Optimal Sensor and Actuator Scheduling in Sampled-Data Control of Spatially Distributed Processes⁎

Proceedings, 10th IFAC International Symposium on Proceedings, 10th of IFAC International Symposium on Advanced Control Chemical Processes Proceedings, 10th IFAC International Symposium on Proceedings, 10th of IFAC International Symposium on Advanced Control Chemical Processes Shenyang, Liaoning, July 25-27, 2018 online at www.sciencedirect.com Advanced Control of China, Chemical Processes Available Advanced of Chemical Processes Shenyang,Control Liaoning, China, July 25-27, 2018 Proceedings, 10th IFAC International Symposium on Shenyang, Liaoning, China, July 25-27, 2018 Shenyang, Liaoning, China, July 25-27, 2018 Advanced Control of Chemical Processes Shenyang, Liaoning, China, July 25-27, 2018

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IFAC PapersOnLine 51-18 (2018) 327–332

Optimal Sensor and Actuator Scheduling Optimal Sensor and Actuator Scheduling Optimal Sensor and Actuator Scheduling Optimal Sensor andControl Actuator Scheduling Sampled-Data of Spatially Sampled-Data Control of Spatially Optimal Sensor and Actuator Scheduling Sampled-Data Control of ⋆⋆ Sampled-Data Control of Spatially Spatially Distributed Processes ⋆⋆ Distributed Processes Sampled-Data Control of Spatially Distributed Processes Distributed Processes Da Xue and Nael H. El-Farra ∗∗⋆ Distributed Processes

