Bounding Set Research Articles

Guaranteed steady state bounds for uncertain (bio-)chemical processes using infeasibility certificates

Analysis and safety considerations of chemical and biological processes require complete knowledge of the set of all feasible steady states. Nonlinearities, uncertain parameters, and discrete variables complicate the task of obtaining this set. In this paper, the problem of outer-approximating the region of feasible steady states, for processes described by uncertain nonlinear differential algebraic equations including discrete variables and discrete changes in the dynamics, is addressed. The calculation of the outer bounds is based on a relaxed version of the corresponding feasibility problem. It uses the Lagrange dual problem to obtain certificates for regions in state space not containing steady states. These infeasibility certificates can be computed efficiently by solving a semidefinite program, rendering the calculation of an outer bounding set computationally feasible. The derived method guarantees globally valid outer bounds for the feasible steady states. The method is exemplified by the analysis of a simple chemical reactor showing parametric uncertainties and large variability due to the appearance of bifurcations characterising the ignition and extinction of a reaction.

A Revised Method of Attainable Region Construction Utilizing Rotated Bounding Hyperplanes

An improved method of candidate attainable region (AR) construction, based on an existing bounding hyperplanes approach, is presented. The method uses a plane rotation about existing extreme points, rather than a plane translation, to eliminate unachievable regions from an initial bounding set. The algorithm is shown to be faster, but has currently only been implemented for two-dimensional constructions and has been extended to include the construction of candidate ARs that involve nonisothermal kinetics in concentration and concentration−time space.

Bounding set calculation for neural network-based output feedback adaptive control systems

Evaluating the bounding set of dynamic systems subject to direct neural-adaptive control is a critical issue in applications where the control system must undergo a rigorous verification process in order to comply with certification standards. In this paper, the boundedness problem is addressed for a comprehensive class of uncertain dynamic systems. Several common but unnecessary approximations that are typically performed to simplify the Lyapunov analysis have been avoided in this effort. This leads to a more accurate and general formulation of the bounding set for the overall closed loop system. The conditions under which boundedness can be guaranteed are carefully analyzed; additionally, the interactions between the control design parameters, the ‘Strictly Positive Realness’ condition, and the shape and dimensions of the bounding set are discussed. Finally, an example is presented in which the bounding set is calculated for the neuro-adaptive control of an F/A-18 aircraft, along with a numerical study to evaluate the effect of several design parameters.

Natural frequency intervals for vibrating systems with polytopic uncertainty

We consider the problem of locating the natural frequencies of uncertain systems whose describing matrices are functions of an unknown parameter vector which is included in an assigned bounding set. We face what we call the weak frequency interval detection problem of determining the smallest interval which includes all possible frequencies. We show that if the system matrices depend affinely on the parameter vector, whose bounding set is a compact polyhedron, then this problem requires the solution of a finite number of eigenvalue problems associated with the vertices of such a polyhedron. Unfortunately, detecting the intervals associated with all the natural frequencies (strong frequency interval detection problem), cannot rely on this property, so that one must resort to Monte Carlo methods or numerical optimization to find them. We show that the strong version is solvable “exploring the vertices only” under some stronger assumptions. In the case in which the uncertainty bounding set is not defined by linear inequalities, not even the extremal frequencies can be associated with the vertices of the admissibility domain. Then again, numerical approach is necessary unless we accept to merge the original system in a larger one of an “affine nature”. Finally, we present as an application the study of structures with uncertain mass distribution.

Guaranteed Steady-State Bounds for Uncertain Chemical Processes

Abstract Abstract Analysis and safety considerations of chemical and biological processes frequently require an outer approximation of the set of all feasible steady-states. Nonlinearities, uncertain parameters, and discrete variables complicate the calculation of guaranteed outer bounds. In this paper, the problem of outer-approximating the region of feasible steady-states, for processes described by uncertain nonlinear differential algebraic equations including discrete variables and discrete changes in the dynamics, is adressed. The calculation of the outer bounding sets is based on a relaxed version of the corresponding feasibility problem. It uses the Lagrange dual problem to obtain certificates for regions in state space not containing steady-states. These infeasibility certificates can be computed efficiently by solving a semidefinite program, rendering the calculation of the outer bounding set computationally feasible. The derived method guarantees globally valid outer bounds for the steady-states of nonlinear processes described by differential equations. It allows to consider discrete variables, as well as switching system dynamics. The method is exemplified by the analysis of a simple chemical reactor showing parametric uncertainties and large variability due to the appearance of bifurcations characterising the ignition and extinction of a reaction.

On the solution of the trajectory survival problem for a nonlinear dynamical system

Nonlinear control systems possessing the flatness property are encountered in many applied mathematical models. In this paper, a trajectory survival problem is considered for a specific nonlinear system that possesses the above property. A method based on the properties of the system is proposed for constructing a control that solves the trajectory survival problem when the controlled object moves to the goal set within a bounding set containing an obstacle. Results of numerical calculations of the control and the trajectory of a system with a given initial position are presented.

