namics Conclusions are given in Section 4 2 Redesign by state feedback We refer the reader to Sontag [2, 3] for the denitions of class K, K and KL fun

Robust Global Stabilization with Input Unmodeled Dynamics: An ISS Small-Gain Approach Λ Zhong-Ping Jiang y Murat Arcak z Abstract: This paper addresses the global asymptotic stabilization of nonlinear systems in the presence of unmodeled dynamics appearing at the input The unmodeled dynamics are restricted to be minimum-phase and relative degree zero On the basis of nonlinear small-gain arguments, we design a static feedback control law to achieve global asymptotic stabilization In the absence of full state information an observer-based control design is developed Introduction The robust stabilization problem for nonlinear systems with input unmodeled dynamics has been studied in the last decade Krstić et al [9], and Jiang et al [6] have studied linear unmodeled dynamics, and proposed redesigns that guarantee boundedness of the closed-loop signals These results have been extended to nonlinear unmodeled dynamics by PralyandWang [], and Jiang and Mareels [5] All these studies impose a small-gain condition on the unmodeled dynamics Alternative passivation redesigns by Jankovic et al [4], Krstić [8] and Hamzi and Praly [3] do not require the small-gain condition, but, instead, restrict the unmodeled dynamics to be strictly passive Arcak and Kokotović [] replaced small-gain and passivity assumptions with the less restrictive requirement that the unmodeled dynamics subsystem be stable, relative degree zero, and minimum-phase Λ This work of the rst author was supported in part by the US National Science Foundation under Grant INT- 9983 The work of the second author was supported in part by National Science Foundation under grant ECS- 982346 and the Air Force Ofce of Scientic Research under grant F4962-95--49 y Dept of Electrical Engineering, Polytechnic University, Brooklyn, NY 2, USA zjiang@controlpolyedu z Dept of Electrical & Computer Engineering, University of California, Santa Barbara, CA 936, USA murat@seideleceucsbedu For nonlinear unmodeled dynamics, studied in [2], the minimum-phase requirement is replaced by a robust stability property for the zero dynamics The redesigns of [, 2] are based on a dynamic control law which requires a priori knowledge about the stability margin of the unmodeled dynamics This paper presents a static redesign which further relaxes the restrictions on the unmodeled dynamics subsystem The key idea of the redesign is to use small-gain assignment tools developed by Jiang et al [] in combination with a change of variables which transforms the input unmodeled dynamics to state-driven unmodeled dynamics The resulting control law guarantees boundedness of closed-loop solutions, and their convergence to a compact set around the origin which can be rendered arbitrarily small If, in addition, the zero dynamics are locally exponentially stable (LES), then we guarantee global asymptotic stability (GAS) and LES for the closed-loop system In the absence of full state information an observer-based variant of the design is developed for a subclass of the systems considered An appealing feature of our redesign is that no a priori knowledge of stability margin is assumed about the underlying unmodeled dynamics While, as in previous redesigns, the unmodeled dynamics are assumed to be relative degree zero and minimum phase, in this paper they are not restricted to be stable This means that the result can be employed to simplify control designs for systems in which a certain subsystem can be treated as unmodeled dynamics Thus, the redesign serves also to enlarge the classes of systems for which standard design methods such as backstepping and forwarding are applicable In Section 2 we present our redesign via statefeedback In Section 3, we present a dynamic output-feedback design with input unmodeled dy-

