Optimization-based decoupling controller design for discrete systems

Chemical Engineering Science 56 (2001) 4695–4710

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Optimization-based decoupling controller design for discrete systems
;

Dennis D. Sourlas ∗;1 Department of Chemical Engineering & Intelligent Systems Center, 143 Schrenk Hall, University of Missouri-Rolla, Rolla, MO 65409-1230, USA Received 13 March 2000; received in revised form 26 December 2000; accepted 23 January 2001

Abstract Decoupling has been studied as a means of improving control loop behavior in the presence of strong interactions, particularly in high-purity distillation. The design of optimal multivariable decoupling controllers is the main topic of this work. The stabilizing and decoupling controllers are described through the Youla parametrization and linear equalities. Then, for given time-domain performance envelopes, a numerical optimization method based on exact penalty functions and linear programming is presented that identi5es the multivariable controller that decouples the closed loop and optimizes its performance. It is also shown that this approach can be extended to encompass traditional decoupling techniques based on (possibly low order) decouplers. Finally, the problem of decoupling is discussed in relation to simultaneous and decentralized multivariable control where it is demonstrated that approximate decoupling may be a necessary compromise when there are not enough degrees of freedom. ? 2001 Elsevier Science Ltd. All rights reserved. Keywords: Optimization; Mathematical; Process control; Stability; Decoupling; Linear systems

1. Introduction Strong interactions in multivariable processes create di;culties in maintaining desirable setpoints for all controlled variables under setpoint changes or load disturbances. Decoupling, a technique that aims at reducing closed loop interactions, has been proposed as solution to this problem. Evidence from industrial practice has indicated that various forms of decoupling can improve the operation of multivariable distillation processes under automatic control (Freitas, Campos, & Lima, 1994). One instance of the problem, the diagonal decoupling problem, aims at the development of feedback control laws that diagonalize the setpoint-to-output closed loop map. A diagonally decoupling controller allows to change the
Part of this work was originally presented as paper 45c at the 1996 Annual AIChE Meeting.

This work was supported by an award from the University of Missouri Research Board. ∗ Tel.: +1-573-341-5525; fax: +1-573-341-4377. E-mail address: [email protected] (D. D. Sourlas). 1 URL: http:==www.umr.edu= ∼ dsourlas.

operating point for one output without aBecting the remaining outputs. Depending on the desired structure of the closed loop I=O map one may de5ne variants of the decoupling problem such as: block-diagonal decoupling, triangular and block-triangular decoupling (see Section 2). The decoupling problem was 5rst discussed in Morgan (1964) within a state feedback context. Since then a large portion of the published works has focused on the development of necessary and su;cient conditions for the existence of decoupling controllers. Several of the proposed approaches achieve decoupling via state feedback and discuss the construction of such decoupling compensators (see for example: Falb & Wolovich, 1967; Gilbert, 1969; Morse & Wonham, 1970a; Va5adis & Karkanias, 1997). These approaches have also been extended to the triangular decoupling problem (for example in Morse & Wonham, 1970b; Commault & Dion, 1983). Necessary and su;cient conditions for the existence of stabilizing static output feedback compensators are given in Wang and Davidson (1975) and Paraskevopoulos and Koumboulis (1995) for regular and singular plants, respectively. In the context of one degree of freedom

0009-2509/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 1 ) 0 0 1 1 8 - X

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compensation dynamic output feedback compensation, the concept of strict adjoint was proposed by Hammer and Khargonekar (1984) in order to develop necessary and su;cient conditions for the existence of stabilizing decoupling controllers. Using the fractional representation theory, the complete parameterization of all decoupling stabilizing controllers within the two degrees of freedom control framework has been presented in Desoer and GMundeNs (1985, 1986) and GMundeNs and Desoer (1990). This parametrization was subsequently extended to non-square systems by GMundeNs (1990). Parametrizations of all decoupling stabilizing compensators for one degree of freedom feedback control systems have been proposed for square plants and non-square plants in Lin and Hsieh (1991, 1993) and Lin (1995). These conditions do not require stable coprime factorizations but are given in terms of scalar polynomials that should satisfy interpolation conditions. Design approaches for decoupling control systems were strongly motivated by the need to achieve tight two-point control in distillation columns. Elimination of strong interactions through decoupling could improve product consistency and also minimize energy consumption, especially in high-purity column where the interactions are strongest. In this context, a decentralized 2 × 2 controller (typically PI or PID) is 5rst designed. The decoupler is an additional dynamical element that is subsequently added to the feedback loop and aims at the minimization of interactions. Early works on decoupling focused on the assessment of diBerent decoupler design methods (ideal decoupling, simpli5ed decoupling, one-way decoupling) as a means to achieve two-point control in high-purity columns, mainly through simulation (see for example the works of Luyben, 1970; Weischedel & Avoy, 1980). Through case studies it was demonstrated that the so-called simpli5ed decoupling and one-way decoupling were more promising. The impact of controller selection on the performance of decouplers was discussed by Waller (1974). Arkun, Manousiouthakis, and Palazoglu (1984) presented an analysis tool that was used to assess the robustness of decouplers. This approach was later extended to the case of decoupling stabilizing controllers (Manousiouthakis & Arkun, 1984). Tsiligiannis and Svoronos (1988, 1989) considered the design of decoupling controllers through the use of dynamic interactors. The so-called “highly structured stability margin” (Figueroa, Desages, Romagnoli, & Palazoglu, 1991) was employed to 5rst analyze the robust stability of the control loops with decouplers and to subsequently design approximate, low order robust decouplers (Figueroa, Barton, & Romagnoli, 1994). In this work the decoupler has the same structure as in simpli5ed decoupling, but lead-lag dynamic elements are considered. The aforementioned stability and design approaches can identify decoupling stabilizing controllers. However,

they do not consider closed loop optimal performance under decoupling control. Over the last decade several works have focused on performance optimization studies in the context of decoupling control. Within a two degree of freedom control framework and using the appropriate decoupling stabilizing controller parametrizations (Desoer & GMundeNs, 1985, 1986; GMundeNs & Desoer, 1990) frequency domain optimization was presented by Lee and Bongiorno (1993). Within an LQR approach Xia, Rao, Sun, and Ying (1993) discussed the approximate decoupling problem. In this work the undesirable interaction terms are included as penalty terms in the quadratic objective function of the LQR model tracking optimization problem. The design of approximately decoupling controllers under uncertainty was formulated and solved within the ‘1 robust control framework by Queh and Loh (1997). The undesirable interaction terms are weighted and augmented in the objective function of the standard ‘1 robust controller design problem. The problem is solved through a sequence of non-linear, non-convex optimization problems. The level of decoupling that is achieved by this method depends on the level of uncertainty and the weights used for the interaction terms. The objective of this paper is to establish a framework capable of answering a host of questions associated with the design of decoupling systems, especially in the context of chemical engineering practice. Some of the related issues are: (a) What is the closed loop performance that can be achieved under decoupling control? This is related to the best achievable performance problem and provides a measure of the performance degradation, that decoupling control causes, relative to the optimal multivariable controller. (b) How can one improve existing control systems through the design of decouplers? The addition of decouplers in a control loop has been used in chemical engineering practice since the existing control structure is maintained. Then, it is relevant to determine the best decoupler structure and dynamics so that the interactions are minimized (or eliminated) with the least penalty in achievable performance. (c) How can one design reliable controller–decoupler con5gurations under controller order and controller structure constraints? The interaction of the decoupler with the controller necessitates retuning the controller which may be restricted in terms of structure (decentralized control) or complexity (PI or PID dynamics). The resulting loop should also be capable of maintaining stability in the case of decoupler failures. Thus one should simultaneously select the feedback controller, decoupler pair. (d) What is the trade-oB between approximate decoupling and robustness, or control structure, in

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in the setpoint subvector u1i only aBect the output subvector y2i , i = 1; : : : ; n. In an analogous manner, the block-triangular decoupling problem is de5ned as:

