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Part III Decentralized control for MIMO processes 1

PART III DECENTRALIZED CONTROL FOR MIMO PROCESSES

3.1 PRELIMINARY CONSIDERATIONS 3.1.1 GENERAL CONCEPTS From RGA analysis, we know that:

1. The difference from single-loop behavior increases as the interaction, as indicated by the magnitude of the off-diagonal gains, increases.

2. Whether the manipulated variables move a greater or lesser amount in multivariable control compared with single-loop depends on the direction of the set point changes. Note that this behavior is different from a single-loop case, in which only the input magnitude is important.

3. System with one-way interaction, deviates less from single-loop behavior than the systems with moderate to strong transmission interaction.

Consequently, the major differences between single-loop and interactive steady-state multiloop systems are:

1. The values of the manipulated variables that satisfy the desired controlled variables must be determined simultaneously.

2. Differences between single-variable and multivariable behavior increase as the transmission interaction increases.

3. The sensitivity of the adjustments in manipulated variables to model changes can be much greater in multiloop systems than in single-loop systems.

There are two basic multivariable control approaches. The first is a straightforward extension of single-loop control to many controlled variables in a single process. This is termed multiloop control and has been applied with success for many decades. The second main category is coordinated or centralized control, in which a single control algorithm uses all measurements to calculate all manipulated variables simultaneously.

For decentralized controller design, we have to consider the effects of interaction on multivariable system behavior, assuming that the process has a controllable input-output selection. The goal for decentralized controller design is to find out how the responses of a control system are influenced by interaction. The design of multiple single-loop controllers for multivariable systems proceeds in two stages:

1. Judicious choice of loop pairing

2. Controller tuning for each individual loop

After control configuration (loop paring) been selected, the first step is to derive the transfer function for the multiloop feedback control system and determine the main differences from single-loop control.

3.1.2 LOOP DECOMPOSITION

KNU/EECS/ELEC835001 Dr. Kalyana Veluvolu


Suppose that decentralized controllers were to be implemented on the system in Figure 3.1.1.

Fig. 3.1.1 Decentralized control for MIMO processes

For a given input and output pairing, the control action in a specific loop can be further related to set-point change and its output to form the feedback control. For instance, when the decentralized system is diagonally paired, then for any )()( sysu ll − loop we have

( ) ( )[ ( ) ( )]l cl l lu s g s r s y s= − (3.1.1)

For an arbitrary pairing scheme, focusing on the ( ) ( )j iu s y s− loop and assuming that only this loop undergoes set-point changes, we thus have

1,

( ) ( ) ( ) ( ) ( ) ( )n

i ij j il cl ll l j

y s g s u s g s g s y s= ≠

= − ∑ (3.1.2)

Successively using eq. (3.1.2) to express all the outputs in terms of )(su j , we find the transmittance between the control action )(su j and its own output to be

)()()()()( susasusgsy jijjiji += (3.1.3)

and the perturbations caused by )(su j to other loops to be

ksusdsy jkjk ∀= ),()()( (3.1.4)

The detailed expression for )(saij and )(sdij defined in eqs (3.1.3) and (3.1.4), respectively, can be readily obtained using the signal-flow graph technique.

Note that )(saij in eq. (3.1.3) represents the additional dynamics in the )()( sysu ij − loop resulting from other control loops, and that eq. (3.1.4) separates it from the interaction-free transfer function of the loop. Therefore, )(saij can be defined as the absolute interaction.

A multivariable control system composed of interacting individual loops with hidden structure, as shown in Fig. 3.1.1 can be further decomposed into separate SISO loops with interactions explicitly shown. Figure 3.1.2 shows the physical structure of a particular loop.



Fig 3.1.2 Interacting Individual Loops for Decentralized Control

Example 3.1.1 Consider a general diagonally paired 22× system as shown in Fig. 3.1.3.

Fig. 3.1.3 TITO System

The overall control action and the perturbation transmittances in loop 11 yu − can be shown to be

2 12 21111

1 2 22

( ) ( ) ( )( ) ( )( ) 1 ( ) ( )

c

c

g s g s g sy s g su s g s g s

= −+

(3.1.5a)

and

2 2121

1 2 22

( ) ( )( )( ) 1 ( ) ( )c

y s g sd su s g s g s

= =+

(3.1.5b)

respectively.

The absolute interaction in loop 1 can be found from eq. (7) as

2 12 2111

2 22

( ) ( ) ( )( )1 ( ) ( )c

c

g s g s g sa sg s g s

= −+

(3.1.5c)



Outputs in response to changes in ul are shown in Fig. 3.1.4 with interaction, interaction-free, and induced perturbation terms structurally separated out:

• The perturbation from the first loop to the second loop is the only source of interaction in the original loop (first loop).

• The perturbation from loop 1 to loop 2 is amplified by the cross transfer function of the process and controller 2 and then fed back to loop 1 as interaction, i.e.

11 2 12 21( ) ( ) ( ) ( )ca s g s g s d s= − (3.1.5d)

• The interaction in loop 1 depends on controller 2, the effects of tuning controller 2 has effect on loop 1 as a result of interaction. The controller design also depends on the tuning the other controller.

Fig. 3.1.4 TITO System Decomposition

A similar analysis shows that for loop 2

1 12 2122 1 12 12

1 11

( ) ( ) ( )( ) ( ) ( ) ( )1 ( ) ( )c

cc

g s g s g sa s g s g s d sg s g s

= − = −+

and the perturbation caused by controller 2 to loop 1 is

1212

1 11

( )( )1 ( ) ( )c

g sd sg s g s

=+

By decomposition, the control loop could be considered a single-loop system; however, changes in any manipulated variable caused by the controller would affect other output variables because of interaction.

The transfer functions can be determined from block diagram manipulation.



