Adaptive Dynamic Inversion via Time-Scale Separation

Transcription

1 Proceedings of the 45th IEEE Conference on Decision & Control Manchester Grand Hyatt Hotel San Diego, CA, USA, December 3-5, 26 Adaptive Dynamic Inversion via Time-Scale Separation Naira Hovakimyan, Eugene Lavretsky and Chengyu Cao Abstract This paper presents an adaptive dynamic inversion method for uncertain nonaffine-in-control systems. The adaptive dynamic inversion controller is defined as a solution of a fast dynamical equation, which achieves time-scale separation between a state predictor and the controller dynamics. Lyapunov based adaptive laws ensure that the state predictor tracks the actual nonlinear system with bounded errors. As a result, the system state tracks the reference model with bounded errors. Benefits of the proposed design method are demonstrated using Van der Pol dynamics with nonlinear control input. I. INTRODUCTION Dynamic inversion is one of the most popular methods for controlling affine-in-control minimum phase nonlinear systems [9], []. The main challenge in extending the methodology to nonaffine systems has to do with the fact that in the latter case the zero dynamics are not always well defined [5]. Several methods for particular classes of nonaffine systems have been reported in [], [4], [2]. Adaptive methods for controlling nonaffine systems have been presented in [] [3], [5] [7], [2]. In this paper, using the main result from [6], [9] and tools from singular perturbations theory [], we present a dynamic inversion methodology for minimum-phase uncertain nonaffine systems with well-defined zero dynamics. To simplify the presentation, first we review the main result from [8]. The methodology in [8] invokes fast dynamics to invert the system, and hence relies on the time-scale separation property between the system dynamics and the dynamics of the inverting controller. The main idea can be illustrated by considering a scalar dynamical system of the form: ẋ = fx, u, x = x, t, where f is a known function of the system state x and the control input u. Assuming that is bounded away from zero for x, u Ω x Ω u R R, whereω x, Ω u are compact sets, the control objective is to find u such that the state of the system tracks a bounded reference input rt C from an arbitrary initial condition x Ω x.in[8], we prove that such a controller can be determined via the solution of the fast dynamics: ɛ u = sign fx, u+ This material is based upon work supported by the United States Air Force under Contracts No. FA and FA955-4-C-47. Naira Hovakimyan is with Aerospace & Ocean Engineering, Virginia Polytechnic Institute & State University, Blacksburg, VA , e- mail: nhovakim@vt.edu Eugene Lavretsky is with Phantom Works, The Boeing Company, U.S.A. eugene.lavretsky@boeing.com Chengyu Cao is with Aerospace & Ocean Engineering, Virginia Polytechnic Institute & State University, Blacksburg, VA , chengyu@vt.edu kx rt ṙt, k>, where ɛ. If >b >, then for minimum phase nonlinear systems in normal form with well-defined zero dynamics, Tikhonov s theorem can be used to prove bounded tracking of a desired reference system [8]. In [3], under a set of restrictive assumptions, the method is extended to uncertain systems, and the adaptive counterpart of the method is developed. Assuming that f in is unknown, the methodology in [3] considers linear-inparameters neural network approximation of the unknown nonlinearity on a compact set of possible initial conditions fx, u =W Φx, u+εx, u, where Φx, u is a bounded continuous regressor, εx, u <ε, while W is a vector of unknown constants, so that the unknown system dynamics can be presented as ẋt =W Φxt,ut + εxt,ut x = x. 2 A state predictor using a series parallel model for the dynamics in 2 is introduced as ˆxt =Ŵ tφxt,ut aˆxt xt with a>, ˆx = ˆx, leading to the prediction error dynamics ėt = aet+ W tφxt,ut εxt,ut, where et =ˆxt xt, W t = Ŵ t W and e = ˆx x. The projection type adaptive law Ŵ t = ΓProj Ŵ t, Φxt,utet, Ŵ = W, 3 where Γ > is the adaptation gain, ensures boundedness of the parameter errors. The control design method from [8] is applied to force the state of the predictor track the reference input rt: Ŵ t Ŵ tφxt,ut Φx, u ɛ ut = sign aet+kˆxt rt ṙt,k >. Following the framework of [8], one needs to require that sign Ŵ t Φx,u be constant, which in its turn is crucial for exponential stability of the boundary layer system required by Tikhonov s theorem. In general, constant sign for Ŵ t Φx,u is not easy to verify. In fact, it is not guaranteed in the most general case. In [3], we consider a special class of nonlinear systems for which a regressor vector can be introduced to ensure the constant sign for Ŵ t Φx,u. That assumption however implies that the nonaffine nature of the system dynamics is weak as compared to its affine counterpart with respect to control efficiency. In this paper, we relax the restrictive assumption of [3] and extend the methodology to uncertain systems of more general class. We recall the main result from [6] that /6/$2. 26 IEEE. 75 Authorized licensed use limited to: UNIVERSITY OF CONNECTICUT. Downloaded on February 2, 29 at 9:58 from IEEE Xplore. Restrictions apply.

