IN [1], an Approximate Dynamic Inversion (ADI) control

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1 1 On Approximate Dynamic Inversion Justin Teo and Jonathan P How Technical Report ACL09 01 Aerospace Controls Laboratory Department of Aeronautics and Astronautics Massachusetts Institute of Technology May 8, 009 Abstract Approximate Dynamic Inversion has been established as a method to control minimum-phase, nonaffine-incontrol systems [1] In this report, we re-state the main results of [1], clarify some minor notational errors, and prove the same results in an expanded form In the large, the main results of [1] still stand The development follows [1] closely, and no novelty is claimed herein The purpose of this report is mainly to supplement our existing results in [] [4] that rely heavily on the results of [1] Index Terms Dynamic inversion, feedback linearization, approximate I INTRODUCTION IN [1], an Approximate Dynamic Inversion ADI) control law was proposed that drives a given minimum-phase nonaffine-in-control system towards a chosen stable reference model The control signal was defined as a solution of fast dynamics, and Tikhonov s Theorem [5, Theorem 11, pp ] in singular perturbation theory was used to show that the control signal approaches the exact dynamic inversion solution, and that the system state approaches and maintains within an arbitrarily close neighborhood of the state of a chosen reference model when the controller dynamics are made sufficiently fast Previous results in [] [4] rely heavily on the results of [1] This report re-state the main results of [1], clarify some minor notational errors, and prove the same results in an expanded form The main purpose is to supplement previous results in [] [4] Importantly, no novelty of any form is claimed herein The main results of [1] are Theorems and 3 in [1]), which establish the ADI method for single-input-single-output SISO) and multi-input-multi-output MIMO) nonlinear systems respectively These correspond in the present report to J Teo is a graduate student with the Aerospace Controls Laboratory, Department of Aeronautics & Astronautics, Massachusetts Institute of Technology, Cambridge, MA 0139, USA csteo@mitedu) J How is director of Aerospace Controls Laboratory and Professor in the Department of Aeronautics & Astronautics, Massachusetts Institute of Technology, Cambridge, MA 0139, USA jhow@mitedu) Theorems 4 and 5 respectively The primary differences in the statement of these theorems between [1] and the present report are the first and third technical assumptions, arising from some minor) notational errors, but nonetheless can lead to confusion or erroneous interpretations We will show explicitly that the assumptions stated in Theorems 4 and 5 herein leads correctly to the desired results Another key difference between [1] and the present report is in the part of the proof which verifies the second technical assumption It was claimed in [1] that the second technical assumption implies, together with [5, Lemma 46, pp 176], that the reduced system of the associated singular perturbation model is input-to-state exponentially stable, which is stronger than the conclusion of [5, Lemma 46, pp 176] We will prove a stronger version of [5, Lemma 46, pp 176] as Lemma 1 in Section II-B, to justify the claim However, to establish Lemma 1, some required intermediate results which are strengthened versions of corresponding results in [5] must be established, which are presented in Section II-B Subtle differences between [1] and the present report will be mentioned in passing, together with some clarifications The report will proceed as follows We recall Tikhonov s Theorem from singular perturbation theory, which is the basis of the ADI method, in Section II-A Strengthened versions of corresponding results in [5] are presented in Section II-B, leading to the sought Lemma 1 The main result for SISO systems and its extension to MIMO systems are presented in Section III and IV respectively The final section concludes this report II PRELIMINARIES Here, we present Tikhonov s Theorem from singular perturbation theory and some strengthened versions of corresponding results in [5] These will be needed to establish the main results of ADI

