Observer-based numerical schemes

Abstract

Numerical analysis and control theory are fundamental areas in engineering and applied mathematics. This thesis explores three concepts from control theory used to enhance classical numerical solvers. The proposed improvement integrates a sampled-data Luenberger observer with a conventional numerical solver in a switched system framework. The method employs the numerical solver when its updates are sufficiently accurate and switches to using process samples to drive an observer when they are not. The switching mechanism is governed by an energy inequality based on a Lyapunov function, potentially triggering sampling as needed. Stability proofs and error estimates utilize input-to-state style stability techniques. This new numerical scheme can handle step sizes significantly larger than those required for the stability of the traditional numerical solver. Additionally, the hybrid approach of switching between the sampled-data observer and the numerical solver can reduce the frequency of sampling needed for accurate observer-based state estimation of the process. In this context, this thesis has the following aims. Firstly, it combined any numerical scheme with sampled-data Luenberger observer in a new hybrid scheme based on switching conditions. The scheme uses the numerical scheme when scheme updates are good enough but switches to an observer driven by process samples when not. A Lyapunov function-based energy inequality determines switching. Thus, the switching condition is central to the hybrid observer-based numerical scheme. The idea underpinning this switching condition is to use a Lyapunov function for the observer as an energy function for the Euler scheme. Loosely speaking, energy for the observer’s solutions will decrease, and we only use the Euler scheme when its energy also decreases. In this sense, the Lyapunov function for the observer becomes a Lyapunov function for the overall hybrid scheme. The switching condition partitions the state space into sections or regions where we use Euler scheme and where we use the observer. Depending on the system and the scheme’s parameters, the region where Euler method is used may be large, small, or even null. Secondly, the aim is to extend the generalized hybrid scheme with a higher-order approximation of the Taylor exponential. We generalize the switched system by using Runge-Kutta. After that, this hybrid ODE solver is constructed by combining the Euler and Luenberger observer to switch from the numerical scheme to the observer when the numerical scheme produces inadequate results. Underpinning our approach is a λ-tracking-based sampled-data observer that invokes a λ dead zone. The resulting hybrid algorithm is a time-stepping numerical scheme. The gains and sampling periods in the sampled- data observer are tuned using a λ-tracking approach. Using a sampled-data observer allows process measurements to be only available at some discrete times, while adaptive tuning allows the gains and sampling times to adjust automatically to each other rather than being subject to design. Finally, an alternative switching approach is considered: switching from observer to Euler based on λ and μ strips.