Modern Control System EKT 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

Laplace Transform’s table for common functions Function, Laplace Transform Unit Impulse, 1 Unit step, Unit ramp, Exponential, Sine, Cosain, Damped sine, Damped cosain, Damped ramp,