Invariant Set Theory Research Articles

Constrained State Regulation Problem of Descriptor Fractional-Order Linear Continuous-Time Systems

This paper deals with the constrained state regulation problem (CSRP) of descriptor fractional-order linear continuous-time systems (DFOLCS) with order 0<α<1. The objective is to establish the existence of conditions for a linear feedback control law within state constraints and to propose a method for solving the CSRP of DFOLCS. First, based on the decomposition and separation method and coordinate transformation, the DFOLCS can be transformed into an equivalent fractional-order reduced system; hence, the CSRP of the DFOLCS is equivalent to the CSRP of the reduced system. By means of positive invariant sets theory, Lyapunov stability theory, and some mathematical techniques, necessary and sufficient conditions for the polyhedral positive invariant set of the equivalent reduced system are presented. Models and corresponding algorithms for solving the CSRP of a linear feedback controller are also presented by the obtained conditions. Under the condition that the resulting closed system is positive, the given model of the CSRP in this paper for the DFOLCS is formulated as nonlinear programming with a linear objective function and quadratic mixed constraints. Two numerical examples illustrate the proposed method.

PiP-X: Online feedback motion planning/replanning in dynamic environments using invariant funnels

Computing kinodynamically feasible motion plans and repairing them on-the-fly as the environment changes is a challenging, yet relevant problem in robot navigation. We propose an online single-query sampling-based motion re-planning algorithm using finite-time invariant sets, commonly referred to as “ funnels”. We combine concepts from nonlinear systems analysis, sampling-based techniques, and graph-search methods to create a single framework that enables feedback motion re-planning for any general nonlinear dynamical system in dynamic workspaces. A volumetric network of funnels is constructed in the configuration space using sampling-based methods and invariant set theory; and an optimal sequencing of funnels from robot configuration to a desired goal region is then determined by computing the shortest-path subgraph (tree) in the network. Analyzing and formally quantifying the stability of trajectories using Lyapunov level-sets ensures kinodynamic feasibility and guaranteed set-invariance of the solution paths. Though not required, our method is capable of using a pre-computed library of motion primitives to speedup online computation of controllable motion plans that are volumetric in nature. We introduce a novel directed-graph data structure to represent the funnel-network and its inter-sequencibility; helping us leverage discrete graph-based incremental search to quickly rewire feasible and controllable motion plans on-the-fly in response to changes in the environment. We validate our approach on a simulated cart-pole, car-like robot, and 6DOF quadrotor platform in a variety of scenarios within a maze and a random forest environment. Using Monte Carlo methods, we evaluate the performance in terms of algorithm success, length of traversed trajectory, and runtime.

Robust Shared Control for Four-Wheel Steering Considering Driving Comfort and Vehicle Stability

Although the four-wheel steering system expands the flexibility of vehicle control, it also brings the problem of difficult coordination between driver comfort and vehicle stability. To this end, this paper proposes robust coordinated control for a four-wheel steering (4WS) vehicle considering driving comfort and vehicle stability. First, the vehicle dynamics model is constructed to reflect the lateral motion characteristics of a 4WS vehicle. Then, the driver model is coupled into the 4WS vehicle model to describe the driver’s handling characteristics. To suppress the system perturbation caused by the uncertainties of driver behavior and vehicle states, the Takagi-Sugeno fuzzy robust control method is developed to design the human-machine co-driving system. Moreover, the robust positive invariant set theory is used to guarantee the stability and safety constraints of the vehicle. Finally, the proposed human-machine shared robust control for 4WS vehicle is verified through the driving simulator platform. The results indicate that the fuzzy robust shared control approach comprehensively improves the driving comfort, vehicle stability, and path tracking.

