ℋ︁2 suboptimal estimation and control for nonnegative dynamical systems

Linear matrix inequalities (LMIs) provide a powerful design framework for linear control problems. In this paper, we use LMIs to develop ??₂ (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 ??₂ suboptimal estimators, static controllers, and dynamic controllers for nonnegative dynamical systems. The resulting ??₂ 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.     

LQ‐optimal control of positive linear systems

The LQ₊ problem, i.e. the finite‐horizon linear quadratic optimal control problem with nonnegative state constraints, is studied for positive linear systems in continuous time and in discrete time. Necessary and sufficient optimality conditions are obtained by using the maximum principle. These conditions lead to a computational method for the solution of the LQ₊ problem by means of a corresponding Hamiltonian system. In addition, the necessary and sufficient conditions are proved for the LQ₊‐optimal control to be given by the standard LQ‐optimal state feedback law. Then sufficient conditions are established for the positivity of the LQ‐optimal closed‐loop system. In particular, such conditions are obtained for the problem of minimal energy control with penalization of the final state. Moreover, a positivity criterion for the LQ‐optimal closed‐loop system is derived for positive discrete‐time systems with a positively invertible (dynamics) generator. The main results are illustrated by numerical examples. Copyright © 2010 John Wiley & Sons, Ltd.     

Full delayed state feedback pole assignment of discrete‐time time‐delay systems

This paper studies the problem of pole assignment of discrete‐time time delay system with delayed state feedback. The problem is solved in this paper by requiring that the maximal delay in the feedback equals the maximal delay of the open‐loop system. A necessary and sufficient condition guaranteeing the existence of a solution is presented. By using the augmentation technique, the pole assignment problem is then transformed to the problem of solving a linear matrix equation such that certain conditions are satisfied. To solve the linear equation problem, when the desired closed‐loop eigenvalues are not prescribed, a parametric approach using real arithmetic is presented by using polynomial matrices associated with the system matrices. When the desired closed‐loop eigenvalues are prescribed, singular value decomposition can be adopted to solve the linear matrix equation. Both approaches can provide full degree of freedom, which can be further utilized to accomplish some other design objects. The robust pole assignment problem is considered to demonstrate the advantages of the method. Numerical examples are employed to illustrate the effectiveness of the proposed approaches. Copyright © 2009 John Wiley & Sons, Ltd.     

The non-linear beam via optimal control with bounded state variables

The non-linear beam with bounded deflection is considered as an optimal control problem with bounded state variables. The theory of necessary optimality conditions leads to boundary value problems with jump conditions which are solved by multiple-shooting techniques. A branching analysis is performed which exhibits the different solution structures. In particular, the second bifurcation point is determined numerically. The bifurcation diagram reveals a hysteresis-like behaviour and explains the jumping to a different state at this bifurcation point.     

Global cluster synchronization in finite time for complex dynamical networks with hybrid couplings via aperiodically intermittent control

In this article, we investigate the cluster synchronization in finite time for nonlinear complex dynamical networks (CDNs) with hybrid couplings via aperiodically intermittent control. First, a new lemma with respect to the convergence in finite time is developed for the nonnegative continuous functions. Second, the classification controller, which is dependent of the coupling of the dynamic nodes among different clusters, is designed to realize the cluster synchronization goal. Third, by means of the Lyapunov functional approach, the inequality analysis technique, and the proposed lemmas, the global cluster synchronization conditions in finite time are addressed in terms of linear matrix inequalities (LMIs) for CDNs via discontinuous feedback control scheme with integral terms. Moreover, the global cluster synchronization conditions in finite time is also achieved in the form of LMIs for CDNs with hybrid couplings via aperiodically intermittent control. In addition, the settling time, which is closely related to the topological structure of networks and the maximum ratio of the rest width to the aperiodic time span, is estimated accurately. Finally, the effectiveness of theoretical results is verified by two simulation examples.     

