MPC PID2.pdf

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1 INTRODUCTION

Proportional-integral-derivative (PID) controllers and

model Predictive Control (MPC) are two control

algorithms commonly found in industrial applications. Thethree parameters commonly found in industrial

applications. The three parameters of a PID controller

must be tuned to the process to obtain a satisfactory

closed-loop process performance.

Over the years, amounts of techniques have been

suggested for tuning of the PID parameters. Explicit

relations for tuning PID controllers were proposed in

(Zieglers and Nichols, 1942). Astrom and Hagglund (1988)

developed a feedback auto tuning of PID controllers. Over

15 years ago, because of its simplicity and efficiency, relay

based auto tuning methods have been integrated into

commercial controllers, which have been successful in

many process control applications [1]. The structure ofPID controllers is discussed in with respect to self-tuning

algorithms and automatic selection of structures.

On the other hand, MPC method has become one of the

most popular control methods both in industry and

academia. The MPC provides an analytical solution; it can

deal with unstable and non-minimum phase plants. The

MPC is an optimal method which incorporation of

weighting of control increments in the cost functions.Because of wide application of the PID controller, many

researchers have attempted to use advanced control

techniques such as optimal control and MPC to restrict the

structure of these controllers to retrieve the PID controller

[2].Rivera et al . (1986) introduced an IMC based PID

controller design for a first order process model. Chien

(1988) extended IMC-PID controller design to cover thesecond order process model. Morari and Zafiriou (1989)

have shown that Internal Model Control (IMC) leads to

PID controller for virtually all models common in

This work is supported by National Nature Science Foundation under

Grant 60974126 and Jangsu Province Nature Science Foundation under

Grant BK2009094

industrial practice [3]. In wang et al. (2000) a least square

algorithm was used to compute the closest equivalent PIDcontroller to an IMC design and a frequency response

approach is adopted. However, the design is still

ineffective when applied to time-delay system. Marques

and Fliess (2000) have developed a simple approach forPID control of linear continuous systems based on flat

output trajectory generation [4]. Important characteristics

of model predictive methods have been combined with

PID control properties by considering flatness based predicted trajectory. The real time results showed that their

methods are applicable and efficient.

In the industrial process control field, due to the aftereffect

phenomena occurred in the real processes’ inner dynamics,

the variety of transport delays, communication delays

involved by the sensors and transducers, and the

computational time, time-delay effects are inevitable.

Nowadays, the performance of traditional controllers is notsatisfactory. Therefore, the design and application of the

advanced controller which takes the time-delay into

account are necessary [5].

MPC is a form of control in which the current control

action is obtained by solving on-line, at each sampling

instant, a finite horizon open-loop optimal control

problem, using the current state of the plant as the initial

state; the optimization yields an optimal control sequence

and the first control in this sequence is applied to the plant.

On the basis of model predictive control, we unify the PID

control method to form a new predictive control algorithm

by changing objective function, enable it has generalized

structural feature of proportion (P), integral (I), and thederivative (D).There is also a certain similarity between

the parameter setting and the conventional PID parameter

variation[6]. The PID controllers are widely applied in the

industrial field. PID controller has many advantages, such

as simple principle, good robustness and have the strong

generality and in general use with function and so on.

However, in the time-delay system, while the normalized

time delay constant is bigger 0.6), a PID controller is

almost ineffective[7]. An improved PID controller based

Tuning of the PID Controller Based on Model Predictive ControlXiaoping Ma

1, Xin Song

1, Huijun Li

1,2, Jieru Niu

1

1. College of information and electric engineering, China University of Mining and Technology

XuZhou, JiangSu 221008, ChinaE-mail: [email protected].

