Robust stability of moving horizon estimation for non‐linear systems with bounded disturbances using adaptive arrival cost

Simultaneous moving horizon estimation and control for nonlinear systems subject to bounded disturbances

“…When $$$ \varphi =0 $$$, $$$ {\Psi}_{EC,k,{N}_e+{N}_c}:= {\Psi}_{C,k,{N}_c} $$$ and problem (3) becomes a model predictive control problem with terminal cost 33 . On the other case, when $$$ \varphi =1 $$$, $$$ {\Psi}_{EC,k,{N}_e+{N}_c}:= {\Psi}_{E,k,{N}_e} $$$ and problem (3) becomes a moving horizon estimation problem 10,12,14,34 . In these cases, the optimization problem (3) has only one objective function and the separation principle needs to be applied since the estimator and the controller are implemented independently.…”

“…This is a property of i‐IOSS systems, 36 where the parameters and structure of functions $$$ \overline{\Phi}\left(\cdotp \right) $$$, $$$ {\pi}_w\left(\cdotp \right) $$$ and $$$ {\pi}_v\left(\cdotp \right) $$$ depend on the stage costs $$$ {\ell}_e\left({\hat{w}}_{j\mid k},{\hat{v}}_{j\mid k}\right) $$$ and arrival cost employed in problem (3), and they should be derived for each choice of $$$ {\ell}_e\left(\cdotp \right) $$$ and $$$ {\Gamma}_{k-{N}_e}\left(\cdotp \right) $$$ 12‐14 . In this work, we use the adaptive arrival cost proposed by Sanchez et al 11 …”

“…Recent results regarding MHE for nonlinear systems are given for robust stability and estimate convergence properties by Alessandri et al, 8,9 Garcia‐Tirado et al 10 and Sanchez et al, 11 among others. In recent years, several results have been obtained for different MHE formulations, advancing from idealistic assumptions, like observability and vanishing disturbances, to realistic situations like detectability and bounded disturbances 12‐14 . Zou et al 15 are concerned with the MHE problem for networked linear systems with unknown inputs under dynamic quantization effects.…”

“…We show that the proposed controller results in closed–loop trajectories along which the states remain bounded. These results rely on two assumptions: the first assumption requires that the optimization criterion includes the adaptive arrival cost proposed by Sanchez et al 11 This assumption allows us to ensure the boundedness of the state estimate and to obtain a bound for the estimation error set if the parameters of the estimation problem are properly chosen 14 . The second assumption requires that the backward (estimation) and forward (control) horizons are sufficiently large so that enough information is obtained to find state estimates and control inputs compatible with the system dynamics, noises and constraints.…”

States and Parameters Estimation for Induction Motors Based on a New Adaptive Moving Horizon Estimation

“…In this work, the approach proposed for the update of the arrival cost function, based on an adaptive algorithm, is formulated as an update of the following elements: P k−N and 􏽥 x k−N . In this case, the update of the initial state 􏽥 x k−N will be done by a so-called smooth update, which implies that once the estimate has left the estimate horizon, the state constraints will not change [29][30][31].…”