Pole-placement self-tuning control of nonlinear Hammerstein system and its application to pH process control

Zhiyun Zou*,Dandan Zhao,Xinghong Liu,Yuqing Guo,Chen Guan,Wenqiang Feng,Ning Guo

China Research Institute of Chemical Defense,Beijing 102205,China

Keywords:Nonlinear system Hammerstein model Pole placement Self tuning control pH control

ABSTRACT By taking advantage of the separation characteristics of nonlinear gain and dynamic sector inside a Hammerstein model,a novel pole placement self tuning control scheme for nonlinear Hammerstein system was put forward based on the linear system pole placement self tuning control algorithm.And the nonlinear Hammerstein system pole placement self tuning control(NL-PP-STC)algorithm was presented in detail.The identification ability of its parameter estimation algorithm of NL-PP-STC was analyzed,which was always identifiable in closed loop.Two particular problems including the selection of poles and the on-line estimation of model parameters,which may be met in applications of NL-PP-STC to real process control,were discussed.The control simulation of a strong nonlinear pH neutralization process was carried out and good control performance was achieved.

1.Introduction

Since Well stead and Zarrop put forward the pole-placement selftuning control method of linear system[1],lots of research work had been done for it[2,3].Up to now,the pole-placement self-tuning control theory of linear system has been developed well,and it has achieved some successful applications in the control of chemical reactor[4],distillation column[5],boiler,furnace,ultrasonic motor[6]and so on.At the same time,lots of theoretical analysis on the pole-placement selftuning control algorithms of linear system has been carried out.

However,the linear pole-placement self-tuning control algorithm has only a little adaptability to mild nonlinear systems.Its control performance will be much deteriorated[7]when it is applied to the control of strong nonlinear systems commonly encountered in chemical production processes[8].

A Hammerstein model consists of a static nonlinear sector followed in series by a linear dynamic element.It corresponds to processes with linear dynamics but a nonlinear gain,and it can adequately represent many of the nonlinearities commonly encountered in industrial processes such as pH neutralization processes[9,10].The Hammerstein model is particularly useful in representing the nonlinearities of a process without introducing the complications associated with general nonlinear operators.Due to the static nature of its nonlinearity,this static nonlinearity can be effectively removed from the control problem,allowing to easily extend linear control algorithms to nonlinear Hammerstein systems[11,12].Anbumani had proposed an adaptive minimum variance control algorithm for nonlinear Hammerstein system[13],but its application was much limited due to the fact that it was unsuitable for non-minimum phase system and also it required large range fluctuation of control actions.

In this paper,through utilizing the separation characteristics of nonlinear gain and dynamic sector inside a Hammerstein model[11,12],a novel pole placement self tuning control scheme for nonlinear Hammerstein system was put forward based on the linear system pole placement self tuning control algorithm[1–3].Then,the nonlinear Hammerstein system pole placement self tuning control(NL-PP-STC)algorithm was presented in detail.The identification ability of its parameter estimation algorithm of NL-PP-STC was analyzed,which was always identifiable in closed loop.Two particular problems including the selection of poles and the on-line estimation of model parameters,which may be met in applications of NL-PP-STC to real process control,were discussed.Finally,the control simulation of a strong nonlinear pH neutralization process was carried out and good control performance was achieved.

2.NL-PP-STC Principle and Algorithm

2.1.Hammerstein model

The structure of a Hammerstein model is illustrated in Fig.1 and it could be described by Eqs.(1)and(2).

Fig.1.Structure of a Hammerstein model.

2.2.Pole placement control principle of Hammerstein system

The principle of pole placement for Hammerstein system is presented in Fig.2 according to the general pole placement control requirements and the characteristics of Hammerstein model[9–14].

In Fig.2,in order to transfer the nonlinear system pole placement control problem into a more simple linear system pole placement control problem,the nonlinear compensation sectoris used to compensate the nonlinear gainof the Hammerstein system.yr(k)is the set point value and x′(k)represents the intermediate variable.The three sectors H(z?1),and G(z?1)are used to assign the system poles to the desired points,which can be taken as:

where nf,ngand nhare the orders of F(z?1),G(z?1)and H(z?1),fi(i=0,1,…,nf),gi(i=1,2,…,ng)and hi(i=0,1,2,…,nh)are parameters.

2.3.Pole placement control algorithm of Hammerstein system

2.3.1.When model is known

Suppose Hammerstein model is already known and then taking=p，=rii=1，2，…，p（）the system output can be obtained from Fig.2.

