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Boundary adaptive synchronization of networked PDEs with adaptive parameter estimators
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Boundary adaptive synchronization of Boundary adaptive synchronization of Boundary adaptive synchronization of networked PDEs with adaptive parameter networked PDEs with adaptive parameter networked PDEs with adaptive parameter estimators estimators estimators
Michael A. Demetriou ∗∗ Michael A. Demetriou ∗∗ Michael Michael A. A. Demetriou Demetriou ∗ ∗ Aerospace Engineering Program, Mechanical Engineering Aerospace Engineering Program, Mechanical Engineering ∗ Department, Worcester Polytechnic Institute, Worcester, MA 01609, ∗ Aerospace Engineering Program, Mechanical Engineering Aerospace Engineering Program, Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester, MA 01609, USA (e-mail: [email protected]). Department, Worcester Polytechnic Institute, Worcester, Department, Worcester Polytechnic Institute, Worcester, MA MA 01609, 01609, USA (e-mail: [email protected]). USA (e-mail: [email protected]). USA (e-mail: [email protected]). Abstract: This work considers the synchronization control of networked PDEs that have Abstract: This workand considers control of networked PDEs that have boundary observation control.the Thesynchronization PDEs are assumed identical, with information exchange Abstract: This considers the synchronization control of PDEs have Abstract: This work workand considers the control of networked networked PDEs that that have boundary observation control. Thesynchronization PDEs are assumed identical, withininformation exchange dictated by a complete directed graph. Parametric uncertainties entering the boundary, result boundary observation observation and and control. control. The The PDEs PDEs are are assumed assumed identical, identical, with with information information exchange exchange boundary dictated by a complete directed graph. Parametric uncertainties entering in the boundary, result in structured perturbation termsgraph. that may render uncertainties the open loopentering system in unstable. Additionally dictated by a a complete complete directed graph. Parametric uncertainties entering the boundary, boundary, result dictated by directed Parametric the result in structured perturbation terms that may render the openare loopimpeding system in unstable. Additionally constant disturbance terms entering at the boundary state regulation. To in structured perturbation terms that may render the open loop system unstable. Additionally in structured perturbation terms that may render the open loop system unstable. Additionally constant disturbance terms entering aatprecompensator the boundary isare impeding state each regulation. To address these parametric uncertainties, utilized to stabilize networked constant disturbance terms entering at the boundary are impeding state regulation. To constantthese disturbance terms entering aatprecompensator the boundary isare impeding state each regulation. To address parametric utilized to stabilize networked PDE. Another componentuncertainties, of the controller provides an on-line cancellation of each the parametric address these parametric parametric uncertainties, precompensator is utilized utilized to stabilize stabilize each networked address these uncertainties, aa precompensator is to networked PDE. Another component of the controller provides an on-line cancellation of the parametric uncertainties attemptingof cancel them provides through their adaptive estimate, of attenuate PDE. Anotherbycomponent component oftothe the controller provides an on-line on-line cancellation ofand thetoparametric parametric PDE. Another controller an cancellation the uncertainties by attempting to cancel themadaptive through estimates their adaptive estimate, and to attenuate the effects of boundary disturbances, their are utilized in the controller. To uncertainties by attempting to cancel them through their adaptive estimate, and to attenuate uncertainties by attempting to canceltheir themadaptive through estimates their adaptive estimate, andcontroller. to attenuate the effects of boundary disturbances, are utilized in the To provide a robustness in the adaptive estimation of the parametric uncertainties and the boundary the effects of boundary disturbances, their adaptive estimates are utilized in the controller. To the effects of boundary disturbances, their adaptive are utilized in the To provide a robustness in the adaptive of the estimates parametric and controller. the boundary disturbances, the standard adaptive estimation laws are complemented with uncertainties a consensus protocol term that provide a robustness robustness in the the adaptive estimation of the the parametric parametric uncertainties and the the boundary boundary provide a in estimation of uncertainties and disturbances, the standard adaptive laws are complemented with a consensus protocol term that penalizes the pairwise disagreement theare adaptive estimates.with Finally, to enforce synchronization disturbances, the standard standard adaptiveof laws complemented with consensus protocol term that that disturbances, the adaptive laws complemented aa consensus protocol term penalizes the pairwise disagreement of theare adaptive estimates. Finally, toutilizes enforce synchronization of all networked PDEs, a synchronization controller is included which only the outputs penalizes the pairwise disagreement of the adaptive estimates. Finally, to enforce synchronization penalizes the pairwise disagreement of the adaptive estimates. Finally, to enforce synchronization of all networked PDEs,PDE a synchronization controller is included which utilizes onlyThese the outputs communicating systems weighted by adaptive synchronization gains. gains, of networked PDEs, a controller is which only the of all all PDEs,PDE a synchronization synchronization controller is included included which utilizes utilizes onlyThese the outputs outputs of all networked communicating systems weighted by adaptive synchronization gains. gains, derived from Lyapunov redesign methods, include a consensus term for added robustness and of all all communicating communicating PDE PDE systems systems weighted weighted by by adaptive adaptive synchronization synchronization gains. gains. These These gains, gains, of derived from Lyapunov redesign methods, include anetworked consensus diffusion term for PDEs added with robustness and to enforce convergence. Simulation studies of five boundary derived from Lyapunov redesign methods, include a consensus term for added robustness and derived from Lyapunov redesign methods, include consensus diffusion term for PDEs added with robustness and to enforce convergence. Simulation of five anetworked boundary observation and control are included.studies to enforce convergence. Simulation studies of five networked diffusion PDEs with boundary to enforce convergence. Simulation studies of five networked diffusion PDEs with boundary observation and control are included. observation and and control control are are included. included. observation © 2016, IFAC (International Federationsystems; of Automatic Control) Hosting by Elsevier Ltd. All stabilization. rights reserved. Keywords: Distributed parameter Adaptive synchronization; Boundary Keywords: Distributed parameter systems; Adaptive synchronization; Boundary stabilization. Keywords: Distributed Distributed parameter parameter systems; systems; Adaptive Adaptive synchronization; synchronization; Boundary Boundary stabilization. stabilization. Keywords: 1. INTRODUCTION to counter/negate the effects of both parametric uncer1. INTRODUCTION to counter/negate the effects ofcan both parametric uncertainties and disturbances. Thisof be achieved by uncerincor1. to counter/negate the both parametric 1. INTRODUCTION INTRODUCTION to counter/negate the effects effects ofcan both parametric tainties and disturbances. Thisdisturbance be achieved by uncerincorThe synchronization and consensus problems have been tainties porating the estimates of the and structured and disturbances. This can be achieved by incortainties and disturbances. This can be achieved by incorThe synchronization and consensus problems have been porating the estimates of the disturbance and structured examined primarily for finite dimensional dynamical sys- porating perturbation in the controller. The synchronization and consensus problems have the of The synchronization and consensus problems have been been porating the estimates estimates of the the disturbance disturbance and and structured structured examined primarily for finite dimensional dynamical sysperturbation in the controller. tems. Migrating to for thefinite infinite dimensional case posed examined primarily for finite dimensional dynamical sys- perturbation in examined primarily dimensional dynamical sysperturbation in the the controller. controller. tems. Migrating to the infinite dimensional case posed This work considers such networked PDE systems with some problems, due to dimensional the functional analytic tems. Migrating to case posed work perturbation considers such networked PDE systems with tems. Migrating mainly to the the infinite infinite dimensional caseanalytic posed This some problems, mainly due to the functional structured arising due to coupling terms at work considers networked PDE systems with framework for themainly stability and analysis. An This This work perturbation considers such such networked systems with some problems, due to the analytic structured arising due toPDE coupling terms at some problems, mainly dueand to convergence the functional functional analytic framework for the stability convergence analysis. An the boundary and with constant disturbances also at the structured perturbation arising due to coupling terms at added difficulty came with the communication perturbation arising due to coupling terms at framework for stability and convergence analysis. the boundary and written with constant disturbances also at the framework for the the stability andinter-agent convergence analysis. An An structured added difficulty came with the inter-agent communication boundary. When is abstract form, inalso terms of boundary and with disturbances at exchange wherecame full-state transmission created a large the the boundary and written with constant constant disturbances also at the the added difficulty with the inter-agent communication boundary. When is abstract form, in terms of added difficulty came with the inter-agent communication exchange whereand full-state transmission created a agent large boundary. evolution equations in functional spaces, they reduce to When written is abstract form, in terms of communication computational burden, as each boundary. When written is abstract form, in terms of exchange where full-state transmission created a large equations in functional they input reduceand to exchange whereand full-state transmission created large evolution communication computational burden, as eacha agent infinite dimensional systems withspaces, collocated evolution equations functional spaces, they reduce reduce to to is described by aand PDE system. evolution equations in in functional spaces, they communication computational burden, as each agent infinite dimensional systems with collocated input and communication and computational burden, as each agent is described by a PDE system. output operators whose structured perturbation is written infinite dimensional systems with collocated input and infinite dimensional systems with collocated input and is described by a PDE system. output operators whose