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Control of plant-wide systems using dynamic supply rates
Automatica 50 (2014) 44–52
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Control of plant-wide systems using dynamic supply rates✩ Michael J. Tippett, Jie Bao 1 School of Chemical Engineering, The University of New South Wales UNSW, Sydney, NSW 2052, Australia
article
info
Article history: Received 24 August 2011 Received in revised form 6 July 2012 Accepted 30 August 2013 Available online 20 November 2013 Keywords: Process control Distributed control Linear networks Linear output feedback H∞ control
abstract This paper presents a framework for analysis of plant-wide processes from a network perspective. Using the concept of dissipativity, the conditions for plant-wide input–output stability and performance are developed, based on the dissipativity of individual subsystems and the topology of the network of the plantwide process. Dynamic supply rates, expressed as quadratic differential forms, are proposed not only to render dissipativity based analysis less conservative but also allow the dynamic plant-wide performance criteria to be specified in terms of desired closed loop supply rates. The links between the plant-wide supply rate, finite L2 gain in an extended input–output space and weighted H∞ norm are explored in this paper. These results lay a foundation for a supply rate-centric approach to plant-wide distributed control. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction Largely driven by increasingly tight economic and environmental requirements, modern chemical plants are becoming increasingly complex, often with dozens of process units with material recycling and energy integration. From a process control point of view, recycling streams can be understood as positive feedback loops within the process network, which have a deleterious impact on control performance (Luyben, Tyréus, & Luyben, 1998). These strong interactions between process units are a key feature of plant-wide process control problems, and are a challenge to control practice due to high sensitivity to disturbances and possible plant-wide instability (Kumar & Daoutidis, 2002). Another important challenge in plant-wide process control is the scale of the problem, which can make centralized control systems computationally difficult or infeasible (Skogestad, 2004). Some existing approaches for control of plant-wide processes deal with the interactions between unit processes as uncertainties, as presented in Grosdidier and Morari (1986), Samyudia,
✩ This work is partially supported by the ARC Discovery Project DP1093045. The first author would like to acknowledge the financial support of the Australian Postgraduate Award, as well as the ESA and UNSW Excellence Awards provided by UNSW. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Denis Dochain under the direction of Editor Frank Allgöwer. E-mail addresses: [email protected] (M.J. Tippett), [email protected] (J. Bao). 1 Tel.: +61 2 9385 6755; fax: +61 2 9385 5966.
0005-1098/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.automatica.2013.09.028
Lee, and Cameron (1994), Swarnakar, Marquez, and Chen (2009) and Skogestad and Morari (1989). A decentralized controller is then designed to stabilize the plant-wide system subject to these uncertainties based on robust control theory. This approach can greatly simplify implementation of the control system. To help facilitate this, interaction measures for plant-wide process systems have been developed, e.g. Bristol (1966), Cheng and Li (2010) and Manousiouthakis, Savage, and Arkun (1989). These interaction measures are useful for input–output pairing and estimating the efficacy of decentralized control schemes. However, this approach is inherently conservative as the known interactions are treated as unknown. Distributed control structures have gained attention in recent years. For example, an approach requiring minimal communication between controllers is presented in Goodwin, Haimovich, Quevedo, and Welsh (2004); the optimal controller network topology is determined in Langbort and Gupta (2009); distributed estimation and model predictive control methods are developed in Mercangöz and Doyle (2007) and Vadigepalli and Doyle (2003). Distributed model predictive control has received much attention recently, e.g. Liu, Chen, Muñoz de la Peña, and Christofides (2010) and Stewart, Venkat, Rawlings, Wright, and Pannocchia (2010). One promising approach to plant-wide analysis is based on a network perspective, decomposing a plant-wide process into individual process units interacting through a network with a known topology (e.g., Bao, Jillson, & Ydstie, 2007; Rojas, Setiawan, Bao, & Lee, 2009; Ydstie, 2002). A key advantage of this approach is its scalability, as it allows analysis and control design based on simpler subsystems and their interconnections.
M.J. Tippett, J. Bao / Automatica 50 (2014) 44–52
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Dissipativity theory, introduced in Willems (1972), has become an important analysis and control design tool. It is particularly suitable for the network approach to plant-wide control, as the plant-wide stability and performance can be determined based on the dissipativity of individual subsystems and the network topology, thus simplifying control design. A dynamical system with input u ∈ Rp , output y ∈ Rq (both with compact support) and state x ∈ Rn respectively is said to be dissipative if there exists a function defined on the input and output variables, called the supply rate s(u, y) and positive semidefinite (at least once differentiable) function defined on the state space, called the storage function V (x) ≥ 0, such that
2. Dissipativity, stability and performance
V˙ (x(t )) ≤ s (u(t ), y(t )) .
with w = yT uT
(1)
The following quadratic (Q , S , R)-type supply rate is commonly used: s(u, y) = yT Qy + 2yT Su + uT Ru,
(2)
where Q = Q T , S and R = RT are real matrices. Results on the application of dissipativity theory to large-scale systems analysis and decentralized control design has been reported in the literature. For example, in Moylan and Hill (1978), Rojas, Setiawan, Bao, and Lee (2008) and Vidyasagar (1981) where dissipativity is used as an enabling tool for stability and operability analysis of large-scale systems. In Hangos, Alonso, Perkins, and Ydstie (1999), a special case of dissipativity, passivity, is used to show the stability of plant-wide nonlinear process systems. This is achieved using the links between passivity and the mass and energy balances underlying process systems. These links were developed in Alonso and Ydstie (1996) and Ydstie and Alonso (1997), where thermodynamics is used to develop physically motivated storage functions. These concepts are used for plant-wide control in Farschman, Viswanath, and Ydstie (1998). However, the above passivity/dissipativity approaches can be very conservative, as the constant supply rates shown in (2) can only provide a coarse description of the systems dynamic features. This work presents a new approach to plant-wide stability and performance analysis and design based on network decomposition and the dissipativity of the subsystems. Both the plant-wide stability and performance specifications (in the form of weighted H∞ norms) are achieved by encoding them as desired closed loop supply rates. Controllers can then be designed to achieve these supply rates, and hence, the desired closed-loop stability and performance properties. Central to the proposed approach is the use of dynamic supply rates, expressed as quadratic differential forms (QDFs). As the QDFs represent a more general form of dissipativity (Pendharkar & Pillai, 2008; Willems & Trentelman, 1998), they capture more detailed dynamic properties and therefore lead to far less conservative dissipativity based conditions than those based on constant supply rates (e.g., in (2)), as shown in Tippett and Bao (2013a). The QDFs also allow the dynamic plant-wide performance criteria to be specified in terms of desired closed loop supply rates as was done in our previous work (Tippett & Bao, 2013b). In the proposed approach, the models of individual process units are only used to validate their dissipativity. All analysis and control design is performed based on the dissipativity of the process units rather than detailed process models. This leads to a supply rate-centric approach to plant-wide distributed control that can deal with arbitrary process network topologies and control structures. The remainder of this paper is structured as follows. In Section 2, a brief overview of dissipativity theory is presented. A dissipativity based decomposition of the plant-wide system and controller network is presented in Section 3. In Section 4 plantwide stability and performance results based on this decomposition are presented.
The key concept used in this paper is dissipativity, which is defined in the context of behavioral systems theory as follows: Definition 1 (Willems & Trentelman, 1998). A controllable system is said to be dissipative with storage function Qψ , and supply rate Qφ , if: ∞
Qφ (w) d t ≥ Qψ (w),
(3)
−∞
for all allowable trajectories of the system with compact support,
T
, a vector of the inputs and outputs.
In this paper we are concerned with the case where Qψ , the storage function, is positive semidefinite. This is equivalent to halfline dissipativity on R− (Willems & Trentelman, 1998). Note that half-line dissipativity on R− implies dissipativity in the general sense in of Definition 1. In this context Qφ and Qψ are quadratic differential forms as shown below: Qφ (w) =
kmax lmax k d k=0 l=0
dt
w k
T
φkl
dl dt
w , l
(4)
where φkl are constant, symmetric, matrices and are the coefficient matrices of the two-variable polynomial matrix φ(ζ , η). The d T dt
indeterminates ζ and η refer to
d and dt respectively. kmax and T d d and dt . The degree of the supply lmax are the highest order of dt rate is defined as max kmax lmax . The matrix is said to
(
,
)
φ(ζ , η)
induce the QDF (Willems & Trentelman, 1998). Definition 2 (Willems & Trentelman, 1998). Let φ ∈ Rq×q [ζ , η], with φ symmetric. The QDF Qφ (w) is called positive, denoted by φ > 0, if Qφ (w) ≥ 0 for all w, and the only w for which Qφ (w) = 0 is w = 0. A QDF is negative definite, φ < 0, if and only if −φ > 0. It worth pointing out that the derivative a of QDF is itself a QDF, d d i.e., dt Qφ = Q d φ , where dt φ = (ζ +η)φ . In simple terms, QDFs are dt
quadratic functions of the inputs and outputs and a finite number of their derivatives. They can be understood as an extension of the commonly used (Q , S , R) shown in (2) that include derivative terms. Because of this, QDF supply rates capture more detailed system information than constant supply rates, allowing for more in depth analysis and less conservative results. The extended variables of input u and output y will be used throughout this paper and are defined as follows:
yˆ =
y
T
u
dy
T
dt
uˆ =
T
du dt
T
...
...
dn˜ y
T T
dt n˜
dm˜ u
T T
(5)
dt m˜
˜ less than or equal to the order of the for some integers n˜ and m system in y and u respectively. Using these variables, the quadratic from (4) can be rewritten as: Qφ
y u
T T yˆ yˆ yˆ Q˜ = ˆ φ˜ ˆ = ˆ u u u S˜ T
S˜ R˜
yˆ . uˆ
(6)
The block matrix φ˜ is referred to as the coefficient matrix of φ , as its entries are the constant coefficient matrices of the polynomials of the indeterminates ζ and η in φ(ζ , η). Methods for determining dissipativity of linear systems with QDF supply rates and storage functions include Pick matrix and frequency domain conditions presented in Willems and Trentelman (1998) and a Linear
46
M.J. Tippett, J. Bao / Automatica 50 (2014) 44–52
Matrix Inequality (LMI) based algorithm presented in Belur and Trentelman (2002). The conditions for asymptotic stability is given below: Proposition 1 (Willems & Trentelman, 1998). An autonomous linear system is asymptotically stable if and only if there exists a QDF storage function Qψ induced by ψ ∈ Rq×q [ζ , η] such that for all trajectories ˙ < 0. of the system, ψ ≥ 0 and ψ The asymptotic stability is a notion of input–output stability used in the remainder of this paper. It implies that for vanishing input, the output and its derivatives asymptotically converge to zero. This is equivalent to the classical definition of input–output stability. It implies the asymptotic Lyapunov stability of the state space of the minimal realization (or any realization with the zero state detectability condition) of the input–output system. The stability condition based on QDF supply rates is given below: Theorem 1. A system which is dissipative with positive semidefinite storage function Qψ with respect to supply rate
T Q˜ yˆ Qφ (ˆy, uˆ ) = ˜S T ˆu
S˜ R˜
yˆ uˆ
(7)
is asymptotically stable with finite L2 gain from uˆ to yˆ
∥ˆy∥2 ≤ γ ∥ˆu∥2
(8)
Given a linear system, its dissipativity with respect to dynamic supply rates can be linked to its weighted H∞ norm. As shown in Belur and Trentelman (2004), the dissipativity of a singleinput–single-output linear system G(s) with respect to the supply rate Qφ (y, u) induced by:
φ(ζ , η) =
−n(ζ )n(η) 0
0
γ d(ζ )d(η) 2
,
φ ∈ R2×2 [ζ , η] n(s)
implies that the L∞ norm ∥WG∥L∞ ≤ γ , where W (s) = d(s) . This result is extended to the multiple-input–multiple-output case with H∞ norm and without a block-diagonal condition on φ(ζ , η), as shown below: Theorem 2. A linear system y = G(s)u with u ∈ Rp and y ∈ Rq , which is dissipative with respect to a QDF supply rate induced by the polynomial matrix φ(ζ , η)
φ(ζ , η) =
Q (ζ , η) S T (ζ , η)
S (ζ , η) , R(ζ , η)
(13)
with positive semidefinite storage function and Q (ζ , η) < 0 (where the dimensions of Q (ζ , η), S (ζ , η) and R(ζ , η) are p × p, p × q and q × q respectively), satisfies the following H∞ norm condition:
N G ≤ γ , α ∞
(14) 1
where, N (jω) = Qˆ 2 (−jω, jω), with Qˆ (−jω, jω) = −Q (−jω, jω) and a scalar α(ω) > 0 satisfying
with
1 1 γ = ∥(−Q˜ )− 2 ∥2 ∥(−Q˜ )− 2 S˜ ∥2 + α α 2 I ≥ R˜ − S˜ T Q˜ −1 S˜
(9)
γ2
(10)
2
α 2 (ω)I ≥ R(−jω, jω) + S T (−jω, jω)Qˆ −1 (−jω, jω)S (−jω, jω),
if Q˜ < 0.
