Design of Model Driven Cascade PID Controller Using Quantum Neural Network

Yourui Huang Yiming Tian Liguo Qu

Anhui University of Science and Technology Huainan 232001, China hyr628@163.com

Keywords: Quantum Neural Network PID Controller Internal Model Control

Abstract: PID optimal parameters selection has been extensively studied, In order to improve some

strict performance requirements for complex system. A new design scheme of PID controllers is

proposed in this paper. This scheme is designed based on the IMC which is a kind of the model

driven controllers. The internal model consists of the full model. The full model is deigned by using

the quantum neural network. The PID control system is first constructed for the augmented system

which is composed of the controlled object and the internal model. Simulation and experiment

results show that the method outperforms the typical model-based approach despite its simplicity

and it is therefore suitable to implement in distributed control system as well as in single-station

controller.

1 Introduction

Proportion-Integral-Derivative(PID) controllers are the controllers most adopted in industry

due to the good cost/benefit ratio they are able to provide[1]. In fact, they can provide satisfactory

performances for a wide range of processes, despite their case of use. Automatic tuning techniques

have been developed to help the operators to select appropriate values for the parameters in such a

way that less and less specific knowledge is required to use them[6-7]. However, it is well known

that the performances of these controllers much depend, in addition to the tuning of the PID

parameters, to the appropriate implement of those additional functionalities, such as anti-windup,

set-point filtering, feedforward, and so on. Methodologies for the effective design of such a features

are nowadays easier and easier to implement, due to the increase of computational power available

in distributed control systems as well as in single-station controllers.

In recent years, the so-called model driven controller has been proposed, where the full model

which describes the controlled object in detail is inserted in the control system. The model driven

controller is attractive, because the control structure is simple and it has the high robustness for

system uncertainties. As one of the model driven controllers, the internal model control(IMC) has

been proposed[2].The main motivation in this study is to consider a new design scheme of nonlinear

controllers based on the idea of the IMC which is a kind of the model driven controller. Concretely,

a model driven cascade PID controller is proposed for nonlinear systems. The internal model is

designed by using the quantum neural network[3-5]. According to the newly proposed control

scheme, it is not necessary to switch some models. Furthermore, the information about the sign of

the system Jacobian is not also required.

This paper is organized as follows. The model driven PID controller is firstly explained, and

followed by the introduction of the quantum neural network. In section 3, the ideology of model

drive cascade PID controller is formulated in detail. Finally, the behavior of the newly proposed

control scheme is examined on simulation example.

Advanced Materials Research Vols. 108-111 (2010) pp 1486-1491Online available since 2010/May/11 at www.scientific.net© (2010) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMR.108-111.1486

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2 Quantum Neural Network

2.1 Quantum Bit

The bit expression in quantum computer is presented with a quantum bit (qubit). For the qubit,

the state 0 represents 0 , and the state 1, 1 . The qubit with the superposition. of two states is

shown as follows.

10 (1)

where α and β are the complex number called the probability amplitude.In the field of quantum

mechanics, the probabilities that 0 and 1 are observed become the square of the absolute value

for α and β, respectively. Here α and β satisfy the following relation.

122 (2)

2.2 Quantum Neural Network

In this section, a novel learning model of quantum neural network is presented. Figure 1 shows

the network model according to the qubit circuit. jx (j=1,2,3 ,m) in the input layer and

ky (k=1,2,,n) in the output layer represent neurons. The input-output properties of neurons in each

layer are concretely exhibited as the following description. For the suffix with the top here, a neuron

in the input layer represents I, and in the output layer, O.

x1

x2

x3

y

InputLayer OutputLayer Fig. 1. Model for quantum neural network

(i) Input-output of neurons in the input layer

The output function of neurons in the input layer is written as follows.

mm xu2

(3)

)( mm ufs (4)

sincos)( ief i (5)

where ix has the input 1,0 . The output function f(x) corresponds to Eq.5. For the input 0, the input

to the network contains the input of 0 , since 0

mu holds and the phase exists on the real axis.

For the input 1, the input to the network corresponds to the input 1 , because 2/

mu holds and

the phase exists on the imaginary axis.

(ii) Input-output of neurons in the output layer

According to Eq.4 and Eq.5, The output function of neurons in the output layer is presented as

follows.

j

n

j

iiio seeev j

1

0

11,1,0

(6)

j

n

j

iSio

k seev kjokkk

1

,1,1 (7)

o

kk

o

k vgu arg2

(8)

o

k

o

k ufs (9)

Where )(xg is the sigmoid function in the following equation

xexg

1

1)( (10)

Advanced Materials Research Vols. 108-111 1487

3 PID Controller Design

3.1 Outline

The block diagram of the proposed control system is shown in Fig.2. The internal model which is

composed with the full model is firstly designed. The full model describes the controlled object as

exactly as possible. The full model is strictly designed. Next, the inner PID control system is

constructed for the augmented system, whose PID parameters are adjusted based on the

pole-assignment. Then, the control system surrounded by the dashed and dotted line realizes

)()( twtya . Here, the control objective is to make y(t) follow the reference signal. The steady state

error occurs because )()( tyty a . In order to remove the steady state error, PID controller is further

designed for the control system surrounded by the dashed and dotted line. That is, the cascade PID

control system is constructed. In Fig.2, )1(: 1 Z denotes the differential operator.

