Time-scaling Transformation Research Articles

A switched system approach for the direct solution of singular optimal control problems

Optimal control problems where the control variable appears linearly in the formulation lead to singular control arcs that introduce analytical and numerical complications during the solution process. In this work, the switching behavior (switching between singular and nonsingular control arcs) is exploited to solve singular optimal control problems using a direct method. The problem is reformulated as a switched system optimal control problem where each control arc defines a different subsystem, i.e., lower and upper limits and a singular arc. The switching between control arcs is handled using binary functions that depend on time. The switched system is embedded into a larger family by relaxing the binary functions to avoid integer variables. The variable switching times at which the control switches between arcs are mapped to fixed points in a new time domain using a time-scaling transformation. A direct transcription method along with control parametrization is applied to transform the embedded problem into a nonlinear programming formulation that contains few variables and constraints. A penalty term is introduced to obtain binary solutions for the optimal selection of control arcs. The proposed reformulation avoids the use of mesh refinement techniques and continuation methods, and allows to solve challenging problems in very short CPU times with high accuracy.

DATT-NGRU: a novel deep learning model with data augmentation for daily stock indexes prediction

PurposeThe stock indexes are an important issue for investors, and in this paper good trading strategies will be aimed to be adopted according to the accurate prediction of the stock indexes to chase high returns.Design/methodology/approachTo avoid the problem of insufficient financial data for daily stock indexes prediction during modeling, a data augmentation method based on time scale transformation (DATT) was introduced. After that, a new deep learning model which combined DATT and NGRU (DATT-nested gated recurrent units (NGRU)) was proposed for stock indexes prediction. The proposed models and their competitive models were used to test the stock indexes prediction and simulated trading in five stock markets of China and the United States.FindingsThe experimental results demonstrated that both NGRU and DATT-NGRU outperformed the other recurrent neural network (RNN) models in the daily stock indexes prediction.Originality/valueA novel RNN with NGRU and data augmentation is proposed. It uses the nested structure to increase the depth of the deep learning model.

Multi-objective optimal control of bioconversion process considering system sensitivity and control variation

This paper aims to determine a time-varying dilution rate of the feed medium which maximizes mean productivity while minimizing both system sensitivity and control cost in glycerol continuous culture. A multi-objective optimal control problem subjected to a nonlinear control system and its auxiliary system is proposed and transformed into a finite-dimensional large-scale multi-objective optimization problem via discretization and time-scaling transformation. A multi-objective competitive swarm optimization algorithm is constructed to solve this transformed problem based on pairwise competition, mutation operation and environment selection. The Pareto front obtained shows the distribution of the optimal target value which indicates the rationality and feasibility of the control system under the approximate optimal strategy. Numerical simulations verify the robustness of the optimal control system and the effectiveness of our numerical solution method. The value of IGD shows the good accuracy and distribution of the proposed multi-objective optimization algorithm.

Sequential adaptive switching time optimization technique for optimal control problems

The control parameterization method used together with time-scaling transformation is an effective approach to transform optimal control problems into optimal parameter selection problems. The approximate problems can then be solved by gradient-based optimization methods. However, the conventional time-scaling transformation requires that all the control component functions switch simultaneously, which can be undesirable in practice. In this paper, we introduce a novel technique where the switching times for each of the control component functions can be adaptively selected. To demonstrate the effectiveness and flexibility of the proposed approach, two example problems with control input and terminal state constraints are solved using the method proposed. Numerical results show that the new method achieves much better objective values with some increase in computation time compared to the conventional time-scaling transformation technique.

Joint state estimation of power battery based on UKPF

Accurate estimation of power battery state is an important function in battery management system. In order to accurately estimate power battery SOC and SOH and improve the performance of long-term estimation of battery SOC, a joint estimation method of power battery state based on UKPF was proposed in this paper. The particle filter algorithm was added on the basis of the unscented Kalman filter algorithm, and the particle filter algorithm was optimized by the unscented Kalman filter algorithm, which improved the particle degradation problem and improved the accuracy of battery state estimation. Based on the time scale transformation, the battery state estimation was completed, and the SOC and SOH were estimated at short and long time scales, respectively. The SOH estimation results were updated to the model parameters for SOC estimation. The results show that the joint estimation method can accurately estimate battery SOC and SOH with an error of less than 3%.

