LFT Distributed Systems in Input-Output Form *

Proceedings of the 20th World Congress The International Federation of Congress Automatic Control Control Proceedings of 20th The International Federation of Automatic Proceedings of the the 20th World World Congress Proceedings of the 20th World Congress Toulouse, France, July 9-14, 2017 The International Federation of Automatic Control Available online at www.sciencedirect.com Toulouse, France, July 9-14, 2017 The International Federation of Automatic The International Federation of Automatic Control Control Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017

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IFAC PapersOnLine 50-1 (2017) 11409–11414 Distributed Controller Design Design for for LPV/LFT LPV/LFT Distributed Controller Distributed Controller Design for Distributed Controller Design for LPV/LFT Distributed Systems Controller Design for LPV/LFT LPV/LFT Distributed in Form Distributed Systems in Input-Output Input-Output Form ⋆ ⋆⋆ Distributed Systems in Input-Output Form Distributed Systems in Input-Output Form Distributed Systems in Input-Output Form ∗ ∗∗ ∗

Qin Qin Liu Liu ∗∗ Qin Liu ∗∗ Qin Qin Liu Liu

Hossam Javad Velni Hossam S. S. Abbas Abbas ∗∗ Javad Mohammadpour Mohammadpour Velni ∗∗ ∗∗ ∗∗∗ ∗∗Werner ∗ Hossam S. Abbas Javad Mohammadpour Velni ∗∗∗ Herbert ∗∗ Hossam S. Abbas Javad Mohammadpour Velni Hossam S. Herbert Abbas Werner Javad Mohammadpour Velni ∗ ∗∗∗ ∗∗∗ Herbert Werner Herbert Herbert Werner Werner ∗∗∗ ∗ ∗ Complex Systems Systems Control Control Laboratory, Laboratory, College College of of Engineering, Engineering, University University of of Complex ∗ ∗ Complex Systems Control Laboratory, College of Engineering, University of Georgia, Athens, GA 30602 USA (e-mail: [email protected]; ∗ Complex Systems Control Laboratory, College of Engineering, University Georgia, Athens, GA 30602 USA (e-mail: [email protected]; ComplexGeorgia, SystemsAthens, ControlGA Laboratory, College of Engineering, University of of 30602 USA (e-mail: [email protected]; [email protected]) Georgia, Athens, Athens, GA [email protected]) 30602 USA USA (e-mail: (e-mail: [email protected]; [email protected]; Georgia, 30602 ∗∗ ∗∗ Electrical [email protected]) Department, Electrical [email protected]) Department, Faculty Faculty of of Engineering, Engineering, Assiut Assiut [email protected]) ∗∗ ∗∗ Electrical Engineering Faculty of Engineering, Assiut University, Assiut, 71515, Department, Egypt (e-mail: (e-mail: [email protected]) ∗∗ Electrical Engineering Department, Faculty of Engineering, Assiut University, Assiut, 71515, Egypt [email protected]) Electrical Engineering Department, Faculty of Engineering, Assiut ∗∗∗ University, Assiut, 71515, Egypt (e-mail: [email protected]) ∗∗∗ Institute of Control Systems, Hamburg University of Technology, University, Assiut, 71515, Egypt (e-mail: [email protected]) Institute of Control Systems, Hamburg University of Technology, University, Assiut, 71515, Egypt (e-mail: [email protected]) ∗∗∗ ∗∗∗ Institute of University Hamburg, 21073, Germany (e-mail: ∗∗∗ Institute of Control Control Systems, Hamburg University of of Technology, Technology, Hamburg, 21073, Systems, GermanyHamburg (e-mail: [email protected]) [email protected]) Institute of Control Systems, Hamburg University of Technology, Hamburg, 21073, Germany (e-mail: [email protected]) Hamburg, 21073, Germany (e-mail: [email protected]) Hamburg, 21073, Germany (e-mail: [email protected]) Abstract: Abstract: This This paper paper considers considers the the controller controller design design problem problem for for parameter-varying parameter-varying distributed distributed systems, systems, Abstract: This paper considers the controller design problem parameter-varying systems, whose time/space-varying dynamics can be be characterized characterized byfor temporal/spatial lineardistributed parameter-varying Abstract: This paper considers the controller design problem for parameter-varying distributed systems, whose time/space-varying dynamics can by temporal/spatial linear parameter-varying Abstract: This paper considers the controller design problem for parameter-varying distributed systems, whose time/space-varying dynamics can be characterized by temporal/spatial linear parameter-varying (LPV) models defined at the spatially-discretized subsystems. Assuming a rational functional depenwhose time/space-varying dynamics can be characterized by temporal/spatial linear parameter-varying (LPV) models defined at the spatially-discretized subsystems. Assuming a rational functional depenwhose time/space-varying dynamics can be characterized by temporal/spatial linear parameter-varying (LPV) models defined at the spatially-discretized subsystems. Assuming a rational functional dependence on the scheduling parameters, the distributed LPV model in linear fractional transformation (LPV) on models defined at at the the spatially-discretized subsystems. Assuming a rational rational functional dependence the scheduling parameters, the distributed LPV model in linear fractional transformation (LPV) models defined spatially-discretized subsystems. Assuming a functional dependence on the scheduling parameters, the distributed LPV model in linear fractional transformation (LFT) form renders analysis and synthesis conditions at the subsystems level with the application dence on the scheduling parameters, the distributed LPV model in linear fractional transformation (LFT) form renders analysis and synthesis conditions LPV at themodel subsystems levelfractional with the transformation application of of dence on the scheduling parameters, the distributed in linear (LFT) form renders and synthesis conditions at the subsystems level with the application of Finsler’s lemma and analysis the full full block block S-procedure technique. The designed distributed distributed controller inherits (LFT) form renders analysis and synthesis conditions at the subsystems level with the application of Finsler’s lemma and the S-procedure technique. The designed controller inherits (LFT) form renders analysis and synthesis conditions at the subsystems level with the application of Finsler’s lemma and the full block S-procedure technique. The designed distributed controller inherits the interconnected structure of the plant and has a (predefined) fixed structure. Simulation results using aa Finsler’s lemma and the full block S-procedure technique. The designed distributed controller inherits the interconnected structure of the plant and has a (predefined) fixed structure. Simulation results using Finsler’s lemma and the full block S-procedure technique. The designed distributed controller inherits the interconnected structure of the plant and has a (predefined) fixed structure. Simulation results using spatially-varying heat equation demonstrate the satisfactory performance of the proposed control design the interconnected structure of the plant and has a (predefined) fixed structure. Simulation results using spatially-varying equation demonstrate the satisfactory performance of theSimulation proposed control designaaa the interconnectedheat structure of the plant and has a (predefined) fixed structure. results using spatially-varying heat equation demonstrate the satisfactory satisfactory performance of the the proposed proposed control control design method. spatially-varying heat equation demonstrate the performance of method. spatially-varying heat equation demonstrate the satisfactory performance of the proposed control design design method. method. method. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: parameter-varying (LPV) (LPV) systems, linear fractional fractional transformation transformation (LFT), (LFT), distributed distributed Keywords: Linear Linear parameter-varying systems, linear Keywords: Linear parameter-varying parameter-varying (LPV)controller systems, linear linear fractional fractional transformation transformation (LFT), (LFT), distributed distributed systems, input-output form, fixed-structure Keywords: Linear (LPV) systems, systems, input-output form, fixed-structure controller Keywords: Linear parameter-varying (LPV)controller systems, linear fractional transformation (LFT), distributed systems, input-output input-output form, fixed-structure fixed-structure systems, form, controller systems, input-output form, fixed-structure controller spatial-scheduling parameters, parameters, aa temporal/spatial temporal/spatial LPV LPV model model 1. 