namics Conclusions are given in Section 4 2 Redesign by state feedback We refer the reader to Sontag [2, 3] for the denitions of class K, K and KL functions and an input-to-state stable (for short, ISS) system A perturbed" version of ISS is given in [] and is dened as follows We use j j to mean the Euclidean norm and k kto mean the L norm Denition A nonlinear control system _x = f(x; u), with x 2 IR n and u 2 IR m, is said to be input-to-state practically stable (ISpS) if, for any bounded input u and any initial condition x(), the solution x(t) exists for every t and satises jx(t)j» (jx()j;t) + fl(kuk) + d () where is of class KL, fl is of class K and d When d =in (), ISpS becomes ISS To make the main features of our redesign more apparent we rst consider a scalar system _x = f(x)+g(x)v (2) _ο = A(ο)+bu (3) v = c(ο)+du ; (4) where ο 2 IR p is the state of unmodeled dynamics driven by the input u 2 IR It is assumed that f, g are known smooth functions with f() =, and g(x) g > ; 8 x 2 IR: (5) A( ) and c( ) are unknown functions vanishing at the origin, b is an unknown vector and d is an unknown constant The stability properties to be analyzed are with respect to the origin (x; ο) = (; ), which is an equilibrium for the open-loop system (2)-(4) The nominal system _x = f(x)+g(x)u is globally asymptotically stabilizable (GAS) by a smooth control law u = ff(x) Our task is to redesign this nominal control law tomake it robust against the destabilizing effect of the unmodeled dynamics Thus, a new control law u = #(x) is to be designed to guarantee GAS for the system (2)-(4) The admissible unmodeled dynamics are characterized by the following assumptions: (A) The unmodeled dynamics subsystem has relative degree zero, that is, d 6= Furthermore, the sign of d is known, say, d> (A2) There exists a known smooth function ^c( ) of class K such that jc(ο)j» ^c(jοj) ; 8ο 2 IR p : (6) Our nal assumption requires a robust stability property for the zero dynamics of the ο-subsystem (3)-(4), that is _z = A(z) d bc(z) =:A (z): () (A3) The zero-dynamics subsystem () disturbed by (w ;w 2 ) _z = A (z + w ) + w 2 (8) is ISS with respect to the input (w ;w 2 ), that is, there exist class K functions fl ( ) andfl 2 ( ) and a class KL function ( ; ) such that the solutions of (8) satisfy jz(t)j» (jz()j;t)+fl (kw k)+fl 2 (kw 2 k) (9) Remark Clearly, (A3) is satised for linear unmodeled dynamics A(ο) = Aο, c(ο) = cο that are minimum-phase and relative degree zero Proposition Under Assumptions (A), (A2) and (A3), there exists a smooth function #(x) such that the system (2) in closed-loop with u = #(x) +~u is ISpS with ~u as input Furthermore, if the rst-order approximation of _z = A (z) is asymptotically stable at z =,then we can render the closed-loop system ISS, and at the same time, LES when ~u = Proof Let b := b, and introduce the variable d μο = ο b Z x which, from (2)-(3), is governed by ds () g(s) Z _μο = A ( ο μ x + b g(s) ds) b f(x) g(x) : () Substituting () into the x-subsystem (2) yields Z _x = f(x)+g(x) du + c( ο μ x ds) + b g(s)

Thus, we have converted the input unmodeled dynamics ο-subsystem into system () driven only by x This important observation allows us to use gain assignment theorems [] to design a robust controller u = #(x) We rst consider the system (), with x regarded as its input Because of (A3), the ο-system μ R is ISS with respect to the inputs w = b x ds and f (x) w 2 = b From (5), we have g(x) b Z x g(s) g(s) ds» jb xj (2) g f(x) b g(x)» jb j ^f(jxj) (3) for some smooth, class K function ^f( ) The inequalities (2) and (3) together with (A3) imply that the solutions μ ο(t) of()satisfy j μ ο(t)j» (j μ ο()j;t) + fl (kxk) (4) where fl (kxk) = fl ( jb j g kxk) + fl 2 (jb j ^f(kxk) From (4) and (2), it follows that the ο-system μ is also