Auxiliary State Variables Research Articles

Constrained optimal control problem of oncolytic viruses in cancer treatment

Oncolytic viruses are genetically modified viruses that selectively infect and destroy cancer cells while leaving normal and healthy cells intact. Most previous studies did not consider realistic constraints such as daily or total injection amount of oncolytic viruses; consequently, a gap exists between the research results and the real-world setting. This study aimed to investigate a constrained optimal control problem in tumor treatment utilizing oncolytic viruses, based on a mathematical model. This constraint reflects the practical scenario where the total dosage of oncolytic viruses that can be administered is limited. We introduced auxiliary state variables to address these limitations and employed the penalty function method. Furthermore, we established the existence of an optimal solution for the optimal control problem and derived the corresponding optimality system. Numerical simulations demonstrate the effectiveness of an approach that involves initiating treatment with a high dosage and gradually reducing it over time. Additionally, we confirmed that applying the optimal control strategy leads to a significant reduction in the number of cancer cells compared with uniform dosing, even when the same quantity of virus is administered.

A distributed robust state estimation method based on alternating direction method of multipliers for integrated electricity‐heat system

AbstractIntegrated electricity‐heat system (IEHS) has been paid more and more attention in recent years for its advantage in improving energy efficiency, reducing carbon emissions and increasing renewable energy penetration. To ensure the safety, reliability and economic operation of IEHS, several centralised state estimation (SE) methods for IEHS have been proposed. However, power systems and heat systems often belong to different management entities, and there are industrial barriers such as information privacy, operational differences, and target differences between them, which leads to less applicability of centralised SE methods. In addition, the robustness of existing distributed SE methods for IEHS is not satisfactory. To this end, a distributed robust state estimation (DRSE) model for IEHS based on the alternating direction method of multipliers (ADMM) is proposed. Firstly, by introducing auxiliary state variables and measurements, a robust linear SE model based on weighted least absolute values (WLAV) is proposed. Then, second‐order cone constraints composed of auxiliary state variables are added to the SE model, leading a SOCP‐based robust SE model. Finally, the ADMM algorithm is used to solve the proposed SOCP‐based robust SE model. Simulations demonstrate that the proposed method has higher estimation accuracy in both general and strongly correlated adverse data tests and also can ensure data privacy, good robustness and high estimation accuracy. This indicates that the method proposed has good robustness and solves the problem of weak robustness of existing distributed static state estimation methods.

Observability analysis and optimization for angles-only navigation of distributed space systems

Angles-only navigation methods are compelling for distributed space systems (DSS) such as swarms and constellations. However, complex dependencies between state observability and system parameters present a challenging design problem. This paper proposes a unified angles-only observability analysis and design framework enabling designers of a DSS to 1) analytically determine whether its orbit state is observable, 2) numerically estimate its expected navigation performance, and 3) intelligently optimize the system to meet navigation requirements. First, a new system measurement topology representation is proposed for which analytic orbit observability can be assessed via a set of graphical conditions. Second, methods for numeric estimation of the achievable state covariance are augmented with auxiliary state variables, dynamics uncertainty, and measurement availability constraints. Third, a system cost function is developed and the topological and numeric methods are placed within a quasi-Newton optimization framework to enable automatic system design. The optimization is applied to a distributed science swarm and a space situational awareness constellation in lunar orbit. Both scenarios converge to a global cost minimum and output designs that achieve user requirements under realistic measurement conditions and constraints. Combined analytic and numeric methods therefore presents a powerful tool for design of angles-only DSS.

Neuro-adaptive control for searching generalized Nash equilibrium of multi-agent games: A two-stage design approach

This paper investigates the generalized Nash equilibrium searching problem of multi-agent games over weight-unbalanced directed networks. In this problem, the feasible action set of each agent is not only constrained by private convex sets and shared coupling equality, but also constrained by its own uncertain dynamics. Moreover, the local cost function is related to both his own actions and others, and each agent is only allowed access to neighbor information. To address this problem, we propose a neuro-adaptive control strategy based on two-stage design. In the first stage of design process, an approximator with adaptive-gain is designed to generate a virtual trajectory that converges to a necessary Nash equilibrium, the compensator based on consensus-tracking is used to obtain non-neighbor information and several auxiliary state variables are used to exchange information with local neighborhoods on a digraphs to deal with equality constraints. In the second stage of the strategy, we further developed a neuro-adaptive tracking controller based on one-step design for tracking the virtual trajectory. In the controller design, the virtual control variables and the actual control laws are obtained in a collective way, without repeating the design process. We introduce a variable called the minimum learning parameter to reduce the number of online learning parameters of the neural network. By using tools from variational inequality theory and Lyapunov stability theory, we prove that the proposed control strategy can drive all agents reach Nash equilibrium. Finally, the effectiveness of control strategy is verified by simulation example.

