Further Results on Adaptive Robust Periodic Regulation

Proceedings of the 7 American Control Conference Marriott Marquis Hotel at Times Square New York City USA July -3 7 ThA5 Further Results on Adaptive Robust Periodic Regulation Zhen Zhang Andrea Serrani Abstract In this paper we extend previous results for the linear output regulation problem with parameter-dependent periodic exosystems to higher relative degree minimum-phase plant models By means of a dynamic extension and a global change of coordinates the original system can be put in an error system such that an adaptive periodic internal model approach can be adopted Moreover the overall system can be put in a suitable form for which robust backstepping design is applicable Persistence of excitation is not required for the applicability of the method Simulation results illustrate the proposed methodology I INTRODUCTION A recent research direction in output regulation of linear and nonlinear systems regards the extension of the classes of exosystems that can be dealt with internal model-based design These latter include parameter-dependent linear systems [7] [] special classes of nonlinear systems [] [8] and linear periodic time-varying exosystems [] More recently an adaptive robust internal model-based design for linear minimum-phase systems driven by parameter dependent periodic exosystems has been proposed in [] where due to space limitation only the construction of robust regulators for unitary relative degree plant models has been given The aim of this paper is to extend the result in [] to higher relative degree plant models As shown in [] for minimum-phase systems under appropriate observability condition a non-minimal realization for the construction of the internal model is instrumental in robust regulator design which does not require the persistence of excitation (PE) for error convergence In this paper we provide a solution via a dynamic extension to transform the original system into a suitable form hence an adaptive periodic internal model can be added and the robust regulator design of the overall system can be dealt with by using a robust backstepping design The paper is organized as follows: In Section II we give the formulation of the problem In Section III we transform the plant model into a particular error system via dynamic extension and make some assumptions on parameter dependent system immersion In Section IV the adaptive periodic internal model in [] is adopted and the robust regulator design of the overall system is dealt with by using robust adaptive backstepping with tuning functions The proposed solution is applicable for minimum-phase plant models with relative degree higher than one This work has been supported by the AFRL/AFOSR Collaborative Center of Control Science under Grant no F3365---354 The authors are with the Department of Electrical and Computer Engineering The Ohio State University Columbus OH Corresponding author Email: serrani@eceosuedu II PROBLEM FORMULATION In this paper we consider an output regulation problem in the form ẇ = S(tσ)w ẋ = A (µ)x + A (µ)y + P (tµ)w + B (µ)u ẏ = A (µ)x + a (µ)y + P (tµ)w e = y + Q(tµ)w () where (xy) R n R denotes the state of the plant u R the control input e R the regulated error and w R s the state of the exosystem The parameter vectors µ R p and σ R satisfy µ K µ and σ K σ respectively where K µ R p and K σ R are given compact