Output Regulation of Uncertain Nonlinear Systems with Nonlinear Exosystems

Output Regulation of Uncertain Nonlinear Systems with Nonlinear Exosystems Zhengtao Ding Manchester School of Engineering, University of Manchester Oxford Road, Manchester M3 9PL, United Kingdom zhengtaoding@manacuk Abstract An adaptive control algorithm is proposed for output regulation of uncertain nonlinear systems in output feedback form under disturbances generated from nonlinear exosystems A new nonlinear internal model is proposed to generate the desired input term for suppression of the disturbances The proposed internal model design is based on boundedness of the disturbance, high gain design and saturation and the designed internal model is capable to tackle disturbances in any specified initial conditions The uncertainty in the system allows the all the parameters to be unknown, except the high frequency gain, and it is specified in a vector of constant unknown parameters, which is tackled using nonlinear dominant functions and an adaptive control coefficient The proposed control algorithm ensures the global convergence of the state variables to the invariant manifold, which implies that the measurement or the tracking error approaches to zero asymptotically Index Terms Output regulation, Internal model principle, Uncertainty, Nonlinear systems, Adaptive control, Backstepping I INTRODUCTION Output regulation concerns with stabilization of dynamic systems as well rejecting the disturbances or tracking the desired trajectories The output regulation problem is well posed and solved for linear systems in [], [2] For nonlinear systems, an important contribution to output regulation is reported in [3], [4] that the necessary and sufficient conditions for the existence of a local full information solution are specified as that the linearized system is stabilizable and there exists a certain invariant manifold A semiglobal extension to these results for a class of feedback linearizable systems is reported in [5] using a saturated high-gain observer [6] Output regulation of error feedback is solved [7], [8] with the application of system immersion technique The uncertainty parameterized by unknown constant parameters are treated as special cases of exogenous signals and the solution, extended from the error feedback regulation, is referred to as structurally stable regulation A semiglobal adaptive output feedback control is presented in [9] for nonlinear systems represented by inputoutput models, using a high-gain observer Global solutions for output regulation using state or partial state feedback are shown for strict feedback systems in [] and for extended strict feedback systems in [] Global output regulation for output feedback system is reported in [2] Global adaptive output regulation for the output feedback systems is shown in [3] A common assumption in the global output regulation results via measurement feedback [2], [3] is that the exosystem is linear The linear dynamics of the exosystem are convenient for the transformation and reformulation of the exosystem model so that a suitable internal model can then be designed for output regulation [4], [5], [6] In this paper, we consider global output regulation of uncertain nonlinear systems with a class of nonlinear exosystems The difficulty encountered in designing internal models for output regulation is that the disturbances are not measured directly, and they are always mixed with the state variables of the system The nonlinear dynamics in the exosystem make the design of the internal