Ind. Eng. Chem. Res. 2010, 49, 7957–7971
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A Quasi-decentralized Approach for Networked State Estimation and Control of Process Systems Yulei Sun and Nael H. El-Farra* Department of Chemical Engineering and Materials Science, UniVersity of California, DaVis, One Shields AVenue, DaVis, California 95616-5294
A quasi-decentralized state estimation and control architecture for plants with limited state measurements and distributed, interconnected units that exchange information over a shared communication network is developed in this work. The objective is to stabilize the plant while minimizing unnecessary network resource utilization and communication costs. The networked control architecture is composed of a family of local control systems that transmit their data in a discrete (on/off) fashion over the network. Each control system includes a state observer that generates estimates of the local state variables from the measured outputs. The estimates are used to implement the local feedback control law and are also shared over the network with the control systems of the interconnected units to account for the interactions between the units. To reduce the exchange of information over the network as much as possible without sacrificing stability, dynamic models of the interconnected units are embedded in the local control system of each unit to provide it with an estimate of the evolution of its neighbors when data are not transmitted through the network. The state of each model is then updated using the state estimate generated by the observer of the corresponding unit and transmitted over the network when communication is re-established. An explicit characterization of the maximum allowable update period (i.e., minimum cross communication frequency between the distributed control systems) needed to enforce the desired stability and performance properties is obtained in terms of the plant-model mismatch, controller, and observer design parameters. The design and implementation of the developed architecture are illustrated using a chemical plant example and compared with other possible networked control strategies. The comparison reveals that the lack of full state measurements imposes limitations on the maximum allowable update period, even if the models used to recreate the plant units’ dynamics are accurate. It is also shown that the quasi-decentralized control architecture is more robust than a centralized networked control structure, with respect to communication suspension. 1. Introduction Traditionally, the problem of designing control systems for plants with multiple distributed interconnected units has been studied within either the centralized or decentralized control frameworks. In centralized control, all measurements are collected and sent to a central unit for processing, and the resultant control commands are then sent back to the plant. In decentralized control, on the other hand, the plant is decomposed into many simpler subsystems (typically based on functional and/or time-scale differences of the unit operations) with interconnections, and several local controllers are connected to each subsystem with no signal transfer taking place between the different local controllers. Both control structures have been the subject of significant research work aimed at investigating their advantages and limitations, and developing methods for overcoming these limitations (see, for example, refs 1-7 and the references therein). Other notable examples of recent works on the control of process networks include the analysis and stabilization of process networks based on passivity and concepts from thermodynamics,8-10 the development of agent-based systems to control spatially distributed reactor networks,11-13 the design of plantwide fault-tolerant control systems,14 the analysis and control of integrated process networks using time-scale decomposition and singular perturbation techniques,15-17 and the design of distributed model predictive control systems.18-20 An approach that provides a compromise between the complexity of traditional centralized control schemes, on one * To whom correspondence should be addressed. Tel.: (530) 7546919. Fax: (530) 752-1031. E-mail:
[email protected].
hand, and the performance limitations of decentralized control approaches, on the other, is quasi-decentralized control, which refers to a control strategy in which most signals used for control are collected and processed locally, while some signals (the total number of which is kept to a minimum) are transferred between the local units and controllers to account for the interactions between the different units and minimize the propagation of disturbances and process upsets from one unit to another. A key consideration in the design and implementation of quasidecentralized control systems is how to integrate and account for communication issues in the formulation and solution of the plantwide control problem. While dedicated point-to-point links offer a reliable medium for communication between the distributed control systems (and have been successfully implemented for years), the increasing complexity and costs of installation and maintenance associated with this communication architecture, as well as the lack of flexibility for real-time reconfiguration in the event of device or equipment failure, represent significant drawbacks especially for large-scale plants with complex interconnections. These limitations have been a major driving force for the increased reliance in the process industries in recent years on sensor and control systems that are accessed over shared communication networks rather than dedicated point-to-point links. For example, the convergence of recent advances in actuator and sensor manufacturing, wireless communications, and digital electronics has produced low-cost wireless sensors and actuators21-23 that can be installed for a fraction of the cost of wired devices24 and offer unprecedented flexibility ranging from high density sensing and actuation capabilities to deployment in areas where wired
10.1021/ie1000746 2010 American Chemical Society Published on Web 04/19/2010