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Da Xue and Nael H. El-Farra ∗∗ Da Da Xue Xue and and Nael Nael H. H. El-Farra El-Farra ∗ ∗ of California, Davis,H.CA 95616 USA (e-mail: ∗ UniversityDa Xue and Nael El-Farra ∗ University of California, Davis, CA 95616 USA (e-mail: Davis, CA 95616 USA (e-mail: ∗ University of California, [email protected]) University of California, Davis, CA 95616 USA (e-mail: [email protected]) [email protected]) ∗ [email protected]) University of California, Davis, CA 95616 USA (e-mail: Abstract: This work presents [email protected]) optimization-based methodology for the placement and Abstract: This This work work presents presents an an optimization-based optimization-based methodology methodology for for the placement placement and and Abstract: scheduling of measurement sensors and control actuators in spatially-distributed processes Abstract: This work presents an optimization-based methodology for the the placement and scheduling of measurement sensors and control actuators in spatially-distributed processes scheduling of measurement and in with low-order dynamics andsensors discretely-sampled outputmethodology measurements. a processes sampledscheduling of measurement and control control actuators actuators in spatially-distributed spatially-distributed Abstract: This work presents an optimization-based for Initially, the placement and with low-order low-order dynamics andsensors discretely-sampled output measurements. measurements. Initially, aa processes sampledwith dynamics and discretely-sampled output Initially, sampleddata observer-based controller, with an inter-sample modelmeasurements. predictor, is designed based on an with low-order dynamics and discretely-sampled output Initially, a sampledscheduling of measurement sensors and control actuators in spatially-distributed processes data observer-based controller, with an inter-sample model predictor, is designed based on an data observer-based controller, with inter-sample model predictor, based on approximate finite-dimensional system captures the infinite-dimensional system’s data observer-based controller, with an anthat inter-sample model predictor, is is designed designed based on an an with low-order dynamics and discretely-sampled output measurements. Initially, a dominant sampledapproximate finite-dimensional system that captures the infinite-dimensional system’s dominant approximate system captures the infinite-dimensional system’s dominant dynamics. Anfinite-dimensional explicitcontroller, characterization ofinter-sample the interdependence between is thedesigned stabilizing locations approximate finite-dimensional system that captures the infinite-dimensional system’s dominant data observer-based with anthat model predictor, based on an dynamics. An explicit characterization of the interdependence between the stabilizing locations dynamics. An explicit characterization of interdependence between the stabilizing locations of the sensors and actuators andsystem the maximum allowable sampling period obtained. Based on dynamics. Anfinite-dimensional explicit characterization of the the interdependence between theis locations approximate that captures the infinite-dimensional system’s dominant of the the sensors sensors and actuators and the the maximum maximum allowable sampling period isstabilizing obtained. Based on of and actuators and allowable sampling period obtained. Based on this characterization, acharacterization constrained finite-horizon optimization problem is is formulated to obtain of the sensors and actuators and the maximum allowable sampling period is obtained. Based on dynamics. An explicit of the interdependence between the stabilizing locations this characterization, a constrained finite-horizon optimization problem is formulated to obtain this characterization, a finite-horizon optimization problem to obtain thethe sensor andand actuator locations, theallowable corresponding sampling that optimally this characterization, a constrained constrained finite-horizon optimization problem isperiod, formulated to obtain of sensors actuators and thetogether maximum sampling periodis isformulated obtained. Based on the sensor sensor and actuator actuator locations, together the corresponding corresponding sampling period, that optimally optimally the and locations, together the sampling period, that balance the tradeoff between the control performance requirements on the one hand, and the the sensor and actuator locations, together the corresponding sampling period, that optimally this characterization, a constrained finite-horizon optimization problem is formulated toand obtain balance the tradeoff between the control performance requirements on the one hand, the balance the tradeoff between the performance requirements on one hand, and the demand forand reduced sampling, on control the other. Thecorresponding objective function penalizes both the control balance the tradeoff between the performance requirements on the the one hand, and the the sensor actuator locations, together the sampling period, that optimally demand for reduced reduced sampling, on control the other. The objective objective function penalizes both the control demand for sampling, on the other. function penalizes both the control performance cost, expressed in terms of performance theThe response speed and the control effort, and the demand for reduced sampling, on the other. The objective function penalizes both the control balance the tradeoff between the control requirements on the one hand, and the performance cost, expressed in terms of the response speed and the control effort, and the performance cost, expressed in terms of the response speed and the control effort, and the samplingfor cost, expressed in terms ofthe theother. sampling frequency. The optimization problem is solved performance cost, expressed in terms of the response speed and the control effort, and the demand reduced sampling, on The objective function penalizes both the control sampling cost, cost, expressed expressed in in terms of of the sampling sampling frequency. frequency. The The optimization optimization problem problem is solved sampling is solved in a receding horizon fashion, athe dynamic policy that varies sensor and actuator sampling cost,cost, expressed in terms terms of the theto sampling frequency. The optimization problem is solved performance expressed inleading terms of response speed and the the control effort, and the in aa receding receding horizon fashion, leading to a dynamic dynamic policy that varies the sensor and actuator actuator in horizon fashion, leading to a policy that varies the sensor and spatial placement, together with the sampling period, over time. The developed methodology is in a receding fashion, leading tosampling a dynamic policy that the sensor and actuator sampling cost,horizon expressed in terms of the frequency. The varies optimization problem is solved spatial placement, together with the sampling period, over time. The developed methodology is spatial placement, together with the period, over time. The developed methodology is illustrated through anfashion, application tosampling ato simulated diffusion-reaction process example. spatial placement, together with the sampling period, over time. The developed methodology is in a receding horizon leading a dynamic policy that varies the sensor and actuator illustrated through an application to a simulated diffusion-reaction process example. illustrated through an to aa simulated diffusion-reaction process example. illustrated through an application application toofsampling simulated diffusion-reaction process example. spatial together with the period, over time.byThe developed © 2018, placement, IFAC (International Federation Automatic Control) Hosting Elsevier Ltd. Allmethodology rights reserved.is Keywords: through Sampled-data control, sensor and actuator placement, receding horizon illustrated an application to a simulated diffusion-reaction process example. Keywords: Sampled-data control, sensor and actuator placement, receding horizon Keywords: Sampled-data Sampled-data control, sensor and and actuator actuator placement, receding receding horizon horizon optimization, scheduling, control, spatially-distributed systems.placement, Keywords: sensor optimization, scheduling, spatially-distributed systems. optimization, scheduling, spatially-distributed systems. optimization, scheduling, control, spatially-distributed systems.placement, receding horizon Keywords: Sampled-data sensor and actuator 1. INTRODUCTION proach was to guarantee closed-loop stability with minimal optimization, scheduling, spatially-distributed systems. 1. INTRODUCTION proach was to guarantee closed-loop stability with minimal 1. proach was to stability minimal sampling frequency. Thisclosed-loop is an appealing goalwith in situations 1. INTRODUCTION INTRODUCTION proach was to guarantee guarantee closed-loop stability with minimal sampling frequency. This is an appealing goal in situations sampling frequency. This is an appealing goal in situations where technological constraints on the measurement sensDespite the extensive body of research work on control sampling frequency. This is an appealing goal in situations 1. INTRODUCTION proach was to guarantee closed-loop stability with minimal technological constraints on the measurement sensDespite the the extensive extensive body body of of research research work work on on control control where where technological constraints on the measurement sensDespite ing techniques make it difficult or costly to collect measureof spatially-distributed systems, the problem of designing where technological constraints on the measurement sensfrequency. This is an or appealing goal in situations Despite the extensive body of research workofon control sampling ing techniques make it difficult costly to collect measureof spatially-distributed systems, the problem designing ing techniques make it difficult or costly to collect measureof spatially-distributed systems, the problem of designing ments frequently. To this end, an exact characterization sampled-data feedback controllers for spatially-distributed ing techniques make it difficult or costly to collect measuretechnological constraints on the measurement sensof spatially-distributed systems, the problem designing Despite the extensive body of research workofon control where ments frequently. To this end, an exact characterization sampled-data feedback controllers for spatially-distributed ments frequently. To end, exact sampled-data feedback controllers for spatially-distributed the maximum allowable sampling period required for processes has received attention (e.g., see of ments frequently. Toit this this end, an exact characterization ing techniques make difficult oran costly to characterization collect measuresampled-data feedbackrelatively controllers forproblem spatially-distributed of spatially-distributed systems,limited the of designing of the maximum allowable sampling period required for processes has received relatively limited attention (e.g., see of the maximum allowable sampling period required for processes has received relatively limited attention (e.g., see stabilization was obtained in the earlier studies and found Logemann et al. (2005); Cheng et al. (2009); Fridman and of the frequently. maximum allowable sampling period required for To this in end, an exact characterization processes has received relatively limited attention (e.g.,and see ments sampled-data feedback controllers for (2009); spatially-distributed stabilization was obtained the earlier studies and found Logemann et al. (2005); Cheng et al. Fridman stabilization was obtained in the earlier studies and found Logemann et