A PI Control Configuration for a Class of MIMO Processes - Industrial

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Ind. Eng. Chem. Res. 2001, 40, 1186-1199

A PI Control Configuration for a Class of MIMO Processes Jose Alvarez-Ramirez* and Rosendo Monroy-Loperena Departamento de Ingenieria de Processos e Hidraulica, Universidad Autonoma Metropolitana-Iztapalapa, Apartado Postal 55-534, Mexico D.F., 09340 Mexico

This paper presents a PI control configuration for a class of multi-input multi-output (MIMO) plants. The control design procedure consists of a modeling error estimator based on a reducedorder observer structure and a nominal feedback function. The estimated modeling error signals are used in the feedback loop to counteract the effects of uncertain parameters. It is shown that this design procedure leads to a classical multivariable PI control configuration with novel parametrization of the controller gains. An advantage of the proposed control design is that easy tuning procedures can be designed. In addition, several special two-input two-output PI control configurations with decentralized structure can be obtained. As an application example, the separation control in an integrated three-product (Petlyuk) distillation column is studied. Sufficient conditions to achieve decentralized regulation in terms of the steady-state gain matrix are provided. Numerical simulations are used to show the effectiveness of the proposed PI controller under measurement dead times and uncertain parameters. 1. Introduction More than five decades after its adoption, the classical PI controller remains one of the most popular and most widely used controllers in the chemical and process industries. Its general properties with regard to effectiveness, simplicity, and conditions of applicability are well recognized.1,2 For single-input single-output (SISO) processes, several configurations have been proposed, and various tuning methods have been developed. The most popular tuning methods are those based on Ziegler-Nichols loop tuning.3 Frequency domain and pole placement techniques have been successfully used.3 Self-tuning techniques have a strong theoretical foundation,4 and efficient implementations based on relay techniques have been derived.1 Inherently multi-input multi-output (MIMO) processes are frequently encountered in the industry. Interactions usually exist between control loops, which account for the renowned difficulty in their control compared with SISO processes. Important examples of MIMO processes are distillation systems5 and FCC plants.6 Even in the control of MIMO processes, PI control is very popular. Many multi-loop controller design methods have been reported in the literature. The BLM method7 extended the Ziegler-Nichols rule in order to include a detuning factor, which determines the tradeoff between stability and performance of the system. Because individual controllers are designed for their respective loops by first ignoring all interactions, the BML method is too conservative to exploit process structures and characteristics for best performance achievement. Specialized for two-input two-output processes (TITO), modified Ziegler-Nichols methods have been reported,8,9 which exploit the structure of interactions of particular processes. On the other hand, decentralized PI control is one of the most common control schemes for interacting MIMO plants in the chemical and process industries. The main * Corresponding author. E-mail: [email protected]. Fax: +52-5-724-4900.

reason is its relatively simple structure, which is easy to understand and to implement.10 The number of tuning parameters is 2n, where n is the number of inputs and outputs, whereas in full-matrix PI control, there are 2n2 parameters. Even for moderately sized processes, this is a significant reduction. In the presence of actuator or sensor failure, decentralized control is easy to stabilize manually because the failure affects only one loop. In this paper, we present a novel PI control configuration for MIMO processes that is derived from modeling error compensation techniques. Under the assumption that the nominal plant model is relative degree one with respect to all of input-output interactions, we construct an observer to estimate the modeling error signals and use this estimation in a feedback function to counteract the adverse effects of uncertainties. This control design leads naturally to a multivariable PI control with an antireset windup scheme in which the control gain and integral time matrices are parametrized in terms of a desired closed-loop stable matrix and a single parameter that reflects the rate of modeling error estimation. In the case of TITO processes, we exploit the structure of the input-direction matrix to provide decentralized PI configurations with guaranteed robust stability. An advantage of the proposed control design is that easy tuning procedures can be designed. As an application example, the separation control in an integrated three-product (Petlyuk) distillation column is studied. Sufficient conditions to achieve decentralized regulation in terms of the steady-state gain matrix are provided. Numerical simulations are used to show the effectiveness of the proposed TITO PI controller under measurement dead times and uncertain parameters. The paper is organized as follows. Section 2 presents the plant models. Section 3 presents the control design. A stability analysis of the resulting controlled process is made based on Lyapunov functions. Sufficient conditions for the robust stabilization under decentralized PI control for TITO plants are also discussed in this section. Section 4 presents an application of our results

10.1021/ie9905308 CCC: $20.00 © 2001 American Chemical Society Published on Web 01/20/2001

Ind. Eng. Chem. Res., Vol. 40, No. 4, 2001 1187

in the control of Petlyuk distillation systems. Conclusions are presented in section 5. 2. Plant Models

and the resulting closed-loop system is given by

y˘ ) Acy z˘ ) E1y + E2z

We will say that a given matrix is stable if all of its eigenvalues have negative real parts. Assume that a nominal model of the class of MIMO plants to be controlled is given as follows:

y˘ ) Ay + Dz + Bu z˘ ) E1y + E2z

(1)

where y ∈ Rn and u ∈ Rn are the plant outputs and inputs, respectively; z ∈ Rm are the internal states of the plant; and A ∈ Rn×n, B ∈ Rn×n, D ∈Rn×m, E1 ∈ Rm×n, and E2 ∈ Rm×m are given matrices. For the plant, we make the following assumptions: (A1) Only the plant output y is available for measurements. (A2) The elements of the matrices A, B, and D are uncertain. An estimate B h of the matrix B is available. (A3) (Minimum-phase assumption) E2 is a stable matrix. (A4) B is a full-rank matrix. It is noted that the plant in eq 1 is minimum-phase and, as a consequence of assumption A4, the relative degree vector is (1, ..., 1) ∈ Rn. Simple models for the two-point control of high-purity distillation columns5,11 and FCC processes6 can be described as in eq 1. 3. Control Design

