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We already designed a model based observer in Lecture 5 for a magnetic levitation train.

In this lecture, a model based observer and a controlle r will be designed to a single-link robot.

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We already designed a model based observer in Lecture 5 for a magnetic levitation train.

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  1. In this lecture, a model 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. We already designed a model based observer in Lecture 5 for a magnetic levitation train. 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 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 an 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

  3. Remember that we used this maglev train example as a tool to demonstrate an important property of nonlinear systems : Finite Escape Time ! Linear systems has a useful principle named Separation Principle, which means that one can design observer and controller separately for a linear system. If the observer and controller are stable individually, then the overall system is also stable. But this is not the case for nonlinear systems. In a nonlinear system, even if the observer and controller are stable separately, the output of the overall system may escape to infinity in a finite time. We demonstrated it with an example (see the notes of lecture 5). For this reason, we have to take both observer and controller dynamics into account during the stability analysis. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 We will start with a simple example, and then we will design an observer and a backstepping controller for a single-link robotic manipulator.

  4. Consider a simple two-dimensinoal system in the form of ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Control objective is to drive x1 to zero by using the control input signal, u, even though x2 is not available for measurement. Observer Design Let’s design a model based observer for unmeasurable state: where is the estimation of . Just like the parameter estimation error, we define a state estimation error to quantify the observer performance:

  5. Note that ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Observation Error Dynamics (to be used in composite stability analysis)

  6. To prove the stability of the observer, following Lyapunov function can be used. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Its time derivative is This means that the state estimation error goes to exponentially.

  7. Controller Design ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 The term will be used to damp the effect of appears in x1 dynamics. Damping coefficient

  8. Substituting the observer dynamics, , from the designed observer into h dynamics yields ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Divide both sides of the equation above by (1+d1), which is the coefficient of . Design the control input signal as

  9. Composite Stability Analysis ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013

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

  11. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Select GES !!! ASITCLSAB

  12. Now we will design an observer for a single-link robot by using the same algorithm. The unmeasurable state is velocity of the robotic arm. As said before, we could get velocity measurement by differentiating the position signal but this would lead a noisy signal. For this reason we will design an observer for velocity. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013

  13. 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.

  14. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 By selecting the state variables as we get the state-space representation as

  15. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Control objective is to drive x1 to a desired trajectory, x1d, by using the control input signal, τ, even though x2 is not available for measurement. Observer Design Let’s design a model based observer for unmeasurable state: where is the estimation of . Just like the parameter estimation error, we define a state estimation error to quantify the observer performance:

  16. Note that ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Observation Error Dynamics (to be used in composite stability analysis)

  17. To prove the stability of the observer, following Lyapunov function can be used. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Its time derivative is This means that the state estimation error goes to exponentially.

  18. Controller Design ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 The term will be used to damp the effect of appears in edynamics. Damping coefficient

  19. Substituting the observer dynamics, , from the designed observer into h dynamics yields ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Divide both sides of the equation above by (Ke+d1), which is the coefficient of . Design the control input signal as

  20. Composite Stability Analysis ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013

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

  22. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Select GES !!! ASITCLSAB

  23. 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 (Nonlinear Damping ) (?)

  24. Exercise: Can you make this controller adaptive? ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013

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