2. Institute of Automation Chinese Academy of Science Beijing 100190, China.

E-mail: [email protected]

Abstract: In this paper an algorithm of PID controller based on model predictive control is derived. With this algorithm the three parameters of a PID controller are tuned to the process to obtain a satisfactory closed-loop process performance. Using this algorithm for

time-delay system is proposed to enhance the real-time performance and the reliability of process control. Analyzed the three parameters

tuned of a simple PID form, the effect is not ideal. A new structure is discussed which provides an effective way of to solve this problem,which introduces feedback from actuator output to the controller. The new structure with constraint provides an effective way of

modeling and control of process. It was suitable to be applied to the real industrial process field.Key Words: predictive control, PID controller, time-delay system, algorithm

1091978-1-4244-5182-1/10/$ 26.00 c 2010 IEEE



on MPC can meet the control requirements of thetime-delay systems.

2 MODEL PREDICTIVE CONTROL

Model Predictive Control (MPC) has an unusual history,

with separate strands of development in system theory

(where it is generally referred to as receding horizoncontrol), generalized predictive control (where the initialintention was the improvement of adaptive control) and in

process control where its almost unique ability to handlehard constraints has led to its wide scale adoption inindustry, making it the industry standard in some

important application areas. Indeed it might be justifiably

being argued that the importance of model predictivecontrol derives primarily from its industrial success, a factthat delineates it from other design procedures that are, in

general, theoretically motivated [4].

MPC is an advanced control algorithm based on model,which is divided into different types according to thedifferent model it used, such as Dynamic Matrix Control

(DMC) Model Algorithmic Control (MAC)

Generalized Predictive Control (GPC) Generalized

Predictive Control of Poles (GPP) and Inferential Control(IC) etc [3].Prediction and control in different level makes the system

design more flexible. Based on practical feedback

information, this iterative optimization has strongadaptability for uncertainties such as modeling errors andenvironment disturbance. Taking the control incremental

sequence into account in the objective function, the MPC

applies to the long time-delay, non-minimum phase, aswell as non-linear processes, and conditions to bettercontrol effect. Therefore it has attached great importance

of the control engineering, presented a number of newalgorithms and won tremendous successes on industry [8].

A MPC controller is composed of 3 parts: predictive model

rolling optimization and feedback correction. Outputerror model (OE) was used to construct a MPC in this paper.Suppose the pulse transfer function of a time-delay process

as follows:

)(1

)()(

2

2

1

1

2

2

1

10 k u z a z a z a

z b z b z bb z k y

n

n

m

m

d

−−−

−−−−

+++

++++=

(1)Suppose that =1-z -1, then

1

1

2

2

1

1

2

2

1

1

1

1

)1()(

−−

+

−−−

−−−−

+++++=

∆++++=

n

n

n

n

n

n

z a z a z a z a

z a z a z a z A

(2)Therefore, formula (3) can be deduced according to (1) and

(2) on k+1 time.

)(

)1()(

)1()1()(

10

11

m jd k ub

jd k ub jd k ub

n jk ya jk ya jk y

m

n

−+−∆+

−+−∆++−∆=

−−+++−+++ +

(3)

The predictive model is OE model in the process, so we

need to deduce the algorithm of the predictive controller

based on this model. Assume the predictive length is P ,

control length is C , which satisfies condition C P ,

Define Y U A and B as:

=

++

+++

++

++

++

+

−+

=

2

1

)(

)1(

)(

)2(

)1(

)(

)(

Y

Y

P d k y

C d k y

C d k y

d k y

d k y

d k y

nd k y

Y

=

+∆

+∆

+∆

∆

−+∆

=

2

1

)(

)2(

)1(

)(

)1(

U

U

C k u

k u

k u

k u

mk u

U

][

100

010

001

21

)1(11

11

11

A A

aaa

aaa

aaa

A

n P P nn

nn

nn

=

=

++×+

+

+

][

0

00

00

21

)(

011

011

B B

b

bbbb

bbbb

B

C m P C P

mm

mm

=

=

+×−

−

−

There, A1 is a P (n+1) matrix, A2 is a P P matrix, B1 is a P m matrix and B2 is a P C matrix.

