Abstract

Considering the varying inertia and load torque in high speed and high accuracy servo systems, a novel discrete second-order sliding mode adaptive controller (DSSMAC) based on characteristic model is proposed, and a command observer is also designed. Firstly, the discrete characteristic model of servo systems is established. Secondly, the recursive least square algorithm is adopted to identify time-varying parameters in characteristic model, and the observer is applied to predict the command value of next sample time. Furthermore, the stability of the closed-loop system and the convergence of the observer are analyzed. The experimental results show that the proposed method not only can adapt to varying inertia and load torque, but also has good disturbance rejection ability and robustness to uncertainties.

1. Introduction

The growth and development of our modern society is heavily dependent on the development, sustenance, and improvement of control systems. In recent years, there has been an increasing demand for high precision controllers in servo systems which should have fast tracking behavior, good disturbance rejection ability, and robustness under uncertainties. However, to improve the performance of servo systems is a formidable task due to several reasons, such as the variation of system inertia and load torque, the nonlinearities caused by backlash, friction, other uncertain disturbances, and variable unknown command signal. The above factors cannot be predicted beforehand, which makes the problem more challenging.

Design of controllers considering backlash and friction has received a lot of attention. Problems like tracking errors, self-excited vibration, delays, and steady state errors may arise in this situation. A recent survey paper [1] summarized the backlash models and the compensation methods. Different mathematical models have been developed to describe the backlash phenomenon, such as dead-zone model and hysteresis model [1]. Friction modeling and compensation have been studied extensively in the past few decades. Dynamic friction models, mainly including the Dahl model [2] and the LuGre model [3], have been proposed and shown to be more beneficial. The majority of previous studies have addressed either the friction compensation problem or the backlash compensation problem. Very few papers dealt with them together. The control of systems in the presence of backlash and friction was discussed in [4, 5]. In [4], the neural network model was adopted, and the experiment was carried out under the condition of low velocity and constant inertia and load torque. In [5], fuzzy logic-based algorithm was used to reduce the effect of backlash and friction, and simulation results were presented to show the effectiveness of the controller. In [6], an adaptive fuzzy speed controller based on the backstepping method was designed by considering the existence of parameter variations and load disturbances; experiments indicated that the designed controller performed with better adaptability and robustness compared with the proportion integration (PI) controller. However, parameter adjustment is relatively complex.

According to the above situations, controllers of time-varying servo systems should have fast reaction characteristic and high precision tracking behavior. So an adaptive sliding mode controller is chosen to be applied in the servo system. Focusing on control algorithm, sliding mode control (SMC) is a nonlinear control method which is capable of suppressing the influence of model perturbations and external disturbances [7]. The analysis and applications of high-order SMC have become the research hotspot [8–10], especially the second-order SMC owing to its simple structure and easy realization [11–15]. In [11], the second-order SMC was combined with the fuzzy identification to make it converge in a finite period of time without chattering, and the adaptive robust compensation was also employed to improve precision. In [12], high-order sliding mode controller was proposed for PMSM speed control. Nonsingular terminal sliding modes were designed and used to eliminate the chattering and stabilize the system. In [13], a second-order sliding mode controller was designed for output tracking of mobile manipulators to improve the tracking performance and reduce the chattering.

To improve the performance of the controller under conditions of backlash, friction, and variable inertia and load torque, we should find a SMC with good adaptive ability. Our motivation is to find a proper modeling method and an effective control algorithm to follow the command signal with high acceleration and velocity. Moreover, very few researches of servo systems with variable inertia and load torque were carried out. This motivates us to carry out the present work to use a new modeling method called characteristic modeling which is proposed by Wu et al. [16]. The main advantage of characteristic modeling is that the system can be described by a second-order time-varying difference equation, and the model parameters are determined beforehand within a fairly small range, which is beneficial to fast parameter convergence of online identification [16, 17]. And all the uncertainties and nonlinearities can be included in the characteristic model. In addition to modeling method, proper control algorithm is also very important in the presence of uncertainties and nonlinearities. In the following sections, a novel adaptive discrete second-order sliding mode controller based on characteristic model is proposed, and a command observer is adopted to predict command value. In addition to simulations, experimental results show that the proposed method can obtain good performance in servo systems with varying inertia and load torque and has good adaptive ability and robustness.

