Definition and Property Analysis of State Entropy in Control Systems

Information entropy is an important measure that quantifies the uncertainty associated with random variables, and has been widely used in signal processing, machine learning, wireless communication, etc. For a dynamical control system, its behavior can also be measured by information entropy, once we assume its initial state follows a certain distribution. This measure is defined as state entropy, which forms a new basic property of control systems, except for stability, robustness, and adaptability. This paper systematically studies the definition and property of state entropy for deterministic control systems, and introduces the mechanism of its time evolution in continuous-time, discrete-time, linear and nonlinear cases. We show that the time derivative of state entropy is the integral of trace of control Jacobian matrix over the initial state distribution, where the control Jacobian matrix is the full derivative of successive state with respect to the initial state. For control systems with constant value of divergence of its vector field, the evolution of state entropy is calculated without the knowledge of precise trajectory, as the trace is independent from the initial distribution. The variation of state entropy is exclusively determined by the positivity of the trace of control Jacobian matrix. We also derive the mathematical form of state entropy for discrete-time control systems. Our work provides a new angle to understand the behavior of dynamical control systems.