Time-scaling Transformation Research Articles

Time Scale Transformation in Bivariate Pearson Diffusions: A Shift from Light to Heavy Tails

Heavy-tailed Pearson diffusions provide a natural alternative to well-known Ornstein–Uhlenbeck and Cox–Ingersoll–Ross processes in applications that require addressing heavy-tailed behavior. In this paper, all three heavy-tailed Pearson diffusions, having inverse gamma, Fisher–Snedecor and Student stationary distributions, are constructed via an absolutely continuous time-change process employed in a specific functional transformation of CIR or OU. Moreover, time-change rates in stochastic clocks are continuous functionals of the CIR process.

Fractional impulsive controller design of fractional-order fuzzy systems with average dwell-time strategy and its application to wind energy systems

In this study, the issue of adaptive impulsive control of a permanent magnet synchronous generator (PMSG)-based wind energy system (WES) with an average dwell-time strategy is investigated using a Takagi–Sugeno (T-S) fuzzy model described by fractional-order differential equations. A T-S fuzzy model is employed to describe the nonlinear fractional-order PMSG (FOPMSG) model, an average dwell time, an average impulsive condition, a Lyapunov function, and a less conservative algebraic inequality criterion that guarantees stabilization for the considered nonlinear system. First, the mathematical model of the FOPMSG in the n−m reference frame is transformed into a dimensionless chaotic system through affine transformation and time scale transformation. Then, the stabilization problem of the nonlinear chaotic FOPMSG model is considered to validate the proposed sufficient conditions, leading to the derivation of a new stability criterion for the FOPMSG model, instead of addressing the general problem to validate the proposed result. Subsequently, the desired control gains can be obtained to ensure the stabilization of the addressed closed-loop system. By combining adaptive and impulsive control, the model under consideration can be stabilized for any target dynamics. Finally, we demonstrate the efficiency and feasibility of the suggested approach through numerical simulations and comparative results.

Global dynamics of a simple model for wild and sterile mosquitoes.

There are known methods to manage the population dynamics of wild and sterile mosquitoes by releasing genetically engineered sterile mosquitoes. Even if a two-dimensional system of ordinary differential equations is considered as a simple mathematical model for developing release strategies, fully understanding the global behavior of the solutions is challenging, due to the fact that the probability of mating is ratio-dependent. In this paper, we combine a geometric approach called the time-scale transformation and blow-up technique with the center manifold theorem to provide a complete understanding of dynamical systems near the origin. Then, the global behavior of the solution of the two-dimensional ordinary differential equation system is classified in a two-parameter plane represented by the natural death rate of mosquitoes and the sterile mosquito release rate. We also offer a discussion of the sterile mosquito release strategy. In addition, we obtain a better exposition of the previous results on the existence and local stability of positive equilibria. This paper provides a framework for the mathematical analysis of models with ratio-dependent terms, and we expect that it will theoretically withstand the complexity of improved models.

Robust [formula omitted] stabilization for systems with uncertain input time-delay

This paper presents a robust H∞ stabilization approach for actively controlled systems that include uncertain input time-delay. The approach comprises two essential steps: (1) converting a controlled system with uncertain time-delay to have only deterministic time-delay through a time-scale transformation; and (2) applying the Chebyshev spectral continuous time approximation (CHsCTA) method and defining appropriate extension matrices to transform the system so there are no time-delays. A robust H∞ control design is implemented on the transformed system, which facilitates the derivation of a robust H∞ controller. An iterative algorithm is proposed for computing the feedback gain matrix, which overcomes the challenges posed by the nonlinear coupling terms when solving the matrix inequalities. Case studies validate the effectiveness of the proposed control algorithm, which, through comparison with existing control algorithms, is shown to be less conservative.

