Constraint Optimization of dead-time processes using Smith-Predictor

Rameez Hayat1, Nisar Ahmed2

faculty of Electronic Engineering GIK Institute of Engineering Sciences and Technology

Swabi, Pakistan 2F acuity of Electronic Engineering

GIK Institute of Engineering Sciences and Technology Swabi, Pakistan

rameezhayat@hotmail.com, nisarahmed@giki.edu.pk

Abstract—This paper presents a modified Smith-Predictor to optimize disturbance-rejection performance with constraints on robustness and reference-tracking. A 2-degree-of-freedom Smith-Predictor is used to deal with the set-point tracking and disturbances separately while ensuring a certain level of maximum sensitivity. So the paper covers three aspects of control system that are the set-point tracking, robustness and disturbance rejection. Constraint optimization algorithm is used to achieve the desired parameters of the controllers. A second order approximate model using Ziegler-Nichols dynamics characterization is used to get a generalized mathematical model for any process that includes dead time. Robustness is achieved by setting a maximum value of sensitivity function while taking optimized values of the controllers for the disturbance-rejection.

Index Terms—Smith-predictor, Constraint optimization, Disturbance rejection, Robustness, Set-point tracking.

I. INTRODUCTION

Many of the real world processes have long delay time and a close-loop transfer function of a dead-time process by using the conventional PI/PID controller would not give a satisfactory performance because of infinite number of poles as discussed in [7, 13]. To overcome this problem Smith-predictor [13] was introduced by Smith in 1959 that works on predictions. In this technique an approximate model of the original process without delay is fed back to the controller so that the closed-loop system would not have any concern with the dead-time and the delay is just added at the output. So far there are many modified version of Smith-predictor as in [9]-[12], In [9, 10] a modified smith-predictor (MSP) is used that have a double-controller scheme which isolates the reference-point tracking from the disturbance rejection and thus both can be improved separately. In this paper an approximate model [1] of the process is first obtained that is required for the feedback loop. The second reason for the estimated model is that some times the mathematical model of the process is not available or the process is so complex that approximate model can be a better option to obtain the desired controller parameters. The modified Smith-Predictor [9] is shown in Fig. 1 where R, D, Y, Gp, Gp* , Gci and Gci are the reference-input, input-disturbance, output, original process model, estimated

process model and the two controllers respectively. In this paper all the equations are in Laplace transform and '(s)' is ignored for convenience. The transfer functions for the block-diagram of Fig. 1 are:

Y = GclGpe~ds 1 + Gc2Gp*e~d*s

R 1 + GclGp* 1 + Gc2Gpe_ds C }

Y G„e_ds

- = (Z) D 1 + Gc2Gpe_ds ^ J

As mentioned above the set-point tracking is completely separated from the disturbance rejection but with a condition that the estimated model must match the original model. And a simple pole placement technique can be used to obtain the desired pole location as done in this paper. A PID (Gci) controller is used to obtain the controller parameters for the pole-placement technique. The poles are obtained for the approximated model and then tested on the original processes that show very close and resembling performance with the estimated model. The main priority in designing a controller depends on the system, e.g a servo-motor need a good reference-tracking but many a times disturbance-rejection is the first priority and this paper takes the disturbances as the main priority. For this purpose a PI (Gc2) controller that has constraints on maximum sensitivity is taken (as done in [4, 5] for a conventional PI/PID control system). The sensitivity function for the double-controller scheme MSP is also derived in this paper which is used in the optimization algorithm. Similar to the PID controller, the PI controller parameters are also obtained for the approximated model and then tested on the original processes. One advantage of using an approximate model is its easiness in obtaining the controller parameters for any process by just varying few values of the process transfer function while using a simulation tools. Section 2 explains the design technique used in this paper. Section 3 consists of design procedure and examples of dead time process. Section 4 shows the simulation results and in the end are the conclusion and references.

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Figure 1 : Double-controller scheme MSP

II. DESIGN FORMULATION.

The design technique used in this paper is such that it covers many aspects of a modern control system as explained below. The system is analysed in the following order.

