Robust Controllers for Regular Linear Systems with Infinite ...

ROBUST CONTROLLERS FOR REGULAR LINEAR SYSTEMS WITH INFINITE-DIMENSIONAL EXOSYSTEMS

arXiv:1604.01501v1 [math.OC] 6 Apr 2016

LASSI PAUNONEN Abstract. We construct two error feedback controllers for robust output tracking and disturbance rejection of a regular linear system with nonsmooth reference and disturbance signals. We show that for sufficiently smooth signals the output converges to the reference at a rate that depends on the behaviour of the transfer function of the plant on the imaginary axis. In addition, we construct a controller that can be designed to achieve robustness with respect to a given class of uncertainties in the system, and present a novel controller structure for output tracking and disturbance rejection without the robustness requirement. We also generalize the internal model principle for regular linear systems with disturbances from the boundary and for controllers with unbounded input and output operators. The construction of controllers is illustrated with an example where we consider output tracking of a nonsmooth periodic reference signal for a two-dimensional heat equation with boundary control and observation, and with periodic disturbances on the boundary.

1. Introduction The purpose of this paper is to construct controllers for robust output regulation of a regular linear system [35, 36, 33] (1a)

x(t) ˙ = Ax(t) + Bu(t) + Bd d(t),

(1b)

y(t) = CΛ x(t) + Du(t)

x(0) = x0 ∈ X

on an infinite-dimensional Banach space X. The main goal in the control problem is to achieve asymptotic convergence of the output y(t) to a given reference signal yref (t) despite external disturbance signals d(t). In addition, it is required that the controller is robust in the sense that output tracking is achieved even under perturbations and uncertainties in the operators (A, B, Bd , C, D) of the plant. The class of regular linear systems facilitates the study of robust output tracking and disturbance rejection for many important classes partial differential equations with boundary control and observation with corresponding unbounded operators B, Bd and C [9, 14, 40, 23]. In this paper we continue the work on designing robust controllers for regular linear systems begun recently in [24]. The reference signal yref (t) and the disturbance signals d(t) considered in the robust output regulation problem are assumed to be generated by an 2010 Mathematics Subject Classification. 93C05, 93B52 (47D06) . Key words and phrases. Robust output regulation, regular linear systems, controller design, feedback, stability. 1

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exosystem of the form (2a)

v(t) ˙ = Sv(t),

(2b)

d(t) = Ev(t)

(2c)

yref (t) = −F v(t).

v(0) = v0 ∈ W

In the case where the exosystem (2) is a system of ordinary differential equations on a finite-dimensional space W , the class of reference and disturbance signals consists of finite linear combinations of trigonometric functions and polynomially increasing terms. In this paper we concentrate on output tracking and disturbance rejection for a general class of nonsmooth periodic and almost periodic reference and disturbance signals. Such exogeneous signals can be generated with infinite-dimensional exosystems on the Hilbert space W = ℓ2 (C) where S = diag(iωk )k∈Z is an unbounded diagonal operator containing the frequencies {ωk }k∈Z that are present in the signals yref (·) and d(·). In particular, any continuous τ -periodic signal can be gen2kπ erated with an exosystem of the form (2) where S = diag i τ k∈Z [17]. Output tracking and disturbance rejection of nonsmooth signals with high accuracy have applications in the control of motor and disk drive systems and in power electronics [10]. Output tracking of signals generated by an infinite-dimensional exosystem have been studied using state space methods in [18, 15, 26, 23, 28], and using frequency domain techniques in [30, 39, 20], as well as extensively under the name repetitive control [16, 38]. As the main results of the paper we introduce two methods for constructing a regular error feedback controller of the form (3a) (3b)

z(t) ˙ = G1 z(t) + G2 (y(t) − yref (t)),

u(t) = KΛ z(t).

z(0) = z0 ∈ Z

to achieve robust output tracking and disturbance rejection for the regular linear system (1). The internal model principle of linear control theory states that in order to solve the robust output regulation problem it is both necessary and sufficient for the feedback controller (3) to include a suitable number of independent copies of the dynamics of the exosystem (2) and to achieve closed-loop stability. This fundamental characterization of robust controllers was originally presented for finite-dimensional linear systems by Francis and Wonham [13] and Davison [11] in 1970’s, and it was later generalized for infinite-dimensional linear systems with finite and infinitedimensional exosystems in [26, 28]. The internal model principle also implies that the robust controllers always tolerate a class of uncertainties and inaccuracies in the parameters G2 and K and in certain parts of the operator G1 of the controller (3). This property can be exploited in controller design as it sometimes allows the use of approximations in defining G1 , G2 , and K, provided that the internal model property is preserved and the closed-loop system achieves the necessary stability properties. The two robust controllers constructed in this paper utilize two different internal model based structures that are naturally complementary to each other. The first construction is based on a new block triangular controller structure that was first introduced in [25, 24] for control of regular linear systems with finite-dimensional exosystems. The second controller uses the

ROBUST CONTROLLERS FOR REGULAR LINEAR SYSTEMS

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observer based structure that was used to solve the robust output regulation problem for an infinite-dimensional exosystem in [15] in the case where the plant had bounded input and output operators. In this paper we generalize both of the controller structures to accommodate for an infinite-dimensional internal model and for unbounded operators B, Bd , and C in the system to be controller. In particular, this requires the use of new techniques in the analysis of the well-posedness and stability of the resulting closed-loop systems. In particular, the construction we present allows the use of unbounded feedback and output injection operators in achieving exponential stability of the pairs (A, B) and (C, A) of the plant. This generalization facilitates robust output tracking and disturbance rejection for new classes of systems such as undamped wave equations that can not be stabilized exponentially with bounded feedback and output injection operators. The two controllers that we construct end up possessing slightly differing properties. In the case of an infinite-dimensional exosystem the reference and disturbance signals are required to have a certain minimum level of smoothness in order for the robust output regulation problem to be solvable, and the exact level depends on the behaviour of the transfer function P (λ) = CΛ R(λ, A)B + D of the plant at the frequencies λ = iωk of the exosystem [19, 15]. More precisely, faster growth of the norms kP (iωk )† k of the Moore–Penrose pseudoinverses of P (iωk ) as |k| → ∞ leads to a higher minimal level of smoothness for yref (t) and d(t). Our results demonstrate that the required level of smoothness is in general lower in the case of the new controller structure than in the case of the observer based controller structure. Moreover, the new controller structure can be used in a situation where the plant has a larger number of inputs than outputs, whereas the construction of the second observer based controller requires that the input and output spaces of the plant are isomorphic. Since the output operator C of the plant is in general unbounded, the regulation error e(t) = y(t) − yref (t) is not guaranteed to converge to zero as t → ∞. Because of this, in this paper the convergence of the regulation error is instead considered in the sense that Z t+1 (4) ky(s) − yref (s)kds → 0, as t → ∞. t

R t+ε This condition is equivalent to requiring that the averages 1ε t ke(s)kds of the error over the intervals [t, t + ε] for any fixed ε > 0 converge to zero as t → ∞. The assumption that the exosystem (2) is infinite-dimensional further leads to situation where the regulation error is not guaranteed to decay at an exponential rate. However, it has been observed in [29, 7] that under suitable assumptions on the plant (1) with bounded operators B and C it is possible to achieve rational decay of the regulation error for sufficiently smooth reference and disturbance signals. In this paper we present new results that establish a priori decay rates for the regulation error for sufficiently smooth reference and disturbance signal. The results we present are based on a new method for nonuniform stabilization of the infinite-dimensional internal model in the controller, and on the subsequent analysis of the closed-loop system using recent results on nonuniform stability of semigroups of operators [21, 4, 6, 3]. The theory presented in this

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paper in particular generalizes the existing results on rational decay rates of the regulation error by facilitating the study of general subexponential decay rates. Beyond obtaining decay rates for the regulation error for the particular controllers we introduce a general methodology for applying the theory of nonuniform stability of semigroups in the study of regular linear control systems. The following theorem presents a simplified version of the main result regarding the nonuniform decay rates of the regulation error. The result demonstrates that the rate of decay of the regulation error is dependent on the rate of growth of the norms kP (iωk )† k as |k| → ∞. For the detailed assumptions on the system (1), the exosystem (2) and the controller (3), see Section 2. A more general version of Theorem 1.1 is presented in Section 6. Here we denote by kxkD(A) = kAxk + kxk the graph norm of a linear operator A. Theorem 1.1. Assume B and Bd are bounded, supk∈Z kR(iωk , A)k < ∞, and the pair (CΛ , A) can be stabilized with bounded feedback. If there exist M, α0 > 0 such that kP (iωk )† k ≤ M (1+|ωk |α0 ) for all k ∈ Z, then for any α > 2α0 + 1 the controllers in this paper can be construced in such a way that 1 Z t+1 α e log t kx0 kD(A) + kz0 kD(G1 ) + kv0 kD(S) ke(s)kds ≤ Me t t for some Mee > 0 and for all v0 ∈ D(S), x0 ∈ D(A) and z0 ∈ D(G1 ). If kP (iωk )† k ≤ M eα0 |ωk | for some M, α0 > 0 and for all k ∈ Z, then for any α > α0 the controllers can be construced in such a way that Z t+1 αMee ke(s)kds ≤ kx0 kD(A) + kz0 kD(G1 ) + kv0 kD(S) log t t e for some Me > 0 and for all v0 ∈ D(S), x0 ∈ D(A) and z0 ∈ D(G1 ).

The new controller structure used in this paper was originally introduced in [25] to solve the robust output regulation problem in the situation where robustness is only required with respect to a given class O0 of perturbations. In such a situation it is possible that the internal model in the controller can be replaced with a reduced order internal model containing smaller numbers of copies of some of the frequencies of the exosystem [27]. In this paper we also generalize the controller achieving robustness with respect to a given class O0 of perturbations presented in [25, 24] for regular linear systems with infinite-dimensional exosystems. Finally, we also use the new controller structure to construct a novel controller that achieves output tracking and disturbance rejection without the additional requirement for robustness with respect to perturbations in the parameters of the plant. Controller design for robust output regulation for regular linear systems with finite-dimensional exosystems has been studied previously in [31, 24]. The output tracking and disturbance rejection for regular linear systems without the robustness requirement have been studied in [23, 37]. In this paper we also extend the characterization of the controllers achieving output regulation via the regulator equations and the internal model principle for


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regular linear systems in the situation where the operator Bd is unbounded and the error feedback controller (3) is a regular linear system. These results generalize the ones in [28, 31] where Bd was assumed to be bounded and in [23] where feedforward control was studied. We illustrate the construction of controllers by considering output tracking and disturbance rejection for a two-dimensional heat equation with boundary control and observation. We design a controller that achieves robust output tracking of a nonsmooth periodic reference signal and rejects periodic disturbances on the boundary of the domain. The main contributions of the paper can be summarized as follows. • Construction of two controllers for robust output regulation of regular linear systems with infinite-dimensional exosystems. • Detailed study of nonuniform stability of the resulting closed-loop systems and nonuniform decay rates of the regulation error e(t). • Construction of a controller achieving robustness with respect to a predefined class O0 of perturbations in the case of an infinitedimensional exosystem. • A novel controller for output tracking and disturbance rejection without the robustness requirement. • Generalization of the regulator equations and the internal model principle for regular linear systems with an unbounded operator Bd and for controllers with unbounded input and output operators. • A number of auxiliary results related to the nonuniform stability and stabilization of the infinite-dimensional internal model. The paper is structured as follows. In Section 2 we state the standing assumptions on the plant, the exosystem, the controller, and the closedloop system. In Section 3 we formulate the main control problems studied in the paper and present a generalization of the internal model principle. The robust controller based on the new controller structure is constructed in Section 4. Subsequently, the same structure is used to construct a controller with a reduced order internal model in Section 4.1 and a controller for output regulation without the robustness requirement in Section 4.2. The observer based controller is constructed in Section 5. In Section 6 we study the nonuniform decay rates for the state of the closed-loop system and the regulation error. The example on robust output tracking and disturbance rejection for a two-dimensional heat equation is studied in Section 7. Section 8 contains concluding remarks. Appendix A contains auxiliary results on the stability of block triangular systems. 2. The System, The Controller and The Closed-Loop System If X and Y are Banach spaces and A : X → Y is a linear operator, we denote by D(A), N (A) and R(A) the domain, kernel and range of A, respectively. The space of bounded linear operators from X to Y is denoted by L(X, Y ). If A : X → X, then σ(A), σp (A) and ρ(A) denote the spectrum, the point spectrum and the resolvent set of A, respectively. For λ ∈ ρ(A) the resolvent operator is given by R(λ, A) = (λ − A)−1 . The inner product on a Hilbert space is denoted by h·, ·i. For two sequences (fk )k∈Z ⊂ X and (gk )k∈Z ∈ R+ we denote kfk k = O(gk ) if there exist Mg , Ng > 0 such that

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kfk k ≤ Mg gk whenever |k| ≥ Ng . Similarly, for two functions f : I ⊂ R → X and g : R+ → R+ we denote kf (t)k = O(g(t)) if there exist Mg , Tg > 0 such that kf (t)k ≤ Mg g(t) whenever |t| ≥ Tg . We denote f (t) . g(t) and fk . gk if there exist M1 , M2 > 0 such that f (t) ≤ M1 g(t) and fk ≤ M2 gk for all values of the parameters t and k. We consider a linear system (1) where x(t) ∈ X is the state of the system, y(t) ∈ Y is the output, and u(t) ∈ U the input. Here d(t) ∈ Ud denotes the disturbance signal to the state of the plant. The spaces X and Ud are Banach spaces and U and Y are Hilbert spaces. We assume that A : D(A) ⊂ X → X generates a strongly continuous semigroup T (t) on X. For a fixed λ0 > ω0 (T (t)) we define the scale spaces X1 = (D(A), k(λ0 − A)·k) and X−1 = (X, kR(λ0 , A)·k) (the completion of X with respect to the norm kR(λ0 , A)·k) [34],[12, Sec. II.5]. We assume the input and output operators of the plant are such that B ∈ L(U, X−1 ), C ∈ L(X1 , Y ), and the feedthrough operator satisfies D ∈ L(U, Y ). Also the extensions of the operator A and the semigroup T (t) to the space X−1 are denoted by A : X ⊂ X−1 → X−1 and T (t), respectively. Throughout the paper we assume that the plant (A, [B, Bd ], C, D) in (1) is a regular linear system [35, 36, 33, 32]. In particular, the operators B ∈ L(U, X−1 ), Bd ∈ L(Ud , X−1 ) and C ∈ L(X1 , Y ) are admissible with respect to A and D ∈ L(U, Y ). We define the Λ-extension CΛ of C as CΛ x = lim λCR(λ, A)x λ→∞

with D(CΛ ) consisting of those x ∈ X for which the limit exists. If C ∈ L(X, Y ), then CΛ = C. Since (A, [B, Bd ], C, D) is a regular linear system, we have R(R(λ, A)[B, Bd ]) ⊂ D(CΛ ) for all λ ∈ ρ(A), and the transfer functions P (λ) and Pd (λ) from u ˆ to yˆ and from dˆ to yˆ, respectively, are given by [33, Sec. 4] P (λ) = CΛ R(λ, A)B + D Pd (λ) = CΛ R(λ, A)Bd for all λ ∈ ρ(A). Finally, we define XB = D(A) + R(R(λ0 , A)B) ⊂ D(CΛ ), XBd = D(A)+R(R(λ0 , A)Bd ) ⊂ D(CΛ ), and X(B,Bd ) = XB +XBd = D(A)+ R(R(λ0 , A)[B, Bd ]) all of which independent of the choice of λ0 ∈ ρ(A). Assumption 2.1. The pairs (A, B) and (C, A) are exponentially stabilizable and detectable, respectively, in the sense that there exist admissible operators K ∈ L(X1 , U ) and L ∈ L(Y, X−1 ) for which C A, B, L, Bd , K is a regular linear system and the semigroups generated by (A + BKΛ )|X and (A + LCΛ )|X are exponentially stable.


