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Novel Fuzzy Preview Controller for Rotary Inverted Pendulum under Time Delays

Kavirayani Srikanth1, and Gundavarapu V Nagesh Kumar2

1Department of Electrical and Electronic Engineering, Gayatri Vidya Parishad College of Engineering (Autonomous), Visakhapatnam, India, 2Department of Electrical and Electronic Engineering, Vignan’s Institute of Information Technology, Visakhapatnam, India
Correspondence to: Kavirayani Srikanth (kavirayanisrikanth@gmail.com)
Received March 17, 2017; Revised June 16, 2017; Accepted August 2, 2017.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

A novel fuzzy preview controller based on look up tables is proposed for control design of a rotary inverted pendulum represented with integrated time delay in system matrices. The proposed control achieved control of all system states with predefined standard requirements. The advantage of preview control helps in conservation of energy as the control input acts upon after a lookup making the system robust even under the impact of custom designed system time delays that were incorporated into the system. The proposed method shows the influence of time delay can be countered effectively by integrating the delay into the system matrix and then using the novel fuzzy granular preview control.

Keywords : Time delay, Preview control, Rotary inverted pendulum, Granular computing
1. Introduction

The rotary inverted pendulum has been involved as the test bed in the control domain for many years, the reason being the mechanism of having a rotary servo controlling the pendulum over the upright equilibrium position which is making the system a complex dynamic plant which becomes an example in control education. The applications of the system have been seen in literature in mapping the logic applied to the control problem in the fields of humanoid robot walking, gesture control, segway transport, satellite launch and other potential areas in which the model can be evolved into a handy tool for various defense and military applications.

The rotary inverted pendulum has a control problem has been investigated by researchers earlier in various capacities of studies on normal controllability, observability of states to design of linear quadratic controllers. Jadlovska and Sarnovsky [1] have studies using the state dependent algebraic ricatti equation as in [1] where comparative analysis was done with other classical techniques. Srikanth and Kumar [2] have applied the condition of time delay to the state space model by taking a integrated model involving the states and time delay and have shown the stability margin using particle swarm optimization. Lhee et al. [3] have designed a fuzzy logic controller similar to an sliding mode control considering with dead zone parameters, however the time delay is not integrated into the system dynamic model directly. Birla and Swarup [4] have investigated the case of a preview based controller only for a linear inverted pendulum based on evolutionary algorithms.

This paper proposes a new method of controlling the rotary inverted pendulum with a novel fuzzy preview controller with granular computing which is efficient in terms of energy usage. The proposed model considered is generated by row and column generation for incorporating the time delay component into the system definition and then a lookup based fuzzy control is applied to the plant model which gives a better control in specifications of peak overshoots, settling time when compared to the real time performance observed over a Quanser rotary inverted pendulum controlled with pole placement with fixed poles.

2. Mathematical Model

The state model defined in Eqs. (3) and (4) is obtained from [2] using the basic dynamic equations defined in Eqs. (1) and (2).

$M0θ¨0+M1θ1 cos θ0θ¨1-M2 sin θ0 cos θ0θ˙12-M3g sin θ0=0,$$M1 cos θ0θ¨1+M4+(M4 sin2 θ0)θ¨1-M1 sin θ0θ¨02+2M2 sin θ0 cos θ0θ˙0θ˙1=τ,$

where τ in Eq. (2) refers to the control input which is applied to the shaft of the arm and Mi (i = 0, 1, 2, 3, 4) in Eqs. (1) and (2) are positive system parameters defined as

$M0=I1+l12m1,$$M1=m1l1L2,$$M2=l12m1,$$M3=l1m1,$$M4=I2+l22m2+L22m1.$

The mathematical model for the rotary inverted pendulum is taken directly as in [2] which is an integrated model of the system with time delay given by the generic form of

$X˙=AX+BU,$

where X representing 5 states for the translation and rotation of the arm and the pendulum with 4 states that represent the system model by Eqs. (8) and (9). One additional state that represents the time delay is integrated into the system. The model is reconfigured with the delay embedded in order to make the system a minimum state variable model which makes the unified representation easier and decoupling the system into various canonical forms easier. The output equation is

$Y=CX+DU.$

The fuzzy control block diagram model that is proposed is represented in

As shown in Figure 1, the important parameters that are playing a key role are the error and the error rate which represent two inputs to the fuzzy preview controller which lookups the values based on which a decision is made and the output is passed on to the controller for amplification of the signal which is then passed as input to the plant. The feedback path has a LQR controller which does the state feedback control. The fuzzy granular based preview controller has a faster rule explosion which results in efficient control.

A weighted average method is adopted to calculate the hierarchical fuzzy controller with type-1 fuzzy controller and type-2 fuzzy controller.

$u=wtype1*utype1fuzzy+wtype2*utype2fuzzy.$

As given in Eq. (10), where granulation is causing refinement in the way the control effort is used for the smoother control of the plant model. The incremental control action is not only a function of error and rate of error but also the time delay component which makes it a nonlinear controller which is more efficient than a liberalized LQR controller in terms of the control effort for reduced oscillations.

