Ho PID controller Stabilization for StableProcesses with Time Delay

Linlin Ou, Danying Gu, Weidong Zhang, Yunze CaiDepartment of Automation, Shanghai Jiao Tong University, Shanghai, 200240, P. R. China

Abstract-The H_ PID controller has been analyticallydeveloped on the basis of modern H_ optimal control theoryand it can be tuned by only one control parameter in terms ofgiven quantitative performance and robustness. This paperconsiders the problem of stabilizing a stable process with timedelay using the H_ PID controller of this kind. Making use ofthe extended Hermite-Biehler Theorem applicable toquasi-polynomials, the stabilizing set of the H_ PIDparameters is determined for open-loop stable plant with timedelay. Simulation results not only demonstrate the validity ofthe obtained stabilizing set but also show that the H_ PIDcontroller parameter can be tuned in a convenient manner.

I. INTRODUCTION

Time delay is often involved in most of industrialprocesses. It is considerably difficult to control such

processes effectively. In order to solve the problems, allsorts of control strategies have been developed, amongwhich PID controllers and Smith predictors are mostrepresentative. Smith predictor is sensitive to modelmismatch and has poor disturbance rejection capability[1-3], while PID controller not only allows processengineers to operate in a convenient manner but also has thesufficient ability to solve many practical control problems[4]. Thus, PID controller is used most widely in industrialprocesses. However, traditional PID controller settingmethods can not give satisfactory answers to the followingkey problems on PID controller design: on the one hand,these approaches are ineffective for plants with long timedelay, on the other hand, most of them have not consideredrobustness problem, i.e. since there does not exist explicitrelationship between the three setting parameters andsystem performance specification, the process engineers donot know how to tune the parameters effectively to meet therequired performance specification and whether the tunedsystems have good robustness when there exists uncertaintyin plants. In order to overcome the above deficiencies, theH infinity (H) PID controller was developed in [5]. Themodified PID controller is designed analytically on thebasis of H_ optimal theory and robust control theory. Itcan be not only used on the plants with long time delay but

This work was supported by National Science Foundation ofChina (60274032) and Science and Technology Rising-Starprogram of Shanghai (04QMH1405).Address: Shanghai Jiao Tong University, Dongchuan Road 800,Shanghai 200240, P. R. ChinaE-mail: oulinlingsjtu.edu.cn

also tuned using only one parameter in terms of givenquantitative performance and robustness.

During the design procedure of the H_ PID controller,rational approximation was employed [5] such that thereexists a probability that the derived controller cannotstabilize the original plants, although it is able to make theapproximate ones stable. Hence, certain constraint has to beimposed on the single adjustable parameter 2. However, itis a very considerable complicated problem to determinethe stabilizing range of A. The main reason is that theclosed-loop characteristic equation of the control systeminvolves time delay. In [6], the conservative stabilizingrange of the H_ PID controller was given fromexperience, i.e. 2 > 0.0650. The objective of this paper isto provide a complete theoretical solution to thestabilization problem of H_ PID controllers for stableprocesses with time delay. Using the extendedHermite-Biehler Theorem applicable to quasi-polynomials[7], the conclusion is drawn that H_ PID controllers canstabilize the stable processes with time delay when thesingle control parameter 2 is larger than 0.073506.Moreover, numerical simulation examples imply that ithelps to guide the process engineers to tune the parameter2 effectively in practical industrial control so as to avoidoccurrence of the circumstance that the systems areunstable when H_ PID controllers are implemented.

The paper is organized as follows. In Section 2, theproblem of stabilizing the stable time-delay process withH_ PID controller is formulated. By using the extension ofthe Hermite-Biehler Theorem, the detailed stabilizationprocedure is investigated and the stabilizing range of thefree parameter contained in H_ PID controller is derivedin Section 3. Simulations are presented in Section 4 todemonstrate the validity of the resulted stabilizing range.Finally, Section 5 provides some concluding remarks.