A. Approximation.

The first step is to approximate the process model by a second order Ziegler-Nichols model [1], The advantage of this technique is that it requires very little information about the process, only the ultimate gain ku and ultimate frequency cou

and the angle 0 where the phase of the process is -180° and yet shows a very accurate approximation, all the parameters mentioned above can be acquired experimentally. Let the approximate model Gmis given by:

e-Ls G m ~~ as2 + bs + c (3)

where

tPm

a = ku((jL>u - A + ACOS(CJL>UL))/(AWU2)

b = (ku sin(wuL))/(wu) c = ku(wu - A)/A min|cpm-cp|

4(oùu - A) sin(ü)uL) + Asin(2wuL) + 2AwuL 4(oùu - A) cos(wuL) + Acos(2wuL) + 3A

wukuGp(0) A =

l + k u G p ( 0 )

Inserting the values of ku, u)u and 0 that are obtained experimentally one get the estimated model Gm.

B. Set-point tracking

From (1) if the approximate model is assumed to be perfectly matched with the original model than the input-output relation is

Y = Gc lGpe-ds

R 1 + GclGp* (4 )

The set-point-tracking involves only one controller Gcl; a PID controller can easily be designed by placing the poles in the desired position. Using (3) and (4)

Y R

kds2 + ks + k¡ as3 + (b + kd)s2 + (c + k)s + k¡ (5)

Where kd, k and kt is the derivative, proportional and integral gain respectively. The procedure would be first to take a desired transfer-function and than set the values of the PID gains in such a way that it matches the desired transfer-function

C. Disturbance rejection

There are many ways of measuring the disturbances like integral error IE, integral absolute error IAE etc. For the Modified Smith-Predictor disturbances can be minimized by maximizing Gc2 as is evident from (2).

| G C | 2 = k2 + G ) 2 (6a)

Disturbances are in general of zero frequency so (6a) can be written as

(6b)

All that one required to suppress the disturbances is by maximizing integral gain. Another way to prove the dependence of the disturbances on integral gain is explained in [8]

D. Sensitivity

The sensitivity function for a control system which is the ratio of the percent change in the total transfer function to the percent change in the process transfer function is given below

9 t / T S = dG,

' / G

(7)

T is the closed-loop transfer function and Gp is the process transfer function, after some calculations the sensitivity function for the modified smith-predictor of the block diagram of Fig. 1 is

S = 1 + G C 2 G p e - ds

(8)

Nyquist-plot is used as a tool to visualize that the maximum sensitivity Ms should not be exceeded from a specified value to maintain a certain level of robustness.

E. Noise filter

A first order noise filter is also used with the 1st

controller Gcl. The advantage of using the noise filter is to make the controller a proper function and secondly to suppress any kind of noise that is normally of high frequency.

III. PROCEDURE

The approximation of the process by a second order model and the reference-tracking is straightforward and can easily be achieved by using (3) and (5). To maximize the integral gain under the constraints of maximum sensitivity Ms some optimization technique must be used. From (8)

f(k, ki, w) = | l + Gc2Gpe-

For maximum sensitivity

f(k, ki, OÙ) > 1/MS2

(9)

(10)

Thus to have a robust control system equation 9 must be maximized to get a minimum sensitivity such that the system should not exceed a certain maximum sensitivity Ms. As shown in [2]

df df df df = — dk + — dkj + — dw = 0

ok okj o OÙ (11)

Using Nyquist-plot of Gc2Gpe-ds the function (see Appendix A)

3f(k, ki, OÙ)

du> = 0 (12)

Similarly for maximum integral gain, dkt = 0 (see reference [2]). Applying these conditions on (10)

equation would be just a function of o>. The unknown variable o> can be acquired by using some numerical tools. Newton-Raphson method or bisection method can be used to find the unknown parameter co quite easily.

IV. SIMULATIONS

The design procedure for the constraint optimization in this paper consists of the following steps.

• 2nd order approximation using modified Ziegler-Nichols dynamic characteristics.

• Set-point tracking using Pole-Placement technique.

• Finding 2nd controller (Gc2) parameters by using the constraint algorithm.

