Dev. Chem. Eng. Mineral Process., 9(1RI, pp.25-31,2001.

Robust Digital PI Control for Uncertain Time-delay Systems in Process Industry

Xiang Liu*, Wenhai Wang and Youxian Sun National Laboratoly of Industrial Control Technology, Institute of Industrial Process Control, Zhejiang Universiv, Hangzhou, 3 I0027, P.R. China

A new robust digital PI controller for a class of typical systems with uncertain time delays is developed for use in process industry. Using this Controller the satisfactory locations of poles of the closed-loop systems will change slowly with the varied time delays, and the robust stability is guaranteed for the time vaned time-delay on a large scale. Simulation results show that the robust controller also works for the systems in the presence of unmodelled dynamics.

Introduction Time delay systems are often encountered in process control practice. For dealing with such a system, the most popular controller in use today is the well-known Smith predictor [l]. The key significance of the predictor is that it can exactly eliminate the time delay in the characteristic polynomial of the closed-loop system. However the time delay in an industrial process control system is usually time variant which may not be canceled exactly. This may cause the closed-loop system to be unstable. To deal with this problem, Matausek et al. [3] presented a modified Smith predictor based on the work done in [2] which can treat a slightly unmatched time delay by on-line adjustment of the long dead-time internal model parameter. For the same purpose Ho- Wang et al. [4] presented a fsesuency domain tuning approach for a PI controller which can simultaneously achieve exact gain and phase margin for a generally stable linear system. Unfortunately both of these approaches have not dealt with the question of how to guarantee robust stability of a genetic process system with a specific perturbation region of the time delay for a set-point input signal. Tian et al. [5 ] also presented a double-controller scheme for control of processes with dominant delay, and showed that the proposed scheme is superior to the Smith predictor in the presence of large uncertainty in process dynamics.

