Simultaneous state and unknown input set-valued observers for quadratically constrained nonlinear dynamical systems

In this article, we propose fixed-order set-valued (in the form of (Formula presented.) -norm hyperballs) observers for several classes of quadratically constrained nonlinear dynamical systems with unknown input signals that simultaneously/jointly find bounded hyperballs of states and unknown inputs that include the true states and inputs. Necessary and sufficient conditions in the form of linear matrix inequalities (LMIs) for the stability (in the sense of quadratic stability) of the proposed observers are derived for ((Formula presented.))-quadratically constrained (((Formula presented.))-QC) systems, which includes several classes of nonlinear systems: (I) Lipschitz continuous, (II) ((Formula presented.))-QC* and (III) linear parameter-varying (LPV) systems. This new quadratic constraint property is at least as general as the incremental quadratic constraint property for nonlinear systems and is proven in the paper to embody a broad range of nonlinearities. In addition, we design the optimal (Formula presented.) observer among those that satisfy the quadratic stability conditions and show that the design results in uniformly bounded-input bounded-state (UBIBS) estimate radii/error dynamics and uniformly bounded sequences of the estimate radii. Furthermore, we provide closed-form upper bound sequences for the estimate radii and sufficient conditions for their convergence to steady state. Finally, the effectiveness of the proposed set-valued observers is demonstrated through illustrative examples, where we compare the performance of our observers with some existing observers.