As a consequence of assumption A3, the above linear system is stable. In fact, because both matrices Ac and E2 are stable, this implies that the triangular matrix

(

A c 0n E1 E2

)

is also stable. Although the control objective is achieved via the feedback controller (eq 5), it cannot be implemented because the modeling error signal η(t) is unknown. However, it is noted that the dynamics of the signal η(t) can be reconstructed from measurements of the output. In fact, η(t) ) y˘ (t) - A h y(t) - B h u(t). Our idea is to construct an observer to approximate the modeling error signal η(t) and use this observation to design the computed control input

uc ) B h -1[(Ac - A h )y - η j]

(6)

where η j (t) is the estimate of the modeling error signal η(t). We choose the observer (where η˘ , unknown, is not included)

j - η) η j˘ ) LMe(η

(7)

The control problem is to design an output feedback control to regulate the plant output y in a desired reference value yr. Without loosing generality, we will assume that yr ) 0. Let a desired closed-loop behavior be given as

where L > 0 is a tuning parameter and Me ∈ Rn×n is a stable matrix. To implement eq 7, we consider the fact that η(t) ≡ y˘ (t) - A h y(t) - B h u(t). Then

y˘ ) Acy

j - y˘ + A hy + B h u) η j˘ ) LMe(η

(2)

where Ac ∈ Rn×n is a stable matrix. The matrix Ac is provided by the designer and reflects the desired characteristics of the controlled plant output (e.g., settling time, output interactions). In this way, if decoupled output dynamics are required, then Ac ) diag(ac,1, ac,2, ..., ac,n), with ac,j < 0 for j ) 1,2, ..., n. According to assumptions A1 and A2, we define the modeling error vector η ∈ Rn as def

η ) ∆Ay + Dz + ∆Bu

(3)

where ∆A ) A - A h and ∆B ) B - B h , where A h is an estimate of A. In the worst-case design, we can take A h ) 0. Notice that the internal state z was included in the modeling error as it is not available for measurement. In principle, provided that the system in eq 1 is observable, we could use a state observer to estimate the unmeasured state z; however, this would make the size of the resulting controller unnecessarily large. The plant in eq 1 can be rewritten as

y˘ ) A hy + η + B hu z˘ ) E1y + E2z

(4)

The ideal feedback controller that leads to the desired closed-loop behavior is

h -1[(Ac - A h )y - η] uid ) B

(5)

-1 -1 j + y, so that the We introduce the vector w def ) L Me η above system is equivalent to

w˘ ) A hy + B hu + η j η j ) LMe(w - y)

(8)

which can be initialized as follows. Because η(t) is unknown, we take the initial estimate η j (0) ) 0, such that w(0) ) y(0). Notice that u in eq 8 is the actual control input acting on the plant (eq 1). Remark 1. Regarding the proposed feedback controller, the following must be stressed: (1) The dynamical system (eq 7) plays the role of a reduced-order observer12,13 when the modeling error signal η is seen as an extended state vector of the plant (eq 1). The application of reduced-order observers for the control of chemical processes is a rather recent event. Such applications are related to the implementation of nonlinear controllers based on state estimation.14-17 In a recent paper, we used an observer-based scheme to estimate unknown chemical kinetics and then used this estimate in an inverse-dynamics controller for a class of SISO plants.18 In this way, the estimator proposed before can be seen as an extension to a class of MIMO plants. (2) The feedback controller in eqs 6 and 8 can be seen as an adaptive controller, where the input signal (eq 6) is computed adaptively according to the dynamics of the estimated modeling error signal η j (t). Contrary to

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Ind. Eng. Chem. Res., Vol. 40, No. 4, 2001

traditional adaptive control structures where adaptation is made in parameters,19 in our case adaptation is made in signals. (3) If the computed control input uc is not subjected to deviations (e.g.,saturations, dead-zones), then u ) uc. Using eq 6, the estimator (eq 8) becomes equivalent to the following system:

w˘ ) Acy η j ) LMe(w - y)

(9)

In this way, we can see the estimation of the modeling error signal η(t) as a problem of model matching where the idea is to compute the signal w(t) to match the desired closed-loop behavior (eq 2). This is a common tool in robust control designs.20,21 3.1. Structure of the Proposed Feedback Controller. In this part, we will show that the proposed control design approach leads to well-known feedback control structures. In fact, we will show that the feedback controller in eqs 6 and 8 is equivalent to a multivariable PI control law with antireset windup (ARW) scheme. We can rewrite eq 8 as

η j ) LMe(w - y)

∫0 y(σ) dσ + Bh ∫0 [u(σ) - u (σ)] dσ}

LMe{-y + Ac

y˘ ) Acy + B h -1e z˘ ) E1y + E2z

t

-1

h {(Ac - A h + LMe)y - LMeAc u )B h LMeB

c

(

def A 0 Mc ) Ec En×n 1 2

(11)

