In this lecture, an adaptive controller for Tumor Growth Problem will be designed.

1. In this lecture, an adaptive controller for Tumor Growth Problem will be designed. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013

2. We have already designed some controllers for Tumor Growth Problem. Remember the system model: ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 • p(t)>0 and q(t)>0 for all time instants (Biologically realistic domain).

3. Also remember the last slide of Lecture 4: Challenge ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Note that q appears at denominator. Can you still apply the backstepping procedure to this system? If no, why? If yes, how?

4. If we had such a dynamics then backstepping design would be very standard and easy since we would write the first state equation in the form ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 and we would add and subtract a virtual control input, say qd, to the right-hand side as and finally we would define a backstepping variable before designing the virtual control input. But currently we have The idea is quite simple: Add and subtract instead of and define the backstepping variable as

5. To observe the performance of the controller to be designed, a tracking error signal, e(t), can be defined as The error system dynamics are generated by first differentiating this equation as ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Adding and subtracting 1/qd, a virtual control input signal (in other words, a desired trajectory for 1/q and hence q(t)), to the right-hand side of the equation above, and defining a new error variable z(t) as results in Now the virtual control input, 1/qd can be designed as

6. We now proceed to describe the dynamics for z(t). ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013

7. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013

8. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Design the control input as which yields

9. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 ASITCLSAB Exponential stability is achieved. Let’s modify this controller to an adaptive controller.

10. EMK Adaptive ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Exponential Stability Asymptotic Stability with

11. Now we will simulate this controller. Before the simulation, let me state an important issue which creates a basis for the next lecture. Consider a scalar system in the form ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 where x is a state variable, a and b are some constants and, finally, u is a control input to the system. One can write this system in linearly parameterized form as where and . Then one can design the control input signal and adaptation rule. Now consider the following scalar system : Can you write this system in linearly parameterized form? Can you design the adaptive control law for this system?

12. Answers to these questions are quite clear : NO ! Adaptive control is a very powerful tool but it is not a magical tool that can be applied to the all kinds of the systems. For some kind of systems just like the previous one, we should find some alternative control design approaches. Robust control provides an alternative approach for control design. There is no need for robust control to have a linearly parameterizable system. Instead, robust control design procedure handles the system by writing it in the form ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 where f (x) is a function containing state(s) and unknown parameters. The only need for robust control is to have a known bound for f (x) like Next week (actually the week after spring break) we will design and implement some robust controllers for PMDC motor.