∀ω.
Proof. See Appendix.
Proof. See Appendix.
The above result is extended to general interconnected systems in Section 4. A negative feedback system of a process with supply rate given by with supply rate induced by (6) and a controller
Example 2. Consider a heat exchanger model (Rojas et al., 2009):
φc (ζ , η) = Q˜ cl =
Qc (ζ , η) ScT (ζ , η)
Q˜ + R˜ c S˜ T − S˜c
Sc (ζ , η)
Rc (ζ , η)
S˜ − S˜cT R˜ + Q˜ c
is stable if:
< 0.
(11)
Example 1. Consider a system with transfer function G(s) = 0.5(1−0.1s) . As this system is non-minimum phase, a constant 100s+1 (Q , S , R) supply rate has R ≥ 0. The stability condition in Moylan and Hill (1978) (Qcl < 0) requires the Q matrix of the controller Qc < 0 to ensure closed loop stability. This implies the controller must have finite gain, not allowing integral control. Using Theorem 1, it can be shown that the system is half-line dissipative with respect to the supply rate induced by
−0.61 − 0.06ζ η 0.5
0.5 −0.001 + 0.29ζ η
(12)
and that a PI controller with Kp = 0.05, Ki = 0.4 is half-line dissipative with respect tothe supply rate induced by θ (ζ , η) = −0.3ζ η 0.5
1 s2
×
The following example shows the dynamic supply rates reduce the conservativeness in dissipativity based stability analysis.
φ(ζ , η) =
G(s) =
0.5 0.53˙ + 0.06˙ ζ + 0.06˙ ζ
. Combining the process and controller
supply rates it is clear that (11) is negative definite. Thus, the closed loop system is asymptotically stable, showing that integral action is admitted.
+ 48.22s + 412.1 11.08s + 411.3 04172
0.8107 . 37.08s + 411.7
It can be shown that it is half-line dissipative with respect to the supply rate induced by
φ(ζ , η) =
−(0.1ζ + 1)(0.1η + 1)I2 02×2
02×2 γ 2 (10ζ + 1)(10η + 1)I2
,
for any γ ≥ 1. According to Theorem 2, the system satisfies the .1s+1 . weighted H∞ norm ∥w G∥∞ ≤ 1 with w(s) = 010s +1 These stability and performance results form the basis of the plantwide stability and performance conditions (in Section 4) based on the dissipativity of the overall system. The plant-wide dissipativity is determined from the dissipativity of the individual processes and controllers and their network structures as presented in Section 3. 3. Network perspective of plant-wide process control The problem considered here is the analysis and control of large-scale systems with subsystems interconnected through a fixed network topology to meet given plant-wide H∞ performance criteria. An area of application is chemical process systems, where individual units are interconnected through mass and energy flows. An example of such a system is the reactor–separator process depicted in Fig. 1. The plant-wide system consists of two
M.J. Tippett, J. Bao / Automatica 50 (2014) 44–52
47
Fig. 3. Partitioning of controller inputs and outputs.
Fig. 1. Reactor–separator system.
Fig. 4. Network view of plant-wide process with distributed control. Fig. 2. Partitioning of process manifest variables.
unit processes interconnected through a fixed network topology, including a recycling stream. The recycling can induce strong interactions between the process units. The aim is to formulate scalable conditions that the control system must satisfy to ensure the plant-wide closed loop system achieves a given weighted H∞ norm condition. In this work, only unidirectional flows in the process network are considered. This is justified in practice where flows between units are usually forced by pumps or compressors. As shown in Fig. 2, the input to each process unit is partitioned into three components: the input from interconnected units uP , external disturbances d and manipulated input uL . Meanwhile the output of each unit consists of all interconnecting and measured outputs. Using the example in Fig. 1, the inputs of the CSTR can be partitioned into interconnecting inputs (inlet flowrates of components), disturbances (inlet temperature of water in jacket) and manipulated input (flowrate of water in jacket). To consider distributed control, the controllers are represented as two port systems with both local and remote inputs and outputs as shown in Fig. 3. The local signals directly interface with the process whilst the remote signals interface with other controllers. The controller input includes local sensor output uC and information from remote controllers uR . The controller output includes the signals sent to local actuator yL and to other controllers yR . To again relate this to the process depicted in Fig. 1, the local controller for the reactor has local input and output of the temperature in the reactor and jacket flowrate respectively. In addition, it sends and receives signals to/from the distillation column controller which it uses to adjust its local manipulated variable. The entire plant-wide process system of n process units with n controllers is depicted in Fig. 4. Each process unit is stacked diagonally to form the large system not including the interconnection
topology, P˜ . The inputs and outputs of this system are the vectors consisting of the inputs and outputs of each system concatenated with one another. The output of this diagonal system is y = col(y1 . . . yn ). Signals uP , uL d and uL are defined in a similar way. The topology of the interconnections between process units is represented by the matrix Hp . Due to the assumption of time invariant interconnections, Hp is a constant matrix with elements of either 0 or 1. Using the example in Fig. 1, Hp =
0 I
I 0
. The
structure of this Hp implies that the output of the first process is the input into the second system, and the output of the second process is the input into the first system. Fp and FI select the interconnecting and measured outputs respectively. As such, Fp and FI are constant matrices with elements of either 0 or 1. The controller network is represented in a similar way. The individual controllers are stacked diagonally to form C˜ (a block diagonal system with the ith diagonal block being the ith controller) as the overall controller without interconnections. The topology of the controller network is captured by matrix Hc , in a similar manner as Hp . This network representation of the control system allows a unified approach to plant-wide process control, encompassing decentralized, distributed and fully centralized control. In the case of decentralized control, Hc = 0 while fully centralized control is achieved by setting Hc as a full matrix. For the distributed control cases, Hc is sparse. With appropriate changes, all developments in the remainder of this paper then follow for all control structures once the necessary changes have been made without increasing the complexity of control design. Assume that a QDF supply rate of the ith process is given as Qφi (uP , d, uL , y), which is induced by a symmetric two-variable polynomial matrix conformally partitioned as:
φi (ζ , η) =
Qi (ζ , η) SiT (ζ , η)
Si (ζ , η) , Ri (ζ , η)
(15)
48
M.J. Tippett, J. Bao / Automatica 50 (2014) 44–52
with
Qi (ζ , η) = Q (ζ , η);
Si (ζ , η) = SI (ζ , η)
RII (ζ , η) Ri (ζ , η) = RTId (ζ , η) RTIL (ζ , η)
RId (ζ , η) Rdd (ζ , η) RTdL (ζ , η)
Sd (ζ , η)
SL (ζ , η)
RIL (ζ , η) RdL (ζ , η) . RLL (ζ , η)
Q(ζ , η) S T (ζ , η)
Φ (ζ , η) =
with
(16)
Sci . Rci
Qc ScT
Θ (ζ , η) =
Sc Rc
Qcill Qcirl
Qcilr Qcirr
Rcill Rcirl
Rci
Rci =
Sci =
Scill Scirl
Scilr Sci
and
Lemma 1. Consider the interconnected system as shown in Fig. 4. If the collection of (unconnected) process units P˜ is dissipative with respect to supply rate QΦ (induced by (16)) and storage function QΨ , and the collection of unconnected controllers C˜ is dissipative with respect to the supply rate QΘ (induced by (18)) and storage function QΣ , then the plant-wide system from all disturbances d to all process
T
output and controller output ypw = yT , yTL , yTR , is dissipative with storage function Qν = QΨ +Σ and supply rate Qµ and induced by
Γ11 (ζ , η) µ(ζ , η) = T Γ12 (ζ , η) X11
Γ11 (ζ , η) = XT12 XT13
X12 X22 XT23
X22 = RLL + Qcll , X23 = Qclr + Sclr Hc ,
(18)
where the superscript l relates to signals to/from local actuators/sensors and the superscript r to signals to/from remote controllers. Then, Φ (ζ , η) and Θ (ζ , η) can be calculated from φi (ζ , η) and θi (ζ , η) based on their definition. The storage function of P˜ , Ψ (ζ , η) (C˜ , Σ (ζ , η)), is the sum of the n individual process (controller) storage functions ψi (ζ , η) (σi (ζ , η)), i = 1 . . . n. The relation between the storage function of the ith process and controller, induced by ψi (ζ , η) and σi (ζ , η), and the storage function of P˜ and C˜ , induced by Ψ (ζ , η) and Σ (ζ , η), is completely analogous to that of the supply rates. The interconnection relations are shown in Fig. 4. The effects of interactions between all process and control units are represented by their input–output properties as captured by their dissipativity, and in particular, their supply rates and the network topologies rather than their detailed models. This enables the entire plantwide system to be modeled as the interconnection of two networks of processes and controllers. The supply rate of the overall plantwide system with interconnections between process units and the controller network can be determined using the following lemma.
FI ,
Proof. The result is easily shown by adding the two dissipation inequalities, QΦ ≥ QΨ˙ and QΘ ≥ QΣ˙ . Then substituting in the interconnection relations to eliminate all variables except the external inputs and process and controller outputs.
, Rcilr , rr
X13 = Sclr + HcT Rclr
T
X33 = Qcrr + Scrr Hc + HcT Scrr + HcT Rcrr Hc .
can be defined in a similar way to the supply rate of P˜ , with Qc = diag(Qc1 , . . . , Qcn ), Sc = diag(Sc1 , . . . , Scn ), Rc = diag(Rc1 , . . . , Rcn ). Due to the distributed control structure, these submatrices are structured as follows:
Qci =
(17)
The supply rate of C˜ , induced by
X12 = SL + FpT HpT RIL + FIT Scll ,
T
Qci SciT
X11 = Q + SI Hp Fp + FpT HpT SIT + FpT HpT RII Hp Fp , +FIT Rcll FI , T
S (ζ , η) , R(ζ , η)
with Q = diag(Q1 , . . . , Qn ), S = diag(S1 , . . . , Sn ) and R = diag(R1 , . . . , Rn ). Assume that the supply rate of the ith controller is induced by
θi (ζ , η) =
0
Γ22 (ζ , η) = Rdd ,
Define the supply rate of system P˜ as a QDF induced by the matrix:
Sd + FpT HpT RId T , Γ12 (ζ , η) = RdL
Γ12 (ζ , η) , Γ22 (ζ , η) X13 X23 , X33
(19)
(20)
Once the interconnection topologies Hp , Hc are known together with the filters FI and Fp , conditions (19) and (20) are linear in the supply rates of the individual processes and controllers. This allows the conditions on the plant-wide dissipativity to be formulated as LMI constraints, which can be efficiently solved. Using the plantwide dissipativity properties developed in this section, plant-wide stability and performance conditions based on those in Section 2 are now presented. 4. Plant-wide stability and performance 4.1. Plant-wide stability For the process and control networks shown in Fig. 4, the plantwide stability can be determined by the following proposition. Proposition 2. Consider the plant-wide system with control depicted in Fig. 4. Assume that the plant-wide system from all disturbances d
T
to all process and controller output ypw = yT , yTL , yTR is dissipative with respect to the supply rate Qµ (w) induced by the polynomial
Γ11 (ζ , η) T Γ12 (ζ , η)
Γ12 (ζ , η) Γ22 (ζ , η)
matrix µ(ζ , η) =
(as per Lemma 1) and the
storage function induced by ν(ζ , η). The plant-wide system is asymptotically stable (in the sense of Proposition 1) with finite L2 gain from the extended variable of ˆ to the extended variable of the plant-wide output, the disturbance, d, yˆ pw (as defined in (5)), i.e.
