Schematic of the implementation of the full-state feedback control in the full linear (top) and full non linear (bottom) simulations. The entire state is first projected onto the unstable eigenvectors and the stable subspace of the balanced modes in order to compute the reduced-order state xr . The state is then multiplied by the gain K, computed based on the reduced-order model using LQR to obtain the control input .
Schematic of the implementation of the observer-based feedback control in the linear (top) and nonlinear (bottom) simulations. The control input u and the sensor measurements y are used as inputs to the observer, which reconstruct the reduced-order state . This state is then multiplied by the gain Kr to obtain the control input u. Both the controller and the observer gain Kr and L are computed based on the reduced-order model.
Actuator localized at the middle of the square plate and modeled as a distribution of the external potential that is added to the system. It is determined by the function , where and is a given parameter.
(Left) The ion vorticity ζ and (right) density fluctuations n of the B-matrix defined in Eq. (14) . These two quantities are going to be the initial conditions of the nonlinear, full linear, and reduced model of the MHW equations.
Representation of the two unstable eigenvectors of the linearized equations. The left part represents its real part, the right its imaginary part.
Ion vorticity and density fluctuation (in color) of the full non linear MHW equations at three successive times with the B-matrix as the initial condition.
The output correspond to the density fluctuation that occurs in the center of the square geometry with no control applied on the system.
Error for balanced truncation ( ), balanced POD ( ), POD ( ), and upper and lower bound for the model reduction scheme.
Full Linear model with 2 eigenvalues in the RHP.
Full Linear model with 4 eigenvalues in the RHP.
Full Linear model with 4 eigenvalues in the RHP: phase space plot.
Full Linear model with 4 eigenvalues in the RHP: inside basin of attraction case.
Output feedback: 4 RHP poles/Full density sensed.
Output feedback: 4 RHP poles/4 density points sensed only.
Nyquist diagram of the loop gain of the input sensitivity function for the unstable case with two right half plane eigenvalues.
Nyquist diagram of the loop gain of the input sensitivity function for the unstable case with four right half plane eigenvalues.
Summary of the 3 systems that will be reduced then stabilized with only one actuator: for fixed α and μ, only κ is varied and obtain 3 different cases with 2, 4, or 8 right half plane (unstable) eigenvalues.
Summary of the 3 new reduced systems. r is the dimension of the stable reduced subsystem.
GM and PM deduced from the loop gain of the sensitivity function.
Article metrics loading...
Full text loading...