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The reference and disturbance signals are generated by an exosystem of the form v(0) = v0 ∈ W

(5a)

v(t) ˙ = Sv(t),

(5b)

d(t) = Ev(t)

(5c)

yref (t) = −F v(t)

on a separable Hilbert space W = ℓ2 (Z; C). We denote by {φk }k∈Z the canonical orthonormal basis of W . The operator S : D(S) ⊂ W → W is defined as X Sv = iωk hv, φk iφk , k∈Z

v ∈ D(S) =

(

) X 2 2 |ωk | |hv, φk i| < ∞ , v∈W k∈Z

where the eigenvalues {iωk }k∈Z ⊂ iR of S are distinct and have a uniform gap, i.e., inf k6=l |ωk − ωl | > 0. The operator S generates an isometric group TS (t) on W . We finally assume that E ∈ L(W, X) and F ∈ L(W, Y ) are Hilbert–Schmidt operators, i.e., (Eφk )k∈Z ∈ ℓ2 (Ud ) and (F φk )k∈Z ∈ ℓ2 (Y ). All the results presented in this paper also trivially apply to the situation where W = Cq for some q ∈ N and S = diag(iωk )qk=1 is a diagonal matrix. Assumption 2.2. The operators K and L in Assumption 2.1 can be chosen in such a way that PK (iωk ) = (CΛ + DKΛ )R(iωk , A + BKΛ )B + D PL (iωk ) = CΛ R(iωk , A + LCΛ )(B + LD) + D are surjective for all k ∈ Z. The functions PK (iωk ) and PL (iωk ) are the transfer functions of the stabilized systems (A + BKΛ , B, CΛ + DKΛ , D) and (A + LCΛ , B + LD, CΛ , D), respectively. Since surjectivity of the transfer function is invariant under state feedback and output injection, Assumption 2.2 is in particular satisfied whenever iωk ∈ ρ(A) and P (iωk ) is surjective, and if it satisfied for some K and L, then it is satisfied for all stabilizing operators K and L. In this paper we will solve the robust output regulation problem with a dynamic error feedback controller z(t) ˙ = G1 z(t) + G2 e(t),

u(t) = Kz(t)

z(0) = z0 ∈ Z,

on a Banach space Z. We assume that G1 : D(G1 ) ⊂ Z → Z generates a semigroup on Z, and G2 ∈ L(Y, Z−1 ) and K ∈ L(Z1 , U ) are admissible, and (G1 , G2 , K) is a regular linear system. We denote by KΛ the Λ-extension of K and for λ0 ∈ ρ(G1 ) we define ZG2 = D(G1 ) + R(R(λ0 , G1 )G2 ) ⊂ D(KΛ ). Also the extension of G1 to the space Z−1 is denoted by G1 : Z ⊂ Z−1 → Z−1 . The system and the controller can be written together as a closed-loop system on the Banach space Xe = X × Z. This composite system with state

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xe (t) = (x(t), z(t))T can be written on X−1 × Z−1 as x˙ e (t) = Ae xe (t) + Be v(t),

xe (0) = xe0 ,

e(t) = CeΛ xe (t) + De v(t), T where e(t) = y(t) − yref (t) is the regulation error, xe0 = (x0 , z0 ) , Ce = CΛ , DKΛ , De = F , A BKΛ Bd E Ae = , Be = . G2 CΛ G1 + G2 DKΛ G2 F h i We also denote Be0 = B0d G02 ∈ L(Ue , X−1 × Z−1 ) where Ue = Ud × Y so h i that Be = Be0 E F . We choose the domain of Ae as ) ( Ax + BKΛ z ∈ X x (6) D(Ae ) = . ∈ XB × ZG2 z G1 z + G2 (CΛ x + DKΛ z) ∈ Z

Theorem 2.3. The closed-loop system (Ae , Be , Ce , De ) is a regular linear system. Moreover, for every λ ∈ ρ(Ae ) we have R(Be0 ) ⊂ R(λ − Aee ) and

R(λ, Ae )Be0 = R(λ, Aee )Be0 i h BKΛ : Xe → X−1 × Z−1 with domain D(Aee ) = where Aee = G2ACΛ G1 +G 2 DKΛ X(B,Bd ) × ZG2 .

Proof. We will show that the system (Ae , Be0 , Ce ) is regular. This will immediately imply that (Ae , Be , Ce , De ) is regular as well. Consider a regular ˆ B, ˆ C, ˆ D) ˆ with linear system (A, A 0 ˆ = B Bd 0 , CˆΛ = CΛ 0 , D ˆ = D 0 0 , Aˆ = , B 0 G1 0 0 G2 0 KΛ 0 0 0

and with state x ˆ = (x, z)T , output yˆ = (y, u)T , and input u ˆ = (u, d, e)T . The operator   0 I ˆ = 0 0 K I 0 h i ˆ B, ˆ C, ˆ D) ˆ since I − D ˆK ˆ = I −D is an admissible feedback operator for (A, 0 I is boundedly invertible [36]. Due to the results in [36] the closed-loop system ˆ is regular linear system and has the resulting from output feedback with K form A BKΛ B Bd 0 C DKΛ D 0 0 , , Λ , , G2 CΛ G1 + G2 DKΛ G2 D 0 G2 0 KΛ 0 0 0 where the system operator has the domain given in (6). This immediately implies that the system (Ae , Be0 , Ce ) is regular. Moreover, the resolvent identities in [36, Prop. 6.6] and a straightforward computation shows that for λ ∈ ρ(Ae ) with large Re λ we have R(λ, Ae )Be0 = R(λ, Aee )Be0 .

Together with the resolvent identity this implies the last claim of the theorem.


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2.1. The Class of Perturbations. In this paper the parameters of the plant are perturbed in such a way that the operators A, B, Bd , C, and D ˜ ⊂ X → X, B ˜ ∈ L(U, X ˜−1 ), B ˜d ∈ L(U, X ˜−1 ), C˜ ∈ are changed to A˜ : D(A) ˜ 1 , Y ), and D ˜ ∈ L(U, Y ), respectively. Here X ˜ 1 and X ˜ −1 are the scale L(X ˜ ˜ ˜ B ˜d ], C, ˜ D) ˜ is spaces of X related to the operator A. We assume that (A, [B, a regular linear system. Moreover, the operators E and F are perturbed in ˜ ∈ L(W, X) and F˜ ∈ L(W, Y ) are Hilbert–Schmidt. For such a way that E ˜ ˜B ˜ +D ˜ and P˜d (λ) = C˜Λ R(λ, A) ˜B ˜d λ ∈ ρ(A) we denote by P˜ (λ) = C˜Λ R(λ, A) the transfer functions of the perturbed plant. Definition 2.4. The class O of considered perturbations consists of opera˜ B, ˜ B ˜d , C, ˜ D, ˜ E, ˜ F˜ ) satisfying the above assumptions. In particular, tors (A, the class O contains the nominal plant, i.e., (A, B, Bd , C, D, E, F ) ∈ O. We denote the operators of the closed-loop system consisting of the per ˜ e = F˜ and ˜ Λ ,D turbed plant and the controller by C˜e = C˜Λ , DK ˜ Λ ˜d E ˜ A˜ BK B ˜ ˜ . Ae = Be = ˜ Λ , G2 F˜ G2 C˜Λ G1 + G2 DK 3. The Robust Output Regulation Problem The main control problem studied in this paper is defined in the following. The Robust Output Regulation Problem. Choose (G1 , G2 , K) in such a way that the following are satisfied: (a) The semigroup Te (t) generated by the closed-loop system operator Ae is strongly stable. (b) For all initial states xe0 ∈ Xe and v0 ∈ W the regulation error satisfies Z t+1 (7) ke(s)kds → 0 as t → ∞. t

˜ B, ˜ B ˜d , C, ˜ D, ˜ E, ˜ F˜ ) ∈ (c) If (A, B, Bd , C, D, E, F ) are perturbed to (A, O in such a way that the perturbed closed-loop system is strongly stable, then for all initial states xe0 ∈ Xe and v0 ∈ W the regulation error satisfies (7).

The necessity for studying the asymptotic behaviour of the regulation error through the convergence of the integrals (7) arises from the operators C and K being unbounded. If these two operators are bounded, then the condition (7) is equivalent to ke(t)k → 0 as t → ∞ (see the proof of Theorem 3.3 for details). In Section 6 we will in addition consider rates of convergence of the integrals in (7). Definition 3.1. We call a controller robust with respect to a given perturba˜ B, ˜ B ˜d , C, ˜ D, ˜ E, ˜ F˜ ) ∈ O for which the perturbed closed-loop system tion (A, is strongly stable if for these perturbed operators the regulation error satisR t+1 fies t ke(s)kds → 0 as t → ∞ for all xe0 ∈ Xe and v0 ∈ W .

Remark 3.2. Although not considered explicitly in the problem, the controllers that solve the robust output regulation problem also tolerate perturbations in the operators G2 and K of the controller as long as the closed-loop

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stability and any additional conditions are satisfied. This is an advantage in control design, because it sometimes allows the operators K and G2 to be replaced with sufficiently accurate approximations. The control problem without the robustness requirement is referred to as the output regulation problem. The Output Regulation Problem. Choose the controller (G1 , G2 , K) in such a way that parts (a) and (b) of the robust output regulation problem are satisfied. The following theorem characterises the controllers solving the output regulation problem in terms of the solvability of the so-called regulator equations [13, 8, 15]. The assumption that the Sylvester equation ΣS = Ae Σ+Be has a solution Σ ∈ L(W, Xe ) satisfying R(Σ) ⊂ D(CeΛ ) in particular guarantee that for all initial states v0 ∈ W of the exosystem the closed-loop state and the regulation error have well-defined behaviour as t → ∞. Theorem 3.3. Assume the controller (G1 , G2 , K) stabilizes the closed-loop system strongly in such a way that the Sylvester equation ΣS = Ae Σ + Be on D(S) has a solution Σ ∈ L(W, Xe ) satisfying R(Σ) ⊂ D(CeΛ ). Then the following are equivalent: (a) The controller (G1 , G2 , K) solves the output regulation problem. (b) The regulator equations (8a)

ΣS = Ae Σ + Be

(8b)

0 = CeΛ Σ + De on D(S) have a solution Σ ∈ L(W, Xe ) satisfying R(Σ) ⊂ D(CeΛ ).

Proof. Similarly as in the proof of [28, Lem. 4.3] the Sylvester equation (8a) implies that Z t Te (t − s)Be TS (s)vds = ΣTS (t)v − Te (t)Σv 0

for all v ∈ D(S), and since the operators on both sides of the equation are in L(W, Xe ) and D(S) is dense in W , the identity holds for all v ∈ W . Thus for all xe0 ∈ Xe and v0 ∈ W the mild state of the closed-loop system has the form (9)

xe (t) = Te (t)(xe0 − Σv0 ) + ΣTS (t)v0 .

Since the closed-loop system is regular, we have that Te (t)(xe0 − Σv0 ) ∈ D(CeΛ ) for almost all t ≥ 0, and the property R(Σ) ⊂ D(CeΛ ) implies ΣTS (t)v0 ∈ D(CeΛ ). For almost all t ≥ 0 the regulation error is thus given by (10a) (10b)

e(t) = CeΛ xe (t) + De v(t) = CeΛ Te (t)(xe0 − Σv0 ) + (CeΛ Σ + De )TS (t)v0 .

If (b) is satisfied, then the regulator equations (8) have a solution. The formula (10) and the regulation constraint CeΛ Σ + De = 0 imply that e(t) = CeΛ Te (t)(xe0 − Σv0 ) for almost all t ≥ 0. Since Ce is admissible with respect


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R1 to Te (t), there exists κ > 0 such that 0 kCeΛ Te (s)xk2 ds ≤ κ2 kxk2 for all x ∈ Xe . Since Te (t) is strongly stable, we have Z 1 Z t+1 kCeΛ Te (s)Te (t)(xe0 − Σv0 )kds ke(s)kds = 0

t

≤ κkTe (t)(xe0 − Σv0 )k → 0

as t → ∞. Thus the controller solves the output regulation problem on W . It remains to show that (a) implies (b). Assume the controller solves the output regulation problem and let Σ ∈ L(W, Xe ) be a solution of (8a). We will show that CeΛ Σ+De = 0. The condition (7) and the formula (10) imply Z t+1 k(CeΛ Σ + De )TS (s)v0 kds t Z t+1 Z t+1 ke(s)kds → 0 kCeΛ Te (s)(xe0 − Σv0 )kds + ≤ t

t

as t → ∞. This must be true for all v0 ∈ W , and thus if we choose v0 = φk for any k ∈ Z, then Z t+1 Z t+1 |eiωk s |ds k(CeΛ Σ + De )TS (s)v0 kds = k(CeΛ Σ + De )φk k · 0← t

t

= k(CeΛ Σ + De )φk k,

which implies (CeΛ Σ + De )φk = 0. Since k ∈ Z was arbitrary and {φk }k∈Z is a basis of W , we have CeΛ Σ + De = 0, and thus Σ is a solution of the regulator equations (8). Even in a more general situation where the state space W of the exosystem is a general Banach space and S is a generator of a general strongly continuous semigroup TS (t) on W , the state of the closed-loop system and the regulation error have the formulas given in (9) and (10), respectively, whenever the Sylvester equation ΣS = Ae Σ + Be has a solution Σ ∈ L(W, Xe ) satisfying R(Σ) ⊂ D(CeΛ ). In this situation the regulator equations are also sufficient for the solution of the robust output regulation problem. The regulator equations also become necessary whenever the semigroup TS (t) satisfies a condition of the form Z t+1 t→∞ kQTS (s)v0 kds −→ 0 ⇒ Qv0 = 0 t

for any Q ∈ L(W, Y ) and v0 ∈ W . This is also in particular true for all finite-dimensional exosystems on W = Cq with σ(S) ⊂ C+ as well as for infinite blockdiagonal exosystems. The following lemma presents a sufficient condition for the solvability of ΣS = Ae Σ + Be .

Lemma 3.4. Assume the closed-loop system is strongly stable, {iωk }k∈Z ⊂ ρ(Ae ), and (11)

(R(iωk , Ae )Be φk )k∈Z ∈ ℓ2 (Xe ).

Then the Sylvester equation ΣS = Ae Σ + Be on D(S) has a unique solution Σ ∈ L(W, Xe ) satisfying R(Σ) ⊂ D(CeΛ ).

12

LASSI PAUNONEN

Proof. Define Σ : D(Σ) ⊂ W → Xe by X Σv = hv, φk iR(iωk , Ae )Be φk , k∈Z

v ∈ D(Σ).

