Robust sliding mode control of uncertain nonlinear systems

ELSEVIER

SYST|Mg CONTROL UTT|RS Systems & Control Letters 32 (1997) 75-90

Robust sliding mode control of uncertain nonlinear systems 1 Xiao-Yun Lu*, Sarah K. Spurgeon Control Systems Research, Department of Engineerin9, University of Leicester, Leicester UK LE1 7RH

Received 13 March 1996; received in revised form 23 June 1997

Abstract

A dynamic sliding mode controller design method is proposed for multiple input-output systems with additive uncertainties. A previous result on the stability of triangular systems is generalised to the case of uniform ultimate boundedness of controlled triangular systems. This is used to prove the stability of the overall closed-i0op system. The uncertain system with appropriately chosen sliding mode control is shown to be ultimately bounded if the zero dynamics of the nominal system are uniformly asymptotically (exponentially) stable. The design method is demonstrated with two examples. © 1997 Elsevier Science B.V. Keywords. Robustness; Sliding mode control; Dynamic feedback; Zero dynamics; Uniform ultimate boundedness

1. Introduction

Sliding mode control is believed to be robust in the control of linear and nonlinear uncertain systems [10]. However, the robu,;tness analysis of sliding mode schemes, particularly dynamic schemes [6, 9], needs further consideration for the case of uncertain nonlinear systems. Here nonlinear systems which are in differential input-output form with additive uncertainties and where the relative order is not necessarily equal to the system order are considered. The uncertainties present in the model are assumed to be cone-bounded. The existence of solution and stability analysis o f the closed-loop system are based on Filippov's work [1] and a generalised Lyal:,unov theorem. To study the stability o f the closed-loop system, the stability results for triangular systems in [12] are generalised to a sufficient condition of ultimate boundedness for such systems. When additive uncertainty is present, the closed-loop system can achieve uniform ultimate boundedness for arbitrarily given e > 0 if a particular sliding reachability condition is adopted. The design method is demonstrated with two examples. The first model is in differential I-O form with a non-trivial zero dynamics given by the Van der Pol equation. The second example considers the control o f gas jet actuators (Euler equation) with uncertainties; this is a state space model which can easily be eliminated into a differential I-O system. The following notation will be used throughout: N~(xo) = {x ~ n~n I IIx - xoll <~},

* Corresponding author. Tel.: +44-116-252 2567; e-mail: [email protected]. I Work supported by UK EPSRC (grant reference GR/J08362). 0167-6911/97/$17.00 ~:) 1997 Elsevier Science B.V. All rights reserved PH S 01 67-69 11 (97)00061-3

76

X.-E Lu, S.K. SpurgeonlSystems & Control Letters 32 (1997) 75-90

where I1' I] is the Euclidian norm. },min(B) and ),max(B), respectively, denote the minimum and maximum eigenvalues of the matrix B. lr denotes the unit matrix of dimension r. The c l a s s - Y function ~(r) is defined on O<~r<<.rl such that c~(r) is continuous, strictly increasing and ~ ( 0 ) - - 0 . The class-J~#L,a function [3(r,s) is defined on O<.r<~rl,O<~s<~sl. For fixed s, fl(r,s)EcU with respect to r and for each fixed r, fl(r,s) is monotone decreasing to zero as s increases, rl,sl may be ec [3].

2. Background 2.1. Differential equations with discontinuous riyht-hand side Due to the presence of uncertainty which may be discontinuous and the use of a discontinuous control action in sliding mode design, differential equations with discontinuous right-hand side must be considered. Such equations are extensively studied in [1, 7]. The main results in [1] will be cited here. Consider the system

~c= f(x, t)

(2.1)

where f ( . , .): [~n x N--~ N satisfies the following condition: Condition B: There exists an open set Q c Nn, 0E Q, such that f ( . , .) is defined everywhere and is Lebesgue measurable almost everywhere in Q x N. For any T ~>0, there exists a Lebesgue integrable function Br(t) such that [[f(x,t)ll ~
2.2. Uncertain systems under consideration Here uncertain systems of the following form are considered

y(~') =q91(y, fi, t ) + Al(f,t), • ..

(2.2)

y(pn,,) = ~pp(f, fi, t) + Ap(~, t ), where f i = ( u l , . . . , u l ~') . . . . . urn,. . ))U '"l~f. . T and . Y'=(Yl . . . .. . . . .y ] n , - l ) n i/> 1. The uncertainties are Lebesgue measurable and satisfy

IlAi(~,t)ll<~pi[l~ll+¢i,

pi~>0, li~>0, i = 1 . . . . . p.