al. (2005); Cheng et al. (2009); Fridman and to depend on the spatial placement of the measurement Blighovsky (2010) for some notable exceptions). This is an stabilization was obtained in the earlier studies and found of the maximum allowable sampling period required for Logemann et al. (2005); Cheng et al. (2009); Fridman and processes has received relatively limited attention (e.g., see to depend on the spatial placement of the measurement Blighovsky (2010) (2010) for for some some notable notable exceptions). exceptions). This This is an an to depend the spatial placement of the Blighovsky sensors andon control actuators. Anearlier implication ofand thisfound findimportant problem given the prevalent use ofFridman digital depend on theobtained spatial placement of studies the measurement measurement stabilization was in the Blighovsky (2010) for some notable This is issenan to Logemann et al. (2005); Cheng et al.exceptions). (2009); and sensors and control actuators. An implication of this findimportant problem given the prevalent use of digital sensensors and control actuators. An of this findimportant problem given the prevalent use of digital digital senis that one could use this characterization identify sors and controllers insome industrial control systems. To date, sensors andon control actuators. An implication implication ofto this findto depend the spatial placement of the measurement important problem given the prevalent use of Blighovsky (2010) for notable exceptions). This issenan ing ing is that one could use this characterization to identify sors and and controllers controllers in industrial industrial control systems. To date, date, ing is that one could use this characterization to identify sors in control systems. To the set of feasible sensor and actuator locations that can the overwhelming majority of studies on the analysis and ing is that one could use this characterization to identify sensors and control actuators. An implication of this findsors and controllers in industrial control systems. To date, important problem given the prevalent use of digital senset of feasible sensor and actuator locations that can the overwhelming overwhelming majority majority of of studies studies on on the the analysis analysis and and the the set of feasible sensor and actuator locations that can the achieve stabilization with minimal sampling frequencies. design of sampled-data control systems have focused prithe setthat of feasible sensor actuator locations can ing is one could use and this characterization tothat identify the overwhelming majority of studies onsystems. the analysis and sors andofcontrollers in industrial control To date, achieve stabilization with minimal sampling frequencies. design sampled-data control systems have focused priachieve stabilization with minimal sampling frequencies. design of sampled-data control systems have focused priWhile this is important from a sampling cost savings marily on lumped parameter systems modeled by ordinary achieve stabilization with minimal sampling frequencies. the set of feasible sensor and actuator locations that can design of sampled-data control systems focused pri- While this is important from a sampling cost savings the overwhelming majority ofsystems studies onhave the analysis and marily on lumped parameter modeled by ordinary While this is from aa sampling cost savings marily on lumped parameter systems modeled by ordinary standpoint, theimportant resulting actuator and sensor placement differential equations (e.g., see Chen and Francis (1995); While this is important from sampling cost savings achieve stabilization with minimal sampling frequencies. marily on lumped parameter systems modeled by ordinary design of sampled-data control systems have focused prithe resulting actuator and sensor placement differential equations equations (e.g., (e.g., see see Chen Chen and and Francis (1995); (1995); standpoint, standpoint, the resulting actuator and placement differential not the impact discrete measurement Nesic and Grune (2005); Husee et Chen al. (2007); Naghshtabrizi standpoint, theimportant resulting actuator and sensor sensor placement While thisconsider is from of a sampling cost savings differential equations (e.g., and Francis Francis (1995); does marily on lumped parameter systems modeled by ordinary does not consider the impact of discrete measurement Nesic and Grune (2005); Hu et al. (2007); Naghshtabrizi does not consider the impact of discrete measurement Nesic and Grune (2005); Hu et al. (2007); Naghshtabrizi sampling on closed-loop performance. It is well known, for et al. (2008); Fujioka (2009); Hu and El-Farra (2011)). does not on consider the impact of discrete measurement the resulting actuator and sensor placement Nesic and Grune (2005); Husee et Chen al. (2007); Naghshtabrizi differential equations (e.g., and Francis (1995); standpoint, sampling closed-loop performance. It is well known, for et al. (2008); Fujioka (2009); Hu and El-Farra (2011)). sampling on closed-loop performance. It is well known, for et al. (2008); Fujioka (2009); Hu and El-Farra (2011)). example, that faster sampling rates are typically required sampling on closed-loop performance. It is well known, for does not consider the impact of discrete measurement et (2008); Fujioka (2009); and(2007); El-Farra (2011)). for example, that faster sampling rates are typically required Nesic andtoGrune (2005); Hu Hu et al. Naghshtabrizi Anal. effort address the sampled-data control problem example, that faster sampling rates are typically required to enhance the control system performance. Ultimately, An effort to address the sampled-data control problem for example, that faster sampling rates are typically required sampling on closed-loop performance. It is well known, for et al. (2008); Fujioka (2009); Hu and El-Farra (2011)). enhance the control system performance. Ultimately, An effort sampled-data control for spatially-distributed systems was undertaken in a series to enhance the control system performance. Ultimately, An effort to to address address the the sampled-data control problem problem for to there is a that need to resolve the inherent conflict that arises spatially-distributed systems was undertaken undertaken in aa series series to enhance the control system performance. Ultimately, example, faster sampling rates are typically required there is a need to resolve the inherent conflict that arises spatially-distributed systems was in of earlier (e.g., Yao and (2011, there is need to resolve the inherent conflict that arises spatially-distributed systems wasEl-Farra undertaken in a2012)). series An effort toworks address the sampled-data control problem for between tight restrictions sampling ratesUltimately, which are of earlier earlier works (e.g., Yao and El-Farra (2011, 2012)). there is aathe need to resolvesystem the on inherent conflict that arises to enhance the control performance. the tight restrictions on sampling rates which are of (e.g., and El-Farra (2011, Theearlier main works idea was toYao include an approximate finitebetween the tight restrictions on sampling rates which are of works (e.g., Yao and El-Farra (2011, 2012)). spatially-distributed systems was undertaken in a2012)). series between required toneed maintain the the control system performance on The main idea was to include an approximate finitebetween the tight restrictions on sampling rates which are there is a to resolve inherent conflict that arises required to maintain the control system performance on The main idea was to include an approximate finitedimensional model of the infinite-dimensional system in required to maintain the control system performance on The main idea was to include an approximate finiteof earlier works (e.g., Yao and El-Farra (2011, 2012)). the one hand, and the usual demand for limited sampling dimensional model model of of the the infinite-dimensional infinite-dimensional system system in in between required to maintain the control system performance on the tight restrictions on sampling rates which are the one hand, and the usual demand for limited sampling dimensional the controller to provide estimates of the dominant slow the one hand, and the usual demand for limited sampling dimensional model of to theinclude infinite-dimensional system in frequency The main idea was anofapproximate finitewhich is desired to minimize measurement costs. the controller to provide estimates the dominant slow the one hand, and the usual demand for limited sampling required to maintain the control system performance on frequency which is desired to minimize measurement costs. the controller to provide estimates of the dominant slow states that compensate the unavailability of output which is to measurement costs. the controller to provide estimates of the dominant slow dimensional model of thefor infinite-dimensional system in frequency states that compensate compensate for the unavailability unavailability of output output frequency which is desired desired to minimize minimize measurement costs. the one hand, and theconsiderations, usual demand for limited sampling states that for the of Motivated by these we present in this measurements between sampling times, and to update the states that compensate for the times, unavailability of output the controller to provide estimates of the slow Motivated by these considerations, we present in this measurements between sampling anddominant to update update the frequency which is desired to minimize measurement costs. Motivated by in measurements between sampling times, and to the an optimization-based approachwe for present the integration model states using the actual measurements atupdate each samMotivated by these these considerations, considerations, we present in this this measurements between sampling times, and to the work states that compensate for the unavailability of output work an optimization-based approach for the integration model states using the actual measurements at each samwork an optimization-based approach for the integration model states the measurements at each of control and actuator/sensor scheduling spatiallypling time. A using key objective of this model-based control ap- work an optimization-based approach for present thein integration Motivated by these considerations, we in this model states using the actual actual measurements atupdate each samsammeasurements between sampling times, and to the of control and actuator/sensor scheduling in spatiallypling time. A key objective of this model-based control apof control and actuator/sensor scheduling spatiallypling time. A key of model-based approcesses with discretely-sampled output mea⋆ of control and actuator/sensor scheduling in spatiallywork an optimization-based approach for thein integration pling time. A using key objective objective of this this model-based control ap- distributed model states actual measurements atcontrol each samFinancial support bythe NSF, CBET-1438456, is gratefully acknowldistributed processes with discretely-sampled output mea⋆ distributed processes with discretely-sampled output measurements. The objective is to simultaneously optimize Financial support by NSF, CBET-1438456, is gratefully acknowl⋆ distributed processes with discretely-sampled output meaedged. control and actuator/sensor scheduling in spatiallyFinancial by NSF, CBET-1438456, is gratefully acknowlpling time.support A key objective of this model-based control ap- of surements. The objective is to simultaneously optimize ⋆ Financial support by NSF, CBET-1438456, is gratefully acknowledged. surements. The objective is to simultaneously optimize edged. surements. processes The objective is to simultaneously optimize distributed with discretely-sampled output mea-

