Optimal Port Allocation for Nonlinear Distributed Parameter Systems*

2nd IFAC Workshop on Control of Systems Governed 2nd IFAC Workshop Control of Systems Governed by Partial Differentialon Equations 2nd IFAC on Control 2nd IFAC Workshop Workshop on Control of of Systems Systems Governed Governed by Partial Differential Equations June 13-15, 2016. Bertinoro, Italy Available online at www.sciencedirect.com by Partial Differential Equations by Partial Differential Equations June 13-15, 2016. Bertinoro, Italy June June 13-15, 13-15, 2016. 2016. Bertinoro, Bertinoro, Italy Italy

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Optimal Port Allocation for Optimal Port Allocation for Optimal Port Allocation for  Optimal Port Allocation for Nonlinear Distributed Parameter Systems  Nonlinear Distributed Parameter Systems Nonlinear Nonlinear Distributed Distributed Parameter Parameter Systems Systems 

Gou Nishida ∗∗ Noboru Sakamoto ∗∗ ∗∗ Gou Nishida ∗∗ Noboru Sakamoto ∗∗ Gou Gou Nishida Nishida Noboru Noboru Sakamoto Sakamoto ∗∗ ∗ ∗ Graduate School of Information Science and Engineering, Tokyo ∗ Graduate School of Information Science and Engineering, Tokyo ∗ Graduate School Science and Institute of Technology, 2-12-1 W8-1, O-okayama, Meguro-ku,Tokyo Tokyo, Graduate School of of Information Information Science and Engineering, Engineering, Tokyo Institute of Technology, 2-12-1 W8-1, O-okayama, Meguro-ku, Tokyo, Institute of Technology, 2-12-1 W8-1, O-okayama, Meguro-ku, 152-8552, JAPAN. (e-mail: [email protected]) Institute of152-8552, Technology, 2-12-1(e-mail: W8-1, O-okayama, Meguro-ku, Tokyo, Tokyo, JAPAN. [email protected]) ∗∗ 152-8552, JAPAN. (e-mail: [email protected]) of Mechatronics, Faculty of Science and Engineering, 152-8552, JAPAN. (e-mail: [email protected]) ∗∗ Department ∗∗ Department of Mechatronics, Faculty of Science and Engineering, ∗∗ Department of Faculty of and Nanzan University, 18 Yamazato-cho, Shyowa-ku, 466-8673, Department of Mechatronics, Mechatronics, Faculty of Science ScienceNagoya, and Engineering, Engineering, Nanzan University, 18 Yamazato-cho, Shyowa-ku, Nagoya, 466-8673, Nanzan University, 18 Yamazato-cho, Shyowa-ku, Nagoya, JAPAN. (e-mail: [email protected]) Nanzan JAPAN. University, 18 Yamazato-cho, Shyowa-ku, Nagoya, 466-8673, 466-8673, (e-mail: [email protected]) JAPAN. JAPAN. (e-mail: (e-mail: [email protected]) [email protected]) Abstract: This paper proposes an optimal way of allocating collocated input-output pairs for Abstract: This paperparameter proposes an optimal allocating collocated input-output pairs for Abstract: This proposes optimal way of allocating input-output pairs for stabilizing distributed systems. Weway firstof introduce finite-dimensional reduced model Abstract: This paper paperparameter proposes an an optimal way ofintroduce allocatingaa collocated collocated input-output pairs for stabilizing distributed systems. We first finite-dimensional reduced model stabilizing distributed parameter systems. We first introduce a finite-dimensional reduced model from sampled initial responses of the systems via the POD (Proper Orthogonal Decomposition)stabilizing distributed parameter systems. We first introduce a finite-dimensional reduced model from sampled initial responses of the systems via the POD (Proper Orthogonal Decomposition)from sampled initial responses of systems the (Proper Decomposition)Galerkin method. Next, optimal gains of the via stabilizing controller for the reduced systems are from sampled initial responses of the the systems via the POD PODcontroller (Proper Orthogonal Orthogonal Decomposition)Galerkin method. Next, optimal gains of the stabilizing for the reduced systems are Galerkin method. Next, optimal gains of the stabilizing controller for the reduced systems designed by the stable manifold method that is an exact numerical solver of Hamilton-Jacobi Galerkin method. Next,manifold optimal method gains of that the stabilizing controller forsolver the reduced systems are are designed by the stable is an exact numerical of Hamilton-Jacobi designed byFinally, the stable stable manifoldthree method that is is methods an exact exact derived numerical solver of shape Hamilton-Jacobi equations.by we present allocation from state matching, designed the manifold method that an numerical solver of Hamilton-Jacobi equations. Finally, we present three allocation methods derived from state shape matching, equations. we three allocation methods state shape matching, dissipation Finally, enhancement, and their mixed evaluation, and derived we show from that the optimal allocations equations. Finally, we present present three allocation methods derived from state shape allocations matching, dissipation enhancement, and their mixed evaluation, and we show that the optimal dissipation enhancement, and their mixed evaluation, and we show that the optimal allocations can be associated with energy controls in terms of port representations. dissipation enhancement, and their mixed evaluation, and we show that the optimal allocations can be associated with energy controls in terms of port representations. can be with controls in of representations. can be associated associated with energy energy controls in terms terms Control) of port port Hosting representations. © 2016, IFAC (International Federation of Automatic by Elsevier Ltd. All rights reserved. Keywords: Optimal control, Model reduction, Nonlinear systems, Distributed parameter Keywords: Optimal control, Model reduction, Nonlinear systems, Keywords:Hamilton-Jacobi Optimal control, control, Model reduction, reduction, Nonlinear Nonlinear systems, systems, Distributed Distributed parameter parameter systems, equations Keywords: Optimal Model Distributed parameter systems, Hamilton-Jacobi equations systems, Hamilton-Jacobi equations systems, Hamilton-Jacobi equations 1. INTRODUCTION are sampled. In this study, we clarify an optimal way how 1. INTRODUCTION are sampled. In this study, we clarify an optimal how 1. INTRODUCTION are sampled. In we clarify an way we decide minimum positions allocating ports.way 1. INTRODUCTION are sampled. In this this study, study, we of clarify an optimal optimal way how how we decide minimum positions of allocating ports. decide minimum positions of ports. This paper presents