Discrete Second-Order Sliding Mode Adaptive Controller Based on Characteristic Model for Servo Systems

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 proposedmethod not only can adapt to varying inertia and load torque, but also has good disturbance rejection ability and robustness to uncertainties.


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 timevarying 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) 2 Journal of Control Science and Engineering 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][9][10], especially the second-order SMC owing to its simple structure and easy realization [11][12][13][14][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.

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.
(1) The system is a SISO system.
(2) The order of the control input () is 1.
(3) (⋅) = 0 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.
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  1 (),  2 (),  0 (), and  1 () are the time-varying parameters of the model, which can be estimated online.And the parameter range can be determined as If system (2) is a minimum phase system, the  1 ()(−1) 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 as where (), (), , ,   , 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.  , θ  , θ  ,   ,   ,   , θ  , θ  ,   , 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. 1 ,  2 ,  1 , ,   , 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. 1 and  2 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 θ  denote the Coulomb friction, the maximum static friction, and the Stribeck speed, respectively. and  2 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.
To verify the effectiveness of the proposed characteristic model, the simulations are performed in MATLAB environment.The scheme of verifying the characteristic model  The results of verifying the characteristic model are shown in Figures 3, 4, and 5. f1 (), f2 (), and ĝ0 () are identification values of the time-varying parameters  1 (),  2 (), and  0 (), respectively, in (4).
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.Journal of Control Science and Engineering 7

Controller Design and Stability Analysis
The command value of next sample time   ( + 1) is used in the proposed DSSMAC controller later in the chapter; the following command observer is designed to obtain the value of   ( + 1).Assume that the command signal could be expressed with a second-order difference equation as where   () =  1 (), |()| < ,  > 0, 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 as where  > 0 and  > 0 are the parameters to be designed and x () =   () − x () denotes the estimation error ( = 1, 2).Subtract ( 7) from ( 6) to obtain Define X() = [ x1 () x2 ()]  ; then, the above equation can be written as X( + 1) =  X() + (), where Define  =  − 1 and  = ; then, Consider a matrix  as where 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  =   > 0: 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.

Figure 1 :
Figure 1: Structure diagram of the servo system.

Figure 2 :
Figure 2: Scheme of verifying the characteristic model.

3 ( 3 (
f1 () and f2 () e) Error between smoothed dynamic model output and characteristic model output f) Error between dynamic model output and characteristic model output