Parameter estimation methods for biological systems

The inverse problem of modeling biochemical processes mathematically from measured time course data falls into the category of system identification and parameter estimation. Analyzing the time course data would provide valuable insights into the model structure and dynamics of the biochemical system. Based on the types of biochemical reactions, such as metabolic networks and genetic networks, several modeling frameworks have been proposed, developed and proved effective, including the Michaelis-Menten equation, the Biochemical System Theory (BST), etc. One bottleneck in analyzing the obtained data is the estimation of parameter values within the system model. As most models for molecular biological systems are nonlinear with respect to both parameters and system state variables, estimation of parameters in these models from experimental measurement data is thus a nonlinear estimation problem. In principle, all algorithms for nonlinear optimization can be used to deal with this problem, for example, the Gauss-Newton iteration method and its variants. However, these methods do not take the special structures of biological system models into account. When the number of parameters to be determined increases, it will be challenging and computationally expensive to apply these conventional methods. In this research, several methods are proposed for estimating parameters in two classes of widely used biological system models: the S-system model and the linear fractional model (LFM), by utilizing their structure specialties. For the S-system, two estimation methods are designed. 1) Based on the two-term structure (production and degradation) of the model, an alternating iterative least squares method is proposed. 2) A separation nonlinear least squares method is proposed to deal with the partially linear structure of the model. For the LFM, two estimation methods are provided. 1) The separation nonlinear least squares method can also be adopted to treat the partially linear structure of the LFM, and moreover a modified iterative version is included. 2) A special strategy using the separation principle and the weighted least squares method is implemented to turn the cost function into a quadratic form and thus the estimates for parameters can be analytically solved. Simulation results have demonstrated the effectiveness of the proposed methods, which have shown better performance in terms of estimation accuracy and computation time, compared with those conventional nonlinear estimation methods

A Formally Verified Conflict Detection Algorithm for Polynomial Trajectories

In air traffic management, conflict detection algorithms are used to determine whether or not aircraft are predicted to lose horizontal and vertical separation minima within a time interval assuming a trajectory model. In the case of linear trajectories, conflict detection algorithms have been proposed that are both sound, i.e., they detect all conflicts, and complete, i.e., they do not present false alarms. In general, for arbitrary nonlinear trajectory models, it is possible to define detection algorithms that are either sound or complete, but not both. This paper considers the case of nonlinear aircraft trajectory models based on polynomial functions. In particular, it proposes a conflict detection algorithm that precisely determines whether, given a lookahead time, two aircraft flying polynomial trajectories are in conflict. That is, it has been formally verified that, assuming that the aircraft trajectories are modeled as polynomial functions, the proposed algorithm is both sound and complete

Nonlinear zero-sum differential game analysis by singular perturbation methods

A class of nonlinear, zero-sum differential games, exhibiting time-scale separation properties, can be analyzed by singular-perturbation techniques. The merits of such an analysis, leading to an approximate game solution, as well as the 'well-posedness' of the formulation, are discussed. This approach is shown to be attractive for investigating pursuit-evasion problems; the original multidimensional differential game is decomposed to a 'simple pursuit' (free-stream) game and two independent (boundary-layer) optimal-control problems. Using multiple time-scale boundary-layer models results in a pair of uniformly valid zero-order composite feedback strategies. The dependence of suboptimal strategies on relative geometry and own-state measurements is demonstrated by a three dimensional, constant-speed example. For game analysis with realistic vehicle dynamics, the technique of forced singular perturbations and a variable modeling approach is proposed. Accuracy of the analysis is evaluated by comparison with the numerical solution of a time-optimal, variable-speed 'game of two cars' in the horizontal plane

Hybridization of Nonlinear and Mixed-Integer Linear Programming for Aircraft Separation With Trajectory Recovery

International audienceThe approach presented in this article aims at finding a solution to the problem of conflict-free motion planning for multiple aircraft on the same flight level with trajectory recovery. One contribution of this work is to develop three consistent models, from a continuous-time representation to a discrete-time linear approximation. Each of these models guarantees separation at all times as well as trajectory recovery, but they are not equally difficult to solve. A new hybrid algorithm is thus developed in order to use the optimal solution of a mixed integer linear program as a starting point when solving a nonlinear formulation of the problem. The significance of this process is that it always finds a solution when the linear model is feasible while still taking into account the nonlinear nature of the problem. A test bed containing numerous data sets is then generated from three virtual scenarios. A comparative analysis with three different initialisations of the nonlinear optimisation validates the efficiency of the hybrid method