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Showing papers in "Optimal Control Applications & Methods in 2009"


Journal ArticleDOI
TL;DR: In this article, a nonlinear programming formulation of the optimal control problem with delays in state and control variables is presented. But the Lagrange multipliers associated with the programming problem provide a consistent discretization of the advanced adjoint equation for the delayed control problem.
Abstract: Optimal control problems with delays in state and control variables are studied. Constraints are imposed as mixed control–state inequality constraints. Necessary optimality conditions in the form of Pontryagin's minimum principle are established. The proof proceeds by augmenting the delayed control problem to a nondelayed problem with mixed terminal boundary conditions to which Pontryagin's minimum principle is applicable. Discretization methods are discussed by which the delayed optimal control problem is transformed into a large-scale nonlinear programming problem. It is shown that the Lagrange multipliers associated with the programming problem provide a consistent discretization of the advanced adjoint equation for the delayed control problem. An analytical example and numerical examples from chemical engineering and economics illustrate the results. Copyright © 2008 John Wiley & Sons, Ltd.

238 citations


Journal ArticleDOI
TL;DR: Two different model reduction methods are introduced and compared, and the so‐called structured Hankel singular values are used in the methods, and indicate how important states in the subsystems are with respect to a chosen input–output map for the entire interconnected system.
Abstract: The problem of model reduction of linear systems with certain interconnection structure is considered in this paper To preserve the interconnection structure between subsystems in the reduction, special care needs to be taken This problem is important and timely because of the recent focus on complex networked systems in control engineering Two different model reduction methods are introduced and compared in this paper Both methods are extensions to the well-known balanced truncation method Compared with earlier work in the area these methods use a more general linear fractional transformation framework, and utilize linear matrix inequalities Furthermore, new approximation error bounds that reduce to classical bounds in special cases are derived The so-called structured Hankel singular values are used in the methods, and indicate how important states in the subsystems are with respect to a chosen input-output map for the entire interconnected system It is shown how these structured Hankel singular values can be used to select an approximation order Finally, the two methods are applied to a model of a mechanical device

145 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered delay-dependent H∞ control for singular Markovian jump systems with time delay and designed a state feedback controller, which guarantees that the resultant closed-loop system is not only regular, impulse free and stochastically stable, but also satisfies a prescribed H∆ performance level for all delays no larger than a given upper bound in terms of linear matrix inequality (LMI) approach.
Abstract: The problem of delay-dependent H∞ control is considered for singular Markovian jump systems with time delay. The aim of the problem is to design a state feedback controller, which guarantees that the resultant closed-loop system is not only regular, impulse free and stochastically stable, but also satisfies a prescribed H∞ performance level for all delays no larger than a given upper bound in terms of linear matrix inequality (LMI) approach. A strict LMI condition is developed to guarantee the existence of the desired state feedback controller. An explicit expression for the desired controller is also given. Numerical examples show the effectiveness of the proposed methods. Copyright © 2008 John Wiley & Sons, Ltd.

88 citations


Journal ArticleDOI
TL;DR: In this article, sufficient conditions for the existence of a state-feedback controller for a class of discrete-time linear systems with random abrupt changes and unknown transition probabilities but varying between known bounds for each mode.
Abstract: This paper deals with the class of discrete-time linear systems with random abrupt changes and unknown transition probabilities but varying between known bounds for each mode. The ℋ∞ control problem of this class of systems is revisited and new sufficient conditions are developed in the linear matrix inequality (LMI) setting to design the state-feedback controller that stochastically stabilizes the system under consideration and at the same time guarantees the disturbance rejection with a desired level γ . Sufficient conditions for existence of the state-feedback controller are developed. It is shown that the addressed problem can be solved if the corresponding developed LMIs are feasible. Numerical examples are employed to show the usefulness of the proposed results. Copyright © 2008 John Wiley & Sons, Ltd.

