Auxiliary Optimization Research Articles

Dynamic Output Feedback Robust MPC Using General Polyhedral State Bounds for the Polytopic Uncertain System With Bounded Disturbance

AbstractThis paper considers the dynamic output feedback robust model predictive control (MPC) for a system with both polytopic model parametric uncertainty and bounded disturbance. For this topic, the techniques for handling the unknown true state are crucial, and the strict guarantee of the input/output/state constraints favors replacing the true state by its bound in the optimization problems. The previous utilized polyhedral bounds, constructed by virtue of the error signals which are some linear combinations of the true state, the estimated state and the output, are generalized, where a bias item is utilized. Based on this unified bounding approach, new techniques for handling the unknown true state are given for both the main and the auxiliary optimization problems. As before, the main optimization problem calculates the control law parameters conditionally, and the auxiliary optimization problem determines the time to refresh these parameters. By applying the proposed method, the augmented state of the closed‐loop system is guaranteed to converge to the neighborhood of the equilibrium point. A numerical example is given to illustrate the effectiveness of the new method.

Research and Application of Auxiliary Optimization Technology of Power Grid Accident Processing Based on the Mode of Regulation and Control Integration

Accident processing is the most important link of the scheduling of daily monitoring. The improvement of intelligent level is of great significance for improving the efficiency of accident processing scheduling, shortening the time of accident processing and preventing further deterioration of accidents. According to features of accident processing scheduling, this paper puts forward an integrated framework of aid decision-making of online accident processing based on large power grid, and carries out a study from five aspects, namely integrated information support platform, risk perception in advance, online fault diagnosis, aid decision-making afterwards and visual display, so as to conduct real-time tracking on operating state of power grid, eliminate potential safety hazards of power grid and upgrade power grid from “manual analysis” scheduling to “intelligent analysis” scheduling.

An Iterative Minimization Formulation for Saddle Point Search

This paper proposes and analyzes an iterative minimization formulation for searching index-1 saddle points of an energy function. We give a general and rigorous description of eigenvector-following methodology in this iterative scheme by considering an auxiliary optimization problem at each iteration in which the new objective function is locally defined near the current guess. We prove that this scheme has a quadratic local convergence rate in terms of number of iterations, in comparison to the linear rate of the gentlest ascent dynamics [W. E and X. Zhou, Nonlinearity, 24 (2011), pp. 1831--1842] and many other existing methods. We also propose the generalization of the new methodology for saddle points of higher index and for constrained energy functions on the manifold. Preliminary numerical results on the nature of this iterative minimization formulation are presented.

Optimization Design on Triangle Plate of Auxiliary Frame of Mixer Truck

In order to solve the question of crack of the auxiliary frame of mixer truck, especially with the life of the triangle welding position is low, the triangle support of auxiliary frame optimization design is put forward Firstly, three dimensional structure model of auxiliary frame is established by Pro/E and the model is imported into HyperMesh software for meshing. The solver RADIOSS is used to calculate the maximum stress and the positions at the different full load conditions. Including the static, left front wheel lifted up by 210 mm, the right rear wheels (double) lifted up by 210 mm and the left front and right rear wheels lifted simultaneously up by 210 mm. Then the experimental vehicle test is performed. The test results and simulation results are compared and show a good consistency and the accuracy of the finite element model is validated. Then the three optimization design schemes of the triangle are put forward and the maximum stress of the optimal auxiliary frame are calculated in contrast with the original stress. The scheme 2 is found to be the most optimal, which provides an effective method for solving the fatigue of the auxiliary frame.

Indirect Optimization of Three-Dimensional Multiple-Impulse Moon-to-Earth Transfers

This paper illustrates an indirect method to optimize multiple-impulse trajectories from circular lunar orbit to Earth. Optimization is performed in the circular restricted three-body problem, and the necessary optimality conditions are found through optimal control theory. In order to overcome the difficulty of initial adjoints estimation, a homotopic approach, which is based on an auxiliary optimization problem with known solution, is developed; this approach proves to be robust and efficient. Examples are presented for a range of lunar orbit orientations to assess the impact on velocity impulse requirements. Optimization results for trajectories with different number of impulses are also compared. The developed procedure can support fast and accurate evaluation of the transfer costs for Moon-to-Earth trajectories both in nominal conditions and for contingency plans.

