Positively Invariant Research Articles

Time-Varying Tube-Based Output Feedback Robust MPC for T–S Fuzzy Systems

For Takagi–Sugeno fuzzy systems subject to inexact membership functions, bounded disturbances, and noises, an output feedback robust model predictive control (RMPC) approach with time-varying robust tubes is investigated. The membership functions errors are bounded within convex sets via the properties of zonotopes and interval matrices. An offline table stores a series of structures that include nested robust positive invariant sets with the corresponding nominal feedback controller gains, ancillary controller gains, and observer gains. According to bounds of real-time estimation error sets, the time-varying structures in the offlined table are searched. Then, the output feedback RMPC problem with time-varying tightened constraints on inputs and states is optimized to stabilize the nominal system. The output feedback RMPC approach can not only update bounds of the estimation errors and uncertain terms resulting from inexact membership functions, but also reduce the computational burden. The proposed RMPC algorithm with recursive feasibility guarantees the robust stability of the controlled systems.

Terminal‐cost design for model predictive control with linear stage‐costs: A set‐theoretic method

AbstractThis article presents a method for designing stabilizing terminal‐costs for linear model predictive controllers whose stage‐costs are a mixture of weighted 1‐norms and ∞‐norms of the system states and inputs. This problem is abstracted by replacing the 1/∞‐norm stage‐costs with convex Minkowski functions. We show that a convex Minkowski terminal‐cost is a control Lyapunov function if and only if the dual of its defining set satisfies a set‐inclusion condition. Furthermore, we show that this set‐inclusion is satisfied by all robust positive invariant sets of a certain dual system. In particular, we show that the minimal convex set that satisfies the set‐inclusion produces the optimal terminal‐cost which is the cost‐to‐go of the corresponding unconstrained infinite‐horizon optimal control problem with convex Minkowski stage‐costs. To reduce the number of constraints in the MPC, we show that the defining set of the Minkowski function can be used as an invariant terminal‐constraint. Finally, we demonstrate our terminal‐cost design procedure for three case studies; a double‐integrator, a galvanometer laser scanner, and a nanosatellite.

Formal Calculation of Positive Invariant Sets for the Lorenz Family Combining Lyapunov Approaches and Quantifier Elimination

AbstractThe Lorenz family is a class of dynamic systems which is parameterized in one variable and includes both the classic Lorenz system and the Chen system. Both systems (and therefore the Lorenz family) can show chaotic behaviour. For different questions of system analysis, e.g. for the estimation of various types of dimensions, it is helpful to be able to determine bounds on the attractor. One typical approach ist the usage of Lyapunov functions to describe positive invariant sets. However, this approach usually requires a good knowledge of the system to perform the necessary calculations. For polynomial systems, these calculations can be carried out using a computation technique known as quantifier elimination. In this contribution we describe this approach and employ it to calculate bounds on the Lorenz family.

Nonexistence of global solutions for a class of viscoelastic wave equations

<p style='text-indent:20px;'>We consider a class of nonlinear evolution equations of second order in time, linearly damped and with a memory term. Particular cases are viscoelastic wave, Kirchhoff and Petrovsky equations. They appear in the description of the motion of deformable bodies with viscoelastic material behavior. Several articles have studied the nonexistence of global solutions of these equations due to blow-up. Most of them have considered non-positive and small positive values of the initial energy and recently some authors have analyzed these equations for any positive value of the initial energy. Within an abstract functional framework we analyze this problem and we improve the results in the literature. To this end, a new positive invariance set is introduced.</p>

Interplay Between Reproduction and Age Selective Harvesting Delays of a Single Population Non-Autonomous System

This paper is concerned with an analysis of the dynamics of a non-autonomous, single population age based growth model with harvesting formulation. First, we derive sufficient conditions for permanence and positive invariance. Then, by constructing a scalar function, namely the Lyapunov function, we arrive at a suitable criterion for global attractivity. With the help of Brouwer fixed point and continuation theorems, we obtain constraints for the existence of a positive periodic solution. Then we prove that there exists only one solution which is almost periodic in nature that is distinct from every other solution. Further, we carry out a numerical simulation to support the analytical findings.

Optimization approach to the constrained regulation problem for linear continuous-time fractional-order systems

Abstract This paper deals with the Constrained Regulation Problem (CRP) for linear continuous-times fractional-order systems. The aim is to find the existence conditions of linear feedback control law for CRP of fractional-order systems and to provide numerical solving method by means of positively invariant sets. Under two different types of the initial state constraints, the algebraic condition guaranteeing the existence of linear feedback control law for CRP is obtained. Necessary and sufficient conditions for the polyhedral set to be a positive invariant set of linear fractional-order systems are presented, an optimization model and corresponding algorithm for solving linear state feedback control law are proposed based on the positive invariance of polyhedral sets. The proposed model and algorithm transform the fractional-order CRP problem into a linear programming problem which can readily solved from the computational point of view. Numerical examples illustrate the proposed results and show the effectiveness of our approach.

