A Variable-Parameter-Model-Based Feedforward Compensation Method for Tracking Control

Dailin Zhang, Zining Wang, and Masayoshi Tomizuka,

Abstract—Base on the accurate inverse of a system, the feedforward compensation method can compensate the tracking error of a linear system dramatically. However, many control systems have complex dynamics and their accurate inverses are difficult to obtain. In the paper, a variable parameter model is proposed to describe a system and a multi-step adaptive seeking approach is used to obtain its parameters in real time. Based on the proposed model, a variable-parameter-model-based feedforward compensation method is proposed, and a disturbance observer is used to overcome the influence of the model uncertainty. Theoretical analysis and simulation results show that the variable-parametermodel-based feedforward compensation method can obtain better performance than the traditional feedforward compensation.

I. Introduction

GENERALLY, feedforward compensation is used to reduce the tracking error of a control system, and then the control accuracy can be improved [1]–[4]. The inverse of a system is commonly obtained from the system model, and the compensation values of the feedforward compensator are calculated according to the inverse and the planned trajectory.

Many methods were proposed to identify system models and achieve their inverses. A simple method is to use the parameter identification to obtain a system model, and then its inverse can be obtained by inversing the system model. Least squares (LS) and recursive least squares (RLS) [5], [6] are effective to identify the transfer function of a linear system. In[1], the system model of a permanent magnet linear motor(PMLM) was obtained by the RLS identification method and an adaptive feedforward component based on the inverse dominant linear model was used to reduce the tracking error.In [7], an inverse Preisach model was used for feedforward compensation of hysteresis compliance and the model was identified from drive experiments. In [8], measured data in every task was used for system identification and the feedforward controller could be updated after each task. In[9], a multi-model adaptive preview control using a set of augmented systems was proposed to enhance the feedforward performance.

The accurate model is difficult to obtain in a real control system with complex dynamics, so the effectiveness of feedforward compensation may be limited. Some methods are able to avoid the difficulties to build accurate models. For example, ILC (iterative learning control) adjusts its control signal to a control system in every iteration by using feedback information from previous iterations, which can improve the control accuracy without knowing the accurate system model[10]–[13]. In effect, ILC can find the perfect inverse of a system for a repetitive trajectory, which makes ILC be able to compensate for the disturbances optimally [14]–[16]. ILC,however, performs badly in the systems with uncertain factors. For example, the change of reference trajectory will result in varying disturbances, and changing the moment of inertia will result in the change of system model. Both uncertainties above will result in performance deterioration.

The paper proposes to use a variable parameter model to describe a system with uncertain factors and achieve the inverse of the system in real time. The basic idea is to calculate the parameters of the variable parameter model by solving the strict equality between variable parameter model output and actual output.

The contribution of the paper includes: Firstly, a variable parameter model with constraints is proposed and a multi-step adaptive seeking approach is used to obtain the parameters of the proposed model in real time. The multi-step adaptive seeking approach can obtain the optimal parameter in every control period by adaptive control approaches [17], [18].Secondly, a variable-parameter-model-based feedforward compensation method is proposed. The proposed variableparameter-model-based feedforward compensation method can reduce the tracking error effectively. Thirdly, a disturbance observer is used to compensate for the model uncertainty, which helps to reduce the influence of disturbances.

The remaining of the paper is organized as follows. Section II gives the problem formulation, Sections III and IV propose the variable parameter model and variable-parameter-modelbased feedforward compensation method, respectively.Section V presents the experimental results. Section VI gives the conclusion and future works.

II. Problem Formulation

A single input single output system is depicted in Fig. 1.The plant is described as a fixed transfer functionP(z), and the disturbances at timekared(k). The control configuration consists of a feedback controllerC(z) and a feedforward controllerF(z). With the inputr(k), the output isy(k), and then the tracking errore(k) can be calculated. The closed-loop system output can be calculated by

whereuf f(k) is the feedforward compensation value at timek.

Fig. 1. A system with a fixed transfer function is changed to a variable parameter system.