By [15, Lem. 6] the operator Σ ∈ L(W, Xe,−1 ) is the solution of the Sylvester equation ΣS = Ae Σ + Be , and condition (11) clearly implies Σ ∈ L(W, Xe ). To show R(Σ) ⊂ D(CeΛ ), we note that the resolvent identity R(iωk , Ae ) = R(1 + iωk , Ae ) + R(1 + iωk , Ae )R(iωk , Ae ) implies (hv, φk iCeΛ R(iωk , Ae )Be φk )k∈Z ∈ ℓ1 (Y ). For all v ∈ W and λ > 0 we have X λCe R(λ, Ae )Σv0 = hv0 , φk iλCe R(λ, Ae )R(iωk , Ae )Be φk k∈Z

=

X λhv0 , φk i

CeΛ R(iωk , Ae )Be φk − λ − iωk k∈Z X −→ hv0 , φk iCeΛ R(iωk , Ae )Be φk

CeΛ R(λ, Ae )Be0

X λhv0 , φk i Eφk k∈Z

λ − iωk

F φk

k∈Z

as λ → ∞ since (Ae , Be0 , Ce ) is regular and since E and F are Hilbert– Schmidt. Thus Σv ∈ D(CeΛ ) by definition. 3.1. The Internal Model Principle. We conclude this section by presenting the internal model principle that characterizes the controllers solving the robust output regulation problem. We use the following two alternate definitions for an internal model. Definition 3.5. A controller (G1 , G2 , K) is said to satisfy the G-conditions if (12a) (12b)

R(iωk − G1 ) ∩ R(G2 ) = {0}

N (G2 ) = {0}.

∀k ∈ Z,

Definition 3.6. Assume dim Y < ∞. A controller (G1 , G2 , K) is said to incorporate a p-copy internal model of the exosystem S if for all k ∈ Z we have dim N (iωk − G1 ) ≥ dim Y. Our first result characterizes the robustness of a controller with respect ˜ B, ˜ B ˜d , C, ˜ D, ˜ E, ˜ F˜ ) ∈ O. The internal model to individual perturbations (A, principle is presented in Theorem 3.8. Theorem 3.7. Assume the controller solves the output regulation problem. ˜ B, ˜ B ˜d , C, ˜ D, ˜ E, ˜ F˜ ) ∈ O be such that the perturbed closed-loop system Let (A, ˜ and ΣS ˜ = A˜e Σ ˜ +B ˜e has a solution. is strongly stable, {iωk }k∈Z ⊂ ρ(A), ˜ ˜ ˜ ˜ D, ˜ E, ˜ F˜ ) if and Then the controller is robust with respect to (A, B, Bd , C, only if for every k ∈ Z the equations (13a) (13b)

˜ k − F˜ φk P˜ (iωk )KΛ zk = −P˜d (iωk )Eφ

(iωk − G1 )zk = 0

have solutions zk ∈ D(G1 ). If (13) have a solution zk ∈ D(G1 ), then it is unique.


13

Proof. The proof can be completed exactly as the proof of [28, Thm. 5.1] (where we replace E by Bd E) due to the fact that by Theorem 2.3 we have ˜e φk . ˜e φk = R(iωk , A˜e )B R(iωk , A˜e )B e Theorem 3.8. Assume (G1 , G2 , K) stabilizes the closed-loop system strongly in such a way that {iωk }k∈Z ⊂ ρ(Ae ) and (R(iωk , Ae )Be φk )k∈Z ∈ ℓ2 (Xe ). If the controller satisfies the G-conditions in Definition 3.5, then it solves the robust output regulation problem. The controller is guaranteed to be robust with respect to all perturbations for which the perturbed closed-loop system ˜e has is stable, {iωk }k∈Z ⊂ ρ(A˜e ) and the Sylvester equation ΣS = A˜e Σ + B a solution Σ ∈ L(W, Xe ) satisfying R(Σ) ⊂ D(C˜eΛ ). Moreover, if {iωk }k∈Z ⊂ ρ(A), then the following hold. (a) The controller solves the robust output regulation problem if and only if it satisfies the G-conditions. (b) If dim Y = p < ∞, the controller solves the robust output regulation problem if and only if it incorporates a p-copy internal model of the exosystem, i.e., dim N (iωk − G1 ) ≥ p for all k ∈ Z. In both cases the controllers are guaranteed to be robust with respect to perturbations for which the perturbed closed-loop system is stable, {iωk }k∈Z ⊂ ˜ ∩ ρ(A˜e ) and the Sylvester equation ΣS = A˜e Σ + B ˜e has a solution ρ(A) Σ ∈ L(W, Xe ) satisfying R(Σ) ⊂ D(C˜eΛ ). Proof. Since (R(iωk , Ae )Be φk )k∈Z ∈ ℓ2 (Xe ), we have from Lemma 3.4 that ΣS = Ae Σ + Be has a solution Σ ∈ L(W, Xe ) such that R(Σ) ⊂ D(CeΛ ). ˜ B, ˜ B ˜d , C, ˜ D, ˜ E, ˜ F˜ ) ∈ O be such that {iωk }k∈Z ⊂ ρ(A˜e ) and ΣS = Let (A, ˜e has a solution Σ = (Π, Γ)T ∈ L(W, Xe ) satisfying R(Σ) ⊂ D(C˜eΛ ). A˜e Σ+ B ˜e φk implies (iωk − A˜e )Σφk = B ˜e φk . By If k ∈ Z, then ΣSφk = A˜e Σφk + B T Lemma 2.3 we have (Πφk , Γφk ) ∈ X(B, ˜B ˜d ) × ZG2 and ˜d Eφ ˜ k ˜ ˜ B (iωk − A)Πφ k − BKΛ Γφk ˜ Λ Γφk = G2 F˜ φk . −G2 C˜Λ Πφk + (iωk − G1 )Γφk − G2 DK

˜ Λ Γφk + F˜ φk ), and The second line implies (iωk − G1 )Γφk = G2 (C˜Λ Πφk + DK ˜ e φk = C˜Λ Πφk + DK ˜ Λ Γφk + F˜ φk = the G-conditions further imply C˜eΛ Σφk + D 0. Since k ∈ Z was arbitrary and {φk }k∈Z is a basis of W , we have that ˜ e = 0. Since the perturbations in O were arbitrary, we have from C˜eΛ Σ + D Theorem 3.3 that the regulation errors for the nominal and the perturbed R t+1 systems satisfy t ke(s)kds → 0 as t → ∞. ˜ the rest Under the assumptions {iωk }k∈Z ⊂ ρ(A) and {iωk }k∈Z ⊂ ρ(A), of the theorem can be proved using Theorem 3.7 exactly as in [28]. The following lemma presents sufficient conditions for the invariance of the G-conditions. Lemma 3.9. If the operators (G1 , G2 ) satisfy the G-conditions, and if K : D(K) ⊂ Z → Y is such that N (iωk − G1 ) ⊂ N (K) for all k ∈ Z, then also R(iωk − G1 + G2 K) ∩ R(G2 ) = {0} for all k ∈ Z. Proof. Let w = (iωk − G1 − G2 K)z = G2 y for some k ∈ Z, z ∈ D(K) and y ∈ Y . This implies (iωk − G1 )z = G2 (y + Kz) ∈ R(iωk − G1 ) ∩ R(G2 ), and we thus have z ∈ N (iωk − G1 ). Due to our assumptions we then also

14

LASSI PAUNONEN

have Kz = 0 and w = (iωk − G1 )z = G2 y, which finally imply w = 0 due to (12a). Lemma 3.10. Assume the controller (G1 , G2 , K) satisfies the G-conditions ˜ B, ˜ B ˜d , C, ˜ D, ˜ E, ˜ F˜ ) ∈ O. If k ∈ Z and iωk ∈ ρ(A) ˜ ∩ ρ(A˜e ), then and let (A, ˜ BKz ˜ k+B ˜d Eφ ˜ k) R(iωk , A)( ˜ R(iωk , Ae )Be φk = zk where zk ∈ Z is the unique element such that zk ∈ N (iωk − G1 ) and ˜ k − F˜ φk . P˜ (iωk )Kzk = −P˜d (iωk )Eφ

˜e φk is the Proof. By Theorem 2.3 we have that (xk , zk )T = R(iωk , A˜e )B T unique element (xk , zk ) ∈ X(B, ˜B ˜d ) × ZG2 satisfying ˜ k = BK ˜ Λ zk + B ˜d Eφ ˜ k (iωk − A)x ˜ (iωk − G1 )zk = G2 (CΛ xk + DKΛ zk + F˜ φk ). ˜ BK ˜ Λ zk + B ˜d Eφ ˜ k ) and the G-conditions in DefiniThus xk = R(iωk , A)( tion 3.5 imply zk ∈ N (iωk − G1 ) and ˜ Λ zk + F˜ φk = P˜ (iωk )KΛ zk + P˜d (iωk )Eφ ˜ k + F˜ φk . 0 = C˜Λ xk + DK

4. The New Controller Structure In this section we construct a robust controller using the general internal model based structure introduced in [24, 25]. The construction of the controller is completed in steps, and the required properties of the parts of the controller are verified in the proof of Theorem 4.1. Step 1◦ : The state space of the controller is chosen as Z = Z0 × X and we choose the structures of the operators (G1 , G2 , K) as G2 G1 G2 (CΛ + DK2Λ ) , G2 = , G1 = L 0 A + BK2Λ + L(CΛ + DK2Λ )

and K = (K1 , −K2Λ ). Due to Assumption 2.1 concerning the stabilizability of (A, B) and the detectability of (C, A), we can choose K2 ∈ L(X1 , U ) and L1 ∈ L(Y, X−1 ) in such a way that (A+BK2Λ )|X and (A+L1 CΛ )|X generate exponentially stable semigroups and the system A, [B, L1 , Bd ], KC2 , D is regular. For λ ∈ C+ we define PL (λ) = CΛ R(λ, A + L1 CΛ )(B + L1 D) + D.

We have from [36, Sec. 7] that (A + L1 CΛ , B + L1 D, CΛ , D) is a regular linear system, and thus supω∈R kPL (iω)k < ∞.

Step 2◦ : The operator G1 is the internal model of the exosystem (2), and it is defined by choosing Z0 = ℓ2 (Y ), and D(G1 ) = { (zk )k∈Z ∈ Z0 | (ωk zk )k∈Z ∈ ℓ2 (Y ) }. G1 = diag iωk IY k∈Z ,

The operator K1 = . . . , K1,−1 , K10 , K11 , . . .) ∈ L(Z0 , U ) is assumed to be Hilbert–Schmidt (i.e., (kK1k k)k∈Z ∈ ℓ2 (C), which is in particular always true if dim U < ∞) and and G2 = (G2k )k∈Z ∈ L(Y, Z0 ). We choose the


15

components K1k ∈ L(Y, U ) of K1 in such a way that PL (iωk )K1k ∈ L(Y ) are boundedly invertible. This is possible since PL (iωk ) are surjective by † k) Assumption 2.2. For example, we can choose K1k = γk kPPL (iω for all † L (iωk ) k 2 k ∈ Z and for a sequence (γk )k∈Z ⊂ ℓ (C) satisfying γk 6= 0 for all k ∈ Z. For more concrete choices of K1k , see Corollary 4.3. If iωk ∈ ρ(A) for some k, then PL (iωk ) = (I − CΛ R(iωk , A)L1 )−1 P (iωk ) implies that PL (iωk )K1k is boundedly invertible if and only if P (iωk )K1k is boundedly invertible. Step 3◦ : We define H ∈ L(Z0 , X) by X Hz = R(iωk , A + L1 CΛ )(B + L1 D)K1k zk , k∈Z

for z = (zk )k∈Z ∈ Z0 .

Step 4◦ : We choose the operator G2 ∈ L(Y, Z0 ) by

G2 = (G2k )k∈Z = (−(PL (iωk )K1k )∗ )k∈Z .

Finally, we define L = L1 + HG2 ∈ L(Y, X−1 ) and choose the domain of the operator G1 to be D(G1 ) = D(G1 ) × D((A + (B + LD)K2Λ + LCΛ )|X ).

The following theorem presents conditions for the solvability of the robust output regulation problem. It should be noted that (kPL (iωk )K1k k)k∈Z ∈ ℓ2 (C) implies k(PL (iωk )K1k )−1 k → ∞ as |k| → ∞, and thus the condition (14) requires that kEφk k and kF φk k decay sufficiently fast as |k| → ∞. Theorem 4.1. If E ∈ L(W, Ud ) and F ∈ L(W, Y ) satisfy (14a) (14b)

kCΛ R(iωk , A + L1 CΛ )Bd kk(PL (iωk )K1k )−1 kkEφk k k∈Z ∈ ℓ2 (C) k(PL (iωk )K1k )−1 kkF φk k) k∈Z ∈ ℓ2 (C),

then the controller with the above choices of parameters solves the robust output regulation problem. The controller is then guaranteed to be robust with respect to all perturbations in O for which the strong closed-loop stability is preserved, {iωk }k∈Z ⊂ ˜ for some N ∈ N, P˜ (iωk )K1k are invertible whenρ(A˜e ), {iωk }|k|≥N ⊂ ρ(A) ever |k| ≥ N , and for which ˜ B ˜d Eφ ˜ k − BK ˜ 1k (P˜ (iωk )K1k )−1 y˜k k kR(iωk , A) (15a) ∈ ℓ2 (C) |k|≥N k(P˜ (iωk )K1k )−1 y˜k k (15b) ∈ ℓ2 (C) |k|≥N

˜ k + F˜ φk . where y˜k = P˜d (iωk )Eφ

In the proof of Theorem 4.1 we will see that if {iωk }k∈Z ⊂ ρ(A), then the conditions (14) can alternatively be replaced with conditions (15) for the nominal plant. However, the condition (14) has the advantage that the condition only involves the resolvent and the transfer function of the stabilized plant. This is an advantage if iωk ∈ / ρ(A) for some k, as is the case in the example in Section 7. In the situation where sup|k|≥N kR(iωk , A)k < ∞

16

LASSI PAUNONEN

for some N ∈ N, also the norms kR(iωk , A)Bk and kR(iωk , A)Bd k are uniformly bounded with respect to k ∈ Z with |k| ≥ N , and the condition (15) simplifies to the following form. Corollary 4.2. If there exists N ∈ N such that {iωk }|k|≥N ⊂ ρ(A) and sup|k|≥N kR(iωk , A)k < ∞, then the conclusions of Theorem 4.1 hold if k(P (iωk )K1k )−1 k(kPd (iωk )Eφk k + kF φk k) |k|≥N ∈ ℓ2 (C)

and the controller is guaranteed to be robust with respect to perturbations in O for which the strong closed-loop stability is preserved, {iωk }k∈Z ⊂ ˜ sup|k|≥N kR(iωk , A)k ˜ < ∞, and ρ(A˜e ),{iωk }|k|≥N ⊂ ρ(A), ˜ k k + kF˜ φk k) k(P˜ (iωk )K1k )−1 k(kP˜d (iωk )Eφ ∈ ℓ2 (C). |k|≥N

It should also be noted that if kCΛ R(iωk , A)L1 k is uniformly bounded for large |k|, then the asymptotic rate of kPL (iωk )† k is at most equal to the rate of kP (iωk )† k. Due to our assumptions the norms kCΛ R(iωk , AL )Bd k are uniformly bounded with respect to k ∈ Z, and it is possible that kCΛ R(iωk , AL )Bd k → 0 as k → ±∞. Thus the summability condition for (kEφk k)k∈Z in (14) is in general weaker or equivalent compared to the summability condition for the sequence (kF φk k)k∈Z . The following corollary presents simplified versions of condition (14) for specific choices of K1 in the cases where kPL (iωk )† k are either polynomially or exponentially bounded.

