Control Principles for Complex System Presentation for Frontiers

Control Principles for Complex System Presentation for Frontiers of Network Science Runyu Lai November 18 th, 2019 Instructor: Professor Boleslaw Szymanski

To form a Control Problem? What is Control: Control theory asks how to influence the behavior of a dynamical system with appropriately chosen inputs so that the system’s output follows a desired trajectory or final state. Input SYSTEM Output Prerequisites of Control: An accurate map of the network that governs the system’s components A quantitative description of the dynamic laws that govern the components’ behavior An ability to influence the states of components Network Science Traditional Theory Topology of real systems Measurement and Modeling Complex Control Principles 2 11/3/2020

Linear System Control Problem State Space Representation Graphical Interpretation State Vertices Input Edges Actuator Nodes Weighted Wiring Diagram Nodes and Controller Complex Control Principles Driver Nodes 3 11/3/2020

Linear System Controllability What is controllability? Initial State Finite Time Final State Kalman’s Criterion Controllability matrix Dimension is small Controllability easily tested. Without knowing detailed values Rank test ill conditioned Dimension is large Sensitive to round-off errors Sensitive to uncertainties in the matrix elements We need to Find rank without numerically calculating Complex Control Principles 4 11/3/2020

Graphical Interpretation System Controllable Complex Control Principles 5 11/3/2020

Structural Controllability What is a structured system: Structural Controllability: Elements in A, B are either fixed zeros or independent free parameters. A system is structural controllable if we can set the nonzero elements in A, B such that resulting system is controllable Structural Controllable? States Strong structural controllability System not structurally controllable if and only if it has inaccessible nodes or dilations. Complex Control Principles 6 11/3/2020

Maximum Matching: A set of edges without common vertices. Matching: A set of directed edges do not share common start or end vertices. Matched Unmatched Complex Control Principles 7 11/3/2020

Minimum Input Theorem Costly and Impractical Control Each Node Individually Find Minimum Input Sufficient to make system fully controllable Maximum Matching: Largest set of edges without common heads or tails. Complex Control Principles 8 11/3/2020

Minimum Controllability Problem Objective Functions Way to ‘Inject’ Input MCP 0 Minimize Number of Nodes MCP 2 Complex Control Principles 9 11/3/2020

Links, Nodes, Controlling Edges Critical Link Redundant Ordinary Critical Node Complex Control Principles Deletion Critical Redundant Deletion Redundant Intermittent Deletion Ordinary 10 11/3/2020

Controllability of Nonlinear Systems Accessibility: Possibility to reach or access an open set of states in the state space from a given initial state. Local controllability: whether the system is controllable in some neighborhood of a given state. Controllability of linearized control system: Controllability properties of its linearization Local nonlinear controllability Around a trajectory: The path the system follows as a function of time Limitation of linearization: Only provide information in the immediate vicinity of an equilibrium point or trajectory. Even if the linearized control system is not controllable, the nonlinear system could be controllable. Complex Control Principles 11 11/3/2020

Controllability of Nonlinear Networked Systems Neuronal Network Motifs Not all symmetries has the same impact Complex Control Principles 12 11/3/2020

Observability What is Observability? Possibility to estimate all unmeasured variables from the measured variables we selected. Kalman’s Rank Condition: Observability Test: Parameter Identification Network Reconstruction Driving Response Copy Synchronization Complex Control Principles Direct Approach 13 11/3/2020

For nonlinear systems, observer design is rather involved and still an open challenge!! Complex Control Principles 14 11/3/2020

Problem Statement O Final State Initial State Goals: Fixed End Points Time-optimal Real-time Challenges in Path Planning Complex Model Dynamics Physical Capacity Complex Control Principles 15 11/3/2020

Model Dynamics Quadcopter Translational Motion: Reference Guidance Quadcopter Control Simplified Dynamics planned state Differential Flatness: Equivalent Navigation Complex Control Principles 16 11/3/2020

Analysis Result of This Network X Y Z Controllable and Observable Complex Control Principles Driver Nodes 17 11/3/2020

Thank You! Any Questions? Complex Control Principles 18 11/3/2020