E L S E V I E R Systems & Control Letters 29 (1996) 51-59

S JTMLS Ih ¢ONTIIOL UTTUS

Further results on global stabilization of discrete nonlinear systems1

W e i Lin*

Department of Systems, Control, and Industrial Engineering, Case Western Reserve University, Cleveland, OH 44106, USA

Received 24 July 1995; revised 10 April 1996

Abstract

In this paper we show how recent advances [ 11,12] in global stabilization of continuous-time nonatiine nonlinear systems with stable free dynamics, can be nontrivially generalized to their discrete-time counterparts, by means of passivity and bounded state feedback. As a consequence, global stabilization results of discrete-time nonlinear systems with triangular structure are established.

Keywords: Discrete-time nonafline systems; Global asymptotic stabilization; Bounded state feedback; Passivity

1. Introduction

Consider a general discrete-time nonlinear control system o f the form

Z: xk+l = f ( x k , uk) -- f o ( xk ) + g(xk ,uk )uk , (1)

where x E R n is the state, uE R m is the control variable, f : R n x R m ~ R n and g : R ~ x R m ~ R n×m are

smooth mappings, with f ( 0 , 0 ) = 0. Throughout we assume the following:

( H I ) There exists a C r (r >>. 2) Lyapunov function V: R n ~ R, which is positive definite and proper, such that the unforced dynamic system of (1) - - Xk+l = f ( x k , O) satisfies V ( f ( x , O ) ) <<. V(x);

(H2) t2 N S = {0}, where

---~ {X E ~n : Z ( f ~ + l ( x ) ) = V( fg (x ) ) , i = O, l . . . . . N } ,

S a_ x ~ N n : g o ( f ~ ( x ) ) = 0, i = 0, 1 . . . . . N ,

with A ( x ) ~ f (x, O ) and go(X)z~ d f / ~u(x, O ), and N is a positive integer.

1 This work was supported in part by the U.S. National Science Foundation under Grant DMS-9634395, and by the Case School of Engineering Career Start-Up Funds.

* Tel.: +1 216 368 4493; fax: +1 216 368 3123; e-mail: linwei@azure.cwru.edu.

0167-6911/96/$12.00 Copyright (~) 1996 Elsevier Science B.V. All rights reserved PII S0167-6911(96)00037-0

52 w. Lin/Systems & Control Letters 29 (1996) 51-59

In a recent paper [9] we established, among many other things, that a discrete-time nonlinear passive system

Zd : xk+ 1 = f (xk , uk ) (2) Yk = h(xk, uk)

with C t (l >1 1) Lyapunov function V(x), which is positive definite and proper, is globally asymptotically stabilizable by uk = - y k if (H2) holds. As an immediate consequence of this result, one has the following theorem which is indeed a discrete analogue of Theorem 2.1 in [9].

Theorem 1. Under assumptions (H1) and (H2), Z is locally asymptotically stabilizable at the equilibrium x = 0 by the smooth state feedback control law

uk = cffxk), ~(0) = 0, (3)

where ~(x) can be solved uniquely from the equation

u + h(x ,u) = 0, ( 4 )

with

If, in addition, Eqs. (4),(5) have a global solution u -- ~(x) which is well-defined on R n, then Z can be globally asymptotically stabilized by (3).

Proof. Choose yk = h(xk, uk) defined in (5) as a dummy output for the system (1). Clearly, the resulting input-output system in the form (2) is passive. As a matter of fact,

~'(xk+~) - V(xk) = V ( A ( x k ) + g(xk, uk)uk) -- V(xk)

= V(J~(xk)) - V(X~)+hT(xk, uk)uk <. y~uk Vuk and Vk >>. O.

By Theorems 5.2 and 5.3 in [9], (1) is asymptotically stabilized by the smooth state feedback uk = ~t(xk), which satisfies equation uk = - Y k =--h(xk, uk). The existence of the local solution u = ~(x) of Eqs. (4),(5) is guaranteed by the implicit function theorem as well as the fact that H ( 0 , 0 ) = 0 and dH(O,O)/~u = I for H(x,u) a=u+h(x,u). []

Theorem 1 provides, in principle, a solution to the problem of stabilization of general discrete-time nonlinear systems with stable free dynamics. However, it has two essential limitations: solving the implicit function equation (4),(5) is required to obtain the state feedback control law ~(x), and no hint is given yet on how to compute u -- ~(x) from (4),(5). Moreover, note that Eqs. (4),(5) do not have, in general, a global solution due to a nature of the nonlinear function equation. For this reason, we should emphasize that Theorem 1 gives only, to some extent, a local solution to the problem under consideration.