(12)

which is a multivariable PI control with control gain Kc ∈ Rn×n and integral time TI ∈ Rn×n matrices given by

h -1(Ac - A h + LMe) Kc ) -B -1 h + LMe) TI ) -L-1A-1 c Me (Ac - A

)

and

∫0 y(σ) dσ -

∫0 [u(σ) - uc(σ)] dσ}

(17)

where

( )

h -1 C1 ) B 0n

t

t

(13)

The integral time matrix TI is well-defined as the matrices Me and Ac are stable, and consequently invertible, matrices. The ARW gain is given by

h -1MeB h KARW ) -LB

(16)

x˘ ) Mcx + C1e

We can use eq 11 in eq 6 to obtain the computed control input c

(15)

Notice that the computed control input is the ideal control input (eq 5) perturbed by the estimation error h -1e). In this way, uc ) uid as term B h -1 (i.e., uc ) uid + B e f 0. The controlled plant is the written as

(10)

where u - uc accounts for differences between the actual and computed control inputs. Therefore t

uc ) B h -1[(Ac - A h )y - η] + B h -1e

T n+m. In compact notation, eq 16 is Let x def ) (y, z) ∈R expressed as

h (u - uc) w˘ ) Acy + B

η j)

systems. Decentralized control is very important in process applications because of its several advantages over a fully multivariable design. These advantages include flexibility in operation, failure tolerance, simplified design, and simplified tuning.10 3.2. Stability Analysis. In the remainder of the work, and for the sake of simplicity in the stability analysis, we will assume that uc ) u. That is, the control input is not subjected to deviations due, for instance, to saturations, dead-zones, etc. j . Then, We introduce the estimation error e def ) η - η the computed control input (eq 6) becomes

On the other hand, the dynamics of the estimation error are governed by (see eq 7)

e˘ ) LMee + η˘

From eq 3, we can compute the time-derivative of the modeling error signal η(t). In fact, we have

h -1e) + D(E1y + E2z) + ∆Bu˘ η˘ ) ∆A(Acy + B

(19)

We can use eq 6 to compute u˘

h )(Acy + B h -1e) - η j] u˘ ) B h -1[(Ac - A h )(Acy + B h -1e) + LMee] )B h -1[(Ac - A

(14)

which generalizes the ARW scheme for SISO plants proposed by Astrom and Hagglund.1 When the actuator saturates, the feedback signal u - uc attempts to drive the input error u - uc to zero by recomputing the integral action such that the controller output uc is exactly at the saturation limit. This prevents the integrator from winding up.1 Remark 2. It is noted that, if A h , Ac, Me, and B h are h are also diagonal matrices, then Kc, TI, and MeB diagonal matrices. In this case, the computed control input (eq 12) becomes a decentralized PID control system consisting of independent SISO controller sub-

(18)

(20)

Using eqs 19 and 20 in eq 18, after some straightforward algebraic manipulations, we obtain

h -1e + ψ(x,e) e ) LMeBB

(21)

where

h -1e) + D(E1y + E2z) + ψ(x,e) ) ∆A(Acy + B h )(Acy + B h -1e)] (22) ∆BB h -1[(Ac - A Notice that ψ(x,e) does not depend on the parameter L. By recalling that x ) (y, z)T, we conclude that there exist


two positive constants ν1 and ν2, independent of L, such that

|y(x,e)| < ν1|x| + ν2|e|

(23)

In this way, the stability of the controlled process is completely determined by the stability of the (x, e) system composed of eqs 17 and 21. The idea for process stabilization is to use a sufficiently large value of L > 0 such that e(t) approaches zero very quickly and up f uid. The main result for the stability of the closed-loop system is established in the following theorem. The proof is given in the Appendix. h -1 is a stable matrix. Theorem 3.1. Assume that MeBB Then, there exists a positive number Lmin such that the closed-loop system of eqs 17 and 21 is stable for all L >Lmin. In other words, the above theorem states that, if the high-frequency matrix B h is chosen adequately such h -1 is stable, then the plant in eq 1 can be that the MeBB robustly stabilized via a multivariable PI controller despite model uncertainties. On the other hand, the h -1 implies that the input directionality stability of MeBB of the closed-loop system is not lost. 3.3. Decentralized Control for TITO Processes. A critical assumption for the stability of the closed-loop system of eqs 17 and 21 was the stability of the matrix h -1. It is noted that, if B h ) B (no uncertainty in MeBB h -1 ) Me, which the control input direction), then MeBB is a stable matrix, and the stability of the controlled process is unconditional. In general applications, this is not the case, and the stability of the uncertain matrix h -1 must be guaranteed. MeBB In some special cases of TITO processes, we can exploit the structure of the matrix B to provide a diagonal estimate B h that guarantees the stability of the h -1 despite strong parametric uncertainties. matrix MeBB Let us choose Me ) -I, so that the stability of the closedloop system of eqs 17 and 21 is conditioned by the stability of the matrix -BB h -1. Assume that the matrix B is described as follows:

(

b b B ) b11 b12 21 22

)

We will consider decentralized control by choosing B h ) -1 , R ), with R * 0 and R * 0. Therefore diag(R-1 1 2 1 2

(

Rb Rb -BB h -1 ) - R1b11 R2b12 1 21 2 22

)