structured perturbation isThe written is described by a PDE Various aspects of thesystem. synchronization and consensus output as a linear combination of the measured states. is prooperators whose perturbation written output operators whose structured structured perturbation isThe written Various aspects of the parameter synchronization and consensus as a linear combination of the measured states. proproblem for distributed systems has been exposed three-part controller consists of a term addressing Various aspects of the synchronization and consensus as a linear combination of the measured states. The proVarious aspects of the synchronization and consensus as a linear combination of the measured states. The proproblem forDemetriou distributed parameter systems been ex- posed three-part controller consists of a term addressing amined by (2009, 2010a,b,c, 2012,has 2013, 2015). the possibly destabilizing effects of the structured perturproblem for distributed distributed parameter systems has been ex- posed three-part controller consists of term problem for parameter systems been exposed three-part controllereffects consists of aastructured term addressing addressing amined by Demetriou (2009, 2010a,b,c, 2012,has 2013, 2015). the possibly destabilizing of the perturOther works considered controllability aspects and converbations by including a precompesator to negate the effects amined by Demetriou (2009, 2010a,b,c, 2012, 2013, 2015). the possibly destabilizing effects of the structured perturamined by Demetriou (2009, 2010a,b,c, 2012, 2013, 2015). the possibly destabilizing effects of thetostructured Other works considered controllability aspects and converbations by including a precompesator negate thepertureffects gence properties Ambrosio and Aziz-Alaoui (2012); Li and bations of the structured perturbation. With this precompensator, Other works considered controllability aspects and converby including a precompesator to negate the effects Other works considered controllability aspects and converbations by including a precompesator to negate the effects gence properties Ambrosio Li and Aziz-Alaoui (2012); Li and of the structured perturbation. With this precompensator, Rao (2013). In particular etAziz-Alaoui al. (2015); Ambrosio thethe system now has an “open-loop” state operator that is gence properties Ambrosio and (2012); of structured perturbation. With this gence properties Ambrosio Li and Aziz-Alaoui (2012); Li Li and and of the structured perturbation. With state this precompensator, precompensator, Rao (2013). In particular et al. (2015); Ambrosio and the system now has an “open-loop” operator that is Aziz-Alaoui (2013); Wu and Chen (2012) examined the stable. This is required to analyze the second component Rao (2013). In particular Li et al. (2015); Ambrosio and the system now has an “open-loop” state operator that Rao (2013). In particular Li et al. (2015); and system now has an “open-loop” state operator that is is Aziz-Alaoui (2013); Wu and Chen (2012) Ambrosio examined the the stable. This is required to analyze the second component synchronization of coupled advection-diffusion PDEs with of the controller which utilizes the adaptive estimates of Aziz-Alaoui (2013); Wu and Chen (2012) examined the stable. This is required to analyze the second component Aziz-Alaoui (2013); Wu and Chen (2012) examined the stable. This is required to analyze the secondestimates component synchronization of coupled advection-diffusion PDEs with of the controller which utilizes the adaptive of full state feedback synchronization controllers. PDEs with both the parametric uncertainties and the constant distursynchronization of advection-diffusion the controller which utilizes adaptive estimates of synchronization of coupled coupled advection-diffusion of thethe controller which utilizes the the adaptive estimates of full state feedback synchronization controllers. PDEs with of both parametric uncertainties and the constant disturbances, in order to adaptively cancel both of them. While full state feedback synchronization controllers. the parametric uncertainties and the constant disturfull state feedback PDEs synchronization controllers. both the parametric uncertainties and the constant disturWhen networked systems with parametric uncer- both bances, in orderPDE to adaptively cancel both of them. aWhile all networked systems are not both yet coupled, new When networked PDEs systems with parametricthe uncerin to cancel of bances, in order orderPDE to adaptively adaptively cancel both of them. them. aWhile While taintiesnetworked and constant disturbances are considered, syn- bances, When PDEs systems with parametric uncerall networked systems are not yet coupled, new When networked PDEs systems with parametric unceradaptive law for the structured perturbation and constant tainties and constant disturbances are considered, the synall networked PDE systems are not yet coupled, a new networked PDE systems areperturbation not yet coupled, a new chronization controllers may not are be able to provide the all tainties and disturbances considered, the adaptive lawcouples for the structured and constant tainties and constant constant disturbances are considered, the synsyndisturbance them via a perturbation consensus protocol in the chronization controllers may not the be effects able toofprovide the adaptive law for the structured and constant adaptive law for the structured perturbation and constant necessary robustness to counter the strucchronization controllers may not be able to provide the disturbance couples them via a consensus protocol in the chronization controllers may not the be effects able toofprovide the disturbance adaptive update laws. This consensus term in the adaptive necessary robustness to counter the struccouples them via aa consensus protocol in in the the couples them consensus tured perturbations parametric uncertainties and disturbance necessary robustness to the of strucadaptive update laws. Thisofvia consensus term protocol in the adaptive necessary robustness from to counter counter the effects effects of the the struclaws serve as a laws. means a weak form ofthe persistence tured perturbations from parametric uncertainties and adaptive update This consensus term in adaptive adaptive update laws. This consensus term in the adaptive disturbances. To address this, an adaptive-based feedback tured perturbations from parametric uncertainties serve as Ioannou a meansand of aSun weak form Finally, of persistence tured perturbations fromthis, parametric uncertainties and and laws of excitation, (1995). a syndisturbances. To address an adaptive-based serve aa means of weak form of laws serve as as Ioannou meansand of aaSun weak form Finally, of persistence persistence linearizing controller is warranted and which will feedback attempt laws disturbances. To address this, an adaptive-based adaptive-based feedback of excitation, (1995). a syndisturbances. To address this, an feedback linearizing controller is warranted and which will attempt of excitation, Ioannou and Sun (1995). Finally, a synof excitation, Ioannou and Sun (1995). Finally, a synlinearizing linearizing controller controller is is warranted warranted and and which which will will attempt attempt
Copyright © 2016, 2016 International Federation of 244Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 International Federation of 244 Automatic Control Peer review under responsibility of International Federation of Automatic Copyright © 2016 International Federation of 244 Copyright © 2016 International Federation of 244Control. Automatic Control 10.1016/j.ifacol.2016.07.448 Automatic Control Control Automatic
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chronization controller that uses adaptive edge-dependent synchronization gains weighting the pairwise differences of the outputs of the networked PDEs is included to enforce synchronization. The mathematical framework is presented in Section 2 and the proposed adaptive controllers are summarized in Section 3, first for the case of the N systems having only a precompensator and adaptive controller. This still couples the systems due to the consensus terms of the adaptive laws. Subsequently the additional component of the synchronization controllers is considered and which further couples all networked PDE systems. Stability and convergence are provided. A numerical example involving five networked PDEs with boundary observation and control is presented in Section 3 to provide insight on the effects of the consensus terms in the adaptive parameter estimation schemes and on the effects of the adaptive gains in the synchronization controllers. Conclusions with future work follow in Section 4. 2. MATHEMATICAL FRAMEWORK The structurally perturbed systems under consideration, include square infinite dimensional systems arising from partial differential equations with uncertainties in the boundary terms, and are given by x˙ i (t) = Axi (t) + BgCxi (t) + Bui (t) + Bd (1) xi (0) = xi0 ∈ H, yi (t) = Cxi (t), i = 1, . . . , N, where g ∈ Rm×m is the unknown parameter in the structured perturbation term BgCxi (t) and d ∈ Rm is a constant disturbance (vector). While all N infinite dimensional systems are assumed identical, their initial condition may differ, i.e. xi0 �= xj0 . The input and output operators, B and C, are of rank m and for this paper are assumed to be collocated, i.e. B = C ∗ , where C ∗ denotes the adjoint of the output operator C. The structured perturbation term BgCxi (t) is typical in PDE systems with boundary observation and control. Such a perturbation may or may not affect the stability of the open-loop system. A representative example is given by the diffusion PDE in one spatial dimension ξ ∈ [0, 1] with Dirichlet-Neumann boundary conditions ∂xi (t, ξ) ∂ 2 xi (t, ξ) = , xi (0, ξ) = xi0 (ξ), ∂t ∂ξ 2 (2) ∂xi (t, 1) − gxi (t, 1) = ui (t) + d, xi (t, 0) = 0, ∂t for i = 1, . . . , N . With the mild abuse of notation for setting x(t) = x(t, ·), the PDEs in (2) can be placed in the form (1) with the state space H = L2 (0, 1), the input operator B = δ(ξ − 1) and the output operator given by 1 Cxi (t) = δ(ξ − 1)xi (t, ξ) dξ = xi (t, 1). 0
Each PDE in (2) has an eigenvalue at the origin for g = 1, and for g > 1 it has positive eigenvalues.
To analyze and quantify the information exchange between the PDE systems in (1), an appropriate communication topology must be defined. A complete directed graph G = (V, E) is assumed to describe the communication topology for the N networked systems in (1). The nodes 245
243
of the graph V = {1, 2, . . . , N } represent the networked systems (1) and the graph edges E ⊂ V × V represent the communication links between them. The set of networked systems that the ith system is communicating with is denoted by Ni = {j : (i, j) ∈ E}. The N × N dimensional graph Laplacian matrix associated with the graph G is denoted by L0 and is given by L0 = D − A, where D is the degree matrix and A the adjacency matrix, Godsil and Royle (2001). 2.1 Problem Statement and Discussion Each controller ui (t) in (1) must achieve three tasks: (1) Stabilization: One component of the controller must ensure that the system is stabilized. (2) Adaptive estimation and compensation: Another component of the controller must use the adaptive estimates of the structured perturbation term and the constant disturbance so that their effects are minimized or cancelled. (3) Synchronization: The last component of the controller must enable the synchronization of all networked systems in (1). The controllers ui (t) must include all three components ui (t) = Uistab (t) + Uiadapt (t) + Uisyn (t),
i = 1. . . . , N.
