Decoupling Multivariable Control with Two Degrees of Freedom

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Decoupling Multivariable Control with Two Degrees of Freedom

Hsiao-Ping Huang, and Feng-Yi Lin

Ind. Eng. Chem. Res., 2006, 45 (9), 3161-3173 • DOI: 10.1021/ie051138z Downloaded from http://pubs.acs.org on November 18, 2008

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Decoupling Multivariable Control with Two Degrees of Freedom

Department of Chemical Engineering, National Taiwan UniVersity, Taipei, Taiwan 10617, Republic of China

A decoupling multivariable control system having two degrees of freedom (2-df) is proposed. The system includes controllers for set-point tracking and for disturbance rejection. Synthesis of an inverse-based multi-input and multioutput (MIMO) decoupler makes use of the zero number deficiency in the adjoint matrix of the process. According to this zero number deficiency, pole(s) and zero(s) are thus introduced to the decoupler to modify the open-loop dynamics and preserve the properness. A new method for synthesizing controllers for disturbance rejection is then presented to design the decoupling controller in the main loop. In addition, model-following and feedforward compensators are incorporated to provide Smith-predictor equivalent capability for point tracking. On the basis of the decoupled open-loop process, the controllers for set-point tracking and disturbance are designed independently. Stability robustness of the system can be tuned with two measures for decoupling and model reduction. Robustness of the system to both sensor failure and actuator failure is also included. Simulation examples are used to illustrate the proposed design method. Results from the simulation show that this proposed method is effective for the MIMO control.

1. Introduction

Most chemical processes are mathematically represented by multi-input and multioutput (MIMO) models. To control these MIMO processes, two major approaches have been adopted. One uses multiple single-input single-output (SISO) controllers to constitute a multiloop control system, while the other uses multivariable controllers to become a multivariable control (MVC) system. Because of the fact that multiloop systems are widely used, interactions among loops have been an important issue. Because of the loop interactions, the design of controllers cannot be carried out independently and also the performances of the controllers are confined to some limits. Different methods for designing controllers known as multiloop systems have been proposed (e.g., refs 1-6, etc.). Although these multiloop SISO controllers work well for some cases, the interactions among the loops cannot be eliminated completely. Theoretically, loop interactions can be eliminated by making use of multivariable controllers, that is, the controllers make adjustment to many process inputs simultaneously with access to many feedback variables. For years, MVC has been an important topic for academic research, and extensive papers and books have been published (e.g., refs 7-20, etc.). Lately, design of multivariable variable controllers has been emphasized on obtaining an inverse of the dynamic process model (e.g., refs 21 and 22). In the general controller of Wang et al.,23_{the loop transfer functions}

are decoupled into several targeting transfer functions, and compensators are designed for step set-point tracking. The resulting system does not perform well in disturbance rejection, which is as important as set-point tracking in process-control applications. Besides, the system designed in that way usually leads to unsatisfactory stability robustness, especially when a Smith predictor is incorporated.22_{The deficiency in disturbance}

rejection of their controller is due to the fact that their system has only one degree of freedom (df) and, hence, can only be optimally designed for one input (i.e., set point or disturbance) but not for both. This awkwardness can be overcome by a control system that has model following and feedforward from the set point.24_{Nevertheless, the designs of model-following}

controllers are highly coupled with the controller in the main

loop, which may then limit the achievable performance.20_On

the other hand, it is also difficult to tune the feedforward controller as mentioned by Åstro¨m and Ha¨gglund.24 _This

coupling difficulty in design for SISO systems has been overcome by a double-controller design presented by Tian and Gao,25,26_{as shown in Figure 1. In this system, the two controllers}

can be designed independently to meet, requiring individual objectives for either set-point tracking or for disturbance rejection. Furthermore, the dead time in the process can also be compensated by incorporating a Smith predictor in their system. Although the configuration works well for SISO systems, it is not trivial to extend it to the control of MIMO systems. The difficulty encountered in the extension is that there needs to be substantial efforts to design two multivariable controllers in two loops, one for set-point tracking and the other for disturbance rejection. Although quite a few methods for MVC design have been reported in the literature with different emphases, it is still not quite mature as far as disturbance rejection is concerned. In this paper, a 2-df multivariable control system is presented. In this 2-df system, incorporated in the main loop is an inverse-based decoupling controller to have good disturbance rejection. In addition, model-following and feed-forward controllers to perform Smith-predictor-type performance for set-point tracking are incorporated. The controllers for two different purposes can be designed independently based on the decoupled open-loop process. Robustness of the system to modeling error as well as integrity is included. Simulation examples are used to illustrate both the design procedures and performances.

2. 2-df Control System for MIMO Processes

The double-controller SISO system of Tian and Gao25 _as

shown in Figure 1 can be substantially simplified to a practical configuration, as shown in Figure 2a. It is easy to show that

* To whom correspondence should be addressed. Tel.: 866-2-2363-8999. Fax: 886-2-2362-3935. E-mail: huanghpc@ntu.edu.tw.

