Analytical Design of Decoupling Internal Model Control (IMC) Scheme

Ind. Eng. Chem. Res. 2006, 45, 3149-3160

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Analytical Design of Decoupling Internal Model Control (IMC) Scheme for Two-Input-Two-Output (TITO) Processes with Time Delays Tao Liu,* Weidong Zhang, and Danying Gu Department of Automation, Shanghai Jiaotong UniVersity, Shanghai 200240, People’s Republic of China

In this paper, a new decoupling control scheme, in terms of an internal model control (IMC) structure, is proposed for two-input-two-output (TITO) processes with time delays. Noteworthy improvement of the decoupling regulation can be achieved for the nominal system output responses, and moreover, either of the system output responses can be quantitatively regulated by virtue of the analytical relationship between the adjustable parameters of the decoupling controller matrix and the nominal system transfer matrix. The ideally optimal controller matrix is analytically derived by proposing the practically desired diagonal system transfer matrix, in terms of the robust H2 optimal performance objective. Because of the difficulty of physical implementation, its practical form is carefully configured according to whether there exist any complex righthalf-plane (RHP) zeros in the process transfer matrix determinant. At the same time, tuning constraints for the proposed decoupling controller matrix to hold the control system robust stability are analyzed in the presence of the process additive and multiplicative uncertainties, and, accordingly, the on-line tuning rule is provided to cope with the process unmodeled dynamics in practice. Finally, illustrative simulation examples are included to demonstrate the remarkable superiority of the proposed method. 1. Introduction In the process industry, two-input-two-output (TITO) processes are mostly encountered multivariable processes, and moreover, a large number of multivariable processes with inputs/ outputs beyond two can be treated as several two-by-two subsystems in engineering practice.1-3 Because of the interaction between the binary process variables, well-developed control methods for single-input-single-output (SISO) processes can hardly be extended to TITO processes.4 Moreover, time delay is a fundamental characteristic of multivariable processes in practice. Its presence in an individual control loop can badly prevent the high gain of the closed-loop controller from being used.5 Many control strategies had been developed for the decoupling regulation of TITO processes with time delays. Based on the robust H2 and H infinity optimal specification, Skogestad and Postlethwaite4 had systematically presented a loop-shaping design methodology within the framework of a conventional unity feedback control structure. In view of the successful application of the Smith predictor (SP) structure for SISO systems with time delays,6 the earlier literature7-10 further extended it to TITO processes with time delays to obtain the delay-free characteristic equation of such a system transfer matrix, and then utilized some previous decoupling control methods developed for linear multivariable systems without time delay. Recently, enhanced control performance, in terms of the multivariable SP structure, had been captured in refs 11 and 12, using some frequency-response specifications such as the ultimate frequency and magnitude/phase margin. In light of the internal model control (IMC) theory,13 refs 14-16 presented some decoupling control methods based on iterative algorithms, all of which were indeed capable of much improved decoupling regulation, in comparison with some previous methods, but at the cost of a considerable computation effort for implementation. According to the frequency response data drawn from relay feedback tests, refs 17-22 have proposed some on-line sequen* To whom correspondence should be addressed. Tel./Fax: +86.21.34202019. E-mail: [email protected].

tial tuning methods, and, meanwhile, refs 23-25 suggested some simultaneous autotuning methods. In fact, many existing decoupling control methods had been based on using a decoupler in front of the process inputs, to obtain the augmented process transfer matrix with diagonal dominance, and then tuned the multiloop controllers by means of some decentralized control methods recently developed, such as the recent literature26-28 based on a static decoupler, i.e., the inverse of the process static gain transfer matrix, and refs 29-32, based on a dynamic decoupler. However, note that the static decoupler has no impact on the dynamic system output responses in essence, and the dynamic decoupler is actually difficult to configure precisely, especially for those TITO processes with large time delays, because of the requirements of physical properness and causality for implementation.29,30 Moreover, note that, although recently enhanced multiloop control methods, such as those discussed in refs 33-38, are capable of achieving remarkably improved system output performance, compared to many existing methods, the controller tuning is aimed at the compromise between the achievable system output performance and the interaction between individual loops, which will inherently lead to decoupling performance degradation, when compared to a control system with a full controller matrix.4,13 Hence, to meet the primary decoupling regulation requirement for a large amount of TITO processes with time delays in industry, decoupling control strategies are preferred to be adopted in practice. In this paper, a new analytical design method of decoupling controller matrix within the frame of a standard IMC structure is proposed for TITO processes with time delays. By proposing the practically desired diagonal system transfer matrix in terms of the robust H2 optimal performance objective, the ideally optimal decoupling controller matrix can be inversely determined. For the convenience of physical realization, its practical form will be closely configured, using a linear fractional approximation approach, based on the mathematical Pade´ expansion. As expected, the computation effort will be relieved on a large scale, in comparison with many existing decoupling control methods based on numerical algorithms. Furthermore, obviously improved tuning capacities for the system output

10.1021/ie051129q CCC: $33.50 © 2006 American Chemical Society Published on Web 03/22/2006

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Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006

process model, which is therefore capable of achieving the desirable nominal system output responses and decoupling regulation and, meanwhile, can be conveniently tuned on-line to cope with the process unmodeled dynamics in practice. For a nominal TITO process, the decoupled system output responses, in terms of the standard IMC structure, should correspond to a diagonal system transfer matrix, i.e., Figure 1. Internal model control (IMC) structure.

responses can be achieved. At the same time, how to evaluate the resultant control system robust stability is analyzed in the presence of the process additive and multiplicative uncertainties that are mostly encountered in practice. For clear interpretation of the proposed method, this paper is organized as follows. Section 2 presents two fundamental decoupling control preconditions for TITO processes. In section 3, the practically desired diagonal system transfer matrix form is analyzed. The ideally desired decoupling controller matrix and its practical form then are analytically derived in section 4. Subsequently, the robust stability constraints for tuning the decoupling controller matrix are analyzed in the presence of the process additive and multiplicative uncertainties, and, correspondingly, the on-line tuning rule is provided in section 5. Several illustrative simulation examples are included in section 6, to demonstrate the superiority of the proposed method. Finally, some conclusions are addressed in section 7. 2. Decoupling Control Preconditions For convenience, a TITO process with time delays is usually identified in engineering practice as

