An iterative Bregman regularization method for optimal control problems with inequality constraints

We study an iterative regularization method of optimal control problems with control constraints. The regularization method is based on generalized Bregman distances. We provide convergence results under a combination of a source condition and a regularity condition on the active sets. We do not assume attainability of the desired state. Furthermore, a priori regularization error estimates are obtained.     

Robustness and robust stability of the active sliding mode synchronization

We have developed relations between uncertainties and signals bounds in one side and the control parameters on the other side in the case of the active sliding mode synchronization. Using Lyapunov stability theorem, we have determined uncertainties levels for which synchronization is achieved for a given set of the control parameters. We have run a nonlinear programming algorithm to determine the control parameters for specific range of the uncertainties. Finally, numerical simulations are presented to verify the derived relations.     

Inexact Iterative Bregman Method for Optimal Control Problems

In this article, we investigate an inexact iterative regularization method based on generalized Bregman distances of an optimal control problem with control constraints. We show robustness and convergence of the inexact Bregman method under a regularity assumption, which is a combination of a source condition and a regularity assumption on the active sets. We also take the discretization error into account. Numerical results are presented to demonstrate the algorithm.     

A semi-smooth Newton method for an inverse problem in option pricing

We present an optimal control approach using a Lagrangian framework to identify local volatility functions from given option prices. We employ a globalized sequential quadratic programming (SQP) algorithm and implement a line search strategy. The linear-quadratic optimal control problems in each iteration are solved by a primal-dual active set strategy which leads to a semi-smooth Newton method. We present first- and second-order analysis as well as numerical results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sufficient quadratic conditions of extremum for discontinuous controls in optimal control problems with mixed constraints

We derive second-order sufficient optimality conditions for discontinuous controls in optimal control problems of ordinary differential equations with initial-final state constraints and mixed state-control constraints of equality and inequality type. Under the assumption that the gradients with respect to the control of active mixed constraints are linearly independent, the sufficient conditions imply a bounded strong minimum in the problem.     

One-dimensional discrete LQR control of compression of the human chest impulsively loaded by fast moving point mass

This paper uncovers some interesting extension of an optimal discrete control methodology partially included in Proceedings and presented at the international conference on “Dynamical Systems Theory and Applications”. There has been applied a scheme for realisation of active control strategy with numerically estimated linear optimal quadratic index of performance in reduction of impact-induced deformation of human chest loaded by a point mass at the central point of upper-torso body. We focused on application of one active element attached between torso’s upper back (looking from posterior direction) and a fixed support. As the practical result we provide values of quality and reaction matrices, some useful deformation and energy dissipation time-characteristics and the resulting shape of control force time-characteristics that would be the demanding one for a hypothetical real implementation.     

Optimal control of impulsive switched systems with minimum subsystem durations

This paper presents a new computational approach for solving optimal control problems governed by impulsive switched systems. Such systems consist of multiple subsystems operating in succession, with possible instantaneous state jumps occurring when the system switches from one subsystem to another. The control variables are the subsystem durations and a set of system parameters influencing the state jumps. In contrast with most other papers on the control of impulsive switched systems, we do not require every potential subsystem to be active during the time horizon (it may be optimal to delete certain subsystems, especially when the optimal number of switches is unknown). However, any active subsystem must be active for a minimum non-negligible duration of time. This restriction leads to a disjoint feasible region for the subsystem durations. The problem of choosing the subsystem durations and the system parameters to minimize a given cost function is a non-standard optimal control problem that cannot be solved using conventional techniques. By combining a time-scaling transformation and an exact penalty method, we develop a computational algorithm for solving this problem. We then demonstrate the effectiveness of this algorithm by considering a numerical example on the optimization of shrimp harvesting operations.     

Effect of the number of active sources on ABR buffer size

Rate-based flow control plays an important role for efficient traffic management of Available Bit Rate(ABR) services. We deal with the problem of the buffer dimension for rate-based ABR control. In this paper, we analyze the Allowed Cell Rate(ACR) of a source and the queue size in a steady state. First, we investigate the effect of the number of active sources on the behavior of the ACR and the maximum queue size. Reflecting the effect of this real scenes, we determine the optimal buffer size and buffer threshold. Furthermore, our analytic results are compared with the case when the effect of the number of active sources is disregarded.     

Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints

This paper deals with the optimal control problem of an ordinary differential equation with several pure state constraints, of arbitrary orders, as well as mixed control-state constraints. We assume (i) the control to be continuous and the strengthened Legendre–Clebsch condition to hold, and (ii) a linear independence condition of the active constraints at their respective order to hold. We give a complete analysis of the smoothness and junction conditions of the control and of the constraints multipliers. This allows us to obtain, when there are finitely many nontangential junction points, a theory of no-gap second-order optimality conditions and a characterization of the well-posedness of the shooting algorithm. These results generalize those obtained in the case of a scalar-valued state constraint and a scalar-valued control.     

Modified projective synchronization of chaotic systems with disturbances via active sliding mode control

We apply the active sliding mode control technique to realize the modified projective synchronization of the chaotic systems. The disturbances are considered both in the drive system and the response system. The sufficient conditions for the modified projective synchronization both the non-identical and identical chaotic systems are presented. The corresponding numerical simulations are provided to illuminate the effectiveness of the proposed active sliding mode controllers.     