Figure 1. Dual-loop system of Tian and Gao.25

10.1021/ie051138z CCC: $33.50 © 2006 American Chemical Society

the system in Figure 2a leads to the following equation:

Without modeling errors between gp,0e-θsand gp,0 /

e-θ*s(i.e.,

gp,0e-θs) gp,0 /

e-θ*s), the response in eq 1 can be written as follows:

Notice that the system in Figure 2a needs only to specify gr,0(s)

for set-point tracking and to design gC(s) for disturbance

rejec-tion. Thus, the design for two controllers in the dual-controller system can be significantly simplified. In the following, the design for the MIMO case will be based on this new control configuration. Consider an m× m system of the following,

where Y(s) and U(s) designate the output and input vectors and

l(s) and GL(s) represent the load and its transfer function matrix

(TFM). Both G(s) and GL(s) are open-loop stable TFMs as per

the following:

As shown in Figure 2b, the new 2-df control system for the MIMO process consists of a diagonal, GC(s) (i.e., gC,i(s), i ) 1, 2,

..., m), and a decoupler, D(s). The GR(s) represents the targeting

TFM for the system to respond to set-point changes, and F(s) is a feedforward compensator. According to the block diagram in Figure 2b, the output of the system can be written as follows:

If G(s)D(s)F(s) is made equal to GR(s), the above equation for Y(s) becomes

where

It is clear to see from eq 6 that Y(s) depends on GR(s) and GC

-(s) separately. As a result, the design for each part can be conducted independently.

Inversing Decoupler. This decoupler in the main loop is used to decouple the multivariable open-loop process into m indi-vidual SISO processes. For this purpose, the process TFM has to be factorized into invertible (i.e., G-(s)) and noninvertible (i.e., G+(s)) parts, that is,

and the decoupler is formulated as

where GF(s) is a prespecified TFM of filter. But the factorization

into the form of eq 8 is not trivial, especially when G(s) has both right-half plane (RHP) zero(s) and different dead times. In general, G(s) is first factorized into two parts:

whereθi) Min{θi1,θi2, ...,θim}.

Provided that G0(s) is invertible, the decoupler can be

designed as

But the factorization of eq 10 does not guarantee that G0(s) is

invertible, because the determinant of G0(s),|G0(s)|, may still

have dead times and RHP zero(s) as factors. Consequently, identification of the RHP zero(s) of |G0(s)| is difficult. To

circumvent the difficulties encountered in the inversion of G0

-(s), the decoupler is given as

where Z(s) ) diag{zi(s)}, adj{G0(s)}) [G0

ji_{(s); i, j ) 1, 2, ...,} m], and G₀ij(s) is the cofactor of g0,ij(s). Notice that each

diagonal element zi(s) is given as a simple and stable transfer

function. The decoupler in eq 12 is thus open-loop stable, since

G(s) is open-loop stable, as has been mentioned. For easy

reference, the decoupled process G(s)D(s) is designated as Q(s), that is,

To implement D(s) of eq 12, each G₀ij(s) is to be replaced by a transfer function in a simpler form, that is,

In other words,Φ(s) is an approximation to adj{G0(s)}. In eq

14, n and p are the number of first-order leads and lags, respectively. In general, p + 2 - n > 0. The parameters in the model (i.e., kA_{, δ, τ}

g,i A

,τ_r,iA,τ_p,1A, andτ_p,2A) can be obtained by

Figure 2. New 2-df control system containing three elements for (a) SISO systems and (b) MIMO systems.

y )gr,0e -θ*s_g Cgp,0e -θs_{+ g} r,0gp,0 / -1_g p,0e -θs 1 + g_Cg_p,0e-θs r + g_L 1 + g_Cg_p,0e-θsl (1) y ) g_r,0e-θsr + gL 1 + g_Cg_p,0e-θsl (2) Y(s) ) G(s)U(s) + G_L(s)l(s) (3) Y(s) ){(I + G(s)D(s)G_C(s))-1G(s)D(s)G_C(s)G_R(s) + (I + G(s)D(s)G_C(s))-1G(s)D(s)F(s)}R(s) + (I + G(s)D(s)G_C(s))-1G_L(s)l(s) (5) Y(s) ) G_R(s)R(s) + G_D(s)l(s) (6) G_D(s) ) (I + G(s)D(s)G_C(s))-1G_L(s) (7) G(s) ) G+(s)G-(s) (8) D(s) ) G--1(s)GF(s) (9) D(s) ){G₀(s)}-1G_F(s) (11) D(s) ) adj{G₀(s)}Z(s) (12) Q(s) ) G(s)D(s) )Θ(s)|G₀(s)|Z(s) (13) φ_ij(s) ) Gˆ₀ji(s) ) kAe-δs

∏

∏

solving the following optimization problem, where P ) [kA_{,δ, τ} g,i A_,τ r,i A_,τ p,1 A_{, andτ} p,2 A_{] andω} fis a frequency

band within which Gˆ₀ij(s) has a good fit to G₀ij(s). The recom-mended value of the frequency band is its bandwidth or critical frequency.

To make each element of D(s) realizable, each diagonal element zi(s) has a number of Nez[zi(s)] excess zeros. This

number is given as the following

where Nep_[φ

ij(s)] is the number of excess poles in φij(s) [i.e., Gˆ0

ji

(s)].

In eq 13,|G0(s)| can be implemented by a reduced order form

of the following,

where φ0_{(s) is the delay free term of} |G

0(s)| and θex is the

excessive time-delay factorizing from the expansion ofφ(s).

