Dynamic Inversion of Multi-input Nonaffine Systems via Time-scale Separation

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1 Proceedings of the 26 American Control Conference Minneapolis, Minnesota, USA, June 4-6, 26 ThC3 Dynamic Inversion of Multi-input Nonaffine Systems via Time-scale Separation Naira Hovakimyan, Eugene Lavretsky and Chengyu Cao Abstract This paper presents an approximate dynamic inversion methodology for a class of multivariable nonaffinein-control systems via time-scale separation The control signal is defined as a solution of fast dynamics, and the coupled system is shown to comply with the assumptions of Tikhonov s theorem from singular perturbations theory Simulations illustrate the theoretical results I INTRODUCTION Dynamic inversion is one of the most popular methods for controlling affine-in-control minimum phase nonlinear systems [] [3] 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 [4] Several methods for particular classes of nonaffine systems have been reported in [5] [8] In [5], nonaffine cascaded systems are considered and a linearizationbased approach is presented using the results from [9] In [6], three classes of nonaffine systems are considered that in addition to nonaffine control term have also linear input, and dynamic backstepping is used for global stabilization In [7], the averaging theory is employed for controlling a special class of nonaffine systems using high frequency control signals Ref [8] solves the control design problem for a class of nonaffine systems that are polynomial in control variable In this paper we present a dynamic inversion methodology for minimum-phase nonaffine systems with welldefined zero dynamics using tools form singular perturbations theory The control signal is computed online as a solution of fast dynamics and presents an extension of the approximate dynamic inversion control methodology reported in [], [] The basic design idea in [], [] relies on time-scale separation between the system and the controller dynamics The latter is designed to approximate the unknown dynamic inversion based control solution, assuming that it exists For example, for a single-input system like ẋ = fx, u, x = x, t, where x R is the system state, u R is the control input, fast dynamics are introduced as follows: ɛ u = sign fx, u + ax, a>, ɛ Assuming that f is a Lipschitz function of its arguments, and that is bounded away from zero for x, u Ω x Ω u R R, whereω x, Ω u are compact sets, we prove that the assumptions of Tikhonov s theorem This material is based upon work supported by the United States Air Force under Contracts No FA , FA955-4-C-47 N Hovakimyan and C Cao are with AOE, Virginia Tech, Blacksburg, VA , {nhovakim, chengyu}@vtedu E Lavretsky is with The Boeing Company, Huntington Beach, CA 92647, eugenelavretsky@boeingcom from singular perturbations theory are satisfied [2] In [], [], we present the methodology also for a very limited class of multivariable systems Specifically, we show that, with the use of the sign in fast dynamics, exponential stability of the boundary layer system required by Tikhonov s theorem can be verified locally via linearization In this paper, we reformulate the fast dynamical equation ɛ u = fx, u +ax, ɛ, and prove that exponential stability of the boundary layer system can be verified via Lyapunov argument Such approach enables extension of this methodology to a much broader class of multivariable systems as compared to the results in [], [] II TIKHONOV S THEOREM In this Section, we recall Tikhonov s theorem from singular perturbations theory see