II. PROBLEM FORMULATION

Practical processes often have complex dynamicresponses, but most of them may be modeled as first-orderstable processes with time delay [8], which are written as:

(1)G(s) Ke= Ois +I

where z is time constant, K is the gain, and 0 is thetime delay. 0, z and K are all positive numbers.

Consider the basic SISO feedback system shown in Fig. 1,where r is the reference input, y is the plant output,

0-7803-9484-4/05/$20.00 ©2005 IEEE 655

G(s) given by (1) is the plant to be controlled, and C(s)is the H_ PID controller developed in [5], which has thefollowing form

C(S) = KC (1+_+T S TFS +])(2)

where

TF = 42+0' TI 2+ F

TD= -TF K-= K(4A 0)

Here, 2 is the positive adjustable parameter.

r e u YC G

Fig. 1. Feedback Control System

The derived closed-loop characteristic equation is

2(S)= 2S2 + (20 +-)s + (I +-s)e-Os2 2

(3)

The objective is to determine the stabilizing set of the singlecontroller parameter 2 for which the closed-loop systemshown in Fig. 1 is stable. Since the closed-loopcharacteristic equation (3) contains the exponential term,the number of its roots may be infinite. This makes theproblem of stabilizing the closed-loop system a formidableone.

III. STABILIZATION ANALYSIS ON H- PID CONTROLLER

In order to solve the problem presented in Section II, wefirst transform the characteristic equation (3) into thefollowing form:

a*(S) =,(s)e0s = [22S2 + (22 + 0)s]eos +(1+ 0 s) (4)2 2

3*(s) is called a quasi-polynomial, which is a polynomialinvolving two variables s and es . Since eos does notpossess any finite zeros, the zeros of 3(s) are identical tothose of 3* (s) , i.e. they are equivalent concerningstabilization analysis [9]. Necessary and sufficientconditions for the stability of 3*(s) are provided by thefollowing theorem [7, 9].

Theorem 1: Let 3* (s) be given by (4), and writea*(jI)= (v)+:jai(n)) where 35(co) and 3i(co)denote the real and imaginary parts of * (jco)respectively. 3*(s) is stable if and only if1) 5r (co) and 5 (co) have only simple real roots andthese roots interlace;2) For some ,

E(co) = ,o) )r(co) )-, (coo )3 (c)o > 0

where r (coo) and 4' (coo) represent the first derivative

with respect to c of 6r (co) and di(c), respectively.Theorem 2: Assume that M and N denote the highest

powers of s and es respectively in 3*(s) . Anappropriate constant q is given to satisfy the requirementthat the coefficients of terms of highest degree in 5r(CW)and 5i(c) do not vanish at co= q. Then the necessaryand sufficient condition for the equations r(co) =0 andd (c) = 0 to have only real roots is that in the intervals

-21yz+ 77 < c< 21yz+ 77, 1 = 1, 2,3,...3r(co) or d (co) has exactly 4/N +M real roots startingwith a sufficiently large I

Substituting s = jco to the equation (4), we have

3r (c) = -(22 + -)cosin(6ow) - 22t02 COs(6o) + 12

8i(co) = (2A +-)CwCOS(OCO)-_,%2CO2 sin(Oco)+-co2 2

(5)

(6)

Taking z= Ow, 6r (co) and d (co) can be transformedinto

-r(z) = -(22+ -)- sin(z) _-2 ( )2 COs(Z) +1

o5 (z) = (22 +- )-cos(z) _-2 ( - )2 sin(z) +-26 6 2

(7)

(8)

The roots of the imaginary part i(c) may be computedaccording to the following equation:

(22+ ) cos(z)-22 2 sin(z)+ =

Then from (9), we have eitherz =O

or

(22+%)0cos(z)12202=12 2

(9)

(10)