1 11 1 2 11 22 12 211

1

2 22 1 2 11 22 12 212

2

2 121

2

1 212

1

( ) ( ) ( ) ( )[ ( ) ( ) ( ) ( )]( )( ) ( )

( ) ( ) ( ) ( )[ ( ) ( ) ( ) ( )]( )( ) ( )

( ) ( )( )( ) ( )

( ) ( )( )( ) ( )

c c c

CL

c c c

CL

c

CL

c

CL

g s g s g s g s g s g s g s g sy sr s g s

g s g s g s g s g s g s g s g sy sr s g s

g s g sy sr s g s

g s g sy sr s g s

+ −=

+ −=

=

=

(3.1.7b)

with the characteristic expression ( )CLg s given by

[ ]111 12

121 22

1 11 2 22 1 2 11 22 12 21

( ) det ( ) ( )

( ) 0( ) ( )det

0 ( )( ) ( )

1 ( ) ( ) ( ) ( ) ( ) ( )[ ( ) ( ) ( ) ( )]

CL C

c

c

c c c c

g s I G s G s

g sg s g sI

g sg s g s

g s g s g s g s g s g s g s g s g s g s

= +

⎡ ⎤⎛ ⎞⎛ ⎞= +⎢ ⎥⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠⎣ ⎦= + + + −

(3.1.6)

When both interaction terms 12 ( )g s and 21( )g s are nonzero, the dynamic response of a single-loop controller between ( ) and ( )i iy s u s depends on all terms in the closed-loop transfer function. As a result, the stability and performance of loop 1 depend on the tuning of loop 2. By a similar argument, the stability and performance of loop 2 depend on the tuning of loop 1. Therefore, the two controllers must be tuned simultaneously to achieved, desired stability and performance.

Owing to inner interactions in a multivariable controlled system, a change of controller parameters of one loop will affect the performance of other ones. The design of SISO controllers for a MIMO process is more difficult because of these interactions. In particular, the design of SISO controllers for highly coupled multivariable processes often leads to poor performance because of a bad choice of manipulated variables, poor specifications and/or poor tuning of the controllers. Many design methods have been reported in literature, which may be roughly divided into four categories:

(1) Detuning methods;

(2) Sequential loop closing methods;

(3) Independent methods;

(4) Simultaneous equation solving methods.

3.2 DETUNING METHODS 3.2.1 RGA BASED DETUNING METHOD A few methods have been proposed for estimating the tuning for multiloop systems without the time-consuming iterations associated with trial and error or the computer computations associated with the optimization approach. The goal of these methods is to provide initial tuning constants that are much closer than single-loop tuning constants to the "best" multiloop values. Naturally,



fine-tuning based on plant experience is still required. Unfortunately, there is no generally accepted method for quickly estimating multiloop tuning. The method given here is selected because it provides insight and introduces some key process-related issues. It also provides a useful correlation for many 2 2× systems; however, it is not easily extended to higher-order systems.

For systems with significant interactions, the readjustment of the tuning can be difficult. To facilitate the tuning, we extend the RGA to include frequency dependant terms by replacing the steady-state gain with corresponding transfer function. Thus the Dynamic RGA (DRGA) is given as:

12 21

11 22

1( ) ( ) ( )1( ) ( )

s g s g sg s g s

λ =−

(3.2.1)

The characteristic equation of the TITO system can be rewritten as:

1 11 2 22

12 211 2 11 22

11 22

1 2 11 221 11 2 22

( ) 1 ( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( ) 1( ) ( )

( ) ( ) ( ) ( )1 ( ) ( ) ( ) ( )( )

CL c c

c c

c cc c

g s g s g s g s g s

g s g sg s gg s g s g sg s g s

g s g s g s g sg s g s g s g ssλ

= + + +

⎛ ⎞+ −⎜ ⎟

⎝ ⎠

= + + +

(3.2.2)

The method takes advantage of simplifications to determine the tuning for three cases of limiting process dynamics for 2 2× systems with PI multiloop controllers. The general approach is to establish how much the PI controller tuning must be changed from single-loop values when applied in a multiloop system.

The basis of the analysis is the closed-loop characteristic expression (3.2.2) divided by 2 221 ( ) ( )cg s g s+ , which does not change the stability limit:

1 2 11 222 22 1 11

2 22

2 22

1 112 22

( ) ( ) ( ) ( )1 ( ) ( ) ( ) ( )( )

1 ( ) ( )( ) ( )1

( )1 ( ) ( )1 ( ) ( )

c cc c

CLc

c

cc

g s g s g s g sg s g s g s g sg s

g s g sg s g s

sg s g sg s g s

λ

λ

+ + +=

+

⎛ ⎞+⎜ ⎟⎜ ⎟= +

+⎜ ⎟⎜ ⎟⎝ ⎠

(3.2.3)

The closed-loop characteristic expression given by equation (3.2.3) determines the stability of the control system. To evaluate potential simplifications, the relative importance of each term must be determined at the critical frequency of the loop. Since the approach is based on stability analysis, which considers only the denominator of the closed-loop transfer function, the same tuning is obtained for all disturbances and set point changes. The method considers three limiting cases for tuning loop

1. loop 1 much faster than loop 2



2. loop 1 much slower than loop 2; and

3. both loops having the same dynamics.

The following analysis considers loop 1, but the same results can be obtained for loop 2 by simply transposing the subscripts.

Loop 1 Much Faster Than Loop 2.

When the loop 1 process is much faster than loop 2, the term 2 22( ) ( )cg j g jω ω is very small at the loop 1 critical frequency because of the tendency of processes to have amplitude ratios that decrease rapidly after the comer frequency. Assuming that 11λ is not a strong function of frequency, as is most often true,

2 22 11

2 22

1 ( ) ( ) 1.01 ( ) ( )

c

c

g j g jg j g j

ω ω λω ω

+=

+

which gives

1 11( ) 1 ( ) ( )CL cg j g j g jω ω ω≈ + (3.2.4)

Therefore, the very fast loop 1 in this case can be tuned like a single-loop controller without interaction. This result confirms a qualitative argument in which we would consider the interaction from the slow loop to be a slow disturbance to the very fast loop 1, which could be tuned using single-loop methods.

