Ind. Eng. Chem. Res. 2008, 47, 4765–4774
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PROCESS DESIGN AND CONTROL Dynamic Operability Analysis for Stable and Unstable Linear Processes Herry Santoso,† Jie Bao,*,† and Peter L. Lee‡ School of Chemical Sciences and Engineering, The UniVersity of New South Wales, Sydney NSW 2052, Australia, and Chancellery, LeVel 4, 55 North Terrace, The UniVersity of South Australia, Adelaide SA 5000, Australia
Dynamic operability analysis determines the achievable control performance for a given process. Based on open-loop models, such analysis is useful in revealing controllability problems in the early process design stages. This paper presents a new approach to dynamic operability analysis based on the concept of passive systems. It is well-known that passive systems are very easy to control. The lack of passivity of a nonpassive process can be quantified by using an input feedforward passivity (IFP) or an output feedback passivity (OFP) index, which measures how much feedforward/feedback is required to render the process passive. A nonminimum phase stable process usually has a shortage of IFP and thus needs a controller with an excessive OFP to ensure closed-loop stability. Excessive OFP of the controller represents the upper limit of the controller gain. Therefore, the shortage of IFP of a stable process can be used to determine its achievable performance. This analysis is then extended to unstable processes using coprime factorization. Numerical methods for dynamic operability analysis for both stable and unstable processes are developed in this paper. 1. Introduction The traditional approach to developing a new process has been to perform the process design and process control problems sequentially. First, the process design engineers construct the process flowsheet based on a steady-state economic analysis. Then the control engineer must devise the control systems to ensure process stability and good dynamic performance that satisfies the operational requirements.15 This staged approach can be problematic, especially when dealing with more complex plant configurations. As little consideration is given to the dynamic operability of the process in the early design stage, the result sometimes is a plant with very poor dynamic characteristics which leads to severe control problems and significant economic penalties.9,17 A better approach is to integrate process design and process control. In the past two decades, integration of process design and control has drawn considerable interest of both industry and academia.17 One approach is to optimize the process design and controller design simultaneously as a single optimization problem (e.g., refs 1, 13, and 18). The decision variables considered include process structures, process design parameters, operating conditions, control structures, and controller parameters. The objective is to maximize the economic benefit subject to the process constraints, disturbances, and uncertainties. This approach makes it possible to analyze and compare many design alternatives as well as to deal with complex multiple trade-offs that usually take places in many design problems. However, the amount of computation required is often extremely high and thus the application of the optimization approach is currently restricted to small-scale problems.17 Another approach is to perform the operability analysis (i.e., to identify the achievable dynamic performance) based on open-
loop process models so that the operability issues can be discovered and rectified in the process design stage. As process operability is fundamentally determined by process design, it can be assessed independently without the prior knowledge of any particular control design.16 In this approach, several quantitative measures of process operability have been proposed to help engineers identify possible operability problems. These include nonminimum phase elements (i.e., time delays11 and right-half-plane (RHP) zeros12), condition numbers,2 process resilience index,5,16 output controllability index,21 disturbance costs,14,22 relative gain array,4 relative disturbance gain,20 and bandwidths of achievable sensitivity functions (based on allpass factorization).24 In this paper, we show how the concept of passive system can be used to perform process dynamic operability analysis. Passive systems are stable and minimum phase and thus are very easy to control. Therefore, the degree of passivity of a process can be used to indicate its operability. In general, the degree of passivity of a process can be quantified by the amount of feedforward and feedback required to render the process passive. To analyze process dynamic operability, a numerical approach is developed to compute the dynamic feedforward and feedback passifying systems, based on which the operability analysis can be performed for stable or minimum phase unstable processes. For nonminimum phase unstable processes, the achievable dynamic performance analysis is developed based on the minimum simultaneous feedforward and feedback required to render the processes stable and minimum phase. This approach is based on coprime factorization and feedforward passivation. The structure of this paper is as follows: The basic concept of passive system is introduced in section 2. The approaches to operability analysis for stable and unstable processes are presented in sections 3 and 4, respectively, followed by the illustrative examples in section 5. 2. Passive Systems
* To whom correspondence should be addressed. E-mail: j.bao@ unsw.edu.au. Fax: +61-2-9385-5966. Tel.: +61-2-9385-6755. † The University of New South Wales. ‡ The University of South Australia.
In this section, the basic concepts of passive systems and passivity index are introduced. A brief description of existing Positive Real Lemma condition is also given.