From the above, it is obvious that one root of the imaginarypart is z = 0. However, it is difficult to obtain the otherroots analytically. Hence, the graphics method is adopted toexamine the location of the roots in (10). Since (10) can beexpressed as

cos(z) + 0/(42 +6) 222- ~~~zsin(z) (42+6)6

The intersection points of the curve

(z) cos(z) + 0/(42 + 0)sin(z)

and

222 z(42+6)6

are just the roots of the equation (10). The positive realroots of (10) are denoted as zj, j= 1,2,..., which arearranged in increasing order of magnitude. Some of theseroots are shown in Fig.2. Substituting s1 = Os into (4), wecan obtain the new quasi-polynomial with M =2 andN = 1 . Then 77 = z / 3 is chosen to satisfy the requirementthat cos(77) is not equal to zero. From Fig.2, it can be seenthat a, (z) has three rational roots in the interval

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[0, 2z - (z / 3)] = [0, 5z / 3], including a root at the origin.oi(z) is an odd function of z, so oi(z) has five realroots in (-5ff! 3, 5vI 3]. Furthermore, there also exists areal root in the interval (5ff I 3,7ff /3] . Then, the totalnumber of its real roots is 4N +M =6 in the interval[-2ff+(ff3),2ff+(f/3)]. From Fig.2, it is easy to seevi(z) has one real root respectively in the interval[21ff + (ff13),2(1 + l)ff+ (f/f3)] and [-2(l + 1)f + (f / 3),-21ff+(f13)] for 1=1,2,.... Therefore, it follows that3- (c) has exactly 41N +M real roots in the intervals[-21ff + (ff3),21ff+ (f/f3)]. According to Theorem 2, theconclusion can be drawn that 3W(z) has only real roots,which is the necessary condition that d*(s) can bestabilized.

Fig. 2. Location of the positive roots involved in equation (10)

In order to find the stabilizing range of the adjustableparameter A, now we consider the real part 6r(z) . Firstly,check whether 5*(s) satisfies Condition 2 of Theorem 1.For some wE(-oo,oo),

E(cto ) = ( Ct)O) Ct)O))- ( Ct)O)6 (C)O ) > O

Taking coz= zO =°, dr(zo) I and >(zo)=0 areobtained. Thus, we have E(zo) = 22/6 +1 > 0, i.e. d* (s)satisfies Condition 2 of Theorem 1.

Then, check Condition 1 of Theorem 1, i.e. the real part6j(co) must have only real roots and the real roots of realpart and imaginary part of 3*(s) interlace. For zo =0,using (7), dr(zo) 1 is obtained, so 63r(zo) > 0. For theother zj, j=1,2,..., the values of the real part may bedenoted as

05~ ~ sn(.2 2 2 sin(z.) 1,(Zj1) = _zj2 [2 (-) + cos(z )(-) + - 2]

6i 6 2z1 z1

Interlacing of the roots of 6r (co) and c(o) isequivalent to dr(ZO) > 0 v JZ1)< ° v (Z2) > ° ,

6. (Z3) < 0 and so on. According to this rule, the followinginequality is derived.

sin(z.) 2 j)A2 2 sin(zj ) 1(-1)i[2 i)(-)+cos(z.)(-) + ( i- 2]<° (11)6i 6 2z1 z1

Finally, we compute the critical value of A/6. For eachj in (11), we may obtain the corresponding range of A/6.The intersection of all the ranges is exactly the area ofA/6 that can make systems stable. The bound of this areais determined by the following equation.