Fig. 1. Unity feedback control system.

multivariable control? When multivariable controllers are used to also achieve decoupling, it is often the case that robustness is achieved at the expense of decoupling ability. Also the use of structurally simple (decentralized=distributed) controllers limits the ability of these controllers to decouple the closed loop. It is therefore relevant to identify controllers that achieve the best compromise between robustness (or structure) and decoupling. The main part of this paper is devoted to answering (a) within a multivariable control framework whereby the controller is designed to achieve diagonal, or triangular, decoupling. This approach encompasses all controller=decoupler combinations that are possible for a given decoupling structure, and is capable of identifying limits of performance. An optimization problem formulation is derived that minimizes a time-domain performance measure and simultaneously accounts for disturbance rejection and setpoint tracking. Closed loop decoupling imposes constraints on the multivariable controller that are expressed in terms of appropriate linear equalities (Section 2). Sections 3 and 4 contain the optimization problem formulation and the proposed problem solution, respectively. An illustrative multivariable controller design example is discussed in Section 5. Finally, in Section 6 it is described how the framework of Sections 2– 4 can be extended to also provide answers to the aforementioned questions (b) – (d). 2. Controller parametrization In this work the focus is on the input–output (I=O) decoupling problem, or setpoint decoupling problem, within a standard unity feedback control con5guration (Fig. 1). Then, the block-diagonal decoupling control problem can be de5ned as follows: Denition 1. For the feedback control interconnection of Fig. 1 consider a partition {u11 ; u12 ; : : : ; u1n } of the input signal u1 (setpoint), and a compatible partition {y21 ; y22 ; : : : ; y2n } of the output signal y2 . Then, a diagonally decoupling controller should be such that changes

Denition 2. Consider the same feedback interconnection and signal partitions as in De5nition 1. Block lower triangular decoupling is achieved when changes in the setpoint subvector u1i only aBect the output subvectors y2j , i 6 j 6 n. Block upper triangular decoupling is achieved when changes in the setpoint subvector u1i only aBect the output subvectors y2j , 1 6 j 6 i. The Youla parametrization of all stabilizing controllers for linear time invariant systems forms the basis for the development of a complete representation of all block decoupling stabilizing controllers. A review of the basic results from the fractional representation theory is given in Appendix A and follows Vidyasagar (1985). Consider a multivariable process represented by the I=O map P that accepts a doubly coprime fractional representation NP ; DP ; N˜ P ; D˜ P , X; Y; X˜ ; Y˜ (Vidyasagar, 1985, Chapter 5). Then the set of all stabilizing controllers S(P) of P is given as S(P) = {(Y − QN˜ P )−1 (X + QD˜ P ); Q ∈ M (S); det(Y − QN˜ P ) = 0} = {(X˜ + DP Q)(Y˜ − NP Q)−1 ; Q ∈ M (S); det(Y˜ − NP Q) = 0}:

(1)

Based on Eq. (1) the closed loop I=O map between the setpoint and the process output in Fig. 1 is parametrized in terms of a stable parameter Q as Hu1 ;y2 = NP (X + QD˜ P ):

(2)

According to Eq. (2), block-diagonal I=O decoupling is achieved iB there exists a parameter Q such that the closed loop map, Hu1 ;y2 , is block-diagonal. This translates into linear equality constraints on Q. Next, these constraints are derived for the 2-channel diagonal decoupling case. Let P be a n × m I=O map and let [u11 u12 ]T denote a partition of the setpoint vector such that u11 ∈ R n1 and u12 ∈ R n2 (n = n1 + n2 ). Also, let NP = [NP1 NP2 ]T be a compatible partition of NP , where NP1 ∈ S n1 ×m and NP2 ∈ S n2 ×m . Also, let X = [X1 X2 ]; Q = [Q1
 D˜ P11 D˜ P12 ˜ ; DP = ˜ DP21 D˜ P22

Q2 ];

where Xi ; Qi ∈ S m×ni ;

D˜ Pij ∈ S ni ×nj ;

i; j = 1; 2:

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Using the above de5nitions the closed loop I=O map Hu1 ;y2 can be expressed as follows:   NP1 (X1 + Q1 D˜ P11 + Q2 D˜ P21 ) NP1 (X2 + Q1 D˜ P12 + Q2 D˜ P22 ) : Hu1 ;y2 = NP2 (X1 + Q1 D˜ P11 + Q2 D˜ P21 ) NP2 (X2 + Q1 D˜ P12 + Q2 D˜ P22 )

(3)

Then, block diagonal decoupling ({(u11 ; y21 ); (u12 ; y22 )}) amounts to

performance problem:

NP1 (X2 + Q1 D˜ P12 + Q2 D˜ P22 ) = 0;

= ˆ

NP2 (X1 + Q1 D˜ P11 + Q2 D˜ P21 ) = 0:

(4)

Existence of a block diagonal decoupling stabilizing controller amounts to satisfaction of the linear equality constraints (4) by a stable parameter Q = [Q1 Q2 ]. In general, a necessary condition for the non-square plant P to accept a diagonally decoupling stabilizing controller (i.e. for Q = [Q1 Q2 ] to exist that satis5es Eq. (4)) within the framework of Fig. 1 is that: m ¿ max{n1 ; n2 }. The set of diagonally decoupling stabilizing controllers is a linearly constrained (through Eq. (4)) subset of S(P). This result applies to both stable and unstable systems and addresses the cases of SISO and block decoupling. Although implicit in nature, this complete description of the decoupling stabilizing controllers can be conveniently used within an optimization-based design framework. This parametrization is equivalent to other parametrization approaches that have been proposed for stable plants under 1 DOF unity feedback control (see GMundeNs & Desoer, 1990, pp. 64 – 66). The two cases of triangular decoupling can be tackled in an exactly similar manner. For example, in the case of block lower triangular decoupling, ({(u11 ; (y21 ; y22 )); (u12 ; y22 )}), the stable parameter Q need to only satisfy one linear equality constraint: NP1 (X2 + Q1 D˜ P12 + Q2 D˜ P22 ) = 0:

(5)

3. Optimization-based controller design Without loss of generality and in order to avoid unnecessary proliferation in notation, the presentation that follows focuses on the 2 × 2 diagonal decoupling case. The decoupling controller is then designed so that it optimizes the achievable closed loop performance in the presence of persistent, bounded, external disturbances (Vidyasagar, 1986). Equivalently, the optimal decoupling controller is such that the size of the uniform envelope that contains the process outputs for all times and for all such disturbances is minimized. This type of performance objective can be quanti5ed in terms of the weighted i∞ norm of the closed loop I=O map between disturbances (u2 ) and outputs (y2 ) (Fig. 1). Hence the optimal decoupling controller design problem is formulated as a best achievable

inf

C ∈ S(P); decoupling

W1 Hu2 ;y2 W2 ∞ ;

(6)

where Hu2 ;y2 is the I=O map between the external disturbances and the output. W1 and W2 are diagonal matrices that represent the uniform bounds on the magnitude of the disturbances and the desirable uniform bound speci5cations on the outputs, respectively: W1 = diag{w11 ; : : : ; w1n };

−1 y2i ∞ 6 w1i ; ∀i;

W2 = diag{w21 ; : : : ; w2m };

u2i ∞ 6 w2i ; ∀i:

(7)

Both W1 and W2 are a priori determined, based on time-domain performance speci5cations. The I=O map Hu2 ;y2 is a non-linear, possibly singular function of the process model, P, and of the controller, C. Based on the Youla parametrization, (1), Hu2 ;y2 can be expressed as an a;ne function of the stable parameter Q subject to the linear decoupling controller parametrization constraints, (4). The optimal performance problem, (6), can then be equivalently rewritten as  = inf W1 NP (Y − QN˜ P )W2 ∞ Q ∈ M (S)

(8)

subject to, NP1 (X2 + Q1 D˜ P12 + Q2 D˜ P22 ) = 0; NP2 (X1 + Q1 D˜ P11 + Q2 D˜ P21 ) = 0: In the remaining presentation it will be assumed that the optimization search in Eq. (8) is performed over a bounded subset of M (S). This is not a restrictive condition, but a rather technical one. Indeed, if the plant has no zeros on the unit circle the parameter Q is bounded. The optimal decoupling performance problem (8) is a linear optimization problem. Existence of decoupling stabilizing controllers implies that the value of the optimization problem (8) is 5nite and vice versa. Upon the solution of Eq. (8) the I=O map QX is identi5ed that optimizes closed loop performance and results in a decoupling stabilizing controller C. The value, , of the optimization problem quanti5es the achievable decoupling controller performance: if  is 5nite one can immediately establish bounds on the range of values of the process outputs for all allowable, bounded disturbances: y2; i ∞ 6 w1;−1i ;

i = 1; 2; : : : ; n:

If  6 1, the process output will not exceed the speci5ed bounds for any allowable disturbance and for all times.