an optimal allocation of collocated we we decide minimum positions of allocating allocating ports. This paper is organized as follows. Section 2 summaThis an allocation of This paper paper presents presents an optimal optimal allocation of collocated collocated input-output pairs called ports (Van der Schaft, 2000) This is organized follows. Section summaThis paper presents an optimal allocation of collocated This paper is as follows. Section 2 summarizes apaper nonlinear optimal as design via 2 stable input-output pairs called ports (Van der Schaft, 2000) This paper is organized organized ascontrol follows. Section 2the summainput-output pairs called ports (Van der Schaft, 2000) in a nonlinear optimal regulator design for distributed rizes a nonlinear optimal control design via the stable input-output pairs called ports (Van der Schaft, 2000) rizes a nonlinear optimal control design via the manifold method for nonlinear distributed parameter sysin optimal for a nonlinear optimal control design parameter via the stable stable in aaa nonlinear nonlinear optimal regulator design for distributed distributed parameter systems. Theregulator controllerdesign is designed by the rizes manifold method for nonlinear distributed sysin nonlinear optimal regulator design for distributed manifold method for for nonlinear distributed parameter systems (Hamaguchi et nonlinear al., 2014, distributed 2015) basedparameter on the PODparameter systems. The controller is designed by the manifold method sysparameter systems. The(Sakamoto controller and is designed designed by the tems stable manifold method Van der by Schaft, (Hamaguchi et al., 2014, 2015) based on the PODparameter systems. The controller is the tems (Hamaguchi et al., 2014, 2015) based on the PODGalerkin method. In Section 3, we first recall a formal stable manifold method (Sakamoto and Van der Schaft, tems (Hamaguchi et al., 2014, 2015) based on the PODstable Sakamoto, manifold method method (Sakamoto and numerical Van der der Schaft, Schaft, 2008; 2012) that is an exact solver Galerkin method. Inof Section 3, we first recall a formal stable manifold (Sakamoto and Van Galerkin method. 3, first recall a port representation PDEs and clarify i.e., 2008; Sakamoto, that an exact method. In Inof Section Section 3, we we first their recallports, a formal formal 2008; Sakamoto, 2012) 2012) that is is of an nonlinear exact numerical numerical solver of Hamilton-Jacobi equations optimalsolver con- Galerkin port representation PDEs and clarify their ports, i.e., 2008; Sakamoto, 2012) that is an exact numerical solver port representation of PDEs and clarify their ports, i.e., collocated input-output pairs (Nishida et al., 2012). Next, of Hamilton-Jacobi equations of nonlinear optimal conrepresentation of PDEs and clarify their2012). ports,Next, i.e., of Hamilton-Jacobi Hamilton-Jacobi equations ofof nonlinear nonlinear optimal con- port trol problems. Although a lot of efforts foroptimal analytically collocated input-output pairs (Nishida et al., of equations concollocated input-output pairs (Nishida et al., 2012). Next, we recall one of boundary energy controls (or passivitytrol problems. Although a lot of efforts for analytically collocated input-output pairs (Nishida et al., 2012). Next, trol problems. Although a lot of efforts for analytically solving the Hamilton-Jacobi equation have been proposed we recall one of boundary energy controls (or passivitytrol problems. Although a lot of efforts for analytically based we recall one boundary controls (or controls) ports,energy the damping can solving equation been recall one of ofusing boundary energy controlsassignment (or passivitypassivitysolving the the Hamilton-Jacobi Hamilton-Jacobi equationethave have been proposed proposed (e.g.,Kunisch et al. (2004); Varshney al. (2009), and see we based controls) using ports, the damping assignment can solving the Hamilton-Jacobi equation have been proposed based controls) using ports, the damping assignment can be regard as an optimal regulator (Nishida et al., 2013). (e.g.,Kunisch et Varshney et see controls) using ports, the damping assignment can (e.g.,Kunisch et al. al. (2004); (2004); Varshney et al. al. (2009), (2009), and see based also Aliyu (2011)), our approach doesn’t use suchand series be regard as an optimal regulator (Nishida et al., 2013). (e.g.,Kunisch et al. (2004); Varshney et al. (2009), and see be regard as an optimal regulator (Nishida et al., 2013). Thus, we can decide distributed control gains by the stable also Aliyu (2011)), our approach doesn’t use such series be regard as an optimal regulator (Nishida et al., 2013). also Aliyu (2011)), our approach doesn’t use such series approximations of solutions. The method for modeling Thus, we can can decide distributed control gains by the the stable stable also Aliyu (2011)), our approach doesn’t use such series Thus, we decide distributed control gains by manifold method. Then, we propose three optimal approximations of method we can decide distributed control gains by theallocastable approximations of solutions. solutions. The method for forbymodeling modeling distributed parameter systems The is implemented a finite- Thus, manifold method. Then, we propose three optimal allocaapproximations of solutions. The method for modeling manifold method. Then, we propose three optimal allocations of ports as optimal problems. Section 4 shows their distributed parameter systems is implemented by a finitemanifold method. Then, we propose three optimal allocadistributed parameter parameter systems is implemented implemented by aa finitefinite- tions dimensional reduction systems of the original infinite-dimensional of ports as optimal problems. Section 44 shows their distributed is by tions of ports as optimal problems. Section shows validity by a numerical experimentation. dimensional reduction of infinite-dimensional of by ports as optimalexperimentation. problems. Section 4 shows their their dimensional reduction of the the original original infinite-dimensional systems via the POD (Proper Orthogonal Decomposition) tions validity a numerical dimensional reduction of the original infinite-dimensional validity systems via the POD (Proper Orthogonal Decomposition) validity by by aa numerical numerical experimentation. experimentation. systems via the POD (Proper Orthogonal Decomposition) -Galerkin method (Holmes et al., 1998; Christofides, 2001). 