[

k11 e-θ11s τ s+1 G(s) ) 11 -θ21s k21 e τ21s + 1

k12 e-θ12s τ12s + 1 k22 e-θ22s τ22s + 1

]

(1)

Hence, the standard IMC structure can be directly adopted for such TITO process models, which is shown in Figure 1, where Gm is the process model. When it exactly matches the actual process, i.e., G ) Gm, there is an open-loop control for the setpoint response, so the system transfer matrix can be simplified as

[

][

g g c c H ) GC ) g11 g12 c11 c12 21 22 21 22

]

(2)

Consequently, the nominal system response will definitely hold stability if the controller matrix C is designed to be stable. However, when there exists an actual process uncertainty, e.g., the process unmodeled dynamics, the system transfer matrix will present the form of

H ) GC[I + (G - Gm)C]-1

(3)

which may be very complex in the presence of various process uncertainties and has a tendency to lose stability in an intangible manner. Therefore, the stable controller matrix C, configured in terms of the nominal process model, may not be able to guarantee the control system robust stability any longer. In fact, how to ascertain all the stabilizing set of C for different process uncertainties is difficult to address and has remained open as yet in the process control community.4,39 In this paper, the attention is focused on tuning the decoupling controller matrix C in an analytical and simple way, according to the referential

H ) GC )

[ ] h1 0 0 h2

(4)

where the diagonal elements h1 and h2 are stable and proper transfer functions. Thus, two fundamental decoupling control preconditions can be ascertained as follows: (1) Both G and C are required to be nonsingular at s ) 0, i.e., det[G(0)] * 0 and det[C(0)] * 0. (2) There is no cross-coupling involved in the tuning of each column controllers of C. Remark 1. For precondition (1), it is the sufficient and necessary condition for the decoupling regulation of a TITO process. The proof is simple: if either det[G(0)] or det[C(0)] were zero, det[H(0)] would be zero, according to eq 2, and thus would contradict the desired diagonal system transfer matrix form shown in eq 4. From eq 1, it can be easily seen that the first decoupling precondition requires k11k22 * k12k21. In this paper, TITO processes with nonsingular static gain transfer matrix, i.e., det[G(0)] * 0, are considered, that is to say, their output responses deserVe to be decoupled in essence. Note that, although some industrial processes can be modeled in the stable form of eq 1, their transfer matrix inVerses seem sensitiVe to the process perturbations. The reason lies in det[G(0)] f 0 in such a situation. Hence, such process modeling should be aVoided for decoupling control design, such as by changing the pairing relationship between the process input and output Variables. As for precondition (2), it is a system operation requirement for regulating TITO processes in engineering practice. The reason can be intuitiVely seen through the transfer matrix multiplication relationship between G and C, obtaining the desired diagonal system transfer matrix shown in eq 4. 3. Desired Diagonal System Transfer Matrix Form Equation 34 shows that, if the desired diagonal system transfer matrix form in the nominal case could be ascertained in the first place, the decoupling controller matrix may be inversely derived as

C ) G-1H )

adj(G) H det(G)

(5)

T where adj(G) ) [Gij]2×2 is the adjoint of G, and Gij denotes the complement minor corresponding to gij in G, i, j ) 1, 2. Note that the process transfer matrix determinant can be formulated as

{

det(G) ) G11G22(1 - G° e-∆θs) (for θ11 + θ22 e θ12 + θ21) (6) e-∆θs (for θ11 + θ22 > θ12 + θ21) -G12G21 1 G°

(

)

where

∆θ ) |θ11 + θ22 - θ12 - θ21| G° )

k12k21 (τ11s + 1)(τ22s + 1) ‚ k11k22 (τ12s + 1)(τ21s + 1)

Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006 3151

Without loss of generality, consider the case where θ11 + θ22 e θ12 + θ21 of some TITO processes. Equations 4 and 5 show that each column controllers of the decoupling controller matrix C are related to the same diagonal element of H. For instance, the first column controllers are in the form of

c11 ) ) c21 )

G11 h det(G) 1 (τ11s + 1)eθ11s 1 h h1 ) 1 G22(1 - G°e-∆θs) k11(1 - G°e-∆θs)

Note that, by proposing the desired diagonal system transfer matrix form as shown in eqs 9 and 10, time domain performance specification of the system output responses can be quantitatively regulated using the adjustable parameters λ1 and λ2. For example, as for those TITO processes without RHP zero of det(G), that is, 1 - G° e-∆θs (if θ11 + θ22 e θ12 + θ21) or 1 e-∆θs/G° (if θ11 + θ22 > θ12 + θ21) has no RHP zero, the desired diagonal transfer functions h1 and h2 shown in eqs 9 and 10 can be respectively simplified as

(7)

G12 G12 h1 h1 ) 11 22 det(G) G G (1 - G° e-∆θs) k21(τ11s + 1)(τ22s + 1) e(θ11+θ22-θ21)s )h1 k11k22(τ21s + 1)(1 - G° e-∆θs) (8)