YP"'"yp(np-1))T with n I q-

"-"

@np=n,

(2.3)

For practical implementation, sometimes it is required that the highest-order derivatives of the control appear linearly in the nominal system as

Y(") = q~a(f, ~, t) u (l~) + (Pb(f,'ff, t) + A(t),

(2.4)

where y ( n ) = ( y ~ ) , . ,y(pnv))T,'~=(U T, U(fl) "~u~([~'). . . . . Um (fl"') )7, A ( t ) = ( d l ( t ) , . . . 1 . .,U(f,--l) . . .,U . . . . . .. . U(m~m--l)) . ...,An(t)) T and ~oa(~,u,t) and q~b(~,u,t) are matrices of appropriate dimension. R e m a r k 2.1. The uncertainty may be due to external uncertainties, internal parameter uncertainties, measurement noise, system identification error, or indeed the elimination procedure used to generate a differential input-output model from a state space model as in [11].

x..-Y. Lu, S.K. Spurgeon/Systems & Control Letters 32 (1997) 75-90

77

Furthermore, suppose that the nominal system of (2.2), with Ai(~, t) = O, i = 1. . . . . p, is a proper differential I - 0 system where p = m, all qi(', ', "), i = 1. . . . . m, are ~l-functions and the Regularity Condition holds.

det Lt3(u~/~,)'

;u(---~))j # 0

(2.5)

V~EN6(0), for some 6 > 0 , Vt>_-0 and generically for ft. From now on all the I-O systems under consideration are assumed to be proper. Definition 2.1. The zero dynamics corresponding to the nominal system is defined as ~01 (0,/~, t) = 0,

(2.6)

. . .

¢pp(0, t), t) = 0. The nominal system is called minimum phase if there exist 6 > 0 and T0 E R ~, t - - f l l + "'" + tim, such that (2.6) is uniformly asymptotically (exponentially) stable for initial condition ~ ( 0 ) E Na(fi0). These zero dynamics are those of the control variables and are a generalisation of the definition in [2] to the multiple input case. They are different from those defined in [4], which relate to the dynamics of the uncontrolled states. The system (2.2) may be expressed in the following generalised controller canonical form (GCCF):

•(1)

=

nl--I

nl

n, = q ~ l ( ( , a , t ) + A l ( ( , t ) , (2.7) ~(m)

~(m)

: ~-(m)

n,,, - - 1

=n,.

~(m) n m : ~l)m((, ~.l,i!) "~ Am((, t), where ((i) = ~[r(i). . . . . ?.(i)~ ~,, : = ( Y i , . . ., y~,,-l)), i : 1. . . . . m and ( = ( ( 0 ) . . . . . ((m))T. 2.3. Dynamic slidin9 mode control As is the case for static sliding mode controller design [ 10], dynamic sliding mode controller design may be divided into two separate phases: (i) Proper choice o f sliding surfaces: Based on the proper differential I-O system, the following choice of sliding surfaces will[ be made: ni

i = 1. . . . . m,

si = j=l

(2.8)

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X.-Y. Lu, S.K. Spurgeon/Systerns & Control Letters 32 (1997) 75 90

where ~ _ , a~i)~]-1 are Hurwitz polynomials with a(~i)= l, i=-1 . . . . . m. For single input systems, this was introduced in [8, 9]. However, the stability of the corresponding closed-loop systems for both single input and multiple input problems has yet to be formally analysed• (ii) Proper choice o f sliding reachability condition: For robust controller design, a strong sliding reachability condition defined as follows is necessary.

Definition 2.2. A general sliding reachability condition : - 7(tc, s),

(2.9)

where s = [s 1,..., Sm]T, and tc = [tel . . . . . ~cl] is a set of constant design parameters such that 7(~c,s) = [~1 (h2, s), ..., 7m(•, S)] T satisfies (1) ~,(~,0) = 0; ( 2 ) 7i(t¢,S) is a continuous function of s if s i c 0 , i = 1.... ,m; (3) 7/Qc,s) is bounded for sEN~(O), i = l , . . . , m ; (4) (2.9) is globally uniformly asymptotically stable. A sliding reachability condition is called strong, if condition (4) is replaced with (41) sTy(n, s) > sTKs for some positive-definite matrix K if s ¢ 0. This condition implies that s--+ 0 globally and at least exponentially.

Example 1. The following continuous sliding reachability condition is strong. Let [~cij] E R mxm be a positivedefinite matrix and ~c0i~>0, i = 1. . . . . m.

/

~c01sat,(sl ) ]

7(to, s) = [~cij]s +

,

(2.10)

k~C0msat,(sm)J where sat,(x), for q > 0 is defined on ~ by 1, sat,7(x)=sat(~)

x>~t/,

= ~ x/q,

Ixl
-1,

x~< -t/.

It is also clear that m s T y ( l~, S) : S T [l'¢ij] S -]- Z t£oiSi satu (si) >1s T [lgij] s. i=1

3. Robust design method Step 1: Choose design parameters to define the sliding surfaces that nl . . . . . nm~ > 1 and nm,+l . . . . . nm,+m2 = 1,ml +m2 = m . For ((i) (O , a 1 ,...,an,_ I) both Hurwitz. This is always possible according to

(2.8). Without loss of generality, suppose i -- 1.... ,ml, c h o o s e ( a (i) I , . . . , a n _(i) l, l ) and [3].

