EIG \ ASSIGN
The EIG\ASSIGN (EIGenstructure ASSIGNment) is a control method that shapes the transient response of a linear time invariant system altering directly its eigenvalues and eigenvectors through the use of feedback [5]. Without explaining carefully the theory of this kind of regulator it's important to remember that, if A is the system dynamic matrix and li \ ui (with i = 1,2,...,#states), is a generic pair eigenvalue\eigenvector so that
A ui = li ui ,
then the self-conjugate set of scalars {li} (eigenvalues) influences the decay\growth rate of the system free transient responses, while the self-conjugate set of vectors {ui} (eigenvectors) influences their shape.
Then, given a self-conjugate set of Desired eigenvalues {lj}D and a corresponding self-conjugate set of Desired eigenvectors {uj}D, the Eig\Assign technique allows to compute the feedback matrix K such that the resulting Closed Loop system has {li}CL Ê {lj}D and the corresponding {ui}CL Ê {uj}A @ {uj}D, where {uj}A represents the best possible set of achievable eigenvalues.
In order to explain the presence of the {uj}A, consider the closed-loop system with the state feedback controller K (the reasoning is the same if we use an output feedback controller)
x'(t) = ( A + BK ) x(t)
uj = inv( I lj - A )BK uj = inv( I lj - A )B hj
· uj must be in the subspace S spanned by the column of the matrix inv( I lj - A )B
·
The number of independent control
variables determines the dimension m of the subspace S in which the achievable eigenvectors must reside: m = rank(B)
then if a uj,D lies precisely in the subspace S, it'll be achieved exactly and uj,A = uj,D, otherwise uj,A will be equal to the projection of uj,D onto S.
In MIMOtool, the following three versions of the EIG\ASSIGN synthesis are available:
1) (pseudo) State Feedback
The standard full state
feedback controller Ksf is computed and, after, it is remapped in an
output feedback controller Kof as indicated in the relative section of
the Synthesis
help page.
2) Output Feedback
3) Constrained Output Feedback
In order to achieve a good trade-off between the dynamic performance and the structural complexity of the controller, this type of control law allows to impose constraints on the feedback matrix Kof so as to reflect physically desirable feedback combinations; the user can fix the constraints specifying the coefficients of a "constraint" matrix CM that reflects the structure of the controller: if CM(i,j) = 0, then Kof(i,j) = 0 and the output j will not feed back to the input i, while if CM(i,j) = NaN (default), there is no constraint on the corresponding coefficient of the controller.
After selecting the type of feedback law and the number of eigenvalues that have to be modified, the user can assign the desired pairs eigenvalue\eigenvector through the edit-fields of the following window:
for each pair, it's necessary to specify the desired eigenvalue (on the top-right) and at least one entry of the desired eigenvector (on the left); only after pressing of the SET button, the parameters are stored, and the corresponding achievable eigenvector is computed and visualized on the right of the window.
The buttons << and >> allow to change the considered eigenvalue\eigenvector pair and only when all the achievable eigenvectors have been computed, the activation of the NEXT button permits to open the subsequent window. It's always possible to modify the parameters of a pair, but the new values become real only with the pressing of the SET button, which computes and visualizes the new achievable eigenvector.
The pressing of the button COMPUTE EIG \ ASSIGN starts the controller computation, then the activation of the two buttons EVALUATION and SIMULATION notifies that the output feedback controller Kof has been correctly computed. If the achievable eigenstructure is linearly dependent (an error message appears on the window), the gain matrix Kof cannot exist and it's necessary to come back to the previous window in order to modify some parameters.
The five buttons on the left allow to visualize the corresponding matrices on the Matlab window: the first three buttons are relative to the desired and achievable eigenstructure, while the others concern the closed loop eigenstructure really obtained by the output feedback controller Kof.
Note: because
of the dimensions of the window, it's possible to carry out this kind of
control only for systems with a total number of eigenvalues less or equal to
18.