D. D. Sourlas / Chemical Engineering Science 56 (2001) 4695–4710

The optimization problem (8) is solved by means of exact penalty functions. The operator constraints of the optimization problem (8) are replaced by equivalent real-valued norm constraints  = inf T1 − T2 QT3 ∞

(9)

Q ∈ M (S)

subject to, H1 (Q)∞ = NP1 (X2 + Q1 D˜ P12 + Q2 D˜ P22 )∞ = 0; H2 (Q)∞ = NP2 (X1 + Q1 D˜ P11 + Q2 D˜ P21 )∞ = 0;

where T1 , T2 , T3 are known stable I=O maps that depend on the weights W1 and W2 , and on the selected fractional representation. The exact penalty function formulation of Eq. (9) is as follows: (!) = inf {T1 − T2 QT3 ∞ Q ∈ M (S)

+!(H1 (Q)∞ + H2 (Q)∞ )};

(10)

where ! is the positive penalty parameter. The objective function of Eq. (10) is 5nite for any 5nite value of !. Indeed, for any stable parameter Q the maps T1 − T2 QT3 , H1 (Q) and H2 (Q) are stable and have 5nite i∞ norm values. Since the feasible set of Eq. (10) is independent of ! and the objective function of Eq. (10) is motonically increasing with respect to !, uniformly for all feasible Q, it is straightforward to establish that ∀!1 ; !2 ¿ 0 such that !1 6 !2 :

(!1 ) 6 (!2 ) 6 :

Compared to Eq. (6), Eq. (10) is still a;ne in the optimization variable Q. In5nite dimensionality and non-diBerentiability are also maintained after this transformation. In Section 4 it will be shown that the solution of Eq. (10) leads to the value of the best achievable decoupling performance according to lim!→∞ (!) = . 4. Solution of the decoupling controller design problem In this section a solution methodology for the optimization problem (6) is proposed, that is based on the penalty function formulation (10). The approach employs the impulse responses of all I=O maps and is based on convergence properties that the optimization problem (10) is shown to possess. Step A: Introduction of impulse response representations Let t1 , t2 , t3 , x1 , x2 , nP1 , nP2 , d˜ P11 , d˜ P12 , d˜ P21 , d˜ P22 be the impulse response matrices of the known I=O maps in Eqs. (9) and (10). Since these maps are stable, their impulse responses are absolutely summable sequences. If, in addition, the coprime factorization is such that the poles of all coprime factors have been placed at the origin, all impulse responses are 5nite (FIR).

4699

The optimization variable Q is replaced by its impulse response matrix q = {qij }i; j=1; 2 , where qij is an absolutely summable sequence (i.e. qij ∈ ‘1 ) for i; j = 1; 2. Since Q is bounded its impulse response sequence will have bounded ‘1 norm. As a result of the use of impulse responses, the optimization problem (10) can be equivalently rewritten as (!) = inf1 t1 − t2 ∗ q ∗ t3 1 q ∈ ‘2×2

+!(h1 (q)1 + h2 (q)1 );

(11)

where ∗ denotes convolution and h1 (q), h2 (q) are the impulse response sequences of H1 (Q); and H2 (Q); respectively, h1 (q) = nP1 ∗ (x2 + q1 ∗ d˜ P12 + q2 ∗ d˜ P22 ); h2 (q) = nP2 ∗ (x1 + q1 ∗ d˜ P11 + q2 ∗ d˜ P21 ); where
 q11 ; q1 = q21 nP1 = [nP11

(12)




 q12 q2 = ; q22

nP12 ];

nP2 = [nP21

nP22 ]:

The properties of Eq. (11) relating to the existence and convergence of its solutions are proven in the appendix. The summary of these results follows: 1 • for any ! ¿ 0 there exists q ∈ ‘2×2 that is optimal for

Eq. (11) (Lemma 9),

• for any sequence {!i }∞ i=1 , !i → ∞ as i → ∞, the corresponding sequence {qi }∞ i=1 of solutions of Eq. (11) 1 such that: is wk ∗ -convergent, i.e. there exists qX ∈ ‘2×2 wk ∗

qi −→ qX as i → ∞ (Lemma 9), and, • for any sequence {!i }∞ i=1 , !i → ∞ as i → ∞, and the corresponding sequence {qi }∞ i=1 of solutions of Eq. (11) it holds that: !i (h1 (qi )1 + h2 (q1 )1 ) → 0 as i → ∞, and the wk ∗ -limit of {qi }∞ i=1 is the impulse response of the solution of Eq. (9) (Theorem 10). The combination of these results also gives as corollary the following which establishes the convergence of the optimization problem values. Corollary 3. Consider the increasing; unbounded sequence of positive numbers {!i }∞ i=1 . Then; (!i ) →  as i → ∞. Although the aforementioned results are asymptotic in nature, in the case of linear equality constraints the exact penalty function formulation converges to the actual value of the constrained problem for some 5nite value of the penalty parameter (Luenberger, 1969, p. 303). As a result, there exists a 5nite value of ! for which the

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D. D. Sourlas / Chemical Engineering Science 56 (2001) 4695–4710

solution of Eq. (11) is the impulse response of the solution of Eq. (9) or (8). Step B: Finite dimensional linear optimization problem formulation The optimization problem (11) is in5nite dimensional. An approximation technique is employed that is shown to lead to linear programs. First, the optimum search in Eq. (11) is restricted within the set of q impulse response matrices that are FIR of length (N + 1), for some given positive and integer value of N : 1 . The obtained optimization problem is as q ∈ "o; N ⊂ ‘2×2 follows: N (!) = inf t1 − t2 ∗ q ∗ t3 1 q ∈ "o; N

+!(h1 (q)1 + h2 (q)1 );

(13)

where, "o; N is the set of FIR matrix sequences with only the 5rst N + 1 elements non-zero. Next, properties of the optimization problem (13) are proven. The combination of these properties with Corollary 3 will establish the following theorem:

optimization problem is obtained:   L L        |s11; k (q)| + |s12; k (q)|      k=0 k=0 N; L (!)= ˆ inf max L L   q ∈ "o; N      | s (q) | + |s22; k (q)|    21; k  

+!

k=0

L

|h1; k (q)| +

k=0

L



k=0

|h2; k (q)| ;

where the subscript k indicates the (k + 1)st element of the corresponding sequence. For FIR coprime factors, and for q ∈ "o; N , the sequences s(q), h1 (q) and h2 (q) are also FIR with no more than the 5rst L + 1 elements non-zero. The index L is a priori known once N and the coprime factors are known. Then, all sequence matrix norms in Eq. (13) can be exactly calculated over the 5nite horizon L and an optimization problem similar to (14) can be formulated such that N; L (!) = N (!). Transformation of Eq. (14) into an LP optimization problem is accomplished through the introduction of new variables as follows:

Theorem 4. The value of the optimal decoupling performance problem can be obtained through the computation of the iterated limit:

N; L (!) = inf & −&ij (k) 6 sij; k (q) 6 &ij (k);

i; j = 1; 2;

 = lim lim N (!):

−'i (k) 6 !hi; k (q) 6 'i (k);

i = 1; 2;

!→∞ N →∞

The global solution of the decoupling performance problem is achieved via solution of a sequence of linear programs. The boundedness condition on q still applies. It is straightforward to establish that N (!) is an upper bound to the value of (!) and that the sequence of values {N (!)}∞ N =0 is non-increasing. Indeed, in our earlier work it has been shown that such a sequence has a limit which is exactly equal to (!) (Sourlas & Manousiouthakis, 1995, Lemma 4:1): Lemma 5. Given ! ¿ 0 it follows that: limN →∞ N (!)= (!). Lemma 5 establishes that determination of the value (!) requires solution of a sequence of 5nite dimensional optimization problems such as Eq. (13). This type of optimization problem can be transformed into an equivalent linear programming problem. The transformation is shown next for the simple 2 × 2 case. Under this condition ˆ 1 −t2 ∗q∗t3 h1 (q) and h2 (q) are scalar and the matrix s(q)=t is a 2 × 2 impulse response matrix. Let sij (q), i; j = 1; 2 be the elements of s(q). If the matrix norms in Eq. (13) are calculated over the 5nite horizon L the following

(14)

k=0

(15)

subject to,

L

k = 0; : : : ; L; k = 0; : : : ; L;

(&i1 (k) + &i2 (k) + '1 (k) + '2 (k)) 6 &;

i = 1; 2:

K=0

Remark 6. If the selected coprime factorization does not consist of FIR factors, the value of the optimal decoupling performance problem is obtained via the computation of the following iterated limit:  = lim lim lim N; L (!): !→∞ N →∞ L→∞

Controller design via ‘1 optimization was proposed by Vidyasagar (1986) and subsequently studied and solved by Dahleh, Pearson and others (see Dahleh & Diaz-Bobillo, 1995). For the general MIMO problem three approaches have been considered (Dahleh & Diaz-Bobillo, 1995, Chapter 12): Finitely many variables (FMV), Finitely many equations (FME) and delay augmentation (DA). These techniques consider the closed loop impulse response sequence as the main optimization variable, and the 5nal solution is obtained through 5nite approximations of increasing dimensionality. The convergence properties of the FMV, FME and DA methods are established in ways analogous to the way convergence properties are established in this work. In this work, the use of the controller parameter Q as

D. D. Sourlas / Chemical Engineering Science 56 (2001) 4695–4710

the main optimization variable in ‘1 optimization (as in Eqs. (8) and (9)) oBers the Yexibility of incorporating various controller speci5cations (decentralization, low order, etc.). Such speci5cations give rise to non-linear equality constraints in terms of the parameter Q (see Sourlas, Edgar, & Manousiouthakis, 1994; Sourlas & Manousiouthakis, 1995). Due to the additional non-linear (or linear) equality operator constraints in the latter problems, an additional level of iterations is required for their solution (relative to FMV, FME, DA). 4.1. Multiobjective decoupling performance Additional performance speci5cations (setpoint tracking, steady state oBset, etc.) can be accommodated within the proposed design framework. A new performance measure should be de5ned in this case. An illustration follows that describes how to formulate the decoupling controller problem so that setpoint tracking and disturbance rejection are simultaneously optimized. Consider the following two types of performance speci5cations: A. Uniform output bounds for the system output signals over all considered bounded disturbances and for all times are desired. This is exactly the type of performance speci5cations that lead to the optimization problem (6), and is quanti5ed through the ∞ norm of the closed loop map: W1 Hu2 ;y2 W2 ∞ , where the weights W1 ; W2 , have been de5ned (7). B. Setpoint tracking is de5ned in terms of a given, bounded setpoint signal that reaches a constant 5nal value. Such a signal can be described through a ∞ . Setpoint trackbounded in5nite sequence u1s ∈ ‘2×1 ing speci5cations are then de5ned as desirable envelope constraints for the error signal e1 (Fig. 1) that corresponds to u1s : the error should remain within a time varying envelope for all times, |e1; i (k)|6 w3;−1i (k);

i = 1; 2;

∀k ¿ 0;

(16)

where w3; i = ˆ {w3; i (k)}∞ k=0 is a bounded sequence with elements that are constant for k larger than a known index. The setpoint tracking speci5cations (16) are satis5ed iB   max sup |w3; i (k)e1; i (k)| = W3 · e1 ∞ i=1;2

The decoupling controller that is simultaneously optimal with respect to the above performance speci5cations corresponds to the solution of a minimization problem inf

C ∈ S(P); decoupling

max{W1 Hu2 ;y2 W2 ∞ ;

W3 · (u1s − Hu1 y2 ∗ u1s )∞ }:

5. Illustrative example A distillation column with side stream is the process considered in this example. A block-diagram of the closed loop is given in Fig. 2. The notation is compatible with Fig. 1. The process model for the distillation column is y2 = Pe2 + Pd u2 ; where y2 is the output vector, e1 is the manipulation vector, u2 is the disturbance vector and P, Pd are the transfer function matrices which correspond to the manipulation and the disturbance, respectively. In this

(17)

ˆ {w3; 1 ; w3; 2 } and “· ” indicates where W3 =diag element-by-element multiplication of sequences. The speci5cations A and B are simultaneously satis5ed iB: max{W1 Hu2 ;y2 W2 ∞ ; W3 · (u1s − Hu1 y2 ∗ u1s )∞ } 6 1: (18)

(19)

Simultaneous optimality in the above discussion refers to optimality of the solution in the context of the multiobjective minimization (19). In this case a controller C is optimal iB any further reduction in one objective, say W1 Hu2 ;y2 W2 ∞ , will only increase the value of the rest, e.g. W3 · (u1s − Hu1 y2 ∗ u1s )∞ . Such controller will be optimal for the given combination of performance weights W1 ; W2 ; W3 , and for the given 5xed input u1s . Additional type B performance speci5cations can be incorporated in this problem formulation. This may entail consideration of more than one setpoint sequences. Furthermore, one may consider 5xed disturbance signals, if, for instance, rejection of step disturbances is among the speci5cations. Each type B signal can have its own speci5cation envelope (weight W3 ). The solution of Eq. (19) is performed in exactly the same manner as described in Section 4. The closed loop map Hu1 y2 can be expressed linearly in terms of the parameter Q, while all the convergence properties discussed in the previous section hold for the optimization problem (19).

k ¿0

=W3 · (u1s − Hu1 y2 ∗ u1s )∞ 6 1;

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Fig. 2. Block diagram for the illustrative example.

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D. D. Sourlas / Chemical Engineering Science 56 (2001) 4695–4710

particular example y, u and d represent the following physical quantities:

 y top composition ; y2 = 2; 1 = y2; 2 side composition

 e2; 1 Top draw = ; e2 = e2; 2 Side draw

 u2; 1 Intermediate reYux duty u2 = = : u2; 2 Upper reYux duty The process model is given as  0:359(z + 2:113) 0:0708(z + 3:683)   z 3 (z − 0:8187) P=  0:340(z + 3:623) z 2 (z − 0:8187)

 0:0654(z + 2:090)  z 3 (z − 0:8007) Pd =   0:276(z + 0:8187)

z 3 (z

− 0:6703)

z 3 (z − 0:8465)  ; 0:599(z + 0:6132)  z 2 (z − 0:8465) 0:104(z + 2:061) z 3 (z − 0:7788) 0:405(z + 0:7788) z 2 (z − 0:6065)

  : 