X.-Y. Lu, S.K. Spurgeon/Systems & Control Letters 32 (1997) 75-90

79

Step 2: Estimate the uncertainty bound as in (2.3) when the system is in the GCCF. Choose 00 and 0 where 0 < 0 < 1, 00 + 0 = 1 and define

p(1)=p(O) ( 1 +

max

.rla!i)l~ ×

l<~j<~ni, l<~i<<.mltl j u

max { ~ ) ~ l<~i<~ml

(3.1) /I '

p : p(o) + 40 Step 3: Choose 7(~.s) = yo(~C.s) + K o sat,(s) in the sliding reachability condition such that sTT0(tC.S) ~sTKs and yo(K.s) is continuous, where K = diag[K1.K2] with K1 E R m~×m~. K2 C ~m2×m: and Ko =diag[kol . . . . . kom] chosen such that Ko > loire and -~o[BD]T[BD]+pIm,

,~min(gl)lm,-

>0,

~min(K2)-p>0 ,

where satn(s) = [sat.(sl ) . . . . . sat.(Sm)]T; D = diag[D1 . . . . . Din, ]T with Di = [0 . . . . . 0. 1]T of dimension ni - 1 for i = 1.... .ml; A =diag[A1 .... .Am,] with Ai the companion matrix of the Hurwitz polynomial y-~]~l a~i)2j_l and A and B satis~r the Lyapunov equation

ATB + BA : - I , - m .

(3.2)

Step 4: Differentiating (2.8) with respect to time t along the trajectories of (2.7) leads to ni--I

Si[(2.7) : Z

ff~i)~Q1 -t-(pi(~,fi,

t)-k-Ai(~,t),

i = 1 . . . . . m.

(3.3)

j=l Now set hi-- 1

Z

aJi)~i)+l + q~i(¢,fi, t ) = -Yoi(K,s) - koi satn(si),

i---- 1,...,m,

(3.4)

j=l where s is as defined in (2.8), to determine the feedback control. Eq. (3.3) becomes = - 7 0 ( ~ , s) - K0 sat,(s) + A(~, t). (/~m)iT Step 5: From (3..4) the highest-order derivatives of the control tu (ill) I . . . . . Um j , can be solved out uniquely if they appear linearly as in (2.4) or locally uniquely by the Implicit Function Theorem as r

u (fl~) = pi(~,'5, t),

i = 1. . . . . m

due to the Regularity Condition. Note that pi(~,'5, t) is a continuous function because q)i is (~1, and the sliding reachability condition (2.10) is continuous. This. dynamic feedback can be realised in canonical form by introducing the pseudo-state

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)L-Y Lu, S.K. Spurgeon/Systems & Control Letters 32 (1997) 75 90

variables as

Zl l) =Z~ 1),

~(1) ~, = pl(~,z,t),

(3.5)

~(m) : z~m), fl,,,--I ~(m) = pm(~,z, t), where z (i) =[,7,1 . ( i ) ,...,Zfi, ( i ) . .)=lUi, .~ U• i .... ,U(if l , - i ) . . ), i = 1..... m; z = ( z (1)..... z(m))T. Eq. (3.5) together with (2.7) m m yields a closed-loop system of dimension ~ i = l ni + ~i=l fli. Step 6: Choose K0 E ~/~ and a 6 > 0 such that, for initial condition K(0)c N~(~0), (1) the Regularity Condition is satisfied; (2) the zero dynamics (2.6) (or (3.5) when ~ = 0 ) are uniformly asymptotically stable; (3) all the initial conditions for (2.7) and (3.5) are compatible.

4. Stability of the closed-loop system This section formally analyses the stability of the closed-loop system. The background to the stability analysis may be outlined as follows. In the ideal sliding mode, the closed-loop system is in triangular form ~=A~+Ao(t),

~=~/('~,z,t),

]]Ao(t)]]<~6, 0~<6<< 1,

where "~ is a vector of dimension n - m. Local uniform ultimate boundedness of the overall system is obtained from the stability of the zero dynamics (i.e. the second subsystem when "~= 0) and the uniform ultimate boundedness by arbitrary ~ > 0 of the first subsystem. 4.1. Uniform ultimate boundedness of triangular systems The sufficient condition for the existence and uniqueness of solutions in the ultimate boundedness theorem in [5] will first be weakened to yield the following sufficient condition. Lemma 4.1. Consider the system (2.1) where f ( t , x ) satisfies Condition B. Let V:N+ xN6(0)---+E be a continuously differentiable function such that

~l(llxll)~< v ( t,x ) <<=2( llxll ), 8V a-7 + O~-Vf(t,x)<~ ox -=3(llxll),

Vllxll ~>#>0, v(t,x) c R+ xN, ff0),

where O~i(" ) ( i = 1,2,3) are class-~f functions and ft<~-~(~l(8)). Then there exist a class-aU,Lf function fi(., .) and a finite time T = T(x(to),#) such that Vllx(t0)]] < ~ - l ( e l ( 6 ) )