2 A Tikhonov s Theorem from Singular Perturbation Theory Consider the standard singular perturbation model [5, Chapter 11] ẋ = ft, x, z, ɛ), ɛż = gt, x, z, ɛ), x0) = ξɛ), z0) = ηɛ), where ɛ is a small positive parameter, and ξɛ), ηɛ) depend smoothly on ɛ Assume that f and g are continuously differentiable in their arguments for all t, x, z, ɛ) [0, ) D x D z [0, ɛ 0 ], where D x R n and D z R m are domains, and ɛ 0 > 0 By the standard singular perturbation model, it is meant that the equation has k 1 isolated real roots z = h i t, x), 1) 0 = gt, x, z, 0) ) i {1,,, k}, for each t, x) [0, ) D x We fix one particular i, and henceforth omit the subscript i Define y = z ht, x) The reduced system is then obtained by setting ɛ = 0, z = ht, x) in the first equation of 1) to get ẋ = ft, x, ht, x), 0), x0) = ξ 0 = ξ0) 3) Let τ = t/ɛ The boundary layer system in the y coordinates in the τ time scale is then given by dy dτ = gt, x, y + ht, x), 0), y0) = η 0 h0, ξ 0 ), 4) where η 0 = η0) The following is the main result needed Theorem 1 Tikhonov [5, Theorem 11, pp ]) Consider the singular perturbation problem of 1) and let z = ht, x) be an isolated root of ) Assume that the following conditions hold for all t, x, z ht, x), ɛ) [0, ) D x D y [0, ɛ 0 ], for some domains D x R n and D y R m which contain their respective origins: 1) On any compact subset of D x D y, the functions f, g, their first partial derivatives with respect to x, z, ɛ), and the first partial derivative of g with respect to t are continuous and bounded, ht, x) and g z t, x, z, 0) have bounded first partial derivatives with respect to their arguments, x t, x, ht, x), 0) is Lipschitz in x, uniformly in t, and the initial data ξɛ) and ηɛ) are smooth functions of ɛ ) The origin is an exponentially stable equilibrium point of the reduced system 3) There exists a continuously differentiable Lyapunov function V : [0, ) D x [0, ) such that W 1 x) V t, x) W x), t, x) + t x t, x)ft, x, ht, x), 0) W 3x), holds for all t, x) [0, ) D x, where W 1, W and W 3 are continuous positive definite functions on D x Let c > 0 be chosen so that {x D x W 1 x) c} is a compact subset of D x 3) The origin is an exponentially stable equilibrium point of the boundary layer system 4), uniformly in t, x) Let R y D y be the region of attraction of the autonomous system dy dτ = g0, ξ 0, y + h0, ξ 0 ), 0), and Ω y be a compact subset of R y Then, for each compact set Ω x {x D x W x) ρc, ρ 0, 1)}, there is a positive constant ɛ such that for all t > 0, ξ 0 Ω x, η 0 h0, ξ 0 ) Ω y, and ɛ 0, ɛ ), the singular perturbation problem of 1) has a unique solution xt, ɛ), zt, ɛ) on [0, ), and xt, ɛ) xt) = Oɛ) holds uniformly for all t [0, ), where xt) is the solution of the reduced system 3) Proof: See [5, Appendix C18, pp ] Proposition 1 See also [5, pp 433], and [1, Remark 1]) If the eigenvalue condition )) g Re λ t, x, ht, x), 0) z k < 0 5) holds for some positive constant k and for all t, x) [0, ) D x, then the origin y = 0 of the boundary layer system 4) is exponentially stable, uniformly in t, x) [0, ) D x, for sufficiently small initial conditions, y0) Proof: Since z = ht, x) is the solution of ), we have gt, x, ht, x), 0) = 0, which shows that y = 0 is an equilibrium point of 4) It remains to show that it is exponentially stable Define gτ, y) = gɛτ, xɛτ), y + hɛτ, xɛτ)), 0),

3 3 so that the boundary layer system 4) can be rewritten as dy dτ = gτ, y), 6) with xɛτ) viewed as an exogenous time-varying signal Then Aτ) = g τ, y) y = g ɛτ, xɛτ), hɛτ, xɛτ)), 0) y=0 z When 5) holds, all eigenvalues of Aτ) have strictly negative real parts for all τ, x) [0, ) D x, so that the origin is an exponentially stable equilibrium point of the linear system dỹ dτ = Aτ)ỹ By [5, Theorem 413], the origin is an exponentially stable equilibrium point of the nonlinear system 6), which translates directly to exponential stability of the origin of the boundary layer system 4) B Other Auxiliary Results Some other intermediate results that will be needed are established here All of these are strengthened versions of corresponding results in [5] The main result needed is Lemma 1, but to establish this, the following are needed Define the closed ball B r as and system B r = {x R n x r}, ẋ = ft, x), 7) where f : [0, ) D R n is piecewise continuous in t and locally Lipschitz in x on [0, ) D, and D R n is a domain that contains the origin Theorem See also [5, Theorem 418, pp 17]) Let D R n be a domain that contains the origin and V : [0, ) D R be a continuously differentiable function such that c 1 x V t, x) c x, t + x ft, x) c 3 x, x µ > 0, t 0 and x D, where c 1, c and c 3 are positive constants, and c 1 < c Take r > 0 such that B r D and suppose that 0 < µ < 8) c1 c r 9) Then, for every initial state xt 0 ) satisfying xt 0 ) c1 c r, there exists T 0 dependent on xt 0 ) and µ) such that the solution of 7) satisfies c c 3 xt) xt 0 ) e c t t 0), t [t 0, t 0 + T ), 10) c 1 c xt) µ, t [t 0 + T, ) 11) c 1 Moreover, if D = R