Prescribed Performance Output Feedback Adaptive Control of Uncertain Strict Feedback Nonlinear Systems

Transcription

1 Milano (Italy August 8 - September, 11 Prescribed Performance Output Feedback Adaptive Control of Uncertain Strict Feedback Nonlinear Systems Artemis K. Kostarigka, George A. Rovithakis Dept. of Electrical & Computer Engineering Aristotle University of Thessaloniki 5414, Thessaloniki, Greece akost@auth.gr, robi@eng.auth.gr Abstract: An output feedback prescribed performance control design is presented for a class of strict feedback uncertain nonlinear systems. To overcome state unavailability an estimator-like system is introduced, which generates rough estimates of the unknown signals, thus avoiding the demanding task of constructing asymptotically convergent observers. Moreover this control approachavoidsposingsevererestrictionsontheunknownsystemnonlinearitiesotherthantheir continuity. To effectively avoid possible division by zero, the designed output feedback controller is of switching type. However, its continuity is guaranteed, thus alleviating any problems related to the existence and uniqueness of solutions. Simulations on a single-link manipulator actuated by a brush dc motor benchmark example illustrate the approach. 1. INTRODUCTION Output feedback control of uncertain nonlinear systems is a challenging task that has gained a lot of attention over the past decades. Giventhat the so-called separation principle is not directly applicable, combined observercontroller approaches were developed, incorporating different kinds of observers for each individual class of nonlinear systems. Available output feedback results refer to: systems in normal form (Ge and Zhang (3; Kim and Calise (7; Freidovich and Khalil (7; Marconi and Praly (8; Andrieu and Praly (8; Du and Chen (9, nonlinear systems with output dependent nonlinearities (Krishnamurthy et al. (; Krishnamurthy and Khorrami (3; Karagiannis et al. (5; Fu (9 and nonlinear systems with state dependent nonlinearities which are either globally Lipschitz or satisfy certain growth conditions (Kaliora et al. (6; Gong and Qian (7; Lei and Lin (7; Andrieu et al. (9b,a. Distinctive cases also exist where specially designed nonlinear observers apply (Maggiore and Passino (5; Abdollahi et al. (6; Stepanyan and Hovakimyan (7; Ding(9.Themaindifficultyofobserver-basedcontrol methodologies is the requirement of achieving convergence to zero of the observation error, a task which gets even more challenging with increased system complexity. Additionally, guaranteeing specific performance characteristics on system response utilizing output feedback, is a highly interesting issue. Under the assumption of full state measurement a novel design procedure, called prescribed performance control (PPC was recently developed (Bechlioulis and Rovithakis (8, 9, 1 to introduce prescribed performance measures both on transient and steady state behavior of system output. Specifically, the output is guaranteed to converge to a predefined arbitrarily small residual set, with rate no less than a prespecified value, exhibiting a maximum overshoot less than a sufficiently small preassigned constant. First results on output feedback PPC were obtained in Kostarigka and Rovithakis (1, assuming strict feedback passivity. In this paper we overcome the restrictions originated from the strict output feedback passivity assumption and present an output feedback PPC design, applicable to uncertain strict feedback nonlinear systems. Besides guaranteeing prescribed performance attributes on the system output, additional novel aspects of this work include a no severe restrictions on the unknown system nonlinearities other than their continuity are posed and b the construction of an asymptotically convergent observer is not required to establish system performance. Alternatively, any