∥ˆypw ∥2 ≤ γ ∥dˆ ∥2 with γ
(21) 1
1
− − = ∥Γ¯ 11 2 ∥2 ∥Γ¯ 11 2 Γ˜ 12 ∥ + α
and α being a positive
T ¯ constant satisfying α 2 I ≥ Γ˜ 22 + Γ˜ 12 Γ11 Γ˜ 12 and Γ¯ 11 = −Γ˜ 11 if the following conditions are satisfied:
−1
(1) Γ˜ 11 , the coefficient matrix of Γ11 (ζ , η), is negative definite and (2) ν(ζ , η) ≥ 0. Proof. The proof follows that of Theorem 2, using the supply rate and storage function for the plant-wide system with control derived in Lemma 1. Remark 1. It is clear from (20) that the dissipativity of the overall system depends on the topologies of the process and controller networks, as well as the dissipativity properties of each system.
M.J. Tippett, J. Bao / Automatica 50 (2014) 44–52
This facilitates the view of the overall system as two interacting networks, for which stability and performance objectives are achieved by a matching of dissipativity properties. Both plant-wide stability and performance conditions can be represented as certain input–output gain conditions, conveniently represented by the dissipativity condition given in Proposition 2. Theorem 1 and Proposition 2 are less conservative than existing dissipativity based stability results in the literature. This is particularly important in control of interconnected plant-wide systems as conservativeness in each subsystem will accumulate at the plantwide level. 4.2. Plant-wide performance Proposition 2 also gives a condition on plant-wide dynamic performance which is represented by the upper bound on the gain between the extended input and output of the plant-wide closed-loop system. As such, Proposition 2 allows both plant-wide stability and performance specifications to be addressed in a single dissipativity condition, which facilitates control design. The gain bound in (21) on the extended input and output space, however, is not necessarily a familiar measure. The link between the above dissipativity condition and weighted H∞ norm bounds on the system in the original input–output space is presented below. The notations ∥.∥L∞ and ∥.∥∞ are used to denote the L∞ and H∞ norms respectively, as defined in Zhou, Doyle, and Glover (1996). Theorem 2 is applied to networked systems by using Lemma 1. This allows the analysis of the plant-wide performance achieved by a control system in terms of weighted H∞ norms. Proposition 3. Consider a plant-wide system as shown in Fig. 4 with the transfer function matrix of the system from all disturbances d to all
T
process and controller outputs ypw = yT , yTL , yTR , denoted as T (s). If T (s) is dissipative with a positive semidefinite storage with respect to the supply rate induced by µ(ζ , η) in Lemma 1 with Γ11 < 0, then it satisfies the following H∞ norm condition:
1 NT ≤ γ , α ∞
(22)
with α(ω) > 0 being a scalar satisfying, for all ω,
γ2 2
α 2 (ω)I ≥ Γ22 (−jω, jω) −1 T + Γ12 (−jω, jω)Γ¯ 11 (−jω, jω)Γ12 (−jω, jω) 1
and N (jω) = Γ¯ 112 (−jω, jω), where Γ¯ 11 = −Γ11 . Proof. The plant-wide dissipativity properties are determined from the individual subsystems and controllers (and the interconnection topologies) using Lemma 1. The result is then shown by applying Theorem 2 to the plant-wide supply rate and storage function. The plant-wide H∞ norm condition given in (22) represents a weighted norm of the mixed-sensitivity function, which can be W1 T1 rewritten as W T ≤ γ where T1 (s) is the system from distur2 2
∞
bance to process output and T2 (s) is the system from disturbance to controller output. A computationally simpler result for our purposes can be obtained for the case where the matrix φ(ζ , η) (or µ(ζ , η)) is block diagonal, this is stated below for the plant-wide case. Proposition 4. Consider a plant-wide system as shown in Fig. 4 with the transfer function matrix of the system from all disturbances d to all
49
T T
process and controller output ypw = yT , yTL , yR , denoted as T (s). If T (s) is dissipative with respect to the supply rate induced by µ(ζ , η) as in Lemma 1 of the form
µ(ζ , η) =
−N T (ζ )N (η) 0
0
γ 2 d(ζ )d(η)I
,
µ ∈ Rq×q [ζ , η]
(23)
then the following L∞ norm condition holds:
∥WT ∥L∞ ≤ γ ,
(24)
where d(s) is a scalar polynomial and N (s) is a polynomial matrix satisfying W (s) = d(1s) N (s). If, in addition, the storage function is positive semidefinite (i.e. half-line dissipativity), then the following weighted H∞ norm condition holds:
∥WT ∥∞ ≤ γ .
(25)
Proof. See Appendix. 4.3. Controller feasibility and synthesis The above results allow the plant-wide performance and stability criteria to be encoded in a desired supply rate that the closedloop plant-wide process T (s) must satisfy. The supply rates that individual controllers must satisfy can be determined from this plant-wide supply rate. Then the controllers that possess such dissipativity conditions can be synthesized individually. The key issue is the feasibility of the controller supply rates that are required for plant-wide stability and performance (i.e., the existence of a controller that possesses such dissipativity). It is well known that for the existence of a non-trivial system satisfying a given supply rate the number of non-negative eigenvalues of the matrix inducing the supply rate is greater than or equal to the dimension of the inputs to the system. One method of ensuring this is to require the controller supply rate to admit a J-spectral factorization. That is, φc (−jω, jω) = F T (−jω)JF (jω) where J is a diagonal matrix with elements equal to 0, 1 or −1. J-spectral factorizability implies the controller supply rate has a constant number of positive and negative eigenvalues (Trentelman & Rapisarda, 1999), equal to the number of positive and negative eigenvalues of J. This can be transformed into an LMI condition as below. Lemma 2. Consider the symmetric single variable polynomial matrix A(ω) = A0 + A2 ω2 + · · · + A2n ω2n
(26)
of degree 2n with the property that Ai = 0 ∀ odd i. Then det(A(ω)) = 0 has no roots ∀ω ∈ R if either: A0 , A2n > 0
(27)
A0 + A2 ≥ 0
(28)
A0 + A2 + A4 ≥ 0
.. . A0 + · · · + A2n−2 ≥ 0 A2n + A2n−2 ≥ 0 A2n + A2n−2 + A2n−4 ≥ 0
.. . A2n + · · · + A2 ≥ 0, Or if (27)–(29) are negative semidefinite. Proof. See Appendix.
(29)
50
M.J. Tippett, J. Bao / Automatica 50 (2014) 44–52
Theorem 3. The matrix φ(ζ , η) =
Q (ζ , η) 0
0 R(ζ , η)
admits a J-
spectral factorization if Q˜ X + X Q˜ ≥ 0
(30)
˜ + Y R˜ ≥ 0 RY
(31)
where X ≥ 0, Y ≤ 0 or X ≤ 0, Y ≥ 0. Proof. Note that φ(ζ , η) admits a J-spectral factorization if Q (ζ , η) and R(ζ , η) do, as the eigenvalues of φ(ζ , η) are the union of the eigenvalues of Q (ζ , η) and R(ζ , η). Using the main result in Carlson and Schneider (1962) Q˜ X + X Q˜ ≥ 0 implies that Q˜ is either positive or negative semidefinite. Which, using Lemma 2 implies that Q (−jω, jω) admits a J-spectral factorization. Using the same argument, the above conditions imply the J-spectral factorizability of R(−jω, jω) and therefore φ(−jω, jω). Combining the above results, the following result provides a convex algorithm (based on LMIs) for determining the required controller dissipativity properties to achieve given levels of plantwide performance. Theorem 4. Consider a plant-wide system as shown in Fig. 4 with the transfer function matrix of the system from all disturbances d to
T
all process and controller output ypw = yT , yTL , yTR , denoted as T (s). Where the ith process is dissipative (with non-negative storage) with respect to the supply rate induced by φi (ζ , η). There exists a distributed controller such that the closed-loop plant-wide system (T (s)) is internally stable and satisfies the norm bound ∥WT ∥∞ ≤ 1 if
Γ˜ 11 < 0
(32)
Q˜ci Xi + Xi Q˜ci ≥ 0
(33)
R˜ci Yi + Yi R˜ci ≥ 0
(34)
for all i, where the ith controller is dissipative rate induced with supply by φci (ζ , η), with coefficient matrix φ˜ ci =
Qci 0
0 Rci
W (jω) =
α(ω)
In this paper, a supply rate centric approach to analysis of plantwide processes is presented. Plant-wide stability and performance criteria are represented in a unified form of desired closed-loop plant-wide dissipativity properties. In this setting, the network of processes is controlled by a network of controllers. Plant-wide stability and performance conditions are based on the dissipativity of individual subsystems and interconnection topologies. This leads to a very general and flexible framework for plant-wide control where different control structures and types, e.g. closed form control (Tippett & Bao, 2013a) or model predictive control (Tippett & Bao, 2013b), can be adopted. In this approach, the individual process models are only used to verify their dissipativity, not directly in control analysis and design. As the supply rate of plant-wide system is a linear function of the supply rates of each individual process unit and controller, the proposed approach allows the plant-wide control and analysis problems to be formulated in a linear manner. It also allows different control structures (with different controller network topologies) to be dealt with in a unified framework. Therefore, this approach is very scalable, suitable for large complex networks. This is in contrast to some existing methods where the structure imposed by decentralized control, or by certain controller network topologies leads to computational difficulties in analysis and control synthesis. Dynamic QDF supply rates are used to capture much more detailed dynamic system properties than traditional constant (Q , S , R) supply rates, and as such, reduce the conservativeness of dissipativity-based analysis. The proposed supply rate centric approach would not be possible based on the classic constant supply rates as they are too coarse to replace process models to represent system dynamic features. Dynamic supply rates also allow the performance criteria to be encoded directly into the dissipativity properties of the system. As a result, both plant-wide stability and performance can be represented using the same dissipativity/ supply rate framework.
. The matrices Xi
and Yi are constant matrices with eigenvalues chosen to ensure the J-spectral factorizability of the controller supply rates, with 1
5. Discussion and conclusion
1 2
(−Γ11 ) (jω),
where α(ω) is a scalar rational function satisfying
(35)
Appendix. Proofs Proof of Theorem 1. Note that dissipativity with respect to (7) implies
γ2 2
α (ω)I > 2
R(−jω, jω) + S T (−jω, jω)Qˆ −1 (−jω, jω)S (−jω, jω). The decision variables in the above LMI problem are the controller supply rates. Proof. The result follows easily by combining the previous results in this paper. To achieve the desired plant-wide performance, the ith local controller must satisfy the supply rate induced by φci (ζ , η). The conditions above ensure the existence of a controller satisfying this supply rate. As the plant-wide supply rate is linear in the process and controller supply rates, the process supply rates can be determined simultaneously, using the LMI condition in Belur and Trentelman (2002). Once the required controller dissipativity properties are known, suitable controllers can be synthesized individually, e.g., by performing a J-spectral factorization of the controller supply rates and subsequently augmenting a ‘seed’ system to synthesize a network of closed form controllers (as presented in Tippett and Bao (2013a)). Another possible way is to implement the dissipativity constraints in model predictive control algorithms to develop a dissipativity based distributed model predictive control framework (as presented in Tippett and Bao (2013b)).