Ilx(t)ll

<~(llx(to)ll,t

Ilx(t)ll ~71(o~2(~)),

-

to), Vt~T.

to <~t
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If all the assumptions hold with 6 = o e , and ~l(') belongs to class .Y£oo, then the inequalities (4.1) hold for any initial state x(to). Furthermore, if ~i(r)~--ki ra, for some positive constants ki and a, then fl(r,s)=kr e x p ( - a s ) with k---(ke/kt )1/a and a = k3/(k2a). The later case will be called exponentially ultimately bounded

with decaying rate a. Proof. That f ( t , x ) satisfies Condition B guarantees the existence and uniqueness of an absolutely continuous solution. Other arguments follow from the Lyapunov direct method and are the same as employed in the proof of Theorem 4.10 in [5]. [] To study the stability of the overall system, the result of [12] on the stability of triangular systems is now generalised to the case of uniform ultimate bounded regulation of such systems. Consider the time-variant nonlinear control system

rbl =Fl(t, wl,u),

w2 =F2(t, wl,w2),

where t E ~+, wi E ~U,, i = 1,2, u E •m, and F2 is continuous with F2( t, 0, 0) = 0, Vt E R+. Suppose u = u( t, wl ) is a feedback control. The closed-loop system is in triangular form

ZI:

Wl=Fl(t, Wl,U(t, wl)),

Z2:

wz=F2(t, Wl,W2).

(4.1)

If the second subsystem is disconnected, the following two systems are obtained: Z~0: ~+l=Fl(t, wl,u(t, wl)),

(4.2)

Z20: w2=Fz(t,O, w2).

(4.3)

Assume that for i = 1,2, there are constants ci > 0 such that (A1) For [Iwll[ "-~,cl,F1 satisfies Condition B; (A2) sup sup

,>~0 IIw,II~c,

Fz(t, wl,w2) - F2(t,O, w2) IIw~ 11

< c~.

Remark 4.1. If in (4.1), F2 is locally Lipschitz with respect to Wl at wl = 0 and uniformly in t, (A2) is satisfied. Theorem 4.1. Suppose (1) Assumptions (A1) and (A2) hoMfor cl = 6o >0,c2 = 62 >0. (2) For any el > 0 sufficiently small there exists a feedback control u = un(wl, t) such that for the closed-

loop system ~'1 = FI (t, wl, un(wl, t) ),

(4.4)

Fx satisfies Condition B and there exists a continuously differentiable function V1 : ~+ × N6o(0) ~ satisfying ~l ([[wl [I) ~< V1(t, Wl ) ~
OF1

OVlFl(t, wl un(wl,t))<~_~3([lWln),

~')1 [(4.4) = "~-~- q- ~W 1

Vllw~ll>~,>0,

v(t,w,)cR+

(4.5)

XN0o(0),

where 0~i(')(i = 1, 2, 3) are classJi ~ functions and el < ~21(~1(60)). (3) The system (4.3) is locally uniformly asymptotically stable for w2(0)c N6:(w°). Then for any ~>G,, there exists 0<61 <~fi0 and a control u = u , ( w l , t ) such that the closed-loop system (4.1) is uniformly ultimately bounded by e for any (wl(0),w2(0))cN6, x N62.

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Proof. Without loss of generality, suppose w ° = 0. According to the Inverse Lyapunov Function Theorem, there exists a 62 > 0 and a continuously differentiable function V2(t,w2) such that, for (4.3),

/~1(11w2[I)~ < v2(t, w2) <~2(llw211), ~V2 OV2F2(t,O, w2)<~-fl3(llw2[[), Ot -4- OW 2

~'2 [2:20 =

1,2,3) are c l a s s - Y functions. Suppose

where t ~ i ( ' ) ( i = sup

V(t, wl)E~+ xNa:(O),

sup

t>~o IIw211~<&

HVw2Vz(t,w2)ll=Rl
Let 0 < 61 ~<30. By assumption (A2), there holds sup sup t~>o Hwill~
F2(t,wl,w2)_--_F2(t,O,w2) =R2
Now consider the derivative of V2 along the trajectory of Z2 :

OV2

OV2 F2(t, wI,w2 ) = I5"2]~:20

f'2 [s~ = --b7 + ~w2 OVa

"F -ff~W2[F2(t' Wl' W2 ) - F 2 ( t ' 0 ' w2)]

<~--~3(IIw211)+RIR211wllI, Ilwlll~ 0 with /31 +/32 ~
If/3 > 0 such that (4.5) is satisfied, where /3l ~< min{61,%l(~l(61))} is implied in (4.5). Consider the Lyapunov function candidate V(t, wl, w 2 ) = Vl(t, wl )+ v V2(t, w2), where v > 0 is a constant to be determined, which is continuously differentiable and positive definite. First suppose that /31 .G<[Iwl(z,)lt < a l and /32 ~< Ilw2(z=)ll ~a2. Then

[(4.1) = ~'1 [Z, ÷ oV2lz2 ~ -~3([[Wl [1) - vfl3(]lw2[[) + --0{3(/31 ) -- Ofi3(g2) ÷

vR1R2l[wlII

oR1R261.