The aforementioned model is obtained from the heavy oil fractionator model given in the shell control problem (Prett & Garcia, 1988, p. 21) and is obtained by sampling every 10 minutes. Next, the design of decoupling controllers for the diagonal decoupling case is illustrated and the performance limitations imposed by the decoupling requirement are quanti5ed. 5.1. Formulation of the decoupling constraints Since the model P is stable, one can use the following parametrization of all stabilizing controllers (see Appendix A): C = Q(I − PQ)−1 = (I − QP)−1 Q; where Q is a 2 × 2 stable map. Employing the above parametrization, the closed loop maps Hu2 y2 : u2 → y2 and Hu1 y2 : u1 → y2 become Hu2 y2 : u2 → y2 ;

Hu2 y2 = Pd − PQPd ;

Hu1 y2 : u1 → y2 ;

Hu1 y2 = PQ:

The decoupling controller parametrization dictates that the stable parameter Q should satisfy linear equality constraints similar to Eq. (4). 5.2. Performance speci=cations Uniform speci=cations: Uniformly bounded disturbances u2; 1 and u2; 2 are considered with magnitude

in the interval [ − 1; +1] at all times. A decoupling controller is sought so that each both process outputs remain within [ − 2; +2], for all times and for all aforementioned disturbances. These speci5cations give rise to the following weights, W1 and W2 , that are employed in the optimization procedure:


 0:5 0 ; W1 = 0 0:5




 1 0 W2 = ; 0 1

Setpoint tracking speci=cations: Two setpoint change scenarios are considered, u1s; a and u1s; b . Each scenario is consists of a unit step change in one of the setpoints:  +1 = = and 0
 0 s; b s; b s; b ∞ : u1 = {u1 (k)}k=0 ; u1 (k) = +1

u1s; a

{u1s; a (k)}∞ k=0 ;

u1s; a (k)




For both cases the error signal envelopes and the corresponding objective function weights are:  k 6 20  [ − 2:; 2:]; 20 ¡ k 6 30 e1; i (k) ∈ [ − 0:4; 0:4];  [ − 0:02; 0:02]; 30 ¡ k   0:5; k 6 20; ⇒ w3; i (k) = 2:5; 20 ¡ k 6 30; i = 1; 2: 

50; 30 ¡ k;

5.3. De=nition of controller design tasks Two instances of the decoupling performance problem will be considered. Case A: The controller is to be designed so that closed loop performance is optimized with respect to the uniform speci5cations alone. An optimization problem similar to Eq. (6) is solved in this case. The penalty function formulation is similar to Eq. (10): (!) = inf {W1 (Pd − PQPd )W2 ∞ + !(P11 Q12 Q ∈ M (S)

+P12 Q22 ∞ + P21 Q11 + P22 Q21 ∞ )}:

(20)

Case B: The controller is to be designed so that uniform closed loop performance and setpoint tracking are simultaneously optimized. An optimization problem similar to Eq. (19) is formulated and solved in this case. The penalty function formulation in this case is (!) = inf {/(P; Pd ; Q) + !(P11 Q12 + P12 Q22 ∞ Q ∈ M (S)

+P21 Q11 + P22 Q21 ∞ )};

(21)

D. D. Sourlas / Chemical Engineering Science 56 (2001) 4695–4710 Table 1 Best achievable performance results Performance value Nominal Decoupling (Case A) Decoupling (Case B)

where ˆ /(P; Pd ; Q)=max

0.80 1.11 1.60



main in the interval 1:60 × [ − 2; +2] = [ − 3:20; 3:20] for all times and for all allowable disturbances. Compared to the nominal case, Case B optimal controller achieves a uniform output envelope that is twice as large. Fig. 4 which depicts the worst-case response that corresponds to maximum peak of y2 . Unit step changes in either of the system’s setpoints will result in tracking errors that

W1 (Pd − PQPd )W2 ∞ ; W3 · ((PQ) ∗ u1s; a )∞ ; W3 · ((PQ) ∗ u1s; b )∞

5.4. Controller design results The decoupling performance results are summarized in Table 1. The nominal case refers to the optimal controller design with respect to Case A, over all stabilizing controllers. Since the optimal performance value is less than 1, feedback stabilizing (not necessarily decoupling) controllers exist that satisfy the performance envelopes of Case A. The actual envelope that contains the closed loop output over all allowable disturbance is for this case: 0:80 × [ − 2; +2] = [ − 1:6; +1:6]. The diagonally decoupling controller design problem, for Case A, has value 1:11 which is greater than 1. Hence, the uniform performance speci5cations cannot be met by any diagonally decoupling controller. In this case, the closed loop output will remain in the range: 1:11 × [ − 2; +2]=[ − 2:22; +2:22] for all times and for all allowable disturbances. This is the tightest uniform envelope achievable via diagonally decoupling control and represents 39% performance degradation relative to the nominal case with respect to Case A speci5cations. The worst-case response of the system that corresponds to the maximum peak of the 5rst output (i.e. y2; 1 : top composition) is shown in Fig. 3. This simulation shows the maximum peak that y2; 1 main attain as a result of an allowable disturbance. This maximum peak is equal to +2:22, as expected. Table 2 summarizes the intermediate results obtained via the penalty function approach. The table contains the values of the ‘1 norm of the impulse response of the oB-diagonal elements of the map Hu1 y2 that correspond to the optimal solution for the corresponding value of the penalty parameter, !. For a diagonally decoupling controller, the magnitude of these elements should be zero which indicates that the decoupling constraints of the original constrained problem (such as in Eq. (9)) are satis5ed. In Case A, the decoupling constraints are satis5ed within truncation error accuracy for ! ¿ 4. The optimal performance value of the decoupling problem for Case B is equal to 1:60 which is greater than 1. This result indicates that the closed loop output will re-

4703



:

lie within the envelope:  k 6 20;  [ − 3:20; 3:20]; 20 ¡ k 6 30; e1; i (k) ∈ [ − 0:64; 0:64];  [ − 0:032; 0:032]; 30 ¡ k: The setpoint tracking error of the optimal closed loop for Case B is shown in Fig. 5 for unit step change in the two setpoints. The two setpoint changes are imposed independently. The steady state setpoint tracking error is equal to 0:032 for both e1; 1 and e1; 2 . The magnitude of each oB-diagonal element of the map Hu1 y2 that corresponds to the optimal solution for each penalty parameter value is given in Table 2. Based on this information it is shown that the decoupling constraints are practically satis5ed for ! ¿ 3.

6. Extensions and other related problems The decoupling problems encountered in chemical engineering often entail features that deviate from the optimization framework that was presented above. Some of these issues are discussed in the next paragraphs and it is demonstrated how the results presented in this work can be extended to solve variations of the decoupling controller design problem. However, the resulting optimization problems for these cases will be non-linear and in5nite dimensional. Although the solution of such optimization problems is feasible, it still remains a challenging computational task. 6.1. Introduction of decouplers in a control loop and approximate decoupling The problem of eliminating (or minimizing) interactions in an existing control loop is often addressed through the introduction of decouplers (Luyben, 1970). The resulting feedback loop is shown in Fig. 6, where the model and the structure of the controller K are maintained after the introduction of the decoupler D. The combined block

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Fig. 3. Case A: worst case closed loop behavior for the optimal decoupling controller.

Table 2 Value of decoupling constraints at the solution of Eq. (11) Penalty

Case A

Case B

parameter


(Hu1 y2 )1; 2
∞


(Hu1 y2 )2; 1
∞


(Hu1 y2 )1; 2
∞


(Hu1 y2 )2; 1
∞

!=1 !=2 !=3 !=4 !=8 ! = 10 ! = 15

0.132 0.124 0.124 0:813 × 10−11 0:565 × 10−9 0:513 × 10−9 0:187 × 10−9

0:106 × 10−8 0:602 × 10−8 0:128 × 10−12 0:278 × 10−12 0:471 × 10−11 0:413 × 10−13 0:176 × 10−9

0.319 0:141 × 10−6 0:184 × 10−10 0:420 × 10−11 0:135 × 10−10 0:196 × 10−10 0:408 × 10−11

0.582 0:632 × 10−1 0:302 × 10−9 0:127 × 10−9 0:717 × 10−11 0:501 × 10−10 0:118 × 10−9

Fig. 4. Case B: worst case closed loop behavior for the optimal decoupling controller.