Loop 1 Much Slower Than Loop 2

When loop 1 is much slower, the term for the fast loop, 2 22( ) ( )cg j g jω ω , would have a very large magnitude at the critical frequency of loop 1, because the amplitude ratio of the integral mode in

2 22( ) ( )cg j g jω ω will have a very high value at a frequency much less than the loop 2 critical frequency. Therefore, 2 22( ) ( ) 1.0cg j g jω ω >> , which leads to the following simplification in the characteristic equation.

2 221 11

11 2 22

111

11

111

11

( ) ( )( ) 1 ( ) ( )( ) ( ) ( )

( )( )( )( )( )

cCL c

c

c

c

g j g jg j g j g jj g j g j

g jg jj

g jg j

ω ωω ω ωλ ω ω ω

ωωλ ω

ωωλ

≈ +

≈

≈

(3.2.5)

As noted, the steady-state relative gain has been used as an approximation for the frequency dependent relative gain. In this case, the gain of the process "seen" by the controller 1 in the multiloop system is changed by 111/λ , I from the single-loop gain ( 11K ). Therefore, the controller gain has to be adjusted by approximately the inverse, 11λ , to maintain the proper stability margin. Since the phase lag is not affected, the integral time is unchanged. Again, this result seems consistent with a qualitative argument that a very fast associated loop would "become part of the process" and affect only the closed-loop process gain.



Since the relative gain can take a wide range of values for different processes, the implication of equation (3.2.5) is that multiloop controller systems with fast and slow processes could have controller gains that change by a large amount from single-loop to multiloop. This situation might be necessary but would generally be undesirable.

Loops 1 and 2 Have the Similar Dynamics For the loops have similar dynamics, define

1 11 1 2 22 2, ,c cg g g gΛ = Λ =

To simplify the stability analysis as, let 1 2Λ = Λ = Λ , the system stability can be determined by the root of following equation,

2 ( )1 2 ( ) 0ssλ

Λ+ Λ + = (3.2.6)

in order to determine the closed-loop stability, rewrite Equation (3.2.6) as 2

2 2

( ) ( ) ( )1 2 ( ) 1 1s s ssλ λ λ λ λ λ λ

⎛ ⎞⎛ ⎞Λ Λ Λ+ Λ + = + +⎜ ⎟⎜ ⎟

+ − − −⎝ ⎠⎝ ⎠

and consider the following two scenarios:

a) 1.0λ > : Since λ is a constant and greater than 1, ultimate frequency is unaffected by the interaction, that is

1 1u I uω ω=

as the factor with 2λ λ− − will always determine the stability limit of the interacting system and the ultimate gain 1u Ik can be calculated from

1

21 11 u Iu Ik g ωλ λ λ= − −

1

2

111

u Iu Ikg ω

λ λ λ− −= (3.2.7)

By the definition of gain margin

1

111

1

u

ukg

ω

= (3.2.8)

Substituting (3.2.8) into (3.2.7), we obtain

( )21 1u I uk kλ λ λ= − − (3.2.9)

b) 1.0λ < : The factor with 2λ λ λ+ − will determine the stability limit of the interacting system and the gain margin 1u IK can be calculated from



1

2

111

u Iu IKg ω

λ λ λ+ −= (3.2.10)

Since 1.0λ < , the term 2λ λ λ+ − is a complex constant. This term affects both phase angle and gain. There is no simple method to relate 1u IK with 1uK . The approximated formulation for FOPDT process can be expressed by

( )1

1 1

11.5tan 11u I u

λω ωπ

−⎛ ⎞−⎜ ⎟= −⎜ ⎟⎜ ⎟⎝ ⎠

(3.2.11)

( )1

1 1

1tan 11u I uk kλλ

π

−⎛ ⎞−⎜ ⎟= −⎜ ⎟

⎜ ⎟⎝ ⎠

(3.2.12)

thus for 1.0λ < , not only the controller gain reduced, but the reset time is increased as well as due to the interaction.

To simplify the calculation, (3.2.11) and (3.2.12) in the region of 0.5 1.0λ< < can be approximated, using least square method, by the following functions:

( )( )

1 1

1 1

0.85 0.1

0.62 0.3u I u

u I uk k

ω λ ω

λ

⎧ = +⎪⎨

= +⎪⎩ (3.2.13)

The correlation of 1 1and u I u u I uk kω ω− − is shown as in Fig. 3.3, where solid line denote the original relation from Equations (3.2.11) and (3.2.12), dotted line represent the Equation (3.2.13)

Fig. 3.2.1 indicates that the linear relation directed from ERGA can be applied to detune the controller parameters and the approximate Equation (3.2.13) provides a more convenient settings compare with Equations (3.2.11) and (3.2.12).



The detuning rules for all three cases are summarized in Table 3.2.1, which provide a guideline for rough tuning of TITO decentralized controller parameters. The method provided in this chapter takes advantage of simplifications to determine the tuning and can be easily accepted by field control engineers. However, fine-tuning based on plant experience or advanced control design methods is still required.

Table 3.1 Effects of interaction on loop 1 Relation Speed of

2 22( ) ( )cG j G jω ω relative to

1 11( ) ( )cG j G jω ω

λ

1u IK and 1uK 1u Iω and 1uω

Compare with single loop response

Much faster >1 1 1u I uK K=

1 1u I uω ω= More sluggish

Much slower >1 1 1u I uK K=

1 1u I uω ω= More sluggish

Almost equal >1 ( )21 1u I uK Kλ λ λ= − − 1 1u I uω ω= More sluggish

Much faster <1 1 1u I uK Kλ= 1 1u I uω ω= More Oscillatory

Much slower <1 1 1u I uK K=

1 1u I uω ω= More Oscillatory

Almost equal <1 ( )1 10.62 0.3u I uK Kφ= + ( )1 10.85 0.1u I uω λ ω= + More Oscillatory

Example 3.2.1. Multiple Loop Tuning for the Wood and Berry Distillation Column

The transfer function:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−

+

+−

+= −−

−−

14.144.19

19.106.6

10.219.18

17.168.12

)( 37

3

se

se

se

se

sG ss

ss

the RGA and NI index

⎥⎦

⎤⎢⎣

⎡−

−=Λ

0.20.10.10.2

and 0NI >

so that 1-1/2-2 loop pairing is indicated.

