CONTROL FOR INTEGRATING PROCESSES BASED ON NEW MODIFIED SMITH PREDICTOR

Zhu Hongdong*, Li Ruixia†, Shao Huihe*

*Department of Automation, Shanghai Jiao Tong University, Shanghai 200030, P.R. China. Email: hongdongzhu@sjtu.edu.cn, Fax:+86-21-62932138.

†Department of Information Engineering, Shandong Jiaotong University, Jinan, Shandong Province, 250023, P.R. China. Email:liruixia-2@163.com.

Keywords: Modified Smith Predictor; Integrating Process; Robust specification; PID control.

Abstract

This paper presents a new modified Smith Predictor called Cascade Smith Predictor (CSP) to control the integrating processes with long dead-time, CSP has two Smith Predictors. The inner predictor keeps the actual process stable and converts the integrating process into a stable one, while the outer smith predictor reject the load disturbance. Combined with a good tuning rule introduced in this paper, the CSP control structure only need to adjust one parameter to control the integrating processes with long dead-time, which make CSP show better performance than that of other Smith predictors.

1 Introduction

In process industry, most processes are stable. Many efficient methods were developed to control this kind of processes in the past decades. In recent years, the control of integrating plus dead-time processes has attracted much attention because of the great difficulty. The most widely used controller is still PID in the process control field. More than 95% of the control loops are of PID controller [5]. Many researchers were devoted to obtain the tuning formulae, such as ZN tuning rule, Refined ZN rule, the optimal tuning rules based on the non-convex optimization and robust specification, and the tuning rule based on Internal Model control and H∞ optimal control. However, when the processes exhibit large dead-time the PID controller should be detuned to keep the robust stability of the closed-loop system. But the performance will be degraded because there is no predictive part in PID control structure. In 1957, Smith proposed a simple yet powerful control strategy – Smith Predictor (SP) in [10]. Although the performance of the closed-loop system with SP is highly enhanced, offset problem will emerge when a load disturbance is injected into the integrating system. Therefore, many kinds of modification to the original Smith Predictor were proposed to control the integrating processes plus long dead-time. Watanabe & Ito proposed a modified Smith predictor in [7], which is called Process Model Control. In 1994, Astrom, Hang & Lim

presented a two-degree-of-freedom (TDOF) modified Smith Predictor [6]. Three parameters should be tuned to obtain the desired performance. Matausek & Micic also proposed a modified Smith Predictor to control the integrating plus dead-time processes in 1996 and improved its performance in 1999 [8,9]. In 1998, a new Smith Predictor that can control the stable, integrating, and unstable processes by setting the three PID controllers in the structure, was presented by Majhi & Atherton in [11]. Then they proposed an auto-tuning method for this kind of modified Smith Predictor in 2002 [12]. Kaya & Atherton presented a PI-PD Smith predictor to control the process with long dead-time and gave an autotuning method [1, 2], respectively. Normey-Rico & Camacho proposed a new dead-time compensator to control the integrating processes [3, 4]. However, the controllers proposed by Normey-Rico & Camacho and Matausek & Micic are only fit for the integrating plus dead-time (IPDT) processes. Although Matausek & Micic [9] proposed a rule to reduce the high order integrating model into the IPDT model and corresponding tuning rule, the response is too lag (see [2]). In this paper, we proposed a Cascade Smith predictor (CSP). It can be used to control IPDT and the integrating first order plus dead-time (IFOPDT) processes. There are two cascaded Smith predictors in CSP. The inner-loop of CSP keeps the actual process stable and converts the integrating process to the first-order plus dead-time (FOPDT) process. The outer-loop rejects the disturbance. A powerful tuning rule is also proposed. The tuning parameters have physical meanings and are familiar to the process engineers. This proposed method achieves better results comparing with other ones. This paper is organized as follows. In section 2, we introduce the idea of CSP. In section 3, we give a powerful tuning rule for the IPDT processes and the IFOPDT processes. In section 4, we present the simulation results comparing with the other methods. At the last section, some conclusions are given.

2 Cascade Smith Predictor

The two-degree-freedom modified Smith Predictor proposed by Astrom & Hang [6] is shown on Fig.1. M(s) rejects the disturbance and keeps the actual process stable. M(s) should have the integrating factors to eliminate the state-state error. If M(s) is directly chosen as a PI controller, the integrating time constant should be large to keep the closed-loop stable. Thus, the recovery time will be huge. In [6], in order to obtain

Control 2004, University of Bath, UK, September 2004 ID-219

the desired disturbance rejection performance, three parameters, k1, k2, and k3, should be selected elaborately. However, the paper did not give a systematic method to choose these parameters. The main difficulty in the design of M(s) is to trade off between the closed-loop stability and the reduction for load disturbance. Due to the existence of dead-time, M(s) should be detuned to keep the closed-loop system stable. If M(s) is chosen as the proportion controller with the appropriate proportion, the closed-loop system will be stable but can not reject the load disturbance.