Thus, if 61 is chosen such that 0<61 <

/~3(/32) RIR------2'

then

~'7"[(4.1) < 0 . Then suppose

~

Ilwlcz,)ll
0 V I = (+

q- 63Wl 1'1 ~, wl,un(wl,t)) ~R3.

Then

/;'[(4.l) = l)l [S, ÷ O' V2 [2:2 ~ 0 such that v(fl3(g2) -

RIR2gl ) - R3 > O.

(4.6)

X.-)~ Lu, XK. Spurgeon/Systems & Control Letters 32 (1997) 75-90

83

Note that such a choice is always possible since el <~61
r>l(4 ~) <0. By Lemma 4.1, condition (1) implies that there exists T1 >~0 such that Uwl(z)(t)UTl as long as Ilwl(z,)(o)ll <61. The above arguments thus prove that, as long as Ilwl(z,)(t)[ I <61 and Uw2(z~)(o)ll <62, w2 is uniformly ultimately bounded by e2, i.e. there exists T2 ~>0 such that Ilw2(z)11< e2 if t > T2. Let T = max{T1, T2}, then [[(WI(Z),W2(z))II < [[WI(Z)[[ q-

[[w2(z)[[ <~1 + ~2 ~ T,

i.e. for the chosen feedback control u = un(wl, t), the system (4.1) is uniformly ultimately bounded by ~. []

4.2. Stability of the uncertain sliding mode control system First note that (2.8) determines a linear coordinate transformation between ~+-~ (~,s), where "~=(~(11)..... ~(J) Y(") n l - - l ' ' ' ' ' ~ l T(") ' ' ' ' ' ~ n m - - l " ~T" The uncertainty class (2.3) in the C-coordinate can be written as [l~i(~,t)nL ~ pill~[I + l .

i = 1 . . . . . m.

This leads to the estimation IIA(C, t)l[ ~
84

X.-Y.

Lu, S.K. Spuroeon/Systems & Control Letters 32 (1997) 75-90

and (4.7)

IIA(C,t)II = HA(~,s,t)ll ~
max {[a)i),} max { n / ~ - l } ) . 1<~i<~ml,l<~j<~ni 1<~i<<,ml

Using si to replace ~(i~ in (3.3) for i = 1,... ,m, it follows that

hi-- 1 "~i1(2.7) = Z j=l

._(i)r(i) j ~j+l + q~(~,~,t) + Ai(~, s, t)

ni--2 -E Uj _(i)~.(i) -~j+l

I _(i) -[- "ni--I

ni--1 ) ~ (i)~,(i)~ _ + ~Oi(~,s,fi, t) + Ai(~,s,t), Si ~ a) ;)

i---- 1,...,m.

j=l

(4.8) The system (2.7) can be equivalently written as

"~= A'~ + Ds'1',

(4.9)

= - 7o(x,s) " K o sat,(s) + A(-~,s,t), where s (0 ---[sl . . . . ,Sm,] T, and Ko = diag[k01,... ,k ore]. A, D, 7o(rC,s) and sat,(s) are the same as in Step 3. Remark 4.2. The uncertainty enters the closed-loop dynamics only through the s-state. This is because the uncertainty is matched to the highest-order derivatives of the control. Note that during ideal sliding, i.e. s--0, the closed-loop dynamics are wholly determined by the choice of sliding surface parameters. Lemma 4.2. Consider the system (4.9). Suppose (1) sTTo(~C,s) >>.sTKs where K = diag[Kl, K2] such that

)~min(KI)Im~-[O-~[BD]T[BD]Wplm,] >0,

(4.1o)

~min(g2 ) - p > 0,

where Oo and p are the same as in (3.1) of Step 2 and B and D are the same as in Step 3. (2) Ko satisfies

KO > loire, where lo = ~ (3) Let A=

00/n -- ra

12. -[BD]

],

-[BD] T (().rain(K1) - p)Im, )

go = m i n { 2 m i n ( A ) , (~min(K2) - p ) , )-min(K0 -

loire)}.

For arbitrary e.o> 0, choose q -- e2ao/mlo > 0 in Step 4.

)L-Y. Lu, S.K. SpurgeonlSystems& ControlLetters32 (1997) 75-90

85

Then (4.9) is uniformly ultimately bounded by ~o. Furthermore, (-(,s) enters any ball N~L (~1 >eo) exponentially and globally with the rate

(7~--

2 max{2ma,:(B), 1/2}e~el2"

Proof. First note that (4.9) satisfies Condition B for the uncertainty class (4.7). Consider the Lyapunov function candidate tbr (4.9):

N~

VI = (B~, ~) +

![sTs,

(4.1 1 )

where B satisfies the Lyapunov equation (3.2). Now differentiating (4.11) along the trajectories of (4.9) and noting (4.7), we obtain

t~ 1<4%= -i1~112 +

2~TBDa ¢~) -

sT~0(~,s,t)

~< -I1~112+2"(TBDs(0 - - s T K s

--sTKo

-- sTKo sate(s) + sYA(~,s, t)

satn(s)+ Ilsll(P¢~)ll~ll÷ p(°>llsll + 10).