Corollary 4.3. The following hold. (a) If there exists α > 0 such that kPL (iωk )† k = O(|ωk |α ) and if (γk )k∈Z ∈ ℓ2 (C), then the choice (16)

K1k = γk

PL (iωk )† kPL (iωk )† k

solves the robust output regulation problem for operators E and F that satisfy |ωk |α (kCΛ R(iωk , A + L1 CΛ )Bd kkEφk k + kF φk k) ∈ ℓ2 (C). |γk | k∈Z If we in particular choose γk = |ωk |−β for some β > 1/2 whenever ωk 6= 0, then |ωk |α /|γk | = |ωk |α+β whenever ωk 6= 0. (b) If there exists α > 0 such that kPL (iωk )† k = O(eα|ωk | ) and (γk )k∈Z ∈ ℓ2 (C) is arbitrary, then the choice (16) solves the robust output regulation problem for operators E and F that satisfy ! eα|ωk | (kEφk k + kF φk k) ∈ ℓ2 (C). |γk | k∈Z

We begin the proof of Theorem 4.1 by considering the stability properties of the semigroup generated by G1 − G2 G∗2 . In the case where dim Yk < ∞ the strong stability of the semigroup follows from [5], but we also need the results for an infinite-dimensional Yk .


17

Lemma 4.4. Assume U and P Yk for 2k ∈ Z are Hilbert spaces. Consider Z = { (zk )k∈Z ∈ ⊗k∈Z Yk | k∈Z kzk kYk < ∞ } with inner product hz, vi = P0 k∈Z hzk , vk iYk for z = (zk )k and v = (vk )k . Assume {iωk }k∈Z has no finite accumulation points, and let G1 = diag(iωk IYk )k∈Z on Z0 with domain D(G1 ) = { (zk )k∈Z ∈ Z0 | (ωk zk )k ∈ Z0 } and G2 = (G2k )k∈Z ∈ L(U, Z0 ). If the components G2k ∈ L(U, Yk ) of G2 are surjective and G∗2k have closed ranges for all k ∈ Z, then the semigroup generated by G1 − G2 G∗2 is strongly stable and iR ⊂ ρ(G1 − G2 G∗2 ). Moreover, if G2k are boundedly invertible for all k ∈ Z, then kR(iωk , G1 − G2 G∗2 )G2 k = kG−1 2k k for all for k ∈ Z. Proof. Since G1 generates a contraction semigroup, the same is true for G1 − G2 G∗2 , and thus σ(G1 − G2 G∗2 ) ⊂ C− . The strong stability of the semigroup will follow from the Arendt–Batty–Lyubich–V˜ u Theorem [1, 22] once we show iR ⊂ ρ(G1 − G2 G∗2 ). Let iω ∈ iR be such that ω 6= ωk for all k ∈ Z. We have iω ∈ ρ(G1 ), and if I + G∗2 R(iω, G1 )G2 is boundedly invertible, then the Woodbury formula (17a) (17b)

R(iω, G1 − G2 G∗2 ) = R(iω, G1 )

− R(iω, G1 )G2 (I + G∗2 R(iω, G1 )G2 )−1 G∗2 R(iω, G1 )

implies that iω − G1 + G2 G∗2 has a bounded inverse. Since G∗2 R(iω, G1 )G2 is bounded and skew-adjoint, we have 1 ∈ ρ(−G∗2 R(iω, G1 )G2 ), and this implies iω ∈ ρ(G1 − G2 G∗2 ). Assume now iω = iωn for some n ∈ Z. We will first show that there exists c > 0 such that k(iωn − G1 + G2 G∗2 )zk ≥ ckzk for all z ∈ D(G1 − G2 G∗2 ). If this is not true, there exists a sequence (zk )k∈N ⊂ D(G1 − G2 G∗2 ) such that kzk k = 1 for all k ∈ N and k(iωn − G1 + G2 G∗2 )zk k → 0 as k → ∞. Since iωn − G1 is skew-adjoint, we have k(iωn − G1 + G2 G∗2 )zk k ≥ |h(iωn − G1 + G2 G∗2 )zk , zk i| ≥ |Reh(iωn − G1 + G2 G∗2 )zk , zk i)| = kG∗2 zk k2 ,

and thus kG∗2 zk k → 0 as k → ∞. For every k ∈ N denote zk = zk1 + zk2 where zk1 ∈ R(iωn − G1 ), zk2 ∈ N (iωn − G1 ), and 1 = kzk k2 = kzk1 k2 + kzk2 k2 . There exists c1 > 0 such that k(iωn − G1 )zk1 k ≥ c1 kzk1 k for all k ∈ N. Thus c1 kzk1 k ≤ k(iωn − G1 )zk1 k = k(iωn − G1 )zk k

≤ k(iωn − G1 + G2 G∗2 )zk k + kG2 kkG∗2 zk k → 0

as k → ∞. Moreover, kG∗2 zk2 k ≥ k(G∗2n )† k−1 kzk2 k, and

k(G∗2n )† k−1 kzk2 k ≤ kG∗2 zk2 k ≤ kG∗2 zk k + kG∗2 zk1 k → 0

as k → ∞. We have now shown that zk1 → 0 and zk2 → 0, but this contradicts the assumption that kzk k2 = 1 for all k ∈ N, and thus shows that iωn − G1 + / σp (G1 + G2 G∗2 ), G2 G∗2 is lower bounded. In particular we now have iωn ∈ ∗ and that the range of G1 + G2 G2 is closed. Finally, the Mean Ergodic Theorem [2, Sec. 4.3] implies that the range of G1 + G2 G∗2 is dense, and we thus have iωn ∈ ρ(G1 + G2 G∗2 ). The structure of G1 and the assumption that the components G2k of G2 are boundedly invertible imply that N (G2 ) = {0} and R(iωk − G1 ) ⊕

18

LASSI PAUNONEN

R(G2 ) = Z0 for all k ∈ Z. Let k ∈ Z and u ∈ U , and denote z = R(iωk , G1 − G2 G∗2 )G2 u. Then R(iωk − G1 ) ∩ R(G2 ) = {0} and N (G2 ) = {0} imply (iωk − G1 )z = 0 ∗ (iωk − G1 )z = G2 (u − G2 z) ⇔ u = G∗2 z.

Thus z = (zl )l∈Z ∈ N (iωk − G1 ) is such that zl = 0 for all l 6= k, and −∗ zk = G−∗ 2k u which further implies kzkZ0 = kG2k ukY . Since u ∈ U was arbitrary, this implies kR(iωk , G1 − G2 G∗2 )G2 k = kG−1 2k k. Proof of Theorem 4.1. We can complete the proof by showing that the controller (G1 , G2 , K) constructed in Section 4 has the following properties: (i) The controller (G1 , G2 , K) is a regular linear system and it satisfies the G-conditions in Definition 3.5. (ii) The operator H is the unique solution of the Sylvester equation (18)

HG1 = (A + L1 CΛ )H + (B + L1 D)K1 ,

and satisfies R(H) ⊂ D(CΛ ) ∩ D(K2Λ ), and G1 + G2 (CΛ H + DK1 ) generates a strongly stable semigroup. (iii) The semigroup generated by Ae is strongly stable and σ(Ae ) ⊂ C− . (iv) There exists Me ≥ 0 such that kR(iωk , Ae )Be φk k ≤ Me max kEφk k, kF φk k, k(PL (iωk )K1k )−1 k(kCΛ R(iωk , A + L1 CΛ )Bd kkEφk k + kF φk k) . ˜e φk k) ∈ ℓ2 (C) if and only if (15) are satisfied. (v) (kR(iωk , A˜e )B The property that the controller (G1 , G2 , K) is a regular linear system on Z = Z0 × X follows from the results in [36, Sec. 7] since the controller is part of a system obtained from a regular linear      K1 −K2Λ 0 0  G1 0 , G2 0 ,  0 CΛ  , 0 D  , L B 0 A 0 0 0 K2Λ with admissible output feedback

0 I 0 ˆ K= . 0 0 I

It is also straightforward to verify that for all λ ∈ ρ(G1 ) (19)

R(λ, G1 )G2 = R(λ, G1e )G2

where G1e : D(G1e ) ⊂ Z → Z0 × X−1 is the operator G1 with domain D(G1e ) = D(G1 ) × X(B,L) . We will now show that (G1 , G2 , K) satisfies the G-conditions. The operator G2 is of the form G2 = (G2k )k∈Z , and its components G2k = −(PL (iωk )K1k )∗ are boundedly invertible for all k ∈ Z. This implies N (G2 ) = {0}, and also further shows that N (G2 ) = {0}. Let n ∈ Z be arbitrary and assume (w, v)T = (iωn − G1 )(z, x)T = G2 y ∈ R(iωn − G1 ) ∩ R(G2 ) for some (z, x)T ∈ Z × X and y ∈ Y . The property (19) implies that (z, x) ∈ D(G1 ) × X(B,L) and (iωn − G1 )z − G2 CK x w G2 y = = v Ly (iωn − A − BK2Λ − LCK )x


19

where CK = CΛ + DK2Λ . The first line implies (iωn − G1 )z = G2 (CK x + y), which means that in particular G2n (CK x + y) = (iωn − iωn )zn = 0. Since G2n is injective, we must have CK x + y = 0. Using this, the second line above implies (iωn − A − BK2Λ )x = L(CK x + y) = 0, and since iωn ∈ ρ((A+BK2Λ )|X ), we must have x = 0. Using this we get y = −CK x = 0, and thus also (w, v)T = G2 y = 0. This concludes that R(iωn −G1 )∩R(G2 ) = {0}. Since n ∈ Z was arbitrary, the controller satisfies the G-conditions. The results in [36, Sec. 7] show that the system (A+L1 CΛ , B+L1 D, CΛ , D) is regular, and since the semigroup generated by A + L1 CΛ is exponentially stable, there exists M ≥ 0 such that kPL (iωk )k ≤ M and kR(iωk , A + L1 CΛ )(B + L1 D)k ≤ M for all k ∈ Z. Using these estimates it is easy to see that the operators G2 and H satisfy G2 ∈ L(Y, Z0 ) and H ∈ L(Z0 , X). The resolvent identity in [36, Prop. 6.6] also shows that R(λ, A + L1 CΛ )R(B + L1 D) ⊂ XB . We will now show that H(D(G1 )) ⊂ XB and that H is the solution of the Sylvester equation (18). Denote AL = A + L1 CΛ and BL = B + L1 D for brevity. Let z = (zk )k∈Z ∈ D(G1 ). Using the resolvent identity −1 R(iωk , AL ) = −A−1 L + iωk AL R(iωk , AL ) we have X Hz = R(iωk , AL )BL K1k zk k∈Z

=− =

X

A−1 L BL K1k zk +

k∈Z −A−1 L BL K1 z

+

X

iωk A−1 L R(iωk , AL )BL K1k zk

k∈Z −1 AL HG1 z

∈ X(B,L)

since (iωk R(iωk , AL )BL K1k zk ) ∈ ℓ2 (X). This implies H(D(G1 )) ⊂ X(B,L) , and that H is the solution of the Sylvester equation (18). The uniqueness of the solution follows from the fact that for any n ∈ Z and z = (zk )k∈Z such that zk = 0 for k 6= n equation (18) implies Hz = R(iωn , AL )BL K1n zn . If z ∈ Z and λ > 0 then we can see analogously as in the proof of Lemma 3.4 that X λCR(λ, AL )Hz = λCR(λ, AL )R(iωk , AL )BL K1k zk k∈Z

−→

X

CΛ R(iωk , AL )BL K1k zk

k∈Z

as λ → ∞ since (AL , BL , C) is regular and since (K1k zk )k∈Z ∈ ℓ2 (U ). Thus R(H) ⊂ D(CΛ ) and analogously it can be shown that R(H) ⊂ D(K2Λ ). These properties imply that we can define C1 = CΛ H + DK1 ∈ L(Z0 , Y ) and X X CΛ Hz + DK1 z = CΛ R(iωk , AL )BL K1k zk + DK1k zk k∈Z

=

X

k∈Z

PL (iωk )K1k zk .

k∈Z

Thus G2 = −C1∗ . The fact that K1k were chosen so that PL (iωk )K1k are boundedly invertible imply that the components G2k of G2 are boundedly invertible for all k ∈ Z. We thus have from Lemma 4.4 that the semigroup

20

LASSI PAUNONEN

generated by G1 + G2 C1 = G1 − G2 G∗2 is strongly stable, iR ⊂ ρ(G1 + G2 C1 ) and kR(iωk , G1 + G2 C1 )G2 k = k(PL (iωk )K1k )−1 k for all k ∈ Z. We will now show that the closed-loop system is strongly stable and iR ⊂ ρ(Ae ). With the chosen controller (G1 , G2 , K) the operator Ae becomes   A BK1 −BK2Λ  G2 CΛ Ae = G2 CΛ G1 + G2 DK1 LCΛ LDK1 A + BK2 + LCΛ with domain D(Ae ) equal to (  x D(Ae ) =  z1  ∈ XB × D(G1 ) × X(B,L) x1 ) Ax + BK1 z1 − BK2Λ x1 ∈ X . (A + BK2Λ + LCΛ )x1 + LCΛ x + LDK1 z1 ∈ X

If we choose a similarity transform Qe ∈ L(X × Z0 × X) 

I Qe =  0 −I

0 I H

 0 0  = Q−1 e , −I

we can define Aê = Qe Ae Q−1 e on X × Z0 × X. Analogously as in the proof of [24, Thm. 12] we can see that (  x D(Aê ) =  z1  ∈ XB × D(G1 ) × XL x1 ) Λ Λ Λ (A + BK2 )x + B(K1 − K2 H)z1 + BK2 x1 ∈ X (A + L1 CΛ )x1 ∈ X where we have denoted XL = D(A)+R(R(λ0 , A)L) = D(A)+R(R(λ0 , A)L1 ). For xe = (x, z1 , x1 )T ∈ D(Aê ) a direct computation using L = L1 + HG2 , C1 = CΛ H + DK1 , and HG1 z1 = (A + L1 CΛ )Hz1 + (B + L1 D)K1 z1 yields  x  z1 Aê xe = Qe Ae  −x + Hz1 − x1 

  (A + BK2Λ )x + B(K1 − K2Λ H)z1 + BK2Λ x1 =  (G1 + G2 (CΛ H + DK1 ))z1 − G2 CΛ x1  (A + L1 CΛ )x1    Λ x A + BK2 B(K1 − K2Λ H) BK2Λ    z 0 G1 + G2 C1 −G2 CΛ = 1 . x1 0 0 A + L 1 CΛ

Since G1 + G2 C1 is strongly stable and iR ⊂ ρ(G1 + G2 C1 ), and since (A + BK2Λ )|X and (A + L1 CΛ )|X generate exponentially stable semigroups and K2Λ H ∈ L(Z0 , U ), we have from Lemma A.1 that the semigroups generated by Aê is strongly stable and iR ⊂ ρ(Aê ). Due to similarity, the same is true for Ae , and we thus conclude that the closed-loop system is strongly stable.