It is supposed by pole placement that

where Q(z?1)and P(z?1)is a polynomial of z?1.In order to make the calculation simple,let Q(z?1)=F(z?1).Combining Eqs.(3)and(4)can get Eq.(5).

Making the corresponding coefficients of power of z at both sides of Eq.(5)equal,a group of linear equations could be achieved.Furthermore,take nf=nb+d,ng=na,np≤na+nb?nc+d(here npis the order of P(z?1)),the only real root of fi(i=0,1,…,nf)and gi(i=1,2,…,ng)could be found,and then F(z?1)and G(z?1)could be achieved.

Substituting Eq.(5)into Eq.(3),we have

In order to assign the closed loop poles of system output to set point value at P(z?1),and make the system out put have no steady state offset,we can take

Fig.2.Scheme of Hammerstein system pole-placement control.

where[P(z?1)|z=1]and[B(z?1)|z=1]indicates the value of P(z?1)and B(z?1)at z=1.

From Fig.2 it is obtained

Substitute the achieved values of F(z?1),G(z?1)and H(z?1)into Eq.(8)to calculate x′(k),and then using numerical calculation method[12]to solve the nonlinear equationthe value of control action u(k)could be achieved.

2.3.2.Case of self tuning

2.3.2.1.Control algorithm.When the parameters of the Hammerstein model are unknown or are time-varying at some degree,the model orders and parameters could be estimated on line using system identification method,and then find the control action through pole placement control algorithm.

Then,the Hammerstein model of the controlled system got from identification method is as:

The pole placement problem of the transfer function of y(k)to e(k)could be simplified as the pole placement problem of the transfer function of y(k)to n(k)by appropriate arrangement.Combining appropriate system identification method and the above pole placement control algorithm when model is already known,the calculation steps of the nonlinear Hammerstein system pole placement self tuning control algorithm could be presented as follows.

Step 3:From F(z?1)x′(k)=H(z?1)yr(k)? G(z?1)y(k)to find the value of x′(k).

Step 4:Using numerical calculation method to solveand get the value of control action u(k).

The nonlinear Hammerstein system pole placement self tuning control algorithm could be implemented through computer's continuous sampling and iterative calculation according to above four calculation steps.

2.3.2.2.Analysis of identification ability.At the above pole placement self tuning control algorithm,the on line closed loop identification of a system's Hammerstein model should be carried out,and thus the identification ability of the above algorithm should be analyzed.

Extend Eq.(8)at time step k?d?1,we have

Assuming the orders and the delay time of the Hammerstein model have achieved using open loop identification method,the system parameters identification model can be obtained from Eq.(9)

In order to make the system identifiable in closed loop,all elements inside the data vector XT(k)should not be linearly interrelated.When the change of yr(k)is frequent,it makes persistent drive to the system.It is known from Eq.(10)that all elements inside the data vector XT(k)are not linearly interrelated and the system is identifiable.When yr(k)has no change,it is found from Eq.(10)that only when

then all elements inside the data vector XT(k)are still not linearly interrelated and the system is identifiable.

nf=nb+d≥nb,ng=na≥na?d is already selected within the above pole placement algorithm,which satisfies the requirements of Eq.(12).Thus,the nonlinear Hammerstein pole placement self tuning control system is always closed loop identifiable.

On the other hand,intuitively,when the system has got enough drive and the controller's parameters are selected appropriately,the above pole placement control algorithm should be converged and the convergence area should be much wide[15,16].However,because of the nonlinearity of the system,the convergence analysis problem is quite complicated and it will be studied in the future.

3.Two Problems in Application of NL-PP-STC

3.1.Selection of poles

The selection of assigned system poles has no general rules and the common used method[2,3,17]is to select two main system poles as the two roots of Eq.(13).

In Eq.(13),ξ is a decaying factor and ω is the natural frequency.They are determined by the system dynamical response requirements.Tsis the sampling period.Other system poles could be assigned near the origin point of the unit circle at z plane.In order to have better system dynamical response,the selection of poles should not make the constant of P(z?1)to be 0,and should make np≥nf.