Optimal reduced‐order LQG synthesis: Nonseparable control and estimation design

The optimal linear‐quadratic‐Gaussian synthesis design approach and the associated separation principle are investigated for the case where the observer design model is a reduced model of the underlying system model. Performance of the resulting reduced‐order controller in the full‐state system environment is formulated in terms of an augmented state vector consisting of the system state vector and the reduced model state vector. Considering explicitly separated linear control and estimation laws, a calculus of variations/Hamiltonian approach is used to determine the necessary conditions for the optimal controller and observer gains for the simplified algorithms. Results show that the optimal gains are not separable, ie, the optimal controller and observer gains are coupled and cannot be computed independently. Numerical examples of an infinite‐horizon and finite‐horizon control and estimation large‐scale multiagent system problem clearly show the advantages of using the nonseparable coupled solutions.     

A gradient flow approach to computing a non-linear discrete time quadratic optimal feedback gain matrix

In this paper we propose an approach to solving infinite planning horizon quadratic optimal regulator problems with linear static state feedback for discrete time systems. The approach is based on solving a sequence of approximate problems constructed by combining a finite horizon problem with an infinite horizon linear problem. A gradient-flow algorithm is derived to solve the approximate problems. As part of this, a new algorithm is derived for computing the gradient of the cost functional, based on a system of difference equations to be solved completely forward in time. Two numerical examples are presented.     

Direct and indirect methods in optimal control with state constraints and the climbing trajectory of an aircraft

In this article, the minimum time and fuel consumption of an aircraft in its climbing phase are studied. The controls are the thrust and the lift coefficient and state constraints are taken into account: air slope and speed limitations. The application of the maximum principle leads to parameterize the optimal control and the multipliers associated with the state constraints with the state and the costate and leads to describe a multipoint boundary value problem, which is solved by multiple shooting. This indirect method is the numerical implementation of the maximum principle with state constraints and it is initialized by the direct method, both to determine the optimal structure and to obtain a satisfying initial guess. The solutions of the boundary value problems we define give extremals, which satisfy necessary conditions of optimality with at most 2 boundary arcs. Note that the aircraft dynamics has a singular perturbation but no reduction is performed.     

Computational optimal control of the terminal bunt manoeuvre—Part 2: minimum‐time case

This is the second part of a paper studies trajectory shaping of a generic cruise missile attacking a fixed target from above. The problem is reinterpreted using optimal control theory resulting in a minimum flight time problem; in the first part the performance index was time‐integrated altitude. The formulation entails non‐linear, two‐dimensional (vertical plane) missile flight dynamics, boundary conditions and path constraints, including pure state constraints. The focus here is on informed use of the tools of computational optimal control, rather than their development. The formulation is solved using a three‐stage approach. In stage 1, the problem is discretized, effectively transforming it into a non‐linear programming problem, and hence suitable for approximate solution with DIRCOL and NUDOCCCS. The results are used to discern the structure of the optimal solution, i.e. type of constraints active, time of their activation, switching and jump points. This qualitative analysis, employing the results of stage 1 and optimal control theory, constitutes stage 2. Finally, in stage 3, the insights of stage 2 are made precise by rigorous mathematical formulation of the relevant two‐point boundary value problems (TPBVPs), using the appropriate theorems of optimal control theory. The TPBVPs obtained from this indirect approach are then solved using BNDSCO and the results compared with the appropriate solutions of stage 1. The influence of boundary conditions on the structure of the optimal solution and the performance index is investigated. The results are then interpreted from the operational and computational perspectives. Copyright © 2007 John Wiley & Sons, Ltd.     

A framework for robust quadratic optimal control with parametric dynamic model uncertainty using polynomial chaos

We propose a framework tailored to robust optimal control (OC) problems subject to parametric model uncertainty of system dynamics. First, we identify a generic class of robust objective kernels that are based on the class of deterministic quadratic objectives. It is demonstrated how such kernels can be expressed as a function of the stochastic moments of the state and how the objective terms relate to the robustness and performance of the optimal solution. Second, we engage the generalized polynomial chaos (gPC) framework to propagate uncertainty along the state trajectory. Integrating both frameworks makes way to reformulate the problem as a deterministic OC problem in function of the gPC expansion coefficients that can be solved using existing methods. We apply the framework to solve the problem of robust optimal startup behavior of a nonlinear mechanical drivetrain that is subject to a bifurcation in its dynamics.     

