Application Of Control Theory Research Articles

Book Reviews

It is always good to have some work with you on a boring transatlantic flight, and it is even better if this work is a review for our book reviews section. The featured review by John W. Cain---seemingly done in this situation---gives a very detailed and well-structured insight into Daniel B. Forger's book on Biological Clocks, Rhythms, and Oscillations. As John mentions, one may even learn from this book an optimal control approach to shifting the human circadian clock, which seems like a very down-to-earth application of optimal control theory. The review recommends the book as “a treasure trove of good advice and a broad overview of issues that one should consider in the mathematical study of biological oscillators.” When you land after the transatlantic flight in Europe, you may by ready for some Mathematical Tapas served by Jean-Baptiste Hiriart-Urruty. Two volumes of the tapas are reviewed in this issue by John Haigh, who seemingly devoured them. Also our second review, by John A. Adam on Ian Stuart's book The Beauty of Numbers in Nature: Mathematical Patterns and Principles from the Natural World, is a very detailed and in-depth discussion. He praises this book impressively and even recommends it as a “slow-time-release” gift for his grandson. I'll try it with my own kids and we'll see. We have five more insightful reviews on a wide range of topics in this issue: complementarity in electronics, optimization and differentiation, direct methods for sparse matrices, history of mathematics, and optimal spacecraft trajectories.

Variable Gain Predictive PN Guidance for Interception of High Speed Re-entry Targets

A new guidance philosophy by introducing a trajectory dependent variable guidance gain in the predictive PN guidance has been proposed in this paper. The predictive PN guidance uses nonlinear models of interceptor and target to compute zero effort miss and time to go, while guidance gain computation utilizes a formula obtained by application of optimal control theory for a weighted control minimization of a linearized geometry PN guidance problem, where weight in the cost function could be any positive definite explicit function of time. The specific problem considered in this paper is interception of high speed re-entry ballistic target at various altitudes, this guidance logic shapes the guidance demand according to the maximum capability of the vehicle in such a fashion that almost full guidance correction happens during initial phase of the interceptor flight where it has sufficient aerodynamic capability. Hence the guidance demands in latter phase of the trajectory is almost negligible, where the interceptor capabilities are very low. The simulation has been done and improvement of this guidance against PN and Predictive PN guidance are shown. This guidance is named as “variable gain predictive PN guidance (VGPRPN)”.

Controllability under positivity constraints of semilinear heat equations

In many practical applications of control theory some constraints on the state and/or on the control need to be imposed. In this paper, we prove controllability results for semilinear parabolic equations under positivity constraints on the control, when the time horizon is long enough. As we shall see, in fact, the minimal controllability time turns out to be strictly positive. More precisely, we prove a global steady state constrained controllability result for a semilinear parabolic equation with $C^1$ nonlinearity, without sign or globally Lipschitz assumptions on the nonlinear term. Then, under suitable dissipativity assumptions on the system, we extend the result to any initial datum and any target trajectory. We conclude with some numerical simulations that confirm the theoretical results that provide further information of the sparse structure of constrained controls in minimal time.

On the issue of effective protection of objects from vibration and shock

A brief review of the research on the solution of the problems of protecting objects from vibration and shock effects is performed. The ways to find the optimal values of vibration isolators are shown on the basis of the application of control theory, including with the use of feed- forward control in systems of shockproof vibration isolation. The technique of designing quasi-invariant systems of vibration protection of objects is described and their high efficiency at various kinds of disturbing effect is shown, established as a result of numerical modeling and experimental researches.

CONTROL THEORY APPLICATIONS TO OPERATIONS SYSTEMS, SUPPLY CHAIN MANAGEMENT AND INDUSTRY 4.0 NETWORKS

Uncertainty, feedback cycles and dynamics are major challenges in production and logistics systems, supply chains, and Industry 4.0 networks. From this perspective, control theory is an important research avenue in operations and supply chain management. We briefly survey the applications of control theory to engineering and management problems in supply chains and operations areas during the period 1960-2017. Our analysis is based on bridging the fundamentals of control and systems theory to supply chains and operations management. Although operations and supply chain systems resemble control systems, they have some peculiarities which do not allow a direct application of control theoretic methods. The required modifications and possible limitations are discussed. In this setting, further development of interdisciplinary approaches to supply chain optimization is argued.

