Fuzzy Controller Design Based on Fuzzy Lyapunov Stability

1. Fuzzy Controller Design Based on Fuzzy Lyapunov Stability Stjepan Bogdan University of Zagreb • Fuzzy Lyapunov stability • Fuzzy numbers and fuzzy arithmetic • Cascade fuzzy controller design • Experimental results • ball and beam • 2DOF airplane • Fuzzy Lyapunov stability and occupancy grid – implementation to formation control

2. Fuzzy Lyapunov stability operator can define stabilizing (allowed) and destabilizing (forbidden) actions in linguistic form QUESTION : if we replace a crisp mathematical definition of Lyapunov stability conditions with linguistic terms, can we still treat these conditions as a valid test for stability? Answer to this question was proposed by M. Margaliot and G.Langholz in “Fuzzy Lyapunov based approach to the design of fuzzy controllers” and L.A. Zadeh in “From computing with numbers to computing with words”.

3. Fuzzy Lyapunov stability 2nd order system Lyapunov function sample: dx1/dt=x2 and dx2/dt~u pos*pos + pos*u = neg => u = ?

4. Fuzzy numbers and fuzzy arithmetic • linguistic terms in a form of fuzzy numbers • fuzzy number - fuzzy set with a bounded support + convex and normal membership function μς(x): • triangular fuzzy number (L-R fuzzy number):

5. Facts against intuition in fuzzy arithmetic: Fuzzy zero ? Fuzzy numbers and fuzzy arithmetic • fuzzy arithmetic

6. Fuzzy numbers and fuzzy arithmetic Definition: greater then or equal to

7. Cascade fuzzy controller design Known facts about the system: - the range of the beam angle θ is ±π/4, - the range of the ball displacement from center of the beam is ±0.3[m] - the ball position and the beam angle are measured. Even though we assume that an exact physical law of motion is unknown, from the common experience we distinguish that the ball acceleration increases as the beam angle increases, and that angular acceleration of the beam is somehow proportional to the applied torque.

8. Cascade fuzzy controller design Task: determine fuzzy controller that stabilizes the system - consider the Lyapunov function of the following form: 4 state variables, 3 linguistic values each  81 rules Observe each of two terms separately and

9. Cascade fuzzy controller design Observe each of two terms separately and only 9+9=18 rules

10. Experimental results – ball and beam Experimental results – ball and beam

11. Experimental results – ball and beam

12. Experimental results – ball and beam

13. Experimental results – 2 DOF airplane

14. Fuzzy Lyapunov stability and occupancy grid – implementation to formation control Wifibot – Robosoft, France I2C bus Ethernet SC12 (BECK) IR sensors encoders Web cam DCS-900

15. Fuzzy Lyapunov stability and occupancy grid – implementation to formation control Visual feedback – web cam DCS-900 320:240 or 640:480 46o 75o Wide angle lens (Sony 0.6x)

16. formation definition - graph(Desai et al.) Formation requires increasing order of IDs! set of predefined rules for formation change possible collisions during formation change Fuzzy Lyapunov stability and occupancy grid – implementation to formation control markers fuzzy controllers

17. Fuzzy Lyapunov stability and occupancy grid – implementation to formation control • Occupancy grid with time windows: • each cell represents resource used by mobile agents, • formation change => path planning and execution for each mobile agent => missions (with priorities?), • one mobile agent per resource is allowed => dynamic scheduling => time windows. Wedge formation to T formation b – 32 => 55 (43,54) c – 34 => 51 (33,42) d – 51 => 33 (52,43) e – 55 => 53 (54)

18. Fuzzy Lyapunov stability and occupancy grid – implementation to formation control b – 32 => 55 (32,43,54,55) c – 34 => 51 (34,33,42,51) d – 51 => 33 (51,52,43,33) e – 55 => 53 (55,54,53) 43, 54 - shared resources

19. Fuzzy Lyapunov stability and occupancy grid – implementation to formation control