The remainder of this paper is organized as follows. Section 2 gives the original dynamic model and the characteristic model of the servo system, and the effectiveness of the characteristic model is verified by simulations. In Section 3, a command observer is designed, a discrete second-order sliding mode adaptive controller based on characteristic model is proposed, and the stability of the closed-loop system is analyzed. Section 4 presents the experimental results. Finally, the conclusion is given in Section 5.

2. Problem Description and System Modeling

The characteristic modeling is based on the dynamics characteristics and control performance requirements of the plants, rather than being only based on accurate plant dynamics analysis. For the same input, a plant characteristic model is equivalent to its practical plant in output. In a dynamic process, the output error can be maintained within a permitted range. In the steady state, their outputs are equal. A characteristic model is different from the reduced-order model of a high-order system. It compresses all the information of the high-order model into several characteristic parameters. Above all, the characteristic modeling makes it possible to design a feasible low-order intelligent controller for various complicated plants with nonlinearities and uncertainties.

The nonlinear system is expressed asDefine , , and , ; then, (1) can be rewritten as The nonlinear system (2) is assumed to be as follows.(1)The system is a SISO system.(2)The order of the control input is 1.(3) when all the bounded variables and are equal to zero.(4) is continuous differentiable to all the variables and , and all partial derivatives are bounded.(5), where is the sample period and is a positive constant.

Lemma 1 (see [16]). If system (2) satisfies the above assumptions, the characteristic model of the system can be established by a second-order slow time-varying difference equation as where , , , and are the time-varying parameters of the model, which can be estimated online. And the parameter range can be determined as , , , and .

If system (2) is a minimum phase system, the in (3) can be neglected. The characteristic model of the system can be written as

The research object is a typical servo system driven by gears; when the system is encountered with backlash and friction, the nonlinear component could be described by a continuous differentiable approximate function to make the model smooth and satisfies the above assumptions [6]. The dynamic equation of the system is expressed aswhere , , , , , and denote the armature voltage, armature current, the resistance, the inductance, the back electromotive force (EMF) coefficient, and the torque coefficient of the motor, respectively. , , , , , , , , , and denote the angular displacement, the angular velocity, the angular acceleration, the rotational inertia, the viscous friction coefficient of the motor, and the load, respectively. , , , , , and denote the elastic torque between the motor and the load, the friction torque, the stiffness coefficient, the gear ratio, the torque of the motor, and the load, respectively. and denote the dead-zone function and the Stribeck friction function, respectively [18, 19]. and denote the relative angular displacement between the motor and the load and half of the backlash width, respectively. , , and denote the Coulomb friction, the maximum static friction, and the Stribeck speed, respectively. and are undetermined parameters reflecting the approximation degree. The structure of the system is shown in Figure 1. ACR, ASR, and APR are PI current controller, PI speed controller, and the proposed position controller, respectively, and the structure in the dashed box represents the dynamic model of the servo system. is the variable position command signal which is unknown beforehand, and the sine signal is often adopted as the testing command signal in practical servo systems.

Figure 1 

Structure diagram of the servo system.

To verify the effectiveness of the proposed characteristic model, the simulations are performed in MATLAB environment. The scheme of verifying the characteristic model is shown in Figure 2. The output of the dynamic model is adopted as the standard output. The recursive least squares (RLS) method with forgetting factor is adopted to estimate the time-varying parameters online. is the system input. and are the dynamic model output and the characteristic model output, respectively. is the error between dynamic model output and characteristic model output. and represent the variables and , respectively, in (4).

Figure 2 

Scheme of verifying the characteristic model.

The system parameters are ,  H,  V/krpm,  N·m/A,  kg·m²,  kg·m²,  N·m/rad,  N·m/krpm,  N·m/krpm, , , ,  N·m,  N·m,  rad/s, and .

The results of verifying the characteristic model are shown in Figures 3, 4, and 5. , , and are identification values of the time-varying parameters , , and , respectively, in (4).

