Stability Analysis of Nonlinear Repetitive Control Schemes

Forwarding design for stabilization of a coupled transport equation-ODE with a cone-bounded input nonlinearity

“…An estimation of the decay rates should be studied so that to have a better insight in how the parameters of the controller can improve or not the performances of the closed-loop system. Finally, we believe that this preliminary work may open the route to different approaches in the context of periodic regulation [2], repetitive control [9] and feedback stabilization in presence of input/output delays.…”

“…Recall that the ω-limit set of the initial condition (z 0 , w 0 ), denoted by ω(z 0 , w 0 ), is the set of all (z , w ) ∈ D(A) such that there exists an increasing sequence of time (t n ) n 0 such that z(t n ) → z and w(t n ) → w in L 2 (0, 1) as n goes to infinity. We are going to prove that ω(z 0 , w 0 ) reduces to (0, 0), so that (z, w) converges to the origin in the R × L 2 (0, 1)topology since it is bounded according to (9). Since (z, w) is uniformly bounded in time according to (9) and the canonical inclusion D(A) → X is compact according to the Sobolev inclusion theorems, the positive orbit {(z(t), w(t)), t ∈ R + } is precompact in X.…”

“…We are going to prove that ω(z 0 , w 0 ) reduces to (0, 0), so that (z, w) converges to the origin in the R × L 2 (0, 1)topology since it is bounded according to (9). Since (z, w) is uniformly bounded in time according to (9) and the canonical inclusion D(A) → X is compact according to the Sobolev inclusion theorems, the positive orbit {(z(t), w(t)), t ∈ R + } is precompact in X. Therefore, according to the LaSalle's invariance principle for infinite-dimensional systems (see [27,Theorem 3.1]), ω(z 0 , w 0 ) is a non-empty compact subset of X that is invariant to the flow of (6).…”

Repetitive control design based on forwarding for nonlinear minimum-phase systems

“…Such a constraint, however, strongly restricts the class of systems to which a RC-scheme can be applied, that is nonlinear systems which are strictly input passive (in other words, with a direct feedthrough term). We refer to [1] for a proof for linear systems where it is shown that exponential stability of a (continuous-time) linear system incorporating a pure delay in the RC-scheme can be achieved only for systems having zero-relative degree between the input and the regulated output; alternatively, see [15,16] for a proof in the context of nonlinear systems based on dissipativity operators.…”

Adaptive output regulation for linear systems via discrete-time identifiers

“…The multivariable linear regulator, introduced by Francis, Wonham and Davison in [1,2,3], boasts a very important robustness property relative to uncertainties in the plant: the regulation errors are ensured to vanish despite arbitrarily large perturbations in the plant dynamics as far as they do not destroy linearity and closed-loop stability. Roughly speaking, this robustness property is a consequence of the fact that if the closed-loop system is linear, and its origin is exponentially stable whenever the input is not present, its steady state is completely governed by the exosystem (interestingly, this is also true in infinite dimension [4]). As far as linearity and expo-nential stability hold, the internal model embedded in the regulator is still able to generate the ideal "feedforward" control law that is needed to keep the regulation errors to zero at the steady state, and uncertainties in the plant just reflect into the right initialization of the internal model, which would be anyway unknown.…”