sets The matrix-valued functions S( ) P ( ) P ( ) and Q( ) are assumed to be smooth The plant is assumed to be robustly minimum-phase with respect to the output y in a sense to be specified in Section III We restrict our study to the case of systems whose relative degree is greater than one as the case of unitary relative degree has been studied in [] Therefore we assume that the system () has a uniform relative degree r with respect to the output y for all µ K µ Assumption : The following condition A (µ)a i (µ)b (µ) = holds for i = r 3 and every µ K µ In addition as in [] the exosystem is assumed to satisfy the following properties: Assumption : ) There exists T > such that S(t + Tσ) = S(tσ) P (t + Tµ) = P (tµ) P (t + Tµ) = P (tµ) and Q(t + Tµ) = Q(tµ) for all t all µ K µ and all σ K σ ) The solution w(t;w σ) of () satisfies w(t + T;w σ) = w(t;w σ) for any (w σ) R s K σ and all t We consider the adaptive robust regulation problem defined in [] that is to design a continuous T -periodic dynamic error-feedback controller of the form ζ = F(t ˆθ)ζ + G(t ˆθ)u ˆθ = ϕ(tζe ˆθ) u = u(tζ e ˆθ) with state ζ R ν and tunable parameter vector ˆθ R κ such that trajectories of the closed-loop system ()-() originating () -444-989-6/7/$5 7 IEEE 594 Authorized licensed use limited to: University of Minnesota Downloaded on November 3 8 at :5 from IEEE Xplore Restrictions apply

ThA5 from any initial condition (w x y ζ ˆθ ) R s+n+ν+κ are bounded and satisfy lim e(t) = for all µ K µ and all σ K σ t III DYNAMIC EXTENSION Consider the following augmentation of the plant dynamics ẋ = A (µ)x + A (µ)y + P (tµ)w + B (µ)u ẏ = A (µ)x + a (µ)y + P (tµ)w ξ = Aξ + B u where ξ = col(ξ ξ r ) R r A = B = d d d d r and d is are the coefficients of a Hurwtz polynomial i = r Let C be such that ξ = Cξ It is well known [5] [6] that there exists a parameter-dependent coordinates transformation x = x D(µ)ξ such that (3) can be transformed into the following form x = A (µ) x + A (µ)y + d(µ) Cξ + P (tµ)w ẏ = A (µ) x + a (µ)y + b(µ) Cξ + P (tµ)w where b(µ) is assumed to be greater than b > for all µ K µ The additional change of coordinates puts system (3) in the form z = x d(µ) b(µ) y ż = Ā(µ)z + Ā(µ)y + P (tµ)w ẏ = A (µ)z + ā (µ)y + b(µ) Cξ + P (tµ)w ξ = Aξ + Bu The robust minimum-phase plant property in Section II is stated as follows: Assumption 3: There exist a symmetric positive definite matrix P(µ) and positive constants p p > such that p I P(µ) p I P(µ)Ā(µ) + Ā (µ)p(µ) I for all µ K µ The above assumption entails that the zero dynamics of the unforced augmented system (5) (ie w = ) ż = Ā(µ)z is globally exponentially stable for all µ K µ [] Then there exists a T -periodic solution Ξ σ (tµ) of the Sylvester differential equation (see []) Ξ σ (tµ) + Ξ σ (tµ)s(tσ) = Ā(µ)Ξ σ (tµ) Ā(µ)Q(tµ) (3) (4) (5) By changing coordinates as z = z Ξ σ (tµ)w the augmented system (5) is put in the form z = Ā(µ) z + Ā(µ)e ė = Ā(µ) z + ā (µ)e + b(µ)[ξ R σ (tµ)w] where R σ (tµ) = The exosystem with output b(µ) [ Ā(µ)Ξ σ (tµ) + ā (µ)q(tµ) P (tµ) Q(tµ) Q(tµ)S(tσ)] ẇ = S(tσ)w y w = R σ (tµ)w is assumed to satisfy the strong immersion property (see [ Assumption 3] for details) so that system (7) is immersed to a system in observer form [] η = A o (tσ)η y η = Cη in which ā q (tσ) ā q (tσ) A o (tσ) = C = ā (tσ) ā (tσ) for some smooth T -periodic functions ā (tσ) ā q (tσ) Assumption 3: There exist an integer m N a smooth T -periodic