Global output regulation through singularities Yuh Yamashita Nara Institute of Science and Techbology Graduate School of Information Science Takayama 8916-5, Ikoma, Nara 63-11, JAPAN yamas@isaist-naraacjp Alberto Isidori Washington University Department of Systems Science and Mathematics One Brookings Drive, St Louis, MO 6313, USA isidori@zachwustledu Abstract In this paper, global output regulation problem using state-feedback for nonlinear systems with no relative degree is concidered We introduce an invariant manifold on which tracking error tends to zero, and through which we derive a coordinate transformation of state By means of the forwarding design method, it is shown that a simple feedback on the new coordinate attains the global output regulation through singularities 1 Introduction In this paper, we present a new method of the global output regulation through singularities Tracking control for nonlinear systems that have no relative degree has been addressed by the approximating method[1] and the switching method, but the exact tracking problem for such systems has been a very difficult problem On the other hand, these systems may be stabilized to an equilibrium point using forwarding[, 5, 6] or backstepping In this work, we convert the tracking problem to a stabilization problem, in the framework of output regulation Output regulation of nonlinear systems has been studied actively in recent years The reference signals to be tracked are generated by a stable autonomous system, so called exo-system The first paper[7] to examine the nonlinear output-regulation problem considers local stability alone Global or semi-global stability for minimum-phase systems was considered in following papers[8, 9, 1] These studies assumed the system is minimum-phase, thus needing a relative degree We extend the target systems of global output regulation using state feedback to the system with no relative degree We introduce new sort of PDE that defines an invariant manifold on which tracking error tends to zero Using linear-growth conditions and some assumptions, it is proven that a use of feedback such that the state tends to the manifold attains the global output regulation for such a system, by means of the forwarding method 2 Problem statements We consider a tracking problem in a single-input singleoutput affine system in the following form: ẋ 1 = x 2 ẋ ρ = x ρ+1 + φ x, ξu ẋ ρ+1 = ψx, ξ+φ 1 x, ξu ξ = ηx, ξ 1 y = hx =x 1 2 x 1,,x ρ+1,ξ T T R n denotes a state vector, u Ran input, and y Ran output The system has an equilibrium point x =,ξ = such that ψ, = and η, = Note that we do not assume that φ x, ξ is nonzero for all x, ξ, which means the relative degree of the system may not exist We rewrite the system 1 shortly as d x = fx, ξ+gx, ξu 3 dt ξ The reference signal is generated by an exo-system ẇ = sw y d = γw 5 The exo-system has an equilibrium point w = which satisfies s = and γ = The exo-system is assumed to be Lyapunov stable, ie there exists Uw such that Uw > w, U = Uw w γw 6 The purpose of control is to find a controller with which tracking error e = y y d converges to zero, and all state variables of 1 are bounded The state may pass through the point such that φ x, ξ =, therefore we address an output regulation problem through singularities