D. D. Sourlas / Chemical Engineering Science 56 (2001) 4695–4710

4705

Fig. 5. Case B: setpoint tracking error under unit step setpoint changes (Left: u1s; a , Right: u1s; b ).

C =DK ˆ is equivalent to the feedback controller of Fig. 1 and should belong to the set S(P). As a result, DK can be parametrized in terms of a stable parameter (Q) that should satisfy additional constraints that express the fact that (i) K is already 5xed, and, (ii) D may be structurally or otherwise constrained. These additional constraints are developed next. The element DK is a stabilizing controller for P and K is known, together with a right coprime factorization:  (1) DK ∈ S(P)⇒DK = (Y − QN˜ P )−1 (X + QD˜ P )  K is known : K = N D−1 ; N ; D ∈ M (S)  K

K

K

where

K

⇒ (Y − QN˜ P )DNK = (X + QD˜ P )DK ;

(22)

where Q ∈ M (S); det(Y − QN˜ P ) = 0. The stable parameter Q in the last operator constraint (22) is identical to the Q-parameter discussed in the earlier portions of the paper (such as in Eq. (8)). Hence, by augmenting Eq. (22) as a constraint in Eq. (8) one can search for the “best” decoupler block D. If only stable decouplers are sought, the dynamic elements of D can be replaced by their (absolutely summable) impulse response sequences, and Eq. (22) can be reduced to a 5nite set of quadratic scalar equalities. The resulting optimization problem is similar to optimization problems that arise in the study of the simultaneous stabilization and performance problem. The iterative algorithm discussed in Section 4 can solve these optimization problems (see Sourlas & Manousiouthakis, 1999). Within the same decoupler design problem, one may consider decouplers with an a-priori bound on the order of their dynamical elements. Low order decouplers have been studied in (Figueroa et al., 1994). In the 2 × 2 case the decoupler D that has structure similar to the simpli5ed decoupler (see Luyben, 1970) and also has upto 0-order dynamic elements, will be represented as follows:

 1 D12 1 U12 =V12 = D= D21 1 U21 =V21 1
 V12 V21 U12 V21 = (V V )−1 ; U21 V12 V12 V21  12 21     V −1 U

Fig. 6. Controller–decoupler feedback control con5guration.

UJ =

0

VJ =

i=0 0

 (i) i aJ z

z +

0−1 i=0

z0 ;  (i) i bJ z

z0 ;

J = {12}; {21}:

Then, the operator constraint (22) becomes (Y − QN˜ P )UNK = (X + QD˜ P )DK V;

(23)

where U , V consist of FIRs of order 0. Again this last constraint should 5rst be reduced to its scalar constituent relations, and then attached to Eq. (8) in order to formulate the optimization problem that will search over all decouplers with the speci5ed structure and order. For higher dimensional systems, symbolic manipulations will facilitate the generation of the 5nal U and V parameters. In cases where order and/or structure constraints are imposed on the decoupler, it is not always possible to obtain diagonal decoupling. In those cases, one could consider triangular decoupling (that was mentioned earlier), or approximate decoupling. In the latter case the decoupling constraints are weighted and subsequently incorporated in the objective function to obtain the following optimization problem that is the equivalent of Eq. (9) under the current scenario:   W1 NP (Y − QN˜ P )W2 ∞ = inf U; V; Q ∈ M (S) W dec H1 (Q)∞ ; W dec H2 (Q)∞ (24)

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D. D. Sourlas / Chemical Engineering Science 56 (2001) 4695–4710

subject to, (Y − QN˜ P )UNK = (X + QD˜ P )DK V; where H1 (Q); H2 (Q) have been de5ned in Eq. (9). The scalar W dec is the weight on the undesirable interactions. Increases in the magnitude of W dec will result in decouplers that are stabilizing, feature smaller interactions (in terms of the ∞ norm) at the expense of performance deterioration. One can incorporate frequency-dependent (dynamic) weights in the place of W dec . In this way one can shape the frequency-domain response of the interactions in any desirable manner. 6.2. Reliable controller–decoupler design The performance of the controller–decoupler con5gurations described in the previous paragraph is limited by the existing controller K. If a controller design is not in place, the two blocks, K and D, should be designed simultaneously in order to eliminate the aforementioned limitations. The resulting pair should possess good performance characteristics as well as reliability. In particular, it is required that the controller stabilizes the system when the decoupler is disabled. For reliability both DK and K must be stabilizing for P. The Youla parametrization then gives (1) DK ∈ S(P) ⇒ DK = (Y − QN˜ P )−1 (X + QD˜ P ) K ∈ S(P) ⇒ K = (X˜ + DP R)(Y˜ − NP R)−1



⇒ (Y − QN˜ P )D(X˜ + DP R)

=(X + QD˜ P )(Y˜ − NP R)

(25)

with R; Q ∈ M (S), det(Y − QN˜ P ) = 0; det(Y˜ − NP R) = 0. Equality (25) is cubic since it involves three (3) “unknown” parameters Q; R and D. The 5rst one is identical to the parameter Q discussed in Section 2 and the decoupling constraints (4), (5) are expressed in terms of this variable. To address the reliable controller–decoupler design task, the operator equality constraint (25) should be incorporated as a constraint within the optimization problem (8). Solution will proceed through iterations as outlined in Section 4. Analogous cubic optimization problems have been considered and solved in the case of SISO model reduction (Sourlas, 1998, 1999, 2000). Depending on the requirements of the problem that is being studied several straightforward modi5cations can be made in the problem formulation described in the previous paragraph. Additional constraints can be imposed on the parameter R if the controller K is to be decentralized (Manousiouthakis, 1993; Sourlas & Manousiouthakis, 1995). If it is important to also optimize the performance under decoupler failures the

weighted ∞-norm of the corresponding closed loop (which is a;ne in R alone) can be included in the objective function. Depending on the remaining degrees of freedom, it may not be possible to simultaneously satisfy decentralization and decoupling for the desirable structure of D and K. Approximate decoupling can be pursued in this case through a modi5ed objective function as before (see (24)). 6.3. Robustness, decentralization and decoupling in multivariable controller design The parametrization of the decoupling stabilizing controllers in terms of the linear equalities (4) or (5), is compatible with similar parametrizations of decentralized and simultaneously stabilizing controllers (Manousiouthakis, 1993; Sourlas & Manousiouthakis, 1995, 1999). All these parametrizations originate from the same Youla parametrization (1) (Vidyasagar, 1985) and, as a result, are expressed in terms of the same stable parameter Q. The immediate consequence is that one may describe the sets of controllers that have multiple properties (decentralized and decoupling, etc.) by simultaneously considering the corresponding parametrizations as constraints in an optimal controller design problem analogous to Eq. (8). However, special attention should be paid to the possibility that simultaneous consideration of such constraints may reduce (or eliminate) the available degrees of freedom. The following illustration elucidates this point: Example 7. Let us consider the case of simultaneous stabilization and decoupling of two 2 × 2. In this case there are eight degrees of freedom that correspond to the scalar entries that comprise the two Q-parameters (one for each plant; see Sourlas & Manousiouthakis, 1999) in this problem. Four simultaneous stabilization constraints arise in this case, and additional two pairs of diagonal decoupling constraints similar to Eq. (4) (one for each plant). For this problem there is zero degree of freedom. To overcome this problem one may relax the diagonal decoupling requirement for both plants and replace it with triangular decoupling which results in two degrees of freedom. Alternatively, diagonal decoupling may be enforced for one plant and approximate decoupling for the other. The preceding discussion establishes that the proposed multivariable decoupling controller design framework can be extended to the design of (i) decentralized and (possibly approximate) decoupling controllers and (ii) simultaneously stabilizing and (possibly approximate) decoupling controllers. Careful degrees of freedom analysis is required in order to avoid optimization problems without any degrees of freedom. To increase the degrees of freedom, one may elect one of the many possibilities including: approximate decoupling, triangular instead

D. D. Sourlas / Chemical Engineering Science 56 (2001) 4695–4710

of diagonal decoupling and in the case of highly dimensional systems block decoupling instead of SISO decoupling.

