1. Fuzzy Control - Example. Adriano Joaquim de Oliveira Cruz NCE e IM, UFRJ adriano@nce.ufrj.br ©2002

2. @2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Control Example 2 Inverted Pendulum n The classic inverted pendulum problem is an interesting case in control theory =u(t)

3. @2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Control Example 3 Mathematical Description -ml 2 d 2 /dt 2 +(mlg)sin( ) = = u(t) n m=mass of the pole, located at the tip n l = length of the pole. = deviation angle from vertical in clockwise direction. n u(t) = torque applied in the counter clockwise direction.

4. @2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Control Example 4 Mathematical Description -ml 2 d 2 /dt 2 +(mlg)sin( )= =u(t) Assuming x 1 = and x 2 =d /dt n Consider x 1 and x 2 the state variables n dx 1 /dt=x 2 n dx 2 /dt=(g/l)sin(x 1 ) - (1/ml 2 )u(t) For small angles sin( ) = dx 2 /dt=(g/l) - (1/ml 2 )u(t)

5. @2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Control Example 5 Mathematical Description dx 2 /dt=(g/l) - (1/ml 2 )u(t) Choosing l=g and m=180/( g 2 ), the linearized and discrete-time state-space equations are: x 1 (t+1) = x 1 (t)+x 2 (t) x 2 (t+1) = x 1 (t) + x 2 (t) –u(t)

6. @2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Control Example 6 Problem domain n Input variables x 1 [-2 o 2 o ] = deviation angle from vertical in the clockwise direction (degrees) x 2 [-5 dps =5 dps]= rate of the deviation angle (degrees per second dps) n Output variable –u [-24 24] = Torque applied to the pole in the counter clockwise direction

7. @2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Control Example 7 Input x 1 defined

8. @2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Control Example 8 Input x 2 defined

9. @2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Control Example 9 Output u defined

10. @2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Control Example 10 Fuzzy Rules x 1 \ x 2PZN PPBPZ ZPZN NZNNB