The dynamics of an n -link robot manipulator can be written as

1. In this lecture, a filter based observer and a controller will be designed to a single-link robot. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013

2. Last week we designed a model-based observer and controller to stabilize a single-link robot without velocity measurement. Remember that the observer dynamics was a direct copy of the unmeasurable state’s equation, which the only difference was that we used estimated value of the unmeasurable state. As an alternative approach to observer design, we will deign a filter-based observer for the same system in this lecture. The reason to seek some alternative ways to design a nonlinear observer is because sometimes it may be difficult, or almost impossible, to design a model based observer to estimate unmeasurable state. An important point is that the filter-based control does not directly aim to estimate the unmeasurable state(s). The main idea behind the filter-based control is to design a stabilizing controller despite the existence of unmeasurable state(s), by using some filtering terms. At the end of the lecture, we will see that this approach can be also used to solve the top problem in control design: Controller + State Observer + Uncertain Parameter Estimator. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013

3. An observer is designed basically to estimate an unmeasurable state. For example, an induction motor has two unmeasurable states, which are the flux components. Actually there are some flux sensors but they produce noisy and unreliable signals. For this reason, it is better to estimate flux instead of measuring it. Same case occurs in velocity measurement for an electrical motor. To get velocity feedback, a control designer can either differentiate the position signal produced by an encoder, or directly use a velocity sensor. Differentiation always leads noisy signal, and velocity sensor does not produce so sensitive and reliable signals. In this lecture we will design a filter-based observer to estimate the velocity of the single-link robot manipulator. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013

4. The dynamics of an n-link robot manipulator can be written as where ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Without loss of generality, we consider a single-link (n=1) robot manipulator for simplicity. For a single-link robotic arm, M(q) is the inertia of the arm, H is the viscous friction, and G(q) is the gravitational torque due to weight of the arm.

5. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Control objective is to drive arm position, q, a desired trajectory, qd, by using the control input signal, which is torque, τ. Then the tracking error signal can be defined as

6. Remember that our usual way was to design a controller is to define another error variable as ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 to be able to substitute the system dynamics into the error dynamics. But, note that, we will not be able to put this error variable, r, into our control input signal since it contains an unmeasurable term, which is the time derivative of the tracking error. For this reason, we have to find an alternative way to define an error signal.

7. Let’s define a new error variable as where ef is constructed through a filter formulation as ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013

8. Note that ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 (1) We already wrote the tracking error dynamics in the previous slide: (2) We will use these two final dynamics in the stability analysis.

9. Controller Formulation Investigating η dynamics yields ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 (3) To design the control input signal, τ, let’s use Lyapunovtype arguments

10. Select the Lyapunov candidate as ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Take the time derivative of it and substitute the Eqs. (1), (2), and (3) into yielded expression:

11. Design the control input signal as ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 and select where kn is a nonlinear damping coefficient. Then The underlined term is the only term does not promote stability. We have to find a way to damp this term. But, before this, let’s check if we are really able to put ef into control input signal, in other words, if it is measurable. Following side note explains this issue.

12. Side Note :Remember the filter dynamics ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Time derivative of the filter dynamics is Taking the Laplace transform of both sides yields

13. Go back to the analysis: Define ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 At this point, if we are able to define an upper bound for , then we can use the same arguments we used in the robust control design. In this way, we will be able to define an upper bound for , and, finally, we will be able to show the boundedness of with a continuously decreasing upper bound. To be able define an upper bound for , Mean Value Theorem helps us. Following side note reminds this theorem.

14. Side Note : Theorem: (Mean Value Theorem) Let be a continuous function on the closed interval [a,b], and differentiable on the open interval (a,b). Then there exist some c in (a,b) such that ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 • This theorem says that if the function we defined in the previous slide is • differentiable, and • uniformly continuous • then we will be able to define an upper bound for it since we will have a bounded f ’(c) at each point in the domain. • fd is already differentiable and uniformly continuous since it is desired one (we select it differentiable and continuous). The question is if f is differentiable and uniformly continuous. Let’s assume it is differentiable (this is an assumption, not fact). To be able to say that f is uniformly continuous, we have to say that it is Lipschitz. Since we assumed it is differentiable, let’s say it is uniformly continuous, which means it satisfies the following Lipschitz condition. is Lipschitzon Aif there exist M>0 s.t.for all

15. Then, by assuming f is differentiable, and by using Mean Value Theorem, we can define an upper bound as ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 At this point let’s define a new variable Then we can rewrite our Lyapunov function as We can also rewrite its time derivative as

16. Let Then ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 By using the lemma where a and b are positive constants and x and y are random variables, we can write a form you are very familiar with !

17. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Stability result is Semi Globally Uniformly Ultimately Bounded. The reason to say “Semi” is due to dependency of a control gain (kn1) on the initial conditions.

18. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Linearization Adaptive Control Backstepping Robust Control Observer + Controller +Estimator Observer + Controller (Nonlinear Damping )