Stability analysis of uncertain genetic sum regulatory networks

This paper addresses the problem of establishing robust stability of uncertain genetic networks with sum regulatory functions. Specifically, we first consider uncertain genetic networks where the regulation occurs at the transcriptional level, and we derive a sufficient condition for robust stability by introducing a bounding set of the uncertain nonlinearity. We hence show that this condition can be formulated as a convex optimization through polynomial Lyapunov functions and polynomial descriptions of the bounding set by exploiting the square matricial representation (SMR) of polynomials which allows to establish whether a polynomial is a sum of squares (SOS) via a linear matrix inequality (LMI). Then, we propose a method for computing a family of bounding sets by means of convex optimizations. It is worthwhile to remark that these results are derived in spite of the fact that the variable equilibrium point cannot be computed as being the solution of a system of parameter-dependent nonlinear equations, and is hence unknown. Lastly, the proposed approach is extended to models where the regulation occurs at different levels and both mRNA and protein dynamics are nonlinear.

An LMI approach for robust stability of genetic networks

This paper addresses the problem of establishing robust stability of uncertain genetic networks with SUM regulatory functions. For these networks we derive a sufficient condition for robust stability by introducing a bounding set of the uncertain nonlinearity, and we show that this condition can be formulated as a linear matrix inequality (LMI) optimization obtained via the square matricial representation (SMR) of polynomials by adopting polynomial Lyapunov functions and polynomial descriptions of the bounding set. Then, we propose a method for computing a family of bounding sets by means of convex optimizations. It is worthwhile to remark that these results are derived in spite of the fact that the variable equilibrium point cannot be computed being the solution of a system of parameter-dependent nonlinear equations, and is hence unknown.

Variational sphere set approximation for solid objects

We approximate a solid object represented as a triangle mesh by a bounding set of spheres having minimal summed volume outside the object. We show how outside volume for a single sphere can be computed using a simple integration over the object’s triangles. We then minimize the total outside volume over all spheres in the set using a variant of iterative Lloyd clustering that splits the mesh points into sets and bounds each with an outside volume-minimizing sphere. The resulting sphere sets are tighter than those of previous methods. In experiments comparing against a state-of-the-art alternative (adaptive medial axis), our method often requires half as many spheres, or fewer, to obtain the same error, under a variety of error metrics including total outside volume, shadowing fidelity, and proximity measurement.

Reliable Localization Using Set-Valued Nonlinear Filters

We propose a novel methodology for reliable localization of an autonomous mobile robot navigating in an unstructured environment using noisy absolute measurements from its exteroceptive sensors. A new deterministic filtering technique is introduced, which is based on the recursive computation of a bounding set that is guaranteed to contain the true state of the system, despite process and observation noise, and taking into explicit consideration uncertainties due to the linearization error. The proposed set-valued nonlinear filter relies on a two-step prediction-correction structure, with each step requiring the solution of a particular convex optimization problem. The method is illustrated by simulation on a localization problem for a nonholonomic rover, and it is compared with the standard extended Kalman filter approach.

Bounds for the effective constitutive relation of a nonlinear composite

For a nonlinear composite, a bound on its effective energy density does not induce a corresponding bound on its constitutive relation, because differentiating a bound on a function does not automatically bound its derivative. In this work, a method introduced by Milton and Serkov for bounding directly the constitutive relation is refined by employing a linear comparison material, in a way similar to that employed by the present authors to obtain bounds of ‘Hashin–Shtrikma’ type for the effective energy of a nonlinear composite. The original Milton–Serkov approach produces bounds with a close relationship to the classical energy bounds of Voigt and Reuss type. The bounds produced in the present implementation are closely related to bounds of Hashin–Shtrikman type for the composite. Indeed, in the case of a linear composite, the present method delivers the Hashin–Shtrikman bounds in their generalized form, valid for any two–point statistics. It is demonstrated for nonlinear examples that the approximate constitutive relation that is obtained by differentiating the energy bound can be on the boundary of the bounding set for the exact constitutive relation, but a simple counterexample is presented to show that this is not always the case.

Localization of a Mobile Robot Using Relative Bearing Measurements

In this paper, the problem of recursive robot localization based on relative bearing measurements is considered, where unknown but bounded measurement uncertainties are assumed. A common approach is to approximate the resulting set of feasible states by simple-shaped bounding sets such as, e.g., axis-aligned boxes, and calculate the optimal parameters of this approximation based on the measurements and prior knowledge. In the novel approach presented here, a nonlinear transformation of the measurement equation into a higher dimensional space is performed. This yields a tight, possibly complex-shaped, bounding set in a closed-form representation whose parameters can be determined analytically for the measurement step. It is shown that the new bound is superior to commonly used outer bounds.

