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.     

Approximation of integro-differential equations associated with piecewise deterministic process

Aim of this paper is to present an approximation scheme for optimal control problems of piecewise deterministic processes and corresponding integro-differential Hamilton–Jacobi–Bellman equations. The method is based on a discrete dynamic programming approach. We discretize the continuous process and the cost functional obtaining a discrete time optimal control problem. The corresponding dynamic programming equation gives an approximation of the integro-differential equation. The main feature of the method is the uniform convergence to the value function of the continuous control problem, which can be characterized as the unique weak solution (in viscosity sense) of the dynamic programming equation. Moreover, under appropriate assumptions, an error estimate on the truncation error is derived. It is worth noting that the method provides approximate feedback controls at any point of the grid without extra computations. An application of the approximation scheme to the numerical solution of an optimal control problem for a storage process is also detailed. © 1997 John Wiley & Sons, Ltd.     

A numerical method for computing optimal controls in feedback and digital forms and its application to the blowing‐venting control system of manned submarines

On the basis of the classical variational reformulation of optimal control problems, we introduce a numerical scheme for solving those problems where the goal is the computation of optimal controls in feedback and digital forms defined on a discrete time mesh. The algorithm reduces the computation of such controls to solving a suitable nonlinear mathematical programming problem where the unknowns are the controls and slope of the state variable of the original problem. The motivation for this study comes from the real‐world engineering problem which consists of maneuvering a manned submarine by using the blowing‐venting control system of the ballast tanks of the vehicle. After checking the proposed algorithm in an academic example, we apply it to the maneuvering problem of submarines whose mathematical model includes a state law which is composed of a system of twenty‐four nonlinear ordinary differential equations. Numerical results illustrate the performance of the numerical scheme. Copyright © 2012 John Wiley & Sons, Ltd.     

Boundary feedback control of a vibrating beam subject to a displacement constraint

The problem of controlling the dynamic response of a beam by means of boundary feedback is solved. The control is applied at one of the supports of the beam by means of a torque actuator in order to keep the maximum deflection within specified bounds. Since the location and time of the maximum deflection depend on the control itself, the feedback is determined by the solution of a min-max problem. The theory is illustrated by a numerical example involving an impulsively loaded beam, and the effectiveness of the proposed control is studied numerically.     

Direct-Forcing Immersed Boundary Method for Mixed Heat Transfer

A direct-forcing immersed boundary method (DFIB) with both virtual force and heat source is developed here to solve Navier-Stokes and the associated energy transport equations to study some thermal flow problems caused by a moving rigid solid object within. The key point of this novel numerical method is that the solid object, stationary or moving, is first treated as fluid governed by Navier-Stokes equations for velocity and pressure, and by energy transport equation for temperature in every time step. An additional virtual force term is then introduced on the right hand side of momentum equations in the solid object region to make it act exactly as if it were a solid rigid body immersed in the fluid. Likewise, an additional virtual heat source term is applied to the right hand side of energy equation at the solid object region to maintain the solid object at the prescribed temperature all the time. The current method was validated by some benchmark forced and natural convection problems such as a uniform flow past a heated circular cylinder, and a heated circular cylinder inside a square enclosure. We further demonstrated this method by studying a mixed convection problem involving a heated circular cylinder moving inside a square enclosure. Our current method avoids the otherwise requested dynamic grid generation in traditional method and shows great efficiency in the computation of thermal and flow fields caused by fluid-structure interaction.     

Temperature Feedback and Collagen Cross-Linking in Argon Laser Vascular Welding

A preliminary single-animal study of in vivo argon laser vascular welding was conducted using a canine model. The effects of temperature feedback control and saline drip cooling on patency and collagen cross-linking were investigated. The surface temperature at the centre of the laser spot was monitored using a two-colour infrared thermometer. The surface temperature was limited by either a saline drip or feedback control of the laser. Acute patency was evaluated and collagen cross-link assays were performed. Though both protocols yielded successful tissue fusion, welds maintained at a surface temperature of 50°C using feedback control had an elevated cross-link count compared to controls, whereas tissues irradiated without feedback control experienced a cross-link decrease. Simulations using the LATIS (LAser-TISsue) computer code suggest that drip-cooled procedures achieve significantly higher temperatures beneath the tissue surface than temperature feedback-controlled procedures. Differences between the volumetric heating associated with drip-cooled and feedback-controlled protocols may account for the different effects on collagen cross-links. Covalent mechanisms may play a role in argon laser vascular fusion. Paper received 20 August 1997; accepted in final form 16 September 1997.     