Using (p(1))2

2

p(1)ll~llllsll ~6'11~[12+ --TU-Ilsll, Ilsll ~< ~

Is, I =

ST sign(S),

i=1

vl 1~49>-< -I1~11z +

0

]

(2min(K2))Im2

]s

"("~min(gl))Ira, 2~WgOs ~) - s T

0 ((p(1))2

--sTKo sat,(s) +

•

i

0[[~]]2 +

k-W-

+ p(O)~ [Ls[[2 +sT[IoI] sign(s) ,/

-((2min(K1) - p)Im,)

L [BD]T

s (1)

--(S'2))T((~min(g2)--p)Im2)S(2)-- (~(ko#-lo)lstzl "4-~v (~-~kov-lo) Isv[), where /~ and v stand for the subscripts where satisfy

O01n-m -[BD] T

-[BD]

Is~l~>~ and Isvl<7,

respectively. Thus, suppose KI and K2

]

((,~-min(gl) - p)lm,)J > 0 ,

(4.12)

,~min(K2) -- p > 0.

If all the si satisfy ]si]>)7, then I211(4.9)~< -

~0(11~112+ Ilsl12)
(4.13)

86

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Lu, S.K. Spurgeon/Systems & Control Letters 32 (1997) 75 90

Otherwise,

ISvl /1 ~< -~oll(~,s)ll 2 + ~ Is,,llo~ - ~oll(~,s)ll 2 + into, < o,

v It(~,s)ll > ~ . ~ .

Thus, choosing q =E~ao/mlo, the system (4.9) is ultimately bounded by eo = ~ if (4.12) is satisfied. It is clear that V is radially unbounded. Thus, (4.9) is globally uniformly ultimately bounded by eo if (4.12) is satisfied. Furthermore, let el > c0,

v, t~4.9>~< -

~oll(~,s)ll 2 +

mlo~l m/oq

~< -~oll(~,s)l/: ÷ ~ll(~,s)ll 2, II(~,s)ll ><. L e t a] =

m/oq/e~. T h e n

al
m/orl/e~. T h u s ,

V11(4.9)~< - (~0 - ~r,)ll(~,s)ll 2,

II(~,s)ll ~>c,.

(4.14)

It is clear that 1

1

min {,~min(B),~ } ]1(~,s )112~ VI ~ max {/~max(B),~ } l]('~,s)H 2. It is deduced from Lemma 4.1, (4.13) and (4.14) that (~,s) enters the ball N~:, exponentially with decaying rate mlotl(e~ - e~) 2 max{)~max(B), 1/2}%2e2"

(aO -- (71)

2 max{2max(B), 1/2}

It is necessary to show (4.10) implies (4.12) to complete the proof. Let v = [vT, v[] ¢ 0, vl E N ", v2 C Nm, and H = [BD] T, J = OoI~-m, Q = ()~min(Kl) - p)Im]. Consider --H

If v2 = 0, 0 = vTJv] > 0 because J is positive-definite. If v2 ¢ 0,

'0 = vTJ/)I /)2T/-/V,-- /)THTv2 ÷ uTQt~2 = *.)TJuI -- 2t~TsTt)2 ÷ t)TQu2 > vTJv, - 2vTIHTv2 ÷ v~[HJ IHT]v> Let P denote the positive-definite matrix such that J

=

p2

.

Then

> vTpPvl - 2vTPP 1HTD2 ÷ v T [ H p - I p - ' H T ] v 2 = {[Pv] - (HP-I)Tv2], [Pvl - (HP-~)Tv2]) >~0. This completes the proof.

[]

Theorem 4.2. Consider the system (2.2) with the uncertainties (2.3). For any 9iven 8 > O, if the corresponding zero dynamics (2.6) is uniformly asymptotically stable, then there exists a feedback control resultin9 from the design method in Section 3, such that the closed-loop system is uniformly ultimately bounded by 8.

~ - K Lu, XK. Spurgeon/Systerns & Control Letters 32 (1997) 75 90

87

Proof. Note that the closed-loop systems (2.7), (3.5) and (4.9), (3.5) are equivalent in stability. By L e m m a 4.2, for arbitrarily given el > 0 , there exists a control u = u ~ ( t , ( , K ) such that (4.9) is uniformly ultimately bounded by /31. Thas, if /31 is small enough, the closed-loop system (2.7), (3.5) resulting from the design method in Section 3, is ultimately bounded by /3 according to Theorem 4.1. This completes the proof. []

Corollary 4.1. I f there are no derivatives of the control u present in the proper I - 0 system, the above result is globally exponer.!tial. Proof. This is obv:ous from Lemma 4.2 as no zero dynamics are involved.