77 citations


Journal ArticleDOI
TL;DR: In this paper, a method is proposed to systematically transform a constrained optimal control problem (OCP) into an unconstrained OCP, which can be treated in the standard calculus of variations.
Abstract: A method is proposed to systematically transform a constrained optimal control problem (OCP) into an unconstrained OCP, which can be treated in the standard calculus of variations. The considered class of constraints comprises up to m input constraints and m state constraints with well-defined relative degree, where m denotes the number of inputs of the given nonlinear system. Starting from an equivalent normal form representation, the constraints are incorporated into a new system dynamics by means of saturation functions and differentiation along the normal form cascade. This procedure leads to a new unconstrained OCP, where an additional penalty term is introduced to avoid the unboundedness of the saturation function arguments if the original constraints are touched. The penalty parameter has to be successively reduced to converge to the original optimal solution. The approach is independent of the method used to solve the new unconstrained OCP. In particular, the constraints cannot be violated during the numerical solution and a successive reduction of the constraints is possible, e.g. to start from an unconstrained solution. Two examples in the single and multiple input case illustrate the potential of the approach. For these examples, a collocation method is used to solve the boundary value problems stemming from the optimality conditions. Copyright © 2009 John Wiley & Sons, Ltd.

66 citations


Journal ArticleDOI
TL;DR: In this paper, delay-dependent exponential stability criteria are presented for switched systems consisting of a family of stable and unstable subsystems with interval time-varying delay, and sufficient conditions for such criteria are obtained and formulated in terms of linear matrix inequalities.
Abstract: Delay-dependent exponential stability criteria are presented for switched systems consisting of a family of stable and unstable subsystems with interval time-varying delay. Two cases with regard to such delay are considered: one is that time-varying delay function is differentiable and bounded and the other is that time-varying delay function is continuous and bounded. It is very difficult to analyze the stability of such systems due to the existence of time delay and unstable subsystems. By introducing some free-weighting matrices, constructing the new Lyapunov–Krasovskii functional and taking advantage of the average dwell time technique, not only is this difficulty overcome but also sufficient conditions for such criteria are obtained and formulated in terms of linear matrix inequalities. Numerical examples are provided to demonstrate the effectiveness and feasibility of the proposed approaches

47 citations


Journal ArticleDOI
TL;DR: In this article, the authors investigated to which extent the excellent performance of symplectic integrators for long-time integrations in astronomy and molecular dynamics carries over to problems in optimal control.
Abstract: SUMMARY For general optimal control problems, Pontryagin’s maximum principle gives necessary optimality conditions, which are in the form of a Hamiltonian differential equation. For its numerical integration, symplectic methods are a natural choice. This article investigates to which extent the excellent performance of symplectic integrators for long-time integrations in astronomy and molecular dynamics carries over to problems in optimal control. Numerical experiments supported by a backward error analysis show that for problems in low dimension close to a critical value of the Hamiltonian, symplectic integrators have a clear advantage. This is illustrated using the Martinet case in sub-Riemannian geometry. For problems like the orbital transfer of a spacecraft or the control of a submerged rigid body, such an advantage cannot be observed. The Hamiltonian system is a boundary value problem and the time interval is in general not large enough so that symplectic integrators could benefit from their structure preservation of the flow. Copyright q 2008 John Wiley & Sons, Ltd.

47 citations


Journal ArticleDOI
TL;DR: In this article, the authors define MP-pseudo-invex multiobjective optimal control problems and prove that the set of optimal solutions of these problems coincides with the optimal solution of a related scalar problem.
Abstract: We introduce the class of MP-pseudoinvex multiobjective optimal control problems. We show that the concept of MP-pseudoinvexity is a sufficient condition of optimality and, further, that problems such that every control process satisfying Pontryagin's maximum principle is an optimal process are necessarily MP-pseudoinvex problems. Moreover, a sub-class of the MP-pseudoinvex problems, which we call MP-invex multiobjective optimal control problems, is defined. We prove that the set of optimal solutions of MP-invex multiobjective problems coincides with the set of optimal solutions of a related scalar problem. Copyright © 2008 John Wiley & Sons, Ltd.

24 citations


Journal ArticleDOI
TL;DR: In this article, an output feedback controller that is both stable and has an H∞ norm strictly less than a specified value is presented. But the controller is designed to achieve absolute stabilization with a specified level of disturbance attenuation, and the main result involves solving a state feedback version of the problem by solving a Riccati equation dependent on a set of scaling parameters.
Abstract: This paper presents a new approach to the robust H∞ control of an uncertain system via an output feedback controller that is both stable and has an H∞ norm strictly less than a specified value. The uncertain systems under consideration contain structured uncertainty described by integral quadratic constraints. The controller is designed to achieve absolute stabilization with a specified level of disturbance attenuation. The main result involves solving a state feedback version of the problem by solving an algebraic Riccati equation dependent on a set of scaling parameters. Then two further algebraic Riccati equations are solved, which depend on a further set of scaling parameters. The required controller is constructed from the Riccati solutions. Copyright © 2008 John Wiley & Sons, Ltd.

