Proceeding of the IEEE International Conference on Robotics and Biomimetics (ROBIO) Shenzhen, China, December 2013
On the Interval Type-2 Fuzzy Logic Control of Ball and Plate System Umar Farooq, Jason Gu and Jun Luo
Abstract—A simple interval type-2 fuzzy gain scheduling controller is designed for the stabilization and reference tracking of ball and plate system. The controller employs plate angles as the premise variables for gain scheduling and its stability is guaranteed through a set of linear matrix inequalities. MATLAB simulations are performed to validate the proposed controller where it is also compared with pole placement and type-1 fuzzy logic controllers. It is shown that the proposed controller has better response and disturbance rejection capability and is robust to measurement noise and errors.
I. INTRODUCTION Ball and plate system belongs to a class of under actuated systems and is widely used to study the control algorithms [1-2]. Of the four degrees of freedom, the two plate angles are actuated by the DC motors while the other degrees of freedom including position of the ball in the x and ydirection, are function of the plate angles. Also, it is an open loop unstable system as the position of the ball becomes unbounded when the plate is tilted around either of its axes. The coupling amongst the two axes is still another constraint which adds to the system complexity. Owing to have these dimensions, the platform is quite popular in the control community for the validation of control schemes. Various techniques ranging from simple PID to complex nonlinear control have been proposed for the control of ball and plate system [3-7]. However, the presence of noises and disturbances often contribute towards the performance degradation of these controllers. On the contrary, fuzzy logic can perform better than classical controllers even in the presence of unmodeled disturbances. Governed by a three stage process including fuzzification, rule base evaluation and defuzzification, fuzzy logic utilizes human knowledge about the plant to generate control input for it. The widely used fuzzy controllers are Mamdani and Takagi-Sugeno. They differ with respect to the inference mechanism and they both yield good results. However, it is often difficult to prove the stability of Mamdani fuzzy controllers which can be a concern while comparing them with classical techniques. This study therefore uses Takagi-Sugeno inference process to design controller for ball and plate system as the closed form expression of such controllers can be obtained to prove the global closed loop stability of the Umar Farooq and Jason Gu are with Department of Electrical and Computer Engineering, Dalhousie University Halifax, N. S., Canada (email:
[email protected],
[email protected]). Jun Luo is with School of Mechatronic Engineering and Automation, Shanghai University, China (e-mail:
[email protected]).
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system [8-9]. The use of fuzzy logic alone and in conjunction with other techniques to control the ball and plate system is also reported [10-13]. A hierarchical fuzzy control scheme is described in [10] for trajectory planning and tracking of ball and plate system. The control inputs for the system are generated by combining the fuzzy trajectory tracking and fuzzy supervision controllers where the fuzzy tracking controller employs four premise variables each described by two fuzzy sets to schedule amongst sixteen linear quadratic controller gains for each axis while the fuzzy supervision controller ensures that a certain distance is maintained from the side walls when the ball is moving towards goal point and it does so by describing both the difference from the side walls and corresponding correction term using five fuzzy sets. The fuzzy trajectory planning controller generates the desired trajectory points along the path from the source to destination and is optimized using genetic algorithm. A fuzzy sliding mode controller is introduced in [11] that employ an intermediate variable to decouple the eighth order ball and plate system into two sets. A SISO fuzzy logic controller based on the signed distance method to reduce the number of rules is then designed for each sub-system. Another design of control scheme for ball and plate system is presented in [12] that combine the bangbang, PD and Mamdani fuzzy logic controllers to generate the control commands for each plate axis. The fuzzy controller uses error and change in error as inputs. A switching function based on the magnitude of the tracking error selects the proper control action to be applied. Although type-1 fuzzy logic controller or simply the fuzzy logic controller shows better performance, an uncertainty exists in defining the membership functions for fuzzy sets. To overcome the associated vagueness, the membership functions are blurred to define a footprint of uncertainty (FOU). The resulting fuzzy sets are termed as type-2 fuzzy sets where the degree of belongingness of a certain variable to a type-2 fuzzy set is also fuzzy instead of a crisp value as in case of type-1 fuzzy set. The relatively less complex version of type-2 fuzzy set is an interval type-2 fuzzy set where the degree of membership is contained in a certain range corresponding to lower and upper membership functions defining the fuzzy set [14]. The applications of interval type-2 fuzzy controllers to real world problems have demonstrated their ability to better handle the uncertainties and noises as compared to type-1 counterparts [15-16]. This paper presents the design of a simple interval type-2 fuzzy logic controller for stabilization and tracking of ball and plate system. The schematic diagram of system is shown
2250
in Fig. 1. The plate angles around the axes
θ x and θ y are
selected as premise variables for scheduling amongst the controller gains that have been found using pole placement technique for different operating regions. The closed loop stability of the system is ensured using Lyapunov conditions. The proposed controller is validated through simulations in MATLAB environment and is compared with type-1 fuzzy and pole placement controllers. In the sections that follow; ball and plate model, interval type-2 fuzzy logic controller and results are presented.