The PI controller developed in this paper is a robust digital controller which can stabilize a class of systems with time variant time delays on a large scale. The advantage of using a PI controller is to increase stability margins andor reduce steady-state errors. Thus it is reasonable that a PI controller is able to stabilize some process control systems with time-variant time delays, such as inertia orintegral inertia time delay systems, etc. The overall robust control system possesses the robust

~~~

*Author for correspondence ( e-mail: xliu@i@c.zju.edu.cn),

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X. Liu, W. Wang and I! Sun

stability for the uncertain time delay even in the presence of model led dynamics.

easier to implement physically. Digital controllers are more useful than those in the continuous case since they are

Definition, Theorem and System Description

The z-transform, f i ( z ) , of a sequence, p = ( p i KO , is defined by j ( z ) = p i z i

With tbis definition, a stable transfer function has all of its poles outside the unit disc of the complex plane. The symbol z denotes the unit delay operator and the

polynomial, i , of order k is given by E = c x i z ' .

m

i=O

k .

i=O

In process industrial systems, a typical time delay system is prescribed to be one of the elements of the following system set.

I 1.

ke-= keqS ke-" Ts +1 Ts 's(Ts +1)

{G(s) I G ( s ) E H,H : -,-

where the parameters, r ,T, T, and T,, are all positive real and k is real not equal to zero.

When the sampling period is chosen as T, > 0 , the set of the characteristic roots of the system (1) with the zero-order holder is

{z I z E Z,Z : ~ , e ~ o ' ~ , eTo'q, eTo'T2 1 . . .(2)

It is obvious that the elements of the set (2) are all located at the outside of the unit disc, except the element 1 for an arbitrary sampling period T, > o . The time delay e-* is replaced by zd with d = ht(-) when it is discrete. Therefore

there exists the following inequality relation between the real z and the positive integer d:

7

To

dT, 5 t c ( d + 1)To . . .(3)

This illustrates that the discrete time delay may be unmatched even if it has no perturbation. Theorem: If the real polynomial is = &zd + M i ( 1 - z ) , where

b = b, + b,z + ... + b,zm ,Ci = a. + alz + ... + a,z" , the positive integer d >O , and all of the characteristic roots of ci situate at the outside of the unit disc, then there exists a large positive real numberhf, > o that when M 2 M, the polynomial i has all of its characteristic roots outside the unit disc if i( I)& 1) > 0 . Proof: (i) First, it will be proved that the root of 1 near 1 in the real axis is at the outside of the unit disc. Let zo = 1 + + 0, when M > 0 . Then

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Robust Digital PI Control for Uncertain Time-Delay Systems

Because all of the roots of 2 are situated at the outside

. . .(4)

the unit dsc, we obtain i ( i ) # 0 , By M,, > 0 it is easy to deduce that E, > 0 ifand only if Ci(l)&l) > 0 , i.e., the root zo situates at the outside of the unit disc.

(ii) Then we shall derive that the other roots of disc.

are also the poles outside the unit

is inside the unit dsc i.e., Izl 1 I 1( z1 f 1 ). By Assume that a root z1 of

i ( z , > = 0 >

lib, >zP I = I&, #P I = M \ ( I - z1 ) h ( Z I )I . . . . ( 5 )

Also by I i ( z , )zP I I l&zl )I , We have M ](I- z1 )ci(zl 11 I I&zl )I. Notethat 1-zl 2 0 , b ( z l ) f 0 ,

. . .(6)

p(Z1 )I Therefore, M I

From the analysis above, the contradiction result can be derived if we choose

10 - 21 1 4 3 )I .

It is easy to see from the equality ( 5 ) that the value of M is relative to the positive integer d. Furthermore the larger is the d the larger the M for every z outside the unit disc. Thus let E > 0, E + 0 , it is reasonable to assume that 1+ E is the root of f whch is the nearest root to the unit disc. By the theorem above the following equality is derived.

.. &1)( 1 + & ) d o

&b( 1) M o =

According to the analysis in the Theorem, it is evident that all of the roots of the

27

X. Liu, W. Wang and Y. Sun

polynomial f are situated at the outside of the unit disc for every positive integer d whch satisfy the inequality d I d o if we choose M= M , in accordance with the equality (7).

Robust Digital PI Controller A single loop digital control system is shown as Figure 1. The transfer function G, (s) is a zero-order holder. The sampling period is To . ? is the z-transform of the specific set-point r.

Figure 1. A single loop digital control system.

The general form of a digital PI controller can be written as

- k ̂ k p ( d 0 +d,z) D=-- - 1 - 2 1 - z

. . .(8)

It is easy to deduce that the z-transformation of the system output y(s) shown in Figure 1 is given by:

b“ where (GG h j = : . By the equalities (8) and (9) the characteristic equation is

obtained a

(1 - z)r; + G Z d = 0 ...( 10)

The parameters of the controller 6 should be selected to cancel the characteristic root at 1+ E . We stipulate that all of the poles of a ̂ are situated at the outside of the unit

disc. By the theorem above, if the parameter k p is chosen as:

...( 1 1)