In order to solve this dilemma, we decouple the disturbance rejection function of M(s) by introducing an outer Smith Predictor to reject the load disturbance. Therefore, the function of M(s) is reduced to keep the process stable and can be designed easily. The new structure is shown on Fig.2, where 0 ( ) LsG s e− is the real process, Gm0(s) is the delay-free part of the process model, Lm is the estimated dead-time of the actual process, Pg(s) is a first-order model which is the delay-free part of the inner-loop model, Gc(s), Gc1(s), and Gc2(s) are the outer-loop controller, the inner-loop controller, and the stabilizing controller, respectively. We call this control structure “Cascade Smith Predictor (CSP)”. It has slight difference from the conventional cascade controller used in the process controlling. The term “cascade control” in process controlling means that two cascade controllers control two different cascade physical variables, while two SPs in CSP control the same process variables. The inner Smith Predictor stabilizes the actual process and transforms the integrating process to the FOPDT process. The load disturbance can be reduced by the outer-loop Smith predictor. The simplified CSP is shown on Fig.3. It is similar with the control schemes proposed in [1,11], but CSP has two advantages. Firstly, in nominal case the transfer function from the set point r to the process output y can be easily tuned to FOPDT dynamic in controlling the IPDT or the IFOPDT processes while the previous two control structures can only be tuned to second order plus dead-time (SOPDT) dynamic. Therefore, in our control scheme, only one parameter λ is directly related to the performance of closed-loop system. Secondly, with the powerful tuning rule, the performance and robustness of CSP are better than that of the previous two control schemes. In nominal case, the transfer functions from setpoint r, and disturbance d to process output y can be given by:

1 0

0 1

( )1 (1 )

Lsc c myr

m c c

G G GG s eG G G

−=+ +

(1)

0 1 0

0 1 2 0

1 (1 )( )1 (1 ) 1

mL sLsm c m c c

yd Lsm c c c m

G e G G G G eG sG G G G G e

−−

−

+ + −=

+ + + (2)

According to Equation (2), the characteristic equation of the closed-loop system is composed of two parts: 0 11 (1 ) 0m c cG G G+ + = and 2 01 0Ls

c mG G e−+ = . The former part can be stabilized by properly designing Gc(s) and Gc1(s). The stability of the second part is only related to Gc2(s). Then, it is convenient to design Gc(s), Gc1(s), and Gc2(s) independently.

3 Tuning Rule for proposed CSP

In process control fields, the IPDT and IFOPDT processes are two typical integrating processes. Assuming that they can be represented by IPDT model:

0( ) m mL s L smm

KG s G e es

− −= = (3)

and IFOPDT model:

0( )( 1)

m mL s L smm

m

KG s G e es T s

− −= =+

(4)

then the integrating high order plus dead-time process can be approximated by (2) through the model reduction. There are three controllers, Gc(s), Gc1(s), and Gc2(s) to be tuned in our proposed control scheme. Firstly, we discuss the tuning rule for Gc2(s) since Gc(s), Gc1(s), and Gc2(s) can be designed independently. The two goals to design Gc2(s) are to keep the roots of 2 01 0Ls

c mG G e−+ = on the left part of complex plane and guarantee enough phase margin Φm and amplitude margin Am of the closed-loop system with the loop transfer function 2 0( ) mL s

c mL s G G e−= .

G0(s) Lse−

+

−

Fig.3. Simplified Cascade Smith Predictor.

d

Gc1(s) r y

－

+

−

Gc2(s)

Gc(s)

Inner-loop Smith Predictor

Outer-loop Smith Predictor

－ －

Gm0 (s)

mL se−

+

−

Fig.2. Original Cascade Smith Predictor.

+ Gc1(s) r y

mL se−

−

+

−

Gc2(s)

Gc(s)

Pg(s)

mL se−

+

− Inner-loop Smith Predictor

Outer-loop Smith Predictor

−−

Gm0 (s)

G0(s) Lse−

−

LsK es

− +

−

Fig.1. TDOF Smith Predictor proposed by Astrom & Hang.