⋆ edged. Financial support by NSF, CBET-1438456, is gratefully acknowlsurements. The objective is to simultaneously Copyright © 2018, 2018 IFAC 321 Hosting edged. 2405-8963 IFAC (International Federation of Automatic Control) by Elsevier Ltd. All rights reserved. Copyright © © 2018 IFAC 321 Copyright ©under 2018 responsibility IFAC 321Control. Peer review of International Federation of Automatic Copyright © 2018 IFAC 321 10.1016/j.ifacol.2018.09.321 Copyright © 2018 IFAC 321

optimize

2018 IFAC ADCHEM 328 Shenyang, Liaoning, China, July 25-27, 2018 Da Xue et al. / IFAC PapersOnLine 51-18 (2018) 327–332

the control system performance and the sampling costs by dynamically scheduling the sensor and actuator locations, as well as the sampling period, subject to the appropriate stability constraints. The rest of the paper is organized as follows. Following some preliminaries in Section 2, an observer-based controller that relies on an inter-sample model predictor and a fixed sensor-actuator placement is introduced in Section 3, along with a parametric characterization of its closed-loop stability region. This characterization is incorporated as a constraint within a finitehorizon optimization problem that is formulated in Section 4, where the cost function includes explicit penalties on the response speed, the control action and the sampling frequency. A receding horizon strategy for implementing the optimization problem solution is then devised in Section 5 to determine the optimal placement and scheduling of the sensors and actuators and the optimal sampling rate. Finally, a simulation case study is presented in Section 6. 2. PRELIMINARIES We consider spatially-distributed processes modeled by highly-dissipative infinite-dimensional systems. An example of this class of systems, which is introduced in this section to illustrate the subsequent theoretical development, are systems described by parabolic PDEs: n ∑ ∂x ¯ ¯ ∂2x = α 2 + βx ¯+ω bi (z)ui ∂t ∂z i=1 (1) ∫ π yi (t) = qi (z)¯ x(z, t)dz, i ∈ {1, 2, · · · , l} 0