2. SUMMARIES OF MODEL REDUCTION AND systems via the POD (Proper Orthogonal Decomposition) -Galerkin method (Holmes et Christofides, 2. SUMMARIES OF MODEL REDUCTION AND -Galerkin method (Holmes et al., al., 1998; 1998; Christofides, 2001). The reduction derives a minimum number of basis2001). func2. SUMMARIES OF REDUCTION AND CONTROL DESIGN -Galerkin method (Holmes et al., 1998; Christofides, 2001). 2.NONLINEAR SUMMARIESOPTIMAL OF MODEL MODEL REDUCTION AND The reduction derives aa minimum number of basis funcNONLINEAR OPTIMAL CONTROL DESIGN The reduction derives minimum number of basis functions from time responses of the system; therefore, the NONLINEAR OPTIMAL CONTROL DESIGN The reduction derives a minimum number of basis funcNONLINEAR OPTIMAL CONTROL DESIGN tions from time responses of the system; therefore, the tions from time responses of the system; therefore, the method is a time specialized control particular set of initial tions from responses of for theaa system; therefore, the 2.1 Objective control systems method specialized control set Objective control systems method is is a aand specialized control for for a particular particular set of of initial initial conditions their responses under the assumption that 2.1 2.1 method is a specialized control for a particular set of initial 2.1 Objective Objective control control systems systems conditions and their responses under the assumption that conditions and their responses under the assumption that controlled responses close to the original free responses. Let X be a real Hilbert Space. Let X �� be the dual space conditions and their responses under the assumption that controlled responses close to the original free responses. � be the dual space Let X be a real Hilbert Space. Let X controlled responses close to the original free responses. Moreover, we assumed that a finite number of inputs Let X be a real Hilbert Space. X be dual of X. Then, the inner product and norm in space X is controlled responses close to the original free responses. Let X Then, be a real Hilbert Space. Let Let X �the be the the dual space Moreover, we assumed that a finite number of inputs of X. the inner product and the norm in X is Moreover, we assumed that a finite number of inputs with particular distribution profiles at some fixedof positions of X. Then, the inner product and the norm in X denoted by (·, ·) and �·� , respectively. The dual pairing X X Moreover, we assumed that a finite number inputs of X. Then, the and inner product and theThe norm inpairing X is is with particular distribution profiles at some fixed positions denoted by (·, ·) �·� , respectively. dual X X with particular distribution profiles at some fixed positions and the full observability of the spatial domain. However, denoted by by (·, ·) ·)(·, and �·� respectively. The dual pairing is denoted .X Let V and H be real separable X ·) X  ×X with particular distribution profiles at some fixed However, positions denoted by (·, and �·� ,, respectively. The dual pairing X ·) X and the full observability of the spatial domain.  is denoted by (·, . Let V and H be real separable X  that ×X . Let and last the two full observability observability of the spatial domain. However, the assumptions of arethe notspatial satisfactory to However, practical is denoted by ·) and H be separable Hilbert spaces such V ⊂V V is ×X . Let and the full domain. is denoted by (·, (·, ·)X V H. andWe H assume be real real that separable X  that ×X V ⊂ the last two assumptions are not satisfactory to practical Hilbert spaces such H. We assume that V is the last two assumptions are not satisfactory to practical situations, because it is better that optimal input positions Hilbert spaces such that V ⊂ H. We assume that dense in H and the injection of V ⊂ H is compact. Please the last twobecause assumptions are not satisfactory to positions practical Hilbert spaces such that V ⊂ H. We assume that V V is is situations, it is better that optimal input dense in H and the injection of V ⊂ H is compact. Please situations, because it is better that optimal input positions are systematically given and almost all actual observations dense in H and the injection of V ⊂ H is compact. Please see (Lions, 1971) for detail. situations, because it is better that optimal input positions dense in H and the injection of V ⊂ H is compact. Please are systematically given and almost all actual observations are systematically systematically given given and and almost almost all all actual actual observations observations see see (Lions, (Lions, 1971) 1971) for for detail. detail. are see for detail.  This We (Lions, consider1971) the nonlinear evolution equation work was supported by JSPS Grants-in-Aid for Scientific  We consider the nonlinear evolution equation This work was supported by JSPS Grants-in-Aid for Scientific  Research (C) No. 26420415, and JSPS Grants-in-Aid forfor Challenging We consider the nonlinear evolution equation This work was supported by JSPS Grants-in-Aid Scientific  d We consider(y(t), the nonlinear evolution equation This work was 26420415, supportedand byJSPS JSPSGrants-in-Aid Grants-in-Aid for Scientific Research (C) No. ϕ) + α(y(t), ϕ) + (Ly(t), ϕ)H d Exploratory Research No. 26630197. N. Sakamoto for wasChallenging supported H Research (C) No. 26420415, and JSPS Grants-in-Aid for Challenging d Research (C) No. 26420415, and JSPS Grants-in-Aid for Challenging dt (y(t), ϕ) + α(y(t), ϕ) + (Ly(t), ϕ) d Exploratory Research No. 26630197. N. Sakamoto was supported H (y(t), ϕ) + α(y(t), ϕ) + (Ly(t), ϕ)H by Nanzan University Pache ResearchN.Subsidy I-A-2 forsupported the 2015 Exploratory Research No. 26630197. Sakamoto was H H dt (y(t), ϕ) + α(y(t), ϕ) + (Ly(t), Exploratory Research No. 26630197. N.Subsidy Sakamoto was supported H H dt +(N (y(t)), ϕ)V  ,V = (Bu(t), ϕ)Vϕ) ,V (1a) by Nanzan University Pache Research I-A-2 for the 2015 academic year. dt by Nanzan University Pache Research Subsidy I-A-2 for the 2015  ,V = (Bu(t), ϕ)V  ,V +(N (y(t)), ϕ) (1a) by Nanzan University Pache Research Subsidy I-A-2 for the 2015 V  ,V = (Bu(t), ϕ)V  ,V academic year. +(N (y(t)), ϕ) (1a) V academic year. +(N (y(t)), ϕ)  = (Bu(t), ϕ)  (1a)