The goal of this paper is to address these issues and to develop a novel approach for the design of a state feedback control law that globally asymptotically stabilizes the system (1), without solving (4),(5) explicitly. The work presented here represents a continuation of a line of work started in [9,11,12], where both local and global stabilization theorems were established for continuous-time general nonlinear systems with stable free dynamics. In this paper we prove that as in the continuous-time case, a discrete-time nonlinear system with general structure (1) is globally asymptotically stabilizable by smooth state feedback as long as the unforced dynamic system of (1) is Lyapunov stable and appropriate controllability-like rank conditions characterized by the fimctions f (x ,O) and 8f/du(x,O) are satisfied. This is done by explicitly constructing a bounded state feedback control law for the discrete-time system (1). The idea for the construction of such a globally stabilizing bounded state feedback law comes from the recent works [11,12]. Although the results of the paper

w. LinlSystems & Control Letters 29 (1996) 51-59 53

are parallel to our recent advances in global stabilization of general continuous-time nonlinear systems with stable free dynamics [11, 12], the proof of the results is different (as explained in [3, 14]) and the analogy is far from obvious. In particular, it can be seen from the next section that the construction of a discrete-time state feedback control law is more subtle than in the continuous-time case.

The preliminary version of this work was presented at the 34th IEEE CDC [8].

2. Main results

An implication of Theorem 1 in Section 1 is that the controllability-like rank condition, characterized by f2 N S={0}, is sufficient for a general nonlinear system (1) with stable free dynamics to be locally stabilizable. When global stabilizability issue is concerned, an implicit condition in addition to (H1),(H2), namely, the existence of a global solution ~(x) of Eqs. (4),(5), seems to be necessary in order to draw global stabilizability from Theorem 1.

Motivated by the recent works [11, 12], we show in this section how the implicit condition of Theorem 1 can be removed, and how a global stabilization theorem can be built for discrete-time nonlinear systems satisfying (HI) and (H2), by means of the bounded feedback design technique [11, 12] combined with the state feedback control schemes proposed in [2, 3, 14]. To this end, we introduce some notations to be used in the sequel. Let j ~ (x )= f (x ,O) and go(x)= df/au(x,O) as defined in the last section. If f0: R n ~ R n is a vector function, by f0g(x) we mean that fok(x)= J~(fok-l(x))Vk = 1,2 . . . . . with f ° ( x ) = x . In other words, f0 k denotes the kth iterate composition of the mapping 3~. Let V: R n ---+ R be a real-scalar function of x E R n and p: R m ~ R m a vector function, 8V(x)/ax is a covector with dimension 1 x n and ~p/a~ an m x m matrix. From (1) and (5), it is clear that V( f (x ,u ) ) can be represented as

V(f(x , u)) = V(fo(x) + g(x, u)u) - V(fo(x)) + ht(x, u)u. (6)

Proceeding essentially the same argument, h(x, u)E R m can be further decomposed as

h(x,u) - h(x,O) = ~ u ~ Rt(x,u)u, (7)

with R: R ~ × R m ~ R re×m, being a C~-2(r >_-2) map. With the aid of (5)-(7), we are able to state and prove the main theorem in this paper which can be

viewed as a discrete analogue of Theorem 3.3 in [11].

Theorem 2. Suppose a discrete-time nonlinear system (1) satisfies (H1) and (H2 ). Then the equilibrium x = 0 of (1) is globally asymptotically stabilizable by arbitrarily small bounded state feedback. In particular, for any fl E (0, 1 ), a state feedback control law bounded by fl is given by

/7 (~v/~l,:lo<x,) g0(xk)) T 0 </7 < 1, (8) u(xk) = 1 + / : ( x k ) 1 + II~z/a~l~=fo<x~)go(xk)ll ='

where p(x) is any C l (l >>, O)function satisfying

max IIR(x,u)ll <. p(x). (9) Ilull ~< 1

Proof. Choose V(x) satisfying (HI) as a Lyapunov function for the closed-loop system (1)-(8). Along the trajectory of (1)-(8) ,

