Robust control of non-linear affine systems
Yew-Wen Liang, Der-Cherng Liaw
*Department of Electrical and Control Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan, ROC
Abstract
Issue of robust control for non-linear affine systems with uncertainties appearing in both drift and non-drift terms are presented.Based on Lyapunov function approach, control laws are proposed to guarantee uniformly asymptotic stability of the equilib-rium point.To facilitate the design and simplify the checking procedure, stabilization control for the uncertain systems possess asymptotic stabilizable nominal time-invariant driftness terms are proposed for the demonstration of robust design.
Ó 2002 Elsevier Science Inc.All rights reserved.
Keywords: Robust control; Uncertain systems; Lyapunov functions; Matched and mismatched uncertainties
1. Introduction
In the recent years, robust stabilization of non-linear systems have been widely discussed (see e.g. [1,3–5,8,10–13,15]). For instance, Gutman [5] devel-oped discontinuous min–max controllers to asymptotically stabilize matched-type uncertain dynamics.Corless and Leitmann [4] employed the same approach to design continuous state feedback controllers for guaranteeing uniform ultimate boundedness of matched-type uncertain system trajectories. Barmish et al.[1] introduced the concept of practical stabilizability and pro-posed stabilizing controllers for systems with matched-type uncertainties via Lyapunov stability.Using a differential-geometric approach, Kravaris and
*
Corresponding author.
E-mail address:dcliaw@cc.nctu.edu.tw(D.-C. Liaw).
0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc.All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 2 ) 0 0 1 2 9 - 7
Palanki [8] designed a class of stabilizing controllers for matched-typed un-certain dynamical systems.Qu [11] introduced the concept of equivalently matched uncertainties and investigated global asymptotic stabilization of a class of non-linear dynamical systems with so-called ‘‘equivalently matched uncertainties’’.He also employed the backstepping design method to the ro-bust control of the uncertain systems with generalized matching condition [12]. Wang et al.[15] employed differential geometric feedback linearization to deal with robust stabilization of uncertain systems with mismatched time-varying uncertainties.
The main goal of this paper is to study the robust control of non-linear affine systems.The uncertainties are supposed to appear in both drift and driftless terms of the system dynamics.Those uncertainties appearing in the drift term are assumed to be in equivalently matched-type as in [11], however, those appearing in the input-related matrix are allowed to be more general than those in [11].Moreover, in this paper we study both local and global cases.To facilitate the proposed design, we also study the robust control of the uncertain systems without assuming the stability of the nominal drift part.This might give a guide to the construction of Lyapunov function for the implementation of control laws.
The paper is organized as follows.In Section 2, we briefly introduce the considered uncertain systems and some basic assumptions of [11].An as-sumption is then introduced to relax the requirements of the equivalently matched-type uncertainties appearing in driftless part.It is followed by the design of control laws for the uncertain system having asymptotic stabilizable nominal driftless part.An illustrative example is also given to demonstrate the use of the main results.Finally, Section 3 gives the conclusions.
2. Robust stabilization of the uncertain systems
Consider a class of non-linear affine systems with uncertainties as given by _xxðtÞ ¼ f ðxðtÞ; tÞ þ Df ðxðtÞ; qðtÞ; tÞ þ fgðxðtÞ; tÞ þ DgðxðtÞ; qðtÞ; tÞgu ð1Þ with xðt0Þ ¼ x0.Here, t2 R denotes time, xðtÞ 2 Rnis the state vector, u2 Rmis
the control vector, qðtÞ 2 Rp
is the uncertainty and fð; Þ, Df ð; ; Þ, gð; Þ and Dgð; ; Þ, respectively, are known vectors and matrix functions with appro-priate dimensions.We decompose the uncertainties into matched and mis-matched parts (for definition, see e.g., [1]) as
Dfðx; q; tÞ ¼ gðx; tÞDfmðx; q; tÞ þ Dfmmðx; q; tÞ ð2Þ
and
For system (1), Assumptions 1 and 2 below are introduced to guarantee the existence of classical solution.
Assumption 1. The uncertainty qðÞ : R ! Rp, is Lebesgue measurable and its
values qðtÞ lie within a pre-specified compact set Q Rp for all t2 R.
Assumption 2. The functions fð; Þ : Rn R ! R, Df ð; ; Þ : Rn Rp R !
Rn and Dgð; ; Þ : Rn Rp R ! Rn mare all continuous.
