An evaluation of the multivariate dispersion charts with estimated parameters under non‐normality

Various charts such as |S|, W, and G are used for monitoring process dispersion. Most of these charts are based on the normality assumption, while exact distribution of the control statistic is unknown, and thus limiting distribution of control statistic is employed which is applicable for large sample sizes. In practice, the normality assumption of distribution might be violated, while it is not always possible to collect large sample size. Furthermore, to use control charts in practice, the in‐control state usually has to be estimated. Such estimation has a negative effect on the performance of control chart. Non‐parametric bootstrap control charts can be considered as an alternative when the distribution is unknown or a collection of large sample size is not possible or the process parameters are estimated from a Phase I data set. In this paper, non‐parametric bootstrap multivariate control charts |S|, W, and G are introduced, and their performances are compared against Shewhart‐type control charts. The proposed method is based on bootstrapping the data used for estimating the in‐control state. Simulation results show satisfactory performance for the bootstrap control charts. Ultimately, the proposed control charts are applied to a real case study.     

Input-to-state stability of Runge–Kutta methods for nonlinear control systems

In the paper we introduce input-to-state stability (ISS) of Runge–Kutta methods for control systems. The ISS properties of Runge–Kutta methods are studied for linear control systems and nonlinear control systems, respectively. The previously reported results in literature are special cases of ISS of Runge–Kutta methods.     

Approximate controllability of a class of semilinear degenerate systems with boundary control

This paper concerns a class of control systems governed by semilinear degenerate equations with boundary control in one-dimensional space. The control is proposed on the ‘degenerate’ part of the boundary. The control systems are shown to be approximately controllable by Kakutani's fixed point theorem.     

Comparison of control charts for Poisson count data in health‐care monitoring

Statistical surveillance is a noteworthy endeavor in many health‐care areas such as epidemiology, hospital quality, infection control, and patient safety. For monitoring hospital adverse events, the Shewhart u‐control chart is the most used methodology. One possible issue of the u‐chart is that in health‐care applications the lower control limit (LCL) is often conventionally set to zero as the adverse events are rare and the sample sizes are not sufficiently large to obtain LCL greater than zero. Consequently, the control chart loses any ability to signal improvements. Furthermore, as the area of opportunity (sample size) is not constant over time, the in‐control and out‐of‐control run length performances of the monitoring scheme are unknown. In this article, on the basis of a real case and through an intensive simulation study, we first investigate the in‐control statistical properties of the u‐chart. Then we set up several alternative monitoring schemes with the same in‐control performances and their out‐of‐control properties are studied and compared. The aim is to identify the most suitable control chart considering jointly: the ability to detect unexpected changes (usually worsening), the ability to test the impact of interventions (usually improvements), and the ease of use and clarity of interpretation. The results indicate that the exponentially weighted moving average control chart derived under the framework of weighted likelihood ratio test has the best overall performance.     

Boundaries of the receding horizon control for interconnected systems

In recent years, the finite-horizon quadratic minimization problem has become popular in process control, where the horizon is constantly rolled back. In this paper, this type of control, which is also called the receding horizon control, is considered for interconnected systems. First, the receding horizon control equations are formulated; then, some stability conditions depending on the interconnection norms and the horizon lengths are presented. For -coupled systems, stability results similar to centralized systems are obtained. For interconnected systems which are not -coupled, the existence of a horizon length and a corresponding stabilizing receding horizon control are derived. Finally, the performance of a locally computed receding horizon control for time-invariant and time-varying systems with different updating intervals is examined in an example.     

Formulation of pseudoparabolic boundary control problems

Boundary control of a pseudoparabolic problem is studied. The trace of a solution y(T; ?) corresponding to a control ? is shown to belong to $\text{L}{}^2\text{(Ω)}$ and continuity properties are established. Two concrete examples are considered and regularity properties of the optimal control are determined in each case.     