Distributed prescribed-time leader–follower formation control of surface vehicles with unknowns and input saturation

In this paper, the prescribed-time leader–follower formation control problem is solved for surface vehicles (SVs) suffering from input saturation and unknowns including uncertain dynamics and external disturbances. Based on the information of neighboring vehicles, a distributed prescribed-time observer for estimating states of the leader is proposed. By virtue of input saturation, the saturation errors and system unknowns are observed by a reduced-order prescribed-time estimator. Moreover, to realize the satisfactory formation error constraints, a time scale transformation function based prescribed-time prescribed performance function is proposed, together with error transformations, the transient performance of formation errors is improved. Lyapunov stability theorem and backstepping method prove that the closed-loop system is prescribed-time stable. Simulations are given to illustrate the effectiveness of proposed theoretical results.

A Joint Estimation Method Based on Kalman Filter of Battery State of Charge and State of Health

In a battery management system, the accurate estimation of the battery’s state of health (SOH) and state of capacity (SOC) are vital functions. The traditional estimation methods have limitations. To accurately estimate the SOC and SOH of power battery and improve the performance of the long-term estimation of a battery’s SOC, a joint estimation method based on a Kalman filter is proposed in this work. First, a second-order RC equivalent circuit model of a ternary lithium battery was built, whose parameters were identified online, and the model’s accuracy was verified. Then, the battery data under actual working conditions were collected. The SOC and SOH were estimated based on the Kalman filter algorithm, and the simulation was implemented using MATLAB. Finally, according to a time scale transformation, the battery state was jointly estimated, the SOC was estimated at a short-time scale, the SOH was estimated at a long-time scale, and the SOH estimation results were updated to the model parameters for SOC estimation. The results show that the accuracy of the method is very good, and it can effectively improve estimation accuracy and ensure batteries’ long-term estimation performance.

Transition trajectory generation with time-scaled trigonometric series control parameterization

This paper is concerned with shaping the time behavior of controls for generating transition trajectories. Parameterizing controls with trigonometric series enables infinite differentiability with respect to time, which is very appealing in real operations. In practice, one may desire the controls to have a specific distribution over time. However, to achieve this usually relies on judicious selection of the cost function. To this end, this paper introduces a time-scaling transformation for the trigonometric series. It yields an intuitive and flexible way to design the controls tailored to the desired shape of a transition maneuver. The process does not rely on additional design effort for the cost function. The proposed method is applied to generating a transition trajectory that converges to the desired condition earlier in comparison to the unscaled formulation. Numerical results demonstrate the effectiveness of the proposed method.

Research on the Life Prediction Method of Meters Based on a Nonlinear Wiener Process

Due to the high reliability of present meters, it is difficult to obtain the failure time of meters through accelerated life tests. Based on the failure data of the accelerated life test, this paper studies the mathematical model based on the Wiener process and establishes the degradation model of the instrument by the maximum likelihood to estimate the parameters of the Wiener model. With full consideration of the possible nonlinear effects in modeling, the time scale transformation method is used to study and obtain the reliability life prediction model of smart meters based on nonlinear data. Finally, the reliability life prediction model of meters is verified and evaluated through the example data of the accelerated life test of smart meters. Compared with the conventional method, this method has less error in calculating the reliability, greatly saves test time, and has a higher accuracy than estimating the lifetime model using the Wiener process directly.

Use of the Taylor theorem to predict kinetic curves in an arbitrary mechanism

This work serves as a proof of concept by reporting the development of a novel method for the solution of kinetic rate equations based on Taylor polynomials. The key observation is that mass action type rate equations are autonomous ordinary differential equations, which give the first derivative of a concentration as the polynomial of the concentrations of all substances appearing in the system. Two time scale transformations (based on first order and second order formal kinetics) are introduced to provide bounded polynomials in a way that the polynomial nature of the ordinary differential equation obtained with the transformed variable is conserved. Three chemical systems are used to test the performance of the method. The first system is a series of two consecutive reactions in which the first process is second order and the second process is first order with respect to its sole reactant. Kinetic curves in this scheme could be described by this method with a suitably chosen time transformation and a 100-degree polynomial with an accuracy better than 0.1%. The second system is a series of three reactions containing an autocatalytic step, it produces three extrema on the kinetic curve of one of the species (oligooscillation). The third system is the parallel-consecutive bimolecular reaction, for which high simulation accuracy was achieved with minimal effort. The simulations show that the time transformation used in the calculations has a central role, and the second order transformations typically works better than the other one. Also, the examples confirm that extrema on the kinetic traces are the problems.