∞
yˆ T Q yˆ + 2yˆ T S uˆ + uˆ Ruˆ dt ≥ 0.
(A.1)
−∞
Let Qˆ = −Q˜ . If Q˜ < 0, we can then complete the square to obtain
∞
1
1
Qˆ 2 yˆ − Qˆ − 2 S uˆ
2
∞
−∞
uˆ T R + S T Qˆ −1 S uˆ dt .
dt ≤ −∞
By applying the reverse triangle inequality and defining α as a constant such that α 2 I ≥ R + S T Qˆ −1 S we then obtain 1 1 ∥Qˆ 2 yˆ ∥22 − ∥Qˆ − 2 S uˆ ∥22 ≤ α 2 ∥ˆu∥22
(A.2)
which reduces to ∥ˆy∥2 ≤ γ ∥ˆu∥2 with
1 1 γ = ∥Qˆ − 2 ∥2 ∥Qˆ − 2 S ∥2 + α
(A.3)
which is the stated result. To show stability, introduce the storage function Qψ (u, y) > 0. For vanishing input the dissipation inequality becomes Qψ˙ (0, y) ≤ yˆ T Q˜ yˆ .
(A.4)
As Q˜ < 0, this implies Qψ˙ (0, y) < 0, which by Proposition 1 implies asymptotic stability.
M.J. Tippett, J. Bao / Automatica 50 (2014) 44–52
Proof of Theorem 2. The half-line dissipativity of the system coupled with Q (ζ , η) < 0 as per Theorem 1 implies asymptotic stability of the system. Dissipativity implies
∞
Qφ (u, y) dt ≥ 0.
(A.5)
−∞
As compactly supported trajectories are assumed, use Parseval’s theorem to obtain
∞
Qφ (¯u, y¯ ) dω ≥ 0,
−∞ ∞
(A.6)
y¯ T Q (−jω, jω)¯y + 2y¯ T S (−jω, jω)¯u
+ u¯ T R(−jω, jω)¯u dω ≥ 0
(A.7)
where u¯ and y¯ denote the Fourier transforms of u and y respectively. To simplify notation we drop the frequency dependence. Defining Qˆ = −Q and using Q < 0, complete the squares: ∞
1
1
Qˆ 2 y¯ − Qˆ − 2 S u¯
T
1
1
∞
u¯ T R + S T Qˆ −1 S u¯ dω.
(A.8)
−∞
Letting F T F = R+S T Qˆ −1 S and using the reverse triangle inequality, 1
1
∥Qˆ 2 y¯ ∥ ≤ ∥Qˆ − 2 S u¯ ∥ + ∥F u¯ ∥. As ∥F u¯ ∥ ≥ ∥Qˆ
− 12
(A.9)
S u¯ ∥, we write
1
∥Qˆ 2 y¯ ∥ ≤ 2∥F u¯ ∥.
(A.10)
Using α defined in the Theorem statement we obtain 1
∥Qˆ 2 y¯ ∥ ≤ γ α∥¯u∥,
(A.11)
which is equivalent to a L∞ norm condition on the system G(s). Using the assumption that the system has positive semidefinite storage, the system is asymptotically stable according to Proposition 1. Therefore, the following H∞ condition is valid:
1 1 Qˆ 2 (ω)G ≤ γ α ∞
∥N (jω)¯ypw ∥ ≤ γ ∥d(jω)d¯ ∥.
(A.16)
Since d(jω) is scalar, we have
∥N (jω)¯ypw ∥ ≤ γ |d(jω)|∥d¯ ∥ 1 ≤ γ ∥d¯ ∥ N ( j ω)¯ y pw d(jω) 1 N ( j ω) T ( j ω) ≤ γ, d(jω)
(A.17) (A.18)
(A.19)
¯ (jω), which is the required L∞ norm where y¯ pw (jω) = T (jω)d condition. Note that the storage function Qν (d, ypw ) is positive semidefinite. The instantaneous form of the dissipation inequality requires Qµ (d, ypw ) ≥ Qν˙ (d, ypw ).
(A.20)
For a vanishing input, this becomes:
Qˆ 2 y¯ − Qˆ − 2 S u¯ dω
−∞
≤
This leads to
L∞
−∞
51
(A.12)
2 d ypw ≥ Qψ˙ (0, ypw ), − N dt
(A.21)
which implies Qψ˙ (0, ypw ) < 0. According to Proposition 1, the plant-wide system, is asymptotically stable, in which case the L∞ and H∞ norms are equivalent. Proof of Lemma 2. The argument is constructed based on the positive (semi-)definite case. The negative (semi-)definite case is completely analogous. It is clear from the form of the polynomial that if all coefficient matrices are positive semidefinite (and the A0 and A2n terms are definite) then the polynomial is strictly positive ∀ω ∈ R. This is true because the polynomial has only even powers of ω. Suppose that A2 is non-positive semidefinite (and all other coefficient matrices are positive semidefinite). Then for |ω| < 1, A0 + A2 ≥ 0 implies xT (A0 + A2 ) x ≥ 0
∀x ∈ R •
xT A0 x + xT A2 x ≥ 0 xT A0 x + xT A2 xω2 ≥ 0
(as |ω| < 1 and A0 > 0)
xT A0 + A2 ω2 x ≥ 0
where y¯ (jω) = G(jω)¯u(jω).
A0 + A2 ω2 ≥ 0.
Proof of Proposition 4. The system T (s) is the system from d to ypw . Dissipativity of T (s) implies
Given A0 > 0, A0 + A2 ≥ 0 it can then be shown by the same reasoning that A0 + A2 + A4 ≥ 0 implies A0 + A2 ω2 + A4 ω4 ≥ 0 for |ω| < 1 even if A4 is non-positive semidefinite. Applying this iteratively, it is clear that Condition (27) and Condition (28) imply that A(ω) > 0 ∀ |ω| < 1. Similarly for |ω| ≥ 1, A2n + A2n−2 ≥ 0 implies
∞
−
yTpw N T
yTpw N T −∞
d
d
∞
γ d d
≤ −∞
N
d
d dt
d
ypw
dt
d dt
N
dt
2 T
dt
∞
dt
−∞
+ γ 2 dT d
d
d
d dt ≥ 0,
dt
d
(A.13)
dt
xT A2n ω2n x + xT A2n−2 xω2n−2 ≥ 0 d dt .
(A.14)
As compactly supported trajectories are assumed, take the Fourier ¯ and y¯ pw and apply Parseval’s transform of d and ypw , denoted d theorem to obtain:
∞
y¯ Tpw N T (−jω) N (jω) y¯ pw −∞
∞
γ 2 d¯ T d (−jω) d (jω) d¯ dt .
≤ −∞
∀x ∈ R•
xT A2n x + xT A2n−2 x ≥ 0
ypw d
xT (A2n + A2n−2 ) x ≥ 0
(A.15)
(A.22)
xT A2n ω2n + A2n−2 ω2n−2 x ≥ 0
A2n ω2n + A2n−2 ω2n−2 ≥ 0. As in (A.22) |ω| ≥ 1 ⇒ ω2n ≥ ω2n−2 and A2n > 0. As above, using this reasoning iteratively, it can be seen that Condition (27) and Condition (29) imply that A(ω) > 0 ∀ |ω| ≥ 1. Thus we can conclude that A(ω) > 0 ∀ω ∈ R which in turn implies det(A(ω)) = 0 has no roots ∀ω ∈ R. As the roots of det(A(ω)) occur only when the inertia of A(ω) changes, which it clearly cannot if A(ω) > 0 ∀ω ∈ R.
52
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References Alonso, A. A., & Ydstie, B. E. (1996). Process systems, passivity and the second law of thermodynamics. Computers & Chemical Engineering, 20, 1119–1124. Bao, J., Jillson, K.R., & Ydstie, B.E. (2007). Passivity based control of process networks. In Proc. DYCOPS (pp. 67–72). Belur, M.N., & Trentelman, H.L. (2004). Synthesis of dissipative behaviors with dynamics in the weighting matrices. In Proc. 16th international symposium on mathemarical theory of networks & systems. Leuven. Belur, M. N., & Trentelman, H. L. (2002). Algorithmic issues in the synthesis of dissipative systems. Mathematical and Computer Modeling of Dynamical Systems, 8(4), 407–428. Bristol, E. H. (1966). On a new measure on interaction for multivariable process control. IEEE Transactions on Automatic Control, 11, 133–134. Carlson, D. H., & Schneider, H. (1962). Inertia theorems for matrices: the semidefinite case. American Mathematical Society. Bulletin, 68(5), 479–483. Cheng, Y., & Li, S. (2010). A new pairing method for multivariable processes. Industrial & Engineering Chemistry Research, 49(13), 6115–6124. Farschman, C. A., Viswanath, K. P., & Ydstie, B. E. (1998). Process systems and inventory control. AIChE Journal, 44(8), 1841–1857. Goodwin, G. C., Haimovich, H., Quevedo, D., & Welsh, J. (2004). A moving horizon approach to networked control system design. IEEE Transactions on Automatic Control, 49(9), 1427–1445. Grosdidier, P., & Morari, M. (1986). Interaction measures for systems under decentralized control. Automatica, 22(3), 309–319. Hangos, K. M., Alonso, A. A., Perkins, J. D., & Ydstie, B. E. (1999). Thermodynamic approach to the structural stability of process plants. AIChE Journal, 45(4), 802–816. Kumar, A., & Daoutidis, P. (2002). Nonlinear dynamics and control of process systems with recycle. Journal of Process Control, 12(4), 475–484. Langbort, C., & Gupta, V. (2009). Minimal interconnection topology in distributed control design. SIAM Journal on Control and Optimisation, 48(1), 397–413. Liu, J., Chen, X., Muñoz de la Peña, D., & Christofides, P. D. (2010). Sequential and iterative architectures for distributed model predictive control of nonlinear process systems. AIChE Journal, 56, 2137–2149. Luyben, W. L., Tyréus, B. D., & Luyben, M. L. (1998). Plantwide process control. NY: McGraw-Hill. Manousiouthakis, V., Savage, R., & Arkun, Y. (1989). Synthesis of decentralised process control structures using the concept of block relative gain. AIChE Journal, 32(6), 991–1003. Mercangöz, M., & Doyle, F. (2007). Distributed model predictive control of an experimental four-tank system. Journal of Process Control, 17(3), 297–308. Moylan, P. J., & Hill, D. J. (1978). Stability criteria for large-scale systems. IEEE Transactions on Automatic Control, 23(2), 143–149. Pendharkar, I., & Pillai, H. K. (2008). Systems with sector bound nonlinearities: a behavioral approach. Systems & Control Letters, 57(2), 112–122. Rojas, O. J., Setiawan, R., Bao, J., & Lee, P. L. (2008). On dissipativity, passivity and dynamic operability of nonlinear processes. Journal of Process Control, 18(5), 515–526. Rojas, O. J., Setiawan, R., Bao, J., & Lee, P. L. (2009). Dynamic operability analysis of nonlinear process networks based on dissipativity. AIChE Journal, 55(4), 963–982. Samyudia, Y., Lee, P. L., & Cameron, I. T. (1994). A methodology for multi-unit control design. Chemical Engineering Science, 49(23), 3871–3882. Skogestad, S. (2004). Control structure design for complete chemical plants. Computers & Chemical Engineering, 28, 219–234. Skogestad, S., & Morari, M. (1989). Robust performance of decentralized control systems by independent designs. Automatica, 25(1), 119–125.