混沌记忆系统的自适应反馈控制和轨迹跟踪控制

研究了混沌记忆系统的自适应反馈控制和基于反馈线性化的轨迹跟踪控制问题.首先,通过绘制系统的时域波形图和混沌吸引子图验证系统的复杂的动力学行为;然后,分别应用自适应反馈控制方法和基于反馈线性化的轨迹跟踪控制方法设计控制器,对系统施加控制;最后,通过数值仿真验证控制器的有效性.     

Approximation of sparse controls in semilinear equations by piecewise linear functions

Semilinear elliptic optimal control problems involving the $L^1$ norm of the control in the objective are considered. A priori finite element error estimates for piecewise linear discretizations for the control and the state are proved. These are obtained by a new technique based on an appropriate discretization of the objective function. Numerical experiments confirm the convergence rates.     

A discontinuous interpolated finite volume approximation of semilinear elliptic optimal control problems

In this article, we describe a discontinuous finite volume method with interpolated coefficients for the numerical approximation of the distributed optimal control problem governed by a class of semilinear elliptic equations with control constraints. The proposed distributed control problem involves three unknown variable: control, state and costate. For the approximation of control, we have adopted three different methodologies: variational discretization, piecewise constant and piecewise linear discretization, while the approximation of state and costate variables is based on discontinuous piecewise linear polynomials. As the resulted scheme is non‐symmetric, optimize‐then‐discretize approach is used to approximate the control problem. Optimal a priori error estimates in suitable natural norms for state, costate and control variables are derived. Moreover, numerical experiments are presented to support the derived theoretical results. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2090–2113, 2017     

On the Error Estimates for the Finite Element Approximation of a Class of Boundary Optimal Control Systems

In this article, we consider an application of the abstract error estimate for a class of optimal control systems described by a linear partial differential equation (as stated in Numer. Funct. Anal. Optim. 2009; 30:523–547). The control is applied at the boundary and we consider both, Neumann and Dirichlet optimal control problems. Finite element methods are proposed to approximate the optimal control considering an approximation of the variational inequality resulting from the optimality conditions; this approach is known as classical one. We obtain optimal order error estimates for the control variable and numerical examples, taken from the literature, are included to illustrate the results.     

Pontryagin's Principle of Mixed Control-State Constrained Optimal Control Governed by Fluid Dynamic Systems

This article deals with Pontryagin's minimum principle for optimal control problems governed by an abstract fluid dynamical system with mixed control-state constraints. Two techniques are mainly used to obtain our results: ?-perturbation for admissible control set and diffuse perturbation for admissible control. Using these results, the necessary conditions for L ²-local optimal solution of our control problems can be studied under some assumptions. In the end of this article, these results are applied to some special fluid dynamical systems which contain the fluid systems driven by its boundary and magnetohydrodynamics(MHD).     

A posteriori error estimates of hp spectral element methods for optimal control problems with L2-norm state constraint

In this paper, we investigate a distributed optimal control problem governed by elliptic partial differential equations with L²-norm constraint on the state variable. Firstly, the control problem is approximated by hp spectral element methods, which combines the advantages of the finite element methods with spectral methods; then, the optimality conditions of continuous system and discrete system are presented, respectively. Next, hp a posteriori error estimates are derived for the coupled state and control approximation. In the end, a projection gradient iterative algorithm is given, which solves the optimal control problems efficiently. Numerical experiments are carried out to confirm that the numerical results are in good agreement with the theoretical results.

     

Poisson and Diffusion Approximation of Stochastic Master Equations with Control

Quantum Trajectories are solutions of stochastic differential equations. Such equations are called Stochastic Master Equations and describe random phenomena in the continuous measurement theory of Open Quantum System. Many recent developments deal with the control of such models, i.e. optimization, monitoring and engineering. In this article, stochastic models with control are mathematically and physically justified as limits of concrete discrete procedures called Quantum Repeated Measurements. In particular, this gives a rigorous justification of the Poisson and diffusion approximations in quantum measurement theory with control.     

