Hybrid Fuzzy Control of Nonlinear Inverted Pendulum System
Abdulbasid Ismail Isa, Mukhtar Fatihu Hamza, Mustapha Muhammad
Hybrid Fuzzy Control of Nonlinear Inverted Pendulum System
Abdulbasid Ismail Isa, Mukhtar Fatihu Hamza, and Mustapha Muhammad
Department of Electrical and Electronics Engineering, Usmanu Danfodiyo, Sokoto, Nigeria
Department of Mechatronics Engineering, Bayero University, Kano, Nigeria [email protected] [email protected] [email protected],ng
Abstract —Complexity and nonlinear behaviors of inverted pendulum system make its control design a very challenging task. In this paper, a hybrid fuzzy adaptive control system using model reference approach is designed to control inverted-pendulum system. First, Lagrange model is used to develop the mathematical model of the system. Moreover, an adaptive fuzzy control system is developed to achieve position control and later simultaneous control of position and pendulum angle in the same control loop. The control algorithm is investigated to achieve control of objective of reference tracking, disturbance rejection and robustness to parameter variation. The performance of the proposed control scheme was compared with PID and LQR controllers, which numerical simulation show that the proposed control scheme provides high-performance dynamic characteristics and is robust with regard to parametric variations, disturbance and reference tracking.
Keywords—Inverted pendulum, nonlinear model, Adaptive control, Fuzzy, LQR, PID I. I NTRODUCTION
Inverted pendulum system (IPS) is used extensively for teaching and research purposes, because of its nonlinear and unstable dynamic behaviours. These properties makes its control design a very challenging and a good benchmark of testing various control methods, ranging from conventional control to intelligent based control methods. Effective control design is very essential in an inverted pendulum system, for its application in humanoid walking control robots[1][2][3][4][5]. A lot of researchers have worked in addressing control problems associated of with the system. Hassan and Ling [6] proposed a PID controller for linear model of IPS to achieve independent control of cart position and angular position, the control design gives a very good steady state response. Jing et al. research cited as [7] “Design and Simulation of Fractional Order Controller for An Inverted Pendulum System” proposed a fractional order based PID controller for position and angular controls on linear model of IPS outperforms conventional PID control in terms of less overshoot and faster convergence. Yim et al. [8] applied PID control to stabilize IPS, the stability was achieved by careful tuning of controller parameters. Andrew [9] applied state feedback controller to achieve rod angle stabilization during strategic movement of the cart. Kumar et al. [10]
