7 O

ctob

er 2

014

X = convex {ai}= åM

i=1

q iai: q i ³ 0, å q i = 1{ } (5.3)

then we have the result of testing ® nite vertices.

Theorem 8: If X is given as (5.3) , and for all ai, i = 1, . . . ,M, (3.1) is satis® ed,then for all a Î X , (5.1) satis® es Assumption 1 and the H¥ problem of (5.1) can besolved. Furthermore, if we take X as cuboid (interval family) , that is

X 1 = {a Î Rn: ai Î [a-

i ,a+i ], i = 1, . . . ,N}

Testing 2N vertices is enough. If we take it as diamond

X 2 = a Î Rn: å ri|ai - a0

i | £ 1 ri > 0{ }it is enough to test 2N vertices, that is

a01, . . . ,a0

i- 1,a0i 6

1ri,a0

i+1, . . . ,a0N( )

T

, i = 1, . . . ,N (5.4)

Remark 6:

(1) The perturbed radius of X 2 can be determined independently by its verticessince from (5.4) it is obvious that any vertex of X 2 has no relation with oneanother. Moreover, the above results can be used straightforwardly in thelinear case.

(2) For a given bounded set X in practice, we may try to ® nd a suitable convexpolytope hull, which contains X , and then the above ® nite test method can beapplied to the polytope, instead of the original set X .

(3) In fact, the result for mixed uncertainties with both the perturbed parameterand disturbance of higher order can be obtained, and it is shown that ourcontrol law is robust.

6. Conclusions

The problem of nonlinear H¥ control deserves to be studied deeply in bothnonlinear control theory and its applications on account of its importance. Thework in the paper is concerned with this problem and the related L 2 gain problem.In particular, a concrete H¥ controller is provided for a special class of systems,homogeneous systems, and the design is global. Then, two extended and applicationcases are considered: higher order disturbance (by homogeneous approximation) andperturbed parameter (by ® nite test) related to nonlinear H¥ control. Moreover, withthe results of homogeneous systems in references such as Rosier (1992), Ryan (1995)and Hermes (1996), further study on H¥ control of homogeneous systems can becarried on.

Acknowledgments

The authors wish to thank the reviewers for their very helpful comments andcorrections. This work is supported by the National Climbing Project of China.

Nonlinear H¥ control of homogeneous systems 91D

ownl

oade

d by

[D

UT

Lib

rary

] at

22:

33 0

7 O

ctob

er 2

014

References

Andreini, A., Bacciotti, A., and Stefani, G., 1988, Global stabilizability of homogeneousvector ® elds of odd degree. Systems Control Letters, 10, 251± 156.

Barmish, B. R., 1994, New Tools for Robustness of Linear Systems (New York: Macmillan).Bartlett, A. C., Hollot, C. V., and Huang, L., 1988, Root locations of an entire polytope

of polynomials: it su� ces to check the edges. Mathematics of Control, Signals andSystems, 1, 61± 71.

Basar, T., and Bernhard, P., 1991, H¥ -optimal Control and Related Minimax DesignProblemsÐ A Dynamic Game Approach (Boston: BirkhaÈ user).

Green, M., and Limbeer, D. J. N., 1995, L inear Robust Control (Englewood Cli� s, NJ:Prentice Hall).

Hahn, W., 1976, Stability of Motion (Berlin± New York: Springer).Hermes, H., 1996, Resonance, stabilizing feedback controls, and regularity of viscosity

solutions of Hamilton± Jacobi± Bellman equations. Mathematics of Control, Signals,and Systems, 9, 59± 72.

Isidori, A., and Kang, W., 1995, H¥ control via Measurement feedback for generalnonlinear systems. IEEE Transactions on Automatic Control, 40, 466± 471.

Pan, Z., and Basar, T., 1996, Adaptive controller design for tracking and disturbanceattenuation in parametric-strict-feedback nonlinear systems. IFAC, San Francisco,pp. 323± 328.

Rosier,L.,1992, Homogeneous Lyapunov function for homogeneous continuous vector ® eld.Systems Control L etters, 19, 467± 473.

Ryan, E. P., 1995, Universal stabilization of a class of nonlinear systems with homogeneousvector ® elds. Systems Control L etters, 26, 177± 184.

Sepulchre, R., and Aeyels, D., 1996, Homogeneous Lyapunov functions and necessaryconditions for stabilization. Mathematics of Control, Signals, and Systems, 9, 34± 58.

Sorabia, P., 1996, H¥ control of nonlinear systems: di� erential games and viscosity solutions,SIAM Journal of Control Optization, 35, 1071± 1097.

van der Schaft, A. J., 1992, L 2 gain analysis of nonlinear systems and nonlinear H¥ control.IEEE Transactions of Automatic Control, 37, 770± 784.

Vidyasagar, M., and Vannelli, A., 1982, New relationships between input± output andLyapunov stability, IEEE Transactions Automatic Control, 27, 481± 483.

92 Nonlinear H ¥ control of homogeneous systemsD

ownl

oade

d by

[D

UT

Lib

rary

] at

22:

33 0

7 O

ctob

er 2

014