A unified feasible direction interior approach to the minimization of linear and general objective functions subject to linear constraints

A modification of Snyman's interior feasible direction method for linear programming is proposed and the method is also extended to problems where the objective function is non-linear. The method attempts to identify the optimal bounding set of active constraints. In the modified algorithm the successive interior steps in the identifying cycle are no longer constrained to be in the place of constant objective function value, but are computed to ensure improvement in the objective function for any non-zero step taken within the cycle. The method is also extended to non-linear objective functions by allowing for line searches within the interior and along bounding hypersurfaces. A formal unified algorithm is presented and the method is illustrated by its successful application to a number of simple problems from different categories.

Set simulations for quadratic systems

In this paper, we study the problem of propagating in time a bounding set for the state of a class of nonlinear quadratic systems. The sequence of bounding sets is called the set simulation of the system, and conveys useful information about the stability and qualitative behavior of the possible time responses of the system. Numerically, efficient recursive algorithms are presented for the specific cases when the bounding sets are orthotopes or ellipsoids.

Robust Stabilization of Uncertain Continuous-Time and Discrete-Time Systems

A method for designing a feedback control law to stabilize a class of uncertain linear both continuous-time and discrete-time systems is presented. The systems under consideration contain uncertain parameters whose values are known only to within a given compact bounding set. The state-space approach based on the solution of the Riccati equation is taken. The new properties of the non-standard Riccati equation are derived. These properties have allowed to develop algorithm of calculation of weighting matrices of output and external disturbance vectors for synthesis of LQG or H∞-regulators. Such regulators (designed for the nominal system) provide stability of the closed systems for the class of objects with uncertain parameters.

Robustness Analysis for Nonlinear Systems With Interval System Matrix

This paper deals with the estimation of a robust domain of attraction for nonlinear systems with unstructured uncertainties in the system matrix. For this goal a piecewise constant parameter-dependent Lyapunov function is used. This type of a Lyapunov function is based on dividing the uncertainty bounding set into a finite number of partitions, yielding a common Lyapunov function for each partition.

Piecewise constant Lyapunov functions for the construction of robust regions of attraction

This paper deals with the estimation of a robust domain of attraction for nonlinear systems with structured uncertainties. For this goal a piecewise constant parameter-dependent Lyapunov function is used. This type of a Lyapunov function is based on dividing the uncertainty bounding set into a finite number of partitions, yielding a common Lyapnov function for each partition. A numerical example is included. Copyright © 1999 John Wiley & Sons, Ltd.

Robust control with a terminal state constraint

This paper introduces a problem of robust control with a terminal state constraint. This problem involves finding an output feedback controller to steer the initial state of a given system into an ellipsoidal bounding set in a specified time. The underlying system in this problem is a time-varying uncertain linear system in which the uncertainty satisfies an integral quadratic constraint. Necessary and sufficient conditions are given for the solution to this problem in terms of a pair of parameter-dependent Riccati differential equations.

Memoryless stabilization of uncertain linear systems including time-varying state delays

The problem of stabilizing a class of uncertain time-delay systems via memoryless linear feedback is examined. The systems under consideration are linear systems with time-varying state delays. They also contain uncertain parameters (possibly time-varying) whose values are known only to within a prescribed compact bounding set. The main contribution given is to enlarge the class of time-delay systems for which one can construct a stabilizing memoryless linear feedback controller. Within this framework, a novel notion of robust memoryless stabilizability is first introduced via the method of Lyapunov functionals. Then a sufficient condition for the stabilizability is proposed. It is shown that solvability of a parameterized Riccati equation can be used to determine whether the time-delay system satisfies the sufficient condition. If there exists a positive definite symmetric solution satisfying the Riccati equation, a suitable memoryless linear feedback law can be derived.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Robust linear quadratic optimal control for systems with linear uncertainties

Abstract This paper presents two simple but effective algorithms for selecting the weighting matrices needed in designing linear quadratic optimal control for systems with linear uncertainties. By utilizing the Lyapunov stability criterion, it is shown that the optimal state feedback control law designed for the nominal system can stabilize the uncertain system, provided the uncertainties satisfy the so-called matching conditions and are within a given bounding set. The methods are tested for three examples, and the results show that the current methods have wider application ranges than some approaches treating similar problems in the literature. Furthermore, this paper considers the case in which matching conditions are not exactly satisfied. A measure of mismatch is adopted from the literature, and a threshold on the mismatch is found to guarantee that the optimal controller designed for the system without the mismatching uncertainties is effective when the mismatching part is added. Finally, a method for adjusting the weighting matrix of the state variables in the cost function is suggested to facilitate the computation of the threshold value