Obviously, it can be found from eqs 7 and 8 that, if the desired diagonal transfer function h1 were not to include an equivalent time delay to offset θ11, the controller c11 will inevitably behave in a predictive manner and so does c21 if θ11 + θ22 > θ21, which, in fact, will violate the causal law in nature. It can be physically understood through the phenomenon that either of the binary process outputs can only begin to track the corresponding setpoint input after certain process time delay. Besides, if the polynomial 1 - G° e-∆θs in the denominators of eqs 7 and 8 contain any right-half-plane (RHP) zeros in the complex plane, h1 is required to include these RHP zeros so that c11 and c21 will not be bundled with them as unstable poles. Based on the aforementioned constraint analysis and the robust H2 optimal performance objective,4,13 the desired diagonal transfer function h1 is proposed in the form of

h1 )

e-θ1s

n

∏ λ s + 1 i)1 1

( ) -s + zi s + z/i

(9)

where λ1 is an adjustable parameter used for meeting the system response requirement for the first process output y1 and θ1 ) max {θ11, θ11 + θ22 - θ21}, and zi is the RHP zero of 1 G°e-∆θs and n denotes its RHP zero number, and z/i denotes the complex conjugate of zi. Hence one of the first column controllers of C can be implemented in a proper and rational form, and meanwhile, the other controller can be practically implemented in series with a specified dead-time compensator, so that independent regulation of the first process output y1 can be realized. Following a similar analysis, the desired diagonal transfer function h2 is proposed in the form of

h2 )

e -θ2s

n

∏ λ s + 1 i)1 2

( ) -s + zi s + z/i

(10)

where λ2 is an adjustable parameter used for meeting the system response requirement for the second process output y2, and θ2 ) max {θ22, θ11 + θ22 - θ12}. As for the case where θ11 + θ22 > θ12 + θ21 of other TITO processes, the desired diagonal transfer functions h1 and h2 can be proposed the same as given previously, respectively. The only difference is that θ1 ) max {θ12, θ12 + θ21 - θ22}, θ2 ) max {θ21, θ12 + θ21 - θ11} and zi is the RHP zero of 1 e-∆θs/G° and n denotes its RHP zero number.

h1 )

1 e-θ1s λ1s + 1

h2 )

1 e-θ2s λ2s + 1

By performing the inverse Laplace transform, yield the time domain response forms of the binary process outputs as

y1(t) ) y2(t) )

{ {

0 1-e 0

-(t-θ1)/λ1

1-e

-(t-θ2)/λ2

(for t e θ1) (for t > θ1) (for t eθ2) (for t > θ2)

It is thus demonstrated that there is no overshoot in either of the nominal system output responses, and their time domain response specifications can be quantitatively achieved by respectively tuning the adjustable parameters λ1 and λ2. For instance, define the rise time tr1 to be the period from the moment that a unit step change of the first set-point input is added to the moment that the first process output y1 reaches 90% of its final value, it can be therefore figured out as tr1 ) 2.3026λ1 + θ1, and similarly, the rise time tr2 of the second process output y2 can be derived as tr2 ) 2.3026λ2 + θ2. Hence, it is convenient to tune the adjustable parameters λ1 and λ2 respectively to obtain the desirable process output responses, which, in fact, are implemented by each column controllers of the decoupling controller matrix C, and, therefore, will be further interpreted in the following section. 4. Decoupling Controller Matrix Design According to the desired diagonal system transfer functions h1 and h2 shown in eqs 9 and 10, the ideally optimal decoupling controller matrix C can be inversely determined, using eq 5. However, there actually exist some difficulties for implementing G-1. Especially when there actually exist some RHP zeros of det(G), it can be found from eq 5 in combination with eqs 9 and 10 that there will exist RHP zero-pole canceling in each controller of the ideally optimal decoupling controller matrix, which will cause it to behave unstably. Therefore, its practical form is required to configure pertinently. For clear interpretation, the design procedure is proposed in terms of two cases: Case 1 is that there exists no RHP zero of det(G), and Case 2 is in the opposite situation, both of which are presented, in detail, in the following two subsections. 4.1. Case 1. It can be seen from eq 6 that no RHP zero of det(G) indicates that 1 - G° e-∆θs (if θ11 + θ22 e θ12 + θ21) or 1 - e-∆θs/G° (if θ11 + θ22 > θ12 + θ21) has no RHP zero. Consequently, 1/(1 - G° e-∆θs) or 1/(1 - e-∆θs/G°) is a stable transfer function. First, consider the case where θ11 + θ22 e θ12 + θ21 for some TITO processes. Substituting eq 9 into eq 7 yields

c11 )

τ11s + 1

e-(θ1-θ11)s k11(1 - G° e-∆θs) λ1s + 1 ‚

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Figure 2. Positive feedback control unit.

For the convenience of physical implementation, rearrange it in the form of

c11 )

(τ11s + 1) e-(θ1-θ11)s k11(λ1s + 1)

‚F

(11)

where F ) 1/(1 - G° e-∆θs). Obviously, the first part of c11 shown in eq 11 can be practically implemented using a conventional first-order lead-lag controller in series with a deadtime compensator, and the second part F can be implemented using a positive feedback unit, as shown in Figure 2. Note that the positive feedback unit shown in Figure 2 holds internal stability because its transfer function has no RHP pole and G° is doubly proper and stable. Similarly, the other three controllers of the decoupling controller matrix can be derived as

k21 (τ11s + 1)(τ22s + 1) e-(θ1+θ21-θ11-θ22)s ‚F ‚ c21 ) k11k22 (τ21s + 1)(λ1s + 1) (12) c12 ) -

k12 (τ11s + 1)(τ22s + 1) e-(θ2+θ12-θ11-θ22)s ‚F ‚ k11k22 (τ12s + 1)(λ2s + 1) (13) c22 )

(τ22s + 1) e-(θ2-θ22)s k22(λ2s + 1)

‚F

(14)

Then, for the case where θ11 + θ22 > θ12 + θ21 for other TITO processes, each column controllers of the decoupling controller matrix can be determined similarly, as

c11 ) -

k22 (τ12s + 1)(τ21s + 1) e-(θ1+θ22-θ12-θ21)s ‚ ‚F k12k21 (τ22s + 1)(λ1s + 1) (15) c21 ) c12 )

c22 ) -

(τ12s + 1) e-(θ1- θ12)s k12(λ1s + 1) (τ21s + 1) e-(θ2-θ21)s k21(λ2s + 1)

