L 1 Adaptive Controller for a Class of Systems with Unknown

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1 28 American Control Conference Westin Seattle Hotel, Seattle, Washington, USA June -3, 28 FrA4.2 L Adaptive Controller for a Class of Systems with Unknown Nonlinearities: Part I Chengyu Cao and Naira Hovakimyan Abstract This paper presents a novel adaptive control methodology for a class of uncertain systems in the presence of time-varying unknown nonlinearities. The adaptive controller ensures uniformly bounded transient and asymptotic tracking for system s both input and output signals simultaneously. The performance bounds can be systematically improved by increasing the adaptation rate. Part II extends the results to a class of systems in the presence of unmodeled dynamics. I. INTRODUCTION This paper extends the results of [], [2] to a class of uncertain systems in the presence of time-varying and state dependent unknown nonlinearities. We prove that subject to a set of mild assumptions the system can be transformed into an equivalent linear system with time-varying unknown parameters and disturbances. For the latter, we extend the methodology from [], [2], which ensures uniformly bounded transient response for system s both input and output signals simultaneously, in addition to stable tracking. The L norm bounds for the error signals between the closed-loop adaptive system and the closed-loop reference system can be systematically reduced by increasing the adaptation rate. The paper is organized as follows. Section II gives the problem formulation. In Section III, the L adaptive control architecture is presented. Stability and uniform performance bounds are presented in Section IV. In Section V, simulation results are presented, while Section VI concludes the paper. Throughout this paper, I indicates the identity matrix of appropriate dimension, Hs L denotes the L gain of Hs, x L denotes the L norm of xt, x t L denotes the truncated L norm of xt at the time instant t, and x 2 and x indicate the 2- and - norms of the vector x respectively. Some of the proofs are in included in the Appendix. II. PROBLEM FORMULATION Consider the following system dynamics: ẋt = A m xt + b ωut + fxt,t, yt = c xt, x = x, x R n is the system state vector measurable, u R is the control signal, y R is the regulated output, b,c R n are known constant vectors, A m is a known n n Hurwitz matrix, ω is an unknown constant, and f : R n R R is an unknown nonlinear function. Research is supported by AFOSR under Contract No. FA The authors are with Aerospace & Ocean Engineering, Virginia Polytechnic Institute & State University, Blacksburg, VA , chengyu, nhovakim@vt.edu Assumption : [Semiglobal Lipschitz condition] For any δ >, there exist L δ > and B > such that fx,t f x,t L δ x x, f,t B, for all x δ and x δ uniformly in t. Assumption 2: [Known sign for control effectiveness] There exist upper and lower bounds ω u > ω l > such that ω l ω ω u. Assumption 3: [Semiglobal uniform boundedness of partial derivatives] For any δ >, there exist d fx δ >, and d ft δ > such that for any x δ, the partial derivatives of fx,t are piece-wise continuous and bounded fx,t d fx δ, fx,t d ft δ. x t The control objective is to design a full-state feedback adaptive controller to ensure that yt tracks a given bounded reference signal rt both in transient and steady state, while all other error signals remain bounded. III. L ADAPTIVE CONTROLLER In this section we develop an adaptive control architecture for the system in that permits complete transient characterization for both ut and xt. The design of L adaptive controller involves a strictly proper transfer function Ds and a gain k R +, which leads to a strictly proper stable Cs = ωkds + ωkds with DC gain C =. The simplest choice of Ds is Ds = s, which yields a first order strictly proper Cs in the following form Cs = ωk s+ωk. Let Hs = si A m b, and r t be the signal with its Laplace transform si A m x. Since A m is Hurwitz and x is finite, r L is finite. For the proof of stability and uniform performance bounds the choice of Ds and k needs to ensure that there exists ρ r such that Gs L < ρ r k g CsHs L r L 2 r L /ρ r L ρr + B, 3 Gs = Hs Cs, and k g = c A m b. 4 Remark : We notice that the upper bound in 3 is a consequence of the semiglobal Lipschitz property of fx,t, stated in Assumption. If fx,t is globally Lipschitz with uniform Lipschitz constant L, then lim ρ r ρ r /8/$ AACC. 493 Authorized licensed use limited to: UNIVERSITY OF CONNECTICUT. Downloaded on February 2, 29 at 9:44 from IEEE Xplore. Restrictions apply.