Scalable clock synchronization analysis: A symmetric noncooperative output feedback tubes-MPC approach

In the cyber-physical environment, the clock synchronization algorithm is required to have better expansion for network scale. In this paper, a new measurement model of observability under the equivalent transformation of minimum mean square error ( MMSE ) is constructed based on basic measurement unit ( BMU ) , which can realize the scaled expansion of MMSE measurement. Based on the state updating equation of absolute clock and the decoupled measurement model of MMSE-like equivalence, which is proposed to calculate the positive definite invariant set by using the theoretical-practical Luenberger observer as the synthetical observer, the local noncooperative optimal control problem is built, and the clock synchronization system driven by the ideal state of local clock can reach the exponential convergence for synchronization performance. Different from the problem of general linear system regulators, the state estimation error and state control error are analyzed in the established affine system based on the set-theory-in-control to achieve the quantification of state deviation caused by noise interference. Based on the BMU for isomorphic state map, the synchronization performance of clock states between multiple sets of representative nodes is evaluated, and the scale of evaluated system can be still expanded. After the synchronization is completed, the state of perturbation system remains in the maximum range of measurement accuracy, and the state of nominal system can be stabilized at the ideal state for local clock and realizes the exponential convergence of the clock synchronization system.

Positively invariant subset for non-densely defined Cauchy problems

This study develops a generalized notion of sub tangential condition to establish the positive invariance of a closed subset under the semiflow generated by a semi-linear non densely defined Cauchy problem. We also remark that the sufficient condition for the positivity of the semiflow implies our sub tangentiality condition. In other words, our sub tangential condition is more powerful since it can be used to show the positive invariance of a much larger class of closed subset. As an illustration we apply our results to an age-structured equation in Lp space which is only defined on a closed subset of Lp.

Field-of-view-constrained impact angle control guidance with error convergence before interception considering speed changes

An impact angle control guidance law is proposed for stationary target interception considering missile's field-of-view limit and speed changes. The proposed impact angle control guidance law is structured as a biased proportional navigation with a time-varying bias. The proposed guidance law does not involve any switching logic for maintaining lock-on; hence, the guidance command is continuous during the entire engagement. Unlike the most existing studies, the proposed method guarantees that the impact angle error converges to zero before interception without the constant-speed assumption. To realize these desirable properties, the positive invariance of the bounded look angle interval and the change of independent variable are utilized. Numerical simulations are conducted to demonstrate the performance of the proposed guidance law.

Robust Motion Planning for Uncertain Systems With Disturbances Using the Invariant-Set Motion Planner

The invariant-set motion planner uses a collection of safe sets to find a collision-free path through an obstacle-filled environment [1] – [4] . This article extends the invariant-set motion planner to systems with persistently varying disturbances and parametric model uncertainty. This is accomplished by replacing the previously used positive invariant sets with robust positive invariant sets. Since the model uncertainty obfuscates the relationship between the invariant sets in the state space, and the references and obstacles in the output space, we reformulate the dynamics in velocity form so that the system output appears directly in the modified system state. Since the persistently varying disturbances will prevent the closed-loop system from converging to the desired reference, we introduce a new robust connection rule where references are connected when the invariant set of one reference contains the minimal volume robust invariant-set of another. In addition, we bound the time required to transition between invariant sets to ensure safety when the obstacles are moving. By parameterizing the invariant sets using a precomputed input-to-state Lyapunov function, we reduce the real-time computational complexity of our motion planner. The robust invariant-set motion planner is demonstrated for an automated highway driving case study.

The impact of toxins on competition dynamics of three species in a polluted aquatic environment

<p style='text-indent:20px;'>Accurately assessing the risks of toxins in polluted ecosystems and finding factors that determine population persistence and extirpation are important from both environmental and conservation perspectives. In this paper, we develop and study a toxin-mediated competition model for three species that live in the same polluted aquatic environment and compete for the same resources. Analytical analysis of positive invariance, existence and stability of equilibria, sensitivity of equilibria to toxin are presented. Bifurcation analysis is used to understand how the environmental toxins, plus distinct vulnerabilities of three species to toxins, affect the competition outcomes. Our results reveal that while high concentrations lead to extirpation of all species, sublethal levels of toxins affect competition outcomes in many counterintuitive ways, which include boosting coexistence of species by reducing the abundance of the predominant species, inducing many different types of bistability and even tristability, generating and reducing population oscillations, and exchanging roles of winner and loser in competition. The findings in this work provide a sound theoretical foundation for understanding and assessing population or community effects of toxicity.

A class of numerical methods for determining the inclusion relation of positive invariant sets in linear dynamic systems

Positive invariant sets play an important role in the system theory and a:lication of dynamic systems. Firstly, in order to judge the inclusion relation between positive invariant sets of systems, the equivalent conditions for judging the inclusion relation of special positive invariant sets are derived for two special positive invariant sets. Three specific positive invariant subsets are discussed, and the verification of the inclusion relation of positive invariant sets can be transformed into a class of numerical problem in the form of optimization, which simplify the conditions of readily implemented. Finally, numerical examples are given to illustrate our conclusion.