Subharmonic Bifurcation for a Non-smooth Double Pendulum with Unilateral Impact

A unilateral impact double pendulum model with hinge links is constructed to detect subharmonic bifurcation for the high dimensional non-smooth system. The non-smooth and nonlinear coupled factors lead a barrier for high dimensional conventional nonlinear techniques. By introducing reversible transformation and energy time scale transformation, the system is expressed as a smooth decoupling form of energy coordinates. Thus, the concept of subharmonic Melnikov function is extended to high-dimensional nonsmooth systems, and the influence of impact recovery coefficient on the existence of subharmonic periodic orbits of double pendulum is revealed. The efficiency of the theoretical results is verified by phase portraits, time process portraits and Poincaré section.

Optimal index and averaging principle for Itô–Doob stochastic fractional differential equations

In this paper, a class of Itô–Doob stochastic fractional differential equations (Itô–Doob SFDEs) models are discussed. Using the time scale transformation method, we consider the averaging principle of the transformed equations and establish the relevant results. At the same time, we find that the optimal index for the original Itô–Doob SFDEs can be determined, the selection of such index is similar to the classical stochastic differential equations model.

A time-scale transformation approach to prescribed-time stabilisation of non-holonomic systems with inputs quantisation

This article addresses the problem of global stabilisation using state feedback for a kind of uncertain non-holonomic systems in chained form. Two distinct characteristics of this study are that the considered system suffers from the input-quantised actuator, and the system states are expected to converge to zero in prescribed finite time. To address these, a time-scale transformation, that can change the original stabilisation into the finite-time stabilisation of the transformed one, is first introduced. Then, under the new framework of equivalent transformation, a quantised state feedback controller that achieves of the performance requirements is developed with the aid of the recursive technique. Finally, simulation results of a unicycle-type mobile robot are provided to confirm the efficacy of the proposed approach.

Optimal control of renewable biohydrogen production: A switched system approach*

Biohydrogen produced from microorganisms such as cyanobacteria is a promising low cost, sustainable and environmentally friendly energy source. Recent studies have shown that high biohydrogen yield can be obtained from Cyanothece sp. ATCC 51142 in a fed-batch reactor. This system has been accurately described with a modified Droop model that can be used for optimization studies. Searching for the optimal operating conditions and the switching time from batch to fed-batch operation, such that the biohydrogen production is maximized, leads to a challenging singular optimal control problem. In this study, a novel reformulation based on the theory of switched systems and time-scaling transformation is proposed to address the switching of the operating modes and the optimal control structure. Solutions are found by solving an embedded optimal control problem that can be solved efficiently as a nonlinear programming problem. No mesh refinement is required to capture the switching times. Smooth optimal control profiles and clear switching structures that maximize the biohydrogen yield were found for two types of control parametrization.

High-Resolution Real-Time Imaging Processing for Spaceborne Spotlight SAR With Curved Orbit via Subaperture Coherent Superposition in Image Domain

With the increasing of application requirements, the high-resolution real-time imaging processing of the spaceborne spotlight synthetic aperture radar (SAR) has been developed. Since the traditional real-time imaging algorithms have the problems that the range model has errors and the two-dimensional (2-D) space-variance of the equivalent velocity caused by the curved orbit cannot be effectively eliminated. Thus, this article proposes a high-resolution real-time imaging algorithm for spaceborne spotlight SAR with curved orbit via subaperture coherent superposition in image domain. In this article, the echo data are first divided into subapertures to avoid the azimuth spectrum aliasing. After that, the 2-D space-variance of the equivalent velocity caused by the curved orbit can be eliminated by the method of azimuth time scale transformation, higher order phase compensation, and introducing phase transition function. Then, the dechirp function is applied for the subaperture signals to obtain the partial-resolution subaperture images. Finally, these partial-resolution subaperture images are coherently superposed in the image domain to obtain the final full-resolution image of the whole echo data. Moreover, the proposed algorithm improves the real-time performance by adopting the idea that the subaperture data recording and subaperture real-time imaging processing are synchronized, which greatly accelerates the acquisition of the final full-resolution imaging result. At the end of this article, the simulations and the real-time performance analysis are performed to validate the proposed algorithm.