The Euler approximation in state constrained optimal control

We analyze the Euler approximation to a state constrained control problem. We show that if the active constraints satisfy an independence condition and the Lagrangian satisfies a coercivity condition, then locally there exists a solution to the Euler discretization, and the error is bounded by a constant times the mesh size. The proof couples recent stability results for state constrained control problems with results established here on discrete-time regularity. The analysis utilizes mappings of the discrete variables into continuous spaces where classical finite element estimates can be invoked.

     

A retrospective view of fuzzy control systems

One of the most active areas for the application of fuzzy set theory has been in process control. We trace the development of this research, from the first papers by Zadeh to current efforts in both theory and practice. We review some themes that appear in this corpus and suggest some directions for future work. In particular, we suggest that by adopting some concepts from artificial intelligence, existing approaches to fuzzy control system design could be significantly enhanced.     

Optimization Techniques for Solving Elliptic Control Problems with Control and State Constraints: Part 1. Boundary Control

We study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints. In a first part we consider boundary control problems with either Dirichlet or Neumann conditions. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. It is shown that a recently developed interior point method is able to solve these problems even for high discretizations. Several numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang-bang and singular controls. The necessary conditions of optimality are checked numerically in the presence of active control and state constraints.     

The Pontryagin maximum principle and a unified theory of dynamic optimization

The Pontryagin maximum principle is the central result of optimal control theory. In the half-century since its appearance, the underlying theorem has been generalized, strengthened, extended, proved and reinterpreted in a variety of ways. We review in this article one of the principal approaches to obtaining the maximum principle in a powerful and unified context, focusing upon recent results that represent the culmination of over thirty years of progress using the methodology of nonsmooth analysis. We illustrate the novel features of this theory, as well as its versatility, by introducing a far-reaching new theorem that bears upon the currently active subject of mixed constraints in optimal control.     

On Quantitative Stability in Optimization and Optimal Control

We study two continuity concepts for set-valued maps that play central roles in quantitative stability analysis of optimization problems: Aubin continuity and Lipschitzian localization. We show that various inverse function theorems involving these concepts can be deduced from a single general result on existence of solutions to an inclusion in metric spaces. As applications, we analyze the stability with respect to canonical perturbations of a mathematical program in a Hilbert space and an optimal control problem with inequality control constraints. For stationary points of these problems, Aubin continuity and Lipschitzian localization coincide; moreover, both properties are equivalent to surjectivity of the map of the gradients of the active constraints combined with a strong second-order sufficient optimality condition.     

Comparative study on control concepts of a robot manipulator with multiple-link/joint flexibilities

If one is dealing with active vibration suppression on a highly nonlinear flexible system, various techniques are needed. On the one hand a suitable dynamic model of the system is required. And on the other hand intelligent model based control concepts are necessary for active vibration damping. We deal with a basic model, where the flexibilities are approximated with linear springs and dampers, a so called lumped element model (LEM). For the control design we propose a control structure with two degrees of freedom (2DoF) for solving the tracking problem, based on the LEM. Such an approach allows designing the feedforward part independently of the feedback part. Hereby the feedforward control is based on the flatness approach, while for the feedback control several strategies are studied using acceleration- and gyrosensor-measurements. The contribution is completed with a validation by measurements from a very fast trajectory on an articulated robot with two flexible links and three elastic joints. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal Investment in Heterogeneous Capital and Technology Under Restricted Natural Resource

We consider the optimal control of nonlinear integral equations with endogenous delay and state constraints, which describe a developing economy subjected to resource constraints. The economy invests in new resource-efficient technologies, invests in new capital, and scraps obsolete capital. We derive the optimality condition and determine long-term asymptotically exponential trajectories that optimally combine scrapping the dirtiest capital and developing new clean technologies. Next, we study the short-term dynamics of the model and show that it leads to a sustainable growth with active resource constraint.     

A factorization with update procedures for a KKT matrix arising in direct optimal control

Quadratic programs obtained for optimal control problems of dynamic or discrete-time processes usually involve highly block structured Hessian and constraints matrices, to be exploited by efficient numerical methods. In interior point methods, this is elegantly achieved by the widespread availability of advanced sparse symmetric indefinite factorization codes. For active set methods, however, conventional dense matrix techniques suffer from the need to update base matrices in every active set iteration, thereby loosing the sparsity structure after a few updates. This contribution presents a new factorization of a KKT matrix arising in active set methods for optimal control. It fully respects the block structure without any fill-in. For this factorization, matrix updates are derived for all cases of active set changes. This allows for the design of a highly efficient block structured active set method for optimal control and model predictive control problems with long horizons or many control parameters.     

Active disturbance rejection control based on weighed-moving-average-state-observer

In this work, we are concerned with the boundary stabilization of a one-dimensional anti-stable wave equation corrupted by a boundary disturbance. We firstly propose, by weighed moving average technique, a state observer to make an estimation of the disturbance. Secondly, we design a control by the active disturbance rejection strategy to stabilize the system.