A global Implicit Function Theorem without initial point and its applications to control of non-affine systems of high dimensions

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1 J. Math. Anal. Appl. 313 (2006) A global Implicit Function Theorem without initial point an its applications to control of non-affine systems of high imensions Weinian Zhang a,, Shuzhi Sam Ge b a Department of Mathematics, Sichuan University, Chengu, Sichuan , PR China b Department of Electrical an Computer Engineering, National University of Singapore, Singapore , Singapore Receive 18 November 2003 Available online 15 September 2005 Submitte by Steven G. Krantz Abstract Control system esign for non-affine systems is a ifficult problem because of the lack of mathematical tools. The key to the problem is solving for an implicit function but the known results for implicit functions are not applicable for higher imensional systems except for single-input an single-output systems. In this paper, a new version of a global implicit function theorem in higher imension is presente an prove. This result can be applie to show the controllability of a class of non-affine multi-input an multi-output (MIMO) system so that approximation base control system esign can be applie with ease Elsevier Inc. All rights reserve. Keywors: Implicit function; Global existence; State feeback control 1. Introuction The implicit function problem is to solve for the implicitly efine functions u = g(x) from a functional equation f(x,u)= 0. The basic an well-known version of the Implicit Function Theorem is as follows. Supporte by NSFC grants nos an , TRAPOYT an China MOE Research grants. * Corresponing author. aresses: matzwn@126.com, matzwn@sohu.com (W. Zhang) X/$ see front matter 2005 Elsevier Inc. All rights reserve. oi: /j.jmaa

2 252 W. Zhang, S.S. Ge / J. Math. Anal. Appl. 313 (2006) Lemma 1 (Implicit Function Theorem). Assume that f : R n R m R m, V U R n R m is a neighborhoo of (x 0,u 0 ) R n R m, an that f is continuous on V U an is continuously ifferentiable in the secon variable u U for each fixe x V.If f(x 0,u 0 ) = 0, et u f(x 0,u 0 ) 0, (1) then there exist a neighborhoo V 0 V of x 0 an a unique continuous mapping g : V 0 R m such that f(x,g(x)) = 0 an g(x 0 ) = u 0. Moreover, if aitionally f is continuously ifferentiable, then the obtaine g is also continuously ifferentiable. The Implicit Function Theorem an its variations are very important tools in the stuy of nonlinear control, mechanics an other engineering applications (see, for example, [3,6 9,16]). Along the path of research in nonlinear sciences, many upate requirements an concepts such as nonsmoothness, measurability, analyticity, convexity, Jacobian vanishing, egeneracy an approximation are raise, an in fact, many of these are practical problems. Those requirements motivate many generalizations of the Implicit Function Theorem in various spaces an manifols [1,4,5,17,20]. The book [18] by S.G. Krantz an H.R. Parks gives an extensive survey on progress of implicit function theorems. In control system esign, most systems consiere are affine in the control u for technical reasons, an, as sai in [10], no effective metho for controlling non-affine systems exists in the literature at the present stage, especially for multi-input an multi-output systems. A single-input single-output (SISO) non-affine nonlinear system was iscusse in [8], where the input u an output y are both one-imensional, using the basic result Implicit Function Theorem (Lemma 1) which only efines the function (mapping) g locally an requires assuming the existence of the point (x 0,u 0 ) which satisfies f(x 0,u 0 ) = 0. The existence of a continuous ieal control input u requires a (semi-)global version of the Implicit Function Theorem. The neee global result was actually given later in [10] without the stringent conition f(x 0,u 0 ) = 0 an a control system was constructe using neural networks although they can be constructe using any other approximation tools as state in [9]. Global existence for the implicit function problem is also interesting an many versions of global implicit function theorem have been given. In 1981, I.W. Sanberg [22] iscusse the problem on convex open subset V an open subset U of norme linear spaces an provie a necessary an sufficient conition for existence an uniqueness of g : V U which satisfies f(x,g(x)) = 0. This conition requires that (S1) for some x 0 V there is exactly one u 0 U such that f(x 0,u 0 ) = 0, (S2) f is locally solvable for u, an (S3) for each S A there is a T B, where A an B are special families of compact subsets of V an U respectively, such that f(x,u)= 0 with (x, u) S U implies x T. In 1985 Ichiraku [13] prove a topological global implicit function theorem. Fix θ W, W ametric space, an let G f,θ := {(x, u) X Y : f(x,u)= θ} for the continuous map f : X Y W, where X, Y are a simply connecte metric space an a globally path-connecte metric space, respectively. If (I1) for every x X the map u f(x,u)is locally homeomorphism, an (I2) each sequence (x n,u n ) G f,θ with (x n ) convergent has a convergent subsequence,