for instance Theorem 2 on page 439 of [2] Consider the problem of solving the so-called singularly perturbed system { ẋt =ft, xt,ut,ɛ, x = ξɛ, 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 the system in be in standard form, that is = gt, x, u, has k isolated real roots u = h i t, x, i {,,k} for each t, x [, ] D x Choose one particular i and drop i henceforth Let vt, x = u ht, x The system given by ẋt =ft, xt,ht, xt,, x = ξ 2 is called the reduced system, and the system given by dv dτ = gt, x, v+ht, x,, v = η h,ξ 3 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 ɛ due to Tikhonov Theorem : Consider the system in 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 D x R n, D v R m, containing 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, /6/$2 26 IEEE 3594

2 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 conditions given by ξ, η are smooth functions of ɛ A2 The origin is an exponentially stable equilibrium point of the reduced system 2 There exists a Lyapunov function V :[, D x [, that W x V t, x W 2 x, t, x+ t x t, xft, x, ht, x, W 3x for all t, x [, D x, where W,W 2,W 3 are continuous positive definite functions on D x, and let c be a nonnegative number such that {x D x W x c} is a compact subset of D x A3 The origin is an exponentially stable equilibrium point of the boundary layer system in 3, 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, the system in has a unique solution x ɛ on [, and x ɛ t x t = Oɛ holds uniformly for t [,, where x t denotes solution of the reduced system 2 Remark : A3 can be verified via Lyapunov argument: if there is a Lyapunov function such that c v 2 V t, x, v c 2 v 2 4 v gt, x, v + ht, x c 3 v 2, 5 for all t, x, v [, D x D v, then A3 is satisfied III TRACKING DESIGN FOR SINGLE INPUT SYSTEMS Consider the following nonlinear single-input system in normal form: ẋ t = x 2 t ẋ r t = x r t ẋ r t = fxt,zt,ut żt = ζxt,zt,ut, with x = x, z = z, for x, z, u D x D z D u, where D x R r, D z R n r and D u R are domains containing their respective origins Here [x t z t] denotes the state vector of the system, xt = [x t x r t] R r, ut is the control input, r is the relative degree of the system, and f : D x D z D u R, ζ : D x D z D u R n r are continuously differentiable functions of their arguments Furthermore, assume that is bounded away from zero for x, z, u Ω x,z,u D x D z D u,whereω x,z,u is 6 a compact set of possible initial conditions; that is, there exists b >, such that >b In addition, assume that the function f cannot be inverted explicitly with respect to u Let the reference model dynamics be given by: ẋ r t =A r x r t+b r rt, x r = x r,, where rt is continuous and bounded reference input signal, x r t =[x r, t x r,r t] 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 = a a 2 a r, B r = b Let et =xt x r t be the tracking error signal Then the open loop error dynamics are given by: ė = F e + x r t,z,ut A r x r t B r rt 7 ż = ζe + x r t,z,ut 8 where F x, z, u =[x 2,,x r,fx, z, u] Ideal dynamic inversion based control is the solution of fx, z, u = a r x r a 2 x 2 a x + br 9 with respect to u, which results in the exponentially stable closed-loop tracking error dynamics ėt =A r et Since 9 cannot be solved explicitly for u, we construct an approximation of the ideal dynamic inversion controller by introducing the following fast dynamics: ɛ ut = pt, e, z, uft, e, z, u, u = u, where pt, e, z, u = e + x rt,z,u, ft, e, z, u = fe+x r t,z,u+a r e r +x r,r t+ +a e +x r, t brt Let u = ht, e, z be an isolated root of ft, e, z, u = Its existence is guaranteed by the assumption > b > Then the corresponding reduced system for the dynamics in 7, 8 is given by: ėt = A r et, żt = ζx r t+et,zt,ht, et,zt 2 The boundary layer system is given by: dv = pt, e, z, v + ht, e, zft, e, z, v + ht, e, z dτ Application of Theorem leads to the following result Theorem 2: 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 partials with respect to e, z, u, and the first partial of f with respect to t are continuous and bounded, ht, e, z and t, e, z, u have 3595