3.2 Full model design

The full model which describes the controlled object as exactly as possible is designed by using

the quantum neural network. The final output is used the probability which is observed the state 1 .

As the imaginary part represents the probabilistic amplitude of the state 1 , the output is the square

of the absolute value in the following equation.

y = Im( o

ns )Im( o

ns ) (11)

For learning in the qubit neuron, the gradient descent is used in this study. The evaluation

function is presented as follows.

E = 2

12

1

M

pp

t

p yy (12)

where M is the number of sample data, t

py is the desired output, and py is the final output of

neurons.

In order to decrease the value of the evaluation function E, θ and δ are updated as follows.

θ(t +1) = θ(t) + ∆θ(t) (13)

δ(t +1) = δ(t) + ∆δ(t) (14)

Subsequently ∆θ and ∆δ are calculated as follows.

∆θ(t +1) = −η

E (15)

∆δ(t + 1) = −η

E (16)

where η is the learning constant.

Tsf

△TI

KPf

△TDf

TS1

++

+ +

--

r(t)w(t)

Main Control System

Ts

△TI

KP

△TD

TS

++

+

-

+

-System

Full Model1

y(t)

y(t)+

-

ya(t)

u(t)

Fig.2 Block diagram of the proposed control system

4 Experiment Results

In order to evaluate the effectiveness of the newly proposed scheme, simulation examples for the

nonlinear system are considered. In the example, the effectiveness of the proposed tuning method is

compared with the classic tuning methods, Ziegler- Nichols method. Four typical control systems

1488 Progress in Measurement and Testing

were chosen to verify the adaptation and robustness of the proposed controller. The transfer

functions in the four control system are given as follows.

Function1 (two-order system):

2

1( )

1.25 1G s

s s

;

Function2 (High-order system): 1

( )(1 )(1 0.125 )(1 0.1 )(1 0.8 )

G ss s s s

;

Function3 (Time-order system): 2

( ) , 4.5, 1.01

sLG s e T LT s

;

Function4 (High-order and time-delay system):

( ) , 1.0(1 )(1 0.01 )(1 0.25 )(1 0.4 )

sLeG s L

s s s s

.

Simulation experiments were executed for each of the functions in Matlab7.0. The input of the

system is a unit step signal in these cases. The parameters of the QSEA are set to 20 iterations. The

unit step responses of the four control systems are shown in Fig.3-Fig.6.

From these figures, we can find that using the proposed PID controller, the overshoot, settling

time and rise time of the unit step response are reduced greatly compared with the Ziegler-Nichols

method for each of the four functions.

Fig.3 The unit step response of Function1

Fig.4 The unit step response of Function2

Advanced Materials Research Vols. 108-111 1489

Fig.5 The unit step response of Function3

Fig.6 The unit step response of Function4

5 Conclusions

In this paper, a new design scheme of model driven cascade PID controllers using the quantum

neural network for nonlinear systems has been proposed. The proposed scheme is based on the idea

of the internal model control, and the internal model is composed of the full model. The full model

is designed using the quantum neural network. The quantum neural network plays a role of

compensating nonlinearities which cannot be expressed by using the linear model. According to the

newly proposed control scheme, there is a strong advantage such that a priori information with

respect to the system Jacobian is not required. Especially, the advantage gives us a suggestion that

the newly proposed scheme enables us to deal with uncertain time-delay systems, and it is useful in

implementing the proposed scheme to real systems. The investigation on this point is currently in

our work.

1490 Progress in Measurement and Testing

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[4]Gruska,J. Quantum computing. McGraw-Hill(1999)

[5] Matsui,N., Takai,M., Nishimura,H. A network model based on qubit-like neuron corresponding

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[6]Han, K.H., Kim, J.H. A Quantum-inspired evolutionary algorithms with a new termination

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Advanced Materials Research Vols. 108-111 1491

Progress in Measurement and Testing 10.4028/www.scientific.net/AMR.108-111 Design of Model Driven Cascade PID Controller Using Quantum Neural Network 10.4028/www.scientific.net/AMR.108-111.1486

DOI References

[6] Han, K.H., Kim, J.H. A Quantum-inspired evolutionary algorithms with a new termination criterion, He

gate, and two-phase scheme. IEEE Trans. Evol. Comput., (2004) 156-169

doi:10.1109/TEVC.2004.823467