Positively Invariant Research Articles

On the Design of Constrained PI-Like Output Feedback Tracking Controllers via Robust Positive Invariance and Bilinear Programming

In this paper, we design a novel PI-like output feedback tracking controller for linear systems subject to state and input constraints. The proposed solution exploits the internal model principle to design via robust positive invariance and the Extended Farkas’ Lemma, a constrained set-point tracking controller. The considered polyhedral setup results in a bilinear optimization design problem that can be solved offline, allowing to simultaneously determine the sets of admissible set-points and plant’s initial conditions. The proposed solution guarantees asymptotic offset-free set-point tracking, local stability, and constraints fulfillment. Simulation results show the effectiveness of the proposed design and contrast it with an alternative scheme.

Delay-Dependent Invariance of Polyhedral Sets for Discrete-Time Linear Systems

In this paper we propose a delay-dependent analysis of the positive invariance property with respect to linear discrete-time systems with delayed states. An appropriate model transformation is employed, together with a matrix parametrization, which allow the derivation of delay-dependent invariance conditions of polyhedral sets with respect to the transformed model. We then show that such conditions imply the confinement of the state trajectories of the original system in the set, as long as the initial states satisfy additional constraints related to the system dynamics. The characterization of this set of admissible initial conditions gives rise to the proposition of a less conservative definition of set-invariance. We illustrate through numerical examples the fact that, under the proposed definition, confinement of state trajectories in the set can be achieved even though it is not invariant according to the classical definition.

Invariant Configuration-Space Bubbles for Revolute Serial-Chain Robots

This letter adapts the invariant-set motion planner (ISMP) for robot motion planning. We derive control invariant subsets of configuration-space bubbles lifted into the state-space. The resulting sets guarantee collision avoidance since they are both constraint admissible and control invariant. We present a command governor that enforces the positive invariance of the constraints in closed-loop. This governor can be used to transform any nominal tracking controller into a constraint enforcing controller. We use these control invariant sets to quantify a relationship between velocity and control authority that enables collision avoidance. We demonstrate our invariant-sets through an illustrative numerical example.

Model Predictive Control for Linear Systems Under Relaxed Constraints

This article considers model predictive control (MPC) for linear systems under relaxed constraints. The main novelty of our proposal is the introduction, and an adequate use, of the terminal dynamics of the slack variable associated with relaxed constraints. The proposed MPC under relaxed constraints retains computational efficiency of the traditional MPC, while it guarantees positive invariance and exponential stability over an enlarged domain of attraction. The design method is also illustrated in a step-by-step manner by an academic example.

Stable invariance of the restricted Lie algebra structure of Hochschild cohomology

We show that the restricted Lie algebra structure on Hochschild cohomology is invariant under stable equivalences of Morita type between self-injective algebras. Thereby we obtain a number of positive characteristic stable invariants, such as the $p$-toral rank of $\mathrm{HH}^1(A,A)$. We also prove a more general result concerning Iwanaga-Gorenstein algebras, using a more general notion of stable equivalences of Morita type. Several applications are given to commutative algebra and modular representation theory. These results are proven by first establishing the stable invariance of the $B_\infty$-structure of the Hochschild cochain complex. In the appendix we explain how the $p$-power operation on Hochschild cohomology can be seen as an artifact of this $B_\infty$-structure. In particular, we establish well-definedness of the $p$-power operation, following some -- originally topological -- methods due to May, Cohen and Turchin, using the language of operads.

On the Stability of Unconditionally Positive and Linear Invariants Preserving Time Integration Schemes

On the Stability of Unconditionally Positive and Linear Invariants Preserving Time Integration Schemes

A stochastic differential equation SIS model on network under Markovian switching

We study a stochastic SIS (susceptible-infected-susceptible) epidemic dynamics on network, under the effect of a Markovian regime-switching. We first prove the existence of a unique global positive solution, and find a positive invariant set for the system. Then, we find sufficient conditions for a.s. extinction and stochastic permanence, showing also their relation with the stationary probability distribution of the Markov chain that governs the switching and with the network topology. We provide an asymptotic lower bound for the time average of the sample-path solution under the conditions ensuring stochastic permanence. From this bound, we are able to prove the existence of an invariant probability measure if the condition of stochastic permanence holds. Under a different condition, we prove the positive recurrence and the ergodicity of the regime-switching diffusion.