‚F

(16)

‚F

(17)

the system output responses will come to the ideal case, i.e., h1 ) e-θ1s and h2 ) e-θ2s; that is to say, the binary process outputs y1 and y2 will reach the Values of set-point inputs just after the process time delays θ1 and θ2, respectiVely, whereas each column controllers of the decoupling controller matrix will no longer be proper and thus cannot be physically realized. In fact, when λ1 and λ2 are tuned to small, the system output responses y1 and y2 will become faster, but the output energy of the decoupling controller matrix and its corresponding actuators will be required to be larger, which has a tendency to surpass their output capacities in practice, and besides, more aggressiVe dynamic behaVior of the system output responses will occur in the presence of the process uncertainty. On the contrary, tuning λ1 and λ2 to large will slow down the system output responses but the output energy of the decoupling controller matrix and its corresponding actuators will be required to be smaller, and, accordingly, less-aggressiVe dynamic behaVior of the system output responses will appear in the presence of the process uncertainty. Based on a large amount of simulation analysis, it is suggested to tune λ1 and λ2 respectiVely within the range of (2-10)θ1 and (2-10)θ2 in the first place. If the resultant system output responses are not satisfactory, then, by monotonically Varying λ1 and λ2 on-line, the best compromise between the nominal system response performance and the output capacities of the decoupling controller matrix and its corresponding actuators can be conVeniently obtained. 4.2. Case 2. When there exist some RHP zeros of det(G), it can be seen from eq 6 that all of them are exactly the RHP zeros of 1 - G° e-∆θs (if θ11 + θ22 e θ12 + θ21) or 1 - e-∆θs/ G° (if θ11 + θ22 > θ12 + θ21). Hence, the RHP zero number of det(G) can be ascertained by observing the Nyquist curve of -G° e-∆θs (or -e-∆θs/G°). Note that the number of the Nyquist curve encircling the point (-1, j0) in the complex plane is equal to the RHP zero number of det(G), considering that it has no RHP pole. Alternatively, all of the RHP zeros of det(G) can be numerically ascertained by solving 1 - G° e-∆θs ) 0 (or 1 e-∆θs/G° ) 0) with some mathematical software packages such as MATLAB and MATHEMATICA. For the case where θ11 + θ22 e θ12 + θ21 for some TITO processes, substituting the desired diagonal system transfer functions eqs 9 and 10 into eq 5 yields

c11 )

k11 (τ12s + 1)(τ21s + 1) e-(θ2+θ11-θ12-θ21)s ‚F ‚ k12k21 (τ11s + 1)(λ2s + 1) (18)

n

(s + ∏ i)1

k11(λ1s + 1) c21 ) -

k21

‚

k11k22

c12 ) -

k12 k11k22

‚D

(τ11s + 1)(τ22s + 1) e-(θ1+θ21-θ11-θ22)s (τ21s + 1)(λ1s + 1)

‚

n

c22 )

(s + z/i ) ∏ i)1

k22(λ2s + 1)

‚ D (21)

(s + z/i ) ∏ i)1

(τ22s + 1) e-(θ2-θ22)s n

‚ D (20)

(s + z/i ) ∏ i)1

(τ11s + 1)(τ22s + 1) e-(θ2+θ12-θ11-θ22)s (τ12s + 1)(λ2s + 1)

(19)

z/i )

n

e-∆θs/G°).

where F ) 1/(1 Note that G° is doubly proper and stable, as is 1/G°. Hence, F can be stably implemented using a control unit similar to that shown in Figure 2. Remark 2. From the proposed controller design formulas shown in eqs 11-18, it is obserVed that each column controllers of the decoupling controller matrix are exactly tuned by a single adjustable parameter. Consequently, there exists no crosscoupling in tuning each column controllers. Besides, it can be easily examined that the resultant decoupling controller matrix is indeed nonsingular at s ) 0. Note that, when the adjustable parameters λ1 and λ2 are tuned to zero, eqs 9 and 10 show that

(τ11s + 1) e-(θ1-θ11)s

‚D

(22)


where

where n

D)

(-s + zi) ∏ i)1

d2 , a ) d1b0 + d0b1, a0 ) d0b0 d1 1

b1 ) (23)

1 - G° e-∆θs

Obviously, the first part of c11-c22 shown in eqs 19-22 can be respectively implemented using a conventional lead-lag controller in series with a dead-time compensator if necessary, but the second part (D) cannot be reliably implemented, because of the RHP zero-pole canceling, which, however, cannot be directly removed. Therefore, a rational approximation form is required to copy it out for physical implementation. In view of the analytical approximation method that was effectively developed for obtaining a rational high-order controller form in the recent literature,40 the mathematical Pade´ expansion is therefore utilized to construct the linear fractional approximation formula, i.e.,

Remark 3. Equation 23 shows that G°e-∆θs has a tendency to decay much faster, in contrast to the rational numerator polynomial, as s f ∞. Hence, it is confirmable that high accuracy can be attained using a rational linear fractional approximation for D. Basically, D can be roughly approximated as n

D)

∏ i)1

n

(-s + zi)

1 - G°(0)

)

(-s + zi) ∏ i)1 1 - [k12k21/(k11k22)]

U

DU/V )

aisi ∑ i)0 (24)

V

bjsj ∑ j)0 where U and V are the user-specified orders to achieve the desirable system performance specification in practice, and the constant coefficients ai (i ) 1, 2, ..., U) and bj (j ) 1, 2, ..., V) are determined by the following two matrix equations:

[

[][

d0 a0 d a1 ) ·1 ·· ·· · aU dU

dU dU+1 ·· · dU+V-1

0 d0 ·· · dU-1

dU-1 dU ·· · dU+V-2

··· ··· ··· ···

0 0 ··· dU-2

dU-V+1 dU-V+2 ·· · dU

][ ] ][ ] [ ] ··· ··· ··· ···

0 0 ·· · dU-V

b1 b2 )·· · bV

b0 b1 ·· · bV

dU+1 dU+2 ·· · dU+V

(25)