Stewart, B. T., Venkat, A. N., Rawlings, J. B., Wright, S. J., & Pannocchia, G. (2010). Cooperative distributed model predictive control. Systems & Control Letters, 59, 460–469. Swarnakar, A., Marquez, H., & Chen, T. (2009). Multi-loop control synthesis for unstable systems and its application: an approach based on the µ interaction measure. International Journal of Robust & Nonlinear Control, 19, 1721–1744. Tippett, M. J., & Bao, J. (2013a). Dissipativity based distributed control synthesis. Journal of Process Control, 23(5), 755–766. Tippett, M. J., & Bao, J. (2013b). Distributed model predictive control based on dissipativity. AIChE Journal, 59(3), 787–804. Trentelman, H. L., & Rapisarda, P. (1999). New algorithms for polynomial J-spectral factorization. Mathematical Control Signals Systems, 12, 24–61. Vadigepalli, R., & Doyle, F. (2003). A distributed state estimation and control algorithm for plantwide processes. IEEE Transactions on Control Systems Technology, 11, 119–127. Vidyasagar, M. (1981). Input–output analysis of large-scale interconnected systems. Berlin: Springer-Verlag. Willems, J. C. (1972). Dissipative dynamical systems, part I: general theory. Archive for Rational Mechanics and Analysis, 45, 321–351. Willems, J. C., & Trentelman, H. L. (1998). On quadratic differential forms. SIAM Journal on Control and Optimisation, 36(5), 1703–1749. Ydstie, B. E. (2002). New vistas for process control: integrating physics and communication networks. AIChE Journal, 48(3), 422–426. Ydstie, B. E., & Alonso, A. A. (1997). Process systems and passivity via the clausiusplanck inequality. Systems & Control Letters, 30(5), 253–264. Zhou, K., Doyle, J., & Glover, K. (1996). Robust and optimal control. New Jersey: Prentice Hall.
Michael J. Tippett received a B.E. in Industrial Chemistry/B.Com. in Business Economics with first class honors and the University Medal from the University of New South Wales (UNSW), Sydney, Australia in 2009. He is currently pursuing his Ph.D. in the area of distributed control systems in the School of Chemical Engineering at UNSW. His current research interests include: distributed and decentralized control, adaptive control, model predictive control, dissipativity-based analysis and control and their applications to chemical processes.
Jie Bao obtained his B.Sc. and M.Sc. degrees in Electrical Engineering from Zhejiang University, China, in 1990 and 1993 respectively. In 1998, he received the Ph.D. degree in Chemical Engineering (Process Control) from the University of Queensland, Australia. He spent one year at University of Alberta as a postdoctoral fellow and then joined the faculty at the University of New South Wales (UNSW), Sydney, Australia. He is currently a Full Professor in the School of Chemical Engineering, UNSW. His research interests include distributed and decentralized control, robust control, fault-tolerant control, dissipativity-based process control and control of industrial processes including aluminum smelting, mineral processing, membrane separation and flow batteries. He is an Associate Editor of the Journal of Process Control.
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Automatica journal homepage: www.elsevier.com/locate/automatica
Control of plant-wide systems using dynamic supply rates✩ Michael J. Tippett, Jie Bao 1 School of Chemical Engineering, The University of New South Wales UNSW, Sydney, NSW 2052, Australia
article
info
Article history: Received 24 August 2011 Received in revised form 6 July 2012 Accepted 30 August 2013 Available online 20 November 2013 Keywords: Process control Distributed control Linear networks Linear output feedback H∞ control
abstract This paper presents a framework for analysis of plant-wide processes from a network perspective. Using the concept of dissipativity, the conditions for plant-wide input–output stability and performance are developed, based on the dissipativity of individual subsystems and the topology of the network of the plantwide process. Dynamic supply rates, expressed as quadratic differential forms, are proposed not only to render dissipativity based analysis less conservative but also allow the dynamic plant-wide performance criteria to be specified in terms of desired closed loop supply rates. The links between the plant-wide supply rate, finite L2 gain in an extended input–output space and weighted H∞ norm are explored in this paper. These results lay a foundation for a supply rate-centric approach to plant-wide distributed control. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction Largely driven by increasingly tight economic and environmental requirements, modern chemical plants are becoming increasingly complex, often with dozens of process units with material recycling and energy integration. From a process control point of view, recycling streams can be understood as positive feedback loops within the process network, which have a deleterious impact on control performance (Luyben, Tyréus, & Luyben, 1998). These strong interactions between process units are a key feature of plant-wide process control problems, and are a challenge to control practice due to high sensitivity to disturbances and possible plant-wide instability (Kumar & Daoutidis, 2002). Another important challenge in plant-wide process control is the scale of the problem, which can make centralized control systems computationally difficult or infeasible (Skogestad, 2004). Some existing approaches for control of plant-wide processes deal with the interactions between unit processes as uncertainties, as presented in Grosdidier and Morari (1986), Samyudia,
✩ This work is partially supported by the ARC Discovery Project DP1093045. The first author would like to acknowledge the financial support of the Australian Postgraduate Award, as well as the ESA and UNSW Excellence Awards provided by UNSW. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Denis Dochain under the direction of Editor Frank Allgöwer. E-mail addresses: [email protected] (M.J. Tippett), [email protected] (J. Bao). 1 Tel.: +61 2 9385 6755; fax: +61 2 9385 5966.
0005-1098/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.automatica.2013.09.028
Lee, and Cameron (1994), Swarnakar, Marquez, and Chen (2009) and Skogestad and Morari (1989). A decentralized controller is then designed to stabilize the plant-wide system subject to these uncertainties based on robust control theory. This approach can greatly simplify implementation of the control system. To help facilitate this, interaction measures for plant-wide process systems have been developed, e.g. Bristol (1966), Cheng and Li (2010) and Manousiouthakis, Savage, and Arkun (1989). These interaction measures are useful for input–output pairing and estimating the efficacy of decentralized control schemes. However, this approach is inherently conservative as the known interactions are treated as unknown. Distributed control structures have gained attention in recent years. For example, an approach requiring minimal communication between controllers is presented in Goodwin, Haimovich, Quevedo, and Welsh (2004); the optimal controller network topology is determined in Langbort and Gupta (2009); distributed estimation and model predictive control methods are developed in Mercangöz and Doyle (2007) and Vadigepalli and Doyle (2003). Distributed model predictive control has received much attention recently, e.g. Liu, Chen, Muñoz de la Peña, and Christofides (2010) and Stewart, Venkat, Rawlings, Wright, and Pannocchia (2010). One promising approach to plant-wide analysis is based on a network perspective, decomposing a plant-wide process into individual process units interacting through a network with a known topology (e.g., Bao, Jillson, & Ydstie, 2007; Rojas, Setiawan, Bao, & Lee, 2009; Ydstie, 2002). A key advantage of this approach is its scalability, as it allows analysis and control design based on simpler subsystems and their interconnections.
M.J. Tippett, J. Bao / Automatica 50 (2014) 44–52
45
Dissipativity theory, introduced in Willems (1972), has become an important analysis and control design tool. It is particularly suitable for the network approach to plant-wide control, as the plant-wide stability and performance can be determined based on the dissipativity of individual subsystems and the network topology, thus simplifying control design. A dynamical system with input u ∈ Rp , output y ∈ Rq (both with compact support) and state x ∈ Rn respectively is said to be dissipative if there exists a function defined on the input and output variables, called the supply rate s(u, y) and positive semidefinite (at least once differentiable) function defined on the state space, called the storage function V (x) ≥ 0, such that
2. Dissipativity, stability and performance
V˙ (x(t )) ≤ s (u(t ), y(t )) .
with w = yT uT
(1)
The following quadratic (Q , S , R)-type supply rate is commonly used: s(u, y) = yT Qy + 2yT Su + uT Ru,
(2)
where Q = Q T , S and R = RT are real matrices. Results on the application of dissipativity theory to large-scale systems analysis and decentralized control design has been reported in the literature. For example, in Moylan and Hill (1978), Rojas, Setiawan, Bao, and Lee (2008) and Vidyasagar (1981) where dissipativity is used as an enabling tool for stability and operability analysis of large-scale systems. In Hangos, Alonso, Perkins, and Ydstie (1999), a special case of dissipativity, passivity, is used to show the stability of plant-wide nonlinear process systems. This is achieved using the links between passivity and the mass and energy balances underlying process systems. These links were developed in Alonso and Ydstie (1996) and Ydstie and Alonso (1997), where thermodynamics is used to develop physically motivated storage functions. These concepts are used for plant-wide control in Farschman, Viswanath, and Ydstie (1998). However, the above passivity/dissipativity approaches can be very conservative, as the constant supply rates shown in (2) can only provide a coarse description of the systems dynamic features. This work presents a new approach to plant-wide stability and performance analysis and design based on network decomposition and the dissipativity of the subsystems. Both the plant-wide stability and performance specifications (in the form of weighted H∞ norms) are achieved by encoding them as desired closed loop supply rates. Controllers can then be designed to achieve these supply rates, and hence, the desired closed-loop stability and performance properties. Central to the proposed approach is the use of dynamic supply rates, expressed as quadratic differential forms (QDFs). As the QDFs represent a more general form of dissipativity (Pendharkar & Pillai, 2008; Willems & Trentelman, 1998), they capture more detailed dynamic properties and therefore lead to far less conservative dissipativity based conditions than those based on constant supply rates (e.g., in (2)), as shown in Tippett and Bao (2013a). The QDFs also allow the dynamic plant-wide performance criteria to be specified in terms of desired closed loop supply rates as was done in our previous work (Tippett & Bao, 2013b). In the proposed approach, the models of individual process units are only used to validate their dissipativity. All analysis and control design is performed based on the dissipativity of the process units rather than detailed process models. This leads to a supply rate-centric approach to plant-wide distributed control that can deal with arbitrary process network topologies and control structures. The remainder of this paper is structured as follows. In Section 2, a brief overview of dissipativity theory is presented. A dissipativity based decomposition of the plant-wide system and controller network is presented in Section 3. In Section 4 plantwide stability and performance results based on this decomposition are presented.
The key concept used in this paper is dissipativity, which is defined in the context of behavioral systems theory as follows: Definition 1 (Willems & Trentelman, 1998). A controllable system is said to be dissipative with storage function Qψ , and supply rate Qφ , if: ∞
Qφ (w) d t ≥ Qψ (w),
(3)
−∞
for all allowable trajectories of the system with compact support,
T
, a vector of the inputs and outputs.
In this paper we are concerned with the case where Qψ , the storage function, is positive semidefinite. This is equivalent to halfline dissipativity on R− (Willems & Trentelman, 1998). Note that half-line dissipativity on R− implies dissipativity in the general sense in of Definition 1. In this context Qφ and Qψ are quadratic differential forms as shown below: Qφ (w) =
kmax lmax k d k=0 l=0
dt
w k
T
φkl
dl dt
w , l
(4)
where φkl are constant, symmetric, matrices and are the coefficient matrices of the two-variable polynomial matrix φ(ζ , η). The d T dt
indeterminates ζ and η refer to
d and dt respectively. kmax and T d d and dt . The degree of the supply lmax are the highest order of dt rate is defined as max kmax lmax . The matrix is said to
(
,
)
φ(ζ , η)
induce the QDF (Willems & Trentelman, 1998). Definition 2 (Willems & Trentelman, 1998). Let φ ∈ Rq×q [ζ , η], with φ symmetric. The QDF Qφ (w) is called positive, denoted by φ > 0, if Qφ (w) ≥ 0 for all w, and the only w for which Qφ (w) = 0 is w = 0. A QDF is negative definite, φ < 0, if and only if −φ > 0. It worth pointing out that the derivative a of QDF is itself a QDF, d d i.e., dt Qφ = Q d φ , where dt φ = (ζ +η)φ . In simple terms, QDFs are dt
quadratic functions of the inputs and outputs and a finite number of their derivatives. They can be understood as an extension of the commonly used (Q , S , R) shown in (2) that include derivative terms. Because of this, QDF supply rates capture more detailed system information than constant supply rates, allowing for more in depth analysis and less conservative results. The extended variables of input u and output y will be used throughout this paper and are defined as follows:
yˆ =
y
T
u
dy
T
dt
uˆ =
T
du dt
T
...