partial system knowledge is incorporated to design a state estimator-like system provided that the nominal system formulated in this way constitutes a stabilizable system. A key property of the state estimator system is the generation of only rough estimates of the unavailable states, thus substantially reducing the complexity of the overall design. All the uncertain terms appearing in our analysis are modeled with the use of neural network approximators, which have been proven very efficient in nonlinear systems identification and control (Narendra and Parthasarathy (199; Sanner and Slotine (199; Chen and Liu (1994; Chen and Khalil (1995; Jaganathan and Lewis (1996; Ge and Hang (1996; Liu and Chen (1993; Yesildirek and Lewis (1995; Polycarpou and Ioannou (199; Polycarpou (1996; Rovithakis and Christodoulou (1994, ; Rovithakis (1, 4; Kostarigka and Rovithakis (9. Copyright by the International Federation of Automatic Control (IFAC 65

2 Milano (Italy August 8 - September, 11 To avoid possible division by zero, the proposed controller is of switching type. However, efforts have been devoted to guaranteeing its continuity, thus alleviating problems connectedtoexistenceofsolutionsandchatteringphenomena. The paper is organized as follows: The Prescribed Performance Output Feedback Control (PPOFC problem is statedinsection,whilesection3isfocusedonthedesign of an adaptive output feedback neural network controller, capable of guaranteeing prescribed performance of the output tracking error, as well as the boundedness of all other closed loop signals. Simulation results on a singlelink manipulator actuated by a brush dc (BDC motor benchmark example are provided in Section 4. Finally, we conclude in Section 5.. PROBLEM STATEMENT Let us consider the SISO system (Σ: ẋ i =(f i ( x i +δf i ( x i +(g i ( x i +δg i ( x i x i+1,i=1,...,n 1 ẋ n =(f n ( x n +δf n ( x n +(g n ( x n +δg n ( x n u y=x 1 where x i := [x 1... x i ] R i,i = 1,...,n is the state, y R is the measured output, u(t R is the control input, f i ( x i, g i ( x i, i = 1,,...,n, are locally Lipschitz, known functions, while δf i ( x i,δg i ( x i, i = 1,,...,n, are uncertain terms denoting the lack of the exact system knowledge.thestatex = x n isassumednottobeavailable for measurement. System (Σ can be written in compact form as ẋ = Ax+f(x+ f(x+(g(x+ g(xu y = Cx wheref(x = f (x Ax,withf (x = [f 1 ( x 1 +g 1 ( x 1 x,..., f n ( x n ] and A R n n a constant matrix. Moreover f(x = [δf 1 ( x 1,..., δf n ( x n ],g(x = [...1] g n ( x n R n, g(x = [...1] δg n ( x n R n, C = [1...] R n. Assumption 1. The nominal system (Σ n of (1, with f(x = g(x =, is state feedback stabilizable and the pair (A,C is observable. Our goal is to design a controller utilizing only measurable signals, capable of regulating the output tracking error e(t = y(t y r (t to a neighborhood of zero with prescribed performance, while guaranteeing the uniform boundedness of all other signals in the closed loop. For the desired output trajectory y r (t we assume that it is bounded with bounded first order time derivatives. To overcome the unavailability of x(t a state estimatorlike system is constructed incorporating the nominal system knowledge. Specifically we define: ˆx = Aˆx+f(ˆx+g(ˆxu+K(y ˆx 1 ( with K R n a constant gain vector. Since (A,C is an observable pair, there exists a constant gain K to guarantee that Ā = A KC is a Hurwitz matrix. Moreover, since (Σ n is stabilizable, the existence of a CLF Vˆx (ˆx and a continuous control law u (ˆx is concluded, such that Vˆx (ˆx [Aˆx+f(ˆx+g(ˆxu (ˆx] (3 ˆx (1 As it will be clarified in the next section, ˆx as produced from ( is not required to accurately estimate the unknown x. Instead only a very rough estimation suffices. This is why ( is defined as a state estimator-like system. The Prescribed Performance Output Feedback