A novel successive convexification framework with symplectic pseudospectral solutions for nonlinear optimal control problems of constrained pantograph delay systems

This paper proposes a novel symplectic pseudospectral iteration framework for the nonlinear optimal control problem (OCP) of a pantograph delay system with inequality constraints. Traditional numerical algorithms for OCPs with pantograph delay systems have predominantly focused on unconstrained scenarios, neglecting constrained problems. Therefore, we propose a novel symplectic pseudospectral iteration framework for constrained problems. First, we employ the successive convexification (SCvx) method to transform the original problem into a series of linear quadratic (LQ) problems, addressing nonlinearity. Then, leveraging time-scale transformations and parametric variational principles, we derive the first-order necessary conditions (FONCs) for these convexified problems, including a Hamiltonian two-point boundary value problem (HTBVP) with stretching terms and a linear complementarity problem. To handle pantograph time delays effectively, we employ a local Legendre-Gauss-Lobatto (LGL) pseudospectral method with proportional grid discretization. Finally, employing a symplectic indirect algorithm, we develop a symplectic pseudospectral method (SPM) for solving the transformed LQ problems, converting them into a system of sparse linear algebraic equations and a linear complementarity problem. Validation of the framework is performed using diverse illustrative examples, demonstrating strong agreement between SCvx and SPM in computational efficiency. Numerical experiments confirm the sparsity of the core matrix and robustness to initial guesses.

Asynchronous switching and sparse discretization with time scaling for control vector parameterization optimal control

This work introduces a novel approach for solving open-loop Optimal Control Problems (OCPs) using multiple time grids, sparse discretization and a time scaling transformation. Control Vector Parameterization (CVP) method parameterizes control variables to create a finite-dimensional problem. The Variable Time Nodes Control Vector Parameterization (VTNCVP) discretization strategy allows each control component an independent time grid, offering enhanced input design flexibility and potentially yielding improved outcomes. A novel time scaling technique for OCPs with multiple time grids is presented that is easier to comprehend than existing methods and can circumvent some numerical difficulties. Additionally, Multiple Grid (MG)–Sparse Variable Time Nodes (MG-SVTN) is presented. This novel discretization strategy employs asynchronous switching with user-specified numbers of subdivisions for input controls, potentially yielding comparable objective function values to VTNCVP while reducing the dimensionality of decision variables. Two example problems with path constraints are solved using the methods.

A new computational approach for optimal control of switched systems

The combination of the time-scaling transformation and control parameterization has proven to be an effective approach in addressing optimal control problems involving switching systems with predefined subsystem sequences. However, this approach has certain limitations. First, the number of control switchings is required to be no less than the number of subsystem switchings. Second, the switching of the subsystem must be accompanied by the switching of the control. Third, this scheme introduces many hyperparameters, leading to combinatorial explosion. To address these drawbacks, we introduce a novel computational approach such that the control switching can be independent of subsystem switching. The superiority of this novel approach can be clearly observed from the solutions obtained using the proposed method for solving two illustrative examples.

Sequential time scaling transformation technique for time-delay optimal control problem

The time-scaling transformation technique used within the computational framework of control parameterization serves as an effective method for the optimization of control switching time as well as the control value for time-delay optimal control problems. However, the conventional time-scaling transformation stipulates that the switching times for all control components must be identical, which may not be feasible in real-world scenarios. To address this limitation, we introduce the sequential time-scaling transformation technique for solving time-delay optimal control problems. By sequentially applying control parameterization and time-scaling transformation to each of the control variable, the new technique enables the configuration of a varying number of segments for the control, and the switching time for each control component can be adaptively selected. Numerical results demonstrate that our approach not only achieves superior cost performance but also offers enhanced flexibility in control strategy.

Trajectory-tracking Control of Two-wheeled Mobile Robot with a Cascaded Time-scale Transformation

Trajectory-tracking Control of Two-wheeled Mobile Robot with a Cascaded Time-scale Transformation

Neuroadaptive tracking control of uncertain nonlinear systems with spatiotemporal constraints

This study investigates the neuroadaptive tracking control problem for a class of strict-feedback nonlinear systems with spatiotemporal constraints. An adaptive neural network-based control system is developed to alleviate the effects of modeling uncertainties and external disturbances. In particular, the proposed method ensures that the system tracking error has a predefined performance boundary (spatial constraint). Moreover, using a novel time-scale transformation method, uncertain nonlinear systems can achieve a prescribed finite-time convergence to a time-varying scaling function in the pointing position (temporal constraint). Finally, the efficiency of the proposed method is verified with two simulation examples.