Develops linear quadratic regulator control for IPS position and angular controls, in which the result showed LQR superiority as compared to state feedback and PID controllers. Sethi et al [11] compares model reference based Adaptive PID and fractional order for angular control of IPS, the result showed that fractional order based adaptive PID controller in terms of less overshoot, settling time and MSE Moreover, a lot of researches on intelligent control are done to achieve stabilisation control of an inverted pendulum system. Lukuman and Magaji [3] . .Suresh et al. [12] developed fuzzy based model reference adaptive (MRAC) controller to control angular position of linear model of IPS, performance of the proposed controller was compared with conventional MRAC controllers, the result showed that (FMRAC) outperformed the conventional controllers in terms of delay time, rise time and settling time. Saifzul et al. [13]
Propose self-erecting control on real system using T-S fuzzy with Adaptive Neuro-fuzzy inference designed based linear model using 16 fuzzy rules, the result showed the system’s stability was guaranteed and experimental results proves effectiveness of the method. Fallahi et al . [14] demonstrated the effectiveness of Adaptive base NN PID controller in pendulum angle control of IPS over conventional PID controller in terms reference tracking and disturbance rejection. Mishra et al. [15] Compares fractional order PID and full order PID controllers optimized with GA for pendulum angle position of IPS, the performance of the controllers was assessed based minimization of the objective functions, and the result showed that fractional order based PID outperformed the full order one. The controllers studied above were applied on linear model of IPS; there performance might be limited around the operation points. Manis et al. [16] presents a robust control strategy for IPS with uncertain disturbances using sliding mode control (SMC), integral based sliding mode (ISMC) controller was used to address tracking angle control of the system, the numerical simulation showed that ISMC is more robust to disturbance and more accurate as compared to conventional SMC super twisting SMC. Parsad et al. [17] applied LQR and PID to control nonlinear dynamical model of an inverted pendulum system, in which the simulation results shows that LQR has a comparative advantage over the PID based control system. Similarly, intelligent controllers were applied on nonlinear inverted pendulum system. For example, Ahmad et al. [18] Designed interval type-2 fuzzy logic controller (IT2-FLC) for nonlinear IPS and compares it with type-1 fuzzy PD controller for pendulum angle control. The simulation result showed that interval type-2 fuzzy logic controller has a good performance over wide range of uncertainties and external disturbances. The goal of this paper was to design effective control scheme that can guarantee robustness external disturbance/parameter variation and reference tracking. This paper is organised as follows: Section II presents the structure and mathematical model of Inverted pendulum system and linearization at operating condition. System’s control assessment is investigated in Section III, in order to buttress the control efficiency of the proposed scheme, linear quadratic regulator and PID control design is presented in Section IV so as to compare the performance of the prosed control method. Numerical simulation results of an inverted-pendulum system under the possible occurrence of uncertainties are provided to demonstrate the robust control performance of the proposed control system in Section V. Result of numerical simulation will be presented in section VI, while Conclusions are drawn in Section VII. II. M ATHEMATICAL