3 bounded first derivatives wrt their arguments, e, z as functions of t, e, z, ht, e, z are Lipschitz in e, z, uniformly in t B2 The origin is an exponentially stable equilibrium point of the system żt =ζx r t,zt,ht,,zt The map e, z ζe + x r t,z,ht, e, z is continuously differentiable 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 the boundary layer system 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 = p,e,z,v + h,e,z 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, has a unique solution x ɛ t on [, and x ɛ t =x r t+oɛ holds uniformly for t [T, Proof We need to verify that Assumptions A, A2, A3 in Theorem are satisfied B clearly implies that A holds We now show that Assumption A2 holds Assumption B2 implies see Lemma 46, page 76 of [2], that the system żt =ζx r t+et,zt,ht, x r 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 46 of [2], it follows that γρ =cρ, forsome constant c> Using the fact that the unforced system ż = ζx r,z,ht,,z has as an exponentially stable equilibrium point, it can be seen from the proof of Lemma 46 of [2] that βρ, t =kρ exp ωt for some positive constants k and ω Thus the solution to the reduced system, 2 satisfies et e c exp ω t and zt e + z c 2 exp ω t for all t and for some ω > Hence, the origin, is an exponentially stable equilibrium point of, 2 From a converse Lyapunov theorem Theorem 44 on pages of [2], 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 t t, e, z + e,zv Ft, e, z w 3 e, z 2, where Ft, e, z = [ A r e ζ e + x r,z,ht, e, z ] We note that any positive c can be chosen in A2 of Theorem, and so a compact Ω e,z {e, z D e,z W 2 e, z ρc, <ρ<} can be chosen to be any subset of D e,z To show that A3 holds, consider the candidate Lyapunov function V t, e, z, v = 2 f 2 t, e, z, v + ht, e, z Since u = ht, e, z is an isolated root of ft, e, z, u =,then V t, e, z, = Consequently, ft, e, z, v + ht, e, z = v pt, e, z, ξ + ht, e, zdξ, which implies V t, e, z, v + ht, e, z = v ft, e, z, χ + ht, e, zpt, e, z, χ + ht, e, zdχ Substituting ft, e, z, v + ht, e, z into the expression for V t, e, z, v + ht, e, z, it follows that V t, e, z, v + ht, e, z = v pt, e, z, χ + ht, e, z χ pt, e, z, ξ + ht, e, zdξ dχ v χ = pt, e, z, χ + ht, e, z pt, e, z, ξ + ht, e, z dξdχ 3 Assumption B3 implies that pt, e, z, v + ht, e, z b Then it follows from 3 that v χ V t, e, z, v + ht, e, z b 2 dξdχ = b2 v Assumption B implies that there exists b > such that pt, e, z, v + ht, e, z b Then from 3 v x V t, e, z, v + ht, e, z b 2 dξdχ = b2 v Inequalities 4, 5 verify 4 To verify 5, consider pt, e, z, v + ht, e, zft, e, z, v + ht, e, z v = pt, e, z, v + ht, e, zft, e, z, v + ht, e, z pt, e, z, v + ht, e, zft, e, z, v + ht, e, z = p 2 t, e, z, v + ht, e, zf 2 t, e, z, v + ht, e, z Therefore B3 and 4 lead to 5 Thus, the boundary layer system has exponentially stable equilibrium at the origin Hence, Theorem applies and it follows that for each compact set Ω e,z D e,z there exists a positive constant ɛ such that for all e,z Ω e,z, u h,e,z Ω v, <ɛ<ɛ, the system of equations given by 6, has a unique solution x ɛ, z ɛ on [, and x ɛ t =x r t + Oɛ, z ɛ t =z r t+oɛ hold uniformly for t [T,, where z r is the solution of ėt = A r et, żt = ζx r t+et,zt,ht, et,zt, and T is such that expta r x x r, ɛ IV EXTENSION TO SYSTEMS WITH MULTIPLE INPUTS Consider the following multi-input nonaffine nonlinear system in its normal form: ẋ k, t ẋ k,rk t = ẋ k,rk t x k, t x k,rk t x k,rk t + f k xt,zt,ut żt =ζxt,zt,ut, k {,,m},