By making use of eq 17, eq 13 can be rewritten into the following,

where

The diagonal TFM,Θ*_{(s), consists of effective apparent dead}

times. The same apparent dead times appear also in GR(s). By

reallocating the pole(s) and zero(s) inφ(s), zi(s) provides the

availability to modify undesirable dynamic characteristics in |G0(s)| and, thus, can improve the dynamics resulting from some

large time constants or excessive lags.

The decoupler D(s) is thus implemented via the transfer functions of the following:

From eq 16, Nez_[z

i(s)] is computed. Then, the pole(s) and

zero-(s) polynomials of zi(s) can be synthesized by simply reallocating

the pole(s) and zero(s) of φ0_{(s). For example, the number} Nez_[z

i(s)] of the smallest poles inφ0(s) may constitute the zero

polynomial of zi(s).

Disturbance Rejection. With the process TFM given above and from the block diagram in Figure 3, the process output in response to a disturbance l(s) becomes

Since the main loop has been decoupled, multiple SISO control elements (gC,i(s), i ) 1, ..., m) can be designed independently.

From eq 18,

and

where

The controller design for gC,i(s) becomes a SISO control

problem. In literature, synthesis of controllers for single-loop systems was usually aimed at set-point tracking, until recently Chen and Seborg27_{provided a design for disturbance rejection.}

But here, the controller will be synthesized in a more explicit way. Notice that qˆi(s) has been factorized into two parts, (i.e., qˆ_i0(s) and e-θi*s). In the same way, so has g_L,i(s),

where g_L,i0(s) and e-Lisare the delay-free parts and the delay parts of gL,i(s), respectively. It is assumed that the open-loop

transfer function from the load input to the output is represented by a first-order plus dead time (FOPDT) model of the following:

To synthesize gC,i(s), a targeting response to a step load input

can be assigned so that

where g_d,i0(s) has two functional forms to define the desired load response for yi. They are

or

Once g_d,i0(s) is specified, gC,i(s) can be synthesized accordingly.

That is,

By applying the first-order Pade’s approximation for e-θi*sin eq 30, the controller gC,i(s) can be given as

where τf,i is the filter time constant and has a default value

as 0.05θi*. In the following, the parameters of [gd,i 0

(s)]1

and [gd,i 0

(s)]2 regarding different values of τL,i in eq 26

will be computed so as to give minimum errors of yiin terms

of an integral of the absolute error (IAE) measure. That is,

y_i(s) ) gL,i(s) 1 + q_i(s)g_C,i(s)l(s)≈ g_L,i(s) 1 + qˆ_i(s)g_C,i(s)l(s) (23) qˆ_i(s) )φ0(s)z_i(s) e-θi*s) qˆ i 0 e-θi*s ₍₂₄₎

g_L,i(s) ) g_L,i0 (s) e-Lis (25)

g_L,i(s) ) kL,ie -Lis τL,is + 1 (26) y_i(s) ){g_L,i(s)[1 - g_d,i(s)]}l(s) )_{g L,i(s)[1 - gd,i 0 (s) e-θi*s]}l(s) (27) [g_d,i0 ]₁(s) ) cs + 1 as2+ bs + 1 (28) [g_d,i0 (s)]₂) 1 bs + 1 (29) g_C,i(s) ) 1 qˆ_i0(s) g_d,i0 (s) 1 - g_d,i0 (s) e-θi*s (30) g_C,i(s) )gf,i(s) qˆ_i0(s) g_d,i0 (s)(1 +θi * s/2) (1 +θi * s/2) - g_d,i0(s)(1 -θi * s/2) (31) g_f,i(s) ) 1 (τ_f,is + 1)n (32) Nez[z_i(s)] ) min{Nep[φ_ij(s)], j ) 1, 2, ..., m} (16) φ(s) ) kDe-θexs

∏

∏

where|h|i* is the peak gain of the ith decoupled single-loop

system:

Typically, the values of these|h|i*are assigned in the range of

1.2-2.0 for stability robustness consideration. It is found that [g_d,i0(s)]1is good for the case where 2 e τL,i/θi*e 100 and [

g_d,i0(s)]2is recommended for the case where 0.1 e τL,i/θi*< 2,

respectively. The resulting parameters of [g_d,i0(s)]1in terms of τL,i/θi*as a parameter are shown in Figure 4. These data are

correlated into the following functional form,

where

The resulting data are given in Table 1. On the other hand, the parameters of [g_d,i0(s)]2are given in Table 2. With those data in

the tables, the controller can be synthesized accordingly to meet different values of|h|i*. If qˆi

0_{(s) includes RHP zeros, (i.e.,Π(1}

- νis),νi> 0), they will be treated as additional dead times by

using the Pade’s approximation. So, these RHP zeros become

where

Thus, qˆ_i0(s) containing additional dead time is represented as

Set-point Tracking. As mentioned, the inverse-based con-troller in the main loop decouples the open-loop process into multiple SISO open-loop processes. As a result, the feedforward compensator F(s) can be synthesized by the following,

where

so that G(s)D(s)F(s) will be made equal to GR(s) and the output

of the system becomes

whereΘ*_{(s) is given by eq 19. Therefore, for set-point tracking,}

the desired response can be obtained by specifying the elements of

G_R0(s) (i.e., g_R,i0 (s)) with a transfer function in the form of eq 43:

τd,i and ζ are the time constant and damping coefficient,

respectively, of the second-order model in eq 43. Notice that [

qˆ_i0]+](s) contains all the RHP zeros in qˆ_i0(s). The standard second-order part in eq 43 describes those usual performance requirements, such as overshoot and settling time, etc. The term (τd,1s + 1)nis to provide the necessary high-frequency roll-off

rate required. After the desired closed-loop transfer function has

Figure 3. Multivariable decoupling control system.

