Minimal input selection for robust control

This paper studies the problem of selecting a minimum-size set of input nodes to guarantee stability of a networked system in the presence of uncertainties and time delays. Current approaches to input selection in networked dynamical systems focus on nominal systems with known parameter values in the absence of delays. We derive sufficient conditions for existence of a stabilizing controller for an uncertain system that are based on a subset of system modes lying within the controllability subspace induced by the set of inputs. We then formulate the minimum input selection problem and prove that it is equivalent to a submodular optimization problem, leading to polynomial-time algorithms with provable optimality bounds. We show that our approach is applicable to different types of uncertainties, including additive and multiplicative uncertainties in the system matrices as well as uncertain time delays. We demonstrate our approach in a numerical case study on the IEEE 39-bus test power system.

[1]  C. H. Lee.  Solution Bounds of the Continuous and Discrete Lyapunov Matrix Equations , 2004 .

[2]  Abhimanyu Das,et al.  Submodular meets Spectral: Greedy Algorithms for Subset Selection, Sparse Approximation and Dictionary Selection , 2011, ICML.

[3]  G. Dullerud,et al.  A Course in Robust Control Theory: A Convex Approach , 2005 .

[4]  Nasir Uddin Ahmed.  Controllability and Stabilizability , 2006 .

[5]  George J. Pappas,et al.  Minimal actuator placement with optimal control constraints , 2015, 2015 American Control Conference (ACC).

[6]  Chi-Tsong Chen,et al.  Linear System Theory and Design , 1995 .

[7]  Soummya Kar,et al.  On the complexity of the constrained input selection problem for structural linear systems , 2014, Autom..

[8]  George J. Pappas,et al.  Minimal Reachability is Hard To Approximate , 2019, IEEE Transactions on Automatic Control.

[9]  Bram de Jager,et al.  Efficient selection of inputs and outputs for robust control , 1999 .

[10]  Albert-László Barabási,et al.  Controllability of complex networks , 2011, Nature.

[11]  Zhipeng Liu,et al.  Towards Scalable Voltage Control in Smart Grid: A Submodular Optimization Approach , 2016, 2016 ACM/IEEE 7th International Conference on Cyber-Physical Systems (ICCPS).

[12]  Basel Alomair,et al.  Minimizing Convergence Error in Multi-Agent Systems Via Leader Selection: A Supermodular Optimization Approach , 2013, IEEE Transactions on Automatic Control.

[13]  George J. Pappas,et al.  Minimal reachability problems , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[14]  Yang Zhang,et al.  Design of Wide-Area Damping Controllers for Interarea Oscillations , 2008, IEEE Transactions on Power Systems.

[15]  Fuzhen Zhang.  Matrix Theory: Basic Results and Techniques , 1999 .

[16]  I. Kamwa,et al.  Control loops selection to damp inter-area oscillations of electrical networks , 2002, IEEE Power Engineering Society Summer Meeting,.

[17]  J. Doyle,et al.  Essentials of Robust Control , 1997 .

[18]  Alexandros G. Dimakis,et al.  Scalable Greedy Feature Selection via Weak Submodularity , 2017, AISTATS.

[19]  S. Pequito,et al.  On the Exact Solution of the Minimal Controllability Problem , 2015 .

[20]  Soummya Kar,et al.  Minimum cost constrained input-output and control configuration co-design problem: A structural systems approach , 2015, 2015 American Control Conference (ACC).

[21]  Charles E. Heckler,et al.  Applied Multivariate Statistical Analysis , 2005, Technometrics.

[22]  Alexander Olshevsky,et al.  Minimal Controllability Problems , 2013, IEEE Transactions on Control of Network Systems.

[23]  Vijay Vittal,et al.  Design of wide-area damping control robust to transmission delay using μ-synthesis approach , 2014, 2014 IEEE PES General Meeting | Conference & Exposition.

[24]  John Lygeros,et al.  On Submodularity and Controllability in Complex Dynamical Networks , 2014, IEEE Transactions on Control of Network Systems.

[25]  Zhipeng Liu,et al.  MinGen: Minimal generator set selection for small signal stability in power systems: A submodular framework , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).