Synthesis identification analysis for closed loop system

WANG Jianhong and RICARDO A.Ramirez-Mendoza

1.School of Electronic Engineering and Automation,Jiangxi University of Science and Technology,Ganzhou 343100,China;2.School of Engineering and Sciences,Tecnologico de Monterrey,Monterrey 64849,Mexico

Abstract:The existing theories for closed loop identification with the linear feedback controller are very mature.To apply the existed theories directly in the control field,we propose a new idea about replacing the original unknown and nonlinear feedback controller with one approximated linear controller,while guaranteeing the equivalent property for the obtained closed loop system.Based on some statistical correlation functions,one condition is derived to show the equivalent property between the approximated linear controller and the original nonlinear controller.The detailed explicit form,corresponding to the approximated linear controller,is also constructed.Furthermore,to give a complete analysis for closed loop identification,the cost function is rewritten as one extended expression,being convenient to understand.Then spectral estimation is introduced to identify the unknown plant in the closed loop system.Finally,the proposed theories are verified by one simulation example.

Keywords:synthesis analysis,closed loop system,simplified function,equivalent controller.

1.Introduction

Two commonly used structures exist for automatic control systems,i.e.,open loop structure and closed loop structure.The open loop structure is one simplest structure,as no feedback effect exists between the controller and the considered plant,expect the forward effect.It means that in the open loop system,the output of the considered plant has no effect on the control input.In case of little influence from the internal and external disturbance,this open loop system can be adopted.The most important characteristic for the closed loop system is that the positive effect and the feedback effect exist simultaneously between the controller and the considered plant,i.e.,a direct effect from the output of the considered plant is imposed on the control input.The goal of the feedback loop is to collect the output information and then compare it with the original excitation input.This error signal,coming from the collected output signal and the original excitation input,can directly act on the controller.The controller performs a certain calculation on the error signal to produce a control effect,so that the output of the considered plant will converge to the expected references.The mission of the closed loop structure is mainly using the negative feedback to reduce system errors.

Currently,there are lots of research on closed loop system identification,for example,in [1],the identification method and the asymptotic problem of linear systems were analyzed in time domain.Among them,various identification methods are the theoretical basis for our identification problem.The identification of the linear and nonlinear systems is considered in the frequency domain [2],that is used to avoid the aliasing effect,coming from the Fourier transform.In [3],various optimization methods in engineering practice were presented,such as Newton method,tangent plane method,and binding method.A frequency domain identification idea was proposed to solve the problem of missing data for discrete-time systems in [4],where the unknown parameter vectors and missing data were identified at the same time through minimizing one criterion function.In case of the unknown but bounded noise.One bounded error identification was proposed to identify the unknown systems with time varying parameters.Then one feasible parameter set was constructed,which included the unknown parameter with a given probability level.In [5],the feasible parameter set was replaced by one confidence interval,as this confidence interval could accurately describe the actual probability that the future predictor will fall into the constructed confidence interval.The problem of constructing this confidence interval is solved by a linear approximation/programming approach [6],which can identify the unknown parameter only for the linear regression model.According to the obtained feasible parameter set,the midpoint or center can be deemed as the final parameter estimation.Furthermore,a unified framework for solving the center of the confidence interval is modified to satisfy the robustness.This robustness corresponds to other external noises,such as outlier and unmeasured disturbance [7].The above mentioned identification strategy is called set membership identification,dealing with the unknown but bounded noise.There are two kinds of descriptions on external noise.One is probabilistic description for statistical noise,and the other is deterministic description for unknown but bounded noise [8].For the probabilistic description on external noise,its probabilistic density function is known in advance.However,for the deterministic description on the external noise,the only information about noise is bound,so this deterministic description can relax the strict assumption on the probabilistic description.Within the framework of the deterministic description on external noise,set membership identification is adjusted to design controllers with two degrees of freedom [9].Set membership control was applied to design feedback control in a closed loop system with the nonlinear system in [10,11],where the considered system was identified by the set membership identification,and the obtained system parameter would benefit the prediction output.Generally,more specifically,one special kind of direct data direct control strategyvirtual reference feedback tuning is proposed to design controllers for the closed loop system with two degrees of freedom [12],where the zonotope parameter identification algorithm is used to give the guaranteed interval for control parameters in presence of bounded disturbance.In[13],the idea of dynamic programming was given to analyze the minimum principle and stability condition for data driven model predictive control respectively.For the common closed loop system,a spectral analysis estimation for the unknown plant is given,and the cost function is reduced to its more simplified form [14].It corresponds to the data driven estimation,i.e.,system identification.One gap between system identification and model predictive control was alleviated in [15],where the popular machine learning and function estimation theory were combined to get the nonparametric estimation for the prediction of the output value.