3. Results

The system matrices representing the model of the integrated rotary inverted pendulum model that were taken during the process are given as follows taken from [2] are as follows:

$A=[0010000010039.2-14.5200081.78-13.9800-4/Tuo2/Tuo000],B=[0025.5424.590].$

A is an identity matrix and B is a null matrix for the model of the Rotary pendulum taken from [2]. The system definitions were obtained by values taken from Appendix A. The system is as such obtained from the dynamic model of the rotary inverted pendulum with an integration of the time delay component added to the system as a fifth state other than the four states which define the vertical and horizontal motions of the plant. The plant model has two rotations (α, θ) which are varied in order to achieve the control. The velocity components of the same were as well considered in the system. The system model dynamics are developed from a standard Quanser product of the rotary inverted pendulum as taken in [2].

The gain values that have been calculated from the classical linear quadratic controller are given as Gain Matrix = [−101.0896175.5642 – 20.172523.46261.3030];

The initial conditions for all the cases is taken as Xo = [pi/60000];

It is assumed that the initial conditions for the pendulum assign the pendulum to a position where it is in the vicinity of the upright equilibrium position ensuring that the case studies only the stabilization about the upright equilibrium position and not the control problem of swing up from the downward equilibrium position which is another case study found in literature.

The various test cases tested as per the block diagram represented in Figure 1 is shown below where the analysis is done for Cases 1 to 12. The error values variations resulted in control action where the pendulum was stabilized about the upright equilibrium position.

The various plots that have been obtained are given below in Figures 2 and 3. In Figure 2, the control effort was successful in the stabilization of the pendulum and Figure 3 we present a counter example case where the failure of the stabilization occurs which is indicative that the failure is also possible if the constraints on the system definition limits are violated. The fuzzy lookup table surface plot is as shown below in Figure 4. Table 1 indicates the rule explosion taking the two inputs of error and the derivative of error. The two values are passed in order to have a faster rule explosion when compared with the case of taking only the rule. The result clearly shows the feasibility of solution using the lookup definitions made which act as the rule base.

It can be observed clearly the status of control for the system from swing up to upright equilibrium in Figures 5 and 6 where the switching happens between stable and unstable regions for the laboratory model of the Quanser rotary inverted pendulum which is an actual experiment conducted on the test bed.

Figure 7 indicates the performance of the RIP in real time on an actual experiment conducted and comparing this with Figure 2 we can clearly see that the peak overshoots are reduced and the settling time are improved.

4. Conclusions

The analysis on the rotary inverted pendulum has clearly indicated that the stabilization about upright equilibrium could be achieved with the preview based fuzzy controller with granular computing. This can be used for the analysis of high end dynamic systems wherein the control action required can be achieved by preview first which is an efficient method when compared to other methods. The reason a preview control is better when compared with other methods is it minimizes the computation time involved in the process as a preview would guide the system faster to convergence than a system without preview.

Acknowledgements

The authors would like to thank the management of Gayatri Vidya Parishad College of Engineering (Autonomous), and GI-TAM University for providing the necessary research facilities. We thank RGUKT, Nuzvid, INDIA for conduct of experiments on the real-time test bed of the rotary inverted pendulum.

Figures
Fig. 1.

System model with fuzzy control.

Fig. 2.

Case 1 output states.

Fig. 3.

Case 9 of output states (unstable).

Fig. 4.

Lookup table classification.

Fig. 5.

Typical real time performance of RIP with offset.

Fig. 6.

Typical stabilization failure case in real time.

Fig. 7.

Swing up and stabilization in real time [2].

TABLES

### Table 1

Test case of error variations

Case IDEEdotStandard refControl action
120.002.001Success
22.002.001Success
320.2.001Fails
42.002.001Fails
52.2.001Fails
62.021Fails
72.0021Success
8.002.0021Fails
9.002.000021Fails
10.002.00002.001Fails
112020.001Fails
12200020.001Fails
13200020001Fails

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Biographies

Kavirayani Srikanth was born in Visakhapatna, India. He obtained his Master of Science in electrical engineering from University of Missouri-Columbia, USA, specializing in intelligent systems. He is currently a part-time research scholar pursuing doctoral program in GITAM, Visakhapatnam, India. He holds the position of assistant professor in electrical and electronics engineering in Gayatri Vidya Parishad College of Engineering (Autonomous), Visakhapatnam, India. His research interests include intelligent systems, robotics, control and automation. He has to his credit over 10 journals and nearly 20 conference publications.

Gundavarapu V. NageshKumar was born in Visakhapatnam, India, in 1977. He received his B.E. degree from GITAM, Visakhapatnam, and his M.E. degree from Andhra University, Visakhapatnam. He received his doctoral degree from JNT University, Hyderabad. He is currently working as a Professor in the Department of Electrical and Electronic Engineering, Vignan’s Institute of Information Technology, Visakhapatnam. His research interests include evolutionary computation and FACTS devices.

June 2018, 18 (2)