The test examples used in this paper consists of variety of processes. These examples are

Gi =

G, =

(s + l ) 2 0 ' G, =

-5s

1 - 1.4s -3.5s (s + 1) 20

3 (s + l)(0.5s + l)(0.2s + l)(0.1s + 1)'

1 G4 = (1.46s + 0.999)2

-5.07s

The first step is to find the 2nd order approximate model (3) of all the processes. The results using MATLAB are:

Table 1. ESTIMATED MODEL PARAMETERS

processes a b c L GI 15.031 7.059 1 13.125 GZ 1.345 1.991 1 5.500 G3 0.784 1.747 1 5.800 G 4 2.131 2.920 1 5.070

df

Now there are three unknowns k, ki and o> and three non-linear equations 10, 12 and 13. Putting the approximate model parameters (3) for Gpe-ds in the above equations and solving (10) and (13)

k = bsin(ooL) - (c - aoo2) cos(ooL) (14)

ki = co[sin(coL)(c - aco2) + b cos(coL) - V(c - aco2)2 + b2/Ms (15)

For every co there are two values for the integral gain. The desired one will be the larger value of the integral gain but the problem is that it will make the overall system unstable because there are no constraints on the poles of the system So to avoid the system to be unstable the lower value of integral gain is taken. Now putting (14) and (15) into (12), the new

The parameters for Gcl are acquired by using equation (5), the desired poles locations in this paper are set according with Ç = 0.707 and Tp =5 seconds. These values are set by the designer to have a desired input response. There are three poles in equation (5). Two poles are set for the values of peak time and damping ratio and the third poles is set with real part five times away (negative axis) from the first two roots. So the total desired characteristic equation is

A= s 3 + 4.4s2 + 4.73s + 2.48 (16)

The values for the PID controller are given in table 2 that set the total set-point response to be equal to (16). The simulation results for a step input is shown in Fig. 2 (ignoring the delay), it is worth noting that these parameters are evaluated for the approximate model. The simulation results for step input using the same values for the original process are shown in Fig. 3.

Table 2. PID CONTROLLER PARAMETERS

kd k ki G I 5 9 . 0 4 7 0 . 0 4 3 7 . 2 8

G2 3 . 9 2 5 . 3 7 3 . 3 3 G3 1 . 7 0 2 . 7 1 1 . 9 4

G4 6 . 4 5 9 . 0 9 5 . 2 8

Figure 2: Unit-step response using approximate models

Figure 3: Unit-step response using original models

The 2nd controller (Gc2) parameters are evaluated using equation 10, 14 and 15. The values are set according to the specified value of maximum sensitivity that should not be exceeded. The Nyquist-plot for the test examples are shown in Fig. 4.

Figure 4: Nyquist-Plot of four processes

The values of PI controller are shown in table 3 for both the approximate model and the original processes. From the table below it is quite clear that the parameters of the PI controller

for both the estimated model and the original model are very close and almost equal. So to avoid heavy mathematical calculations for very complex systems such as the first system in the test processes, one can use the approximate model to design the PI controller.

Table 3 . CONTROLLER G C 2 PARAMETERS

Ü) k ki Tf GI 0.087 0.194 0.025 0.004 Ci 0.087 0.191 0.025 0.004 G2* 0.232 0.174 0.065 0.025 G2 0.223 0.174 0.062 0.025 G3* 0.231 0.175 0.065 0.033 G3 0.255 0.163 0.072 0.033 G4* 0.222 0.205 0.066 0.025 C4 0.222 0.205 0.066 0.025

Tf is the time constant for the noise filter used with the first controller Gcl. Note that noise filter has no effect on the second controller Gc2 and thus do not change the robustness or the disturbance-rejection performance of the system. The step input response with a step disturbance response for each of the four processes are shown in the figures below: (original transfer function in solid lines, approximate transfer function in dashed lines)

Figure 5: Unit-step input response of

Figure 6: Unit-step input response of C2

Figure 7: Unit-step input response of C3

but

dî _ f(w + do) - f(w) d u du

f(w + do) - f(w) = 0

(17a)