model more difficult We propose a novel design of a high-gain internal model which exploits the boundedness of the disturbances and the high gain dominance with delicately defined saturation levels The proposed internal model design is then used together with nonlinear adaptive control techniques to provide a solution to the output regulation with nonlinear exosystems The proposed control algorithm guarantees the convergence of the state variable to the invariant manifold, which also implies that the tacking error converges to zero II PROBLEM FORMULATION We consider a single-input-single-output nonlinear system which can be transformed into the output feedback form with ẋ = A c x + φ(y, w, a) + bu y = Cx e = y q(w) () A c =, b = C = [ ], b ρ where x R n is the state vector, u R is the control, y R is the output, e is the measurement output, a R q and b R n are vectors of unknown parameters, φ(y, w, a) is a smooth vector field with each element being polynomials of b n

its variables and satisfying φ(, w, a) =, q is an unknown polynomial of w, w R m are disturbances, and they are generated from an exosystem ẇ = s(w) (2) Assumption : The system is minimum phase, ie, the polynomial B(s) = n i=ρ b is n i is Hurwitz, and the high frequency gain b ρ is known Remark : We assume that b ρ is known to simplify the presentation In the case of unknown b ρ, a Nussbaum gain may be designed in a similar way as in [7] Assumption 2: The flows of vector field s(w) are bounded and converge to periodic solutions The adaptive output regulation problem that we are going to solve is to find a finite dimensional system = ν(, e(t)), R s, u = u(, e(t)) (3) such that for every x() R n, w() Ω R m, () R s, x(t), (t) and u(t) are bounded t, and lim t e(t) = III STATE TRANSFORMATION For the system () with relative degree ρ >, the following filter is introduced [8] ξ = λ ξ + ξ 2 ξ ρ = λ ρ ξ ρ + u (4) where λ i > for i =,, ρ are the design parameters Define the filtered transform z = x [ d d ρ ]ξ (5) where ξ = [ξ ξ ρ ] T, di R n for i =,, ρ and they are recursively generated by d ρ = b and d i = [A c + λ i+ I] d i+ for i = ρ 2,, The system () is then transformed to z = A c z + φ(y, w, a) + dξ y = C z (6) where d = [A c + λ I] d It can be shown that d = b ρ and D(s) := n i= ρ d i s n i = B(s) (s + λ i ) (7) i= With ξ as the input, the system (6) is with relative degree one and minimum phase We introduce another state transform to extract the internal dynamics of (6) with z R n given by z = z 2:n d 2:n d y (8) where ( ) 2:n refers to the vector or matrix formed by the 2nd row to the nth row With the coordinates {z, y}, (6) is rewritten as ż = Dz + ψ(y, w, θ) ẏ = z + ψ y (y, w, θ) + b ρ ξ (9) where the unknown parameter vector θ = [a T, b T ] T, and D is the companion matrix of d given by d 2 /d D =, d n /d and ψ(y, w, θ) = D d 2:n d y + φ 2:n (y, w, a) d 2:n d φ (y, w, a) ψ y (y, w, θ) = d 2 d y + d 2:n d φ (y, w, a) Notice that D is Hurwitz, from Assumption and (7), and that the dependence of d on b is reflected in the parameter θ in ψ(y, w, θ) and ψ y (y, w, θ) and it is easy to check that ψ(, w, θ) = and ψ y (, w, θ) = IV INTERNAL MODEL The output regulation problem considered in this paper is well posed if the following assumption is satisfied Assumption 3: There exists an invariant manifold π(w) R n satisfying π(w) s(w) = Dπ(w) + ψ(q(w), w, θ) () w Based on Assumption 3, we have q(w) w s(w) = π (w) + ψ y (q(w), w, θ) + b ρ α () With ξ bing viewed as the input, α is the feedforward term used for output regulation to tackle the disturbances, and it given by α(w, θ) = b ρ [ q(w) w s(w) π (w) ψ y (q(w), w, θ)] (2) We now introduce the last transform based on the invariant manifold with z = z π (3) Finally we have the model for the control design z = D z + ψ ė = z + ψ y + b ρ (ξ α(w)) ξ = λ ξ + ξ 2 ξ ρ = λ ρ ξ ρ + u (4)

