136 research outputs found
Delineating Parameter Unidentifiabilities in Complex Models
Scientists use mathematical modelling to understand and predict the
properties of complex physical systems. In highly parameterised models there
often exist relationships between parameters over which model predictions are
identical, or nearly so. These are known as structural or practical
unidentifiabilities, respectively. They are hard to diagnose and make reliable
parameter estimation from data impossible. They furthermore imply the existence
of an underlying model simplification. We describe a scalable method for
detecting unidentifiabilities, and the functional relations defining them, for
generic models. This allows for model simplification, and appreciation of which
parameters (or functions thereof) cannot be estimated from data. Our algorithm
can identify features such as redundant mechanisms and fast timescale
subsystems, as well as the regimes in which such approximations are valid. We
base our algorithm on a novel quantification of regional parametric
sensitivity: multiscale sloppiness. Traditionally, the link between parametric
sensitivity and the conditioning of the parameter estimation problem is made
locally, through the Fisher Information Matrix. This is valid in the regime of
infinitesimal measurement uncertainty. We demonstrate the duality between
multiscale sloppiness and the geometry of confidence regions surrounding
parameter estimates made where measurement uncertainty is non-negligible.
Further theoretical relationships are provided linking multiscale sloppiness to
the Likelihood-ratio test. From this, we show that a local sensitivity analysis
(as typically done) is insufficient for determining the reliability of
parameter estimation, even with simple (non)linear systems. Our algorithm
provides a tractable alternative. We finally apply our methods to a
large-scale, benchmark Systems Biology model of NF-B, uncovering
previously unknown unidentifiabilities
On Functional Observability and Structural Functional Observability
In this paper, functional observability, functional detectability, and
structural functional observability are revisited and investigated. It is found
that a PBH-like rank condition for functional observability and a similar one
for functional detectability in the literature are not always sufficient.
Conditions on the system matrices to establish their sufficiency are given. A
new concept of modal functional observability is proposed, leading to necessary
and sufficient conditions for functional observability and detectability in the
general case. Afterward, structural functional observability is redefined
rigorously from a generic perspective. A new and complete graph-theoretic
characterization for structural functional observability is proposed. Based on
these results, the problems of selecting the minimal sensors to achieve
functional observability and structural functional observability are shown to
be NP-hard. Nevertheless, polynomial-time algorithms are given based on the
supermodular set functions to find approximation solutions. A corollary of our
results is that, verifying the structural target controllability of state
variables is possible in polynomial time, where is the system state
dimension, a problem that may be hard otherwise.Comment: This is an early version of a collaborative work with Professor
Tyrone Fernando of University of Western Australia, Australia, and Professor
Mohamed Darouach of Universit de Lorraine, France. The co-authors' names are
scheduled to be added in an updated versio
Target Controllability and Target Observability of Structured Network Systems
The duality between controllability and observability enables methods
developed for full-state control to be applied to full-state estimation, and
vice versa. In applications in which control or estimation of all state
variables is unfeasible, the generalized notions of output controllability and
functional observability establish the minimal conditions for the control and
estimation of a target subset of state variables, respectively. Given the
seemly unrelated nature of these properties, thus far methods for target
control and target estimation have been developed independently in the
literature. Here, we characterize the graph-theoretic conditions for target
controllability and target observability (which are, respectively, special
cases of output controllability and functional observability for structured
systems). This allow us to rigorously establish a weak and strong duality
between these generalized properties. When both properties are equivalent
(strongly dual), we show that efficient algorithms developed for target
controllability can be used for target observability, and vice versa, for the
optimal placement of sensors and drivers. These results are applicable to
large-scale networks, in which control and monitoring are often sought for
small subsets of nodes.Comment: Codes are available in GitHub
(https://github.com/montanariarthur/TargetCtrb
Model reduction of controlled Fokker--Planck and Liouville-von Neumann equations
Model reduction methods for bilinear control systems are compared by means of
practical examples of Liouville-von Neumann and Fokker--Planck type. Methods
based on balancing generalized system Gramians and on minimizing an H2-type
cost functional are considered. The focus is on the numerical implementation
and a thorough comparison of the methods. Structure and stability preservation
are investigated, and the competitiveness of the approaches is shown for
practically relevant, large-scale examples
Koopman Kernel Regression
Many machine learning approaches for decision making, such as reinforcement
learning, rely on simulators or predictive models to forecast the
time-evolution of quantities of interest, e.g., the state of an agent or the
reward of a policy. Forecasts of such complex phenomena are commonly described
by highly nonlinear dynamical systems, making their use in optimization-based
decision-making challenging. Koopman operator theory offers a beneficial
paradigm for addressing this problem by characterizing forecasts via linear
time-invariant (LTI) ODEs, turning multi-step forecasts into sparse matrix
multiplication. Though there exists a variety of learning approaches, they
usually lack crucial learning-theoretic guarantees, making the behavior of the
obtained models with increasing data and dimensionality unclear. We address the
aforementioned by deriving a universal Koopman-invariant reproducing kernel
Hilbert space (RKHS) that solely spans transformations into LTI dynamical
systems. The resulting Koopman Kernel Regression (KKR) framework enables the
use of statistical learning tools from function approximation for novel
convergence results and generalization error bounds under weaker assumptions
than existing work. Our experiments demonstrate superior forecasting
performance compared to Koopman operator and sequential data predictors in
RKHS.Comment: Accepted to the thirty-seventh Conference on Neural Information
Processing Systems (NeurIPS 2023
A direct design procedure for linear state functional observers
We propose a constructive procedure to design a Luenberger observer to estimate a linear multiple linear state functional for a linear time-invariant system. Among other features the proposed design algorithm is not based on the solution of a Sylvester equation nor on the use of canonical state space forms. The design is based on the solution set of a linear equation and a realization method. The consistency of this equation and the stability of the observer can be used as a functional observability test
Passivity Enforcement via Perturbation of Hamiltonian Matrices
This paper presents a new technique for the passivity enforcement of linear time-invariant multiport systems in statespace form. This technique is based on a study of the spectral properties of related Hamiltonian matrices. The formulation is applicable in case the system input-output transfer function is in admittance, impedance, hybrid, or scattering form. A standard test for passivity is first performed by checking the existence of imaginary eigenvalues of the associated Hamiltonian matrix. In the presence of imaginary eigenvalues the system is not passive. In such a case, a new result based on first-order perturbation theory is presented for the precise characterization of the frequency bands where passivity violations occur. This characterization is then used for the design of an iterative perturbation scheme of the state matrices, aimed at the displacement of the imaginary eigenvalues of the Hamiltonian matrix. The result is an effective algorithm leading to the compensation of the passivity violations. This procedure is very efficient when the passivity violations are small, so that first-order perturbation is applicable. Several examples illustrate and validate the procedure
On unknown-input functional observability of linear systems
Finding the least possible order of a stable Unknown-Input Functional Observer (UIFO) has always been a challenge in observer design theory. A practical recursive algorithm is proposed in this technical note to design a minimal multi-functional observer for multi-input multi-output (MIMO) linear time-invariant (LTI) systems with unknown-inputs. The concept of unknown-input functional observability is introduced,and it is used as a certificate of the convergence of our algorithm. The proposed procedure looks for a number of additional auxiliary functions to be augmented to the original functions desired for reconstruction. The resulting UIFO is proper, and minimal (of minimum possible order). Moreover, the algorithm does not need the system to be unknown-input observable. A numerical example shows the procedure as well as the effectiveness of the proposed algorithm
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