21

To guarantee the solvability of the Sylvester equation ΣS = Ae Σ + Be we need to estimate kR(iωk , Ae )Be φk k for k ∈ Z. We have from Theorem 2.3 that R(iωk , Ae )Be φk = R(iωk , Aee )Be φk where Aee is the operator Ae with the domain D(Aee ) = X(B,Bd ) ×ZG2 = X(B,Bd ) ×D(G1 )×X(B,L) . This further implies that for any k ∈ Z the element xe = R(iωk , Ae )Be φk is obtained with ê from the solution of the triangular system xe = Q−1 e x (iωk − Aêe )ˆ xe = Qe Be φk

where Aêe is the operator Aê with domain D(Aêe ) = X(B,Bd ) × D(G1 ) × D(CΛ ). A direct estimate using exponential stability of (A + BK2Λ )|X and (A + L1 CΛ )|X and the admissibility properties of the operators B, Bd , C, L1 , and K2 , shows that for all k ∈ Z we have kR(iωk , Aêe )Qe Be φk k . max kEφk k, kF φk k, kR(iωk , G1 + G2 C1 )G2 k(kCΛ R(iωk , AL )Bd kkEφk k + kF φk k) .

The condition (14) for the solvability of the robust output regulation problem now follows from kR(iωk , G1 + G2 C1 )G2 k = k(PL (iωk )K1k )−1 k. Finally, we will show that under the additional assumptions on the per˜e φk k)|k|≥N ∈ turbations the conditions (24) are equivalent to (kR(iωk , A˜e )B 2 ˜ ℓ (C). Due to the assumption {iωk }k∈Z ⊂ ρ(Ae ) this is further equivalent ˜e φk k)k∈Z ∈ ℓ2 (C). Let k ∈ Z be such that |k| ≥ N . We to (kR(iωk , A˜e )B begin by characterizing N (iωk − G1 ). Let zk = (z1k , xk1 )T ∈ N (iωk − G1 ) for some k ∈ Z and denote CK = CΛ + DK2Λ . Then using the fact that (G1 , G2 ) satisfy the G-conditions in Definition 3.5 we get   CK xk1 = 0 k k 0 (iωk − G1 )z1 − G2 CK x1 = ⇔ (iωk − G1 )z1k = 0 0 (iωk − A − BK2Λ )xk1 − LCK xk1  (iωk − A − BK2Λ )xk1 = 0 and since iωk ∈ ρ((A + BK2Λ )|X ), we have zk = (z1k , 0)T where z1k ∈ N (iωk − G1 ). This immediately implies that the restriction of the operator P˜ (iωk )K to the subspace N (iωk − G1 ) is given by (P˜ (iωk )K)|N (iωk −G1 ) = P˜ (iωk )K1k and it is boundedly invertible by assumption. If we denote ˜ k + F˜ φk , then by Lemma 3.10 we have y˜k = P˜d (iωk )Eφ   ˜ B ˜d Eφ ˜ k − BK ˜ 1k (P˜ (iωk )K1k )−1 y˜k ) R(iωk , A)( ˜e φk =   R(iωk , A˜e )B z˜k 0

where z˜k = (˜ zkl )l∈Z ∈ Z0 is such that z˜kk = −(P˜ (iωk )K1k )−1 y˜k and z˜kl = 0 for ˜e φk k)|k|≥N ∈ ℓ2 (C) all l 6= k. This immediately implies that (kR(iωk , A˜e )B if and only if (15) are satisfied. 4.1. Controller with a Reduced Order Internal Model. In this section we modify the internal model in the controller in Section 4 to design a controller that is robust with respect to a predefined class O0 ⊂ O of perturbations. In particular, this construction generalizes the results in [25] for regular linear systems and for infinite-dimensional exosystems. In this section we assume that {iωk }k∈Z ⊂ ρ(A) and P (iωk ) are boundedly invertible for all k ∈ Z. In particular we then have that either both U

22

LASSI PAUNONEN

and Y are infinite-dimensional, or dim Y = dim U . The class of admissible perturbations O0 ⊂ O may be chosen freely, but it is assumed that all its ˜ B, ˜ B ˜d , C, ˜ D, ˜ E, ˜ F˜ ) ∈ O0 are such that iωk ∈ ρ(A) ˜ and perturbations (A, P˜ (iωk ) is boundedly invertible for all k ∈ Z. For such O0 the construction of the reduced order internal model begins by defining n ˜ ˜ ˜ ˜ ˜ Sk = span P˜ (iωk )−1 (CR(iω k , A)Bd Eφk + F φk ) o ˜ B, ˜ B ˜d , C, ˜ D, ˜ E, ˜ F˜ ) ∈ O0 ⊂ U (A, and pk = dim Sk for k ∈ Z. The number of copies of each frequency iωk of the exosystem that we include in the reduced order internal model is equal to pk . Let ( ) X 2 Z0 = (zk )k∈Z zk ∈ Yk ∀k and kzk k < ∞ k∈Z

where Yk = Cpk if pk < dim Y and Yk = Y if pk = dim Y or pk = ∞. Define G1 : D(G1 ) ⊂ Z0 → Z0 by ) ( X 2 2 |ωk | kzk k < ∞ , G1 = diag iωk IYk k∈Z , D(G1 ) = (zk )k∈Z ∈ Z0 k∈Z

and choose K1 ∈ L(Z0 , U ) in such a way that K1 = (K1k )k∈Z , where K1k ∈ L(Yk , U ) are such that  pk if pk < dim Y   γk Qk ∈ L(C , U ) K1k = P (iωk )−1  ∈ L(Y, U ) if pk = dim Y or pk = ∞  γk kP (iωk )−1 k

where for each k ∈ Z we have Qk = [u1k , . . . , upkk ] ∈ L(Cpk , U ) and where k ⊂ U are bases of the subspaces Sk normalized in such a way that {ulk }pl=1 supk kQk kL(Cpk ,U ) < ∞, and (γk )k∈Z ∈ ℓ2 (C) where γk > 0. We then have that (kK1k k)k∈Z ∈ ℓ2 (C) and thus K1 ∈ L(Z0 , U ). Finally, we define G2 = (−(PL (iωk )K1k )∗ )k∈Z ∈ L(Y, Z0 ). The structure and the rest of the parameters of (G1 , G2 , K) are chosen as in the beginning of Section 4. Theorem 4.5. Assume P (iωk ) are invertible for all k ∈ Z. Assume further that E ∈ L(W, Ud ) and F ∈ L(W, Y ) satisfy (20a) kR(iωk , A) Bd Eφk − BP (iωk )−1 yk k k∈Z ∈ ℓ2 (C) γk−1 kQ†k P (iωk )−1 yk k (20b) ∈ ℓ2 (C) k∈J (20c) γk−1 kP (iωk )−1 kkyk k k∈Z\J ∈ ℓ2 (C)

where yk = Pd (iωk )Eφk + F φk and where J ⊂ Z is the set of indices for which Yk 6= Y . Then the controller with the above choices of parameters solves the robust output regulation problem for the class O0 of perturbations.


23

In particular, the controller is robust with respect to all perturbations in O0 for which the strong closed-loop stability is preserved, {iωk }k∈Z ⊂ ρ(A˜e ), and ˜ B ˜d Eφ ˜ k−B ˜ P˜ (iωk )−1 y˜k k kR(iωk , A) (21a) ∈ ℓ2 (C) k∈Z γk−1 kQ†k P˜ (iωk )−1 y˜k k (21b) ∈ ℓ2 (C) k∈J γk−1 kP (iωk )−1 kkP (iωk )P˜ (iωk )−1 y˜k k (21c) ∈ ℓ2 (C) k∈Z\J

˜ k + F˜ φk . where y˜k = P˜d (iωk )Eφ

Proof. The strong stability of the closed-loop system and the property iR ⊂ ρ(Ae ) can be verified similarly as in the proof of Theorem 4.1 since we again have C1 = CΛ H + DK1 = −G∗2 and the components G2k ∈ L(Y, Yk ) of G2 are either boundedly invertible or surjective finite rank operators. ˜ B, ˜ B ˜d , C, ˜ D, ˜ E, ˜ F˜ ) ∈ O0 and k ∈ Z be arbitrary and denote Let (A, ˜ ˜ ˜ y˜k = Pd (iωk )Eek +F ek . We begin by showing that the equations (13) in Theorem 3.7 have a solution for every k ∈ Z, i.e., there exists zk ∈ N (iωk − G1 ) ˜ such that yk . If k ∈ Z is such that Yk = Y , we can choose z P (iωk )KΛ zk = −˜ l l 1k = 0 for l 6= k and zk = 0 with z1k = (z1k )l∈Z ∈ Z0 such that z1k 1 −1 −1 k ∈ N (iωk − G1 ) z1k = − γk kP (iωk ) kP (iωk )P˜ (iωk ) y˜k . Then clearly z1k 0 and z P˜ (iωk )KΛ 1k = P˜ (iωk )K1k z1k = −P˜ (iωk )P˜ (iωk )−1 y˜k = −˜ yk . 0 It remains to consider the situation Yk 6= Y . We have P˜ (iωk )−1 y˜k ∈ Sk = l ) R(Qk ) by definition. Choose zk = z1k with z1k = (z1k l∈Z such that 0 † ˜ 1 −1 p k l z = 0 for all l 6= k and z1k = − γk Qk P (iωk ) y˜k ∈ Yk = C k . Then clearly 1k z1k ∈ N (iωk − G1 ) and 0 z k yk . P˜ (iωk )KΛ 1k = P˜ (iωk )K1k z1k = −P˜ (iωk )Qk Q†k P˜ (iωk )−1 y˜k = −˜ 0 Thus the equations (13) in Theorem 3.7 have a solution for all k ∈ Z. ˜e has a solutions To prove that the Sylvester equation ΣS = A˜e Σ + B satisfying R(Σ) ⊂ D(C˜eΛ ) whenever (21) holds, we will construct the solution Σ = (Π, Γ)T explicitly. Let k ∈ Z, choose zk as above and define ˜ BK ˜ Λ zk + B ˜d Eφ ˜ k ) and Γφk = zk . Then the properties Πφk = R(iωk , A)( ˜ k − F˜ φk Γφk = zk ∈ N (iωk − G1 ) and P˜ (iωk )KΛ zk = −˜ yk = −P˜d (iωk )Eφ imply ˜ ˜ (iωk − A)Πφ k − BKΛ Γφk ˜ (iωk − Ae )Σφk = ˜ Λ Γφk ) (iωk − G1 )Γφk − G2 (C˜Λ Πφk + DK ˜d Eφ ˜ k B ˜ = ˜ k ) = Be φk , −G2 (P˜ (iωk )KΛ zk + P˜d (iωk )Eφ ˜e φk . By Lemma 3.4 we have Σ ∈ L(W, Xe ) and or ΣSφk = A˜e Σφk + B ˜e φk k)k∈Z ∈ ℓ2 (C). If k ∈ Z R(Σ) ⊂ D(C˜eΛ ) if (kΣφk k)k∈Z = (kR(iωk , A˜e )B

24

LASSI PAUNONEN

is such that Yk = Y , then ˜ BK ˜ 1k z k + B ˜d Eφ ˜ k )k + kz k k kΣφk k . kΠφk k + kΓφk k = kR(iωk , A)( 1k 1k −1 ˜ ˜ B ˜d Eφ ˜ k−B ˜ P˜ (iωk )−1 y˜k )k + kP (iωk )P (iωk ) y˜k k = kR(iωk , A)( γk kP (iωk )−1 k−1

On the other hand, if k ∈ Z is such that Yk = Cpk , then ˜d Eφ ˜ k )k + kz k k ˜ BK ˜ 1k z k + B kΣφk k . kR(iωk , A)( 1k 1k

† ˜ −1 ˜ B ˜d Eφ ˜ k−B ˜ P˜ (iωk )−1 y˜k )k + kQk P (iωk ) y˜k k . = kR(iωk , A)( γk

Thus (kΣφk k)k∈Z ∈ ℓ2 (C) is satisfied if (21) are satisfied. In the case of the nominal plant the conditions (21) reduce to (20). 4.2. A New Controller Structure for Output Regulation. Proceeding in a similar manner as in the construction of the controller with a reduced order internal model in Section 4.1 we can also use the new controller structure to achieve output tracking and disturbance rejection in the situation where the controller is not required to be robust with respect to any perturbations to the parameters of the plant. The construction below includes a single copy of the internal model of the exosystem into the controller. In this section we assume {iωk }k∈Z ⊂ ρ(A), but it is not required that Assumption 2.2 on the surjectivity of the transfer functions is satisfied. Instead, we only need to assume that yk ∈ R(P (iωk )) for all k ∈ Z, where yk = Pd (iωk )Eφk + F φk . Let G1 to have the same general structure as in Section 4. Choose Z0 = W = ℓ2 (C) and G1 = S = diag(iωk )k∈Z , and choose K1 = (K1k )k∈Z : Z0 → U in such a way that K1k = γk kuukk k where (γk )k∈Z ∈ ℓ2 (C) with γk 6= 0 and where uk ∈ U are chosen in such a way that ( P (iωk )uk = yk if yk 6= 0 uk ∈ / N (P (iωk ))

if

yk = 0.

where yk = Pd (iωk )Eφk + F φk . If P (iωk ) has a closed range, we can in particular choose uk = P (iωk )† yk . We have (kK1k k)k∈Z ∈ ℓ2 (C) and thus K1 ∈ L(Z0 , U ). We define G2 = (−(PL (iωk )K1k )∗ )k∈Z ∈ L(Y, Z0 ). The rest of the parameters of (G1 , G2 , K) are chosen as in the beginning of Section 4. Theorem 4.6. Assume E ∈ L(W, Ud ) and F ∈ L(W, Y ) satisfy (22a) (22b) (22c)

(kR(iωk , A)Bd Eφk − R(iωk , A)Buk k)k∈J ∈ ℓ2 (C) γk−1 kuk k k∈J ∈ ℓ2 (C)

(kR(iωk , A)Bd Eφk k)k∈Z\J ∈ ℓ2 (C)

where yk = Pd (iωk )Eφk + F φk and where J ⊂ Z is the set of indices for which yk = 6 0. Then the controller with the above choices of parameters solves the output regulation problem. Proof. We have PL (iωk )K1k 6= 0 for all k ∈ Z. The strong stability of the semigroup generated by Ae and iR ⊂ ρ(Ae ) can be shown similarly as in the proof of Theorem 4.1 since we again have C1 = CΛ H + DK1 = −G∗2 , and the components G2k ∈ L(Y, C) of G2 are rank one and nonzero.


25

It remains to show that the regulator equations (8) have a solution Σ ∈ L(W, Xe ) satisfying R(Σ) ⊂ D(CeΛ ). We will construct the solution Σ = (Π, Σ)T explicitly by defining Πφk and Γφk . Let k ∈ Z. If yk 6= 0, then define kuk k l ) l k Γφk = z1k where z1k = (z1k l∈Z with z1k = 0 for all l 6= k and z1k = − γk . 0 Then Γφk ∈ N (iωk − G1 ). Moreover, define Πφk = R(iωk , A)(BKΓφk + Bd Eφk ) ∈ X(B,Bd ) . Then CeΛ Σφk + De φk = CΛ Πφk + DKΛ Γφk + F φk

= P (iωk )KΛ Γφk + Pd (iωk )Eφk + F φk = −P (iωk )uk + yk = 0.