18 citations


Journal ArticleDOI
TL;DR: It is shown that the result obtained in a static context, namely, that the Pareto-optimal cooperative solution can be attained as a normalized noncooperative Rosen equilibrium, can be generalized to a dynamic setting.
Abstract: We consider a finite-horizon two-player differential game where the players face a common terminal environmental constraint. Our aim is to verify whether the result obtained in a static context, namely, that the Pareto-optimal cooperative solution can be attained as a normalized noncooperative Rosen equilibrium, can be generalized to a dynamic setting. We show that this is not the case. However, we provide a taxation mechanism of players' emissions that forces them to implement the less-polluting cooperative strategies. Copyright © 2008 John Wiley & Sons, Ltd.

17 citations


Journal ArticleDOI
TL;DR: In this article, the optimal control problem of discrete-time switched systems is formulated as an optimization problem involving both continuous and discrete-valued variables, and an algorithm that combines the discrete filled function method and a descent method is developed for solving this problem.
Abstract: In this paper, we consider the optimal control problem of discrete-time switched systems. This problem is formulated as an optimization problem involving both continuous and discrete-valued variables. It can be transformed into a discrete optimization problem. A metric in the space of switching sequences is introduced and an appropriate discrete filled function is constructed. Then, an algorithm that combines the discrete filled function method and a descent method is developed for solving this problem. For illustration, some numerical examples are solved. Copyright © 2009 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this article, a new approach was proposed to achieve noise-to-state stabilization in probability for stochastic recurrent neural networks driven by the noise of unknown covariance. But this approach is based on the Lyapunov technique, inverse optimality, differential game theory and the Hamilton-Jacobi-Isaacs equation.
Abstract: In this paper, we extend our previous research results regarding the stabilization of recurrent neural networks from the concept of input-to-state stability to noise-to-state stability, and present a new approach to achieve noise-to-state stabilization in probability for stochastic recurrent neural networks driven by the noise of unknown covariance. This approach is developed by using the Lyapunov technique, inverse optimality, differential game theory, and the Hamilton–Jacobi–Isaacs equation. Numerical examples demonstrate the effectiveness of the proposed approach. Copyright © 2008 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: An easy‐to‐implement dynamic optimization technique based on sequential quadratic programming method and control vector parameterization approach is provided and gives better results in shorter computation times.
Abstract: Determination of the optimal aeration profile for an activated sludge system in which nitrification and denitrification take place sequentially in a single reactor (alternating aerobic–anoxic) is an attractive optimization problem because of complexities involved in, and high computational times required for solution. The rigorous dynamic modeling and start-up simulation of such a system, together with aeration profile optimization by an evolutionary algorithm (EA), were tackled in a previous study. In this paper an easy-to-implement dynamic optimization technique based on sequential quadratic programming method and control vector parameterization approach is provided. In comparison with EA, the proposed algorithm gives better results in shorter computation times. Copyright q 2009 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this article, an information geometric algorithm is used to solve the distribution control problem of the output determined by the control input only, where the authors consider the distribution of the outputs only.
Abstract: In this paper, we use an information geometric algorithm to solve the distribution control problem. Here, we consider the distribution of the output determined by the control input only. We set up two manifolds that are formed by the B-spline functions and the system output probability density functions, and we call them the B-spline manifold(B) and the system output manifold(M), respectively. Moreover, we call the new designed algorithm natural gradient-projection algorithm. In the natural gradient step, we use natural gradient algorithm to obtain an optimal trajectory of the weight vector on the B-spline manifold from the viewpoint of information geometry. In the projection step, we project the selected points on B onto M. The coordinates of the projections on M give the trajectory of the control input u. Copyright © 2008 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this paper, an enhanced particle swarm optimization (EPSO) method was proposed to solve the DELD problem of units with valve-point effects and prohibited operating zones, where feasibility-based rules and heuristic search strategies with priority list based on probability were devised to handle complex constraints effectively.