subject to the boundary and initial conditions: (2) x ¯(0, t) = x ¯(π, t) = 0, x ¯(z, 0) = x ¯0 (z) where x ¯(z, t) ∈ R is the process state variable, z ∈ [0, π] is the spatial coordinate, t ∈ [0, ∞) is the time, ui ∈ R is the i-th manipulated input, n is the number of manipulated inputs, bi (·) is the actuator distribution function, yi is the i-th measured output, qi (·) is the i-th sensor distribution function, and x ¯0 (z) is a smooth function of z. The PDE of (1), subject to the boundary and initial conditions of (2) can be formulated as an infinite-dimensional system with the following state-space representation: x(t) ˙ = Ax(t) + Bu(t), x(0) = x0 (3) y(t) = Qx(t)

where x(t) = x ¯(z, t), t > 0, z ∈ [0, π], is the state function defined on an appropriate Hilbert space, H = L2 (0, π), endowed with the proper inner product and norm; A is the differential operator; B and Q are the input and output operators, respectively; u = [u1 · · · un ]T , y = [y1 · · · yl ]T , and x0 = x ¯0 (z). Applying modal decomposition techniques, the infinite-dimensional system can be decomposed as follows: x˙ s = As xs + Bs u, xs (0) = Ps x0 (4) x˙ f = Af xf + Bf u, xf (0) = Pf x0 (5) (6) y = Qs xs + Qf xf where xs = Ps x is the state of a finite-dimensional system that describes the evolution of the slow (possibly unstable) eigenmodes; xf = Pf x is the state of an infinitedimensional system that captures the evolution of the fast (stable) eigenmodes; and Ps and Pf are the orthogonal projection operators, where As = Ps A, Bs = Ps B, Af = Pf A, and Bf = Pf B. To simplify the presentation of the 322

main results, we assume that the number of actuators and sensors is the same and equal to the number of the dominant modes (i.e., n = l = m). Based on the above decomposition, the following approximate finite-dimensional system which describes the temporal evolution of the slow (possibly unstable) eigenmodes can be derived using Galerkin’s method: a˙ s = As as + Bs (za )u(t), y¯s = Qs (zs )as (7) where as = [a1 · · · am ]T ∈ Rm ; ai is the amplitude of the i−th eigenmode; y¯s is the output; As is an m × m diagonal matrix of the form As = diag{λj } containing the slow eigenvalues in the spectrum of A; Bs is an m × m input matrix whose individual elements are parameterized by the locations of the control actuators za ; Qs is an invertible m × m output matrix whose individual elements are parameterized by the locations of the measurement sensors zs (the invertibility assumption can be ensured by proper selection of the sensors’ locations). The reduced-order system of (7) will be used in the next section to design a model-based output feedback controller that achieves stabilization using discretely-sampled output measurements. 3. OBSERVER-BASED CONTROL USING AN INTER-SAMPLE MODEL PREDICTOR We consider an output-feedback control structure in which the output measurements collected by the sensors are available to the controller only at discrete sampling times. A dynamic state observer is used to generate an estimate of the state which is used to compute the control action. Owing to the unavailability of output measurements between sampling times, an inter-sample model predictor is used to generate an estimate of the output between sampling times. This estimate is used by the observer and is re-set using the actual output when it becomes available at the next sampling time. The implementation of this combined control and update strategy is given by: u(t) = Kη(t), t ∈ [tk , tk+1 ) �s u(t) + L(� � s η(t)) �s η(t) + B ys (t) − Q η(t) ˙ = A ˙� � � as (t) + Bs (za )u(t) as (t) = As � (8) � s (zs )� as (t) y�s (t) = Q � −1 ¯s (tk ), k ∈ {1, 2, · · · } � as (tk ) = Q s y tk+1 = tk + ∆ �s , B �s and Q � s are where η is the state of the observer; A � s , respectively; K approximate models of As , Bs and Q and L are the feedback controller and observer gains, respectively; � as and y�s are the state and output of the inter-sample model predictor; tk denotes the k-th sampling time, and ∆ > 0 is the sampling period. For simplicity, we � s = Qs (i.e., no uncertainty in zs ). assume that Q

The following theorem provides an exact characterization of the closed-loop stability properties under the observerbased controller of (8) with a periodic sampling strategy. The proof is conceptually similar to the one in Garcia et al. (2014) and is omitted for brevity. Theorem 1. Referring to the closed-loop system of (7)-(8), the origin is exponentially stable if and only if: �s , B �s (za ), As , Bs , Q � s (zs ))] < 1, (9) λmax [M (K, L, ∆, A where λmax is the largest eigenvalue magnitude of the matrix M which is given by:

2018 IFAC ADCHEM Shenyang, Liaoning, China, July 25-27, 2018 Da Xue et al. / IFAC PapersOnLine 51-18 (2018) 327–332

M = Is eΛ∆ Is , where

(10)

] Im×m Om×m Om×m Is = Om×m Im×m Om×m (11) Om×m Om×m Om×m where Im×m is the m × m identity matrix, Om×m is the m × m zero matrix, and Λ is the closed-loop matrix given by:   Bs K Om×m As �s �s K − LQ � s −LQ �s  �s + B A Λ =  LQ (12) �s ) �s )K �s (As − A (Bs − B A [

Remark 1. The stability condition of (9) provides an explicit characterization of the interdependence between the process and model parameters, the controller and observer gains, the sampling period, and the sensor and actuator locations in influencing closed-loop stability. For example, for fixed model and controller parameters, this characterization can be used to determine how the maximum allowable sampling period depends on the spatial placement of the sensors and actuators. This characterization is important as it provides the basis for the optimization formulation introduced in the next section. 4. AN OPTIMIZATION-BASED FORMULATION FOR SENSOR AND ACTUATOR PLACEMENT While the stability condition in Theorem 1 provides a useful tool that can be used to analyze how different sensor and actuator locations influence the size of the maximum allowable sampling period, it leaves open the question of how to best place the sensors and actuators in a way that optimally balances the desire for reduced sampling frequency with the demand for improved control system performance. Furthermore, the implementation of the observer-based controller of (8) involves the use of a constant sampling rate which is not always the best choice, especially in the presence of unexpected external disturbances and changing operating conditions. In this section and the next, we describe an optimization-based formulation that aims to address both problems. We consider the following finite-horizon optimization problem in which the sampling period, the sensor and actuator locations at any given time are determined by minimizing the following cost functional: min J(zsk , zak , ∆k )

k ,∆ zsk ,za k

where, J = Jp + Js ∫ tk +H Jp = [� ysT (τ )Wy y�s (τ ) + uT (τ )Wu u(τ )]dt Js =

∫

329

where M is defined in (10)-(12), tk , is the k−th sampling time, zsk  zs (tk ), zak  za (tk ), H denotes the optimization horizon, Wy , Wu are positive-definite weighting matrices that represent the penalties on the model output and the control action, respectively, and β(∆k , w∆ ) > 0 is the penalty on the sampling frequency, which is taken to be a smooth function of the sampling period ∆k , and w∆ is a positive weighting on the sampling penalty. Referring to the above optimization formulation, the decision variables are the sensor locations, zsk , the actuator locations zak , and the sampling period, ∆k , which must be chosen at time tk to satisfy the closed-loop stability condition of (9), which captures the feasible region for the optimization problem. The solution to the optimization problem at a given time tk yields the optimal triple (zsk , zak , ∆k ) that minimizes the total cost J over the given horizon H. Remark 2. While the optimal actuator/sensor placement problem for spatially-distributed systems has been the subject of significant prior work (e.g., see Antoniades and Christofides (2000); Demetriou and Kazantzis (2004); Armaou and Demetriou (2006); Iftime and Demetriou (2009)), the emphasis of previous approaches has been either on meeting certain controllability and observability requirements, or on optimizing the performance of the closed-loop system (e.g., in terms of the response speed and control effort) without consideration of the effects or costs associated with discrete measurement sampling. The formulation proposed in (13) represents a departure from conventional approaches by aiming to balance control performance considerations with the demand for reduced sampling. Specifically, the objective function in (13) consists of a performance cost, Jp , which imposes penalties on the model output and the control action, and a sampling cost, Js , which penalizes frequent sampling. A simple choice for the sampling cost is to set β = w∆ /∆, which ensures that faster sampling rates incur larger penalties. Note that Jp is expressed in terms of the model output since the actual output is unavailable between sampling times. Remark 3. The choice of the various penalty weights in the optimization formulation should be made to reflect the relative significance of the different costs. For example, imposing larger penalties on the sampling frequency suggests an increased emphasis on the sampling cost relative to the control performance. On the other hand, for systems where sampling costs are negligible, more emphasis is placed on the controller performance. 5. DYNAMIC ACTUATOR/SENSOR SCHEDULING USING RECEDING HORIZON OPTIMIZATION

tk tk +H

β(∆k , w∆ )dt

tk

subject to: � s (z k )� y�s (t) = Q s as (t), tk ≤ t < tk + H ˙� � �s (z k )u(t), tk ≤ t < tk + H as (t) + B as (t) = As � a

�s (zak )u(t) + L(� � s (zsk )η(t)), �s η(t) + B ys (t) − Q η(t) ˙ =A t k ≤ t < tk + H u(t) = Kη(t), tk ≤ t < tk + H �s , B �s , As , Bs , Q � s )] < 1 λmax [M (K, L, ∆k , A

(13) 323

Referring to the implementation of the optimization formulation in (13), one possible strategy is to solve the problem once at the beginning of the horizon, apply the resulting optimal sensor and actuator locations and sampling period over the entire horizon length, and then repeat the optimization when the end of the horizon is reached. While this strategy reduces the computational load associated with repeated on-line optimization (especially if H is large), it does not take into account the fact that the cost function is dependent on the model state trajectory, and the fact that the model state is to be updated at each

