"Efficient numerical
methods for moving horizon estimation"
Niels Haverbeke (K.U. Leuven, ESAT-SCD)
Abstract:
In model-based predictive
control strategies, accurate estimates of the current state and model
parameters are required in order to predict the future system behavior
for a given control realization. One particularly powerful approach for
constrained nonlinear state estimation is Moving Horizon Estimation
(MHE). In MHE past measurements are reconciled with the model response
by optimizing states and parameters over a finite past horizon. The
basic strategy is to use a moving window of data such that the size of
the estimation problem is bounded by looking at only a subset of the
available data and summarizing older data in one initial condition
term. This also establishes an exponential forgetting of past data
which is useful for time-varying dynamics.
Compared to other state estimation approaches, MHE offers many
advantages following from its formulation as a dynamic optimization
problem. Inequality constraints on the variables (states, parameters,
disturbances) can be included in a natural way and the nonlinear model
equation is directly imposed over the horizon length. Empirical studies
show that MHE can outperform other estimation approaches in terms of
accuracy and robustness. In addition to these well-known advantages,
the framework of MHE allows for formulations different from the
traditional (weighted) least-squares formulation.
The greatest impediment to a widespread acceptance of MHE for real-time
applications is still its associated computational complexity. Despite
tremendous advances in numerical computing and Moore’s law,
optimization-based estimation algorithms are still primarily applied to
slow processes. In this work, we present fast structure-exploiting
algorithms which use robust and efficient numerical methods and we
demonstrate the increased performance and flexibility of nonlinear
constrained MHE. MHE problems are typically solved by general purpose
(sparse) optimization algorithms. Thereby, the symmetry and structure
inherent in the problems are not fully exploited. In addition, the
arrival cost is typically updated by running a (Extended) Kalman filter
recursion in parallel while the final estimate covariance is computed
from the derivative information. In this thesis, Riccati based methods
are derived which effectively exploit the inherent symmetry and
structure and yield the arrival cost update and final estimate
covariance as a natural outcome of the solution process. The primary
emphasis is on the robustness of the methods which is achieved by
orthogonal transformations.
When constraints are imposed, the resulting quadratic programming (QP)
problems can be solved by active-set or interior-point methods. We
derive modified Riccati recursions for interior-point MHE and show that
square-root recursions are recommended in this context because of
numerical conditioning. We develop an active-set method which uses the
unconstrained solution obtained from Riccati recursions and employs a
Schur complement technique to project onto the reduced space of active
constraints. The method involves non-negativity constrained QPs for
which an efficient gradient projection method is proposed. We implement
the algorithms in efficient C code and demonstrate that MHE is
applicable to fast systems.
These QP methods are at the core of solution methods for general convex
and nonlinear MHE as is demonstrated. Convex formulations are
investigated for robustness to outliers and abrupt parameter changes.
Furthermore, the methods are embedded in a Sequential Quadratic
Programming strategy for nonlinear MHE. One application has been of
particular interest during this doctoral research: estimation and
predictive control of blood-glucose at the Intensive Care Unit (ICU).
For this application reliability and robustness of the estimates as
well as of the numerical implementations are crucial. We evaluate an
MHE based MPC control strategy and show its potential for this
application.
Promotors: Prof. B. De Moor, Prof. M. Diehl
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