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devices may be difficult or impossible to deploy (such as inside waterways and high-temperature areas in oil refineries). Augmenting existing process control and monitoring systems with networked sensors and actuators promises to expand the capabilities of the existing control technology beyond what is feasible with point-to-point architectures alone. These are appealing goals that coincide with the recent calls over the past few years (see, for example, refs 25 and 26) for expanding the traditional process control and operations paradigm in the direction of smart plant operations. To harness the potential of networked sensors and actuators in the implementation of quasi-decentralized control structures, there is a need to address the fundamental challenges introduced by this technology from a control point of view. One such issue is the inherent tradeoff between the achievable control quality and the extent of network resource utilization. Maximizing the control performance requires continuous (or at least frequent) collection of data and disseminating it broadly to the target control systems. On the other hand, the limited resources of a shared communication medium (e.g., limited battery power in a wireless sensor/actuator network), suggest that sensing and communication should be reduced to aggressively conserve network resources and extend the lifetime of the network as much as possible. Realizing the potential of networked sensors and actuators to improve process control requires the development of effective strategies for characterizing and managing this tradeoff. The challenges introduced by the integration of communication networks in control systems have been the subject of a significant and growing body of research work (see, for example, refs 27-35 for some results and references in this area). However, the majority of research studies on networked control systems have focused on single-unit processes using a centralized control architecture. A centralized control structure implemented over a network, however, is not always the best choice for controlling a large-scale plant. By comparison, results on networked control of multiunit plants with tightly interconnected units have been limited. In an effort to address this problem, in ref 36, we developed in a quasi-decentralized networked control architecture that enforces close-loop stability with minimal cross communication between the constituent subsystems. The main idea was to embed in the local control system of each unit a set of dynamic models that provide the local controller with estimates of the states of the neighboring units, to be used when state information is not transmitted over the network. Both the control and communication laws, in this case, were derived under the assumption that the full state of each unit is available for measurement. In most practical applications, direct measurements of the full state are seldom available. The lack of full state measurements has important implications that must be considered both at the local control level and at the plantwide communication level. Another motivation to study this problem is that limiting the number of measurements can also help conserve network resources by reducing the number of deployed sensors as long as the states can be reliably estimated from key plant measurements. Motivated by these considerations, in this work, we develop a quasi-decentralized output feedback control architecture for multiunit plants with limited state measurements and tightly interconnected units that exchange information over a shared (possibly wireless) communication network. We address the problem of designing an integrated state estimation, control, and communication policy that requires minimal communication between the units without sacrificing closed-loop stability. To
this end, we embed in the local control system of each unit a set of dynamic models that provide an approximation of the interactions between the given unit and its neighbors in the plant when communication is suspended over the network. To address the lack of full state measurements, an appropriate state observer is included in the local control system of each unit to generate estimates of the local state variables from the measured outputs. The estimates are used to implement the local state feedback controllers and are also transmitted over the plantwide communication network to update the state of the corresponding model embedded in the interconnected subsystems when communication is re-established. The rest of the paper is organized as follows. Following some mathematical preliminaries in Section 2, the networked quasi-decentralized control structure is presented in Section 3. The closed-loop system is then cast as a switched system and its stability and performance properties are analyzed in Section 4, leading to an explicit characterization of the maximum allowable update period (i.e., minimum crosscommunication frequency) between the control systems in terms of the accuracy of the models and the choice of the control laws and state observers. Finally, the proposed framework is illustrated in Section 5, using a simulation example, and concluding remarks are given in Section 6. 2. Preliminaries 2.1. Plant Description. We consider a large-scale distributed plant composed of n interconnected processing units and represented by the following state-space description: n
x˙1 ) A1x1 + B1u1 +
∑A
1jxj,
y1 ) C1x1
j)2 n
x˙2 ) A2x2 + B2u2 +
∑
A2jxj,
y2 ) C2x2
j)1,j*2
l
(1)
l n-1
x˙n ) Anxn + Bnun +
∑A
njxj,
yn ) Cnxn
j)1
(2) (pi T pi where xi :) [x(1) i xi ... xi )] ∈ R denotes the vector of process state variables associated with the ith processing unit, pi is the number of state variables in the ith unit, yi :) [yi(1) yi(2) ... yi(qi)]T ∈ Rqi, and ui :) [ui(1) ui(2) ... ui(ri)]T ∈ Rri denote the vector of measured outputs and manipulated inputs associated with the ith processing unit, respectively. The term xT denotes the transpose of a column vector x; Ai, Bi, Aij, and Ci are constant matrices. The interconnection term Aijxj, where i * j, describes how the dynamics of the ith unit are influenced by the jth unit in the plant. Note from the summation notation in eq 1 that each processing unit generally can be connected to all the other units in the plant. Note also that, although each subsystem is referred to as a unit for the sake of simplicity, each subsystem can comprise a collection of unit operations, depending on how the plant is decomposed. Referring to the plant of eq 1, we consider a quasidecentralized networked control structure in which each unit has a dedicated local control system where the sensors and actuators are connected to the controller through a dedicated (wired) communication network. The local control systems, in turn, communicate with each other through a shared (possibly wireless) communication network over which data are exchanged between the units. Our main objective is to devise an integrated control and communication strategy that stabilizes the individual subsystems (and the overall plant) at the origin