A V ( x ) = v ( fo (x ) + g ( x , u ) u ) - V (x ) = v ( f o ( x ) ) - V ( x ) + h t ( x , u ) u

= V(fo(x)) - V(x) + hT(x,O)u + uTR(x,u)u. (10)

54 W. Lin/Systems & Control Letters 29 (1996) 51-59

According to (5), hT(x,0)= dV/~l~=fo(x ) Oo(x). Hence,

AV(x) <~ V(fo(x)) - V(x) + ~ oo(x)u + Ilull=llR(x,u)N. (11) o£~ fo(X)

By construction,

Ilu(x)ll ~< ½/7 </~ < 1,

R e c a l l that R(x,u) defined by (7) is of class of C r-2 (r/> 2). Thus the function

O(x) ~- max IIR(x,u)ll Ilull-< 1

is well-defined and at least continuous on •n. So it is always possible to find a C t (l I> 0) bounding function p(x) such that

IIg(x,u)ll ~ O(x) <. p(x) whenever Ilull ~< 1,

With this in mind, we deduce from (11) and (8),(9) that

AV(x) <<. V(fo(x)) - V(x) + ~-~ Oo(X)U(X) + Ilu(x)ll2p(x) ct~ f o(x )

/7 II ~v/a~l~=fo(x) 00(x)lP V(fo(x))- V(x)+ - -

I -4-p2(x) 1 + Itdv/~l~=fo(x>oo(x)ll 2

[ ' × -1 + p(x)(1 + p2(x))(1 + II ov/o~l~=focx> oo(x)ll =)

,a II ~Vl~,l~=fo~x> oo(x)ll = V(fo(x)) - V(x) + 1 + p2(x~) 1 + IlaVla~l,=loOOOo(x)ll 2 ( -1 + ~) <~ o.

Therefore, the closed-loop system is Lyapunov stable. To prove asymptotic stability, we set AV(x )=0 . This, in turn, implies that

dV I y0(xk)=0, u ( x O = 0 and V(fo(xk))=V(xk) V k = 0 , 1 ~t=f0(xk )

The last relation results in Vxo = x E R n,

c3V ~=f0~+'(x) V(fok+l(x))=V(fok(X)) and ~ #0(f0k(x))=0 V k = 0 , 1 , 2 . . . . . (12)

where xk = f0k(x) is a trajectory of the unforced dynamic system xk+l = J~(xk) starting from x(0) = x. Thus the equilibrium x = 0 of the closed-loop system is globally asymptotically stable. The conclusion follows from the hypothesis 1"2 N S = {0} and LaSalle's invariance principle. []

Using the result described in Theorem 2, it is possible to resolve, under a weaker condition, the global stabilization problem of discrete-time affine systems

Xk+l = Jb(xk) + Oo(Xk)Uk, (13)

which was previously considered in [2, 3, 14, 15].

Corollary 1. Suppose a discrete-time affine nonlinear system (13) satisfies (H1) and (H2). Then there exists a bounded state feedback control law which renders the equilibrium x = 0 of (13) globally asymptotically

W. Linl Systems & Control Letters 29 (1996) 51-59 55

stable• In particular, a typical state feedback control law is given by (8), with any C t (I >~ O) function p(x) satisfying

max IIR(x,u)ll ~ p(x), (14) Ilu[I ~< 1

for R(x, u) = gT(x)( f l 02 V/Oa 2 i~=fo(x)+go(x)uO dO)go(x).

Proof. In the affine case, it is straightforward to deduce from (5) and (7) that

00t2 i~t=fo(x)+Oo(X)U 0 dO go(x).

Thus, Corollary 1 follows immediately from Theorem 2. []

Remark 1. In the previous papers [2, 3, 14], we proved that under the conditions (H1) and (H2), an affine discrete-time nonlinear system is locally asymptotically stabilizable by smooth state feedback. If, in addition to (HI) and (H2), the following assumptions hold, i.e.,

(H3) V(fo(x)+ go(x)u) is quadratic in u; (H4) V is either convex or 02V/O~21~=fo(x). >1 0,

the system (13) is globally asymptotically stabilizable by smooth state feedback as well. Corollary 1 shows that the conditions (H3) and (H4) are unnecessary and can indeed be removed for global stabilizability, thus filling a gap between feedback stabilization of continuous-time affine systems with stable free dynamics [1,4-7] and their discrete counterparts [2, 3, 14].