1. INTRODUCTION INTRODUCTION spatial-scheduling 1. INTRODUCTION spatial-scheduling parameters, aa temporal/spatial LPV model can be used to capture the varying system dynamics over the spatial-scheduling parameters, temporal/spatial LPV model 1. INTRODUCTION can be used to capture the varying system dynamics over the 1. INTRODUCTION spatial-scheduling parameters, a temporal/spatial LPVover model can be used to capture the varying system dynamics temporal/spatial scheduling parameter set. can be used to capture the varying system dynamics over the In temporal/spatial scheduling parameter set. dynamics over the In the the last last decade, decade, much much attention attention has has been been paid paid to to address address can be used to capture the varying system the temporal/spatial scheduling parameter set. In the last decade, much attention has been paid to address temporal/spatial scheduling parameter set. the control problem of large-scale spatially distributed systems, In the last decade, much attention has been paid to address the control problem of large-scale spatially distributed systems, temporal/spatial scheduling parameter set. Most research work on distributed control relies on the stateIn the last problem decade, of much attention has been paid tosystems, address Most research work on distributed control relies on the statethe spatially distributed whose distributed can captured using difthe control control problemdynamics of large-scale large-scale spatially distributed systems, whose distributed dynamics can be be captured using partial partial dif- space Most work on control relies on the staterepresentation of the and considers fullthe control problem of large-scale spatially distributed systems, Most research work distributed control relies on space research representation of distributed the plant plant model, model, considers fullresearch work on on distributed control and relies on the the statestatewhose distributed dynamics can be captured using partial differential equations (PDEs), see e.g., Jovanovic (2004); Bamieh whose distributed dynamics can be captured using partial dif- Most ferential equations (PDEs), see e.g., Jovanovic (2004); Bamieh space representation of the plant model, and considers fullorder controller design, i.e., controller order is equal equal or larger larger whose distributed dynamics can be captured using partial difspace representation of the plant model, and considers fullorder controller design, i.e., controller order is or space representation of the plant model, and considers fullferential equations (PDEs), see e.g., Jovanovic (2004); Bamieh et al. (2002); D’Andrea and Dullerud (2003); Stewart (2000). ferential equations (PDEs), see e.g., Jovanovic (2004); Bamieh et al. (2002); D’Andrea and Dullerud (2003); Stewart (2000). order controller design, i.e., controller order is equal or larger than the plant order, see D’Andrea and Dullerud (2003) for ferential equations (PDEs), see e.g., Jovanovic (2004); Bamieh order controller design, i.e., controller order is equal or larger than the plant order, see D’Andrea and Dullerud (2003) for order controller design, i.e., controller order is equal or larger et al. (2002); D’Andrea and Dullerud (2003); Stewart (2000). Due to the underlying large system order, a centralized conet al. (2002); D’Andrea and Dullerud (2003); Stewart (2000). Due to the underlying large system order, a centralized conthan the plant order, see D’Andrea and Dullerud (2003) for distributed control design of LTSI systems, see Wu (2003) and et al. (2002); D’Andrea and Dullerud (2003); Stewart (2000). than the plant order, see D’Andrea and Dullerud (2003) for distributed control design of LTSI systems, see Wu (2003) and the plant order, see of D’Andrea and Dullerud (2003) and for Due large aa centralized controller this class is numerically infeasible to Due to toof the underlying large system system order, centralized controller ofthe thisunderlying class of of systems systems is often often order, numerically infeasible to than distributed control design LTSI systems, see Wu (2003) Liu and Werner (2016) for gain-scheduled control of LTSV Due to the underlying large system order, a centralized condistributed control design of LTSI systems, see Wu (2003) and Liu and Werner (2016) for gain-scheduled control of LTSV distributed control design of LTSI systems, see Wu (2003) and troller of this class of systems is often numerically infeasible to solve; moreover, it is sensitive to actuator and sensor failure. troller of this class of systems is often numerically infeasible to solve; moreover, it is sensitive to actuator and sensor failure. Liu and Werner (2016) for gain-scheduled control of LTSV systems. The importance importance of gain-scheduled controller design designcontrol in input-output input-output troller of this classit of systems istooften numerically infeasible to systems. Liu and (2016) of The of controller in Liu and Werner Werner (2016) for for gain-scheduled control of LTSV LTSV solve; and failure. solve; moreover, moreover, it is is sensitive sensitive to actuator actuator and sensor sensor failure. systems. The importance of controller design in input-output (I-O) form is threefold. Firstly, the experimentally identified solve; moreover, it is sensitive to actuator and sensor failure. systems. The importance of controller design in input-output Instead of preserving the continuous nature in space, an ef(I-O) form is threefold. Firstly, the experimentally identified Instead of preserving the continuous nature in space, an ef- systems. The importance of controller design in input-output (I-O) form experimentally identified are in I-O see and Werner for Instead of continuous nature space, an (I-O) form is threefold. Firstly, the experimentally identified fective was proposed in Dullerud models are is in threefold. I-O form, form,Firstly, see Liu Liuthe and Werner (2013) (2013) for the the Instead framework of preserving preserving the continuous nature in in and space, an efef- models fective framework wasthe proposed in D’Andrea D’Andrea and Dullerud (I-O) form is threefold. Firstly, the experimentally identified Instead of preserving the continuous nature in space, an efmodels are in I-O form, see Liu and Werner (2013) for the the black-box identification of temporal/spatial LPV I-O models. fective framework was proposed in D’Andrea and Dullerud models are in I-O form, see Liu and Werner (2013) for (2003), where the overall system can be treated as the interblack-box identification of temporal/spatial LPV I-O models. fective framework was proposed in D’Andrea and Dullerud (2003), where the overall system can be treated as the intermodels are in I-O form, see Liu and Werner (2013) for the fective framework was proposed in D’Andrea and Dullerud black-box identification of temporal/spatial LPV I-O models. Secondly, it has been demonstrated in T´ o th et al. (2012) that (2003), where the overall system can be treated as the interblack-box identification of temporal/spatial LPV I-O models. connection of spatially-discretized subsystems, and each subSecondly, it has been demonstrated in T´ o th et al. (2012) that (2003), where where the overall overall system system can can be treated treatedand as each the interinterconnection of spatially-discretized subsystems, sub- black-box identification of temporal/spatial LPV I-O models. (2003), the be as the Secondly, it has been demonstrated T´ oits th et al. (2012) that converting an I-O into state-space repreconnection of each Secondly, it demonstrated in T´ th et that system is with and capabilities. when converting an LPV LPV I-O model model in into repreconnection of spatially-discretized spatially-discretized subsystems, and each subsub- when system is equipped equipped with actuating actuatingsubsystems, and sensing sensingand capabilities. Secondly, it has has been been demonstrated in T´ooits th state-space et al. al. (2012) (2012) that connection of spatially-discretized subsystems, and each subwhen converting an LPV I-O model into its state-space representation, the phenomenon of dynamic dependence occurs and system is equipped with actuating and sensing capabilities. when converting converting an LPV LPV I-O I-O model into its state-space state-space repreThe resulting localized subsystems of much smaller order can can when sentation, the phenomenon of dynamic dependence occurs and system is equipped with actuating and sensing capabilities. The resulting localized subsystems of much smaller order an model into its