A Zonotopic-Based Watermarking Design to Detect Replay Attacks

This paper suggests the use of zonotopes for the design of watermark signals. The proposed approach exploits the recent analogy found between stochastic and zonotopic-based estimators to propose a deterministic counterpart to current approaches that study the replay attack in the context of stationary Gaussian processes. In this regard, the zonotopic analogous case where the control loop is closed based on the estimates of a zonotopic Kalman filter (ZKF) is analyzed. This formulation allows to propose a new performance metric that is related to the Frobenius norm of the prediction zonotope. Hence, the steady-state operation of the system can be related with the size of the minimal Robust Positive Invariant set of the estimation error. Furthermore, analogous expressions concerning the impact that a zonotopic/Gaussian watermark signal has on the system operation are derived. Finally, a novel zonotopically bounded water-mark signal that ensures the attack detection by causing the residual vector to exit the healthy residual set during the replay phase of the attack is introduced. The proposed approach is illustrated in simulation using a quadruple-tank process.

Ultimate Boundedness and Finite Time Stability for a High Dimensional Fractional-Order Lorenz Model

In this paper, the global attractive set (GAS) and positive invariant set (PIS) of the five-dimensional Lorenz model with the fractional order derivative are studied. Using the Mittag-Leffler function and Lyapunov function method, the ultimate boundedness of the proposed system are estimated. An effective control strategy is also designed to achieve the finite time stability of this fractional chaotic system. The corresponding boundedness and control scheme are numerically verified to show the effectiveness of the theoretical analysis.

Multiple asymptotical [formula omitted]-periodicity of fractional-order delayed neural networks under state-dependent switching

This paper presents theoretical results on multiple asymptotical ω-periodicity of a state-dependent switching fractional-order neural network with time delays and sigmoidal activation functions. Firstly, by combining the geometrical properties of activation functions with the range of switching threshold, a partition of state space is given. Then, the conditions guaranteeing that the solutions can approach each other infinitely in each positive invariant set are derived. Furthermore, the S-asymptotical ω-periodicity and the convergence of solutions in positive invariant sets are discussed. It is worth noting that the number of attractors increases to 3n from 2n in a neural network without switching. Finally, three numerical examples are given to substantiate the theoretical results.

Viscosity subsolutions of Hamilton–Jacobi equations and invariant sets of contact Hamilton systems

This paper aims at providing a dynamic perspective of viscosity subsolutions to contact Hamiltonian–Jacobi equations H(x, ∂xu, u) = 0 on connected, closed manifold M. Based on implicit variational principles, we focus on the connection between positive invariant sets under Hamiltonian flow and viscosity subsolutions. We apply the connection to give a new necessary and sufficient condition for the existence of viscosity solutions of the stationary equation from the dynamical view. Finally, we discuss the multiplicity of viscosity solutions and give several illustrative examples.

Path-Based Stability Analysis for Monotone Control Systems on Proper Cones

In this article, we study positive invariance and attractivity properties for nonlinear control systems, which are monotone with respect to proper cones. Monotonicity simplifies such analysis for specific sets defined by the proper cones. Instead of Lyapunov functions, a pair of so-called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">paths</i> in the state space and input space play important roles. As applications, our results are utilized for analysis of asymptotic stability and also input-to-state stability on proper cones. The results are illustrated by means of examples.

Dynamic analysis of the e-SITR model for remote wireless sensor networks with noise and Sokol-Howell functional response

Wireless Sensor network (WSN) is used for a wide range of uses, including defense, healthcare, environmental tracking etc. In this work, we have formulated a Susceptible-infected-terminally infected-recovered (SITR) model to investigate the offensive worm dynamics in remote sensor networks utilizing Sokol-Howell functional response. The positive invariance and boundedness of the model have been investigated in the proposed work. In addition to that, the existence of possible equilibrium points with feasible conditions has been evaluated. The local stability of all equilibrium points and the global stability of the interior steady state of nodes are measured with the Rougth-Hurwitz criteria, Lasalle’s invariance principle, and Bendixson-Dulac criteria. The impact of noise on the proposed model is being investigated. The numerical imitations are carried out, which strengthens the diagnostic discoveries. Our numerical simulation reveals that growth rate, internode interference coefficients, and crashing rates are critical to the model's dynamics.