θy
z
x
θx
y
II. BALL AND PLATE MODEL Ball and plate system is controlled by two DC motors mounted in an orthogonal fashion which provide the desired plate angles to position the ball at any desired point on the plate. Using the Lagrangian approach, the dynamic model of ball and plate system can be derived as [13]:
Fig.1. Schematic of Ball and Plate Model
•2 • • J b ⎞ •• ⎛ + − − m x mx my θ θ x x θ y + mg sin θ x = 0 (1) ⎜ r 2 ⎟⎠ ⎝
⎡• ⎤ ⎢ x1 ⎥ ⎢• ⎥ ⎡ x2 ⎤ ⎢ x2 ⎥ ⎢ ⎥ 2 ⎢ • ⎥ ⎢ B( x1 x4 + x4 x5 x8 − g sin x3 ) ⎥ ⎢ x3 ⎥ ⎢ ⎥ x4 ⎢• ⎥ ⎢ ⎥ • ⎢ x4 ⎥ ⎢ 0 ⎥ X = ⎢• ⎥ = ⎢ ⎥ x6 ⎢ x5 ⎥ ⎢ ⎥ ⎢ • ⎥ ⎢ B ( x x 2 + x x x − g sin x ) ⎥ 5 8 1 4 8 7 ⎢ x6 ⎥ ⎢ ⎥ x ⎢ ⎥ ⎢ 8 ⎥ ⎢• ⎥ ⎢ ⎥ 0 ⎦ ⎢ x7 ⎥ ⎣ ⎢• ⎥ ⎣ x8 ⎦
•2 • • J b ⎞ •• ⎛ + − − m y mx my θ θ y x θ y + mg sin θ y = 0 (2) ⎜ r 2 ⎟⎠ ⎝ ••
• •
••
τ x = ( J p + J b + mx 2 ) θ x + 2mx x θ x + mxy θ y •
•
(3)
• •
+ m x y θ y + mx y θ y + mgx cos θ x ••
• •
••
τ y = ( J p + J b + mx 2 ) θ y + 2my y θ y + mxy θ x •
•
(4)
• •
+ m x y θ x + mx y θ x + mgy cos θ y Where,
x is the ball position along x-axis, y is the ball
position along y-axis,
T
θ x is the plate tilt along x-axis, θ y is
the plate tilt along y-axis,
⎡1, 0, 0, 0, 0, 0, 0, 0 ⎤ Y=⎢ ⎥ X ⎣0, 0,0, 0,1, 0, 0, 0 ⎦
τ x is the torque applied to x-axis,
τ y is the torque applied to y-axis, m is the mass of the ball,
X = ( x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 )
T
J p is the inertia of the plate.
= m+
(6)
Where,
r is the radius of the ball, J b is the inertia of the ball and
Let, B
(5)
Jb , the ball and plate system can be r2
(7)
T
• • • • = ⎛⎜ x, x, θ x , θ x , y, y, θ y , θ y ⎞⎟ ⎝ ⎠
described in state space form as:
By assuming low velocity and acceleration of the plate
⎛
•
rotation ⎜ θ x
⎝
• ⎞