+ d

K r y

mL se−

1/ s

−

+

−

M(s)

e1

d

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For the IPDT process (3), both the PD and the P controller can stabilize G(s). There are various methods to design Gc2(s). In [8], the P controller was used and the corresponding tuning rule was given. [9] used a PD controller and obtained good results. We also adopt the PD controller, and the parameters are chosen as in [9]. Gc2(s) is a practical PD controller with the following form:

2 22

2

( 1)( )( / 1)

c dc

d

K T sG sT Ns

+=

+ (5)

Here, N is chosen as in [10]. According to [9], Td2 and Kc2 are chosen as follows:

2

2 2 2 2

/ 2(1 ) ( / 2 )

d M

mc

m M m

T L

KK L

απ

α π α

=−Φ

=− + −Φ

(6)

where LM equals Lm when controlling the IPDT process (3). Matausek & Micic suggested a and Φm to be 0.4 and 64o, respectively. For the IFOPDT process (4), we also use the PD controller, but the tuning rule is based on another robust specification. Here we introduce a robust specification γ[13], which is defined as:

0

1 max Re[ ( )]L jω

ωγ ≤ ≤∞= (7)

where the quantity γ is the simple inverse of the maximum absolute real part of the loop transfer function. The reasonable values of γ are in range from 1.5 to 2.5. According to [13], the relationship between the new robust specification γ, the phase margin Φm and the amplitude margin Am are

arccos(1/ )m

m

A γγ

>Φ >

(8)

Here, Gc2(s) has the same form as (5). Since N=10, we conveniently use the following approximate form:

2 2 2( ) ( 1)c c dG s K T s≈ + (9)

If we choose Td2= Tm, 22 0( ) m mL s L sm c

c mK KL s G G e e

s− −= = , The

maximum of the absolute real part of the loop transfer

function 2( ) mL sm cK KL s es

−= is

20max Re[ ( )] m c mL j K K Lω

ω≤ ≤∞

= (10)

Then, Gc2(s) is chosen as:

2 2 21( ) ( 1) ( 1)c c d mm m

G s K T s T sK Lγ

= + = + (11)

When γ is chosen as 1.5, 2, and 2.5, the corresponding phase margin Φm are about 51.8 o, 61.4 o, and 67.1 o, respectively. In this paper, we choose γ as 2. Remarks:

1) It is should be noted that Φm is not the phase margin of the closed-loop system given by Fig.3. Φm is that of the system with the loop transfer function 2 0( ) mL s

c mL s G G e−= . 2) For the IFOPDT process, the form of Gc2(s) can be

practically used as in (5) because N is large (=10). In

fact, all the PD controllers used in CSP have practical form as in (5) in simulation because the ideal PD controller cannot be implemented practically.

Secondly, we introduce the tuning rule for Gc(s) and Gc1(s). For the IPDT process (1), Gc1(s) is a P controller and Gc(s) is a PI controller. It is our tuning principle to tune the transfer functions from the setpoint r to the process output y to the first order plus dead-time dynamics:

1( )1

mL syrG s e

sλ−=

+ (12)

Assume that Gc(s) and Gc1(s) have the following forms:

1 1

1( ) (1 )

( )

c ci

c c

G s KT s

G s K

= +

= (13)

Let

11

1,

cm

c i

KK

K Tλλ

=

= = (14)

Then, the tuning principle (12) can be achieved. In this case, the transfer function from the disturbance d to the process output y is

2

2 2 2

1( )1

( / 1) ( / 1) ( 1)

mm

m

L sL sm

yd

dL s

d m c d

K es eG ss s

s T Nss T Ns K K T s e

λλ

−−

−

+ −=

++

+ + +

i

i (15)

For the IFOPDT model (4), make Gc1(s) be a PD controller and Gc(s) be a PI controller:

11 1 1 1

1

1( ) (1 )

1( ) ( 1)/10 1

c ci

dc c c d

d

G s KT sT sG s K K T s

T s

= +

+= ≈ +

+

(16)

Our tuning principle is same as (12). Then,

1 11 ,

1,

c d mm

c i

K T TK

K Tλλ

= =

= = (17)

Therefore, the transfer function from the disturbance d to the process output y is

2

2 2

1( )1 ( 1)( / 1)

( / 1)

mm

m

L sL sm

ydm

dL s

d m c

K es eG ss s T s

s T Nss T Ns K K e

λλ

−−

−

+ −=

+ ++

+ +

i

i (18)

Since0

lim ( ) 0ydsG s

→= , the constant load disturbance can be

rejected by the closed-loop system.