10.1021/ie070599c CCC: $40.75 2008 American Chemical Society Published on Web 06/14/2008
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Definition 1 (Passive Systems23). A linear system with m inputs and m outputs and n states is passive if and only if its transfer function matrix G(s) ) C(sI - A)-1B + D is positive real (PR), or equivalently: • Re[λi(A)] e 0 for i ) 1,..., n • G(jω) + G(jω)/ g 0 for all real ω, jω * λi(A) • Imaginary eigenvalues of A are nonrepeated and the residue matrix at those eigenvalues is Hermitian and nonnegative definite. The system is said to be strictly passive or strictly positive real (SPR) if • Re[λi(A)] < 0 for i ) 1,..., n • G(jω) + G(jω)* > 0 for all real ω, jω * λi(A), where G(jω)* denotes the complex conjugate transpose of G(jω). From the definition, it is clear that passive systems represent a class of minimum phase systems with relative degree no larger than one. For linear systems, this implies that the phase shift is within [-90°, 90°]. Therefore, the negative feedback system of a strictly passive system and a passive system is asymptotically stable because the phase lag of the open loop system never exceeds -180°.19 This implies that strictly passive processes are very easy to control. They can be controlled by any passive controller with any positive gain. This also implies that the degree of passivity of a process may be used to indicate its operability. A stable process can always be rendered passive by a constant feedforward element. Similarly, a minimum phase unstable process can always be rendered passive by a constant feedback element. Therefore, the minimum feedforward or feedback required to render the process passive can be used to indicate the degree of passivity of the process, as the input feedforward passivity (IFP) and the output feedback passivity (OFP), respectively: Definition 2. A system H is said to be • IFP(ν) if system H with a constant negative feedforward νI is passive. • OFP(F) if system H with a constant positive feedback FI is passive. If ν > 0 (or F > 0), then the system is in excess of input feedforward (or output feedback) passivity. In this case, the system is said to be strictly input (or output) passive. If ν < 0 (or F < 0), then the system is in deficit of input feedforward (or output feedback) passivity. From the above definition, it can be seen that PR systems are input feedforward passive, and SPR systems are stable and strictly input feedforward passive. In a negative feedback configuration, the lack of passivity of one system can be compensated by the excess of passivity of another system such that the closed-loop system is stable: Theorem 1.19 Consider two linear systems in the negative feedback interconnection. If system G1 is asymptotically stable and IFP(ν) and system G2 is OFP(F). Then the closed-loop system is asymptotically stable if ν + F > 0. If a stable process is not passive and a certain amount of feedforward is required to passify the process, then it is said that the process has a shortage of IFP. In this case, a controller which has a certain amount of excess OFP is needed to ensure the closed-loop stability. As the excessive OFP is related to the corresponding system gain,10 this implies that the upper limit of the controller gain and thus the achievable performance of the closed-loop system is limited by the IFP index of the process. For a minimum phase unstable process, the degree of passivity of the process is usually associated with the lack of OFP. Thus, a controller with an excess of IFP is needed to stabilize the closed-loop system. This implies that the lower bound of the
Figure 1. Feedback control system.
Figure 2. Framework for Operability Analysis of Stable Processes
controller gain and thus the minimum control action that should be provided to the system is determined by the OFP index of the process. In a more complex situation of a nonminimum phase unstable process, both feedforward and feedback systems are needed to determine the process operability. This will be further discussed in Section 4. The passivity of a process can be determined by using the following Positive Real Lemma: Theorem 2 (Positive Real Lemma25). Consider a stable linear system with transfer function matrix G(s). Assume its state space realization (not necessarily a minimal realization) is given by G(s) :) (A, B, C, D). System G (s) is positive real if there exist P ) P T g 0, Q, and W such that ATP + PA ) -QTQ
(1)
B P+W Q)C
(2)
T
T
D+D )W W (3) Equations 1–3 can be rewritten as the following algebraic Riccati equation (ARE): T
T
ATP + PA + (PB - CT)(D + DT)-1(BTP - C) ) 0 (4) T Finding P ) P g 0 that satisfies eq 4 is equivalent to finding a feasible P ) PT g 0 for the following linear matrix inequality (LMI):3
[
]
