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Coupled Tank System

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A= [A 0;-C 0] B=[B 0]’ C=[C 0] D=[0][pic 17]

Fig.3 Initial Response to Pole placement- Integral


If the system is fully observable, a state estimator may be used to provide estimated states for use in feedback control. The state estimator is known as observer. The control law is:

u = -K [pic 18]

[pic 19]

Fig.4 Initial Response to Observer


Where the state of the system has been augmented by an integrator state, the compensator is given by:

= + y[pic 20][pic 21][pic 22][pic 23]

u = [pic 24]

Assuming the system is observable, the output of the estimator C can be compared to the output of the system, and any difference between them may be multiplied by a gain vector and added to the state estimator dynamics. Therefore:[pic 25]

e = Y – C [pic 26]

= C (X-)[pic 27]

Multiplying this error by a gain vector L, the desired state error correction term is formed, which can then be added to the dynamics of the estimator to form:

= A + Bu - LC(X-)[pic 28][pic 29][pic 30]

= (A-LC) + Bu + LCX[pic 31]

Since the state estimate must converge to the controlled state faster than the state itself can change, the eigenvalues of A-LC should be placed farther to the left than the eigenvalues of A-BK. A good rule of thumb is to make the estimator dynamics at least twice as fast as the controlled system dynamics.

= (A-BK-LC) + Ly[pic 32][pic 33]

u = -K[pic 34]

[pic 35]

Fig.5 Initial Response to Observer with Integral


LQR stands for Linear Quadratic Regulator.

For optimal control, a controller is sought where the controller gain K is determined by solving a linear quadratic regulator (LQR) problem. This controller seeks to provide a control effort u that minimizes a Lagrangian cost function:

J = [pic 36]

Solution for K can be given by,

K = R-1 BT P

Where P is the unique, symmetric, positive definite solution to the steady-state algebraic Riccati equation:

PA + AT P – PBR-1 BT P + Q = 0

The minimum value of the cost function is based on the initial state x0, and is given by:

Jmin = [pic 37]

The response to initial states after introducing LQR control is plotted in the below figure.

[pic 38]

Fig.5 Initial Response to LQR Controller


The response of the system to initial conditions with the introduction of LQR controller with integral action has been plotted.[pic 39]

Fig.6 Initial Response to LQR with Integral


This Project work has analysed and applied modern control design methods to coupled tank system. Full state feedback control for pole placement was applied - first without integral effort, then with an integrator added. Output compensation using state estimation techniques with observers was then discussed and simulated. LQR and LQR plus integral techniques were then used to design converter that were optimized with respect to quadratic performance indices based on state and control effort transients.


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