Nonlinear Feasibility Problem Research Articles

Bookings in the European gas market: characterisation of feasibility and computational complexity results

As a consequence of the liberalisation of the European gas market in the last decades, gas trading and transport have been decoupled. At the core of this decoupling are so-called bookings and nominations. Bookings are special long-term capacity right contracts that guarantee that a specified amount of gas can be supplied or withdrawn at certain entry or exit nodes of the network. These supplies and withdrawals are nominated at the day-ahead. The special property of bookings then is that they need to be feasible, i.e., every nomination that complies with the given bookings can be transported. While checking the feasibility of a nomination can typically be done by solving a mixed-integer nonlinear feasibility problem, the verification of feasibility of a set of bookings is much harder. The reason is the robust nature of feasibility of bookings-namely that for a set of bookings to be feasible, all compliant nominations, i.e., infinitely many, need to be checked for feasibility. In this paper, we consider the question of how to verify the feasibility of given bookings for a number of special cases. For our physics model we impose a steady-state potential-based flow model and disregard controllable network elements. For this case we derive a characterisation of feasible bookings, which is then used to show that the problem is in coNP for the general case but can be solved in polynomial time for linear potential-based flow models. Moreover, we present a dynamic programming approach for deciding the feasibility of a booking in tree-shaped networks even for nonlinear flow models. It turns out that the hardness of the problem mainly depends on the combination of the chosen physics model as well as the specific network structure under consideration. Thus, we give an overview over all settings for which the hardness of the problem is known and finally present a list of open problems.

Mixed H∞ and passive control for singular systems with time delay via static output feedback

In this paper, the mixed H∞ and passive control problem is studied for singular systems with time delay. A sufficient condition is proposed to ensure the considered system to be regular, impulse-free and stable with mixed H∞ and passivity performance level. Then the criterion is adopted to design a static output feedback controller in terms of bilinear and linear matrix inequalities. An algorithm is established to cast the nonlinear feasibility problem into a sequential minimization problem. Finally, a numerical example is presented to show the effectiveness of the proposed methods.

Controller Verification and Parametrization Subject to Quantitative and Qualitative Requirements

Verifying if a process controller achieves a desired goal regarding safety specifications or performance is an important task in practice. This work presents a method for controller verification and parametrization of uncertain polynomial discrete-time systems with closed-loop requirements. Apart from quantitative constraints, also qualitative requirements, which are not directly linked to a specific time or amplitude, are considered. For formalizing these constraints, we employ linear temporal logic formulas and polynomial inequalities. Uncertainties can be considered in the input, the output, the initial conditions and the model parameters to account e.g. for model plant mismatch and noise, described as unknown-but-bounded variables. We combine the requirements and the system dynamics into a nonlinear feasibility problem to verify the controller and determine admissible controller parametrization. This problem is solved by relaxing it to a mixed-integer linear program. The relaxation procedure guarantees that the derived set of possible parametrization fulfill the quantitative and qualitative requirements of the closed-loop behavior despite the present uncertainties. The proposed method is illustrated by verifying and parametrizing a controller for a two tank system.

Validation of nominations in gas network optimization: models, methods, and solutions

In this article, we investigate methods to solve a fundamental task in gas transportation, namely the validation of nomination problem: given a gas transmission network consisting of passive pipelines and active, controllable elements and given an amount of gas at every entry and exit point of the network, find operational settings for all active elements such that there exists a network state meeting all physical, technical, and legal constraints. We describe a two-stage approach to solve the resulting complex and numerically difficult nonconvex mixedinteger nonlinear feasibility problem. The first phase consists of four distinct algorithms applying mixedinteger linear, mixedinteger nonlinear, nonlinear, and methods for complementarity constraints to compute possible settings for the discrete decisions. The second phase employs a precise continuous nonlinear programming model of the gas network. Using this setup, we are able to compute high-quality solutions to real-world industrial instances that are significantly larger than networks that have appeared in the mathematical programming literature before.

