Disconnected plant network

The following example was originally studied in the PhD Dissertation of Fu Lin. The plant graph contains n = 50 randomly distributed nodes in a region of 10 times 10 units. Two nodes are neighbors if their Euclidean distance is not greater than units. We examine the problem of adding edges to a plant graph which is not connected and solve the sparsity-promoting optimal control problem (P) using graphsp_IP_w.m for controller graph with m = 1094 potential edges. This is done for 200 logarithmically-spaced values of $gamma in [10^{-3},2.5]$ using the path-following iterative reweighted algorithm that employs the weighted ell_1 norm as a proxy for inducing sparsity

$| w circ x |_1 ; mathrel{mathop:}= ; displaystyle{sum_{l , = , 1}^{m}} ; w_l , |x_l|.$

Following Candes, Wakin, and Boyd ’08, we set the weights to be inversely proportional to the magnitude of the solution to (SP) at the previous value of gamma ,

$w_l^+ ; = ; frac{1}{|x_l| ,+,varepsilon }$

where $varepsilon = 10^{-3}$ is introduced to ensure that the weights are well-defined when x_l = 0 . For $gamma = 10^{-3}$ , we initialize weights using the optimal centralized vector of the edge weights x_c . Topology identification is followed by the polishing step that computes the optimal vector of the controller edge weights.

The optimal centralized vector of the edge weights x_c contains both negative and positive elements.

The optimal centralized vector of the edge weights x_c .

The number of nonzero elements in the vector of the controller edge weights decreases and the closed-loop performance deteriorates as gamma increases. In particular, we show the optimal tradeoff curve between the ${cal H}_2$ performance loss (relative to the optimal centralized controller) and the sparsity of the vector .

$displaystyle{{rm Sparsity~level!:} ;; frac{{bf card} , (x)}{{bf card} , (x_c)} times 100% ;; {rm vs} ;; gamma}$

For gamma = 0 , the optimal centralized vector of edge weights x_c is populated with nonzero elements.

Increased emphasis on sparsity induces controller graphs with smaller number of nonzero elements.

$displaystyle{{rm Performance~loss!:} ;; frac{J , - , J_c}{J_c} times 100% ;; {rm vs} ;; gamma}$

Relative to the optimal centralized controller, performance of the optimal sparse controller deteriorates gracefully with increased emphasis on sparsity.

$displaystyle{{rm Performance~vs~Sparsity!:} ;; frac{J , - , J_c}{J_c} times 100% ;; {rm vs} ;; frac{{bf card} , (x)}{{bf card} , (x_c)} times 100%}$

The optimal tradeoff curve between the ${cal H}_2$ performance loss (relative to the optimal centralized controller) and the sparsity of the vector .

Topologies of the plant (blue lines) and controller graphs (red lines) for four values of gamma . As expected, larger values of gamma yield sparser controller graphs. Since the plant graph has three disconnected subgraphs, at least two edges in the controller are needed to make the closed-loop network connected.

gamma = 0.02

gamma = 0.09

gamma = 0.63

gamma = 2.5

For gamma = 2.5 , only four edges are added. Relative to the optimal centralized vector of the controller edge weights x_c , the identified sparse controller in this case uses only 0.37% of the edges, i.e.,

${bf card} , (x)/{bf card} , (x_c) ; = ; 0.37%$

and achieves a performance loss of 82.13% ,

Here, x_c is the solution to (P) with gamma = 0 and the pattern of non-zero elements of is obtained by solving (P) with gamma = 2.5 via the path-following iterative reweighted algorithm.