# Robust ESO Two-Degree-of-Freedom Control Design for Permanent Magnet Synchronous Motor

Authors: Jian-Ping Wen; Bing-Gang Cao; Jun-Yi Cao
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

A robust two-degree-of-freedom control scheme is proposed for permanent magnet synchronous motor (PMSM) using extended state observer (ESO). The robustness is achieved based on the ESO. Parameter perturbation and external disturbances in PMSM drive system are treated as disturbance variable, and then the motion model of PMSM is transformed into an extended state model by introducing this disturbance variable. To estimate the disturbance variable, an ESO is constructed. Estimator is compensated into the control system to improve robustness and adaptability of 2DOF controller against parameter perturbation and external disturbances. The effectiveness of the proposed control scheme is demonstrated with simulation results.


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## Body

## 1. Introduction

The PMSM plays an important role in industrial application due to its high efficiency, high power density, low inertia, no need for maintenance, and high air gap magnetic density [1, 2]. To achieve the high-performance control, the vector control of PMSM drive is developed. In the vector control technique, the Proportional-integral (PI) controller provides the efficient solution to real-world control problems and is applied to many industrial applications [3, 4]. However, the PI controller had many problems in high-performance applications requiring fast and precise speed response, quick speed recovery under any disturbances, and insensitivity to the machine parameters, and it cannot give good command tracking and load regulation property simultaneously because the closed loop zeros cannot be placed arbitrarily. The speed controller that uses a conventional PI controller has poor robustness to nonlinearity, strong coupling, and dynamic uncertainty of PMSM drive systems [5, 6]. Many robust control techniques are proposed to overcome the disturbances of systematic nonlinearity and systematic uncertainty in order to achieve fast speed response and robustness and adaptability to the parameter variations [7–15]. According to the output error correction, observer used in these strategies can be fall into two categorical types: classical state observer and variable structure observer. The classical state observer such as the Luenberger observer using output error linear correction and all state variable feedback control can only be used in linear and certain systems and fail to nonlinear and uncertain systems. The variable structure observer uses nonsmooth structure to improve its robustness to both system uncertainties and measurement errors but with chattering phenomenon. To overcome the drawbacks of the PI controller, 2DOF controller is investigated to treat the command tracking and disturbance regulation specifications, separately. However, the 2DOF controller is sensitive to the machine parameters. When the parameter variation exceeds certain range, performances of the 2DOF controller become worse.The extended state observer uses the extended state variable to observe the system uncertainties and external disturbances. The control system is compensated by the estimator of the extended state variable to improve its robustness.In this paper, a robust ESO-2DOF control scheme is introduced. Regarding parameter perturbation and external disturbances as system disturbance variable, which are observed by the ESO and compensated into 2DOF controller to cancel influences of the disturbances, simulation results show that the proposed strategy effectively improves the robustness of the 2DOF controller as well as good command tracking and strong disturbance rejection characteristic simultaneously.


## 2. Mathematical Model of PMSM

On the basis of assumptions that the stator windings generate sinusoidal magnetic field, air gap is uniform and saturation is negligible. With reference to synchronous rotating reference frame, the voltage and torque equations of an IPMSM may be expressed as follows [16]:(2.1)ud=Rsid+Lddiddt-ωrLqiq,uq=Rsiq+Lqdiqdt+ωr(Ldid+ψf),(2.2)Te=1.5np[ψfiq+(Ld-Lq)idiq].Equation of motion(2.3)dωrdt=1J(npTe-npTL-Bωr),
where ud and uq are stator d- and q-axis voltages; Ld and Lq are stator d- and q-axis inductances; Rs is resistance of the stator windings; idand iq are statord- and q-axis currents; Ψf is amplitude of the flux induced by the permanent magnets of the rotor in the stator phases; np is number of pole pairs; Te is electromagnetic torque; ωr is angular speed of the motor; J  is inertia of moment; TL is the load torque; B is viscous friction coefficient.