2 k g CsHs L r L r L /ρ r L + B = L, and the upper bound in 3 degenerates into Gs L < /L, which is the same as the one derived in [3] for systems with constant unknown parameters. We consider the following state predictor or passive identifier for generation of the adaptive laws: ˆxt = A mˆxt + b ˆωtut + ˆθt xt + ˆσt ŷt = c ˆxt, ˆx = x. 5 The adaptive estimates ˆωt, ˆθt, ˆσt are defined as: ˆθt = Projˆθt, xt x tpb, ˆθ = ˆθ ˆσt = Projˆσt, x tpb, ˆσ = ˆσ 6 ˆωt = Projˆωt, x tpbut, ˆω = ˆω xt = ˆxt xt, R + is the adaptation gain, P is the solution of the algebraic equation A mp +PA m = Q, Q >, and the projection operator ensures that the adaptive estimates ˆωt, ˆθt, ˆσt remain inside the compact sets [ω l, ω u ], [ θ b, θ b ], [ σ b, σ b ], respectively, with θ b, σ b defined as follows θ b = L ρ, σ b = B + ǫ, 7 ǫ is an arbitrary positive constant and ρ = ρ r + β, 8 with β being an arbitrary positive constant that satisfies Gs L L ρ <. We notice that 3 implies that Gs L L ρr <. Since L ρ depends continuously on ρ, then Gs L L ρ < can always be satisfied if β is small enough. Remark 2: In the following analysis we demonstrate that ρ r and ρ characterize the domain of attraction of the closed loop reference system yet to be defined and the system in respectively. We notice that since β can be set arbitrarily small, ρ can approximate ρ r arbitrarily closely. The control signal is generated through gain feedback of the following system: χs = Ds rs, us = kχs, 9 k R + is introduced in 2, and rs is the Laplace transform of rt = ˆωtut + ˆθt xt + ˆσt k g rt. The complete L adaptive controller consists of 5, 6 and 9 subject to the L -gain upper bound in 3. IV. ANALYSIS OF L ADAPTIVE CONTROLLER A. Closed-loop Reference System We now consider the following closed-loop reference system with its control signal and system response being defined as follows: ẋ ref t = A m x ref t + b ωu ref t + fx ref t,t u ref s = Cs/ωk g rs r ref s 2 y ref t = c x ref t, x ref = x, 3 r ref s is the Laplace transformation of the signal r ref t = fx ref t,t, and k g is introduced in 4. The next Lemma establishes the stability of the closed-loop reference system in -3. Lemma : For the closed-loop reference system in - 3 subject to the L -gain upper bound in 3, if then x ρ r, 4 x ref L < ρ r, 5 u ref L < ρ ur, 6 ρ r is introduced in 3 and ρ ur = Cs/ω L ρ r L ρr + B + k g r L. B. Equivalent Linear Time-Varying System In this section we demonstrate that the nonlinear system in can be transformed into an equivalent linear system with unknown time-varying parameters. To streamline the subsequent analysis, we need to introduce several notations. Let γ be the desired performance bound for x L, and β be an arbitrary positive constant verifying the following upper bound γ Cs L Gs L L ρ γ + β < β, 7 β was introduced in 8. Further, let γ 2 = Cs ω ρ u = ρ ur + γ 2, 8 L ρ γ + Cs L ω c o H o s c o γ.9 L It follows from Lemma 4 in [] that there exists c o R n such that c o Hs = N ns N d s, 2 the order of N d s is one more than the order of N n s, and both N n s and N d s are stable polynomials. We will prove that by increasing the adaptation rate, one can arbitrarily reduce γ. Lemma 2: If the truncated L norms of xt and ut verify x t L ρ, u t L ρ u, 2 then there exist differentiable θτ and στ with uniformly bounded derivatives over τ [, t] such that θτ < θ b, 22 στ < σ b, 23 fxτ,τ = θτ xτ + στ. 24 Proof. Using the definitions in 7, we have fx, L ρ x + B < θ b x + σ b, 25 which implies that there exist θ and σ such that θ < θ b, σ < σ b Authorized licensed use limited to: UNIVERSITY OF CONNECTICUT. Downloaded on February 2, 29 at 9:44 from IEEE Xplore. Restrictions apply.

