  In this paper, we propose fixed-order set-valued (in the form of l2-norm
hyperballs) observers for some classes of nonlinear bounded-error dynamical
systems with unknown input signals that simultaneously 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 ($\mathcal{M},\gamma$)- Quadratically Constrained
(($\mathcal{M},\gamma$)-QC) systems, which includes several classes of
nonlinear systems: (I) Lipschitz continuous, (II) ($\mathcal{A},\gamma$)-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
$\mathcal{H}_{\infty}$ 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 condition 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.
