to Parseval theorem, this norm is determined by the following way:
,
where is trace of matrix; is matrix conjugated to; is the integral of the error squared under input signal which is equal to-function.index can be calculated with the help of the controllability or observability gramians. The function gram allows to calculate Gram function for an estimation of system controllability, which is called controllability gramian. Controllability gramian is applied to research controllability properties of the system models set in state space, and also for construction of their minimal realizations. It is more convenient for calculations, than controllability matrix.gramian of this system is determined with the integral:
.
In MatLab: Gc = gram (A, B) gramian is positive defined if and only if the pair of matrices is controlled. Controllability gramian is calculated with the solution of the continuous Lyapunov equation:
.
In MatLab: Gcc = lyap (A, B * B ') .
Observability gramian of the system is determined with the integral:
.
In MatLab: Go = gram (A, C) . gramian is positive defined if and only if the pair of matrixes is observed. Observability gramian is calculated with the solution of the continuous Lyapunov equation:
.
In MatLab: Goo = lyap (A, C * C) . is that the matrix should be stable, ie for continuous models all eigenvalues ​​should have negative real parts.is illustrated in MATLAB using the following commands: gram (controllability gramian), lyap (the solution of Lyapunov equation), < i> trace (a trace of matrix).
Gc = gram (A, B);% or Gc = lyap (A, B * B ');
J = sqrt (trace (C * Gc * C '));
% or J = sqrt (trace (B '* Go * B)), where Go is observability gramian.
-norm can also be determined in MatLab as follow:
h2_n = normh2 (A, B, C, D)
-norm accepts the value which is equal to infinity in the following cases:
- the model is unstable or neutral;
- the continuous model has a nonzero matrix D .
Program code:
H2op_loop = normh2 (A0, B0, C0, D0)% task 1.10
Results of the program:
H2op_loop =
.6137
II. Synthesize the real PD-compensator ( ) which would guarantee desired phase margin at gain crossover frequency :
Theoretical information:
Synthe...