The stabilization must be achieved with a robust static output feedback controller. One assumes that a bound on the uncertain parameter g is available and in this case the stabilization component of the controller takes the form Uistab (t) = −g ∗ yi (t), i = 1, . . . , N. (3) Such a controller results in x˙ i (t) = A − Bg ∗ C xi (t) + Bui (t) + BgCxi (t) + Bd, xi (0) = xi0 ∈ H, yi (t) = Cxi (t), i = 1, . . . , N, where the “nominal” state operator Am A−Bg ∗ C is the generator of an exponentially stable C0 semigroup Curtain and Zwart (1995). This implicitly assumes that the triple (A, B, C, ) is statically stabilizable Oostveen (2000). The adaptive estimation component, which includes an adaptive cancellation part, is given by Uiadapt (t) = − gi (t)yi (t) − d i (t), (4)
where g i (t) denotes the ith adaptive estimate of the unknown g and d i (t) denotes the ith adaptive estimate of the unknown d. Application of (3), (4) in (1) results in x˙ i (t) = A − Bg ∗ C xi (t) + Bui (t) (5) −B g i (t) − g Cxi (t) − B d i (t) − d ,
xi (0) = xi0 ∈ H, yi (t) = Cxi (t), i = 1, . . . , N. At this stage, one must provide adaptive laws for g i (t) and d i (t) so that each of the N systems in (5) is well-posed and stable. The parameter adaptive laws may be the ones presented in Curtain et al. (2003), which do not interact with each other and provide N independent estimates g i (t) and d i (t), or implement the distributed adaptive parameter estimations scheme developed in Demetriou (2011). The latter would couple the N systems via the consensus protocol included in the adaptive laws for g i (t) and d i (t).
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Finally, to address synchronization, a control component should employ a consensus protocol based on only the output signals yi (t) of each system in (1). The structure of the consensus protocol may use fixed (constant) synchronization gains or adaptive gains, whose updates will depend on the level of agreement between communicating agents. Thus, the synchronization controllers have the form � � � Uisyn (t) = − αij (t) yi (t) − yj (t) , i = 1, . . . , N. (6) j∈Ni
One now must provide the update laws for the gains αij (t). The application of (3), (4) and (7) in (1) results in � x˙ i (t) = Am xi (t) − B αij (t)yij (t) j∈Ni
� � � � −B g�i (t) − g Cxi (t) − B d�i (t) − d xi (0) = xi0 ∈ H, yi (t) = Cxi (t), i = 1, . . . , N,
(7)
where yij (t) yi (t) − yj (t) denotes the pairwise difference of the outputs of each networked system in (1). The synchronization of the N systems in (1) is achieved when each pairwise state difference converges to zero via lim �xi (t) − xj (t)�H = 0, ∀i, j. t→∞
3. MAIN RESULTS
By denoting the adaptive estimates of the unknown parameter g and constant disturbance d by g�i (t) and d�i (t), respectively, the second component of the controller Uiadapt (t) as given in (4), results in the closed-loop systems gi (t)Cxi (t) − B d�i (t). (8) x˙ i (t) = Am xi (t) − B� where the parameter errors are defined via g�i (t) = g�i (t) − g, d�i (t) = d�i (t) − d, i = 1, . . . , N
Remark 1. Even though the parameter g in the structured perturbation and the constant disturbance d are common to all N systems (1), N estimates are proposed in (8) in order to ensure that all such N estimates agree and implement a distributed estimation scheme to provide robustness of the learning scheme. It remains to propose the adaptive laws for updating g�i (t) and d�i (t), and subsequently examine the well-posedness and convergence of the resulting adaptive system. Lemma 1. Consider each of the systems in (1) with the controllers (3), (4) ui (t) = −g ∗ yi (t) − g�i (t)yi (t) − d�i (t),
N � i=1
V˙ i =
N �
�xi (t), (Am + A∗m )xi (t)�H
i=1
� � � � � � T (t) L0 ⊗ Im G(t) � −2G − 2δ�T (t) L0 ⊗ Im δ(t), where d�1 (t) g�1 (t) � = .. . � G(t) = ... , d(t) . g�N (t) d�N (t) N � i=1
where
V˙ i ≤ −µ�x(t)�2H � � � � � � T (t) L0 ⊗ Im G(t) � −2G − 2δ�T (t) L0 ⊗ Im δ(t),
x1 (t) � �N x(t) = ... , H = H . xN (t) While one can show N � 2 2 � � Vi ≥ �x(t)�2 + min{βi }�G(t)� + min{κi }�d(t)� ,
H
i=1
i
F
i
F
exponentially stability cannot be inferred. The fact that the graph Laplacian L0 has a zero eigenvalue, results in N �
V˙ i ≤ −µ�x(t)�2H ≤ 0.
i=1
(9)
Using arguments based on Gr¨onwall’s lemma produces the boundedness of all signals. However, to show convergence of the states xi (t) to zero, one must use the analog of Barbˇalat’s lemma for infinite dimensional systems. Following the exposure in Curtain et al. (2003), the state convergence can be shown. Parameter convergence is achieved without persistence of excitation only when the all-to-all inter-agent connectivity is assumed, Demetriou (2011).
(10)
3.2 Adaptive robust synchronizing controller design for networked systems
j∈Ni
� �� �� ˙ d�i (t) = κi yi (t) − λi d�i (t) − d�j (t) ,
Following Demetriou (2011), it can easily be shown that the derivative along (8), (9), (10) is given by
Using the fact that Am + A∗m ≤ −µI then
3.1 Adaptive robust controller design for single system
and the consensus adaptive laws � �� �� g�˙ i (t) = βi yi2 (t) − γi g�i (t) − g�j (t) ,
Proof. The well-posedness is based on the arguments presented in Curtain et al. (2003) for adaptive estimation of single systems and in Demetriou (2011) for the use of parameter adaptive laws with consensus. Due to space limitations, portion of the stability proof is provided. To examine the stability of (8), (9) and (10), the aggregate of all Lyapunov functionals for each system must be considered since the adaptive laws (9), (10) couple the N systems in (8). Consider the Lyapunov functionals N N � � � � � 1 1 � �xi (t)�2 + Tr g�iT (t)� gi (t) + d�Ti d�i (t) . Vi = βi κi i=1 i=1
j∈Ni
where βi , γi , κi , λi are user-defined adaptive gains. Then the systems (8), (9) and (10) are well-posed, with lim �xi (t)�H = 0, i = 1, . . . , N, t→∞
and all other signals are bounded. 246
The controllers (3) and (4) provide stabilization and adaptive parameter estimation with inter-agent interaction, as enforced by the consensus protocols in (9), (10). To ensure synchronization, another component must be included. Adding the synchronization component (6), produces the
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interacting networked systems with adaptation of the synchronization gains x˙ i (t) = Am xi (t) − B gi (t)Cxi (t) − B d i (t) αij (t) yi (t) − yj (t) , −B j∈Ni
g˙ i (t) = βi yi2 (t) − γi
j∈Ni
gi (t) − gj (t)
,
(11)
˙ di (t) − dj (t) , di (t) = κi yi (t) − λi j∈N
i α˙ ij (t) = νi yi (t)yij (t) − αij (t) ,
where νi are user-defined adaptive gains Ioannou and Sun (1995). Convergence and well-posedness results can be obtained as in Lemma 1. The Lyapunov functionals now include an additional term N N 1 �xi (t)�2 + Tr g iT (t) Vi = gi (t) β i i=1 i=1 1 2 1 T αij (t) . + di di (t) + κi νi j∈Ni
The derivative along (11) produces N i=1
V˙ i ≤ −µ�x(t)�2H − 2
N
2 αij (t)
i=1 j∈Ni
T (t) L0 ⊗ Im G(t) −2G − 2δ T (t) L0 ⊗ Im δ(t).
Using the previously established bounds of the graph Laplacian L0 , we have N
V˙ i ≤ −µ�x(t)�2H − 2
i=1
N
2 αij (t) ≤ 0.