Bilinear matrix inequality approaches to robust guaranteed cost control for uncertain discrete‐time delay system

The robust guaranteed cost control problem for uncertain discrete‐time delay system is considered in this paper. Sufficient conditions for the existence of the robust guaranteed cost controllers via memoryless state feedback and static output feedback are expressed as bilinear matrix inequality (BMI). Furthermore, the design methods of optimal robust guaranteed cost controllers, which minimize the upper bound of a given quadratic cost function are presented. Alternate iterative algorithms are proposed to solve the nonconvex optimization problems with BMI constrains. A numerical example is given to illustrate the effectiveness of the proposed methods.Copyright © 2012 John Wiley & Sons, Ltd.     

Robust guaranteed cost control for time‐delay fractional‐order neural networks systems

In this paper, we investigate the problem of guaranteed cost control of uncertain fractional‐order neural networks systems with time delays. By employing the Lyapunov‐Razumikhin theorem, a sufficient condition for designing a state‐feedback controller which makes the closed‐loop system asymptotically stable and guarantees an adequate cost level of performance is derived in terms of bilinear matrix inequalities. Two numerical examples are given to show the effectiveness of the obtained results.     

Robust delay‐range‐dependent stabilization for Markovian jump systems with mode‐dependent time delays and nonlinearities

This paper discusses the robust stabilization problem for a class of Markovian jump systems with nonlinear disturbances and time delays, which are time‐varying in intervals and depend on system mode. By exploiting a new Lyapunov–Krasovskii functional, which takes into account the range of delay and by making use of novel techniques, mean‐square exponential stability result is proposed. Based on the obtained stability condition, a sufficient condition for state feedback controller, which stabilizes system and maximizes the bound on nonlinear perturbations is derived in terms of linear matrix inequalities involving a convex optimization. Finally, illustrative examples are presented to show the benefits and effectiveness of the proposed approaches. Copyright © 2009 John Wiley & Sons, Ltd.     

Decentralized output‐feedback stabilization for interconnected discrete systems with unknown delays

The decentralized feedback stabilization problem of a class of nonlinear interconnected discrete‐time systems is considered. This class of systems has unknown‐but‐bounded state‐delay and uncertain nonlinear perturbations satisfying quadratic constraints that are functions of the overall state and delayed state vectors. A decentralized output feedback scheme is proposed and analyzed such that the overall closed‐loop system guarantees global delay‐dependent stability condition, derived in terms of local subsystem variables. Incorporating feedback gain perturbations, new resilient decentralized feedback scheme is subsequently developed. The proposed approach is formulated within the framework of convex optimization over linear matrix inequalities. Simulation results illustrate the effectiveness of the proposed decentralized output‐feedback controllers. Copyright © 2009 John Wiley & Sons, Ltd.     

Multiobjective fault‐tolerant fixed‐order/PID control of multivariable discrete‐time linear systems with unmeasured disturbances

In this paper, a multiobjective fault‐tolerant fixed‐order output feedback controller design technique is proposed for multivariable discrete‐time linear systems with unmeasured disturbances. Initially, a multiobjective fixed‐order controller is designed for the system by transforming the problem of tuning the parameters of the controller into a static output feedback problem and solving a mixed H₂/H_∞ optimization problem with bilinear matrix inequalities. Subsequently, the fixed‐order controller is used to construct the closed‐loop system and an active fault‐tolerant control scheme is applied using the input/output data collected from the controlled system. Motivated by its popularity in industry, the proposed method is also used to tune the parameters of proportional‐integral‐derivative controllers as a special case of structured controllers with the fixed order. Two numerical simulations are provided to demonstrate the design procedure and the flexibility of the proposed technique.     

A 2D‐based approach to guaranteed cost repetitive control for uncertain discrete‐time systems

This paper presents a two‐dimensional (2 D)‐based approach to the problem of guaranteed cost repetitive control for uncertain discrete‐time systems. The objective is to design a control law such that the closed‐loop repetitive control system is robustly stable and a certain bound of performance criteria is guaranteed for all admissible uncertainties. It is shown first how the proposed repetitive control scheme can be equivalently formulated in the form of a distinct class of 2 D system. Then, sufficient conditions for the existence of guaranteed cost control law are derived in terms of linear matrix inequality (LMI), and the control law matrices are characterized by the feasible solutions to this LMI. Moreover, an optimization problem is introduced to efficiently solve the optimal guaranteed cost control law by minimizing the upper bound of the cost function. The proposed approach is applicable not only to SISO systems, but also to MIMO systems. Two numerical examples are provided to demonstrate the effectiveness of the proposed controller design procedures. Copyright © 2010 John Wiley & Sons, Ltd.     

