âNational University of Singapore. Faculty of Engineering, Department of Electrical and Computer Engineering. 10 Kent Ridge Crescent, Singapore, 119260.
A revised Terminal Sliding Mode Controller Design for Servo Implementation Khalid Abidi∗ , and Jian-Xin Xu∗ ∗
National University of Singapore Faculty of Engineering, Department of Electrical and Computer Engineering 10 Kent Ridge Crescent, Singapore, 119260
Abstract— Terminal Sliding Mode (TSM) control is known for its high gain property nearby the vicinity of the equilibrium while retaining reasonably low gain elsewhere. This is desirable in digital implementation where the limited sampling frequency may incur chattering if the controller gain is overly high. In this work we integrate a linear switching surface with a terminal switching surface. The mixed switching surface can be designed according to the precision requirement. The analysis, simulations and experimental investigation show that the mixed SMC design outperforms the linear SMC as well as the pure TSMC.
II. P ROBLEM F ORMULATION A. System Properties Consider the following continuous-time model of piezomotor driven linear stage
I. I NTRODUCTION
= x2 (t)
x˙ 2 (t)
= −
y(t)
SLIDING MODE control is a powerful technique that has been successfully used for the control of the linear and nonlinear systems. In order to design sliding mode control systems, a switching surface or a sliding mode is defined first, and a sliding mode controller is then designed to drive the system state variables to the sliding mode so that the desired convergence property can be obtained in the sliding mode, which is not affected by any modeling uncertainties and/or disturbances [1]. Reviewing the history of the development of the sliding mode control systems [2]-[4], it can be found that linear sliding mode has been widely used to describe the desired performance of closed loop systems, that is, the system state variables reach the system origin asymptotically in the linear sliding mode. Although the parameters of the linear sliding mode can be adjusted such that the convergence rate may be arbitrarily fast, the system states in the sliding mode cannot converge to zero in finite time. Recently, a new technique called terminal sliding mode control has been developed in [1] to achieve finite time convergence of the system dynamics in the terminal sliding mode. In [5]-[7], the first-order terminal sliding mode control technique is developed for the control of a simple second-order nonlinear system and an 4th-order nonlinear rigid robotic manipulator system with the result that the output tracking error can converge to zero in finite time. In this paper, a revised terminal sliding mode control law is developed. It is shown that the new method can achieve better performance than with the linear SM or pure TSM. To validate the proposed method simulation and experiments are conducted on a piezo-motor system. The paper is organized as follows. The problem formulation is presented in §2. Appropriate terminal sliding surface and SMC are designed in §3. In §4, numerical and experimental results will be presented. Conclusions are given in §5.
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x˙ 1 (t)
kf v kf 1 x2 (t) + u(t) − f (x, t) m m m
(1)
= x1 (t)
where x1 is the position, x2 is the velocity, u is the voltage input, and f (x, t) is the friction disturbance and is assumed bounded such that |f (x, t)| ≤ fmax . The constants m, kf v , and kf are the nominal mass, damping, and force constants respectively. The objective is to design a TSM such that the output, y(t), of system (1) will converge to a desired reference trajectory r(t) in finite time. III. TSM C ONTROLLER D ESIGN In this section we will discuss the TSM controller design. The controller will be designed based upon an appropriate selection of a Lyapunov function. Further, the closed-loop system will be analyzed to derive the stability conditions. This section will conclude with a discsussion on the tracking errorbound. A. Controller Design Consider the terminal sliding surface defined below, σ = c1 e + c2 e˙ + c3 ep
(2)
where e = r − y is the tracking error, σ is the sliding function, and c1 , c2 , c3 , p are design constants. Before we proceed with the controller design, we need to rewrite the system (1) in terms of the tracking error e. Consider the new state z = e, ˙ it can be shown that the system can be rewritten as
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e(t) ˙
= r˙ − x2 (t) = z(t)
z(t) ˙
=
kf v kf kf v 1 z(t) − u(t) + r¨ − r˙ + f (x, t) m m m m
(3)
Now, select the Lyapunov function V (t) = 12 σ 2 . The derivative of the Lyapunov function is given as V˙ = σ σ˙ = σ(c1 e˙ + c2 z˙ + pc3 ep−1 z).
(4)
selecting c2 = 1 and cc24 fmax in order to understand better the effects of having a linear SM incorporated with a TSM. From Fig.1 we see that increasing c1 and decreasing p lead to a smaller error bound. Note that if the linear term was absent,
To achieve V˙ = −c4 σ < 0, where c4 is a constant, we set 2
c1 e˙ + c2 z˙ + pc3 ep−1 z = −c4 σ
(5)
and substitute the expression of z˙ from (3) into (5) to obtain kf v kf kf v 1 c1 z + c2 z− u + r¨ − r˙ + f + pc3 ep−1 z (6) m m m m = −c4 σ. From (7) the control can be derived as m kf v c1 kf v c3 p−1 c4 u = r¨ − r˙ + + +p e z+ σ kf m c2 m c2 c2 1 + f. (7) kf Note that the disturbance f is unknown so the controller will be revised to the form c1 c3 m kf v kf v c4 r˙ + + p ep−1 z + σ . u= r¨ − + kf m c2 m c2 c2 (8) Ideally the term cc24 σ in (8) is actually cc24 sign(σ), however, since the controller will be implemented in a digital environment the switching term will not be used as it can cause chattering. Once the controller is designed, we need to examine the stability.