3.2.On-line iterative estimation of model parameters

At the above self tuning algorithm,the parameters of the nonlinear Hammerstein model should be iteratively estimated on-line.Through some appropriate adjustment to the Hammerstein model,the parameter estimation problem of nonlinear Hammerstein model could be changed to two linear model parameter estimation ones and they could be solved using iterative least square estimation method.In order to strength the adaptability of the parameter estimation algorithm to the time-varying properties of the controlled system and to get good estimation results at both oscillation and steady state,the iterative least square estimation method with forgetting factor could be used.

Referring to[16,18,19],the iterative least square estimation algorithm of Hammerstein model with forgetting factor could be derived as follows.

Here it requires that λmin≤ λ(k)≤ 1 and λminis a constant.

4.Simulation of PH Process Control

In order to demonstrate the effectiveness of the above NL-PP-STC algorithm,simulation experiments of the NL-PP-STC algorithm are carried out for the control of a pilot plant pH process[20]as shown in Fig.3.Here soda is 0.1 mol·L?1NaCO3and acid is 0.1 mol·L?1HCl.

A Hammerstein model of the pH process[20]is

Fig.3.A pilot plant pH process.

where u(k)=um(k)?us,y(k)=ym(k)?ys,um(k)is the electric current to adjust the valve position of the acid flow and ym(k)is the pH value of the neutral liquid.us=2.5 mA,ys=5.713 pH and the sampling period Ts=30 s.The subscript s indicates the steady state value and subscript m shows the measured value.

In simulation,taking nf=4,ng=2 and P(z?1)=1 ?1.2z?1+0.49z?2?0.078z?3+0.004z?4,the following simulation results are achieved.

4.1.Initial start-up

Let the set point value(SV)of pH changed from 3.713 to 7.713 and from 7.713 to 3.713 in square wave form and start up the nonlinear pole placement self tuning controller,the achieved pH control response is shown in Fig.4.It can be seen that the pH value could track the change of set point value quite well in short time.

Fig.4.Initial starts up response of NL-PP-STC system.

It is also found that the estimated values of system parameters quickly track to real values and finally converge to the nearby of the real values.As an example,Fig.5 shows the convergence process of parameter.

Fig.5.convergence process of parameter

4.2.When the nonlinear property changed

In order to investigate the adaptability of NL-PP-STC algorithm to system's time-varying property,let r3change 10%from 1.15 to 1.265.The simulation control result of NL-PP-STC is illustrated in Fig.6.From Fig.6,it can be seen that after a very short time of small amplitude oscillations,late the set point tracking results are quite satisfactory.On the other hand,the estimated value of r3begins from 1.15 and quickly converges to the nearby of 1.265.

Fig.6.pH control result when the system parameter r3 changed.

4.3.After convergence of estimated system parameters

After the estimated system parameters converged,let the set point value of pH change from 5.713 to 7.0.The pH control responses are presented in Fig.7.The control result of NL-PP-STC is compared with the control result of a nonlinear PID controller(NL-PID).From Fig.7,it is found that NL-PP-STC has short set point tracking time and has no overshoot,and its control response is better than NL-PID.

Fig.7.pH control result after convergence of estimated system parameters.

5.Conclusions

The above theoretical analysis and control simulation results show that the nonlinear Hammerstein system pole placement selftuning control(NL-PP-STC)algorithm put forward by this paper has good control performance when it is applied to the kind of nonlinear systems which can be fully described by Hammerstein model such as pH neutralization process.

As some adaptive parameter estimation algorithms such as iterative least square parameter estimation algorithm with forgetting factor are used in NL-PP-STC algorithm,NL-PP-STC has good adaptability to system's time-varying properties.When developing NL-PP-STC algorithm,the pole placement requirements of both outputs to disturbance transfer function and to set point transfer function are considered at the same time.Thus,NL-PP-STC has a better set point tracking and disturbance rejecting ability,which are the specific characteristics of NL-PPSTC and are still lack by other pole placement self tuning algorithms.

As the computation of NL-PP-STC algorithm is not complex,it has a very good future for industrial process application.

Nomenclature

d time delay

e(k) white noise sequence at time step k

k time step

p order of the nonlinear static gain of Hammerstein model

rimodel parameters of the nonlinear static gain of Hammerstein model

u(k) system input at time step k

x(k) intermediate variable of Hammerstein model at time step k

y(k) system output at time step k

yr(k) set point reference value at time step k

z time shift factor

λminminimum value of forgetting factor

λ(k) forgetting factor of least square estimation algorithm at time step k

Superscripts

^ estimated value of a variable

' calculated value of an intermediate variable of Hammerstein model

Subscripts

r reference value of set point