Service-oriented distributed software complex for evaluation and multi-criteria analysis of reliability and survivability of on-board equipment of small satellites: Russian and Belarusian segments

The paper presents solutions to the problems of estimation, analysis and forecasting of indicators of reliability, survivability and efficiency of on-board equipment (BE) of small satellites (SS) in standard and specified conditions of operation. A distributed software complex for multi-criteria evaluation, analysis and prediction of the values of reliability and survivability indicators of SS BE was developed taking into account the factors of outer space, as well as control of their reconfiguration at different stages of the life cycle (SPIIRAS). The developed complex of methodologies and software tools for estimating the reliability of SS BE (UIIP) was integrated as a distributed segment into the SPIIRAS software complex. The developed software systems are based on using the applied control theory of structural dynamics of complex objects, as well as the developed methods of estimation, analysis and forecasting of indicators of reliability, survivability and efficiency of SS BE, and database mining approach.

Development of electric cart for improving walking ability — application of control theory to assistive technology

This paper explains the development ofan electric cart that helps the elderly maintain or improve their physical strength. Unlike commercially available ones, it has a pedal unit that providessome exercise for a user in training his lower limbs. An impedance model describes the feeling of pushingthe pedals. The largest pedal load is determined based on a pedaling experiment. An $H_{\infty}$ controller isdesigned for each of the largest pedal load and virtually no load. A control law, which is based on the concept ofdynamic parallel distributed compensation, is designed using the rating of perceived exertion of a driveras a criterion to choose a pedal load between the largest and almost zero. Five university students andtwelve elderly people participated experiments to verify the system design and the validity of the system.

Incremental stability of Lur’e systems through piecewise-affine approximations

Lur’e-type nonlinear systems are virtually ubiquitous in applied control theory, which explains the great interest they have attracted throughout the years. The purpose of this paper is to propose conditions to assess incremental asymptotic stability of Lur’e systems that are less conservative than those obtained with the incremental circle criterion. The method is based on the approximation of the nonlinearity by a piecewise-affine function. The Lur’e system can then be rewritten as a so-called piecewise-affine Lur’e system, for which sufficient conditions for asymptotic incremental stability are provided. These conditions are expressed as linear matrix inequalities (LMIs) allowing the construction of a continuous piecewise-quadratic incremental Lyapunov function, which can be efficiently solved numerically. The results are illustrated with numerical examples.

Teaching Robotics in Secondary School: Examples and Outcomes

Robotics and control theory education has become an important part of secondary and high school education nowadays. The paper gives some examples that help to show control theory application in a simple way: linefollowing and manipulator based tasks. Some real projects of students who studied robotics this way are described.

Discrete Linear Time Invariant Analysis of Digital Fluid Power Pump Flow Control

A fundamental part of a digital fluid power (DFP) pump is the actively controlled valves, whereby successful application of these pumps entails a need for control methods. The focus of the current paper is on a flow control method for a DFP pump. The method separates the control task concerning timing of the valve activation and the task concerning the overall flow output control. This enables application of linear control theory in the design process of the DFP pump flow controller. The linearization method is presented in a general framework and an application with a DFP pump model exemplifies the use of the method. The implementation of a discrete time linear controller and comparisons between the nonlinear model and the discrete time linear approximation shows the applicability of the control method.

EXPERIMENTAL AND NUMERICAL MODELING OF FLUID-SEABED INTERACTION IN SHALLOW WATER

For the design and the construction of the defence structures against beach erosion, there is a need to predict and un-
 derstand the evolution in time of the shape of the sea sandy bed, using different numerical models and control theories. In the present paper a numerical model based on shallow water equations, which is an application of control theory to the evolution of sandy bed, is used in order to propose a formulation for the wave motion based on fluid and structure coupling, using minimization principles. Furthermore, measurements from a physical experiment are used in order to verify the accuracy of the model. The experiment was carried out in a multi directional wave basin at the SOGREAH (LHF facility, G-INP, France) and provided extensive measurements and detailed analysis of combined hydrodynamics and morphodynamics, suggesting a subtle interplay between several feedback mechanisms associated to wavedriven rip current circulations, wave nonlinearities, sediment transport, and seabed evolution. Using the numerical model, wave characteristics and depth profiles have been calculated and compared to experimental results.