input-to-output stable (IOS) from the input x to the output y = c( ο μ R + b x precisely, ds) g(s) More jy (t)j» ios (j μ ο()j;t) + fl ios (kxk) (5) where ios (jο()j;t) μ = ^c(2 (jο()j;t)) μ and fl ios (kxk) = ^c(2fl (kxk) + 2jb j g kxk) Now, consider the x-system (2), rewritten as _x = f(x)+g(x)(du + y ) : (6) The control law u = #(x) +~u will be chosen in such a way that the x-system is ISS with respect to the new inputs y and ~u, that is, there exists aclasskl function ( ; ) and a class K function fl x ( ) suchthat jx(t)j» x (jx()j;t)+fl x (ky k)+fl x (dk~uk): () The ISS gain fl x ( ) that will be assigned by the control law u = #(x)+~u is to satisfy the small-gain condition 2fl x f 2fl ios (s)» s for all s s, or, equivalently, 2fl ios (2s)» fl x (s) ; 8 s :5s : (8) When s > in (8), fl x ( ) can be taken as a smooth, class K function that is linear around zero Let ff(x) = ( kx f(x))=g(x) with k > The time derivative of V (x) = 2 x2 along solutions of (6) satises _V = kx 2 + xg(x)(du + y ff(x)) : (9) Let ^ff be a smooth function satisfying that jff(x)j»jxj^ff(x) and let» be a positive constant such that»d Choose the following smooth control law #(x) =»x^ff(x) 2fl x (jxj)sign(x) : (2) Then, substituting u = #(x)+~u in (9) gives _V» kx 2 + jxjg(x) jy + d~uj 2fl x (jxj) : (2) Because of Sontag's ISS algorithm [2], the ISS property () follows from (2) Thanks to the small-gain condition (8) between fl ios and fl x, the proof of the rst statement in Proposition is completed with the help of Theorem 2 in [] The second statement can be proved as in [, Corollary 23] by taking advantage of the LES condition on the rst-order approximation of _z = A (z) Indeed, using the LES property of the linearization of _z = A (z), a direct application of [, Lemma A2] yields that fl and fl 2 can be picked as K functions being linearly bounded near the origin This implies that fl and fl ios are linearly bounded near the origin Hence, (8) holds with s =, leading to the desired ISS property Finally, the LES property of the entire closed-loop system follows from the fact that the linearization of the entire closed-loop system is asymptotically stable for sufciently large k and» 2 Corollary If the unmodeled dynamics are linear, minimum-phase and relative degree zero, then there exists a robust controller u = #(x) that achieves GAS and LES for the system (2) We now proceed with higher-dimensional systems of the form _X = F (X; x) _x = f(x; x)+g(x; x)v (22) _ο = A(ο)+bu v = c(ο)+du

where X 2 IR n, x 2 IR, ο 2 IR p is the state of the unmodeled dynamics system, g(x; x) g > for all (X; x), and A( ), b, c( ) and d satisfy Assumptions (A)-(A3) Proposition 2 Consider system (22) If there exists a smooth function ff( ), with ff() =, such that _X = F (X; ff(x)) is GAS at X =, then there exists a robust controller #(X; x) such that the entire system in closed-loop with u = #(X; x) +~u is ISpS with respect to the input ~u Furthermore, if the rst-order approximation of both _X = F (X; ff(x)) and _z = A (z) are LES at the origin, then we can render the closed-loop system ISS, and at the same time, LES when ~u = Proof From Theorem in [3], the GAS property for the X-system implies the existence of a smooth real-valued function K( ), K() =, and a smooth globally invertible function, with < jg(x)j», such that _X = F (X; K(X) + u) is ISS with input u As in the proof of Proposition, we introduce a change of variables x = x K(X) (23) Z x μο = ο b ds: (24) g(x; s) The mapping (X; x; ο)! (X; x; ο) μ is a global diffeomorphism preserving the origin In the new coordinates (X; x; ο), μ the system (2) is rewritten as _X = F (X; K(X)+x) _x = f (X;K(X)+x) @ @X ( ) @ @X ( K(X) +[(K(X)+x) i ) F (X; K(X)+x)+ g(x;k(x)+x) ψ Z du + c( ο μ K(X)+x! ds + b ) g(x;s) ψ Z K(X)+x _μο = A μο + b b f (X;K(X)+x) g(x;k(x)+x) := A ( μ ο + w ) + w 2 ds g(x;s) Thanks to (A3), the