represystem is equipped with actuating and sensing capabilities. sentation, the phenomenon of dynamic dependence occurs and requires non-trivial treatment. This phenomenon applies to the The resulting localized subsystems of much smaller order can sentation, the phenomenon of dynamic dependence occurs and be exploited for distributed controller design. If the spatiallyrequires non-trivial treatment. This phenomenon applies to the The resulting localized subsystems of much smaller order can be exploited for distributed controller design. If the spatiallysentation, the phenomenon of dynamic dependence occurs and The resultingfor localized subsystems of much smaller order can requires non-trivial treatment. This phenomenon applies to the state-space realization of temporal/spatial LPV I-O models as be exploited distributed controller design. If the spatiallyrequires non-trivial treatment. This phenomenon applies to discretized subsystems are linear and invariant under temporal state-space realization of temporal/spatial LPV I-O models as be exploited exploitedsubsystems for distributed distributed controller design. If Ifunder the spatiallyspatiallydiscretized are linear and invariant temporal requires non-trivial treatment. This phenomenon applies to the the be for controller design. the state-space realization of temporal/spatial LPV I-O models as well. Thirdly, the I-O framework motivates fixed-structure condiscretized subsystems are linear and invariant under temporal state-space realization of temporal/spatial LPV I-O models and spatial translation, the system is called linear timeand well. Thirdly, the I-O framework motivates fixed-structure condiscretized subsystems are linear and invariant under temporal and spatial subsystems translation, are thelinear system called linear and state-space realization of temporal/spatial LPV I-O models as as discretized andis invariant undertimetemporal well. Thirdly, the I-O framework motivates fixed-structure controller design, which has predefined temporal and spatial order and spatial translation, the system is called linear timeand well. Thirdly, the I-O has framework motivates fixed-structure conspace-invariant (LTSI). However, However, most distributed systems in trollerThirdly, design, the which predefined temporal and spatial order and spatial spatial translation, translation, the system systemmost is called called linear timeand space-invariant (LTSI). distributed systems in well. I-O framework motivates fixed-structure conand the is linear timeand troller design, Compared which has has to predefined temporal and spatial spatial order and the full-order controller space-invariant (LTSI). However, most distributed distributed systems in design, which predefined temporal and order practice exhibit linear and (LTSV) properand structure. structure. the state-space state-space full-order controller space-invariant However, most systems in troller practice exhibit (LTSI). linear timetimeand space-varying space-varying (LTSV) propertroller design, Compared which has to predefined temporal and spatial order space-invariant (LTSI). However, most distributed systems in and structure. Compared to the state-space full-order controller design, a fixed-structure controller of smaller order can largely practice exhibit linear timeand space-varying (LTSV) properand structure. Compared to the state-space full-order controller ties, due to boundary conditions, non-uniform physical propdesign, a fixed-structure controller of smaller order can largely practice exhibit linear timetimeand space-varying space-varying (LTSV) properties, dueexhibit to boundary conditions, non-uniform(LTSV) physicalproperprop- and structure. Compared to the state-space full-order controller practice linear and design, aa fixed-structure of smaller order can largely the computational complexity. Its become more ties, boundary physical propdesign, controller of order can erties, etc. address the and/or variation of reduce the computationalcontroller complexity. Its benefits benefits become more ties, due due toTo boundary conditions, non-uniform physical properties, etc.to To address conditions, the temporal temporalnon-uniform and/or spatial spatial variation of reduce design, a fixed-structure fixed-structure controller of smaller smaller order can largely largely ties, due to boundary conditions, non-uniform physical propreduce the computational complexity. Its benefits become more significant when dealing with large order systems. Recent reerties, etc. To address the temporal and/or spatial variation of reduce the the computational computational complexity. Its benefits benefits become more LTSV systems, the linear parameter-varying (LPV) technique, significant when dealing with large order systems. Recent reerties, etc. To address the temporal and/or spatial variation of LTSV systems, the linear parameter-varying (LPV) technique, reduce complexity. Its become more erties, etc. To address the parameter-varying temporal and/or spatial variation of sults significant when dealing with large order systems. Recent reon I-O controller design can be found in Wollnack et al. LTSV systems, the linear (LPV) technique, significant when dealing with large order systems. Recent rewhich was first introduced in Shamma (1988) to cope with sults on I-O controller design can be found in Wollnack et al. LTSV systems, the linear parameter-varying (LPV) technique, which was first introduced in Shamma (1988) to cope with significant when dealing with large order systems. Recent reLTSV systems, the linear parameter-varying (LPV) technique, sults on I-O controller design can be found in Wollnack et al. (2013), Wollnack and Werner (2015b), Wilfried et al. (2007), which was first introduced in Shamma (1988) to cope with sults on I-O controller design can be found in Wollnack et time-varying systems, can be extended in a systematic way to (2013), Wollnack and Werner (2015b), Wilfried et al. (2007), which was first introduced in Shamma (1988) to cope with time-varying systems, can beinextended in(1988) a systematic way to sults on I-O controller design can be found in Wollnack et al. al. which was first introduced Shamma to cope with (2013), Wollnack and Werner (2015b), Wilfried et al. (2007), etc. time-varying systems, can extended aa systematic to (2013), Wollnack and Werner (2015b), Wilfried et al. (2007), solve analogous problems of If LTSV time-varying systems, can be be extended in insystems. systematic way to etc. solve analogous problems of distributed distributed systems. If an anway LTSV (2013), Wollnack and Werner (2015b), Wilfried et al. (2007), time-varying systems, can be extended in a systematic way to etc. solve problems distributed If LTSV system can functions of solve analogous analogous problems of of as distributed systems. If an an and/or LTSV etc. system can be be parametrized parametrized as functionssystems. of temporaltemporaland/or etc. Inspired by by the the work work in in Wollnack Wollnack and and Werner Werner (2015a), (2015a), where where solve analogous problems of distributed systems. If an LTSV Inspired system can be parametrized as functions of temporaland/or system can be parametrized as functions of temporaland/or Inspired by the work in Wollnack and Werner (2015a), where controller design for polytopic LPV systems of system can be parametrized as functions of temporal- and/or distributed Inspired by the work in Wollnack and Werner (2015a), where distributed controller design for polytopic LPV systems of ⋆ Inspired by controller the work indesign Wollnack and WernerLPV (2015a), where ⋆ This distributed for polytopic systems of This work work was was supported supported by by the the German German Research Research Foundation Foundation (DFG) (DFG) affine dependence on scheduling parameters has been addistributed controller design for polytopic LPV systems of affine dependence on scheduling parameters has been ad⋆ distributed controller design for polytopic LPV systems of through the research research fellowshipby Li the 2763/1-1. This work was supported German Research Foundation (DFG) ⋆ through the fellowship Li 2763/1-1. This work was supported by the German Research Foundation (DFG) affine dependence on scheduling parameters has been ad⋆ affine dependence on scheduling parameters has been adThis work was supported by the German Research Foundation (DFG) affine dependence on scheduling parameters has been adthrough the research fellowship Li 2763/1-1. through the research fellowship Li 2763/1-1.