[]

5. Examples In this section, two pertinent examples are used to show the logic of the design method. The first example has a nontrivial zero dynamics given by the Van del Pol equation. The second example is the control of gas jet actuators with uncertainty using two controls. This example also shows how to use the design method to control a state space model. In both examples, the one-dimensional random variable in Matlab, rand(l), is used in the uncertainty description to demonstrate the robustness of the method. Strictly speaking, it is not Lebesgue measurable. However it is practically integrable in Matlab.

5.1. A nonlinear model with non-trivial zero dynamics Consider the following nonlinear model: = u sin(p) + (1 + y 2 ) / / + uy + u + #ti(u 2 - 1 ) + A,

(5.1)

where A = y sin(p) + rand(1 ) is the uncertainty. Its corresponding zero dynamics are obtained by setting y = y = j ) = 0 as +u+pfi(u 2-1)=0. This is the Van der Pol equation and is uniformly asymptotically stable for kt<0 with (u(0))2 + 0i(0))2 < 1. Thus (5.1) is a minimum phase system. Step 1 : Choose sliding surface s = ay + y with a = 2. 3 Step 2: Choose 0 0 = ~ 10 = ~ . From M l < ~ [ y [ + l o p (°) 1 , 1 o = l , p ( 1 ) = 3 ~ p = 4 .

Step

3: A = [2] : , B = [ - ¼ ] . Then

{(1/Oo)[BD]T[BD]+ p} = 4 ¼ .

Thus choose k = 5 > 4 ¼ , k 0 = 1.5 > 10 = 1.

Step 4: The controller is solved from ~4= ks + k0sat,(s) as / / = - { k s + k0 sat,(s) + u sin(p) + uy + u + #ti(u 2 - 1)}/(1 + y2). Simulation results are shown in Fig. 1 with (y(0), y(0), u(0), fi(0)) = (-0.52, 1.3, 0.7, -0.51 ), ~, = -0.7, e = 0.5. It is noted that the four-dimensional system would be uniformly asymptotically stable if there were no uncertainty A.

5.2. Control of gas jet actuators with uncertainties The angular velocity control of a gas jet actuator with uncertainties is modelled by the Euler equations:

Al(x,t),

(5.2)

~2uz+A2(x,t),

(5.3)

21 = J~lX2X3 "~- ~lUl + 22=,-~2XlX3 --

23=d3x2xl,

X.-E Lu, S.K. Spurgeon/Systems & Control Letters 32 (1997) 75 90

88

........ }......... ~......... i ......... ~......... ~ ......... i......... ~......... ~......... ~.........

......... } ......... ! ......... } ......... } ......... } ........ i ......... ! ......... ! .......

-1

0

2

4

i

6

.................... "......... ~..................

8

10

12

14

16

18

20

~.l-....,...:~...~ : .......i.................... !..........i .........::..........!......... i........

0

2

4

6

8

10 t

12

14

16

18

20

Fig. 1. Minimum phase system with uncertainty; ultimate boundedness is achieved.

~'l = 0.5, d 2 = 0.2174, d 3 = - 0.6471; ~l ¢ 0, 72 ¢ 0. The nominal system is strongly accessible and feedback linearisable if Xl ~ 0 or x2 ~ 0. The uncertainty is represented by

Al(X,t)=O.3xasinxl + 0.5 rand(l),

zJ2(x,t)=O.2x3cosx2 + 0.5rand(I).

The control task is to track a constant trajectory (0, h, 0), h = 0.5. The error dynamics are el = d l ( e 2 +

h)e3 +

~lUl ~-

Al(e,t),

e2 = d2eae3 + c¢2u2 + A2(e,t),

(5.4)

e3 = ag3(e2 + h)el, with (el, e2, e 3 ) - - ( X l , X 2 - h,x3). Choose fictitious outputs Yl = e3, Ye = e2. The corresponding I-O system is Yl = ag3(Y2 + h)[aglyl(Y2 + h) + ~lUl] +

Yl

/~ d 2 d 3 ( yYlPI+ 2 h) + ~2u2) +

~1,

Y2 = ~C~'2 Yl))l -I- ~2U2 -]- A2, d3(Y2 + h)

.~ ' AI = a~C3(Y2+ h)A1 + (Y2Yl+ h____a2

A2 = a2.

To estimate the uncertainty bound in the (Yl, 3)1,)'2) coordinates consider the infinite strip

9=

(yl,~,yz)l-

l<<.Sq <~l,-~-<~y2<~

.

(5.5)

With (Yl, 3)1, Y2)E 9 , the new uncertainty bound is obtained as [511 ~<0.546]yll + 1.243,

1521~0.21y~l +0.5.

(5.6)

Note that this effectively restricts the initial condition choice of )71 and y2 and that the coordinate transformation carried out is Yl =x3, -Yl = ~'3xlx2, Y2 =x2 - h.