above μ ο-system is ISS with respect to the inputs (w ;w 2 )! Then, it follows that the ο-subsystem μ is ISS with X and x considered as inputs For notational simplicity, denote z = (X; ο) μ Thus, as a cascadeinterconnection of two ISS systems, the z-system is ISS with respect to the input x That is, there exist a class KL function z ( ; ) and a class K function fl z ( ) such that jz(t)j» z (jz()j;t)+fl z (kxk): (25) The rest of the proof is similar to the proof of Proposition 2 3 Redesign by output-feedback In this section, we deal with a class of dynamically input-perturbed output feedback systems which are described by _z = q(z;y) _x = x 2 + f (z;y) _x n = v + f n (z;y) _ο = A(ο)+bff(y)u v = c(ο)+dff(y)u y = x (26) where y = x 2 IR is the output, z 2 IR p, x 2 IR n, ο 2 IR p 2,andA( ), b, c( ) and d are as in (2), and fulll Assumptions (A2) and (A3) We assume that ff :IR! IR is a globally invertible function It is further assumed that (A) The z-subsystem of (26) is ISS when y is considered as its input (A8) For each» i» n, there exist two known smooth, class K functions ' i ( ) and ' i2 ( ) such that jf i (z;y)j» ' i (jzj) + ' i2 (jyj) : (2) Our control task is to nd a dynamic outputfeedback lawofthetype u = #(y; χ) ; _χ = $(y; χ) (28) which globally asymptotically stabilizes (26) at the origin We treat the cases n = and n > separately When n =,a slight modication of the design in the previous section yields a static

output-feedback control law u = #(y) This result is stated below, whose proof is omitted due to the page limitation Theorem Under Assumptions (A2), (A3), (A) and (A8), if there is a known constant ff > such that ff(y) ff for all y, then we can nd a smooth static output-feedback law u = #(y) that drives the output y to an arbitrarily small neighborhood of the origin while guaranteeing the boundedness of all closed-loop signals Furthermore, if the rst-order approximation of both _z = q(z;) and ο _ μ = A ( ο) μ are asymptotically stable, then the origin of the closed-loop system is GAS When n>, output-feedback global stabilization is achieved by restricting the unmodeled dynamics subsystem to be linear (A9) There exist a constant matrix μ A and a constant vector μc such thata(ο) = μ Aο and c(ο) =μcο In addition, d> and μ A bμc=d is an asymptotically stable matrix We further assume that ff(y) is known, and, without loss of generality, ff(y) = Theorem 2 Under Assumptions (A), (A8) and (A9), for the system (26) with n> and ff(y) =, we can nd a smooth dynamic output-feedback law (28) that drives the output y to an arbitrarily small neighborhood of the origin while guaranteeing the boundedness of all closed-loop signals Furthermore, if the rst-order approximation of _z = q(z;) is asymptotically stable, then the origin of the closed loop system (26) and (28) is GAS Proof Our design starts with the change of variables ο μ = P n ο i= i x i with n = b and d i = A i μ 8i = 2;:::;n: Then, direct computation yields ψ! nx _μο =(μ A bμc=d) ο+ μ μa y ( n μc + I p ) i f i (z;y) i= where I p is the p p identity matrix Hence, the system (26) can be rewritten as _z = q(z;y) _x = Fx+ dg(u +μc μ ο=d)+f(z;y) _μο = ( μ A bμc=d) μ ο +( μ A y P n i= ( n μc + I p ) i f i (z;y)) y = x (29) where f = (f ;:::;f n ) T, F and G are constant matrices dened as F = 2 6 4 3 I n 5 ; G = μc μc 2 μc n 2 6 4 3 5 (3) Dening ex = d Px with P = (p ij) a lowertriangular matrix of order n satisfying that p ii = and p ij =foreach» i» n and j>i, it is not difcult to show that the values of p ij, i>j, can be chosen so that Letting PFP = 2 6 4 p 2 ψ i (z;y) = p i+; y + ψ n (z;y) = μc y + 3 p n I n 5 : (3) μc nx j= 8» i» n, it follows that ix j= p ij f j (z;y) (32) p nj f j (z;y) (33) _ex = A c ex + G(u +μc μ ο) + ψ(z;y)=d (34) where A c is the n n matrix with ones on the upper-diagonal and zeros elsewhere Note that the nominal