Robust iterative learning model predictive control for repetitive motion of maglev planar motor

This paper presents a robust iterative learning model predictive control (RILMPC) scheme for repetitive trajectory tracking of a magnetically levitated (maglev) planar motor. The motivation lies in the improvement of tracking performance and disturbance rejection ability for the maglev system. Relying on the past error in the repetitive motion, the model discrepancy is persistently compensated with iterations to improve the prediction accuracy. The cost function that serves as the upper bound of the tracking error is then derived from the past control trajectory based on the Lipschitz property of disturbances. In the proposed RILMPC scheme, a minimal robust positive invariant set is introduced into the MPC optimisation to cope with unmodeled dynamics and disturbances. Furthermore, it is theoretically shown that the perturbed closed-loop system is stable, and the proposed RILMPC scheme is recursively feasible with perfect tracking performance under some conditions. Finally, comparative experiments carried out on a maglev planar motor demonstrate that the proposed control strategy possesses satisfactory transient/steady-state tracking performance and robustness against disturbances.

Hybrid variables-dependent event-triggered model predictive control subject to polytopic uncertainties

This paper focuses on the model predictive control (MPC) issue where the measurable state is released under an event-triggered mechanism (ETM) to implement MPC over an infinite horizon. A new dynamic ETM (DETM) is devised to conserve network resources, which contains an additive internal dynamic variable (IDV), a multiplicative adaptively adjusting variable, a time-varying weighting matrix and several flexible scalars. The MPC problem is formulated as a ‘min–max’ optimisation problem (OP), where a hard constraint and robust positive invariant set on the predictive state/IDV are considered simultaneously. By resorting to a Lyapunov-like function that depends on the IDV, we put forward an auxiliary OP with matrix-inequality-based constraints. By the feasible solutions of such an auxiliary OP, the feedback gain matrix is designed which ensures the asymptotic stability of the closed-loop system. Two examples are presented to demonstrate the validity of the devised DETM and the DETM-based MPC algorithm. The study verifies that the devised DETM has advantages over an existing counterpart in conserving network resources while achieving the desired performance.

Hierarchical Optimization-Based Model Predictive Control for a Class of Discrete Fuzzy Large-Scale Systems Considering Time-Varying Delays and Disturbances

In this manuscript, model predictive control for class of discrete fuzzy large-scale systems subjected to bounded time-varying delay and disturbances is studied. The considered method is Razumikhin for time-varying delay large-scale systems, in which it includes a Lyapunov function associated with the original non-augmented state space of system dynamics in comparison with the Krasovskii method. As a rule, the Razumikhin method has a perfect potential to avoid the inherent complexity of the Krasovskii method especially in the presence of large delays and disturbances. The considered large-scale system in this manuscript is decomposed into several subsystems, each of which is represented by a fuzzy Takagi–Sugeno (T-S) model and the interconnection between any two subsystems is considered. Because the main section of the model predictive control is optimization, the hierarchical scheme is performed for the optimization problem. Furthermore, persistent disturbances are considered that robust positive invariance and input-to-state stability under such circumstances are studied. The linear matrix inequalities (LMIs) method is performed for our computations. So the closed-loop large-scale system is asymptotically stable. Ultimately, by two examples, the effectiveness of the proposed method is illustrated, and a comparison with other papers is made by remarks.

Relating Cut and Paste Invariants and Tqfts

Abstract In this paper, we are concerned with a relation between TQFTs and the cut and paste SKK invariants introduced by Karras, Kreck, Neumann, and Ossa. Cut and paste SKK invariants are functions on the set of smooth manifolds whose values on cut and paste equivalent manifolds differ by an error term depending only on the glueing diffeomorphisms. Here we investigate a surprisingly natural group homomorphism between the group of invertible TQFTs and the group of SKK invariants and describe how these groups fit into an exact sequence. We conclude in particular that all positive real-valued SKK invariants can be realized as restrictions of invertible TQFTs.

Kleene Algebra to Compute Invariant Sets of Dynamical Systems

In this paper, we show that a basic fixed point method used to enclose the greatest fixed point in a Kleene algebra will allow us to compute inner and outer approximations of invariant-based sets for continuous-time nonlinear dynamical systems. Our contribution is to provide the definitions and theorems that will allow us to make the link between the theory of invariant sets and the Kleene algebra. This link has never be done before and will allow us to compute rigorously sets that can be defined as a combination of positive invariant sets. Some illustrating examples show the nice properties of the approach.

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.