Ind. Eng. Chem. Res., Vol. 49, No. 17, 2010
while simultaneously: (1) minimizing the unnecessary utilization of the communication network resources, and (2) accounting for the lack of complete state measurements within each unit. In the next section, we describe the design procedure of a quasidecentralized estimation and control strategy that meets the desired objectives by relying on a collection of process models and state observers that compensate for the lack of the needed information. 2.2. Illustrative Example. In this section, we introduce a benchmark example of a plant composed of interconnected units with recycle. The example will be used in subsequent sections to demonstrate the application of the quasi-decentralized output feedback control system design methodology described in the next section. To this end, we consider a plant composed of two well-mixed, nonisothermal continuous stirred-tank reactors (CSTRs) with interconnections, where three parallel irreversible elementary exothermic reactions of the following form occur: k1
A 98 B k2
A 98 U k3
A 98 R where A is the reactant species, B is the desired product, and U and R are undesired byproducts. The feed to CSTR 1 consists of two streams: one containing fresh A at flow rate F0, molar concentration CA0, and temperature T0, and another containing recycled A from the second reactor at flow rate Fr, molar concentration CA2, and temperature T2. The feed to CSTR 2 consists of the output of CSTR 1, and an additional fresh stream feeding pure A at flow rate F3, molar concentration CA03, and temperature T03. The output of CSTR 2 is passed through a separator that removes the products and recycles unreacted A to CSTR 1. Because of the nonisothermal nature of the reactions, a jacket is used to remove/provide heat to both reactors. Under standard modeling assumptions, a plant model of the following form can be derived: T˙1
)
C˙A1 ) T˙2
)
C˙A2 )
F0 Fr (T - T1) + (T2 - T1) + V1 0 V1
3
∑ G (T )C i
A1
+
i)1
F0 Fr (C - CA1) + (CA2 - CA1) V1 A0 V1 F1 F3 (T - T2) + (T03 - T2) + V2 1 V2
1
3
∑ R (T )C i
1
A1
+
Q2 FcpV2
i)1
3
∑ G (T )C i
2
A2
i)1
F1 F3 (C - CA2) + (CA03 - CA2) V2 A1 V2
Q1 FcpV1
3
∑ R (T )C i
2
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Table 1. Process Parameters and Steady-State Values for the Chemical Reactors of eq 2 parameter
value 4.998 m3/h 39.996 m3/h 30.0 m3/h 34.998 m3/h 1.0 m3 3.0 m3 8.314 kJ/(kmol K) 300.0 K 300.0 K 4.0 kmol/m3 2.0 kmol/m3 -5.0 × 104 kJ/kmol -5.2 × 104 kJ/kmol -5.4 × 104 kJ/kmol 3.0 × 106 h-1 3.0 × 105 h-1 3.0 × 105 h-1 5.0 × 104 kJ/kmol 7.53 × 104 kJ/kmol 7.53 × 104 kJ/kmol 1000.0 kg/m3 0.231 kJ/(kg K) 457.9 K 1.77 kmol/m3 415.5 K 1.75 kmol/m3
F0 F1 F3 Fr V1 V2 R T0 T03 s CA0 s CA03 ∆H1 ∆H2 ∆H3 k10 k20 k30 E1 E2 E3 F cp Ts1 s CA1 Ts2 s CA2
three steady states: two locally asymptotically stable and one s s unstable at (Ts1, CA1 , Ts2, CA2 ) ) (457.9 K, 1.77 kmol/m3, 415.5 K, 1.75 kmol/m3). The control objective is to stabilize the plant at the (open-loop) unstable steady state. Operation at this point is typically sought to avoid high temperatures, while simultaneously achieving reasonable conversion. The manipulated variables for the first reactor are chosen to be the heat input rate Q1 and the inlet reactant concentration CA0, while Q2 and CA03 are used as manipulated variables for the second reactor. Only the temperatures of the two reactors are assumed to be available as measurements. 3. Resource-Aware Networked State Estimation and Control Architecture 3.1. Distributed Output Feedback Controller Synthesis. To realize the desired quasi-decentralized control structure, the first step is to synthesize for each unit a controller that stabilizes the states at the origin in the absence of communication constraints (i.e., when the control systems are connected via ideal point-to-point connections). Specifically, we consider control laws of the following form: n
ui(x) ) Kixi +
∑
Kijxj
(3)
j)1,j*i
A2
i)1
(2) where Ri(Tj) ) ki0 e(-Ei)/(RTj), Gi(Tj) ) (-∆Hi)/(Fcp)Ri(Tj), for j ) 1, 2. Tj, CAj, Qj, and Vj denote the temperature of the reactor, the concentration of A, the rate of heat input to the reactor, and the reactor volume, respectively, with subscript “1” denoting CSTR 1. ∆Hi, ki, Ei, for i ) 1, 2, 3, denote the enthalpies, pre-exponential constants and activation energies of the three reactions, respectively, cp and F denote the heat capacity and density of fluid in the reactor. For the typical values of the process parameters given in Table 1, the plant with Q1 ) Q2 ) s s 0, CA0 ) CA0 , CA03 ) CA03 (where the superscript “s” denotes the operating steady state) and a recycle ratio of r ) 0.5 has
where Kixi is the local feedback component responsible for stabilizing the ith subsystem in the absence of interconnections, and Kijxj is a “feedforward” component that compensates for the effect of the jth neighboring subsystem on the dynamics of the ith unit. Note that the implementation of the control law of eq 3 as written requires the availability of state measurements both from the local subsystem being controlled (xi) and from the connected units (xij), which are seldom available in practice. Considering this, a state observer is designed for each local controller to generate estimates of the local state variables from the local measured outputs, and it is combined with the state feedback law of eq 3 to yield an output feedback controller of the following form:
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Ind. Eng. Chem. Res., Vol. 49, No. 17, 2010 n
ui ) Kijxi +
∑
Kijjxj
j)1,j*i
(4)
n
x¯˙i ) (Ai - LiCi)xji +
∑
Aijjxj + Biui + Liyi
j)1,j*i