The characteristic polynomial of the above matrix is

s2 + (R1b11 + R2b22)s + R1R2(b11b22 - b12b21) ) 0 Thus, R1b11 + R2b22 > 0 and R1R2(b11b22 - b12b21 > 0 are necessary and sufficient conditions for the stability of the matrix -BB h -1 and, consequently, for the stability of the closed-loop system of eqs 17 and 21, provided that L > Lmin. It is noted that, because B is invertible by assumption, its determinant b11b22 - b12b21 * 0. The following cases, for which decentralized control is guaranteed to work, are very important from a practical viewpoint. (a) b11b22 - b12b21 < 0, b11 > 0, b12 < 0, b21 > 0, b22 < 0, R1 > 0, and R2 < 0. This structure of signs of the matrix B arises in the LV control of distillation col-

Figure 1. Schematic diagram of a Petlyuk distillation system.

umns.5 Then, by virtue of Theorem 1, stability of controlled distillation columns under decentralized PI -1 control is assured if B h ) diag(R-1 1 , R2 ) is selected with sign(R1) ) sign(b11) ) +1 and sign(R2) ) sign(b22) ) -1. Notice that the information required for robust LV regulation of distillation columns is minimal, namely, only the sign of the diagonal elements of the matrix B. (b) b11b22 - b12b21 > 0, b11 > 0, b12 < 0, b21 < 0, b22 > 0, R1 > 0, and R2 > 0. We will see in the next section that this structure of signs of the matrix B arises in the two-point control of Petlyuk distillation systems. Thus, by virtue of Theorem 1, stability of controlled Petlyuk distillation systems under decentralized PI control is -1 assured if B h ) diag(R-1 1 , R2 ) is selected with sign(R1) ) sign(b11) ) +1 and sign(R2) ) sign(b22) ) +1. As in the above case, the information required to attain robust stabilization is minimum; namely, only the sign of the diagonal elements of the matrix B. It should be stressed that the above are not the only cases for which decentralized PI control guarantees robust stability of the controlled process. However, the cases discussed above are very important in the control of distillation processes because they display the closedloop integrity property,10 which requires the process to remain stable in the face of arbitrary sensor or actuator failures. The closed-loop integrity property can be easily established from the fact that only the sign of the diagonal elements of B is required for robust stability. 3.4. Implications for Tuning. The above properties of the proposed PI control configuration suggest a tuning procedure according to the following steps: (i) Choose the desired closed-loop matrix Ac (see eq 2). If decoupled output dynamics is a control objective, choose Ac ) diag-1 -1 (-τ-1 c,1 , -τc,2 , ..., -τc,n), where τc,j > 0 is the desired closed-loop time constant in the jth loop. (ii) Determine the matrix B h that such that -BB h -1 is a stable matrix. In general, this is not an easy problem because its solution involves stability estimation under parametric uncertainties in the matrix B. Although several results (e.g., edge-like theorems) can be found in the literature,22 there is a lack of results for practical applications. Decentralized control can lead to less complex stability conditions. In the TITO cases discussed above, choose

1190


Figure 2. Step response of the Petlyuk column under a (0.5% disturbance in the control inputs.

sign(R1) ) sign(b11) and sign(R2) ) sign(b22). From a h 11 and R-1 h 22, practical viewpoint, choose R-1 1 ) b 2 ) b h 22 can where the estimated diagonal elements b h 11 and b be obtained from step and frequency responses of the plant. (iii) Because the observer gain L has frequency -1 for a physically meaningful tununits, take τe def ) L ing, where τe is the estimation time-constant. Reduce τe up to a point where either a satisfactory transient response is attained or further reductions cause deterioration in the response rather than improvements. Decreasing τe will increase the sensitivity of the feedback loop, so that better disturbance rejection capabilities are obtained. However, excessively small values of the estimation time constant might amplify measurement noise and excite unmodeled high-frequency dynamics. Although these tuning operations are in accordance with engineering intuition, we must note the following: (1) The tuning of the closed-loop time constants τc,j, j ) 1, 2, ..., n is particularly easy to carry out in view to the fact that the response of the jth controlled loop is faster

with smaller values of τc,j. (2) Once the matrices Ac ) -1 -1 h have been chosen, the diag(-τ-1 c,1 , τc,2 , ..., τc,n) and B control gain (eq 13) depends only on one parameter, namely, the estimation time constant τe. In view of the monotonic property showed in the proof of Theorem 1 (see the Appendix), it is clear that reductions of τe will increase the sensitivity function of the loops. Hence, the minimum allowable value of τe (i.e., the maximum allowable value of L) is constrained by the bandwidth of measurement noise, unmodeleded dynamics, and samplings. Detuned PI control is a possible remedy for this situation.4 A detuned version of the PI controller (eq 12) is

Kc(I + T-1 I /s)Q(s) where s denotes d/dt (or the Laplace domain variable) and Q(s) is the detuning factor with Q(0) ) I. Q(s) is beneficial in that it improves robustness against unmodeled dynamics and measurement noise, but has the disadvantage that it is not clear how to choose it. For


Figure 3. Response of the Petlyuk column under full-matrix PI control with B h ) Kτo and τe ) L-1 ) 45 min.