where ψ = ψ(y, w, θ) ψ(q(w), w, θ) ψ y = ψ y (y, w, θ) ψ y (q(w), w, θ) To solve the problem, we need an assumption on the structure of the exosystem Assumption 4: There exists an immersion of the exosystem η = F η + ϕ(α) α = Hη (5) where η R r and F =, H = Remark 2: An example of exosystem in (5) is a Van de Pol equation η = η 2 η 3 + η η 2 = η Since the state in the internal model η is unknown, we design the internal model ˆη = (F ggkh)(ˆη b ρ ggke) + ϕ(ˆη b ρ ghgke) + ggkξ (6) where g is a design parameter, G = diag{, g,, g r }, k R r is chosen such that F o = F kh is Hurwitz, ϕ( ) : R r R r is a decentralized C ρ soft saturation function with l u,i () if ϕ i () M l,i ϕ i () = ϕ i () if ϕ i () < M l,i (7) l l,i () if ϕ i () M l,i The saturation levels are chosen such that max w Ω ϕ i(α(w)) < M l,i < M u,i < max w Ω, R ϕ i(α(w) ) (8) with M u,i denoting another positive real design parameter The functions l u,i and l l,i, for i =,, r, are (ρ ) order Hermite-Birkoff interpolation functions [9] between the points of ( p,i, M l,i ) and ( p,i + i, M u,i ), and between ( n,i, M l,i ) and ( n,i i, M u,i ) respectively with i being a positive real design parameter, and p,i and n,i being the values such that ϕ i ( p,i ) = M l,i and ϕ i ( n,i ) = M l,i The Hermite-Birkoff interpolation functions ensure that the derivatives match up to the specified order The boundary conditions for ( p,i, M l,i ) are the derivatives of ϕ i at p,i, and at the upper point ( p,i + i, M u,i ), the derivatives are set to zero Similar settings apply to the points ( n,i, M l,i ) and ( n,i i, M u,i ) Therefore the soft saturation function ϕ i has its values bounded in T, [ M u,i, M u,i ] and it has continuous derivatives up to order ρ Remark 3: If ϕ i is upper bounded or lower bounded, then the respective soft saturation function is not needed Note that the right hand side of (8) can be infinity In such a case, we only need to ensure max w Ω ϕ i (α(w)) < M l,i < M u,i If we define the auxiliary error it can be shown that η = η ˆη + b ρ ggke (9) η = (F ggkh) η + ϕ(η ) ϕ(η η ) +b ρ ggk( z + ψ y ) (2) To analyze the property of the internal model, we define a scaled auxiliary error It can be shown that ζ = G η (2) ζ = g(f kc)ζ + G [ϕ(η ) ϕ( η η )] +b ρ gk( z + ψ y ) (22) To complete the internal model design, we need a result given in the following lemma Lemma : For a smooth ϕ i and the saturation function defined in (8), there exists an upper bound of the expression ϕi(η) ϕi(η ), for all w Ω, R, and η = α(w) being a smooth function of w Proof: Let us consider in two cases Case, ϕ i (α(w) ) M l,i From the definition of M l,i we have max w Ω ϕ i (α(w)) < M l,i Since ϕ i is continuous, which is implied by the smoothness, there exists a positive real value ɛ, such that > ɛ In this case we have ϕ i (η ) ϕ i (η ) < 2M u,i ɛ (23) Case 2, the saturation function is not activated, ie, ϕ i (α(w) ) < M l,i In this case, we have Hence, we have ϕ i (η ) ϕ i (η ) = ϕ i (η ) ϕ i (η )) ϕ i (η v) = dv v [ ] ϕi (η + s) = dv (24) s s= v = ϕ i (η ) σ(ϕ i (η )) [ ] ϕi (η + s) dv (25) s s= v