V ,V academic year. Copyright © 2016 International Federation of 286 2405-8963 © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2016 International Federation of 286 Automatic Control Copyright 2016 International Federation of 286 Peer review© under International Copyright © 2016 responsibility International of Federation of Federation of Automatic 286Control. Automatic Control Automatic Control 10.1016/j.ifacol.2016.07.455 Automatic Control

V ,V

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for all ϕ ∈ V with the initial condition y(0) = y0 ∈ H, (1b) where y(t) ∈ V is the state, u(t) ∈ Rm is the control input and B : Rm → V � is a continuous linear operator, α : V × V → R is the symmetric bilinear continuous form that is coercive, i.e., there exists a constant κ > 0 such that α(v, v) ≥ κ�v�2V for all v ∈ V , L : V → H is the self-adjoint continuous linear operator, and N : V → V � is the continuous nonlinear operator mapping such that N (0) = 0 and it’s Fr`echet derivative N � satisfies N � (0) = 0. 2.2 Problem setting for optimal control design of nonlinear infinite-dimensional systems In this paper, we assume that the evolution equation in (1a) with the initial condition (1b) has a solution. Let us consider the following optimal regulator design for the infinite-dimensional system. Definition 2.1. An optimal control input is determined by the optimal control problem minimizing the cost functional  ∞ (Qy(t), y(t))H + uT (t)Ru(t)dt (2) J = 0

over all trajectories of the system (1a) with (1b), where Q is a positive semi-definite linear operator, and R a positive definite matrix. 2.3 Proper orthogonal decomposition

We obtain samples of state distributions from a free response y(t) of the system (1a, 1b) with u(t) = 0 for making a set of basis functions that represents the dynamics of the system. The set of the samples is called a snapshot set, and we defined it as the set of functions S = {yk | 1 ≤ k ≤ n, yk ∈ X}, where yk = y(tk ) is the sample at each discrete time tk for k = 1, · · · , n, X is a separable Hilbert space, and n is the number of elements in S. Let us consider the optimal solution ϕ maximizing the objective function �(yk , ϕ)2 �k max (3) ϕ∈X �ϕ�2 as an optimal orthogonal basis for representing the snapshot set S, where we have defined the average of images of a map f : X → R or X → X by n 1 �f (yk )�k = f (yk ). (4) n k=1