2 45th IEEE CDC, San Diego, USA, Dec. 3-5, 26 enables approximation of positive valued functions by a set of radial basis functions RBFs with positive coefficients. This consequently leads to consideration of a specialized set of basis functions and appropriate choice of adaptive laws such that in ensemble they verify the exponential stability property of the boundary layer system required by Tikhonov s theorem. The paper is organized as follows. In Section II, we recall some preliminaries from approximation theory and Tikhonov s theorem from singular perturbation theory. We give our main result on tracking a given reference signal for single input systems in Section III. A relevant simulation example is given in Section IV. II. MATHEMATICAL PRELIMINARIES A. Preliminaries on Approximation Theory In this section, we recall the main result from [6], [9]. Let CR r denote the usual space of continuous maps f : R r R. Theorem : [6] Let K : R r R be an integrable bounded function such that K is continuous and Kxdx,andS K be the set of functions R r qx = M x zi w i K, 4 δ where M > is a positive integer, δ >, w i R, and z i R r. Then for any continuous function f : R r R, for any ε> and for any compact subset Ω R r there exists a q S K such that q f Ω <ε, where Ω is the indicator function of Ω R n such that { if x Ω Ω x = if x Ω. The proof in [6] gives an explicit expression for the coefficients w i in 4. It shows that the coefficients are directly proportional to the corresponding function values on the compact set Ω. Consequently if fx > for all x Ω, then w i >. We also notice that the Gaussians given by x xci φ =e x xc 2 i σ i 2,wherex ci is the center, while σ i σ i is the width parameter, represent one particular choice of K. B. Preliminaries on Singular Perturbations Theory For proving our main result we will use Tikhonov s theorem on singular perturbations, which we recall below see for instance Theorem.2 on page 439 of []. Consider the problem of solving the system { ẋt =ft, xt,ut,ɛ, x = ξɛ Σ : 5 ɛ ut =gt, xt,ut,ɛ, u = ηɛ, where ξ : ɛ ξɛ and η : ɛ ηɛ are smooth. Assume that f and g are continuously differentiable in their arguments for t, x, u, ɛ [, ] D x D u [,ɛ ], where D x R n and D u R m are domains, ɛ >. In addition, let Σ be in standard form, i.e. =gt, x, u, has k isolated real roots u = h i t, x, i {,...,k} for each t, x [, ] D x. We choose one particular i, which is fixed. We drop the subscript i henceforth. Let vt, x =u ht, x. In singular perturbations theory, the system given by Σ : ẋt =ft, xt,ht, xt,, x = ξ, 6 is called the reduced system, and the system given by Σ b : dv dτ = gt, x, v+ht, x,, v = η h,ξ 7 is called the boundary layer system, where η = η and ξ = ξ, t, x [, D x are treated as fixed parameters. The new time scale τ is related to the original time t via the relationship τ = t. The following result is due ɛ to Tikhonov []. Theorem 2: Consider the singular perturbation system Σ given in 5, and let u = ht, x be an isolated root of gt, x, u, =. Assume that the following conditions are satisfied for all [t, x, u ht, x,ɛ] [, D x D v [,ɛ ] for some domains D x R n and D v R m,which contain their respective origins: A. On any compact subset of D x D v, the functions f, g, their first partial derivatives with respect to x, u, ɛ, and the first partial derivative of g with[ respect to t are ] continuous and bounded, ht, x and g t, x, u, have bounded [ first derivatives ] with respect to their arguments, x t, x, ht, x is Lipschitz in x, uniformly in t, and the initial data given by ξ and η are smooth functions of ɛ. A2. The origin is an exponentially stable equilibrium point of the reduced system Σ given by equation 6. There exists a Lyapunov function V :[, D x [, that satisfies W x V t, x W 2 x V V t t, x+ x t, xft, x, ht, x, W 3x for all t, x [, D x,wherew,w 2,W 3 are continuous positive definite functions on D x,andletc be a nonnegative number such that {x D x W x c} is a compact subset of D x. A3. The origin is an equilibrium point of the boundary layer system Σ b given by equation 7, which is exponentially stable uniformly in t, x. Let R v D v denote the region of attraction of the autonomous system dv dτ = g,ξ,v + h,ξ,, andlet Ω v be a compact subset of R v. Then for each compact set Ω x {x D x W 2 x ρc, <ρ<}, there exists a positive constant ɛ such that for all t, ξ Ω x, η h,ξ Ω v and <ɛ<ɛ, Σ has a unique solution x ɛ on [, and x ɛ t x t =Oɛ holds uniformly for t [,, wherex t denotes the solution of the reduced system Σ in 6. The following Remark will be useful in the sequel []. 76 Authorized licensed use limited to: UNIVERSITY OF CONNECTICUT. Downloaded on February 2, 29 at 9:58 from IEEE Xplore. Restrictions apply.