2 sin(A.)2+ ( )()2 + sin(z1) 1 (12)6 6 2z1 z12

Consider that z1 in (12) are also the roots of the equation(10). Rewriting (10) and combining it with (12) yield a setof equations that are expressed as

ACSz() 2 sin(z) Ai sin(z) Icos(z)(-) +2 (-)+ 2__

I-6 12z z12Z 2-sin(z)(-) + 2cos(z)(-) + -cos(z) + - =0

Simplifying (13), we haveA-

I[1+cos(z)-2 ]

6 4 z

(13)

(14)

Since equation (14) is an even function, only the positivevalues of z are considered. Substituting (14) to (12), thepositive values of z are presented, i.e.

zi = 1.5708, Z2 = 2.4333, Z3 = 4.7124, Z4 = 7.5501and so on. Then, using the relation between A/6 and zin (14), shown in Fig.3, the maximal value of A/6satisfying (13) is derived, which equals to 0.0735 whenZ = Z2 = 2.4333.

z

Fig. 3. Plot of A/I in equation (14) as a function of z

In order to satisfy (10) and (11), A/6 must be largerthan the critical value 0.0735. Thus, in terms of the twoconditions in Theorem 1, the quasi-polynomial d*(s) isstable only when A/6>0.0735. As a consequence, themain result is achieved.

Theorem 3: The condition under which the H_ PIDcontroller is able to stabilize the closed-loop systemcontaining stable first-order plant with time-delay is that the

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value of the adjustable parameter 2 must be tuned to belarger than 0.07350, i.e. A > 0.07350

IV. SIMULATION

In the following, a numerical example will be providedto check the resulted stabilizing range of the single controlparameter in the H- PID controllers.Example Consider a typical papermaking control system.

There are numerous control objectives in the system, suchas basis weight, moisture content, stream pressure, etc., ofwhich the most important is basis weight, i.e., the weight ofone square meter of paper. The dynamic model of the papermachine for basis weight control is as follows [10].

G(s) 5.15e-28sG()=1.8s+1IFor it, the following H- PID controller is yielded:

C( ) =2.52S2 + 3.2s + I

5.15[22s2 + (22 + 1.4)s]

(15)

(16)

From (15), it is seen that the time delay equals to 2.8. Aunit-step setpoint is introduced at t = 0 . The nominalsystem responses are shown in Fig.4 when A2/6 is givendifferent values. Fig.4 (a) shows that the closed-loopprocess output distracts from the desired value at all timessince the value of is chosen to be 0.1988, which equalsto 0.0710 beyond the stabilizing range. However, forA=0.3 (larger than 0.0750 ), Fig.4(c) shows that theoutput can converge at the desired value, although theconvergence time is very long. Fig.4 (b) shows that, theoutput response is basically located in the transitional stagefrom divergence to convergence when is given thecritical value 0.2058. The same results have been gained forother first-order stable processes with time delay. Thus, thesimulation results imply that the derived stabilizing range isvalid. Then, in terms of Theorem 3, only needs to betuned starting from the lower bound 0.2058 during thesetting of H_ PID controller with the transfer function(16), avoiding the time-consuming operation for the checkof the system stability.

V. CONCLUSIONS

In this paper, the stabilization problems of the H_ PIDcontroller have been discussed. The stabilizing range of thesingle adjustable parameter in such the H_ PID controllerhas been derived. By using the extension of theHermite-Biehler Theorem to quasi-polynomials, theanalytical process of stabilization using the H_ PIDcontroller is presented. Simulations demonstrate the validityof the achieved stabilizing result, which is a need of HPID controller studies. In order to stabilizing first-orderstable plants with time-delay using H_ PID controllers,A/6 must satisfy the condition presented in Theorem 3.Furthermore, the result not only provides convenient for thesetting of H_ PID controllers, but also helps to avoid theclosed-loop systems becoming unstable.

(b)

t(s)

(c)

Fig. 4. Response of the nominal systems for:

(a) A=0.1988, (b) A=0.2058, and (c) A=0.3

Thus, the single adjustable parameter plays theextremely important role in H- PID controllers. It relatesdirectly not only to the nominal performance and robustnessof the systems but to the stabilization of the systems. Thestabilization analysis can also be extended to solve theproblems of stabilizing integral processes and unstableprocesses with H- PID controllers or H2 PID controllers.

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