where jxi is the observer-generated estimate of the state of the ith subsystem and Li is the observer gain (chosen so that Ai LiCi is Hurwitz). Notice that, because of the coupling between the dynamics of the various subsystems, the implementation of the state observer in eq 4 requires knowledge of the state estimates of the connected subsystems. Notice also that the observer generating jxi from yi resides in the local control system of the ith subsystem, while the observer generating jxj is located in the jth unit (i.e., on the other side of the shared communication network). Based on this, and without loss of generality, in the remainder of this paper, we will consider the case when jxi is assumed to be available to the local controller of unit i continuously, while jxj is available only at times when the information is transmitted through the communication network. This is justified by the fact that the local information is transmitted over a dedicated wired network (where communication disruptions are minimal), whereas the transmission of information from the neighboring units involves using the shared medium, which is resource-constrained and potentially less reliable (e.g., a wireless sensor network). Remark 1: When comparing the state and output feedback controllers of eqs 3 and 4, we observe that, while a choice of Kij ) 0 reduces the full state feedback control strategy to a fully decentralized one (with no signal transfer needed across the communication network), a similar choice does not do so in the output feedback control strategy, because of the fact that the local observer requires estimates of the states of the other plant units (which must be transmitted over the network) to generate an estimate of the local state. Remark 2: The requirement that the controller and observer gains be stabilizing in the absence of network constraints (i.e., when there is continuous transmission of measurements between the units) is a desirable feature in the sense that it decouples the control and communication design tasks from each other and offers the designer the flexibility to choose the desired control law independently of the characteristics of the communication medium deployed. However, as will be discussed later on, this requirement is not necessary to ensure stability of the networked closed-loop plant when the communication network is present (i.e., when communication is periodically suspended) as long as the frequency of communication suspension is not too large and reasonably accurate models are used for compensation during periods of communication suspension. Remark 3: Note that, although a simple Luenberger observer is used to illustrate the design and implementation of the quasidecentralized output feedback control architecture, this choice is not unique and other explicit observer designs can be used instead. The only requirement is that the observer possess an explicit evolution equation relating the evolution of the state estimate explicitly to the plant matrices, the output, and the observer design parameters. As will be seen in the next section, this feature will allow obtaining an explicit expression for the closed-loop system and subsequently deriving both necessary and sufficient conditions for closed-loop stability in terms of the observer gain. This, in turn, will allow deriving transparent criteria for tuning the observer gain to achieve closed-loop stability.
3.2. Quasi-decentralized Implementation over Networks. One way to reduce the unnecessary utilization of the plantwide communication network is to reduce the transfer of information (in this case, jxj) between the control systems without sacrificing closed-loop stability. To this end, a dynamic model of each connected unit is included in the control system of the ith unit to provide it with an estimate of the evolution of the states of those units when data are not sent over the network. This allows the control systems of the neighboring units to send their data at discrete time instants rather than continuously. “Feedforward” from one unit to another is then performed by updating the state of each model using the observer-generated estimates of the corresponding unit transmitted at discrete time instances. Between consecutive transmission times, the control action for each unit relies on a collection of models that are embedded in the local control system and are running for a certain period of time. A schematic of this model-based control architecture is shown in Figure 1. Within this architecture, the control law for each unit is implemented as follows: n
ui(t) ) Kijxi(t) +
∑
i ) 1, 2, ..., n
Kijxˆij(t),
j)1,j*i n
∑
x¯˙i(t) ) (Ai - LiCi)xji(t) +
Aijxˆij(t) + Biui(t) + Liyi(t)
j)1,j*i
xˆ˙ ij(t) ) Aˆjxˆij(t) + Bˆjuˆij(t) + Aˆjijxi(t)
(5)
n
+
∑
Aˆjlxˆil(t),
t ∈ (tk, tk+1)
l)1,l*i,l*j n
uˆij(t) ) Kjxˆij(t) + Kjijxi(t) +
∑
Kjlxˆil(t),
t ∈ (tk, tk+1)
l)1,l*i,l*j
xˆij(tk) ) jxj(tk),
j ) 1, ..., n, j * i, k ) 0, 1, 2, ...
where jxi is the estimate of xi generated by the local observer, xˆji is the estimate of xj provided by a dynamic model of the jth unit involving Aˆj, Bˆj, and Aˆjl, which are constant matrices. The fact that jxi appears directly in the model of the jth unit follows from (1) the structure of the plant whereby the ith unit feeds material and energy back into unit j and (2) the fact that the observer-generated estimates of xi are assumed to be available continuously to the local control system of the ith unit. In the case when data of jxi are not available continuously (i.e., if they are exchanged over the shared network instead), then a model of the ith unit will have to be added to estimate the evolution of xi for the times that the data are not available. Note also that the models used by the ith controller to re-create the behavior of the neighboring units do not necessarily match the actual dynamics of those processes (i.e., generally, Aˆj * Aj, Bˆj * Bj, Aˆjl * Ajl). Furthermore, a choice of Aˆj ) O, Bˆj ) O, Aˆjl ) O corresponds to the special case where, between consecutive transmission times, the corresponding model acts as a zeroorder hold by keeping the last available estimates from neighboring units until the next ones are available from the network. This representation allows the performance of the zeroorder hold and that of the model-based compensation schemes (see the simulation study for such a comparison) to be compared. Remark 4: An important feature that distinguishes the control structure of Figure 1 from centralized networked control structures (see, for example, ref 29) is that the communication medium in Figure 1 is shared by multiple interconnected units and connects the sensor/observer suite of each unit with the control systems of the other units. Therefore, the network serves as a “feedforward” path over which updates from the neighboring units are transmitted (i.e., the information communicated is
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Figure 1. Schematic representation of the integrated quasi-decentralized networked state estimation and control architecture.
used only to update the feedforward component of each controller and not the entire control law). In a centralized networked control system, on the other hand, the communication network is inserted between the sensors/observer on one side and a single controller on the other, thus providing a feedback path through which updates of the entire plant state information are transmitted and used by the controller. An implication of this architectural difference is that, although in the centralized structure no information from the plant will be available to the controller during intervals of communication suspension (and thus the control action will be based solely on the model forecasts), some plant information in the form of local measurements and state estimates will be available to each controller in the quasi-decentralized control structure, and the control action will be based on a combination of the data received from the local sensors/observer suite, as well as the embedded models’ forecasts. The continuous availability of (at least) partial measurements from the plant units enhances the robustness of the plant to communication suspensions, which helps to reduce network resource utilization further (see the simulation example for a demonstration of this point). Remark 5: Under the assumption that the plant state is completely observable from the outputs, an alternative strategy (relative to that given by eq 5 and depicted in Figure 1) that can be considered is to include, within each local control system, an observer for the full plant state. This would allow the exchange of only the output data (instead of the state estimates) between the units at each discrete communication time. For large-scale plants with a large number of interconnected units, however, this approach would increase the computational load in each control subsystem, because a high-order observer would have to be run at each node. By contrast, the quasi-decentralized structure of eq 5 reduces this computational burden by including within each unit only an observer for the local state. 4. Analysis of Networked Closed-Loop Stability and Performance 4.1. Characterizing the Maximum Allowable Update Period. In this section, we analyze the stability properties of the networked closed-loop system to obtain an explicit characterization of the maximum allowable update period, h :) tk+1
- tk, which determines the minimum rate at which the control system of a given unit should receive observer estimates from the other units through the plantwide communication network to update the corresponding model state. To simplify the analysis, we consider the case when the update periods are constant and the same for all the units (i.e., we require that all units communicate their information concurrently every h time units). Extensions to cases when the update period is timevarying and when the different units transmit their data at different rates are possible; these are the subject of other research work. To formulate the networked closed-loop system, we define the following estimation errors as: eij )
{
jxj - xˆij 0
(for j * i) (for j ) i)
i, j ) 1, 2, ..., n
(6)
where eji represents the difference between two different estimates of xj: one generated by the local observer in the jth control system and the other generated by the model of the jth unit embedded in the local control system of the ith unit. Note that, because the observer estimates of xi (xji) are assumed to be available to the local control system of the ith unit at all times, we always have eii ) 0. Introducing the augmented vectors ej :) [(e1j )T (ej2)T · · · (ejn)T]T, e :) [e1T e2T · · · enT]T, x :) [x1T x2T · · · xnT]T, xj :) [xj1T jx2T · · · jxnT]T, it can be shown that the overall closedloop plant of eqs 1 and 5 can be formulated as a switched system of the following form: x(t) ) Λ11x(t) + Λ12x¯(t) + Λ13e(t) x¯˙(t) ) Λ21x(t) + Λ22x¯(t) + Λ23e(t) e˙(t) ) Λ31x(t) + Λ32x¯(t) + Λ33e(t), e(tk) ) 0 k ) 0, 1, 2, ...
t ∈ (tk, tk+1)
(7)
where the plant and observer states evolve continuously in time and the estimation errors are reset to zero at each transmission instance, because the state of each model in each unit is updated every h seconds. Referring to eq 7, Λ11, Λ12, Λ21, and Λ22 are m × m matrices; Λ13 and Λ23 are m × mn matrices; Λ31 and Λ32 are mn × m matrices; and Λ33 is an mn × mn matrix (where m ) ∑ni)1pi and pi is the dimension of the ith state vector. These matrices are linear combinations of Ai, Bi, Aij, Aˆi, Bˆi, Aˆij, Ki, Kij, and Li,
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which are the matrices used to describe the dynamics, the models, the control laws, and the local state observers of the different units. The explicit forms of these matrices are omitted for brevity but can be obtained by substituting eq 5 into eq 1 (see the simulation study in the next section for the explicit forms of these matrices in the case of a two-unit plant). Defining the augmented state ξ(t) :) [xT(t) xjT(t) eT(t)]T, we can rewrite the closed-loop dynamics of the overall plant as follows: ξ˙ (t) ) Λξ(t),
t ∈ [tk, tk+1)
ξ(tk) ) [x (tk) x¯ (tk) 0]T, T
T
k ) 0, 1, 2, ...
