Oblique Projected Dynamical Systems and Incremental Stability Under State Constraints

Sufficient Conditions for Output Reference Tracking for Nonlinear Systems: a Contractive Approach

“…Then, there exists a positive real number κ > 0 such that, for any κ ∈ (0, κ ), the closed-loop system (4), ( 5), with ψ defined as in (16), admits a C 1 function P : R n+1 → R (n+1)×(n+1) , a positive continuous function µ : R n+1 → R >0 and positive real numberp, p ∈ R >0 satisfying (9) and…”

“…Proof. Consider the closed-loop system (4), with ϕ, Γ defined as in ( 5), and ψ selected as in (16). For r = 0, the origin of ( 4) is an equilibrium, and because of Theorem 3, it is locally exponentially stable.…”

Constrained State Estimation for Nonlinear Systems: A Redesign Approach Based on Convexity

“…2) The state of the redesigned observer O m remains in X for all forward times when initialized in X . As it is often the case in designs with constraints (see, e.g., [40], [42], [47], [54]), the solutions presented in the sequel rely on a key convexity assumption, as formalized next.…”

“…The correction term M defined in (14) may also be interpreted as an element of the normal cone to the level set of c in the metric defined by P (t), thus suggesting a link with the projected dynamics used in [47] for constant metric P .…”

“…Instead, several projection-based approaches have been developed for specific classes of systems and observer designs. In the context of continuous-time systems, projectionbased solutions have been proposed for high-gain observers in [40], for Lur'e type systems in [46], and within the framework of projected dynamical systems in [47] for exponentially convergent observers with linear correction terms. Projection methods are also very well-known in the context of constrained Kalman filters (KF) for discrete-time systems with the socalled equality constrained KF (ecKF), which was proved to provide a smaller state error covariance than the unconstrained algorithm.…”