The feedforward controller can be described by

Substituting (2) into (1), we have

According to Stearnset al. [11], the ideal feedforward controller is

If the disturbance is zero and will be considered in the following uncertainty portion, the obtained inverse of the system has three possible solutions.

Solution 1:Substituting (4) into (3), we have

From (5), it can be concluded that the tracking errore(k)=r(k)?y(k)will be reduced to zero with the feedforward controller.

Solution 2:If the ideal feedforward controller is not available because the number of zeros is bigger than that of poles, the feedforward controller can be described by

The feedforward controller becomes

And the reference inputr(k) becomesr(k+d) by previewingr(k)dtime steps. Obviously,F(z) is realizable and the feedforward compensation effect remains unchanged.

Solution 3:IfP(z) has an unstable zero, zero phase error tracking control (ZPETC) can be used to reduce the tracking error [19]. In this method, the feedforward tracking control is designed as

whereGZPETC(z) is the zero phase error tracking controller,is the denominator ofandare two parts of the numerator, which contain cancelable and uncancelable zeros, respectively.is a scale.

Equation (8) does not produce the perfect inverse, but can help reduce the tracking error effectively. Because the variable parameter model is the main point, the paper simply focuses on (5) and (7).

When the plant has realtime changing parameters and the system becomes a variable parameter one,F(z) should follow the corresponding changes in order to ensure the last term in(3) be equal to one.

The variable parameter system is shown in Fig.1. The variable parameter plant isP1(z,θ) instead ofP(z) and the corresponding feedforward controller isF(z,θ) , where θ is a variable parameter. For every value of θ, the ideal feedforward controller can be obtained according to (4), (7),and (8)

At every control period, the parameter θ should be identified and the corresponding variable parameter function is obtained.the identified function ofP1(z,θ) is supposed to bePv(z,θ),which is a variable parameter model shown in Fig. 2. The variable parameter model means the identified transfer functionPv(z,θ) will be changing with the parameter θ.GZPETC(z,θ)is the zero phase error tracking controller with variable parameter θ. And the uncertainty in the identification process will be resolved by the model uncertainty compensation based on disturbance observer in Section IV.For example, the disturbanced(k) will be considered as an uncertainty portion in the paper.

Fig. 2. variable parameter model.

Remark 1:If a variable parameter modelPv(z,θ), where θ=[θ1,θ2,...,θn]is a variable parameter and its the initial value is θ(0)=[11...1], has the same input and output asP1(z,θ) at every discrete timek,Pv(z,θ) is the perfect description ofP1(z,θ).

Therefore, if we can find a variable parameter modelPv(z,θ)meeting the requirements in Remark 1 in every sampling instant, the feedforward controllerF(z,θ) will result in a very perfect tracking performance according to (4)–(9). In the next section, the variable parameter model will be proposed.

III. The Variable Parameter Model With Constraints

A variable parameter modelPv(z,θ) can have the same output asP1(z,θ) by tuning the variable parameter θ, but realtime performance to obtain the optimal parameters ofPv(z,θ) is very difficult to be guaranteed becausePv(z,θ) is a complex variable parameter system. This paper proposes a variable parameter model with constraints and a multi-step adaptive seeking approach to obtain the optimal parameters ofPv(z,θ)in real time.

The constraints of the variable parameter model are set to limit the variation range of parameters, which ensure the optimal parameters be achieved within a limited seeking range[20].

In this section, a structure combining a variable gain and an identified transfer function is used to simplify the variable parameter model. And the multi-step adaptive seeking approach computes the optimal parameters within the constrained range in real time.

A. Variable Parameter Model With Constraints

As shown in Fig. 2, the object under study isP1(z,θ). No disturbance is applied and the influence of disturbance will be discussed in the next section. Given a reference trajectory, we can get the outputy. At the same time, a parallel identification systemQSis set up with the variable parameter modelPv(z,θ) . ExceptPv(z,θ), the identification systemQShas the same structure as the given system and the output is assumed to beya(k). Then the identification error is

At every period, the parameters ofPv(z,θ) need to be sought becausePv(z,θ) is a variable parameter model.