(8)
where Λ )
[
Λ11 Λ12 Λ13 Λ21 Λ22 Λ23 Λ31 Λ32 Λ33
]
(9)
To analyze closed-loop stability, we must express the plant response as a function of the update period. Following techniques similar to those used in ref 36, it can be shown that the system described by eq 8 with the initial condition ξ(t0) ) [xT(t0) xj T(t0) 0]T ) ξ0 has the following response: ξ(t) ) eΛ(t-tk)(IoeΛhIo)kξ0 for t ∈ [tk, tk+1)
(10)
with tk+1 - tk ) h, where
[
Im×m Om×m Om×mn Om×mn Io ) Om×m Im×m Omn×m Omn×m Omn×mn
]
Im×m is the m × m identity matrix and Om×mn is the m × mn zero matrix. This response is similar in form to that obtained under full-state feedback but differs in that the augmented vector ξ now includes the observer states and that the matrices Λ and Io possess different structures that reflect the presence of the state observers. This allows a direct application of the result of Theorem 1 in ref 36 to derive the necessary and sufficient conditions for closed-loop stability. Specifically, it can be shown (see the proof of Theorem 1 in ref 36) that the zero solution of the system of eq 8, ξ ) [xT xj T eT]T ) [0 0 0]T, is globally exponentially stable if and only if the eigenvalues of the test matrix described by eq 11,
[
][
]
Im×m Om×m Om×mn Im×m Om×m Om×mn Om×mn eΛh Om×m Im×m Om×mn M(h) ) Om×m Im×m Omn×m Omn×m Omn×mn Omn×m Omn×m Omn×mn (11) are strictly inside the unit circle (i.e., r(M(h)) < 1, where r(M) is the spectral radius of the matrix (r(M) ) limkf∞|Mk|1/k)). Remark 6: By examining the structure of the test matrix M, it can be seen that the minimum stabilizing rate at which each control system in the network must communicate with the remainder of the plant is parametrized by (1) the degree of mismatch between the dynamics of the units and the models used to describe them, (2) the choice of the control laws (both the feedback and feedforward components), and (3) the selection of the observer gains. Therefore, the stability criterion can be used to compare different models, controllers, and state observers, in terms of their robustness, with respect to communication suspension (i.e., which ones require data updates less frequently
than others). These interdependencies are illustrated in the simulation study presented in the next section. Remark 7: Note that the control systems of the various units do not need to be designed separately from one another and can instead be obtained from a single centralized design. For example, if the plant is of a manageable size where a centralized design is feasible, one can first design a centralized controller/ observer for the entire non-networked plant (i.e., assuming that the units can exchange their data continuously over dedicated links), and then decompose it according to the structure of eq 4. The implementation of the various controllers can then be performed in a quasi-decentralized fashion in which the components requiring only local data are implemented directly using those data (which are available continuously), while the components that require data from the other units are implemented using the models’ estimates until the needed data are acquired over the network at discrete communication times to reset the models’ estimates. Therefore, the quasi-decentralized architecture provides a way to implement any centralized control and estimation scheme with minimal cross communication between the control subsystems. In this sense, the quasidecentralized nature of the architecture stems not from the way the controllers are designed per se, but rather from the way they are implemented. 4.2. Incorporating Performance Considerations. In addition to stability considerations, the performance of the networked closed-loop plant under disturbances is of major concern. Our objective in this section is to assess the performance of the networked closed-loop system under disturbances and characterize its dependence on the update period to determine a suitable communication rate that ensures minimal influence of the disturbances. To this end, we can rewrite the dynamics of the ith processing unit in the following form: n
x˙i ) Aixi + Biui + yi ) Cixi zi ) Fixi + Giui
∑
Aijxj + Eiw
(12)
j)1,j*i
where w is the disturbance input, zi is the performance output signal of interest, and Ei are constant matrices. Following some manipulations similar to those performed in the previous section, the overall networked closed-loop plant can be formulated as follows: ξ˙ (t) ) Λξ(t) + Hw(t), t ∈ [tk, tk+1) x(tk) ξ(tk) ) jx(tk) , k ) 0, 1, 2, ...; h ) tk+1 - tk 0 z(t) ) Nξ(t)
[ ]
(13)
where H ) [ET O O]T, E ) [E1T E2T · · · EnT]T, z is the augmented vector of performance outputs, and N is a constant matrix that depends on Fi and Gi. Following ref 37, we use the extended H2 norm as the performance index. This is basically an H2-like norm that is suitable for analyzing periodic systems and captures the 2-norm of the performance output when an impulse disturbance is introduced in the input at t ) t0 (see ref 37 for the theoretical details and for other types of performance measures that can be used). In this case, the closed-loop response of the system to an impulse disturbance w ) δ(t - t0) can be expressed explicitly as z(t) ) NeΛ(t-tk)(IoeΛh)kH,
t ∈ [tk, tk+1)
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Figure 2. Time evolution of the temperature (top left), reactant concentration (top right), rate of heat input (bottom left), and feed concentration (bottom right) for the first reactor under the output feedback controller of eq 16, with a continuous exchange of information between the two reactors. The dashed profiles depict the evolution of the state estimates generated by the local observer of the first reactor.
and the extended H2 norm, |G|H2, is given by |G| H2 ) trace(HTXH)1/2
(14)
where X is the solution of the discrete Lyapunov equation: M(h)TXM(h) - X + Wo(0, h) ) 0 with M(h) ) IoeΛh, and Wo(0, h) is the observability Gramian computed as Wo(0, h) ) T ∫0h eΛ tNTNeΛtdt. By examining eq 14, it can be seen that the performance will depend on the update period (as well as the models’ accuracy and controller/observer design parameters) through its dependence on X, which, in turn, is dependent on the test matrix M.