By tuning the individual loops with the other loops open,



1 112.8( ) ( )

16.7 1

sey s u ss

−

=+

3

2 219.4( ) ( )

14.4 1

sey s u ss

−−=

+

Ziegler-Nichols tuning leads to the following parameters:

Loop 1 Loop 2 80.01 =cK 15.02 −=cK 26.31 =Iτ 35.92 =Iτ

With the other loop open, these tuning parameters lead to the quite good individual dosed-loop responses to set-point changes.

If we close both loops and make the same set-point change in y1 (no set-point change is desired for y2) using these tuning parameters, we see that the process is nearly unstable due to loop interactions and oscillates continuously, as shown below:



Thus, we must detune both loops. Since the dynamic of the two loops are almost the same and λ=2, the original controller gains 1cK and 2cK must be detuned by

( )2 (2 2) 0.59λ λ λ− − = − ≈

The new parameters are:

Loop 1 Loop 2 47.01 =cK 088.02 −=cK 26.31 =Iτ 35.92 =Iτ

The process is stable and has better closed-loop response. They could be further improved by continued adjust of parameters.

3.2.2 BIGGEST LOG MODULUS TUNING Since the multivariable BLT method is directly analogous to the classical Nyquist stability criterion method for SISO systems. We first brief review the SISO case:

The closed-loop characteristic equation for a SISO system is

1 ( ) ( ) 0cg s g s+ = (3.2.14)



Where, g is the open-loop process transfer function and gc is the feedback controller transfer function. A Nyquist (or Bode or Nichols) plot of ( ) ( )cg i g iω ω is made as the frequency ω goes from zero to infinity. The number of encirclements of the (-1, 0) point is equal to the number of roots of the closed-loop characteristic equations that lie in the right half of the s plane if the open-loop system is stable. Thus, if the (-1, 0) point is encircled, the closed-loop system is unstable.

The farther away from the (-1, 0) point, the system is more stable. One commonly used measure of the distance of the ( ) ( )cg i g iω ω contour from the (-1, 0) point is the maximum closed-loop log modulus (Lc

max). The closed-loop log modulus is the magnitude of the closed-loop servo-transfer function.

20 log1

c

c

ggLcgg

=+

(3.2.15)

A commonly used specification is +2 dB for Lcmax. A value of the controller gain (Kc) is selected,

and L, is plotted as a function of frequency. The maximum value of Lc is determined, and if it is less than +2 dB, the gain is increased. This procedure is repeated until the biggest log modulus for all frequencies is +2 dB. A Nichols chart can be used alternatively to transform ( ) ( )cg j g jω ω information into Lc results.

For the n × n multivariable process, with the transfer function elements in ( )G s are of the form

1

2 3 4

( 1)( 1)( 1)( 1)

ijD sij ij

ijij ij ij

K s eg

s s sτ

τ τ τ

−+=

+ + + (3.2.17)

The feedback controller matrix ( )cG s is diagonal since we are using n SISO controllers

1

2

3

( ) 0 0 ..... 00 ( ) 0 ... 00 0 ( ) ... 0

( ). . . .. . . .0 0 0 ... ( )

c

c

cc

cn

g sg s

g sG s

g s

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(3.2.18)

where,

1( ) (1 )ci ciIi

g s Ksτ

= + (3.2.19)

The design procedure is given as follows:

1. Calculate the Ziegler-Nichols setting for each individual loop. The ultimate gain Ku and ultimate frequency uω of each diagonal transfer function ( )iig s are calculated in the classical SISO way. (Review the background material for determine ultimate gain and ultimate frequency of a transfer function: To find ultimate gain Ku and ultimate frequency

uω of each loop, a initial value of the frequency is guessed. The phase angle is calculated,



and frequency is varied until the phase angle is equal to 180°− . This frequency is uω . The ultimate gain is the reciprocal of the real part of Gii at uω ).

2. Specify a factor F, it is typical vary from 2 to 5. The gains of all feedback controllers (Kc) are calculated by dividing the Ziegler-Nichols gain (KZN) by the factor ,F

iZNci

KK

F= (3.2.20)

where,

2.2

i

i

uZN

KK = (3.2.21)

The integral constant ( Iτ ) of all controllers are calculated by multiplying the Ziegler-Nichols reset times ( ZNτ ) by the same factor ,F

iIi ZNFτ τ= (3.2.22)

where,

21.2iZN

ui

πτω

= (3.2.23)

The F factor can be considered a “detuning” factor which is applied to all loops. The larger the value of F, the more stable the system will be but the more sluggish will be the set-point and load responses. The method outlined below yields a value of F which gives a reasonable compromise between stability and performance in multivariable systems.

3. Fro the closed-loop system -1( ) [ ] ( )c cY s I GG GG R s= + (3.2.24)

Since the inverse of a matrix has the determinant of the matrix in the denominator, the closed-loop characteristic equation of the multivariable system is the scalar equation.

det (I + ) = 0 cGG

Define a new function W(s),

( ) 1 det( ( ) ( ))cW s I G s G s= − + + (3.2.25)

The quantity W/(l + W) will be similar to the closed-loop servo-transfer function for a SISO loop /(1 )c cgg gg+ . Therefore, based on intuition and empirical grounds, a multivariable closed-loop log modulus Lcm are defined.

20 log1

=+cmWL

W (3.2.26)



The proposed tuning method is based on varying the factor F until the “biggest log modulus” max( )cmL is equal to some reasonable number. Thus, the proposed tuning method is called the

“biggest log modulus tuning” (BLT).

To determine the criterion of max( )cmL for different dimensional processes, recall that +2dB is a good choice which gives reasonable time domain responses for set-point and load disturbances.

