Positively Invariant Research Articles

Mathematical analysis of a COVID-19 model with double dose vaccination in Bangladesh

COVID-19 is an infectious disease that kills millions of people each year and it is a major public health problem around the globe. The current COVID-19 situation is still now concerning, though the vaccination program is running. In this study, we considered a COVID-19 model with a double-dose vaccination strategy to control the current outbreak situation in Bangladesh. The fundamental qualitative analysis of this mathematical model has been performed. The conditions of positive invariance, boundedness with suitable initial conditions were analyzed. We have estimated the basic reproduction number (R0) for disease transmission and determined that our model contains two equilibrium points: the disease-free equilibrium and a disease-endemic equilibrium. We used the Routh-Hurwitz criteria to determine the stability of the equilibria. The disease will be eradicated from the community if R0 < 1, otherwise the disease persists in the population. To support the qualitative analysis of our model, we performed numerical simulations using MATLAB routine and estimated model parameters. Sensitivity analysis is used to explore the association for Mild and Critical cases concerning the corresponding model parameters. We observed that the most significant parameter to spread the virus is the transmission rate. The numerical simulations showed that a full dose vaccination program significantly reduces the mild and critical cases and has potential impact to eradicate the virus from the community. The information that we generated from our analysis may help the public health professionals to impose the best strategy effectively to control the outbreak situation of the virus in Bangladesh.

Tractable robust MPC design based on nominal predictions

Many popular approaches in the field of robust model predictive control (MPC) are based on nominal predictions. This paper presents a novel formulation of this class of controller with proven input-to-state stability and robust constraint satisfaction. Its advantages are: (i) the design of its main ingredients are tractable for medium to large-sized systems, (ii) the terminal set does not need to be robust with respect to all the possible system uncertainties, but only for a reduced set that can be made arbitrarily small, thus facilitating its design and implementation, (iii) under certain conditions the terminal set can be taken as a positive invariant set of the nominal system, allowing us to use a terminal equality constraint, which facilitates its application to large-scale systems, and (iv) the complexity of its optimization problem is comparable to the non-robust MPC variant. We show numerical closed-loop results of its application to a multivariable chemical plant and compare it against other robust MPC formulations.

Characterizing positively invariant sets: Inductive and topological methods

We present two characterizations of positive invariance of sets under the flow of systems of ordinary differential equations. The first characterization uses inward sets which intuitively collect those points from which the flow evolves within the set for a short period of time, whereas the second characterization uses the notion of exit sets, which intuitively collect those points from which the flow immediately leaves the set. Our proofs emphasize the use of the real induction principle as a generic and unifying proof technique that captures the essence of the formal reasoning justifying our results and provides cleaner alternative proofs of known results. The two characterizations presented in this article, while essentially equivalent, lead to two rather different decision procedures (termed respectively LZZ and ESE) for checking whether a given semi-algebraic set is positively invariant under the flow of a system of polynomial ordinary differential equations. The procedure LZZ improves upon the original work by Liu, Zhan and Zhao (Liu et al., 2011). The procedure ESE, introduced in this article, works by splitting the problem, in a principled way, into simpler sub-problems that are easier to check, and is shown to exhibit substantially better performance compared to LZZ on problems featuring semi-algebraic sets described by formulas with non-trivial Boolean structure.

Artificial Pancreas: Avoiding Hyperglycemia and Hypoglycemia for Type One Diabetes

The objective of this work is to automatically regulate glycemia of Type 1 Diabetes Mellitus (T1DM) avoiding both hyperglycemia and hypoglycemia risks. A positive state feedback controller was designed previously to regulate Blood Glucose Concentration (BGC) in the fasting phase maintaining the system in the positively invariant set (PIS). The drawback of this positive controller is that when tested in the postprandial phase it couldn’t avoid hyperglycemia. Therefore, in this work, the positive state feedback controller was developed to avoid both hypoglycemia and hyperglycemia maintaining the system inside the PIS. Meal disturbance is estimated by a sliding mode perturbation observer to be included in the control law. Such that meal effect is canceled early enough preventing glycemia from raising to hyperglycemia, but the positivity of the new controller isn’t guaranteed. Therefore, a hybrid controller is designed to switch to the previous positive controller whenever the new controller has a negative action. A positive control is essential in this problem since the control input (insulin) can only be infused and it cannot be taken back from the bloodstream in case of any overdoses. The hybrid positive controller is tested in silico on five virtual T1DM patients. The results shown that the average percentage of time for glycemia over 180mg/dl (3.6%), normal range (80-120mg/dl) (78.2%), and below (80mg/dl) (0%) from overall simulation time. In conclusion, the hybrid positive control law succeeded to maintain the system inside the PIS avoiding hypoglycemia and preventing hyperglycemia keeping BGC in normal range rejecting meal disturbance.