Figure 4. (a), (b), (c) Relations between the parameters of [gd,i 0

(s)]1and

τL/θi*with different values of|h|i*.

{

}

∫

∑

been specified, the feedforward element F(s) is synthesized as [Qˆ0_(s)]-1_G

R0(s). The control structure becomes the one as shown

in Figure 2b. Notice that g_R,i0 (s) must include the RHP zeros of

qˆ_i0and g_R,i0 (s)/qˆ_i0(s) must be proper.

3. Stability and Robustness

Under the presented 2-df control framework, the stability of the system is determined by the main feedback loop. Because each element of the process G(s) in eq 4 is an open-loop stable function, the decoupler D(s) in eq 12 is designed to be open-loop stable. The nominal stability of the system is guaranteed by the following two conditions:

whereσj denotes the largest singular value. Because the nominal

process (i.e., G) has been decoupled and GC(s) is also diagonal

and is designed by the synthesis method, condition (1) is satisfied as those in dealing with a number of m single-loop systems. Because of the approximation made in eq 14, Qˆ (s) may not equal|G0(s)|Θ(s)Z(s) exactly. As a result, a model error

(i.e., ∆Q(s)) originating from this approximation can be estimated without difficulty. In general,∆Q(jω) is very small

in the frequency range concerned for nominal stability, and the second condition can easily be satisfied.

As for stability robustness to modeling error of G(s), consider the control system in Figure 5, where the real process is presented as

and

where the perturbation∆(s) is bounded on l(jω). The system

will be robustly stable if

where

Thus, by selecting an adequate |h|i*, the controller GC(s) is

synthesized also to satisfy the robust stability in eq 46. As mentioned, to ensure that eq 46 is met, two types of mod-eling deficiencies must be taken into consideration. One is due to the approximations for both adj{G0(s)}and|G0(s)|. The other

is due to the modeling error of the process. Because of the aforementioned approximations, a transfer function matrix for the effectiveness of decoupling is considered as the following:

On the basis of Γ(s), an index is defined to indicate the effectiveness of decoupling,

whereωg,iis the frequency where|γii(jωg,i)| ) 0.707.

Notice that D_jid emphasizes the discrepancy of the off-diagonal elements of Γ(s) from zero. This value is mainly determined by the bandwidth of good approximation of φij(s)

to the cofactor G₀ji. If the value is too high to be satisfactory, the model order of φ(s) can be increased. In other words, D_jid serves as a tuning factor to improve the stability robustness of the system. For good stability robustness, it is recommended that D_jidis no greater than 0.3.

On the other hand, an index (i.e., p,i) which measures the

discrepancy between the desired and the estimated transfer functions form set point to output is also defined as follows,

where

Table 1. Correlation between the Parameters of [gd,i 0_](s)]

1andτL/θi*

for Different|h|i*Values

|h|i* y R p1 p2 p3 p4 1.2 a* _{3 4.821× 10}-5 _{-0.011 12 0.948 2 13.5} b* _{3 4.504× 10}-6 _{-0.001 104 0.1 6.349} c* _{3 9.575× 10}-6 _{-0.002 282 0.201 8 3.397} 1.3 a* _{2.3 4.359× 10}-4 _{-0.038 43 1.243 5.411} b* _{2.3 6.103× 10}-5 _{-0.005 981 0.207 4 4.14} c* _{2.3 1.292× 10}-4 _{-0.012 16 0.414 2 2.314} 1.4 a* _{2 0.001 039 -0.057 53 1.16 2.831} b* _{2 2.353× 10}-4 _{-0.013 52 0.279 3.053} c* _{2 4.494× 10}-4 _{-0.026 01 0.544 9 1.82} 1.5 a* _{1.7 0.002 122 -0.082 05 1.142 1.59} b* _{1.7 7.599× 10}-4 _{-0.027 83 0.370 5 2.373} c* _{1.7 0.001 276 -0.050 09 0.708 8 1.472} 1.6 a* _{1.6 0.002 536 -0.081 5 0.943 4 1.055} b* _{1.6 0.001 034 -0.034 61 0.387 7 1.964} c* _{1.6 0.001 814 -0.062 43 0.739 4 1.321} 1.7 a* _{1.5 0.002 867 -0.079 91 0.799 5 0.727 7} b* _{1.5 0.001 503 -0.041 7 0.398 2 1.675} c* _{1.5 0.002 572 -0.075 03 0.763 5 1.215} 1.8 a* _{1.4 0.003 261 -0.080 16 0.702 8 0.497 1} b* _{1.4 0.001 974 -0.047 7 0.403 5 1.455} c* _{1.4 0.003 384 -0.087 2 0.786 6 1.126} 1.9 a* _{1.3 0.003 758 -0.0817 5 0.634 4 0.331 8} b* _{1.3 0.002 416 -0.054 19 0.413 6 1.271} c* _{1.3 0.004 192 -0.100 2 0.816 3 1.041} 2.0 a* _{1.3 0.002 832 -0.063 59 0.498 0.271 6} b* _{1.3 0.003 421 -0.062 79 0.411 2 1.138} c* _{1.3 0.005 213 -0.107 3 0.787 4 1.022}