Abstract: Dynamic economic load dispatch (DELD), which plays an important role in power systems operation, is cast as a complex non-linear constrained optimization problem. It has non-smooth and non-convex characteristic when generation unit valve-point effects and prohibited operating zones are taken into account. The operating region of the units having prohibited zones is broken into isolated feasible sub-regions, which results in multiple decision spaces for the DELD problem. This paper proposes an enhanced particle swarm optimization (EPSO) method to solve the DELD problem of units with valve-point effects and prohibited operating zones. In the proposed EPSO method, feasibility-based rules and heuristic search strategies with priority list based on probability are devised to handle complex constraints effectively. Furthermore, the effects of three crucial parameters on the performance of the EPSO for DELD problem are studied as well. The feasibility and effectiveness of the proposed method are demonstrated for three examples and the test results are compared with those of other methods reported in the literature. It is shown that the proposed method is capable of yielding higher-quality solutions. Copyright © 2008 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this paper, the problem of intercepting a maneuvering target by a guided interceptor is considered, where the interceptor gets the information on the relative velocity and the target lateral acceleration with time delays.
Abstract: The problem of intercepting a maneuvering target by a guided interceptor is considered. It is assumed that the interceptor gets the information on the relative velocity and the target lateral acceleration with time delays. This problem is formulated as a zero-sum differential game with information delays. By using the uncertainty set concept, this differential game is transformed to a new perfect information differential game with a delayed dynamics. The solution of this game is derived, yielding the interception strategy. Illustrative examples are presented. Copyright © 2008 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this paper, the authors apply principles from robustness to a situation where the decision maker is a bank owner and the decision rule determines the optimal provisioning strategy for loan losses.
Abstract: The concept of robustness started emerging in financial and economics literature in the late 1990s. In particular, principles from robust control theory have been used by economic decision makers to investigate the fragility of decision rules across a range of economic models. In line with this tendency, our article applies principles from robustness to a situation where the decision maker is a bank owner and the decision rule determines the optimal provisioning strategy for loan losses. In this regard, we recognize that bank provisions are made for debts that have been identified as impaired or non-performing. Our first objective is to formulate a dynamic banking loan loss model involving a provisioning portfolio consisting of provisions for expected losses and loan loss reserves for unexpected losses. Here, unexpected loan losses and provisioning for expected losses are modeled via a compound Poisson process and an exponential Levy process, respectively. Historical evidence from Organization for Economic Corporation and Development countries assists in confirming some of the modeling choices made. This setup naturally leads to a finite-horizon provisioning problem that may be solved via a mixed optimal/robust control approach involving a constraint for risk. Our investigation concludes with a brief analysis of some of the robustness issues and suggestions for topics of possible future research. Copyright © 2008 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this paper, the relaxed non-quadratic stability conditions, fuzzy observer designs and ∞ controller designs for discrete-time Takagi-Sugeno fuzzy systems based on a relaxed approach in which fuzzy Lyapunov functions are used.
Abstract: This paper investigates the relaxed non-quadratic stability conditions, fuzzy observer designs and ∞ controller designs for discrete-time Takagi-Sugeno fuzzy systems based on a relaxed approach in which fuzzy Lyapunov functions are used First, a new relaxed condition of non-quadratic stability is presented, which is shown to be useful in designing fuzzy controller and observer Second, new fuzzy observers based on the relaxed non-quadratic stability conditions have been proposed Then, a sufficient linear matrix inequality (LMI)-type condition is proposed to guarantee the existence of the ∞ controllers based on the fuzzy observers designed It is shown that the controller and observer parameters can be obtained by solving a set of LMIs that are numerically feasible with commercially available software Finally, the effectiveness and less conservativeness of the proposed approach are demonstrated by two examples