(26)

where dk (k ) 0, 1, ..., U + V) are the constant coefficients of each term in the mathematical Maclaurin expansion series of D shown in eq 23, i.e.,

dk )

1 dkD lim k k! sf0 ds

(for k ) 0, 1, ..., U + V)

(27)

and b0 should be taken as

b0 )

{

1 (for bj g 0) -1 (for bj < 0)

(28)

Note that eqs 25 and 26 can be transparently derived by substituting eq 24 into the mathematical Maclaurin expansion series of D shown in eq 23 and then comparing the constant coefficients of each complex variable with the same index at both sides. For instance, letting U ) V ) 1 yields the first-order approximation formula in the form of

D1/1 )

a1s + a0 b1s + b0

(29)

ObViously, increasing the approximation order will result in a much-better approximation leVel. Note that there possibly exist infinitely many RHP zeros of det(G), which may be identified by obserVing whether [k12k21τ11τ22/(k11k22τ12τ21)] > 1 (if θ11 + θ22 e θ12 + θ21) or [k11k22τ12τ21/(k12k21τ11τ22)] > 1 (if θ11 + θ22 > θ12 + θ21), because the Nyquist curVe of -G°e-∆θs(or -e-∆θs/G°) will encircle the origin infinitely many times with the radius of k12k21τ11τ22/(k11k22τ12τ21) (or k11k22τ12τ21/(k12k21τ11τ22) if θ11 + θ22 > θ12 + θ21) as ω f ∞ if there exist infinitely many RHP zeros of det(G). GiVen that off-dominant RHP zeros of a system transfer function haVe little impact on the achieVable system performance,1,4 it is suggested to utilize those dominant RHP zeros of det(G) to propose the desired diagonal system response transfer functions shown in eqs 9 and 10, so that the practical decoupling controller matrix can be analytically deriVed in a simple manner, but in exchange for certain system performance degradation. Hence, it is dependent on the user choice of the dominant RHP zero number to make the compromise between the achieVable system performance and the calculation complexity of the decoupling controller matrix and its cost of utilization. In addition, note that the choice of b0 shown in eq 28 is intended to keep all of bj (for j ) 0, 1, ..., V) the same sign, to exclude the possibility of any RHP zeros from being enclosed in the denominator of eq 24. EVen so, such a high-order approximation (i.e., V g 3) may be inVolVed with RHP poles, which can be identified using the Routh-Hurwitz stability criterion. Therefore, it is suggested to use the RouthHurwitz stability criterion to identify the stability of such a highorder approximation before using it in practice, to achieVe much better system performance. NeVertheless, such a linear fractional approximation form, in terms of V e 2, can be reliably implemented without identification and, therefore, is primarily recommended for use in practice, for simplicity. As for the case where θ11 + θ22 > θ12 + θ21 for other TITO processes, following a similar design procedure as above, each column controllers of the decoupling controller matrix can be derived as

c11 ) -

k22 k12k21

‚

(τ12s + 1)(τ21s + 1) e-(θ1+θ22-θ12-θ21)s n

(τ22s + 1)(λ1s + 1)

(s + z/i ) ∏ i)1

‚D

(30)

3154


c21 )

(τ12s + 1) e-(θ1-θ12)s n

k12(λ1s + 1)

c12 )

(τ21s + 1) e-(θ2-θ21)s n

k21(λ2s + 1)

c22 ) -

k11 k12k21

‚

(s + ∏ i)1

‚D

(31)

‚D

(32)

z/i ) Figure 3. Perturbed control system with additive uncertainty.

(s + z/i ) ∏ i)1

(τ12s + 1)(τ21s + 1) e-(θ2+θ11-θ12-θ21)s n

(τ11s + 1)(λ2s + 1)

∏ i)1

(s +

Figure 4. Unified perturbed system form.

‚D

z/i ) (33)

Thus, the transfer matrix from V to U is obtained as

TA ) -C[I + (G - Gm)C]-1

(38)

Given the fact that G ) Gm for the nominal control system, eq 38 can be therefore simplified as

where

TA ) -C

n

D)

(-s + zi) ∏ i)1 1 - (e

-∆θs

/G°)

Correspondingly, D can be as well implemented using the analytical approximation formula proposed in eq 24.

Undoubtedly, TA holds stability, because the decoupling controller matrix C has been designed to be stable. According to the small gain theorem, the perturbed system with additive uncertainty holds robust stability if and only if

|C|∞ < 5. Control System Robust Stability Analysis Obviously, according to the stable decoupling controller matrix C proposed in Section 4, the resultant IMC structure holds internal stability for the nominal system response of a stable TITO process. However, there usually exist the process unmodeled dynamics in engineering practice, and, therefore, how to evaluate the resultant control system robust stability must be addressed, so that the tuning constraints for holding the control system stability can be ascertained in the presence of the process uncertainty. There are three types of process uncertainty that are mostly encountered in practice, i.e., the additive uncertainty, multiplicative input, and output uncertainties, all of which are required to cope with properly during the system operation. Note that many other types of process unstructured or structured uncertainties can be lumped into the above-mentioned process uncertainties to cope with in practice.4 First, consider the process additive uncertainty shown in Figure 3. It describes the actual process family ΠA ) {G ˆ A(s): G ˆ A(s) ) G(s) + ∆A}, where ∆A is assumed to be stable. In fact, it can be viewed as parameter perturbation of the process transfer matrix. For the convenience of robust stability analysis, the perturbed control system with additive uncertainty can be rearranged into the standard T-∆ structure, as shown in Figure 4. In this figure, T represents the transfer matrix connecting the outputs of ∆A with its inputs. Figure 3 shows that

U ) CE

(34)

E ) R - (Y - GmU)

(35)

Y ) GU + V

(36)