vector valued function β : R R m and a continuous ( reparameterization ) σ θ R qm where θ = θ q θ q θ θi R m i = q such that ā i (tσ) = θ i β(t) i = q Note that since the reparameterization of σ is continuous there exists a compact set K θ R qm such that θ K θ for all σ K σ The new parameterization allows one to rewrite the matrix A o (tσ) in (8) with a minor abuse of notation as A o (tσ) = A b Θβ(t)C =: A o (tθ) where the matrix A b R q q is in Brunovsky form and Θ R q m collects the vectors θ i IV ADAPTIVE BACKSTEPPING DESIGN WITH TUNING FUNCTION Since the term R σ (tµ)w in (6) is not available for feedback we must reconstruct it asymptotically via an appropriate internal model It was shown in [] that for the unitary relative degree case a certainty-equivalence internal model could be chosen as ζ = Fζ + G(t)u im (9) y im = H(θ)ζ with state ζ = col(ζ ζ ) R q(m+) and matrices ( ) ( ) F G F = G(t) = F G (t) (6) (7) (8) 595 Authorized licensed use limited to: University of Minnesota Downloaded on November 3 8 at :5 from IEEE Xplore Restrictions apply

ThA5 H(θ) = ( H H (θ) ) l q l q l l F = l q I l I l I β(t) I F = G (t) = I G = ( ) H = L H (θ) = θ and L = ( ) l q l q l is an arbitrarily outputinjection gain such that A b L C is Hurwitz When the relative degree is greater than one the internal model (9) can not be implemented directly As usual we treat ξ as a virtual input so that the internal model (9) can be modified as ζ = Fζ + G(t)ξ () y im = H(θ)ζ As shown in [] there exists a T -periodic smooth matrixvalued function pair (Σ(tθ)H(θ)) satisfying Σ(tθ) + Σ(tθ)S(tµ) = [F + G(t)H(θ)]Σ(tθ) R σ (tµ) = H(θ)Σ(tθ) After introducing the change of coordinates system (6)-() reads as where χ = ζ Σ(tθ)w G(t)e () b(µ) χ = Fχ + J (tµ) z + J (tµ)e z = Ā(µ) z + Ā(µ)e ė = Ā(µ) z + a a (µ)e + b(µ)h(θ)χ + b(µ)[ξ H(θ)ζ] J (tµ) = b(µ) G(t)Ā(µ) J (tµ) = b(µ) [FG(t) G(t)ā (µ) Ġ(t)] a a (µ) = ā (µ) + H(θ)G(t) The above system is rewritten in a more compact form as ż a = A a (tµ)z a + A a (tµ)e ė = A a (θµ)z a + a a (µ)e + b(µ)[ξ H(θ)ζ] () where ( ) ( ) F A a J (tµ) (tµ) = A a J (tµ) Ā (µ) (tµ) = Ā (µ) A a (θµ) = ( b(µ)h(θ) Ā (µ) ) and z a = col(χ z) It is known that the upper subsystem (z a e) of () can be globally stabilized by a virtual input ξ = k e + H(θ)ζ for k sufficiently large Plus the structure of the ξ-dynamics and the availability of (ζeξ) for feedback allow one to use adaptive robust backstepping design (see [3]) In what follows we illustrate how this method can be used for the robust regulator design For the sake of simplicity we set v = u d ξ d ξ 3 d r ξ r and hence ξ r = v Let θ = ˆθ θ denote the estimation error and observe that H(ˆθ)ζ = H(θ)ζ θ ζ Since the system ż a = A a (tµ)z a is uniformly exponentially stable there exists a continuously differentiable periodic and positive definite function-valued matrix Q a (tµ) for all t and all µ K µ satisfying [9 Theorem 78] ρ I Q a (tµ) ρ I A a (tµ)q a (tµ) + Q a (tµ)a a (tµ) + Q a (tµ) I where ρ and ρ are finite positive constants Choose now the Lyapunov-like function candidate for system (z a e) as V = z aq a (tµ)z a + e + b(µ) γ θ θ Step : Introducing ξ = ξ α and setting ξ = e in which the first stabilizing function α will be assigned later The error-dynamics reads as ξ = A a (µθ)z a +a a (µ) ξ +b(µ)[ ξ +α H(θ)ζ] (3) while the derivative of V is V z az a + z aq a (tµ)a a (tµ) ξ + b(µ) γ θ ˆθ +a a (µ) ξ + ξ A a (µθ)z a + ξ b(µ)[ ξ + α H(θ)ζ] Function α is used as a control to stabilize (3) with respect to V For simplicity of notation in what follows we drop the arguments of matrices when this does not cause confusion The derivative of V reads as V z az a [k b(µ) a a ] ξ + z a(q a A a + A a ) ξ + b(µ) ξ ξ + b(µ) γ θ ( ˆθ γ ξ ζ ) where the first tuning function is τ = γ ξ ζ and the first stabilizing function is α (ζe θ) = ke + H(ˆθ)ζ By choosing k sufficiently large ie k b [c + 4( c ) (λ + ρ λ ) + λ ] where A a λ A a λ a a λ and a positive number c one obtains V c z az a c ξ + b(µ) ξ ξ + b(µ) γ θ ( ˆθ τ ) (4) 596 Authorized licensed use limited to: University of Minnesota Downloaded on November 3 8 at :5 from IEEE Xplore Restrictions apply

ThA5 System (3) in the new coordinate of ξ reads as ξ = [k b(µ) a a (µ)] ξ + b(µ) ξ b(µ)ζ θ + A a z a Step i: Follow the above procedure and introduce ξ i+ = ξ i+ α i with the stabilizing function α i to be chosen later yielding the ξ i -dynamics ξ i = ξ i+ α i j= ζ α i ˆθ α i ξ j+ α i ξ ξ j ˆθ α i t Use α i to stabilize system (z a ξ ξ ξ i ) with respect to V i = V i + ξ i Then V i c i z az a c i ξ j + b(µ) γ θ ( ˆθ τi + γ ξ i α i ζ i ) ( θ )( θ τi ) + ξ i [ ξ i + ξ i+ + α i α i j= ζ α i t α i ˆθ α i ξ j+ ] ξ ξ α i i j [b(µ) ξ + N (z a ξ )] Choose the i-th tuning function as τ i = τ i γ α i and the i-th stabilizing function as ξ i ζ α i (eξ ξ i ζ θt) = ξ i k i ξi + α i ζ + α i + α i t ˆθ τ α i i + ξ j+ ξ j The resulting V i is i ( j= ˆθ )( α i γζ ) V i c i z az a c i ξ j + ξ i ξi+ k i ξ i ξ α i i [N (z a ξ ) + b(µ) ξ ] + [ b(µ) γ θ ˆθ ]( ˆθ τi ) By choosing k i sufficiently large Vi reads as ˆθ (5) where < c i c i The ξ i dynamics reads as ξ i = ξ i k i ξi + ξ i+ + b(µ) θ α i ζ α i ˆθ ( ˆθ τi ) ( ˆθ ) α i γζ α i [N (z a ξ ) + b(µ) ξ ] Step r: Now we are in position to design the input v to stabilize the full system with respect to V r = V r + ξ r Our goal is to make V r non-positive Then V r c r z az a c r ξ j + b(µ) γ θ ( ˆθ τr + γ ξ r +v ( r ζ ) + θ ( ˆθ τr ) + ξ r [ ξ r ζ + α t + α ˆθ ξ r [N (z a e) + b(µ) ξ ] Choose the last tuning function as and the input as ˆθ + j= τ r = τ r γ ξ r ζ ξ j ξ j+ )] v = v ξ r k r ξr + ζ + + α t ˆθ τ r r + ξ j+ ( ξ j θ )(γ ξ r ζ ) j= where the choice of v will be clear in the sequel The resulting V r is V r c r z az a c r ξ j + [ b(µ) γ θ Noticing that r + ξ r ]( ˆθ τr ) k r ξ r + ξ r v θ ξ r [N (z a ξ ) + b(µ) ξ ] c r z az a z aa a A a + c r A a A a ξ r = c r (z a + ξ r ) (z a + A a c r ( ) ξ r c r α ξ r ) i V i c i z az a c i ξ j + ξ i ξi+ + [ b(µ) γ θ i ˆθ ]( ˆθ τi ) (6) 597 c r ξ ξ [ (kb(µ) a a )] ξ r = c r ( ξ + [ (kb(µ) a a c r )] ξ r ) + [ (kb(µ) a a c )] ( ) ξ r r Authorized licensed use limited to: University of Minnesota Downloaded on November 3 8 at :5 from IEEE Xplore Restrictions apply

ThA5 c r ξ ξ α b(µ) ξ r = c r ( ξ + b(µ) c r an obvious choice of v is given by ξ r ) + b(µ) c r ( ξ ) ξ r v = ( ) [λ + (k b + λ ) + b c r ] ξ r Finally the update law is chosen as ˆθ = τ r = γ( ξ Therefore the actual feedback input u is )ζ u = ξ r k r ξr + ζ + + ˆθ t θ r + ξ j+ ( ξ j= j θ )( ξ r γζ ) ( ) ξ r [λ + (kb + λ ) + b c r ] +d ξ + d ξ 3 + + d r ξ r Now it is shown that Vr is non-positive that is The resulting ξr is V r c r z az a c r ξ j k r ξ r (7) ξ r = ξ r k r ξr + b(µ) θ r ζ ( θ ) (γ ζ ) [N (z a ξ ) + b(µ) ξ ] ( ) [λ + (kb + λ ) + b c r ] ξ r A Stability analysis We are in position to prove the main result of