Modern Control System EKT 308

Modern Control SystemEKT 308 Modeling in state space Modeling Physical Systems

State Variable Model • Modern Control Theory is based on state variable. • Can handle • multiple-input multiple output • Linear, nonlinear • Time variant or invariant. State: The smallest set of variables (called state variables) such that knowledge of these variables at , together with knowledge of the input for , completely determines the behavior of the system at any time State Vector : State variables representing a system state form a vector called state vector.

State Variable Model (contd…) State-space: The n-dimensional space spanned by the n state vectors is called the state-space. Any state can be represented by a point in the state space.

State-Space Equation Variables: Input variables, output variables and state variables

State-Space Equation (contd…) Let us define

State-Space Equation (contd…)

Example

Scilab program A = [0, -2;1, -3]; disp (A); B=[2;0]; disp(B); C=[0 3]; disp(C); t = 0:0.01:10; len = length(t); //u = sin(t); //u = t; u = ones(1, len); dt = 0.01; x = [0;0]; y = zeros(1, len); for idx=1:len xdot = A*x+B*u(idx); x = x + xdot * dt; y(idx)=C * x; end figure (1); plot(t, y);

Mechanical Systems • Mechanical systems are governed by Newton's Laws of motion. • There are three basic elements that comprise a mechanical system. These are • Mass, • Damping(friction), and • Spring. • Newton's Second law: Force = mass x acceleration

Summary

Modern Control System EKT 308