MODEL
AND
LINERASATION OF INVERTED
PENDULUM
SYSTEM The model of inverted pendulum cart was obtained using Lagrange method, which is one of the modelling methods used for dealing with complex systems.
Fig. 1.
Inverted Pendulum System[17]
The generalized coordinate of the system are 𝑥(𝑡) and 𝜃(𝑡) . Therefore, Lagrange equation of the system is by:
𝐿 = (𝑀 + 𝑚)𝑥̇ (𝑡) − 𝑚𝑙𝑥̇(𝑡)𝜃̇(𝑡) 𝑐𝑜𝑠 𝜃 (𝑡) + 𝑚𝑙 𝜃̇ (𝑡) + 𝑚𝑔𝑙 𝑐𝑜𝑠 𝜃 (𝑡) + 𝐹𝑥(𝑡) (1) Dynamics of the system can be obtained by applying: 𝑑𝑑𝑡 ( ∂𝐿∂𝑥̇(𝑡) ) − ∂𝐿∂𝑥(𝑡) = 0 𝑥(𝑡) dynamics can by written as: 𝑥̈(𝑡) = 𝐹+𝑚𝑙𝜃̈ (𝑡) 𝑐𝑜𝑠 𝜃(𝑡)−𝑚𝑙𝜃̇ (𝑡) 𝑠𝑖𝑛 𝜃(𝑡)(𝑀+𝑚) (3) Similarly, 𝜃(𝑡) dynamics can be written by applying: 𝑑𝑑𝑡 ( ∂𝐿∂𝜃̇ (𝑡) ) − ∂𝐿∂𝜃(𝑡) = 0 (4) Therefore 𝜃(𝑡) dynamics can by written as: 𝜃̈(𝑡) = −𝑚𝑔𝑙 𝑠𝑖𝑛 𝜃(𝑡)+𝑚𝑙𝑥̈(𝑡) 𝑐𝑜𝑠 𝜃(𝑡)−𝑚𝑙𝑥̇(𝑡)𝜃̇ (𝑡) 𝑠𝑖𝑛 𝜃(𝑡)𝑚𝑙 (5) Equation (3) and (5) can be simplified as: { 𝑥̈(𝑡) = (𝐹−𝑚𝑙𝜃̇ (𝑡) 𝑠𝑖𝑛 𝜃(𝑡))𝑙−𝑚𝑔𝑙 𝑠𝑖𝑛 𝜃(𝑡) 𝑐𝑜𝑠 𝜃(𝑡)−𝑚𝑙𝑥̇(𝑡)𝜃̇ (𝑡) 𝑠𝑖𝑛 𝜃(𝑡) 𝑐𝑜𝑠 𝜃(𝑡)(𝑙(𝑀+𝑚)−𝑚𝑙 𝑐𝑜𝑠 𝜃(𝑡)) 𝜃̈(𝑡) = (𝑚𝑙 𝑐𝑜𝑠 𝜃(𝑡)(𝐹−𝑚𝑙𝜃̇ (𝑡) 𝑠𝑖𝑛 𝜃(𝑡)))+(−𝑚𝑔𝑙 𝑠𝑖𝑛 𝜃(𝑡)−𝑚𝑙𝑥̇(𝑡)𝜃̇ (𝑡) 𝑠𝑖𝑛 𝜃(𝑡))(𝑀+𝑚)((𝑀+𝑚)𝑚𝑙 −(𝑚𝑙) 𝑐𝑜𝑠 𝜃(𝑡)) (6) Our major concern is to keep the pendulum in the upright position around 𝜃(𝑡) = 0 , the linearization might be considered about this upright about equilibrium point. The linear model for the system around the upright equilibrium point is derived by simply linearization of the nonlinear system given in 𝜃(𝑡) = 0 𝑠𝑖𝑛 𝜃 (𝑡) → 𝜃(𝑡) 𝑐𝑜𝑠 𝜃 (𝑡) → 1 𝜃̇ (𝑡) → 0 Therefore equation (6) becomes {𝑥̈(𝑡) =
𝐹−𝑚𝑔𝜃(𝑡)𝑀 𝜃̈(𝑡) =
𝐹−(𝑀+𝑚)𝑔𝜃(𝑡)(𝑀𝑙)
The resultant linear model can be represented as: 𝐱̇(𝑡) = 𝐀𝐱(𝑡) + 𝐁𝐮(𝑡)𝐲(𝑡) = 𝐂𝐱(𝑡) + 𝐃𝐮(𝑡)} (7) Let 𝐱(𝑡) = [𝜃(𝑡) 𝜃̇(𝑡) 𝑥(𝑡) 𝑥̇(𝑡)]
Therefore, the system’s equation can be written in a compact form as:
𝐀 = [ 0 1 0 0 −(𝑀+𝑚)𝑀𝑙 −𝑔𝑚𝑀
𝐁 = [0 ] 𝑇 𝐂 = [1 0 0 00 1 0 00 0 1 00 0 0 1]
𝐃 = [0 0 0 0] 𝑇 Table I showed the inverted pendulum system parameters.
TABLE I. S YSTEM P ARAMETERS
Parameter Value
Acceleration due to gravity, 𝑔 -2 Mass of the bob 𝑚 𝑀 𝑙 III.
SYSTEM’S
CONTROL
ASSESMENT
The system’s behaviour was asssessed based on it’s parameters shown in Table I, and it was found to be state controrablle, observable and unstable. IV.
LINEAR
QUADRATIC
REGULATOR
AND
PID
CONTROL
𝐽 = ∫ (𝐱 𝑇 (𝑡)𝐐𝐱(𝑡) + 𝐮 𝑇 (𝑡)𝐑𝐮(𝑡)) ∞ 𝑑𝑡 (8) Where 𝐐 and 𝐑 are positive semi-definite and positive definite matrices respectively. The LQR gain vector 𝐊 is given by: 𝐊 = 𝐑 −1 𝐁 𝑇 𝐏 (9) Where 𝐏 is a positive definite symmetric constant matrix obtained from the solution of matrix algebraic Riccati equation (ARE) 𝐀 𝑇 𝐏 + 𝐏𝐀 − 𝐏𝐁𝐑 −1 𝐁 𝑇 𝐏 + 𝐐 = 0 (10) The overall optimal control law is given by: 𝑢(𝑡) = −𝐊𝐗(𝑡) + 𝐍𝐫(𝑡) (11) Fig. 2 showed the SIMULINK model of LQR control applied to nonlinear inverted pendulum system. The −𝐊𝐗(𝑡) component of equation (11) is used to stabilised system. 𝐍 : Scaling matrix 𝐊 : State feedback matrix Fig. 2.
