Design Case Studies | ![]() ![]() |
MIMO LQG Design
Start with the complete two-axis state-space model Pc
derived above. The model inputs and outputs are
Pc.inputname ans = 'u-x' 'u-y' 'w-ex' 'w-ix' 'w_ey' 'w_iy' P.outputname ans = 'x-gap' 'y-gap' 'x-force' 'y-force'
As earlier, add low-pass filters in series with the 'x-gap'
and 'y-gap'
outputs to penalize only low-frequency thickness variations.
Next, design the LQ gain and state estimator as before (there are now two commands and two measurements).
k = lqry(Pdes(1:2,1:2),eye(2),1e-4*eye(2)) % LQ gain est = kalman(Pdes(3:4,:),eye(4),1e3*eye(2)) % Kalman estimator RegMIMO = lqgreg(est,k) % form MIMO LQG regulator
The resulting LQG regulator RegMIMO
has two inputs and two outputs.
Plot its singular value response (principal gains).
Next, plot the open- and closed-loop time responses to the white noise inputs (using the MIMO LQG regulator for feedback).
% Form the closed-loop model cl = feedback(Pc,RegMIMO,1:2,3:4,+1); % Simulate with lsim using same noise inputs lsim(Pc(1:2,3:6),':',cl(1:2,3:6),'-',wxy,t)
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The MIMO design is a clear improvement over the separate SISO designs for each axis. In particular, the level of /
thickness variation is now comparable to that obtained in the decoupled case. This example illustrates the benefits of direct MIMO design for multivariable systems.
![]() | Cross-Coupling Between Axes | Kalman Filtering | ![]() |