For MIMO processes, the best tuning criterion for this method is

max( ) 2 cmL n= (3.2.27)

For n= 1, this reduces to the classical +2dB criterion. For a 4 × 4 system, a max( )cmL of 8 should be used. These empirical findings suggest that the higher the order of the system, the more under damped the closed-loop system must be to achieve reasonable responses. Note that this procedure guarantees that the system is stable with all controllers on automatic and that each individual loop will be stable if all others are on manual. Thus, a portion of the integrity question is automatically answered.

The method weighs each loop equally; i.e., each loop is detuned by the same factor F. If it is more important to keep tighter control of some variable than others, the method can be easily modified by using different weighting factors for different controlled variables. The less-important loop could be detuned more than the more-important loop.

The simplicity of this method is its major advantage. Nevertheless, the disadvantage results from the fact that loop performance and stability can not be clearly defined through the detuning procedures. Generally, this class method only provides reasonable preliminary controller settings with guaranteed closed-loop stability.

Example 3.2.1. Wood and Berry binary distillation column plant 3 8.1

7 3 3.4

12.8 18.9 3.8( ) ( )1 16.7 1 21 1 14.9 ( )( ) ( )6.6 19.4 4.9

1 10.9 1 14.4 1 13.2

s s s

Ds s s

B

e e eX s R ss s s F sX s S se e e

s s s

− − −

− − −

⎡ ⎤ ⎡ ⎤−⎢ ⎥ ⎢ ⎥⎡ ⎤ ⎡ ⎤+ + +⎢ ⎥ ⎢ ⎥= +⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦⎢ ⎥ ⎢ ⎥+ + +⎣ ⎦ ⎣ ⎦

.

where ( )DX s and ( )BX s are the overhead and bottom compositions of methanol, respectively; R is the reflux flow rate and S is the steam flow to the reboiler; F is the feed flow rate, a load disturbance. It is a typical TITO plant with strong interactions and significant time delays.

Using BLT method, each loop is also detuned by same factor 4. The resultant PI controllers are listed in the Table. The close loop responses for the controller is as shown in the Figure, where the unit set-points change in r1 at t=0 and r2 at t=800.

Controller for Wood and Berry

Controller Kc,ii ,i iiτ

Loop 1 0.375 8.29

Loop 2 -0.075 23.6



Close loop response for Wood and Berry by using BLT method

3.3 INDEPENDENT METHODS 3.3.1 STRUCTURE DECOMPOSITION The structure of decentralized control system is given as in Fig. 3.1.3.

The transfer function seen by controllers 1( )cg s and 2 ( )cg s are 1( )g s 2 ( )g s where 1( )g s and

2 ( )g s are given by:

12 21 21 11

22 2

12 21 12 22

11 1

( ) ( ) ( )( ) ( )1 ( ) ( )

( ) ( ) ( )( ) ( )1 ( ) ( )

c

c

c

c

g s g s g sg s g sg s g s

g s g s g sg s g sg s g s

= −+

= −+

(3.3.1)

The tuning of one controller depends on the other one controller. The system can then be separated into two SISO systems with a set-point change on one loop is seen as a disturbance by the other.

Different approximations can be used for 1( )g s and 2 ( )g s .

1. If the transfer functions of 11( )g s and 22 ( )g s contain no unstable zero or a delay shorter than )()( 2112 sLsL + , where L represents the process delay, a possible approximation at

frequencies lower than the cross-over frequency ( coω ) is:



12 211 11 22 2

22

12 212 22 11 1

11

( ) ( )( ) ( ) for ( ) ( ) 1( )

( ) ( )( ) ( ) for ( ) ( ) 1( )

c

c

g s g sg s g s g s g sg s

g s g sg s g s g s g sg s

= − >

= − > (3.3.2)

This makes the tuning easier since the transfer function seen by one controller is independent of the other controller. The process dynamics on the regulated variable depends primary on the dynamic of the manipulated variable where the set-point change occurred.

Fig. 3.3.1. Explicit Decomposed TITO Process

2. If the transfer functions of 11( )g s or 22 ( )g s contain an unstable zero or a delay longer than )()( 2112 sLsL + , the following approximation can be used.

12 211 11

22

12 212 22

11

( ) ( )( ) ( )

( ) ( )( ) ( )

g s g sg s g sK

g s g sg s g sK

= −

= − (3.3.3)

where 11K and 22K are, respectively, the gains of 11( )g s and 22 ( )g s .

3.3.2 GAIN AND PHASE MARGIN DESIGN After obtaining the equivalent processes, many SISO PID tuning methods can be used. Here, we introduce a gain and phase margin method. For an SOPDT model,

02

2 1

( )1

Lsbg s ea s a s

−=+ +

(3.3.4)

the standard PID controller is of the form



( ) ic p d

Kg s K K ss

= + +

which can be rewritten as 2 2

( ) d p ic

K s K s K As Bs Cg s ks s

+ + + += = (3.3.5)

where dA K k= , pB K k= and iC K k= .

By selecting 2 1, and 1A a B a C= = = in Equation (3.3.5), the open loop transfer function becomes

0( ) ( ) sLc

bg s g s k es

−= (3.3.6)

Denote the gain and phase margins as Am and Φm , and their crossover frequencies as ω p andωg , respectively, we have

arg ( ) ( )c g gg j g jω ω π⎡ ⎤ = −⎣ ⎦ (3.3.7a)

( ) ( ) 1m c g gA g j g jω ω = (3.3.7b)

( ) ( ) 1c p pg j g jω ω = (3.3.7c)

arg ( ) ( )m c p pg j g jπ ω ω⎡ ⎤Φ = + ⎣ ⎦ (3.3.7d)

Substitute Equation (3.3.6) into Equations (3.3.7a)-( 3.3.7d), we obtain

2g L πω = 0

gmA

kbω

=

0 pkb ω= 2m pLπ ωΦ = −

which results

1(1 )2m

mAπ

Φ = − (3.3.8a)

02 m

kA Lbπ

= (3.3.8b)

Some possible gain and phase margin selections are given in Table 3.3.1

Table 3.3.1: Typical Gain and Phase Margin Values

mA mΦ k

2 4π 04Lbπ



3 3π 06Lbπ

4 3 8π 08Lbπ

The PID parameters are given by

1

02

12

p

im

d

K aK

A LbK a

π⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(3.3.9)

Example 3.3.1. Example 3.2.1 Continued for independent design method

1. Find the equivalent transfer function 7

12 211 11

223 9

12 212 22

11

( ) ( ) 12.8 6.43(14.4 1)( ) ( )( ) 16.7 1 (21.0 1)(10.9 1)

( ) ( ) 19.4 9.75(16.7 1)( ) ( )( ) 14.4 1 (21.0 1)(10.9 1)

s s

s s

g s g s e s eg s g sg s s s s

g s g s e s eg s g sg s s s s

− −

− −

+= − = −

+ + +

− += − = +

+ + +

which are not FOPTD transfer functions, a model reduction is required which has been assigned as an exercise.