A Wiener path integral technique for determining the stochastic response of nonlinear oscillators with fractional derivative elements: A constrained variational formulation with free boundaries

A technique based on the concept of Wiener path integral (WPI) is developed for determining approximately the joint response probability density function (PDF) of nonlinear oscillators endowed with fractional derivative elements. Specifically, first, the dependence of the state of the system on its history due to the fractional derivative terms is accounted for, alternatively, by augmenting the response vector and by considering additional auxiliary state variables and equations. In this regard, the original single-degree-of-freedom (SDOF) nonlinear system with fractional derivative terms is cast, equivalently, into a multi-degree-of-freedom (MDOF) nonlinear system involving integer-order derivatives only. From a mathematics perspective, the equations of motion referring to the latter can be interpreted as constrained. Second, to circumvent the challenge of increased dimensionality of the problem due to the augmentation of the response vector, a WPI formulation with mixed fixed/free boundary conditions is developed for determining directly any lower-dimensional joint PDF corresponding to a subset only of the response vector components. This can be construed as an approximation-free dimension reduction approach that renders the associated computational cost independent of the total number of stochastic dimensions of the problem. Thus, the original SDOF oscillator joint PDF corresponding to the response displacement and velocity is determined efficiently, while circumventing the computationally challenging task of treating directly equations of motion involving fractional derivatives. Two illustrative numerical examples are considered for demonstrating the reliability of the developed technique. These pertain to a nonlinear Duffing and a nonlinear vibro-impact oscillators with fractional derivative elements subjected to combined stochastic and deterministic periodic loading. Note that alternative standard approximate techniques, such as statistical linearization, need to be significantly modified and extended to account for such cases of combined loading. Remarkably, it is shown herein that the WPI technique exhibits the additional advantage of treating such types of excitation in a straightforward manner without the need for any ad hoc modifications. Comparisons with pertinent Monte Carlo simulation data are included as well.

Unscented Kalman filter for airship model uncertainties and wind disturbance estimation.

An airship is lighter than an air vehicle with enormous potential in applications such as communication, aerial inspection, border surveillance, and precision agriculture. An airship model is made up of dynamic, aerodynamic, aerostatic, and propulsive forces. However, the computation of aerodynamic forces remained a challenge. In addition to aerodynamic model deficiencies, airship mass matrix suffers from parameter variations. Moreover, due to the lighter-than-air nature, it is also susceptible to wind disturbances. These modeling issues are the key challenges in developing an efficient autonomous flight controller for an airship. This article proposes a unified estimation method for airship states, model uncertainties, and wind disturbance estimation using Unscented Kalman Filter (UKF). The proposed method is based on a lumped model uncertainty vector that unifies model uncertainties and wind disturbances in a single vector. The airship model is extended by incorporating six auxiliary state variables into the lumped model uncertainty vector. The performance of the proposed methodology is evaluated using a nonlinear simulation model of a custom-developed UETT airship and is validated by conducting a kind of error analysis. For comparative studies, EKF estimator is also developed. The results show the performance superiority of the proposed estimator over EKF; however, the proposed estimator is a bit expensive on computational grounds. However, as per the requirements of the current application, the proposed estimator can be a preferred choice.

Novel Heterogeneous Mode-Dependent Impulsive Synchronization for Piecewise T-S Fuzzy Probabilistic Coupled Delayed Neural Networks

This article investigates the heterogeneous impulsive synchronization for T-S fuzzy probabilistic coupled delayed neural networks (CDNNs) with mode-dependent parameters and piecewise membership functions. To begin with, a novel CDNNs model with adjustable coupling strength and probabilistic coupling delays is designed to ensure the accuracy of the CDNNs model. Meanwhile, the generalized isolated node with four types of mismatched parameters, named heterogeneous isolated delayed neural network, is first considered to extend the synchronization problem. Then, the mode-dependent fuzzy rules are introduced to design the novel model, which implies that switching signals and fuzzy processes are interdependent and can share information to communicate. To improve the hybrid controller’s reliability, the mode-dependent impulses are also developed here, in which the impulsive effects with different properties can occur at any moment in the switching interval. The exponential synchronization conditions are derived by means of the method of auxiliary state variables, Lyapunov–Krasovskii functional, average switching dwell period, and mode-dependent average impulsive dwell period. Moreover, the improved mode-dependent piecewise approximated membership functions are proposed to reduce the main results’ conservatism. Finally, a numerical example is provided to illustrate the effectiveness of the main results.