The next stabilization result can be deduced from a combination of Theorem 2 with Theorem 2.4 or Corollary 2.5 presented in [13].

T h e o r e m 3. Consider a discrete-time nonlinear system of the form

x 1 xk+l = f ( k , ~ k ) ,

~1 = ~,, k+l

(15) 7--1

~¢+1 : Vk•

Suppose the subsystem xk+l = f(xk,~lk) satisfies (H1) and (H2). Then, the overall system (15) /s globally asymptotically stabilizable by smooth state feedback. In particular, a typical smooth state feedback control law is given by

l)k V(Xk, ~i . . . . . ~Yk ) 1 2 = = u ( f { f [ f ( . . , f ( f ( xk , ~k), ~k)'..), ~j,~- 1 ], ~j~ }),

where u(.) is defined as in (8)•

(16)

Proof. By assumption and Theorem 2, there exists a smooth state feedback control law

( OV/O~l~:fo<x~) go(Xk ) )W k = U ( X k ) = - -

1 + p2(xk) 1 + II Ov/O~l==fo<x~)go(xk)ll 2

56 W. LinlSystems & Control Letters 29 (1996) 51-59

such that the subsystem

xk+l = f(xk, ¢~) = f(xk, U(Xk))

is globally asymptotically stable at the equilibrium x = O. Using Theorem 2.4 in [13], we conclude that the system with an "integrator"

Xk+l = f(xk, ~1), ~1 2 k+l = ~k'

is globally asymptotically stabilizable as well, by means of the smooth state feedback control law

~2k = ~2(xk, ~lk ) = u( f (xk, ~ ) ).

By essentially using the inductive argument as in the proof of Corollary 2.5 in [13], it is straightforward to prove that the overall system (16) can be globally asymptotically stabilized by the smooth state feedback law (16). []

As an immediate consequence, we have the following corollary.

Corollary 2. Consider a discrete-time nonlinear system of the form

xk+l = f(xk, ~),

: (17) ?-1 1 ?-1

~k+ 1 ~- eYk -~- ~7-1 (Xk, ~k . . . . ' ~k )'

ok + . . . . .

with f ( 0 , 0 ) = 0 and c~i(O ..... O) = O, i = 1 .... ,7. Suppose the subsystem xk+l = f ( xk ,~ ) satisfies (HI) and (H2). Then, the overall system (17) is olobally asymptotically stabilizable by smooth state feedback as well.

3. Two illustrative examples

In this section, we use two examples to illustrate how the theory developed so far can be applied to solve the problem of global stabilization via bounded state feedback, for discrete-time nonaffine nonlinear systems with stable free dynamics.

Example 1. Consider a planar nonaffine nonlinear system

El: Xk+l=yk+(XkCOSyk)u~ yk+l=--Xk+sin(xkuk).

(18)

Choose V(X)= l[IX[[2 ~--- l ( x 2 +y2) as a Lyapunov function for the system El. Then it is immediate to check that V(fo(X))= V(X). Hence hypothesis (H1) holds. Moreover, a direct calculation gives

0 = O-~ a=fo(x)Oo(X)=[y - x ] [x0] = - x 2,

O= ~-~=f~(x)OO(fo(X))=[-x _y](Oy]=_y2.

W. LinlSystems & Control Letters 29 (1996) 51-59 57

ThUS g2 n S = {0}. By Theorem 2, X1 is globally asymptotically stabilized by the smooth state feedback control law

1 1 x 2 u(Xk) = u(xk, yk) -- 2 1 + p2(Xk) (1 +x4) ' (19)

where p ( X ) = p(x,y) is any C l ( l /> 0) function satisfying the inequality (9), which can be computed as follows.

Note that in the present example,

AV(X) = ½[(y + (xcosy)u4) 2 + ( - x + sin(xu)) 2 - (x 2 + y2)] = _x2u + R(X,u)u 2,

where

x(xu - sin(xu)) + ½ sin2(xu) R(X, u) = u2

From (20), we obtain

max IIR(X,u)ll <<. max (Ixl x u - s i n ( x u ) I Ilul[ ~< l lul ~< 1 "~

+ (xy cos y)u 2 + ½(x cos y)2u6.

1 2) + ~ + Ixycosyl + l (xcosy)2

(20)

(21)

is globally asymptotically stable at the equilibrium (x, 41 . . . . . ~r) = 0.