2018 IFAC ADCHEM Shenyang, Liaoning, China, July 25-27, 2018 Da Xue et al. / IFAC PapersOnLine 51-18 (2018) 327–332 330

sampling time. It may also limit the ability of the process to respond in a timely fashion to changes in operating conditions by limiting the frequency of feedback from the process. These considerations call for a receding horizon implementation strategy, where (13) is solved on-line at each tk , and the resulting sensor and actuator locations, zsk and zak , together with the corresponding stabilizing sampling period, ∆k , are implemented only for a single sampling period within the horizon. By the end of the sampling interval at t = tk + ∆∗k , a new sampled output measurement is sent from the sensor to the controller and used to update the model state as follows: ∗ ∗ (zsk , zak , ∆∗k )

(zs (t), za (t), ∆(t)) = = arg min  as (tk + ∆∗k ) =

2 3 4 5 6 7 8 9 10

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k ,∆ zsk ,za k  −1 (z k )¯ Q y s s s (tk

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With the updated model state, the optimization problem ∗ ∗ of (13) is re-solved to determine (zak+1 , zsk+1 , ∆∗k+1 ) at12 t = tk+1 = tk + ∆∗k . The newly obtained solution is then implemented for t ∈ [tk+1 , tk+1 + ∆∗k+1 ) and the problem is re-solved at end of the sampling period. Algorithm 1 summarizes the on-line receding horizon strategy. Remark 4. The repeated on-line optimization approach described in Algorithm 1 leads to a time-varying schedule for moving the sensor and actuator locations as well as the sampling rate. Compared with the observer-based controller in (8) with a constant sampling rate, the timevarying scheduled sensor and actuator locations and sampling rates obtained through the optimization formulation allow the process to adaptively respond to unexpected changes in operating conditions. Remark 5. The receding horizon implementation of the optimization formulation bears resemblance to the implementation of MPC policies, where a finite-horizon optimization problem is re-solved at every sampling time, and the optimal solution is implemented for the duration of the first sampling period only, at the end of which new output measurements are obtained and fed back to the controller to update the model state and correct possible deviations in the model state trajectory. This feedbackbased approach is appealing since it provides some robustness to process-model mismatch and changes in operating conditions due to disturbances. Note, however, that unlike conventional MPC schemes, the frequency at which the optimization problem is re-solved is not fixed. It s rather determined online as part of the optimization problem solution. Furthermore, closed-loop stability is guaranteed through the constraint in (9) and is therefore independent of the horizon length. Remark 6. From an implementation standpoint, the horizon length should be chosen such that H > ∆. A suitable choice for H can be made on the basis of the characterization of the feasible region in terms of za , zs and ∆ obtained prior to online implementation. Specifically, by analyzing the λmax (M ) over a specified range of sensor and actuator locations, an upper bound on the feasible sampling period can be obtained. In other words, the maximum allowable sampling period, ∆max , can be viewed as a function of zs and za , where the stability condition that λmax (M ) < 1 is satisfied. Then by considering a range of possible sensor and actuator locations, a corresponding range of possible sampling periods could be estimated and used to choose 324

characterize ∆max (zs , za ) based on the stability condition of (9); specify the penalty weights Wy , Wu and w∆ ;  −1 y¯s (t0 ); initialize ys (t0 ) = y¯s (t0 ),  as (t0 ) = η(t0 ) = Q s solve (13) to determine zs0 , za0 , and ∆0 ; place zs = zs0 and za = za0 ; start system operation; sample output measurements y¯s (t) at t = ∆1 ; set k = 1; solve equation (13) to determine zsk , zak and ∆k ; relocate zs = zsk and za = zak ; while tk ≤ t < tk + ∆k do zs (t) = zsk and za (t) = zak ; u(t) = K(η(t)); end if t = tk + ∆k then sample output measurements y¯s (t);  −1 y¯s (t); update model state  as (t) = Q s k = k + 1; goto step 9 end Algorithm 1. A receding horizon implementation strategy for actuator/sensor scheduling.

a suitable horizon length, where ∆max serves as a lower bound on H (see Section 6 for an analysis of the impact of the horizon length). 6. SIMULATION EXAMPLE We consider a non-isothermal diffusion-reaction process described by the following parabolic PDE: ∂x ¯ ¯ ∂2x = + [(βT + θ1 )γe−γ − βU ]¯ x + βU b(z)u(t) (14) ∂t ∂z 2 subject to Dirichlet boundary conditions as in (2), where x ¯ is a dimensionless process temperature, the manipulating input u is a dimensionless temperature of the cooling medium, βT = 80.0, γ = 2.0, βU = 1.66 are nominal process parameters, θ1 = 0.01 is a parametric uncertainty in the heat of reaction, and b(z) is the actuator distribution function. It can be verified that the operating steadystate x(z, t) = 0 (with u = 0) is unstable. The control objective is to stabilize the temperature profile at this unstable, spatially uniform steady-state by manipulating the temperature of the cooling medium with minimal output measurement sampling. A point control actuator and a single point measurement sensor (with finite support) are assumed to be available. We consider the first eigenvalue of the differential operator to be dominant and use Galerkin’s method to derive an uncertain ODE model which is used for the synthesis of the controller. It was verified that the controller under continuous sampling stabilizes the closed-loop system. In the following simulation studies, the controller and observer gains are chosen as K = 15 and L = 50, respectively. 6.1 Characterizing the closed-loop stability region Following the proposed methodology, the first step is to characterize the stability region in terms of the sensor and

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actuator locations and the sampling period. We begin by analyzing the effect of the sensor location on closed-loop stability, the sampling and performance costs when the actuator is placed at the middle of the spatial domain at za = π/2.