## 3. Design of ESO-2DOF Controller

### 3.1. Extended State Observer

The function of state observer is to reconstruct system state that bases on known inputs and measured outputs. The control systems generally have some disturbances such as parameter perturbation, unmodeled-dynamic, and measurement error. The ESO is used to estimate the uncertainty of system and external disturbances. By compensating the estimator into the control system, the nonlinear and uncertain system can be approximated linearization and certainty. As an example, the procedure of developing ESO for first-order system is given.A class of first-order system is described as follows:(3.1)dx1dt=f(x1,t,w(t))+bu,y=x1,
where x1 is state variable; t is time variable; w(t) is unknown disturbance; f(x1,t,w(t)) is unknown function; b is system parameter, which denotes the known part of system; u is control input; y is output signal.Let functionf(x1,t,w(t)) be the extended state variable of system, let x2=f(x1,t,w(t)), and let unknown disturbance w(t) be the differential of extended state variable. Equation (3.1) can be rearranged into following equation(3.2)ẋ1=x2+bu,ẋ2=w(t),y=x1.Ifx2 is bounded, the ESO of (3.2) can be defined as(3.3)e=z1-y,ż1=z2-β01f1(e)+bu,ż2=-β02f2(e),
where β01 and β02 are system gains; f1(e) and f2(e) are correcting function, which satisfy e·f1(e)>0 and e·f2(e)>0,∀e≠0.Iffi(e) (i= 1, 2) in (3.3) is replaced with e, the model denoted by (3.3) has the transitional Luenberger observer form. If fi(e)=e+kisign(e), the model denotes variable structure observer.The extended state variablez2 is the real-time action of the unknown disturbances [17]. z1 is estimator of x1.


### 3.2.2DOF Controller

The 2DOF controller does not consist of two individual Proportional-integral-derivative (PID) controllers, but it can set independently the parameters of each PID controller to achieve good command tracking and disturbance rejection characteristic simultaneously.This paper uses advanced 2DOF controller [18]. It is shown in Figure 1.Figure 1
2DOF Control Configuration.In Figure1, H(s) is a compensation unit; Gc(s) is a primary controller; Gc′(s) is a derivative unit; Gp(s) is the controlled object; α,γ are 2DOF proportional and derivative coefficients, separately; r is reference signal.For the PMSM regulation speed system, the expressions are given by(3.4)Gc(s)=Kp(1+1Tis),H(s)=11+βTis,Gc′(s)=kpTds1+ηTds,
where β is 2DOF integral coefficient; 1/η is derivative gain; Kp is proportional gain; Ti is integral time; Td is derivative time.The transfer function of 2DOF controller can be obtained as(3.5)C(S)=Kp(α+αγTdS1+ηTdS+1TiS-(1-α)(β-1)1+βTiS).From Figure1, it is seen that the advanced 2DOF controller mainly consists of reference signal filter and PID controller. The adjustable parameters of the reference signal filter are α,β, and γ, the change of which influences the command tracking performance of advanced 2DOF controller. The dynamic processes are controlled by the PID controller, the parameters of which can be set according to the traditional PID adjusting methods.


### 3.3. Robust ESO-2DOF Controller

To overcome the impact of disturbances to the system, this paper uses the ESO to estimate and compensate the disturbances into the 2DOF controller. Thus, motion model of PMSM is transformed into the extended state model.Substituting (2.2) into (2.3) yields(3.6)ω̇r=1.5npψfJiq+λ,
where λ=[(Ld-Lq)npidiq-npTL-Bωr]/J.The variableλ is regarded as the extended state variable of system. Let the differential of λ be g, and (3.6) can be rewritten in the following form:(3.7)ω̇r=λ+1.5npψfJiq,λ̇=g.Referring to (3.3), one can obtain the ESO of (3.7):(3.8)e=z1-ωr,ż1=z2-β01fal(e,α01,δ)+b0u,ż2=-β02fal(e,α02,δ),
where 0<α01<1; 0<α02<1; δ is the length of linear segment of the fal(·) function which is given by (3.9); ωr is the output rotating speed of the PMSM; b0 denotes known part of model, the expression of which is 1.5npψf/J; z1 is estimator of speed; z2 is estimator of the extended state variable.For (3.8), the correcting function adopts the fal(·) to achieve quickly smooth convergence property and good stability at equilibrium point:(3.9)fal(e,α,δ)={|e|αsign(e),|e|>δ,eδ1-α,|e|≤δ,
where sign(·) is a symbolic function.The error dynamic model of (3.8) can be described as(3.10)e1=z1-ωr,e2=z2-λ,ė1=e2-β01fal(e1,α01,δ),ė2=g-β02fal(e1,α02,δ).If|g|<w0, that is,  the disturbances are bounded, one can obtain the following:(3.11)limt→∞|z1(t)-ωr(t)|<β01(w0β02)1/α02,(3.12)limt→∞|z2(t)-λ(t)|<β02·(w0β02)α01/α02.The extended state variablez2 can better estimate the disturbances by adjusting the parameters of (3.11) and (3.12).By using the extended state variable, the estimator of the disturbances can be obtained. The ESO-2DOF controller is developed through compensating the disturbances into 2DOF controller. In the light of the 2DOF controller structure, the estimator is compensated before the primary controller.The diagram of the ESO-2DOF controller is shown in Figure2. r is speed reference.Figure 2
2DOF Control Configuration Base on ESO.


