Network with 100 unstable nodes
We next present the Matlab code and the computational results obtained
using lqrsp.m. In this example, we use the weighted norm to promote
sparsity. We set , and select logarithmicallyspaced
points for .
Matlab code
N = 100;
L = 10;
load data/positions
Aii = [1 1; 1 2];
Bii = [0; 1];
n = size(Bii,1);
Aij = eye(n);
B1 = kron(eye(N), Bii);
B2 = B1;
A = zeros(2*N,2*N);
dismat = zeros(N,N);
for i = 1:N
for j = i:N
if i == j
A( (i1)*n + 1 : i*n, (j1)*n + 1 : j*n ) = Aii;
else
dismat(i,j) = sqrt( norm( pos(i,:)  pos(j,:) )^2 );
dismat(j,i) = dismat(i,j);
A( (i1)*n + 1 : i*n, (j1)*n + 1 : j*n ) = Aij / exp( dismat(i,j) );
A( (j1)*n + 1 : j*n, (i1)*n + 1 : i*n ) = Aij / exp( dismat(j,i) );
end
end
end
Q = eye(2*N);
R = eye(N);
options = struct('method','wl1','gamval',logspace(2,log10(68.6649),48), ...
'rho',100,'maxiter',10,'blksize',[1 1],'reweightedIter',1);
tic
solpath = lqrsp(A,B1,B2,Q,R,options);
toc
Computational results
Sparsity vs. performance

Number of nonzero elements in the feedback gain decreases with .

In the absence of sparsity constraints, i.e., at , the optimal controller
is obtained from the positive definite solution of the algebraic Riccati equation

Performance loss:

The above results demonstrate that the optimal sparse feedback gain,
with of nonzero elements relative to the centralized feedback gain ,
introduces performance loss of compared to .
Communication graphs of distributed controllers
As increases, the communication architecture of distributed controllers becomes sparser.
Note that the communication graph does not have to be connected since the subsystems are

Communication graph of the distributed controller for


Communication graph of the distributed controller for


Communication graph of the distributed controller for

Danger of truncation
