Simultaneous state estimation and control for nonlinear systems subject to bounded disturbances

TL;DR

This work develops a jointly optimized output-feedback scheme for nonlinear systems subject to bounded disturbances by unifying moving horizon state estimation (MHE) and model predictive control (MPC). The controller solves a receding-horizon problem that simultaneously estimates the initial state and disturbances while computing future control actions, using backward (estimation) and forward (control) windows whose lengths are tied to stability guarantees under i-IOSS detectability. A multi-objective formulation with an adjustable balance parameter $\\varphi$ encompasses pure MHE, pure MPC, and their simultaneous combination; stability is ensured by ensuring the backward window is sufficiently long to achieve accurate state estimation and the forward window is long enough to guarantee controllability, with a contractive bound on the total cost. Theoretical results are complemented by two numerical examples (a nonlinear scalar system and a van der Pol oscillator) showing that the simultaneous approach yields better robustness and often lower computational burden than separate estimation and control. Overall, the paper provides a practical framework for robust, bounded-disturbance control of nonlinear systems with output feedback, linking horizon lengths to stability and proposing concrete guidelines for implementation.

Abstract

In this work, we address the output--feedback control problem for nonlinear systems under bounded disturbances using a moving horizon approach. The controller is posed as an optimization-based problem that simultaneously estimates the state trajectory and computes future control inputs. It minimizes a criterion that involves finite forward and backward horizon with respect the unknown initial state, measurement noises and control input variables and it is maximized with respect the unknown future disturbances. Although simultaneous state estimation and control approaches are already available in the literature, the novelty of this work relies on linking the lengths of the forward and backward windows with the closed-loop stability, assuming detectability and decoding sufficient conditions to assure system stabilizability. Simulation examples are carried out to compare the performance of simultaneous and independent estimation and control approaches as well as to show the effects of simultaneously solving the control and estimation problems.

Figures (6)

• Figure 1: ${\omega}(N_c)$ for $\delta=1, \Delta^w_c=10^{-1}$ and constraints sets \ref{['Constr_S01']} (\ref{['figure:example1 omega-a']}) and \ref{['Constr_S02']} (\ref{['figure:example1 omega-b']}) for independent (red) and simultaneous (blue) approaches.
• Figure 2: Evolution of system output for different initial conditions, difference between trajectories and i-IOSS bound.
• Figure 3: Evolution of system output for $N_c=20$ (\ref{['figure:larger control horizons-a']}) and $N_c=70$ (\ref{['figure:larger control horizons-b']}), with $N_e=30$ for independent MHE and MPC (red line) and simultaneous MHE--MPC (blue dotted line).
• Figure 4: MSE of $100$ simulations for different values of $N_e$, $N_c$ and $\epsilon$
• Figure 5: Two realizations of $x_1$ and $x_2$ for $\epsilon=0.1$, $N_e=2$ (\ref{['figure:example 2, epsilon=1 x1 and x2-a']} - \ref{['figure:example 2, epsilon=1 x1 and x2-b']}), $N_e=20$ (\ref{['figure:example 2, epsilon=1 x1 and x2-c']} - \ref{['figure:example 2, epsilon=1 x1 and x2-d']}) and $N_c=35$.

Theorems & Definitions (3)

• Remark 1
• Definition 1
• Theorem 1