Dynamic Pinning Synchronization of Fuzzy-Dependent-Switched Coupled Memristive Neural Networks With Mismatched Dimensions on Time Scales

This article addresses the problem of dynamic pinning synchronization of fuzzy-dependent-switched (Fds) coupled memristive neural networks (CMNNs) with mismatched dimensions on time scales. To begin with, the probabilistic coupling delays, time scales, mismatched dimensions, and function projective synchronization rules are considered to design the novel CMNNs to improve the reliability and generalization ability of the model. Then Fds rules and dynamic pinning control (DPC) method are adopted to design the CMNNs, which can effectively promote the information exchange between the switching signals and the fuzzy processes and can improve the utilization of the communication bandwidth between the nodes of CMNNs. Meanwhile, the method of constructing auxiliary state variables is adopted here to deal with the presented model, so that the coupled and isolated systems with different dimensions can realize information exchange and data sharing. This method also provides a solution for researchers by using low-dimensional systems to estimate or synchronize high-dimensional systems. Moreover, by means of Lyapunov–Krasovskii functional, auxiliary orthogonal matrix, and some inequality processing techniques, the conditions of modified function projective synchronization for Fds CMNNs are derived via the DPC on time scales. Finally, two numerical examples are provided to illustrate the effectiveness of the main results.

Airship aerodynamic model estimation using unscented Kalman filter

An airship model is made-up of aerostatic, aerodynamic, dynamic, and propulsive forces and torques. Besides others, the computation of aerodynamic forces and torques is difficult. Usually, wind tunnel experimentation and potential flow theory are used for their calculations. However, the limitations of these methods pose difficulties in their accurate calculation. In this work, an online estimation scheme based on unscented Kalman filter (UKF) is proposed for their calculation. The proposed method introduces six auxiliary states for the complete aerodynamic model. UKF uses an extended model and provides an estimate of a complete state vector along with auxiliary states. The proposed method uses the minimum auxiliary state variables for the approximation of the complete aerodynamic model that makes it computationally less intensive. UKF estimation performance is evaluated by developing a nonlinear simulation environment for University of Engineering and Technology, Taxila (UETT) airship. Estimator performance is vali dated by performing the error analysis based on estimation error and 2-σ uncertainty bound. For the same problem, the extended Kalman filter (EKF) is also implemented and its results are compared with UKF. The simulation results show that UKF successfully estimates the forces and torques due to the aerodynamic model with small estimation error and the comparative analysis with EKF shows that UKF improves the estimation results and also it is more suitable for the under-consideration problem.

Robust fault estimation and fault-tolerant control for nonlinear Markov jump systems with time-delays

The problem of the robust fault estimation and robust fault-tolerant control, for a class of nonlinear time-delayed Markov jump systems (MJSs) with both actuator and sensor faults, was studied. Firstly, by extending the system state, the actuator fault state vector, the sensor fault state vector and the original system state vector were extended to auxiliary state variables, and the original system was extended to a generalized system. Then, for the singular system, a generalized observer was designed to achieve the simultaneous estimation of its actuator faults, sensor faults and original system state. In addition, based on the observer estimation, a state feedback fault-tolerant controller was designed to make the closed-loop system stable and meet certain performance indicators. By solving linear matrix inequality, sufficient conditions for the existence of generalized observer and fault-tolerant controller were given. Finally, a numerical example and a practical example were given to demonstrate the effectiveness of the proposed approach.

A Robust State Estimation Method Based on SOCP for Integrated Electricity-Heat System

With the application and development of combined heat and power (CHP) techniques, the coupling of electrical and heat energy is gradually increasing. State estimation (SE) is essential to achieve the comprehensive and accurate operating status in an integrated electricity-heat system (IEHS). This article proposes a robust SE method based on second-order conic programming (SOCP) for IEHS. Firstly, auxiliary state variables and measurements are introduced to obtain the exact linear measurement equations for IEHS. Then, a robust SE model based on weighted least absolute values (WLAV) for IEHS is proposed. To further improve the estimation performance, the constraint conditions satisfied by auxiliary state variables are relaxed into second-order cone constraints and added to the proposed WLAV model, leading a SOCP-based robust SE method. Simulations demonstrate that the proposed method ensures a global optimal solution and has good robustness and high estimation accuracy.