From the smoothness of ϕ i, the integration in (25) exists for any finite value of, and therefore the integral exists for ɛ For > ɛ, we still have ϕ i (η ) ϕ i (η ) < 2M u,i ɛ (26) Combining both the cases, we conclude the proof of the lemma With the result shown in Lemma, we complete the design of the internal model by setting the gain g as where ḡ = max i g > 2λ max (P F )rḡ + 2 (27) max w Ω, R ϕ i (η ) ϕ i (η )) (28) and P F is a positive definite matrix, satisfying P F (F kh) + (F kh) T P F = I V CONTROL DESIGN If the system () is of relative degree one, then ξ in (4) is the control input For the systems with higher relative degrees, adaptive backstepping will be used to find the final control output u from the desirable value of ξ Supposing that ˆξ is desirable value for ξ, we define From (4) we have ˆξ = b ρ ξ (29) ė = z + ξ + b ρ ξ b ρ η + ψ y (3) where ξ = ξ ˆξ Sine the nonlinear functions involved in ψ and ψ y are polynomials with ψ(, w, θ) = and ψ y (, w, θ) =, w is bounded, and the unknown parameters are constants, it can be shown that ψ < r z ( e + e p ) (3) ψ y < r y ( e + e p ) (32) where p is a known positive integer, depending on the polynomials in ψ and ψ y, and r z and r y are unknown positive real constants We now design the virtual control ˆξ as, with c >, ξ = c e ˆl (e + e 2p ) + b ρˆη gk e (33) where ˆl is an adaptive coefficient with ˆl () = Using (9), we have the resultant error dynamics ė = c e ˆl (e + e 2p ) + z +b ρ ξ b ρ η + ψ y (34) The adaptive law is given by ˆl = γ l (e 2 + e 2p ) (35) where γ l is a positive real design parameter If the relative degree ρ =, we set u = ˆξ For ρ >, adaptive backstepping can be used to obtain the following results: ( ˆξ 2 = b ρ e c ξ l ˆξ ) 2 ξ e + ˆξ e [b ρ(ξ ˆη ) + gk e] + ˆξ ˆη ˆη + ˆξ ˆl (36) ˆl ˆξ i = ξ i 2 c i ξi l i ( ˆξ i e + ˆξ i e [b ρ(ξ ˆη ) + gk e] + ˆξ i ˆη ) 2 ξi ˆη + ˆξ i ˆl ˆl for i = 3,, ρ where ξ i = ξ i ˆξ i for i =,, ρ 2, c i and l i, i = 2,, ρ, are positive real design parameters, and τ i, for i =,, ρ 2, are tuning functions Finally we design the control input as u = ˆξ ρ (37) VI STABILITY ANALYSIS In this section, we shall establish the boundedness of all the variables and the convergence to zero of the measurement output We start with the analysis of the internal model by defining V = ζ T P F ζ (38) Using (22), we have V = gζ T ζ + 2ζ T P f {G [ϕ(η ) ϕ(η η )] +b ρ gk( z + ψ y )} gζ T ζ + 2λ max (P F )rḡζ T ζ +2 b ρ g P F k ζ ( z + ψ y ) ζ T ζ + (b ρ g P F k ) 2 ( z + ψ y ) 2 ζ T ζ + 2(b ρ g P F k ) 2 z 2 +2(b ρ g P F k γ y ) 2 ( e + e p ) 2 ζ T ζ + 2(b ρ g P F k ) 2 z 2 +4(b ρ g P F k γ y ) 2 (e 2 + e 2p ) (39) Define a Lyapunov function candidate V = β V + β 2 z T P z z + ρ 2 [e2 + ξ i 2 + γ l (l ˆl ) 2 ] (4) i= where β and β 2 are two positive reals, P z is a positive definite matrix satisfying P z D + D T P z = I With the design of ˆξ i, for i =,, ρ, the dynamics of ξ i can be easily evaluated From the dynamics of z in (4) and

the dynamics of η in (2), virtual controls and adaptive laws designed in the previous section, we have the derivative of V as V = β [ ζ T ζ + 2(b ρ g P F k ) 2 z 2 +4(b ρ g P F k γ y ) 2 (e 2 + e 2p )] +β 2 [ z T z + 2 z T P z ψ] +(ˆl l )(e 2 + e 2p ) c e 2 ˆl (e 2 + e 2p ) +e z + e ψ y eb ρ ζ ρ + i= [ c i ξ2 i k i ( ˆξ i e )2 ξ2 i ξ i ˆξ i e z ξ i ˆξ i e ψ y + ξ i ˆξ i e b ρζ ] (4) The stability analysis can be proceeded by using the inequalities 2xy < rx 2 + y 2 /r or xy < rx 2 + y 2 /(4r) for x >, y > and r being any positive real, to tackle the cross terms between the variables z, ζ, e, ξi, for i =,, ρ It can be shown that there exists a sufficiently big positive real β and then a sufficiently big positive real β 2 finally the sufficient big l such that the following result holds V 2 β ζ T ζ 4 β ρ 2 z T z c e 2 c