through the research fellowship Li 2763/1-1. Copyright © 2017 11901 Copyright 2017 IFAC IFAC 11901Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 11901 Copyright © 2017 IFAC 11901 Peer review under responsibility of International Federation of Automatic Control. Copyright © 2017 IFAC 11901 10.1016/j.ifacol.2017.08.1803

Proceedings of the 20th IFAC World Congress 11410 Qin Liu et al. / IFAC PapersOnLine 50-1 (2017) 11409–11414 Toulouse, France, July 9-14, 2017

dressed, this paper considers a rather general class of LTSV systems whose coefficients are rational functions of scheduling parameters—a case where the polytopic LPV synthesis technique can not be applied. The rational functional dependence leads to the representation of the LPV model in linear fractional transformation (LFT) form–the so-called LPV/LFT model. Of interest here is the distributed controller design that inherits the interconnected structure of the plant, such that analysis and synthesis conditions are developed at the subsystem level— independent of the number of subsystems. Furthermore, the distributed controller can accommodate fixed structure. It is well known that the fixed-structure controller design leads to nonconvex synthesis conditions that can be formulated in terms of bilinear matrix inequalities (BMIs). With the application of the full block S-procedure technique introduced in Scherer (2001) and further imposing constraints on the multiplier, it will be shown that it suffices to check a finite number of BMI constraints in the parameter set. Furthermore, The paper is organized as follows: Section 2 recaps two lemmas. Section 3 presents the multidimensional I-O models that define the interconnected dynamics of both open- and closedloop LTSV systems at the subsystem level, as well as their LPV/LFT representations. Analysis and synthesis conditions are derived in Section 4. A numerical example is employed in Section 5 to demonstrate the performance of the proposed controller design technique. Conclusions are drawn in Section 6. 2. PRELIMINARIES This section revisits Finsler’s Lemma (see de Oliveira and Skelton (2001)) and the full block S-procedure technique (see Scherer (2001)). They will be employed later to derive the analysis and synthesis conditions proposed in this paper. Lemma 1. (Finsler’s Lemma). Let x ∈ Rn , Q ∈ Sn and B ∈ Rm×n such that rank(B) < n. The following statements are equivalent: i) xT Qx < 0, ∀Bx = 0, x �= 0. ii) ∃X ∈ Rn×m : Q + XB + B T X T < 0. Lemma 2. (Full Block S-Procedure). Given a quadratic matrix inequality RT (Θ)M R(Θ) < 0, (1) with the gain-scheduled matrix R(Θ) written in a lower LFT form � � R11 R12 , (2) R(Θ) = Θ ⋆ R21 R22 and the structured scheduling block Θ ∈ Θ where Θ denotes a compact set and Θ = {Θ : diag{θ1 Ir1 , . . . , θnθ Irnθ }, |θi | < 1, i = 1, . . . , nθ },