Flexible Performance-Based Neural Network Control for Mechanical Systems Under DoS Attack and Event Triggering

In this work, the flexible performance-based control scheme is presented for the networked uncertain mechanical system under the quasi-periodic denial-of-service (DoS) attack and the appointed event-triggering (ET) mechanism. This scheme features the capability of releasing and recovering the prescribed performance in light of the activation and sleeping of the DoS attack. These functionalities attribute to the introduction of a self-regulation auxiliary system that is developed based on the theories of the positive system and the finite-time stability. Furthermore, the neural network (NN) is utilized to approximate the system uncertainty, and the noncertainty-equivalent structure is adopted to construct the update law of the NN. Then, the influence of the quasi-periodic DoS attack and the appointed ET mechanism on uncertainty approximation are totally isolated. Based on the Lyapunov function method and the invariant set theory, the boundedness of the closed-loop system is theoretically proved. Finally, both the numerical and experimental illustrations are presented to demonstrate the effectiveness and the benefits of the developed control scheme.

On Consistency, Viability, and Admissibility in Constrained Ensemble and Hierarchical Control Systems

Several control architectures, such as decentralized, distributed and hierarchical control, have been elaborated over the past decades for controlling systems composed of a set of subsystems. However, computational complexity and constraint satisfaction are still challenging tasks. We present an approach to control an ensemble of similar heterogeneous systems with input and state constraints via an identical control input. This control input is globally admissible and computed based on an aggregated system that reflects the overall behavior of the ensemble. To limit the computational complexity of the control task, the aggregated system is designed such that its dimension is independent of the number of subsystems. To guarantee viability, i.e. state constraint satisfaction for all times, appropriate consistency conditions are derived based on invariant set theory. The presented approach is illustrated with a numerical example.

Data-Driven Fault Compensation Tracking Control for Coupled Wastewater Treatment Process

Dear Editor, This letter is concerned with the data-driven fault compensation tracking control for a coupled wastewater treatment process (WWTP) subject to sensor faults. Invariant set theory is introduced to eliminate the completely bounded and differentiable conditions of coupled non-affine dynamics and to explicitly express the control inputs. An adaptive fault compensation mechanism is constructed to accommodate the effects of sensor faults. By employing a cubic absolute-value Lyapunov criteria, it is shown that all the signals are bounded and the tracking error converges to an adjustable neighborhood near the origin. Experiment studies are executed on a standardized platform of WWTP to illustrate the effectiveness of the proposed strategy.

Set-based fast gradient projection algorithm for model predictive control of grid-tied power converters

Model Predictive Control (MPC) has attracted much attention and is widely used in power electronics. However, implementing the MPC algorithm is still a difficult task due to the fast dynamics of power converters and strict time constraints. In this paper, a computationally efficient MPC algorithm for grid-tied power converters based on the fast gradient projection method and invariant set theory is proposed. The algorithm is implemented and tested through hardware-in-the-loop simulations using Texas Instruments digital signal processors and Xilinx Field Programmable Gate Arrays platforms.

Decentralized Adaptive Fault-Tolerant Cooperative Control for Multiple UAVs with Input Saturation and State Constraints

This paper proposes a fault-tolerant cooperative control (FTCC) scheme for multiple UAVs in a distributed communication network against input saturation, full-state constraints, actuator faults, and unknown dynamics. Firstly, by considering physical limitations, an auxiliary control signal is designed to simplify the analysis process. Secondly, to avoid the difficulty in the back-stepping design caused by full-state constraints, virtual control signals are constructed to transform constrained variables, while the dynamic surface control is adopted to avoid the phenomenon of “differential explosion.” Thirdly, a disturbance observer (DO) is designed to estimate the unknown uncertainty caused by parameter uncertainty and actuator fault. Moreover, a recurrent wavelet fuzzy neural network (RWFNN) is used to compensate for the estimation errors of DO. Finally, it is proved that all states are uniformly ultimately bounded (UUB) by Lyapunov and invariant set theory. The effectiveness of the proposed scheme is further demonstrated by the simulation results.

