Nonnegative Orthant Research Articles

ℋ︁2 suboptimal estimation and control for nonnegative dynamical systems

AbstractLinear matrix inequalities (LMIs) provide a powerful design framework for linear control problems. In this paper, we use LMIs to develop ℋ︁2 (sub)optimal estimators and controllers for nonnegative dynamical systems. Specifically, we formulate a series of generalized eigenvalue problems subject to a set of LMI constraints for designing ℋ︁2 suboptimal estimators, static controllers, and dynamic controllers for nonnegative dynamical systems. The resulting ℋ︁2 suboptimal controllers guarantee that the closed‐loop plant system states remain in the nonnegative orthant of the state space. Finally, a numerical example is provided to demonstrate the efficacy of the proposed approach. Copyright © 2008 John Wiley & Sons, Ltd.

Some new existence, sensitivity and stability results for the nonlinear complementarity problem

In this work we study the nonlinear complementarity problem on the nonnegative orthant. This is done by approximating its equivalent variational-inequality-formulation by a sequence of variational inequalities with nested compact domains. This approach yields simultaneously existence, sensitivity, and stability results. By introducing new classes of functions and a suitable metric for performing the approximation, we provide bounds for the asymptotic set of the solution set and coercive existence results, which extend and generalize most of the existing ones from the literature. Such results are given in terms of some sets called coercive existence sets, which we also employ for obtaining new sensitivity and stability results. Topological properties of the solution-set-mapping and bounds for it are also established. Finally, we deal with the piecewise affine case.

Global stability analysis for SEIS models with n latent classes

We compute the basic reproduction ratio of a SEIS model with n classes of latent individuals and bilinear incidence. The system exhibits the traditional behaviour. We prove that if R(0) < or = 1, then the disease-free equilibrium is globally asymptotically stable on the nonnegative orthant and if R (0) > 1, an endemic equilibrium exists and is globally asymptotically stable on the positive orthant.

Memoryless Control to Drive States of Delayed Continuous-time Systems within the Nonnegative Orthant

The stabilization problem for the class of linear continuous-time systems with fixed but unknown delay is solved when the additional condition that the states are nonnegative is studied. In particular, the synthesis of state-feedback controllers is solved by giving necessary and sufficient conditions in terms of Linear Programs. The solution is also extended to stabilization by bounded control (including nonnegative control) and stabilization under uncertainty.

Neural Network Adaptive Control for Discrete-Time Nonlinear Nonnegative Dynamical Systems

Nonnegative and compartmental dynamical system models are derived from mass and energy balance considerations that involve dynamic states whose values are nonnegative. These models are widespread in engineering and life sciences, and they typically involve the exchange of nonnegative quantities between subsystems or compartments, wherein each compartment is assumed to be kinetically homogeneous. In this paper, we develop a neuroadaptive control framework for adaptive set-point regulation of discrete-time nonlinear uncertain nonnegative and compartmental systems. The proposed framework is Lyapunov-based and guarantees ultimate boundedness of the error signals corresponding to the physical system states and the neural network weighting gains. In addition, the neuroadaptive controller guarantees that the physical system states remain in the nonnegative orthant of the state space for nonnegative initial conditions.

Algorithmic copositivity detection by simplicial partition

We present new criteria for copositivity of a matrix, i.e., conditions which ensure that the quadratic form induced by the matrix is nonnegative over the nonnegative orthant. These criteria arise from the representation of the quadratic form in barycentric coordinates with respect to the standard simplex and simplicial partitions thereof. We show that, as the partition gets finer and finer, the conditions eventually capture all strictly copositive matrices. We propose an algorithmic implementation which considers several numerical aspects. As an application, we present results on the maximum clique problem. We also briefly discuss extensions of our approach to copositivity with respect to arbitrary polyhedral cones.

Generalizations of Khovanskiĭ's theorems on the growth of sumsets in Abelian semigroups

We show that if P is a lattice polytope in the nonnegative orthant of R k and χ is a coloring of the lattice points in the orthant such that the color χ ( a + b ) depends only on the colors χ ( a ) and χ ( b ) , then the number of colors of the lattice points in the dilation nP of P is for large n given by a polynomial (or, for rational P, by a quasipolynomial). This unifies a classical result of Ehrhart and Macdonald on lattice points in polytopes and a result of Khovanskiĭ on sumsets in semigroups. We also prove a strengthening of multivariate generalizations of Khovanskiĭ's theorem. Another result of Khovanskiĭ states that the size of the image of a finite set after n applications of mappings from a finite family of mutually commuting mappings is for large n a polynomial. We give a combinatorial proof of a multivariate generalization of this theorem.