Control (PPOFC problem 1 for (Σ is solvable if there exist a possibly parameterized output feedback control law to render the states x(t,ˆx(t of (1, ( bounded and the output tracking error e(t to satisfy λ l ρ(t e(t λ u ρ(t, t (4 for some constants λ l,λ u satisfying M < λ l < λ u < 1,in case e( (5 1 < λ l < λ u < M,in case e( (6 wherem isaconstantdefinedinbechlioulisandrovithakis (8. To proceed let us define x = x ˆx and the corresponding error system x=ā x+f(x f(ˆx+ f(x+(g(x g(ˆxu+ g(xu (7 To achieve prescribed performance we consider the error transformationε(t = T (BechlioulisandRovithakis ( e(t ρ(t (8 and differentiate it with respect to time to obtain ε=r((f 1 ( x 1 +δf 1 ( x 1 +(g 1 ( x 1 +δg 1 ( x 1 x ν (8 where r = ( T ρ 1 e ρ and ν = ẏ r + e ρ ρ. Lets consider the augmented system (Σ a that consists of (, (7 and (8. Following the arguments originally stated in Bechlioulis and Rovithakis (8, the following theorem introduces necessary and sufficient conditions for the PPOFC problem. Theorem 1. Consider system (1 with a stabilizable nominal system (Σ n and its corresponding, well defined, augmented system (Σ a. The PPOFC problem for (1 admits a solution if and only if there exists a continuous output feedback control law such that all closed loop signals of (Σ a are uniformly bounded. The proof of Theorem 1 is omitted, but can be easily obtained using the same steps as in Theorem of Bechlioulis and Rovithakis (8. 3. MAIN RESULTS Let s consider the following Lyapunov candidate function for the augmented system (Σ a. V( x,ˆx,ε = 1 x P x+vˆx (ˆx+ 1 ε (9 wherep isapositivedefinitematrixandvˆx (ˆxisaCLFfor (. Differentiating V( x,ˆx,ε along the solutions of (Σ a yields: V( x,ˆx,ε= 1 x Q x+ x P f(x f(ˆx+ f(x+g(xu g(ˆxu+ g(xu+ Vˆx ˆx [Aˆx+f(ˆx+g(ˆxu+K(y ˆx 1] +εr(f 1 ( x 1 +δf 1 ( x 1 +(g 1 ( x 1 +δg 1 ( x 1 x ν (1 where Q is a positive definite matrix satisfying Ā P + PĀ = Q. 1 Due to page limitation prescribed performance preliminaries are omitted. However, detailed description can be found in Bechlioulis and Rovithakis (1. 651

3 Milano (Italy August 8 - September, 11 Thetermεr(f 1 ( x 1 +δf 1 ( x 1 +(g 1 ( x 1 +δg 1 ( x 1 x ν can be trivially decomposed as follows: εr(f 1 ( x 1 +δf 1 ( x 1 +(g 1 ( x 1 +δg 1 ( x 1 x ν = f k (x 1,ˆx,ε+ ( κ 1 +κ x + k (x 1 (11 where f k (x 1,ˆx,ε = εr(f 1 ( x 1 +g 1 ( x 1 ˆx ν + ˆx + 1 (εr 4 + (εr 8κ (1+ g1( x1 is a term containing known κ 1 quantitiesofmeasurablearguments, k (x 1 = ( 1 δf1 (x 1 + δg 1 (x δg 4κ 1 (x 1 4 is an uncertain term and κ 1,κ are positive design constants. Using (11 and defining fg (x,ˆx,u = f(x f(ˆx + f(x + (g(x g(ˆxu + g(xu, V becomes: V( x,ˆx,ε q x + x P fg (x,ˆx,u+ Vˆx ˆx [Aˆx +f(ˆx+g(ˆxu+k(y ˆx 1 ]+f k (x 1,ˆx,ε+ k (x 1 (1 where q = q κ 1 κ, with q the minimum eigenvalue of Q. From this moment on, in order to simplify its notation, we will drop the arguments of fg (x,ˆx,u. Like this, using the identity x P fg κ 3 x + 1 P 4κ fg and defining 3 q = q κ 3, (1 becomes: V( x,ˆx,ε q x + 1 4κ P fg +f k (x 1,ˆx,ε 3 + Vˆx ˆx [Aˆx+f(ˆx+g(ˆxu+K(y ˆx 1]+ k (x 1 Adding and subtracting the term Vˆx ˆx g(ˆx u (ˆx, where u is the controller defined via (3, we obtain V( x,ˆx,ε q x +F(ˆx,y,ε+G(ˆxu+H(ˆx,y+ P fg 4κ 3 where F(ˆx,y,ε = Vˆx ˆx [K(y ˆx 1 g(ˆxu ] + k (x 1 +f k (x 1,ˆx,ε, G(ˆx = Vˆx ˆx g(ˆx and H(ˆx,y = Vˆx ˆx [Aˆx +f(ˆx+g(ˆxu ]arecontinuousscalarfunctions.however, owing to (3 H(ˆx,y. Therefore, V becomes V( x,ˆx,ε q x +F(ˆx,y,ε+G(ˆxu+ P fg 4κ 3 (13 SinceF(ˆx,y,ε,G(ˆxcontainunknownterms,theideaisto approximate them by suitable neural network structures. Applying the neural network Density Property (Cybenko (1989weconcludetheexistenceofconstantbutunknown weightvectorswf RL 1,WG RL,continuousregressor terms Z F (ˆx,y,ε,Z G (ˆx and ω F (ˆx,y,ε,ω G (ˆx denoting the approximation errors such that: F(ˆx,y,ε = WF Z F (ˆx,y,ε+ω F (ˆx,y,ε G(ˆx = WG (14 Z G (ˆx+ω G (ˆx (ˆx,y,ε Ω ac Ωˆx Ω x Ω ε R n R n R where Ωˆx,Ω x,ω ε are some arbitrary compact sets. Such a substitution is possible owing to the smoothness and the boundedness of F(ˆx,y,ε, G(ˆx, (ˆx,y,ε Ω ac. Moreover, on the generic compact set Ω ac the approximation errors can be suitably bounded as ω F (ˆx,y,ε ω F, ω G (ˆx ω G, where ω F, ω G are some unknown bounds. 3.1 Controller design Consider the following controller u(t = u σ (t, with u σ (t= b (ˆx,ŴG a(ˆx,y,ε,ŵf +η( ε, ˆx +δ u d σ (15 which switches according to a switching strategy σ(t, taking values in I,II with d I = b(ˆx,ŵg,d II = δb. In (15 a(ˆx,y,ε,ŵf = Ŵ F Z F(ˆx,y,ε R, b(ˆx,ŵg = ŴG Z G(ˆx R, while η( is a K function and δ u,δ b > are design constants. Furthermore, Ŵ F,ŴG are estimates of WF,W G defined in (14 which are generated through the update laws: Ŵ F = (ŴF +ϕ F (j +Z F (ˆx,y,ε (16 Ŵ G = ŴG +Z G (ˆxu (17 In (16 ϕ F (j is a strictly increasing sequence satisfying lim j ϕ F (j, with the index j =,1,,..., denoting the number of times the following equality is satisfied: H = a(ˆx,y,ε,ŵf+η( ε, ˆx + δ u c j+1 b(ˆx,ŵg a(ˆx,y,ε,ŵf +η( ε, ˆx +δ u δb = (18 while σ(t = II with c > 1. Moreover, ϕ F (j = when σ(t = I. Before we proceed and analyze the proposed control law let us first present the switching strategy σ(t. Switching Strategy.WedefinethesetsR 1 = b(ˆx,ŵg δ b and R = (ˆx,ŴG : (ˆx,ŴG : b(ˆx,ŵg > δ b. We initiate at (ˆx(,ŴG( R and allocate σ(t = I whenever (ˆx,ŴG R. The switching variable σ(t remains in that state until (ˆx,ŴG R 1, at which time σ(t = II. Consequently, σ(t remains unaltered until (ˆx,ŴG R. Summarizing, the switching strategy is defined as follows: I, if (ˆx, Ŵ σ(t = G R (19 II, if (ˆx,ŴG R 1 Lemma. The controller (15-(19 with strictly positive and bounded regressors Z F (ˆx,y,ε >, guarantee (ˆx,y,ε Ω ac, where Ω ac is some compact set: 1 H whenever u = u II ; the sequence ϕ F (j = c j with c > 1 is bounded. Remark: Lemma proves that H whenever u = u II and that j is finite, thus eliminating any issues related to the existence and uniqueness of the solution of (16 that may be caused owing to its discontinuous right-hand side (Polycarpou and Ioannou (199. Moreover,j is piecewise continuouswithrespecttotime.hence,thesolutionof(16 exists and is continuous, which together with the fact that u I = u II at b(ˆx,ŵg = δ b, lead to the continuity of u. Lemma 3. Thecontinuouscontroller(15-(19guarantees, (ˆx,y,ε Ω ac, that ŴF,ŴG,u(t L. Both lemmas can be proved following the analysis presentedinkostarigkaandrovithakis(9.therefore,the actual proofs are omitted. 3. Output feedback stabilization analysis Let us consider the Lyapunov like function L( x,ˆx,ε, W F, W G = V( x,ˆx,ε+ 1 W F + 1 W G ( 65

4 Milano (Italy August 8 - September, 11 where V( x,ˆx,ε is defined in (9 and W F, W G are parameter errors defined as W F = ŴF W F, W G = ŴG W G. The following proposition is necessary. Proposition 4. Whenever x Ω x R n with Ω x an arbitrarily large compact set and (ˆx,y,ε Ω ac, the term fg (x,ˆx,u is upper bounded as follows fg (x,ˆx,u b where b > is an unknown, not necessarily small constant. Proof. Since f(,g(,δf(,δg( are continuous functions, they will be bounded for bounded arguments. Moreover, owing to Lemma 3, u L (ˆx,y,ε Ω ac. Thus, whenever x Ω x, (ˆx,y,ε Ω ac : fg (x,ˆx,u f(x + f(ˆx + f(x +( g(x + g(ˆx u + g(x u b (1 where b is an unknown, not necessarily small constant. The properties of the proposed controller are summarized in the next theorem. Theorem 5. Consider (1 with (Σ n satisfying Assumption 1 and the corresponding augmented system (Σ a. Whenever x Ω x and (ˆx,y,ε Ω ac, the continuous output feedback controller (15-(19 solves the PPOFC problem for (Σ. Proof. According to Theorem 1 we need to establish the boundedness of ( x,ˆx,ε. In that respect wedistinguish two cases. Case1(σ(t = I : Differentiating(alongthesolutions