Note that, it is known that continuous and piecewise continuous functions are two kinds of Lebesgue measurable functions (e.g., [14]). Regarding the nominal system and the uncertainties, [11] introduced the following three as-sumptions:
Assumption 3. The origin of the uncontrolled nominal model _xx¼ f ðx; tÞ of system (1) is locally uniformly asymptotically stable.In particular, there exists a neighborhood Xx, a smooth function (i.e., continuously differentiable
func-tion) V :Xx R ! Rþ and continuous, strictly increasing functions
ci: Rþ ! Rþ, i¼ 1; 2; 3, with
cið0Þ ¼ 0; i¼ 1; 2; 3 ð4Þ
lim
r!1ciðrÞ ¼ 1; i¼ 1; 2 ð5Þ
such that for allðx; tÞ 2 Xx R,
c1ðkxkÞ 6 V ðx; tÞ 6 c2ðkxkÞ ð6Þ and oVðx; tÞ ot þ r T xVðx; tÞf ðx; tÞ 6 c3ðkxkÞ: ð7Þ
Herek k denotes the Euclidean norm and rT
xVðx; tÞ denotes the transpose of
the column vectorrxVðx; tÞ ¼oxoVðx; tÞ.
Assumption 4. There exist two known, non-negative continuous functions emðx; tÞ and emmðx; tÞ such that
(i) kDfmðx; q; tÞk 6 emðx; tÞ and
(ii) krT
xVðx; tÞDfmmðx; q; tÞk 6 emmðx; tÞ and emmðx; tÞ=krTxVðx; tÞgðx; tÞk is
uni-formly bounded with respect to t.Here, the two scalar functions emðx; tÞ
and emmðx; tÞ are assumed to be uniformly bounded with respect to t.
Assumption 5. The uncertainties of system (1) appear in non-drift part satis-fying the following condition:
inf ðx;q;tÞ kmin I þ1 2ðDgmðx; q; tÞ þ Dg T mðx; q; tÞÞ kr T xVðx; tÞDgmmðx; q; tÞk krT xVðx; tÞgðx; tÞk P g >0: ð8Þ
Here, I and kminðÞ denote, respectively, the identity matrix and the smallest
eigenvalue of a symmetric matrix.
Note that, the uncertainties Dfmmðx; q; tÞ in Assumption 4 and Dgmmðx; q; tÞ in
Assumption 5 are referred as the so-called ‘‘equivalently matched-type’’ uncer-tainties [11].From Assumptions 4 and 5,rT
xVðx; tÞDfmmðx; q; tÞ and rTxVðx; tÞ
Dgmmðx; q; tÞ must vanish when rTxVðx; tÞgðx; q; tÞ ¼ 0.
Under Assumptions 1–5, [11] proposed a class of control laws that makes the origin of the closed-loop uncertain system globally asymptotically stable. However, the stability conclusion can be derived under a more general as-sumption as given in Asas-sumption 6 below.In addition, both local and global cases will also be considered in this study.Details are given as follows.
First, we introduce the next assumption, which can also be found in ([13], condition 2.3) and ([2], Assumption 5) for a more general version. However, Qu only studied the global result and Chen only obtained practical stability result.
Assumption 6. Dgðx; q; tÞ is uniformly bounded with respect to time and there exists an g > 0 such that
DTxVðx; tÞDgðx; q; tÞgTðx; tÞr
xVðx; tÞ P ðg 1Þ krTxVðx; tÞgðx; tÞk 2
ð9Þ for all (x; q; t) with x2 Xxand q2 Xq, where Xxis a neighborhood of x¼ 0 and
Xq denotes the region of the uncertainty parameter q.
To study the relationship between Assumption 5 and 6, we show in Lemma 1 below that condition (8) implies condition (9).
Lemma 1. Condition (8) implies Condition (9).