Identification of low-SNR Coulter signals based on strongly coupled Lorenz-like oscillators

A Coulter prototype experiment is established and innovatively combined with coupled Lorenz-like oscillators to identify low-signal-to-noise ratio (SNR) Coulter signals. Coulter signals are collected by a Wheatstone bridge and fitted by Gaussian pulse signals. Using numerical simulation, low-SNR Gaussian pulses are added to the oscillators, and the synchronization mutation phenomena in the chaotic oscillators performed by the time scale transformation are analyzed. The maxima in the phenomena are used to detect the low-SNR pulses and construct an identification formula for different particle sizes. The nonlinear circuit of the oscillators is designed and fabricated. Then, it is applied to process low-SNR Coulter signals in the experiment. By selecting the maxima in the output signals from the circuit, according to the identification formula containing the correction coefficient, the particle size distribution can be obtained. The nonlinear circuit can effectively process low-SNR Coulter signals, which provides a good foundation for reducing the lower limit of detection in Coulter signals.

Optimal switching control of 1,3-propanediol fed-batch production with a cost on smooth feeding rate variation

In this paper, we propose a nonlinear switched control system for microbial 1,3-propanediol (1,3-PD) fed-batch production, where the smooth feeding rate of glycerol as well as the switching instants between batch process and feeding process are taken as control variables. To maximize 1,3-PD yield and minimize the fluctuation of feeding rate, we then present an optimal switching control problem with a cost on the smooth feeding rate variation and subject to path constraints reflecting the operational requirements. For solving this problem, we convert it to a series of parameter optimization problems by using a novel control parameterization scheme, a time-scaling transformation and a constraint transcription technique. An analytical expression of the total variation for the smooth feeding rate is also derived. On this basis, we develop a parallel Particle Swarm Optimization (PSO) algorithm together with the gradient-based optimization to solve the resulting problem. Finally, numerical simulations indicate that sacrificing a small amount of 1,3-PD can reduce the feeding rate variation significantly.

Global Prescribed-Time Stabilization for a Class of Uncertain Feedforward Nonlinear Systems

The problem of global prescribed-time stabilization (PTS) is addressed in this brief for a class of uncertain feedforward nonlinear systems. To achieve this task, a new time-scale transformation (TST) is first introduced, which has the good property that it changes the original global PTS into the global asymptotic stabilization of the transformed one but does not changes the system structures. Then, under the new framework, both state and output feedback controllers are developed to achieve the requiring performances with the help of the non-recursive technique. At last, simulation results of a nonlinear liquid level control resonant circuit (NLLCRC) system are provided to show the validity of the given approach.

A reliability analysis method based on the mixed correlated competition model considering multi-performance degradation and sudden failures

Complex products usually fail due to competing multi-performance degradation and sudden failure modes. The correlations between competing failure modes, including among multi-performance degradation failures and between degradation and sudden failures, must be considered in reliability analysis. This paper proposes a general reliability analysis method that considers such cases of complex mixed correlated competition. Firstly, a nonlinear Wiener process model based on a time-scale transformation function is applied to model performance degradation failure, and a multi-performance-degradation correlated-competition model based on a Copula function is constructed, which considers the correlated competition between multi-performance degradation failures. Secondly, a three-parameter Weibull distribution is used to model sudden failures, and an evaluation method based on nonlinear planning-maximum likelihood estimation is proposed to estimate its parameters. Thirdly, a mixed correlated-competition model of multi-performance degradation and sudden failures is proposed for product reliability evaluation. Finally, a servo system experiment is used for demonstration. The experimental results demonstrate that the proposed method achieves higher evaluation accuracy and has wider applicability than existing methods, which ignore mixed correlated competition.

An enhanced control parameterization technique with variable switching times for constrained optimal control problems with control-dependent time-delayed arguments and discrete time-delayed arguments

In this paper, we consider an optimal control problem whose dynamic system consists of control-dependent time-delayed arguments in the control and discrete time-delayed arguments in the state. Recently, control parameterization methods using hybrid time-scale transformation techniques have been developed to optimize both the control parameters and their switching times when the dynamic system consists of discrete time-delayed arguments only. This transformation maps the variable switching times in the original time horizon to fixed switching times in the new time horizon. Thus, the purpose of this paper is to extend the above technique to solve problems with both control-dependent time-delayed arguments and discrete time-delayed arguments. Using the above technique, we convert both the delayed state vector and the delayed control vector into non-delayed vectors in the new time horizon; hence both the control parameters and their switching times can be optimized. The gradients of the objective functions and the constraint functions with respect to the switching times and the controls are obtained. Two numerical examples, one of which is a practical problem involving the optimal transmission of messages in an internet, are solved to demonstrate the efficiency of our method.