This means that ϕ is the most parallel function to each element in S in the sense of mean square. Then, the stationary condition of the variational problem of minimizing J(ϕ) = �(yk , ϕ)2 �k − λ(�ϕ�2 − 1) under the assumption �ϕ�2 = 1 can be written as follows (Holmes et al., 1998): Rϕ = λϕ, (5) where λ ∈ R is non-negative and R : X → X is a compact operator defined by Rϕ = �(yk , ϕ)yk �k . The maximum value of (3) is equivalent to the maximum eigenvalue of (5). The existence of an orthonormal basis consisting of eigenvectors in (5) is guaranteed by HilbertSchmidt theorem. The orthogonal basis obtained from this procedure is called a POD basis. Then, there exist the associated eigenvalues λ1 ≥ λ2 · · · ≥ 0. 287

285

2.4 Galerkin projection and reduced order models By using the Galerkin projection and the POD method, we can derive a finite-dimensional reduced system from the infinite-dimensional system (1a, 1b). By describing any signal y with the POD basis as l  y(t) = ai (t)ϕi ,

(6)

i=1

the finite-dimensional reduced system is given as a˙ = Ar a + Nr (a) + Br u,

(7) T

where we definite vector a by a(t) = [a1 (t), · · · , al (t)] ∈ Rl , the matrices Ar ∈ Rl×l , Br ∈ Rl×m and the nonlinear function Nr (a) ∈ Rl are defined by (8) (Ar )ij = −α(ϕi , ϕj ) − (Lϕi , ϕj )H , (9) (Br )ij = (bi , ϕj )V  ,V ,     l  (Nr (a))i = N ak ϕk , ϕi , (10) k=1

V  ,V

respectively, and the map bk : R → V is given by Bu =  m k=1 bk uk , where the subscript ( · )ij means the (i, j)element of a matrix, and δi,j is the Kronecker delta. The initial value a0 = a(0) is determined by (a0 )i = (y0 , ϕi )H for i = 1, · · · , l. �

The order l is determined by the criterion whether a POD basis can sufficiently describe the original response. For example, we can use the energy ratio E(l) := l n j=1 λj / j=1 λj between primal l POD basis functions and the sampled response, namely E(l) ≈ 1. 2.5 Optimal problem for reduced order systems The problem in Definition 2.1 can be transformed into the following problem for the reduced order system (7). Proposition 1. [Hamaguchi et al. (2014)] An optimal control input for the reduced system (7) is determined by the optimal control problem minimizing the cost functional  ∞ J= aT (t)Qa(t) + uT (t)Ru(t)dt (11) 0

over all trajectories of the system (7) with the initial condition, where Q is a semi-positive definite matrix, and R is a positive definite matrix.

The optimal controller derived from the problem in Proposition 1 can be written as u(a) = −1/2R−1BrT p(a), (12)

where p = (∂V /∂a)T is determined by a stabilizing solution of the Hamilton-Jacobi equation (HJE): 1 H(a, p) = pT fr (a) + aT Qa − pT Br R−1 BrT p = 0, (13) 4 and fr = Ar a + Nr (a). Then, V is a value function defined as the minimum of (7) and, and V is said to be a stabilizing solution if p(0) = 0 and 0 is an asymptotical stable equilibrium of the vector field fr (a)−1/2Br R−1 BrT p of the controlled system. In conventional approaches, (13) is attempted to be solved directly with respect to V by using series approximations. On the other hand, the stable manifold method used in this study can find sufficient exact numerical solutions (see Appendix A).

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3. MAIN RESULT In this section, we first confirm that the product of the input u and the output y˙ has the physical dimension of power, and controls consisting of such a pair possess a certain optimality in terms of port representation. Next, we propose two ways and their mixed way of allocating the collocated input-output pairs for stabilizing distributed parameter systems. 3.1 Port representation for PDEs We first recall a formal port representation of PDEs and their ports (Nishida et al., 2012). Port representations are originated from Hamiltonian systems in analytical physics. A Hamiltonian is equivalent to the total energy of the system. Thus, a power balance between internal energy changes and interactions with environment through ports can be introduced from the system representations. Furthermore, this relation can be rephrased by the Stokes theorem in the case of distributed parameter systems, i.e., internal energy changes can be transformed into energies flowing through the boundary in some case. Therefore, the port representations are used for boundary energy controls (Duindam et al., 2009). However, boundary energy controls cannot be applied to all systems of PDEs, because they are not always written as Hamiltonian systems. The applicability can be systematically checked by the following results. Theorem 3.1. (Nishida et al. (2014)). Let us consider a system Δ dy = Δ1 dy 1 + · · · + Δl dy l = 0 of PDEs defined on a vertically star-shaped domain over the domain Z in the bundle J 2r M , where M is the base manifold with the local coordinates x = (x0 , x1 , · · · , xm ), J 2r M is the 2r-th order jet bundle that is the space of state functions yI (x) including up to 2r-th order derivatives with respect to x, I is the multi-index that describes all variations of higher order derivatives, and dy i for 1 ≤ i ≤ l is the differential 1-form that is equivalent to the infinitesimal variation of y i . Then the system can be described as follows: Δ = ΔE + ΔD , (14) E E E ∞ 2r l D where Δ = (Δ1 , · · · , Δl ) ∈ (C (J M )) , Δ = D ∞ 2r l ∗ ∗ (ΔD 1 , · · · , Δl ) ∈ (C (J M )) , D ΔE = D ΔE , D ΔD �= D ΔD , and we have defined the operator D Δ and its adjoint operator D∗Δ for a test function f = (f1 , · · · , fl ) ∈ (C ∞ (J 2r M ))l as the following (l × l)-matrix operator with (a, b)-components for 1 ≤ a ≤ l and 1 ≤ b ≤ l: r � ∂Δa DI fb , (15) (D Δ )ab (fb ) = ∂ubI |I|=0 � � r � ∂Δb (16) (−1)|I| DI f (D∗Δ )ab (fb ) = b . ∂uaI |I|=0