Proof. Suppose condition (8) holds.Then multiplying (8) by krT xVðx; tÞg
ðx; tÞk2, we have for all (x; q; t), kmin I þ1 2ðDgmðx; q; tÞ þ Dg T mðx; q; tÞÞ krT xVðx; tÞgðx; tÞk 2 gkrT xVðx; tÞgðx; tÞk 2 PkrT xVðx; tÞDgmmðx; q; tÞk krTxVðx; tÞgðx; tÞk: ð10Þ From the Cauchy–Schwartz Inequality, we have jvT
1v2j 6 kv1k kv2k for any
vectors v1; v22 Rn.Also, it is known that kminðAÞ kvk26vTAv for any A¼
rT xVðx; tÞgðx; tÞfI þ Dgmðx; q; tÞggTðx; tÞrxVðx; tÞ grT xVðx; tÞgðx; tÞg Tðx; tÞr xVðx; tÞ P rT xVðx; tÞDgmmðx; q; tÞgTðx; tÞrxVðx; tÞ: ð11Þ Here, we have used the fact that 1
2½Dgmðx; q; tÞ þ Dg T mðx; q; tÞ ¼ Dgmðx; q; tÞ þ 1 2½Dg T mðx; q; tÞ Dgmðx; q; tÞ with DgTmðx; q; tÞ Dgmðx; q; tÞ an antisymmetric
matrix and that xTBx¼ 0 for all x if B is an antisymmetric matrix.Rearranging
these terms, we have rT xVðx; tÞfgðx; tÞDgmðx; q; tÞ þ Dgmmðx; q; tÞggTðx; tÞrxVðx; tÞ Pðg 1ÞrT xVðx; tÞgðx; tÞg Tðx; tÞr xVðx; tÞ: ð12Þ
Since Dgðx; q; tÞ ¼ gðx; tÞDgmðx; q; tÞ þ Dgmmðx; q; tÞ, the assertion is then
proved.
Lemma 1 shows that Assumption 6 is more general than Assumption 5.It is noted that the converse of Lemma 1 is not true.An example is given in Ex-ample 1 below.
Example 1. Consider system (1) with xðtÞ, uðtÞ, qðtÞ 2 R2, gðx; tÞ ¼ diagfx 1; x32g
and fðx; tÞ ¼ 0.Also, the uncertainties given by (2) and (3) are
Dfðx; q; tÞ ¼ 0; ð13Þ Dgmðx; q; tÞ ¼ q1 2 2 q2 for q1; q2P 3 4 ð14Þ and kDgmmðx; q; tÞk 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x4 1þ x82 q : ð15Þ For q1¼ q2¼ 1, we have kmin I þ1 2½Dgmðx; q; tÞ þ Dg T mðx; q; tÞ ¼ 0: ð16Þ
It follows that there does not exist an g > 0 such that condition (8) holds. However, we can show that Assumption 6 holds for such system.To see this, choose
Vðx1; x2Þ ¼12ðx21þ x 2 2Þ:
It follows that rT xVðx; tÞDgðx; q; tÞg Tðx; tÞr xVðx; tÞ ¼ rT xVðx; tÞgðx; tÞDgmðx; q; tÞgTðx; tÞrxVðx; tÞ þ rT xVðx; tÞDgmmðx; q; tÞgTðx; tÞrxVðx; tÞ P q1x41þ q2x82þ 4x 2 1x 4 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 1þ x22 q ðx4 1þ x 8 2Þ Pðg 1Þ ðx4 1þ x 8 2Þ ¼ ðg 1Þ krT xVðx; tÞgðx; tÞk 2
for all x around the neighborhood Xx¼ fxj kxk 6 1=4g of the origin for
0 < g <3
2, which implies that Assumption 6 holds.
To achieve certain stability performance, several control laws have been proposed (see e.g. [3,4,8,11]). Among these control laws, in order to achieve maximum control effort regarding the Lyapunov function VðxÞ for the nominal system, they are usually chosen in the form of
uðx; tÞ ¼ wðx; tÞgTðx; tÞr
xVðx; tÞ; ð17Þ
where wðx; tÞ is a non-negative scalar function to be determined by compen-sating the effect of uncertainties.Under the control law given by (17) and the decomposition of uncertainties given (2) and (3), the time derivative of VðxÞ along the trajectories of the uncertain system (1) is calculated as
_ V V ¼ oVðx; tÞ ot þ r T xVðx; tÞ½f ðx; tÞ þ gðx; tÞDfmðx; q; tÞ þ Dfmmðx; q; tÞ wðx; tÞrT xVðx; tÞ½gðx; tÞ þ Dgðx; q; tÞg Tðx; tÞr xVðx; tÞ 6 c3ðkxkÞ þ emðx; tÞkrT xVðx; tÞgðx; tÞk þ emmðx; tÞ gwðx; tÞkrT xVðx; tÞgðx; tÞk 2 ð18Þ Here, we have used Assumption 6 in the last inequality above.Choose wðx; tÞ to satisfy wðx; tÞ Pemðx; tÞ kr T xVðx; tÞgðx; tÞk þ emmðx; tÞ g krT xVðx; tÞgðx; tÞk 2 ð19Þ It follows that _ V V 6 c3ðkxkÞ ð20Þ
We hence have the next results.