instance, if measurement noise is the problem, Q(s) should be chosen as a low-pass filter. On the other hand, if unmodeled dynamics are important, Q(s) should be chosen as a led-lag filter.4,21 Remark 3. In this paper, we focused on the class of plants described by eq 1. The proposed control design procedure relies strongly on the high-gain estimation of the modeling error signal η(t). It is a well-known fact20,21 that the presence of nonminimum-phase components limits the achievable performance of the closedloop plant. In our case, this limitation is represented by the existence of a maximum value Lmax for the estimation parameter L. That is, the closed-loop system can undergo unstability phenomena for excessively large values of the estimation parameter L. It should be stressed that such performance limitations can not be modified by feedback and so they cannot be modified by any (adaptive or not) control strategy. The assumption of relative degree one can be relaxed by extending the observer scheme (eq 7) to estimate unmeasured states (see, for instance, Alvarez-Ramirez23 for theoretical results on the SISO case). However, the Hurwitz h -1 (see Theorem 3.1) condition for the matrix MeBB

involves the analysis of quite complex matrix structures, such that it is not easy to establish a general result. We have some results for special cases, which will be published elsewhere. 4. Example: Control of Petlyuk Distillation Columns The separation of more than two components by continuous distillation has traditionally been accomplished by arranging columns in series. An alternative design consists of an ordinary column shell with the feed and sidestream product draw divided by a vertical wall though a set of trays. This implementation offers savings in investment as well as operating costs and is usually denoted as a Petlyuk column. Compared to an ordinary distillation column, the Petlyuk column has many more degrees of freedom in both operation and design. This makes the design of both the column and its control system more complex.24 Chavez et al.25 found that the Petlyuk column has five degrees of freedom at steady state, and they found that a multiplicity of steady-state solutions can occur when three purities in each of the products are specified. Wolff and Skogestad24

1192


Figure 4. Response of the Petlyuk column under full-matrix PI control with B h ) Kδ/τo and τe ) L-1 ) 45.

found that the Petlyuk column can present steady-state “holes” in certain operating regions for which it is not possible to achieve the desired product specifications. They also studied the decentralized control of the Petlyuk column for three-point and four-point conditions. Simple PI controllers were used, which proved to give acceptable performance. In this section, we will use the control design in the above section to propose a suitable control design for two- and three-point control of the Petlyuk column. Our result will show that the LV control of two purities in the Petlyuk column can be accomplished with a decentralized PI controller and that the control problem is not more difficult than that in ordinary distillation columns. The case study corresponds exactly to case IV of Weyburn and Seader.26 The feed is liquid at its bubble point with a flowrate of 1000 kmol/h: 200 kmol/h of benzene, 400 kmol/h of toluene, and 400 kmol/h of o-xylene. At the bubble point, the temperature and pressure of the mixture are 383.4 K and 101.33 kPa, respectively. The following general assumptions are made: (1) The reflux divider and all stages, except the

condenser and reboilers, are considered to be adiabatic. (2) Heat-transfer rates for condensers and reboilers are never specified. (3) The system operates at atmospheric pressure (101.33 kPa) with zero pressure drop throughout the system. (4) The configuration of the system, i.e., number of stages and locations of feeds, side streams, and interlinks, is known. (5) All products streams leave the system as saturated liquids. (6) The model description considers only the material balance of the system, assuming constant molar flows and ideal behavior, as the mixture studied (benzene, toluene, and o-xylene) is nearly ideal. Accordingly, the separation factor was computed from Raoult’s law by using the Antoine equation for the vapor pressure. The system consists of a 20-stage prefactionator interlinked with a 32-stage fractionator. The latter includes a total condenser and a partial reboiler. The stage locations for the feed, products, and interlinks are all included in Figure 1. All model equations are solved simultaneously via an RKF method with variable stepsize. The Petlyuk column has five steady-state degrees of freedom,26 which were consumed by the following


Figure 5. Response of the Petlyuk column under decentralized PI control with B h taken as the diagonal elements of K/τo and τe ) L-1 ) 45 min.

specifications: molar purity of benzene in the distillate, xD 1 ) 95%; molar purity of toluene in the middle (side) product, xS2 ) 90%; molar purity of o-xylene in the bottoms, xB3 ) 95%; total bottom flowrate, 380 kmol/h; and reflux ratio values in the range from 4.525 to 5.75. If we assume that benzene does not appear in the bottom and o-xylene does not appear in the distillate, then the purity specifications can be achieved by specifying any value of the total bottom flowrate in the range from 377.7 to 421.04 kmol/h, as can be shown by a mass balance. The value of 380 kmol/h and a reflux ratio of 5.75 were chosen to simulate the column presented by Wayburn and Seader.26 4.1. Simple Dynamical Model. The nominal operating point studied has L ) 1050.0 kmol/h, V ) 2060 S kmol/h, RL ) 0.476, RV ) 0.44, xD 1 ) 0.943, x2 ) 0.916, B and x3 ) 0.967. We will consider LV control of the Petlyuk column. The reflux (L) is used to control the top composition (xD 1 ), and the boilup (V) is used to control the bottom composition (xB3 ). Following standard ideas from single-column distillation processes, a

simple one-time-constant model can be described as follows:11,27

y˘ ) (-y + Ku)/τo

(24)

B T where y ) (xD and u ) (L, V)T in deviation 1 , x3 ) variables. In addition, τo is the open-loop dominant time-constant and K ∈ R2×2 is the process gain matrix. According to the system in eq 1, we have A ) diag-1 (-τ-1 o , -τo ) and B ) K/τo. The plant parameters τo and K can be readily estimated from step responses (see Figure 2). We obtained the following estimates: τo ) 30 h and