LQR controlled System
While, This controller is commonly developed using three terms namely, proportional term, differential term and integral term combined together in a linear form[19]. The PID model in time domain is given as follows: 𝑢(𝑡) = 𝐾 𝑝 𝑒(𝑡) + 𝐾 𝑖 ∫ 𝑒(𝜏) 𝑡0 𝑑𝜏 + 𝐾 𝑑 𝑑𝑒(𝑡)𝑑𝑡 (12) Where 𝑈(𝑡) and 𝑒(𝑡) are control and error signals respectively. Similarly, 𝐾 𝑝 , 𝐾 𝑖 and 𝐾 𝑑 are proportional, integral and derivative constants respectively. The proportional term of PID reduces error due to disturbance; integral term eliminates steady-state error and the derivative term dampens the dynamic response, and hence improving the system stability. Fig. 3 showed PID control of cart’s position, while Fig. 4 depicts simultaneous control of PID of cart’s position and pendulum angle of the system. V. HYBRID
ADAPTIVE
FUZZY
CONTROLLER
DESIGN This section consist madel based adaptive control design and PI-D based fuzzy controller A. Adaptive Control Design
Generally, adaptive controller is a kind of a controller that can modify its behaviour in response to changes in the dynamics of the process and disturbance.
Fig. 3.
PID control of cart’s Position
Fig. 4.
PID control of cart’s and pendulum angle Positions
This research designed model reference based adaptive control, this control strategy addresses the control problem based on specific a given reference model. Now let 𝑦(𝑡) : Plant output response 𝑦 𝑚 (𝑡) : Reference model output response for a given control action The model error 𝑒 𝑚 (𝑡) is given by: ( ) ( ) ( ) m m e t y t y t = − (13) The adjustable parameter equation based on the given cost function 𝐽 is given by: 𝑑𝜃(𝑡)𝑑𝑡 = −𝛾 ∂𝐽(𝑡)∂𝑒(𝑡) ∂𝑒 𝑚 (𝑡)∂𝜃(𝑡) = −𝛾𝑒(𝑡) ∂𝑒 𝑚 (𝑡)∂𝜃(𝑡) (14) The chosen control law is given by: 𝑢(𝑡) = 𝐾 𝑝 𝜆 (𝑡) + 𝐾 𝑖 ∫ 𝜆 (𝑡) 𝑡0 𝑑𝑡⏟ 𝐼𝑛𝑝𝑢𝑡1 + 𝐾 𝑑 𝑑𝜆 (𝑡)𝑑𝑡 ⏟ 𝑖𝑛𝑝𝑢𝑡2 (15) Where: 𝜆 (𝑡) = 𝜃 (𝑡)𝑟(𝑡) − 𝜃'(𝑡)𝑦(𝑡) 𝜆 (𝑡) = 𝜃 (𝑡)𝑟(𝑡) − 𝜃'(𝑡)𝑦(𝑡) 𝜆 (𝑡) = 𝜃 (𝑡)𝑟(𝑡) − 𝜃'(𝑡)𝑦(𝑡) In laplace form, the plant output can be written as: 𝑦(𝑠) = 𝐺(𝑠)𝑢(𝑠) (16) Therefore: 𝑦(𝑠) =
𝐺(𝑠)(𝐾 𝑝 𝜃 (𝑠)𝑟(𝑠)+ 𝐾𝑖𝑠 𝜃 (𝑠)𝑟(𝑠)+𝐾 𝑑 𝑠𝜃 (𝑠)𝑟(𝑠))(1+3𝐺(𝑠)𝜃'(𝑠)) (17) We can apply equation(13), to obtain error model as: 𝑒(𝑠)= 𝐺(𝑠) (𝐾 𝑝 𝜃 (𝑠)𝑟(𝑠) + 𝐾𝑖𝑠 𝜃 (𝑠)𝑟(𝑠) + 𝐾 𝑑 𝑠𝜃 (𝑠)𝑟(𝑠))(1 + 3𝐺(𝑠)𝜃'(𝑠)) 𝐺 𝑚 (𝑠)𝑟(𝑠) (18) Based on the above equations adjustable parameters can be written as: 𝑑𝜃 (𝑡)𝑑𝑡 ≈ −𝛾 𝑝 𝑒 𝑚 (𝑡)𝑦 𝑚𝑑𝜃 (𝑡)𝑑𝑡 ≈ −𝛾 𝐼 𝑒 𝑚 (𝑡)𝑦 𝑚𝑑𝜃 (𝑡)𝑑𝑡 ≈ −𝛾 𝐷 𝑒 𝑚 (𝑡)𝑦 𝑚𝑑𝜃'(𝑡)𝑑𝑡 ≈ −𝛾'𝑒 𝑚 (𝑡)𝑦 𝑚 𝐺 𝑚 (𝑠)} (19) B. Fuzzy Control Design
Basic fuzzy controller consists of four functional basic blocks. These blocks include fuzzification, rule base, inference mechanism and defuzzificationA fuzzification is a conversion of crisp inputs into fuzzy membership values that are used in the rule base in order to execute the related rules so that an output can be generated, while inference mechanism represents the expert’s decision making in interpreting and applying knowledge about how to control the plant. A defuzzification interface converts the conclusions of inference mechanism into the crisp control input for the process. A block diagram of fuzzy control system is shown in Figure 6. In this controller, variables are divided into input and output. This controller uses two input variables (𝑃 + 𝐼)(𝑘) and Change in error ∆𝑒(𝑘) one output variable 𝑢(𝑘) (𝑃 + 𝐼)(𝑘) and ∆𝑒(𝑘) are Negative Big NB, Negative Medium NM, Negative Small NS, Zero Z, Positive Small PS, Positive Medium PM and Positive Big PB. Fig. 5 shows membership function diagram.