Moreover, since Γφk ∈ N (iωk −G1 ) we have that CΛ Πφk +DKΛ Γφk = −F φk implies (iωk − A)Πφk − BKΛ Γφk (iωk − Ae )Σφk = = Be φk , (iωk − G1 )Γφk − G2 (CΛ Πφk + DKΛ Γφk )

or equivalently ΣSφk = Ae Σφk + Be φk . Thus Σ satisfies the regulator equations (8) on the subspace span{φk }. Alternatively, if k ∈ Z is such that yk = 0, we choose Γφk = 0 and Πφk = R(iωk , A)Bd Eφk . Then we can similarly see that CeΛ Σφk +De φk = 0 and ΣSφk = Ae Σφk +Be φk . It remains to show that Σ is bounded and R(Σ) ⊂ D(CeΛ ). By Lemma 3.4 the operator Σ has these properties if (kΣφk k)k∈Z = (kR(iωk , Ae )Be φk k)k∈Z ∈ ℓ2 (C). If yk 6= 0, we have k k k + Bd Eφk )k + kz1k kΣφk k . kΠφk k + kΓφk k = kR(iωk , A)(BK1k z1k

= kR(iωk , A)Bd Eφk − R(iωk , A)Buk k +

kuk k γk

and similarly if yk = 0, then k k kΣφk k . kR(iωk , A)(BK1k z1k + Bd Eφk )k + kz1k k = kR(iωk , A)Bd Eφk k.

We thus have (kΣφk k)k∈Z ∈ ℓ2 (C) due to the conditions (22). By Theorem 3.3 the controller solves the output regulation problem. 5. The Observer-Based Controller In this section we construct an observer-based controller that solves the robust output regulation problem. The construction is again done in parts and the required properties of the component operators are verified in the proof of Theorem 5.1. For this controller structure it is necessary to assume that PK (iωk ) are boundedly invertible for all k ∈ Z. This in particular implies dim U = dim Y if the plant has a finite number of inputs or outputs. Step 1◦ : We begin by choosing the state space of the controller as Z = Z0 × X, and choosing the structure of (G1 , G2 , K) as G1 0 G2 G1 = , G2 = , −L (B + LD)K1 A + BK2Λ + L(CΛ + DK2Λ ) and K = (K1 , K2Λ ). By Assumption 2.1 we can choose K21 ∈ L(X1 , U ) and Λ )| and (A + LC )| generate L ∈ L(Y, X−1 ) in such a way that (A + BK21 X Λ X exponentially stable semigroups and the system A, [B, L, Bd ], KC21 , D is regular. We define Λ Λ PK (λ) = (CΛ + DK21 )R(λ, A + BK21 )B + D,

Λ λ ∈ ρ(A + BK21 ).

26

LASSI PAUNONEN

We assume that PK (iωk ) are boundedly invertible for all k ∈ Z.

Step 2◦ : The operators (G1 , G2 ) make up the internal model of the exosystem (2), and they are defined by choosing Z0 = ℓ2 (Y ), and D(G1 ) = { (zk )k∈Z ∈ Z0 | (ωk zk )k∈Z ∈ ℓ2 (Y ) } G1 = diag iωk IY k∈Z ,

We choose G2 = (G2k )k∈Z in such a way that the components G2k ∈ L(Y ) are boundedly invertible and (kG2k k)k∈Z ∈ ℓ2 (C). In particular, it is possible to choose G2k = g2k IY , where (g2k )k∈Z ∈ ℓ2 (C) and g2k 6= 0 for all k ∈ Z. For more concrete choices of G2k , see Corollary 5.3. Step 3◦ : We define H : R([B, Bd ]) ⊂ X−1 → Z0 such that Λ Λ )x)k∈Z )R(iωk , A + BK21 Hx = (G2k (CΛ + DK21

∀x ∈ R([B, Bd ]).

Λ , [B, B ], C + DK Λ , D) Since we have from [36, Sec. 7] that (A + BK21 Λ d 21 is a regular linear system, it is immediate that H is well-defined and H ∈ L(X, Z0 ).

Step 4◦ : We choose the operator K1 ∈ L(Z0 , U ) in such a way that X K1 = − (G2k PK (iωk ))∗ zk k∈Z

Λ + K H ∈ L(X , U ), which is for all z = (zk )k∈Z ∈ Z0 . We define K2 = K21 1 1 an admissible observation operator for A and K2Λ = K2 . Finally, we choose the domain of the operator G1 as n D(G1 ) = (z1 , x1 ) ∈ D(G1 ) × X(B,L) o (A + LCΛ )x1 + (B + LD)(K1 z1 + K2Λ x1 ) ∈ X .

Theorem 5.1. If E ∈ L(W, Ud ) and F ∈ L(W, Y ) satisfy 2 kPK (iωk )−1 G−1 k (kEφ k + kF φ k) (23) ∈ ℓ2 (C), k k 2k k∈Z

then the controller with the above choices of parameters solves the robust output regulation problem. The controller is guaranteed to be robust with respect to all perturbations in O for which the strong closed-loop stability is preserved, {iωk }k∈Z ⊂ ρ(A˜e ), ˜ for some N ∈ N, P˜ (iωk ) are invertible whenever |k| ≥ N , {iωk }|k|≥N ⊂ ρ(A) and for which ˜ B ˜d Eφ ˜ k−B ˜ P˜ (iωk )−1 y˜k k kR(iωk , A) (24a) ∈ ℓ2 (C) |k|≥N k k(G2k PK (iωk )∗ )−1 (I − K2Λ RL BL )P˜ (iωk )−1 y˜k k (24b) ∈ ℓ2 (C) |k|≥N k kRL BL P˜ (iωk )−1 y˜k k (24c) ∈ ℓ2 (C) |k|≥N

˜ k + F˜ φk , Rk = R(iωk , A + LCΛ ) and BL = B + LD. where y˜k = P˜d (iωk )Eφ L

If {iωk }k∈Z ⊂ ρ(A), it is again possible to replace the condition (23) with (24) for the nominal plant. In the situation where sup|k|≥N kR(iωk , A)k < ∞, also the norms kR(iωk , A)Bk and kR(iωk , A)Bd k are uniformly bounded with respect to k ∈ Z with |k| ≥ N , and the conditions in Theorem 5.1 simplify to the following form.


27

Corollary 5.2. If there exists N ∈ N such that {iωk }|k|≥N ⊂ ρ(A) and sup|k|≥N kR(iωk , A)k < ∞, then the conclusions of Theorem 4.1 hold if k(G2k PK (iωk )∗ )−1 kkP (iωk )−1 k(kPd (iωk )Eφk k + kF φk k) |k|≥N ∈ ℓ2 (C)

and the controller is guaranteed to be robust with respect to perturbations in O for which the strong closed-loop stability is preserved, {iωk }k∈Z ⊂ ρ(A˜e ), ˜ sup|k|≥N kR(iωk , A)k ˜ < ∞, and {iωk }|k|≥N ⊂ ρ(A), ˜ k k + kF˜ φk k) k(G2k PK (iωk )∗ )−1 kkP˜ (iωk )−1 k(kP˜d (iωk )Eφ ∈ ℓ2 (C). |k|≥N

In the following we present concrete parameter choices in the situations where the norms kPK (iωk )−1 k are polynomially or exponentially bounded.

Corollary 5.3. The following hold. (a) If there exists α > 0 such that kPK (iωk )−1 k = O(|ωk |α ) and if (γk )k∈Z ∈ ℓ2 (C), then the choice

(25)

G2k = γk IY

solves the robust output regulation problem for operators E and F that satisfy |ωk |2α (kEφk k + kF φk k) ∈ ℓ2 (C). |γk |2 k∈Z

If we in particular choose γk = |ωk |−β for some β > 1/2 whenever ωk 6= 0, then |ωk |2α /|γk |2 = |ωk |2(α+β) whenever ωk 6= 0. (b) If there exists α > 0 such that kPK (iωk )−1 k = O(eα|ωk | ) and if (γk )k∈Z ∈ ℓ2 (C) is arbitrary, then the choice (25) solves the robust output regulation problem for operators E and F that satisfy ! e2α|ωk | (kEφk k + kF φk k) ∈ ℓ2 (C). |γk |2 k∈Z

Proof of Theorem 5.1. We complete the proof of the theorem by showing that (G1 , G2 , K) has the following properties: (i) The controller (G1 , G2 , K) is a regular linear system and it satisfies the G-conditions in Definition 3.5. (ii) The operator H is the unique solution of the Sylvester equation (26)

Λ Λ G1 H = H(A + BK21 ) + G2 (CΛ + DK21 )

on X(B,L) and G1 + B1 K1 generates a strongly stable semigroup. (iii) The semigroup generated by Ae is strongly stable and σ(Ae ) ⊂ C− . (iv) There exists Me ≥ 0 such that 2 kR(iωk , Ae )Be0 k ≤ Me (1 + kPK (iωk )−1 G−1 2k k )(kEφk k + kF φk k).

˜e φk k) ∈ ℓ2 (C) if and only if (24) are (v) (kR(iωk , A˜e )B The controller (G1 , G2 , K) being a regular linear system Sec. 7] if we consider the regular linear system G1 0 G2 0 0 0 0 K1 K2Λ , , , 0 A −L B L 0 D 0 CΛ

satisfied. follows from [36, 0 , 0

28

LASSI PAUNONEN

and apply output feedback with the admissible feedback operator   0 0 ˆ = I 0 . K 0 I Moreover, we have that for all λ ∈ ρ(G1 )

R(λ, G1 )G2 = R(λ, G1e )G2

(27)

where G1e : D(G1e ) ⊂ Z → Z0 × X−1 is the operator G1 with domain D(G1e ) = D(G1 ) × X(B,L) . We will now show that the controller satisfies the G-conditions. Since the components G2k of the operator G2 were assumed to be boundedly invertible, we have N (G2 ) = {0}, and further N (G2 ) = {0}. Let n ∈ Z be arbitrary and let (w, v)T = (iωn − G1 )(z, x)T = G2 y ∈ R(iωn − G1 ) ∩ R(G2 ) for some (z, x)T ∈ Z ×X and y ∈ Y . We have from (27) that (z, x)T ∈ D(G1 )×X(B,L) and (iωn − G1 )z w G2 y = = v −Ly (iωn − A − LCΛ )x − (B + LD)(K1 z + DK2Λ x) We have w = (wk )k∈Z ∈ Z0 and the structure of G1 implies that necessarily 0 = wn = G2n y. Since G2n is invertible, we must have y = 0, which further implies (w, v)T = G2 y = 0 and concludes that R(iωk − G1 ) ∩ R(G2 ) = {0}. Next we will show that H is the unique solution of (26). Denote AK = Λ and C = C + DK Λ for brevity. If x ∈ X A + BK21 K Λ (B,Bd ) , then 21 X kiωk G2k CK R(iωk , AK )xk2 k∈Z

=

X kG2k CK x − G2k CK R(iωk , AK )AK xk2 < ∞, k∈Z

since (kG2k k) ∈ ℓ2 (C) and supk∈Z kCK R(iωk , AK )AK xk < ∞ for x ∈ X(B,Bd ) . This implies H(X(B,Bd ) ) ⊂ D(G1 ). In particular H(XB ) ⊂ D(G1 ), and a direct computation shows that X G1 Hx − HAK x = G2k CK R(iωk , AK )(iωk − AK )x = G2 CK x k∈Z

for all x ∈ XB . Thus H is a solution of (26). On the other hand, if H = (Hk )k∈Z is a solution of (26), then necessarily iωk Hk = Hk AK +G2k CK , which implies that the solution is unique. The operator B1 = HB + G2 D ∈ L(U, Z0 ) is well-defined and of the form B1 = (B1k )k∈Z , where (28)

B1k = G2k CK R(iωk , AK )B + G2k D = G2k PK (iωk ).

Thus B1 = −K1∗ . The fact that G2k were chosen to be invertible imply that the components B1k of B1 are boundedly invertible for all k ∈ Z. We thus have from Lemma 4.4 that the semigroup generated by G1 + B1 K1 = G1 − B1 B1∗ is strongly stable and iR ⊂ ρ(G1 + B1 K1 ). Moreover, since


29

B1 = −K1∗ , we can see as in the proof of Lemma 4.4 that

−1 kK1 R(iωk , G1 + B1 K1 )k = kR(iωk , G1 + B1 B1∗ )B1 k = kB1k k

= kPK (iωk )−1 G−1 2k k

for all k ∈ Z. We will now show that the closed-loop system is strongly stable and iR ⊂ ρ(Ae ). When the controller (G1 , G2 , K) is chosen as suggested, we have that   A BK1 BK2Λ  G2 DK2Λ Ae =  G2 CΛ G1 + G2 DK1 Λ −LCΛ BK1 A + BK2 + LCΛ with domain

D(Ae ) =

 x  z1  ∈ XB × D(G1 ) × X(B,L) x1 ) n Ax + BK1 z1 + BK2Λ x1 ∈ X . (A + BK2Λ + LCΛ )x1 − LCΛ x + BK1 z1 ∈ X

(

Choose a similarity transform Qe ∈ L(X × Z0 × X)   −I 0 0 Qe =  H I 0 = Q−1 e −I 0 I

on X × Z0 × X. With a similar approach as and define Aê = Qe Ae Q−1 e in [24, Thm. 12] we can again see that (  x D(Aê ) =  z1  ∈ XB × D(G1 ) × XL x1 ) Λ )x − BK z − BK Λ x ∈ X (A + BK21 1 1 2 1 (A + LCΛ )x1 ∈ X Λ + For any xe = (x, z1 , x1 )T ∈ D(Aê ) a direct computation using K2Λ = K21 Λ )x Λ K1 H, B1 = HB + G2 D, and G1 Hx = H(A + BK21 )x + G2 (CΛ + DK21 yields      Λ −x A + BK21 −BK1 −BK2Λ x Λ ˆ      z1  . Ae xe = Qe Ae Hx + z1 = 0 G1 + B1 K1 B1 K2 −x + x1 0 0 A + LCΛ x1

Since the operator G1 + B1 K1 is strongly stable and iR ⊂ ρ(G1 + B1 K1 ), Λ )| and since (A + BK21 X and (A + LCΛ )|X generate exponentially stable semigroups, we have from Lemma A.1 that the semigroups generated by Aê is strongly stable and iR ⊂ ρ(Aê ). Due to similarity, the same is true for Ae , and we thus conclude that the closed-loop system is strongly stable. To estimate kR(iωk , Ae )Be0 k for k ∈ Z we again note that Theorem 2.3 implies R(iωk , Ae )Be0 = R(iωk , Aee )Be0 where Aee is the operator Ae with the domain D(Aee ) = X(B,Bd ) × ZG2 = X(B,Bd ) × D(G1 ) × X(B,L) . For all k ∈ Z