Distributed Nash Equilibrium Seeking in an Aggregative Game on a Directed Graph

An aggregative game with local constraint sets is studied in this article, where each player's cost function is dependent on the aggregation function that is unavailable to all players. To compute the Nash equilibrium (NE) point of the game in a distributed manner, the players are endowed with several auxiliary state variables that are used to estimate the aggregation function by exchanging their estimates with local neighbors on a directed graph. In the two cases with strongly connected weight-balanced and weight-unbalanced directed graphs, respectively, NE seeking strategies are proposed by the interconnection of projected gradient-play with average consensus dynamics. The proposed algorithms are proved to be able to reach the NE point by using tools from variational inequality theory and Lyapunov stability theory. Finally, an example is simulated to demonstrate the theoretical results.

A Hybrid Model of Opinion Dynamics With Memory-Based Connectivity

Given a social network where the individuals know the identity of the other members, we present a model of opinion dynamics where the connectivity among the individuals depends on both their current and past opinions. Thus, their interactions are not only based on the present states but also on their past relationships. The model is a multi-agent system with active or inactive pairwise interactions depending on auxiliary state variables filtering the instantaneous opinions, thereby taking the past experience into account. When an interaction is (de)activated, a jump occurs, leading to a hybrid model. The proven stability properties ensure that opinions converge to local agreements/clusters as time grows. Simulation results are provided to illustrate the theoretical guarantees.

Estimation of Airship Aerodynamic Forces and Torques Using Extended Kalman Filter

An airship is a lighter than air, aerial vehicle whose model is based on dynamic, aerodynamic, aerostatic and propulsion forces and torques. Apart from other, aerodynamic forces and toques are difficult to measure. In this work, an estimation scheme for aerodynamic forces and torques based on the Extended Kalman Filter (EKF) is presented. It is assumed that the airship attitude and position estimates are available. EKF estimates the airship body axes linear and angular velocities and aerodynamic forces and torques. As the method measures a complete aerodynamic model instead of measuring its individual parameters by utilizing minimum auxiliary state variables, it is computationally non-intensive and can provide online aerodynamic model information that can be used in controller implementation in a real-time environment. Nonlinear simulation environment is developed for the experimental airship and EKF performance is evaluated. For validating the estimator's performance, 3-σ uncertainty bounds and error analysis, estimator convergence analysis and it's closed-loop simulations with Sliding Mode Controller have been performed. The simulation results show that EKF successfully estimates the airship states and aerodynamic forces and torques with minimum estimation error enhancing the model-based nonlinear controller performance.

LQ Synchronization of Discrete-Time Multiagent Systems: A Distributed Optimization Approach

The linear quadratic (LQ) synchronization problem for multiagent systems is solved by developing a distributed algorithm. It is the first time in the literature to formulate the LQ synchronization problem consisting of auxiliary synchronization state variables in discrete time with a finite horizon. The solution to this LQ synchronization problem is first considered in a centralized setting by leveraging connections to an alternating direction method of multiplier and then extended to a distributed setting, in which the individual agent's control inputs can be computed independently, thereby making the solution scalable. Numerical examples for both homogeneous and heterogeneous multiagent systems are given to demonstrate the effectiveness of the proposed method.

Pricing bounds and bang-bang analysis of the Polaris variable annuities

This paper studies the no-arbitrage pricing of the ‘Polaris Income Plus Daily’ structured in the ‘Polaris Choice IV’ variable annuities recently issued by the American International Group. Distinct from the withdrawal benefits studied in the literature, Polaris allows the income base to ‘lock in’ the high water mark of the investment account over a certain monitoring period which is related to the timing of the policyholder's first withdrawal. By prudently introducing certain auxiliary state and decision variables, we manage to formulate the pricing model under a Markovian stochastic optimal control framework. By a slight modification of the fee structure, we show the existence of a bang-bang solution to the stochastic control problem: the optimal withdrawal strategy is among a few explicit choices. We consequently design a novel Least Squares Monte Carlo (LSMC) algorithm to approach the optimal solution. Convergence results are established for the algorithm by applying the theory of nonparametric sieve estimation. Compared with existing LSMCs, our algorithm possesses a number of advantages such as memory reduction, preservation of convexity and monotonicity of the continuation value, reducing computational cost of the tuning parameter selection, and evading extrapolation of the value function estimate. Finally, we prove that the obtained pricing result works as an upper bound of the no-arbitrage price of Polaris with the real fee structure. Numerical experiments show that this upper bound is fairly tight.

