Sufficient Conditions For Exogenous Input Estimation In Nonlinear Systems

A large class of nonlinear plants subject to unknown but bounded exogenous inputs (disturbance inputs and senso r noise) is considered. A convex programming method for the design of state observers for these plants is proposed. Thes e observers can be used to estimate the exogenous inputs to som e specified degree of accuracy. Additionally, conditions gua r nteeing estimation of the disturbance inputs with arbitrary accuracy are presented. The effectiveness of the proposed observers is illustrated with a numerical example. I. I NTRODUCTION In noisy environments, it is imperative to design observers for dynamical systems which perform at pre-specified performance levels. From a practical viewpoint, it is desirabl e to estimate the state while simultaneously reconstructing the unknown (exogenous) inputs. For example, in secure network s, unknown input reconstruction is useful for the mitigation o f attack signals. Another example could be the reconstructio n of unmodeled physiological inputs in biomedical systems. Unknown input reconstruction for linear systems has been studied using linear observers in [1]–[3], and using slidin g mode observers in [4]–[6]. Other unknown input observer architecture for globally Lipschitz nonlinear systems are p oposed in, for example, [7]–[13]. Extensions to monotone nonlinearities and slope-restricted nonlinearities are d iscussed in [14]–[16]. In this paper, we present a systematic framework for the design of observers for a class of nonlinear dynamical syste ms in the presence of bounded disturbance inputs and bounded sensor noise. We lump the disturbance input and sensor noise into a single exogenous input. The class of nonlinearities u nder consideration are characterized by a set of symmetric matri ces. Ourcontributions include: (i) a convex programming framework for designing observers for nonlinear systems wi th exogenous inputs; (ii) providing performance guarantees a nd explicit bounds on the unknown input reconstruction error; (iii) providing conditions for unknown input reconstructi on in nonlinear systems with arbitrary accuracy; and, (iv) for linear error dynamics, demonstrating that our proposed LMI s are a generalization of existing conditions for unknown inp ut 1 A. Chakrabarty (chakraa@purdue.edu) and S. H. Żak (zak@purdue.edu) are affiliated with the School of Electrical and Computer Eng ineering, Purdue University, West Lafayette, IN. 3 G.T. Buzzard (buzzard@purdue.edu) is affiliated with the De partment of Mathematics, Purdue University, West Lafayette, IN. 2 M.J. Corless (corless@purdue.edu) is affiliated with the Sc hool of Aeronautics and Astronautics, Purdue University, West Laf ayette, IN. 4 A.E. Rundell (rundell@purdue.edu) is affiliated with the We ldon School of Biomedical Engineering, Purdue University, West Lafaye tte, IN. observers. Proofs omitted in this paper for space considera t ons are available at [17]. II. N OTATION We denote byR the set of real numbers, and R the set of realn×m matrices. For any matrixP , we denoteP as its transpose, and‖P‖ as the maximum singular value of P . For any vectorv ∈ R, we consider the norm‖v‖ = √ v⊤v. For a bounded functionv(·) : R → R, we consider the norm ‖v(·)‖∞ = supt ‖v(t)‖. For a symmetric matrixM = M, we use the star notation to avoid rewriting symmetric terms, that is,