We can apply boundary energy controls to the subsystem ΔE . However, for systems including ΔD , we can use distributed energy controls with the following power balance. Proposition 2. (Nishida et al. (2014)). For the subsystem ΔE = 0, there exist Lagrangians L ∈ Ωv0 (J 2r M ) from which the subsystem can be derive, where Ω i is the space of (vertical) i-forms. Then, at an instantaneous time, the power balance equation of (14) is given by 288

∂E = ∂t

� � Z

eap fpa

+

r−1 �

a eaqk fqk

+

ead fda

k=0

�

dx,

(17)

where 1 ≤ a ≤ l, 0 ≤ |K| ≤ r − 1 means the multi-index with respect to spatial variables, |J| = k, and we have defined ⎧ � � ∂L ⎪ a a |Kt| a ⎪ (f , e ) = (−1) DKt a , ut , ⎪ ⎪ ∂u ⎨ p p � Kt � ∂L (18) a a a (f , e ) = Dt uJ , , ⎪ a ⎪ ⎪ qk qk � ∂u ⎪ J � ⎩ a a (fd , ed ) = −uat , −ΔD a . by using L such that dv L = ΔE du ∈ Ωv1 (J 2r M ), where dv is the vertical exterior differential operator that means a variational differential operator, and DI is the total differential with respect to I.

When there is no ΔD in systems, the dissipative term ead fda in (17) disappears and the remaining integrands can be � transformed into those of the boundaryD integration dx. However, the system (1a) includes Δ as follows. ∂Z Proposition 3. Let Z be a spatial domain where the system (1a) is defined. We assume that Z is contractible to the origin. Let E be the total energy of the system. Then, the system satisfies the power balance � � � dE = y˙ + (−Dx )|I| yI + Ly + N (y) − Bu y˙ dx, dt Z (19) where Dx is the differential operator with respect to x, yI is the |I|-th derivative with respect to x with the evenorder |I|, and we assumed α(y(t), ϕ) = ((−Dx )|I| yI , ϕ). Proof. (19) can be introduced from (17) with (18) of the system. Indeed, the second term α(y(t), ϕ) under the assumption α(y(t), ϕ) = ((−Dx )|I| yI , ϕ), the third term (Ly(t), ϕ)H , and a part of the fourth term (N (y(t)), ϕ)V  ,V can be defined by the following stationary conditions of variational calculus, i.e., they are in ΔE : ∂L dv Lα (yI ) = dyI = yI dyI ∂y � �� � I = −Di yI dyI\i + Di yI dyI\i for Lα (yI ) =

= · · · = (−1)|I| (DI yI ) dy

(20)

∂L dy = Ly dy ∂y

(21)

(1/2)yI2 ,

dv LL (y) = for LL (y) = (L/2)y 2 ,

∂LN dy := N E (y) dy (22) ∂y for LN (y), we have split the nonlinear term N (y) = N E (y) + N D (y) with the variational part N E (y) ∈ ΔE and the non-variational part N D (y) ∈ ΔD (see (14)). These relations can be also checked in terms of the selfadjointness: D∗ΔE = D ΔE . dv LN (y) =

3.2 Optimality of energy controls As we have seen in the previous section, we cannot directly apply boundary energy controls derived from the power balance to the system (1a) because of ΔD included in the system. However, distributed energy controls, i.e., full

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controls on some domain using distributed inputs are still valid. Here, let us consider the damping assignment with distributed controls in the system. We first recall the damping assignment using port representations can be regard as an optimal regulator. Proposition 4. (Nishida et al. (2013)). The control input u = (R)−1 Qy in (19) has the (inverse) optimality in the sense of the minimize the cost L = Qy 2 + Ru2 , where Q and R are smooth weight functions. Hence, the optimal gain (R)−1 Q can be calculated as a solution of Hamilton-Jacobi equations by the stable manifold method (see Appendix A).