(a) and

(b)

(c) Characteristic model output

(d) Error between dynamic model output and smoothed dynamic model output

(e) Error between smoothed dynamic model output and characteristic model output

(f) Error between dynamic model output and characteristic model output

(a) and
(b)
(c) Characteristic model output
(d) Error between dynamic model output and smoothed dynamic model output
(e) Error between smoothed dynamic model output and characteristic model output
(f) Error between dynamic model output and characteristic model output

Figure 3 

Verification results when  rpm.

(a) and

(b)

(c) Characteristic model output

(d) Error between dynamic model output and smoothed dynamic model output

(e) Error between smoothed dynamic model output and characteristic model output

(f) Error between dynamic model output and characteristic model output

(a) and
(b)
(c) Characteristic model output
(d) Error between dynamic model output and smoothed dynamic model output
(e) Error between smoothed dynamic model output and characteristic model output
(f) Error between dynamic model output and characteristic model output

Figure 4 

Verification results when  rpm.

(a) and

(b)

(c) Characteristic model output

(d) Error between dynamic model output and smoothed dynamic model output

(e) Error between smoothed dynamic model output and characteristic model output

(f) Error between dynamic model output and characteristic model output

(a) and
(b)
(c) Characteristic model output
(d) Error between dynamic model output and smoothed dynamic model output
(e) Error between smoothed dynamic model output and characteristic model output
(f) Error between dynamic model output and characteristic model output

Figure 5 

Verification results when  rpm.

According to Figures 3, 4, and 5, the error between smoothed dynamic model and dynamic model and the error between characteristic model and the dynamic model are both very small, and the steady state error between characteristic model and dynamic model caused by the use of continuous differentiable approximate function is also very small and is considered in the following section. The results indicate that the characteristic model can properly describe the electromechanical system.

3. Controller Design and Stability Analysis

The command value of next sample time is used in the proposed DSSMAC controller later in the chapter; the following command observer is designed to obtain the value of . Assume that the command signal could be expressed with a second-order difference equation aswhere , , , and is the sample period. Equation (6) can describe most command signals with acceleration, and indicated that the acceleration is bounded.

The command observer is designed aswhere and are the parameters to be designed and denotes the estimation error (.

Subtract (7) from (6) to obtain

Define ; then, the above equation can be written as , where

Define and ; then,

Consider a matrix aswhere

Then, through a fundamental algebra operation, we get , where is a unit matrix.

The parameters and are chosen to satisfy the following condition, which gives :where

Theorem 2. If observer (7) is applied to system (6) and and are chosen to satisfy condition (13), then the observation error is bounded stable.

Proof. Take as the Lyapunov equation; then,Obviously, when ; then, the set is an invariant set. If is outside the set , then it will finally come back inside set ; that is, is bounded stable.

The characteristic model of the system can be written as follows:where , , and are the estimation of , , and and denotes the model identification error. represents , which denotes the output of the model, and represents , which denotes the output of the designed controller or the control input of the plant; define as the tracking error; then, where denotes the variable unknown position command value of next sample time, is the estimation value of , and denotes the command estimation error.

Define . Assuming , where is a positive constant, then substituting (16) into (17), we obtain

The sliding function is defined as , and the second-order sliding function is defined as , where the parameters and are the parameters to be designed. When reaching the sliding mode surface, we obtain ; that is, ; then, (18) becomes

The equivalent control is obtained from (19) aswhere is the command estimation value, which can be obtained from (7).

The switching control is designed aswhere is the parameter to be designed; then, combining (20) and (21), the total control action is designed as

Theorem 3. If controller (22) is applied to system (16) and , , , then the closed-loop system is stable.

Proof. Take (22) into (18) to obtainEquation (23) can be rewritten asSubtract (24) from (23) to obtainthat is,such thatDefine the quasi-sliding mode band width ; according to (24) when , we havethat is,According to (23), (24), and (29), we obtainWhen , (29) becomesIf , (27) becomes .If , according to (23) and (31), we haveThe verification of the three conditions (27), (30), and (32) proves the existence of the convergent quasisliding mode, and the quasisliding mode is reached within a finite number of steps. Therefore, the closed-loop system is stable.