Solving eqs 34-36 yields

U ) C[I + (G - Gm)C]-1R - C[I + (G - Gm)C]-1V (37)

(39)

1 |∆A|∞

(40)

However, the robust stability constraint shown in eq 40 is not analytical and the H infinity norm computation effort is considerably large. Thus, considering the equivalent relationship between the small gain theorem and the spectral radius stability criterion, which can be theoretically derived using the generalized MIMO Nyquist stability theorem,4 the equivalent robust stability constraint can be obtained as

F(C∆A) < 1, ∀ ω

(41)

Hence, it is practical to use eq 41 to evaluate the robust stability of the IMC structure intuitively, in the face of the process additive uncertainty, that is, observing whether the magnitude plot of the left side of eq 41 falls below unity for all ω ∈ [0, +∞), which can be graphically performed using some commercial control software packages, such as the MATLAB robust control toolbox.41 As for the process multiplicative input and output uncertainties that respectively describe the actual process family ΠI ) {G ˆ I(s): G ˆ I(s) ) G(s)(I + ∆I)} and ΠO ) {G ˆ O(s): G ˆ O(s) ) (I + ∆O)G(s)}, where ∆I and ∆O are assumed to be stable, the transfer matrixes that connect their outputs and inputs can be respectively derived as

TI ) -C[I + (G - Gm)C]-1G

(42)

TO ) -GC[I + (G - Gm)C]-1

(43)

Then, following a similar analysis, obtain the spectral radius robust stability constraints:

F(CG∆I) < 1, ∀ ω

(44)

F(GC∆O) < 1, ∀ ω

(45)


Figure 5. Achievable system response performance for a Wood-Berry distillation column.

Hence, for a specified multiplicative uncertainty bound of ∆I (or ∆O), in practice, the control system robust stability can be intuitively evaluated by observing whether the magnitude plot of the left side of eq 44 (or eq 45) falls below unity for all ω ∈ [0, +∞). In this way, the admissible tuning range of the adjustable parameters λ1 and λ2 of the decoupling controller matrix can be numerically ascertained. To cope with the process uncertainty in practice, it is suggested to monotonically increase the adjustable parameters λ1 and λ2 of the decoupling controller matrix on-line, so that the nominal system response will be gradually slowed in exchange for better system robust stability, which will be illustrated in the following simulation example 1. 6. Simulation Tests

[

]

Example 1. Consider the well-known Wood-Berry distillation column process:42

12.8 e-s -18.9 e-3s + 1 21s + 1 G ) 16.7s -7s -19.4 e-3s 6.6 e 10.9s + 1 14.4s + 1

Note that there is θ11 + θ22 ) 4 < θ12 + θ21 ) 10, and the Nyquist stability criterion can be used to confirm that there is no RHP zero of det(G). Hence, by employing the analytical design formulas proposed in eqs 11-14, the decoupling controller matrix

C)F‚

[

16.7s + 1 12.8(λ1s + 1) 0.0266(16.7s + 1)(14.4s + 1) e-4s (10.9s + 1)(λ1s + 1)

-0.0761(16.7s + 1)(14.4s + 1) e-2s (21s + 1)(λ2s + 1) -(14.4s + 1) 19.4(λ2s + 1)

]

can be obtained, where

F)

1 0.5023(16.7s + 1)(14.4s + 1) -6s 1e (21s + 1)(10.9s + 1)

Correspondingly, it can be implemented using the control unit shown in Figure 2. For illustration, two groups of simulation tests are performed, by setting λ1 ) 2 and λ2 ) 4, and λ1 ) 4 and λ2 ) 6, respectively. The simulation results are shown in Figure 5 by adding a unit step change at t ) 0 and t ) 100, respectively, to the binary set-point inputs. Note that the simulation solver option is chosen as ode5 (Dormand-prince) and the step size is fixed at a value of 0.02 throughout this paper.

3156


Figure 6. System output responses according to the second-order controller approximation.

Figure 5 shows that the binary process output responses have been absolutely decoupled from each other, and there exists no overshoot in the set-point responses. Moreover, according to the time domain system response analysis given in section 3, the rise time of set-point responses can be conveniently obtained as tr1 ) 2.3026λ1 + 1 for the process output y1, and tr2 ) 2.3026λ2 + 3 for the process output y2. Hence, it is convenient to tune the adjustable parameters λ1 and λ2 on-line, to achieve the desired binary system output responses. Note that tuning λ1 and λ2 aims at the tradeoff between the achievable system response performance and the output capacities of the decoupling controller matrix and its corresponding actuators, as interpreted in Remark 2. Figure 5 has illustrated the compromise effect; that is, the binary control output energy, in terms of the smaller tuning parameters λ1 ) 2 and λ2 ) 4, is required to be more aggressive than that of λ1 ) 4 and λ2 ) 6. Moreover, the binary control outputs u1 and u2 are observed to be somewhat oscillatory during the stage of dynamic response, which has a tendency to wear out the corresponding actuators in a quick fashion for some industrial processes. The reason lies in F of the decoupling controller matrix C, of which the denominator is involved with a time delay factor that will unfavorably trigger the decoupling controller matrix to yield oscillatory output signals. The phenomenon may be not admissible for some industrial and chemical processes. Therefore, it is suggested to utilize the analytical approximation formulas proposed in eqs 24-28 to copy out F for implementation. For instance, a secondorder approximation formula can be therefore obtained as

F2/2 )