3 45th IEEE CDC, San Diego, USA, Dec. 3-5, 26 Remark : Assumption A3 can be locally verified by linearization. Let ϕ denote the map v gt, ξ, v+ht, ξ,ɛ. It can be shown that [ if] there exists ω > such that the Jacobian matrix ϕ v satisfies the eigenvalue condition [ ] ϕ Re λ t, x, ht, x, ω < for all t, x v [, D x, then Assumption A3 is satisfied. III. TRACKING CONTROLLER A. Problem formulation Consider the following nonlinear single-input system in normal form: ẋt = Axt+Bfxt,zt,ut,x = x, żt = ζxt,zt,ut, z = z, where [x z ] denotes the state vector of the system, x = [x x r ] R r, u is the control input, r is the relative degree of the system, x, z, u D x D z D u,andd x R r, D z R n r and D u R are domains containing their respective origins, f : D x D z D u R, ζ : D x D z D u R n r are continuously differentiable unknown functions of their arguments, while A and B correspond to the controllable canonical normal form representation of the nonlinear system dynamics, i.e. A = , B =.. Assumption : Let be bounded away from zero for x, z, u Ω x,z,u D x D z D u,whereω x,z,u is a compact set of possible initial conditions; i.e. there exists b > such that >b. Let the reference model dynamics be given as ẋ r t = A r x r t+b r rt,x r = x r,, where rt is a continuous bounded reference input signal, x r =[x r, x r,r ] R r is the state of the reference model, and the Hurwitz matrix A r and the column vector B r have the following structure:. A r =..., B r =.. a a 2... a r b The control objective is to design a tracking control law to ensure that xt x r t as t, while all other error signals remain bounded. B. RBF approximation Let gx, z, u = x, z, u. Following Assumption, gx, z, u >b > for all x, z, u Ω x,z,u D x D z D u. Without loss of generality, let gx, z, u >b >. Let F denote the family of all continuous functions fx, z, u in Ω x,z,u monotonic with respect to u, i.e. for all f F we have gx, z, u >b >. 8 For all possible M, N, θ i R and all possible positive w i R + consider the following family of functions φ : Ω x,z,u R: where φx, z, u =φ x, z+φ 2 x, z, u, 9 φ x, z θ Φ x, z = N θ i e y yc i y yc i δ i 2 y [x z ],δ i >, φ 2 x, z, u w Φ 2 x, z, u M = e χ χc j χ χc j σ j 2 dξ, w j j= χ [x z ξ],σ j >, where ξ is introduced to denote the integration variable, Φ x, z is the vector of Gaussians independent of u, while Φ 2 x, z, u is the vector of the integrals of the Gaussians dependent upon u. The vectors y ci [x c i zc i ], i =,,N, χ cj =[x c j zc j u cj ], j =,,M represent the fixed centers of the basis, δ i, σ j are the fixed width parameters, while θ i R, w j R + are the unknown constant parameters. We denote this family of functions by S. It is straightforward to verify from 9- that for all φx, z, u S sgn φ x, z, u φ2 =sgn x, z, u >. 2 We note that F CR n+ and S CR n+. A straightforward corollary from Theorem is given by the following proposition. Proposition : S is dense in F uniformly with respect to norm. Proof. First we notice that for any fx, z, u from the class F the following representation is true: fx, z, u =fx, z, + ξ= gx, z, ξdξ. 3 Since fx, z, CR n+, it follows from Theorem that fx, z, can be approximated arbitrarily closely by a φ type function from over x, z Ω x,z, i.e. fx, z, φ x, z ε, where ε >. Since gx, z, u C + R n+, it follows from Theorem that for arbitrary ε 2 > there exists positive valued q + S G such that for x, z, u Ω x,z,u D x D z D u one has gx, z, u q + x, z, u ε 2, which leads to the following upper bound ξ= ξ= ξ= gx, z, ξ q + x, z, ξ dξ gx, z, ξ q + x, z, ξ dξ ε 2 dξ ε 2 ξ= dξ = ε 2 u Authorized licensed use limited to: UNIVERSITY OF CONNECTICUT. Downloaded on February 2, 29 at 9:58 from IEEE Xplore. Restrictions apply.