A hybrid feedback control strategy for autonomous waypoint transitioning and loitering of unmanned aerial vehicles

We consider the problem of autonomously controlling a fixed-wing aerial vehicle to visit a neighborhood of a pre-defined waypoint, and when nearby it, loiter around it. To solve this problem, we propose a hybrid feedback control strategy that unites two state-feedback controllers: a transit controller capable of steering or transitioning the vehicle to nearby the waypoint and a loiter controller capable of steering the vehicle about a loitering radius. The aerial vehicle is modeled on a level flight plane with system performance characterized in terms of the aerodynamic, propulsion, and mass properties. Thrust and bank angle are the control inputs. Asymptotic stability properties of the individual control algorithms, which are designed using backstepping, as well as of the closed-loop system, which includes a hybrid algorithm uniting the two controllers, are established. In particular, for this application of hybrid feedback control, Lyapunov functions and hybrid systems theory are employed to establish stability properties of the set of points defining loitering. The analytical results are confirmed numerically by simulations.

Portfolio Optimization for a Large Investor Controlling Market Sentiment Under Partial Information

We consider an investor faced with the utility maximization problem in which the risky asset price process has pure-jump dynamics affected by an unobservable continuous-time finite-state Markov chain, the intensity of which can also be controlled by actions of the investor. Using the classical filtering theory, we reduce this problem with partial information to one with full information and solve it for logarithmic and power utility functions. In particular, we apply control theory for piece-wise deterministic Markov processes (PDMP) to our problem and derive the optimality equation for the value function and characterize the value function as the unique viscosity solution of the associated dynamic programming equation. Finally, we provide a toy example, where the unobservable state process is driven by a two-state Markov chain, and discuss how investor's ability to control the intensity of the state process affects the optimal portfolio strategies as well as the optimal wealth under both partial and full information cases.

Combining pressure and temperature control in dynamics on energy landscapes

Complex systems from science, technology or mathematics usually appear to be very different in their specific dynamical evolution. However, the concept of an energy landscape with its basins corresponding to locally ergodic regions separated by energy barriers provides a unifying approach to the description of complex systems dynamics. In such systems one is often confronted with the task to control the dynamics such that a certain basin is reached with the highest possible probability. Typically one aims for the global minimum, e.g. when dealing with global optimization problems, but frequently other local minima such as the metastable compounds in materials science are of primary interest. Here we show how this task can be solved by applying control theory using magnesium fluoride as an example system, where different modifications of MgF2 are considered as targets. In particular, we generalize previous work restricted to temperature controls only and present controls which simultaneously adjust temperature and pressure in an optimal fashion.

Algorithm for Overcoming the Curse of Dimensionality For Time-Dependent Non-convex Hamilton–Jacobi Equations Arising From Optimal Control and Differential Games Problems

In this paper we develop a parallel method for solving possibly non-convex time-dependent Hamilton---Jacobi equations arising from optimal control and differential game problems. The subproblems are independent so they can be implemented in an embarrassingly parallel fashion, which usually has an ideal parallel speedup. The algorithm is proposed to overcome the curse of dimensionality (Bellman in Adaptive control processes: a guided tour. Princeton University Press, Princeton, 1961; Dynamic programming. Princeton University Press, Princeton, 1957) when solving HJ PDE . We extend previous work Chow et al. (Algorithm for overcoming the curse of dimensionality for certain non-convex Hamilton---Jacobi equations, Projections and differential games, UCLA CAM report, pp 16---27, 2016) and Darbon and Osher (Algorithms for overcoming the curse of dimensionality for certain Hamilton---Jacobi equations arising in control theory and elsewhere, UCLA CAM report, pp 15---50, 2015) and apply a generalized Hopf formula to solve HJ PDE involving time-dependent and perhaps non-convex Hamiltonians. We suggest a coordinate descent method for the minimization procedure in the Hopf formula. This method is preferable when even the evaluation of the function value itself requires some computational effort, and also when we handle higher dimensional optimization problem. For an integral with respect to time inside the generalized Hopf formula, we suggest using a numerical quadrature rule. Together with our suggestion to perform numerical differentiation to minimize the number of calculation procedures in each iteration step, we are bound to have numerical errors in our computations. These errors can be effectively controlled by choosing an appropriate mesh-size in time and the method does not use a mesh in space. The use of multiple initial guesses is suggested to overcome possibly multiple local extrema in the case when non-convex Hamiltonians are considered. Our method is expected to have wide application in control theory and differential game problems, and elsewhere.