n, then 10) and 11) hold for any initial state xt 0 ), with no restriction on how large µ is Proof: Let ρ = c 1 r and η = c µ and define Ω t,η = {x B r V t, x) η}, Ω t,ρ = {x B r V t, x) ρ} Since η < ρ by 9), we have Ω t,η Ω t,ρ A boundary point x of Ω t,η satisfies either x = r > c /c 1 µ > µ by 9), or c µ = η = V t, x) c x which implies x µ Similarly, a boundary point x of Ω t,ρ satisfies either x = r > µ or c 1 r = ρ = V t, x) c x which implies x c1 /c r > µ by 9) Hence on all boundary points of Ω t,η and Ω t,ρ, we have x µ so that V t, x) is negative by 8), and all solutions starting in Ω t,η or Ω t,ρ cannot leave them Since c xt 0 ) ρ by assumption, we have V t 0, xt 0 )) c xt 0 ) ρ xt 0 ) Ω t0,ρ Then, xt) Ω t,ρ for all t t 0 A solution starting in Ω t,ρ must enter Ω t,η in finite time because in the set Ω t,ρ \ Ω t,η, V satisfies V t, x) c 3 µ < 0 The foregoing inequality implies that V t, xt)) V t 0, xt 0 )) c 3 µ t t 0 ) ρ c 3 µ t t 0 ), which shows that V t, xt)) reduces to η within the time interval [t 0, t 0 +ρ η)/c 3 µ )] For a solution starting inside Ω t,η, inequality 11) holds for all t t 0, since for any xt 0 ) Ω t,η, inequality c 1 xt) V t, xt)) η = c µ holds for all t t 0, which implies 11) with T = 0 For a solution starting inside Ω t,ρ but outside Ω t,η, let t 0 + T be the first time it enters Ω t,η For all t [t 0, t 0 + T ], V c 3 x c 3 c V Hence, by the Comparison Lemma [5, Lemma 34, pp 10], V t, xt)) satisfies V t, xt)) V t 0, xt 0 ))e c 3 c t t 0), t [t 0, t 0 + T ], which gives for all t [t 0, t 0 + T ], c 1 xt) V t, xt)) V t 0, xt 0 ))e c 3 c t t 0),

4 4 c xt 0 ) e c 3 c t t 0), yielding 10) If D = R n, then r can be chosen arbitrarily large, and any initial state xt 0 ) can be included in the set {x R n x c1 c r} We will need the definition of input-to-state exponential stability Consider the system ẋ = ft, x, u), 1) where f : [0, ) R n R m R n is piecewise continuous in t and locally Lipschitz in x and u Definition 1 See also [6] and [5, Definition 47, pp 175]) The system 1) is said to be input-to-state exponentially stable if there exist a class K function γ and positive constants k and λ such that for any initial state xt 0 ) and any bounded input ut), the solution xt) exists for all t t 0 and satisfies ) xt) k xt 0 ) e λt t0) + γ sup uτ) 13) t 0 τ t For definitions and properties of class K, K and KL functions, see [5, Section 44] Theorem 3 See also [5, Theorem 419, pp 176]) Let V : [0, ) R n R be a continuously differentiable function such that c 1 x V t, x) c x, t + x ft, x, u) c 3 x, x c 4 u > 0, for all t, x, u) [0, ) R n R m, where c 1, c, c 3, c 4 are positive constants with c 1 < c Then the system 1) is inputto-state exponentially stable, and its solution satisfies 13) with c k =, λ = c 3 c, γr) = c 4 r c 1 c c 1 Proof: By applying the global version of Theorem, we find that the solution xt) exists and satisfies ) c xt) xt 0 ) e c 3 c t t 0) + sup c 4 uτ), c 1 τ t 0 for all t t 0 Since xt) depends only on uτ) for τ [t 0, t], the supremum on the right-hand side of the above inequality can be taken over [t 0, t], which yields 13) Lemma 1 See also [5, Lemma 46, pp 176]) Suppose ft, x, u) is continuously differentiable and globally Lipschitz in x, u), uniformly in t If the unforced system of 1), namely ẋ = ft, x, 0), 14) has a globally exponentially stable equilibrium point at the origin x = 0, then the system 1) is input-to-state exponentially stable Its solution satisfies 13), and γ can be chosen to be a linear function for some positive constant c γr) = cr, Remark 1 Observe that all assumptions of Lemma 1 are identical to those of Lemma 46 in [5, pp 176], but the conclusion is stronger, namely of input-to-state exponential stability, with γ of 13) being a linear function Note that not all class K functions can be bounded above by a class K ) linear function, eg γr) = tanr) for r [0, π ) Proof: View the system 1) as a perturbation of the unforced system 14) The Converse Lyapunov Theorem [5, Theorem 414, pp ] shows that the unforced system 14) has a Lyapunov function V t, x) that satisfies c 1 x V t, x) c x, t + x ft, x, 0) c 3 x, x c 4 x, 15) for some positive constants c 1, c, c 3, c 4, with c 1 < c, globally Due to the uniform global Lipschitz property of f, the perturbation term satisfies ft, x, u) ft, x, 0) L u, for some Lipschitz constant L > 0, for all t t 0 and all x, u) The derivative of V along solutions of 1) satisfies V = t + ft, x, 0) + x x c 3 x + c 4 L x u ft, x, u) ft, x, 0)), To use the term c 3 x to dominate c 4 L x u for large x, we rewrite the foregoing inequality as V c 3 1 θ) x c 3 θ x + c 4 L x u, where θ 0, 1) Then, V c 3 1 θ) x, x c 4L u, c 3 θ for all t, x, u) Hence, the conditions of Theorem 3 are satisfied with c 1 = c 1, c = c, c 3 = c 3 1 θ), c 4 = c 4L c 3 θ We conclude that the system is input-to-state exponentially