3 and fx, = θ x + σ. 27 We construct the trajectories of θτ and στ according to the following dynamics: [ ] [ θτ dfxτ,τ ] = A θτ d xτ η, 28 στ [ A η = xτ σ b στ θ b θτ ], 29 with the initial values being bounded according to 26. The determinant of A η is: deta η = xτ θ b θτ + σ b στ. 3 For any τ [, t, t is an arbitrary constant or, if θτ < θ b, στ < σ b, 3 it follows from 3 that deta η τ over [, t. Hence, it follows from 28, 29 that d θτ xτ + στ = dfxτ,τ, 32 στ σ b στ = θτ 33 θ b θτ for any τ [, t. Using the initial condition from 27, we can integrate to obtain the left. θτ xτ + στ = fxτ,τ, τ [, t34 στ t σ b στ = θτ, 35 θ b θτ lim τ with τ approaches t from στ Next we calculate, assuming that σ b στ στ < σ b. Since στ may cross zero, we divide [, t into countable subsets: [,t, [t,t 2,..., [t I, t σt = = σt I =, and write στ σ b στ = i=,..,i ti t i στ σ b στ στ + = lim t I σ b στ signστlnσ b στ signσlnσ b σ. 36 From 35 and 36, we have lim signστlnσ b στ signσlnσ b σ = 37 lim signθτlnθ b θτ signθlnθ b θ. In what follows, we prove by contradiction. If is not true, since θτ and στ are continuous, it follows from 26 that there exists t [, t] such that either while i ii lim b, or 38 lim b, 39 θτ < θ b, στ < σ b, τ [, t. 4 i In this case we have lim signθτlnθ b θτ =. 4 Since it is obvious that signσlnσ b σ and signθlnσ b θ are bounded, it follows from 37 that lim signστlnσ b στ =, and hence lim στ = σ b. 42 It follows from 34 that lim θτ xτ + στ = lim fxτ,τ, which along with 38 and 42 implies that lim fxτ,τ = fx t, t = θ b x t + σ b. 43 From Assumption it follows that fx t, t L ρ x t + B = θ b x t + σ b ǫ, which contradicts 43, and therefore 38 is not true. ii Following the same steps as above, one can derive a contradicting argument to 39. Since 38, 39 are not true, then the relationships in 22, 23 hold. Eq. 24 follows from 22, 23 and 34 directly. Further, notice from that bounded xτ and uτ imply bounded right hand side for the system dynamics, and hence bounded ẋτ over [,t]. In the light of Assumption 3, dfxτ,τ d xτ then dfxτ,τ and d xτ are bounded, although the derivative may not be continuous. Since θτ is bounded, θτ d xτ is bounded. From 22, 23 it follows that deta η τ, and therefore we conclude from 28 that θτ and στ are bounded. This concludes the proof. If 2 holds, Lemma 2 implies that the system in can be rewritten over τ [, t] as: ẋτ = A m xτ + bωuτ + θτ xτ + στ, yτ = c xτ, x = x, 44 θτ, στ are unknown time-varying signals subject to the upper bounds 22, 23 for all τ [, t], while their derivatives for all τ [, t] are subject to θτ d θ ρ,ρ u <, στ d σ ρ,ρ u <. 45 Remark 3: We notice that though Lemma 2 proves the existence and boundedness of θτ, σt, their continuity is not guaranteed. The reason is that d xτ can be piecewise continuous due to the definition of the norm. Thus, in 28 the right-hand sides can be piece-wise continuous functions of t. 495 Authorized licensed use limited to: UNIVERSITY OF CONNECTICUT. Downloaded on February 2, 29 at 9:44 from IEEE Xplore. Restrictions apply.