## 3.1. Extended State Observer

The function of state observer is to reconstruct system state that bases on known inputs and measured outputs. The control systems generally have some disturbances such as parameter perturbation, unmodeled-dynamic, and measurement error. The ESO is used to estimate the uncertainty of system and external disturbances. By compensating the estimator into the control system, the nonlinear and uncertain system can be approximated linearization and certainty. As an example, the procedure of developing ESO for first-order system is given.A class of first-order system is described as follows:(3.1)dx1dt=f(x1,t,w(t))+bu,y=x1,
where x1 is state variable; t is time variable; w(t) is unknown disturbance; f(x1,t,w(t)) is unknown function; b is system parameter, which denotes the known part of system; u is control input; y is output signal.Let functionf(x1,t,w(t)) be the extended state variable of system, let x2=f(x1,t,w(t)), and let unknown disturbance w(t) be the differential of extended state variable. Equation (3.1) can be rearranged into following equation(3.2)ẋ1=x2+bu,ẋ2=w(t),y=x1.Ifx2 is bounded, the ESO of (3.2) can be defined as(3.3)e=z1-y,ż1=z2-β01f1(e)+bu,ż2=-β02f2(e),
where β01 and β02 are system gains; f1(e) and f2(e) are correcting function, which satisfy e·f1(e)>0 and e·f2(e)>0,∀e≠0.Iffi(e) (i= 1, 2) in (3.3) is replaced with e, the model denoted by (3.3) has the transitional Luenberger observer form. If fi(e)=e+kisign(e), the model denotes variable structure observer.The extended state variablez2 is the real-time action of the unknown disturbances [17]. z1 is estimator of x1.


## 3.2.2DOF Controller

The 2DOF controller does not consist of two individual Proportional-integral-derivative (PID) controllers, but it can set independently the parameters of each PID controller to achieve good command tracking and disturbance rejection characteristic simultaneously.This paper uses advanced 2DOF controller [18]. It is shown in Figure 1.Figure 1
2DOF Control Configuration.In Figure1, H(s) is a compensation unit; Gc(s) is a primary controller; Gc′(s) is a derivative unit; Gp(s) is the controlled object; α,γ are 2DOF proportional and derivative coefficients, separately; r is reference signal.For the PMSM regulation speed system, the expressions are given by(3.4)Gc(s)=Kp(1+1Tis),H(s)=11+βTis,Gc′(s)=kpTds1+ηTds,
where β is 2DOF integral coefficient; 1/η is derivative gain; Kp is proportional gain; Ti is integral time; Td is derivative time.The transfer function of 2DOF controller can be obtained as(3.5)C(S)=Kp(α+αγTdS1+ηTdS+1TiS-(1-α)(β-1)1+βTiS).From Figure1, it is seen that the advanced 2DOF controller mainly consists of reference signal filter and PID controller. The adjustable parameters of the reference signal filter are α,β, and γ, the change of which influences the command tracking performance of advanced 2DOF controller. The dynamic processes are controlled by the PID controller, the parameters of which can be set according to the traditional PID adjusting methods.