7. Concluding remarks The best achievable performance problem has been formulated for input–output decoupling controllers. Solution of this problem is achieved through a search over the set of all decoupling stabilizing controllers. A convenient, linear parametrization of all decoupling stabilizing controllers is developed in this work and when employed in the aforementioned problem formulation it gives rise to in5nite dimensional linear equality constraints. Exact penalty functions are employed and a solution methodology is proposed that identi5es the best achievable decoupling performance with respect to the given set of time-domain performance speci5cations. The solution algorithm requires solution of a sequence of linear optimization problems. This sequence is shown to converge to the solution of the decoupling performance problem, if a decoupling stabilizing controller exists. The computational methodology presented in this work can be employed to quantify performance degradation due to decoupling. As a result, it can serve as a detailed decoupling structure screening tool during controller synthesis and design. It can also be used to assess potential improvements in control system performance compared to existing decoupling control schemes. The general framework discussed in the paper can provide the foundation for the analysis and design of decoupling control systems through the use of decouplers. These are constrained versions of the general decoupling problem since certain structural and=or dynamic properties of the controller=decoupler combinations are a-priori speci5ed. This approach has been studied extensively and has been considered in industrial applications. It is demonstrated that one can formulate optimization problems that simultaneously identify controller and decoupler pairs with structure and=or order constraints such that undesirable closed loop interactions are eliminated (or minimized) and closed loop performance is optimized. In this case, the optimization problems are non-linear and their solution requires solution of highly dimensional non-linear optimization problems. Further research in the areas numerical approximation and large scale optimization methods will enhance the engineer’s ability to solve these classes of decoupling control problems.

Acknowledgements The author thanks Mr. Yu Wang for assisting with the programming required to solve the example problem.

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Appendix A A.1. Review of coprime factorizations A linear, causal, multivariable process P (Fig. 1) is represented by a corresponding Input–Output (I=O) map, P, or a transfer function matrix P(z). Also, M (S) represents the set of stable, causal, I=O maps. The set of n × m members of M (s) will be devoted by S n×m . Then, P (or P(z)) accepts a left coprime factorization (D˜ P ; N˜ P ) and a right coprime factorization (NP ; DP ) iB −1 P = NP DP−1 = D˜ P N˜ P ;

(A.1)

where NP ; DP ; D˜ P ; N˜ P ∈ M (S), and the pairs (D˜ P ; N˜ P ) and (NP ; DP ) are left coprime (lc) and right coprime (rc), respectively. In the case of linear, 5nite dimensional systems, coprimeness amounts to no unstable transmission zero coincidences between the maps of the coprime pair. The algebraic condition for coprimeness is that stable causal I=O maps X; Y; X˜ ; Y˜ exist such that (NP ; DP ) are rc ⇔ ∃X; Y ∈ M (S) : XNP + YDP = U;

(D˜ P ; N˜ P ) are lc ⇔ ∃X˜ ; Y˜ ∈ M (S) : D˜ P Y˜ + N˜ P X˜ = U;

where U is the unit in M (S) (i.e. stable with stable and causal inverse). The eight maps NP ; DP ; D˜ P ; N˜ P ; X; Y; X˜ ; Y˜ constitute a doubly coprime factorization of P iB

 Y X DP −X˜ · = I: (A.2) −N˜ P D˜ P NP Y˜ Given P and a controllable and observable state space realization of P, the eight maps that constitute the above doubly coprime factorization can be obtained through the solution of a state feedback controller and a state observer problem (Vidyasagar, 1985, p. 83). Given a doubly coprime factorization of P, one can parametrize all feedback stabilizing controllers C (see Fig. 1) C = (Y − QN˜ P )−1 (X + QD˜ P ) = (Y − QN˜ P )(X˜ + DP Q) (A.3) for any Q ∈ M (S). The above expressions, (A.3), identify all stabilizing controllers for a given process. Illustration: Consider the special case where P is a stable I=O map. Then, a doubly coprime factorization can be constructed as NP = N˜ P = P; X = 0;

Y = 0;

DP = I; Y˜ = 0:

D˜ P = I;

X˜ = 0; (A.4)

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D. D. Sourlas / Chemical Engineering Science 56 (2001) 4695–4710

Then, any stabilizing controller is given by −1

−1

C = (I − QP) Q = Q(I − PQ) ;

Q ∈ M (S): (A.5)

Although parametrization (A.5) is based on a particular factorization, it still describes all stabilizing controllers C for the stable process P. Appendix B B.1. Properties of the exact penalty function formulation The material in this appendix is a special case of similar proofs in Sourlas and Manousiouthakis (1999) and are given for completeness. Lemma 8. The objective function of Eq. (11) is wk ∗ -sequentially lower semicontinuous. Proof of Lemma 8. It su;ces to show that for any 1 wk ∗ -convergent sequence {qi }∞ i=1 , qi ∈ ‘2×2 , such that wk ∗

1 , one has qi −→ qX ∈ ‘2×2

t1 − t2 ∗ qX ∗ t3 1 + !(h1 (q) X 1 + h1 (q) X 1 ) i→∞

(B.1)

In this context wk ∗ -convergence of the matrix sequences amounts to wk ∗ -convergence of each (‘1 ) sequence that comprises an entry in the matrix qi :

 q q qX qX wk ∗ qi = i; 11 i; 12 −→ qX = 11 12 qi; 21 qi; 22 qX21 qX22 wk ∗

iB qi; jk −→ qXjk ∈ ‘1 ;

j; k = 1; 2

∗

as i → ∞: {qi }∞ i=1 ,

Next, such a wk -convergent sequence, with wk ∗ -limit qX is considered. The proof proceeds by showing that each term in the objective function is in itself wk ∗ -slsc. This amounts to proving that each of the sequences {t1 − t2 ∗ qi ∗ t3 }∞ i=1 , ∞ ∗ {h1 (qi )}∞ and { h (q ) } , is wk convergent and 2 i i=1 i=1 wk

∗

t1 − t2 ∗ qi ∗ t3 −→ t1 − t2 ∗ qX ∗ t3 ; wk ∗

X h1 (qi ) −→ h1 (q);

wk ∗

h2 (qi ) −→ h1 (q) X

as i → ∞:

• The space of absolutely summable sequences, ‘1 , is the

dual of the space of bounded sequences that converge to zero, c0 : ‘1 = (c0 )∗ (Conway, 1990, p. 76). Hence, 1 any sequence u = {u(i)}∞ i=0 ∈ ‘ de5nes a bounded linear functional on the Banach space c0 . In other words, for any in5nite sequence c = {c(i)}∞ i=0 ∈ c0 the following quantity is 5nite: ∞ c; u= ˆ c(i)u(i) ¡ ∞: i=0

1 • The sequence of ‘1 sequences {ui }∞ i=1 ; ui ∈ ‘ has a

wk ∗ -limit uX iB ∀c ∈ c0 :

(B.2)

Indeed, once Eq. (B.2) has been shown, Theorem 9ii in (Yoshida, 1980, p. 125) gives

lim c; h1 (qi ) − h1 (q) X  = 0:

i→∞

c; h1 (qi ) − h1 (q) X  

 qi; 11 − qX11 ∗ d˜ P12 qi; 21 − qX21
 ! q − qX12 ∗ d˜ P22 + i; 12 qi; 22 − qX22

=c; nP11 ∗ (qi; 11 − qX11 ) ∗ d˜ P12  +c; nP12 ∗ (qi; 21 − qX21 ) ∗ d˜ P12  + c; nP11 ∗ (qi; 12 − qX12 ) ∗ d˜ P22  + c; nP12 ∗ (qi; 22 − qX22 ) ∗ d˜ P22  where nP11 ; nP12 ; qi; 11 ; qi; 12 ; qi; 21 ; qi; 22 ; qX11 ; qX12 ; qX21 ; qX22 ; d˜ P12 ; d˜ P22 ∈ ‘1 . Through straightforward calculations one can show that c; nP11 ∗ (qi; 11 − qX11 ) ∗ d˜ P12  = a; (qi; 11 − qX11 ) ∗ d˜ P12 

= b; (qi; 11 − qX11 );

a = {a(i)}∞ i=0

i→∞

which proves the lemma.