According to (3), we assume j=d+1,d+2, d+P we can

obtain liner equations, and these can be shown as:

)( 11112222 Y AU BU BY A −+= (4)

Because A2 is invertible matrix, we convert (4) into:

)( 1111

1

222

1

22 Y AU B AU B AY −+= −−

(5)

Because of matrix G=A2-1 B2 and vector f=A2

-1(B1U 1-A1Y 1 )

Then we could get the OE prediction equation:

1092 2010 Chinese Control and Decision Conference



f GU Y += 22 (6)

Y 2 and U 2is the predictive output vector and input

increment predictive output respectively in this equation.That is:

+∆

+∆

+∆

=

++

++

++

=

)(

)1(

)1(

,

)(

)2(

)1(

22

C k u

k u

k u

U

P d k y

d k y

d k y

Y

(7)

Formula (6) shows the relationship between future inputand future output on current system status. Process actualoutput couldn’t keep consistent with the set value at any

time due to the real system restrict input signals,. We needintroduce the performance specification:

2222

2

1

2

1

)()(

)]([

)]()([

SU U Y RQY R

jk u s

id k yid k r q J

C

j

i

P

i

i

τ τ +−−=

+∆+

++−++=

=

=

(8)

, where R is the set-point vector whose dimension is P , Q is

diagonal positive definite weighting matrix with P P

dimension used for weighting error signal, S is a positive

definite diagonal weighting matrix with C C dimension

used for weighting input signal.

According to formula (8), the objective function J can beconverted into a standard quadratic form which only

contains input vector U 2:

M U L HU U J ++= 2222

1 τ τ (9)

Expression form of H , L and M as follows:

−−=

−−=

+=

)()(

)(2

)(2

f RQ f R M

QG f R L

S QGG H

τ

τ τ

τ

(10)

The constraint conditions of the system must be considered.Suppose output constraint, control input constraint and

control constraint expressed as follows:

∆≤≤∆

−

−

−

−

≤

−

−

×+×

×

×+

max2min

1)(2min1

1max

min

max

2

)(2)(

)(

U U U

U I k u

I k uU

Y f

f Y

U

V

V

G

G

C P c

C

C C P

(11)

, where

1

11

1

1

×

×

=

C

C

I

V

C C

=

×111

011

001

3 PID AND PREDICTIVE CONTROL

The PID controller is widely applied in industrial field.

Apart from its simple structure and relatively easy tuning,one of the main reasons for its popularity is that it provides

the ability to remove offset by using integral action. Itimproves the performance robustness in the steady state

against noise and uncertainties. Moreover, since PIDcontrollers are so widely used, one might expect that thestructure should arise naturally given reasonable

assumptions on system internal dynamics and control performance specifications [5].This is the continuous form of a PID controller thatcommonly used in industrial process control field:

)()1

11()( s E

sT

sT

sT G sU

F

D

I +++=

(12)After discretization, the formula can be converted as:

)()1

1

11()(

1

k e z

z

hT

T

z T

hGk u

hT

T

F

D

F

F

+−

−

++

−+= .

Converted it into the increment form as:

)()1

211()(

1

211

1

1 k e z

z z

hT

T z

T

h z Gk u

hT

T

F

D

F

F −

+

−−

−−

−

+−

+−+−=∆ ,

where =1-z -1

It can be converted into a discretization incremental form

as follows:

1

2

1 1

)(}))(

)((

)2

(

)1({)(

−

−

−

+−

+−

+

−+

−−

+−

++

+

−=

z hT

T

k e z hT

T

T hT

hT T G

z T

hT

hT

T

hT

T G

hT

T Gk u

F

F

F

D

I F

I F

I

I

F

F

F

D

F

D

(13)

We can assume the pulse transfer function is formula (1). ,We need to analyze formula (4) further in order to explorethe relationship between PID controller and predictivecontrol. Without considering the restrict condition, the

optimal input control variable U 2opt should accord with the below formula:

02)

(2

211

1

2

11

1

222

1

22

1

2

=+−−

+

−

−−−

opt

opt

SU RY A A

U B AU B AQ B A τ

(14)We can obtain the optimal incremental control value:

][

)(])()[(

11

1

211

1

2

2

1

2

1

2

1

22

1

22

U B AY A A RQ

B AS B AQ B AU opt

−−

−−−−

−+⋅

+= τ τ

(15)

Let D = [1 0 0 0]. The predictive controller gives the

control action at time k+1.