The proposed multi-step adaptive seeking approach can iteratively runQSand tune its model parameters. The goal of the multi-step adaptive seeking approach is to obtain the optimal parameters ofPv(z,θ) to letyabe equal toy, that is, to let every ε(k) be equal to zero. Constraints of a variable parameter model will be used to accelerate the seeking process by restricting the seeking range of parameters.

Assumption 1:It is assumed that every variable parameter of the variable parameter model has an upper and lower bound,which is to reduce the seeking range and guarantee the system stability. In other words, it can be described by

wherei= 1 ,2,...,n, andandare upper and lower bounds of theith parameter θi, respectively.

It is supposed that there are multiple steps of seeking process to achieve the optimal valueθ. The optimal value ofθat every timekcan be obtained by the seeking approach

where θ?(k) is the optimal value at timek. At thejth step of seeking process, θx,j(k)=[θ1,j(k)θ2,j(k)...θn,j(k)], where θn,j(k) is the value of thenth parameter θn, is the value of the parameterθat timek.J(θx,j(k)) is the cost function of θx,j(k)

where εj(k) is the identification error at thejth step of seeking process, respectively.

Considering the real-time performance and complexity, too many variable parameters are not practical. In the proposed variable parameter model, the identified transfer functionP(z)is used as a fixed plant and a variable gain is used to tune the output, so only one variable parameter is used in the variable parameter model.

The variable parameter model is described as

wheref(θ) is a variable proportional gain, and θ includes only one variable parameter θ1, that is,f(θ)=θ1.

B. Multi-step Variable Parameter Seeking Approach

A multi-step variable parameter seeking approach is proposed to seek the optimal parameterf(θ) in (13). The goal is to adjust the variable proportional gainf(θ) by a multi-step adaptive seeking approach to lety(k) equalya(k) at every momentk. Here “multi-step” means the optimal parameter θ(k)is achieved through multiple calculations within one control period.

From Fig. 2, it can be seen that the given system and the parallel identification systemQShave the same controllerC(k). With the same reference input, there exists

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The outputs of the two systems are

Substituting (14) into (16), it can be concluded thatPv(z,θ)will be equal toP1(z) only ify(k) equalsya(k) at every timek.

f(θ) can be described by

At timek, the actual output isy(k), and the seeking approach is to run the variable parameter systemQSin parallel to adaptively find the optimal value of θ(k). At thejth step, θ(k) is supposed to be equal to θ1,j(k),ya(k) can be obtained at the step, then εj(k) is obtained. The optimal value of θ (k) can be obtained by

where θ?(k) is the optimal value, εj(k) is thejth identification error.

Algorithm 1 Multi-step variable parameter seeking approach

Thejth parameter update law is

θ1,j(k) and εj(k) are recorded at every step and θ?(k) is obtained by selecting θ1,j(k) when εj(k) is smaller enough.

In order to reduce the number of steps for obtaining the optimal value θ?(k) , the last parameter value θ(k?1) can be used as the present initial value, because the system does not change dramatically during a very small period.mvalues withinare tested to seek an optimal value θ(k?1)+θopt(k). Then the initial value can be set as θ(k?1)+θopt(k) and the parameter θ (k) can be obtained by

In order to avoid local minima, θopt(k) can be replaced by random values everyhsteps in Algorithm 1 under the conditions that (18) is met and θ(k?1)+θopt(k) is within

It can be seen that the variable parameter model in (13) and its parameter seeking approach are simple and practical.

IV. Variable-Parameter-Model-Based Feedforward Compensation Method

With the built variable parameter model, the feedforward compensation in Fig. 1 becomes variable-parameter-modelbased feedforward compensation.

A. Variable-Parameter-Model-Based Feedforward Compensation Method

The proposed variable-parameter-model-based feedforward compensation method is shown in Fig. 3. Compared with Fig. 1,P(z)andF(z) are replaced byPv(z,θ) andF(z,θ).

Fig. 3. The variable-parameter-model-based feedforward compensation.

With the variable parameter model in (13), the feedforward controller can be obtained by

The basic variable-parameter-model-based feedforward compensation method usesF(z,θ) in (21) to compensate the tracking error.