3 Transformation of system We make the following assumption that is one of the solvability conditions for a local version of the tracking problem Assumption 1 There exist functions πw R n and cw which satisfy fπw + gπwcw = πw sw 7 w π 1 w =γw 8 Let π A w denote π 1 w,,π ρ w T, π B w = π ρ+1 w, and π C w =π ρ+2 w,,π n w T Moreover, we assume the existence of another invariant manifold x ρ+1 = π x 1,,x ρ,ξ,w including x T,ξ T T = πw Assumption 2 There exist functions π x, ξ, w R and c x, ξ, w Rsuch that ψ x, π x, ξ, w,ξ+φ 1 x, π x, ξ, w,ξc x, ξ, w ρ 1 π = x i+1 + π {π x, ξ, w x i=1 i x ρ + φ x, π x, ξ, w,ξc x, ξ, w} + π ξ η x, π x, ξ, w,ξ+ π w sw π x, ξ, w+φ x, π x, ξ, w,ξc x, ξ, w π ρ w sw+ x 1 π 1 w+ + ρ 1 x ρ π ρ w = 9 1 c π A w,π C w,w=cw 11 π π A w,π C w,w=π B w 12 φ 1 x, z, ξ π φ x, z, ξ x ρ 13 x denotes x 1,,x ρ T, and s are constants such that polynomial is Hurwitz s ρ + ρ 1 s ρ 1 + + 1 s + 1 Equations 9,1 mean that x ρ+1 π x, ξ, w =is an invariant manifold under the input u = c x, ξ, w, and on the manifold the error dynamics become d ρ e dt ρ + d ρ 1 e ρ 1 dt ρ 1 + + 1ė + e = 15 Under these assumptions, the system 1 can be transformed to x 1 = x 2 x ρ 1 = x ρ x ρ = x 1 ρ 1 x ρ + z + φ x, z, ξ, wũ + c x, ξ,w φ x,, ξ, w c x, ξ,w ż = ψ x, z, ξ, w ψ x,, ξ,w + φ 1 x, z, ξ, wũ + c x, ξ,w φ 1 x,, ξ, w c x, ξ,w π {z + x φ x, z, ξ,wũ + c x, ξ,w 16 ρ φ x,, ξ, w c x, ξ,w} π ξ { η x, z, ξ, w η x,, ξ,w} ξ = η x, z, ξ, w η,,,w x = x π A w 17 ξ = ξ π C w 18 π x, ξ,w =π x + π A w, ξ + π C w,w 19 z = x ρ+1 π x, ξ,w 2 φ x, z, ξ,w = φ x + π A w,z+ π x, ξ,w, ξ 21 + π C w φ 1 x, z, ξ,w = φ 1 x + π A w,z+ π x, ξ,w, ξ 22 + π C w ψ x, z, ξ, w = ψ x + π A w,z+ π x, ξ,w, ξ 23 + π C w η x, z, ξ, w = η x + π A w,z+ π x, ξ,w, ξ 2 + π C w c x, ξ,w =c x + π A w, ξ + π C w,w 25 ũ = u c x, ξ,w 26 By expanding φ, φ 1, ψ, and η as φ x, z, ξ, w = φ x,, ξ,w+p x, z, ξ,wz 27 φ 1 x, z, ξ, w = φ 1 x,, ξ,w+p 1 x, z, ξ,wz 28 ψ x, z, ξ, w = ψ x,, ξ, w+p 2 x, z, ξ, wz 29 η x, z, ξ, w = η,,,w+e 1 ξ,w ξ + E 2 x, ξ,w x + E 3 x, z, ξ, 3 wz, the system 16 becomes x 1 = x 2 x ρ 1 = x ρ x ρ = { x 1 ρ 1 x ρ } + z + φ x, z, ξ,wũ + c x, ξ,wp 31 x, z, ξ,wz
ż = θ x, z, ξ, wz { + φ 1 x, z, ξ, w π } φ x, z, x ξ, w ũ ρ ξ = E 1 ξ,w ξ + E 2 x, ξ,w x + E 3 x, z, ξ, wz, θ x, z, ξ, w =P 2 x, z, ξ, w π π x ρ ξ E 3 x, z, ξ, w { + c x, ξ,w P 1 x, z, ξ,w π } P x, z, x ξ, w ρ 32 If one can stabilize 31 globally, then global output regulation problem for system 1,2 is solvable By applying a feedback ũ = θ x, z, ξ, wz + v φ 1 x, z, ξ, w π x ρ φ x, z, ξ, w the system 31 is converted to 33 ξ = E 1 ξ,w ξ + E 2 x, ξ,w x + E 3 x, z, ξ, wz 3 1 x = 1 x ρ 1 35 + + v δ x, z, ξ,wz ζ x, z, ξ, w ż = v 36 ζ x, z, ξ, w = φ x, z, ξ, w φ 1 x, z, ξ, w π x ρ φ x, z, ξ, w δ x, z, ξ, w =1+ c x, ξ,wp x, z, ξ,w ζ x, z, ξ, wθ x, z, ξ, w 37 38 The system 3,35,36 is not a feedforward system because ξ is included in equation 35 However, the terms including ξ vanish in 35 when z = and v = Therefore, a method similar to forwarding design can be applied to this system Global output regulation with state feedback In this section, global stabilization of the system 3, 35,36 will be discussed, after we prove the following lemma: Lemma 3 Consider the system ẋ = fx, w+gx, z, wz ż = qz ẇ = sw 39 ẇ = sw is stable in the definition of Lyapunpov, ẋ = fx, w is globally asymptotically stable with a radially unbounded Lyapunov function W x, w which satisfies W x, w > x, W,w=, fx, w+ sw < x, 1 x w x x Kw W x, w, 2 Kw >, for x M>, and ż = qz is globally asymptotically stable and locally exponentially stable with a radially unbounded Lyapunov function Uz Moreover, gx, z, w satisfies linear-growth condition with respect to x Then, x T,z T T converges to the origin Proof: First of all, boundedness of x will be shown Because