i=1 j∈Ni
Due to the specific nature of the adaptive laws of the synchronization gains, we have asymptotic convergence of αij (t) to zero. This is stated in the following. Lemma 2. Consider the systems in (1) with the controllers (3), (4) and (6) resulting in the interacting networked systems (11). The systems (11) are well-posed, with lim �xi (t)�H = 0, i = 1, . . . , N t→∞
and lim αij (t) = 0,
t→∞
i = 1, . . . , N, j ∈ Ni ,
with all other signals bounded. Proof. The well-posedness easily follows from Lemma 1 and arguments from the relevant works Demetriou (2013, 2015). While the parameter estimates gi (t) and di (t) remain bounded, they cannot be guaranteed to converge to the true values without imposing either a persistence of excitation or an all-to-all connectivity. The latter is required since the graph Laplacian L0 has a zero eigenvalue which does not allow any negative terms of � gi (t)�F and N �d i (t)�2 in i=1 V˙ i . 4. EXAMPLE
We consider N = 5 networked diffusion PDEs in one spatial dimension as represented in (1) but without the constant disturbance terms 247
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∂ 2 xi (t, ξ) ∂xi (t, ξ) =α , xi (0, ξ) = xi0 (ξ), i = 1, . . . , 5, ∂t ∂ξ 2 ∂xi (t, 1) − gxi (t, 1) = ui (t) xi (t, 0) = 0, ∂t where the thermal diffusivity is set to α = 0.1 and m = 1, i.e. g is a scalar. The output of each networked PDE is yi (t) = xi (t, 1) and in this case one has that the input and output operators are collocated, i.e. B = C ∗ = δ(ξ − 1). The unknown boundary parameter was set to g = 1.1α. It can be shown that the open loop system is unstable when the unknown boundary parameter g ≥ α. In fact, for g = α, the operator A + gBC has an eigenvalue at the origin. Since the exact value of the boundary parameter is not known, but the upper bound |g| ≤ 2α is known, then the stabilizing controller Uistab (t) in (3) is as Uistab (t) = −2αyi (t), i = 1, . . . , 5. The corresponding adaptive parameter controller in (4) is Uiadapt = − gi (t)yi (t), i = 1, . . . , 5. The updates for the adaptive estimate of the boundary parameter are given by the consensus adaptive law (9). The last component of the control law is given by the synchronization controller (6). The last remaining adaptation provides a rule for updating the gains αij and is given by (11). To simulate the above networked PDEs, a finite elementbased approximation scheme is used. A total of 40 linear splines, H¨ollig (2003) that are modified to account for the (Dirichlet-Neumann) boundary conditions are used. When each PDE is viewed in weak form with test functions set equal to each of the approximating linear splines, then a system of 40 ordinary differential equations arises. The spatial integrals which are required to compute the matrix representations of the state, input and output operators are computed using a composite two-point GaussLegendre quadrature rule, Celia and Gray (1992). The resulting semidiscrete system of 40 differential equations was subsequently integrated numerically over the interval [0, 10]s using the stiff ODE solver from the Matlab ODE library, routine ode23s which is based on a 4th order Runge-Kutta scheme. The initial conditions for the 5 networked PDEs were x1 (0, ξ) = 39.4 sin(1.3πξ) exp(−7ξ 2 ), x2 (0, ξ) = 12.6 sin(2.1πξ) cos(1.5πξ), x3 (0, ξ) = 7.6 sin(3.6πξ) exp(−7ξ 2 ), x4 (0, ξ) = 2.5 sin(5πξ) exp(−ξ 2 ), x5 (0, ξ) = −26.2 sin(5πξ) exp(−7(ξ − 0.5)2 ). The adaptive gains βi = 0.01, γi = 75 and the adaptive consensus gains were νi = 0.5, i = 1, . . . , 5. The initial conditions for parameter estimates were chosen as g1 (0) = 0.05, g2 (0) = 0.125, g3 (0) = 0.15, g4 (0) = 0.18 and g5 (0) = 0.03. Finally, the initial conditions for the synchronization gains were given by αij (0) = 0.5, i = 1, . . . , 5, j ∈ Ni . Figure 1 depicts the evolution of the adaptive estimates gi (t) along with the true value g = 0.11. The average value 5 of the adaptive estimates gave (t) = 15 i=1 gi (t) is shown to converge to the steady state value of limt→∞ gave (t) = 0.1085 indicating a parameter convergence. The evolution of the adaptive synchronization gains αij (t) is depicted in Figure 2, where it is observed that all gains asymptotically converge to zero, as theoretically predicted. The evolution
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Fig. 3. Evolution of L2 (0, 1) norm of mean state xm (t, ξ) using adaptive synchronization gains αij (t) and constant synchronization gains αij .
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Fig. 1. Evolution of adaptive parameter estimates g i (t). 7
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(12)
is depicted in Figure 3. In addition to the adaptive synchronization gains, the case of constant synchronization gains was also considered and presented in Figure 3. While one can perform an optimization to find the optimal constant synchronization gains as was presented in Demetriou (2015), in the simulation the constant gains were chosen to be equal to the initial conditions for the adaptive synchronization gains, namely αij = 0.5, i = 1, . . . , 5, j ∈ Ni . It is observed that in the case of adaptive synchronization gains, the norm of the mean state converges to zero faster than the case with constant αij . To further examine the effects of the proposed controllers on synchronization, the spatial distribution of the mean state (12) for two time instances (t = 5s and t = 10s) is presented in Figure 4 for both the case of adaptive synchronization gains αij (t) and constant synchronization gains αij . In both instances, the mean state converges (pointwise) to zero faster with the adaptive synchronization gains than the constant ones. 248
0
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Fig. 4. Spatial distribution of the mean state xm (t, ξ) vs the spatial variable at t = 5s and at t = 10s. Table 1 summarizes the effects of the synchronization controllers. As a measure of synchronization, one considers the deviation from the mean δi (t, ξ) = xi (t, ξ) − xm (t, ξ), i = 1, . . . , 5. The spatial L2 (0, 1) norm of the aggregate deviation from the mean is given by 5 1 5 2 2 �δ(t)� = δi2 (t, ξ) dξ. �δi (t, ξ)� = i=1
0
i=1
Evaluating the L2 (0, t) (time) norm of the aggregate deviation from the mean, computed by 10 5 1 10 �δ(t)�2 dt = δi2 (t, ξ) dξ dt, 0
0
i=1
0
provides a single piece of information on the success of synchronization. In Table 1, one observes that when adaptation of the synchronization gains is implemented, then the above norm is 5.9230, whereas when constant synchronization gains are used, it has the value of 6.1843,
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case
value
adaptive αij (t)
5.9230
constant αij
6.1843
Table 1. L2 (0, 10) norm of the aggregate deviation from the mean �δ(t)�2 . indicating an improved agreement (synchronization) when adaptive gains are used. 5. CONCLUSION An adaptive scheme to stabilize and synchronize networked diffusion PDEs with boundary observation and control has been proposed. The networked PDEs were assumed to be identical, differing on their initial conditions, and could exchange information through a communication topology described by a complete directed graph. Structured perturbation culminating from parametric coupling of the boundary control and observation were included in the networked PDEs and which resulted on positive eigenvalues, thus necessitating the use of a stabilizing boundary controller. Additionally, constant disturbances were assumed to enter through the boundary. The controllers consisted of three parts: one part, which served as a precompensator, used static output feedback to stabilize the open loop system. The second part used the adaptive estimates of the structured perturbation parameter and of the constant disturbance in a feedback stabilization format to cancel their effects. The adaptive laws included additional terms that coupled the PDEs through this adaptations; such a coupling took the form of a consensus protocol and which aided in the parameter convergence. The final part of the controller included a consensus protocol consisted of the pairwise differences of the outputs of all communication PDEs, weighted by adaptive edge-dependent synchronization gains. Stability and convergence properties were analyzed and a numerical example provided further insights on the use of adaptation in the consensus protocols in parameter and synchronization convergence. An immediate extension of the above results would involve networked systems with not only different initial conditions, but also different operators (Ai , Bi , Ci ). However this will come at the expense of stringent requirements on each of the networked systems that each must satisfy the matching condition Am = Ai − Bi gi∗ Ci , i = 1, . . . , N . This is currently being investigated by the author. ACKNOWLEDGEMENTS The author gratefully acknowledges financial support from the AFOSR, grant FA9550-12-1-0114. REFERENCES Ambrosio, B. and Aziz-Alaoui, M. (2012). Synchronization and control of coupled reaction-diffusion systems of the fitzhugh-nagumo type. Computers & Mathematics with Applications, 64(5), 934 – 943. Ambrosio, B. and Aziz-Alaoui, M.A. (2013). Synchronization and control of a network of coupled reactiondiffusion systems of generalized Fitzhugh-Nagumo type. 249
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In Conference on Structural Nonlinear Dynamics and Diagnosis—2012, volume 39 of ESAIM Proc., 15–24. Celia, M.A. and Gray, W.G. (1992). Numerical methods for differential equations. Prentice Hall, Englewood Cliffs, NJ. Curtain, R.F. and Zwart, H.J. (1995). An Introduction to Infinite Dimensional Linear Systems Theory. SpringerVerlag, New York. Curtain, R.F., Demetriou, M.A., and Ito, K. (2003). Adaptive compensators for perturbed positive real infinitedimensional systems. Int. J. Appl. Math. Comput. Sci., 13(4), 441–452. Demetriou, M.A. (2009). Natural consensus filters for second order infinite dimensional systems. Systems & Control Letters, 58(12), 826 – 833. Demetriou, M.A. (2010a). Enforcing and enhancing consensus of spatially distributed filters utilizing mobile sensor networks. In Proc. of the 49th IEEE Conference on Decision and Control. Demetriou, M.A. (2010b). Measure of disagreement of spatially distributed filters for distributed parameter systems. In Proc. of the 2010 American Control Conference. Baltimore, Maryland, USA. Demetriou, M.A. (2012). Enforcing consensus on adaptive parameter estimation of structurally perturbed infinite dimensional systems. IEEE Trans. on Automatic Control, 57(12), 3147–3152. Demetriou, M.A. (2013). Synchronization and consensus controllers for a class of parabolic distributed parameter systems. Systems & Control Letters, 62(1), 70 – 76. Demetriou, M. (2011). Adaptive consensus filters for collocated infinite dimensional systems. In Proc. of the 50th IEEE Conference on Decision and Control and European Control Conference, 597–602. Orlando, FL. Demetriou, M. (2015). Adaptation and optimization of synchronization gains in the regulation control of networked distributed parameter systems. IEEE Trans. on Automatic Control, 60(8), 2219–2224. Demetriou, M.A. (2010c). Design of consensus and adaptive consensus filters for distributed parameter systems. Automatica, 46(2), 300 – 311. Godsil, C. and Royle, G. (2001). Algebraic graph theory, volume 207 of Graduate Texts in Mathematics. SpringerVerlag, New York. H¨ollig, K. (2003). Finite Element Methods with B-Splines. SIAM, Frontiers in Appl. Math., Philadelphia, PA. Ioannou, P.A. and Sun, J. (1995). Robust Adaptive Control. Prentice Hall, Englewood Cliffs, NJ. Li, T. and Rao, B. (2013). Exact synchronization for a coupled system of wave equations with dirichlet boundary controls. Chinese Ann. of Mathematics, Ser. B, 34. Li, W., Chen, T., Xu, D., and Wang, K. (2015). Synchronization of delayed coupled reaction-diffusion systems on networks. Mathematical Methods in the Applied Sciences, 38(11), 2216–2228. Oostveen, J. (2000). Strongly Stablizable Distributed Parameter Systems. SIAM, Frontiers in Appl. Math., Philadephia, PA. Wu, K. and Chen, B.S. (2012). Synchronization of partial differential systems via diffusion coupling. IEEE Trans. on Circuits and Systems I: Regular Papers, 59(11), 2655–2668.
