Consistency tests in guaranteed simulation of nonlinear uncertain systems with application to an activated sludge process

In this paper, interval arithmetic simulation techniques are presented to determine guaranteed enclosures of the state variables of both continuous and discrete-time systems with uncertain but bounded parameters. In nonlinear uncertain systems axis-parallel interval boxes are mapped to complexly shaped regions in the state space that represent sets of possible combinations of state variables. The approximation of each region by a single interval box causes an accumulating overestimation from time-step to time-step, usually called the wrapping effect. The algorithm presented in this paper minimizes the wrapping effect by applying consistency techniques based on interval Newton methods. Subintervals that do not belong to the exact solution at a given time can be eliminated in order to give a tighter but still conservative approximation of the exact solution. Additionally, efficient splitting and merging strategies are employed to limit the number of subintervals. The proposed algorithm is applied to the simulation of an activated sludge process in biological wastewater treatment.     

Robustly complete synthesis of sampled-data control for continuous-time nonlinear systems with reach-and-stay objectives

Formal methods are becoming favorable for control and verification of safety-critical systems because of the rigorous model-based computation. Relying on an over-approximated model of the original system behaviors, formal control synthesis algorithms are not often complete, which means that a controller cannot necessarily be synthesized even if there exists one. The main result of this paper shows that, for continuous-time nonlinear systems, a sample-and-hold control strategy for a reach-and-stay specification can be synthesized whenever such a strategy exists for the same system with its dynamics perturbed by small disturbances. Control synthesis is carried out by a fixed-point algorithm that adaptively partitions the system state space into a finite number of cells. In each iteration, the reachable set from each cell after one sampling time is over-approximated within a precision determined by the bound of the disturbances. To meet such a requirement, we integrate validated high-order Taylor expansion of the system solution over one sampling period into every fixed-point iteration and provide a criterion for choosing the Taylor order and the partition precision. Two nonlinear system examples are given to illustrate the effectiveness of the proposed method.     

Hybrid output regulation for linear impulsive systems with aperiodic jumps: A discrete-time feedback controller design approach

This paper is concerned with the problem of hybrid output regulation for a class of linear impulsive systems with aperiodic jumps. Firstly, by leveraging time-dependent Lyapunov function technique and impulsive control theory, sufficient conditions for achieving output regulation are obtained in state feedback case. Then, the results are extended to error feedback case by constructing an impulsive observer. In this framework, two novel hybrid controllers are designed. Such controllers only need the discrete-time system state or error signal for feedback. The complete procedures for controller designs are also presented. Finally, two illustrative examples, including a numerical example and an LC circuit, are given to show the validity and applicability of the proposed control laws.     

A new approach to the Pontryagin maximum principle for nonlinear fractional optimal control problems

In this paper, we discuss a new general formulation of fractional optimal control problems whose performance index is in the fractional integral form and the dynamics are given by a set of fractional differential equations in the Caputo sense. The approach we use to prove necessary conditions of optimality in the form of Pontryagin maximum principle for fractional nonlinear optimal control problems is new in this context. Moreover, a new method based on a generalization of the Mittag–Leffler function is used to solving this class of fractional optimal control problems. A simple example is provided to illustrate the effectiveness of our main result. Copyright © 2016 John Wiley & Sons, Ltd.     

An optimal control approach to the design of periodic orbits for mechanical systems with impacts

In this paper we study the problem of designing periodic orbits for a special class of hybrid systems, namely mechanical systems with underactuated continuous dynamics and impulse events. We approach the problem by means of optimal control. Specifically, we design an optimal control based strategy that combines trajectory optimization, dynamics embedding, optimal control relaxation and root finding techniques. The proposed strategy allows us to design, in a numerically stable manner, trajectories that optimize a desired cost and satisfy boundary state constraints consistent with a periodic orbit. To show the effectiveness of the proposed strategy, we perform numerical computations on a compass biped model with torso.     

A discrete approach to designing optimal hedging-point control policies for production-inventory systems with general stochastic behavior

The hedging-point policy for a production-inventory system is investigated under the effect of probabilistic machine breakdowns and repairs assuming general discrete distributions for the repair time and the time to failure. Using a methodology whereby inventory levels can assume only discrete values, an optimal safety stock size that minimizes the total expected cost per unit time is determined.     

A cone complementarity linearization approach to robust controller design for continuous-time piecewise linear systems with linear fractional uncertainties

This paper investigates the robust H_∞ control problem for uncertain continuous-time piecewise systems by using the piecewise continuous Lyapunov function. The uncertainties of the systems under consideration are expressed in a linear fractional form. A strict linear matrix inequality approach is developed to obtain stability condition and H_∞ performance. The H_∞ controller design problem is solved by exploiting the cone complementarity linearization (CCL) method, which can be cast into an iterative minimization problem subject to LMI constraints. Finally two examples are given to illustrate the application of the proposed approach.     

A unified approach to the feasible point method type for nonlinear programming with linear constraints under degeneracy and the convergence properties

In this paper, we introduce a general class of feasible point algorithms for solving nonlinear programming problems with linear constraints under degeneracy. Many wellknown algorithms are its special cases. Also, many new algorithms can be derived from this general class. Under rather simple conditions, we prove the global convergence property for this class of algorithms.This research was partially supported by the National Natural Science Foundation of China. The first author's research was partially also supported by the National Science Foundation of the United States of America under Grant DCR-86-96135.