Comparative study of RPSALG algorithm for convex semi-infinite programming

The Remez penalty and smoothing algorithm (RPSALG) is a unified framework for penalty and smoothing methods for solving min-max convex semi-infinite programing problems, whose convergence was analyzed in a previous paper of three of the authors. In this paper we consider a partial implementation of RPSALG for solving ordinary convex semi-infinite programming problems. Each iteration of RPSALG involves two types of auxiliary optimization problems: the first one consists of obtaining an approximate solution of some discretized convex problem, while the second one requires to solve a non-convex optimization problem involving the parametric constraints as objective function with the parameter as variable. In this paper we tackle the latter problem with a variant of the cutting angle method called ECAM, a global optimization procedure for solving Lipschitz programming problems. We implement different variants of RPSALG which are compared with the unique publicly available SIP solver, NSIPS, on a battery of test problems.

A new warmstarting strategy for the primal-dual column generation method

This paper presents a new warmstarting technique in the context of a primal-dual column generation method applied to solve a particular class of combinatorial optimization problems. The technique relies on calculating an initial point and on solving auxiliary linear optimization problems to determine the step direction needed to fully restore primal and dual feasibilities after new columns arrive. Conditions on the maximum size of the cuts and on a suitable initial point are discussed. Additionally, the strategy ensures that the duality gap of the warmstart is bounded by the old duality gap multiplied with a (small) constant, which depends on the relation between the old and modified problems. Computational experiments demonstrate the gains achieved when compared to a coldstart approach.

Time-response-based evolutionary optimization

Solutions to engineering problems are often evaluated by considering their time responses; thus, each solution is associated with a function. To avoid optimizing the functions, such optimization is usually carried out by setting auxiliary objectives (e.g. minimal overshoot). Therefore, in order to find different optimal solutions, alternative auxiliary optimization objectives may have to be defined prior to optimization. In the current study, a new approach is suggested that avoids the need to define auxiliary objectives. An algorithm is suggested that enables the optimization of solutions according to their transient behaviours. For this optimization, the functions are sampled and the problem is posed as a multi-objective problem. The recently introduced algorithm NSGA-II-PSA is adopted and tailored to solve it. Mathematical as well as engineering problems are utilized to explain and demonstrate the approach and its applicability to real life problems. The results highlight the advantages of avoiding the definition of artificial objectives.

Distributed filtering with randomly occurring uncertainties over sensor networks: the channel fading case

This paper is concerned with the distributed filtering problem for a class of uncertain stochastic systems with fading channels over sensor networks. The norm-bounded uncertainty enters into the system in a random way, and the Bernoulli distributed white sequence is introduced to govern the random occurrences of such uncertainties. The lossy sensor network suffers from the phenomenon of fading measurements that are described by a modified łth-order Rice fading model in which the channel coefficients can have any probability density function on the interval [0, 1]. Through available output measurements from not only the individual sensor but also its neighboring sensors, a sufficient condition is established for the desired distributed filter to ensure that the filtering dynamics is exponentially mean-square stable and the prescribed performance constraint is satisfied. The distributed filter gains are characterized by solving an auxiliary convex optimization problem. Finally, a simulation example is provided to show the effectiveness of the proposed filtering scheme.

Markov Games with Frequent Actions and Incomplete Information

We study a two-player, zero-sum, stochastic game with incomplete information on one side in which the players are allowed to play more and more frequently. The informed player observes the realization of a Markov chain on which the payoffs depend, while the non-informed player only observes his opponent's actions. We show the existence of a limit value as the time span between two consecutive stages vanishes; this value is characterized through an auxiliary optimization problem and as the solution of an Hamilton-Jacobi equation.

