Min-Max Design of FIR Digital Filters by Semidefinite Programming - Applications of Digital Signal Processing - page 205

% H-infinity optimal (delayed) inverse FIR filter Q(z) which minimizes
% J(Q) = ||(QP-z^(-n))W||,
% the maximum frequency gain of (QP-z^(-n))W.
% This design uses SDP via the KYP lemma.
%
% Inputs:
% P: Target stable linear system in SS object

% W: Weighting stable linear system in SS object

% N: Order of the FIR filter to be designed

% n: Delay (this can be omitted; default value=0);

%
% Outputs:
% q: The optimal FIR filter coefficients

% gmin: The optimal value

%

if nargin==3
n=0

end

%% Initialization
z = tf('z');
T1 = -z^(-n) * W;
T2=P * W;
[A1,B1,C1,D1]=ssdata(T1);
[A2,B2,C2,D2]=ssdata(T2);
n1 = size(A1,1);
n2 = size(A2,1);

%% FIR filter to be designed
Aq = circshift(eye(N),-1);
Aq(N,1) = 0;
Bq = [zeros(N-1,1);1];

%% Semidefinite Programming
A = [A1, zeros(n1,n2), zeros(n1,N);
zeros(n2,n1), A2, zeros(n2,N);
zeros(N,n1),Bq * C2, Aq];
B = [B1;B2;Bq * D2];

NN = size(A,1);

X = sdpvar(NN,NN,'symmetric');
alpha_N1 = sdpvar(1,N);
alpha_0 = sdpvar(1,1);
gamma = sdpvar(1,1);