gx, z, w satisfies linear-growth condition with respect to x, there exist G z,w and G 1 z,w such that gx, z, w G z,w+g 1 z,w x 3 By evaluating the value of W Ẇ = {fx, w+gx, z, wz} + x w sw gx, z, wz x x {G z,w z + G 1 z,w x z } W x, wkwg z,w+g 1 z,w z, x max1,m is obtained Because of the exponential stability of ż = qz and the stability of ẇ = sw, there exist functions γ x,w >, γ 1 x,w >, K max w > and a positive constant such that Kw K max w 5 G z,w z γ z,w exp t 6 G 1 z,w z γ 1 z,w exp t 7 Therefore, Ẇ γ 2 z,w exp tw = K max w{γ z,w + γ 1 z,w} exp tw, x max1,m 8
is obtained, which derives W x, w exp γ 2 z,w exp τdτ W x,w { } γ2 z,w = exp 1 exp t W x,w γ2 z,w exp W x,w 9 This equation shows that W xt,wt is bounded The value of xt is also bounded because the function W is radially unbounded Secondly, consider the following integral: Ψx,z,w = xτ,wτ gxτ,zτ,wτzτ dτ 5 xτ The existence of Ψ can be proved by the boundedness of xt, wt and exponential stability of ż = qz Indeed, there exists γ 3 x,z,w such that x gx, z, w z 51 γ 3 x,z,w exp t, and thus the integral can be evaluated as Ψx,z,w γ 3 x,z,w = γ 3x,z,w exp τ dτ 52 We will show that the function Ψ belongs to C -class Let x t,z t,w t denote the trajectory of system 39 for an initial state X =x T,z T,w T T For given ɛ>, there exists T such that gx, z, wz T x dt < ɛ, 53 x T,z T,z T T X < 1 because the trajectories satisfy 51 Moreover, from the continuity of the trajectory with respect to the initial state, we can show the existence of <d 1 such that T x gx, z, wz x,w gx,z,w z x < ɛ 2, xt,z T,z T T X <d dt 5 Hence, for given ɛ there exists d such that x gx, z, wz x,w gx,z,w z x dt <ɛ, x T,z T,z T T X <d, 55 which means the continuity of the function Ψ Adding the function Ψ to the Lyapunov functions W and U makes a candidate of the Lyapunov function for the system 39 V x, z, w =W x, w+uz+ψx, z, w 56 The identical equation W xs,xs = W xt,wt + s t Ẇ xτ,zτ,wτ dτ 57 implies { W x, w+ψx, z, w = lim W xs,ws s s } fxτ,wτ + t x w swτ dτ 58 This equation shows that V x, z, w Because 58 holds, V x, z, w = means Uz =andw x, w + Ψx, z, w = Hence, V x, z, w = implies z = and x = by the definition of Ψ Therefore, V x, z, w is positive definite with respect to x and z Let us prove V x, z, w is radially unbound with respect to x and z It is obvious from 58 that V tends to infinity when z tends to infinity So, we only have to show that W +Ψ x The derivative of the right-hand side of 58 can be evaluated as follows: Ẇ fx, w sw = gx, z, wz x w x x G x, z, w z + G 1 x, z, w x z x γ z,w + γ 1 z,w x exp t = x [{γ z,w + γ 1 z,w} x +1 x γ z,w] exp t [ γ z,w + γ 1 z,w x x ] + γ z,w max x 1 x exp t 59
The above inequality becomes Ẇ fx, w x w sw [γ z,ww + γ 5 z,w] exp t when x M [γ 6 z,w + γ 5 z,w] exp t when x <M 6 γ x,z = 61 γ z,w + γ 1 z,wk max w γ 5 x,z = γ z,w max x <1 x 62 γ 6 x,z = γ z,w + γ 1 z,w sup x M x x 63 Therefore, W x,w + Ψx,z,w exp γ W x,w γ 5 γ 6 67 is established, which means that W +Ψ tends to infinity when x tends to infinity Hence, V x, z, w is radially unbounded with respect to x and z Finally, by construction V = U z qz+ x fx, w+ sw 68 w holds, so the system is globally asymptotically stable with respect to x and z The simple version of this lemma without w is a wellknown theorem We apply this lemma to the global stabilization problem for 3,35,36, with several assumptions We can integrate 6 as W exp γ 1 expt W x,w + exp γ expτ expt γ 5 exp τ+ x fxτ,wτ exp + + w swτ dτ W x,w γ 5 fxτ,wτ + x w swτdτ when xτ M τ t, γ W W x,w + γ 5 + γ 6 exp