Theorem 1. Consider the uncertain system (1) satisfying Assumptions 1–4 and 6. Then the origin is locally (resp., globally if Xx¼ Rn) uniformly asymptotically
To relax the possible discontinuity of the control law, [11] used the relation _
V
V 6 c3ðkxkÞ þ 2ebtinstead of Inequality (20) for controller design.With a
slight modification of QuÕs approach [11], we propose the control law (17) with wðx; tÞ being given by wðx; tÞ ¼ e2 mðx; tÞ gðemðx; tÞkrTxVðx; tÞgðx; tÞk þ ebtÞ þ e 2 m mðx; tÞ gðemmðx; tÞebtÞ krTxVðx; tÞgðx; tÞk 2 if x6¼ 0; 0 otherwise; 8 > > > > < > > > > : ð21Þ
where and b are two positive constants determined by the designer to achieve desired stability performance.Note that, by Assumption 4, e2
m mðx; tÞ= krT xVðx; tÞgðx; tÞk 2
is uniformly bounded with respect to t and rT
xVð0; tÞ ¼ 0
since Vðx; tÞ is a locally positive definite function.These imply that uðx; tÞ is continuous everywhere and
kuðx; tÞjj 6emðx; tÞ g þ emmðx; tÞ gkrT xVðx; tÞgðx; tÞk : ð22Þ
We then have the next result
Theorem 2. Suppose the uncertain system (1) satisfying Assumption 1–4 and 6. Then the origin is locally (resp., globally if Xx¼ Rn) asymptotically stable if the
control laws uðx; tÞ are chosen in the form of (17) with wðx; tÞ in (21) and sat-isfying condition (A.2).
Proof. The proof is analogous to those of [11] with a slight modification. Details are given in Appendix A.
It is observed from Theorems 1 and 2 that the designed stabilizing control laws strongly depend on the given Lyapunov function Vðx; tÞ for the nominal drift part fðx; tÞ.However, in general, there is no guideline for the construction of Lyapunov function.To facilitate the design of control laws and simplify the checking procedure, we can rewrite the uncertain system (1) as
_xxðtÞ ¼ f ðx; tÞ þ Df ðx; q; tÞ þ fgðxÞ þ D~ggðx; q; tÞgu; ð23Þ where D~ggðx; q; tÞ ¼ gðx; tÞ þ Dgðx; q; tÞ gðxÞ.That is, we extract time-invariant part gðxÞ from gðx; tÞ and put the remaining time-varying part into the un-certain term.In the following, instead of requiring the uniformly asymptotic stability assumption on the nominal drift part fðx; tÞ, we assume that
is locally asymptotically stabilizable at the origin.As given in [9], such an as-sumption is equivalent to Asas-sumption 7 below.
Assumption 7. There exists a neighborhood Xx of the origin x¼ 0 and a
smooth locally positive definite function VðxÞ satisfying rT
xVðxÞgðxÞ 6¼ 0 for all x 2 Xxn f0g: ð25Þ
Remark 1. Note that, although Assumption 7 is a special case for (ii) of As-sumption 4, it might provide a guideline for the construction of Lyapunov function VðxÞ for some special cases.One of the results concerning the exis-tence of a quadratic positive definite function VðxÞ satisfying condition (25) can be found in [9].The results are obtained from TaylorÕs series expansion of gðxÞ and the determination of the local definiteness of a defined scalar function.It was shown that, for instance, there exists a VðxÞ satisfying condition (25) for the linear driftless system _xx¼ Bu (i.e., gðxÞ ¼ B) if B is a square non-singular matrix.Also, condition (25) is satisfied for the bilinear driftless system _xx¼Pmi¼1uiBix with Pmi¼1ðxTBixÞ
2
>0 for all x6¼ 0, where x 2 Rn, u
i2 R and
Bi2 Rn n.In particular, the condition ofP m i¼1ðx TB ixÞ 2 >0 is guaranteed when one of the matrices Biis definite.For details, please refer to [9].