3. INTERCONNECTED LTSV SYSTEMS In this paper, we are interested in distributed systems equipped with a spatially distributed array of collocated actuator-sensor pairs. The attachment of actuator-sensor pairs induces spatial discretization, such that the global system can be considered as the interconnection of localized subsystems, each capable of actuating and sensing. For the simplicity of presentation, we consider subsystems of one spatial dimension and with single input and single output. But the framework is rather general and can be applied to cope with systems of multiple spatial dimensions (see D’Andrea and Dullerud (2003)). 3.1 Plant Model in LPV/LFT I-O Form After the spatial discretization, a parameter-varying distributed system consists of Ns subsystems G, whose dynamics are governed by two-dimensional I-O model (in the sense that all involved signals are functions of time and one-dimensional space) A(θt , θs , zt , zs )y(k, s) = B(θt , θs , zt , zs )u(k, s), (5) nθt nθs where θt ∈ R , θs ∈ R are temporal and spatial scheduling parameters, respectively, y(k, s) ∈ Rny and u(k, s) ∈ Rnu are the output and input signals of subsystem s at time instant k, respectively. For the purpose of clarity, we consider subsystems of single input and single output. Furthermore, zt and zs are forward temporal and spatial shift operators, respectively, e.g., zt−1 zs y(k, s) = y(k − 1, s + 1), A and B are gain-scheduled polynomials and defined as na ma � � a(ik ,is ) (θt , θs )zt−ik zs−is , (6a) A=1+ B=

nb �

ik =1 is =−ma mb �

b(jk ,js ) (θt , θs )zt−jk zs−js ,

(6b)

jk =1 js =−mb

where na , ma , nb , and mb are the indices of time- and spaceshifted outputs and inputs, respectively, and a(ik ,is ) , b(jk ,js ) are the scheduled weighting coefficients. The implicit representation of (5) is written as � � � � ¯ t , θs ) −B(θ ¯ t , θs ) ξy (k, s) = 0, A(θ (7) ξu (k, s) where A¯ ∈ Rnξy and B¯ ∈ Rnξu are the parameter-dependent coefficient vectors of A and B, respectively, ξy (k, s) ∈ Rnξy and ξu (k, s) ∈ Rnξu are the conformably temporally- and spatially-shifted outputs and inputs, respectively.

We consider the class of systems, where A and B have rational dependence on scheduling parameters θt and θs . By pulling the where ri denotes the multiplicity of the scheduling parameters θi . scheduling parameters out, each subsystem can be represented as the interconnection of an LTSI system G augmented by local The inequality (1) holds if and only if there exists a full-block feedback with its own temporal and spatial scheduling blocks multiplier Π such that as shown in Fig. 1. The LFT representation of (7) is given by   � � R11 R12 M 0    [∗] 0 I  < 0, (3) � � ξy (k, s) 0 Π ¯0 B ¯p 0 ξu (k, s) R21 R22 A¯0 −B = 0, (8) ¯ ¯ ¯ C0 −D0 Dp −I  p(k, s)  and for any Θ ∈ Θ � � q(k, s) Θ [∗]Π ≥ 0. (4) where p(k, s) and q(k, s) denote the input and output of the I scheduling block, respectively (see (10) below for the definition of the matrices in (8)). Assume that temporal and spatial variThe transpose of a matrix (or a vector) is denoted as ∗. 11902

Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Qin Liu et al. / IFAC PapersOnLine 50-1 (2017) 11409–11414

ations are decoupled, i.e., the spatial properties of subsystems do not change in time. Thus, the scheduling block satisfies � �� � � � pt (k, s) qt (k, s) Θt = or p(k, s) = Θq(k, s), (9) ps (k, s) Θs qs (k, s) with pt and qt ∈ RnΘt , ps and qs ∈ RnΘs , Θt ∈ Θt and Θs ∈ Θs , where qt and qs , pt and ps are the inputs and outputs of the temporal and spatial uncertainty channels, respectively. Θt and Θs are the structured temporal and spatial uncertainties of size nΘt and nΘs , respectively. Θt and Θs are two compact sets with the uncertainties structured in diagonal matrices form as Θt = {Θt : diag{θt1 Irθt , . . . , θtnt Irθtn }, |θti | < 1, i = 1, . . . , nt }

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as the functional dependence of coefficients ak and bk on the scheduling parameters can be predefined by the designer.

By closing the loop between the plant and the controller subsystem, the generalized plant of a controlled subsystem, where the weighting filters Ws and Wks are included to shape sensitivity and control sensitivity of the closed-loop system, respectively, is shown in Fig. 2. Defining the coefficient vectors A¯k and B¯k of the controller K and those of the filters Ws and Wks in a similar way as the plant, the LPV implicit representation of the generalized plant takes the form    ¯  ξy (k, s) A −B¯ 0 0 0  ξ (k, s)  0 −B¯k   u 1   B¯k A¯k 0 t B s s s  ξz1 (k, s) ¯ ¯ ¯ 0 A 0 −B  Θs = {Θs : diag{θs1 Irθs1 , . . . , θsns Irθsn }, |θsi | < 1, i = 1, . . . , ns }, (13) ξz2 (k, s) s ¯ ks 0 −A¯ks 0 0 B ξw (k, s) where rθti and rθsi denote the multiplicity of scheduling pak k rameters θti and θsi , respectively. := R(θ , θ , θ , θ )ξ(k, s) = 0. t

pt (k, s−1) ps (k, s−1)

Θt Θs−1

qt (k, s−1)

pt (k, s)

qs (k, s−1)

ps (k, s)

Θt Θs

G

qt (k, s)

pt (k, s+1)

qs (k, s)

ps (k, s+1)

G

qt (k, s+1)

Θt Θs+1

qs (k, s+1)

s

t

s

Note that the weighting filters can in general be parameterdependent as the plant and the controller. For the sake of presentation simplicity, we consider here only parameter-invariant filters Ws and Wks . z1 (k, s)

Ws

G

Wks r(k, s)

Figure 1. Time-/space-varying interconnected systems in LFT representation. Assume the well-posedness of the LFT representation (8) and ¯ p Θ)−1 for all Θ ∈ Θ. The (9), i.e., the existence of (I − D LPV model (5) can be recovered by applying the lower LFT definition, i.e., � � � � � � ¯0 + B ¯0 . ¯ p Θ)−1 C¯0 −D ¯p Θ(I − D A¯ −B¯ = A¯0 −B (10) Note that in the rest of the paper, a calligraphic uppercase letter denotes a gain-scheduled matrix, whereas a regular uppercase letter denotes a constant matrix. 3.2 Fixed-Structure Controller A fixed-structure controller design (as opposed to full-order controller) that inherits the interconnected properties of the plant is considered here. The controller dynamics defined at the subsystem level can be expressed as Ak (θtk , θsk , zt , zs )u(k, s)+B k (θtk , θsk , zt , zs )e(k, s) = 0. (11) In case of a reference tracking problem, the control error is defined as e(k, s) = r(k, s) − y(k, s), where the reference is given as r(k, s). Controller polynomials Ak and B k are scheduled by controller scheduling parameters θtk and θsk as k

k

k

A =1+

na �

ma �

ak(ik ,is ) (θtk , θsk )zt−ik zs−is ,

(12a)

ik =1 is =−mk a nk b k

B =

�

k

mb �

bk(jk ,js ) (θtk , θsk )zt−jk zs−js .