Table 2. Correlation between the Parameters of [gd,i 0

(s)]2andτL/θi*

for Different|h|i*Values

|h|i* b* 1.2 3.499 4 1.3 2.056 4 1.4 1.352 7 1.5 0.937 9 1.6 0.663 7 1.7 0.466 8 1.8 0.314 4 1.9 0.184 1

(1) G_Cstabilizes Qˆ (s) in a simple closed loop.

(2)σj{-GC(jω)[I + Qˆ (jω)GC(jω)] -1_} e 1 Max ω {σj[∆Q(jω)]} ; ∀ω∈ [0,∞]

Figure 5. Additive uncertain control system.

G˜ (s) ) G(s) + ∆(s) (44) σ j(∆(jω)) e |l(jω)| (45) σj[M(jω)]|l(jω)| < 1, ω ∈ [ 0,∞] (46) M(s) ) -D(s)G_C(s)[I + G(s)D(s)G_C(s)]-1 (47) Γ(s)) G(s)D(s)[Qˆ0(s)]-1G_R0(s)){γij; i, j ) 1, 2, ..., m} (48) D_jid) max ω

(

)

{

}

The recommended value of the above p,i is <0.3. Based on

these criteria, the desired decoupling performance, loop per-formance, and robust stability can be achieved by tuning the parameters of GR(s) and GC(s).

Robustness to Sensor Failure. There are two types of sensor failure considered here. One is that some sensor is broken, and the other is that some sensor malfunctions with an additional bias. In the case of a sensor being broken, the characteristic function of the system becomes

where I˜i_{denotes an identity matrix with its ith diagonal element}

being removed. Since G(s)D(s)GC(s) is a diagonal matrix, the

product of G(s)D(s)GC(s)I˜ibecomes a diagonal submatrix. As

a result, if the nominal case is stable, the system will remain stable, too. Consequently, any sensor broken in this system will not cause stability problem. On the other hand, if some sensor malfunctions with additional biases, it is equivalent to the case where the system has additional but constant unknown inputs as disturbances entering along with the feedback. Since the main loop is designed for dealing with disturbance, these disturbances will not jeopardize the stability in this case.

Robustness to Actuator Failure. In the case of actuator failure, the characteristic equation to be considered becomes

where I˜i_{denotes an identity matrix with the ith diagonal element}

being removed. To retain the system integrity under the condi-tions of an actuator being broken, GCin the main loop should

be able to stabilize the remaining system. This can be achieved by assigning the proper value to each|h|i*. With such a GC, the

system still has to be robust to the possible aforementioned modeling errors. For this purpose, eq 46 must also apply to each of the following Mi_{, that is,}

4. Illustration with Control of 2× 2 or 3 × 3 MIMO Processes

In literature, there are quite a few MIMO processes studied for process control. A few of such processes, including those of 2× 2 and 3 × 3, are used as illustrations of control using the presented design method. The results are compared with those using other methods.

Example 1. Consider the Wood and Berry28_{2× 2 process.}

The transfer function matrices of this process are given as the following:

First, G(s) is factorized into two parts:

The Bode’s diagrams of |G0(s)| are as shown in Figure 6. A

rational approximation for |G0(s)| is found and given in the

following:

For comparison, the frequency response of this model is also plotted on the same figure.

On the basis of eq 55, the adjoint of G0(s) is

Because each cofactor of G0(s) has FOPDT dynamics, the

num-ber of excess poles for each (i.e., Nep_[G 0 ji

]) is one. By eq 16, the number of excess zeros for each element of Z(s), Nez_(z

i), is one.

Thus, the following simple transfer function is assigned to each of zi(s) to compensate for undesired poles in the determinant of G0(s).

Thus, the decoupler can be obtained by eq 20. The result is given in Table 3. By the given decoupler, the decoupled open-loop process Qˆ (s) is also given in Table 4. Notice that the Θ(s) ) Θ*_{(s) in this case. As required, the desired closed-loop}

trans-fer function G_R0(s) (see Table 4) together with the following are specified:

Notice that the above values are so chosen that the system can sustain its stability under one broken actuator.

By the resulting values ofτL,1/θ1*(14.9) andτL,2/θ2*(4.4),

the following [g_d,10 (s)]1 and [gd,2 0

(s)]1 are thus obtained by

making use of the data in Table 1:

and

The results of design are summarized in Tables 4 and 5. det{I + G(s)D(s)G_C(s)I˜i}) 0 (52)

det{I + GI˜iDG_C}) 0 (53)

Mi(s) ) - I˜iD(s)G_C(s)[I + G(s)I˜iD(s)G_C(s)]-1 (54)

G(s) )

[

]

[

]

[

]

[

]

Figure 6. Bode’s diagrams of|G0(s)| and φ(s) for Wood and Berry (WB)

(2× 2) process. φ(s) ) -123.58(1.67s + 1) (24.75s + 1)(8.61s2+ 4.52s + 1) (57) adj{G₀(s)})

[

]

Simulation results for a step set-point change and a unit-step disturbance input are given in Figures 7 and 8, respectively. The results are compared with those from the multivariable

Smith predictor of Wang et al.22 _{in Figures 7 and 8. The}

maximum singular values of M(s) in eq 46 of the two systems are compared in Figure 9.