2010 Chinese Control and Decision Conference 1093



1111)1( U wY w Rwk u U Y R ++=+∆ (16),

1

1

1)1(

1

1

)(

)2(

)1(

)(

)1()(

)(

)2(

)1(

×

×+

×

∆

−+∆

−+∆

=

+

+−+

−+

=

++

++

++

=

m

n

P

k u

mk u

mk u

U

d k y

nd k ynd k y

Y

P d k r

d k r

d k r

R

Where w RwY1and wU1 are row vectors with weighted

coefficient RY 1 and U 1 are column vectors. Also we can

get expression of Y 1 is:

1)(

0

11

01

)(

)2(

)1(

)(

)1(

)(

×+×+

∆

−+−∆

−+−∆

+

+−

−

=

d mn k u

md k u

md k u

B

k y

nk y

nk y

AY

(17),

where A0 is (n+1) (n+1) matrix and B0 is (n+1)(m+d)

matrix.

Substitution formula (17) into formula (16), we can getthis function:

∆

−+∆

−+∆

+

∆

−+−∆

−+−∆

+

+−

−

+

++

++

++

=+∆

)(

)2(

)1(

)(

)2(

)1(

w

)(

)1()(

)(

)2()1(

)1(

10Y1

01

k u

mk u

mk u

w

k u

md k u

md k u

B

k y

nk ynk y

Aw

P d k r

d k r d k r

wk u

U

Y R

(18)

Simplification of the (18), we got the equation:

)()()()(

)()()1(

11

1

k u z C k y z C

P d k r z C k u

U Y

R

∆++

++=+∆

−−

−

(19)

It shows that at time k+1 the optimal input control actiongiven by the predictive controller relate to not only the

output setting at the moment between time k+d+1 and

time k+d+P , but also the process input increment at the

period of time k-d+1-m to time k . then we can convertformula (19) into a common form of the PID controller,

that is:

)()(1

)(

)()(1

)()()(

11

11

11

1111

k e z C z

z C z

P d k r z C z

z C z z C z k u

U

Y

U

Y

P d

R

−−

−−

−−

−−−−−−

−−

++−

+=∆

(20)

We find the limitation of the PID control from formula

(20). According to (20), the current output increment of thecontroller should be related on all previous deviations e,

and also relate to the output setting in an interval. But theordinary PID controller only has three adjusting

parameters which are much less than the parameters in the

formula (20). The transfer functions of the set value to the

process input increment are different from the error

transfer functions of the error to the process input

increment. Based on PID we could get better effects if weintroduce set value into controller output by a suitable

feedforward loop. In order to get PID parameters from

formula (12), we reduced the transfer function order to

formula (21):

1

1

2

2

1

10

1 −

−−

+

++

z

z z

α

β β β (21)

So we can get the PID parameters by comparing thecoefficients of formula (21) and formula (12).

4 SIMULATION

For a simple time-delay system:

se s

sG 20

201

1)( −

+=

(22)Then we dispersed it by zero-order holder. Therefore we

get the Pulse Transfer Function model (sample time is 1

second):

1

21

9512.01

04877.0)(

−

−

−=

z z H

(23)Respectively PID parameters tuned by Zieglers-Nichol

(ZN), Endometrial Control (IMC), andMPC in this paper

are obtained at the table 1:

Table 1. The different tuning methods of PID controller

parameters

Tuning method

of PID

K T1 TD TF

Zieglers-Nichol 1.33 31.0 7.74 -

IMC 0.935 30.5 6.48 -

MPC 0.467 18.44 0.005 0.539In this table we use predictive length and control length of

the predictive control are P=8 and C=1.