5 5 stable with solution satisfying 13), and c k =, λ = c 31 θ), γr) = c 4L c r c 1 c c 3 θ c 1 Hence γcan be chosen to be a linear function γr) = cr, with c c4l c c 3θ c 1 III TRACKING DESIGN FOR MINIMUM-PHASE NONAFFINE-IN-CONTROL SISO SYSTEMS Consider an n-th order SISO nonaffine-in-control system of constant and well-defined) relative degree ρ, expressed in normal form φ ρ) = fx, z, u), x0) = x 0, ż = gx, z, u), z0) = z 0, 16) defined for all x, z, u) D x D z D u with D x R ρ, D z R n ρ and D u R being domains containing the origins The partial) state x is defined as x = [φ, φ, φ ),, φ ρ 1) ] T, and φ q) denotes the q-th time derivative of φ The state vector of the system is [x T, z T ] T, u is the control input, and f : D x D z D u R, g : D x D z D u R n ρ are continuously differentiable functions of their arguments To ensure that its relative degree is constant and well-defined, assume that is bounded away from zero for all x, z, u) D x D z D u That is, there exists b 0 > 0 such that b0 for all x, z, u) D x D z D u This implies that sign ) { 1, +1} is a constant, and that ψ : u fx, z, u) is a bijection for every fixed x, z) D x D z Additionally, assume that the function f cannot be explicitly inverted with respect to u Remark That ψ is a bijection means that the inverse of f with respect to u exist for every fixed x, z) D x D z By f being not explicitly invertible with respect to u, it is meant that an analytical expression for u in terms of x, z, and the evaluation of f at x, z, u) cannot be written This happens for example, when f is a transcendental equation in u, like fx, z, u) = sinu) + u The problem is to design a controller so that x tracks the state of a chosen ρ-th order stable linear reference model φ ρ) r + a rρ 1) φ ρ 1) r + + a r1 φr + a r0 φ r = b r r, 17) where x r = [φ r, φ r, φ ) r,, φ ρ 1) r ] T R ρ is its state, r is a continuously differentiable reference input signal with bounded time derivative ṙ, and x r 0) = x r0 is some chosen initial state, possibly with x r0 = x 0 Stability of the reference model requires that all roots of the characteristic equation s ρ + a rρ 1) s ρ a r1 s + a r0 = 0 lie in the open left half complex plane, denoted by C Define the tracking error φ e = φ φ r and error vector e = x x r = [φ e, φ e, φ ) e,, φ e ρ 1) ] T R ρ, and choose the desired stable error dynamics φ ρ) e + a eρ 1) φ ρ 1) e + + a e1 φe + a e0 φ e = 0, 18) with initial condition defined by e0) = e 0 = x 0 x r0 Similarly, stability of the desired error dynamics requires that all roots of s ρ + a eρ 1) s ρ a e1 s + a e0 = 0 lie in C Observe that in [1], a ei was set equal to a ri for i {0, 1,, ρ 1} This is a minor extension of [1] that allows the error dynamics to be specified independently of the reference model dynamics For notational convenience in the sequel, define a r = [a r0, a r1,, a rρ 1) ] T, a e = [a e0, a e1,, a eρ 1) ] T, α = sign ) As observed above, α { 1, +1} is a constant The openloop time-varying) error dynamics are then given by the system φ ρ) e = fe + x r t), z, u) + a T r x r t) b r rt), ż = ge + x r t), z, u), 19) with initial conditions e0) = e 0, z0) = z 0 Observe that time variance in 19) is induced by the external signals x r t) and rt) only We want to apply Theorem 1 to the system 19) with an appropriate controller to be specified The ideal dynamic inversion control is found by solving fe + x r t), z, u) + a T r x r t) b r rt) = a T e e 0) for u, resulting in the exponentially stable closed-loop tracking error dynamics 18) Since 0) cannot in general) be solved explicitly for u, an approximation of the dynamic inversion controller is constructed by introducing fast dynamics where ɛ u = α ft, e, z, u), u0) = u 0, 1) ft, e, z, u) = fe + x r t), z, u) + a T r x r t) b r rt) + a T e e Here, ɛ is a positive controller design parameter, chosen