τ + fxτ,wτ + x w swτ dτ W x,w γ 5 γ 6 + fxτ,wτ + x w swτdτ when xτ <M τ t Combining 6 and 65 generates an inequality W fxτ,wτ + x w swτdτ exp γ W x,w γ 5 γ 6 6 65 66 The system dynamics restricted on x =,z = is written as ξ = E 1 ξ,w ξ 69 The above dynamics correspond to the zero-error dynamics of I/O-linearizable systems φ = Assumption The system 69 is globally asymptotically stable with a Lyapunov function W ξ,w, ie there exists a Lyapunov function W ξ,w such that W,w = 7 W ξ,w >, ξ 71 ξ E 1 ξ,w ξ + w sw <, ξ 72 Moreover, there exists a positive function K w and a positive constant M such that ξ ξ <K ww, ξ M 73 Let us consider the stability of the dynamics restricted on z ξ = E 1 ξ,w ξ + E 2 x, ξ,w x 1 x = 1 x = A x 7 ρ 1 To guarantee the stability of the system 7, we assume the following linear-growth condition:
Assumption 5 The cross term E 2 x, ξ,w x satisfies linear-growth condition with respect to ξ, ie there exist positive functions b x, w, b 1 x, w such that E 2 x, ξ,w b 1 x, w ξ + b x, w 75 Under these assumptions the following theorem is established Theorem 6 Under assumptions 1, 2,, and 5, the system 7 is globally asymptotically stable Proof: The proof is straightforward from lemma 3 The Lyapunov function of 7 is V x, ξ,w =W ξ,w E 2 x + ξ E 1 ξ,w ξdτ + 1 2 xt P x 76 P is a matrix satisfying PA+ A T P = I 77 To stabilize the dynamics of z, we adopt a feedback v = kz k is a positive constant Further assumptions are required for global stability Assumption 7 There exist M 1 and K 1 w such that the Lyapunov function V satisfies V x, ξ x ξ K 1 wv x, ξ,w, 78 x, ξ M 1 References [1] J Hauser, S Sastry, and P Kokotović: Nonlinear control via approximate input-output linearization: The ball and beam example, IEEE Trans on Autom Cont, AC-37, pp 392 398, 1992 [2] C J Tomlin and S S Sastry: Switching through Singularities, Proc of the 36th IEEE CDC San Diego, pp 1 6, 1997 [3] R Hirschorn and J Davis: Output tracking for nonlinear systems with singular points, SIAM J of Cont and Optim, Vol25, No3, pp 57 557, 1987 [] A R Teel: A nonlinear small gain theorem for the analysis of control systems with saturation, IEEE Trans on Autom Cont, AC-1, pp 1256 127, 1996 [5] F Mazenc and L Praly: Adding integrations, saturated controls, and stabilization for feedforward systems, IEEE Trans on Autom Contr, AC-1, pp 1559 1578, 1996 [6] R Sepulchre, M Janković, and P Kokotović: Constructive Nonlinear Control, Springer, London, 1997 [7] A Isidori and C I Byrnes: Output regulation of nonlinear systems, IEEE Trans on Autom Cont, AC- 35, pp 131 1, 199 [8] H K Khalil: Robust servomechanism output feedback controllers for feedback linearizable systems, Automatica, Vol3, pp 1587 1599, 199 [9] A Isidori: A remark on the problem of semiglobal nonlinear output regulation, IEEE Trans on Autom Cont, AC-2, pp 173 1738, 1997 [1] A Serrani and A Isidori: Global output regulation for a class of nonlinear systems, Proc of the 38th IEEE CDC Phoenix, pp 119 1195, 1999 Assumption 8 The functions ζ, δ and E 3 linear-growth condition with respect to x and ξ satisfy Now, the main theorem is obtained Theorem 9 Under the assumptions 1, 2,, 5, 7, and 8, the global output regulation problem is solved by a feedback v = kz k is a positive constant Proof: Under the feedback, z tends to zero exponentially Therefore, global stability is obvious from lemma 3 5 Conclusion We presented a method for global output regulation for nonlinear systems that have no relative degree Future work will extend this method to error feedback case