Proceedings of the 2013 International Conference on Machine Learning and Cybernetics, Tianjin, 14-17 July, 2013
ZD AND ZG CONTROLLERS FOR EXPLICIT AND IMPLICIT TRACKING OF PENDULUM WITH SINGULARITY FINALLY CONQUERED YU-NONG ZHANG, CHEN PENG, XIAO-TIAN YU, YONG-HUA YIN, YING-BIAO LING School of Information Science and Technology, Sun Yat-sen University, Guangzhou 510006, China E-MAIL:
[email protected],
[email protected],
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Abstract: Zhang dynamics (ZD) and gradient dynamics (GD) are both powerful methods. Based on a pendulum system, this paper investigates both of the explicit and implicit tracking control using the ZD method. For solving the singularity-containing implicit tracking problems, this paper overcomes the singularities by using the ZD method in combination with the GD method (i.e., the ZG method). Analyses and simulations of an explicit tracking example and two implicit tracking examples show the superiority of the ZD and ZG methods.
the desired output yd (t). As an example, the pendulum system [10] considered in this paper can be described by the equations: x˙ 1 (t) = x2 (t), g k 1 u(t), x˙ 2 (t) = − sin x1 (t) − x2 (t) + l m ml2
(1) (2)
where x1 (t) and x2 (t) are the state variables, and x1 (t) corresponds to θ (see Figure 1). The rod is assumed to be rigid and of length l with zero mass, connecting with a bob of mass m. Let g be the acceleration of gravity, θ the angle subtended by the rod and the vertical axis through the pivot point, and k Keywords: the coefficient of friction of the joint. The control variable u(t) Zhang dynamics (ZD); Gradient dynamics (GD); ZG (Zhang- corresponds to a torque applied to the pendulum. gradient) controller; Pendulum; Explicit and implicit tracking In this paper, both of the explicit and implicit trackingcontrol problems of the pendulum system (1)-(2) are formulated and investigated. On one hand, the explicit tracking control 1. Introduction and pendulum system of the pendulum system is solved by ZD. On the other hand, ZD and GD are exploited together to solve the singularityAs two powerful methods, Zhang dynamics (ZD) and gradicontaining implicit tracking-control problems of the pendulum ent dynamics (GD) have recently been studied and compared system. [1]–[4]. While the GD method is based on a scalar-valued nonnegative energy function and has been designed originally for solving constant (or to say, time-invariant) problems, the ZD method is based on an indefinite matrix/vector-valued erθ ror function (termed Zhang function or Zhangian) and has been l proposed since 2001 for the online solution of various timevarying problems. In addition, the GD method is associated with explicit dynamics, while the ZD method is generally depicted in implicit dynamics [1]–[4]. As shown later in this paper, by using the ZD method in combination with the GD method [i.e., the ZG (Zhang-gradient) method], singularities in the implicit tracking control problems can be overcome. mg Tracking control is widely encountered in engineering [5][9]. Traditionally and generally speaking, the tracking-control Figure 1. A pendulum system considered problem of a system is to design a controller in terms of the input u(t) for the system such that the actual output y(t) tracks < = 0 > 1 2 ? 3 @ ;4 5 6 /7 8 9 :.A -B + ,C *# D $ % )"E (F !& G 'K JH L M IN ~ O }P |m Q u w v{zyxtR sln ko p q rS jT U g h V ia b d ce W f_ `^]X \Y Z [