73.648s2 + 51.077s + 2.01 150.662s2 + 32.283s + 1

Substituting it into the above decoupling controller matrix C and also by letting λ1 ) 4 and λ2 ) 6, the simulation results shown in Figure 6 are obtained. This figure still shows that the binary process output responses are completely decoupled with

each other, and, meanwhile, the binary control outputs u1 and u2 become well-smoothed while the system performance seems affected only slightly. Note that better system performance can be obtained using such a high-order approximation formula. For comparison, three recently developed decoupling control methods26,28,31 are used here, all of which have already demonstrated their advantages over many existing methods by simulation results for the commonly used Wood-Berry column example. Based on the idea of using a static decoupler, Lee’s method26 proposed a decoupling controller matrix composed of proportional-integral (PI) controllers, and Astrom’s method28 presented an enhanced tuning method of decentralized PI controllers. Wang’s method31 utilized a dynamic decoupler, in conjunction with decentralized proportional-integral-differential (PID) controllers for the binary process. In the proposed method, let λ1 ) 4 and λ2 ) 6 to obtain the similar set-point tracking speed with these decoupling control methods. By adding a unit step change at t ) 0 and t ) 150, respectively, to the binary set-point inputs, and an inverse step change of load disturbance with a magnitude of 0.1, to both of the process inputs at t ) 300, one can obtain the simulation results provided in Figure 7. It is clearly shown that the proposed method has resulted in obviously improved system output responses and decoupling regulation. Note that better nominal system performance for the set-point tracking and load disturbance rejection can be conveniently obtained on-line in the proposed method by gradually decreasing the adjustable parameters λ1 and λ2. To compare the control system robust stability, assume that the static gains of the first column elements of the process transfer matrix are actually 20% larger and those of the second column elements 30% larger, while all of the time delays and time constants of the process transfer matrix are actually 20% larger, which is utilized to represent severe process parameter uncertainty that may be actually encountered, because of the process unmodeled dynamics. According to the robust stability analysis provided in section 5, the magnitude plot of spectral


Figure 7. Nominal system responses for example 1.

Figure 8. Magnitude plots of spectral radius for the perturbed systems.

radius for identifying the control system robust stability can be drawn using eq 41, as shown in Figure 8. It is obvious that the peak value (the dotted line in Figure 8) is much smaller than unity, which indicates that the proposed control system is capable of good robust stability. Correspondingly, the perturbed system output responses are provided in Figure 9. This figure demonstrates that the proposed control system holds robust stability well. Note that the binary control outputs have varied little and, thus, are omitted, for the sake of brevity. In fact, better robust stability can be conveniently obtained to cope with the severe process parameter uncertainty in the proposed control system by monotonically increasing the adjustable parameters λ1 and λ2 on-line. To further demonstrate the robust stability of the proposed decoupling control system, assume that there actually exists the process multiplicative input uncertainty, ∆I ) diag[(s + 0.3)/ (s + 1), (s + 0.3)/(s + 1)], which can be practically viewed as a situation in which the binary process inputs, fed by the corresponding actuators, increase by up to 100% uncertainty at

high frequencies and by almost 30% uncertainty within the lowfrequency range. In the other case, assume that there actually exists the process multiplicative output uncertainty ∆O ) diag[ -(s + 0.2)/(2s + 1), -(s + 0.2)/(2s + 1)], which can be roughly regarded to be the binary process measurements obtained from the corresponding output sensors decrease by up to 50% uncertainty at high frequencies and by almost 20% uncertainty within the low-frequency range. According to the robust stability constraints (eqs 44 and 45 given in section 5), the magnitude plots of spectral radius for identifying the control system robust stability are also drawn in Figure 8 (the dashed-dotted and solid lines), both of which indicate that the proposed control system facilitates good robust stability. Correspondingly the perturbed system output responses are provided in Figure 10. Once again, it is observed that the proposed system output responses have held stability well and remained entirely decoupled in the presence of the assumed process multiplicative input and output uncertainties. Example 2. Consider a TITO process with time delays, as studied by Wang et al.:31

[

1.68 e-2s -0.51 e-7.5s (32s + 1)2(2s + 1) (28s + 1)2(2s + 1) G) -1.25 e-2.8s 4.78 e-1.15s (43.6s + 1)(9s + 1) (48s + 1)(5s + 1)

]

In Wang’s paper, the first-order process transfer matrix model was identified as

[

-0.5332 e-19.5838s 1.7171 e-14.8791s 67.7099s + 1 48.3651s + 1 Gm ) -1.2585 e-8.4505s 4.7861 e-4.9768s 48.7805s + 1 49.7512s + 1

]

3158


Figure 9. Perturbed system responses for example 1 due to the process parameter perturbation.

Figure 10. Perturbed system responses for example 1 due to the process multiplicative uncertainties.

Hence, it is utilized to derive the decoupling controller matrix in the proposed method. The Nyquist curve of det(Gm) is drawn in Figure 11. It is observed that there exist infinitely many RHP zeros of det(Gm). By numerically solving det(Gm) with some mathematical toolboxes such as MATLAB, it can be determined that there exists only one dominant RHP zero at s ) z1 ) 0.0129, which is therefore used to propose the desired system transfer matrix. Note that there is θ11 + θ22 ) 24.5606 > θ12 + θ21 ) 23.3296. Using the analytical design formulas given as eqs 24-28 and 30-33, the decoupling controller matrix can be obtained:

[

2.2148(48.3651s + 1)(48.7805s + 1) -(48.7805s + 1) (49.7512s + 1)(77.5194s + 1)(λ1s + 1) 1.2585(77.5194s + 1)(λ2s + 1) C)D‚ -0.2467(48.3651s + 1)(48.7805s + 1) e-4.7047s (48.3651s + 1) e-3.4737s 1.7171(77.5194s + 1)(λ1s + 1) (67.7099s + 1)(77.5194s + 1)(λ2s + 1)

]

where

D)

1 - 77.5194s 1.1809(48.3651s + 1)(48.7805s + 1) -1.231s 1e (67.7099s + 1)(49.7512s + 1)

For simplicity, the first-order approximation formula shown in eq 29 is proposed to copy out D involved with a RHP zero-pole canceling for physical implementation, i.e., D1/1 ) -(373.2751s + 5.5271)/(4.4191s + 1). For comparison, tune the adjustable parameters λ1 ) 40 and λ2 ) 100 to obtain the similar set-point tracking speed with Wang’s method. By adding a unit step change at t ) 0 and t ) 1500, respectively, to the binary set-point inputs, and an inverse step change of load disturbance with a magnitude of 0.1 to both of the process inputs at t ) 3000, obtain the simulation results, as shown in Figure 12. There clearly is no overshoot in the binary system output responses by using the proposed method, and the peak values of the binary control outputs are much smaller. Note that better system response performance may be captured in the proposed method by using a high-order approximation formula for D involved with a RHP zero-pole canceling or decreasing the adjustable parameters on-line. To compare the control system robust stability, assume that the static gains of the first column elements of the actual process transfer matrix are actually 10% larger and those of the second column elements 20% larger, while all of the time delays are actually 10% larger. The perturbed system responses are shown in Figure 13, which also demonstrates that the propose control system is capable of apparently improved robust stability.