ATP + PA PB - CT e0 (5) BTP - C - (D + DT) The positive real condition can be checked easily by solving LMI (5) using any semidefinite programming tools (e.g., the MATLAB LMI Toolbox). 3. Operability Analysis of Stable Processes Consider the feedback control system shown in Figure 1, where G(s) is a stable process transfer function, with m inputs, m outputs, and n states; Gd(s) is a disturbance transfer function; and K(s) is a multivariable controller. Assume the state space representation for the process G(s) and its disturbance Gd(s) be given by the following equations:
{
x˙ ) Ax + Bu + Ed y ) Cx + Du + Fd
(6)
Ind. Eng. Chem. Res., Vol. 47, No. 14, 2008 4767 -1
G(s) ) C(sI - A) B + D
(7)
-1
Gd(s) ) C(sI - A) E + F
(8)
For stable linear processes, the process inputs can be scaled using a constant nonsingular matrix W so that the positive real condition is satisfied at steady state: G(0)W + WTG(0)T > 0 (9) -1 One obvious choice of W is G(0) if G(0) is nonsingular. It should be noted that scaled process G(s)W may not be passive as the positive real condition may not hold at frequencies other than ω ) 0. In this case, a dynamic feedforward compensator Gff(s) is needed to passify the scaled process, i.e. Gp(s) ) G(s)W + Gff(s)
(10)
such that Gp(s) is strictly positive real. If Gp(s) is obtained with a minimum feedforward Gff(s) (i.e., a Gff(s) with the minimum infinity norm), it can be understood as the passive approximation of the nonpassive process G(s). In terms of closed-loop stability, adding a feedforward Gff(s) to the scaled process G(s)W is equivalent to adding a feedback loop to the passive controller C(s). From the passivity index point of view, this means the scaled process G(s)W has a shortage of IFP, and thus needs a controller that has an excess OFP to ensure the closed-loop stability. Observe that a dynamic feedforward compensator Gff(s) is used instead of a constant feedforward element as originally defined in Definition 2. This allows development of a less conservative controllability analysis approach. In this case, Gff(s) represents the dynamic IFP index of the process. In theory, C(s) may have an infinitely large gain without causing instability to the system. In this case, the best performance is achieved and the final controller K(s) (see Figure 2) becomes K(s) ) WGff(s)-1
achieve. The bandwidth of the sensitivity functions in (15) and (16) is determined by the passive approximation of the process Gp(s), or the dynamic feedforward compensator Gff(s) via eq 10. The key issues in this approach is to find the passive approximation of a process. Consider a nonpassive stable process G(s) :) (A, B, C, D). Therefore, the scaled system is G(s)W :) (A, BW, C, DW). Assume the passified process Gp(s) :) (Ap, Bp, Cp, Dp). As the feedforward passivation only needs to change the zero dynamics of the scaled process, we can assume
Observe that the upper limit of the controller gain and thus the achievable performance of the closed-loop system is limited by Gff(s). For example, since the scaled process is positive real at steady state, the value of Gff(0) should be zero resulting in an infinite controller gain to achieve perfect control performance at steady state. The achievable sensitivity function S(s) and the complementary sensitivity function T(s) of the closed-loop system can be estimated as follows: S(s) ) [I + G(s)WGff(s)-1]-1 ) I - G(s)WGp(s)-1 -1
T(s) ) I - S(s) ) G(s)WGp(s)
(13) (14)
The achievable set point tracking performance then can be deduced from the bandwidth of the maximum singular value profile of the sensitivity function S(jω): σ[S(jω)] ) σ[I - G(jω)WGp(jω)-1]
(15)
while the disturbance attenuation performance can be deduced from the bandwidth of the maximum singular value profile of the corresponding disturbance sensitivity function Sd(jω), given as σ[Sd(jω)] ) σ[S(jω)Gd(jω)]
(16)
The larger the bandwidth of these sensitivity functions is, the faster the dynamic response the feedback control system can
Bp ) BW
(18)
Dp ) (Cp - C)Ap-1Bp + DW
(19)
Such a Dp can always be found if G(0) is nonsingular. Based on this, the passive approximation Gp(s) of a given nonpassive process G(s) can be estimated by solving an optimization problem of finding the minimum infinity norm of Gff(s), i.e. ||Gff||∞ < γ
(20)
subject to the constraint that Gp(s) is positive real. This leads to the following LMI problem. Problem 1 min γ
(21)
P ) PT > 0
(22)
P,X,Cp
subject to:
[
The controller gain is limited by the following condition: (12)
(17)
Furthermore, since it is desired to have offset free control at steady state, it is required that Gp(0) ) G(0)W (i.e., Gff(0) ) 0) resulting in:
(11)
σ(K(jω)) e σ(W)/σ _(Gff(jω))
Ap ) A
[
ATp P + PAp BTp P - Cp
PBp - CTp -(Dp + DTp )
]
0 ATp X + XAp BTp X Cp - C
(Cp - C)
T
XBp
(23)
]
(24)
T [(Cp - C)A-1 0 T (Cp - C) I
(29)