Determination of feeding rate ratio in coupled fed-batch fermentation with pH feedback considering limited number of switches

This paper focuses on binary optimal control of fed-batch fermentation of glycerol by Klebsiella pneumoniaewith pH feedback considering limited number of switches. To maximize the concentration of 1,3-propanediol at terminal time, we propose a binary optimal control problem subjected to time-coupled combinatorial constraint with the ratio of feeding rate of glycerol to that of NaOH as control variables. Based on time-scaling transformation and discretization, the binary optimal control problem is first transformed into a mixed binary parameter optimization problem consisting of not only continuous variables but also binary variables, which is then divided into two subproblems via combinatorial integral approximation decomposition. Finally, a novel fruit fly optimizer with modified sine cosine algorithm and adaptive maximum dwell rounding are applied to solve the obtained subproblems numerically. Numerical results show the rationality and feasibility of the proposed method.

Geometric singular perturbation of a nonlocal partially degenerate model for <i>Aedes aegypti</i>

<p style='text-indent:20px;'>This paper is devoted to investigate the existence of traveling wave solutions for a partially degenerate <i>Aedes aegypti</i> model with nonlocal effects. By taking specific kernel forms and time scale transformation, we transform the nonlocal model into a singularly perturbed system with small parameters. A locally invariant manifold for wave profile system is obtained with the aid of the geometric singular perturbation theory, and then the existence of traveling wave solutions is proved provided that the basic reproduction number <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{R}_0>1 $\end{document}</tex-math></inline-formula> through utilizing the Fredholm orthogonal method. Furthermore, we study the asymptotic behaviors of traveling wave solution with the help of asymptotic theory. The methods used in this work can help us overcome the difficulty that the solution map associated with the system is not compact. Numerically, we perform simulations to demonstrate the theoretical results.</p>

A new switching time optimization technique for multi-switching systems

<p style='text-indent:20px;'>For a multi-switching system, it consists of multiple parallel switching systems. The optimal control problem controlled by a multi-switching system is to determine the optimal time sequence for each of the parallel switching systems. The time-scaling transformation is a well-known switching time optimization approach which has been widely used in various problem settings. However, the time-scaling transformation requires that these parallel switching systems have the same number of subsystems and are designed to switch simultaneously. Thus, it is not applicable for solving optimal control problems of general multi-switching systems. This paper presents a new technique for optimizing the switching times of multi-switching systems.</p>

Drift estimation by timescale transformation

In this study, we consider a model free control technique for a class of minimum phase nonlinear systems with unknown dynamics. We propose a timescale transformation approach that can be used to estimate the drift and the high frequency gain of the output dynamics of nonlinear systems with a well defined relative degree. The proposed controller provides an effective output regulation mechanism that avoids the need for direct disturbance model estimation and high gain output feedback controllers. A simulation study is conducted to demonstrate the effectiveness of the proposed approach.

Robust suboptimal feedback control for a fed-batch nonlinear time-delayed switched system

The optimal control strategy constructed in the form of a state feedback is effective for small state perturbations caused by changes in modeling uncertainty. In this paper, we investigate a robust suboptimal feedback control (RSPFC) problem governed by a nonlinear time-delayed switched system with uncertain time delay arising in a 1,3-propanediol (1,3-PD) microbial fed-batch process. The feedback control strategy is designed based on the radial basis function to balance the two (possibly competing) objectives: (i) the system performance (concentration of 1,3-PD at the terminal time of the fermentation) is to be optimal; and (ii) the system sensitivity (the system performance with respect to the uncertainty of the time-delay) is to be minimized. The RSPFC problem is subject to the continuous state inequality constraints. An exact penalty method and a novel time scaling transformation approach are used to transform the RSPFC problem into the one subject only to box constraints. The resulting problem is solved by a hybrid optimization algorithm based on a filled function method and a gradient-based algorithm. Numerical results are given to verify the effectiveness of the developed hybrid optimization algorithm.