Delay-Dependent Polyhedral Invariant Sets for Continuous-Time Linear Systems

In this paper, we analyse the positive invariance of polyhedral sets with respect to the trajectories of linear dynamical systems represented in terms of delay-differential equations. An appropriate model transformation is employed, together with a matrix parametrization which allows exploiting system's structure by decoupling delay-dependent modes from delay-independent ones. Then, positive invariance conditions are obtained with explicit dependence on the delay, seen as a parameter. Connections and equivalence of positive invariance between the original and the transformed systems are established. The derived conditions are used to tackle the problem of computing a feedback control law which makes a given polyhedron positively invariant with respect to an input-delayed linear system. Illustrative examples complete the presentation.

Coverage control of mobile sensor networks with directional sensing.

Control design of mobile sensors for coverage problem is addressed in this paper. The mobile sensors have non-linear dynamics and directional sensing properties which mean the sensing performance is also affected by the pointing directions of the sensors. Different from the standard optimal coverage problem where sensors are assumed to be omni-directional ones, orientation angles of the directional sensors should also be controlled, other than the position control, to achieve the coverage purpose. Considering also the non-linear dynamics of the mobile sensors, new control methodology is necessarily developed for the coverage problem with directional sensors. In the approach proposed, an innovative gradient based non-smooth motion controller is designed for the mobile sensors with unicycle dynamics. With the proposed controllers, the states of sensors will always stay in an positive invariant set where the gradient of the performance valuation function is well-defined if they are initialized within this set. Moreover, the sensors' states are proved to converge to some critical point where the gradient is zero. Simulation results are provided to illustrate the performance of the proposed coverage control strategy.

Analysis of an Eco-Epidemiological Model Considering Effect of Harvesting and Prey Refuge

In this work, we analyze an eco-epidemiological model with the disease in the prey, considering a constant proportion of harvesting of either species or a prey refuge. The positive invariant set, the conditions of existence, and locally asymptotically stability of the equilibria are studied using the stability theory of ordinary differential equation. The global stability of border equilibria by constructing Lyapunov functions and permanence of the system by comparison theoremare proved. The numerical simulation further proved the correctness of the theoretical analysis. The result indicates that overfishing would lead to population extinction and a reasonable fishing strategy should keep the coexistence of populations.

The global attractive sets and synchronization of a fractional-order complex dynamical system

&lt;abstract&gt;&lt;p&gt;This paper presents a chaotic complex system with a fractional-order derivative. The dynamical behaviors of the proposed system such as phase portraits, bifurcation diagrams, and the Lyapunov exponents are investigated. The main contribution of this effort is an implementation of Mittag-Leffler boundedness. The global attractive sets (GASs) and positive invariant sets (PISs) for the fractional chaotic complex system are derived based on the Lyapunov stability theory and the Mittag-Leffler function. Furthermore, an effective control strategy is also designed to achieve the global synchronization of two fractional chaotic systems. The corresponding boundedness is numerically verified to show the effectiveness of the theoretical analysis.&lt;/p&gt;&lt;/abstract&gt;

Bistable dynamics on a tick population equation incorporating Allee effect and two different time-varying delays

&lt;p style='text-indent:20px;'&gt;We study the bistable dynamic behaviors for a tick population model involving Allee effect and multiple different time-varying delays. Utilizing some basic inequality techniques and dynamics theory, the positive invariant sets and exponential stability conditions of the zero equilibrium and larger positive equilibrium for the addressed model are presented. In addition, some numerical examples are shown to verify the correctness and novelty of the theoretical results.&lt;/p&gt;