4 C. Tracking error signal Lemma 3: For the system in and the L adaptive controller in 5, 6 and 9, for any t such that 2 holds, we have θ m ρ,ρ u x t L λ min P, 46 θ m ρ,ρ u 4θ 2 b + 4σ 2 b + ω u ω l 2 +4 λ maxp λ min Q θ bd θ ρ,ρ u + σ b d σ ρ,ρ u. 47 D. Transient and Steady-State Performance Theorem : Consider the reference system in -3 and the closed-loop L adaptive controller in 5, 6, 9 subject to 3. If x ρ r, 48 and the adaptive gain is chosen to verify the lower bound: we have: > θ mρ,ρ u λ min Pγ 2, 49 x L γ, 5 x x ref L < γ, 5 u u ref L < γ 2, 52 γ and γ 2 are defined in 7 and 9. It follows from 49 that arbitrarily small γ can be obtained by increasing the adaptive gain. V. SIMULATIONS Consider the dynamics of a single-link robot arm rotating on a vertical plane: I qt + Fqt, qt,t = ut, 53 qt and qt are the measured angular position and velocity, respectively, ut is the input torque, I is the unknown moment of inertia, Fqt, qt,t is an unknown nonlinear function that lumps the forces and torques due to gravity, friction, disturbance and other external sources. The control objective is to design ut to achieve tracking of a bounded reference input rt by qt, r L. Let x = [x x 2 ] = [q q]. The system in 53 can be presented in the canonical form: ẋt = A m xt + bωut + fxt,t, yt = c xt, [ ] b = [ ], c = [ ], A m =,.4 ω = /I, and fxt,t = [.4]xt Fx 2 t,x t,t. Let ω = /I = and Fx 2 t,x t,t = x 2 2t + x 2 t + sin.2t, so that the compact sets can be conservatively chosen as ω l =.5, ω u = 2, θ b = 2,σ b =. In the implementation of [ the L adaptive controller, ] we set Q = 2I, and hence P = Time t a x t solid, ˆx t dashed, and rtdotted time t b Time-history of ut Fig.. Performance of L adaptive controller for Fx 2 t, x t, t = x 2 2 t + x2 t + sin.2t Time t a x t solid, ˆx t dashed, and rtdotted time t b Time-history of ut Fig. 2. Performance of L adaptive controller for Fx 2 t, x t, t = x 2 2 t + x2 t + sin5t Time t a x t solid, ˆx t dashed, and rtdotted time t b Time-history of ut Fig. 3. Performance of L adaptive controller for Fx 2 t, x t, t = x 2 2 t + x2 t + sin2t 496 Authorized licensed use limited to: UNIVERSITY OF CONNECTICUT. Downloaded on February 2, 29 at 9:44 from IEEE Xplore. Restrictions apply.

5 We need to verify the condition in 3. Letting s Ds = /s, we have Gs = s+ωkhs, Hs = s [ s 2 +.4s+ s 2 +.4s+ ], and we choose conservative L ρ = 2. One can numerically verify that for ωk > 3 the upper bound Gs L L ρ < holds. Since ω >.5, we set k = 6. We set the adaptive gain c =. The simulation results of the L adaptive controller are shown in Figures a-b for reference input r = cos.5t. Next, without any retuning of the controller we consider different nonlinearity Fx 2 t,x t,t = x 2 2t + x 2 t + sin5t. The simulation results are shown in 2a-2b. Finally, we consider much higher frequencies in the nonlinearity: Fx 2 t,x t,t = x 2 2t + x 2 t + sin2t. The simulation results are shown in 3a-3b. We note that the L adaptive controller guarantees smooth and uniform transient performance in the presence of different unknown nonlinearities without requiring any retuning. We also notice that x t and ˆx t are almost the same in Figs. a, 2a and 3a. VI. CONCLUSION A novel L adaptive control architecture is presented that has guaranteed transient response in addition to stable