Supermeasured: Violating Bell-Statistical Independence Without Violating Physical Statistical Independence

Bell’s theorem is often said to imply that quantum mechanics violates local causality, and that local causality cannot be restored with a hidden-variables theory. This however is only correct if the hidden-variables theory fulfils an assumption called Statistical Independence. Violations of Statistical Independence are commonly interpreted as correlations between the measurement settings and the hidden variables (which determine the measurement outcomes). Such correlations have been discarded as “fine-tuning” or a “conspiracy”. We here point out that the common interpretation is at best physically ambiguous and at worst incorrect. The problem with the common interpretation is that Statistical Independence might be violated because of a non-trivial measure in state space, a possibility we propose to call “supermeasured”. We use Invariant Set Theory as an example of a supermeasured theory that violates the Statistical Independence assumption in Bell’s theorem without requiring correlations between hidden variables and measurement settings (physical statistical independence).

A Comparative Study of Robust MPC and Stochastic MPC of Wind Power Generation System

In this paper, a complete comparison analysis of two advanced control algorithms, namely robust model predictive control (MPC) and stochastic MPC, is performed in order to optimize the operation of a wind power generation system (WPGS). The power maximization often conflicts with the mechanical load experienced by the turbine in the full-load region (i.e., the higher the power extracted, the higher the load) under the wind speed disturbance, thereby leading to high maintenance cost resulting from the fatigue damage. Thus, a typical 5 MW wind turbine operating in a high-speed region is considered to guarantee system security and economy. The robust MPC is designed by utilizing the min–max framework to track steady-state optimum operating reference trajectory with the deterministic constraint of output power, while the stochastic MPC is constructed by incorporating the invariant set theory to also ensure the system security subjecting to the probabilistic constraint of output power. The relation between the constraints and the implications on optimal performance are also studied. Comprehensive simulations on a mechanism model and FAST simulator are carried out to demonstrate the validation of the two control methods under various scenarios. It is discovered that when wind speed in the near future can be predicted and utilized in controller design, the stochastic MPC can effectively reduce the maintenance cost by suppressing the constraint violation rate compared to robust MPC with a similar energy utilization due to the incorporation of the stochastic characteristics of wind speed.

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.

Asymptotic Stabilization of Delayed Linear Fractional-Order Systems Subject to State and Control Constraints

Studies have shown that fractional calculus can describe and characterize a practical system satisfactorily. Therefore, the stabilization of fractional-order systems is of great significance. The asymptotic stabilization problem of delayed linear fractional-order systems (DLFS) subject to state and control constraints is studied in this article. Firstly, the existence conditions for feedback controllers of DLFS subject to both state and control constraints are given. Furthermore, a sufficient condition for invariance of polyhedron set is established by using invariant set theory. A new Lyapunov function is constructed on the basis of the constraints, and some sufficient conditions for the asymptotic stability of DLFS are obtained. Then, the feedback controller and the corresponding solution algorithms are given to ensure the asymptotic stability under state and control input constraints. The proposed solution algorithm transforms the asymptotic stabilization problem into a linear/nonlinear programming (LP/NP) problem which is easy to solve from the perspective of computation. Finally, three numerical examples are offered to illustrate the effectiveness of the proposed method.

Nodal solutions of semi-linear elliptic equations with dependence on the gradient

In the present paper we prove the existence of a nontrivial nodal solution of a semi-linear elliptic equation depending on the gradient. Our approach makes use of the theory of invariant sets of descending flow to localize the solution.

Nontrivial solutions for nonlinear discrete boundary value problems of the fourth order

We study the existence of multiple nontrivial solutions for nonlinear fourth order discrete boundary value problems. We first establish criteria for the existence of at least two nontrivial solutions of the problems and obtain conditions to guarantee that the two solutions are sign-changing. Under some appropriate assumptions, we further prove that the problems have at least three nontrivial solutions, which are positive, negative, and sign-changing, respectively. We include two examples to illustrate the applicability of our results. Our theorems are proved by employing variational approaches, combined with the classic mountain pass lemma and a result from the theory of invariant sets of descending flow.