(12b)

jk =1 js =−mk b

The controller scheduling block can be chosen as a copy or a function of the plant scheduling block. This issue will be further discussed in Section 4. In this context, fixed-structure means that the polynomial orders nka , mka , nkb , and mkb , as well

e(k, s)

K(θtk , θsk , zt , ts )

u(k, s)

z2 (k, s)

y(k, s)

G(θt , θs , zt , ts )

Figure 2. Generalized plant of a controlled subsystem. By pulling the plant and controller scheduling blocks out as shown in Fig. 3, the generalized plant in LFT form is expressed as  ξ (k, s)  y ¯  ξu (k, s)  ¯ ¯ A0 −B0 0 0 0 Bp 0 0 0   ¯ k A¯k 0 ¯k 0 B ¯ k 0 0 ξz1 (k, s) 0 −B B 0 0 0 p   ξ (k, s)  ¯s   ¯ s 0 0 0 0  z2 0 A¯s 0 −B B   = 0, ξ (k, s) r   ks ks ¯ ¯  0 −A 0 0 0 0 0  p(k, s)   0 B   C¯ −D  ¯0 0 ¯ p 0 −I 0  k  0 0 D 0   p (k, s) k ¯k k k ¯ ¯ ¯   D0 C0 0 0 −D0 0 Dp 0 −I q(k, s) q k (k, s) (14) with the scheduling channel partitioned as � � � � p(k, s) q(k, s) =Υ k , (15) pk (k, s) q (k, s)

and Υ ∈ Υ, where Υ := diag{Θt , Θs , Θkt , Θks }, Υ is an augmented compact set defined as Υ := diag{Θt , Θs , Θkt , Θks }. Define a matrix M as  ¯0 A¯0 −B ¯ k A¯k B 0 �  ¯0s � B 0 M11 M12 = M= ¯ ks  0 B M21 M22   C¯0 −D ¯0 ¯ k C¯ k D 0

0

0 0 0 ¯k 0 0 −B 0 s ¯ ¯s A 0 −B 0 −A¯ks 0 0 0 0 ¯k 0 0 −D 0

¯p B 0 0 0 ¯p D 0

The lower LFT computation recovers R in (13), i.e., R(Υ)ξ(k, s) = (Υ ⋆ M )ξ(k, s) = 0,

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Proceedings of the 20th IFAC World Congress 11412 Qin Liu et al. / IFAC PapersOnLine 50-1 (2017) 11409–11414 Toulouse, France, July 9-14, 2017

�

Θt Θs−1

�

�

G

Θt Θs

�

G

Θkt

Θks−1

�

K �

Θkt

Θks

K �

�

Θkt

Θks+1

� Pt 0 P =P = , Pt > 0, det(Ps ) �= 0, (20a) 0 Ps    P 0 Π3   Π4   0 −P    I 0   Π5   [∗]    Π  < 0, (20b) 2 0 −γ I  6    0 F (Υ)  I  R(Υ) F T (Υ) 0 with Π3 = diag{Π2t , Π2+ , Π1− }, Π4 = diag{Π1t , Π1+ , Π2− }, � �T Π5 ξ(k, s) = z1T (k, s) z2T (k, s) and Π6 ξ(k, s) = r(k, s). Proof 1. Let the Lyapunov function candidate be chosen as ∞ � V (k) = (21) xTt (k, s)Pt xt (k, s). T

x− (k, s)

x− (k, s − 1)

K

Θs+1

�

G

x+ (k, s + 1)

x+ (k, s)

�

� Θt

�

Figure 3. Closed-loop system in LFT form. where ⋆ denotes the star product of two matrices, i.e., Υ ⋆ M := M11 + M12 Υ(I − M22 Υ)−1 M21 . In (14), each signal vector ξi (k, s) (i = y, u, z1, z2 , r) can be partitioned into three parts as �T � T T T (k, s) ξi,+ (k, s) ξi,− (k, s) , (18) ξi (k, s) = ξi,t

where ξi,t (k, s) denotes the temporal variables, ξi,+ (k, s) and ξi,− (k, s) denote the positive and negative spatial variables received from neighboring subsystems, respectively. A statelike vector x(k, s) ∈ Rnx can then be extracted from ξ(k, s) as � � � � xt (k, s) xt (k + 1, s) x(k, s) = x+ (k, s) , ∆x(k, s) = x+ (k, s + 1) , x− (k, s − 1) x− (k, s) where xt , x+ and x− denote the temporal, the positive and negative spatial states, respectively. Subsystems exchange information through spatial states as shown in Fig. 3. The augmented shift operator ∆ is defined as ∆ = diag{zt , zs , zs−1 }. Selection matrices Π1 = diag{Π1t , Π1+ , Π1− } and Π2 = diag{Π2t , Π2+ , Π2− } are properly determined such that x(k, s) = Π1 ξ(k, s), ∆x(k, s) = Π2 ξ(k, s).

�

s=−∞

The positive-definiteness of the Lyapunov function V (k) requires Pt > 0. Stability of the closed-loop system is assured if the summed energy of the overall system dissipates, i.e., V (k + 1) − V (k) < 0. (22) It can be easily derived that the summed spatial energy over space cancels out (see Wu (2003)), i.e., � � �� � ∞ � � x+ (k, s) x (k, s + 1) [∗] Ps + = 0, −[∗] Ps x− (k, s − 1) x− (k, s) s=−∞

(23) where Ps is indefinite due to the spatial non-causality, i.e., Ps has both positive and negative eigenvalues.