This formulation is easier to solve and can be used when the state space representation of the system is not minimal. Once the passive approximation Gp(s) is obtained, the achievable dynamic performance can be estimated using eqs 15 and 16. The proposed method also provides a different perspective to operability analysis compared to the classical internal model control (IMC) framework.16 In the IMC framework, it is argued that an effective feedback controller attempts to “invert” the process model. Thus, any process characteristics that prevent this inversion (e.g., the nonminimum phase elements) represent an inherent limitation to the closed-loop system performance. In the presence of these limiting process characteristics, it is suggested that the process transfer function G(s) be factorized into a minimum phase part G-(s), which is invertible, and a nonminimum phase part G+(s), which is not invertible, such that G+(s) has a gain of unity at steady state and G-(s) is realizable and stable. If there are no constraints on the controller gain, the best achievable control performance then is given by the IMC controller Kc(s) ) G-(s)-1.16,24 The subsystems in Figure 2 can be regrouped so that the feedback system can be equivalently represented in an extended IMC structure shown in Figure 3, where GM(s) is the process model and Kc(s) is the IMC controller. Since the process G(s) is stable, the closedloop stability of the IMC structure is only determined by the stability of the IMC controller Kc(s). Observe that the IMC controller Kc(s) consists of a feedback loop of a passive system Cc(s) and a strictly passive system Gp(s) and a constant scaling matrix W. According to the passivity theorem, Cc(s) may have an infinitely large gain to give the best control performance without causing instability. In this case, the IMC controller Kc(s) is equal to Kc(s) ) WGp(s)-1
(30)
resulting in the same sensitivity function S(s) and complementary sensitivity function T(s) as described in eqs 13 and 14. From this extended IMC formulation, it can be seen that, instead of factorizing the process transfer function into a minimum phase part G-(s) and a nonminimum phase part G+(s), in the proposed passivity-based operability analysis, the process transfer function G(s) is decomposed into a passive subsystem Gp(s), which is invertible, and a nonpassive subsystem -Gff(s). The process invertibility then is determined by its IFP index, which measures how far the process is from being passive. 4. Operability Analysis of Unstable Processes Many existing open-loop analysis methods can only be used for stable processes (e.g., refs 16 and 24). However, in control
practice, the operability of unstable processes is often a bigger concern to most chemical and control engineers. In this section, the operability analysis for unstable processes will be developed using the concept of input feedforward passivation. Consider the same feedback control system shown in Figure 1, where G(s) is a nonminimum phase unstable process. For this type of process, both input feedforward and output feedback should be used simultaneously to passify the process. Since this feedforward and feedback are not independent of each other, it is difficult to determine their values simultaneously. To overcome this problem, the unstable process G(s) is first left coprime ˜ (s) such that factorized into two stable subsystems N˜(s) and M ˜(s) (31) G(s) ) ˜ M(s)-1N ˜ ˜ (s) System N(s) contains all the RHP zeros of G(s) while M should contain all the RHP poles of G(s). The coprimeness ˜ (s) and M ˜ (s) should not have common RHP zeros implies that N that may result in pole-zero cancelations. Mathematically, the following Bezout identity is satisfied: Definition 3 (Left Coprime Factorization25). Two matrices ˜ and N˜ in RH∞ are left coprime over RH∞ if they have the M same number of rows and if there exist matrices Xl and Yl in RH∞ such that
[]
˜ N ˜] Xl ) M ˜X + N ˜Y ) I [M l l Yl
Coprime factorization is not unique. One way of finding coprime factorization of a particular system is by using a normalized coprime factorization: Theorem 3 (Normalized Left Coprime Factorization25). Consider a linear system with transfer function matrix G(s). Let (A, B, C, D) be a state space realization of G(s). Define R ) I + DTD > 0 ˜ ) I + DDT > 0 R
(33)
(34) Suppose (C, A) is detectable and (A, B) has no uncontrollable modes on the imaginary axis. Then there is a normalized left ˜ (s)-1N˜(s) coprime factorization G(s) ) M
[
]
B + LD ˜(s) N ˜(s)] : ) A + LC L [M ∈ RH∞ -1⁄2 -1⁄2 ˜-1⁄2 ˜ ˜ R D R C R (35)
where ˜-1 L ) -(BDT + YCT)R and
[
Y ) Ric
˜-1C)T -CTR ˜-1C (A - BDTR -BR-1BT
˜-1C) -(A - BDTR
(36)
]
g0 (37)
After applying the normalized left coprime factorization to the unstable process G(s), we now have two stable subsystems ˜ (s). The passive approximation of each subsystem N˜(s) and M then can be estimated separately as shown in Figure 4. Observe that N˜(s) is a nonminimum phase stable subsystem. This subsystem can be passified by adding a dynamic feedforward N˜ff(s) to the scaled subsystem N˜(s)WN˜, i.e.: ˜ (s) ) N ˜(s)W˜ + N ˜ (s) N p N ff
Figure 3. Extended internal model control (IMC) structure.