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Boundary adaptive synchronization of Boundary adaptive synchronization of Boundary adaptive synchronization of networked PDEs with adaptive parameter networked PDEs with adaptive parameter networked PDEs with adaptive parameter estimators estimators estimators
Michael A. Demetriou ∗∗ Michael A. Demetriou ∗∗ Michael Michael A. A. Demetriou Demetriou ∗ ∗ Aerospace Engineering Program, Mechanical Engineering Aerospace Engineering Program, Mechanical Engineering ∗ Department, Worcester Polytechnic Institute, Worcester, MA 01609, ∗ Aerospace Engineering Program, Mechanical Engineering Aerospace Engineering Program, Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester, MA 01609, USA (e-mail: [email protected]). Department, Worcester Polytechnic Institute, Worcester, Department, Worcester Polytechnic Institute, Worcester, MA MA 01609, 01609, USA (e-mail: [email protected]). USA (e-mail: [email protected]). USA (e-mail: [email protected]). Abstract: This work considers the synchronization control of networked PDEs that have Abstract: This workand considers control of networked PDEs that have boundary observation control.the Thesynchronization PDEs are assumed identical, with information exchange Abstract: This considers the synchronization control of PDEs have Abstract: This work workand considers the control of networked networked PDEs that that have boundary observation control. Thesynchronization PDEs are assumed identical, withininformation exchange dictated by a complete directed graph. Parametric uncertainties entering the boundary, result boundary observation observation and and control. control. The The PDEs PDEs are are assumed assumed identical, identical, with with information information exchange exchange boundary dictated by a complete directed graph. Parametric uncertainties entering in the boundary, result in structured perturbation termsgraph. that may render uncertainties the open loopentering system in unstable. Additionally dictated by a a complete complete directed graph. Parametric uncertainties entering the boundary, boundary, result dictated by directed Parametric the result in structured perturbation terms that may render the openare loopimpeding system in unstable. Additionally constant disturbance terms entering at the boundary state regulation. To in structured perturbation terms that may render the open loop system unstable. Additionally in structured perturbation terms that may render the open loop system unstable. Additionally constant disturbance terms entering aatprecompensator the boundary isare impeding state each regulation. To address these parametric uncertainties, utilized to stabilize networked constant disturbance terms entering at the boundary are impeding state regulation. To constantthese disturbance terms entering aatprecompensator the boundary isare impeding state each regulation. To address parametric utilized to stabilize networked PDE. Another componentuncertainties, of the controller provides an on-line cancellation of each the parametric address these parametric parametric uncertainties, precompensator is utilized utilized to stabilize stabilize each networked address these uncertainties, aa precompensator is to networked PDE. Another component of the controller provides an on-line cancellation of the parametric uncertainties attemptingof cancel them provides through their adaptive estimate, of attenuate PDE. Anotherbycomponent component oftothe the controller provides an on-line on-line cancellation ofand thetoparametric parametric PDE. Another controller an cancellation the uncertainties by attempting to cancel themadaptive through estimates their adaptive estimate, and to attenuate the effects of boundary disturbances, their are utilized in the controller. To uncertainties by attempting to cancel them through their adaptive estimate, and to attenuate uncertainties by attempting to canceltheir themadaptive through estimates their adaptive estimate, andcontroller. to attenuate the effects of boundary disturbances, are utilized in the To provide a robustness in the adaptive estimation of the parametric uncertainties and the boundary the effects of boundary disturbances, their adaptive estimates are utilized in the controller. To the effects of boundary disturbances, their adaptive are utilized in the To provide a robustness in the adaptive of the estimates parametric and controller. the boundary disturbances, the standard adaptive estimation laws are complemented with uncertainties a consensus protocol term that provide a robustness robustness in the the adaptive estimation of the the parametric parametric uncertainties and the the boundary boundary provide a in estimation of uncertainties and disturbances, the standard adaptive laws are complemented with a consensus protocol term that penalizes the pairwise disagreement theare adaptive estimates.with Finally, to enforce synchronization disturbances, the standard standard adaptiveof laws complemented with consensus protocol term that that disturbances, the adaptive laws complemented aa consensus protocol term penalizes the pairwise disagreement of theare adaptive estimates. Finally, toutilizes enforce synchronization of all networked PDEs, a synchronization controller is included which only the outputs penalizes the pairwise disagreement of the adaptive estimates. Finally, to enforce synchronization penalizes the pairwise disagreement of the adaptive estimates. Finally, to enforce synchronization of all networked PDEs,PDE a synchronization controller is included which utilizes onlyThese the outputs communicating systems weighted by adaptive synchronization gains. gains, of networked PDEs, a controller is which only the of all all PDEs,PDE a synchronization synchronization controller is included included which utilizes utilizes onlyThese the outputs outputs of all networked communicating systems weighted by adaptive synchronization gains. gains, derived from Lyapunov redesign methods, include a consensus term for added robustness and of all all communicating communicating PDE PDE systems systems weighted weighted by by adaptive adaptive synchronization synchronization gains. gains. These These gains, gains, of derived from Lyapunov redesign methods, include anetworked consensus diffusion term for PDEs added with robustness and to enforce convergence. Simulation studies of five boundary derived from Lyapunov redesign methods, include a consensus term for added robustness and derived from Lyapunov redesign methods, include consensus diffusion term for PDEs added with robustness and to enforce convergence. Simulation of five anetworked boundary observation and control are included.studies to enforce convergence. Simulation studies of five networked diffusion PDEs with boundary to enforce convergence. Simulation studies of five networked diffusion PDEs with boundary observation and control are included. observation and and control control are are included. included. observation © 2016, IFAC (International Federationsystems; of Automatic Control) Hosting by Elsevier Ltd. All stabilization. rights reserved. Keywords: Distributed parameter Adaptive synchronization; Boundary Keywords: Distributed parameter systems; Adaptive synchronization; Boundary stabilization. Keywords: Distributed Distributed parameter parameter systems; systems; Adaptive Adaptive synchronization; synchronization; Boundary Boundary stabilization. stabilization. Keywords: 1. INTRODUCTION to counter/negate the effects of both parametric uncer1. INTRODUCTION to counter/negate the effects ofcan both parametric uncertainties and disturbances. Thisof be achieved by uncerincor1. to counter/negate the both parametric 1. INTRODUCTION INTRODUCTION to counter/negate the effects effects ofcan both parametric tainties and disturbances. Thisdisturbance be achieved by uncerincorThe synchronization and consensus problems have been tainties porating the estimates of the and structured and disturbances. This can be achieved by incortainties and disturbances. This can be achieved by incorThe synchronization and consensus problems have been porating the estimates of the disturbance and structured examined primarily for finite dimensional dynamical sys- porating perturbation in the controller. The synchronization and consensus problems have the of The synchronization and consensus problems have been been porating the estimates estimates of the the disturbance disturbance and and structured structured examined primarily for finite dimensional dynamical sysperturbation in the controller. tems. Migrating to for thefinite infinite dimensional case posed examined primarily for finite dimensional dynamical sys- perturbation in examined primarily dimensional dynamical sysperturbation in the the controller. controller. tems. Migrating to the infinite dimensional case posed This work considers such networked PDE systems with some problems, due to dimensional the functional analytic tems. Migrating to case posed work perturbation considers such networked PDE systems with tems. Migrating mainly to the the infinite infinite dimensional caseanalytic posed This some problems, mainly due to the functional structured arising due to coupling terms at work considers networked PDE systems with framework for themainly stability and analysis. An This This work perturbation considers such such networked systems with some problems, due to the analytic structured arising due toPDE coupling terms at some problems, mainly dueand to convergence the functional functional analytic framework for the stability convergence analysis. An the boundary and with constant disturbances also at the structured perturbation arising due to coupling terms at added difficulty came with the communication perturbation arising due to coupling terms at framework for stability and convergence analysis. the boundary and written with constant disturbances also at the framework for the the stability andinter-agent convergence analysis. An An structured added difficulty came with the inter-agent communication boundary. When is abstract form, inalso terms of boundary and with disturbances at exchange wherecame full-state transmission created a large the the boundary and written with constant constant disturbances also at the the added difficulty with the inter-agent communication boundary. When is abstract form, in terms of added difficulty came with the inter-agent communication exchange whereand full-state transmission created a agent large boundary. evolution equations in functional spaces, they reduce to When written is abstract form, in terms of communication computational burden, as each boundary. When written is abstract form, in