978-1-4799-0260-6/13/$31.00 © 2013 IEEE 777
Proceedings of the 2013 International Conference on Machine Learning and Cybernetics, Tianjin, 14-17 July, 2013
2. ZD controller for explicit tracking
TABLE 1. PARAMETERS SETTING OF THE PAPER Parameter m g l k γz γg (for z1g1 in Section 3)
In the explicit tracking control, for example, we have the actual output y(t) = x1 (t), which is to track a desired output (or to say, path) yd (t), e.g., yd (t) = sin t or yd (t) = sin 2t cos t. Thus, the 1st Zhangian [2]-[4] can be constructed as z1 (t) = y(t) − yd (t) = x1 (t) − yd (t) ∈ R. By applying the ZD design formula [2]-[4]
Value 1 (kg) 9.81 (N/kg) 1 (m) 0.5 (kg/s) 3.9 1000
3. ZG controller for implicit tracking
z˙1 (t) = −γz z1 (t)
with the design parameter γz > 0 ∈ R used to scale the conThe previous section shows the effectiveness of the ZD vergence rate of the ZD solution, we have method alone applied to the explicit tracking control of the pendulum system (1)-(2). Moreover, to solve the singularityx2 (t) − y˙ d (t) = −γz (x1 (t) − yd (t)) . containing implicit tracking-control problems, the ZD method can also be used in combination with the GD method, which Then, by constructing the 2nd Zhangian as produces the ZG (Zhang-gradient) method. Two implicit tracking-control examples are thus given as below to show the z2 (t) = x2 (t) − y˙ d (t) + γz (x1 (t) − yd (t)) effectiveness and superiority of the ZG method. and applying the ZD formula z˙2 (t) = −γz z2 (t), the ZD controller in the form of u(t) for the explicit tracking of the pen- Example 1 dulum system (1)-(2) with y(t) = x1 (t) can be obtained as Let us consider a simple and introductory implicit trackingu(t) = mgl sin x1 (t) + (kl2 − 2γz ml2 )x2 (t) control example, where the actual output y(t) = x1 (t)x2 (t) + + ml2 y¨d (t) + 2γz ml2 y˙ d (t) (3) x2 (t), together with the desired path yd (t) being equation (5) or (6) to be tracked. Using the ZG method in this case, a ZG 2 2 − γz ml (x1 (t) − yd (t)). controller is developed to overcome the singular point(s). By constructing the 1st Zhangian as Since controller (3) is obtained by applying the ZD method twice and with no part of the GD method, it is also termed a z1 (t) = y(t) − yd (t) = x1 (t)x2 (t) + x2 (t) − yd (t) z2g0 controller. Besides, let us define the output tracking error E(t) as and applying the ZD design formula z˙1 (t) = −γz z1 (t), a function h(t) can be defined as E(t) = |y(t) − yd (t)|. (4) h(t) = ml2 (z˙1 (t) + γz z1 (t))
For simulations, two desired paths are chosen, i.e., yd (t) = sin t
(5)
yd (t) = sin 2t cos t.
(6)
= (x1 (t) + 1)u(t) + ml2 (x22 (t) − y˙ d (t)) − (x1 (t) + 1)mgl sin x1 (t) − (x1 (t) + 1)kl2 x2 (t) + γz ml2 y(t) − yd (t) .
and
Initial states are set as x1 (0) = x2 (0) = 0, and other parame- Evidently, h(t) should theoretically be zero. Therefore, a ZD ters are set as shown in Table 1. The corresponding simulation controller of z1g0 type (or to say, a conventional controller) results are shown in Figure 2. From Figure 2(a) and (b), it would be derived from h(t) = 0, which is can be seen that, some time after the control process starts, 2 2 1 u(t) = − ml (x2 (t) − y˙ d (t)) the actual output y(t) converges to the desired path yd (t). In (x1 (t) + 1) addition, it follows from Figure 2(c) and (d) that the tracking (7) + γz ml2 (y(t) − yd (t)) errors E(t) firstly increase a little bit and then converge to 0, which substantiates the efficacy of the z2g0 controller. + mgl sin x1 (t) + kl2 x2 (t).