Journal ArticleDOI
TL;DR: In this paper, a differential oligopoly game with advertising is investigated, where different dynamics occur between two groups of agents, the former playing a competitive Nash game and the latter cooperating as a cartel.
Abstract: A differential oligopoly game with advertising is investigated, where different dynamics occur between two groups of agents, the former playing a competitive Nash game and the latter cooperating as a cartel. Sufficient conditions for stability and a qualitative analysis of the profit ratio and social welfare at equilibrium are provided. A threshold value for the size of the competitive fringe is pointed out by a suitable numerical simulation. Copyright © 2009 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this article, the authors explore the link between status-seeking and the exploitation of a common property exhaustible resource and find that a higher degree of status-consciousness leads to greater excessive consumption, and lower capital accumulation.
Abstract: This paper explores the link between status-seeking and the exploitation of a common property exhaustible resource. The extracted resource is an input in the production of a final good. The other input is man-made capital, which is a second state variable. Extraction requires efforts. Economic agents derive utility not only from absolute consumption, but also from relative consumption, because the latter is a signal of relative status. We consider a differential game involving n infinitely lived agents. We compare the Markov-perfect Nash equilibrium of this game with the outcome under cooperation. We find that the degree of status-consciousness has important impact on the Markov-perfect Nash equilibrium. A higher degree of status-consciousness leads to greater excessive consumption, and lower capital accumulation. If extraction is costless, status-consciousness has no impact on the extraction/resource-stock ratio. However, with costly extraction, higher status-consciousness reduces this ratio. Copyright © 2009 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this article, two methods are proposed to recover uniform complete controllability for the chained system, one involves a global singularity-free state scaling transformation, the other is based on a time transform, and both of them require an innovative design of dynamic control component for its subsystem.
Abstract: In this paper, the inverse optimal control designs for chained systems are investigated. The presented designs are based on the thorough study of controllability of chained systems. Particularly, two methods are proposed to recover uniform complete controllability for the chained system. One involves a global singularity-free state-scaling transformation, the other is based on a time transform, and both of them require an innovative design of dynamic control component for its subsystem. Using either of the approaches, the chained system is mapped into a controllable linear time-varying system for which control can systematically be designed to ensure exponential convergence or asymptotic stability. Both state-feedback and output-feedback designs are presented and literally shown to be inversely optimal. Simulation results are used to verify the effectiveness of the proposed controls. Copyright © 2008 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this paper, the management of a coastal aquifer under seawater intrusion (SWI) using distributed control methods is analyzed, where the aquifer's state is taken as the water head elevation (vis-a-vis sea level, say), which varies with time and in space.
Abstract: We analyze the management of a coastal aquifer under seawater intrusion (SWI) using distributed control methods. The aquifer's state is taken as the water head elevation (vis-a-vis sea level, say), which varies with time and in space since extraction, natural recharge and lateral water flows vary with time and in space. The water head, in turn, induces a temporal-spatial SWI process, which changes the volume of fresh water in the aquifer. Under reasonable conditions we show that the optimal state converges to a steady-state process that is constant in time. We characterize the optimal steady-state process in terms of a standard control problem (in space) and offer a tractable algorithm to solve for it. Copyright © 2009 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: The optimal control problem is solved numerically to obtain an optimal efficacy of Ribavirin in a combination treatment of ribavirin with interferon, where the efficacy of the latter has a clinically validated functional form.
Abstract: A mathematical model to represent hepatitis C is presented. An objective functional keeping biomedical goals in mind is formulated. The optimal control problem is solved numerically to obtain an optimal efficacy of ribavirin in a combination treatment of ribavirin with interferon, where the efficacy of the latter has a clinically validated functional form. Copyright © 2009 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this article, a combined control and design problem of a cold store with three possible refrigeration technologies (i.e., mechanical refrigeration, ventilation and evaporative cooling) is studied.
Abstract: The design of controlled processes is a combined optimal control and design problem (OCDP). Literature on solving large OCDPs is rare. This paper presents an algorithm for solving large OCDPs. For this algorithm system dynamics, objective function and their first-order derivatives must be continuous in the state, control and design parameters. The algorithm is successfully applied to the combined control and design problem of a cold store with three possible refrigeration technologies: mechanical refrigeration, ventilation and evaporative cooling. As a result, insight into cost effectiveness of the refrigeration technologies is generated. It is concluded that for this cold store in the Netherlands evaporative cooling is too expensive. Ventilation is economically viable if the cold store is to be used in January only. In case the cold store is to be operated all year then it is most economical to rely on mechanical refrigeration only and use the overcapacity during most part of the year to shift refrigeration to low-tariff hours. Copyright © 2008 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this paper, a linear time-invariant dynamic inversion autopilot with guaranteed robustness is proposed, using the generalized stability margin to choose the dynamics that will be inverted (A and B matrices), and McFarlane-Glover loop shaping to select the C matrix of the inverted plant, the desired dynamics, and a robust state estimator.