Convergence of Map Seeking Circuits

The map-seeking circuit (MSC) is an explicit biologically-motivated computational mechanism which provides practical solution of problems in object recognition, image registration and stabilization, limb inverse-kinematics and other inverse problems which involve transformation discovery. We formulate this algorithm as discrete dynamical system on a set Δ=? ?=1 L Δ(?), where each Δ(?) is a compact subset of a nonnegative orthant of ? n , and show that for an open and dense set of initial conditions in Δ the corresponding solutions converge to either a vector with unique nonzero element in each Δ(?) or to a zero vector. The first result implies that the circuit finds a unique best mapping which relates reference pattern to a target pattern; the second result is interpreted as "no match found". These results verify numerically observed behavior in numerous practical applications.

Multiplicative Updates for Nonnegative Quadratic Programming

Many problems in neural computation and statistical learning involve optimizations with nonnegativity constraints. In this article, we study convex problems in quadratic programming where the optimization is confined to an axis-aligned region in the nonnegative orthant. For these problems, we derive multiplicative updates that improve the value of the objective function at each iteration and converge monotonically to the global minimum. The updates have a simple closed form and do not involve any heuristics or free parameters that must be tuned to ensure convergence. Despite their simplicity, they differ strikingly in form from other multiplicative updates used in machine learning. We provide complete proofs of convergence for these updates and describe their application to problems in signal processing and pattern recognition.

Generalizations of Khovanskiĭ's theorem on growth of sumsets in abelian semigroups

We show that if P is a lattice polytope in the nonnegative orthant of Rk and χ is a coloring of the lattice points in the orthant such that the color χ(a+b) depends only on the colors χ(a) and χ(b), then the number of colors used on the lattice points lying in nP is for large n given by a polynomial (or, for rational P, by a quasipolynomial). This unifies a classical result of Ehrhart on lattice points in polytopes and a result of Khovanskiĭ on sumsets in semigroups. We also prove a strengthening of multivariate generalizations of Khovanskiĭ's result.

Neural Network Adaptive Output Feedback Control for Intensive Care Unit Sedation and Intraoperative Anesthesia

The potential applications of neural adaptive control for pharmacology, in general, and anesthesia and critical care unit medicine, in particular, are clearly apparent. Specifically, monitoring and controlling the depth of anesthesia in surgery is of particular importance. Nonnegative and compartmental models provide a broad framework for biological and physiological systems, including clinical pharmacology, and are well suited for developing models for closed-loop control of drug administration. In this paper, we develop a neural adaptive output feedback control framework for nonlinear uncertain nonnegative and compartmental systems with nonnegative control inputs. The proposed framework is Lyapunov-based and guarantees ultimate boundedness of the error signals. In addition, the neural adaptive controller guarantees that the physical system states remain in the nonnegative orthant of the state space. Finally, the proposed approach is used to control the infusion of the anesthetic drug propofol for maintaining a desired constant level of depth of anesthesia for noncardiac surgery.

Dynamic graphs

Dynamic graphs are defined in a linear space as a one-parameter group of transformations of the graph space into itself. Stability of equilibrium graphs is formulated in the sense of Lyapunov to study motions of positive graphs in the nonnegative orthant of the graph space. Relying on the isomorphism of graphs and adjacency matrices, a new concept of dynamic connective stability of complex systems is introduced. A dynamic interaction coordinator is added to complex interconnected system to ensure that the desired level of interconnections between subsystems is preserved as a connectively stable equilibrium of the overall system despite uncertain structural perturbations. It is shown how the coordinator can be designed to adaptively adjust the interconnection levels in order to assign a prescribed state of the complex multi-agent system as a stable equilibrium point. The equilibrium assignment is achieved by the action of the coordinator which solves an optimization problem involving a two-time-scale system; the coordinator action is slow compared to the fast dynamics of the agents. Polytopic connective stability of the multi-agent systems with a coordinator is established by the concept of vector Lyapunov functions and the theory of M -matrices.

Nonnegative Moore–Penrose inverses of Gram operators

This paper is concerned with necessary and sufficient conditions for the nonnegativity of Moore–Penrose inverses of Gram operators between real Hilbert spaces. These conditions include statements on acuteness (or obtuseness) of certain closed convex cones. The main result generalizes a well known result for inverses in the finite dimensional case over the nonnegative orthant to Moore–Penrose inverses in (possibly) infinite dimensional Hilbert spaces over any general closed convex cone.