Thus, (22) is equivalent to � �� � � ∞ � xt (k + 1, s) xt (k, s) � x+ (k, s) [∗] P x+ (k, s + 1) − [∗] P x− (k, s − 1) x− (k, s) s=−∞ � �� � � ∞ � � P 0 Π3 [∗] ξ(k, s) < 0. (24) = 0 −P Π4 s=−∞

4. CONTROLLER SYNTHESIS CONDITION In this section, analysis and synthesis conditions defined at the subsystem level are derived. Compared to a centralized control scheme, the computational complexity of distributed controller design is reduced to the size of a single subsystem. With the application of the full block S-procedure, it will be shown that it suffices to check a finite number of constraints within the parameter set. 4.1 Analysis Condition Quadratic performance of the global system is taken as the performance measure. An asymptotically stable system is said to have quadratic performance γ (γ > 0) if �� � � ∞ � ∞ � −γI 0 r(k, s) ≤ 0. (19) [∗] 0 1 I z(d, s) γ k=1 s=−∞

The analysis condition that establishes stability and quadratic performance of the closed-loop system is formulated as follows. Theorem 1. A system in the form of (17) is asymptotically stable and has quadratic performance γ, if there exist a structured matrix P and a gain-scheduled matrix F (Υ), such that for any Υ ∈ Υ,

After incorporating quadratic performance into the stability by applying the S-procedure, the LMI representation of the bounded real lemma at the subsystem level can be formulated as    P 0 Π3  0 −P  Π4  [∗]  ξ(k, s) < 0. (25) I 0  Π5  Π6 0 −γ 2 I Given a fixed value of Υ, the equality (17) and inequality (25) are components of item i) in Finsler’s Lemma. Invoking its equivalent statement item ii) implies that there exists a matrix F(Υ) for each fixed Υ, such that (20b) holds. 4.2 Synthesis Conditions Although Theorem 1 establishes asymptotic stability and quadratic performance of the closed-loop system, condition (20b) requires checking an infinite number of constraints inside the variation space Υ. Assuming a rational dependence of F (Υ) on Υ, its quadratic LFT form is written as F(Υ) = T T (Υ)Fˆ TF (Υ) (26) F

where Fˆ denotes a constant matrix. Under the assumption of well-posedness, the gain-scheduled matrix TF (Υ) can be expressed as TF (Υ) = TF 11 + TF 12 Υ(I − TF 22 Υ)−1 TF 21 . (27)

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The synthesis condition is derived with the application of the full block S-procedure as follows: Theorem 2. Consider a distributed LPV/LFT plant model in the form of (8) and (9). There exists a fixed-structure controller which guarantees well-posedness, asymptotic stability and quadratic performance γ of the closed-loop system in the form of (14) and (15), if there exist a structured matrix P , a matrix Fˆ , and a multiplier X such that � � Pt 0 T P =P = , Pt > 0, det(Ps ) �= 0, (28a) 0 Ps  P 0  0 −P  I 0   0−γ 2 I [∗]   0 Fˆ   Fˆ 0

X

� � Ω ≥ 0, [∗] X I T

(28c)

(29)

where

S11 S21

� I 0 0 , = 0 TF 11 0 0 0 TF 11 � � 0 TF 21 0 , = 0 0 TF 21

Fixed-structure controller design results in BMI constraints (28b). A DK iteration based approach has demonstrated its efficiency in Wollnack and Werner (2015a) and can be applied: 1. Given a stabilizing controller (via initialization at the first iteration, or found in Step 2), optimize γ over P , Fˆ and X; 2. Given P , Fˆ and X found in Step 1, optimize γ over the controller parameters; 3. Go back to Step 1 until γ is minimized.

(28b)

with Ω defined as Ω := diag{Υ, Υ, Υ}, and T11 = S11 N11 , T12 = [S12 S11 N12 ] , � � � � S N S S N T21 = 21 11 , T22 = 22 21 12 , N21 0 N22 �

With the use of (D, G)-scaling, under the assumption that |θti | < 1 and |θsi | < 1, the multiplier condition (28c) is then trivially fulfilled. It suffices to consider only (28a) and (28b) to solve for the desired controller.



    T11 T12   0 I  < 0,  T T  21 22 

�

0

0 0

�

S12 = TF 12 , 0 TF 12 � � 0 T S22 = F 22 , 0 TF 22

(30) (31)

� � T T , N21 = M21 , (32) N11 = ΠT3 ΠT4 ΠT5 ΠT6 I M11 � � T T N12 = 0 0 0 0 0 M12 , N22 = M22 . (33) Proof 2. The proof follows by applying the full Block Sprocedure to (20b), and is omitted due to lack of space. Note that the application of the full block S-procedure separates the augmented scheduling block Υ from (20b), such that the main condition (28b) is parameter independent, whereas the multiplier condition (28c) still involves an infinite number of constraints in the parameter set. Imposing additional constraints on the multiplier X can reduce (28c) into a finite number of constraints at the price of increased conservatism. With X22 > 0 and the controller scheduling block being a copy of the plant one, i.e., Θkt = Θt and Θks = Θs , (28c) is convexified. It is then sufficient to only check (28c) at the vertices of Ω. Furthermore, condition (28c) can be rendered trivially fulfilled when (D, G)scaling (see Dettori and Scherer (2001)) is used. This is realized by imposing commutability between Ω and components of X, i.e., X ∈ X , where the multiplier set X is defined as � � � � � X11 X12 X = X ∈ R2nΥ ×2nΥ ��X = X T = , T X12 X22

t,k s,k t s t , Xij , Xij , Xij }, Xij ∈ RnΘt ×nΘt , Xij = diag{Xij t,k s Xij ∈ RnΘs ×nΘs , Xij ∈R

nΘk ×nΘk t

t

s,k , Xij ∈R

T Xij Ω = ΩXij , i, j = 1, 2, X11 = −X22 < 0, X12

nΘk ×nΘk

s , � = −X12 . s

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5. NUMERICAL EXAMPLE To demonstrate the performance of the proposed distributed control design method, the heat equation of one spatial dimension in Wollnack and Werner (2015a) is taken as a numerical example, i.e., ∂y(k, s) ∂ 2 y(k, s) − κ(θs ) = u(t, s), (34) ∂t ∂x2 where y(k, s) denotes the temperature, and u(k, s) denotes the control input. The thermal diffusivity κ(θs ) varies with respect to spatial scheduling parameter θs . The application of the finite difference method to (34) discretizes the PDE both in time and space. Let the sampling time be chosen as Tt = 0.01 s and the overall system of length 10 m be spatially discretized into 41 subsystems, i.e., s = {−20, . . . , 20}. The resulting sampling space is then Ts = 0.25 m. The scheduling parameter θs is scheduled by the spatial variable s � � 3 3 (35) θs = tanh( s − 10) + tanh(− s − 10) + 2 /2. 4 4 The thermal diffusivity has a rational functional dependence on θs , i.e., κ(θs ) = θs + θs2 . (36) The LFT formulation of the plant subsystem model gives ¯p = [1 1] , ¯0 = [0 Tt 0 0] , B A¯0 = [1 −1 0 0] , B � � � � 0 2α −α −α ¯ 0000 ¯ ¯ , D0 = , Dp = [0 0] , (37) C0 = 0 0 0 0 0000 with α =

Tt Ts2

and Θs = diag{θs , θs }.