The control effects of the 3 groups tuning method of PID

are shown in the figure1.




Fig 1.Comparison control effects of PID controller with different parameters

Figure 1shows that the PID parameters tuned by the

predictive control algorithm is not superior to the other

methods. On the contrary, predictive control algorithmneeds a large weight value of input increment; otherwise

the system will be concussion or even divergent. The result

is that the response speed of the PID controller is too slow.

Figure 2 shows the control variables of 3 kinds of PIDtuning method.

Fig 2. The output from 3 different parameters PID controller

In this section, the validity of the designed MPC system is

verified through the control experiments. Fig.2 shows theexperimental results with 3 different parameters PID

controller.

5 CONSTRAINT CONDITION

In the absence of constraints, the optimal minimization of

(20) is a function of the current state, the previous input,and the desired setpoints and a feedback matrix can be

computed offline in order to achieve fast implementation

online. However, in general, the constrained minimization

of (20) can be highly computational intensive.

The problem is that the output of PID controller is

restricted by actuator ability, the controller has none path

to receive input signals in the real process. The real process

output and the actuator output are not the same, because of

the non-linear constraints of actuator, but the controller

still use the actuator output instead of the real output, so

the state variable in the PID controller and the real state are

mismatched. Therefore, methods for explicitly solving the

multi-parametric quadratic programming problem offline

have to be considered [9].

An effective way to solve this problem is to introduce

feedback from actuator output to the controller.

Then the controller with constraint condition is designed.

Fig 3. The implementation of private control algorithm with constraintcondition

The control effect of the controller using the figure 3

structure is shown in figure 4

Fig 4. Control effect of PID controller with constraint condition

From figure 4 we see that the controller in figure 3

structure can get a better effect. The structure has its

advantages: we don’t need to solve some quadratic

programming problem while the system still efficient

almost as same as MPC with constraint condition. Because

the structure’s calculating process is simple. It only needs

to set some parameters but not needs to solve any

optimization problem. It means that the predictive control

used in some controllers has poor computing capability

(e.g. PLC)6 CONCLUSION

There exist inevitable time-delay effects in industrial

process. A PID controller based on MPC is discussed in

this paper. A simple PLC model of predictive control

system was analyzed. With the dynamic experiments and

operator’ experience, its model was established, which

properly represents the behavior of the process and shows

superior to other modeling method. A PLC controller is

constituted, and then after simulation on the model. The

obtained results are not good enough but because the

controller has no path to receive input signals in the real

process. So we need to improve the constraint conditions.

A new structure is discussed which provides an effectiveway of modeling and control to a complex industrial

process. Meanwhile, with the development of the PID andthe application of MPC, PID controller based on MPC

would play a more and more important role in the

complicated industrial processes.

REFERENCES

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2010 Chinese Control and Decision Conference 1095



[2] M. Lazar, W.P.M.H. Heemels, S. Weiland, Stabilization Conditions

for Model Predictive Control of Constrained PWA Systems, 43rd

IEEE Conference on Decision and Control, December 14-17, 2004.

[3] Xi Yugeng, Predictive Control, National Defense Industry Press,

Beijing, June 1993

[4] Kim, J.-S.; Yoon, T.-W.; Shim, H.; Seo, J.H, Switching adaptiveoutput feedback model predictive control for a class of

input-constrained linear plants, Control Theory & Applications,

IET Volume 2, Issue 7, July 2008, 573 - 582

[5] Vladimir Havlena, Jiri Findejs, Application of model predictive

control to advanced combustion control, Control EngineeringPractice, January 2004.

[6] Wei Guo; Wei Wang; Xiaohui Qiu, An Improved Generalized

Predictive Control Algorithm Based on PID, Intelligent

Computation Technology and Automation (ICICTA), Volume1, 20-22 Oct. 2008

[7] Clarke, D.W. “Application of Generalized Predictive Control to

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MPC PID2.pdf