Example 2. Consider a multiinput nonaffine nonlinear system

Xk + 1 -~- Z 2 --~ U k

S2 : Yk+l = Yk + vk zk+l = 2zk sin(uk + xkykvk)

Clearly, Z2 is of the form (1) with X & [x y z] T and

J ~ ( X ) : ' o l ( x ) = ~-~fu(X'0): 2z ~ ' 92(X)=-~o(X'0)= 2xyz

Let V(X) = ½(x z + y: + z 4) for the system Zz. It is easy to verify that (HI) is satisfied• In fact,

V(fo(X)) = ½(z 4 + y2) <~ V(X),

v(~ (x ) )=v (x ) =~ x=0.

system

xk+l = Yk + (xk cos yk)(~l) 4

Yk+l = --Xk + sin(xk ~1) ¢1 = ~2

~1 : k+l

<~ [xlelXl + le2]X[ + y2 + (xcos y)2 ~< (2 +x2)e l+x2 -~- y2 + (xcos y)2.

Hence p(xk, Yk) = (2 + ~)el+X~ + y2 + (xk cos yk) 2 is a bounding function that does the job. Moreover, by Theorem 3 there is a smooth state feedback law (16) with (19), such that the following

58

Moreover, a simple manipulation shows that

W. LinlSystems & Control Letters 29 (1996) 51-59

The last three equations clearly imply 12 f3 S = {0}. Therefore, it follows from Theorem 2 that 272 is globally asymptotically stabilized by (8), which in the current case is equal to [] 1

uk = ~ z (22) /)k (1 "~ p2(Xk))(l q-Z4k + y2) Lykj ,

with an appropriate bounding function p ( X ) - p(x ,y ,z ) satisfying (9). In the current example, observe that

A V ( X ) = ½[((z 2 + u) 2 + (y + v) 2 + (2z sin(u + x y v ) ) 4) - (x 2 + y2 +z4)]

----- -½x 2 + [z 2 ylU + UTR(X, U)U,

where U ~ [u v] T and

( ~ (2z)4shl4([1-xY]g)~ R(X, U) ~- + 2[IUII z ] I2x2, (23)

from which it follows that

max llR(X,u)ll ~< max (1 +(2z) 4 {sin([1 xy]U)'~2"~ IJUH~ | HUH~ 1 . \ Ilfl) / ]

~< 1 + (2z)4e 21111 xy]lj = 1 + (2z)4e 2(l+x2y2)'/2 <~ 1 + 16z4e (xzy2+2).

Thus we can choose p(Xk)= 1 + 16z~e (x~y]+2) for (22). The resulting smooth state feedback law (22) renders the equilibrium ( x , y , z ) = 0 of 272 globally asymptotically stable.

Obviously, it follows from Theorem 3 that the system (21) with a number of "integrators" is smoothly stabilizable as well.

Remark 2. According to Corollary 1 and the arguments in Example 2, we also conclude that the following affine nonlinear system

Xk+l = z2 + uk Z3 : Yk+l = Yk + Vk (24)

zk+l = 2Zk(Uk + xkykvk)

is globally asymptotically stabilizable by the smooth state feedback law of the form (22) with any C t ( l /> 0) function p(X) satisfying the condition (14). It is not difficult to prove that for the system 273, a possible choice of the bounding function is

p(xk, yk,zk) = 1 + 16za(x2y~ + 1) 2.

Finally, it should be emphasized that the results given in [2, 3, 14] cannot be applied to the affine system (24), simply because the particular Lyapunov function under consideration, namely, V = ½(x 2 + y2 + z4), does not

W. Linl Systems & Control Letters 29 (1996) 51-59 59

satisfy the assumptions (H3) and (H4). Therefore, Corollary 1 has refined the previous results proposed in [2, 3, 14].

4. Conclusions

We have proved that a discrete-time general nonlinear system with stable free dynamics is globally asymp- totically stabilizable by an arbitrarily small bounded state feedback, provided the system satisfies suitable controllability-like rank conditions characterized by the functions f(x, O) and Of/Ou(x,O). A bounded state feedback control law has been constructed explicitly. This work generalized the recent advances in global stabilization of general continuous-time nonlinear systems [11, 12] to their discrete-time counterparts, thus complementing the global stabilization results in [11, 12].

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