LQG
(Output Feedback)
This type of synthesis implements the classical LQG (Linear Quadratic Gaussian) control on the working model in presence of process and sensor noise
x'(t) = A x(t) + B u(t) + w(t)
y(t) = C x(t) + D u(t) + v(t)
where w and v (white gaussians) are, respectively, a process noise with mean hw = 0 and power spectral density (of the covariance matrix) Cw = W, and a sensor or measurement noise with mean hv = 0 and power spectral density (of the covariance matrix) Cv = V; considering the following stochastic function
J = E{ integral( xT Q x + uT R u )dt },
where Q and R are the cost matrices for the state and input vectors, the objective of LQG is to find an optimal full-state feedback of the estimated state u(t) = -Kc xes(t) which minimizes J.
The optimal state estimate xes(t) of x(t) is produced, using the sensor measurement y(t), by a Kalman estimator with gain matrix Kf and dynamic
x'es(t) = A xes(t) + B u(t) + Kf ( y(t) - C xes(t) - D u(t) ),
and then the closed loop system becomes
in which the output feedback controller Kof(s) is composed by the observer and the optimal gain matrix Kc.
The LQG subsection is made up of the only following window:
If a matrix has size less or equal to 10x10, the relative push button allows to visualize all its coefficients, and the user is able to modify any of them; instead if a matrix can't be displayed on the window, it's possible to fix only the value T such that mat_name = T*eye(size(mat_name)). The changes introduced become effective only through the function associated with the SAVE MATRIX button that, if the matrix is correct, save it modifying the string of the relative button in [mat_name].
When all the matrices have been saved, the user can start the controller computation pressing the button COMPUTE LQG, which appears on the window, and the activation of the two buttons EVALUATION and SIMULATION notifies that the output-feedback controller Kof has been correctly computed.
Note: Q and W must be symmetric and positive semi
definite, while R and V must be symmetric and positive definite.