2. Apply gain and phase method with the gain margin of 3 and the phase margin of 3π for the approximated models, the parameters of the PID controllers can be obtained as:

1

0.1208( ) 0.4660cG ss

= +

2

0.0343( ) 0.2122cG ss

= − −

The closed-loop system responses for both detuning method (dotted line) and independent design method (solid line) are shown as Figure 3.3.2, where the unit set-points change in y1 (at t=0) and y2 (at t=200), and the unit load disturbance occurs at t=400.



Figure 3.3.2. Closed-loop responses for Example 3.3.1

It can be seen that the performance of both designs are satisfactory. With given gain and phase margin the independent design method results better set-point tracking while the detuning factor method has better performance on load disturbance rejection.

Example 3.3.2. An industrial-scale polymerization reactor,

0.2 0.4 0.4

1 1

0.2 0.4 0.42 2

22.89 11.64 4.234( ) ( )4.572 1 1.807 1 3.445 1 ( )( ) 4.689 5.80 ( ) 0.601

2.174 1 1.801 1 1.982 1

s s s

s s s

e e ey s u ss s s d sy s u se e e

s s s

− − −

− − −

− −⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎡ ⎤ ⎡ ⎤+ + += +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ −⎣ ⎦ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥⎢ ⎥ ⎢ ⎥+ + +⎣ ⎦ ⎣ ⎦

where 1y and 2y are two measurements representing the reactor condition; 1u and 2u are the set-points of two reactor feed flow loops; and load disturbance d is the purge flow rate of the reactor. It is easy to confirm that this process is not column diagonally dominant.

Following the same procedures as in Example 3.3.1, we obtain the parameters of the PID controllers as:

1

0.0094( ) 0.0148cG ss

= +

2

0.0163( ) 0.0238cG ss

= +

Figure 3.3.3 shows the closed-loop responses, where the unit set-points change in y1 (at t=0) and y2 (at t=120), and unit load disturbance occurs at t=240. The closed-loop system with detuning method is unstable, and not shown in the figure.



Figure 3.3.3. Closed-loop responses for Example 3.3.2

3.3.3 INTERNAL MODEL CONTROL (IMC) DESIGN METHOD

This method is an analytical design method for the multiloop PID controllers to give desired closed-loop responses by extending the generalized IMC–PID method for single input/single output (SISO) systems [34] to MIMO systems. In the multiloop feedback system, the closed-loop transfer function matrix H(s) is given by

1( ) ( ) ( ) ( ( ) ( )) ( ) ( ) ( )c cY s H s R s I G s G s G s G s R s−= = + (3.3.10)

where G(s) is the open-loop stable process, ( )cG s is the multiloop controller, and Y(s) and R(s) are the controlled variable vector and the set-point vector, respectively.

By the design strategy of the multiloop IMC controller, the desired closed loop response Ri of the ith loop is typically chosen by

( )( 1) i

i iii n

i i

y G sRr sλ

+= =+

(3.3.11)

Where, Gii+ is the nonminimum part of Gii,

iλ is an adjustable constant for system performance and stability, and

ni is chosen such that the IMC controller would be realizable.

The requirement of Gii+(0)= 1 is necessary for the controlled variable to track its set point.

Let the desired closed-loop response matrix ( )R s be



1 2( ) [ , , , ]nR s diag R R R= … (3.3.12)

To design the multiloop controller ( )cG s such that all the diagonal elements of H(s) resemble those of ( )R s as close as possible over a frequency range relevant to control applications.

( )cG s , with an integral term, can be written in a Maclaurin series as

2 30 1 2

1( ) [ ( )]c c c cG s G G s G s O ss

= + + + (3.3.13)

Note that 0cG , 1cG , and 2cG correspond to the integral, proportional, and derivative terms of the multiloop PID controller, respectively.

Expanding G(s) in a Maclaurin series also gives

2 30 1 2( ) ( )= + + +G s G G s G s O s (3.3.14)

Where 0G =G(0), 1G =G'(0), and 2G =G''(0)/2.

By substituting Eqs. (3.3.13) and (3.3.14) into Eq. (3.3.10) and rearranging Eq. (3.3.10), H(s) is obtained in a Maclaurin series as

1 1 1 20 0 0 0 0 1 1 0 0 0

10 0 0 2 1 1 2 0

1 1 1 30 1 1 0 0 0 0 1 1 0 0 0 0 0

4

( ) ( ) ( ) ( )( )

( ) [

( )( ) ( )( ) ( ) ]

( )

c c c c c

c c c c

c c c c c c c

H s I G G s G G I G G G G G G s

G G G G G G G G

I G G G G G G I G G G G G G G G s

O s

− − −

−

− − −

= − + + + +

+ + + −

− + + + + +

+

(3.3.15)

( )R s can also be expressed in a Maclaurin series as

2 3 4''(0) ''(0)( ) (0) '(0) ( )2 6

R RR s R R s s s O s= + + + + (3.3.16)

Where, (0)R I= , given that Gii+(0) = 1.

By comparing each diagonal element of H(s) and ( )R s in Eqs 2.41 and 2.42 for the first three s terms (s, s2, s3), we can express 0cG , 1cG , and 2cG in terms of the process model parameters and the desired closed-loop response parameters.