Nonlinear Backstepping Control Design for Coupled Nonlinear Systems under External Disturbances

A nonlinear backstepping control is proposed for the coupled normal form of nonlinear systems. The proposed method is designed by combining the sliding‐mode control and backstepping control with a disturbance observer (DOB). The key idea behind the proposed method is that the linear terms of state variables of the second subsystem are lumped into the virtual input in the first subsystem. A DOB is developed to estimate the external disturbances. Auxiliary state variables are used to avoid amplification of the measurement noise in the DOB. For output tracking and unmatched disturbance cancellation in the first subsystem, the desired virtual input is derived via the backstepping procedure. The actual input in the second subsystem is developed to guarantee the convergence of the virtual input to the desired virtual input by using a sliding‐mode control. The stability of the closed‐loop is verified by using the input‐to‐state stable (ISS) property. The performance of the proposed method is validated via numerical simulations and an application to a vehicle system based on CarSim platform.

Distributed optimization for multi-agent systems with constraints set and communication time-delay over a directed graph

This paper is concerned with the problem of distributed optimization for a multi-agent system with constraints set and communication time-delay over a directed graph. The considered cost function is a summation of all local cost functions associated with each agent. Firstly, a novel distributed algorithm is developed to solve such a problem, where auxiliary state variables are also exchanged to compensate the nonzero gradient of local cost function and accelerate the convergence of estimate states to the optimal point. Secondly, the minimizer of distributed optimization of a multi-agent network is determined by the variational inequality in spite of the existence of time delay. Furthermore, delay-dependent and delay-free sufficient conditions on the convergence of states of agents to the optimal point are derived by constructing a new Lyapunov–Krasovskii functional, respectively. Finally, a numerical example and a comparison are provided to validate the obtained results.

Pricing Bounds and Bang-Bang Analysis of the Polaris Variable Annuities

This paper studies the no-arbitrage pricing of the Income Plus Daily structured the Choice IV variable annuities recently issued by the American International Group. Distinct from the withdrawal benefits studied the literature, Polaris allows the income base to lock in the high water mark of the investment account over a certain monitoring period which is related to the timing of policyholder's first withdrawal. By prudently introducing certain auxiliary state and decision variables, we manage to formulate the pricing model under a Markovian stochastic optimal control framework. By a slight modification of the fee structure, we show the existence of a bang-bang solution to the stochastic control problem: the optimal withdrawal strategy is among a few explicit choices. We consequently design a novel Least Squares Monte Carlo (LSMC) algorithm to approach the optimal solution. Convergence results are established for the algorithm by applying the theory of nonparametric sieve estimation. Compared with existing LSMCs, our algorithm possesses a number of advantages such as memory reduction, preserving convexity and monotonicity of the continuation value, reducing the computational cost of the tuning parameter selection, and evading extrapolating value function estimate. Finally, we prove that the obtained pricing result works as an upper bound of the no-arbitrage price of Polaris with the real fee structure. Numerical experiments show that this upper bound is fairly tight.

Modeling of static friction in closed-loop kinematic chains\u2014Uniqueness and parametric sensitivity problems

Problems with uniqueness and high parametric sensitivity of the solution of equations of motion, encountered in the static friction regime, are addressed. Friction in joints of a multibody system with closed-loop kinematic chains is discussed. Three different models of friction are studied: the discontinuous Coulomb model with stiction regime represented in terms of additional constraints; the approximate Coulomb model, smoothed in the vicinity of zero relative velocity; and the LuGre model with presliding displacements represented in terms of auxiliary state variables. Firstly, a rigid body model is investigated. It is shown that in the case of constraint addition approach, problems with uniqueness of solution emerge in the static friction regime. In the case of continuous models of friction, the solution in the stiction regime and its vicinity is highly sensitive to some hardly measurable or arbitrarily chosen parameters of the model of friction. Origins of nonuniqueness and high sensitivity are investigated, and the questionable credibility of the stiction regime simulation results is discussed. Secondly, a simplified model of body and joint elasticity is introduced to investigate the impact of flexibility on the mechanism frictional behavior. It is shown that taking the flexibility into consideration may eliminate the uniqueness and sensitivity problems. Moreover, the quantities that represent flexibility may be regarded as the key factors influencing the results of stiction regime simulation. Five examples are provided to illustrate the presented considerations.