A Set-Based Prognostics Approach for Wind Turbine Blade Health Monitoring

This paper presents a model-based prognostics procedure using a zonotopic Kalman filter in tandem with a zonotopic set-based propagation of degradation, aiding in the quantification of uncertainties associated with prognostics. The prognostics procedure is then applied to the degradation of a wind turbine blade material subjected to a forecasted bounded set description of wind profile. To facilitate an online condition based implementation, an otherwise nonlinear based Kalman filter from the nonlinear wind turbine model is presented in a pseudolinear form, a polytopic linear parameter varying representation, decreasing computational cost and easing in the propagation of the positive invariant zonotopic uncertainty sets to a reachable set that triggers an end of life. Using this information of health, the remaining useful life with its associated uncertainties can be predicted

Positive Psychological Invariants in Family Relationships and Values as the Basis of Personal and Professional Self-Realization of Youth

Positive Psychological Invariants in Family Relationships and Values as the Basis of Personal and Professional Self-Realization of Youth

Dynamics for a non-autonomous fall armyworm-maize interaction model with a saturation functional response.

In this study, we present a non-autonomous model with a Holling type II functional response, to study the complex dynamics for fall armyworm-maize biomass interacting in a periodic environment. Understanding how seasonal variations affect fall armyworm-maize dynamics is critical since maize is one of the most important cereals globally. Firstly, we study the dynamical behaviours of the basic model; that is, we investigate positive invariance, boundedness, permanence, global stability and non-persistence. We then extended the model to incorporate time dependent controls. We investigate the impact of reducing fall armyworm egg and larvae population, at minimal cost, through traditional methods and use of chemical insecticides. We noted that seasonal variations play a significant role on the patterns for all fall armyworm populations (egg, larvae, pupae and moth). We also noted that in all scenarios, the optimal control can greatly reduce the sizes of fall armyworm populations and in some scenarios, total elimination may be attained. The modeling approach presented here provides a framework for designing effective control strategies to manage the fall armyworm during outbreaks.

Reduced-complexity interpolating control with periodic invariant sets

A low-complexity interpolating control scheme based on the concept of periodic invariance is proposed. Periodic invariance allows the state trajectory to leave the controllable invariant set temporarily but return into the set in a finite number of steps. A periodic set with easy representation is considered to reduce the expensive computation of the controllable invariant set. Since this set is not a traditional invariant set, a vertex reachability problem of target sets is solved off-line for each vertex of the set and provides a contractive control sequence that steers the system state back into the original set. Online, the periodic interpolating control (pIC) scheme allows to transition between such periodic invariant sets and an inner set endorsed with positive invariance properties. Proofs of recursive feasibility and asymptotic stability of the pIC are given. A numerical example demonstrates that pIC provides similar performance compared to more expensive optimisation-based schemes.

Positive output controllability of linear discrete–time invariant systems

Abstract This paper studies the output controllability of discrete linear time invariant systems (LTI) with non-negative input constraints. Some geometrical arguments and positive invariance concepts are used to derive the necessary and/or sufficient conditions for the positive output controllability of discrete LTI systems. The paper also provides several academic examples, which support the theoretical results.

Robust fuzzy model predictive control for voltage regulation in islanded microgrids

An effective robust fuzzy model predictive control (RFMPC) method for secondary voltage control in islanded microgrids (μGs) is presented here. In contrast to the existing techniques, which require a detailed model of μG and an ideal communication network between μG central controller and primary local controllers, RFMPC is synthesized for a non-linear model of the μG with various time delays, uncertainties, and bounded disturbances. The famous Takagi-Sugeno fuzzy approach is adopted to approximate the inherently non-linear model of μG by locally linear dynamics. The Lyapunov–Razumikhin functional method is exploited to deal with time delays. In this regard, sufficient conditions are provided in the form of linear matrix inequalities (LMIs). Then, a sequence of control laws corresponding to a set of terminal constraints is computed offline. Doing so, the online stage is reduced to solving a convex problem with LMI constraints considering the sequence of constraint sets obtained in the offline stage, thereby reducing the computational burden significantly. Robust positive invariance and input-to-state stability property concerning communication network deficiency are then speculated. The effectiveness of the proposed RFMPC is verified via a comprehensive suite of simulations in the MATLAB/SimPowerSystems environment.