Although the analytical tuning rules so obtained take the interaction effect of the multiloop system fully into account, some of them (that is, for 1cG and 2cG ) would be too complicated to use it practically and also often show severe inaccuracy because of the inherent limitation of a Maclaurin series for the high-frequency region. However, by using the frequency-dependent characteristics of the closed-loop interactions, these limitations can be avoided to result simple but efficient tuning rules.

Design of proportional gain Kc and derivative time constant Dτ



As indicated from Eq. (3.3.13), the impact of 1cG and 2cG on the PID algorithm becomes more predominant at high frequencies, whereas it is relatively insignificant at low frequencies. Therefore it is desirable for 1cG and 2cG to be designed based on the process characteristics at high frequencies.

Given that ( ) ( ) 1cG j G jω ω at high frequencies, H(s) can be approximated to

1( ) ( ( ) ( )) ( ) ( ) ( ) ( )c c cH s I G s G s G s G s G s G s−= + ≈ (3.3.17)

This feature at high frequencies indicates that 1cG and 2cG can be designed by considering only the main diagonal elements in G(s). In fact, at high frequencies, the transmission interaction is mostly eliminated by low-pass filtering through the processes in the other loops. Therefore, the generalized IMC-PID method for the SISO system can be directly applied to the design of Kci and

Diτ of the multiloop PID controller.

Dropping the off-diagonal terms in G(s), we can obtain the ideal multiloop controller ( )cG s to give the desired closed-loop responses ( )cG s as

1 1( ) ( ) ( )[ ( )]cG s G s R s I R s− −= − (3.3.18)

where 11 22( ) [ , , . . . , ]nnG s diag G G G= .

Thus, the ideal controller ( )ciG s of the ith loop are designed as

1( ) [ ( )]( )

1 ( ) ( ) ( 1) ( )i

i iici n

ii i i ii

Q s G sG sG s Q s s G sλ

−−

+

= =− + −

(3.3.19)

where Qi(s) is the IMC controller given by 1[ ( )] ( 1) inii iG s sλ−− + .

As Gii+(0) is 1, Eq (3.3.19) can be rewritten in a Maclaurin series with an integral term

2 3'(0)1( ) [ (0) '(0) ( )2

iici i i

fG s f f s s O ss

= + + + (3.3.20)

where, ( ) ( )i cif s G s s= .

By truncating the higher-order s terms, the resulting first three terms can be interpreted as the standard PID controller.

Therefore, Kci and KDi of the multiloop PID controller can be obtained by

' (0);''(0) / 2

ci i

Di i ci

K ff Kτ

==

(3.3.21)

Design of integral time constant τ i

Because the integral term (0)cG in the PID control algorithm is dominating at low frequencies, it needs to be designed based on the interaction characteristics at low frequencies. It is clear from Eq (3.3.10) that the loop interactions cannot be neglected at low frequencies. Therefore, (0)cG



should be designed by taking the off-diagonal terms of G(s) into account. Comparing each diagonal element of the first-order s terms in Eqs (3.3.15) and (3.3.16) gives an analytical tuning rule for Iiτ as

1

[ ' (0) ][ (0)]

ii i i ciIi

ii

G n KG

λτ +−

−= − (3.3.22)

The design procedure is to find each controller based on the paired transfer function while satisfying some constrains due to process interactions. In this approach the issue of interaction between loops is examined first, and sufficient conditions are derived for the individual loops that guarantee stability and robust performance. Subsequently, the SISO controllers are designed such that each loop satisfies these sufficient conditions.

Advantage:

• The approach is systematic and with stability guaranteed,

• The settings obtained by this approach can lead to performances corresponding closely to the selected tracking closed-loop specification.

Disadvantages:

• The resulting controllers are not of standard PID structure and an optimization procedure is included, so it is difficult to implement and accepted by control engineers.

• Not straight forward to extend the approach to 3×3 and higher dimension systems.

• The method is potentially conservative, since the information on other controllers is not exploited during the design of a particular controller.

Example 3.3.1. Wood and Berry system: For the diagonal pairing, the resultant PI controller is listed in the Table. The close loop responses is as shown in the Figure, where the unit set-points change in r1 at t=0 and r2 at t=200.

Controller for Wood and Berry

Controller Kc,ii ,i iiτ

Loop 1 0.7248 16.7000

Loop 2 -0.1375 14.4



0 50 100 150 200 250 300 350 4000

0.5

1

1.5

y1

Time

Output Result of y1 (IMC Method For Wood and Berry)

0 50 100 150 200 250 300 350 400-0.5

0

0.5

1

1.5

y2

Time

Output Result of y2 (IMC Method For Wood and Berry)

Close loop response for Wood and Berry by using IMC method

3.4. SEQUENTIAL LOOP CLOSING METHODS In sequential loop closing methods, a controller is designed for a selected input-output pair and this loop is closed. Then a second controller is designed for a second pair while the first controller has been closed. The dynamic interaction of current loop is considered in the closing of next loop, and so on. In this manner, a series controller is designed sequentially. Usually the fastest loop is designed first.

If the process model is not available, the relay auto-tuning approach is a useful and simple method to obtain system frequency information for PI controller design. Because MIMO systems have an infinite number of ultimate points, the relay auto-tuning approach for SISO systems cannot be directly applied, a sequential relay-feedback test can be used to locate the ultimate points of a MIMO system and to design PI controllers. After the frequency response information has been obtained, the Z-N or modified ZN tuning rules can be used to tune the PI/PID controllers. Additional constraints are imposed to compute the controller settings so as to guarantee their nominal stability.

Consider an n×n system ( )G s with the entry ( )ijg s under decentralized control with a known pairing. The closed-loop relationship between iy and iu becomes a bit more complicated than its open-loop one ( iig ):


122 2

, ( ) 1i i iii CL ii

i iiCL

y G G H Gg s gu g

−⎛ ⎞ ⎡ ⎤= = −⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦

i i ,

where the subscript CL stands for closed-loop, 22G is the sub-matrix of the rest part of the system

matrix ( )G s and [ ] 12 22 2 22 2c cH G G I G G −= + is the complementary sensitivity function for the rest

loops.