(17b)

Figure 8: Unit-step input response of G4

V . CONCLUSIONS The optimization technique is applied for the Smith-

predictor that shows remarkable results for dead-time processes. The double controller scheme with the approximate 2nd order model of the original process makes it easy to design the controllers for the dead-time processes that shows good response for both reference-input and for disturbances. Secondly the approximation helps in generalizing the technique for any kind of complex or even unknown processes. Using the double controller Smith-predictor the set- point tracking is achieved very easily by using Pole-Placement method and the minimization of the integral gain under the constraints on maximum sensitivity gives good disturbance-rejection capability providing a certain level of robustness.

APPENDIX

A. PROOF OF r = ° Où) A graphical and intuitive approach is used to show

that df/dui = 0. Shown below is a figure of Nyquist-curve for a general system and a circle with radius 1/Ms centered at -1 that is the critical point on the Nyquist-Plot that touches the Nyquist-curve at only one point.

0.8

0.6

0.4

0.2

ki 0

-0 .2

-0.4

- . 0 6

- . 0 8

-1.8 -1.6 -1.4 -1.2

Figure 9: Nyquist plot

That proves df/diù = 0.

REFERENCES

1. T. B. "Sekara and M. R. Matau'sek, "Revisiting the Ziegler-Nichols process dynamics characterization," J. Proc. Control 20 (2010) pp. 360-363,2010.

2. Q. G. Wang, T. H. Lee, H. W. Fung, Q. Bi and Y. Zhang, "PID tuning for improved performance," IEEE Trans. Contr. Syst. Technol. 7, pp. 457-465,1999.

3. Xiumei Wu, "Modified Smith predictor for disturbance rejection of single sinusoidal signal," 8th IEEE International Conference on Control and Automation, Xiamen, China, June 9-11,2010.

4. H. Panagopoulos, K. J. Aström and T. Hägglund. "Design of PID controllers based on constrained optimization," IEE Proceedings-Control Theory & Applications, 149:1, pp. 32-40,2002.

5. K. J. Aström, H. Panagopoulos, and T. Hägglund, "Design of PI controllers based on non-convex optimization," Automatica, 34, (5). DD. 585-601,1998.

6. A. J. Isaksson and S. F. Graebe, "Derivative filter is an integral part of PID design," IEE Proc. Contr. Theory Appl. 149, pp. 41-45,2002.

7. J. E. Normey-Rico and E. F. Camacho, Control of Dead-time Processes, Springer-Verlag London Limited 2007.

8. Astrom, K. J. and T. Hägglund, PID Controllers: Theory, Design, and Tuning, 2nd edn. Instrument Society of America, 1995, Research Triangle Park, NC.

9. Y. C. Tian and F. Gao, "Double-controller scheme for control of processes with dominant delay," IEE Proc-Control Theory Application, Vol. 145, No. 5, September 1998.

10. K.. J. Astrom, C. C. Hang and B. C. Lim, "A new smith predictor for controlling a process with an integrator and long dead-time," IEEE Trans. Automat. Contr. 39, pp. 343-345,1994.

11. Somanath Majhi, D. P. Atherton, "Obtaining controller parameters for a new Smith predictor using autotuning," Automatica 36, pp. 1651-1658,2000.

12. K. Watanabe and M. Ito, "A process-model control for linear systems with delay," IEEE Trans. Automat. Contr, vol. AC-26, no. 6, pp. 1261-1266, Dec. 1981.

13. O. J. Smith, "A controller to overcome dead time," ISA J. 6, pp. 28-33, 1959.

14. Richard C. Dorf and Robert H. Bishop, Modern Control Systems, Twelfth edn. Prentice Hall.

15. J. G. Ziegler and N. B. Nichols, "Optimum settings for automatic controllers," Trans. ASME 64, pp. 759-768,1942.

1 k -0.8 -0.6 -0.4 -0.2 0

From Fig. 9, /(&>) and /(o> + dco) are both the radius of the circle where do> is a small change in the frequency. So by using numerical method, the forward difference for the above Nyquist-plot is given as: