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1. In this lecture we will compare two linearizing controller for a single-link robot: Linearization via Taylor Series Expansion Feedback Linearization ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013
2. Linear control theory has been predominantly concerned with Linear Time Invariant (LTI) systems of the form with x being a vector of states and A being the system matrix. LTI systems have quaite simple properties such as A linear system has a unique equilibrium point if A is nonsingular; The equilibrium point is stable if all eigenvalues of A have negative real parts, regardless of initial conditions; In the presence of an external input u(t), i.e., with the system has a number of interesting properties. For example a sinusoidal input leads to a sinusoidal output of the same frequency. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Slotine, Li, 1993.
3. Never forget THERE IS NO LINEAR SYSTEMS IN NATURE ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013
4. One of the characteristic properties of nonlinear systems is “Multiple Equilibrium Points” Nonlinear systems frequently have more than one equilibrium point (an equilibrium point is a point where the system can stay forever without moving). This can be seen by the following simple examples. Consider the first order linear system ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Solution of this differential equation is Following figure shows the time variation of this solution for various initial conditions. The system clearly has a unique equilibrium point at x=0. Slotine, Li, 1993.
5. Now consider the following nonlinear systems: Solution of this differential equation is ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Following figure shows the time variation of this solution. The system has two equilibrium points, x=0 and x=1, and its qualitative behavior strongly depends on its initial condition.
6. 1. Linearization of Nonlinear Systems via Taylor Series Expansion General form of an n-dimensional nonlinear system is ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 and of an n-dimensional linear time-invariant system is The linearized form of a nonlinear system can be found as
7. Example: Linearize System ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Eigenvalues = 1,1 Origin is unstable
8. Let’s linearize a single-link robotic manipulator model now. Dynamic model is as follows: ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013
9. By selecting the state variables as ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 we get the state-space representation as follows:
10. By setting the control input signal, u, to zero, let’s find the equilibrium points ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 From the first equation, we get Finally, from the second equation, we get
11. Then the equilibrium points are ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Let’s linearize the system around the origin
12. Remember that the system dynamics is ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Then
13. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 The Jacobian is
14. Since we want to linearize the system around the equilibrium point ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 then
15. Then the linearized form of the system is ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Note that this dynamical model is a general LTI system of the form not viscous friction ! By using MATLAB Symbolic Toolbox, we find the eigenvalues of A matrix as viscous friction !
16. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 Let’s design a simple linear state feedback controller in the form of so that we get In this way, by properly selecting the entries of K vector, we will be able to locate the eigenvalues of newly-created system matrix, (A-BK), to the left-half plane to get stability.
17. But this stability result will be valid only around the small neighborhood of the linearization point, ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 and we will not have a global stability result.
18. There are many algorithms in the literature proposed to find the entries of K vector. The most conventional algorithm can be implemented in MATLAB as follows . ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 % Desired closed-loop poles p1 = -1; p2 = -2; % Entries of K K = place(A,B,[p1 p2]); The control input signal u=-Kx drives the trajectories to the equilibrium point x1=x2=0. If the desired trajectory for position is xd, then the control law is modified as
19. 2. Linearization of Nonlinear Systems via Feedback Linearization Feedback linearization is a nonlinear control method and the control input signal to be designed will contain a nonlinear term. Again consider the general form of a nonlinear system ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013 If we can rewrite the system in the simplest form, i.e., the form that we will not need a coordinate transformation to transform the system into a linear form as then renders the linear time-invariant and controllable system if (A,B) controllable and Let’s see if we can write single link robot dynamics in this form.
20. 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 Then renders
22. Compare the performances of these two controllers via simulation. ECE 893 Industrial Applications of Nonlinear Control Dr. UgurHasirciClemson University, Electrical and Computer Engineering Department Spring 2013