...
dn˜ y
T T
dt n˜
dm˜ u
T T
(5)
dt m˜
˜ less than or equal to the order of the for some integers n˜ and m system in y and u respectively. Using these variables, the quadratic from (4) can be rewritten as: Qφ
y u
T T yˆ yˆ yˆ Q˜ = ˆ φ˜ ˆ = ˆ u u u S˜ T
S˜ R˜
yˆ . uˆ
(6)
The block matrix φ˜ is referred to as the coefficient matrix of φ , as its entries are the constant coefficient matrices of the polynomials of the indeterminates ζ and η in φ(ζ , η). Methods for determining dissipativity of linear systems with QDF supply rates and storage functions include Pick matrix and frequency domain conditions presented in Willems and Trentelman (1998) and a Linear
46
M.J. Tippett, J. Bao / Automatica 50 (2014) 44–52
Matrix Inequality (LMI) based algorithm presented in Belur and Trentelman (2002). The conditions for asymptotic stability is given below: Proposition 1 (Willems & Trentelman, 1998). An autonomous linear system is asymptotically stable if and only if there exists a QDF storage function Qψ induced by ψ ∈ Rq×q [ζ , η] such that for all trajectories ˙ < 0. of the system, ψ ≥ 0 and ψ The asymptotic stability is a notion of input–output stability used in the remainder of this paper. It implies that for vanishing input, the output and its derivatives asymptotically converge to zero. This is equivalent to the classical definition of input–output stability. It implies the asymptotic Lyapunov stability of the state space of the minimal realization (or any realization with the zero state detectability condition) of the input–output system. The stability condition based on QDF supply rates is given below: Theorem 1. A system which is dissipative with positive semidefinite storage function Qψ with respect to supply rate
T Q˜ yˆ Qφ (ˆy, uˆ ) = ˜S T ˆu
S˜ R˜
yˆ uˆ
(7)
is asymptotically stable with finite L2 gain from uˆ to yˆ
∥ˆy∥2 ≤ γ ∥ˆu∥2
(8)
Given a linear system, its dissipativity with respect to dynamic supply rates can be linked to its weighted H∞ norm. As shown in Belur and Trentelman (2004), the dissipativity of a singleinput–single-output linear system G(s) with respect to the supply rate Qφ (y, u) induced by:
φ(ζ , η) =
−n(ζ )n(η) 0
0
γ d(ζ )d(η) 2
,
φ ∈ R2×2 [ζ , η] n(s)
implies that the L∞ norm ∥WG∥L∞ ≤ γ , where W (s) = d(s) . This result is extended to the multiple-input–multiple-output case with H∞ norm and without a block-diagonal condition on φ(ζ , η), as shown below: Theorem 2. A linear system y = G(s)u with u ∈ Rp and y ∈ Rq , which is dissipative with respect to a QDF supply rate induced by the polynomial matrix φ(ζ , η)
φ(ζ , η) =
Q (ζ , η) S T (ζ , η)
S (ζ , η) , R(ζ , η)
(13)
with positive semidefinite storage function and Q (ζ , η) < 0 (where the dimensions of Q (ζ , η), S (ζ , η) and R(ζ , η) are p × p, p × q and q × q respectively), satisfies the following H∞ norm condition:
N G ≤ γ , α ∞
(14) 1
where, N (jω) = Qˆ 2 (−jω, jω), with Qˆ (−jω, jω) = −Q (−jω, jω) and a scalar α(ω) > 0 satisfying
with
1 1 γ = ∥(−Q˜ )− 2 ∥2 ∥(−Q˜ )− 2 S˜ ∥2 + α α 2 I ≥ R˜ − S˜ T Q˜ −1 S˜
(9)
γ2
(10)
2
α 2 (ω)I ≥ R(−jω, jω) + S T (−jω, jω)Qˆ −1 (−jω, jω)S (−jω, jω),
if Q˜ < 0.
∀ω.
Proof. See Appendix.
Proof. See Appendix.
The above result is extended to general interconnected systems in Section 4. A negative feedback system of a process with supply rate given by with supply rate induced by (6) and a controller
Example 2. Consider a heat exchanger model (Rojas et al., 2009):
φc (ζ , η) = Q˜ cl =
Qc (ζ , η) ScT (ζ , η)
Q˜ + R˜ c S˜ T − S˜c
Sc (ζ , η)
Rc (ζ , η)
S˜ − S˜cT R˜ + Q˜ c
is stable if:
< 0.
(11)
Example 1. Consider a system with transfer function G(s) = 0.5(1−0.1s) . As this system is non-minimum phase, a constant 100s+1 (Q , S , R) supply rate has R ≥ 0. The stability condition in Moylan and Hill (1978) (Qcl < 0) requires the Q matrix of the controller Qc < 0 to ensure closed loop stability. This implies the controller must have finite gain, not allowing integral control. Using Theorem 1, it can be shown that the system is half-line dissipative with respect to the supply rate induced by
−0.61 − 0.06ζ η 0.5
0.5 −0.001 + 0.29ζ η
(12)
and that a PI controller with Kp = 0.05, Ki = 0.4 is half-line dissipative with respect tothe supply rate induced by θ (ζ , η) = −0.3ζ η 0.5
1 s2
×
The following example shows the dynamic supply rates reduce the conservativeness in dissipativity based stability analysis.
φ(ζ , η) =
G(s) =
0.5 0.53˙ + 0.06˙ ζ + 0.06˙ ζ
. Combining the process and controller
supply rates it is clear that (11) is negative definite. Thus, the closed loop system is asymptotically stable, showing that integral action is admitted.
+ 48.22s + 412.1 11.08s + 411.3 04172
0.8107 . 37.08s + 411.7
It can be shown that it is half-line dissipative with respect to the supply rate induced by
φ(ζ , η) =
−(0.1ζ + 1)(0.1η + 1)I2 02×2
02×2 γ 2 (10ζ + 1)(10η + 1)I2
,
for any γ ≥ 1. According to Theorem 2, the system satisfies the .1s+1 . weighted H∞ norm ∥w G∥∞ ≤ 1 with w(s) = 010s +1 These stability and performance results form the basis of the plantwide stability and performance conditions (in Section 4) based on the dissipativity of the overall system. The plant-wide dissipativity is determined from the dissipativity of the individual processes and controllers and their network structures as presented in Section 3. 3. Network perspective of plant-wide process control The problem considered here is the analysis and control of large-scale systems with subsystems interconnected through a fixed network topology to meet given plant-wide H∞ performance criteria. An area of application is chemical process systems, where individual units are interconnected through mass and energy flows. An example of such a system is the reactor–separator process depicted in Fig. 1. The plant-wide system consists of two
M.J. Tippett, J. Bao / Automatica 50 (2014) 44–52
47
Fig. 3. Partitioning of controller inputs and outputs.
Fig. 1. Reactor–separator system.
Fig. 4. Network view of plant-wide process with distributed control. Fig. 2. Partitioning of process manifest variables.
unit processes interconnected through a fixed network topology, including a recycling stream. The recycling can induce strong interactions between the process units. The aim is to formulate scalable conditions that the control system must satisfy to ensure the plant-wide closed loop system achieves a given weighted H∞ norm condition. In this work, only unidirectional flows in the process network are considered. This is justified in practice where flows between units are usually forced by pumps or compressors. As shown in Fig. 2, the input to each process unit is partitioned into three components: the input from interconnected units uP , external disturbances d and manipulated input uL . Meanwhile the output of each unit consists of all interconnecting and measured outputs. Using the example in Fig. 1, the inputs of the CSTR can be partitioned into interconnecting inputs (inlet flowrates of components), disturbances (inlet temperature of water in jacket) and manipulated input (flowrate of water in jacket). To consider distributed control, the controllers are represented as two port systems with both local and remote inputs and outputs as shown in Fig. 3. The local signals directly interface with the process whilst the remote signals interface with other controllers. The controller input includes local sensor output uC and information from remote controllers uR . The controller output includes the signals sent to local actuator yL and to other controllers yR . To again relate this to the process depicted in Fig. 1, the local controller for the reactor has local input and output of the temperature in the reactor and jacket flowrate respectively. In addition, it sends and receives signals to/from the distillation column controller which it uses to adjust its local manipulated variable. The entire plant-wide process system of n process units with n controllers is depicted in Fig. 4. Each process unit is stacked diagonally to form the large system not including the interconnection
topology, P˜ . The inputs and outputs of this system are the vectors consisting of the inputs and outputs of each system concatenated with one another. The output of this diagonal system is y = col(y1 . . . yn ). Signals uP , uL d and uL are defined in a similar way. The topology of the interconnections between process units is represented by the matrix Hp . Due to the assumption of time invariant interconnections, Hp is a constant matrix with elements of either 0 or 1. Using the example in Fig. 1, Hp =
0 I
I 0
. The
structure of this Hp implies that the output of the first process is the input into the second system, and the output of the second process is the input into the first system. Fp and FI select the interconnecting and measured outputs respectively. As such, Fp and FI are constant matrices with elements of either 0 or 1. The controller network is represented in a similar way. The individual controllers are stacked diagonally to form C˜ (a block diagonal system with the ith diagonal block being the ith controller) as the overall controller without interconnections. The topology of the controller network is captured by matrix Hc , in a similar manner as Hp . This network representation of the control system allows a unified approach to plant-wide process control, encompassing decentralized, distributed and fully centralized control. In the case of decentralized control, Hc = 0 while fully centralized control is achieved by setting Hc as a full matrix. For the distributed control cases, Hc is sparse. With appropriate changes, all developments in the remainder of this paper then follow for all control structures once the necessary changes have been made without increasing the complexity of control design. Assume that a QDF supply rate of the ith process is given as Qφi (uP , d, uL , y), which is induced by a symmetric two-variable polynomial matrix conformally partitioned as:
φi (ζ , η) =
Qi (ζ , η) SiT (ζ , η)
Si (ζ , η) , Ri (ζ , η)
(15)
48
M.J. Tippett, J. Bao / Automatica 50 (2014) 44–52
with
Qi (ζ , η) = Q (ζ , η);
Si (ζ , η) = SI (ζ , η)
RII (ζ , η) Ri (ζ , η) = RTId (ζ , η) RTIL (ζ , η)
RId (ζ , η) Rdd (ζ , η) RTdL (ζ , η)
Sd (ζ , η)
SL (ζ , η)
RIL (ζ , η) RdL (ζ , η) . RLL (ζ , η)
Q(ζ , η) S T (ζ , η)
Φ (ζ , η) =
with
(16)
Sci . Rci
Qc ScT
Θ (ζ , η) =
Sc Rc
Qcill Qcirl
Qcilr Qcirr
Rcill Rcirl
Rci
Rci =
Sci =
Scill Scirl
Scilr Sci
and
Lemma 1. Consider the interconnected system as shown in Fig. 4. If the collection of (unconnected) process units P˜ is dissipative with respect to supply rate QΦ (induced by (16)) and storage function QΨ , and the collection of unconnected controllers C˜ is dissipative with respect to the supply rate QΘ (induced by (18)) and storage function QΣ , then the plant-wide system from all disturbances d to all process
T
output and controller output ypw = yT , yTL , yTR , is dissipative with storage function Qν = QΨ +Σ and supply rate Qµ and induced by
Γ11 (ζ , η) µ(ζ , η) = T Γ12 (ζ , η) X11
Γ11 (ζ , η) = XT12 XT13
X12 X22 XT23
X22 = RLL + Qcll , X23 = Qclr + Sclr Hc ,
(18)
where the superscript l relates to signals to/from local actuators/sensors and the superscript r to signals to/from remote controllers. Then, Φ (ζ , η) and Θ (ζ , η) can be calculated from φi (ζ , η) and θi (ζ , η) based on their definition. The storage function of P˜ , Ψ (ζ , η) (C˜ , Σ (ζ , η)), is the sum of the n individual process (controller) storage functions ψi (ζ , η) (σi (ζ , η)), i = 1 . . . n. The relation between the storage function of the ith process and controller, induced by ψi (ζ , η) and σi (ζ , η), and the storage function of P˜ and C˜ , induced by Ψ (ζ , η) and Σ (ζ , η), is completely analogous to that of the supply rates. The interconnection relations are shown in Fig. 4. The effects of interactions between all process and control units are represented by their input–output properties as captured by their dissipativity, and in particular, their supply rates and the network topologies rather than their detailed models. This enables the entire plantwide system to be modeled as the interconnection of two networks of processes and controllers. The supply rate of the overall plantwide system with interconnections between process units and the controller network can be determined using the following lemma.
