Consider a scalar system in the form

1. In this lecture, another alternative to deal with the parameter uncertainty in the system model will be provided: Robust Control of PMDC motor ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013

2. Consider a scalar system in the form 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 ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 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?

3. 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 The idea behind the robust control is to introduce the parameter uncertainties as a disturbance to the system and to design a control law so that the system is always “robust” to the disturbances caused by uncertainties.

4. A procedural way for robust control of nonlinear systems has been presented in ECE 874. • But the procedure was for first-order systems. Today we will design a robust controller for PMDC motor and for this reason we need to slightly modify the standard formulation. • Dynamic model of the PMDC motor is as follows: ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 mechanical subsystem electrical subsystem

5. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 We can neglect this term since It has no effect at steady state. In fact, it has effect on system behavior only during transient. Inductance (L) is too small If we neglect this term, then we can write following equality from the electrical dynamics: If we substitute this current expression into the mechanical dynamics, we get

6. Now we can design a robust controller by using this dynamics. Side Note : We can absolutely use our 3 state variable (third-order) model, which is ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 to design a robust controller by using robust backstepping approach or something else. But the second-order model given above will be used as a tool to present the idea behind robust control .

7. Control objective is to drive the motor position , q, to a desired trajectory, qd. As usual, let’s define a tracking error as ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 and investigate its dynamics, Note that we could NOT substitute the system dynamics into the error dynamics, since the dynamics is second order. To do this, let’s define a new tracking error, called “filtered tracking error” as Side Note : If one takes the Laplace Transform of both sides of the equation above, one gets which implies that r is the filtered form of e, as its name suggests. Also note that if r0, then so does e.

8. Investigating the r dynamics yields ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Side note: Remember the general control design formulation for robust control (from ECE874) But in PMDC motor dynamics, the control input signal has a coefficient containing the unknown parameters. We have to get rid of this coefficient to be able to design the control input signal. To do this, we can multiply the both sides of the equation by R/KT. For this reason, we should multiply r dynamics with (JR/KT) to be able to substitute the system dynamics into the r dynamics.

9. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Side Note : It may lead to confusion to add the derivatives of the desired trajectory to the f function since, in general form, we designed the control input signal as (ECE 874) But, in this design, time derivatives of the desired trajectory have some unknown coefficients and for this reason we can not cancel them out by adding some feedforward terms to the control input signal. Then the open-loop r dynamics will be

10. Lyapunov stability analysis method can be used to design the control input signal, u. Select the Lyapunov function as ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 and take its time derivative, which is Now design the control input as where VR is the robust term to be designed. In the following, design of three different versions of this robust term is presented. Before designing it, let me write the expression for the time derivative of the Lyapunov function

11. Sliding Mode Robust Controller Design the robust term as Then the time derivative if the Lyapunov function will be ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 We can upper bound this function as

12. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Global exponential stability is achieved.

13. High Gain Robust Controller • Design the robust term as Then the time derivative if the Lyapunov function will be ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 We can upper bound this function as

14. Case I : If then ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 GES ! (same stability result with sliding mode robust controller) Case II : If then GLOBALLY UNIFORMLY ULTIMATELY BOUNDED with an adjustable upper bound,

15. High Frequency Robust Controller • Design the robust term as Then the time derivative if the Lyapunov function will be ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 We can upper bound this function as

16. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Same stability result with the Case II of robust high gain controller; Globally Uniformly Ultimately Bounded

17. Some Comments The controller is NOT model based. It just need to measurement of state variables. No need to have a linearly parameterizable system. The main problem with the controller is the chattering phenomena. Smaller values of leads lower values of the tracking error but also leads chattering at the same time. This video capture was taken during the experiment of this controller for a low value of . Chattering may damage the system to be controlled. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 As said before, there are some other robust control design techniques like robust backstepping, which may be designed for this system later. p.s. Demonstration of this controller will be at Riggs 25, on Thursday (3-5 pm.)

18. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Backstepping Adaptive Control Linearization Robust Control