3 W. Zhang, S.S. Ge / J. Math. Anal. Appl. 313 (2006) then the projection pr : G f,θ X is a homeomorphism an G f,θ can be seen to be the graph of a continuous map g : X Y such that f(x,g(x)) = θ for all x X. In 1991 Blot [2] introuce an stuie the maximal (nonextenable) implicit functions in a purely topological setting. Let f : X Y W be a continuous map on topological spaces. Fix θ W. Uner the conition of local solvability, Blot stuie relations between solutions of the functional equation f(x,g(x)) = θ an the connecte components of f 1 (θ). These known results are not applicable to the problem of controlling non-affine systems. As in [8], one has no way to check the existence of the pair (x 0,u 0 ) in Sanberg s conition (S1). Checking (S3) an (I2) is also ifficult. These ifficulties were overcome using a result of oneimensional global implicit functions as given in [10] without assuming the existence of a point (x 0,u 0 ) which satisfies f(x 0,u 0 ) = 0. Lemma 2 ([10, Lemma 1] an [9, Lemma 2.8]). Assume that f : R n R R is continuously ifferentiable an there exists a positive constant such that u f(x,u) >>0 for all (x, u) R n R. Then there exists a unique continuous (smooth) function g : R n R such that f(x,g(x)) = 0. As we know, one major ifficulty in controlling non-affine systems comes from the solvability of a global implicit function without assuming the existence of a point (x 0,u 0 ) which satisfies f(x 0,u 0 ) = 0. The above mentione result (Lemma 2) can only be use to iscuss SISO problems (i.e., one-imensional problems). An interesting problem is to generalize Lemma 2 to higher imensions. Unfortunately the metho use in [10] for Lemma 2 cannot be generalize to higher imensions. Actually, Lemma 2 was prove by fining intermeiate value of a continuous function. One can compare values of one-imensional maps via the monotonicity of functions, but this is not the case in high imensional spaces. In this paper we will realize the generalization. Using the fixe point theory, a global implicit function theorem is presente for multi-input an multi-output functions without assuming the existence of a point (x 0,u 0 ) which satisfies f(x 0,u 0 ) = 0. Some corollaries an remarks are given for f : R n R m R k where k m. This global result in high imension can achieve the control of non-affine MIMO systems. We further apply this theorem to such a system, proving the existence of an ieal control input an giving conitions uner which the non-affine MIMO system is controllable. 2. Global Implicit Function Theorem Let [ u f(x,u)] enote the th entry of the Jacobian u f(x,u). Theorem 1. Assume that f : R n R m R m is a continuous mapping an it is continuously ifferentiable in the secon variable u R m. If either [ ] u f(x,u) [ ] ii u f(x,u), (x, u) Rn R m,i= 1,...,m, (2) for a fixe constant >0, then there exists a unique mapping g : R n R m such that f(x,g(x)) = 0. Moreover, this mapping g is continuous. Aitionally, if f is continuously ifferentiable, then the obtaine g is also continuously ifferentiable.