Fig. 5.
Membership Function This research proposed co-opting the PID controller with intelligent adaptive based fuzzy controller so at as to improve the control efficiency and resolution. Simulink diagram of the prosed hybrid controller is shown Fig. 6.Therefore, the overall output of the proposed hybrid controller is given by:
𝑈(. ) = 𝑢(𝑘) + 𝐾 𝑝 𝑒(𝑡) + 𝐾 𝑖 ∫ 𝑒(𝜏) 𝑡0 𝑑𝜏 + 𝐾 𝑑 𝑑𝑒(𝑡)𝑑𝑡 (20) Where: 𝑢(𝑘) = ∑ 𝜇 𝑗 (𝑧 𝑗 ) 𝑧 𝑗 ∑ 𝜇 𝑗 (𝑧 𝑗 ) Fig. 6.
Simulink Block of Hybrid Fuzzy Controller VI.
RESULT
AND
DISCUSSION The proposed control method was investigated via numerical simulations in MATLAB/Simulink. Controllers were design to achieve position tracking control of the cart and simultaneous control operation (position and pendulum angle control), the rejection efficiency and robustness to parametric variations of the prosed control will be assed. However, the weighting matrices are chosen as follows:
𝐐 = [1 0 0 00 9 0 00 0 230 00 0 0 180] and
𝐑 = 1.5
The LQR gain was obtained as:
𝐊 = [2.0960 −1.2221 12.3828 12.7813]
PID controller constants are shown in Table 2 as follows:
TABLE II.
PID
CONSTANTS C ART ’ S P OSITION C ONTROL
Control variable KP Ki KD
Position 0.6 16 10 Velocity 10 8.9 0.009 TABLE III.
PID
CONSTANTS C ART ’ S P OSITION AND P ENDULUM A NGLE C ONTROL
Control variable KP Ki KD
Angle 6.9 0.009 1.4 Position 1 18 1 A. Cart Position Control
The simulation was carried out with a step input signal as the desired position of the cart. Fig. 7 showed system response of the developed position controllers at no disturbance, it can be seen that the three controllers performs satisfactorily in positioning the cart to the desired position. Table IV presents performance of the controllers, it can be seen that the proposed adaptive fuzzy controller outperforms LQR and PID controllers in terms of settling time, overshoot.
TABLE IV. C ART P OSITION C ONTROL A T NO DISTURBANCE -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 100.20.40.60.81 E D eg r ee o f m e m be r s h i p NB NM NS ZR PS PM PB
Performance Indices
Hybrid Adaptive Fuzzy LQR PID Settling Time (Seconds) 6.1772 11.1301 11.5323 Overshoot (%) 0.6216 3.2985 18.0396 Steady state error 0 0.0319 0
Similarly, the velocity response of the cart at no disturbance is shown Fig. 8, controllers efforts was in Table V presents the performance of the controllers. It can be seed that the proposed controller outperforms LQR and PID Controllers respectively, because its velocity becomes constant quickly with less overshoot. The pendulum is disturbed by an external signal simulated in a form of random noise through its control input, while simulating the model. Fig. 9 showed the simulation result, which showed the proposed controller to be more robust to disturbance than the conventional controllers.
