Control objective is clear. Guess what the control input is?

1. This lecture presents an another interesting tool for backsteppingcontrol design: Shoot-the-Moon Game Table. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013

2. The Shoot-the-Moon game may give people the illusion of breaking the laws of physics by rolling a ball “uphill” on two rods. The secret is that the interaction between the shape of the ball and the angle between the two rods forms an extra slope, which combined with the apparent, visible slope of the rods, create a virtual slope for the ball. The ball moves down the virtual slope, which, depending on ball location and rod separation, may be in the opposite direction from the visible slope. Moreover, the virtual slope can be reshaped by manipulating the rods, which is equivalent to changing the force acting on the ball. This is used to control ball position and velocity. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013

3. At the beginning of play, the ball is placed at the lower end. Players adjust the angle between the rods at the free-moving ends in order to move the ball up the slope while avoiding prematurely dropping the ball. Performance is gauged by dropping the ball in one of a series of holes. Holes further from the start point are more difficult to reach and thus are assigned higher scores. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013

4. Control objective is clear. • Guess what the control input is? ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013

5. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 x : position on x-axis θ : angle between the rods (control input) m : mass of the ball g : acceleration of gravity R : radius of the ball r : radius of the rods Rr : effective radius,Rr=(R+r)/2 by Peng, Groff, Burg. 2012

6. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Select the state variables as Then the state space representation will be Control objective is to drive x1 to a desired trajectory, x1d, by using the control input signal, θ.

7. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Define tracking error as Investigating the dynamics of the error yields Add and subtract a virtual control input, ,to the right-hand side of the equation Define and design Then the final dynamics for e will be

8. Investigating h dynamics yields To stabilize the system, one can not directly design θ by using the equation above. For this reason, let call the function containing control input signal as ua, auxiliary control input signal ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Then the final dynamics for h will be Select the Lyapunov candidate as Time derivative of this function is Then one can design the auxiliary control input as which yields GES of EP is achieved.

9. We proved the boundedness of auxiliary control input signal, ua. But ua is not the actual control input signal. The actual control input signal is θ. Remember that We proved the boundedness of both ua and x1. Does it imply ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 is also bounded? Let me ask this question in a more general form: Consider a two-variable function a = f (x,y) Then the question is This is not true in general. It might be true or not depending on f. One can produce many contra-examples. One of this is This function can be bounded for unbounded values of y !

10. Then to be able to say is bounded, we have to examine g. Following is the variation g with respect to θ, for constant x1. It is a monotonically increased function with some lower and upper bounds. Physical nature of the system already puts some bound on θ. Then we can surely say that the proposition is true. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013

11. This completes the stability proof but we still have a very important problem ????????????? How can we calculate the value of θ for any value of ua? Remember that ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 The actual control input signal θ appears at lots of places (I could not count!) How can we extract the value of θ for any given value of ua? Any suggestions?

12. First the symbolic programming tools were used to find an explicit expression for θas a function of x1 andua. We designed uaas ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 At any time instant of the operation, uahas a numeric value. We had assigned uaas By using the Symbolic Toolbox of MATLAB, an analytic expression of θdepending on x1 and ua was searched. But there is no analytical solution !!!

13. Since θ always has a very small value (9° max.), some approximation like were used to find an analytical solution. Failed  Additional approximations Again and again failed  ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013

14. The only way to implement the controller is the use of a Lookup Table. ua values ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 x1 values Each cell of this Lookup Table gives the corresponding θ value.

15. Experimental Results by Peng, Groff, and Burg (2011). Penget al used a very similar controller for this system. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013

16. http://www.youtube.com/watch?v=xPgyNae1H6c&list=UUChmanU5n7_HRKc1wtLJrzA&index=10http://www.youtube.com/watch?v=xPgyNae1H6c&list=UUChmanU5n7_HRKc1wtLJrzA&index=10 ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013

17. Shoot-the-Moon Game Table is also a very good (and complex) example of the systems with Nonholonomic Behavior. Nonholonomic Systems present very interesting behavior does not exists in holonomic systems. Nonholonomic behavior of the Shoot-the-Moon Game Table may be the topic of another lecture of this class. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013