Becausef(θ) consists of only one parameter θ1wherethe corresponding feedforward compensation value can be obtained easily by the inverse of the parameter.To avoid the influence of high-frequency disturbances, a lowpass filterQ(z) is used in (13) and (21), thenf(θ) can be rewritten as

With the proposed variable parameter system shown in Fig. 1,the fixed plantP(z) becomesP1(z,θ). Under the conditions in Remark 1,P1(z,θ) can be identified asPv(z,θ) by the method in Section III. If the disturbanced(k) is zero and will be considered in the next section, (1) becomes

From Fig. 1, we haveuf f(k)=r(k)·F(z,θ). Then substituting it into (23), we have

IfF(z) has Solution 1,F(z)P(z) will be one andy(k) is equal tor(k).

IfF(z) has Solution 2, substituting (6) into (23) and letwe have the same result.

IfF(z) has Solution 3,will be equal to one andy(k) is equal tor(k).

Therefore, the tracking error is theoretically equal to zero.

However, the variable parameter model has difficulties in obtaining the accurate model under the following conditions:1) large disturbances cause too big parameter deviation; 2)parameter value exceeds the upper or lower bounds.

B. Model Uncertainty Compensation Based on Disturbance Observer

To filter disturbances and improve the control accuracy, a disturbance observer [21], [22] can be applied. Based on the variable parameter model, the designed disturbance observer is shown in Fig. 3. It can be seen that the variable parameter modelPv(z,θ) has replacedP1(z,θ) in the disturbance observer. Ifd(k) in (1) is not zero,d(k) is supposed to becomedu(k) withP1(z,θ) replaced byPv(z,θ).du(k) will be compensated by the disturbance observer.

In Fig. 3,Qd(z) is a low-pass filter. Before running the disturbance observer, the variable parameter modelPv(z,θ)should be sought and the feedforward controllerF(z) can be obtained by (21).

It is supposed that the model uncertainty isΔP, which is calculated by

From Fig.3, the output of the disturbance observer can be obtained by

whereu?(k) is the sum of the outputs of the feedback controllerC(z) and feedforward controllerF(z).

From the schematic diagram of the variable-parametermodel-based feedforward compensation shown in Fig. 3,u?(k)can be calculated by

Remark 2:With a disturbance observer helping to observe the model uncertainty,e(k) is close toea(k) if the disturbance is fully compensated, then the variable parameter model is able to acquire the optimal estimation of the systemP1(z,θ).

Proof:Suppose the transfer function from the disturbancedu(k) to the tracking error isin the proposed variableparameter-model-based feedforward compensation in Fig. 3. It is noticed that the transfer function isHDE(z) in the traditional feedforward compensation method. According to Yu and Tomizuka [21], the following relationship exists if the feedforward controller replaces the ILC in the paper

BecauseQd(z) is a low-pass filter,Qd(z) is equal to one andcan be considered as zero when the frequency is low.According to (26) the model uncertainty at low frequency can be estimated by the disturbance observer and then compensated. So the tracking error can be further reduced compared with the traditional feedforward compensation method.

From Fig. 2, it can be concluded thate(k) is close toea(k) if the disturbance observer is applied, so it existsu(k)≈ua(k)andy(k)≈ya(k), which implies that the variable parameter model is able to acquire the optimal estimation ofP1(z,θ). For a traditional disturbance observer used in the paper, a more accurate model will help achieve more accurate estimation of uncertainty. So the variable parameter model works better with the realtime update of the model parameter.

In summary, there are two cases for the variable parameter model. On the one hand, the variable parameter model is theoretically equal toP1(z,θ) under the conditions that the disturbance is small and θ can be sought within its bounds. On the other hand, if the above conditions cannot be met, the model uncertainty resulted from the variable parameter model can be obtained by the disturbance observer. Therefore, the feedforward controller can reduce the tracking error to the greatest degree in the two cases.

