Mass-spring system

mass-spring pic 

Mass-spring system with N masses on a line.

For the system with N masses on a line, the state vector is determined by x , = , left[ ; p^T ~, v^T ; right]^T, where p and v denote the vectors of positions and velocities of all masses. Setting all masses and spring constants to unity and partitioning matrices in the state-space representation conformably with the partition of x yields

      A      ; = ;      left[      begin{array}{cc}      O & I      T & O       end{array}      right],     ~~     B_1     ; = ;     B_2     ; = ;     left[     begin{array}{cc}      O       I      end{array}      right],      ~~      Q ; = ;       left[      begin{array}{cc}      I & O      O & I       end{array}      right],      ~~      R ; = ; 10I.

where I and O are N times N identity and zero matrices, and T is a tridiagonal symmetric Toeplitz matrix with -2 on the main diagonal and 1 on the first upper and lower sub-diagonals, e.g.,

     T      ; sim ;     left[     begin{array}{rrrr}     -2 & 1 & 0 & 0      1 &  -2 & 1 & 0      0 & 1 & -2 & 1      0 & 0  & 1 & -2     end{array}     right].

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.
optimal position gain 

The optimal centralized position feedback gain matrix F_p in the mass-spring system with N = 50 masses. Both F_p and F_v in F_c = [F_p ~, F_v] have almost constant diagonals (modulo edges) and exponential off-diagonal decay.

position gains for middle mass 

Optimal centralized position gains for the middle mass n = 25.
Note exponential decay with the spatial distance.

Localization can be enforced by truncating the optimal centralized controller. This, however, may introduce performance degradation and even instability of the closed-loop system. (Additional information is provided in the Network with 100 unstable nodes example.) Instead of using simple truncation, the alternating direction method of multipliers enforces sparsity in the G-minimization step and improves the quadratic performance in the F-minimization step. This mechanism of alternating between promoting sparsity and optimizing the quadratic performance plays an important role in striking a balance between these two competing objectives.

truncation of position gains 

An example of truncation: all but five largest elements of the optimal centralized position gain are set to zero. The resulting controller will only use the position measurements of the mass on which it acts and of the two neighboring masses to the left and to the right.

For illustration, we use cardinality function, weighted ell_1 norm, and sum-of-logs as the sparsity-promoting penalty functions. For all three cases, the optimal position and velocity feedback gains become diagonal matrices for large values of gamma. It is noteworthy that the optimal sparse feedback gain, with only 2% of nonzero elements relative to the optimal centralized controller F_c, introduces performance loss of only 8% (compared to F_c).

The following links contain Matlab codes and computational results for each case.