i ξ2 i (42) i= The boundedness of V further implies ζ, z, e, ξi L 2 L for i =,, ρ, and the boundedness of ˆl Since the disturbance w is bounded, e, z, ζ L implies the boundedness of y, z and ˆη, which further implies the boundedness of ˆξ and then the boundedness of ξ The boundedness of ˆξ and ξ, together with the boundedness of e, ˆη, ˆl and ˆb ρ implies the boundedness of ˆξ2 and then the boundedness of ξ 2 follows the boundedness of ξ 2 Applying the above reasoning recursively, we can establish the boundedness of ˆξ i for i > 2 to i = ρ We then conclude that all the variables are bounded The boundedness of all the variables implies the boundedness of ζ, z, ė, and ξi, which further implies, together with ζ, z, e, ξi L 2 L and Barbalat s lemma, lim t ζ =, lim t z =, lim t e(t) = and lim t ξi = for i =,, ρ The above analysis proves the following result Theorem : For a system () satisfying the Assumptions to 4, the output regulation problem is globally solvable In particular, the feedback control system consists of the ξ-filters (4), the adaptive internal model (6), the parameter adaptive law (35) and the feedback control (37) VII AN EXAMPLE We use a simple example to illustrate the proposed control design, concentrating on the design of nonlinear internal model Consider a first order system ẏ = θy 2 w + u e = y (43) where θ is an unknown parameter, the disturbance w is generated by ẇ = w 2 + w w 3 ẇ 2 = w (44) with w η, a known constant It is easy to see that q(w) =, π(w) = and α(w) = w (45) From the exosystem and the desired feedforward input α, it can be seen that Assumption 4 is satisfied with w = η, and that the exosystem (44) is in the format of (5), with φ (η ) = η η 3 φ 2 (η ) = η (46) Following (6), we design the internal model [ ] [ ] [ ˆη k ˆη = ] y ˆη 2 k 2 ˆη 2 k 2 y [ φ (ˆη + k ] [ ] y) k φ 2 (ˆη k + u (47) y) k 2 where k = gk, k 2 = g 2 k 2 Since the system is of relative degree one, we design the functions φ i () as σ i (φ i ()) for i =, 2 with σ i being the saturation functions with saturation levels at η 3 η and η respectively The control input is given by where u = cy ˆl (y + y 3 ) + ˆη gk y (48) ˆl = y 2 + y 4, with ˆl () = (49) Finally we decide the value of g From the functions φ i, i =, 2, we have ḡ = + 3 2 η The matrix P F can be decided once the gain k is decided For this simple example, we have [ ] P F = + k2 k k 2 k + k k 2 + (5) k k 2 It is then straightforward to calculate g using (27) with r = 2 For simulation study, we set c =, k = 3, k 2 = 2, and we take η = 2 based on the characteristics of the exosystem With the chosen k and η, we obtain g = 23 In the simulation study, we found that this g value is very conservative, and the results shown in this paper were obtained with g = 2 The system output and input are shown in Figure, while the disturbance generated from the exosystem and its estimate generated from the internal model are shown in Figure 2 As shown in the figures, the internal model successfully reproduces the feedforward control needed after a transient period, and the system output is regulated to zero, as required