## 3.3. Robust ESO-2DOF Controller

To overcome the impact of disturbances to the system, this paper uses the ESO to estimate and compensate the disturbances into the 2DOF controller. Thus, motion model of PMSM is transformed into the extended state model.Substituting (2.2) into (2.3) yields(3.6)ω̇r=1.5npψfJiq+λ,
where λ=[(Ld-Lq)npidiq-npTL-Bωr]/J.The variableλ is regarded as the extended state variable of system. Let the differential of λ be g, and (3.6) can be rewritten in the following form:(3.7)ω̇r=λ+1.5npψfJiq,λ̇=g.Referring to (3.3), one can obtain the ESO of (3.7):(3.8)e=z1-ωr,ż1=z2-β01fal(e,α01,δ)+b0u,ż2=-β02fal(e,α02,δ),
where 0<α01<1; 0<α02<1; δ is the length of linear segment of the fal(·) function which is given by (3.9); ωr is the output rotating speed of the PMSM; b0 denotes known part of model, the expression of which is 1.5npψf/J; z1 is estimator of speed; z2 is estimator of the extended state variable.For (3.8), the correcting function adopts the fal(·) to achieve quickly smooth convergence property and good stability at equilibrium point:(3.9)fal(e,α,δ)={|e|αsign(e),|e|>δ,eδ1-α,|e|≤δ,
where sign(·) is a symbolic function.The error dynamic model of (3.8) can be described as(3.10)e1=z1-ωr,e2=z2-λ,ė1=e2-β01fal(e1,α01,δ),ė2=g-β02fal(e1,α02,δ).If|g|<w0, that is,  the disturbances are bounded, one can obtain the following:(3.11)limt→∞|z1(t)-ωr(t)|<β01(w0β02)1/α02,(3.12)limt→∞|z2(t)-λ(t)|<β02·(w0β02)α01/α02.The extended state variablez2 can better estimate the disturbances by adjusting the parameters of (3.11) and (3.12).By using the extended state variable, the estimator of the disturbances can be obtained. The ESO-2DOF controller is developed through compensating the disturbances into 2DOF controller. In the light of the 2DOF controller structure, the estimator is compensated before the primary controller.The diagram of the ESO-2DOF controller is shown in Figure2. r is speed reference.Figure 2
2DOF Control Configuration Base on ESO.


## 4. Simulation Results and Analysis

In order to verify the validity of the proposed ESO-2DOF controller, a computer simulation model is developed in MATLAB/Simulink software according to Figure2. Table 1 lists the parameters of the tested PMSM.Table 1
Parameters of the PMSM.
NameValueRated power1.5 kWRated speed2800 r/minNumber of poles8Rated current22AStator resistance0.093Ωd-axis inductance0.4 mHq-axis inductance0.5 mHmagnetic flux0.026 WbThe parameters in ESO-2DOF controller fall into two groups. One group isβ01, β02, α01, α02,and δ. These parameters belong to the ESO. The others belong to advanced 2DOF controller. The robustness of the ESO to both disturbance and parametric variation is strong when α01 and α02 are small [19, 20]. α01 and α02 are set to 0.25 and 0.5 in simulation, respectively. δ denotes the length of linear segment of the nonlinear function. Slope of the linear segment can be described by differential of δ. In general, δ is less than 5, or else the performances of nonlinear feedback and disturbance rejection performances of system will get worse. δ is set to 0.05 in simulation.By selecting appropriate values forβ01 and β02, the tracking performances of the state variables in the ESO can be improved. If β01 is too small, the tracking performances of z1 and z2 can be degraded. On the contrary, if β02 has a high value, the steady-state characteristic of system becomes poor.The simulation results show thatβ01 can be set according to the sampling frequency. The recommendation is that β01 is one to two times sampling frequency and β02 is one to ten times β01.Due to limitation of space, only two simulation results are presented to illustrate the characteristics of the proposed ESO-2DOF controller. In the first test condition, a step command speedn=1500 rpm is applied to the PMSM drive starting from rest at t=0 s and with no load. At t=0.04 s, the load disturbance is used. The second test condition is that the moment of inertia is changed into 150% of the original value and the load disturbance is used at t=0.03 s and the unloading is occurred at 0.05 s.Figure3 shows the speed responses with load disturbance at t=0.04 s when the motor speed is 1500 rpn. The fast tracking and disturbance rejection performance of the ESO-2DOF controller is achieved well.Figure 3
Dynamic antidisturbance curves of PMSM system.Figure4 shows the estimator of the disturbance. As can be observed from Figure 5, the ESO can effectively estimate the disturbances.Figure 4
Disturbance curves of system.Figure 5
Dynamic speed response curves of PMSM system with parameter perturbation.Figures5 and 6 show speed response with load disturbance and with unloading when the moment of inertia is changed into 150% of the original value. The results show that the ESO-2DOF controller has strong robustness to parameter perturbation.Figure 6
Dynamic speed response curves of PMSM system with unloading.Figure7 shows the estimator of the unloading disturbance. As can be observed from Figure 7, the ESO can effectively estimate the disturbances.Figure 7
Disturbance curves of system with parameter perturbation.


## 5. Conclusions

To improve the robustness of 2DOF controller, the ESO-2DOF controller is proposed. The parameter perturbation and external disturbances are estimated simultaneously by using the ESO. The estimator is compensated into 2DOF controller to guarantee the strong robustness, the good command tracking, and disturbance rejection characteristic simultaneously. Simulation results are given to demonstrate the validity of the proposed control scheme.



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*Source: 405251-2010-05-30.xml*