4 45th IEEE CDC, San Diego, USA, Dec. 3-5, 26 Since the approximation is considered over the compact set x, z, u Ω x,z,u, then there exist finite ρ>such that u ρ. Hence, ξ= gx, z, ξdξ ξ= q + x, z, ξdξ ρε 2 5 for any arbitrary small ε 2. This implies that the second term in 3 can be approximated arbitrarily closely by a φ 2 type function from. Hence, for arbitrary ε > there exists φx, z, u Ssuch that fx, z, u φx, z, u ε for all x, z, u Ω x,z,u D x D z D u. The proof is complete. Following the statement in Proposition, consider the following approximation for the unknown function fx, z, u in system dynamics 8 via the family of functions φ : Ω x,z,u R so that fx, z, u =W Φx, z, u+εx, z, u, where x, z, u Ω x,z,u, εx, z, u ε, W =[θ w ], Φx, z, u =[Φ x, z Φ 2 x, z, u],andw i s are positive. C. State predictor Consider the following one-step-ahead state predictor using a series parallel model for the dynamics in 8: ˆxt =Aˆxt+BŴ tφxt,zt,ut a s,i e s,i t, 6 with ˆx = x,wheree s t =ˆxt xt is the prediction error signal, a s,i > define a Hurwitz polynomial, while Ŵ t is an adaptive parameter for estimating the unknown constant vector W. Then the prediction error dynamics for the series parallel model in 6 are: ė s t = A s e s t+b W tφxt,zt,ut εxt,zt,ut 7 żt = ζxt,zt,ut 8 with e s =, z = z, W t = Ŵ t W,whereA s has the same structure of A r except for the last row being comprised of the coefficients a s,i instead of a i. The proof of the next lemma follows from the properties of the projection operator [7]. Lemma : The adaptive law Ŵ t = ΓProj Ŵ t, Φxt,zt,ute s tpb, 9 where Proj, denotes the Projection operator [7], P = P > solves the Lyapunov equation A s P + PA s = Q for arbitrary Q >, Γ > is the adaptation gain matrix, ensures that the parametric errors W t are ultimately bounded. Remark 2: Notice that since w i are positive, the compact set in the application of the Projection operator can be selected in a way so that ŵ i t remain positive for all t, i.e. ŵ i t >w i > for all i =,,M. Remark 3: Notice that ultimate boundedness of parametric errors, stated in Theorem, does not imply stability of the overall system. One needs to construct a bounded ut and prove in addition that in the presence of this feedback one of the systems, 8 or 6, remains bounded and achieves the tracking objective. We will apply the methodology from [8] to force the state of the predictor in 6 track the desired reference input. Boundedness of the system state will follow. D. Control design Let et = ˆxt x r t be the tracking error signal between the series parallel model and the reference system. Then the open loop time-varying tracking error dynamics are given by: ėt = F ˆxt,xt,e s t,zt,ut, Ŵ t A r x r t B r rt, e = e 2 żt = ζxt,zt,ut, z = z, 2 where F ˆx, x, e s,z,u,ŵ =[ˆx 2,, ˆx r, Ŵ Φx, z, u i=r a s,ie s,i ]. Dynamic inversion based controller is defined for the state predictor as the solution of Ŵ Φx, z, u a s,i e s,i = a iˆx i + br 22 resulting in the asymptotically stable closed-loop tracking error dynamics ėt =A r et. Since 22 cannot in general be solved explicitly for u, we construct an approximation of the dynamic inversion controller by introducing the following fast dynamics: ɛ u = sign ft, e, z, u, u = u, 23 where ft, e, z, u =Ŵ tφe + x r t e s t,z,u a s,i e s,i t+ a i e i + x r,i t brt.24 Let u = ht, e, z be an isolated root of ft, e, z, u =. The reduced system for 2-2 is given by: ėt =A r et, e = e 25 żt =ζx r t+et e s t,zt,ht, et,zt 26 with z = z. The boundary layer system is: dv dτ = sign ft, e, z, v + ht, e, z. 27 Tikhonov s theorem leads to the following result. Theorem 3: Assume that the following conditions are satisfied for all [t, e, z, u ht, e, z,ɛ] [, D e,z D v [,ɛ ] for some