30

LASSI PAUNONEN

we have that xe = R(iωk , Ae )Be φk is given by xe = Q−1 ê where x ê is the e x solution of the triangular system (iωk − Aêe )ˆ xe = Qe Be φk where Aêe is the operator Aê with domain D(Aêe ) = X(B,Bd ) ×D(G1 )×D(CΛ ). Later in Lemma 6.3 it is shown that there exists a constant MG ≥ 1 such that 2 kR(iωk , G1 − G2 G∗2 )k ≤ M (1 + kPK (iωk )−1 G−1 2k k ). A direct estimate using Λ exponential stability of (A + BK2 )|X and (A + L1 CΛ )|X , the admissibility of B, Bd , C, L, and K2 , and the boundedness of HBd shows that shows that for all k ∈ Z 2 kR(iωk , Aêe )Qe Be φk k . (1 + kPK (iωk )−1 G−1 2k k )(kEφk k + kF φk k)

and thus the conditions (23) imply that (kR(iωk , Ae )Be φk k)k∈Z ∈ ℓ2 (C). Finally, we will show that under the additional assumptions on the per˜e φk k)|k|≥N ∈ turbations the conditions (24) are equivalent to (kR(iωk , A˜e )B 2 ˜ ℓ (C). Since {iωk }k∈Z ⊂ ρ(Ae ), this property is further equivalent to ˜e φk k)k∈Z ∈ ℓ2 (C). Let k ∈ Z be such that |k| ≥ N and (kR(iωk , A˜e )B ˜ ˜ k + F˜ φk . We have from Lemma 3.10 that denote y˜ = Pd (iωk )Eφ ˜ B ˜d Eφ ˜ k + BKz) ˜ R(iω , A)( k ˜e φk = R(iωk , A˜e )B z where z ∈ N (iωk −G1 ) is such that Kz = −P˜ (iωk )−1 y˜. Since (iωk −G1 )z = 0, we have that z = (z1 , x1 )T ∈ D(G1 ) × X(B,L) and denoting BL = B + LD and RL = R(iωk , A + LCΛ ) we have iωk − G1 0 z1 =0 −BL K1 iωk − A − LCΛ − BL K2Λ x1 (iωk − G1 )z1 = 0 ⇔ x1 = RL BL K1 z1 + RL BL K2Λ x1 = RL BL Kz = −RL BL P˜ (iωk )−1 y˜. Thus z1 ∈ N (iωk − G1 ), which implies z1 = (z1l )l∈Z is such that z1l = 0 for all l 6= k and K1 z1 = K1k z1k . Finally, the above equations imply K1k z1k = Kz − K2Λ x1 = (I − K2Λ RL BL )Kz −1 −1 (I − K2Λ RL BL )P˜ (iωk )−1 y˜. (I − K2Λ RL BL )Kz = −K1k and thus z1k = K1k Now   ˜ B ˜d Eφ ˜ k−B ˜ P˜ (iωk )−1 y˜) R(iωk , A)( ˜e φk =   z1 R(iωk , A˜e )B −1 ˜ −RL BL P (iωk ) y˜

and since K1k = −(G2k PK (iωk ))∗ and

kz1 k = kz1k k = k(G2k PK (iωk )∗ )−1 (I − K2Λ RL BL )P˜ (iωk )−1 y˜k ˜e φk k)k∈Z ∈ ℓ2 (C) if and only if the conditions (24) we have that (kR(iωk , A˜e )B are satisfied.


31

6. Nonuniform Stability of the Closed-Loop Semigroup and Decay Rates for e(t) In this section we derive decay rates for the orbits kTe (t)xe0 k of the closedloop system and the regulation error e(t) corresponding to the initial states xe0 ∈ D(Ae ) and v0 ∈ D(S). These results are based on the theory of nonuniform stability of semigroups presented in [21, 4, 6, 3]. It should also be noted that the result presented in the following theorem is not limited to the controller structures used in this paper, but instead it applies to any robust controller for which the closed-loop system satisfies kR(iω, Ae )k = O(g(|ω|)). Theorem 6.1. Assume the controller (G1 , G2 , K) solves the robust output regulation problem and assume g : R+ → [1, ∞) is a monotonically increasing function such that kR(iω, Ae )k ≤ O(g(|ω|)). In particular, for the controllers in Sections 4 and 5 we can choose g(·) for which there exists d 1 Mg , ωg > 0 such that | dω g(ω) | ≤ Mg for all ω > ωg and (a) k(PL (iωk )K1k )−1 k2 ≤ g(|ωk |) for all k ∈ Z in the case of the controller in Section 4. 2 (b) kPK (iωk )−1 G−1 2k k ≤ g(|ωk |) for all k ∈ Z in the case of the controller in Section 5.

(c) k(PL (iωk )K1k )† k2 ≤ g(|ωk |) for all k ∈ Z in the case of the controller in Section 4.1. (d) kuk k2 |γk |−2 kPL (iωk )uk k−2 ≤ g(|ωk |) for all k ∈ Z in the case of the controller in Section 4.2. Then there exists M0 , Me > 0 and 0 < c < 1 such that (29)

kTe (t)xe0 k ≤

Me kAe xe0 k, −1 Mlog (ct)

∀xe0 ∈ D(Ae )

−1 where Mlog is the inverse of the function

Mlog (ω) = M0 g(ω)(log(1 + M0 g(ω)) + log(1 + ω)),

ω > 0.

Furthermore, there exists Mee ≥ 1 such that for all v0 ∈ D(S) and for ˜e0 ∈ D(Ae ) the all initial states of the form xe0 = x ˜e0 − A−1 e Be v0 where x regulation error satisfies (30)

Z

t

t+1

ke(s)kds ≤

Mee kAe x ˜e0 − ΣSv0 k −1 Mlog (ct)

where Σ ∈ L(W, Xe ) is the solution of the regulator equations (8). −1 (t) has particularly simple forms in the most important The function Mlog situtations where g(·) is either a polynomial or an exponential function, i.e., when the norms k(PL (iωk )K1k )−1 k and kPK (iωk )−1 G−1 2k k grow either polynomially or exponentially fast when |ωk | is large. In particular, if g(ω) = −1 (ct) ∼ (t/ log t)1/α [3, Ex. 1.7], and thus (29) ω α for some α > 0, then Mlog

32

LASSI PAUNONEN

and (30) simplify to kTe (t)xe0 k ≤ Me Z

t+1 t

ke(s)kds ≤

Mee

1

log t t

log t t

α

1

α

kAe xe0 k, kAe x ˜e0 − ΣSv0 k.

In addition, if X is a Hilbert space, then it follows from [6] that the logarithm in the above inequalities can be omitted and we have Z t+1 Me Mee kTe (t)xe0 k ≤ 1/α kAe xe0 k, ke(s)kds ≤ 1/α kAe x ˜e0 − ΣSv0 k. t t t

−1 Moreover, if g(ω) = eαω for some α > 0, then Mlog (ct) ∼ α1 log t [4, Ex. 1.6] and (29) and (30) simplify to Z t+1 αMee αMe ke(s)kds ≤ kAe xe0 k, kAe x ˜e0 − ΣSv0 k. kTe (t)xe0 k ≤ log t log t t In the case where the operator Bd and the operator L in the controller are bounded, the convergence rate (30) is achieved for all v0 ∈ D(S) and xe0 ∈ D(Ae ), since in this case we have Be ∈ L(W, Xe ) and A−1 e Be v0 ∈ D(Ae ). The norm on the right-hand side of (30) can then be estimated by

kAe x ˜e0 − ΣSv0 k ≤ kAe xe0 k + kΣkkSv0 k + kBe kkv0 k.

In the further particular case where also the operator B is bounded, we have D(Ae ) = D(A) × D(G1 ) = D(A) × D(G1 ) × D(A) for both of the controller structures. Then an easy estimate using the admissibility of C and K leads to the following corollary. Here kxkD(A) = kAxk + kxk and kz0 kG1 and kv0 kD(S) are defined analogously. Corollary 6.2. If B, Bd , and L are bounded operators and the assumptions ˜ ee > 0 such that for all initial of Theorem 6.1 are satisfied, then there exists M states v0 ∈ D(S), x0 ∈ D(A) and z0 ∈ D(G1 ) Z t+1 ˜e M kx0 kD(A) + kz0 kD(G1 ) + kv0 kD(S) . ke(s)kds ≤ −1e Mlog (ct) t

In particular, for the controllers in Sections 4, 4.1, 4.2 and 5 there exists ˜ e > 0 such that for all initial states v0 ∈ D(S), x0 ∈ D(A) and z0 = M e (z10 , x01 )T ∈ D(G1 ) × D(A) Z t+1 ˜e M ke(s)kds ≤ −1e kx0 kD(A) + kz10 kD(G1 ) + kx01 kD(A) + kv0 kD(S) . Mlog (ct) t Finally, if the operators C and K are bounded, then the proof of Theorem 6.1 shows that the regulation error e(t) converges to zero at a rate Me ˜e0 − ΣSv0 k. ke(t)k ≤ −1e kAe x Mlog (ct)

The nonuniform stability properties described in Theorem 6.1 are based on the following result on the behaviour of the resolvent of the stabilized internal model. In particular Theorem 6.3 implies that the semigroup generated by G1 − G2 G∗2 is nonuniformly stable in the sense of [4, 6, 3].


33

Theorem 6.3. Assume U and P Yk for2 k ∈ Z are Hilbert spaces. Consider Z = { (zk )k∈Z ∈ ⊗k∈Z Yk | k∈Z kzk kYk < ∞ } with inner product hz, vi = P0 k∈Z hzk , vk iYk for z = (zk )k and v = (vk )k . Assume {iωk }k∈Z has no finite accumulation points, and let G1 = diag(iωk IYk )k∈Z on Z0 with domain D(G1 ) = { (zk )k∈Z ∈ Z0 | (ωk zk )k ∈ Z0 } and G2 = (G2k )k∈Z ∈ L(U, Z0 ). Assume further that G2k ∈ L(U, Yk ) of G2 are surjective and G∗2k have closed ranges for all k ∈ Z, and assume g : R+ → [1, ∞) is a monotonically increasing function such that kG†2k k2 ≤ g(|ωk |) for all k ∈ Z, and there exists d 1 Mg , ωg > 0 such that | dω g(ω) | ≤ Mg for all ω > ωg . Then the semigroup ∗ generated by G1 − G2 G2 is strongly stable, iR ⊂ ρ(G1 − G2 G∗2 ) and there exists M > 0 such that (31)

kR(iω, G1 − G2 G∗2 )k ≤ M g(|ω|),

|ω| > ωg .

Proof. The strong stability of the semigroup and iR ⊂ ρ(G1 − G2 G∗2 ) follow from Lemma 4.4. The approach in the remaining part of the proof is inspired by the technique used in [21, Ex. 1–3]. If the estimate (31) does not hold for any M > 0, then there exists (sn )n∈N ⊂ R satisfying |sn | ≥ 1, and |sn | → ∞ as n → ∞, and (zn )n∈N ⊂ D(G1 ) with kzn k = 1 for all n ∈ N such that g(|sn |)k(isn − G1 + G2 G∗2 )zn k → 0

as n → ∞. For all n ∈ N we then also have

0 ← Rehg(|sn |)(isn − G1 + G2 G∗2 )zn , zn i = g(|sn |)kG∗2 zn k2 ,

which further implies p g(|sn |)k(isn − G1 )zn k

≤ g(|sn |)k(isn − G1 + G2 G∗2 )zn k +

p

g(|sn |)kG∗2 zn k → 0

as n → ∞. For each n ∈ N denote mn = arg mink |sn − ωk | ∈ Z. Since d = inf k6=l |ωk − ωl | > 0, we have that |sn − ωk | ≥ d/2 for all k 6= mn . Denote zn = (znk )k∈Z where znk ∈ Y . For any 0 < δ < 1/2 there exists Nδ ∈ N such that for all n ≥ Nδ we have X δ2 ≥ g(|sn |)k(isn − G1 )zn k2 = g(|sn |) |sn − ωk |2 kznk k2 k∈Z

≥ g(|sn |)|sn − ωmn |2 kznmn k2 + g(|sn |)

d2 X k 2 kzn k . 4 k6=mn

Since |sn | → ∞ as n → ∞, we can assume that |ωmn | ≥ ωg + 1 for all n ≥ Nδ . Define yn = (ynk )k∈Z ∈ Z0 such that ynmn = znmn and ynk = 0 for k 6= mn . Then X 4δ2 kznk k2 ≤ 2 g(|sn |)kzn − yn k2 = g(|sn |) (32) d k6=mn

and (33)

kznmn k2 = kzn k2 −

X

k6=mn

kznk k2 ≥ 1 −

4δ2 . d2

34

LASSI PAUNONEN

The above estimates also imply 2

2

δ ≥ g(|sn |)|sn − ωmn |

kznmn k2

2

≥ |sn − ωmn |

4δ2 1− 2 d

and thus |sn − ωmn |2 ≤ δ2 /(1 − 4δ2 /d2 ) ≤ 2δ2 if δ2 < d2 /8 and n ≥ Nδ . This means that the points sn of the sequence approach the points in the set {ωk }k∈Z as n → ∞. Now p p g(|sn |)kG∗2 zn k = g(|sn |)kG∗2 yn + G∗2 (zn − yn )k p 2δ ≥ g(|sn |)kG∗2 yn k − kG2 k d

d 1 due to (32). Since | dω g(ω) | is bounded for ω > ωg and since |sn − ωmn | → 0

as n → ∞, there exists N2 ∈ N such that Using (33) and

k(G∗2mn )† k2

=

kG†2mn k2

g(|sn |) g(|ωmn |)

≥ 1/2 for all n ≥ N2 .

≤ g(|ωmn |) we get

g(|sn |) kz mn k2 k(G∗2mn )† k2 n 1 4δ2 4δ2 g(|sn |) 1− 2 1− 2 ≥ ≥ g(|ωmn |) d 2 d

g(|sn |)kG∗2 yn k2 = g(|sn |)kG∗2mn znmn k2 ≥

for all n ≥ max{Nδ , N2 }. Combining the above estimates we see that for a small enough δ > 0 we thus have p p 2δ g(|sn |)kG∗2 zn k ≥ g(|sn |)kG∗2 yn k − kG2 k d 1 4δ2 2 1 2δ 1− 2 ≥√ − kG2 k > 0 d d 2 p for all n ≥ max{Nδ , N2 }. This contradicts the property g(|sn |)kG∗2 zn k → 0 as n → ∞, and therefore the proof is complete.

Proof of Theorem 6.1. The triangular structure of the operators Aê in the proofs of Theorems 4.1, 4.5, 4.6, and 5.1 imply that if the function g : R+ → [1, ∞) is chosen so that k(PL (iωk )K1k )† k2 ≤ g(|ωk |) for the controllers in Sections 4, 4.1, and 4.2, or so that k(G2k PK (iωk ))† k2 ≤ g(|ωk |) in the case of the controller in Section 5, then kR(iω, Ae )k = O(g(|ω|)).

Moreover, (PL (iωk )K1k )† = (PL (iωk )K1k )−1 in part (a), (G2k PK (iωk ))† = kuk k † −1 in PK (iωk )−1 G−1 2k in part (b), and k(PL (iωk )K1k ) k = |γk | kPL (iωk )uk k part (c). The nonuniform decay rate (29) now follows directly from [4, Thm. 1.5]. It remains to consider the behaviour of the regulation error. Assume ˜e0 ∈ D(Ae ). Then ΣSv0 = v0 ∈ D(S) and xe0 = x ˜e0 − A−1 e Be v0 where x Ae Σv0 + Be v0 implies Ae (xe0 − Σv0 ) = Ae x ˜e0 − Be v0 − ΣSv0 + Be v0 = Ae x ˜e0 − ΣSv0 ∈ Xe and thus xe0 − Σv0 ∈ D(Ae ). Since Ce is admissible with respect to Te (t) R1 there exists κ > 0 such that 0 kCeΛ Te (s)xk2 ds ≤ κ2 kxk2 for all x ∈ Xe .