(32)
(38)
The passive approximation of this subsystem N˜p(s) via feedforward passivation then can be determined by solving Problem 1 or Problem 2 described in section 3.
Ind. Eng. Chem. Res., Vol. 47, No. 14, 2008 4769
˜ (s)N ˜ (s)]-1[I K(s) ) WN˜{[Cˆ(s)-1 + M p ff -1˜ ˜ (s)-1M ˜ (s)}W˜-1 ˜ ˜ ˜ Mp(s)Nff(s)Np(s) Mff(s)]+N p ff M
(43)
When Cˆ(s) is set to infinity, the final controller K(s) is equal to ˜ (s)-1M ˜ (s)-1W˜-1 K(s) ) WN˜N p M ff
Figure 4. Feedforward and Feedback passivation of Unstable Processes.
(44)
This is the controller with the largest controller gain without causing closed-loop instability, from which the achievable performance of the closed-loop system can be estimated. On the other hand, when Cˆ(s) is set to zero, the final controller K(s) is equal to ˜ (s)-1M ˜ (s)W˜-1 K(s) ) WN˜N p ff M
Figure 5. Loop shifting of feedforward and feedback passivation of unstable processes.
˜ (s)-1 is a minimum phase unstable On the other hand, M subsystem. To passify this subsystem, a negative dynamic ˜ ff(s) should be applied to the scaled subsystem feedback M -1 ˜ -1 WM˜ M(s) , i.e. ˜ (s) ) W˜-1M ˜(s)-1[I + M ˜ (s)W˜-1M ˜(s)-1]-1 M p M ff M
(39)
˜ p(s) is strictly passive. Equation 39 can be rearranged such that M to give ˜ (s) ) [M ˜(s)W˜ + M ˜ (s)]-1 M p M ff
(40) -1
˜ (s) via which shows that the passive approximation of M feedback passivation can be determined from the inverse of ˜ (s) via feedforward passivathe passive approximation of M tion. Thus, it can be determined by following the same procedure for feedforward passivation (solving Problem 1 or Problem 2). Observe that a dynamic feedforward N˜ff(s) and a dynamic ˜ ff(s) are used instead of the constant IFF/OFP as feedback M originally defined in Definition 2 so that the controllability analysis is less conservative. After passifying both the numerator and the denominator of the nonminimum phase unstable process using feedforward passivation, we have two passive subsystems ˜ p(s)N˜p(s) is stable and minimum in series. The overall system M phase but usually nonpassive. According to the IMC framework,16 perfect control can be achieved for a stable and minimum phase process by using an IMC controller which is the inverse of the process system, as shown below: ˜ (s)N ˜ (s)W˜-1C(s)W˜]-1 ˜[I + M Kc(s) ) WN˜-1C(s)WM p p N M (41) with C(s) f ∞. In this case, we have ˜ (s)-1M ˜ (s)-1 Kc(s) ) N p p
(42)
By using loop shifting, the system with the feedforward and feedback in Figure 4 can be transformed into an equivalent system shown in Figure 5. Furthermore, without affecting the stability conditions of the closed-loop system, these feedforward and feedback loops can be shifted to the controller Cˆ(s) side as shown in Figure 6. Further simplification can be made by absorbing all the scaling matrices in Figure 6 to the final controller K(s). The result is presented in Figure 7. The final controller K(s) can be written as
(45)
which represents the controller with the smallest controller gain that stabilizes the unstable process. By using controller K(s) given in eq 44, the sensitivity function S(s) and the complementary sensitivity function T(s) of the closed-loop system are ˜ (s)-1M ˜ (s)-1W˜-1]-1 S(s) ) [I + G(s)WN˜N ff p M -1˜
-1
-1 -1
˜ (s) M (s) W˜ ] T(s) ) I - [I + G(s)WN˜N ff p M
(46) (47)
The best set point tracking performance then can be deduced from the bandwidth of the maximum singular value profile of the sensitivity function S(jω) while the disturbance attenuation performance can be deduced from the bandwidth of the maximum singular value profile of the corresponding disturbance sensitivity function Sd(jω) (see eq 16). The larger the bandwidth of these sensitivity functions is, the faster the dynamic response the feedback control system can achieve. Observe that the bandwidth of the sensitivity function and the corresponding ˜ p(s). It disturbance sensitivity function is based on N˜ff(s) and M ˜ ff(s) by substituting M ˜ p(s) using eq 40. can be rewritten in M It should be noted that the above analysis is not directly based on passivity. An extra feedforward element can be used to passify the overall stable and minimum-phase system so that the controllability can be inferred from the passivity condition (similar to section 3). This step leads to only very minor differences in the analysis result at