Fig. 7.
Carts’s Position Response at no disturnace TABLE V. V ELOCITY RESPONSE OF THE CART A T NO D ISTURBANCE
Performance Indices
Hybrid Adaptive Fuzzy LQR PID Settling Time (Seconds) 8.1685 30.7334 13.1383 Overshoot (%) 0.3597 22.7591 7.4024 TABLE VI. C ART P OSITION C ONTROL D UE TO DISTURBANCE
Performance Indices
Hybrid Adaptive Fuzzy LQR PID Settling Time (Seconds) 6.1501 12.6544 11.4962 Overshoot (%) 0.7667 3.3125 18.1397 Steady state error 0 0.0342 0
The effect of the pendulum disturbance was also investigated on the velocity response of the cart. Fig. 10 showed the in fact of controllers on velocity response in Table VI, which showed that the proposed controller is more robust to disturbance than the conventional controllers. The performance of the proposed controller was also compared with that of the conventional controller due to parameteric variation at a step command by increaing the mass of the cart by 20% .Fig. 11 and shows the simulation result, which shows that the LQR controller fails to balance the stabilise the position of the while the proposed controller stablise the cart effectively. The velocity response and performance of the cart was in Fig. 12 and Table VIII respectively. It can be observed that the proposed showed superior rubusteness to paramere variation in comparison to conventional controllers. If fact LQR controlled system cannot produce enough control energy to giveout constant velocity.
Fig. 8.
Velocity Response of the Cart at no disturbance P o s i t i on ( M e t e r s ) Reference SignalLQR Controlled SystemPID Controlled SystemHybrid Adaptive Fuzzy Controlled System V e l o c i t y ( M e t e r s pe r S e c ond s ) Reference SignalLQR Controlled SystemPID Controlled SystemHybrid Adaptive Fuzzy Controlled System
Fig. 9.
Carts’s Position Response due to disturnace
Proceeding numerical showed the effectiveness of the proposed control over the conventional controllers in terms shorter settling time, less overshoot and robustness to disturbance/parameter variation. Fig. 13 and Fig. 14 showed the variation of settling time and overshoot for cart’s position control in accordance with testing conditions and control methods.
TABLE VII. V ELOCITY RESPONSE OF THE CART D UE T O D ISTURBANCE
Performance Indices
Hybrid Adaptive Fuzzy LQR PID Settling Time (Seconds) 12.3356 35.2848 12.9465 Overshoot (%) 7.8724 26.4599 9.2150
Fig. 10.
Velocity Response due to disturnace
Fig. 11.
Cart’s position Response due to parameter variation TABLE VIII. C ART P OSITION C ONTROL D UE TO P ARAMETER V ARIATION
Performance Indices
Hybrid Adaptive Fuzzy LQR PID Settling Time (Seconds) 6.1687 99.6906 11.5230 Overshoot (%) 0.6127 8.4322 18.1814 Steady state error 0 Not stable 0
Fig. 15 and Fig. 16 showed the variation of settling time and overshoot for cart’s velocity response in accordance with testing conditions and control methods. P o s i t i on ( M e t e r s ) Reference SignalLQR Controlled SystemPID Controlled SystemHybrid Adaptive Fuzzy Controlled System V e l o c i t y ( M e t e r s pe r S e c ond s ) Reference SignalLQR Controlled SystemPID Controlled SystemHybrid Adaptive Fuzzy Controlled System P o s i t i on ( M e t e r s ) Reference SignalLQR Controlled SystemPID Controlled SystemHybrid Adaptive Fuzzy Controlled System
Fig. 12.
Velocity Response due to Parameter Variation TABLE IX. V ELOCITY RESPONSE OF THE CART D UE PARAMETER V ARIATION
Performance Indices
Hybrid Adaptive Fuzzy LQR PID Settling Time (Seconds) 8.1870 99.9885 13.1070 Overshoot (%) 0.4014 39.4526 7.5498
Fig. 13.