The distributed controller is assumed to have the following predefined structure � � A¯k (θs ) = 1 ak(1,0) (θs ) ak(1,−1) (θs ) ak(1,1) (θs ) , � � k ¯ k (θs ) = bk (θs ) bk (θs ) bk B (0,0) (1,0) (1,−1) (θs ) b(1,1) (θs ) .

A fixed-structure distributed controller of affine dependence k,1 on θs is considered here, e.g., ak(1,0) (θs ) = ak,0 (1,0) + θs a(1,0) , k,1 where ak,0 (1,0) and a(1,0) are unknown constants and needed to be determined in controller design. Its LFT representation is written�as � � � k,0 k,0 k,0 ¯0k = bk,0 bk,0 bk,0 bk,0 , A¯k0 = 1 a(1,0) a(1,0) a(1,0) , B (0,0) (1,0) (1,0) (1,0) � � k,1 k,1 k,1 k k ¯ ¯ k = 0, ¯ C0 = − 0 a(1,0) a(1,0) a(1,0) , Bp = −1, D p � � k,1 k,1 k,1 k,1 k ¯ =− b D 0 (0,0) b(1,0) b(1,0) b(1,0) ,

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and Θks = θs , i.e., the controller scheduling block is part of the plant scheduling block. Further assume that the matrix F (Υ) is affine in Υ with Υ = diag{Θs , Θks }, i.e., F (Υ) = F0 + ΥF1 . Its quadratic LFT representation in the form of (26) is given by   � � � � 0 I T /2 T 0 F 1 F 11 F 12 , =  I 0 . Fˆ = F1 /2 F0 TF 21 TF 22 I 0 Moreover, (D, G)-scaling has been imposed on the multiplier.

The performance of the proposed controller design approach is evaluated by tracking given reference inputs. Between 0.1 s and 0.3 s, a unit step is given as the reference at subsystems 18 to 23, whereas between 0.5 s and 0.7 s, the reference is switched to -2. Fig. 4 shows a comparison between the given reference and the closed-loop responses at subsystem 21, whereas Fig. 5 shows the 3-D plot of the closed-loop response and the control input. It can be seen that the controller demonstrates a satisfactory performance in terms of reference tracking. (a)

y(k, 21)

1 0 -1 -2 (b)

u(k, 21)

1

0

-1 0

0.2

0.4

0.6

0.8

1

time (s)

Figure 4. (a): comparison between the reference (red solid) and closed-loop responses (blue dashed) at the subsystem 21. (b): the control input at the subsystem 21. 1

u(k, s)

y(k, s)

1 0 -1 -2 20 0 -20 spatial variable s

0

0.5 time (s)

0

-1 1 20

0 -20

spatial variable s

0

0.5

1

time (s)

Figure 5. 3-D plot of the closed-loop response (left) and the control input (right). 6. CONCLUSION In this work, a fixed-structure distributed controller design approach for parameter-varying distributed systems has been developed. We have shown that provided a rational dependence on the scheduling parameters, the implicit I-O representation which describes the localized dynamics at the subsystems can

be characterized in LPV/LFT form and leads to subsystembased synthesis conditions. The designed controller has been evaluated on a spatially-varying heat equation. Stability and a satisfactory performance in terms of reference tracking have been achieved. REFERENCES Bamieh, B., Paganini, F., and Dahleh, M.A. (2002). Distributed control of spatially invariant systems. IEEE Transactions on Automatic Control, 47(7), 1091–1107. D’Andrea, R. and Dullerud, G.E. (2003). Distributed control design for spatially interconnected systems. IEEE Transactions on Automatic Control, 48(9), 1478–1495. de Oliveira, M.C. and Skelton, R.E. (2001). Stability tests for constrained linear systems. In S.O.R. Moheimani (ed.), Perspectives in Robust Control, volume 268 of Lecture Notes in Control and Information Sciences. Springer London. Dettori, M. and Scherer, C.W. (2001). LPV Design for a CD Player: an Experimental Evaluation of Performance. In Proceedings of 40th IEEE Conference on Decision and Control. Jovanovic, M. (2004). Modeling, Analysis, and Control of Spatially Distributed Systems. Ph.D. thesis, University of California, Santa Barbara, CA. Liu, Q. and Werner, H. (2013). Experimental Identification of Spatially-Interconnected Parameter-Invariant and LPV Models for Actuated Beams. In Proceedings of 52nd IEEE Conference on Decision and Control. Liu, Q. and Werner, H. (2016). Distributed identification and control of spatially interconnected systems with application to an actuated beam. Control Engineering Practice, 54, 104– 116. Scherer, C.W. (2001). LPV control and full block multipliers. Automatica, 37(3), 361–375. Shamma, J.S. (1988). Analysis and Design of Gain Scheduled Control Systems. Ph.D. thesis, Massachusetts Institute of Technology. Stewart, G.E. (2000). Two-Dimensional Loop Shaping Controller Design for Paper Machine Cross-directional Processes. Ph.D. thesis, University of British Columbia, Vancouver, Canada. T´oth, R., Abbas, H.S., and Werner, H. (2012). On the statespace realization of LPV input-output models: Practical approaches. IEEE Transactions on Control Systems Technology, 20(1), 139–153. Wilfried, G., Didier, H., Bernussou, J., and Boyer, D. (2007). Polynomial LPV synthesis applied to turbofan engines. In 17th IFAC Symposium on Automatic Control in Aerospace. Wollnack, S., Abbas, H.S., T´oth, R., and Werner, H. (2013). Fixed-Structure LPV Controller Synthesis Based on Implicit Input-Output Representations. In Proceedings of 52nd IEEE Conference on Decision and Control. Wollnack, S. and Werner, H. (2015a). Distributed fixedstructure control of spatially interconnected LTSV systems. In Proceedings of 54th IEEE Conference on Decision and Control. Wollnack, S. and Werner, H. (2015b). Stability and L2 performance analysis of LPV-IO systems using parameterdependent Lyapunov functions. In Proceedings of American Control Conference. Wu, F. (2003). Distributed control for interconnected linear parameter-dependent systems. IEE Proceedings - Control Theory and Applications, 150(5), 518–527.

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