4 254 W. Zhang, S.S. Ge / J. Math. Anal. Appl. 313 (2006) In comparison with Lemma 2, the ifficulty in the proof of Theorem 1 is substantial. Lemma 2 can be prove by the Intermeiate Theorem for continuous functions since it is consiere on R, but in R m there is not a so simple orer between vectors as in R. Proof. We first iscuss the case that [ u f(x,u)] ii > 0 for all i.from(2),wehave [ ] u f(x,u) [ ] ii u f(x,u), (x, u) Rn R m,i= 1,...,m. (3) For each x 0 R n,letc:= f(x 0, 0). Ifc = 0, then we efine g(x 0 ) = 0. If c 0, then we consier the close ball B(O, c /) := {u R m : u c /} in R m, where the norm of vectors is efine by c =max { c 1,..., c m }, where c =[c 1,...,c m ] T. (4) Define a mapping T : B(O, c /) R m by T u = u 1 l f(x 0,u), u R m, (5) where l>0 is a large constant. For u R m, by the Mean Value Theorem, we have f(x 0,u)= f(x 0, 0) + u f(x 0,ξ)u, (6) where ξ u. Thus T u = u 1 l f(x 0,u) = u 1 ( c + 0,ξ)u) l u f(x 1 l c + I 1 l u f(x 0,ξ) u, (7) where I R m m is a unit matrix an enotes the norm of matrices corresponing to the norm efine in (4). Since u f(x 0,u)is continuous, all entries are boune by a constant M>0, i.e., [ ] u f(x 0,u) M, i = 1,...,m, for all u c /. Thus 1 1 [ ] l u f(x 0,u) > 0, i = 1,...,m, ii for large l>m. Therefore, by (3), we have I 1 l u f(x 0,u) { = max 1 1 [ ] i=1,...,m l u f(x 0,u) + 1 [ ] } ii l u f(x 0,u) { = max 1 1 ([ ] i=1,...,m l u f(x 0,u) [ ] )} ii u f(x 0,u) 1 l < 1 (8)

5 W. Zhang, S.S. Ge / J. Math. Anal. Appl. 313 (2006) for all u c /. It follows from (7) that T u 1 ( l c + 1 ) c l = c, u c /, which implies that T maps B(O, c /)) into itself as we take a larger l>m. Furthermore, T efine in (5) is clearly a continuous mapping. Applying (8) again, for arbitrary u 1,u 2 B(O, c /), we obtain T u 1 T u 2 = (u 1 u 2 ) 1 ( f(x0,u 1 ) f(x 0,u 2 ) ) l I 1 l u f(x 0,η) u 1 u 2 ( 1 l ) u 1 u 2, (9) where η B(O, c /) is given by the Mean Value Theorem. This means that T is a contraction in B(O, c /). By Banach s fixe point theorem, there exists a unique u 0 B(O, c /) such that T u 0 = u 0, that is, f(x 0,u 0 ) = 0. Therefore, it is reasonable to efine g(x 0 ) = u 0. Following the above arguments, a corresponence g : R n R m at each x R n is efine uniquely an satisfies that f(x,g(x)) = 0. Moreover, by conition (2) an [19, Result ], the Jacobian matrix u f(x,u)is invertible for all (x, u) Rn R m because it is strictly iagonally ominant uniformly with respect to (x, u) R n R m. Then the local result Lemma 1, the well-known Implicit Function Theorem, implies the continuity of g on R n. Continuous ifferentiability of g is also obtaine similarly. In the case that [ ously, fˆ satisfies (3). As before, we can fin a unique g such that ˆ u f(x,u)] ii < 0 for all i, we turn to consier f(x,u)= f(x,u).obvi- ˆ f(x,g(x)) = 0 for all x R m. Such a g is what we are looking for. If [ u f(x,u)] ii > 0forsomei an < 0 for other i s, with a permutation of components of f, then there is no loss of generality in assuming that [ u f(x,u)] ii > 0fori = 1,...,k an < 0for i = k + 1,...,m.Let f(x,u)= ( f 1 (x,u),...,f k (x, u), f k+1 (x, u),..., f m (x, u) ) T, u= (u1,...,u m ) T, where T means transpose of matrices an vectors. Define ˆ f(x,u)= ( f 1 (x,u),...,f k (x, u), f k+1 (x,u),..., f m (x, u) ) T. (10) It follows from (2) that fˆ satisfies (3). As above we can fin a unique g : R n R m such that f(x,g(x)) ˆ = 0. By the efinition in (10), we see that f(x,g(x))) = Some generalization We are also concerne with cases that the range R m of f in Theorem 1 is not the same as the omain of u. If the range of f is R 1, we have the following interesting an useful corollaries. Corollary 1. Assume that f : R n R m R 1 is a continuous mapping an it is continuously ifferentiable in the secon variable u R m. If either