Global dissipativity of fuzzy cellular neural networks with inertial term and proportional delays

ABSTRACTThis paper is concerned with the global dissipativity of fuzzy cellular neural networks with inertial term and proportional delays. Based on Lyapunov functionals and linear matrix inequality approach, new sufficient conditions are derived to ensure the global dissipativity and global exponential dissipativity of the suggested system. Moreover, the globally exponential attractive sets and positive invariant sets are also presented here. Finally, two numerical examples with its simulations are proposed to illustrate the effectiveness of the obtained results.

Polarity of stability and robust positive invariance

This note reports a theoretically and practically relevant polarity relationship between stability and robust positive invariance for linear dynamics. In particular, the note demonstrates that stability and robust positive invariance analyses for linear dynamics are equivalent in a well defined sense from both theoretical and computational points of view.

Energy efficient convex-lifting-based robust control of a heat exchanger

This paper analyses control performance and energy savings reached by application of four robust control approaches implemented on the laboratory plate heat exchanger. The output temperature of the cold medium is the controlled variable and the flow rate of the hot medium is the manipulated variable. Closed-loop control of the heat exchanger aims to ensure offset-free setpoint temperature tracking. First, the computationally demanding robust model predictive control (RMPC) strategy is investigated. Improvements ensured by implementing three modifications of the convex-lifting-based robust control methods are analysed. These modifications are based on the (i) non-tunable, (ii) tunable, and (iii) multiple tunable robust positive invariant (RPI) sets. Designed robust control approaches significantly improved control performance. For considered control setup, the settling time was reduced up to 70%. Energy savings were increased by 82%, when compared to RMPC. Moreover, the considered methods are proper for real-time implementation as they significantly reduce computational demands. The designed robust convex-lifting-based methods are suitable for industrial hardware with limited computational requirements.

Numerical checking method for positive invariance of polyhedral sets for linear dynamical systems

Numerical checking method for positive invariance of polyhedral sets for linear dynamical systems

A topological method for finding invariant sets of continuous systems

A usual way to find positive invariant sets of ordinary differential equations is to restrict the search to predefined finitely generated shapes, such as linear templates, or ellipsoids as in classical quadratic Lyapunov function based approaches. One then looks for generators or parameters for which the corresponding shape has the property that the flow of the ODE goes inwards on its border. But for non-linear systems, where the structure of invariant sets may be very complicated, such simple predefined shapes are generally not well suited. The present work proposes a more general approach based on a topological property, namely Ważewski's property. Even for complicated non-linear dynamics, it is possible to successfully restrict the search for isolating blocks of simple shapes, that are bound to contain non-empty invariant sets. This approach generalizes the Lyapunov-like approaches, by allowing for inwards and outwards flow on the boundary of these shapes, with extra topological conditions. We developed and implemented an algorithm based on Ważewski's property, SOS optimization and some extra combinatorial and algebraic properties, that shows very nice results on a number of classical polynomial dynamical systems.

Dynamics of a non-autonomous biocontrol model on native consumer, biocontrol agent and their predator

In this paper, we propose and study the dynamics of a non-autonomous biocontrol model concerned with native consumer A, biocontrol agent B and their common predator L. We assume that some parameters in our proposed model are nonnegative functions instead of being bounded below by positive reals. This assumption is more realistic but makes mathematical proofs more challenging. We first study the positive invariance, permanence, and the global attractivity of bounded positive solution and boundary solution of the proposed model with general time-dependent parameters. Then we focus on the dynamics of the model with periodic parameters which include the existence and uniqueness of positive periodic solutions, and the global asymptotic stability of boundary periodic solution. We also explore and discuss the effects of introducing the biocontrol agent to ecosystem through theoretical analysis and numerical simulations. Our results show that introducing the biocontrol agent to ecosystem has positive effects by promoting its biodiversity and coexistence of all species, also potentially has negative effects by eliminating the predator. In addition, our numerical simulations show that (i) the amplitudes of periodic parameters could affect the permanence of non-autonomous periodic system; and (ii) the non-autonomous periodic system may suppress or improve the permanence of its autonomous version but related to the amplitudes of periodic parameters.

Global Dynamics of the Chaotic Disk Dynamo System Driven by Noise

The disk dynamo system, which is capable of chaotic behaviours, is obtained experimentally from two disk dynamos connected together. It models the geomagnetic field and is used to explain the reversals in its polarity. Actually, the parameters of the chaotic systems exhibit random fluctuation to a greater or lesser extent, which can carefully describe the disturbance made by environmental noise. The global dynamics of the chaotic disk dynamo system with random fluctuating parameters are concerned, and some new results are presented. Based on the generalized Lyapunov function, the globally attractive and positive invariant set is given, including a two-dimensional parabolic ultimate boundary and a four-dimensional ellipsoidal ultimate boundary. Furthermore, a set of sufficient conditions is derived for all solutions of the stochastic disk dynamo system being global convergent to the equilibrium point. Finally, numerical simulations are presented for verification.

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.