K)

(

0.00314 -0.00378 -0.00055 0.00069

)

(25)

The condition number of the matrix K is too large, namely, 284.55, which implies that full-matrix PI control will be very sensitive to input uncertainties.27,28 In fact, because B-1 ) τoK-1, the stability of the closedloop system will depend on the input directionality

1194


Figure 6. Response of the Petlyuk column under decentralized PI control with B h taken as the diagonal elements of Kδ/τo and τe ) L-1 ) 45 min.

induced by the steady-state gain matrix K. Sensitivity of the input directionality is exemplified as follows. The inverse of K is given by

K-1 )

(

7876.7 43151.0 6278.5 35845.0

)

(26)

Assume a (15% error in the estimation of the elements of K. For instance, take the perturbed matrix Kδ

Kδ )

(

0.95 × 0.00314 -1.15 × 0.00378 -0.85 × 0.00055 0.9 × 0.00069

)

(27)

and its inverse

K-1 δ )

(

-3454.2 -24180.0 -2600.4 -16593.0

)

(28)

Compared with the real inverse matrix (eq 26), it is noted that the elements of the above approximate inverse are negative, such that input directionality is

lost and closed-loop unstability is expected to be present when full-matrix PI control with Kδ is used to regulate the column operation. 4.2. Numerical Simulations. We have carried out several numerical simulations to illustrate the performance of the proposed PI control configuration. To this end, we have taken Ac ) diag(-0.3333, -0.3333), which corresponds to τc,1 ) τc,2 ) 3 h. The column was started from the nominal conditions and then regulated at the B setpoint (xD 1 , x3 ) ) (0.95, 0.95) At t ) 200 h, the B setpoint was changed to (xD 1 , x3 ) ) (0.99, 0.99), and at t ) 400 h, the feed composition was changed from (0.2, 0.4, 0.4) to (0.25, 0.35, 0.4). Figure 3 presents the response of the distillation column under full-matrix PI control with B h ) K/τo and h -1 ≈ -I, as expected τe ) L-1 ) 45 min. Because -BB from the result in Theorem 1, the response of the controlled column is stable. Moreover, the performance of the full-matrix PI controller is acceptable. Now suppose that a practical unavoidable error27,28 was introduced in the estimation of the steady-state gain


Figure 7. Response of the Petlyuk column under decentralized PI control with B h taken as the diagonal elements of Kδ/τo and three different values of the estimation time constant τe.

matrix K. To this end, assume that B h ) Kδ/τo is taken as the estimated high-frequency gain matrix. As expected from the loss of input directionality in the inverse of B h , the resulting closed-loop system is unstable (see Figure 4). Because the resulting controller is unstable, the control inputs L and V reach their saturation limits, and the setpoints are not tracked. At this point, decentralized control is an alternative for avoiding closed-loop unstabilities10 induced by uncertainties in the steadystate matrix K. Let us determine whether the conditions for decentralized control are met. The determinant of the matrix B ) K/τo is positive (8.76 × 10-8/τo > 0), b11 ) 0.00314/ τo > 0 and b22 ) 0.00068/τo > 0. Consequently, the conditions of case 2 in section 3.2 are satisfied, and unconditionally stable decentralized control of the Petlyuk distillation column is possible with B h ) diag(R-1 1 , -1 R2 )/τo, R1 > 0, and R2 > 0. Figure 5 presents the behavior of the controlled column under the same PI -1 controller parameters as in Figure 4, with R-1 1 and R2

taken as the diagonal elements of the matrix K. As expected, the PI controller is stable and provides an acceptable tracking of setpoints. To demonstrate the unconditional stability property of the decentralized PI controller, Figure 6 presents the behavior of the conand R-1 are taken as the trolled column when R-1 1 2 diagonal elements of the matrix Kδ. Whereas full-matrix PI control led to unstable closed-loop behavior, decentralized PI control produced stable closed-loop behavior with acceptable performance. As the estimation time constant τe ) L-1 becomes smaller, the modeling error signal η(t) is estimated more quickly. In principle, decrements of τe will lead to better closed-loop performance. Figure 7 presents the behavior -1 of the decentralized control (with R-1 1 and R2 taken as the diagonal elements of the matrix Kδ) for three different values of the estimation time constant τe (0.75, 3.0, and 6.0 h), and to better show this phenomenon, we changed τc,1 ) τc,2 ) 5 h, and the time to change the setpoints doubled. It is noted that better performance is obtained for smaller values of τe. Of course, the

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Figure 8. Behavior of the two-point controlled Petlyuk column when a 15-min dead time is present in the input channel.

minimum allowable value of τe is restricted by the size of the unmodeled dynamics (including dead times) and measurement noise. Figure 8 presents the behavior of a controlled column with τc,1 ) τc,2 ) 20 h, τe ) 15 h, and dead time in the input channel of 15 min. This dead time can be considered as induced by measurement sampling. It is observed that acceptable closed-loop behavior is obtained for values of the dead time on the order of τe; however, oscillatory response and even unstabilities are obtained for excessively small τe values. For example, in Figure 9, we present the same column but with a dead time of 30 min. It is concluded from a heuristic viewpoint that it is not recommended to estimate the modeling error signal faster than the input dead times. This is in agreement with engineering intuition.21,22 4.3. Three-Point Control. A drawback of two-point control of the Petlyuk column is that the purity of the medium stream is not regulated, such that under external disturbances (e.g., changes in feed composition) the purity xS2 can be lowered considerably.24 This motivates the use of three-point control where the medium

stream flow rate (S) is used to control the medium purity (xS2 ). For control design, we will consider a simple model as in eq 24. The estimated open-loop time constant τo has the same value as above, and the steadystate gain matrix K ∈ R3×3 is given by