6 256 W. Zhang, S.S. Ge / J. Math. Anal. Appl. 313 (2006) f(x,u 1,...,u m ) u f(x,u 1,...,u m ) i u, (x, u) Rn R m, (11) j for some i = 1,...,m, where >0 is a fixe constant, then there exists a unique mapping g : R n R m such that f(x,g(x)) = 0. Moreover, g is continuous. Aitionally, if f is continuously ifferentiable, then the obtaine g is also continuously ifferentiable. Proof. There is no loss of generality in assuming in (11) that i = 1. Let f := (f 1,...,f m ) T, (12) where f 1 (x, z 1,z 2,z 3,...,z m ) := f(x,z 1,z 2,z 3,...,z m ), f 2 (x, z 1,z 2,z 3,...,z m ) := f(x,z 2,z 1,z 3,...,z m ), f m (x, z 1,z 2,z 3,...,z m ) := f(x,z m,z 1,z 3,...,z m 1 ), that is, except for f 1, all others are efine by the same function f with a permutation of variables. Clearly, f : R n R m R m an z 1 f 2 (x, z 1,z 2,...,z m ) = y 2 f(x,z 2,z 1,...,z m ), z 2 f 2 (x, z 1,z 2,...,z m ) = y 1 f(x,z 2,z 1,...,z m ). Similarly, we calculate the others, an, therefore, we obtain y 1 y 2 y m = z y 2 y 1 y m , y 2 y 3 y 1 where y 1 always appears on the iagonal. Therefore, conition (11) implies that [ ] z [ ] ii z, (x, z) Rn R m,i= 1,...,m, i.e., the corresponing conition in Theorem 1 is satisfie. It follows that there exists a unique mapping g : R n R m such that f(x,g(x)) = 0. Consiering the first a component, we have f(x,g(x)) = 0. Continuity an continuous ifferentiability follow from Theorem 1. This completes the proof. The iea in the proof of Corollary 1 also implies the following generalization to R k where k m, but we have no nee to repeat the same proceure for its proof. Corollary 2. Assume that f : R n R m R k (k m) is a continuous mapping an it is continuously ifferentiable in the secon variable u R m. If either f(x,u 1,...,u m ) u f(x,u 1,...,u m ) is u, s = 1,...,k, (13) j s

7 W. Zhang, S.S. Ge / J. Math. Anal. Appl. 313 (2006) for all (x, u) R n R m, where i 1,...,i k {1,...,m} are istinct an >0 is a fixe constant, then there exists a unique mapping g : R n R m such that f(x,g(x)) = 0. Moreover, g is continuous. Aitionally, if f is continuously ifferentiable, then the obtaine g is also continuously ifferentiable. Note that the generalization to R k where k>mis complicate. For simplicity, assume that f : R n R m R k is a continuous mapping, continuously ifferentiable in the secon variable u R m, such that f(x,u 1,...,u m ) u f(x,u 1,...,u m ) i u, i = 1,...,m, j for all (x, u) R n R m, where >0isafixe constant. Let f(x,u)= (f 1 (x,u),...,f k (x, u)) T. Consier f = (f 1,...,f m ) T. By Theorem 1, there exists a unique mapping g : R n R m such that f(x,g(x)) = 0 an g is continuous. A problem is: Does the unique mapping g satisfy that f i (x, g(x)) = 0 for all i = m + 1,...,k? An explicit counterexample against the case of k>m is the mapping f : R 1 R 1 R 2, efine by f = (f 1,f 2 ) T where f 1 (x, u) = u x, f 2 (x, u) = x u + 1. Obviously, there is a unique function g(x) = x such that f 1 (x, g(x)) = 0. However, f 2 (x, g(x)) Controllable MIMO non-affine system Consier a multi-input multi-output (MIMO) nonlinear system Y (n) = F ( Y,Y (1),Y (2),...,Y (n 1),U ), (14) where Y R m is the measure output, U R m is the control input, Y (i) enotes the ith time erivative of the output Y (i = 1, 2,...,n 1), an F : R m(n+1) R m is an unknown nonlinear vector function. It shoul be note that, unlike most recent results, the nonlinearity F is an implicit function with respect to U. The control objective can be escribe as: Given a esire output Y (t), fin a control U such that the output of the system tracks the esire trajectory with an acceptable accuracy, while all the states an the control remain boune. This problem was iscusse for m = 1 (that is, a SISO system) in [8,9], where the one-imensional result Lemma 2 can be applie. Let X =[X 1,X 2,...,X n ]=[Y,Y (1),...,Y (n 1) ] R m n be the state matrix, where X 1,X 2,...,X n an Y,Y (1),...,Y (n 1) are all m-imensional vectors in column. We may represent system (14) in a state space moel Ẋ j = X j+1, j = 1,...,n 1, Ẋ n = F(X,U), (15) Y = X 1. The following assumptions are mae for system (15): (A1) F(X,U)is C 1 for (X, U) R m(n+1). (A2) [ U F(X,U)] ii [ U F(X,U)] for all (X, U) R m(n+1), i = 1,...,m, where >0isaefinite constant. (A3) The reference signals Y (t), Y (1) (t), Y (2) (n) (t),..., Y (t) are smooth an boune.