Immersed Boundary-Lattice Boltzmann Coupling Scheme for Fluid-Structure Interaction with Flexible Boundary

Coupling the immersed boundary (IB) method and the lattice Boltzmann (LB) method might be a promising approach to simulate fluid-structure interaction (FSI) problems with flexible structures and complex boundaries, because the former is a general simulation method for FSIs in biological systems, the latter is an efficient scheme for fluid flow simulations, and both of them work on regular Cartesian grids. In this paper an IB-LB coupling scheme is proposed and its feasibility is verified. The scheme is suitable for FSI problems concerning rapid flexible boundary motion and a large pressure gradient across the boundary. We first analyze the respective concepts, formulae and advantages of the IB and LB methods, and then explain the coupling strategy and detailed implementation procedures. To verify the effectiveness and accuracy, FSI problems arising from the relaxation of a distorted balloon immersed in a viscous fluid, an unsteady wake flow caused by an impulsively started circular cylinder at Reynolds number 9500, and an unsteady vortex shedding flow past a suddenly started rotating circular cylinder at Reynolds number 1000 are simulated. The first example is a benchmark case for flexible boundary FSI with a large pressure gradient across the boundary, the second is a fixed complex boundary problem, and the third is a typical moving boundary example. The results are in good agreement with the analytical and existing numerical data. It is shown that the proposed scheme is capable of modeling flexible boundary and complex boundary problems at a second-order spatial convergence; the volume leakage defect of the conventional IB method has been remedied by using a new method of introducing the unsteady and non-uniform external force; and the LB method makes the IB method simulation simpler and more efficient.     

A Conservative Numerical Method for the Cahn–Hilliard Equation with Generalized Mobilities on Curved Surfaces in Three-Dimensional Space

In this paper, we develop a conservative numerical method for the Cahn– Hilliard equation with generalized mobilities on curved surfaces in three-dimensional space. We use an unconditionally gradient stable nonlinear splitting numerical scheme and solve the resulting system of implicit discrete equations on a discrete narrow band domain by using a Jacobi-type iteration. For the domain boundary cells, we use the trilinear interpolation using the closest point method. The proposing numerical algorithm is computationally efficient because we can use the standard finite difference Laplacian scheme on three-dimensional Cartesian narrow band mesh instead of discrete Laplace–Beltrami operator on triangulated curved surfaces. In particular, we employ a mass conserving correction scheme, which enforces conservation of total mass. We perform numerical experiments on the various curved surfaces such as sphere, torus, bunny, cube, and cylinder to demonstrate the performance and effectiveness of the proposed method. We also present the dynamics of the CH equation with constant and space-dependent mobilities on the curved surfaces.     

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.     

On Computational Modelling of Strain-Hardening Material Dynamics

In this paper we show that entropy can be used within a functional for the stress relaxation time of solid materials to parametrise finite viscoplastic strain-hardening deformations. Through doing so the classical empirical recovery of a suitable irreversible scalar measure of work-hardening from the three-dimensional state parameters is avoided. The success of the proposed approach centres on determination of a rate-independent relation between plastic strain and entropy, which is found to be suitably simplistic such to not add any significant complexity to the final model. The result is sufficiently general to be used in combination with existing constitutive models for inelastic deformations parametrised by one-dimensional plastic strain provided the constitutive models are thermodynamically consistent. Here a model for the tangential stress relaxation time based upon established dislocation mechanics theory is calibrated for OFHC copper and subsequently integrated within a two-dimensional moving-mesh scheme. We address some of the numerical challenges that are faced in order to ensure successful implementation of the proposed model within a hydrocode. The approach is demonstrated through simulations of flyer-plate and cylinder impacts.     

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.     

Robust H∞ output feedback control for Markovian jump systems with actuator saturation

This paper concerns the output feedback control problem for a class of uncertain continuous‐time Markovian jump systems with actuator saturation. The controller is nonlinear in nature and will be first parameterized in the quasi‐linear parameter varying form. Conditions under which the closed‐loop system is stochastically stable with γ disturbance attenuation are then derived in terms of an LMI approach. The problem of designing an output feedback controller such that the estimate of the domain of attraction is enlarged is then formulated and solved as an optimization problem with LMI constraints. The case where the transition rate matrix of the Markov process is unknown is considered, and the robust H_∞ output feedback controller is then derived. Finally, a numerical example is given to illustrate the effectiveness of the proposed results. Copyright © 2011 John Wiley & Sons, Ltd.     

Gradient flow approach to LQ cost improvement for simultaneous stabilization problem

In this paper we consider LQ cost optimization for the simultaneous stabilization problem. The objective is to find a single simultaneously stabilizing feedback gain matrix such that all closed-loop systems exhibit good transient behaviour. The cost function used is a quadratic function of the system states and the control vector. This paper proposes to seek an optimization solution by solving an ordinary differential equation which is a gradient flow associated with the cost function. Two examples are presented to illustrate the effectiveness of the proposed procedure.