Settling time summary of Cart’s position Control A. Simulateneus Control
This section presents numerical simulation of cart’s position and pendulum angle control in the same control loop. The developed controllers are expected in stabilise pendulum rod vertically up right at a particular reference cart position set at 0.3 meters, the simulation was carried out and system response was shown in Fig. 17 and Table IX showed the performance indices of the developed controllers.
Fig. 14.
Overshoot summary of Cart’s Position Control
Fig. 15.
Settiling time summary of Cart’s Velocity Response Fig. 16.
Overshoot summary of Cart’s Velocity Response
It can be seen hybrid adaptive controller is more efficient in tracking command reference signal, for it shorter settling time and less overshoot.
TABLE X. C ART P OSITION C ONTROL A T NO DISTURBANCE V e l o c i t y ( M e t e r s pe r S e c ond s ) Reference SignalLQR Controlled SystemPID Controlled SystemHybrid Adaptive Fuzzy Controlled System S e tt ili ng t i m e ( S e c ond s ) Summary of Settiling time for Cart's Position Control Hybrid Adaptive Fuzzy ControlLQR ControllerPID Controller O v e r s hoo t ( % ) Summary of Overshoot for Cart's Position Control Hybrid Adaptive Fuzzy ControllerLQR ControllerPID Controller S e tt ili ng t i m e ( S e c ond s ) Summary of Settiling time for Cart's Velocity Response Hybrid Adaptive Fuzzy ControllerLQR ControllerPID Controller O v e r s hoo t ( % ) Summary of Overshoot for Cart's Velocity Response Hybrid Adaptive Fuzzy ControllerLQR ControllerPID Controller
Performance Indices
Hybrid Adaptive Fuzzy LQR PID Settling Time (Seconds) 7.7567 11.5004 8.7765 Overshoot (%) 3.1305 3.2251 29.8675 Steady state error 0 0.0094 0
Fig. 17.
Carts’s Position Response at no disturnace
Pendulum angle response is shown in Fig. 18, it can be observed that hybrid adaptive fuzzy controller quickly stabilises the pendulum in the vertically upright position, followed by PID controller and later LQR. Table X summarised the performances indices of the developed controllers. Fig. 19 showed the cart’s position response under the influence of disturbance. Table XI showed the performance of controllers in the presence of disturbance.
Fig. 18.
Pendulum angle position at no disturbance TABLE XI. P ENDULUM A NGLE CONTROL OF THE CART A T NO D ISTURBANCE
Performance Indices
Hybrid Adaptive Fuzzy LQR PID Settling Time (Seconds) 5.6920 40.5025 8.4096 Overshoot (%) 2.1444 15.9199 2.1795
It wase observed that proposed hybdrid fuzzy controller is more rubust to disturbance as compared PID and LQR controllers.
Fig. 19.
Carts’s Position Response due to disturnace TABLE XII. C ART P OSITION C ONTROL D UE TO DISTURBANCE
Performance Indices
Hybrid Adaptive Fuzzy LQR PID Settling Time (Seconds) 7.7322 11.0007 8.7669 Overshoot (%) 3.1065 3.2343 29.8444 Steady state error 0 0.0097 0 P o s i t i on ( M e t e r s ) Reference SignalLQR Controlled SystemPID Controlled SystemHybrid Adaptive Fuzzy Controlled System P endu l u m A ng l e P o s i t i on ( R ad i an ) Reference SignalLQR Controlled SystemPID Controlled SystemHybrid Adaptive Controlled System P o s i t i on ( M e t e r s ) Reference SignalLQR Controlled SystemPID Controlled SystemHybrid Adaptive Fuzzy Controlled System
Fig. 20.