4 with x = x, z = z,whereu =[u,,u m ], x = [x, x,r x m, x m,rm ],and[x z ] R n The control objective is to design u such that x tracks the state x r of the reference system: ẋ r t =A r x r t+b r rt, t, x r = x r, Here x r =[x r, x r,r x r m, x r m,r m ] is the state of the reference model, r =[r r m ] is a vector of continuous and bounded reference input signals The pair A r,b r is assumed to be in block-diagonal Brunovsky A r, canonical form: A r =,B r = B r, B r,m A r,m, where A r,k,b r,k are of the form: A r,k =,B r,k = a r k, a r k,2 a r k,r k [,,,b k ], k {,,m} We also assume that A r,k are Hurwitz for all k {,,m} Letet =xt x r t be the tracking error The open-loop time-varying error dynamics are given by: ė = F e + x r t,z,ut A r x r t B r rt, 7 ż = ζe + x r t,z,ut, k {,,m}, 8 where F x, z, u =[x,2,,x,r,f x, z, u,,x m,2,,x m,rm,f m x, z, u] For dynamic inversion based control, we seek an m-dimensional solution u of the following system of m equations f x, z, u = a r,x, a r,r x,r + b r f m x, z, u = a r m,x m, a r m,r m x m,rm + b m r m resulting in asymptotically stable closed loop tracking error dynamics ėt =A r et Since this system cannot be solved explicitly, we consider the fast dynamics: ɛ ut = P t, et,zt,utft, et,zt,ut, 9 where ft, e, z, u = f e+ x r t,z,u+a r,x r,t+e, + + a r,r x r,r t+ e,r b r t f m e+ x r t,z,u+a r m,x r m,t+e m, + + a r m,r m x r m,r m t+ e m,rm b m r m t ] and P R m m is the Jacobian matrix with its i th [ row j th column element being defined as: P ij x, z, u = i j x, z, u Let u = ht, e, z be an isolated root of ft, e, z, u = The reduced system for 7,8 is: ėt = A r et, e = e, żt = ζet+x r t,zt,ut, z = z The boundary layer system is: dv dτ = P t, et,zt,v+ ht, e, z ft, e, z, v + ht, e, z 2 Theorem 3: Let the following conditions be 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 m, which contain their respective origins: C On any compact subset of D e,z D v, the functions f, ζ, and their first partials with respect to e, z, u, and the first partial of f with respect to t are continuous and bounded, ht, e, z and t, e, z, u have bounded first partials wrt their arguments, z t, e, z, ht, e, z, e t, e, z, ht, e, z are Lipschitz in e, z, uniformly in t C2 The origin is an exponentially stable equilibrium point of the system żt =ζx r t,zt,ht,,zt The map z,e ζe + x r t,z,ht, z, e is continuously differentiable in z,e, uniformly in t C3 For all x, z D x D z, for all u,u 2 D u the matrix P x, z, u Px, z, u 2 is strictly positive definite, ie there exists c > such that ξ R m ξ P x, z, u Px, z, u 2 ξ 2c ξ 2 2 Then the origin of 2 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 = P,z,e,v + h,z,e f,z,e,v + h,z,e Then for each compact set Ω z,e D z,e there exists a positive constant ɛ and a T > such that for all t, z,e Ω z,e, u h,z,e Ω v and <ɛ<ɛ, the system of equations 6, 9 has a unique solution x ɛ t on [, and x ɛ t =x r t +Oɛ holds uniformly for t [T, Proof Verification of Assumptions A and A2 can be done similar to the lines in the proof of Theorem 2 For verification of Assumption A3, consider the following Lyapunov function candidate for 2: V t, e, z, vτ = 2 f t, e, z, vτ+ht, e, z ft, e, z, vτ+ht, e, z 22 From 2 it follows that for any v,v 2 D v,wehave ξ P t, e, z, v + ht, e, z Pt, e, z, v 2 + ht, e, zξ 2c ξ 2 23 Since ft, e, z, ht, e, z =, then V t, e, z, = Therefore V t, e, z, v = V t, e, z, sds, 24 L 3597