y η & estimate 2 8 6 4 2 2 2 4 6 8 2 4 6 8 2 time(s) u 2 2 4 6 8 2 4 6 8 2 4 6 8 2 time(s) 5 5 5 Original Fig System output y and input u Estimate 5 2 4 6 8 2 4 6 8 2 time(s) Fig 2 Disturbance η and its estimate ηˆ VIII CONCLUSIONS We have proposed a control design for output regulation of uncertain systems with nonlinear exosystems The success of the proposed design relies on the design of a new nonlinear internal model, which incorporates in the high gain design with saturation The proposed control design only uses the structural information of the system, and the high frequency gain, and no knowledge is needed of any other system parameters Even though the adaptive control techniques are used for the control design, the unknown system parameters are not estimated, unlike a completely adaptive treatment in [7] The system uncertainties are tackled by using high order polynomials which dominate the uncertainties This robust way of tackling uncertainties makes it possible to deal with the unknown disturbances in the measurement, which is not considered in [7] With one coefficient being made adaptive, other control design coefficients involved in the proposed algorithm can be any positive reals The proposed control algorithm ensures the convergence of state variables to the invariant manifold globally, and the measurement output approaches zero asymptotically REFERENCES [] E J Davison, The robust control of a servomechanism problem for linear time-invariant multivariable systems, IEEE Trans Automa Contr, vol 2, no, pp 25 34, 976 [2] B A Francis, The linear multivariable regulator problem, SIAM J Contr Optimiz, vol 5, pp 486 55, 977 [3] A Isidori and C I Byrnes, Output regulation of nonlinear systems, IEEE Trans Automa Contr, vol 35, no 2, pp 3 4, 99 [4] J Huang and W J Rugh, On a nonlinear multivariable servomechanism problem, Automatica, vol 26, no 6, pp 963 972, 99 [5] H K Khalil, Robust servomechanism output feedback controllers for a class of feedback linearizable systems, Automatica, vol 3, no, pp 587 599, 994 [6] F Esfandiari and H K Khalil, Output feedback stabilisation of fully linearizable systems, International Journal of Control, vol 56, pp 7 37, 992 [7] A Isidori, Nonlinear Control Systems, Springer-Verlag, Berlin, 3 edition, 995 [8] C I Byrnes, F D Priscoli, and A Isidori, Regulation of Uncertain Nonlinear Systems, Birhäuser, Boston, 997 [9] H K Khalil, Adaptive output feedback control of nonlinear systems represented by input-output models, IEEE Trans Automa Contr, vol 4, no 2, pp 77 88, 996 [] R Marino, P Tomei, I Kanellakopoulos, and P V Kokotovic, Adaptive tracking for a class of feedback linearizable systems, IEEE Trans Automat Contr, vol 39, pp 34 39, 994 [] R A Freeman and P V Kokotovic, Tracking controllers for systems linear in unmeasured states, Automatica, vol 32, no 5, pp 735 746, 996 [2] A Serrani and A Isidori, Global robust output regulation for a class of nonlinear systems, Systems & Control Letters, vol 39, pp 33 39, 2 [3] Z Ding, Global output regulation of uncertain nonlinear systems with exogenous signals, Automatica, vol 37, pp 3 9, 2 [4] V O Nikiforov, Adaptive non-linear tracking with complete compensation of unknown disturbances, European Journal of Control, vol 4, pp 32 39, 998 [5] Z Ding, Global output regulation of a class of nonlinear systems with unknown exosystems, in Proceedings of 4th IEEE Conference on Decision and Control, Orlando, USA, 2, pp 65 7 [6] X Ye and J Huang, Decentralized adaptive output regulation for large-scale nonlinear systems, in Proceedings of IFAC Symposium on Nonlinear Control Design, St Peterburg, Russian, 2, pp 656 66 [7] Z Ding, Universal disturbance rejection for nonlinear systems in output feedback form, IEEE Trans Automatic Control, vol 48, no 7, pp 222 226, 23 [8] R Marino and P Tomei, Global adaptive output feedback control of nonlinear systems, part i: Linear parameterization, IEEE Trans Automat Contr, vol 38, pp 7 32, 993 [9] A Quarteroni, R Sacco, and A Saleri, Numerical Mathematics, Springer, New York, 2