= c; [nP11 nP12 ] ∗

h1 (q) X 1 6 lim inf h1 (qi )1 h2 (q) X 1 6 lim inf h2 (qi )1 ;

(B.4)

Employing (12) one gets

where

i→∞

(B.3) wk ∗

t1 − t2 ∗ qX ∗ t3 1 6 lim inf t1 − t2 ∗ qi ∗ t3 1 ; i→∞

lim c; ui − u = 0:

i→∞

X According First it will be shown that h1 (qi ) −→ h1 (q). to the de5nition of wk ∗ -convergence, (B.3) it su;ces to show that ∀c ∈ c0 :

6 lim inf t1 − t2 ∗ qi ∗ t3 1 + !(h1 (qi )1

+h2 (qi )1 ):

In the remaining section, the validity of Eq. (B.2) is demonstrated. The presentation employs the property of ‘1 which is the dual of the Banach space c0 , and the definition of bounded linear functionals on this space.These necessary concepts are brieYy reviewed:

b = {b(i)}∞ i=0

and and

a(i) = b(i) =

∞ k=i ∞ k=i

c(k)nP11 (k − i); a(k)d˜ P12 (k − i):

(B.5)

D. D. Sourlas / Chemical Engineering Science 56 (2001) 4695–4710

Furthermore, a; b ∈ c0 . Indeed, the sequence a is bounded: |a(i)|6nP11 1 sup|c(k)| ¡ ∞: k ¿i

Also, the limit of the sequence a is zero: lim |a(i)|6nP11 1 lim sup|c(k)| = 0:

i→∞

i→∞ k ¿ i

Similar arguments establish that b ∈ c0 . In view of wk ∗ the de5nition of qi (qi11 −→ qX11 ), the de5nition of wk ∗ -convergence (B.3), the fact that b ∈ c0 and Eq. (B.5) one has lim c; nP11 ∗ (qi; 11 − qX11 ) ∗ d˜ P12  i→∞

= lim b; (qi; 11 − qX11 ) = 0: i→∞

Based on this construction the desirable result follows, i.e. Eq. (B.4) is valid. In the same way, one can show that wk ∗ wk ∗ X as i → ∞. Finally, t1 − t2 ∗ qi ∗ t3 −→ t1 − h2 (qi ) −→ h1 (q), t2 ∗ qX ∗ t3 amounts to
 c11 c12 ∈ (c0 )2×2 : ∀c = c21 c22 lim c; (t1 − t2 ∗ qi ∗ t3 ) − (t1 − t2 ∗ qX ∗ t3 ) = 0:

i→∞

This case reduces to the previous one since the linear functional can be expressed in terms of linear functionals that involve scalar quantities, as before c; (t1 − t2 ∗ qi ∗ t3 ) − (t1 − t2 ∗ qX ∗ t3 )

=c11 ; (t2 ∗ (qX − qi ) ∗ t3 )11  + c12 ; (t2 ∗ (qX − qi ) ∗ t3 )12  + c21 ; (t2 ∗ (qX − qi ) ∗ t3 )21  + c22 ; (t2 ∗ (qX − qi ) ∗ t3 )22  Thus, Eq. (B.2) is valid and the proof of the Lemma is complete. Lemma 9. Consider the increasing; unbounded sequence of positive numbers {!i }∞ i=1 . Then; 1 that is the solution 1. for each i; there exist qi ∈ ‘2×2 of Eq. (11); and wk ∗ 1 such that qij −→ qX as j → ∞; 2. there exist qX ∈ ‘2×2 ∗ where {qij }∞ j=1 is a wk -convergent subsequence of the sequence of optimization problem solutions {qi }∞ i=1 . Proof of Lemma 9. The optimization problem (11) is posed over a bounded ball of the dual of a Banach space. Hence its feasible region is wk ∗ compact according to Alaoglu’s theorem (Conway, 1990, p. 130). Furthermore, the value of the optimization problem is 5nite for all 5nite ! values and its objective function is wk ∗ lower semicontinuous (Lemma 8). Hence, Weierstrass (Luenberger, 1969, p. 40) establishes the existence of the solution of Eq. (11).

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The existence of the wk ∗ convergent subsequence is a direct result of the wk ∗ compactness of the feasible region. Theorem 10. Let  and (!) be de=ned as in Eqs. (6) and (10) (or Eq. (11)); respectively. Also consider an increasing sequence of positive numbers {!i }∞ i=1 and the of solutions of Eq. (11); corresponding sequence {qi }∞ i=1 as in Lemma 9. Then; the following statements hold: 1. !i (h1 (q)1 + h2 (q)1 ) → 0 as i → ∞; and 2. if qX is the wk ∗ -limit of a subsequence of the solution sequence {qi }∞ i=1 ; it is also the impulse response of the solution of (6). Proof of Theorem 10. The proof is based on Lemmas 8 and 9 and proceeds according to the proof of Lemma 1 and that of Theorem 1 in (Luenberger, 1969, p. 305,306) with all arguments made in the wk ∗ topology. References Arkun, Y., Manousiouthakis, B., & Palazoglu, A. (1984). Robustness analysis of process control systems. A case study of decoupling control in distillation. Industrial Engineering and Chemical Processing Design and Development, 23, 93–101. Commault, C., & Dion, J. (1983). Transfer matrix approach to the triangular block decoupling problem. Automatica, 19(5), 542–553. Conway, J. B. (1990). A course in functional analysis (2nd ed.). New York: Springer. Dahleh, M., & Diaz-Bobillo, I. J. (1995). Control of uncertain systems: a linear programming approach. Englewood CliBs, NJ: Prentice-Hall. Desoer, C., & GMundeNs, A. N. (1985). Decoupling linear multiinput multioutput plants by dynamic output feedback. Proceedings of the American control conference, Boston, MA (pp. 1010 –1011). Desoer, C., & GMundeNs, A. N. (1986). Decoupling linear multiinput multioutput plants by dynamic output feedback: An algebraic theory. Transactions on Automatic Control, 31(8), 744–750. Falb, P., & Wolovich, W. (1967). Decoupling in the design of multivariable control systems. IEEE Transactions on Automatic Control, 12, 651–659. Figueroa, J., Barton, G., & Romagnoli, J. (1994). Design of simple robust decouplers with joint stability and performance speci5cations. Comparative Chemical Engineering, 18(5), 415– 425. Figueroa, J., Desages, A., Romagnoli, J., & Palazoglu, A. (1991). Highly structured stability margins for process control systems. Comparative Chemical Engineering, 15(7), 493–502. Freitas, M., Campos, M., & Lima, E. (1994). Dual composition control of a debutanizer. ISA Transactions, 33(1), 19–25. Gilbert, E. (1969). The decoupling of multivariable control systems. SIAM Journal of Control, 7, 50–63. GMundeNs, A. N. (1990). Parametrization of all decoupling compensators and all achievable diagonal maps for the unity feedback system. Proceedings of the 29th IEEE Conference Decision and Control. Honolulu, Hawaii (pp. 2492–2493). GMundeNs, A. N., & Desoer, C. A. (1990). Algebraic theory of linear feedback systems with full and decentralized compensators. Heidelberg, NY: Springer. Hammer, J., & Khargonekar, P. (1984). Decoupling of linear systems by dynamic output feedback. Mathematical Systems Theory, 17, 135–157.

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