Linearizing the plant equations around the unstable steady state yields the following system to which the quasi-decentralized estimation and control architecture is applied: x˙1 ) A1x1 + B1u1 + A12x2, x˙2 ) A2x2 + B2u2 + A21x1,
(15)
where xi, ui, and yi are the dimensionless state, manipulated input, and measured output vectors for the ith unit, respectively, defined by
[ ] [ ] T1 - T1s
x1 )
5. Application to Chemical Reactors with Recycle In this section, we revisit the chemical plant example introduced in Section 2.2 to illustrate the implementation of the developed networked state estimation and control system design methodology presented in Section 3. Because of the strong interconnection between the two reactors through the recycle stream, as well as the open-loop instability of the chosen operating point, a fully decentralized control system for this plant may not be capable of stabilizing the plant unless large controller gains (which are undesirable in practice) are used. Therefore, stabilization of each reactor requires that the local control systems exchange measurements to adequately account for the effect of the interactions. Given the need to reduce the utilization rate of the wireless communication medium, however, the information exchange between the two reactors must be kept to a minimum. The tradeoff between these two conflicting objectives can only be balanced by implementing the proposed quasi-decentralized control strategy.
y1 ) C1x1 y2 ) C2x2
T2 - T2s
T1s
T2s
x2 )
s CA1 - CA1
s CA2 - CA2
s CA1
u1 )
[
Q1 CA0 -
s CA0
s CA2
]
u2 )
[
Q2 s CA03 - CA03
]
and y1 )
T1 - T1s T1s
y2 )
T2 - T2s T2s
Ai, Bi, Aij, and Ci are constant matrices, given by A1 )
[
25.2914 4.9707 -78.028 -45.9368
]
A2 )
[
-2.837 1.4157 -22.4506 -24.8828
]
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Ind. Eng. Chem. Res., Vol. 49, No. 17, 2010
[
-6 0 B1 ) 9.45 × 10 0 2.8234
A12 )
[
]
31.7512 0 0 34.6421
]
[
-6 0 B2 ) 3.47 × 10 0 5.7071
A21 )
[
14.6953 0 0 13.469
]
]
C1 ) [1 0 ] C2 ) [1 0 ] 5.1. Characterization of the Networked Plant’s Stability Properties. Following the proposed methodology, a stabilizing output feedback controller of the following form is initially designed for each reactor: ui ) Kijxi + Kijjxj, j * i x¯˙i ) (Ai - LiCi)xji + Liyi + Aijjxj + Biui
(16)
where K1 and K2 were selected by placing the eigenvalues j 1 :) A1 + B1K1 and A j 2 :) A2 + B2K2 at -5 and of both A j 12 ) -1, respectively; K12 and K21 were chosen to force A j j A21 ) O, where Aij ) Aij + BiKij; and L1 and L2 were selected by placing the eigenvalues of both A1 - L1C1 and A2 - L2C2 at -50 and -25, respectively. The closed-loop state and manipulated input profiles depicted in Figure 2 show that, when the observer-generated estimates are communicated continuously between the two units, the output feedback controllers successfully stabilize the plant at the desired steady state. For the sake of brevity, only the profiles of the states and manipulated inputs for CSTR 1 are shown in Figure 2; the closed-loop states and manipulated inputs for CSTR 2 exhibit similar tendencies. However, because the observer estimates from the neighboring reactor can be received only through the communica-
tion network, and to reduce the unnecessary utilization of network resources, a model of the form xˆ˙j ) A˜jxˆj + A˜jijxi, where A˜j ) Aˆj + BˆjKj, A˜ji ) Aˆji + BˆjKji, and Aˆj,Bˆj,Aˆji are estimates of Aj, Bj, and Aji, respectively, is embedded in the local control system of the ith unit, to provide it with an estimate of jxj. Consequently, the local state observer in the ith unit takes the form xj˙i ) (Ai - LiCi)xji + Liyi + Aijxˆj + Biui. The controller for CSTR 1 is implemented as follows: u1 ) K1jx1 + K12xˆ2 x¯˙1 ) (A1 - L1C1)xj1 + L1y1 + A12xˆ2 + B1u1 xˆ˙2 ) A˜2xˆ2 + A˜21jx1
(17)
while the controller of CSTR 2 takes the following form: u2 ) K2jx2 + K21xˆ1 x¯˙2 ) (A2 - L2C2)xj2 + L2y2 + A21xˆ1 + B2u2 xˆ˙1 ) A˜1xˆ1 + A˜12jx2
(18)
The model state is used by the local controller so long as no data from the neighboring unit are transmitted over the network, but it is updated using the observer estimate provided by the local state observer of the other reactor whenever it becomes available from the network. Our objective is to determine the largest update period that guarantees plant stability. To this end, we define the augmented state vector ξ: ξ :) [xT xT eT]T ) [xT1 xT2 jxT1 jxT2
(e11)T (e21)T (e12)T (e22)T]T
Figure 3. (a) Dependence of the maximum eigenvalue magnitude of the test matrix M(h) ) IoeΛhIo (λmax) on the update period for the case when accurate models are used by the local controllers. (b) Contour plot showing the dependence of λmax on model uncertainty in the enthalpy of the first reaction and the update period. (c) Dependence of λmax on the update period for different values of model uncertainty in the enthalpy of the first reaction. The solid line in panel (c) corresponds to the case when a zero-order hold scheme is used to compensate for measurement unavailability.
