Mass-spring system: weighted $Huge{ell_1}$ norm

Mass-spring system with masses on a line.

The weighted ell_1 norm is given by

$g(F) , = , sum_{i, , j} W_{ij} | F_{ij} |.$

Since

${ W_{ij} , = , 1/|F_{ij}| , ~~ F_{ij} , neq , 0 } ~~ Rightarrow ~~ {rm bf{card}} , (F) , = , sum_{i, , j} W_{ij} |F_{ij}|,$

we use the re-weighted scheme of Candes et al. ’08

$W_{ij}^+ , = , frac{1}{|F_{ij}| , + , varepsilon}, ~~~ 0 , < , varepsilon , ll , 1,$

where is the optimal sparse feedback gain from the previous re-weighted iteration.

At gamma = 0 , the solution of the optimal control problem is computed from the positive definite solution of the algebraic Riccati equation. For a fixed value of gamma > 0 , we initialize the re-weighted scheme using the solution from the previous value of gamma . We update the weights until convergence or a maximum number of iterations is reached.

We next show the Matlab code and the computational results obtained using lqrsp.m. We set ${rho = 100, , varepsilon = 10^{-3} }$ and select logarithmically-spaced points for $gamma in [10^{-4}, , 10^{-1}]$ .

Matlab code

% Mass-spring system

% State-space representation of the mass-spring system with N = 50 masses
N = 50;
I = eye(N,N);
Z = zeros(N,N);
T = toeplitz([2 -1 zeros(1,N-2)]);
A = [Z I; -T Z];
B1 = [Z; I];
B2 = [Z; I];
Q = eye(2*N);
R = 10*I;

% Compute the optimal sparse feedback gains
options = struct('method','wl1','gamval',logspace(-4,-1,50),...
        'rho',100,'maxiter',100,'blksize',[1 1],'reweightedIter',5);
tic
solpath = lqrsp(A,B1,B2,Q,R,options);
toc

Computational results

Download Matlab code mass_spring_wl1.m to reproduce these figures.

Sparsity patterns

The number of nonzero elements in decreases with gamma .

The number of nonzero elements in the feedback gain as a function of gamma .
For a system with N = 50 masses, there are total of 5000 elements in .

The number of nonzero sub-diagonals in both position and velocity feedback gains F = [F_p ; ; F_v] decreases with gamma. For large values of gamma both F_p and F_v become diagonal matrices.

Sparsity pattern of the feedback gain matrix F = [F_p ; ; F_v] for $gamma = 10^{-4}$ .

Sparsity pattern of the feedback gain matrix F = [F_p ; ; F_v] for gamma = 0.0105 .

Sparsity pattern of the feedback gain matrix F = [F_p ; ; F_v] for gamma = 0.1000 .

Performance of sparse feedback gains

In the absence of sparsity constraints, i.e., at gamma = 0 , the optimal ${cal H}_2$ controller

$F (0) , := , F_c , = , R^{-1} B_2^T P$

is obtained from the positive definite solution of the algebraic Riccati equation

$A^T P ~ + ~ P , A ~ - ~ P , B_2 , R^{-1} B_2^T P ~ + ~ Q ~ = ~ 0.$

As gamma increases, the feedback gain becomes sparser and the quadratic performance deteriorates.

Sparsity level:

$displaystyle{frac{ {rm bf{card}} , (F) }{ {rm bf{card}} , (F_{c}) }} times 100 %$

Performance loss:

$displaystyle{frac{J(F) ,-, J(F_c)}{J(F_c)}} times 100 %$

The above results demonstrate that the optimal sparse feedback gain, with of nonzero elements relative to the centralized feedback gain F_c , introduces performance loss of only compared to F_c .

Sparsity vs. performance

$begin{array}{cccc} gamma & 0.014 & 0.043 & 0.10 [0.1cm] displaystyle{frac{ {rm bf{card}} , (F) }{ {rm bf{card}} , (F_{c}) }} & 9.36 % & 5.80 % & {bf 1.96} % [0.35cm] displaystyle{ frac{J(F) ,-, J(F_c)}{J(F_c)} } & 0.80 % & 2.31 % & {bf 7.80} % end{array}$

Diagonals of sparse feedback gains

The elements on the main diagonals of the optimal sparse feedback gains become larger as the number of the sub-diagonals of drops. Thus, in order to compensate for communication with smaller number of neighboring masses, each mass places more emphasis on its own position and velocity when forming control action.

Diagonal of the optimal sparse F_p for different values of gamma : $gamma = 10^{-4}~(circ)$ , 0.0281~(+) , 0.1000~(*) .
Blue circles are almost on the top of the main diagonal of the optimal centralized F_p .

Diagonal of the optimal sparse F_v for different values of gamma : $gamma = 10^{-4}~(circ)$ , 0.0281~(+) , 0.1000~(*) .
Blue circles are almost on the top of the main diagonal of the optimal centralized F_v .

Also see computational results obtained using other sparsity-promoting functions.