A finite-time maintenance policy of deteriorating systems with multiple dependent performance characteristics

Condition monitoring and health management of deteriorating systems have attracted significant attentions in reducing failure rate and ensuring normal operation of systems. However, related existing methodologies usually confront difficulty when applied to systems with multivariate performance characteristics (PCs) in the presence of model uncertainty. In this work, a periodic inspection and maintenance policy based on bivariate degradation process is proposed, where each PC is modeled by a gamma process with time-scale transformation. Further, copula functions are used to model the dependency between degradation increments. The uncertainty of model parameters is estimated by the Bayesian Markov chain Monte Carlo (MCMC) algorithm. Subsequently, the maintenance policy is optimized by minimizing the expected cost under parameter uncertainty. The expectation and variance of maintenance cost are computed over a finite-time horizon. Finally, a real example of heavy machine tools is applied to illustrate the proposed maintenance model. Numerical results show that the proposed maintenance policy provides a more robust and accurate solution in maintenance decision-making for deteriorating systems, leading to lower maintenance cost compared with other policies.

Numerical algorithm for optimal control of switched systems and its application in cancer chemotherapy

This paper considers an optimal control problem of switched dynamical systems with control input and system state constraints. Unlike in traditional switched dynamical systems, the switching times cannot be specified directly and they are governed by a state-dependent switching condition. Thus, the existing methods cannot be directly used to solve this problem. To overcome this difficulty, the switching conditions are transformed into a continuous-time inequality constraint by introducing an integer constraint. Further, the original optimal control problem is approximated by using a sequence of constrained non-convex nonlinear parameter optimization problems by using a relaxation method, a control vector parameterization technique, and a time-scaling transformation. Following that, a penalty function-based intelligent optimization algorithm is proposed for obtaining a global optimal solution based on a more effective penalty function method and a more effective intelligent optimization algorithm. The convergence results show that the proposed method is globally convergent. Numerical simulation results show that the proposed method is lower time-consuming, has faster convergence speed, can obtain a better objective function value than the existing typical algorithms, and can achieve a stable and robust performance when considering the small perturbations in constraint conditions or the small perturbations of the model parameters.

Chaotic Analysis of Fractional-Order Permanent Magnet Synchronous Generator for Wind Turbine

Fractional-order wind turbine is a strongly coupled non-linear dynamic system. It mainly studies the significant chaos characteristics such as the complex chaotic motion with fractional order varying. According to the mathematical model of the system, the fractional order Lorenz chaotic equation is established by linear affine transformation and time scale transformation. The theory of Lyapunov stability analysis is adopted to deeply study the development process of the system from stable operation to chaotic motion. The correctness of the chaos characteristics of the system is verified.

심전도신호로부터 시간-스케일 2차원 변환을 가진 딥러닝기반 감정인식

This paper describes the design of ensemble deep models based on time-scale transformation from electrocardiogram (ECG) signals for emotion recognition. As the number of senior citizens living alone increases, emotion robots that can interact with them and other emotion robots are becoming increasingly important. Existing emotion robots usually recognized emotion through images of the user’s facial expressions or voice signals. However, there are many situations where the user’s emotions cannot read under various environments. Therefore, research on recognizing the user’s emotions through ECG signals among different biomedical signals is actively being conducted. The proposed method converts ECG signals into various types of two-dimensional time-scale representations. We then designed a four-stream deep learning model by applying it to an ensemble form and transfer learning. Finally, an experiment was conducted using the ASCERTAIN sentiment database. This database contains data recorded by 58 people with 9 different emotions. Among these emotions, we used six representatives (surprise, happiness, anger, disgust, fear, and sadness). The experimental results revealed that the presented ensemble deep models showed good performance in comparison with each single deep model and the original model without transformation.

A switched system formulation for optimal integration of scheduling and control in multi-product continuous processes

Integrating control decisions into scheduling strategies has been shown to improve process economics. Typical integrated formulations consist of mixed-integer optimization models that pose computational challenges. Often, these models describe the scheduling as a static system whereas the production process is considered dynamic over certain time periods. In this work, we propose an alternative formulation that circumvents the need to deal with integer variables. We regard the scheduling as a dynamic system and incorporate the model of the process unit to formulate the integration as a switched system. We address multi-product continuous processes where each product defines a subsystem with different operating conditions. Variable switching instants and sequences are handled using time-scaling transformation and parametrization of the integer decisions. Due dates are also taken into consideration. The resulting formulation is efficiently treated as a nonlinear programming problem that allows to carry out transitions involving stable and unstable steady states. An iterative procedure to improve the transition times is provided. We solve three case studies under different scenarios. Results show that optimal switching sequences and control profiles that maximize the total profit can be obtained at low computational cost. Solutions provide a dynamic description of the scheduling and production process. The proposed framework can be considered as an alternative to mixed-integer models.