The Polynomial Approach to the LQ non-Gaussian Regulator Problem through Output Injection

Abstract

In this paper, an improved approach for the solution of the regulator problem for linear discrete-time dynamical systems with non-Gaussian disturbances and quadratic cost functional is proposed. It is known that a sub-optimal recursive control can be derived from the classical LQG solution by substituting the linear filtering part with a quadratic, or in general polynomial, filter. However, we show that when the system is not asymptotically stable the polynomial control does not improve over the classical LQG solution, due to the lack of the internal stability of the polynomial filter. In order to enlarge the class of systems that can be controlled, we propose a new method based on a suitable rewriting of the system by means of an output injection term. We show that this allows to overcome the problem and to design a polynomial optimal controller also for non asymptotically stable systems. Numerical results show the effectiveness of the method.

Distributed infinite‐horizon optimal control of continuous‐time linear systems over network

“…Further extensions deserve additional investigation, for example, the introduction of communications delays or packet dropouts, 38,39 link failure, 40 or polynomial techniques for non‐Gaussian noises 41‐43 . Also, other future directions are secure and resilient solutions that account for disturbances, faults, threats, and attacks and/or disturbances.…”

Incremental nonlinear stability analysis of stochastic systems perturbed by Lévy noise

“…For the white noise system (7), the perturbed system and its nominal dynamics are parameterized such that the virtual system and virtual dynamics are established as in (19) and (20) without the shot noise terms 𝜉 𝜇 (t, q), 𝛿𝜉𝜇, and dN(t). Stochastic contraction is defined in Definition 2 of Pham et al, 29 but is only applicable to white noise systems (7).…”

“…Stochastic contraction is defined in Definition 2 of Pham et al, 29 but is only applicable to white noise systems (7). For the virtual system (19), we create a more general definition of stochastic contraction.…”

“…Here, M(t, q) ≜ Θ T Θ(t, q) is the contraction metric described before; the parameterization over 𝜇 is such that ( 19) and ( 20) hold and V(0, q 0 , 𝛿q 0 ) = ‖ ‖ y 0 − x 0 ‖ ‖ . Similar to the direct method of Lyapunov, we analyze the behavior of the system by analyzing the derivative of the Lyapunov-like function V(t, q, 𝛿q) along trajectories of the virtual system (19).…”

Filtering of systems with nonlinear measurements with an application to target tracking

“…Polynomial filters improve over linear filters by recursively projecting the system state on the space of polynomial (rather than linear) functions of the output. Consequently, they are well suited to linear systems with strongly non‐Gaussian noises, which have seen an increasing interest in control engineering applications, digital communications with non‐Gaussian noise components, fault estimation, sensor or actuator faults, multiplicative noises, and bilinear systems . Due to its increased effectiveness in dealing with non‐Gaussian noise arising from the linear representations of nonlinear output maps, the virtual measurement approach was shown to compare favorably with EKF, UKF, and PF.…”