Mathematical Problems in Engineering

Volume 2015, Article ID 316360, 7 pages

http://dx.doi.org/10.1155/2015/316360

## Unknown Disturbance Estimation for a PMSM with a Hybrid Sliding Mode Observer

^{1}Tongji University, Shanghai 200092, China^{2}Zhejiang Textile & Fashion College, Ningbo 315211, China^{3}Wild SC Intelligent Technology CO., LTD, Ningbo 315500, China

Received 7 July 2014; Revised 26 September 2014; Accepted 2 October 2014

Academic Editor: Xudong Zhao

Copyright © 2015 Gang Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A hybrid sliding mode observer that combines high gain feedback and a high-order sliding mode term is developed to identify the time-varying disturbance for a permanent-magnet synchronous motor (PMSM). Based on the measurable current and the position, the unknown disturbance can be identified from the sliding mode term without digital filter effect. It is then used to enhance the robustness of the speed control dynamics. For ease of implementation in real applications, such as DSP and FPGA, the proposed observer is properly designed to avoid complex mathematical operators. Simulation results are given to illustrate the performance of the proposed observer.

#### 1. Introduction

In real industrial application, the unknown disturbance on a permanent-magnet synchronous motor (PMSM) is inevitable and it limits the performance of the controlled processes. For such a situation, a robust observer with high estimation accuracy is required to recover the uncertainty in real time.

The sliding-mode-based observer has been proven to be an effective approach for handling uncertain systems, due to its capability of reconstructing the uncertainties based on the equivalent injection input concept [1–3], and there are many effective results presented during past years [4–13]. In [14], a sliding mode flux observer is developed for an induction motor, and the unknown rotor resistance can be reconstructed from a switching function after the sliding surface driven to zero. In [15, 16], the sliding mode observer is used to identify the back-EMFs for a PMSM. However, such results require a low-pass filter for the uncertainties reconstruction because of the discontinuous switching feedback, and the uncertainty estimation accuracy is greatly dependent on the low-pass filter parameters.

To improve the estimation accuracy by removing the filtering effect, some existing works have suggested the usage of a saturation function to replace the switching term [17–19]. However, this results in a trade-off between the robustness of observer and the estimation accuracy of uncertainty. Another existing method is based on the higher-order sliding mode techniques, by treating the derivatives of the system input as a new control signal, which results in a continuous integral function of the switching term; then the chattering/filtering effect can be avoided [20, 21].

Another problem is that the sliding mode gains have to be chosen large enough to ensure the stability of the observer, which is usually related to the system initial values. To solve this problem, a hybrid observer that combines high gain feedback and higher-order sliding mode observer has been proposed in [22], in which the high gain feedback works to constrain the estimation error within an invariant set regardless of initial values; then the sliding mode gains can be designed to ensure the global stability of the observer. Such hybrid observers have been applied into a series DC motor for the speed estimation and unknown time-varying parameter identification for a DC motor [23].

In this paper, we will consider the unknown disturbance estimation for a permanent-magnet synchronous motor. A hybrid sliding mode observer is developed for the disturbance identification. The most challenging problem is how to properly design the observer algorithm to simplify the observer implementation in real applications, such as in a digital signal processor (DSP) and FPGA. The contribution of this work can be summarized as follows.(1)With the proposed observer, the disturbance can be asymptotically estimated and then used to enhance the robustness of the speed control dynamics. The theoretical analysis and simulation results are given to demonstrate the effectiveness of the proposed observer.(2)A hybrid observer structure is adopted, in which the high gain feedback term works to speed up the convergence time and guarantees the sliding mode gain design is independent of the system initial values.(3)For disturbance identification, the digital filtering effect is successfully avoided without sacrificing robustness of the observer.(4)The twisting sliding mode algorithm is properly integrated in the proposed observer, to make the observer ease for implementation in real application.

The remainder of this paper is organized as follows. Section 2 introduces the mathematical mode of the PMSM and some background results are presented. In Section 3, a speed observer and a disturbance are developed to identify the unknown external load disturbance. In Section 4, some simulations are provided to illuminate the effectiveness of the two proposed observers. Section 5 concludes this paper.

#### 2. Preliminary

##### 2.1. Mathematical Model

The state equation of a surface-mounted magnet brushless AC motor in stationary - reference frame is given by [24]:where is the state current in - reference frame, is the state voltage in - reference frame, is the resistance, is the inductance, is the flux-linkage due to permanent magnet, is the rotor position, is the electrical angular speed, is the rotor moment of inertia, is the viscous-friction coefficient, and is the load torque.

Here, the load torque is assumed to be the system uncertainty that includes the unknown external disturbance, the parameter variations of rotor inertia, and the viscous-friction coefficient.

The target of this paper is to develop a robust and high accuracy observer that can identify the unknown disturbance , based on the measurable current, voltage, and rotor position.

##### 2.2. Background Results

Consider an uncertain nonlinear system in the form ofwhere andare constant matrices; denotes the lumped system uncertainty; is the system input; the nonlinear functions and are smooth Lipschitz vectors with triangular structures, it haswith being the relative degree order between the disturbance and the measurable output .

Then, a hybrid observer that combines high gain feedback with higher-order sliding mode can be designed as [22]where is the high gain linear feedback parameter, given byand is a positive tuning parameter of nonlinear feedback , which is designed as an integral function of a th-order sliding mode term ; denotes the upper bound of .

It has been proven in [22] that the high gain feedback term works to constrain the estimation error to within an invariant set and the nonlinear term will asymptotically track the unknown disturbance without digital filtering effect if we properly design the sliding mode feedback term.

The mechanical dynamics given in (2) can be considered as a second-order nonlinear system in (3), with the relative degree order between the disturbance and the measurable output being two; that is, , . Then, one can design a hybrid observer in the form of (6), with being third-order sliding mode feedback. However, to the best of our understanding, the existing third-order or higher-order sliding mode algorithms involve complex math operations that require more computation resources in real implementation, such as in digital signal processor and FPGA.

In next section, we will design a hybrid observer that is based on the second-order twisting sliding mode algorithm [1]; the most advantage of such sliding algorithm is that it only requires the sign of the first derivative operator, but not its real value. Thus, the proposed observer takes less computation resources and is easy for implementation in digital signal processor.

#### 3. Observer Design

Consider the mechanical dynamic system described in (2); the rotor position is measurable, and the target is to estimate the unknown disturbance , as well as the angular speed .

It is well-known that there are many methodologies to estimate the angular speed based on the measurable position . One common method is based on the back-forward differentiator; that is, , where is the sampling time and subscript denotes the variable value at time . For such method, the estimation error is proportional to sampling time. On the other hand, small sampling time may enlarge the numerical error of , especially at low rotor speed and low position sensor resolution situation.

As mentioned in the previous section, in order to simplify the hybrid observer structure and remove the digital filtering effect, we will consider using the second-order twisting algorithm for the unknown disturbance estimation.

The system diagram is shown in Figure 1. The first observer (speed observer) will be performed to identify the rotor speed without involving the operation of inverse of sampling time, that is, . Then the second observer (disturbance observer) ensures the unknown disturbance can be identified online without digital filtering effect. Finally, one can use the identified disturbance to compensate the control loop to improve the robustness of the control system.