An Extension of Sums of Squares Relaxations to Polynomial Optimization Problems Over Symmetric Cones

This paper is based on a recent work by Kojima which extended sums of squares relaxations of polynomial optimization problems to polynomial semidefinite programs. Let $$\varepsilon$$ and $$\varepsilon_{+}$$ be a finite dimensional real vector space and a symmetric cone embedded in $$\varepsilon$$; examples of $$\varepsilon$$ and $$\varepsilon_{+}$$ include a pair of the N-dimensional Euclidean space and its nonnegative orthant, a pair of the N-dimensional Euclidean space and N-dimensional second-order cones, and a pair of the space of m × m real symmetric (or complex Hermitian) matrices and the cone of their positive semidefinite matrices. Sums of squares relaxations are further extended to a polynomial optimization problem over $$\varepsilon_{+}$$, i.e., a minimization of a real valued polynomial a(x) in the n-dimensional real variable vector x over a compact feasible region $$\{ {\bf x} : b({\bf x}) \in \varepsilon_{+}\}$$, where b(x) denotes an $$\varepsilon$$- valued polynomial in x. It is shown under a certain moderate assumption on the $$\varepsilon$$-valued polynomial b(x) that optimal values of a sequence of sums of squares relaxations of the problem, which are converted into a sequence of semidefinite programs when they are numerically solved, converge to the optimal value of the problem.

Asymptotic analysis, existence and sensitivity results for a class of multivalued complementarity problems

In this work we study the multivalued complementarity problem on the non-negative orthant. This is carried out by describing the asymptotic behavior of the sequence of approximate solutions to its multivalued variational inequality formulation. By introducing new classes of multifunctions we provide several existence (possibly allowing unbounded solution set), stability as well as sensitivity results which extend and generalize most of the existing ones in the literature. We also present some kind of robustness results regarding existence of solution with respect to certain perturbations. Topological properties of the solution-set multifunction are established and some notions of approximable multifunctions are also discussed. In addition, some estimates for the solution set and its asymptotic cone are derived, as well as the existence of solutions for perturbed problems is studied.

Delayed feedback control for a chemostat model

In this paper, we discuss asymptotic properties and numerical simulations of a chemostat model with delayed feedback control. A chemostat model with two organisms can be made coexistent by feedback control of the dilution rate which depends affinely on the concentrations of two organisms [P. De Leenher, H.L. Smith, Feedback control for chemostat models, J. Math. Biol. 46 (2003) 48]. Then the coexistence takes its simplest form; the equilibrium point in the non-negative orthant is globally asymptotically stable. We show that stability of the equilibrium point is changed by ‘time-delay’ caused in controlling the dilution rate after measuring the concentrations of two organisms.

Computer-Algebra Implementation of the Least-Squares Method on the Nonnegative Orthant

Maple procedures for solving the so-called NNLS (Nonnegative Least Squares) problem are described. The NNLS problem is to minimize $$\left\| {Ex - f} \right\|_2$$ subject to $$x \geqslant 0.$$ The solution of an NNLS problem is the crucial step of the conventional algorithm for solving linear least-squares problems with linear inequality constraints. Bibliography: 5 titles.

LMI Approximations for Cones of Positive Semidefinite Forms

An interesting recent trend in optimization is the application of semidefinite programming techniques to new classes of optimization problems. In particular, this trend has been successful in showing that under suitable circumstances, polynomial optimization problems can be approximated via a sequence of semidefinite programs. Similar ideas apply to conic optimization over the cone of copositive matrices and to certain optimization problems involving random variables with some known moment information. We bring together several of these approximation results by studying the approximability of cones of positive semidefinite forms (homogeneous polynomials). Our approach enables us to extend the existing methodology to new approximation schemes. In particular, we derive a novel approximation to the cone of copositive forms, that is, the cone of forms that are positive semidefinite over the nonnegative orthant. The format of our construction can be extended to forms that are positive semidefinite over more general conic domains. We also construct polyhedral approximations to cones of positive semidefinite forms over a polyhedral domain. This opens the possibility of using linear programming technology in optimization problems over these cones.

How Strong Is Strong Regularity?

We compare the notions of regularity and strong regularity of interval matrices. For an n × n interval matrix A we construct 2 n open convex cones, all of them lying in the interior of the nonnegative orthant. It is shown that regularity of A is characterized by nonemptiness of all these cones, whereas strong regularity is characterized by nonemptiness of their intersection.

Adaptive control for nonlinear compartmental dynamical systems with applications to clinical pharmacology

There are significant potential clinical applications of adaptive control for pharmacology in general, and anesthesia and critical care unit medicine in particular. Specifically, monitoring and controlling the levels of consciousness in surgery are of particular importance. Nonnegative and compartmental models provide a broad framework for biological and physiological systems, including clinical pharmacology, and are well suited for developing models for closed-loop control of drug administration. In this paper, we develop a direct adaptive control framework for nonlinear uncertain nonnegative and compartmental systems with nonnegative control inputs. The proposed framework is Lyapunov-based and guarantees partial asymptotic set-point regulation, that is, asymptotic set-point regulation with respect to part of the closed-loop system states associated with the plant. In addition, the adaptive controller guarantees that the physical system states remain in the nonnegative orthant of the state space. Finally, a numerical example involving the infusion of the anesthetic drug propofol for maintaining a desired constant level of consciousness for noncardiac surgery is provided to demonstrate implementation of the proposed approach.