Pendulum angle Position Response due to disturnace TABLE XIII. P ENDULUM A NGLE CONTROL OF THE CART D UE TO D ISTURBANCE
Performance Indices
Hybrid Adaptive Fuzzy LQR PID Settling Time (Seconds) 5.6787 44.0003 8.3552 Overshoot (%) 2.1424 15.8716 2.1714
Fig. 20 dipicts pendulum angle response in the presence disturbance, which showed that hybrid adaptive fuzzy controller quickly stabilises the pendulum in the vertically upright position, followed by PID controller and later LQR. Table XII summarised the performances indices of the developed controllers. Rubustness to parameter variation was assesd by increasing pendulum length by 5% and mass of the cart by15%. Systems’s response shown in Fig. 21 presents controllers performance, which showed the rubustness the proposed control over the conventional control methods due to parametric variation as summarised Table XIII. Pendulum angle response due to parameter shown in Fig. 22, it is observed that hybrid adaptive fuzzy controller quickly stabilises the pendulum in the vertically upright position, followed by PID controller and later LQR as shown Table XIV.
Fig. 21.
Cart’s Position Response due to Parameter Variation TABLE XIV. C ART P OSITION C ONTROL D UE TO P ARAMETER V ARIATION
Performance Indices
Hybrid Adaptive Fuzzy LQR PID Settling Time (Seconds) 7.7143 18.3265 8.8201 Overshoot (%) 3.1351 3.8197 30.8157 Steady state error 0 0.0086 0 Fig. 22.
Pendulum angle Position Response due to parameter variation TABLE XV. V ELOCITY RESPONSE OF THE CART D UE TO P ARAMETER V ARIATION P endu l u m A ng l e P o s i t i on ( R ad i an ) Reference SignalLQR Controlled SystemPID Controlled SystemHybrid Adaptive Fuzzy Controlled System P o s i t i on ( M e t e r s ) Reference SignalLQR Controlled SystemPID Controlled SystemHybrid Adaptive Fuzzy Controlled System P endu l u m A ng l e P o s i t i on ( R ad i an ) Reference SignalLQR Controlled SystemPID Controlled SystemHybrid Adaptive Fuzzy Controlled System
Performance Indices
Hybrid Adaptive Fuzzy LQR PID Settling Time (Seconds) 5.8317 85.2165 8.4312 Overshoot (%) 2.0910 16.0875 2.2572
The numerical simulation showed the effectiveness of the proposed control over the conventional controllers in terms shorter settling time, less overshoot and robustness to disturbance/parameter variation. Fig. 23 and Fig. 24 showed the variation of settling time and overshoot for cart’s position control in accordance with testing conditions and control methods.
Fig. 23.
Settling time summary of Cart’s position Control
Fig. 24.
Overshoot summary for Cart’s position Control
Fig. 25.
Settling time summary of Pendulum Angle position Control
Fig. 26.
Overshoot summary of Pendulum Angle position Control
Fig. 25 and Fig. 26 showed the variation of settling time and overshoot for pendulum angle position control in accordance with testing conditions and control methods. VII.
CONCLUSION
The mathematical model of inverted pendulum system was derived and system control behaviour was also assessed. Adaptive based fuzzy controller was applied on nonlinear model of test system so at invesgitate the control efficiency of proposed method. The performance of the control scheme was invesigated through numerical simulations with MATLAB, the control system was tested at various testing conditions; the result was compared with convetional controllers natabley PID and LQR controllers.The first section presents numerical S e tt ili ng t i m e ( S e c ond s ) Summary of Settiling time Cart's Position Control Hybrid Adaptive Fuzzy ControllerLQR ControllerPID Controller O v e r s hoo t ( % ) Summary of Overshoot for Cart's Position Control Hybrid Adaptive Fuzzy ControllerLQR ControllerPID Controller S e tt ili ng t i m e ( S e c ond s ) Set of Control Tests(1:Without disturbance 2:With disturbance 3:Parameter Variation)Summary of Settiling time for Pendulum Angle Control Hybrid Adaptive Fuzzy ControllerLQR ControllerPID Controller O v e r s hoo t ( % ) Summary of Overshoot for Pendulum Angle Control Hybrid Adaptive Fuzzy ControllerLQR ControllerPID Controller international Journal Of Advances In Engineering & Technology, vol. 2, p. 594, 2012. [3] A. A. H. Chiroma, A. Khan, M. F. Hamza, A. B. Dauda, M. Nadeem, S. Asadullah , et al. , "Utilizing Modular Neural Network for Prediction of Possible Emergencies Locations within point of Interest of Hajj Pilgrimage,"
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