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Fig. 1. Plot (a): Contour plot of λmax (M (zs , ∆)) for za = π/2. Plot (b): Maximum allowable sampling period as a function of sensor location, ∆max (zs ). Plot (c): Dependence of performance and sampling costs on sensor location. Plot (d): Magnified view of the performance cost as a function of zs .

Fig.1(a) shows a contour plot of λmax (M (zs , ∆)) where the region enclosed by the unit-contour line (the uncolored region) is the stable region, while the colored region represents the zone where λmax (M (zs , ∆)) > 1 and therefore is the unstable region. The implication of this plot is that for a specific sensor location, there is a maximum allowable sampling period beyond which instability would occur. Fig.1(b) traces the unit-contour line in Fig.1(a) which defines the maximum allowable sampling period, ∆max (zs ), as a function zs . The shape of ∆max (zs ) is symmetric and suggests that prolonging the maximum allowable sampling period requires placing the sensor closer to the boundaries. Fig.1(c) shows a snapshot of the breakdown of the total cost (defined in (13)) at the first sampling time, t1 = 0.208, when the optimization problem is solved on-line for the first time in the operating stage. The weighting parameters are chosen as Wy = Wu = 150, w∆ = 1, and H = 2 (which is approximately twice the maximum allowable sampling period achievable in the actuator spatial domain). The red line shows the sampling cost, Js , as a function of zs , while the blue curve depicts the control performance cost, Jp , as a function of zs . As expected, the dependence of Js on zs is opposite to that of ∆max (zs ) shown in Fig.1(b) since longer sampling periods translate into lower sampling cost. It can be seen that both the sampling and performance costs exhibit similar qualitative dependence on zs , where the cost is largest at the center, zs = π/2 and decreases as the sensor is moved closer to the boundary (even though the performance cost appears to depend only weakly on zs as can be seen in Fig.1(d)). This suggests that no tradeoff between the two costs exists, and that reducing the total cost favors locating the sensor close to the boundary. This leads to a static (i.e., fixed) sensor placement instead of a dynamic scheduling policy. 325

Plot (a): Contour plot of λmax (M (za , ∆, zs = π/2)). Plot (b): Maximum allowable sampling period as a function of actuator locations, ∆max (za ), for various sensor locations.

6.2 Optimization-based actuator scheduling strategy The stability region characterization in Fig.2 is used in this section to formulate and solve the optimization problem of (13). The weighting parameters and the optimization horizon length are kept the same as in Section 6.1. Note that when the maximum allowable sampling period is used, the closed-loop system is only critically stable. Therefore, to enforce asymptotic stability, the actual sampling period needs to be reduced slightly in practice. In this section, the sampling period used is 0.7∆max . Fig.3 shows the influence of the actuator location on the total cost and its breakdown at t = 0.208 and at t = 4, respectively. It can be seen that, in both cases, minimizing the sampling cost favors moving the actuator closer to the boundary whereas minimizing the performance cost favors moving it away from the boundary, leading to an optimal location in between that minimizes the total cost. Comparing the two plots, it can also be seen that the sampling cost does not change over time (note the different scales), whereas the performance cost decreases as time goes on. This is expected since the sampling cost does not depend on the model state, and therefore is unaffected by the sampling and model updates. But since the controller is designed to be stabilizing, the model output continues to decay over time, leading to smaller performance costs. Moreover, at early times when the model output is far away from its steady state, the performance cost is more dominant, but as the output converges, the sampling cost picks up and becomes more dominant at later times. As a result, the shape of the total cost varies over time, causing the optimal actuator location to vary with time. Fig.4 shows the results when the optimization problem in (13) is solved in a receding horizon fashion. Fig.4(a) depicts the closed-loop state profile when zs = π/10, while Figs.4(b)-(d) compare the closed-loop state trajectory, the optimal actuator schedules, and the sampling frequencies

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Table 1. zs π/2 π/5 π/10

Cost comparisons for different sensor locations.

performance cost 2410.2 2165.2 1865.5

sampling cost 20.63 16.10 13.60

total cost 2430.8 2181.3 1879.1

for various sensor locations. Fig.4(b) shows that the sensor location slightly affects the closed-loop state trajectory. However, it does influence the optimal actuator schedule as can be seen in Fig.4(c); and moving the sensor away from the center helps reduce the sampling frequency as can be seen from Fig.4(d). Regarding the optimal actuator schedules, it can be seen from Fig.4(c) that all schedules start at the middle location which is the one that optimizes control performance (recall that the performance cost dominates over the sampling cost initially). However, as the closed-loop state converges close to the steady-state the sampling cost becomes increasingly important, and as a result the actuator location begins to gradually shift closer to the boundary. This transition occurs earlier (with more frequent switchings thereafter) as the sensor is moved closer to the boundary. Table 1 summarizes the cumulative costs (based on the true response over the entire simulation time) for different sensor locations. Placing the sensor at zs = π/10 reduces both the performance and sampling costs, which is consistent with the discussion in Section 6.1. 326

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