Positively Invariant Research Articles

Robust dynamic positioning of autonomous surface vessels with tube-based model predictive control

This paper proposes two robust dynamic positioning (DP) approaches for autonomous surface vessels when full states are measurable and when only partial states are available with measurement errors. High fidelity nonlinear hydrodynamics are considered which are approximated as linear models in the local area of DP setpoint. The linearization errors are seen as bounded unmodeled dynamics and are accommodated in the tube-based model predictive control (MPC) together with environmental disturbances. The tube-based MPC controller contains all the possible uncertain trajectories in a tube that is based on a precomputed robust positive invariant set and a nominal trajectory solved online. The total controller consists of a feedforward part compensating the predicted environmental forces, a nominal part guiding the vessel towards and stabilizing it at the origin, and an affine feedback part bounding the uncertain vessel trajectory within the tube. Furthermore, an output feedback robust DP controller is proposed by utilizing a simple Luenberger observer to estimate the system states when full states are not available. The resultant estimation and measurement errors are also incorporated in the tube-based MPC. Simulation results show that the proposed full state and output feedback robust DP controllers can achieve the DP goals within system constraints.

Online feedback motion planning for spacecraft obstacle avoidance using positively invariant sets

This paper investigates a novel reliable feedback motion planning algorithm for steering a spacecraft to the desired position while satisfying dynamics and environmental constraints, such as actuator constraints, unexpected state constraints, bounded disturbances, etc. The proposed algorithm, termed safeRRT∗, applies positively invariant (PI) set transitions to an online sampling approach based on the asymptotically optimal rapidly-exploring random tree (RRT∗). A time-varying safe corridor is generated which consists of a sequence of PI sets under the instantaneous constraints. A spacecraft is guaranteed to remain within the corridor when the feedback controllers corresponding to the PI set transitions are executed. Simulations of the spacecraft avoiding obstacles validate the feasibility and effectiveness of the proposed safeRRT∗ algorithm.

Global dissipativity of delayed discrete-time inertial neural networks

This paper addresses global dissipativity for a class of delayed discrete-time inertial neural networks (DINN). With a newly developed discrete-time Halanay inequality, a novel criterion of the global dissipativity for the DINN is established. Moreover, the global robust dissipativity of DINN with bounded parameter uncertainties is also investigated. Meanwhile, some specific estimates of global attractive sets and positive invariant sets are derived. Finally, two simulation examples validate the efficacy of the proposed results.

Error Bounds for PD-Controlled Mechanical Systems Under Bounded Disturbances Using Interval Arithmetic

We present a numerical algorithm based on invariant set theory to evaluate the worst-case performance of PD-controlled mechanical systems affected by bounded disturbances. By performing a specific coordinate transformation, the search and computation of positive invariant sets is simplified. It is shown that, thanks to the preservation of problem structure, the proposed method allows to obtain tight, component-wise bounds on the states, which are especially useful for performance evaluation and tuning of a PD controller. The bounds are formally guaranteed and can be used for safety certification. The method is compared to ultimate boundedness, and the superior results are shown via numerical simulations.

Dynamics for a non-autonomous predator-prey system with generalist predator

In this paper, we study complex dynamics of a non-autonomous predator-prey system with a Holling type II functional response and predator being generalist. We first studied basic dynamics such as boundedness, positive invariance, permanence, non-persistence and globally asymptotic stability. We also provide sufficient conditions for the existence, uniqueness and globally asymptotic stability of positive periodic solutions and boundary periodic solutions of the proposed model when parameters are periodic. We give the integral conditions to prove the extinction of prey and predator and globally asymptotic stability of boundary periodic solutions, and we show that these conditions have more reasonable biological interpretation than those expressed by supremum and infimum of parameters in some literature. We also perform numerical simulations to complement our analytical results and obtain more insights. Some of our numerical simulations indicate that periodic system may promote or suppress the permanence of its autonomous version with parameters being the averages of periodic parameters.

Random exponential attractor for stochastic discrete long wave-short wave resonance equation with multiplicative white noise

We mainly consider the existence of a random exponential attractor (positive invariant compact measurable set with finite fractal dimension and attracting orbits exponentially) for stochastic discrete long wave-short wave resonance equation driven by multiplicative white noise. Firstly, we prove the existence of a random attractor of the considered equation by proving the existence of a uniformly tempered pullback absorbing set and making an estimate on the 'tails' of solutions. Secondly, we show the Lipschitz property of the solution process generated by the considered equation. Finally, we prove the existence of a random exponential attractor of the considered equation, which implies the finiteness of fractal dimension of random attractor.