part of the ex-system (34) is in the controllable canonical form Also note that ex = y=d We follow the global stabilization algorithm in [] to introduce the observer-like dynamic system _b~x i = b ~xi+ + `i(y b ~x ) ;» i» n _b~x n = u + `n(y b ~x ) (35) where the design parameters `i are chosen in such a way that the observation error e = ex b ~x (ie each component ise i = ex i b ~xi ) satises _e = Me L( d )y + Gμcμ ο + ψ(z;y)=d (36) with M an asymptotically stable matrix and L = (`;:::;`n) T In view of (29), (34), (35) and (36), we obtain the following system, which is in a suitable form for our robust control redesign

_z = q(z;y) _μο = ( A μ bμc=d) ο μ +( μ A y P n i= ( n μc + I) i f i (z;y)) _e = Me L( d )y + Gμcμ ο + ψ(z;y)=d _y = db ~x2 + de 2 + ψ (z;y) _b~x 2 = b ~x3 + `2( d )y + `2e _b~x n = u + `n( d )y + `ne : (3) To complete our robust redesign using gain assignment theorems of [], it is important to note that the information of partial state (y;b ~x2 ;:::;b ~xn ) is available to the designer and that the cascaded (z; μ ο;e)-system is ISS when y is considered as the input With these remarks in mind, a repeated application of Proposition gives a robust nonlinear controller #(y;b ~x2 ;:::;b ~xn ) that solves the global output feedback stabilization problem 2 4 Concluding remarks We have presented a redesign that achieves robust stabilization in the presence of input unmodeled dynamics The class of unmodeled dynamics is restricted to be relative degree zero, and to have ISS zero dynamics An interesting feature of our redesign is that, in contrast to previous results, it does not require any a priori knowledge of stability margin on the unmodeled dynamics (In fact, unmodeled dynamics are not required to be stable) The use of our small-gain redesign in conjunction with the observer design in [] led to an output-feedback scheme that ensures robustness against input unmodeled dynamics A new research task would be to compare the closed-loop performance of our static redesign with that of the dynamic redesign in [2] References [] M Arcak and P V Kokotović Further results on robust control of nonlinear systems with input unmodeled dynamics ACC'99, pp 46-465, 999 [2] M Arcak and P V Kokotović Robust nonlinear control of systems with input unmodeled dynamics Syst & Contr Lett, 2 [3] B Hamzi and L Praly, Ignored input dynamics and a new characterization of control Lyapunov functions, ECC'99, Karlsruhe, 999 [4] M Janković, R Sepulchre and P V Kokotović, CLF based designs with robustness to dynamic input uncertainties, Systems and Control Letters, Vol 3, pp 45 54, 998 [5] Z P Jiang and I Mareels A small-gain control method for nonlinear cascaded systems with dynamic uncertainties IEEE Trans Automat Control, 42, pp 292-38, 99 [6] Z P Jiang, I Mareels and J-B Pomet Controlling nonlinear systems with input unmodeled dynamics Proc 35th IEEE Conf Decision Contr, pp 85-86, 996 [] Z P Jiang, A Teel and L Praly Smallgain theorem for ISS systems and applications Mathematics of Control, Signals and Systems, vol, pp 95 2, 994 [8] M Krstić, Stability margins in inverse optimal input-to-state stabilization, ACC'98, pp 648 652, 998 [9] M Krstić, J Sun and P V Kokotović Robust control of nonlinear systems with input unmodeled dynamics IEEE Trans Automat Control, vol 4, no 6, pp 93 92, 996 [] L Praly and ZP Jiang Stabilization by output feedback for systems with ISS inverse dynamics Syst Contr Lett, 2, 9-33, 993 [] L Praly and Y Wang Stabilization in spite of matched unmodeled dynamics and an equivalent denition of input-to-state stability Mathematics of Control, Signals, and Systems, 9, pp -33, 996 [2] E D Sontag Smooth stabilization implies coprime factorization IEEE Trans Automat Control, 34, pp 435-443, 989 [3] E D Sontag Further facts about input to state stabilization IEEE Trans Automatic Control, 35, pp 43 46, 99