5 where assumes integration along the pass L from zero to L v and V t, e, z, v =f t, e, z, v+ht, e, zpt, e, z, v+ ht, e, z Since t, e, z, v + ht, e, z = Pt, e, z, v + ht, e, z, v then ft, e, z, v + ht, e, z = Pt, e, z, s + L ht, e, zds It follows from 24 that V t, e, z, v = Pt, L L Pt, e, z, s + ht, e, zds e, z, s + ht, e, zds, and consequently V t, e, z, v = ds P t, e, z, s + ht, e, z L L Pt, e, z, s + ht, e, zds, 25 where L, L assume integration along the pass L and pass L from zero to v, s, respectively It follows from 23 and 25 that V t, e, z, v L L 2c ds ds = χ 2c dξdχ, and therefore v V t, e, z, v c v 2 26 Assumption C implies that there exists c 2 such that V t, e, z, v c 2 v 2, which verifies 4 The relationships in 2 and 22 imply V t, e, z, v = f Pt, e, z, v + ht, e, zp t, e, z, v + ht, e, zf, which along with 23 implies that V t, e, z, v 2c V t, e, z, v Consequently, the inequality in 26 leads to V t, e, z, v 2c 2 v 2, and this proves that 5 holds with c 3 =2c 2 Hence the boundary layer system has exponentially stable origin Hence Theorem applies and so it follows that the system of equations given by 6, 9 has a unique solution x ɛ, z ɛ on [, and x ɛ t =x r t+oɛ, z ɛ t = z r t+oɛ hold uniformly, where z r denotes the solution of żt =ζx r t+et,zt,ht, et,zt, z = z Finally, we would like to note that B3 it just a particular case of C3, when u R and f : R n R R Lemma : If u R and f : R n R R, then 2 is equivalent to existence of c> such that t, e, z, u c u D u 27 Proof Indeed, in this case 2 is simplified to t, e, z, u t, e, z, u 2 c u,u 2 D u Since 2 this holds u,u 2 D u,then t,e,z,u c for any u and hence t, e, z, u c 28 Letting c = c, the relationship in 28 proves 27 The opposite is straightforward: given 27, it is obvious that t, e, z, u t, e, z, u 2 c 2 We now define a class of systems, which satisfy the condition in C3 For any t, e, z, u [, D e,z D u,letλ min Pt, e, z, u be the minimum eigenvalue of Pt, e, z, u, ie ξ R m Pt, e, z, uξ λ min Pt, e, z, u ξ 29 Then for arbitrary ū R m we have for all ξ R m ξ P t, e, z, ūpt, e, z, ūξ λ 2 min Pt, e, z, ū ξ 2 Let û D u Δt, e, z, ū, û =Pt, e, z, û Pt, e, z, ū 3 Then ξ R m we have Δt, e, z, ū, ûξ λ max Δt, e, z, ū, û ξ, where λ max Δt, e, z, ū, û denotes the maximum eigenvalue of Δt, e, z, ū, û Let Δ max t, e, z, ū max û D u λ max Δt, e, z, ū, û Lemma 2: If there exists ɛ > and at least one ū R m such that + 2Δ max t, e, z, ū+ ɛ λ min Pt, e, z, ū 3 for all t, e, z [, D e,z, then C3 holds Proof For any û, ŭ D u, ξ R m, from 3 we have ξ P t, e, z, ûpt, e, z, ŭξ = ξ Δt, e, z, ū, û+pt, e, z, ū Δt, e, z, ū, ŭ+pt, e, z, ū ξ 32 It follows from 29 that Pt, e, z, ūξ λ min Pt, e, z, ū ξ For ξ,let Pt, e, z, ūξ λξ,t,e,z,ū ξ λ min Pt, e, z, ū Δt, e, z, ū, ŭξ λξ,t,e,z,ū, ŭ ξ Δ max t, e, z, ū 33 Expanding the relationship in 32 implies ξ P t, e, z, ûpt, e, z, ŭξ = ξ Δ t, e, z, ū, ûδt, e, z, ū, ŭ+δ t, e, z, ū, û Pt, e, z, ū+p t, e, z, ūδt, e, z, ū, ŭ +P t, e, z, ūpt, e, z, ū ξ λ 2 λ 3 λ λ 2 + λ 3 +λ 2 ξ 2, 34 where λ = λξ,t,e,z,ū, λ 2 = λξ,t,e,z,ū, ŭ, λ 3 = λξ,t,e,z,ū, ŭ It is easy to verify that λ 2 λ 2 +λ 3 λ λ 2 λ 3 > if λ λ 2 + λ 3 + λ 2 + λ λ 2 λ 3 > 2 It follows from 3 and 33 that there exists c > such that λ λ2+λ3+ λ 2+λ λ 2λ 3 2 c, and hence it follows from 34 that ξ P t, e, z, ûpt, e, z, ŭξ c ξ 2, which verifies the condition in C3 The case of ξ =is trivially satisfied 3598