A Simple Neural Network for Sparse Optimization With $l_1$ Regularization

This paper considers a class of sparse optimization problems with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_1$</tex-math></inline-formula> -norm regularization and general convex constraints, in which the individual functions involved are differential except <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_1$</tex-math></inline-formula> regularization term. Firstly, a sufficient and necessary condition for the subgradients of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_1$</tex-math></inline-formula> -norm is discussed. Subsequently, a sufficient and necessary optimality condition for the considered problem is obtained. According to this condition, a simple neural network with differential equation structure is proposed. Secondly, positive invariance and exponential convergence of state trajectory to the set of equality constraints are studied. In addition, the intermediate state variable is always non-negative when its initial value is so. Moreover, boundedness, global existence and stability in the sense of Lyapunov of state solution to the proposed neural network are guaranteed. Thirdly, the proposed network is globally convergent to an optimal solution of the considered problem from any initial point. At last, sufficient experiments including two numerical experiments, signal recovery, data classification and image restoration problems with real data sets are provided to show the efficiency of this approach.

A multivariate CVaR risk measure from the perspective of portfolio risk management

In this paper, we define a new multivariate conditional Value-at-Risk (MCVaR) risk measure. This MCVaR considers both individual risks and the aggregate risk of a portfolio, but prioritizes the aggregate risk. The new MCVaR risk measure is based on the minimization of the expectation of a multivariate loss function, which balances the shortfall and surplus risks of the aggregate risk and the individual risks in an overall risk of a portfolio. It is shown that the MCVaR risk measure holds the properties of positive homogeneity, translation invariance, subadditivity, and monotonicity under certain conditions. Numerical examples of the MCVaR risk measure are presented to illustrate the effect of dependence among individual risks on the MCVaR.

Constrained regulation problem for continuous-time stochastic systems under state and control constraints

Constrained regulation problem (CRP) for continuous-time stochastic systems is investigated in this article. New existence conditions of linear feedback control law for continuous-time stochastic systems under constraints are proposed. The computation method for solving constrained regulation problem of stochastic systems considered in this article is also presented. Continuous-time stochastic linear systems and stochastic nonlinear systems are focused on, respectively. First, the condition of polyhedral invariance for stochastic systems is established by using the theory of positive invariant set and the principle of comparison. Second, the asymptotic stability conditions in the sense of expectation for two types of stochastic systems are established. Finally, finding the linear feedback controller model and corresponding algorithm of constrained regulation problem for two types of stochastic systems are also proposed by using the obtained condition. The presented model of the stochastic constrained regulation problem in this article is formulated as a linear programming problem, which can be easily implemented from a computational point of view. Our approach establishes a connection between the stochastic constrained regulation problem and positively invariant set theory, as well as provides the possibility of using optimization methodology to find the solution of stochastic constrained regulation problem, which differs from other methods. Numerical examples illustrate the proposed method.

A Robust Impedance Controller for Improved Safety in Human–Robot Interaction

Abstract This paper presents a novel impedance controller modified with a switching strategy for the purpose of improving safety in human–robot interactions. Under normal operating conditions, an impedance controller is enabled when adequate tracking performance is maintained in the presence of bounded disturbances. However, if disturbances are greater than anticipated such that tracking performance is degraded, the proposed controller temporarily switches modes to a control strategy better apt to limit control inputs. With disturbances returning to the prescribed bounds, tracking performance will be restored and the impedance controller will resume for nominal operation. The control parameters are constrained by a few conditions necessary for smooth operation. First, a pair of equality constraints is required for the control signal to be continuous when switching control modes. Second, a Lyapunov analysis is performed to formulate an equality constraint on the control parameters to ensure only a single switch occurs when changing control modes to avert control chatter. Third, a matrix inequality constraint is necessary to ensure a robust positive invariant set is formed for when impedance control is active. Numerical simulations are provided to illustrate the controller and conditions. The simulation results successfully validate the presented theory, demonstrating how the constraints yield a continuous control signal, eliminate switching chatter, and permit robustness to disturbances.

Multi-periodicity of switched neural networks with time delays and periodic external inputs under stochastic disturbances

This paper presents new theoretical results on the multi-periodicity of recurrent neural networks with time delays evoked by periodic inputs under stochastic disturbances and state-dependent switching. Based on the geometric properties of activation function and switching threshold, the neuronal state space is partitioned into 5n regions in which 3n ones are shown to be positively invariant with probability one. Furthermore, by using Itô’s formula, Lyapunov functional method, and the contraction mapping theorem, two criteria are proposed to ascertain the existence and mean-square exponential stability of a periodic orbit in every positive invariant set. As a result, the number of mean-square exponentially stable periodic orbits increases to 3n from 2n in a neural network without switching. Two illustrative examples are elaborated to substantiate the efficacy and characteristics of the theoretical results.