The whole design procedure is as follows:

1. A relay is placed between 1y and 1u while all other loops are on manual. Following the relay-feedback test, a controller can be designed from the ultimate gain and ultimate frequency for loop 1.

2. Perform relay-feedback test between 2y and 2u while loop 1 is on automatic (closed) and other loops still on manual. A controller can also be designed for loop 2 using ultimate gain and ultimate frequency already taking loop one interaction into consideration following the relay-feedback test.

3. Once the controller on the loop 2 is put on automatic, a relay-feedback experiment is performed between 3y and 3u , and controller can be designed.

4. In the same way, all loops will be tested and closed-loop.

1. Relay-feedback experiment is performed between 1y and 1u again. Generally, a new set of tuning constants is found for the controller in loop 1.

2. This procedure is repeated until the controller parameters converge.

Typically, the controller parameters converge in 3~4 relay-feedback tests for 2×2 systems. At each stage, a modified Ziegler-Nichols formula, with a detuning factor f = 2.5, is employed to tune the controller parameters. It is worth to note that the proposed MIMO auto-tuning concept repeats the “identification-design” procedure on equivalent SISO transfer functions.

This class of approach has several advantages.

1. it makes the problem simple by treating the MIMO system as a sequence of SISO systems which the relay-feedback system is proven useful and reliable;

2. it operates in an efficient manner.

3. it can be applied for nonlinear multivariable systems.

However, the method has some fundamental disadvantages:

1) it cannot guarantee convergence even if around the neighborhood of the true solution;

2) loop failure tolerance is not guaranteed automatically when loops designed early fail;

3) the performance of the method depends strongly on which loop is designed first and how the first controller is designed;

4) the iteration procedure is essential because closing the subsequent loops may alter the response of the previously designed loops.



5) there is only a weak tie between the tuning procedure and the loop performance.

Hence, conservative design may result due to the RHP zero on the diagonal which may not be the RHP transmission zeros of the MIMO process. In many cases, the controller will be re-tuned one after the other with all other loops being closed with the controllers obtained in the previous step. This procedure will go on until they converge. For higher dimensional systems, the number of relay-feedback tests will be dramatic large for a possible converge solution.

3.5 SIMULTANEOUS EQUATION SOLVING METHODS In Simultaneous equation solving methods, the equations related to controller parameters are created according to certain performance criterion and solved by some optimization technique. Generally, design of multi-loop controller by way of simultaneous equation solving is numerically difficult. Consider a stable 2×2 process,

1 11 12 1

2 21 22 2

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

y s g s g s u sy s g s g s u s⎡ ⎤ ⎡ ⎤ ⎡ ⎤

=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

.

Assume that proper input-output pairing has been made to the process. The process is to be controlled in a negative feedback configuration by the multi-loop controller:

1

2

( ) 0( )

0 ( )c

cc

g sG s

g s⎡ ⎤

= ⎢ ⎥⎣ ⎦

.

The controller design objective is to find ( )cG s such that both loops achieve satisfactory performance. The equivalent transfer 1( )g s between input 1( )u s and output 1( )y s , 2 ( )g s between input 2 ( )u s and output 2 ( )y s can be obtained as

12 211 11 1

2 22

( ) ( )( ) ( )( ) ( )c

g s g sg s g sg s g s−= −

+ (3.5.1a)

and

21 122 22 1

1 11

( ) ( )( ) ( )( ) ( )c

g s g sg s g sg s g s−= −

+, (3.5.1b)

respectively. It is thus clear that in order to achieve good loop performance, 1( )cg s and 2 ( )cg s should be designed for 1( )g s and 2 ( )g s instead of the plant diagonal element 11( )g s and 22 ( )g s . By applying modified Ziegler-Nichols to the equivalent transfer function 1( )g s and 2 ( )g s , for designing the controller ( ), 1, 2cig s i = such that a given point on the Nyquist curve of

( ), 1,2ig s i = is moved to a desired point. In this way, the optimum settings for the multi-loop controller can be obtained with all the benefits derived from the well-developed SISO tuning.

Let ( )( ) aij

i i i aiA g j r e π ϕω − += = (3.5.2)



be the two given points on the Nyquist curves ( ), 1, 2ig s i = . They are to be moved respectively to the desired points

( )( ) ( ) biji i i ci i biB g j g j r e π ϕω ω − += = .

by ( ), 1, 2cig s i = of PID type, i.e.,

, ,,

1( ) (1 ), 1, 2ci p i d ii i

g s k s is

ττ

= + + = .

Applying the modified Ziegler-Nichols method to each ( )cig s , yields

,( ) (1 ), 1, 2i ici p i i

i i

g s k s is

ω βαβ ω

= + + =

where

2tan( ) 4 tan ( ), 0

21 , 0

tan( )

bi ai i bi aii

ii

ibi ai

ϕ ϕ α ϕ ϕα

αβα

ϕ ϕ

⎧ − + + −⎪ >⎪= ⎨⎪ − =⎪ −⎩

.

It follows from the phase part of the representation of iB that

,( ) (1 tan( )), 1, 2ci p i bi aig j k j iω ϕ ϕ= + − = .

Then, the representation of iB can be written as ( )

,( ) cos( ) aiji i p i bi ai big j k r e π ϕω ϕ ϕ − += − . (3.5.3)

Bringing ( ), 1, 2ig s i = and ( ), 1, 2cig s i = into Equation (3.5.3) gives two equations which are complex and can be broken down to four real equations. Because the number of unknowns equals the number of real equations, intuitively, it is expected that the problem can be easily solved.

Unfortunately this is not the case. The main difficulty lies in the nonlinearities of the equations which are further complicated by the coupling among the equations. In addition, the graph based iterative optimization technique may be complicated for control engineers and the extension to 3×3 or higher dimensional systems is, if not impossible, not straight forward.


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