high frequencies and is practically unnecessary. Another factor that may affect the input output controllability is the relative degree of the process. As shown earlier using the IMC framework, without input constraints, a stable minimum phase process has no control limitation. However, if a stable and minimum phase process has a relative degree larger than 0, the IMC controller, which is the inverse of the process model, is not proper. If ideal derivative action is allowed, then the minimum phase process has to have a relative degree no more than 1 to have a realizable IMC controller. A higher relative degree warrants the use of a low-pass filter in the IMC controller, which implies reduction of achievable performance at high frequencies. In the controllability analysis for stable processes presented in section 3, the passified process GP(s) :) (AP, BP, CP, DP) has a relative degree of 0 (DP * 0). This ensures that the IMC controller KC(s) ) WGP(s)-1 is realizable. In the controllability analysis for unstable processes presented in this ˜P section, both the passified numerator N˜P and denominator M have a relative degree of zero. This ensures that the converted stable and minimum phase system is invertible. 5. Case Studies In this section, two case studies are presented to demonstrate the use of the proposed operability analysis method.
4770 Ind. Eng. Chem. Res., Vol. 47, No. 14, 2008
Figure 6. Framework for operability analysis of unstable processes.
Figure 7. Final control system for unstable processes. Table 1. List of Controlled, Manipulated, and Disturbance Variables
SC FS
LSF LSR
y(s)
u(s)
xD xB xDH xBH xDL xBL xDH xBL xDL xDL xBH xDH
R QR RH QRH RL FH/FL RH QRH RL RL QRH RH
d(s) xF xF
xF xF
5.1. Case Study I: Heat-Integrated Distillation Column. Distillation is probably one of the most popular and important technologies in the process industries. It is used in many chemical processes for separation or purification purposes. To reduce the energy consumption in this highly energy intensive technology, several heat-integrated distillation systems have been proposed. Dynamic operability of these processes is not easy to determine due to the interactions between the process units. Consider the four different heat-integrated distillation systems studied by Chiang and Luyben:7 single column (SC), feed-split (FS) heat-integrated columns, light-split/forward (LSF) heatintegrated columns, and light-split/reverse (LSR) heat-integrated columns. The linearized models of these systems are given in Appendix A in the form as follows: y(s) ) G(s)u(s) + Gd(s)d(s)
(48)
where G(s) and Gd(s) are the process and disturbance transfer functions with time constants measured in minutes; y(s) are the controlled variables; u(s) are the manipulated variables; and d(s) are the disturbance. Table 1 lists the controlled variables, the manipulated variables, and the disturbance of each distillation system where xD, xB, and xF are the distillate, bottom, and feed compositions; R and QR are the reflux flow rates and the reboiler heat duties; and FH/FL is the feed split ratio. Subscripts “ H ” and “L” refer to the high-pressure column and the low-pressure column of the corresponding distillation systems, respectively.
Figure 8. Maximum singular value plots of the sensitivity functions.
Observe that both the process and the disturbance transfer functions contain time delay elements. These time delays can be approximated by rational transfer functions using a Pade´ approximation. By solving Problem 1 described in section 3, the proposed operability measures can be easily calculated to compare the best achievable dynamic control performance of the four distillation systems. Figures 8 and 9 show the maximum singular value plots of the sensitivity functions and the disturbance sensitivity functions of the four distillation systems, while Table 2summarizes the corresponding estimated bandwidths and settling times. From this result, the best achievable dynamic performance for set-point tracking of the four distillation systems in descending order are SC > LSF > LSR > FS (49) while the best achievable control performance for disturbance attenuation are LSR > LSF > SC > FS
(50)
Ind. Eng. Chem. Res., Vol. 47, No. 14, 2008 4771
Figure 9. Maximum singular value plots of the disturbance sensitivity functions.