6 6 sufficiently small to achieve closed-loop stability Observe that 1) relaxes the requirement for exact dynamic inversion while increasing the control in a direction to reduce the discrepancy 0) so as to approach the exact dynamic inversion solution Let u = ht, e, z) be an isolated root of ft, e, z, u) = 0 In accordance with the theory of singular perturbations [5, Chapter 11], the reduced system for 19), 1), obtained by setting ɛ = 0 and u = ht, e, z) is φ ρ) e = a T e e, e0) = e 0, ) ż = ge + x r t), z, ht, e, z)), z0) = z 0 3) With v = u ht, e, z) and τ = t/ɛ, the boundary layer system is dτ = α ft, e, z, v + ht, e, z)) 4) Applying Theorem 1 to 19) and 1) yields the following Theorem 4 Hovakimyan et al [1, Theorem ]) Consider the system 19) and 1), and let u = ht, e, z) be an isolated root of ft, e, z, u) = 0 Assume that the following conditions hold for all t, e, z, u ht, e, z), ɛ) [0, ) D e,z D v [0, ɛ 0 ], for some domains D e,z R n and D v R which contain their respective origins: 1) On any compact subset of D e,z D v, the functions f, g, their first partial derivatives with respect to x, z, u), and rt), ṙt) are continuous and bounded, ht, e, z) and x, z, u) have bounded first partial derivatives with respect to their arguments, and x, z, g x, g z as functions of e + x r t), z, ht, e, z)), are Lipschitz in e and z uniformly in t ) The origin is an exponentially stable equilibrium of the system ż = gx r t), z, ht, 0, z)) The map e, z) ge + x r t), z, ht, e, z)) is continuously differentiable and Lipschitz in e, z) uniformly in t 3) t, e, z) e + x rt), z, ht, e, z)) is bounded from below by some positive number for all t, e, z) [0, ) D e,z Then the origin of 4) is exponentially stable Let R v D v be the region of attraction of the autonomous system dτ = α f0, e 0, z 0, v + h0, e 0, z 0 )), and Ω v be a compact subset of R v Then, for each compact subset Ω e,z D e,z, there exists positive constants ɛ and T such that for all t > 0, e 0, z 0 ) Ω e,z, u 0 h0, e 0, z 0 ) Ω v, and ɛ 0, ɛ ), the system 16), 17), 1) has a unique solution xt, ɛ), zt, ɛ), x r t), ut, ɛ) on [0, ), and xt, ɛ) x r t) = Oɛ) holds uniformly for all t [T, ) Remark 3 The primary differences between Theorem 4 and Theorem of [1] is the first and third technical assumptions For comparison, we recall here these assumptions from [1]: 1) On any compact subset of D e,z D v, the functions f and g 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 and z as functions of t, e, z, ht, e, z)) are Lipschitz in e and z, uniformly in t 3) t, e, z, v) t, e, z, v + ht, e, z)) is bounded below by some positive number for all t, e, z) [0, ) D e,z Note that f and g are not explicitly functions of t and e Proof: The proof proceeds by showing that satisfaction of the assumptions above implies satisfaction of the assumptions of Theorem 1, whose result can be translated to the stated conclusions We identify x, z, y, and ht, x) of Theorem 1, denoted here by x s, z s, y s, and h s t, x s ) respectively for distinction) with quantities in 19) and 1) by x s [e T, z T ] T, z s u, y s v, h s t, x s ) ht, e, z) Also, f and g of Theorem 1 denoted here by f s and g s ) are identified with quantities in 19) and 1) as φ e φ ) e f s R n, φ ρ 1) e fe + x r t), z, u) + a T r x r t) b r rt) ge + x r t), z, u) g s α ft, e, z, u) R Now, translate the first assumption of Theorem 1 Since x r t) is the state of the exponentially stable system 17), x r t) and ẋ r t) are both continuous and bounded if rt) and ṙt) are continuous and bounded To have f s and g s continuous and bounded for any compact subset of D xs D ys requires that f,

7 7 g and rt) be continuous and bounded for any compact subset of D e,z D v Similarly, to have the first partial derivatives of f s and g s with respect to x s, z s, ɛ) continuous and bounded, we require that the first partial derivatives of f and g with respect to x, z, u) be continuous and bounded The first partial derivative of g s with respect to t, corresponds in the present section to the first partial derivative of α ft, e, z, u) with respect to t given by ) α x e + x rt), z, u)ẋ r t) + a T r ẋ r t) b r ṙt) Hence we require x and ṙt) to be continuous and bounded The requirement that h s t, x s ) have bounded first partial derivatives with respect to its arguments translates directly to requiring the same for ht, e, z) Since gs z s t, x s, z s, 0) corresponds to α f t, e, z, u), and given by α e + x rt), z, u), we require that x, z, u) have bounded first partial derivatives with respect to its arguments, and that ṙ is bounded The remaining Lipschitz conditions of Theorem 1 on x s are straightforward Also, since the initial conditions are independent of ɛ, the smoothness conditions are automatically satisfied Summarizing these gives assumption 1 above Next, we show that the second assumption of Theorem 1 holds To show that the origin is an exponentially stable equilibrium point of the reduced system ), 3), we proceed in a manner similar to the proof of Lemma 47 in [5, pp 180] Let t 0 0 be the initial time Clearly, e = 0 is an exponentially stable equilibrium point of ), and its solution satisfies et) k e et 0 ) exp λ e