tracking for uncertain systems in the presence of unknown state and time-dependent nonlinearities. The control signal and the system response approximate the same signals of a closed-loop reference system, which can be designed to achieve desired specifications. In Part II, we present an extension to a class of systems in the presence of unmodeled dynamics, [5]. The results of these papers question the need for the neural network based adaptive control paradigm. REFERENCES [] C. Cao and N. Hovakimyan. Guaranteed transient performance with L adaptive controller for systems with unknown time-varying parameters: Part I. American Control Conference, pages , 27. [2] C. Cao and N. Hovakimyan. Stability margins of L adaptive controller: Part II. American Control Conference, pages , 27. [3] C. Cao and N. Hovakimyan. Design and analysis of a novel L adaptive control architecture, Part II: Guaranteed transient performance. In Proc. of American Control Conference, pages , 26. [4] C. Cao and N. Hovakimyan. Novel L neural network adaptive control architecture with guaranteed transient performance. IEEE Trans. on Neural Networks, 8:6 7, July 27. [5] C. Cao and N. Hovakimyan. L adaptive controller for nonlinear systems in the presence of unmodelled dynamics: Part II. American Control Conference, 28. [6] H. K. Khalil. Nonlinear Systems. Prentice Hall, Englewood Cliffs, NJ, 22. APPENDIX Proof of Lemma. It follows from -3 that x ref s = Gs r ref s + HsCsk grs + si A m x. 54 Example 5.2 in [6] page 99 implies that x reft L Gs L r reft L + k gcshs L r t L + r L. 55 If 5 is not true, since x ref = x < ρ r and x ref t is continuous, there exists t such that x reft L ρ r, 56 x ref t = ρ r Using Assumption and the upper bound in 56, we arrive at the following upper bound r reft L L ρr x reft L + B. 58 Substituting 58 into 55, and noticing that r t L r L, we obtain x reft L Gs L L ρr ρ r + k gcshs L r L + r L + Gs L B. 59 The condition in 3 can be solved for ρ r to obtain the following upper bound Gs L L ρr ρ r + k gcshs L r L + r L + Gs L B < ρ r, 6 which implies that x reft L < ρ r, and contradicts 57. This proves 5. Using 5, it follows from Assumption that r ref L < ρ rl ρr + B. 6 Example 5.2 in [6] page 99 further implies that u ref L < Cs/ω L ρ rl ρr + B + k g r L, 62 which proves 6. Proof of Lemma 3. It follows from 2 that 44 and 45 hold for any τ [, t]. Consider the following Lyapunov function candidate: V xτ, ωτ, θτ, στ = x ω τp xτ + 2 τ + θ 2 τ + σ 2 τ, 63 ωτ ˆωτ ω, θτ ˆθτ θτ, στ ˆστ στ. 64 It follows from 5 and 44 that over [, t] xτ = A m xτ + b ωut + θτ xτ + στ, 65 x =. We can verify straightforwardly that V 4θb 2 + 4σb 2 + ω u ω l 2 θmρ, ρu /. Let t, t] be the first time-instant of the discontinuity of either of the derivatives of ˆθt, ˆσt, θt, σt. Next we prove that V τ θmρ, ρu, τ [, t ]. 66 Using the projection based adaptation laws from 6, one has the following upper bound for V τ: V τ x τq xτ + 2 θτ θτ + στ στ 67 for any τ [, t. The projection algorithm ensures that for all τ [, t and therefore ω l ˆωt ω u, ˆθτ θ b, ˆστ σ b, 68 max ω 2 + θ 2 τ + σ 2 τ τ [,t ωu ω l 2 + 4θb 2 + 4σb 2 / 69 for any τ [, t. If at any τ [, t V τ θmρ, ρu, 7 Authorized licensed use limited to: UNIVERSITY OF CONNECTICUT. Downloaded on February 2, 29 at 9:44 from IEEE Xplore. Restrictions apply.