then there always exists a positive real number v o > 0 such that each root of the characteristic equation (10) is located at the outside of the unit disc under the

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Robust Digital PI Control for Uncertain Time-Delay Systems

con&tion of 0 < v I vo , i.e. the closed-loop system is asymptotically stable. Furthermore if we select a small positive real number E > 0 , it can be obtained by (7) that:

Next, we dustrate the design procedure concretely for the inertia time delay system. In t h s situation the z-transformation of the transfer function G ( s ) in ( 1 ) with the zero-order holder can be written as

The pole of the system is z = eTo'T . Choosing the small positive number E = 0.01, any other small positive parameter is also suitable, and d o + d , z = ( 1 - e-TolT Z ) , then for a nominal time deIay do we obtain

& .. - - 6( 1)( 1 + &)do+,

By the theorem if the magnified coefficient k,in (8) satisfies the inequalities

0 < k , I vo , then the closed loop system has all of its roots outside the unit disc for

every d I d o . S d a r l y , if the system a s ) is a integral time delay system, then the PI controller in

(8) is degenerated into a proportional controller, i.e.,

D, = k , . . .( 15)

where 0 < k , I vo . If the system a s ) is an integral inertia time delay system, then the proportional

controller is:

d, = k , ( l - e -TO ' T z) ...( 16)

with k , = sgn( &l))v and 0 < v I vo .

Because the PI controller have the closed-loop system asymptotically stable for every time delay which satisfy the inequality d s d o , it is a powerfit1 robust controller for uncertain time delay systems.

29

X. Liu, W. Wang and Y. Sun

An Simulation Example In the black liquid evaporation phase of the paper-making process, the model for the physical relationship between the input steam pressure ( m a ) and the output black liquid density (%) is:

62 .5e-”’ 1 + 10s

G ( s ) =

Choosing the sampling period To = 3 min., the z-transformation of model with the zero-order holder is

1 9.59492’ ( G G h ) = 1-0 .8465~

By ( 12) and ( 13), the robust PI controller is derived as:

- D =

The step responses of the closed-loop system with different time delays, i.e.,

9.72 x 10-3(1 - 0.8465 Z) 1 - 2

z =18,9, and 3, are shown in Figure 2.

1 4

3

~

r

3 0 6 0 9 0 1 2 0 1 5 0 1 8 0 2 1 0 2 4 0 2 7 0 : 0

I

Figure 2. Step responses of the density of black liquid.

I 0

Figure 2 shows that the closed-loop system is stable for every time delay r which satisfies r S 18 . Furthermore the closed-loop system enjoys no steady-state error.

Considering the unmodelled dynamics. We let:

D = - 9.7 x 10-3(1 - 0 . 8 ~ )

1 - z

the time delay is 18. Figure 3 shows the effect of the unmodelled dynamics in which the curve 2 is the step response of the system in the presence of the unmodelled dynamics.

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Robust Digital PI Control for Uncertain Time-Delay Systems

I 4 ,

I

Figure 3. Step responses: I . the nominal system; 2. the system with the unmodelled dynamics (T =18).

Simulation results show that the digital control scheme can make the typical time delay process system track the set-point input signal without the steady-state error within any specified perturbed region of the time-varying time delay. The robust schemes to deal with time-delay systems have been presented in [2-51. These schemes have the control systems enjoy different optimal or robust performances. Fung et al. [4] aimed to achieve the satisfactory gain and phase margin of a process using the frequency response of the process. The focus of the method is on the optimal performance of the system. Tian et al. [5] gave out a double-controller scheme that oriented to the robust stability of the overall time delay systems. Compared to these schemes, the features of the scheme in this paper are the stabilizing capacity against the time-varying time delay. This may vary in a specific sufficiently large region, and the simple hgital structure can be implemented in control engineering more easily.

Conclusions A robust PI digital controller for a class of systems with uncertain time delays in process control is proposed. It is demonstrated that the controller gives robust stability and satisfactory performance for the systems with time variant time delays on a large scale.

References 1. Smith, 0. J. 1959. A controller to overcome dead time. ISA J. , 6(2), 28-33. 2. Astrom, K. J. ; Hang, C. C., and Lim, B. C. 1994. A new Smith predictor for controlling a process

with an integrator and long dead-time. IEEE Trans. Automat. Contr. ,39( 2), 343-345. 3. Matausek, M. R., and A. D. Micic, A. D., 1996. A modified Smith predictor for controlling a process

with an integrator and long dead-time. IEEE Trans. Automat. Contr. ,41(8), 1199-1203. 4. Fung, H.; Wang, Q. -G., and Lee, T. -H., 1998. PI tuning in terms of gain and phase margins.

Automatica, 34(9 ), 1145-1 149. 5 . Tian, Y.-C., and Gao, F. 1998. Double-controller scheme for control of processes with dominant

delay. IEE Proc. Control Theory Appl., 145( 5) ,479484.

Received: 20 October 1999; Accepted after revision: 2 June 2000.

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