1 In this lecture we will compare two linearizing controller for a single-link robot: Linearization via Taylor Series Expansion Feedback Linearization

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. Slotine, Li, 1993.

3 Never forget THERE IS NO LINEAR SYSTEMS IN NATURE

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 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 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 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 Eigenvalues = 1,1 Origin is unstable

Let’s linearize a single-link robotic manipulator model now. Dynamic model is as follows:

By selecting the state variables as we get the state-space representation as follows:

By setting the control input signal, u, to zero, let’s find the equilibrium points From the first equation, we get Finally, from the second equation, we get

Then the equilibrium points are Let’s linearize the system around the origin

Remember that the system dynamics is Then

The Jacobian is

Since we want to linearize the system around the equilibrium point then

Then the linearized form of the system is Note that this dynamical model is a general LTI system of the form By using MATLAB Symbolic Toolbox, we find the eigenvalues of A matrix as not viscous friction ! viscous friction !

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.

But this stability result will be valid only around the small neighborhood of the linearization point, 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. % 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 x 1 = x 2 =0. If the desired trajectory for position is x d, 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 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

21 Then renders

22 Compare the performances of these two controllers via simulation.