8 258 W. Zhang, S.S. Ge / J. Math. Anal. Appl. 313 (2006) Uner assumptions (A1), (A2), system (15) inclues the class of affine systems iscusse in [14,15,23] an the class of non-affine SISO system in [10]. Define vectors X an Υ as X = [ Y,Y (1),...,Y(n 1) Υ = X X an efine a filtere tracking error as ], X R m n, E = Υ [λ 1,λ 2,...,λ n 1, 1] T, E R m, (17) where the vector Λ =[λ 1,λ 2,...,λ n 1 ] can be chosen appropriately so that s n 1 + λ n 1 s n 2 + +λ 1 is Hurwitz an, therefore, Υ(t) 0asE(t) 0. Then, the time erivative of the filtere tracking error can be written as Ė = F(X,U) Y (n) (t) + Υ [0 Λ] T. (18) Let E =[e 1,...,e m ] T. Consier a continuous function { 1 exp( e/γ ), e > 0, sat(e) = (19) 1 + exp(e/γ ), e 0, with γ being any small positive constant. As γ 0, sat(e) approaches a step-transition from 1 at e = 0 to 1 at e = 0 + continuously. We have the following result to establish the existence of an ieal control, U, that brings the output of the system to the esire trajectory. Theorem 2. Consier system (15) satisfying assumptions (A1) (A3). There exists an ieal control input U such that Ė = k v E k v sat(e), where sat(e) := [sat(e 1 ),..., sat(e m )] T an k v is a positive constant. Subsequently, Eq. (20) leas to lim t Y(t) Y (t) =0. Proof. Plus an minus k v E + k v sat(e) to the right-han sie of the error equation (18), which gives Ė = F(X,U)+ ν k v E k v sat(e), (21) where ν is efine as ν = k v E + k v sat(e) Y (n) (t) + Υ [0 Λ] T. (22) From assumption (A2) an the fact that ν/u = 0, we know [ ] (F(X,U)+ ν) U [ ] (F(X,U)+ ν) ii U, (X, U) Rm(n+1). By our global Implicit Function Theorem (Theorem 1), there exists a continuous ieal control input U (Z) with Z =[X, ν] T R m(n+1) such that F ( X, U (Z) ) + ν = 0. Uner the action of U, (21) an (23) imply that (20) hols. (16) (20) (23)

9 W. Zhang, S.S. Ge / J. Math. Anal. Appl. 313 (2006) Let ρ = E T E = m j=1 ej 2.IfE(t) is a solution of Eq. (20), then ρ = 2E T E = 2k v ρ 2k v E T sat(e) 2k v ρ, because each e j sat(e j ) 0 an, therefore, E T sat(e) = m j=1 e j sat(e j ) 0. By Comparison Theorem [12], 0 ρ(t) ρ(0) exp( 2k v t) 0 as t +, implying that E(t) 0ast +. Hence lim t Y(t) Y (t) =0 an the proof is complete. 5. Approximation of ieal control input The contraction of map in the proof of Theorem 1 gives a metho to approximate the ieal control input U. As efine in (22), ν is a function of X an not epenent on F because both E an Υ epen on X. Once X R m n is fixe, ν is given. Moreover, γ (k v + 1) λ X X +k v + M Y (k v + 1) λ X +(k v + 1) λ nm Y + k v + M Y, (24) where λ :=max{ λ 1,..., λ n 1, 1} an M Y > 0 is the boun such that all Y, Y (1),..., Y (n 1) are less than or equal to M Y. Although we usually know less about F in control problems, sometimes it is still possible to assume that F is Lipschitzian, or strongly assume that (A4) [ U F(X,U)] M 1 for all (X, U) R m(n+1), where M 1 > 0 is a constant. Construct a sequence {U (j) (X)} such that { U(0) (X) = 0, U (j) (X) = U (j 1) (X) M (F (X, U (25) (j 1)(X)) + ν). This sequence is convergent because the map T in the proof of Theorem 1 is a contraction. As a consequence, we have lim U (j)(x) = U (X, ν). j (26) Furthermore, from (9), we have U (j+1) (X) U (j) (X) (1 M 1 +1 ) U (j)(x) U (j 1) (X). Then by inuction U (j+1) (X) U (j) (X) ( 1 ) j U (1)(X) U (0)(X). M It implies inuctively that U (j+p) (X) U (j) (X) M ( ) j { ( 1 1 ) p } U (1)(X). M M Then, letting p, by (24), we obtain the estimate of error