Inner Approximations of Consistent Parameter Sets by Constraint Inversion and Mixed-Integer Programming

Mathematical modeling has become an indispensable tool in the analysis, prediction and control of chemical and biological systems. However, the estimation of consistent model parametrizations and model invalidation are challenging tasks, but crucial for reliable model-based analysis and prediction. Set-based estimation methods are useful to determine guaranteed outer approximations of consistent parameter sets, i. e. consistent parametrizations are never excluded. However, these conservative outer approximating sets often include inconsistent parametrizations which lead to inconsistent models and hence possibly wrong model-based predictions. This paper proposes a set-based framework to determine inner approximations, i.e. the model is guaranteed consistent with measurement data for all parametrizations from this set. Our approach is based on the reformulation and inversion of measurement data constraints and by imposing nonlinear constraints on binary variables. The relaxation of the mixed-integer nonlinear feasibility problem into a mixed-integer linear feasibility problem allows the inner approximations to be determined efficiently. The applicability of this approach is demonstrated considering a nonlinear biochemical reaction network.

The Dimensional Synthesis of Planar Parallel Cable-Driven Mechanisms Through Convex Relaxations

The wrench-closure workspace (WCW) of parallel cable-driven mechanisms is the set of poses for which any wrench can be produced at the end-effector by a set of positive cable tensions. In this paper, we tackle the dimensional synthesis problem, namely, that of finding a geometry for a planar parallel cable-driven mechanism (PPCDM) whose WCW contains a prescribed workspace. To this end, we first recall a linear program to determine whether a given pose is inside or outside the WCW of a given PPCDM. The relaxation of this linear program over a box leads to a nonlinear feasibility problem that can only be satisfied when this box is completely inside the WCW. We extend this feasibility problem to find a PPCDM geometry whose WCW includes a given set of boxes. These boxes represent the prescribed workspace or an estimate thereof, which may be obtained through interval analysis. Finally, we introduce a nonlinear program through which the PPCDM geometry is changed while maximizing the scaling factor of the prescribed set of boxes. When the optimum scaling factor is greater or equal to one, the WCW of the resulting PPCDM contains the set of boxes.

Combining qualitative information and semi‐quantitative data for guaranteed invalidation of biochemical network models

SUMMARYMathematical models of complex biological networks are valuable to make predictions on system properties and to identify therapeutic targets. However, development, validation and analysis of predictive models is often hampered as absolute and quantitative measurement data are rarely available. Instead, the data are typically uncertain with respect to the measured variable or time, relational due to normalization, or data are given as conditional if‐then observations. Many common approaches for model development and validation cannot deal with such semi‐quantitative data and qualitative information. We present a framework for the guaranteed invalidation and parameter estimation of dynamical models using such data. For this purpose, the data are formally expressed by sets of equalities and inequalities containing binary variables. Then a mixed‐integer nonlinear feasibility problem is constructed, which is subsequently relaxed into a mixed‐integer linear program that can be solved efficiently. A model can be proved inconsistent, that is, invalid with the uncertain and semi‐quantitative/qualitative data, if the solution set of the mixed‐integer linear program is empty. To exemplify the approach, we analyze different models whether they can show adaptation to a step‐input. First, we invalidate all but one model and, second, derive outer‐bounds for those regions in the parameter space of the non‐invalidated model that contain parameterizations for which it is consistent with the data. Copyright © 2012 John Wiley & Sons, Ltd.

Complete Fault Diagnosis Of Uncertain Polynomial Systems

The increase in complexity in process control goes along with an increasing need for complete and guaranteed fault diagnosis. In this contribution, we propose a set-based method for complete fault diagnosis for polynomial systems. It is based on a reformulation of the diagnosis problem as a nonlinear feasibility problem, which is subsequently relaxed into a semidefinite program. This is done by exploiting the polynomial/rational structure of the discrete-time model equations. We assume the measurements of the output and the input to be available as uncertain, but bounded convex sets. The applicability of the method is demonstrated considering a two-tank system subject to multiple faults.

Flatness-Based Feedback Control of an Automotive Solenoid Valve

This brief considers the control of solenoid valve actuators used for gas exchange in internal combustion engines. Although solenoid valves offer performance benefits over traditional camshaft-based valve systems, maintaining low impact velocity is a critical performance requirement. Flatness provides a convenient framework for meeting a number of performance specifications on the valve's end motion. The proposed control design incorporates voltage constraints, nonlinear magnetic effects, and various motion planning requirements. A flat output acts as a design parameter and is parameterized with a spline basis. A nonlinear feasibility problem is solved to obtain optimal spline coefficients such that performance requirements are met. The resulting flat output provides an open-loop control which is augmented with feedback so that a linear stable tracking error system results. The proposed control scheme is demonstrated in simulation and on an experimental testbed. The performance of a proportional-integral controller is compared experimentally to the flatness-based method