4 Simulation Results

Example 1. Consider the integrating process 51( ) sG s e

s−= . (19)

Control 2004, University of Bath, UK, September 2004 ID-219

Almost all the previous works discussed this process and compared the performance and robustness with others. Comparisons among our proposed method, and the controlling methods proposed in [4, 9, 12] in the nominal case and the case where dead-time varies form 5 to 5.5 are proposed. It should be pointed out that the comparisons should be performed in a fair circumstance. That means the fast response is not the only specification because each proposed method has at least one free parameter which is related to the performance of the disturbance rejection and the robustness of the closed-loop system. Unfortunately, some comparisons in previous works are in unfair circumstance. In this paper, we firstly adjust the free parameters of the control methods mentioned above to obtain the similar disturbance rejection performance in nominal case by fixing the free parameters of all controllers and comparing the disturbance rejection performance when dead-time is varied to 5.5. In this example, we choose λ as 4. According to the tuning rule (6) and (14), Kc=1, Ti=4, Kc1=0.25, Kc2=0.1206, Td2=2. The controller parameters in [4] are set as: Kc= 0.1559, Ti= 13.8569, T1=4, Dm=1.6, T0=4.4284. The controller parameters in [9] are set as: Kr= 0.25, Tr= 4, K0=0.1206, Td=2. The controller parameters in [12] are set as: Kf=1, Kp=0.25, Ti= 1, Kd= 0.1047. The responses of the closed-loop system for these controller settings are given in Fig.4 and Fig.5, in nominal case and the model mismatch case, respectively. A unity setpoint change and disturbance of d=-0.1 are introduced at t=0 and 100, respectively. In this example, our result is similar to the ones obtained by [4, 9]. The result obtained by [12] is the worst in the four results because it has large overshoot when a load disturbance is introduced to the closed-loop system. However, the proposed methods in [4,9] are only fit for the controlling IPDT processes. Example 2. Considering the integrating first order process plus dead-time:

6.71( )(10 1)

sG s es s

−=+

, (20)

which was used in [2], since the control structures proposed in [9,4] are only fit for the controlling of the IPDT processes, comparisons are made among the proposed method and the methods proposed in [2] and [12]. The parameters of the proposed controller are set as: λ=3, Kc=1, Ti=3, Kc1=1/3, Kc2= 0.0746, Td2=10. In [12] the controller parameters are set as: λ=3, Kf=2/3, Tf=10, Kp=10/9, Ti= 10, Kd= 0.0781. In [2], Ts=5, Kf=1, Tf=10, Kp=2.5, Ti= 10, Kd= 0.0434, Td= 6.6800. the response to the change of a unity setpoint and the disturbance of d=-0.1 at 100 in nominal case and the model mismatch case (where dead-time of process is perturbed to 7.4) are shown on Fig.6 and Fig.7, respectively. It is evident that the performance of the proposed method is the best among the three controllers.

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

1.2

1.4

Time(s)

Pro

cess

out

put

ProposedMajhi_2000Kaya_2003

Fig. 6. Response for example 3 in nominal case.

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

Time(s)

Pro

cess

out

put

ProposedNormey-Rico_2002Matausek_1999Majhi_2000

Fig. 5. Response for example 1 where dead-time is 5.5.

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

Time(s)

Pro

cess

out

put

ProposedNormey-Rico_2002Matausek_1999Majhi_2000

Fig. 4. Response for example 1 in nominal case.

Control 2004, University of Bath, UK, September 2004 ID-219

5 Conclusions

In this paper, a new modified Smith Predictor, which is called Cascade Smith Predictor, is proposed to control the IPDT processes and the IFOPDT processes. It is composed of two cascade loops: the function of the inner-loop is to stabilize the integrating process and convert it to a generalized first order plus dead-time process, while the function of the outer-loop is to reject the load disturbance. A new robust specification is used when tuning the parameters for the IFOPDT process. Two sets of tuning formulae are also proposed to control the typical IPDT processes and the IFOPDT processes. The simulation results show that: 1) for the IPDT processes, the response of the proposed method is as good as that of the proposed in [9,4] in nominal case and the model mismatch case. 2) For the IFOPDT processes, it is evident that the response of the proposed method is the best.

Acknowledgements

This research was supported by “863” High-Tech Research & Development Program of China, Grant No. 2001-AA413130.

References

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0 50 100 150 2000

0.5

1

1.5

Time(s)

Pro

cess

out

put

ProposedMajhi_2000Kaya_2003

Fig. 7. Response for example 3 where dead-time is 7.4.

Control 2004, University of Bath, UK, September 2004 ID-219