A Non-Smooth Variational Control Framework for Fault Diagnosis and Recovery in Industrial Systems

Abstract

Industrial processes have discontinuities due to switching, saturation of actuators, stiction of valves, friction, degradation of components, and abrupt faults; thus, the assumption of differentiability needed for many fault tolerant control algorithms becomes invalid. In this work, propose a non-smooth variational approach to fault detection and reconstruction as well as post-fault recovery. The uncertain process is modeled by a differential inclusion that includes possible admissible controls, unknown faults, bounded disturbance inputs, and a maximal monotone operator. The fault reconstruction problem is reformulated as a convex variational optimization problem which includes constraints of residual consistency, sparse penalties, and total variation. The optimality conditions are given via the notions of subdifferential and normal cone mapping. Existence and uniqueness of the solution are proved under coercivity and restricted strong monotonicity properties, and a deterministic bound on the reconstruction error is obtained. The proposed primal-dual proximal approach finds the solution to the constrained problem without smoothing the non-smooth components. A numerical example involving a three-tank system under actuator failure, intermittent biases in sensors, parameter uncertainty, and bounded disturbance shows the algorithmic performance. The new approach leads to a normalized estimation error of faults equal to 0.061, to restoration of the controlled variable level after 14.8 s, and convergence of the primal-dual residual to less than 10⁻³ after 113 iterations.