Mixed H2/H∞ robust output feedback sliding mode control

This paper presents an output feedback sliding mode control scheme for uncertain dynamical systems. The design problem is solved in two steps, involving first a state feedback and then an output feedback problem. First, using the null space dynamics, the sliding surface for the unmatched uncertainty is designed. Then, by tuning the sliding surface, a robust controller is constructed for the whole uncertainty; this problem takes the form of static‐output feedback. Based on this, a dynamic output feedback controller for the system augmented with the sliding surface is designed. The synthesis involves the solution of an Linear Matrix Inequality (LMI) and Bilinear Matrix Inequality (BMI) problem; the BMI problem is solved iteratively. The proposed approach is illustrated by applying it to a well‐known robust benchmark problem and also experimentally on a spring mass system with variable stiffness. Simulation and experimental results show that the proposed method outperforms previous approaches in terms of robust performance. Copyright © 2011 John Wiley & Sons, Ltd.     

Polynomial LQG design of subrate digital feedback systems via frequency decomposition

This paper describes the optimal synthesis of digital feedback systems in which the measured variable is sampled at a faster rate than the control is activated. Combining the ‘polynomial equations’ approach with frequency decomposition, the design methodology deals entirely with pulse‐transfer function models and, computationally, requires only the solution of scalar spectral factorization and pole‐placement problems. The relative influence of the ‘subrate’ LQG controller on closed‐loop performance is assessed from results engendered by a series of numerical examples. Copyright © 2000 John Wiley & Sons, Ltd.     

Optimal control of radiative heat transfer in glass cooling with restrictions on the temperature gradient

This paper is motivated by an optimal boundary control problem for the cooling process of molten and already formed glass down to room temperature. The high temperatures at which glass is processed demand to include radiative heat transfer in the computational model. Since the complete radiative heat transfer equations are too complex for optimization purposes, we use simplified approximations of spherical harmonics coupled with a practically relevant frequency bands model. The optimal control problem is considered as a partial differential algebraic equation (PDAE)‐constrained optimization problem with box constraints on the control. In this paper, we augment the objective by a functional depending on the state gradient, which forces a minimization of thermal stress inside the glass. To guarantee consistent and grid‐independent values of the reduced objective gradient at the end of the cooling process, we pursue two approaches. The first includes the temperature gradient with a time‐dependent linearly decreasing weight. In the second approach, we augment the objective functional by the final state tracking and final state gradient optimization. To determine an optimal boundary control, we apply a projected gradient method with the Armijo step size rule. The reduced objective gradient is computed by the continuous adjoint approach. The arising time‐dependent PDAEs are numerically solved by variable step size one‐step methods of Rosenbrock type in time and adaptive multilevel finite elements in space. We present two‐dimensional numerical results for an infinitely long glass block and compare the two different approaches derived to ensure consistency at the end of the cooling process. Copyright © 2011 John Wiley & Sons, Ltd.     

High‐order maximum principles for the stability analysis of positive bilinear control systems

We consider a continuous‐time positive bilinear control system, which is a bilinear control system with Metzler matrices. The positive orthant is an invariant set of such a system, and the corresponding transition matrix is entrywise nonnegative for all time. Motivated by the stability analysis of positive linear switched systems under arbitrary switching laws, we define a control as optimal if it maximizes the spectral radius of the transition matrix at a given final time. We derive high‐order necessary conditions for optimality for both singular and bang–bang controls. Our approach is based on combining results on the second‐order derivative of a simple eigenvalue with the generalized Legendre‐Clebsch condition and the Agrachev–Gamkrelidze second‐order optimality condition. Copyright © 2015 John Wiley & Sons, Ltd.     

