Disturbance Utilization Control Steve Rogers Disturbance Utilization Control (DUC) is similar to DAC except that the estimated disturbance states are all used in the control calculation. That is, DUC takes advantage of all the generated state estimates in the control calculation by utilizing techniques such as LQR, H-infinity, or pole placement. If the plant model is given as dx =( A+δA ) x + Bu+ Fw , y=Cx , the development may proceed as follows. dx
[ ][
̂x A −FH d x̂ = 0 D dt w ̂x a 0 0
][ ] [ ] {
[ ]}
F H a ̂x ̂x B ̂x w + 0 u + L y−[ C G 0 ] ̂x w 0 0 Da ̂x a x̂ a
. L is the observer gain
vector or matrix and may be solved by a Kalman filter, LQR, pole placement, or Ĥx u=−K ̂x w infinity. The minimizing control law is , which may also be solved by ̂x a
[]
LQR, pole placement, or H-infinity. Of principal interest in the DUC development is that all the disturbance states are involved in the control law. The disturbance states may be made of various combinations of disturbances. If it is desired to use DUC as an add-on to a PID controller, as shown in the Simulink diagram below, the above development may be simplified and modified slightly.
Band-Limited White Noise
Dist PID(s) PID
G LTI System
sduc duc
Scope1
A 2 state disturbance estimator may be formulated as follows.
[ ]
w=Hz=[ 1 0 ] z1 , z2
dz =Dz +σ , dt
[ ] [ ][ ] [ ]
d z1 = 0 1 z1 + σ1 dt z2 0 0 z2 σ2
.
The observer and controller may be designed using pole placement, lqr, or other statespace method. A typical matlab script is shown below. A = [0 1;0 0];B = [0 1]';C = [1 Ts];D = 0;pk = [-0.5 -0.6];po = [-5 -6]; Qo = diag(in(1:2));Qk = diag(in(3:4));Ro = in(5);Rk = in(6); K = lqr(A,B,Qo,Ro);L = lqr(A',C',Qk,Rk); sduc = ss(A-C'*L,L',[0 0],D);
The following matlab code was used to design the PID controller. A DUC add-on controller was formulated as shown above. The results are in the plots below.