this paper: Theorem 4: Given the compact sets K µ and K θ there exist k i > i = r such that for any k i k i i = r and any γ > the adaptive controller ζ = Fζ + G(t) Cξ ξ = Aξ + Bu(e ξ ζ t ( ) ˆθ) ē ˆθ = γ W ξ u = ξ r k r ξr + ζ + + ˆθ t θ r + ξ j+ ( ξ j= j θ ) ξ r γζ ( ) ξ [ r λ c r + (kb + λ ) + b] B Aξ where W = ( W W W r ) W = ζ and W i = α i ζ i = r and the pair (F G( )) is given above solves the output regulation problem for () ρ = Proof: Consider the overall closed loop system ẇ = S(tσ)w ż a = A a (tµ)z a + A a (tµ) ξ ξ = A a z a [k b(µ) a a ] ξ + b(µ) ξ + b(µ)w θ ξ i = α i Aa z a + α i (k b(µ) a a ) ξ α i b(µ) ξ ξ i k i ξi + ξ i+ + α r i θ γ W j ξ j i ξ r = θ γw i + b(µ)w i θ j=i+ A a z a + (k b(µ) a a ) ξ b(µ) ξ ξ r k r ξr + b(µ)w r θ r θ γw r ˆθ = γw ξ (8) with ρ = col(θµσ) In more compact form system (8) reads as ẇ a = S a (tσ)w a ẋ a = A a (tw a x a )x a b(µ)w a (tw a x a ) θ θ = γw a(tw a x a )x a (9) where W a (tw a x a ) = ( W W W r) wa = col(θµσw) and x a = col(z a ξ); the matrix-valued function A a (tw a x a ) is obtained from system (8) but is omitted for reason of space We will prove the theorem by showing that for any initial conditions w a () K θ K µ K σ R s x a () R q(m+)+n+r θ() R qm the corresponding trajectory (w a (t)x a (t) θ(t)) of (9) satisfies the following properties: i There exists a compact set Ω which depends on the initial conditions such that (w a (t)x a (t) θ(t)) Ω t ii The trajectory (w a (t)x a (t) θ(t)) is defined for all t and satisfies lim t x a (t) = First we show boundedness A Lyapunov-like function candidate is chosen as V r (tµx a θ) = x au a (tµ)x a + b(µ) γ θ θ for which the derivative of V r is given by(7) where U a (tµ) = diag(q a (tµ) ) This in addition to the fact that V r (tµx a θ) satisfies c (x a θ) V r (tµx a θ) c (x a θ) for some constants c c > and all t µ K µ shows that (w a (t) x a (t) θ(t)) evolves in a compact invariant set Ω which depends only on the initial conditions 598 Authorized licensed use limited to: University of Minnesota Downloaded on November 3 8 at :5 from IEEE Xplore Restrictions apply

ThA5 5 With PE condition Regulated error e(t) Estimation error 5 In absence of PE condition Regulated error e(t) Estimation error 5 5 5 5 5 5 5 5 5 3 35 4 45 5 Time [s] 5 5 5 3 Time [s] Fig Simulation results for Example 4 with PE condition Fig Simulation results in absence of PE condition Next we show attractivity Due to the periodicity of the system (9) and the Lyapunov-like function V r (tµx a θ) we resort to LaSalle s invariance principle for periodic systems [4] Define E = {(ty a ) R Ω : V (tya ) = } with y a = col(w a x a θ) and let M be the union of all trajectories y a (tt y ) with the property that (ty a (tt y )) is in E for all t where y is the initial condition of y a By [4 Theorem 3] the set M is attractive and so the claim is proven We conclude that the output regulation problem is solved since asymptotic convergence of e(t) to the origin is implied by that of x a (t) Example 4: Consider the output regulation problem for system () with relative ( degree r = where ) A = σ sin(t) B = S(tσ) = R(tµ) = σ sin(t) ( r sin(t) ) and σ and r are unknown constants satisfying σ > and r Let Ā (µ) = a (µ) Ā (µ) = a (µ) Ā (µ) = a (µ) ā (µ) = a and b(µ) = b be unknown coefficients ranging over known compact intervals with a > and b > The parameter vector is therefore given by (σ r a a a a b) It can be verified that