Selective Input Adaptation in Parametric Optimal Control Problems involving Terminal Constraints

This paper is concerned with input adaptation in dynamic processes in order to guarantee feasible and optimal operation despite the presence of uncertainty. For optimal control problems having terminal constraints, two sets of directions can be distinguished in the input function space: the so-called sensitivity-seeking directions, along which a small input variation does not affect the terminal constraints, and the complementary constraint-seeking directions, along which a variation does affect the terminal constraints. Two selective input adaptation scenarios are thus possible, namely, adaptation along each set of input directions. This paper proves the important result that the cost variation due to the adaptation along the sensitivity-seeking directions is typically smaller than that due to the adaptation along the constraint-seeking directions

Parametric Sensitivity of Path-Constrained Optimal Control: Towards Selective Input Adaptation

In the context of dynamic optimization, plant variations necessitate adaptation of the input profiles in order to guarantee both feasible and optimal operation. For those problems having path constraints, two sets of directions can be distinguished in the input space at each time instant: the so-called sensitivity-seeking directions, along which a small input variation does not affect the active path constraints; the complementary constraint-seeking directions, along which a variation affects the path constraints. Hence, three selective input adaptations are possible, namely, adaptation along each set of input directions and adaptation of the switching times be- tween arcs. This paper considers parametric variations around a nominal optimal solution and quantifies the influence of these variations on each type of input adaptation

Unified Framework for the Propagation of Continuous-Time Enclosures for Parametric Nonlinear ODEs

Abstract This paper presents a framework for constructing and analyzing enclosures of the reachable set of nonlinear ordinary differential equations (ODEs) using continuous-time setpropagation methods. The focus is on convex enclosures that can be characterized in terms of their support functions. A generalized differential inequality is introduced, whose solutions describe such support functions for a convex enclosure of the reachable set under mild conditions. It is shown that existing continuous-time bounding methods that are based on standard differential inequalities or ellipsoidal set propagation techniques can be recovered as special cases of this generalized differential inequality. A way of extending this approach for the construction of nonconvex enclosures is also described, which relies on Taylor models with convex remainder bounds. This unifying framework provides a means for analyzing the convergence properties of continuous-time enclosure methods. The enclosure techniques and convergence results are illustrated with numerical case studies throughout the paper, including a six-state dynamic model of anaerobic digestion

Exploration of signaling cycles using dynamic optimization

One of the basic characteristics of every living system is the ability to respond to extracellular signals. This is carried out through a limited number of protein-based signaling networks, whose function is not based only on simple transmission of the received signals, but incorporates the processing, encoding and integration of both, external and internal signals. The results than lead to different changes in gene expression, regulate cell growth, differentiation, embryo development, and stress responses in mammalian cells, whereas the malfunction is in correlation with diseases. Commonly observed instance of signal transduction through a series of protein kinase reactions are the kinases of the mitogen activated protein kinase (MAPK) cascades. These pathways are found in almost all eukaryotes and play an important role in controlling different cellular processes, including fundamental functions. In order to understand better the nature of this regulation and to gain greater insight into the mechanisms that determine the function of cells, MAPK cascades have been intensively studied using mathematical modeling and computational simulations. The primary aim is to faithfully describe the system and to be able to predict the system behavior. Synergistically with experimental analysis, reported observations have identified properties of these pathways, such as rapid induction, noise resistance, amplification capability, threshold induction mechanism etc. Here, we investigate one class of approaches for analyzing the relationship between network structure and functional behavior and the overall idea involves applying optimization techniques. By manipulating the desired functional behavior and by monitoring the corresponding parameter values, one can learn how model parameters and functions are related, and then be in a position to discover new design principles. The primary motivation was to explore if there is any trade-off while promoting simultaneously large amplification and fast signal propagation. We identified the competing parameters in the linear tricyclic cascade and their values for the optimal design for minimal response times and given amplification. We also incorporated “ultrasensitivity”, in order to analyze interplay between this steady-state property and dynamic behavior of the system. Special emphasis is placed on the robustness of the resulting tricyclic cascades in the face of variations in kinase and phosphatase concentration ratios

Sensitivity analysis of uncertain dynamic systems using set-valued integration

We present an extension of set-valued integration to enable efficient sensitivity analysis of parameter-dependent ordinary differential equation (ODE) systems, using both the forward and adjoint methods. The focus is on continuous-time set-valued integration, whereby auxiliary ODE systems are derived whose solutions describe high-order inclusions of the parametric trajectories in the form of polynomial models. The forward and adjoint auxiliary ODE systems treat the parameterization error of the original differential variables as a time-varying uncertainty, and propagate the sensitivity bounds forward and backward in time, respectively. This construction enables building on the sensitivity analysis capabilities of state-of-the-art solvers, such as CVODES in the SUNDIALS suite. Several numerical case studies are presented to assess the performance and accuracy of these set-valued sensitivity integrators

Branch-and-lift algorithm for deterministic global optimization in nonlinear optimal control

This paper presents a branch-and-lift algorithm for solving optimal control problems with smooth nonlinear dynamics and potentially nonconvex objective and constraint functionals to guaranteed global optimality. This algorithm features a direct sequential method and builds upon a generic, spatial branch-and-bound algorithm. A new operation, called lifting, is introduced, which refines the control parameterization via a Gram-Schmidt orthogonalization process, while simultaneously eliminating control subregions that are either infeasible or that provably cannot contain any global optima. Conditions are given under which the image of the control parameterization error in the state space contracts exponentially as the parameterization order is increased, thereby making the lifting operation efficient. A computational technique based on ellipsoidal calculus is also developed that satisfies these conditions. The practical applicability of branch-and-lift is illustrated in a numerical example. © 2013 Springer Science+Business Media New York