Table 3. Models of Decouplers for 2× 2 Systems

process Wood and Berry Jerome and Ray

d11(s) -19.4(8.61s 2_{+ 4.52s + 1)} (14.4s + 1)(1.67s + 1) (0.88s + 1)(2.9s + 1) (4s2+ 6s + 1) d12(s} 18.9(8.61s 2_{+ 4.52s + 1) e}- 2s (21s + 1)(1.67s + 1) -0.5(0.88s + 1)(2.9s + 1) e- 2s (2s + 1)(3s + 1) d21(s) -6.6(8.61s 2_{+ 4.52s + 1) e}- 4s (10.9s + 1)(1.67s + 1) -0.33(0.88s + 1)(2.9s + 1) e- 3s (4s + 1)(5s + 1) d22(s) 12.8(8.61s 2_{+ 4.52s + 1)} (16.7s + 1)(1.67s + 1) (0.88s + 1)(2.9s + 1) (s2+ 1.5s + 1)

Table 4. Decoupled Open-Loop Processes, Desired Closed-Loop Transfer Functions, and Feedforward Compensators for 2× 2 Systems

process loop 1 loop 2

Wood and Berry (WB)

qˆi(s) -123.58 e - s (24.75s + 1) -123.58 e- 3s (24.75s + 1) gR,i 0 (s) 1 4s2+ 2.8s + 1 1 4s2+ 2.8s + 1 F(s) -0.008 091 9(24.75s + 1) 4s2+ 2.8s + 1 -0.008 091 9(24.75s + 1) 4s2+ 2.8s + 1

Jerome and Ray (JR)

qˆi(s) 0.835(-s + 1) (1.143s2+ 2.179s + 1)e - 2s 0.835(-s + 1) (1.143s2+ 2.179s + 1)e - 3s gR,i 0 (s) (-s + 1) (3.24s2+ 2.52s + 1) (-s + 1) (3.24s2+ 2.52s + 1) F(s) (1.143s2+ 2.179s + 1) 0.835(3.24s2+ 2.52s + 1) (1.143s2+ 2.179s + 1) 0.835(3.24s2+ 2.52s + 1)

Table 5. Resulting Controllers with Full Forms and Reduced PID Forms for 2× 2 Systems

process loop 1 loop 2

Wood and Berry (WB)

gc,i(s), Full form -65.96s

3_{- 147s}2_{- 30.58s - 1}

886.1s3+ 2447s2+ 164.4s

-250.1s3_{- 214s}2_{- 32.99s - 1}

2489s3+ 4032s2+ 287.7s

gc,i(s), Reduced PID form -0.0658

(

)(

)

(

)(

)

Jerome and Ray (JR)

gc,i(s), Full form 21.91s

4_{+ 55s}3_{+ 45.55s}2_{+ 13.76s + 1}

36.64s4+ 83.94s3+ 49.14s2+ 1.832s

31.98s4+ 76.6s3+ 58.95s2+ 15.87s + 1 65.33s4+ 134.3s3+ 71.61s2+ 2.629s

gc,i(s), Reduced PID form _6.459

(

11.83s+ 2.203s

)(

)

(

)(

)

Example 2. Consider the Jerome and Ray15_{2× 2 process.}

The transfer function matrices of this process are given as the following:

We assume that the disturbance models are

First, G(s) is factorized into two parts:

By the proposed design procedure, the inverse-based controllers and the two-element controllers are given in Tables 4 and 5. The simulation results for a unit-step set-point change and a unit-step disturbance input are shown in Figures 10 and 11, respectively. In addition, the maximum singular values of M(s) at different frequencies are given in Figure 12. It can be seen that the 2-df MV control has good performance and robustness. Example 3. The Tyreus29 ₃ × 3 process as follows is

considered.

Figure 7. Set-point tracking responses for WB (2× 2) process.

Figure 8. Load rejecting responses for WB (2× 2) process.

G(s) )

[

]

[

]

Figure 9. The maximum singular values of M(s) for WB (2× 2) process.

G₀(s) )

[

]

[

]

Assume the dynamic of disturbance is equal to the first column of G(s), and that is

(i) First, partition G(s) into two parts. They are as follows:

(ii) Then, approximated models of |G0(s)| and G0 ij_{(s) are}

obtained to give the discrepancy indices at a level of 0.2. The results are as follows:

Figure 10. Set-point tracking responses for Jerome and Ray (JR) (2× 2) process.

Figure 11. Load rejecting responses for JR (2× 2) process.

G(s))

[

]

[

]

[

]

Figure 12. Maximum singular values of M(s) for JR (2× 2) process.

Θ(s) )

[

]

[

]

Figure 13. Set-point tracking responses for Tyreus (3× 3) process.