Application of Optimal Control Strategies for the Spread of HIV in a Population

This paper presents an application of optimal control theory to assess the effectiveness of control measures on the spread of HIV in a population. This paper formulates and analyzes a deterministic mathematical model with use of condom, screening and therapy as control variables using optimal control theory and Pontryagin’s Maximum Principle. It formulates the appropriate optimal control problem and investigate the necessary conditions for the disease control in order to determine the role of unaware infectives in the spread of HIV using of condom, screening of unaware infective and antiretroviral therapy are used as the control items. The optimality system is derived and solved numerically.

A Mean Field Game Computational Methodology for Decentralized Cellular Network Optimization

In cellular communication networks, quality of service (QoS) is defined as the ratio of the user’s signal level to the system noise plus other agents’ signal interference levels, and is commonly used to measure the performance of a mobile user over the network. In this paper, Nash equilibrium strategies, which minimize a linear combination of the system QoS and the transmitted power in code division multiple access networks, are found via an application of mean field game (MFG) control theory. Computational investigations of decentralized cellular network optimization via the application of nonlinear MFG control theory to this class of problem are presented for downlink and uplink scenarios with uniform and nonuniform agent population with respect to localized and nonlocalized interferences.

Robust control of static Var compensator-based power oscillation dampers using polynomial control in power systems

A static Var compensator (SVC) installed in a power transmission network can be effectively exploited to enhance the damping of low frequency electromechanical oscillations. The application of robust control theory offers more reliable and robust damping controller to achieve desired damping level considering variations in the operating conditions of power system. This paper presents a new approach to design a robust proportional-integral (PI) controller for stabilizing power system oscillations. The variability in operating conditions is captured using an interval polynomial and then, Kharitonov’s theorem is used to design the desired damping controller. The proposed method is based on plotting the stability boundary locus in the ( kp-ki) plane and then computing the stabilizing values of the parameters of a PI controller. Besides stabilization, computation of stabilizing PI controllers that achieve user specified gain margin (Gm), phase margin (Pm) and bandwidth is studied simultaneously. This novel method enables designers to make the convenient trade-off between stability and performance by choosing the proper margins and bandwidth specifications. In addition, the most appropriate stabilizing input signal is selected using Hankel singular value (HSV) and right half plane-zeros (RHP-zeros) for the SVC-based supplementary damping controller. The effectiveness and robustness of the proposed controller are demonstrated using eigenvalue analysis and time-domain simulation for a 16 machine 68-bus test system. The simulations and analysis are implemented in matrix laboratory environment and power system analysis toolbox.

On Feedback Stabilization of Linear Switched Systems via Switching Signal Control

Motivated by recent applications in control theory, we study the feedback stabilizability of switched systems, where one is allowed to chose the switching signal as a function of $x(t)$ in order to stabilize the system. We propose new algorithms and analyze several mathematical features of the problem which were unnoticed up to now, to our knowledge. We prove complexity results, (in)equivalence between various notions of stabilizability, existence of Lyapunov functions, and we provide a case study for a paradigmatic example introduced by Stanford and Urbano in the 1990s.

Single-cell resolution of intracellular T cell Ca2+ dynamics in response to frequency-based H2O2 stimulation.

Adaptive immune cells, such as T cells, integrate information from their extracellular environment through complex signaling networks with exquisite sensitivity in order to direct decisions on proliferation, apoptosis, and cytokine production. These signaling networks are reliant on the interplay between finely tuned secondary messengers, such as Ca2+ and H2O2. Frequency response analysis, originally developed in control engineering, is a tool used for discerning complex networks. This analytical technique has been shown to be useful for understanding biological systems and facilitates identification of the dominant behaviour of the system. We probed intracellular Ca2+ dynamics in the frequency domain to investigate the complex relationship between two second messenger signaling molecules, H2O2 and Ca2+, during T cell activation with single cell resolution. Single-cell analysis provides a unique platform for interrogating and monitoring cellular processes of interest. We utilized a previously developed microfluidic device to monitor individual T cells through time while applying a dynamic input to reveal a natural frequency of the system at approximately 2.78 mHz stimulation. Although our network was much larger with more unknown connections than previous applications, we are able to derive features from our data, observe forced oscillations associated with specific amplitudes and frequencies of stimuli, and arrive at conclusions about potential transfer function fits as well as the underlying population dynamics.