FI ,
Proof. The result is easily shown by adding the two dissipation inequalities, QΦ ≥ QΨ˙ and QΘ ≥ QΣ˙ . Then substituting in the interconnection relations to eliminate all variables except the external inputs and process and controller outputs.
, Rcilr , rr
X13 = Sclr + HcT Rclr
T
X33 = Qcrr + Scrr Hc + HcT Scrr + HcT Rcrr Hc .
can be defined in a similar way to the supply rate of P˜ , with Qc = diag(Qc1 , . . . , Qcn ), Sc = diag(Sc1 , . . . , Scn ), Rc = diag(Rc1 , . . . , Rcn ). Due to the distributed control structure, these submatrices are structured as follows:
Qci =
(17)
The supply rate of C˜ , induced by
X12 = SL + FpT HpT RIL + FIT Scll ,
T
Qci SciT
X11 = Q + SI Hp Fp + FpT HpT SIT + FpT HpT RII Hp Fp , +FIT Rcll FI , T
S (ζ , η) , R(ζ , η)
with Q = diag(Q1 , . . . , Qn ), S = diag(S1 , . . . , Sn ) and R = diag(R1 , . . . , Rn ). Assume that the supply rate of the ith controller is induced by
θi (ζ , η) =
0
Γ22 (ζ , η) = Rdd ,
Define the supply rate of system P˜ as a QDF induced by the matrix:
Sd + FpT HpT RId T , Γ12 (ζ , η) = RdL
Γ12 (ζ , η) , Γ22 (ζ , η) X13 X23 , X33
(19)
(20)
Once the interconnection topologies Hp , Hc are known together with the filters FI and Fp , conditions (19) and (20) are linear in the supply rates of the individual processes and controllers. This allows the conditions on the plant-wide dissipativity to be formulated as LMI constraints, which can be efficiently solved. Using the plantwide dissipativity properties developed in this section, plant-wide stability and performance conditions based on those in Section 2 are now presented. 4. Plant-wide stability and performance 4.1. Plant-wide stability For the process and control networks shown in Fig. 4, the plantwide stability can be determined by the following proposition. Proposition 2. Consider the plant-wide system with control depicted in Fig. 4. Assume that the plant-wide system from all disturbances d
T
to all process and controller output ypw = yT , yTL , yTR is dissipative with respect to the supply rate Qµ (w) induced by the polynomial
Γ11 (ζ , η) T Γ12 (ζ , η)
Γ12 (ζ , η) Γ22 (ζ , η)
matrix µ(ζ , η) =
(as per Lemma 1) and the
storage function induced by ν(ζ , η). The plant-wide system is asymptotically stable (in the sense of Proposition 1) with finite L2 gain from the extended variable of ˆ to the extended variable of the plant-wide output, the disturbance, d, yˆ pw (as defined in (5)), i.e.
∥ˆypw ∥2 ≤ γ ∥dˆ ∥2 with γ
(21) 1
1
− − = ∥Γ¯ 11 2 ∥2 ∥Γ¯ 11 2 Γ˜ 12 ∥ + α
and α being a positive
T ¯ constant satisfying α 2 I ≥ Γ˜ 22 + Γ˜ 12 Γ11 Γ˜ 12 and Γ¯ 11 = −Γ˜ 11 if the following conditions are satisfied:
−1
(1) Γ˜ 11 , the coefficient matrix of Γ11 (ζ , η), is negative definite and (2) ν(ζ , η) ≥ 0. Proof. The proof follows that of Theorem 2, using the supply rate and storage function for the plant-wide system with control derived in Lemma 1. Remark 1. It is clear from (20) that the dissipativity of the overall system depends on the topologies of the process and controller networks, as well as the dissipativity properties of each system.
M.J. Tippett, J. Bao / Automatica 50 (2014) 44–52
This facilitates the view of the overall system as two interacting networks, for which stability and performance objectives are achieved by a matching of dissipativity properties. Both plant-wide stability and performance conditions can be represented as certain input–output gain conditions, conveniently represented by the dissipativity condition given in Proposition 2. Theorem 1 and Proposition 2 are less conservative than existing dissipativity based stability results in the literature. This is particularly important in control of interconnected plant-wide systems as conservativeness in each subsystem will accumulate at the plantwide level. 4.2. Plant-wide performance Proposition 2 also gives a condition on plant-wide dynamic performance which is represented by the upper bound on the gain between the extended input and output of the plant-wide closed-loop system. As such, Proposition 2 allows both plant-wide stability and performance specifications to be addressed in a single dissipativity condition, which facilitates control design. The gain bound in (21) on the extended input and output space, however, is not necessarily a familiar measure. The link between the above dissipativity condition and weighted H∞ norm bounds on the system in the original input–output space is presented below. The notations ∥.∥L∞ and ∥.∥∞ are used to denote the L∞ and H∞ norms respectively, as defined in Zhou, Doyle, and Glover (1996). Theorem 2 is applied to networked systems by using Lemma 1. This allows the analysis of the plant-wide performance achieved by a control system in terms of weighted H∞ norms. Proposition 3. Consider a plant-wide system as shown in Fig. 4 with the transfer function matrix of the system from all disturbances d to all
T
process and controller outputs ypw = yT , yTL , yTR , denoted as T (s). If T (s) is dissipative with a positive semidefinite storage with respect to the supply rate induced by µ(ζ , η) in Lemma 1 with Γ11 < 0, then it satisfies the following H∞ norm condition:
1 NT ≤ γ , α ∞
(22)
with α(ω) > 0 being a scalar satisfying, for all ω,
γ2 2
α 2 (ω)I ≥ Γ22 (−jω, jω) −1 T + Γ12 (−jω, jω)Γ¯ 11 (−jω, jω)Γ12 (−jω, jω) 1
and N (jω) = Γ¯ 112 (−jω, jω), where Γ¯ 11 = −Γ11 . Proof. The plant-wide dissipativity properties are determined from the individual subsystems and controllers (and the interconnection topologies) using Lemma 1. The result is then shown by applying Theorem 2 to the plant-wide supply rate and storage function. The plant-wide H∞ norm condition given in (22) represents a weighted norm of the mixed-sensitivity function, which can be W1 T1 rewritten as W T ≤ γ where T1 (s) is the system from distur2 2
∞
bance to process output and T2 (s) is the system from disturbance to controller output. A computationally simpler result for our purposes can be obtained for the case where the matrix φ(ζ , η) (or µ(ζ , η)) is block diagonal, this is stated below for the plant-wide case. Proposition 4. Consider a plant-wide system as shown in Fig. 4 with the transfer function matrix of the system from all disturbances d to all
49
T T
process and controller output ypw = yT , yTL , yR , denoted as T (s). If T (s) is dissipative with respect to the supply rate induced by µ(ζ , η) as in Lemma 1 of the form
µ(ζ , η) =
−N T (ζ )N (η) 0
0
γ 2 d(ζ )d(η)I
,
µ ∈ Rq×q [ζ , η]
(23)
then the following L∞ norm condition holds:
∥WT ∥L∞ ≤ γ ,
(24)
where d(s) is a scalar polynomial and N (s) is a polynomial matrix satisfying W (s) = d(1s) N (s). If, in addition, the storage function is positive semidefinite (i.e. half-line dissipativity), then the following weighted H∞ norm condition holds:
∥WT ∥∞ ≤ γ .
(25)
Proof. See Appendix. 4.3. Controller feasibility and synthesis The above results allow the plant-wide performance and stability criteria to be encoded in a desired supply rate that the closedloop plant-wide process T (s) must satisfy. The supply rates that individual controllers must satisfy can be determined from this plant-wide supply rate. Then the controllers that possess such dissipativity conditions can be synthesized individually. The key issue is the feasibility of the controller supply rates that are required for plant-wide stability and performance (i.e., the existence of a controller that possesses such dissipativity). It is well known that for the existence of a non-trivial system satisfying a given supply rate the number of non-negative eigenvalues of the matrix inducing the supply rate is greater than or equal to the dimension of the inputs to the system. One method of ensuring this is to require the controller supply rate to admit a J-spectral factorization. That is, φc (−jω, jω) = F T (−jω)JF (jω) where J is a diagonal matrix with elements equal to 0, 1 or −1. J-spectral factorizability implies the controller supply rate has a constant number of positive and negative eigenvalues (Trentelman & Rapisarda, 1999), equal to the number of positive and negative eigenvalues of J. This can be transformed into an LMI condition as below. Lemma 2. Consider the symmetric single variable polynomial matrix A(ω) = A0 + A2 ω2 + · · · + A2n ω2n
(26)
of degree 2n with the property that Ai = 0 ∀ odd i. Then det(A(ω)) = 0 has no roots ∀ω ∈ R if either: A0 , A2n > 0
(27)
A0 + A2 ≥ 0
(28)
A0 + A2 + A4 ≥ 0
.. . A0 + · · · + A2n−2 ≥ 0 A2n + A2n−2 ≥ 0 A2n + A2n−2 + A2n−4 ≥ 0
.. . A2n + · · · + A2 ≥ 0, Or if (27)–(29) are negative semidefinite. Proof. See Appendix.