the pair (S(tσ) R(tµ)) is immersed into a 4-dimensional uniformly observable system with a (tσ) = σ sin (t) + 8σ + 4 a (tσ) = 7σ sin(t)cos(t) a (tσ) = 5 + σ sin (t) and a 3 (tσ) = Hence the realization in observer canonical form reads as ā 3 (tσ) ā (tσ) A o (tσ) = ā (tσ) C = ( ) ā (tσ) with periodic coefficients ā (tσ) = 3σ + 4 σ sin (t) ā (tσ) = 3σ sin(t)cos(t) ā (tσ) = 5 + σ sin (t) and ā 3 (tσ) = As a result the proposed design applies to the given system with θ = σ The controller gains and adaptation gain have been chosen as k = 5 k = L = (54) and γ = 6 respectively The initial conditions of the simulations are z() = e() = ζ() = w() = ( ) and ˆθ() = 54 In Figure at t = 5 s the parameter σ is set from 3 to 4 The results of the regulated error and the estimation error show that this parameter variation does not affect the error convergence or parameter convergence with PE condition holds Compared with Figure in Figure PE condition is not satisfied when the exogenous signals are turned off at t = 8 s the convergence of the regulated error is preserved and the parameter remains bounded but does not converge to its true value These results verify that the proposed solution does not require the PE condition for asymptotic regulation V CONCLUSIONS We have extended the result of the linear periodic output regulation with parameter-dependent periodic exosystems [] to general (arbitrary relative degree) minimumphase plant models In order to use the non-minimal realization for construction of the internal model proposed in [] a dynamic extension is used to transform the original system into an error system Then by means of a coordinates transformation the overall system allows the use of robust backstepping technique for the robust regulator design We emphasize the fact that persistence of excitation is not required for asymptotic regulation of the error REFERENCES [] C I Byrnes and A Isidori Nonlinear internal models for output regulation IEEE T Automat Contr 49():44 7 4 [] A Isidori Nonlinear Control Systems II Springer Verlag 999 [3] M Krstic I Kanellakopoulos and P Kokotovic Nonlinear and adaptive control design Wiley New York 995 [4] JP LaSalle An invariance principle in the theory of stability Differential equations and dynamical systems pages 77 86 967 [5] R Marino and P Tomei Global adaptive observer for systems via filtered transformation IEEE T Automat Contr 37(8):39 45 99 [6] R Marino and P Tomei Global adaptive output-feedback control of nonlinear systems Part I: linear parameterization IEEE T Automat Contr 38():7 3 99 [7] VO Nikiforov Adaptive non-linear tracking with complete compensation of unknown disturbance European Journal of Control 4:3 9 998 [8] F Delli Priscoli Output regulation with nonlinear internal models Systems & Control Letters 53(3-4):77 85 4 [9] W Rugh Linear System Theory Prentice Hall nd edition 996 [] A Serrani A Isidori and L Marconi Semi-global nonlinear output regulation with adaptive internal model IEEE T Automat Contr 46(8):78 94 [] Z Zhang and A Serrani The linear periodic output regulation problem Systems & Control Letters 55(7):58 9 6 [] Z Zhang and A Serrani Robust regulation with adaptive periodic internal models In Proceedings of the 6 IEEE Conference on Decision Conference San Diego LA December 6 599 Authorized licensed use limited to: University of Minnesota Downloaded on November 3 8 at :5 from IEEE Xplore Restrictions apply