X.-Y. Lu, S.K. Spurgeon/Systems & Control Letters 32 (1997) 75 90

89

To investigate local observability: det [ ~(Yl' ))1' Y:-)I =d,h#0. I_ t3(el,e2,e3) Je=0 Thus, the regulation of ~ = ( y l , ) ) l , y 2 ) at ~ = 0 is equivalent to that of e at e = 0 . ( y l , ) ) l , y 2 ) = 0 yields the zero dynamics ~lUl = 0 , ~ 2 U 2 = 0 , which is trivial and uniformly asymptotically stable. The Regularity Condition yields det [ ~?(Ul,u.)/ly,,2,,y2)=o = cqe2 # 0. Thus, the corresponding nominal I-O system is proper. Step 1: It is clea:r that nl = 2 , n2 = 1 =~ml = 1,m2 = 1. Choose the sliding surface s1 = 2 y l q- ill,

$2 ----Y2,

where (2, 1) is Hurwitz. Step 2: From (3.1) and (5.6), for 0 = 3,00 = ¼,/o = 1.34, and p(0)= 0.582,p(1)= 1.746,p = 1.6. Step 3: As in the first example, {(1/Oo)[BD]T[BD]+ p} =0.25 + 1.6 = 1.85. Choose the sliding reachability condition (2.10) such that kol = k02 = 1.5 > lo and k21

, 7:i31

k22]

Step 4: The dynamic controller is obtained in the original coordinate system as -1

Ul -- (XI,~3X~ [kllSl -- k12s2 + kol satn(sl ) + a~C3XlX3 -4- Jffl,~3x22x3 Jr- Jff3Xl(g~/2XlX3 Jr- (X2U2)].

-1

U2 = - - [ k 2 1 s 1 -~- k22s2 q- ko2 sat,1(s2) + ~ 2 X l ] . o(2

Under the coordinaLe transformation, the region @ C ~ 3 in (5.5) is transformed into ~0CR3: ~ 0 = {(Xl,X2,X3)[0"25 <'~X2~<0 . 7 5 , - 1 ~< ~t3XIX 2 <~. 1}.

Step 5: Choose C~l= 0.8333, ~2 = 0.4348, q = 0.1 and x(0) = (1.6, 0.42, - 0 . 5 ) E ~0. The simulation results are shown in Fig. 2.

2L

~

2c

.......... ,s......... .......

0

5

10

15

2°

20

5

10

i!tl...................i................... 0

5

10

15

20

Fig. 2. Uncertain gas jet system.

15

20

90

X.-Y. Lu, S.K. Spurgeon/Systems & Control Letters 32 (1997) 75 90

6. Conclusions

This work considers the robustness of dynamic sliding mode control of nonlinear systems which are in differential input-output form. The additive uncertainties in the model are assumed to satisfy some bounded and/or sector condition (cone-bounded). For the stability analysis, a zero dynamics arises, which are actually the dynamics of the control variables. Because of the presence of uncertainties, the performance which can be achieved is uniformly ultimately bounded instead of asymptotic in practice. However, if this ultimate bound can be rendered arbitrarily small in theory, the uniform asymptotic stability of this type of zero dynamics ensures the closed-loop system, which is in triangular form, is uniformly ultimately bounded. With the proper choice of sliding reachability condition, it is in fact exponentially ultimately bounded. If there are no derivatives of the control present in the proper I-O system, the results are global. References [l] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

A.F. Filippov, Differential equations with discontinuous right hand side, Amer. Math. Soc. Translations, ser. 2, 42 (1964) 199-231. MI Fliess, What the Kalman state variable representation is good for, Proc. IEEE CDC, Honolulu, Hawaii, 1990, pp. 1282-1287. W. Hahn, Stability of Motion, Springer, New York, 1967. A. Isidori, C.I. Byrnes, Output regulation of nonlinear systems, IEEE Trans. Automat. Control AC-35 (2) (1990) I31-140. H.K. Khalil, Nonlinear Systems, Macmillan, New York, 1992. X.Y. Lu, S.K. Spurgeon, Asymptotic feedback linearization and control of non-flat systems via sliding modes, Proc. 3rd European Control Conf., Rome, Italy, 5-8 September 1995, pp. 693-698. B.E. Paden, S.S. Sastry, A calculus for computing Filippov's differential inclusion with application to the variable structure control of robot manipulators, IEEE Trans. Circuits Systems CAS-34 (1) (1987) 73-82. H. Sira-Ramirez, A dynamical variable structure control strategy in asymptotic output tracking problems, IEEE Trans. Automat. Control 38 (4) (1993) 615 620. H. Sira-Ramirez, On the dynamical sliding mode control of nonlinear systems, Int. J. Control 57 (5) (1993) I039-106l. V.I. Utkin, Sliding Modes in Control and Optimization, Springer, Berlin, 1992. A.J. van der Schaft, Representing a nonlinear state space system as a set of higher-order differential equations in the inputs and outputs, Systems Control Lett. 12 (1989) 151-160. M. Vidyasagar, Decomposition techniques for large-scale systems with nonadditive interactions: stability and stabilizability, IEEE Trans. Automat. Control AC-25 (4) (1980) 773-779.