(29)
50
M.J. Tippett, J. Bao / Automatica 50 (2014) 44–52
Theorem 3. The matrix φ(ζ , η) =
Q (ζ , η) 0
0 R(ζ , η)
admits a J-
spectral factorization if Q˜ X + X Q˜ ≥ 0
(30)
˜ + Y R˜ ≥ 0 RY
(31)
where X ≥ 0, Y ≤ 0 or X ≤ 0, Y ≥ 0. Proof. Note that φ(ζ , η) admits a J-spectral factorization if Q (ζ , η) and R(ζ , η) do, as the eigenvalues of φ(ζ , η) are the union of the eigenvalues of Q (ζ , η) and R(ζ , η). Using the main result in Carlson and Schneider (1962) Q˜ X + X Q˜ ≥ 0 implies that Q˜ is either positive or negative semidefinite. Which, using Lemma 2 implies that Q (−jω, jω) admits a J-spectral factorization. Using the same argument, the above conditions imply the J-spectral factorizability of R(−jω, jω) and therefore φ(−jω, jω). Combining the above results, the following result provides a convex algorithm (based on LMIs) for determining the required controller dissipativity properties to achieve given levels of plantwide performance. Theorem 4. Consider a plant-wide system as shown in Fig. 4 with the transfer function matrix of the system from all disturbances d to
T
all process and controller output ypw = yT , yTL , yTR , denoted as T (s). Where the ith process is dissipative (with non-negative storage) with respect to the supply rate induced by φi (ζ , η). There exists a distributed controller such that the closed-loop plant-wide system (T (s)) is internally stable and satisfies the norm bound ∥WT ∥∞ ≤ 1 if
Γ˜ 11 < 0
(32)
Q˜ci Xi + Xi Q˜ci ≥ 0
(33)
R˜ci Yi + Yi R˜ci ≥ 0
(34)
for all i, where the ith controller is dissipative rate induced with supply by φci (ζ , η), with coefficient matrix φ˜ ci =
Qci 0
0 Rci
W (jω) =
α(ω)
In this paper, a supply rate centric approach to analysis of plantwide processes is presented. Plant-wide stability and performance criteria are represented in a unified form of desired closed-loop plant-wide dissipativity properties. In this setting, the network of processes is controlled by a network of controllers. Plant-wide stability and performance conditions are based on the dissipativity of individual subsystems and interconnection topologies. This leads to a very general and flexible framework for plant-wide control where different control structures and types, e.g. closed form control (Tippett & Bao, 2013a) or model predictive control (Tippett & Bao, 2013b), can be adopted. In this approach, the individual process models are only used to verify their dissipativity, not directly in control analysis and design. As the supply rate of plant-wide system is a linear function of the supply rates of each individual process unit and controller, the proposed approach allows the plant-wide control and analysis problems to be formulated in a linear manner. It also allows different control structures (with different controller network topologies) to be dealt with in a unified framework. Therefore, this approach is very scalable, suitable for large complex networks. This is in contrast to some existing methods where the structure imposed by decentralized control, or by certain controller network topologies leads to computational difficulties in analysis and control synthesis. Dynamic QDF supply rates are used to capture much more detailed dynamic system properties than traditional constant (Q , S , R) supply rates, and as such, reduce the conservativeness of dissipativity-based analysis. The proposed supply rate centric approach would not be possible based on the classic constant supply rates as they are too coarse to replace process models to represent system dynamic features. Dynamic supply rates also allow the performance criteria to be encoded directly into the dissipativity properties of the system. As a result, both plant-wide stability and performance can be represented using the same dissipativity/ supply rate framework.
. The matrices Xi
and Yi are constant matrices with eigenvalues chosen to ensure the J-spectral factorizability of the controller supply rates, with 1
5. Discussion and conclusion
1 2
(−Γ11 ) (jω),
where α(ω) is a scalar rational function satisfying
(35)
Appendix. Proofs Proof of Theorem 1. Note that dissipativity with respect to (7) implies
γ2 2
α (ω)I > 2
R(−jω, jω) + S T (−jω, jω)Qˆ −1 (−jω, jω)S (−jω, jω). The decision variables in the above LMI problem are the controller supply rates. Proof. The result follows easily by combining the previous results in this paper. To achieve the desired plant-wide performance, the ith local controller must satisfy the supply rate induced by φci (ζ , η). The conditions above ensure the existence of a controller satisfying this supply rate. As the plant-wide supply rate is linear in the process and controller supply rates, the process supply rates can be determined simultaneously, using the LMI condition in Belur and Trentelman (2002). Once the required controller dissipativity properties are known, suitable controllers can be synthesized individually, e.g., by performing a J-spectral factorization of the controller supply rates and subsequently augmenting a ‘seed’ system to synthesize a network of closed form controllers (as presented in Tippett and Bao (2013a)). Another possible way is to implement the dissipativity constraints in model predictive control algorithms to develop a dissipativity based distributed model predictive control framework (as presented in Tippett and Bao (2013b)).
∞
yˆ T Q yˆ + 2yˆ T S uˆ + uˆ Ruˆ dt ≥ 0.
(A.1)
−∞
Let Qˆ = −Q˜ . If Q˜ < 0, we can then complete the square to obtain
∞
1
1
Qˆ 2 yˆ − Qˆ − 2 S uˆ
2
∞
−∞
uˆ T R + S T Qˆ −1 S uˆ dt .
dt ≤ −∞
By applying the reverse triangle inequality and defining α as a constant such that α 2 I ≥ R + S T Qˆ −1 S we then obtain 1 1 ∥Qˆ 2 yˆ ∥22 − ∥Qˆ − 2 S uˆ ∥22 ≤ α 2 ∥ˆu∥22
(A.2)
which reduces to ∥ˆy∥2 ≤ γ ∥ˆu∥2 with
1 1 γ = ∥Qˆ − 2 ∥2 ∥Qˆ − 2 S ∥2 + α
(A.3)
which is the stated result. To show stability, introduce the storage function Qψ (u, y) > 0. For vanishing input the dissipation inequality becomes Qψ˙ (0, y) ≤ yˆ T Q˜ yˆ .
(A.4)
As Q˜ < 0, this implies Qψ˙ (0, y) < 0, which by Proposition 1 implies asymptotic stability.
M.J. Tippett, J. Bao / Automatica 50 (2014) 44–52
Proof of Theorem 2. The half-line dissipativity of the system coupled with Q (ζ , η) < 0 as per Theorem 1 implies asymptotic stability of the system. Dissipativity implies
∞
Qφ (u, y) dt ≥ 0.
(A.5)
−∞
As compactly supported trajectories are assumed, use Parseval’s theorem to obtain
∞
Qφ (¯u, y¯ ) dω ≥ 0,
−∞ ∞
(A.6)
y¯ T Q (−jω, jω)¯y + 2y¯ T S (−jω, jω)¯u
+ u¯ T R(−jω, jω)¯u dω ≥ 0
(A.7)
where u¯ and y¯ denote the Fourier transforms of u and y respectively. To simplify notation we drop the frequency dependence. Defining Qˆ = −Q and using Q < 0, complete the squares: ∞
1
1
Qˆ 2 y¯ − Qˆ − 2 S u¯
T
1
1
∞
u¯ T R + S T Qˆ −1 S u¯ dω.
(A.8)
−∞
Letting F T F = R+S T Qˆ −1 S and using the reverse triangle inequality, 1
1
∥Qˆ 2 y¯ ∥ ≤ ∥Qˆ − 2 S u¯ ∥ + ∥F u¯ ∥. As ∥F u¯ ∥ ≥ ∥Qˆ
− 12
(A.9)
S u¯ ∥, we write
1
∥Qˆ 2 y¯ ∥ ≤ 2∥F u¯ ∥.
(A.10)
Using α defined in the Theorem statement we obtain 1
∥Qˆ 2 y¯ ∥ ≤ γ α∥¯u∥,
(A.11)
which is equivalent to a L∞ norm condition on the system G(s). Using the assumption that the system has positive semidefinite storage, the system is asymptotically stable according to Proposition 1. Therefore, the following H∞ condition is valid:
1 1 Qˆ 2 (ω)G ≤ γ α ∞
∥N (jω)¯ypw ∥ ≤ γ ∥d(jω)d¯ ∥.
(A.16)
Since d(jω) is scalar, we have
∥N (jω)¯ypw ∥ ≤ γ |d(jω)|∥d¯ ∥ 1 ≤ γ ∥d¯ ∥ N ( j ω)¯ y pw d(jω) 1 N ( j ω) T ( j ω) ≤ γ, d(jω)
(A.17) (A.18)
(A.19)
¯ (jω), which is the required L∞ norm where y¯ pw (jω) = T (jω)d condition. Note that the storage function Qν (d, ypw ) is positive semidefinite. The instantaneous form of the dissipation inequality requires Qµ (d, ypw ) ≥ Qν˙ (d, ypw ).
(A.20)
For a vanishing input, this becomes:
Qˆ 2 y¯ − Qˆ − 2 S u¯ dω
−∞
≤
This leads to
L∞
−∞
51
(A.12)
2 d ypw ≥ Qψ˙ (0, ypw ), − N dt
(A.21)
which implies Qψ˙ (0, ypw ) < 0. According to Proposition 1, the plant-wide system, is asymptotically stable, in which case the L∞ and H∞ norms are equivalent. Proof of Lemma 2. The argument is constructed based on the positive (semi-)definite case. The negative (semi-)definite case is completely analogous. It is clear from the form of the polynomial that if all coefficient matrices are positive semidefinite (and the A0 and A2n terms are definite) then the polynomial is strictly positive ∀ω ∈ R. This is true because the polynomial has only even powers of ω. Suppose that A2 is non-positive semidefinite (and all other coefficient matrices are positive semidefinite). Then for |ω| < 1, A0 + A2 ≥ 0 implies xT (A0 + A2 ) x ≥ 0
∀x ∈ R •
xT A0 x + xT A2 x ≥ 0 xT A0 x + xT A2 xω2 ≥ 0
(as |ω| < 1 and A0 > 0)
xT A0 + A2 ω2 x ≥ 0
where y¯ (jω) = G(jω)¯u(jω).
A0 + A2 ω2 ≥ 0.
Proof of Proposition 4. The system T (s) is the system from d to ypw . Dissipativity of T (s) implies
Given A0 > 0, A0 + A2 ≥ 0 it can then be shown by the same reasoning that A0 + A2 + A4 ≥ 0 implies A0 + A2 ω2 + A4 ω4 ≥ 0 for |ω| < 1 even if A4 is non-positive semidefinite. Applying this iteratively, it is clear that Condition (27) and Condition (28) imply that A(ω) > 0 ∀ |ω| < 1. Similarly for |ω| ≥ 1, A2n + A2n−2 ≥ 0 implies
∞
−
yTpw N T
yTpw N T −∞
d
d
∞
γ d d
≤ −∞
N
d
d dt
d
ypw
dt
d dt
N
dt
2 T
dt
∞
dt
−∞
+ γ 2 dT d
d
d
d dt ≥ 0,
dt
d
(A.13)
dt
xT A2n ω2n x + xT A2n−2 xω2n−2 ≥ 0 d dt .
(A.14)
As compactly supported trajectories are assumed, take the Fourier ¯ and y¯ pw and apply Parseval’s transform of d and ypw , denoted d theorem to obtain:
∞
y¯ Tpw N T (−jω) N (jω) y¯ pw −∞
∞
γ 2 d¯ T d (−jω) d (jω) d¯ dt .
≤ −∞
∀x ∈ R•
xT A2n x + xT A2n−2 x ≥ 0
ypw d
xT (A2n + A2n−2 ) x ≥ 0
(A.15)
(A.22)
xT A2n ω2n + A2n−2 ω2n−2 x ≥ 0
A2n ω2n + A2n−2 ω2n−2 ≥ 0. As in (A.22) |ω| ≥ 1 ⇒ ω2n ≥ ω2n−2 and A2n > 0. As above, using this reasoning iteratively, it can be seen that Condition (27) and Condition (29) imply that A(ω) > 0 ∀ |ω| ≥ 1. Thus we can conclude that A(ω) > 0 ∀ω ∈ R which in turn implies det(A(ω)) = 0 has no roots ∀ω ∈ R. As the roots of det(A(ω)) occur only when the inertia of A(ω) changes, which it clearly cannot if A(ω) > 0 ∀ω ∈ R.
52
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Michael J. Tippett received a B.E. in Industrial Chemistry/B.Com. in Business Economics with first class honors and the University Medal from the University of New South Wales (UNSW), Sydney, Australia in 2009. He is currently pursuing his Ph.D. in the area of distributed control systems in the School of Chemical Engineering at UNSW. His current research interests include: distributed and decentralized control, adaptive control, model predictive control, dissipativity-based analysis and control and their applications to chemical processes.
Jie Bao obtained his B.Sc. and M.Sc. degrees in Electrical Engineering from Zhejiang University, China, in 1990 and 1993 respectively. In 1998, he received the Ph.D. degree in Chemical Engineering (Process Control) from the University of Queensland, Australia. He spent one year at University of Alberta as a postdoctoral fellow and then joined the faculty at the University of New South Wales (UNSW), Sydney, Australia. He is currently a Full Professor in the School of Chemical Engineering, UNSW. His research interests include distributed and decentralized control, robust control, fault-tolerant control, dissipativity-based process control and control of industrial processes including aluminum smelting, mineral processing, membrane separation and flow batteries. He is an Associate Editor of the Journal of Process Control.