terms of exchange where full-state transmission created a large equations in functional they input reduceand to exchange whereand full-state transmission created large evolution communication computational burden, as eacha agent infinite dimensional systems withspaces, collocated evolution equations functional spaces, they reduce reduce to to is described by aand PDE system. evolution equations in in functional spaces, they communication computational burden, as each agent infinite dimensional systems with collocated input and communication and computational burden, as each agent is described by a PDE system. output operators whose structured perturbation is written infinite dimensional systems with collocated input and infinite dimensional systems with collocated input and is described by a PDE system. output operators whose structured perturbation isThe written is described by a PDE Various aspects of thesystem. synchronization and consensus output as a linear combination of the measured states. is prooperators whose perturbation written output operators whose structured structured perturbation isThe written Various aspects of the parameter synchronization and consensus as a linear combination of the measured states. proproblem for distributed systems has been exposed three-part controller consists of a term addressing Various aspects of the synchronization and consensus as a linear combination of the measured states. The proVarious aspects of the synchronization and consensus as a linear combination of the measured states. The proproblem forDemetriou distributed parameter systems been ex- posed three-part controller consists of a term addressing amined by (2009, 2010a,b,c, 2012,has 2013, 2015). the possibly destabilizing effects of the structured perturproblem for distributed distributed parameter systems has been ex- posed three-part controller consists of term problem for parameter systems been exposed three-part controllereffects consists of aastructured term addressing addressing amined by Demetriou (2009, 2010a,b,c, 2012,has 2013, 2015). the possibly destabilizing of the perturOther works considered controllability aspects and converbations by including a precompesator to negate the effects amined by Demetriou (2009, 2010a,b,c, 2012, 2013, 2015). the possibly destabilizing effects of the structured perturamined by Demetriou (2009, 2010a,b,c, 2012, 2013, 2015). the possibly destabilizing effects of thetostructured Other works considered controllability aspects and converbations by including a precompesator negate thepertureffects gence properties Ambrosio and Aziz-Alaoui (2012); Li and bations of the structured perturbation. With this precompensator, Other works considered controllability aspects and converby including a precompesator to negate the effects Other works considered controllability aspects and converbations by including a precompesator to negate the effects gence properties Ambrosio Li and Aziz-Alaoui (2012); Li and of the structured perturbation. With this precompensator, Rao (2013). In particular etAziz-Alaoui al. (2015); Ambrosio thethe system now has an “open-loop” state operator that is gence properties Ambrosio and (2012); of structured perturbation. With this gence properties Ambrosio Li and Aziz-Alaoui (2012); Li Li and and of the structured perturbation. With state this precompensator, precompensator, Rao (2013). In particular et al. (2015); Ambrosio and the system now has an “open-loop” operator that is Aziz-Alaoui (2013); Wu and Chen (2012) examined the stable. This is required to analyze the second component Rao (2013). In particular Li et al. (2015); Ambrosio and the system now has an “open-loop” state operator that Rao (2013). In particular Li et al. (2015); and system now has an “open-loop” state operator that is is Aziz-Alaoui (2013); Wu and Chen (2012) Ambrosio examined the the stable. This is required to analyze the second component synchronization of coupled advection-diffusion PDEs with of the controller which utilizes the adaptive estimates of Aziz-Alaoui (2013); Wu and Chen (2012) examined the stable. This is required to analyze the second component Aziz-Alaoui (2013); Wu and Chen (2012) examined the stable. This is required to analyze the secondestimates component synchronization of coupled advection-diffusion PDEs with of the controller which utilizes the adaptive of full state feedback synchronization controllers. PDEs with both the parametric uncertainties and the constant distursynchronization of advection-diffusion the controller which utilizes adaptive estimates of synchronization of coupled coupled advection-diffusion of thethe controller which utilizes the the adaptive estimates of full state feedback synchronization controllers. PDEs with of both parametric uncertainties and the constant disturbances, in order to adaptively cancel both of them. While full state feedback synchronization controllers. the parametric uncertainties and the constant disturfull state feedback PDEs synchronization controllers. both the parametric uncertainties and the constant disturWhen networked systems with parametric uncer- both bances, in orderPDE to adaptively cancel both of them. aWhile all networked systems are not both yet coupled, new When networked PDEs systems with parametricthe uncerin to cancel of bances, in order orderPDE to adaptively adaptively cancel both of them. them. aWhile While taintiesnetworked and constant disturbances are considered, syn- bances, When PDEs systems with parametric uncerall networked systems are not yet coupled, new When networked PDEs systems with parametric unceradaptive law for the structured perturbation and constant tainties and constant disturbances are considered, the synall networked PDE systems are not yet coupled, a new networked PDE systems areperturbation not yet coupled, a new chronization controllers may not are be able to provide the all tainties and disturbances considered, the adaptive lawcouples for the structured and constant tainties and constant constant disturbances are considered, the synsyndisturbance them via a perturbation consensus protocol in the chronization controllers may not the be effects able toofprovide the adaptive law for the structured and constant adaptive law for the structured perturbation and constant necessary robustness to counter the strucchronization controllers may not be able to provide the disturbance couples them via a consensus protocol in the chronization controllers may not the be effects able toofprovide the disturbance adaptive update laws. This consensus term in the adaptive necessary robustness to counter the struccouples them via aa consensus protocol in in the the couples them consensus tured perturbations parametric uncertainties and disturbance necessary robustness to the of strucadaptive update laws. Thisofvia consensus term protocol in the adaptive necessary robustness from to counter counter the effects effects of the the struclaws serve as a laws. means a weak form ofthe persistence tured perturbations from parametric uncertainties and adaptive update This consensus term in adaptive adaptive update laws. This consensus term in the adaptive disturbances. To address this, an adaptive-based feedback tured perturbations from parametric uncertainties serve as Ioannou a meansand of aSun weak form Finally, of persistence tured perturbations fromthis, parametric uncertainties and and laws of excitation, (1995). a syndisturbances. To address an adaptive-based serve aa means of weak form of laws serve as as Ioannou meansand of aaSun weak form Finally, of persistence persistence linearizing controller is warranted and which will feedback attempt laws disturbances. To address this, an adaptive-based adaptive-based feedback of excitation, (1995). a syndisturbances. To address this, an feedback linearizing controller is warranted and which will attempt of excitation, Ioannou and Sun (1995). Finally, a synof excitation, Ioannou and Sun (1995). Finally, a synlinearizing linearizing controller controller is is warranted warranted and and which which will will attempt attempt
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chronization controller that uses adaptive edge-dependent synchronization gains weighting the pairwise differences of the outputs of the networked PDEs is included to enforce synchronization. The mathematical framework is presented in Section 2 and the proposed adaptive controllers are summarized in Section 3, first for the case of the N systems having only a precompensator and adaptive controller. This still couples the systems due to the consensus terms of the adaptive laws. Subsequently the additional component of the synchronization controllers is considered and which further couples all networked PDE systems. Stability and convergence are provided. A numerical example involving five networked PDEs with boundary observation and control is presented in Section 3 to provide insight on the effects of the consensus terms in the adaptive parameter estimation schemes and on the effects of the adaptive gains in the synchronization controllers. Conclusions with future work follow in Section 4. 2. MATHEMATICAL FRAMEWORK The structurally perturbed systems under consideration, include square infinite dimensional systems arising from partial differential equations with uncertainties in the boundary terms, and are given by x˙ i (t) = Axi (t) + BgCxi (t) + Bui (t) + Bd (1) xi (0) = xi0 ∈ H, yi (t) = Cxi (t), i = 1, . . . , N, where g ∈ Rm×m is the unknown parameter in the structured perturbation term BgCxi (t) and d ∈ Rm is a constant disturbance (vector). While all N infinite dimensional systems are assumed identical, their initial condition may differ, i.e. xi0 �= xj0 . The input and output operators, B and C, are of rank m and for this paper are assumed to be collocated, i.e. B = C ∗ , where C ∗ denotes the adjoint of the output operator C. The structured perturbation term BgCxi (t) is typical in PDE systems with boundary observation and control. Such a perturbation may or may not affect the stability of the open-loop system. A representative example is given by the diffusion PDE in one spatial dimension ξ ∈ [0, 1] with Dirichlet-Neumann boundary conditions ∂xi (t, ξ) ∂ 2 xi (t, ξ) = , xi (0, ξ) = xi0 (ξ), ∂t ∂ξ 2 (2) ∂xi (t, 1) − gxi (t, 1) = ui (t) + d, xi (t, 0) = 0, ∂t for i = 1, . . . , N . With the mild abuse of notation for setting x(t) = x(t, ·), the PDEs in (2) can be placed in the form (1) with the state space H = L2 (0, 1), the input operator B = δ(ξ − 1) and the output operator given by 1 Cxi (t) = δ(ξ − 1)xi (t, ξ) dξ = xi (t, 1). 0
Each PDE in (2) has an eigenvalue at the origin for g = 1, and for g > 1 it has positive eigenvalues.