3.3 Power product between POD-basis and input-output distributions The damping assignment consists of the input u and the observed output y˙ that are functions distributed on the domain Z and their product appeared in (19). However, u and y˙ should be actually discrete and sampled in mechanical implementations of controllers and observers. Hence, this fact is considered to be the validity of introducing POD-Galerkin method. Let us consider the following two ways of determining optimal allocations of input-output positions with particular distributions. The first method is a function shape matching via POD. Let y(x, t) be an output obtained from a full domain observation. Let bi (x) be a given input-output distribution around a point x ∈ Z, where B = [b1 , · · · , bm ]. Note that we assumed that the input and output distributions are same for simplifications. Now, we consider the inner product (y(x, t), bi (x)) that means the degree of the parallelism between y and bi . For an initial response y0 (x, t) and each i, we consider the problem (ϕi (x), bi (x))2 , max x∈Z �bi (x)�2

(23)

where ϕi (x) is the POD basis derived from y0 (x, t). This means that we find a position x such that the most parallel bi (x) to y0 , i.e., the most suitable for canceling the output y0 by the input distribution bi . When we observe the output y as a domain truncation through a window distribution around x, the same problem setting can be used, i.e., we find x that is the most suitable for observing the output y0 by the sensor distribution bi . The second method is a dissipation enhancement in terms of the power balance (19). For a velocity y˙ 0 (x, t) of an initial response and each i, we consider the problem min x∈Z

(ϕti (x, t), bi (x))2 , �bi (x)�2

(24)

where ϕti (x) is the POD basis derived from y˙ 0 (x, t). This means the assignment of controlled damping in which there are less autonomous dissipations in the sense of controlling the power product uy˙ in (19). For practical purposes, we can consider the mixed evaluation function consists of (23) and (24). 289

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4. NUMERICAL EXPERIMENTATION This section shows that the proposed method can decrease cost values of the optimal problem. 4.1 Experimental control model The following Burgers equation is frequently used as a test model for boundary and distributed controls, because it can describe various physical phenomena, e.g., shock wave, traffic stream, and supersonic stream: ∂y ∂2y ∂y −ν 2 +y = B(x)u(t) (25) ∂t ∂x ∂x with the boundary conditions are given by y(0, t) = 0 and ∂y/∂x(π, t) = 0, where we have assumed the setting defined for (1a), y = y(x, t) is the state at x ∈ [0, π] = Z, u(t) is the input, ν is the dynamic viscosity coefficient, and B(x) is the spatial distribution of the control action that is defined as exp{−1/(1 − x ¯2 )} at each control position x ¯ = 0. If we operate the inner product with respect to ϕ to (25), then the second and third terms in the left side of the equation correspond to α(y(t), ϕ) and (N (y(t)), ϕ)V  ,V in (1a), respectively. An l-dimensional reduced model of (25) is given by a˙ = Ar a + Nr (a) + Br u, (26) where Ar , Nr (a) and Br are given by  π ϕ�i ϕ�j dx, (27) (Ar )ij = −ν 0

(Nr (a))ij = [a� (T 1 )ij a, · · · , a� (T k )ij a, 

π

· · · , a� (T l )ij a]� ,

ϕi ϕ�j ϕk dx, (T k )ij = −  π0 ϕi bj dx. (Br )ij =

(28) (29) (30)

0

4.2 Problem settings We used the 2-dimensional reduced system, i.e., l = 2. We compared a linear optimal controller derived from the Riccati equation with the following three nonlinear optimal controllers from the viewpoint of control performances, i.e., cost values of optimal problems defined by (23) and/or (24). Method 1 used equally spaced control points for determining the centers of input distributions. Method 2 used the optimal allocation by using the evaluation function (23). Method 3 used the optimal allocation by using the mixed evaluation function consists of (23) and (24). In methods 2 and 3, we chose two control points according to each optimal problem and the intermediate point between the two optimal point as the auxiliary third control point, because l = 2. The POD basis was derived from the initial response illustrated in Figure 8. 4.3 Numerical results The cost values and the control points used in the numerical experimentations are shown in Table 1. The optimal control points were determined by the pointwise inner products illustrated in Figures 1 and 2, where we divided the spatial domain Z into 100 meshes and we plotted the

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values of the inner products at each mesh point. The actual distributed control inputs applied to the control system are shown in Figures 3–6. The stable manifold and the time response of the controlled system calculated by the stable manifold method is shown in Figures 7 and 8, respectively. Method 3 used the mixed evaluation function consists of (23) and (24) achieved the best performance, i.e., the less cost value and the less magnitudes of control distributions. Table 1. Cost values and control mesh points of control methods Linear

Method 1

Method 2

Method 3

Cost

24.323

17.404

17.024

17.015

Point 1 Point 2 Point 3

17 50 83

17 50 83

88 49 69

89 42 66

Fig. 3. Time response of linear optimal inputs

Fig. 4. Time response of optimal inputs (method 1) Fig. 1. Pointwise inner product between POD basis of y and bi

Fig. 5. Time response of optimal inputs (method 2)

Fig. 2. Pointwise inner product between POD basis of y˙ and bi REFERENCES J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag Berlin Heidelberg, 1971. A. J. van der Schaft, L2 -Gain and Passivity Techniques in Nonlinear Control, 2nd revised and enlarged edition, Springer-Verlag, London, 2000. V. Duindam, A. Macchelli, S. Stramigioli and H. Bruyninckx (Eds.): Modeling and Control of Complex Physical Systems - The Port-Hamiltonian Approach, Springer, 2009. 290