This paper considers the synthesis identification analysis for the closed loop system with the system model and the noise model.All above derivations on the closed loop identification are effective on the condition that the feedback controller must be known or linear.However,this condition does not hold in practice,thus we study the closed loop identification with the nonlinear controller.We use one linear feedback controller to replace the original nonlinear controller and establish one equivalent property in the statistical sense.The equivalent linear controller can be constructed by some statistical correlation functions.

Based on our previous results of the closed loop system identification,in this paper we will do some contributions for the closed loop system identification,i.e.,the synthesis identification analysis for the closed loop system.We analyze this problem from two different points.The first point is about modifying the original cost function to its another simplified form,which is convenient for the later identification process.The second point is replacing the nonlinear feedback controller with one linear feedback controller.Then the existing research on the linear controller design or linear system identification can be applied directly.

2.Closed loop system structure

One closed loop system is given inFig.1,wherey(t) is the closed loop output,u(t) is the input signal,C(z) is a simple linear controller,G0(z) denotes the unknown plant,H0(z) is one noise shape model to deal with the external white noise,r(t) is the external excitation input,e(t) is a white noise with zero mean and variance λ0,the colored noisev(t) passes through that noise filter,andzis the backward shift operator.

Fig.1 Structure of the closed loop system

FromFig.1,we see

Using the basic modern control theory and some mathematical equation computations,we have

Expanding each term in (2),we get

Similarly,define one sensitivity function to reduce the computational complexity.

Using the above defined sensitivity function,we rewrite (3) as follows:

The main problem of the closed loop identification is to identify one unknown parameter vector.

3.Prediction error identification reviewed

The above considered closed loop system can be divided into two kinds,i.e.,parameter identification and nonparameter identification. Here the first case-parameter identification is considered,so firstly we need to parameterize the unknown plant and noise filter,i.e.,(2) is parameterized.

where θ is one unknown parameter vector,which needs to be identified.

The mission of our considered closed loop identification is to estimate the parameter estimatoron the basis of the observed input-output sequenceZN={y(t)whereNis the number of observed data.

From (6),define one step ahead predictiony(t,θ) as

Combining (3) and output prediction (7),we have the error function as follows:

Using the above defined error function,we get the following numerical optimization problem to estimate the unknown parameter vector:

Through using the smooth property with respect to the parameterized prediction error ε(t,θ),the cost function in the above optimization process can be reduced to a finding root process,i.e.,

It means that

where ∂θmeans partial derivative operation about θ,and after some tedious mathematical operations,is expanded as

The detailed computation process about solving the above parameter optimization problem can be referred to many papers,for example,our previously published works [11].This paper is not the identification algorithm,but on other synthesis identification analysis about our considered closed loop system.

4.Synthesis identification analysis

ObservingFig.1again,excitation input {r(t)} is a quasi stationary reference signal,and its spectrum is ϕr(w).Furthermore,through using the condition of that independent and identically distributed white noise with variance λ0,and that external white noisee(t) has power spectrum λ0.Furthermore,colored noisev(t) has power spectrum

where ∗ is the complex conjugate.

From (5),the uncorrelated property holds between excitation input {r(t)} and white noise {e(t)},then we obtain some spectrum relations directly,such as

Similarly,we have the following relation about the output spectrum:

Similarly,we get another form for the cross spectrum:

Observing the above three equations simultaneously,it holds that

The advantage about the above spectral relations concerns estimating the transfer functionG0(z) with its spectral analysis estimation.More specifically,spectral analysis estimationis set as follows:

Taking the limit operation on both sides of (17),whenNis sufficiently large,that spectral analysis estimationapproaches to

Consider the cost function (9),assume the true parameter is in the considered parameter set,it means that one true parameter vector θ0exists to satisfy

As the error function (8) is not dependent of the unknown parameter explicitly,so in order to find its detailed relation with the unknown parameter,we rewrite it as another form,i.e.,

Combining (3) and (18),we easily obtain the following detailed expression as

Substituting (19) into the prediction error (18),we get

From our derivations above,another form for the cost function is rewritten as follows:

It means that we identify the unknown parameter estimations through minimizing the following improved cost function:

Equation (22) tells us that if the numberNof observed data is sufficiently large,the whole minimum process for the improved cost functionVN(θ) is guaranteed to be the global minimum,i.e.,the following relation holds:

It means that

The above obtained result is a simplified form for the classical result.

5.Replace nonlinear controller with linear equivalent controller

ObservingFig.1and (12),feedback controllerC(z) exists in the above derivation.During all the designed problems for the closed loop system,for example,system identification,controller design,and state filter,the detailed form about the feedback controllerC(z) must be needed,for example,PID controller.However,in practice,all plants and controllers are all nonlinear forms,so when the feedback controllerC(z) is one nonlinear form,(12) could not be solved easily,as we do not know the explicit form for the nonlinear controller.

Now the problem of replacing an unknown nonlinear controller with an equivalent linear controller is studied,and the equivalent property is also established.