Bell's theorem, non-computability and conformal cyclic cosmology: A top-down approach to quantum gravity

This paper draws on a number of Roger Penrose's ideas—including the non-Hamiltonian phase-space flow of the Hawking Box, conformal cyclic cosmology, non-computability, and gravitationally induced quantum state reduction—in order to propose a radically unconventional approach to quantum gravity: Invariant Set Theory (IST). In IST, the fundamental laws of physics describe the geometry of the phase portrait of the universe as a whole: “quantum” process is associated with fine-scale fractal geometry, “gravitational” process with larger-scale heterogeneous geometry. With this, it becomes possible to explain the experimental violation of Bell inequalities without having to abandon key ingredients of general relativity: determinism and local causality. Ensembles in IST can be described by complex Hilbert states over a finite set ℂp of complex numbers, where p is a large finite integer. The quantum mechanics of finite-dimensional Hilbert spaces is emergent as a singular limit when p→∞. A small modification to the field equations of general relativity is proposed to make it consistent with IST.

Voltage regulation of buck converter with constant power load: An adaptive power shaping control

In this paper, the problem of voltage regulation of DC–DC buck converter with unknown constant power load (CPL) and input voltage is addressed. A power shaping controller (PSC) based on the passivity theory is proposed to stabilize this system. In addition, borrowing immersion & invariance (I&I) technique, an observer can be designed to estimate CPL and input voltage simultaneously. By combining PSC with I&I observer, an adaptive control scheme is achieved. Besides, the estimate of the domain of attraction is given to rigorously guarantee the convergence property by virtue of the invariant set theory. Then, the locally asymptotic stability of the closed-loop system is established. Finally, the simulation and experimental results show the effectiveness of the proposed control method.

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.

There is no compact metrizable space containing all continua as unique components

We answer a question of Piotr Minc by proving that there is no compact metrizable space whose set of components contains a unique topological copy of every metrizable compactification of a ray (i.e. a half-open interval) with an arc (i.e. closed bounded interval) as the remainder. To this end we use the concept of Borel reductions coming from Invariant descriptive set theory. It follows as a corollary that there is no compact metrizable space such that every continuum is homeomorphic to exactly one component of this space.

Maximal almost disjoint families, determinacy, and forcing

We study the notion of [Formula: see text]-MAD families where [Formula: see text] is a Borel ideal on [Formula: see text]. We show that if [Formula: see text] is any finite or countably iterated Fubini product of the ideal of finite sets [Formula: see text], then there are no analytic infinite [Formula: see text]-MAD families, and assuming Projective Determinacy and Dependent Choice there are no infinite projective [Formula: see text]-MAD families; and under the full Axiom of Determinacy [Formula: see text][Formula: see text] or under [Formula: see text] there are no infinite [Formula: see text]-mad families. Similar results are obtained in Solovay’s model. These results apply in particular to the ideal [Formula: see text], which corresponds to the classical notion of MAD families, as well as to the ideal [Formula: see text]. The proofs combine ideas from invariant descriptive set theory and forcing.

Grid-Supporting Three-Phase Inverters With Inherent Root Mean Square Current Limitation Under Balanced Grid Voltage Sags

In this article, a novel nonlinear droop control method is proposed for three-phase grid-supporting inverters that rigorously guarantee limited rms value of the inverter current and closed-loop system stability under both normal grid conditions and balanced voltage sags. Contrary to the traditional dq frame approaches that align the inverter output voltage with the d-axis, the proposed controller aligns the inverter current with the d-axis resulting in the desired current limitation and detailed closed-loop system stability conditions. Inspired by the recently presented state-limiting PI controller and using nonlinear invariant set theory, it is rigorously proven that the rms value of the inverter current is always limited below a given value, even during transients or faults, without requiring additional adaptive saturation units, as commonly applied in conventional approaches. Furthermore, analytic conditions for the controller parameter selection are provided to ensure asymptotic stability for the entire closed-loop grid supporting inverter for the first time without depending on particular values of the filter and line parameters. To verify the effectiveness of the proposed controller compared to existing current-limiting control methods, extensive simulation and experimental results of a three-phase inverter are provided under a normal grid and under different balanced voltage sag scenarios.