To analyze and quantify the information exchange between the PDE systems in (1), an appropriate communication topology must be defined. A complete directed graph G = (V, E) is assumed to describe the communication topology for the N networked systems in (1). The nodes 245
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of the graph V = {1, 2, . . . , N } represent the networked systems (1) and the graph edges E ⊂ V × V represent the communication links between them. The set of networked systems that the ith system is communicating with is denoted by Ni = {j : (i, j) ∈ E}. The N × N dimensional graph Laplacian matrix associated with the graph G is denoted by L0 and is given by L0 = D − A, where D is the degree matrix and A the adjacency matrix, Godsil and Royle (2001). 2.1 Problem Statement and Discussion Each controller ui (t) in (1) must achieve three tasks: (1) Stabilization: One component of the controller must ensure that the system is stabilized. (2) Adaptive estimation and compensation: Another component of the controller must use the adaptive estimates of the structured perturbation term and the constant disturbance so that their effects are minimized or cancelled. (3) Synchronization: The last component of the controller must enable the synchronization of all networked systems in (1). The controllers ui (t) must include all three components ui (t) = Uistab (t) + Uiadapt (t) + Uisyn (t),
i = 1. . . . , N.
The stabilization must be achieved with a robust static output feedback controller. One assumes that a bound on the uncertain parameter g is available and in this case the stabilization component of the controller takes the form Uistab (t) = −g ∗ yi (t), i = 1, . . . , N. (3) Such a controller results in x˙ i (t) = A − Bg ∗ C xi (t) + Bui (t) + BgCxi (t) + Bd, xi (0) = xi0 ∈ H, yi (t) = Cxi (t), i = 1, . . . , N, where the “nominal” state operator Am A−Bg ∗ C is the generator of an exponentially stable C0 semigroup Curtain and Zwart (1995). This implicitly assumes that the triple (A, B, C, ) is statically stabilizable Oostveen (2000). The adaptive estimation component, which includes an adaptive cancellation part, is given by Uiadapt (t) = − gi (t)yi (t) − d i (t), (4)
where g i (t) denotes the ith adaptive estimate of the unknown g and d i (t) denotes the ith adaptive estimate of the unknown d. Application of (3), (4) in (1) results in x˙ i (t) = A − Bg ∗ C xi (t) + Bui (t) (5) −B g i (t) − g Cxi (t) − B d i (t) − d ,
xi (0) = xi0 ∈ H, yi (t) = Cxi (t), i = 1, . . . , N. At this stage, one must provide adaptive laws for g i (t) and d i (t) so that each of the N systems in (5) is well-posed and stable. The parameter adaptive laws may be the ones presented in Curtain et al. (2003), which do not interact with each other and provide N independent estimates g i (t) and d i (t), or implement the distributed adaptive parameter estimations scheme developed in Demetriou (2011). The latter would couple the N systems via the consensus protocol included in the adaptive laws for g i (t) and d i (t).
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Finally, to address synchronization, a control component should employ a consensus protocol based on only the output signals yi (t) of each system in (1). The structure of the consensus protocol may use fixed (constant) synchronization gains or adaptive gains, whose updates will depend on the level of agreement between communicating agents. Thus, the synchronization controllers have the form � � � Uisyn (t) = − αij (t) yi (t) − yj (t) , i = 1, . . . , N. (6) j∈Ni
One now must provide the update laws for the gains αij (t). The application of (3), (4) and (7) in (1) results in � x˙ i (t) = Am xi (t) − B αij (t)yij (t) j∈Ni
� � � � −B g�i (t) − g Cxi (t) − B d�i (t) − d xi (0) = xi0 ∈ H, yi (t) = Cxi (t), i = 1, . . . , N,
(7)
where yij (t) yi (t) − yj (t) denotes the pairwise difference of the outputs of each networked system in (1). The synchronization of the N systems in (1) is achieved when each pairwise state difference converges to zero via lim �xi (t) − xj (t)�H = 0, ∀i, j. t→∞
3. MAIN RESULTS
By denoting the adaptive estimates of the unknown parameter g and constant disturbance d by g�i (t) and d�i (t), respectively, the second component of the controller Uiadapt (t) as given in (4), results in the closed-loop systems gi (t)Cxi (t) − B d�i (t). (8) x˙ i (t) = Am xi (t) − B� where the parameter errors are defined via g�i (t) = g�i (t) − g, d�i (t) = d�i (t) − d, i = 1, . . . , N
Remark 1. Even though the parameter g in the structured perturbation and the constant disturbance d are common to all N systems (1), N estimates are proposed in (8) in order to ensure that all such N estimates agree and implement a distributed estimation scheme to provide robustness of the learning scheme. It remains to propose the adaptive laws for updating g�i (t) and d�i (t), and subsequently examine the well-posedness and convergence of the resulting adaptive system. Lemma 1. Consider each of the systems in (1) with the controllers (3), (4) ui (t) = −g ∗ yi (t) − g�i (t)yi (t) − d�i (t),
N � i=1
V˙ i =
N �
�xi (t), (Am + A∗m )xi (t)�H
i=1
� � � � � � T (t) L0 ⊗ Im G(t) � −2G − 2δ�T (t) L0 ⊗ Im δ(t), where d�1 (t) g�1 (t) � = .. . � G(t) = ... , d(t) . g�N (t) d�N (t) N � i=1
where
V˙ i ≤ −µ�x(t)�2H � � � � � � T (t) L0 ⊗ Im G(t) � −2G − 2δ�T (t) L0 ⊗ Im δ(t),
x1 (t) � �N x(t) = ... , H = H . xN (t) While one can show N � 2 2 � � Vi ≥ �x(t)�2 + min{βi }�G(t)� + min{κi }�d(t)� ,
H
i=1
i
F
i
F
exponentially stability cannot be inferred. The fact that the graph Laplacian L0 has a zero eigenvalue, results in N �
V˙ i ≤ −µ�x(t)�2H ≤ 0.
i=1
(9)
Using arguments based on Gr¨onwall’s lemma produces the boundedness of all signals. However, to show convergence of the states xi (t) to zero, one must use the analog of Barbˇalat’s lemma for infinite dimensional systems. Following the exposure in Curtain et al. (2003), the state convergence can be shown. Parameter convergence is achieved without persistence of excitation only when the all-to-all inter-agent connectivity is assumed, Demetriou (2011).
(10)
3.2 Adaptive robust synchronizing controller design for networked systems
j∈Ni
� �� �� ˙ d�i (t) = κi yi (t) − λi d�i (t) − d�j (t) ,
Following Demetriou (2011), it can easily be shown that the derivative along (8), (9), (10) is given by
Using the fact that Am + A∗m ≤ −µI then
3.1 Adaptive robust controller design for single system
and the consensus adaptive laws � �� �� g�˙ i (t) = βi yi2 (t) − γi g�i (t) − g�j (t) ,
Proof. The well-posedness is based on the arguments presented in Curtain et al. (2003) for adaptive estimation of single systems and in Demetriou (2011) for the use of parameter adaptive laws with consensus. Due to space limitations, portion of the stability proof is provided. To examine the stability of (8), (9) and (10), the aggregate of all Lyapunov functionals for each system must be considered since the adaptive laws (9), (10) couple the N systems in (8). Consider the Lyapunov functionals N N � � � � � 1 1 � �xi (t)�2 + Tr g�iT (t)� gi (t) + d�Ti d�i (t) . Vi = βi κi i=1 i=1
j∈Ni
where βi , γi , κi , λi are user-defined adaptive gains. Then the systems (8), (9) and (10) are well-posed, with lim �xi (t)�H = 0, i = 1, . . . , N, t→∞
and all other signals are bounded. 246
The controllers (3) and (4) provide stabilization and adaptive parameter estimation with inter-agent interaction, as enforced by the consensus protocols in (9), (10). To ensure synchronization, another component must be included. Adding the synchronization component (6), produces the
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interacting networked systems with adaptation of the synchronization gains x˙ i (t) = Am xi (t) − B gi (t)Cxi (t) − B d i (t) αij (t) yi (t) − yj (t) , −B j∈Ni
g˙ i (t) = βi yi2 (t) − γi
j∈Ni
gi (t) − gj (t)
,
(11)
˙ di (t) − dj (t) , di (t) = κi yi (t) − λi j∈N
i α˙ ij (t) = νi yi (t)yij (t) − αij (t) ,
where νi are user-defined adaptive gains Ioannou and Sun (1995). Convergence and well-posedness results can be obtained as in Lemma 1. The Lyapunov functionals now include an additional term N N 1 �xi (t)�2 + Tr g iT (t) Vi = gi (t) β i i=1 i=1 1 2 1 T αij (t) . + di di (t) + κi νi j∈Ni
The derivative along (11) produces N i=1
V˙ i ≤ −µ�x(t)�2H − 2
N
2 αij (t)
i=1 j∈Ni
T (t) L0 ⊗ Im G(t) −2G − 2δ T (t) L0 ⊗ Im δ(t).