Table 6. Desired Closed-Loop Transfer Functions, Resulting Controllers with Full Forms, and Reduced PID Forms for 3× 3 Systems

process Tyreus subsystem of Alatiqi and Luyben

Loop 1 gR,1 0 (s) 1 25s2+ 7s + 1 1 25s2+ 7s + 1 gc,1(s), Full form 83.45s 3_{+ 187.4s}2_{+ 32.11s + 1} 490s3+ 1366s2+ 20.84s 295s3+ 405.6s2+ 53.07s + 1 327.2s3+ 560s2+ 26.48s

gc,1(s), Reduced PID form 1.46

(

)(

)

(

)(

)

gc,2(s), Reduced PID form 0.2623

(

)(

)

(

)(

)

gc,3(s), Reduced PID form _0.0731

(

11.24s+ 0.7137s

)(

)

(

)(

)

(iii) The compensator Z(s) is used to rearrange the poles-zeros of|G0(s)|, and the resulting Qˆ(s) becomes

(iv) The desired set-point responses are given in Table 6.

(v) The following indices resulted, D₂₁d ) 2.71%, D₃₁d ) 4.02%, D₁₂d ) 16.83%, D₃₂d ) 18.29%, D₁₃d ) 16.0%, and D₂₃d ) 0.35%; and discrepancy indices, p,1) 10.22%, p,1) 7.97%,

and p,1) 5.78%.

(vi) To satisfy robust stability, the value of|h|i*for each loop

is thus selected as 1.3, and gC,i and reduced

proportional-integral-derivative (PID) controllers are obtained as shown in Table 6.

Simulation results for a unit-step change in set point are given in Figure 13. The responses for load rejection are also shown in Figure 14.

Example 4. Consider another subsystem of the 4× 4 process of Alatiqi and Luyben30_{as the following:}

Notice that it is assumed that the disturbance enters along with

u3.

In the same way, the 2-df controllers are designed by the proposed design methods for this case. The desired set-point tracking responses are selected so that the decoupling perfor-mance indices are 0.2. Then, the inverse-based controllers are synthesized to have|h|1*) 1.3 and |h|2*) |h|3*) 1.4. The

results are summarized in Table 6. Simulation results for a unit-step change in set point and for a unit-unit-step change in load are given in Figures 15 and 16, respectively. The results show that performances of the system are superior to other designs or even better, especially in disturbance rejection.

5. Conclusion

In this paper, a 2-df decoupling multivariable control system and method of design have been proposed. It is capable of dealing with both servo tracking and disturbance rejection. The method to decouple loop interactions is based on fundamental linear algebra without an explicit inverse of the process transfer function. For disturbance rejection, decoupling controllers for a decoupled open-loop process are designed using a proposed new method of synthesis. The achievable set-point tracking response can be easily defined by making use of the resulting loop decoupling. Model-following and feedforward controllers are thus designed based on the decoupled process. The system robust stability can be tuned by selecting|h|i*while synthesizing

a diagonal element, GC(s), in the main loop. The selection of

|h|i*can be made by making use of two proposed measures for

decoupling and model reductions. Robustness to the integrity of the system is also included to take account of sensor and actuator failures. Furthermore, the proposed method incorporates dead-time compensation directly in the design. Not only can two objectives be achieved simultaneously while maintaining the minimum loop interactions and desired system robustness, but also their design can be carried out separately. Several

Gˆ₀11(s) ) 30.13 e -0.2504s (29s + 1)(14.97s + 1)(1.507s + 1) (70) Gˆ₀21(s) ) -16.21(1645s + 1) e-4.53s (239.4s2+ 24.75s + 1)(382.7s + 1)(33.80s + 1) (71) Gˆ₀31(s) )14.45(24.78s + 1) e -2.232s (415.4s + 1)(2.766s + 1)3 (72) Gˆ₀12(s) ) 1.09 e -6.139s (21.06s + 1)(0.5274s + 1)(3.264s + 1) (73) Gˆ₀22(s) ) 17.25(20.31s + 1) (62.03s + 1)(34.80s + 1)(5.049s + 1) (74) Gˆ₀32(s) ) 4.849 (64.75s + 1)(0.8686s + 1)(2.238s + 1) (75) Gˆ₀13(s) ) -0.1071(-33.45s + 1) e -6.694s (22.65s + 1)2(1.117s + 1)(5.277s + 1) (76) Gˆ₀23(s) ) -20.48(7.619s + 1) e-2.162s (42.78s + 1)(42.78s + 1)(11.29s + 1)(11.27s + 1) (77) Gˆ₀33(s) ) 0.5485(418.6s + 1) e-0.2888s (66.91s + 1)(345.4s + 1)(2.980s + 1)(1.852s + 1) (78) qˆ₁) 54.77 e -0.97s (24.64s + 1) (79) qˆ₂) 54.77 e -0.68s (24.64s + 1)(1.171s + 1) (80) qˆ₃) 54.77 e -1.85s (24.64s + 1) (81)

Figure 14. Load rejecting responses for Tyreus (3× 3) process.

G(s))

[

]

[

]

examples of different dimensions have been used to illustrate the proposed method and its effectiveness in obtaining good performance and robustness.

Acknowledgment

The work is supported by the National Science Council of Taiwan (NSC-93-2214-E-002-008).

Literature Cited

(1) Luyben, W. L. Simple method for tuning SISO controllers in multivariable systems. Ind. Eng. Chem. Process Des. DeV. 1986, 25, 654.

