State-Feedback Controller Sparsification via Quasi-Norms

In this paper, quasi-norms are utilized to sparsify a pre-given well-performing state-feedback controller stabilizing a linear time-invariant (LTI) system. To do so, an unconstrained optimization problem is firstly formulated which incorporates two terms: (i) the Frobenius norm of difference of the pre-given feedback controller and the one to be designed; (ii) the 0 < q < 1 quasi-norm of the feedback controller to be designed. The former term heuristically features the disturbance attenuation performance and the latter term promotes the sparsity. Next, obtaining an analytic threshold for the sparsity-promoting parameter, the analytic solution of the formulated unconstrained optimization problem is expressed which is basically the designed sparse feedback controller. Throughout the numerical simulations, it is observed that when 0 < q < 1 decreases, the sparsity-performance balance is significantly improved. Furthermore, the proposed method is interestingly capable of being applied to the large-scale systems with thousands of states.

[1]  Mayuresh V. Kothare,et al.  Closed-loop feedback sparsification under parametric uncertainties , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[2]  Kazufumi Ito,et al.  A variational approach to sparsity optimization based on Lagrange multiplier theory , 2013 .

[3]  M. Kothare,et al.  Output Feedback Controller Sparsification via H2-Approximation , 2015 .

[4]  Michel Verhaegen,et al.  Sequential Convex Relaxation for Robust Static Output Feedback Structured Control , 2017 .

[5]  Maryam Babazadeh,et al.  Sparsity Promotion in State Feedback Controller Design , 2017, IEEE Transactions on Automatic Control.

[6]  Nader Motee,et al.  State feedback controller sparsification via a notion of non-fragility , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[7]  Aranya Chakrabortty,et al.  Sparsity-Constrained Mixed $H_{2}/H_{\infty}$ Control , 2018, 2018 Annual American Control Conference (ACC).

[8]  Fu Lin,et al.  Design of Optimal Sparse Feedback Gains via the Alternating Direction Method of Multipliers , 2011, IEEE Transactions on Automatic Control.

[9]  Javad Lavaei,et al.  Transformation of Optimal Centralized Controllers Into Near-Globally Optimal Static Distributed Controllers , 2019, IEEE Transactions on Automatic Control.

[10]  Mikhail V. Khlebnikov,et al.  An LMI approach to structured sparse feedback design in linear control systems , 2013, 2013 European Control Conference (ECC).

[11]  Aranya Chakrabortty,et al.  Structurally Constrained $\ell_{1}$-Sparse Control of Power Systems: Online Design and Resiliency Analysis , 2018, 2018 Annual American Control Conference (ACC).

[12]  Fu Lin,et al.  Augmented Lagrangian Approach to Design of Structured Optimal State Feedback Gains , 2011, IEEE Transactions on Automatic Control.