t t 0 )), 5) for some positive constants k e, λ e, and for all t t 0 With assumption above, Lemma 1 shows that system 3) with e as input, is input-to-state exponentially stable, and its solution satisfies zt) k z zs) exp λ z t s)) + sup c z eζ), 6) s ζ t for some positive constants k z, λ z, c z, and for all t s t 0 Substituting s = t + t 0 )/ into 6) yields ) t + t0 zt) k z z exp λ ) zt t 0 ) + sup c z eζ) 7) t+t 0 ζ t To estimate z t+t0 ), substitute s = t0 and replace t by t+t 0 in 6) to obtain ) t + z t0 k z zt 0 ) exp λ ) zt t 0 ) Using 5), we have + sup c z eζ) 8) t 0 ζ t+t 0 sup c z eζ) c z k e et 0 ), 9) t 0 ζ t+t 0 sup c z eζ) c z k e et 0 ) exp t+t 0 ζ t λ ) et t 0 ) 30) Let the composite state be x ez = [e T, z T ] T Using 7), we obtain x ez t) et) + zt), ) t + t0 et) + k z z exp λ ) zt t 0 ) + sup c z eζ), t+t 0 ζ t which, on substitution of 5) and 30), and using the fact that exp a ) exp a ) for all a R yields x ez t) 1 + c z )k e et 0 ) exp λ ) et t 0 ) ) t + t0 + k z z exp λ ) zt t 0 ) Substitution of 8) into the preceding gives x ez t) 1 + c z )k e et 0 ) exp λ ) et t 0 ) + k z exp λ zt t 0 ) + k z zt 0 ) exp ) sup c z eζ) t 0 ζ t+t 0 ) ), λ zt t 0 ) and using 9) yields x ez t) 1 + c z )k e et 0 ) exp λ ) et t 0 ) + c z k e k z et 0 ) exp λ ) zt t 0 ) + k z zt 0 ) exp λ z t t 0 )) Finally, defining λ ez = 1 min{λ e, λ z }, and using the facts that et 0 ) x ez t 0 ), zt 0 ) x ez t 0 ), we obtain x ez t) k ez x ez t 0 ) exp λ ez t t 0 )), where k ez = 1+c z )k e +c z k e k z +k z, valid for all t t 0 0 This shows that x ez = 0 is an exponentially stable equilibrium point of the reduced system ), 3)

8 8 Hence by a Converse Lyapunov Theorem [5, Theorem 414, pp ], there exists a Lyapunov function V : [0, ) D e,z [0, ) that satisfies where c 1 x ez V t, x ez ) c x ez, t + ˆft, xez ) c 3 x ez, x ez ˆft, x ez ) = φ e, φ ) e,, φ ρ 1) e, a T e e, ge + x r t), z, ht, e, z)) R n It can be seen that the Lyapunov function condition of assumption in Theorem 1 is satisfied with W 1 r) = c 1 r, W r) = c r, and W 3 r) = c 3 r Further, by choosing c sufficiently small, the set {x ez D e,z W 1 x ez ) = c 1 x ez c} can be made compact We conclude that satisfaction of assumption in the current theorem implies satisfaction of assumption in Theorem 1 We will use Proposition 1 to show that the origin is an exponentially stable equilibrium point of the boundary layer system 4), uniformly in t, e, z), so that assumption 3 of Theorem 1 is satisfied Define gt, e, z, u) = α ft, e, z, u), so that the boundary layer system 4) can be rewritten as dτ = gt, e, z, v + ht, e, z)) 31) Then, using the definitions of f in 1) and α, we have ) g f t, e, z, u) = sign t, e, z, u), ) = sign e + x rt), z, u), = e + x rt), z, u), and hence, g t, e, z, ht, e, z)) = e + x rt), z, ht, e, z)) From the preceding, assumption 3 implies that the eigenvalue condition 5) holds Proposition 1 then applies to show that the boundary layer system 31) or 4), has the origin as an exponentially stable equilibrium, uniformly in t, e, z) [0, ) D e,z Thus all assumptions of Theorem 1 are implied by the current assumptions Observe that the set Ω e,z {x ez D e,z W x ez ) = c x ez ρc, ρ 0, 1)} is compact by the choice of c above Then, for each such compact set Ω e,z, there exists a positive constant ɛ such that for all t > 0, e 0, z 0 ) Ω e,z, u 0 h0, e 0, z 0 ) Ω v, and ɛ 0, ɛ ), the system 19), 1) has a unique solution et, ɛ), zt, ɛ), ut, ɛ) on [0, ), and et, ɛ) ēt) = Oɛ), zt, ɛ) zt) = Oɛ), holds uniformly for all t [0, ), where ēt) and zt) are the solutions of the reduced system ) and 3) respectively Since ēt) is the solution of the exponentially stable system ), and x r t) is the solution of system 17), using the definition of e = x x r in the above yields xt, ɛ) x r t) ēt) = Oɛ), xt, ɛ) x r t) expat)e 0 = Oɛ), where A = Rρ ρ a e0 a e1 a eρ 1) Since ) is exponentially stable, A is Hurwitz, so that for any ɛ > 0, there exists T < such that expat)e 0 ɛ, t T Then for all t T, we reach the desired conclusion xt, ɛ) x r t) = Oɛ) Remark 4 Observe that if e 0 = 0, then T can be chosen to be 0, and xt, ɛ) x r t) = Oɛ) holds for all t > 0 This can be achieved by setting x r0 = x 0 See also [1] for further discussions