(2) Yu, C. C.; Luyben, W. L. Design of multiloop SISO controllers in multivariable processes. Ind. Eng. Chem. Fundam. 1986, 25, 498.

(3) Shen, S. H.; Yu, C. C. Use of relay-feedback test for automatic tuning of multivariable systems. AIChE J. 1994, 40, 627.

(4) Bao, J.; Lee, P. L.; Wang, F. Y.; Zhou, W. B.; Samyudia, Y. A New Approach to Decentralized Process Control Using Passivity and Sector Stability Conditions. Chem. Eng. Commun. 2000, 182, 213.

(5) Bao, J.; McLellan, P. J.; Forbes, J. F. A Passivity-Based Analysis for Decentralized Integral Controllability. Automatica 2002, 38, 243.

(6) Huang, H. P.; Jeng, J. C.; Chiang, C. H.; Pan, W. A direct method for multi-loop PI/PID controller design. J. Process Control 2003, 13, 769. (7) Niederlinski, A. A Heuristic Approach to the Design of Linear Multivariable Interacting Control Systems. Automatica 1971, 7, 691.

(8) Rosenbrock, H. H. Computer-Aided Control System Design; Aca-demic Press: New York, 1974.

(9) MacFarlane, A. G. J.; Kouvaritakis, B. A Design Technique for Linear Multivariable Feedback Systems. Int. J. Control 1977, 25, 837.

(10) Horowitz, I. Quantitative Synthesis of Uncertain Multiple Input-Output Feedback Systems. Int. J. Control 1979, 30, 81.

(11) Goodwin, G. C.; Ramadge, P. J.; Caines, P. E. Discrete Time Multivariable Adaptive Control. IEEE Trans. Autom. Control 1980,

AC-25, 449.

(12) Doyle, J. C.; Stein, G. Multivariable Feedback Design: Concepts for a Classical/Modern Synthesis. IEEE Trans. Autom. Control 1981,

AC-26, 4.

(13) Sinha P. K. MultiVariable control: An introduction; Marcel Dekker: New York, 1984.

(14) Arulalan, G. R.; Deshpande, P. B. New Algorithm for Multivariable Control. Hydrocarbon Process. 1986, 65, 51.

(15) Jerome, N. F.; Ray, W. H. High Performance Multivariable Strategies for Systems Having Time Delay. AIChE J. 1986, 32, 914.

(16) Stein, G.; Athans, M. The LQG/LTR Procedure for Multivariable Feedback Control Design. IEEE Trans. Autom. control 1987, AC-32, 105. (17) Agamennoni, O. E.; Rotstein, H.; Desages, A. C.; Romagnoli, J. A. Robust Controller Design Methodology for Multivariable Chemical Processes: Structured Perturbations. Chem. Eng. Sci. 1989, 44, 2597.

(18) Maciejowski, J. M. MultiVariable Feedback Design; Addison-Wesley: New York, 1989.

Figure 15. Set-point tracking responses for subsystem of Alatiqi and Luyben (AL) process.

(19) Figueroa, J. L.; Agamennoni, O. E.; Desages, A. C.; Romagnoli, J. A. Robust Multivariable Controller Design Methodology: Stability and Performance Requirements. Chem. Eng. Sci. 1991, 46, 1299.

(20) Skogestad, S.; Postlethwaite, I. MultiVariable feedback control; John Wiley & Sons: New York, 1996.

(21) Wang, Q. G.; Zou, B.; Lee, T. H.; Bi, Q. Auto-tuning of multivariable PID controllers from decentralized relay feedback. Automatica

1997, 33, 319.

(22) Wang, Q. G.; Zou, B.; Zhang, Y. Decoupling Smith Predictor Design for Multivariable Systems with Multiple Time Delays. Inst. Chem.

Eng. 2000, 78, 565.

(23) Wang, Q. G.; Zhang, Y.; Chiu, M. S. Noninteracting control design for multivariable industrial processes. J. Process Control 2003, 13, 253. (24) Åstro¨m, K. J.; Ha¨gglund, T. PID Controllers: Theory, Design, and

Tuning; ISA Press: Research Triangle Park, NC, 1995; p 284.

(25) Tian, Y. C.; Gao, F. Double-controller scheme for separating load rejection from set-point tracking. Trans. Inst. Chem. Eng. 1998, 76, 445.

(26) Tian, Y. C.; Gao, F. Control of Integrator Processes with Dominant Time Delay. Ind. Eng. Chem. Res. 1999, 38, 2979.

(27) Chen, D.; Seborg, D. E. PI/PID controller design based on direct synthesis and disturbance rejection. Ind. Eng. Chem. Res. 2002, 41, 4807. (28) Wood, R. K.; Berry, M. W. Terminal composition control of a binary distillation column. Chem. Eng. Sci. 1973, 28, 1707.

(29) Tyreus, B. D. Paper presented at the Lehigh University Distillation Control Short Course, Lehigh University, Bethlehem, PA, 1982.

(30) Alatiqi, I. M.; Luyben, W. L. Control of a complex side stream column/stripper distillation configuration. Ind. Eng. Chem. Process Des.

DeV. 1986, 25, 762.

ReceiVed for reView October 12, 2005 ReVised manuscript receiVed January 13, 2006 Accepted February 6, 2006