Minimax Control for a Class of Linear Systems Subject to Disturbances<Superscript>1</Superscript>

A class of linear dynamical control systems subject to uncertain but bounded disturbances is considered. The bounds imposed on the disturbances depend on the control magnitude and grow with the control. This situation occurs if the disturbances are due to the inaccuracy of the control implementation and often takes place in engineering applications such as transportation, aerospace, and robotic systems. Under certain assumptions, the minimax control problem is formulated and solved. Explicit expressions for the optimal control (both open-loop and feedback) are obtained that provide the minimax to the given performance index for arbitrary but bounded disturbances. Examples are given. ¹This work was supported by the Russian Foundation for Basic Research (Project 05-01-00647) and the Grant for Russian Scientific Schools (NSh 1627.2003.1).     

Inverse optimal control and construction of control Lyapunov functions

In this paper, the construction of CLFs for nonlinear systems and a new inverse optimal control law are presented. The construction of a CLF for an affine nonlinear system is reduced to the construction of a CLF for a simpler system, and a new L _g V type control law with respect to a CLF is provided. This control law is a generalization of Sontag’s formula and contains a design parameter. Tuning this parameter gives many suboptimal solutions for the optimization problem. Also, the gain margin and sector margin of the control law are calculated. Examples are provided to illustrate the main theoretical results of the paper. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 61, Optimal Control, 2008.     

The control of pseudohyperbolic systems

Certain problems concerning the control of processes described by pseudohyperbolic-type equations are studied. Inequalities in negative norms are obtained. These inequalities imply the existence and uniqueness of the generalized solution of a boundary-value problem with right-hand sides belonging to a negative space, which makes it possible to consider the optimal problem in the class of generalized effects. The existence of the optimal control is proved. Some examples of pulse optimal control problems are given. Bibliography:5 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 80, 1996, pp. 68–77.     

Decentralized Reliable Control of Interconnected Time-Delay Systems Against Sensor Failures

In this paper, we study the problem of designing decentralized reliable feedback control methods under a class of control failures for a class of linear interconnected continuous-time systems having internal subsystem time-delays and additional time-delay couplings. These failures are described by a model that takes into consideration possible outages or partial failures in every single actuator of each decentralized controller. The decentralized control design is performed through two steps. First, a decentralized stabilizing reliable feedback control set is derived at the subsystem level through the construction of appropriate Lyapunov-Krasovskii functional and, second, a feasible linear matrix inequalities procedure is then established for the effective construction of the control set under different feedback schemes. Two schemes are considered: the first is based on state-measurement and the second utilizes static output-feedback. The decentralized feedback gains in both schemes are determined by convex optimization over linear matrix inequalities. We characterize decentralized linear matrix inequality-based feasibility conditions such that every local closed-loop subsystem of the linear interconnected delay system is delay-dependent robustly asymptotically stable with an γ-level ℒ₂-gain. The developed results are tested on a representative example.     

Asymptotic analysis of Neumann periodic optimal boundary control problem

An optimal boundary control problem in a domain with oscillating boundary has been investigated in this paper. The controls are acting periodically on the oscillating boundary. The controls are applied with suitable scaling parameters. One of the major contribution is the representation of the optimal control using the unfolding operator. We then study the limiting analysis (homogenization) and obtain two limit problems according to the scaling parameters. Another notable observation is that the limit optimal control problem has three controls, namely, a distributed control, a boundary control, and an interface control. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimal singular and chattering modes in the problem of controlling the vibrations of a string with clamped ends

The problem of minimizing the root mean square deviation of a uniform string with clamped ends from an equilibrium position is investigated. It is assumed that the initial conditions are specified and the ends of the string are clamped. The Fourier method is used, which enables the control problem with a partial differential equation to be reduced to a control problem with a denumerable system of ordinary differential equations. For the optimal control problem in the l₂ space obtained, it is proved that the optimal synthesis contains singular trajectories and chattering trajectories. For the initial problem of the optimal control of the vibrations of a string it is also proved that there is a unique solution for which the optimal control has a denumerable number of switchings in a finite time interval.