domains D e,z R n and D v R, which contain their respective origins: B. On any compact subset of D e,z D v, the functions f, ζ, and their first partial derivatives with respect to e, z, u, and the first partial derivative of f with respect to t are continuous and bounded, ht, e, z and t, e, z, u have bounded first derivatives with respect to their arguments, e, z as functions of t, e, z, ht, e, z are Lipschitz in e, z, uniformly in t. 78 Authorized licensed use limited to: UNIVERSITY OF CONNECTICUT. Downloaded on February 2, 29 at 9:58 from IEEE Xplore. Restrictions apply.

5 45th IEEE CDC, San Diego, USA, Dec. 3-5, 26 B2. The origin is an exponentially stable equilibrium point of the system żt =ζx r t e s t,zt,ht,,zt. The map e, z ζe + x r t e s t,z,ht, e, z is continuously differentiable and Lipschitz in e, z, uniformly in t. B3. t, e, z, v t, e, z, v+ht, e, z is bounded below by some positive number for all t, e, z [, D e,z. Then the origin of 27 is exponentially stable. Moreover, let Ω v be a compact subset of R v, where R v D v denotes the region of attraction of the autonomous system dv dτ = sign f,e,z,v+h,e,z. Then for each compact subset Ω z,e D z,e there exists a positive constant ɛ and a T > such that for all t, e,z Ω e,z, u h,e,z Ω v and < ɛ < ɛ, the system of equations 6, 23 has a unique solution ˆx ɛ t on [,, and ˆx ɛ t =x r t+oɛ holds uniformly for t [T,. Proof. We need to verify that Assumptions A, A2, A3 in Tikhonov s Theorem are satisfied. Assumption B clearly implies that A holds. We now show that Assumption A2 holds. Assumption B2 implies see Lemma 4.6, page 76 of [], that the system żt = ζx r t e s t +et,zt,ht, x r t e s t + et,zt with e viewed as the input is input to state stable. Thus, there exist class K and class KL functions γ and β, respectively, such that zt β zt,t t +γ sup t τ t eτ for all t t, t [,. Furthermore, from the proof of Lemma 4.6 of [], it follows that γρ =cρ, for some constant c>. Using the fact that the unforced system ż = ζx r e s,z,ht,,z has as an exponentially stable equilibrium point, it can be seen from the proof of Lemma 4.6 of [] that βρ, t = kρe ωt for some positive constants k and ω. Thus the solution to the reduced system satisfies et e c e ωt and zt x + z c 2 e ω t for all t and for some ω >. Hence, the origin, is an exponentially stable equilibrium point of From a converse Lyapunov theorem it follows that there exists a Lyapunov function V : [, D e,z R such that w e, z 2 V t, e, z w 2 e, z 2 and V t t, e, z + e,z V Ft, e, z w 3 e, z 2, where Ft, e, z = [A r e ζ e + x r t e s t,z,ht, e, z ], []. We note that any positive c can be chosen in A2 of Tikhonov s Theorem, and so Ω e,z {e, z D e,z W 2 e, z ρc, <ρ<} can be any compact subset of D e,z. In the light of Remark it is easy to see that with the definition of the boundary layer system, given by 27, its exponential stability can be verified locally by linearization with respect to v. Indeed, since Φ x, z is independent of u, it follows from 24 that t, e, z, u =ŵ t Φ 2t, e, z, u As stated in Remark 3, since w i are positive, the compact set in the application of the Projection operator can be selected such to ensure that ŵ i t >. Therefore, using the condition. in 2 we conclude that t, e, z, u Φ2 t, e, z, u sgn =sgn >. Thus, the linearization of 27 around its origin implies that the boundary layer system has locally exponentially stable origin. Hence, Tikhonov s theorem applies and so it follows that for each compact set Ω e,z D e,z there exists a positive constant ɛ and such that for all e,z Ω e,z, u h,e,z Ω v and < ɛ < ɛ, the system of equations given by 6, 23 has a unique solution ˆx ɛ, z ɛ on [,, andˆx ɛ t =x r t+oɛ, z ɛ t =z r t+oɛ hold uniformly for t [T,, z r being the solution of ėt = A r et, żt = ζx r t e s t+et,zt,ht, et,zt with e = e,z = z,andt is such that e TAr x e TAr x r, ɛ. Corollary : From Lemma and Theorem 3 it follows that xt tracks x r t with bounded errors. Proof. Indeed, application of Tikhonov s Theorem implies that: ˆx ɛ t =x r t+oɛ holds uniformly for t [T,. Thus, there exists a compact set Ωˆx such that ˆxt Ωˆx for all t. Choosing Ω x,z,u = Ωˆx Ω z Ω u be the set of RBF distribution, standard Lyapunov arguments can be applied to prove that the projection based adaptation law in 9 ensures that the prediction error e s t remains ultimately bounded. Recalling that x r t =ˆxt et =xt+e s t et, we get that as t the tracking error xt x r t is ultimately bounded. IV. SIMULATIONS Consider the following Van der Pol oscillator driven by nonlinear control input: ẋ = Ax + B x + x 2 x 2 +tanhx + u +3 + tanhu 3 +.u, 28 [ ] [ ] where A =, B =, and x =[x x 2 ] R 2 is the state vector available as measurement, u R is the control signal, and x = [ ]. It is easy to see that the system dynamics are invertible, but not in terms of elementary functions. The linear component.u is added to keep the control efficiency bounded away from zero in the entire space of variables. Simulation is performed using] the following reference ] param- [ [ eters A r =, B 2 3 r =, and rt = 2 +e t with zero initial +et 5 +et 3 conditions. The approximation of the nonlinearity is done with the use of 5 RBFs. Among them, 25 are Φ x-type Gaussians distributed over the grid x [ 2, 2], x 2 [ 2, 2] with the step size equal to in both dimensions and the width δ =.Weuse25 Φ 2 x, u-type basis functions, distributed 79 Authorized licensed use limited to: UNIVERSITY OF CONNECTICUT. Downloaded on February 2, 29 at 9:58 from IEEE Xplore. Restrictions apply.

6 45th IEEE CDC, San Diego, USA, Dec. 3-5, 26 over the grid x [ 2, 2], x 2 [ 2, 2], with step size equal to, andu [ 4, 4] with step size equal to 2 and the width set to σ =2in all dimensions. The norm upper bound for the projection operator is selected as W =, the lower bound for the positive weights w is set to., and the adaptation gain is Γ=.2. We use the following state predictor with zero initialization: ˆxt =Aˆxt+Bˆθ tφ xt + ŵ tφ 2 xt,ut [ ] ˆxt xt. 25 Denoting by et =ˆxt x r t the error between the state predictor and the reference system, we have ėt =A r et B r rt + [ κt ], where κt = ˆθ tφ xt + ŵ tφ 2 xt,ut [25 ]ˆxt xt+[2 3] ˆxt. The control signal needs to solve κ =2r. The fast dynamics Tracking Performance: eps =.6 Ref. Model: Ref. Model:2 System State: System State: Fig.. Tracking performance Control Time, sec Fig. 2. Control signal for determining the solution of it are designed as:.2 ut =2rt + [25 ]ˆxt xt [2 3] ˆxt ˆθ tφ xt ŵ tφ 2 xt,ut,u =. Figure shows the closed-loop tracking performance of the reference reference state x r t by actual system state xt, while Figure 2 shows the actual control effort ut. We note that all the states and control variable remain within the domain of RBF approximation. V. CONCLUSIONS In this paper, we presented a new design technique for adaptive dynamic inversion of nonaffine-in-control uncertain systems. Using the main result from [6], we proposed a new family of basis functions that respects the monotonic property with respect to control input of the unknown system dynamics. Using this class of approximators, we designed a state predictor and used tools from singular perturbation theory to achieve the desired tracking objective for the state predictor. With the projection type of adaptive law, standard Lyapunov arguments implied that the state predictor tracks the system state with bounded errors. 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