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Proceedings of the 2013 International Conference on Machine Learning and Cybernetics, Tianjin, 14-17 July, 2013
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It is clear that the the conventional controller (7) can not go through the singularity x1 (t) = −1. Figure 3 shows the results of the conventional controller (7) applied on the implicit tracking of pendulum system (1)-(2) with y(t) = x1 (t)x2 (t)+x2 (t), desired path (5), initial states x1 (0) = −1.5, x2 (0) = 1, and other parameters set as in Table 1. It can be seen from Figure 3 that, as state x1 approaches −1, the control variable u becomes extremely large, which leads to system crash. In contrast, by using the GD method [1]-[3] further, an energy function can be defined as (t) = h2 (t)/2. Then, for the
implicit tracking control of the pendulum system (1)-(2) where y(t) = x1 (t)x2 (t) + x2 (t), a novel ZG controller of z1g1 type in form of u(t) ˙ is designed as u(t) ˙ = −γg
∂(t) = −γg (x1 (t) + 1)h(t), ∂u
(8)
where the design parameter γg > 0 ∈ R is used to scale the convergence rate of the GD solution. Using the same initial states, parameters shown in Table 1 and u(0) = 0, the simulation results of the z1g1 controller (8) applied on the pendulum
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Figure 3. Output-tracking performance and crash of the conventional controller (7) for singularitycontaining implicit tracking of pendulum system (1)-(2) with desired path (5)
system (1)-(2) for desired paths (5) and (6) are shown in Figure the pendulum system (1)-(2) with y(t) = sin x1 (t) cos x2 (t) + 4. From Figure 4 we see that the z1g1 controller (8) and the sin x2 (t) can thus be designed as resultant closed-loop system pass the singularity of x1 = −1. u(t) ˙ = −γg s(t)h(t), (9) Note that the log scale is used in Figure 4 to show the order of magnitude of output tracking error E(t) for better clarity. which overcomes the singularities of s(t) = 0. Using the parameters shown in Table 1 and initial values Example 2 x1 (0) = 2, x2 (0) = 1 and u(0) = 0, the simulation resultAs another example of tracking a path yd (t), a different out- s of the z1g1 controller (9) applied on the implicit tracking of pendulum system (1)-(2) with y(t) = sin x1 (t) cos x2 (t) + put is set as y(t) = sin x1 (t) cos x2 (t) + sin x2 (t). Similarly, sin x2 (t) for desired path (5) are shown in Figure 5. Note that h(t) = ml2 (x2 (t) cos x1 (t) cos x2 (t) − y˙ d (t)) the log scale is used to show the order of magnitude of output 2 tracking error E(t) for better clarity. In Figure 5, to demon+ s(t)(u(t) − mgl sin x1 (t) − kl x2 ) strate the singularity-conquering property of the z1g1 controller + γz ml2 (y(t) − yd (t)), (9) more clearly, a subplot [i.e., Figure 5(d)] is additionally pwhere s(t) = cos x2 (t) − sin x1 (t) sin x2 (t). The novel ZG resented for the curve of s(t) = cos x2 (t) − sin x1 (t) sin x2 (t) controller of z1g1 type in form of u(t) ˙ for implicit tracking of which evidently passes zero. It can easily be seen that, in the
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Figure 4. Output-tracking performance of the z1g1 controller (8) for implicit tracking of pendulum system (1)-(2) with desired paths (5) (corresponding to upper subplots) and (6) (lower subplots)
first two seconds, the z1g1 controller (9) has passed through Acknowledgements the singularity of s(t) = 0 successfully. In other words, the ZG controller derived from the ZG method solves the singularityThis work is supported by the National Natural Science containing implicit tracking-control problem. Foundation of China (under grants 61075121 and 60935001), the Specialized Research Fund for the Doctoral Program of Institutions of Higher Education of China (with project number 4. Conclusions 20100171110045) and also by the National Innovation Training Program for University Students (under grant 201210558042). Based on the pendulum system (1)-(2), this paper has investigated the explicit and implicit tracking control using the ZD References and ZG methods. Specifically, we have shown the effectiveness of the ZD method by applying it to the explicit tracking [1] Y. Zhang, C. Yi, D. Guo, and J. Zheng, “Comparison on control of the pendulum system (1)-(2) at first. Then, we have Zhang neural dynamics and gradient-based neural dynaminvestigated how the singularity problem comes into being in ics for online solution of nonlinear time-varying equathe more general implicit-tracking control and how it leads to tion”, Neural Computing & Applcations, Vol. 20, pp. 1–7, a system crash. Finally, using the ZD method in combination 2011. with the GD method (producing what we call the ZG method, [2] Y. Zhang, Z. Ke, P. Xu, and C. Yi, “Time-varying square i.e., Zhang-gradient method), the singularities appearing in the roots finding via Zhang dynamics versus gradient dynamimplicit tracking control problems have been conquered. Simics and the former’s link and new explanation to Newtonulation results have further substantiated that both ZD and ZG Raphson iteration”, Information Processing Letters, Vol. controllers have good performance for the pendulum system. 110, pp. 1103–1109, 2010.
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