9 9 IV EXTENSION TO MINIMUM-PHASE NONAFFINE-IN-CONTROL MIMO SYSTEMS Consider the n-th order MIMO nonaffine-in-control system expressed in normal form φ ρ1) 1 = f 1 x, z, u), x 1 0) = x 10, φ ρ) = f x, z, u), x 0) = x 0, φ ρm) m = f m x, z, u), x m 0) = x m0, ż = gx, z, u), z0) = z 0, 3) defined for all x, z, u) D x D z D u with D x R ρ, D z R n ρ, and D u R m being domains containing the origins The partial) state x is defined as x = [x 1, x,, x m ] T R ρ, ρ = m i=1 ρ i, with each x i = [φ i, φ i,, φ ρi 1) i ] T R ρi for i {1,,, m}, and φ q) denotes the q-th time derivative of φ The state vector of the system is [x T, z T ] T, u = [u 1, u,, u m ] T R m is the control input, and f i : D x D z D u R for i {1,,, m}, g : D x D z D u R n ρ are continuously differentiable functions of their arguments Define fx, z, u) = [f 1 x, z, u), f x, z, u),, f m x, z, u)] T 33) Assume that the inverse of the function u fx, z, u) exists for each fixed x, z) D x D z, but that it cannot be written in closed form The problem is to design a controller so that x tracks the state of a chosen ρ-th order stable linear reference model described by the following set of linear ordinary differential equations φ ρ1) r1 + a T r1x r1 = b r1 r 1, x r1 0) = x r10, φ ρ) r + a T rx r = b r r, x r 0) = x r0, φ ρm) rm + a T rmx rm = b rm r m, x rm 0) = x rm0, 34) where for each i {1,,, m}, the corresponding vectors are a ri = [a ri0, a ri1,, a riρi 1)] T R ρi and x ri = [φ ri, φ ri,, φ ρi 1) ri ] T R ρi, r i is a continuously differentiable reference input signal with bounded time derivative ṙ i Let r = [r 1, r,, r m ] T Here, ρ i corresponds to those defined for system 3) so that x ri is of the same dimension as x i in 3), and the state of the reference model x r = [x T r1, x T r,, x T rm] T is of the same dimension as x in 3) Stability of the reference model requires that for each i {1,,, m}, all roots of the characteristic equation lie in C s ρi + a riρi 1)s ρi a ri1 s + a ri0 = 0 Define the tracking error e = x x r, which can be decomposed as e = [e T 1, e T,, e T m] T, e i = x i x ri = [φ ei, φ ei,, φ ρi 1) ei ] T, i {1,,, m} Choose the desired stable error dynamics as described by the following set of linear ordinary differential equations φ ρ1) e1 + a T e1e 1 = 0, e 1 0) = e 10 = x 10 x r10, φ ρ) e + a T ee = 0, e 0) = e 0 = x 0 x r0, φ ρm) em + a T eme m = 0, e m 0) = e m0 = x m0 x rm0, 35) where for each i {1,,, m}, the corresponding vectors are a ei = [a ei0, a ei1,, a eiρi 1)] T R ρi Similarly, ρ i corresponds to those defined for system 3) so that e i is of the same dimension as x i in 3), and the state of the desired error dynamics e is of the same dimension as x in 3) Stability of the desired error dynamics requires that for each i {1,,, m}, all roots of the characteristic equation s ρi + a eiρi 1)s ρi a ei1 s + a ei0 = 0 lie in C Similar to the SISO case, observe that this is a minor extension of [1], wherein a eij is set equal to a rij for i {1,,, m}, j {0, 1,, ρ i 1} The open-loop time-varying) error dynamics are then given by the system φ ρ1) e1 φ ρ) e φ ρm) em = f 1 e + x r t), z, u) + a T r1x r1 t) b r1 r 1 t), = f e + x r t), z, u) + a T rx r t) b r r t), = f m e + x r t), z, u) + a T rmx rm t) b rm r m t), ż = ge + x r t), z, u), 36) with initial conditions e0) = e 0, z0) = z 0 Similarly, observe that time variance in 36) is induced by the external signals x r t) and rt) only The ideal dynamic inversion control is found by solving the

10 10 system of m equations f 1 e + x r t), z, u) + a T r1x r1 t) b r1 r 1 t) = a T e1e 1, f e + x r t), z, u) + a T rx r t) b r r t) = a T ee, f m e + x r t), z, u) + a T rmx rm t) b rm r m t) = a T eme m, 37) for u R m, resulting in the exponentially stable closedloop tracking error dynamics 35) Since 37) cannot in general) be solved explicitly for u, an approximation of the dynamic inversion controller is constructed by introducing fast dynamics ɛ u = P ft, e, z, u), u0) = u 0, 38) where P R m m is a chosen constant matrix, and with 33), ft, e, z, u) = fe + x r t), z, u) + A r x r t) B r rt) + A e e, a T r a T r 0 0 A r = Rm ρ, 0 0 a T rm b r b r 0 0 B r = Rm m, 0 0 b rm a T e a T e 0 0 A e = Rm ρ 0 0 a T em Let u = ht, e, z) be an isolated root of ft, e, z, u) = 0 The reduced system for 36), 38), obtained by setting ɛ = 0 and u = ht, e, z) is φ ρ1) e1 = a T e1e 1, e 1 0) = e 10, φ ρ) e = a T ee, e 0) = e 0, φ ρm) em = a T eme m, e m 0) = e m0, ż = ge + x r t), z, ht, e, z)), z0) = z 0 39) With v = u ht, e, z) and τ = t/ɛ, the boundary layer system is dτ = P ft, e, z, v + ht, e, z)) 40) Applying Theorem 1 to 36), 38), and noting the definition of f in 33) yields the following Theorem 5 Hovakimyan et al [1, Theorem 3]) Consider the system 36) and 38), and let u = ht, e, z) be an isolated root of ft, e, z, u) = 0 Assume that the following conditions hold for all t, e, z, u ht, e, z), ɛ) [0, ) D e,z D v [0, ɛ 0 ], for some domains D e,z R n and D v R m which contain their respective origins: 1) On any compact subset of D e,z D