Continuous Control (MIMO) - 10 Flashcards
What is the concept of the desired region of pole placement?
In theory, one can place the poles
anywhere. In practice, one has to
consider
actuator limitations,
stability margin,
minimum damping. This creates a trapezoid shape limited by to vertical lines and two diagonals.
What do the matrices Q and R in LQR do?
The matrices Q and R make it possible to weight the importance of
performance (deviations from the steady state) versus actuator effort.
What is the optimal state feedback in LQR?
K = R^-1B^TP_+
and P_+ follows the Riccati equation
Why should we use observers?
Assuming that the state can be measured, the LQR provides an
optimal performance with respect to a quadratic cost function.
One cannot measure all state variables of most systems or avoids it to
save costs for sensors. In those cases, when the system is observable, we build an observer that reconstructs the
state; this makes it possible to use LQRs. Observers can also be used to detect faults/defects in systems.
What is the observer that minimizes quadratic cost function for the expectation of the
estimation error with gaussian noises ?
Kalman filter tells us that it is a Luenberger observer with gain L = P_+C^TV^-1 and P_+ satisfies the Riccati equation
How does the state matrix look like for the combination of state feedback and observer design? How does the seperation principle enter here?
A = [(A-BK) & BK \ 0 & (A -LC)]
B=[B \ 0]
State is x and x-^x=~x
And know the sepration principle says that the eigenvalues of a controller consisting of state feedback and a Luenberger observer are the union of the eigenvalues of the controlled
system without observer (Ac ) and the ones of the observer (Ao). so det(sI-Aco)=det(sI-Ac)det(sI-Ao).
The eigenvalues of Ao should be left of the ones of Ac in
the complex plane to make observer dynamics faster, which helps in achieving better state estimation performance