C. Advantage Analysis Compared With the Fixed Parameters Feedforward Compensation Method

Considering the model uncertaintyΔP,P1(z,θ) in every control period can be estimated as

In Fig. 3, it is supposed thatΔPis caused bydu(k). The output can be obtained by

With the variable parameter model, ifF(z,θ) has Solution 1 or Solution 2, then (30) becomes

The tracking error can be calculated by

IfF(z,θ) has Solution 3,f?1(θ)inF(z,θ) andf(θ) inPv(z,θ)can be canceled, so the feedforward controller can be considered as a single ZPETC and the tracking error can be reduced. According to the theory of ZPETC, the tracking error can also be estimated by (32) if the frequency ofr(k) meets the requirement of ZPETC.

From (32), it can be concluded that the tracking error is determined bydu(k).

With the variable-parameter-model-based feedforward compensation, the model uncertainty can be reduced to be a very low level by using the variable parameter model, which implies

When the parameters is over the setting constraints, the disturbance-observer-based compensation will help achieve the optimal estimation ofP1(z,θ) according to Remark 2 and model uncertainty is compensated byud(k) obtained by disturbance observer. According to (29),e(k) can be controlled to be a very small value

With the fixed parameters model, the model uncertainty is big if disturbances exist or a model has variable parameters. In this case, the tracking error is determined bydu(k) and can be estimated by (29).

In summary, the above theoretical analysis shows that the variable-parameter-model-based feedforward compensation is able to achieve a smaller tracking error than the fixed parameters model.

V. Illustrative Examples

Servo systems are often used in robots or numerical control machine tools. Two categories of application examples in servo systems are investigated: the one is thatP(z) has a varying proportional gain; the other is that a servo system is influenced by complex disturbances.

Generally, a servo system can be described by [23]

IfP(z) is a fixed transfer function,aiandbjare constants wherei=0,1,...,nandj=0,1,...,m.

If the servo system has variable parameters,aiandbjare varying coefficients and the transfer function will bePv(z,θ),where θ is a vector composed ofaiandbj. But it is timeconsuming to identify the parameters and very difficult to achieve the accurate values of the vector θ. According to Section III, the variable model in (13) can be used if the conditions in Remark 1 are met. As a result, the variable parameter model is described by the product ofP(z) andf(θ),and the model uncertainty is considered asdu(k) which can be compensated by the disturbance observer in Fig. 3.

The plant of a servo systemP(z) is set for a fixed transfer function

The feedback controller is a proportional-derivative (PD)controller, whose transfer function is

whereTsis the sampling period and set for 0.004 s.

The reference input is a sine curve

This section verifies the effectiveness of the variableparameter-model-based feedforward compensation method by analyzing the performance of the variable parameter model and comparing with ILC and a traditional feedforward compensation method. And the advantages of the variableparameter-model-based feedforward compensation are also illustrated by examples with model uncertainty.

A. Performance Analysis of the Variable Parameter Model

To analyze the performance of the variable parameter model the identification system in Fig. 2 is used to estimate the variable parameterf(θ). Becausef(θ) in (13) has only one parameter θ1, the variable parameter model becomes

where θ1is a variable scalar coefficient and

In order to evaluate the performance of the proposed variable parameter model, the variable parameter θ1is set with different changing rules and the torque disturbancedis set as a complex function. The changing rules of the variable parameter θ1are given as follows:

1) a line

2) a sine

3) a sine over the set seeking range

4) the sine in (41) with the following disturbance added

Within the set seeking range θ1∈[0.8,1.4], the variable parameter θ1can be obtained in every timekby the seeking method in Section III. By using (40)–(43), the variable parameters are obtained and the results are shown in Figs. 4(a)–(d), respectively. In the figure, “identified” and “actual”represent the identified and actual parameters, respectively.