v, the functions f, g, their first partial derivatives with respect to x, z, u), and rt), ṙt) are continuous and bounded, ht, e, z) and x, z, u) have bounded first partial derivatives with respect to their arguments, and x, z, g x, g z as functions of e + x r t), z, ht, e, z)), are Lipschitz in e and z uniformly in t ) The origin is an exponentially stable equilibrium of the system ż = gx r t), z, ht, 0, z)) The map e, z) ge + x r t), z, ht, e, z)) is continuously differentiable and Lipschitz in e, z) uniformly in t 3) For every t, e, z) [0, ) D e,z, all the eigenvalues of P e + x rt), z, ht, e, z)) have negative real parts bounded away from zero Then the origin of 40) is exponentially stable Let R v D v be the region of attraction of the autonomous system dτ = P f0, e 0, z 0, v + h0, e 0, z 0 )), and Ω v be a compact subset of R v Then, for each compact subset Ω e,z D e,z, there exists positive constants ɛ and T such that for all t > 0, e 0, z 0 ) Ω e,z, u 0 h0, e 0, z 0 ) Ω v, and ɛ 0, ɛ ), the system 3), 34), 38) has a unique solution xt, ɛ), zt, ɛ), x r t), ut, ɛ) on [0, ), and xt, ɛ) x r t) = Oɛ) holds uniformly for all t [T, ) Proof: Similar to the proof of Theorem 4, we show that satisfaction of the above assumptions imply satisfaction of those of Theorem 1 to get the desired conclusions In the same way as the proof of Theorem 4, it can be shown that the first assumption of Theorem 1 is implied by assumption 1 above For the second assumption, note that the first m equations of 39) represents m decoupled exponentially stable linear time

11 11 invariant systems with composite state e = [e T 1, e T,, e T m] T Hence for each i {1,,, m}, the solutions satisfy e i t) k ei e i t 0 ) exp λ ei t t 0 )), for some positive constants k ei and λ ei, for all t t 0 Using the facts that et) m e i t), i=1 and for all c 1 > c > 0, e i t) et), exp c 1 t t 0 )) exp c t t 0 )), t t 0, we have for all t t 0, m m et) e i t) k ei e i t 0 ) exp λ ei t t 0 )), i=1 et 0 ) i=1 m k ei exp λ ei t t 0 )), i=1 et 0 ) exp λ e t t 0 )) m k ei, i=1 = k e et 0 ) exp λ e t t 0 )), where 0 < λ e = min{λ e1, λ e,, λ em } and k e = m i=1 k ei Hence the verification of assumption proceeds as in the proof of Theorem 4 We will use Proposition 1 to show that the origin of the boundary layer system 40) is an exponentially stable equilibrium point, uniformly in t, e, z) Define gt, e, z, u) = P ft, e, z, u), so that the boundary layer system 40) can be rewritten as dτ Then, taking derivatives, = gt, e, z, v + ht, e, z)) g f t, e, z, u) = P t, e, z, u) = P e + x rt), z, u), and hence, g t, e, z, ht, e, z)) = P e + x rt), z, ht, e, z)) Hence assumption 3 implies that the eigenvalue condition 5) holds, and Proposition 1 applies to show that the boundary layer system has the origin as an exponentially stable equilibrium, uniformly in t, e, z) [0, ) D e,z The stated conclusions follow immediately from Theorem 1 in a similar manner to the proof of Theorem 4 CONCLUSIONS The statements of the ADI method are re-stated with some minor notational corrections, and the proofs are expanded This is to supplement our existing results in [] [4] As such, Theorem 1 and in [3] should be replaced by Theorem 4 and 5 in the present report respectively Also, Theorem 1 in [4] should be replaced by Theorem 4 in the present report ACKNOWLEDGMENTS The authors acknowledge Dr Eugene Lavretsky, cooriginator of the Approximate Dynamic Inversion method, for helpful comments in the preparation of the present report The first author gratefully acknowledges the support of DSO National Laboratories, Singapore Research funded in part by AFOSR grant FA REFERENCES [1] N Hovakimyan, E Lavretsky, and A Sasane, Dynamic inversion for nonaffine-in-control systems via time-scale separation part I, J Dyn Control Syst, vol 13, no 4, pp , Oct 007 [] J Teo and J P How, Equivalence between approximate dynamic inversion and proportional-integral control, in Proc 47th IEEE Conf Decision and Control, Cancun, Mexico, Dec 008, pp [3], Equivalence between approximate dynamic inversion and proportional-integral control, MIT, Cambridge, MA, Tech Rep ACL08-01, Sep 008, Aerosp Controls Lab [Online] Available: [4] J Teo, J P How, and E Lavretsky, On approximate dynamic inversion and proportional-integral control, in Proc American Control Conf, St Louis, MO, Jun 009, to appear [5] H K Khalil, Nonlinear Systems, 3rd ed Upper Saddle River, NJ: Prentice Hall, 00 [6] L Grüne, E D Sontag, and F R Wirth, Assymptotic stability equals exponential stability, and iss equals finite energy gain if you twist your eyes, Syst Control Lett, vol 38, no, pp , Oct 1999