6 θ mρ, ρ u is defined in 47, then it follows from 69 that x τp xτ 4λmaxP λ minq θ bd θ ρ, ρ u + σ b d σρ, ρ u, 7 and hence x τq xτ λminq λ maxp x τp xτ 4 θ bd θ ρ, ρ u + σ b d σρ, ρ u It follows from 22, 23 and 68 that. 72 θτ 2θ b, στ 2σ b 73 for all τ [, t. Since θτ and στ are continuous over [, t, the upper bounds in 45 and 73 lead to the following upper bound: θτ θτ + στ στ Hence, if V τ 2 θ bd θ ρ, ρ u + σ b d σρ, ρ u. 74 θmρ, ρu, then from 67 and 72 we have V τ. 75 θmρ, ρu It follows from 75 that V τ for any τ [, t. Since λ minp xτ 2 x τp xτ V τ, then for any τ [, t xτ 2 θmρ, ρu λ. minp Since V τ is continuous, we further have θ mρ, ρ u xτ, τ [, t]. 76 λ minp Given t [, t] such that V t θmρ, ρu let t 2 t, t] be the next time-instant such that discontinuity of any of the derivatives ˆθt, ˆσt, θt, and σt. Using similar derivations as above, we can prove that θ mρ, ρ u xτ, τ [t, t2]. 77 λ minp Iterating the process until the time instant t, we get θ mρ, ρ u x t L λ min P, 78 which concludes the proof. Proof of Theorem. The proof will be done by contradiction. Assume that 5-52 are not true. Then, since x x ref = γ, u u ref =, and xτ, x ref τ, uτ, u ref τ are continuous, there exists t such that while, xt x ref t = γ, or 79 ut u ref t = γ 2, 8 x x ref t L γ, u u ref t L γ 2. 8 Since 48 holds, 5-6 follow from Lemma directly. Taking into consideration the relationships in 8, 8 and 8, we have x t L ρ, u t L ρ u. Hence, it follows from 49 and Lemma 3 that x t L γ. 82 Let rτ = ωτuτ + θτ xτ + στ, r τ = θτ xτ + στ. It follows from 9 that χs = Dsωus+r s k grs+ rs, rs and r s are the Laplace transformations of signals rτ and r τ. Consequently Ds χs = rs kgrs + rs, + kωds us = kds rs kgrs + rs kωds Using the definition of Cs from 2, we can write ωus = Csr s k grs + rs, 84 and the system in consequently takes the form: xs = Hs Csr s + Csk grs Cs rs + si A m x. 85 Let eτ = xτ x ref τ. It follows from 54 that es = Hs Csr 2s Cs rs, e =, r 2s is the Laplace transformation of the signal r 2τ = θτ xτ x ref τ. 86 Example 5.2 in [6] page 99 gives the following upper bound: e t L Gs L r 2t L + r 3t L, 87 r 3τ is the signal with its Laplace transformation being r 3s = CsHs rs. From the relationship in 65 we have xs = Hs rs, which leads to r 3s = Cs xs, and hence r 3t L Cs L x t L. Since xτ x ref τ xτ x ref τ e t L for any τ [, t], it follows from 7, 22 and 86 that r 2t L L ρ e t L. From 87 we have e t L Gs L L ρ e t L + Cs L x t L. Eq. 82 and the L -gain upper bound from 3 lead to the following upper bound e t L Cs L Gs L L ρ γ, which along with 7 leads to e t L γ β < γ. 88 We notice that from 2 and 84 one can derive us u ref s = Cs ω θ τxs x ref s r 4s, r 4s = rs. Therefore, it follows from Example 5.2 in [6] page 99 Cs ω u u ref t L Lρ ω Cs L x x ref t L + r 4t L. 89 We have r 4s = Cs ω c o Hsc o Hs rs = Cs ω c o Hsc o xs, c o was introduced in 2. Using the polynomials from 2, we can write that Cs Cs N = d s ω c o Hs ω N ns. Since Cs is stable and strictly proper, the complete system Cs is proper and c o Hs stable, which implies that its L gain exists and is finite. Hence, we have r 4t L Cs x L. The upper bound in L ω c o Hsc o 82 leads to the following upper bound: r 4t L Cs ω c o Hs c o γ. 9 L It follows from 88, 89, and 9 and the definition of γ 2 in 9 that u u ref t L L ρ/ω Cs L γ β + Cs ω c o Hs c o γ < γ 2. 9 L We note that the upper bounds in 88 and 9 contradict the equality in 8, which proves The upper bound in 5 follows from 5-52 and 82 directly. 498 Authorized licensed use limited to: UNIVERSITY OF CONNECTICUT. Downloaded on February 2, 29 at 9:44 from IEEE Xplore. Restrictions apply.