H2 controller synthesis using closed-loop parameterizations

Problem description

The plant dynamics are:

                       x = Ax_t + B(u_t + v_t)
                       y = Cx_t + w_t

The orginal problem is as follows

                  min_{K}    ||Q^{1/2}Y||^2 + ||R^{1/2}U||^2
                                  + ||Q^{1/2}W||^2 + ||R^{1/2}Z||^2
              subject to     K internally stabilizes G
                             K \in S

where Y, U denote the closed-loop transfer matrices from w to y and u, and W, Z, denote the closed-loop transfer matrices from v to y and u, S is a binary matrix that encodes the controller structure.

We solve the problem using closed-loop parameterizations, one of them is as follows

          min_{Y,U,W,Z}  ||Q^{1/2}Y||^2 + ||R^{1/2}U||^2
                            + ||Q^{1/2}W||^2 + ||R^{1/2}Z||^2
           subjec to      [I -G][Y W]
                                [U Z]  = [I 0]   (1)
                             [Y W][-G] = [I]
                             [U Z][I]    [0]     (2)
                         Y,U,W,Z \in FIR(N)      (3)
                          Y \in R,  U \in T      (4)

where (1)-(3) encode the internal stability constraint, and (4) encodes the sparsity constraint S using the notion of Sparsity invariance.

Rely on YALMIP to reformulate the above problem into an SDP, then call Mosek/SeDuMi to get a soluton

Syntax

   >>  [K,H2,info] = clph2(A,B,C,Q,R,userOpts)

1. Input variables
   • (A,B,C): system dynamics in discrete time
   • Q: performance weights on output y
   • R: performance weights on input u
2. userOpts is a structure and contains the following options
   • N: Order of FIR approximation (default:8)
   • solver: sedumi, sdpt3, csdp or mosek (default)
   • spa: Distributed control Yes/No (default: 0)
   • S: Sparsity pattern for the controller (default: [])

Related paper

@misc{zheng2019systemlevel,
    title={System-level, Input-output and New Parameterizations of Stabilizing Controllers, and Their Numerical Computation},
    author={Yang Zheng and Luca Furieri and Maryam Kamgarpour and Na Li},
    year={2019},
    eprint={1909.12346},
    archivePrefix={arXiv},
    primaryClass={math.OC}
}