Figure 11. Nyquist curve of the process model determinant.

7. Conclusions To meet the increasing demands for advanced decoupling control strategies for TITO processes with time delays in modern process industry, this paper has proposed a new analytical

Figure 12. Nominal system responses for example 2.

Figure 13. Perturbed system responses for example 2.

decoupling control scheme that is based on a standard IMC structure, which is capable of absolute decoupling regulation for the nominal binary system output responses. By virtue of the analytical design procedure developed for the decoupling controller matrix, the computation effort is relieved on a large scale, in contrast to many existing decoupling control methods that are based on numerical algorithms, and, moreover, there exists a quantitative tuning relationship between the adjustable control parameters and the nominal binary system output responses, which will therefore contribute much convenience to the system operation, from a practical point of view. At the same time, the sufficient and necessary constraints for tuning the adjustable parameters of the decoupling controller matrix to hold the control system robust stability have been addressed in the presence of the process additive and multiplicative input and output uncertainties that are mostly encountered in engineering practice. Correspondingly, a practical way to evaluate the control system robust stability has been presented in terms of the spectral radius stability criterion. In addition, it has noted that tuning the adjustable control parameters aims at

3160


the compromise between the nominal system performance and its robust stability and the output capacities of the decoupling controller matrix and its corresponding actuators, which, in fact, can be conveniently implemented on-line by virtue of the fact that each column controllers of the proposed decoupling controller matrix are respectively tuned by a single adjustable parameter in a monotonic manner. Acknowledgment This work is supported by National Natural Science Foundation of China (60474031), SRFDP (20030248040), Science and Technology Rising-Star Program of Shanghai (04QMH1405), and NCET (04-0383). Literature Cited (1) Seborg, D. E.; Edgar, T. F.; Mellichamp, D. A. Process Dynamic and Control, 2nd Edition; Wiley: New York, 2004. (2) Shinskey, F. G. Process Control System, 4th Edition; McGrawHill: New York, 1996. (3) Luyben, W. L. Process Modelling, Simulation and Control for Chemical Engineers; McGraw-Hill: New York, 1990. (4) Skogestad, S.; Postlethwaite, I. MultiVariable Feedback Control: Analysis and Design, 2nd Edition; Wiley: Chichester, U.K., 2005. (5) Holt, B. R.; Morari, M. Design of resilient processing plantssV. The effect of deadtime on dynamic resilience. Chem. Eng. Sci. 1985, 40 (7), 1229-1237. (6) Smith, O. J. M. Closer control of loops with dead time. Chem. Eng. Prog. Trans. 1957, 53 (5), 217-219. (7) Alevisakis, G.; Seborg, D. E. An extension of the Smith Predictor to multivariable linear systems containing time delays. Int. J. Control 1973, 3 (17), 541-557. (8) Ogunnaike, B. A.; Ray, W. H. Multivariable controller design for linear systems having multiple time delays. AIChE J. 1979, 25 (6), 10431056. (9) Watanabe, K.; Ishiyama, Y.; Ito, M. Modified Smith predictor control for multivariable systems with delays and unmeasurable step disturbances. Int. J. Control 1983, 37 (5), 959-973. (10) Jerome, N. F.; Ray, W. H. High-performance multivariable control strategies for systems having time delays. AIChE J. 1986, 32 (6), 914931. (11) Desbiens, A.; Pomerleau, A.; Hodouin, D. Frequency based tuning of SISO controllers for two-by-two processes. IEE Proc. Control Theory Appl. 1996, 143 (1), 49-56. (12) Wang, Q. G.; Zou, B.; Zhang, Y. Decoupling Smith predictor design for multivariable systems with multiple time delays. Chem. Eng. Res. Des. 2000, 78 (4), 565-572. (13) Morari, M.; Zafiriou, E. Robust Process Control; Prentice Hall: Englewood Cliffs, NJ, 1989. (14) Wang, Q. G.; Zhang, Y.; Chiu, M. S. Non-interacting control design for multivariable industrial processes. J. Process Control 2003, 13 (3), 253265. (15) Dong, J.; Brosilow, C. B. Design of robust multivariable PID controllers via IMC. Proc. Am. Control Conf. 1997, 5, 3380-3384. (16) Jerome, N. F.; Ray, W. H. Model-predictive control of linear multivariable systems having time delays and right-half-plane zeros. Chem. Eng. Sci. 1992, 47 (4), 763-785. (17) Gilbert, A. F.; Yousef, A.; Natarajan, K.; Deighton, S. Tuning of PI controllers with one-way decoupling in 2 × 2 MIMO systems based on finite frequency response data. J. Process Control 2003, 13 (6), 553-567. (18) Toh, W. H.; Rangaiah, G. P. A methodology for autotuning of multivariable systems. Ind. Eng. Chem. Res. 2002, 41 (18), 4605-4615.

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ReceiVed for reView October 10, 2005 ReVised manuscript receiVed February 27, 2006 Accepted February 27, 2006 IE051129Q

Analytical Design of Decoupling Internal Model Control (IMC) Scheme

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