6 V SIMULATIONS Let the system dynamics be given by: ẋt = fxt,ut, x = [ ], where x R 2 is the state vector, u R 2 is the control signal, and fx, u = 5x +tanhu + u tanhu u 5x 2 tanhu + u tanhu u 2 35 It follows directly that Px, u [ is independent of x and ] 7 p 2 p can be expressed as Pu = p 2, +p 6+p p 2 where p =tanh 2 u + u 2 +3,p [ 2 =tanh ] 2 u 2 3 Let 6 P = lim Pū = Then it follows ū,ū 2 [ 6 ] p3 2 p from 3 that Δû, ū = 3 p 4, +p 3 p 3 p 4 where p 3 = tanh 2 û t + û 2 t + 3,p 4 = tanh 2 û 2 t 3 as ū, ū 2 For any ξ = [ξ, ξ 2 ] such that ξ =, we have ξ Δ û, ūδû, ūξ = 2 tanh 2 û t +û 2 t ξ + ξ tanh 2 û 2 t 3 2 ξ 2 2 Therefore, max λ maxδû, ū = max ξ Δ û, ūδû, ūξ < û û, ξ = 6 as ū, ū 2 Hence, lim Δ maxū < 6 Since λ min P = ū, ū 2 6, it can be checked easily that lim λ min Pū + 2Δ max ū > ū, ū 2 It follows from Lemma 2 that the condition in C3 is satisfied for any u R 2 The control objective is to force Fig Trajectories of xt solid lines and r dt dotted lines the system state xt to track the following signal: { +expt 8 r d t = 5 +expt 5 + +expt expt 8 5 +expt 5 + +expt 3 +2 Choose [ the ] reference [ ] matrices as: A r = 2,B 2 r =,x r =[ ] To force the states of the reference model x r t track the desired signal r d t asymptotically, we choose the reference input as rt =2r d t+ṙ d t The fast dynamics are designed as: 2 ut = P ut fxt,ut A r x r t B r rt Fig 2 Trajectories of u i t Fig shows the tracking performance of r d t by the system states Control signals are given in Fig 2 Remark 2: Finally, we note that Lemma 2 should be viewed only as a sufficient condition for the proof of the fact that there is at least one class of systems verifying C3 in Theorem 3 The simulation example, intended to verify the conditions of Lemma 2, appears therefore conservative In a companion paper [2], we verify directly A-A3 of Tikhonov s theorem for a nonaffine pendulum system VI CONCLUSIONS We presented a design methodology for approximate dynamic inversion of multi-input nonaffine systems using time-scale separation The control signal is sought as a solution of a fast dynamical equation that satisfies the conditions of Tikhonov s theorem from singular perturbations theory A class of multivariable nonlinear systems is identified that verifies the required assumptions REFERENCES [] AIsidori Nonlinear Control Systems Springer, 995 [2] HK Khalil Nonlinear Systems Prentice Hall, 22 [3] S Devasia, C Degang, B Paden Nonlinear inversion-based output tracking IEEE Trans Autom Contr, 4:93-942, 996 [4] D Nesic, E Skafidas, IMY Mareels, RJ Evans, Minimum phase properties for input nonaffine nonlinear systems IEEE Trans Autom Contr, 44: , 999 [5] R Sepulchre, M Jankovic and P Kokotovic Constructive Nonlinear Control Springer, 997 [6] D Fontaine, P Kokotovic Approaches to global stabilization of nonlinear system nonaffine in control ACC: , 998 [7] S Luca Analysis and Control of Flapping Flight: from Biological to Robotic Insects PhD Thesis, UC Berkeley, 999 [8] E Moulay, W Perruquetti Stabilization of nonaffine systems: a constructive method for polynomial systems IEEE Trans Autom Contr, 5:52-526, 25 [9] W Lin Global asymptotic stabilization of general nonlinear systems with stable free dynamics via passivity and bounded feedback Automatica, 32: , 996 [] N Hovakimyan, E Lavretsky, A Sasane Stabilization of nonaffine nonlinear systems using time-scale separation 6 th IFAC World Congress, Prague, Czech Republic, 25 [] N Hovakimyan, E Lavretsky, A Sasane Dynamic inversion for nonaffine-in-control systems via time-scale separation: Part I ACC: , 25 [2] A Young, C Cao, N Hovakimyan, E Lavretsky Nonlinear control of a double pendulum system via dynamic inversion and time-scale separation ACC,