Figure 11. Maximum singular value plots of the disturbance sensitivity functions.
Table 2. Estimated Bandwidths and Settling Times for Heat-Integrated Distillation Systems
The result from this approach is different from that reported by Chiang and Luyben,7 which was estimated from the closed-loop step response with a decentralized PI controller tuned using the SVT tuning method. The proposed approach gives the best dynamic performance that can be achieved by any multivariable controller that guarantees closed-loop stability and thus is, understandably, higher than the result presented in ref 7 In practice, one may want to avoid the FS configuration due to the poor dynamic performance it has in both set-point tracking and disturbance rejection cases. Taking into account the energy consumption level, one may also want to avoid the SC system because it consumes almost double the amount of energy that consumes by any other distillation systems.7 Since both the LSR and the LSF systems have a fairly similar dynamic performance and energy level consumption, other design criteria may also be considered in order to decide which configuration that should be selected. 5.2. Case Study II: Two Continuous Stirred Tank Reactor. Consider a two continuous stirred tank reactor (CSTR) process studied by Cao and Yang.6 This process is designed for performing a first-order irreversible exothermic reaction. In order to maintain the temperatures in both tanks (i.e., T1 and T2) at desired values in the presence of cooling-water temperature fluctuations, there are two control configurations that can be selected. In the first configuration (S1), these temperatures are controlled by manipulating both feed flow rates to the tanks (i.e., F1 and F2) while in the second configuration (S2), these temperatures are controlled by manipulating both cooling water flow rates to the tanks (i.e., Fcw1 and Fcw2). The linearized models of these two systems are given in Appendix B in the form as follows:
set point tracking
SC FS LSF LSR
disturbance attenuation
bandwidth (radians/min)
settling time (min)
bandwidth (radians/min)
settling time (min)
0.109 0.075 0.096 0.092
27.5 39.9 31.2 32.6
0.009 0.005 0.011 0.012
332.9 599.1 272.3 249.6
Figure 10. Maximum singular value plots of the sensitivity functions. Table 3. Estimated Bandwidths and Settling Times for CSTR Control Configurations set point tracking
S1 S2
disturbance attenuation
bandwidth (radians/s)
settling time(s)
bandwidth (radians/s)
settling time (s)
6.35 1.29
0.47 2.32
∞ ∞
0 0
It can be seen that the single column (SC) system achieves the best dynamic performance in terms of set-point tracking while the light-split/reverse (LSR) system achieves the best dynamic performance in terms of disturbance rejection. For both cases, the LSF system provides a fairly similar performance compared to the LSR system while the feed split (FS) system always gives the worst performance.
{
x˙ ) Ax + Biu + Ed y ) Cx
(51)
Observe that matrix A has one right half-plane pole at 0.83 which implies that both configurations S1 and S2 are unstable. By using the normalized left coprime factorization, we can represent each unstable system with two stable subsystems ˜ (s). The passive approximation of each stable N˜(s) and M subsystem can be obtained by solving Problem 1 described in section 3. It should be noted that after passivation, the overall systems are not necessarily passive but stable and minimum phase.