Motivated by Assumption 7, Eq.(23) can then be rewritten as _xx¼ gðxÞgTðxÞr
xVðxÞ þ ff ðx; tÞ þ Df ðx; q; tÞ þ gðxÞgTðxÞrxVðxÞg
þ fgðxÞ þ D~ggðx; q; tÞgu: ð26Þ Here, fðx; tÞ does not require to satisfy Assumption 3.Let
f0ðxÞ ¼ gðxÞgTðxÞrxVðxÞ; ð27Þ
then the origin is an asymptotic stable equilibrium point for _xx¼ f0ðxÞ if
As-sumption 7 holds [9].By decomposing the uncertainty Dfðx; q; tÞ and f ðx; tÞ into matched and mismatched parts, we can rewrite Eq.(26) as
_xxðtÞ ¼ f0ðxÞ þ gðxÞDfmðx; q; tÞ þ Dfmmðx; q; tÞ þ fgðxÞ þ D~ggðx; q; tÞgu: ð28Þ
From the above derivation, we then have the next result from Theorem 2. Lemma 2. Suppose the uncertain system (26) satisfying Assumptions 1, 2, 4, 6 and 7 with Dgðx; q; tÞ being replaced by D~ggðx; q; tÞ. Then the origin is locally (resp., globally if Xx ¼ Rn) asymptotically stable if the control laws uðx; tÞ are
chosen in the form of (17) with wðx; tÞ as in (21) and satisfying condition (A.2). For the demonstration of the proposed robust stabilization design, numer-ical results for Example 1 are obtained as depicted in Fig.1.In these
simula-tions, the initial state and the positive definite function VðxÞ are, respectively, chosen to be x0¼ ð0:5; 0:3Þ T and VðxÞ ¼1 2ðx 2 1þ x22Þ to make rTxVðxÞgðxÞ 6¼ 0
in a neighborhood of the origin.The system parameters and the uncertainties are considered as q1¼ q2¼ 1, emmðx; tÞ ¼ 0, emðx; tÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x6 1þ x142 p and Dgmmðx; q; tÞ ¼ 1 2 x4 2sin x1 x21cos x2 x2 1cos x1 x42sin x2 ð29Þ For the stabilization design, we choose g¼ ¼ b ¼ 1.Fig.1(a) and (b), respectively, show the time evolution of the norm of the system state with and without uncertainities.These show that all the system states converge to the origin, which agree with the results of Theorem 2.However, since the closed loop system behaves like a polynomial system with order greater than one, the convergent rate is getting smaller as system states get closer to the origin.
3. Conclusions
This paper has studied the robust stabilization of uncertain non-linear affine systems.The uncertainties considered in this study are more general than that
of the so-called ‘‘equivalently matched-type’’.Since control laws strongly de-pend on the given Lyapunov function, stabilization design is also proposed for uncertain system with asymptotic stabilizable nominal driftless time-invariant terms but not the stabilizability of the nominal drift part to provide a means of the construction of Lyapunov function for fulfilling the design.
Appendix A (Proof of Theorem 2)
By direct calculation, we have from (17), (18) and (21) that _
V
V 6 c3ðkxkÞ þ 2ebt: ðA:1Þ
The constant in (A.1) will be determined later to guarantee the uniformly asymptotic stability of the origin.Choose r0>0 and q > 0 such that Br0 Xx
and q < minkxk¼r0c1ðkxkÞ, where Br0 denote the open ball with radius r0.It
implies that X1¼ D fx 2 Br0:c1ðkxkÞ 6 qg Br0.Define Xt;q¼ D fx 2 Br0jV ðx; tÞ 6 qg. This leads to X2¼ D
fx 2 Br0:c2ðkxkÞ 6 qg 6 Xt;q X1.Choose such that
<1 2x2Xinf1nX2
c3ðkxkÞ: ðA:2Þ
It follows that _VV <0 for all x2 X1n X2and the state will remain inside X1if
it starts inside X2.This shows uniformly stability of the origin.To show the
attractive property of the origin, it is noted that 0 6 VðxðtÞ; tÞ ¼ V ðxðt0Þ; t0Þ þ Rt t0 _ V Vðx; ðsÞ; sÞ ds 6Vðxðt0Þ; t0Þ þRt t0ðc3ðkxðsÞkÞ þ 2e bsÞ ds: ðA:3Þ
This implies that lim t!1 Z t t0 c3ðkxðsÞkÞ ds 6 V ðxðt0Þ; t0Þ þ 2 b e bt0<1: ðA:4Þ
Moreover, the state trajectory xðtÞ is continuous (see e.g., [6]) and bounded since the origin is uniformly stable.The boundedness property of xðtÞ together with Assumptions 1, 2 and 6 imply that xðtÞ is uniformly continuous.Thus, the function c3ðkxðtÞkÞ is also uniformly continuous.Then, from (A.4) and the use
of BabalatÕs Lemma (see e.g., [7]), we have c3ðkxðtÞkÞ ! 0 as t ! 1.It follows
that xðtÞ ! 0 and the results of theorem is hence implied. Acknowledgements
This research was supported by the National Science Council, Taiwan, R.O.C. under Grants NSC 90-2212-E-009-068, NSC 89-2612-E-009-005, NSC
90-2212-E-009-067, NSC 90-2213-E-009-102, and by the Ministry of Educa-tion, Taiwan, R.O.C. under Grant EX-91-E-FA06-4-4.
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