Using the previously established bounds of the graph Laplacian L0 , we have N
V˙ i ≤ −µ�x(t)�2H − 2
i=1
N
2 αij (t) ≤ 0.
i=1 j∈Ni
Due to the specific nature of the adaptive laws of the synchronization gains, we have asymptotic convergence of αij (t) to zero. This is stated in the following. Lemma 2. Consider the systems in (1) with the controllers (3), (4) and (6) resulting in the interacting networked systems (11). The systems (11) are well-posed, with lim �xi (t)�H = 0, i = 1, . . . , N t→∞
and lim αij (t) = 0,
t→∞
i = 1, . . . , N, j ∈ Ni ,
with all other signals bounded. Proof. The well-posedness easily follows from Lemma 1 and arguments from the relevant works Demetriou (2013, 2015). While the parameter estimates gi (t) and di (t) remain bounded, they cannot be guaranteed to converge to the true values without imposing either a persistence of excitation or an all-to-all connectivity. The latter is required since the graph Laplacian L0 has a zero eigenvalue which does not allow any negative terms of � gi (t)�F and N �d i (t)�2 in i=1 V˙ i . 4. EXAMPLE
We consider N = 5 networked diffusion PDEs in one spatial dimension as represented in (1) but without the constant disturbance terms 247
245
∂ 2 xi (t, ξ) ∂xi (t, ξ) =α , xi (0, ξ) = xi0 (ξ), i = 1, . . . , 5, ∂t ∂ξ 2 ∂xi (t, 1) − gxi (t, 1) = ui (t) xi (t, 0) = 0, ∂t where the thermal diffusivity is set to α = 0.1 and m = 1, i.e. g is a scalar. The output of each networked PDE is yi (t) = xi (t, 1) and in this case one has that the input and output operators are collocated, i.e. B = C ∗ = δ(ξ − 1). The unknown boundary parameter was set to g = 1.1α. It can be shown that the open loop system is unstable when the unknown boundary parameter g ≥ α. In fact, for g = α, the operator A + gBC has an eigenvalue at the origin. Since the exact value of the boundary parameter is not known, but the upper bound |g| ≤ 2α is known, then the stabilizing controller Uistab (t) in (3) is as Uistab (t) = −2αyi (t), i = 1, . . . , 5. The corresponding adaptive parameter controller in (4) is Uiadapt = − gi (t)yi (t), i = 1, . . . , 5. The updates for the adaptive estimate of the boundary parameter are given by the consensus adaptive law (9). The last component of the control law is given by the synchronization controller (6). The last remaining adaptation provides a rule for updating the gains αij and is given by (11). To simulate the above networked PDEs, a finite elementbased approximation scheme is used. A total of 40 linear splines, H¨ollig (2003) that are modified to account for the (Dirichlet-Neumann) boundary conditions are used. When each PDE is viewed in weak form with test functions set equal to each of the approximating linear splines, then a system of 40 ordinary differential equations arises. The spatial integrals which are required to compute the matrix representations of the state, input and output operators are computed using a composite two-point GaussLegendre quadrature rule, Celia and Gray (1992). The resulting semidiscrete system of 40 differential equations was subsequently integrated numerically over the interval [0, 10]s using the stiff ODE solver from the Matlab ODE library, routine ode23s which is based on a 4th order Runge-Kutta scheme. The initial conditions for the 5 networked PDEs were x1 (0, ξ) = 39.4 sin(1.3πξ) exp(−7ξ 2 ), x2 (0, ξ) = 12.6 sin(2.1πξ) cos(1.5πξ), x3 (0, ξ) = 7.6 sin(3.6πξ) exp(−7ξ 2 ), x4 (0, ξ) = 2.5 sin(5πξ) exp(−ξ 2 ), x5 (0, ξ) = −26.2 sin(5πξ) exp(−7(ξ − 0.5)2 ). The adaptive gains βi = 0.01, γi = 75 and the adaptive consensus gains were νi = 0.5, i = 1, . . . , 5. The initial conditions for parameter estimates were chosen as g1 (0) = 0.05, g2 (0) = 0.125, g3 (0) = 0.15, g4 (0) = 0.18 and g5 (0) = 0.03. Finally, the initial conditions for the synchronization gains were given by αij (0) = 0.5, i = 1, . . . , 5, j ∈ Ni . Figure 1 depicts the evolution of the adaptive estimates gi (t) along with the true value g = 0.11. The average value 5 of the adaptive estimates gave (t) = 15 i=1 gi (t) is shown to converge to the steady state value of limt→∞ gave (t) = 0.1085 indicating a parameter convergence. The evolution of the adaptive synchronization gains αij (t) is depicted in Figure 2, where it is observed that all gains asymptotically converge to zero, as theoretically predicted. The evolution
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0.2
4.5
0.18
4
0.16
3.5
0.14
3
0.12
2.5
0.1
2
0.08
1.5
|xm(t)|H with adaptive αij(t)
0.06
0.02
0
1
2
3
4
5 Time (sec)
6
7
8
9
m
0
10
0
1
2
3
4
5 Time (sec)
6
7
8
9
10
Fig. 3. Evolution of L2 (0, 1) norm of mean state xm (t, ξ) using adaptive synchronization gains αij (t) and constant synchronization gains αij .
8
α (t) 12
α14(t)
0.2
α (t) 15
adaptive αij(t)
α21(t)
constant αij
0.15
α (t)
6
ij
0.5
Fig. 1. Evolution of adaptive parameter estimates g i (t). 7
H
1
gˆ1 (t) gˆ2 (t) gˆ3 (t) gˆ4 (t) gˆ5 (t) g
0.04
|x (t)| with constant α
34
xm(5,ξ)
α41(t) α43(t)
5
α45(t)
0.05
α51(t) α (t)
4
0.1
54
0
0
0.1
0.2
0.3
0.4 0.5 0.6 Spatial variable ξ
0.7
0.8
0.9
1
0.3
0.4 0.5 0.6 Spatial variable ξ
0.7
0.8
0.9
1
3 0.012 adaptive αij(t)
0.01
2
constant αij
xm(10,ξ)
0.008
1
0.006 0.004
0
0.002
0
1
2
3
4
5 Time (sec)
6
7
8
9
10
0
Fig. 2. Evolution of adaptive synchronization gains αij (t). of the L2 norm of the mean state 5 1 xm (t, ξ) = xi (t, ξ) 5 i=1
(12)
is depicted in Figure 3. In addition to the adaptive synchronization gains, the case of constant synchronization gains was also considered and presented in Figure 3. While one can perform an optimization to find the optimal constant synchronization gains as was presented in Demetriou (2015), in the simulation the constant gains were chosen to be equal to the initial conditions for the adaptive synchronization gains, namely αij = 0.5, i = 1, . . . , 5, j ∈ Ni . It is observed that in the case of adaptive synchronization gains, the norm of the mean state converges to zero faster than the case with constant αij . To further examine the effects of the proposed controllers on synchronization, the spatial distribution of the mean state (12) for two time instances (t = 5s and t = 10s) is presented in Figure 4 for both the case of adaptive synchronization gains αij (t) and constant synchronization gains αij . In both instances, the mean state converges (pointwise) to zero faster with the adaptive synchronization gains than the constant ones. 248
0
0.1
0.2
Fig. 4. Spatial distribution of the mean state xm (t, ξ) vs the spatial variable at t = 5s and at t = 10s. Table 1 summarizes the effects of the synchronization controllers. As a measure of synchronization, one considers the deviation from the mean δi (t, ξ) = xi (t, ξ) − xm (t, ξ), i = 1, . . . , 5. The spatial L2 (0, 1) norm of the aggregate deviation from the mean is given by 5 1 5 2 2 �δ(t)� = δi2 (t, ξ) dξ. �δi (t, ξ)� = i=1
0
i=1
Evaluating the L2 (0, t) (time) norm of the aggregate deviation from the mean, computed by 10 5 1 10 �δ(t)�2 dt = δi2 (t, ξ) dξ dt, 0
0
i=1
0
provides a single piece of information on the success of synchronization. In Table 1, one observes that when adaptation of the synchronization gains is implemented, then the above norm is 5.9230, whereas when constant synchronization gains are used, it has the value of 6.1843,
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case
value
adaptive αij (t)
5.9230
constant αij
6.1843
Table 1. L2 (0, 10) norm of the aggregate deviation from the mean �δ(t)�2 . indicating an improved agreement (synchronization) when adaptive gains are used. 5. CONCLUSION An adaptive scheme to stabilize and synchronize networked diffusion PDEs with boundary observation and control has been proposed. The networked PDEs were assumed to be identical, differing on their initial conditions, and could exchange information through a communication topology described by a complete directed graph. Structured perturbation culminating from parametric coupling of the boundary control and observation were included in the networked PDEs and which resulted on positive eigenvalues, thus necessitating the use of a stabilizing boundary controller. Additionally, constant disturbances were assumed to enter through the boundary. The controllers consisted of three parts: one part, which served as a precompensator, used static output feedback to stabilize the open loop system. The second part used the adaptive estimates of the structured perturbation parameter and of the constant disturbance in a feedback stabilization format to cancel their effects. The adaptive laws included additional terms that coupled the PDEs through this adaptations; such a coupling took the form of a consensus protocol and which aided in the parameter convergence. The final part of the controller included a consensus protocol consisted of the pairwise differences of the outputs of all communication PDEs, weighted by adaptive edge-dependent synchronization gains. Stability and convergence properties were analyzed and a numerical example provided further insights on the use of adaptation in the consensus protocols in parameter and synchronization convergence. An immediate extension of the above results would involve networked systems with not only different initial conditions, but also different operators (Ai , Bi , Ci ). However this will come at the expense of stringent requirements on each of the networked systems that each must satisfy the matching condition Am = Ai − Bi gi∗ Ci , i = 1, . . . , N . This is currently being investigated by the author. ACKNOWLEDGEMENTS The author gratefully acknowledges financial support from the AFOSR, grant FA9550-12-1-0114. REFERENCES Ambrosio, B. and Aziz-Alaoui, M. (2012). Synchronization and control of coupled reaction-diffusion systems of the fitzhugh-nagumo type. Computers & Mathematics with Applications, 64(5), 934 – 943. Ambrosio, B. and Aziz-Alaoui, M.A. (2013). Synchronization and control of a network of coupled reactiondiffusion systems of generalized Fitzhugh-Nagumo type. 249
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