ObservingFig.1again,it is emphasized that the feedback controllerf(·) is not a linear controller,but a nonlinear controller.Then the input for the considered system corresponds to

Observe the right and left sides for the nonlinear controllerf(·),and formulate its input signal and output signal as follows:

To be convenient for well understanding this main essence in Section 5,we plot the originalFig.1again here and modify the feedback controller as a nonlinear feedback controller,then this modified structure is shown inFig.2.

Fig.2 Structure of nonlinear feedback closed loop system

As the nonlinear system or the nonlinear controller is not easily analyzed,and the research on the linear system or the linear controller is mature,we adjust the nonlinear feedback system as the commonly used linear feedback system.It corresponds to constructing a linear feedback controllerC(z) to replace the original nonlinear feedback controllerf(·),and this equivalent property holds in the sense of some statistical probability.

More specifically,denote the output of the linear feedback controllerC(z) asC(z)y(t).Then a measure is used to verify the approximated degree between the linear controllerC(z) and the nonlinear controllerf(·).We set this measure as the following error criterion:

Thus the linear feedback controllerC(z) is chosen to minimize one constructed error criterion functionE[C(z)].The detailed minimization result for that constructed error criterion function is given as Theorem 1.

Theorem 1Set the feedback controller’s input signal asy(t) and its outputf(y(t)) in the considered nonlinear feedback closed loop system,C(z) is a linear feedback controller,and the condition of using the linear feedback controllerC(z) to replace the original nonlinear controlf(y(t)) is

where ϕy,f(τ) denotes the power spectral between input signaly(t) and its corresponding outputf(y(t)) in the above considered nonlinear feedback closed loop system.

ProofSetC1(z) as another linear controller,let

and similarly we also have

Define

Then we have

Using the defined error criterion to get

AsC(z) minimizes the defined error criterionE[C(z)]on the condition thatE[C1(z)]≥E[C(z)] for all the linear time invariant controllerC1(z).The right side is nonnegative for all the linear feedback controllersC1(z)on the condition of the linear part inE(z)y(t) approaching to zero,i.e.,

SubstitutingE(z)y(t) and interchanging the order of integration give

Sincee(t) is an arbitrary impulse response,the coefficient ofe(t) is zero.Then interchanging the order of integration,the above optimality condition becomes

Doing the same operation on (24),we get

Then the proof of Theorem 1 is completed.□

To construct the explicit form for the linear feedback controllerC(z),we expand (26) to its reduced form.

The above mathematical operations are some basic expectation operations.

ObservingFig.2again,we have

To determine the linear controllerC(z),post multiplication of (36) byy(t) and taking the cross correlation operation,we have

wherey(t) is a scalar,so the transpose notation is neglected.Ry(τ),Rry(τ),andRuy(τ) are given as

From (37),it holds that

In case of no excitation input,i.e.,r(t)=0,then the linear controllerC(z) is simplified as

CommentGenerally Theorem 1 gives a condition about the linear feedback controller during the process of replacing the nonlinear feedback controller. Equations(38) and (39) are two detailed forms about the constructed linear controllers.

6.Simulation example

In this section we apply our derived results on one single input and single output system,which is controlled by one feedback controller.The true data generating system is given as

Their corresponding parametrized forms are denoted as

where θ is defined as follows:θ=[a1,a2,a3,a4,a5,b1,b2]T.

A Gaussian white noisee(t) with variance λ0=0.5 is added through the noise filterH0(z).The sampled time isTs=1,the true parameter vector θ0is defined as

The data generating system is operated in a closed loop system with a unit feedback controller.In solving the numerical optimization problem to identify the unknown parameter vector,the initial value for the unknown parameter vector θinitis chosen as

The input variableu(t) has a weight of 1 and the excitation signalr(t) also has one bound,i.e.,

The applied input signal is shown inFig.3,and the output signaly(t) is measured through some physical devices.Fig.4shows the observed output signal.When using the predictor error identification method to identify the unknown parameters,the condition for terminating the recursive method is to guarantee the estimation error

Fig.3 Input signal

Fig.4 Output signal

to be less than a small constant value,for example,0.05.

Based on the observed data,shown inFig. 3andFig.4,these observed data are used to identify the unknown parameter vector θ through minimizing the improved or simplified cost function (22).After getting this parameter estimationand substituting it into the plant and noise filter,the final information about the closed loop system is all known.To compare the true closed loop system and the identified closed loop system,we only show their corresponding output responses inFig.5,where two output responses are given.FromFig.5,we see these two output responses are similar to each other.

Fig.5 Comparison about two output responses

7.Conclusions

This paper introduces our newly results on classical closed loop system identification from two different points.The first point is on modifying the original cost function to its another improved or simplified form.The second point is to replace the nonlinear feedback controller with one linear feedback controller.Then the existed research on linear system identification can be directly applied.Based on our results about system identification and direct data driven control,the next step is around with the combination with the identification and control.