Singular linear systems with delay: ℋ︁∞ stabilization

This paper deals with the class of continuous‐time singular linear systems with time‐delay in the state vector. The stabilization problem of this class of systems using a state feedback controller is tackled. New delay‐dependent sufficient conditions on ??_∞ stabilization are developed. A design algorithm for a memoryless state feedback controller which guarantees that the closed‐loop dynamics will be regular, impulse‐free and stable with γ‐disturbance rejection is proposed. It is shown that the addressed problem can be solved if the corresponding developed linear matrix inequalities (LMIs) with some constraints are feasible. A numerical example is employed to show the usefulness of the proposed results. Copyright © 2007 John Wiley & Sons, Ltd.     

Optimal exploration and consumption of a natural resource—deterministic case

An exploration and consumption model for exhaustible natural resources is used to formulate a deterministic optimal control problem over an infinite time horizon. A dynamic programming approach is employed throughout. The non-linear partial differential equation which results from an application of the necessary conditions is solved analytically by the method of characteristics. Several properties of the solution are discussed.     

Linear optimal control for an aircraft model

A linear optimal proportional plus integral (PI) controller is designed for a lateral aircraft model such that an approximately decoupled closed-loop step response is obtained for the roll angle and side-slip angle while the linear optimal feedback gains are simultaneously minimized. A standard mathematical programming algorithm is employed to calculate the diagonal state weighting matrix Q such that the specifications are satisfied. The synthesis procedure consists of the iterative minimization of a functional where at each iteration step a linear optimal control problem is solved.     

Numerical solution of time-delayed optimal control problems with terminal inequality constraints

In this short communication we consider an approximation scheme for solving time-delayed optimal control problems with terminal inequality constraints. Time-delayed problems are characterized by variables x (t - τ) with a time-delayed argument. In our scheme we use a Páde approximation to determine a differential relation for y (t), an augmented state that represents x (t - τ). Terminal inequality constraints, if they exist, are converted to equality constraints via Valentine-type unknown parameters. The merit of this approach is that existing, well-developed optimization algorithms may be used to solve the transformed problems. Two linear/non-linear time-delayed optimal control problems are solved to establish its usefulness.     

A Second Order Finite-Difference Ghost-Point Method for Elasticity Problems on Unbounded Domains with Applications to Volcanology

We propose a finite-difference ghost-point approach for the numerical solution of Cauchy-Navier equations in linear elasticity problems on arbitrary unbounded domains. The technique is based on a smooth coordinate transformation, which maps an unbounded domain into a unit square. Arbitrary geometries are defined by suitable level-set functions. The equations are discretized by classical nine-point stencil on interior points, while boundary conditions and high order reconstructions are used to define the field variables at ghost-points, which are grid nodes external to the domain with a neighbor inside the domain. The linear system arising from such discretization is solved by a multigrid strategy. The approach is then applied to solve elasticity problems in volcanology for computing the displacement caused by pressure sources. The method is suitable to treat problems in which the geometry of the source often changes (explore the effects of different scenarios, or solve inverse problems in which the geometry itself is part of the unknown), since it does not require complex re-meshing when the geometry is modified. Several numerical tests are successfully performed, which asses the effectiveness of the present approach.     

A Hybrid Immersed Boundary-Lattice Boltzmann Method for Simulation of Viscoelastic Fluid Flows Interaction with Complex Boundaries

In this study, a numerical technique based on the Lattice Boltzmann method is presented to model viscoelastic fluid interaction with complex boundaries which are commonly seen in biological systems and industrial practices. In order to accomplish numerical simulation of viscoelastic fluid flows, the Newtonian part of the momentum equations is solved by the Lattice Boltzmann Method (LBM) and the divergence of the elastic tensor, which is solved by the finite difference method, is added as a force term to the governing equations. The fluid-structure interaction forces are implemented through the Immersed Boundary Method (IBM). The numerical approach is validated for Newtonian and viscoelastic fluid flows in a straight channel, a four-roll mill geometry as well as flow over a stationary and rotating circular cylinder. Then, a numerical simulation of Oldroyd-B fluid flow around a confined elliptical cylinder with different aspect ratios is carried out for the first time. Finally, the present numerical approach is used to simulate a biological problem which is the mucociliary transport process of human respiratory system. The present numerical results are compared with appropriate analytical, numerical and experimental results obtained from the literature.