35

Because Σ is the solution of the regulator equations (8), the proof of Theorem 3.3 and the nonuniform decay of Te (t) imply Z 1 Z t+1 kTe (s)Te (t)(xe0 − Σv0 )kds ≤ κkTe (t)(xe0 − Σv0 )k ke(s)kds = 0

t

≤

κMe κMe kAe (xe0 − Σv0 )k = −1 kAe x ˜e0 − ΣSv0 k. −1 Mlog (ct) Mlog (ct)

7. Robust Control of a Two-Dimensional Heat Equation In this section we construct a controller that achieves robust output tracking and disturbance rejection for a two-dimensional heat equation with Neumann boundary control and boundary measurement. In particular, we consider the equation (34a) (34b) (34c)

xt (ξ, t) = ∆x(ξ, t),

x(ξ, 0) = x0 (ξ)

∂x ∂x (ξ, t)|Γ1 = u(t), (ξ, t)|Γ2 = d(t), ∂n ∂n Z x(ξ, t)dξ, y(t) =

∂x (ξ, t)|Γ0 = 0, ∂n

Γ3

on the unit square ξ = (ξ1 , ξ2 ) ∈ Ω = [0, 1]×[0, 1]. Here u(t) is the Neumann boundary control input and d(t) is the external disturbance signal. The control and disturbance are located on the parts Γ1 and Γ2 of the boundary ∂Ω, where Γ1 = { ξ = (ξ1 , 0) | 0 ≤ ξ1 ≤ 1 } and Γ2 = { ξ = (0, ξ2 ) | 0 ≤ ξ2 ≤ 1/2 } and the remaining part of the boundary is denoted by Γ0 = ∂Ω\(Γ1 ∪Γ2 ). The observation is on the part Γ3 = { ξ = (1, ξ2 ) | 0 ≤ ξ2 ≤ 1 } of the boundary of the square. The controlled heat equation can be written as an abstract linear system ∂x = on X = L2 (Ω) by choosing A = ∆ with D(A) = { x ∈ H 2 (Ω) | ∂n 0 on ∂Ω } and choosing operators B, Bd ∈ L(C, X−1 ) and C ∈ L(X1 , C) such that [9] Z 1 x(0, ξ2 )dξ2 . B = δΓ1 (·), Bd = δΓ2 (·), Cx = 0

We have from [9, Cor. 1] that the controlled heat equation (34) is a regular linear system with D = 0 ∈ C.

7.1. Reference and Disturbance Signals. Our aim is to construct a controller that achieves robust output tracking of a 2π-periodic reference signal yref (t) depicted in Figure 1 despite 2π-periodic disturbance signals. We use the controller structure presented in Section 4. The reference signal and 2π-periodic disturbance signals d(t) can be generated with an infinite-dimensional exosystem on the space W = ℓ2 (Z, C) with frequencies ωk = k for k ∈ Z. If (φk )k∈Z denotes the canonical basis of W and if for some fixed small ε > 0 we choose F ∈ L(W, C) in such a way that yr (k)|k|1/2+ε k 6= 0 F φk = 0 k=0

36

LASSI PAUNONEN

1 0 −1 0

2π

4π

6π

Figure 1. The 2π-periodic reference signal. where yr (k) are the complex Fourier coefficients of the reference signal yref (t), then yref (t) is generated with the initial state v0 = (v0k )k∈Z with v00 = 1 and v0k = |k|−1/2−ε for all k 6= 0. The Fourier coefficients yr (k) of yref (t) satisfy |yr (k)| = O(|k|−3 ), and thus |F φk | = O(|k|−5/2+ε ). 7.2. Stabilization and Controller Parameters. Since 0 ∈ σp (A) the heat equation is unstable. If we choose L1 ∈ L(C, X) and K2 ∈ L(X, C) in such a way that Z 2 x(ξ)dξ, L1 = −π 2 · 1 K2 x = −π Ω

h i Λ where 1(ξ) = 1 for all ξ ∈ Ω, then (A, C K2 , [B, L1 , Bd ]) is a regular linear system and the semigroups generated by A + L1 CΛ and (A + BK2 )|X are exponentially stable. For this example the transfer function of the plant has an explicit formula P (λ) = λ1 for all λ ∈ C+ \ {0}, and a straightforward computation using the identity PL (λ) = CL R(λ, A + L1 CΛ )B = (I − CR(λ, A)L1 )−1 P (λ) shows that 1 PL (λ) = , λ ∈ C+ \ {0}. λ + π2 Since dim Y = 1, we choose the internal model to contain one copy of the exosystem so that Z0 = W = ℓ2 (Z; C) and G1 = S, and as suggested Step 4◦ and in Corollary 4.3, we choose K1 ∈ L(Z0 , C) with components PL (iωk )−1 ik + π 2 γ0 γ0 √ = , 1 + |k|1/2+κ |PL (iωk )−1 | 1 + |k|1/2+κ k2 + π 4 γ0 √ = −(PL (iωk )K1k )∗ = − 1/2+κ (1 + |k| ) k2 + π4

K1k = G2k

where κ > 0 is fixed and small, and γ0 > 0. Then k(PL (iωk )K1k )−1 k = O(|k|3/2+κ ). Moreover, we define L ∈ L(C, X−1 ) in such a way that X L = L1 + HG2 = L1 + R(iωk , A + L1 CΛ )BK1k G2k k∈Z

= −π 2 · 1 − γ02

R(iωk , A + L1 CΛ )B (1 + |k|1/2+κ )2 (π 2 − ik) k∈Z X


37

7.3. Solvability of the Robust Output Regulation Problem. Since supω∈R kR(iω, A)k < ∞, we can use the conditions in Corollary 4.2 to determine the solvability of the output regulation problem. For all k 6= 0 we have |k|(1 + |k|1/2+κ ) |(P (iωk )K1k )−1 | = γ0 For our choice of the reference signal yref (t) we have kF φk k = O(|k|−5/2+ε ), and thus |(P (iωk )K1k )−1 |kF φk k = O(|k|−1+ε+κ ), and |(P (iωk )K1k )−1 |kF φk k k6=0 ∈ ℓ2 (C) whenever ε + κ < 1/2. Since C and Bd are admissible, we have |Pd (iωk )| . |ωk |−1 = |k|−1 for all k 6= 0. Because of this, ˜ k | . |k|1/2+κ |Eφ ˜ k |. |(P (iωk )K1k )−1 ||Pd (iωk )||Eφ

Thus Corollary 4.2 shows that the controller is capable of rejecting any ˜ k satisfying |Eφ ˜ k | . |k|−β disturbance signal that can be expressed with Eφ for any exponent β > 1 + κ and corresponding to the initial state v0 = (v0k )k6=0 with v00 = 1 and v0k = |k|−1/2−ε for k 6= 0. In particular, this ˆ includes any reference signal d(t) whose Fourier coefficients satisfy d(k) = ˜ − β ˜ |k| for any β > 3/2 + ε + κ. 7.4. Robustness Properties of the Controller. By Corollary 4.2 the controller is guaranteed to be robust with respect to perturbations that satisfy the standing assumptions for the class O, for which A˜ remains a generator of an analytic semigroup, the strong closed-loop stability is preserved, and ˜ k k + kF˜ φk k) k(P˜ (iωk )K1k )−1 k(kP˜d (iωk )Eφ ∈ ℓ2 (C) |k|≥N

for some sufficiently large N ∈ N. With our choice of K1 the above condition simplifies to ! |k|1/2+κ ˜ ˜ k | + |F˜ φk |) (|Pd (iωk )||Eφ ∈ ℓ2 (C). |P˜ (iωk )| |k|≥N

In particular, since the operators Bd , E and F do not appear in the operator ˜d , E, ˜ and F˜ Ae , the controller is robust with respect to any perturbations B for which the above summability condition is satisfied. 7.5. Convergence of the Regulation Error. Since |(P (iωk )K1k )−1 |2 = O(|k|3+2κ ), the results in Section 6 imply that the regulation error has a nonuniform convergence rate 1 log t 3+2κ kAe xe0 k kTe (t)xe0 k = Me t

for all intial states xe0 ∈ D(Ae ) of the closed-loop system. Since the initial state v0 corresponding to the reference signal yref (t) is such that v0 ∈ / D(S), Theorem 6.1 does not imply a rate of convergence for the output to this particular reference yref (t). However, the results do guarantee that the

38

LASSI PAUNONEN

1/(3+2κ) R t+1 integrals t ke(s)kds converge at rates of order logt t for initial states v0 ∈ D(S) of the exosystem and for suitable initial states xe0 of the closed-loop system. 7.6. Numerical Approximation and Simulation. The controlled heat equation was simulated using a finite difference approximation with a 16 × 16 grid on the square Ω = [0, 1] × [0, 1]. In the simulation the infinitedimensional exosystem approximated using a 21–dimensional truncation of the operator S. The controller parameters were set to κ = 1/8, γ0 = 12, and ε = 1/10. For simulation the resolvent operators R(iωk , A + L1 CΛ ) appearing in L were approximated numerically using a higher order finite difference approximation with a 41 × 41 grid, and the infinite sum was approximated with a truncation corresponding to the truncation of the exosystem. The behaviour of the controlled system was simulated for an external disturbance signal d(t) = cos(4t) + 21 sin(t). Initial states of the plant and the controller are chosen to be zero. Figures 7.6 and 7.6 depict the output y(t) of the controlled plant on the interval [4π, 12π] and the behaviour of the R t+1 integrals t ke(s)kds, respectively. Finally, Figure 7.6 depicts the state of the controlled system at time t ≈ 20. 1 0 −1 4π

8π

12π

Figure 2. Output y(t) of the controlled plant.

0.1

0.0 0

4π

12π

Figure 3. Behaviour of the integrals

20π

Z

t

t+1

ke(s)kds.


39

1 0 –1 1 1

ξ2

ξ1 0

0

Figure 4. State of the controlled heat equation at time t ≈ 20. 8. Conclusions In this paper we have studied the construction of controllers for output tracking and disturbance rejection for a regular linear system with nonsmooth reference and disturbance signals. In particular, we have constructed two controllers that solve the robust output regulation problem, one controller that achieves robustness with respect to a predefined class O0 of uncertainties, and one controller for output tracking and disturbance rejection without the robustness requirement. The two different controller structures for robust output regulation result in controllers that differ especially in the conditions for the solvability of the robust output regulation problem. In particular, these solvability conditions imposed on E and F are distinctly weaker in the case of the controller constructed in Section 4 than those for the observer-based controller in Section 5. The underlying reason for this difference is that the controller structure in Section 4 allows accurate study of the asymptotic rates of the norms kR(iωk , Ae )Be φk k that in turn guarantee the solvability of the Sylvester equation ΣS = Ae Σ + Be . On the other hand, we saw in Section 6 that the nonuniform convergence rates for the closed-loop system and the regulation error are of the same order for both of the controller types. The controllers constructed in this paper are infinite-dimensional, and the most important topic for further research is to develop methods for approximating them with finite-dimensional error feedback controllers. Appendix A. Stability of Composite Systems Lemma A.1. Assume X1 , X2 , and X3 are Banach spaces and let   A1 ∆1 ∆12 ∆22 ∆3  , A =  0 A2 0 0 A3

on X = X1 × X2 × X3 satisfy the following.

40

LASSI PAUNONEN

(a) The semigroup T1 (t) generated by A1 is exponentially stable and ∆1 and ∆12 are A1 -admissible input operators. (b) The semigroup T2 (t) generated by A2 is strongly stable. (c) The semigroup T3 (t) generated by A3 is exponentially stable and ∆22 and ∆3 are A3 -admissible output operators. Then the semigroup T (t) generated by the operator A with the natural domain is strongly stable. Moreover, if iR ⊂ ρ(A2 ), then iR ⊂ ρ(A) and there exists M ≥ 0 such that kR(iω, A)k ≤ M kR(iω, A2 )k. Proof. The exponential stability of T1 (t) and T3 (t) together with the triangular structure of A imply σ(A) ∩ C+ = σ(A2 ) ∩ C+ and that kR(λ, A)k . kR(λ, A2 )k for all λ ∈ ρ(A2 ) ∩ C+ . The semigroup T (t) generated by A has the form   T1 (t) S1 (t) S2 (t) T2 (t) S3 (t) , T (t) =  0 0 0 T3 (t) where

Z

t

Z

t

T2 (t − s)∆3 T3 (s)x3 ds S3 (t)x3 = 0 0 Z t Z t T1 (t − s)∆12 ∆22 T3 (s)x3 ds T1 (t − s)∆1 S3 (s)x3 ds + S2 (t)x3 = S1 (t)x2 =

T1 (t − s)∆1 T2 (s)x2 ds,

0

0

for all x2 ∈ X2 and x3 ∈ D(A3 ). To show that T (t) is strongly stable it is sufficient to show that S1 (t), S2 (t) and S3 (t) converge to zero strongly as t → ∞. It is easy to see that the second part of S2 (t) decays to zero exponentially since T1 (t) and T3 (t) are exponentially stable. Since T3 (t) is exponentially stable and R∆3 is admissible, it is easy to ∞ see that S3 (t) is uniformly bounded and 0 k∆3 T3 (s)x3 kds < ∞ for all x3 ∈ D(A3 ). Let x3 ∈ D(A3 ) and let M2 > 0 be such that kT2 (t)k ≤ M2 for all t ≥ 0. Then for all t > 0 and 0 < t1 < t we have

Z ∞ Z t1

k∆3 T3 (s)x3 kds. + M kS3 (t)x3 k ≤ T (t − s)∆ T (s)x ds T (t − t ) 2 2 1 3 3 3 1

2 t1 0 R∞ There exists t1 > 0 such that t1 k∆3 T3 (s)x3 kds < ε/(2M ), and for such fixed t1 > 0 the first term on the right-hand side becomes smaller than ε > 0 for all sufficiently large t > t1 . Thus kS3 (t)x3 k → 0 as t → ∞, and since x3 ∈ D(A3 ) was arbitrary and S3 (t) is uniformly bounded, we have that S3 (t) → 0 strongly as t → ∞. To show that S1 (t) and the first part of S2 (t) converge to zero strongly as t → ∞, it is sufficient to consider F (·) : [0, ∞) → L(Y, X1 ) Z t F (t)y = T1 (t − s)∆S(s)yds 0

where ∆ is A1 -admissible and S(·) is a strongly continuous and strongly stable operator valued function such that kS(t)k ≤ MS for all t ≥ 0. There exists κ > 0 such that for all f ∈ C([0, 1], Y ) and t ∈ [0, 1]

Z t

≤ κkf kL2 (0,1) .

T (t − s)∆f (s)ds 1

0


41

Let y ∈ Y and t > 0 be arbitrary, and let t = n + t0 for some n ∈ N0 and t ∈ [0, 1). If M1 , ω > 0 are such that kT1 (t)k ≤ M e−ωt for all t ≥ 0 and if we denote g(k) = kS(· + k)ykL2 (0,1) for k ∈ N0 , then

Z t

n−1 X

g(k)kT1 (t − k − 1)k T (t − s)∆S(s)yds ≤ κg(n) + κ 1

0

k=0

≤ κg(n) + κM eω

n−1 X k=0

e−ω(n−k) g(k) −→ 0

as t → ∞ since g(k) → 0 as k → ∞.

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Department of Mathematics, Tampere University of Technology, PO. Box 553, 33101 Tampere, Finland E-mail address: [email protected]

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