Fig. 1.
in other words c1 = 0, the error would converge to a minimum p1 bound of c14 fmax which in this example is 1 as seen from Fig.1. This shows the benefits of having a combined linear and nonlinear term in the TSM.
B. Convergence Analysis
IV. N UMERICAL AND E XPERIMENTAL I NVESTIGATION
In order to check the stability we will revisit the Lyapunov function V = 21 σ 2 . Substitution of (8) in (4) will lead to V˙ = σ(−c4 σ + c2 f ). From (9) it can be concluded that c2 |σ| ≥ fmax c4
(9)
(10)
which gives a bound on the sliding function σ and the proper selection of the constants c2 and c4 would minimize this bound. However, our main concern is the tracking error e. Consider the function Ve = 12 e2 and (2). The derivative V˙ e is given by c1 c3 1 V˙ e = ee˙ = e − e − ep + σ (11) c2 c2 c2 which provides a minimum bound on the error once the system has entered sliding mode. This bound is the solution of c2 c1 |e| + c2 |ep | = fmax (12) c4 which can be rewritten as |e| =
c2 fmax c4 (c1 + c2 |ep−1 |)
Error bound as a function of c1 and p
(13)
if the exponent is set between 0 < p < 1 then it can be be possible to plot the relationship between e, c1 and p while
Conisder the system with the following nominal parameters: m = 1kg, kf v = 144N and kf = 6N/V olt. This simple linear model does not contain any nonlinear and uncertain effects such as the frictional force in the mechanical part, highorder electrical dynamics of the driver, loading condition, etc., which are hard to model in practice. In general, producing a high precision model will require more efforts than performing a control task with the same level of precision. A. Numerical Investigation The system is simulated at a sampling time of T = 1ms and the disturbance force f acting on the system is modeled simply as 10 if x2 < 0 0 if x2 = 0 f= (14) −10 if x2 > 0. For the new TSM the control parameters are selected as c1 = 50, c2 = 1, c3 = 10, and c4 = 1000 while the exponent p = 21 31 . These values are found after a few trials to get the most optimum performance. The new TSM is compared against an SM controller where c3 = 0 and a TSM controller where c1 = 0. In Fig.2 we see the desired trajectory and the tracking performance of all three controllers. A better idea about the tracking can be found by looking at Fig.3 where the tracking error of all three controllers is plotted. We see that the new
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TSM outperforms the other controllers. From Fig.4 we see that the control signal is almost identical for all three cases.
Fig. 4. Control input comparison of the new TSM, SM and TSM controllers’ performance
Fig. 2. Position trajectory and comparison of the new TSM, SM and TSM controllers’ performance
Fig. 5.
System Block Diagram
Fig. 3. Tracking error comparison of the new TSM, SM and TSM controllers’ performance
B. Experimental Investigation Using the experimental system with the block diagram Fig.5 that was modelled in the previous section, the controller is implemented. For the new TSM the control parameters are initially selected as in the previous section and are retuned as it is expected not to exactly match the experimental case. The tuned parameters are c1 = 70, c2 = 3, c3 = 25, and c4 = 1200 21 . The new TSM is compared against while the exponent p = 31 an SM controller where c3 = 0 and a TSM controller where c1 = 0. In Fig.6 we see the desired trajectory and the tracking performance of all three controllers. Similar to the simulation in Fig.7 we can see that the new TSM outperforms the other controllers, however, the difference between the TSM and the new TSM is around 7 microns with the new TSM going as low as 17 microns for the given 40mm displacement. From Fig.8 we see that the control signal for all three cases.
Fig. 6. Position trajectory and comparison of the new TSM, SM and TSM controllers’ performance
V. CONCLUSION This work presents a revised TSM controller based on a linear SM combined with a TSM. Theoretical investigation shows that the revised controller can provide better performance than with either the pure SM or the pure TSM. Numerical and Experimental comparison with the SM and TSM controllers
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Fig. 7. Tracking error comparison of the new TSM, SM and TSM controllers’ performance
Fig. 8. Control input comparison of the new TSM, SM and TSM controllers’ performance
prove the effectiveness of the proposed method. R EFERENCES [1] Z. Man, and X. H. Yu, ”Terminal sliding mode control of mimo linear systems,” IEEE Transaction on Circuits and Systems I: Fundamental Theory and Applications, vol. 41, pp. 1065-1070, 1997. [2] R. A. Decarlo, S. H. Zak, and G. P. Matthews, ”Variable structure control of nonlinear multivariable systems: A tutorial,” Proc. IEEE, vol. 76, pp. 212-231, 1988. [3] V. Utkin, ”Sliding mode control in discrete-time and difference systems. In A.S.I. Zinober (Eds.)”, Variable Structure and Lyapunov Control, pp.87-107, 1994. [4] K. Abidi, and X.J. Xu, ”On the discrete-time integral sliding mode control,” IEEE Transactions on Automatic Control, vol. 52, pp. 709715, 2007. [5] G. G. Morgan, and U. Ozguner, ”A decentralised variable structure control algorithm for robotic manipulators,” IEEE Journal of Robotics and Automation, vol. 1, pp. 57-65, 1985. [6] C. M. Dorling, and A. S. I. Zinober, ”Two approaches to hyperplane design in multivariable variable structure control systems,” International Journal of Control, vol. 44, pp. 65-82, 1986. [7] H. Khurana, S. I. Ahson, and S. S. Lamba, ”On the stabilization of largescale control systems using variable structure system theory,” IEEE Transactions on Automatic Control, vol. 31, pp. 176-178, 1986.
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