Abstract: Honeywell and others have successfully applied dynamic inversion control to many flight control problems. It is a simple and systematic design procedure that guarantees nominal performance. However, unlike ℋ∞ methods, it does not provide any robustness guarantees. In this paper, we propose a procedure for designing a linear time-invariant dynamic inversion autopilot with guaranteed robustness. We will use the generalized stability margin to choose the dynamics that will be inverted (A and B matrices), and McFarlane–Glover loop shaping to select the C matrix of the inverted plant, the desired dynamics, and a robust state estimator. The proposed procedure is illustrated by designing a pitch axis autopilot that achieves robust performance with a number of Boeing 747 aircraft models. Copyright © 2008 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this paper, an optimal control approach for a robust control design problem of the neutral type time-delay systems, taking parameter uncertainties and state delay into account, is proposed, and a suitable linear state feedback control law is characterized via Lyapunov stability theory.
Abstract: This paper proposes an optimal control approach for a robust control design problem of the neutral type time-delay systems, taking parameter uncertainties and state delay into account. The robust control design problem can be equivalently transformed into an optimal control problem, and the amount of plant uncertainties is indirectly reflected in the performance index. By introducing algebraic manipulations and appropriate uncertainty descriptions, the uncertain dynamical system can not only achieve stability, but can also reach the guaranteed level of performance. A suitable linear state feedback control law is characterized via Lyapunov stability theory to ensure quadratic stability and performance robustness of the closed-loop system. Copyright © 2008 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this paper, an optimal control for a linear system with quadratic performance is obtained using genetic programming (GP) using non-traditional methods, and the obtained GP solution is compared with the traditional Runge-Kutta method.
Abstract: In this paper, optimal control for a linear system with quadratic performance is obtained using genetic programming (GP). The goal is to find the optimal control with reduced calculus effort using non-traditional methods. The obtained GP solution is compared with the traditional Runge–Kutta method. To obtain optimal control, the solution of matrix Riccati differential equation is computed based on grammatical evolution. The accuracy of the solution of the GP approach to the problem is qualitatively better than traditional methods. An illustrative numerical example is presented for the proposed method. Copyright © 2008 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this paper, a trajectory optimization algorithm that generates a quadric control update, which satisfies the constraints and necessary conditions to the second order, is presented, which is designed to solve multistage optimization problems.
Abstract: This paper describes a trajectory optimization algorithm that generates a quadric control update, which satisfies the constraints and necessary conditions to the second order. The algorithm is designed to solve multistage optimization problems. The algorithm is tested against a commercially available Sequential Quadratic Programming algorithm on problems with linear dynamics and linear and nonlinear constraints. This algorithm is a departure from previous methods because it explicitly satisfies the constraints to the second order. Copyright © 2009 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this article, the authors considered the use of optimal control theory in designing radio frequency excitation pulses for magnetic spin systems satisfying Bloch dynamics, which are required in applications of nuclear magnetic resonance to initially transfer sample magnetization vectors to the transverse plane.
Abstract: This paper considers the use of optimal control theory in designing radio frequency excitation pulses for magnetic spin systems satisfying Bloch dynamics. Such pulses are required in applications of nuclear magnetic resonance to initially transfer sample magnetization vectors to the transverse plane. Once transferred, signals released by nuclei as they respond to a static magnetic field normal to the transverse plane are then analyzed and interpreted. Continuous time deterministic optimal control theory is employed to determine time-dependent pulse amplitudes and frequencies that minimize the distance between final magnetization vectors and a chosen target vector. Pulses are designed to excite a range of resonant frequencies and to tolerate miscalibration errors in applied fields. The model presented permits a unified treatment of the control problem as considered by a variety of authors, and a thorough mathematical analysis of the existence, and characteristics of, optimal excitation pulses. Practical numerical algorithms for designing optimal pulses are given, and the effectiveness of the algorithms is illustrated by comparing the pulses that they generate with those commonly used in high-resolution spectroscopy. Copyright © 2008 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this article, a series of generalized eigenvalue problems subject to a set of linear matrix inequalities (LMI) constraints are formulated for designing ℋ2 suboptimal estimators, static controllers, and dynamic controllers for nonnegative dynamical systems.
Abstract: Linear matrix inequalities (LMIs) provide a powerful design framework for linear control problems. In this paper, we use LMIs to develop ℋ2 (sub)optimal estimators and controllers for nonnegative dynamical systems. Specifically, we formulate a series of generalized eigenvalue problems subject to a set of LMI constraints for designing ℋ2 suboptimal estimators, static controllers, and dynamic controllers for nonnegative dynamical systems. The resulting ℋ2 suboptimal controllers guarantee that the closed-loop plant system states remain in the nonnegative orthant of the state space. Finally, a numerical example is provided to demonstrate the efficacy of the proposed approach. Copyright © 2008 John Wiley & Sons, Ltd.