10 260 W. Zhang, S.S. Ge / J. Math. Anal. Appl. 313 (2006) U (X, ν) U (j) (X) M M 0 + γ ( 1 M ( 1 M (k v + 1) λ X +M ) j U (1)(X) ) j ( 1 M where M = M 0 + (k v + 1) λ nm Y + k v + M Y, provie that we assume ) j, (27) (A5) F is boune an F(X,U) M 0 for all (X, U) R m(n+1), where M 0 > 0 is a constant. If we o not know so much information of F as in assumptions (A4) an (A5), the approximation of ieal control input U given by (26) an (27) may not work. In that case, [10] suggests that a neural network [11,21] can be taken as a function approximator which emulates a given nonlinear function up to a small error tolerance, where an analysis of Lyapunov functions for semi-globally uniformly ultimate bouneness will be involve. Remark. The conition (A2) actually guarantees the existence of the ieal control input for a non-affine nonlinear MIMO system, no matter whether it is still consiere to be restrictive in practices. We hope to make a progress with the new theorem of global implicit functions in the control problem of non-affine nonlinear MIMO system or start a new approach to the goal. Acknowlegments The authors are grateful to the referees for their helpful comments an suggestions. References [1] V. Alexanrov, Implicit function theorem for systems of polynomial equations with vanishing Jacobian an its application to flexible polyhera an frameworks, Monatsh. Math. 132 (2001) [2] J. Blot, On global implicit functions, Nonlinear Anal. 17 (1991) [3] D. Briges, C. Calue, B. Pavlov, D. Ştefănescu, The constructive implicit function theorem an applications in mechanics, Chaos Solitons Fractals 10 (1999) [4] G.J. Butler, H.I. Freeman, Further critical cases of the scalar implicit function theorem, Aequationes Math. 8 (1972) [5] B.D. Craven, M.Z. Nashe, Generalize implicit function theorems when the erivative has no boune inverse, Nonlinear Anal. 6 (1982) [6] R. Datko, An implicit function theorem with an application to control theory, Michigan Math. J. 11 (1964) [7] S.S. Ge, C.C. Hang, T. Zhang, Nonlinear aaptive control using neural networks an its application to CSTR systems, J. Process Control 9 (1999) [8] S.S. Ge, C.C. Hang, T. Zhang, Aaptive neural network control of nonlinear systems by state an output feeback, IEEE Trans. Syst. Man Cybernetics Part B 29 (1999) [9] S.S. Ge, C.C. Hang, T.H. Lee, T. Zhang, Stable Aaptive Neural Network Control, Kluwer Acaemic, Boston, [10] S.S. Ge, C. Wang, Aaptive NN control of uncertain nonlinear pure-feeback systems, Automatica 38 (2002) [11] M.M. Gupta, D.H. Rao, Neuro-Control Systems: Theory an Applications, IEEE Neural Networks Council, New York, [12] J.K. Hale, Orinary Differential Equations, Wiley Interscience, New York, [13] S. Ichiraku, A note on global implicit function theorems, IEEE Trans. Circuits Systems 32 (1985) [14] M. Jankovic, Aaptive output feeback control of nonlinear feeback linearizable system, Internat. J. Contr. 10 (1996) 1 18.

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