Fig. 6. Time response of optimal inputs (method 3)

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Fig. 7. Stable manifold of nonlinear optimal regulator (curves: solutions of HJ equation, planes: solutions of Riccati equation)

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G. Nishida and B. Maschke, “Control Lyapunov Function Construction using Symplectic Structure for Partial Differential Equations”, Proc. of International Symposium on Mathematical Theory of Networks and Systems, pp.834–839, 2014. K. Hamaguchi, G. Nishida and N. Sakamoto, Suboptimal Feedback Control of Nonlinear Distributed Parameter Systems by Stable Manifold Method, Proc. of the 19th IFAC World Congress, pp. 11357–11362, 2014. K. Hamaguchi, G. Nishida, N. Sakamoto, Y. Yamamoto, “Nonlinear Optimal Control in Catalytic Process via Stable Manifold Method”, in Proc. of the 5th IFAC Workshop on Lagrangian and Hamiltonian Methods for Non Linear Control, Lyon, July, 2015. Appendix A. STABLE MANIFOLD METHOD

Fig. 8. Time response of system with nonlinear control N. Sakamoto and A. J. van der Schaft, Analytical approximation methods for the stabilizing solution of the Hamilton-Jacobi equation, IEEE Transactions on Automatic Control, Vol. 53, No. 10, pp. 2335–2350, 2008. N. Sakamoto, Case studies on the application of the stable manifold approach for nonlinear optimal control design, Automatica, Vol. 49, No. 2, pp. 568–576, 2013. K. Kunisch, S. Volkwein, and L. Xie, HJB-POD-based feedback design for the optimal control of evolution problems, SIAM Journal on Applied Dynamical Systems, Vol. 3, No. 4, pp. 701–722, 2004. A. Varshney, S. Pitchaiah, and A. Armaou, Feedback control of dissipative PDE systems using adaptive model reduction, AIChE Journal, Vol. 55, No. 4, pp. 906–918, 2009. M. D. S. Aliyu, Nonlinear H∞ -Control, Hamiltonian Systems and Hamilton-Jacobi Equations, CRC Press, 2011. P. Holmes, J. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, 1998. P. D. Christofides, Nonlinear and Robust Control of PDE Systems: Methods and Applications to transport reaction processes, Springer, 2001. G. Nishida and B. Maschke, “Implicit Representation for Passivity-Based Boundary Controls”, Proc. of the 4th IFAC Workshop on Lagrangian and Hamiltonian Methods for Non-Linear Control, Bertinoro, pp. 200207, 2012. G. Nishida, K. Yamaguchi and N. Sakamoto, “Optimality of passivity-based controls for distributed portHamiltonian systems”, Proc. of 9th IFAC Symposium on Nonlinear Control Systems, pp. 146-151, 2013. 291

The partial derivative of a stabilizing solution of the HJE (13) in nonlinear control problems, i.e., p = (∂V /∂a)� can be calculated as the stable manifold of the Hamiltonian system ⎧ 1¯ ⎪ ⎨ a˙ = f − Rp, � �T � 2 �T (A.1) ¯ T ∂q 1 ∂(pT Rp) ⎪ p˙ = − ∂f ⎩ − p+ ∂a 4 ∂a ∂a that is equivalent to the HJE (13), where we have defined ¯ R(x) = Br R−1 BrT and q = aT Qa. We apply a numerical method of calculating the stable manifold with arbitrary precision, called the stable manifold method (Sakamoto and Van der Schaft, 2008) to this problem. In the stable manifold method, the pair of an optimal trajectory a and an optimal feedback gain p is determined by a solution in (a� , p� )-space via the transformation T defined as follows: � � � �� � � I S a −1 a T = , =T , (A.2) P PS + I p p� where the symmetric matrix P is a solution of a Riccati equation, the matrix S is a solution of Lyapunov equation ¯ ), and such a solution F S +SF T = F with F = (A− R(0)P S always exists because F is stable. Indeed, from the Hamiltonian system (A.1) with the transformation (A.2), we obtain the following representation with the blockdiagonalized linear term: � �� � �� � � � � F 0 a˙ a ns (t, a� , p� ) + , (A.3) � = � � � nu (t, a , p ) p˙ p 0 −F T where ns and nu are higher order derivative terms. The stable manifold of the transfromed Hamiltonian system (A.3) can be calculated by the iterative algorithm of finding the sequences of functions, {ak (t, ξ)} and {pk (t, ξ)} such ⎧ that � t ⎪ Ft ⎪ ⎨ ak+1 (t, ξ) = e ξ + eF (t−s) ns (s, ak (s, ξ), pk (s, ξ))ds, � ∞ 0 T ⎪ ⎪ e−F (t−s) nu (s, ak (s, ξ), pk (s, ξ))ds ⎩ pk+1 (t, ξ) = − t

(A.4) for an arbitrary parameter ξ ∈ Rn with the initial conditions a0 (t, ξ) = eF t ξ and p0 (t, ξ) = 0. If the Hamiltonian H(a, p) in (13) is sufficiently close to zero for some k, we consider that the solution satisfies the HJE (13). If the solution for a fixed ξ passes through a given initial state, the calculation is terminated.