Finding the nearest positive-real system
For control theorists and engineers, this provides a practical method to enforce passivity in system models, addressing a long-standing problem with no prior general solution.
The paper introduces the first algorithm to find the nearest positive-real system to a given non-positive-real linear time-invariant system, minimizing perturbations to all system matrices. The algorithm uses a reformulation as a strict port-Hamiltonian system and a fast gradient method, demonstrated on examples.
The notion of positive realness for linear time-invariant (LTI) dynamical systems, equivalent to passivity, is one of the oldest in system and control theory. In this paper, we consider the problem of finding the nearest positive-real (PR) system to a non PR system: given an LTI control system defined by $E \dot{x}=Ax+Bu$ and $y=Cx+Du$, minimize the Frobenius norm of $(Δ_E,Δ_A,Δ_B,Δ_C,Δ_D)$ such that $(E+Δ_E,A+Δ_A,B+Δ_B,C+Δ_C,D+Δ_D)$ is a PR system. We first show that a system is extended strictly PR if and only if it can be written as a strict port-Hamiltonian system. This allows us to reformulate the nearest PR system problem into an optimization problem with a simple convex feasible set. We then use a fast gradient method to obtain a nearby PR system to a given non PR system, and illustrate the behavior of our algorithm on several examples. This is, to the best of our knowledge, the first algorithm that computes a nearby PR system to a given non PR system that (i) is not based on the spectral properties of related Hamiltonian matrices or pencils, (ii) allows to perturb all matrices $(E,A,B,C,D)$ describing the system, and (iii) does not make any assumption on the original given system.