Tracking Control for Directional Drilling Systems Using Robust Feedback Model Predictive Control

A rotary steerable system (RSS) is a drilling technology which has been extensively studied and used for over the last 20 years in hydrocarbon exploration and it is expected to drill complex curved borehole trajectories. RSSs are commonly treated as dynamic robotic actuator systems, driven by a reference signal and typically controlled by using a feedback loop control law. However, due to spatial delays, parametric uncertainties and the presence of disturbances in such an unpredictable working environment, designing such control laws is not a straightforward process. Furthermore, due to their inherent delayed feedback, described by delay differential equations (DDE), directional drilling systems have the potential to become unstable given the requisite conditions. This paper proposes a Robust Model Predictive Control (RMPC) scheme for industrial directional drilling, which incorporates a simplified model described by ordinary differential equations (ODE), taking into account disturbances and system uncertainties which arise from design approximations within the formulation of RMPC. The stability and computational efficiency of the scheme are improved by a state feedback strategy computed offline using Robust Positive Invariant (RPI) sets control approach and model reduction techniques. A crucial advantage of the proposed control scheme is that it computes an optimal control input considering physical and designer constraints. The control strategy is applied in an industrial directional drilling configuration represented by a DDE model and its performance is illustrated by simulations.

Construction of Positive Invariant Sets in a Prey-predator Model with the Holling Type II Nonlinearity

Construction of Positive Invariant Sets in a Prey-predator Model with the Holling Type II Nonlinearity

Stability Analysis for A Class of Linear Hyperbolic System of Balance Laws with Sampled-data Control

Abstract The stability for a class of linear hyperbolic systems with distributed sampled-data controllers is discussed in this paper. The original sampled-data system is firstly transformed into a new equivalent system by modelling the sampling induced error as a reset integrator operator. Then we construct an appropriate Lyapunov function and obtain sufficient conditions for the Re - stability of the system based on linear matrix inequalities (LMIs). A numerical example illustrates our results: when the sampling interval is within the allowable range, the solution of the system converges from the domain of attraction to a positive invariant set.

Dynamical analysis of a fractional-order avian-human influenza epidemic model with logistic growth for avian population

In this paper, a fractional order avian-human influenza epidemic model with logistic growth for avian population is investigated. The dynamical behavior of this model is discussed. We first establish the existence, uniqueness, non-negativity and positive invariance of the solution. Then we analyze the existence of various equilibrium points, and some sufficient conditions are derived to ensure the global asymptotic stability of the disease free equilibrium point and endemic equilibrium point. Finally, we take some numerical simulations to validate the analytical results.

Framework for Studying Stability of Switching Max-Plus Linear Systems

Abstract We propose a framework for studying the stability of discrete-event systems modelled as switching max-plus linear systems. In this framework, we propose a set of notions of stability for generic discrete-event systems in the max-plus algebra. Then we show the loss of equivalence of these notions for switching max-plus linear systems due to the lack of global monotonicity and the accompanying difficulty in rigorous analysis. This serves as a motivation to relax the assumption on monotonicity of the dynamics to positive invariance of max-plus cones. Then we proceed to generalise the notions of stability when the dynamics is restricted to such cones. The stability analysis approach presented in this paper serves as a first step to study the stability of a general class of switching max-plus linear systems.

GeCo: Geometric Conservative nonstandard schemes for biochemical systems

We generalize the nonstandard Euler and Heun schemes in order to provide explicit geometric numerical integrators for biochemical systems, here denoted as GeCo schemes, that preserve both positivity of the solutions and linear invariants. We relax the request on the order convergence of the denominator function for the first-order approximation and we let it depend on the step size also throughout the solution approximating values. The first-order variant is exact on a two-dimensional linear test problem. Moreover, we introduce a class of modified mGeCo(α) schemes and, by tuning the parameter α≥1, we improve the numerical performance of GeCo integrators on some examples taken from the literature. A numerical comparison with BBKS and mBBKS schemes, which are implicit integrators, positive and conserve linear invariants, show the gain in efficiency of both GeCo and mGeCo(α) procedures as they generate similar errors with an explicit functional form.

An observer‐based output feedback robust MPC approach for constrained LPV systems with bounded disturbance and noise

SummaryThe present paper addresses an observer‐based output feedback robust model predictive control for the linear parameter varying system with bounded disturbance and noise subject to input and state constraints. The main contribution is that the on‐line convex optimization problem not only simultaneously optimizes the observer and controller gains to stabilize the augmented closed‐loop system but also incorporates the refreshment of bounds of the estimation error set. The optimization problem steers the nominal augmented closed‐loop system to converge to the origin, and the real augmented closed‐loop system bounded within robust positive invariant set converges to a neighborhood of the origin such that recursive feasibility of the optimization and robust stability of the controlled system are ensured. Two numerical examples are given to illustrate the effectiveness of the method.