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Figure 4. Time evolution of the temperature (top left), reactant concentration (top right), rate of heat input (bottom left), and feed concentration (bottom right) for the first reactor under the quasi-decentralized model-based output feedback control architecture of eqs 17 and 18 when perfect models are embedded in the local control systems. The solid, dashed, and dotted profiles in each plot correspond, respectively, to the cases when the update period is chosen to be smaller, equal to, and larger than the maximum allowable value.
where the estimation errors are given by e11 ) 0, e12 :) jx2 - xˆ2, e21 :) jx1 - xˆ1, e22 ) 0, and let tk (for k ) 0, 1, 2, ...) be the instants when the model states are updated, such that the update period tk+1 - tk ) h is constant and the same for both units. Then, it can be shown that the closed-loop system of eq 15 and eqs 17 and 18 can be cast in the form of eq 8, where ξ(tk) ) [xT1 (tk) xT2 (tk) jxT1 (tk) jxT2 (tk) 0 0 0 0]T (since the model states are updated at tk and e21(tk) ) e12(tk) ) 0, for k ) 0, 1, 2, ...) and
[
Λ11 Λ12 Λ13 Λ ) Λ21 Λ22 Λ23 Λ31 Λ32 Λ33
[
]
[
Λ21 )
[
[
L1C1 O O L2C2
Λ23 )
[
]
Λ22 )
O O j 21 O -A
[
B1K12 B2K2 O O
]
j 12 H1 A j 21 H2 A
j 12 O -A O O
Λ33
[
]
]
]
]
]
j i - LiCi. By examining the above expressions, and and Hi ) A from the fact that M(h) ) IoeΛhIo, with Io given as
[
]
A1 A12 B1 K1 Λ12 ) A21 A2 B2K21 -B1K12 O O Λ13 ) K -B O O 2 21
[ ] [
I4×4 O4×4 O4×8 O Io ) 4×4 I4×4 O4×8 O8×4 O8×4 O8×8
where
Λ11 )
Λ31
O O j 12 - A˜12 H1 - A˜1 A Λ32 ) j 21 - A˜21 H2 - A˜2 A O O O O O O j 12 O O A˜1 -A ) j 21 A˜2 O O -A O O O O
O O L1C1 O ) O L2C2 O O
]
for this example, it can be seen that the eigenvalues of M depend on the mismatch between the models and the reactors, the controller and observer gains, and the update period. In the remainder of this section, we will investigate the interplays between these parameters. Since closed-loop stability of the linearized plant requires all eigenvalues of M to lie within the unit circle, it is sufficient to verify whether the maximum eigenvalue magnitude, denoted as λmax, is 0.01 h (notice that the two profiles intersect at h ) 0.01 h). By comparing the values of the performance indices in Figures 9c and 9d, we can also see that the effects of a given disturbance on T1 are more pronounced than they are on T2, which implies that the temperature of the second reactor recovers faster from the disturbance than does the temperature of the first reactor. The predictions of Figure 9 are confirmed by the closed-loop state profiles shown in Figure 10 (for the sake of brevity, only the temperature profiles are shown). Figures 10a and 10b show that the closed-loop system with h ) 0.02 h exhibits a better response (faster recovery from the disturbance) than that obtained at h ) 0.04 h, when an impulse disturbance is introduced in F0. This is consistent with the fact that the value of the extended H2 at h ) 0.02 h is smaller than it is at h ) 0.04 h. Similarly, when the update period is fixed at h ) 0.02 h, Figures 10c and 10d indicate that a disturbance in F0 has a stronger effect on the closed-loop system than a disturbance in F3, which is also predicted by Figure 9. 6. Conclusion In this work, a methodology for the design of quasidecentralized output feedback controllers for plants with distributed interconnected units and limited state measurements has been presented. The approach is based on a hierarchical architecture in which each unit in the plant has a local control system, with its sensors and actuators connected to the local controller through a dedicated communication network, and, in turn, the local control systems communicate with each other through a shared communication network. A state observer was included within each control system to generate estimates of the local states from the locally measured outputs, and the estimates were transmitted to the rest of the plant through the network. To achieve closed-loop stability with minimal crosscommunication between the units, each control system relied on a set of models of its neighboring units to re-create the states of those units when direct information of their values was not available. The models were updated at discrete times with the observer estimates provided by local state observers to compensate for modeling errors. An explicit characterization of the maximum allowable update period in terms of model uncertainty, controller, and observer design parameters was obtained. The analysis was facilitated by the linear structure of the plants considered, which allowed one to obtain both necessary and sufficient conditions for the stability by applying results from networked control system theory. The stability and performance properties of the developed quasi-decentralized control strategy were illustrated using a simulation example and compared with other networked control strategies.
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ReceiVed for reView January 12, 2010 ReVised manuscript receiVed March 26, 2010 Accepted March 30, 2010 IE1000746