(

0.00314 -0.00378 -2.9268 × 10-5 K ) -0.00055 0.00069 0.000507 0.001127 -0.001427 -0.001639

)

(29)

We consider decentralized control. To this end, B h ) diag-1 -1 -1 -1 -1 (R-1 , R , R )/τ , where R , R , and R are taken as o 1 2 3 1 2 3 the diagonal elements of K. From Theorem 1, the resulting decentralized PI controller will be stable if the h -1 is stable. We obtain the matrix matrix -BB h -1 ) -KK

(

1 -1.2038 -9.321 × 10-3 -KK h -1 ) - -0.7971 1 0.73478 -0.68761 0.87065 1

)

whose eigenvalues are -2.427, -0.56643, and -6.5356 × 10-3. In this way, Theorem 1 implies that the


Figure 9. Behavior of the two-point controlled Petlyuk column when a 30-min dead time is present in the input channel.

decentralized PI control will be stable. Figure 10 presents the behavior of the controlled Petlyuk column under three-point decentralized control with τe )10 min and the following sequence of setpoint changes and disturbances: The setpoint is initially chosen as (xD 1, xB3 , xS2 ) ) (0.95, 0.95, 0.95). At t ) 200 h, the setpoint is B S changed to (xD 1 , x3 , x2 ) ) (0.99, 0.99, 0.99), and at t ) 400 h, the feed composition is changed from (0.2, 0.4, 0.4) to (0.25, 0.35, 0.4). As expected, the decentralized PI control is stable and the tracking of setpoints is acceptable despite of feed composition changes. Remark 4. The superior characteristics of the proposed PI control for the output regulation of Petlyuk distillation processes can be summarized as follows: (a) A model-based tuning procedure is provided. Specifically, the selection of the PI control gains, according to eq 13, is made on the basis of open-loop characteristics (i.e., open-loop time constant and steady-state gain). In addition, the entire tuning procedure depends only on one parameter, namely, the estimation time constant τe ) L-1 > 0. (b) A systematic procedure to check closed-

loop stability under decentralized control is obtained. In fact, it is only necessary to check the Hurwitz stability of the matrix BB h -1, which can be done with 23 existing procedures. (c) The PI control is endowed with a natural ARW scheme to reduce the adverse effects induced by control input saturations on the integral actions. (d) Unstabilities induced by unmodeled dynamics (e.g., the relative degree of the actual plant is higher than one) can be easily detected. In fact, excessively small values of the estimation time constant τe > 0 excite high-frequency dynamics (see Figure 9). A remedy for this unstability phenomenon is to augment the value of the estimation time-constant. In this way, the proposed PI control configuration has the advantage over traditional approaches that the tuning parameter τe ) L-1 > 0 displays monotonic stability properties. That is, the smaller the value of τe > 0, the faster the closedloop response. Hence, detuning of the proposed PI control scheme to account for unmodeled dynamics is easy to carry out just by augmenting the value of the estimation time-constant τe.

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Figure 10. Behavior of the three-point controlled Petlyuk column under three-point decentralized control with τe ) 10 min.

5. Conclusions We proposed a PI control design for MIMO processes. Under the assumption that the nominal plant is relative degree one, the idea is to use inverse control endowed with an observer that provides estimates of the underlying modeling error signals. This control design approach leads to classical multivariable PI controllers with a novel parametrization of the controller gain and integral time matrices. A nice result is that an ARW scheme to confront input saturations arises naturally from our control design approach. We have established the stability of the resulting closed-loop system via Lyapunov functions. Stability conditions has been discussed for the case of decentralized control. In several particular cases, which are important for the control of distillation processes, unconditional stability of the closed-loop system under decentralized PI control has been established. As an application of the proposed control design, we considered the control of a ternary mixture into three

products using a Petlyuk distillation system. We have shown that control of two streams via LV control is possible via decentralized PI controllers. Moreover, the controller presents good performance under setpoint changes and disturbances. In addition, it is interesting to remark that, because the process is ill-conditioned, full-matrix PI control leads to closed-loop unstability in the presence of moderate parameter uncertainty. On the contrary, decentralized PI control shows unconditional stability in the presence of strong uncertainties in the input channel. Appendix Proof of Theorem 1. From assumption A3, Mc is a stable matrix, such that there exists a symmetric and positive-definite matrix Pc ∈ R(m+n)×(m+n) satisfying the Lyapunov equation MTc Pc + PcMc ) -I. By hypothesis, h -1 is also a stable matrix, and there exists a symMeBB metric and positive-definite matrix Pe ∈ Rn×n satisfying the Lyapunov equation MTc Pc + PcMc ) -I. Choose the


quadratic function V(x,e) ) xTPc x + eTPee. The time derivative of V(x,e) along the trajectories of the system of eqs 17 and 21 is

V˙ ) -xTx + xTPcC1e - LeTe + LeTe + eTPeψ(x,e) e -|x|2 - L|e|2 + λmax(Pc)ν3|x||e| + λmax(Pe)|e||ψ(x,e)| where ν3 ) ||C1|| and λmax(Pc) and λmax(Pe) are the maximum eigenvalues of Pc and Pe, respectively. We can use eq 23 to obtain the following estimate:

V˙ e -|x|2 - (L - λmax(Pe)ν2)|e|2 + (λmax(Pc)ν3 + λmax(Pe)ν1)|x||e| We next introduce def

Lmin ) λmax(Pe)ν2 + [λmax(Pc)ν3 + λmax(Pe)ν1]2/4 It is easy to show that V˙ < 0 for all L > Lmin. This completes the proof of the theorem. Literature Cited (1) Astrom, K. J.; Hagglund, T. Automatic Tuning of PID Controllers; ISA: New York, 1988. (2) McMillan, G. K. Tuning and Control Loop Performance: A Practitioner’s Guide; ISA: New York, 1990. (3) D′Azzo, J. J.; Houpis, C. H. Linear Control Systems; McGraw Hill: New York, 1981. (4) Gawthrop, P. J. Self-tuning PID controllers: Algorithms and implementation. IEEE Trans. Autom. Control 1986, 31, 201. (5) Skogestad, S.; Morari, M. LV-control of a high-purity distillation column. Chem. Eng. Sci. 1988, 43, 33. (6) Hovd, M.; Skogestad, S. Procedure for regulatory control structure selection with application to the FCC process. AIChE J. 1993, 39, 1938. (7) Luyben, W. L. Simple method for tuning SISO controllers in mutivariable systems. Ind. Eng. Chem. Process Des. Dev. 1986, 25, 654. (8) Palmor, Z. J.; Halevi, Y.; Krasney, N. Automatic tuning of decentralized PID controllers for TITO processes. Automatica 1995, 31, 1001. (9) Wang, Q.-G.; Lee, T.-H.; Zhang, Y. Multiloop version of the modified Ziegler-Nichols method for two-input two-output processes. Ind. Eng. Chem. Res. 1998, 37, 4725. (10) Campo, P. J.; Morari, M. Achievable closed-loop properties of systems under descentrilized control: Conditions involving the steady-state gain. IEEE Trans. Autom. Control 1994, 39, 932.

(11) Chien, I.-L.; Tang, Y. T.; Chang, T.-S. Simple nonlinear controller for high-purity distillation columns. AIChE J. 1997, 43, 3111. (12) Haddad, W. Optimal reduced-order observer estimators. J. Guidaence, Control Dyn. 1990, 13, 1126. (13) Nikoukhah, R. New methodology for observer design and implementation. IEEE Trans. Autom. Control 1998, 43. (14) Kravaris, C.; Soroush, M. Nonlinear control of a batch polymerization reactor. An experimental study. AIChE J. 1992, 38, 1429. (15) Daoutidos, P.; Kravaris, C. Dynamic output feedback control of minimum-phase multivariable nonlinear processes. Chem. Eng. Sci. 1994, 49, 433. (16) Soroush, M.; Kravaris, C. Nonlinear control of a polymerization CSTR with singular characteristic matrix. AIChE J. 1994, 40, 980. (17) Kan, K. M.; Tade, M. O. Nonlinear control of a simulated industrial evaporation system using a feedback linearization technique with a state observer. Ind. Eng. Chem. Res. 1999, 38, 2995. (18) Aguilar, R.; Gonzalez, J.; Alvarez-Ramirez, J.; Barron, M. A. Control of a fluid catalytic cracking unit based on proportionalintegral reduced-order observer. Chem. Eng. J. 1999, 75, 77. (19) Alvarez-Ramirez, J. Adaptive control of feedback linearizable systems: A modeling error compensation approach. Int. J. Robust Nonlinear Control 1999, 9, 361. (20) Astrom, K. J.; Wittenmark, B. Adaptive Control; AddisonWesley Publishing Co.: Reading, MA, 1989. (21) Morari, M.; Zafiriou, E. Robust Process Control; Prentice Hall: New York, 1989. (22) Maciejowoski, J. M. Multivariable Feedback Design; Addison-Wesley Publishing Co.: Wokingham, U.K., 1989. (23) Bhattacharayya, S. P.; Chapellat, H.; Keel, L. H. Robust Control. The Parametric Approach; Prentice Hall: Upper Saddle River, NJ, 1995. (24) Wolff, E. A.; Skogestad, S. Operation of integrated threeproduct (Petlyuk) distillation columns. Ind. Eng. Chem. Res. 1995, 34, 2094. (25) Chavez, C. R.; Seader, J. D.; Wayburn, T. L. Multiple steady state solutions for interlinked separation systems. Ind. Eng. Chem. Process Des. Dev. 1981, 20, 546. (26) Wayburn, T. L.; Seader, J. D. In Proceedings of the 2nd International Conference on Foundations of Computer-Aided Process Design; CACHE: Ann Arbor, MI, 1984; p 765. (27) Skogestad, S.; Morari, M.; Doyle, J. C. Robust control of ill-conditioned plants: High-purity distillation. IEEE Trans. Autom. Control 1988, 12, 1092. (28) Sagfors, M. F.; Waller, K. V. Multivariable control of illconditioned distillation utilizing process knowledge. J. Process Control 1998, 8, 197.

Received for review July 19, 1999 Revised manuscript received August 21, 2000 Accepted November 8, 2000 IE9905308

A PI Control Configuration for a Class of MIMO Processes - Industrial

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