From the results, it can be seen that the proposed seeking method can obtain the values of the variable parameter accurately. But there is model uncertainty when the parameter θ1is out of the set seeking range. For example, in Fig. 4(c)θ1can be obtained when its value is within the seeking range, but not when its value is over the seeking range, which implies that the model uncertainty exists. In this case, the parameter over the seeking range is set as the boundary value. The disturbance in (43) is complex because it changes with the change of the outputy(k?1). And its result in Fig. 4(d) shows that the identified variable parameter is not fully in accordance with (41), especially at 6.2 s, 18.8 s and 31.4 s.The disturbance causes the inconsistency of the parameters,but when the identified variable parameter is used in Fig. 5(d)the excellent compensation effectiveness verifies that the identified variable parameter matches the real system model.So it can be concluded that the variable parameter is able to be identified when the disturbance is compensated by a disturbance observer.

Fig. 4. Identification of θ1 in the variable parameter model. Within the seeking range, the parameters under different changing rules including (a) a line,(b) a sine curve, (c) a sine curve over the set seeking range, and (d) a sine curve with the disturbance can be obtained.

B. Comparison With a Traditional Feedforward Compensation Method

By using the variable parameters sought in Section V-A, the variable-parameter-model-based feedforward compensation method is used to compensate the tracking error. A traditional feedforward controller shown in Fig. 1 is used to compare the compensation performance. In Fig. 3,Qd(z) andQ(z) are designed as the same second ordered Butterworth low-pass filter whose transfer function is shown as follows:

Fig. 5. The tracking errors. Different conditions, that is, (a) a line, (b) a sine curve, (c) constant 1 with the disturbance and (d) a sine over the set seeking range, (e) with a varying transfer function, are considered.

The compensation results are shown in Figs. 5(a)–(d),respectively. “proposed method”, “traditional method” and“without compensation” represent the tracking errors obtained by the proposed variable-parameter-model-based feedforward compensation method, traditional feedforward method and without compensation, respectively.

From the results, it can be concluded that the variableparameter-model-based feedforward compensation method is much better than the traditional feedforward compensation. Just like the theoretical analysis, the traditional feedforward compensation can compensate the tracking error, but the inaccurate plant model limits its effectiveness. Fig. 5 shows that the proposed variable-parameter-model-based feedforward compensation method has much lower tracking errors than the traditional feedforward compensation. Fig. 5(c) and Fig. 5(d)show that the tracking error can be compensated by the variableparameter-model-based feedforward compensation with disturbance observer, even if the model uncertainty exists, i.e.,the variable parameter is over the set seeking range in Fig. 5(c),and there are the disturbance inputs in Fig. 5(d). Fig. 5(d) uses the obtained parameters in Fig. 4(d), and the result shows the tracking error can be compensated to a very small level, which implies that the disturbance can be overcome by the proposed variable-parameter-model-based feedforward compensation method.

In addition to the variable parameter θ1, a more complex condition is considered in the paper. Under the condition that the variable parameter θ1is set with the rule in (41), at the 6th second, the plant of the servo systemP(z) in (36) is changed as follows:

In this case, numerator polynomial coefficients in (35) are changed except for the varying parameter θ1.

ButP(z) in the variable parameter model (39) remains unchanged. Fig. 5(e) shows that the tracking error by the traditional method becomes bigger after 6 s, but that by the proposed variable-parameter-model-based feedforward compensation method remains a small value. This result further verifies that the proposed method is effective to reduce the tracking error when the parameters of a system are varying.

VI. Conclusion and Future Works

In the paper, a variable-parameter-model-based feedforward compensation method is proposed to reduce the tracking error.Based on the built variable parameter model, the nonlinear plant is constructed as a variable parameter model with constraints. A multi-step adaptive seeking approach is used to obtain the parameter of the variable parameter model, and then the inverse of the system can be calculated by the variable parameter model. Finally, the proposed variableparameter-model-based feedforward compensation method can compensate the tracking error to the greatest degree.

By an example of a servo system, the effectiveness of the variable-parameter-model-based feedforward compensation method is verified.

1) The proposed multi-step adaptive seeking method can obtain the variable parameter accurately;

2) The variable-parameter-model-based feedforward compensation method can achieve smaller tracking errors than a traditional feedforward compensation method, and the disturbance observer can help achieve good effectiveness even when the model uncertainty exists.

In the future, more real systems are expected to use the variable-parameter-model-based feedforward compensation method to reduce the tracking error.