Positive Invariant Sets for Safe Integrated Vehicle Motion Planning and Control

This article describes a method for real-time integrated motion planning and control aimed at autonomous vehicles. Our method leverages feedback control, positive invariant sets, and equilibrium trajectories of the closed-loop system to produce and track trajectories that are collision-free with guarantees according to the vehicle model. Our method jointly steers the vehicle to a target region and controls the velocity while satisfying constraints associated with future motion of surrounding obstacles. We develop a receding-horizon implementation of the control policy and verify the method in both a simulated road scenario and an experimental validation using a scaled mobile robot with car-like dynamics using only onboard sensing. The results show that our method generates dynamically feasible and safe (i.e., collision-free) trajectories in real time, and indicate that the proposed planner is robust to sensing and mapping errors.

Admissible sets for slowly-varying discrete-time systems

This note describes a method to calculate admissible sets for dynamical systems with slowly-varying parameters. A set of initial states is said to be admissible if all state trajectories starting within this set respect safety-critical constraints for all future time instances. For many classes of systems, the Maximal Admissible Set (MAS) equals the Maximal Robust Positive Invariant (MRPI) set. Therefore, much of the literature focus on algorithms that produce (approximations of) the MRPI set. However, for slowly-varying systems it turns out that the MAS can be larger than the MRPI set. A property referred to as the M-proximal contraction property is defined for such systems to state sufficient conditions for sets to be admissible. It is then shown that Algorithm 2 produces an admissible set that is a superset of the MRPI set.

Global dissipativity of high-order Hopfield bidirectional associative memory neural networks with mixed delays

In this paper, the problem of the global dissipativity of high-order Hopfield bidirectional associative memory neural networks with time-varying coefficients and distributed delays is discussed. By using Lyapunov–Krasovskii functional method, inequality techniques and linear matrix inequalities, a novel set of sufficient conditions for global dissipativity and global exponential dissipativity for the addressed system is developed. Further, the estimations of the positive invariant set, globally attractive set and globally exponentially attractive set are found. Finally, two examples with numerical simulations are provided to support the feasibility of the theoretical findings.

Tree-Structured Polyhedral Invariant Set Calculations

This letter provides a description of how hierarchical dependencies between inequalities can be exploited in order to efficiently calculate polyhedral approximations of maximal robust positive invariant sets using geometrically motivated methods. Due to the hierarchical dependencies, the calculations of preimage sets and minimal representations can be alleviated. It is also shown that as a byproduct from the calculations of minimal representations, a stopping criterion is obtained, which means that the commonly used subset test is superfluous.

Weak separation, positivity and extremal Yangian invariants

We classify all positive n-particle NkMHV Yangian invariants in mathcal{N} = 4 YangMills theory with n = 5k, which we call extremal because none exist for n > 5k. We show that this problem is equivalent to that of enumerating plane cactus graphs with k pentagons. We use the known solution of that problem to provide an exact expression for the number of cyclic classes of such invariants for any k, and a simple rule for writing them down explicitly. We provide an alternative (but equivalent) classification by showing that a product of k five-brackets with disjoint sets of indices is a positive Yangian invariant if and only if the sets are all weakly separated.

Robust dissipativity analysis for delayed memristor-based inertial neural network

The issue of robust dissipativity for delayed memristor-based inertial neural networks is studied in this article. By employing some famous inequalities, the results on global dissipativity and global robust dissipativity of the delayed memristor-based inertial neural networks are obtained via generalized Lyapunov functional method. Moreover, the positive invariant sets and globally attractive sets of the discussed system are also given. Without making the variable transformation, the second order functional differential system is investigated directly, which is different from the previous works. Finally, some examples and their numerical simulations are shown to corroborate the effectiveness of obtained results.

Distributed model predictive control approach for cooperative car-following with guaranteed local and string stability

In this paper, a serial distributed model predictive control (MPC) approach for connected automated vehicles (CAVS) is developed with local stability (disturbance dissipation over time) and multi-criteria string stability (disturbance attenuation through a vehicular string). Two string stability criteria are considered within the proposed MPC: (i) the l∞-norm string stability criterion for attenuation of the maximum disturbance magnitude and (ii) l2-norm string stability criterion for attenuation of disturbance energy. The l∞-norm string stability is achieved by formulating constraints within the MPC based on the future states of the leading CAV, and the l2-norm string stability is achieved by proper weight matrix tuning over a robust positive invariant set. For rigor, mathematical proofs for asymptotical local stability and multi-criteria string stability are provided. Simulation experiments verify that the distributed serial MPC proposed in this study is effective for disturbance attenuation and performs better than traditional MPC without stability guarantee.