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

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.

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]:𝑢_𝑑=𝑅_𝑠𝑖_𝑑+𝐿_𝑑𝑑𝑖_𝑑𝑑𝑡−𝜔_𝑟𝐿_𝑞𝑖_𝑞,𝑢_𝑞=𝑅_𝑠𝑖_𝑞+𝐿_𝑞𝑑𝑖_𝑞𝑑𝑡+𝜔_𝑟(𝐿_𝑑𝑖_𝑑+𝜓_𝑓𝑇),_𝑒=1.5𝑛_𝑝[𝜓_𝑓𝑖_𝑞+(𝐿_𝑑−𝐿_𝑞)𝑖_𝑑𝑖_𝑞]. (2.1)(2.2)

Equation of motion𝑑𝜔_𝑟=1𝑑𝑡𝐽(𝑛_𝑝𝑇_𝑒−𝑛_𝑝𝑇_𝐿−𝐵𝜔_𝑟), (2.3) where 𝑢_𝑑 and 𝑢_𝑞 are stator 𝑑- and 𝑞-axis voltages; 𝐿_𝑑 and 𝐿_𝑞 are stator 𝑑- and 𝑞-axis inductances; 𝑅_𝑠 is resistance of the stator windings; 𝑖_𝑑and i_q are stator d- and q-axis currents; Ψ_𝑓 is amplitude of the flux induced by the permanent magnets of the rotor in the stator phases; 𝑛_𝑝 is number of pole pairs; 𝑇_𝑒 is electromagnetic torque; 𝜔_𝑟 is angular speed of the motor; J is inertia of moment; 𝑇_𝐿 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:𝑑𝑥₁𝑑𝑡=𝑓(𝑥₁,𝑡,𝑤(𝑡))+𝑏𝑢,𝑦=𝑥₁, (3.1) 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 function 𝑓(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 equation𝑥̇₁=𝑥₂𝑥̇+𝑏𝑢,₂=𝑤(𝑡),𝑦=𝑥₁. (3.2)

If 𝑥₂ is bounded, the ESO of (3.2) can be defined as𝑒=𝑧₁𝑧̇−𝑦,₁=𝑧₂−𝛽₀₁𝑓₁𝑧̇(𝑒)+𝑏𝑢,₂=−𝛽₀₂𝑓₂(𝑒), (3.3) 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.

If f_i(e) (i = 1, 2) in (3.3) is replaced with e, the model denoted by (3.3) has the transitional Luenberger observer form. If 𝑓_𝑖(𝑒)=𝑒+𝑘_𝑖sign(𝑒), the model denotes variable structure observer.

The extended state variable 𝑧₂ 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 Figure 1, 𝐻(𝑠) is a compensation unit; 𝐺_𝑐(𝑠) is a primary controller; 𝐺^′_𝑐(𝑠) is a derivative unit; 𝐺_𝑝(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 by𝐺_𝑐(𝑠)=𝐾_𝑝1(1+𝑇_𝑖𝑠1),𝐻(𝑠)=1+𝛽𝑇_𝑖𝑠,𝐺^′_𝑐𝑘(𝑠)=^𝑝𝑇_𝑑𝑠1+𝜂𝑇_𝑑𝑠, (3.4) where 𝛽 is 2DOF integral coefficient; 1/𝜂 is derivative gain; 𝐾_𝑝 is proportional gain; 𝑇_𝑖 is integral time; 𝑇_𝑑 is derivative time.

The transfer function of 2DOF controller can be obtained as𝐶(𝑆)=𝐾_𝑝(𝛼+𝛼𝛾𝑇_𝑑𝑆1+𝜂𝑇_𝑑𝑆+1𝑇_𝑖𝑆−(1−𝛼)(𝛽−1)1+𝛽𝑇_𝑖𝑆). (3.5)

From Figure 1, 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) yields𝜔̇_𝑟=1.5𝑛_𝑝𝜓_𝑓𝐽𝑖_𝑞+𝜆, (3.6) where 𝜆=[(𝐿_𝑑−𝐿_𝑞)𝑛_𝑝𝑖_𝑑𝑖_𝑞−𝑛_𝑝𝑇_𝐿−𝐵𝜔_𝑟]/𝐽.

The variable 𝜆 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:𝜔̇_𝑟=𝜆+1.5𝑛_𝑝𝜓_𝑓𝐽𝑖_𝑞,𝜆̇=𝑔. (3.7)

Referring to (3.3), one can obtain the ESO of (3.7):𝑒=𝑧₁−𝜔_𝑟,𝑧̇₁=𝑧₂−𝛽₀₁fal(𝑒,𝛼₀₁,𝛿)+𝑏₀𝑧̇𝑢,₂=−𝛽₀₂fal(𝑒,𝛼₀₂,𝛿), (3.8) where 0<𝛼₀₁<1; 0<𝛼₀₂<1; 𝛿 is the length of linear segment of the fal(⋅) function which is given by (3.9); 𝜔_𝑟 is the output rotating speed of the PMSM; b₀ denotes known part of model, the expression of which is 1.5𝑛_𝑝𝜓_𝑓/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:fal(𝑒,𝛼,𝛿)=|𝑒|^𝛼𝑒sign(𝑒),|𝑒|>𝛿,𝛿^1−𝛼,|𝑒|≤𝛿, (3.9) where sign(⋅) is a symbolic function.

The error dynamic model of (3.8) can be described as𝑒₁=𝑧₁−𝜔_𝑟,𝑒₂=𝑧₂𝑒̇−𝜆,₁=𝑒₂−𝛽₀₁fal(𝑒₁,𝛼₀₁𝑒̇,𝛿),₂=𝑔−𝛽₀₂fal(𝑒₁,𝛼₀₂,𝛿). (3.10)

If |g|<𝑤₀, that is, the disturbances are bounded, one can obtain the following:lim_𝑡→∞|𝑧₁(𝑡)−𝜔_𝑟(𝑡)|<𝛽₀₁(𝑤₀𝛽₀₂)^1/𝛼02,lim_𝑡→∞|𝑧₂(𝑡)−𝜆(𝑡)|<𝛽₀₂𝑤⋅(⁰𝛽₀₂)^{𝛼01/𝛼02}. (3.11)(3.12)

The extended state variable z₂ 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 Figure 2. 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 Figure 2. Table 1 lists the parameters of the tested PMSM.

Table 1 

Parameters of the PMSM.

The parameters in ESO-2DOF controller fall into two groups. One group is 𝛽₀₁, 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 for 𝛽₀₁ 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 that 𝛽₀₁ 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 speed 𝑛=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.

Figure 3 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.

Figure 4 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.

Figures 5 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.

Figure 7 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.