4772 Ind. Eng. Chem. Res., Vol. 47, No. 14, 2008
The proposed operability measures then can be estimated using eq 46. Figures 10 and 11 show the maximum singular value plots of the sensitivity functions and the disturbance sensitivity functions of the two control configurations, while Table 3 summarizes the corresponding estimated bandwidths and settling times. From Figure 10 and Table 3, it is clear that control configuration S1 will provide a better dynamic tracking performance compared to control configuration S2, as S1 can achieve a settling time 5 times smaller than that of S2. In terms of disturbance attenuations, both control systems will provide a very good dynamic performance by rejecting the process’ disturbances very quickly. This result is consistent with the result obtained by Cao and Yang6 using a very different operability analysis approach. 6. Conclusion An operability analysis method that can be used to compare the achievable dynamic control performance of different processes has been proposed in this paper. This work provides new insights into process operability from a passivity perspective. In the proposed analysis approach, by using feedforward passivation, a process transfer function is decomposed into a passive subsystem (or the passive approximation) and a nonpassive subsystem. The process operability then can be inferred from the nonpassive subsystem, which represents how far the process is from being passive. An operability analysis method for unstable processes is also developed based on coprime factorization and feedforward passivation. Acknowledgment This research is partially supported by ARC Discovery Project DP0558755. H.S. acknowledges financial support from AusAID. Appendix A: Heat-Integrated Distillation Column The corresponding process G(s) and disturbance Gd(s) transfer functions for the heat-integrated distillation column are given as follows: Single Column (SC):
[
Feed Split (FS):
-4.44 3.6 (12s + 1)(4s + 1) (15.5s + 1)(2s + 1) G(s) ) -33.4 12.2e-s (23s + 1)(s + 1) (19s + 1)(s + 1) 1.09e-4.5s + 1)2 Gd(s) ) (20s + 1)(3s -0.4s 19.7e (25s + 1)(2s + 1)
[
[
]
]
(52)
(53)
-7.4 0.35 4.45 0 (14s + 1)(4s + 1) (16s + 1)(4s + 1) (25.7s + 1)(2s + 1) -41 17.3e-0.9s 9.2e-0.3s 0 (21s + 1)(s + 1) (17s + 1)(0.5s + 1) 20s + 1 G(s) ) -4.66 3.6 0.042(78.7s + 1) 0.22e-1.2s (17.5s + 1)(4s + 1) (13s + 1)(4s + 1) (13s + 1)(4s + 1) (21s + 1)(11.6s + 1)(3s + 1) -34.5 1.82e-s -6.92e-0.6s 12.2e-0.9s (20s + 1)(s + 1) (21s + 1)(s + 1) (18.5s + 1)(s + 1) 20s + 1 1.02e-4.5s (25s + 1)(2s + 1)2 19.7e-0.3s (25s + 1)(s + 1) Gd(s) ) 0.75e-5s (15.6s + 1)(2s + 1)2 16.61e-0.6s (25s + 1)(2s + 1)
Light-Split/Forward (LSF):
[
[ ]
4.96 -7.5 0 2 (15.4s + 1)(3s + 1) (9.6s + 1) -1.5s -37.4 6.39e 7.1e-0.9s G(s) ) (16.7s + 1)(6s + 1)(s + 1) (17s + 1)(3s + 1) (13.7s + 1)(s + 1) 0.29(-12s + 1)e-5s 5.55 -10.92 (21s + 1)(6s + 1) (15s + 1)(4s + 1) (13s + 1)(6s + 1)3 0.98e-4s (20s + 1)(4s + 1)2 18.6e-s Gd(s) ) (20s + 1)(8s + 1)(2s + 1) 2.44e-7s (38.5s + 1)(5s + 1)2
[
]
]
]
(54)
(55)
(56)
(57)
Ind. Eng. Chem. Res., Vol. 47, No. 14, 2008 4773
[
Light-Split/Reverse (LSR):
-4.6 4.15 -0.25e-1.5s (14.3s + 1)(4s + 1) (14s + 1)(4s + 1) (10s + 1)(10s + 1) -36.4 5.6e-1.5s 7.56e-0.9s G(s) ) (14.5s + 1)(6s + 1) (15.4s + 1)(4s + 1) (12s + 1)(s + 1) 0.27e-2s 5.51 -9 (16s + 1)(4s + 1) (13.5s + 1)(4s + 1) (9.5s + 1)3
[
0.414(-8s + 1) (6s + 1)3(2s + 1) 16.67e-s Gd(s) ) (19s + 1)(6s + 1)(2s + 1) 0.92e-5s (11s + 1)3
]
]
(59)
Appendix B: Two Continuous Stirred Tank Reactors
[
(58)
] ]
The corresponding matrices A, B1, B2, E, and C for the two continuous stirred tank reactor process are given as follows: -17.9751 -295.8655 0 0 0 0 0.0207 0.1889 0.0704 0 0 0 0 0.3879 0.8000 0 0 0 A) 0.0977 0 0 -18.008 -295.8655 0 0 0.0617 0 0.0131 0.0433 0.0589 0 0 0 0 0.3787 -0.6220
(60)
17.8996 -13.7811 0 0 0 0 -0.0131 0.0101 0 0 0 0 0 0 -0.0294 0 0.0137 0 [B1|B2|E] ) 17.8636 17.8636 0 0 0 0 0.0082 0.0082 0 0 0 0 0 0 0 0.0235 0 0.0081
(61)
[
C)
[
0 362.9950 0 0 0 0 0 0 0 0 362.9950 0
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ReceiVed for reView April 27, 2007 ReVised manuscript receiVed March 27, 2008 Accepted April 4, 2008 IE070599C
