INDIAN INSTITUTE OF TECHNOLOGY ROORKEE Design Of Multiloop P and PI controllers based on quadratic optimal approach

2 Project Division LQR Study of LQR and its benefits over pole placement Design of of LQR controller On inverted pendulum which is Single input multi output fourth order system Design of LQR for multiloop P and PI controller using LQR Implementation on real life problem Future Scope

3 LQR v/s Pole placement LQR(Linear Quadratic Regulator) Balances between the acceptable response and the amount of control energy required. Q and R determine the relative importance of the error and the expenditure of this energy. A* P + PA - PBR-1 B* P + Q = 0 Riccati Equation. Pole Placement U=KX : control signal determined by instantaneous state. Necessary condition: Completely State controllable system Objective is to have zero input response and zero output response Also called Regulator (constant input response)

4 Inverted Pendulum 4th order system. objective is to stabilize the pendulum in upright direction. LQR controller is implemented for stabilization. PD controller is used for “swing up”. Controller switched at upright position of pendulum. Euler-lagrangian energy equation are used.

5 Inverted Pendulum

6 1) PD Controller Aims to swing up controller from rest(alpha=180 degree) Kp: proportional to cart position error. Kv: proportional to cart velocity. Assumed desired performance as : damping ratio=0.59 and damping velocity =26 rad/s.

7 Inverted Pendulum 2) LQR Controller Minimize cost function. Q and R matrix are selected by hit and trial method. Determine matrix K such that U=-KX. Activate when angle of pendulum becomes zero.

8 Final results from the taken example Position of cart Velocity of cart

9 Final results from the taken example Angle (alpha) Angular velocity

10 Multiloop P Controller Using LQR Used iterative approach to obtain optimal value of Q and R. For MIMO system Necessary condition : system should be full state controllable. Kp should not have complex eigenvalue. Hamiltonian matrix does not have eigenvalues on imaginary axis.

11 Thank you