of (Σ a and using (13,(14,(1 we obtain: L q x +WF Z F (ˆx,y,ε+WG Z G (ˆxu +ω(ˆx,y,ε,u+ W F W F + W G W G where ω(ˆx,y,ε,u = ωf (ˆx,y,ε + ω G (ˆxu + 1 P κ 3 b. UsingLemma3andemployingthe neuralnetworkdensity Property we conclude that ω(ˆx,y,ε,u W, (ˆx,y,ε Ω ac, where W is an unknown constant. To continue, after adding and subtracting the term Ŵ F Z F(ˆx,y,ε+Ŵ G Z G(ˆxu, L becomes: L q x +Ŵ F Z F (ˆx,y,ε+Ŵ GZ G (ˆxu+W + W F WF Z F (ˆx,y,ε + W G WG Z G (ˆxu Employing the update laws (16-(17 with ϕ F (j = and using the identity W Ŵ = 1 W + 1 Ŵ 1 W, we obtain: L q x +Ŵ F Z F (ˆx,y,ε+Ŵ GZ G (ˆxu+µ I 1 W F 1 ŴF 1 W G 1 ŴG where µ I = W + 1 W F + 1 W G. Using the definitions of a(ˆx,y,ε,ŵf,b(ˆx,ŵg, L becomes: ( L q x + a(ˆx,y,ε,ŵf +b(ˆx,ŵgu W FG +µ I where W ( FG = W F, W. G Applying u(t = ui (t, with u I as defined in (15 with d I, and dropping the negative term δ u, L yields: L q x η( ε, ˆx 1 W FG +µ I ( Hence, L whenever either ( ˆx(t, ε(t > η 1 (µ I or x(t > q 1 (µ I, from which we conclude the boundedness of ( x,ˆx,ε. Case (σ(t = II : Using the weight update laws (16, (17 with ϕ F (j = c j, c > 1 and following similar steps as in Case 1, we arrive at: L q x + ( a(ˆx,y,ε,ŵf+b(ˆx,ŵgu W FG +µ II wherenowµ II = W+ 1 W F + 1 W G + W F ϕ F (N.Inthe expressionofµ II,theconstant W F representstheunknown bound of W F which exists owing to Lemma 3, while N is the maximum j index value which is bounded according to Lemma. If we apply u(t = u II (t, with u II as defined in (15 with d II, we obtain: L q x + a(ˆx,y,ε,ŵf b(ˆx,ŵg a(ˆx,y,ε,ŵf [ ] b(ˆx,ŵg η( ε, ˆx +δu δb W FG +µ II To proceed, notice from Lemma that H. Hence L becomes: L q x η( ε, ˆx 1 W FG +µ II (3 from which we conclude, as in Case 1, the boundedness of ( x,ˆx,ε. Additionally, using (, (3 we arrive at L q x η( ε, ˆx 1 W FG +µ (4 where µ = max(µ I,µ II, from which we conclude that x(t,ˆx(t,ε(t as well as the parameter errors WF, W G, are uniformly ultimately bounded with respect to the sets: E x = x R n : x(t q 1 (µ, Eˆx = ˆx R n : ˆx(t η 1 (µ, E ε = ε R : ε(t η 1 (µ and E W = ( WF, W G ( R L 1 R L : W F, W G µ respectively, thus establishing the uniform boundedness of ( x,ˆx,ε. The results presented are valid x Ω x and (ˆx,y,ε Ω ac, where Ω ac is the compact set in which the approximation capabilities of the linear-in-the-weights neural networks hold and Ω x R n is an arbitrarily large compact set. Such a result can be readily obtained following similar steps an in Bechlioulis and Rovithakis (1. The proof is omitted. 4. SIMULATION RESULTS To illustrate the proposed control scheme, we consider the single-link manipulator actuated by a brush dc (BDC motor benchmark example, whose dynamics is expressed as follows: M m q +C m q +N m sin(q = I (5 L I = RI K B q +V where q R is the link angular position, q R its velocity and q R its acceleration, while I,V R are the motor current and input control voltage respectively. Moreover for the system parameters we consider M m = 1,C m = 1,N m = 1,L =.5,R =.5 and K B = 1. δ b 653

5 Milano (Italy August 8 - September, x q(t (rad / qr(t (rad e(t (rad V (Volt time (sec time (sec time (sec (a (b (c Fig. 1. Performance of the adaptive PPOFC scheme: (a Link angular position (solid line along with desired trajectory (dashed line, (b Output tracking error (solid line along with performance bounds (dashed lines and (c Control effort. Let the system s input and output be the motor voltage V and link angular position q respectively. Using the state variables x 1 = q,x = q and x 3 = I, system (5 can be expressed in the strict feedback form (1 with y = Cx = [1 ]x. We assume that the known parts of the considered system are f(x = [ ] and g(x = [ 1], while the uncertain terms are f(x = [, C M x N M sin(x 1, R L x 3 K B L x ], g(x = [,, 1 L 1] All the known system parts are included in the following linear state estimator-like system: where A = ˆx = Aˆx+Bu+K(x 1 ˆx 1 (6 ], B = [ 1], K = [K 1 K K 3]. [ 1 1 It is not difficult to verify that (A,B is a controllable pair, while (A,C is an observable pair. Therefore, the nominal system (Σ n is both stabilizable and observable and Assumption 1 is satisfied. Moreover, the gain matrix K is chosen as K = [1 1.1], in order to guarantee that Ā = A KC is Hurwitz. We initiate the system at x 1 ( =.,x ( = π,x 3 ( = with initial observer states ˆx 1 ( =.,ˆx ( = π,ˆx 3 ( =. Our purpose is to force the output y = q to track the bounded trajectory y r = π sin(t using only measurable signals. Moreover, for the output tracking error e = y y r we require a steady state of no more than 5 1 3, minimum speed of convergence as obtained by the exponential e 3t and no overshoot. The aforementioned transient and steady state error bounds are prescribed via the performance function ρ(t = ( e 3t and M =. Furthermore, since e( =. ( >, e the output error transformation is chosen ε = T ρ = ln. The control and update laws proposed in ( M+ e ρ 1 e ρ Theorem 5 were implemented using High Order Neural Networks HONNs (Rovithakis (a, whose structure was selected according to a trial and error procedure. The initial values of the weight vectors W F (t R,W G (t R 1, were randomly initialized in [,1]. Furthermore we have selected κ 1 = κ = κ 3 = 1, c =,c = and η( = 1(1 e 1(. Finally, concerning the switching procedure, the following constants have been used: δ u = 1,δ b =. The performance of the proposed adaptive output feedback control scheme is demonstrated in Figures 1(a-1(c. Figure 1(a shows the link angular position along with its reference. The output error satisfies the prescribed performance specifications as Figure 1(b clearly illustrates. Finally the demanded control effort (input control voltage is pictured in Figure 1(c. 5. CONCLUSIONS An output feedback controller for uncertain strict feedback nonlinear systems with continuous state dependent nonlinearities has been presented, capable of guaranteeing prescribed performance bounds on both the transient and the steady state response of the system output. Instead of designing an asymptotically convergent observer, any partial system knowledge was incorporated to design a state estimator-like system, which was used to provide rough estimates of the unavailable states, thus substantially reducing the complexity of the overall design. Under a stabilizability assumption for the nominal part of the system, the proposed adaptive output feedback control architecture established prescribed performance guarantees on the system output, as well as boundedness of all other closed loop signals. To effectively avoid possible division by zero, the designed output feedback controller is of switching type. However, its continuity is guaranteed, thus alleviating any problems related to the existence and uniqueness of solutions. Simulations on a single-link manipulator actuated by a brush dc motor benchmark example illustrated the approach. REFERENCES Abdollahi, F., Talebi, H., and Patel, R. (6. A stable neural network-based observer with application to flexible-joint manipulators. Neural Networks, IEEE Transactions on, 17(1, Andrieu, V. and Praly, L. (8. Global asymptotic stabilization for nonminimum phase nonlinear systems admitting a strict normal form. Automatic Control, IEEE Transactions on, 53(5, Andrieu,V.,Praly,L.,andAstolfi,A.(9a. Asymptotic tracking of a reference trajectory by output-feedback for aclassofnonlinearsystems. Systems & Control Letters, 58(9, Andrieu, V., Praly, L., and Astolfi, A. (9b. High gain observers with updated gain and homogeneous correction terms. Automatica, 45(, Bechlioulis, C.P. and Rovithakis, G.A. (9. Adaptive control with guaranteed transient and steady state 654

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