Portfolio optimization in a regime-switching market with derivatives

We consider the optimal asset allocation problem in a continuous-time regime-switching market. The problem is to maximize the expected utility of the terminal wealth of a portfolio that contains an option, an underlying stock and a risk-free bond. The difficulty that arises in our setting is finding a way to represent the return of the option by the returns of the stock and the risk-free bond in an incomplete regime-switching market. To overcome this difficulty, we introduce a functional operator to generate a sequence of value functions, and then show that the optimal value function is the limit of this sequence. The explicit form of each function in the sequence can be obtained by solving an auxiliary portfolio optimization problem in a single-regime market. And then the original optimal value function can be approximated by taking the limit. Additionally, we can also show that the optimal value function is a solution to a dynamic programming equation, which leads to the explicit forms for the optimal value function and the optimal portfolio process. Furthermore, we demonstrate that, as long as the current state of the Markov chain is given, it is still optimal for an investor in a multiple-regime market to simply allocate his/her wealth in the same way as in a single-regime market.

Wiretap Channels: Implications of the More Capable Condition and Cyclic Shift Symmetry

Characterization of the rate-equivocation region of a general wiretap channel involves two auxiliary random variables: <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">U</i> , for rate splitting and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V</i> , for channel prefixing. In this paper, we explore specific classes of wiretap channels for which the evaluation of the rate-equivocation region is simpler. We show that if the wiretap channel is more capable, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V</i> = <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</i> is optimal and the boundary of the rate-equivocation region is achieved by varying rate splitting <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">U</i> alone. Conversely, we show under a mild condition that if the wiretap channel is not more capable, then <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V</i> = <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</i> is strictly suboptimal. Next, we focus on the class of cyclic shift symmetric wiretap channels. We show that optimal rate splitting <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">U</i> that achieves the boundary of the rate-equivocation region is uniform with cardinality | <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</i> | and the prefix channel between optimal <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">U</i> and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V</i> is expressed as cyclic shifts of the solution of an auxiliary optimization problem over a single variable. We provide a special class of cyclic shift symmetric wiretap channels for which <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">U</i> =φ is optimal. We apply our results to the binary-input cyclic shift symmetric wiretap channels and thoroughly characterize the rate-equivocation regions of the BSC-BEC and BEC-BSC wiretap channels.

Bethe-logarithm calculation using theB-spline method

A method based on the B-spline basis set is developed that significantly simplifies the Bethe-logarithm calculations for the atomic hydrogen. Without any auxiliary optimization and extrapolation, this method not only can calculate the Bethe logarithms of low-lying states to high precision using relatively small basis sets, but can also calculate high-lying Rydberg states efficiently. For the ground state, the Bethe logarithm is calculated to 24 significant figures using 450 B-spline functions. For high-lying Rydberg states, we can reproduce all the results of Jentschura and Mohr [Phys. Rev. A 72, 012110 (2005)]. DOI: 10.1103/PhysRevA.87.022510

Distributed filtering in sensor networks with randomly occurring saturations and successive packet dropouts

SUMMARY This paper is concerned with the distributed filtering problem for a class of nonlinear systems with randomly occurring sensor saturations (ROSS) and successive packet dropouts in sensor networks. The issue of ROSS is brought up to account for the random nature of sensor saturations in a networked environment of sensors, and accordingly, a novel sensor model is proposed to describe both the ROSS and successive packet dropouts within a unified framework. Two sets of Bernoulli distributed white sequences are introduced to govern the random occurrences of the sensor saturations and successive packet dropouts. Through available output measurements from not only the individual sensor but also its neighboring sensors, a sufficient condition is established for the desired distributed filter to ensure that the filtering dynamics is exponentially mean-square stable and the prescribed performance constraint is satisfied. The solution of the distributed filter gains is characterized by solving an auxiliary convex optimization problem. Finally, a simulation example is provided to show the effectiveness of the proposed filtering scheme. Copyright © 2013 John Wiley & Sons, Ltd.

A coordinate descent MM algorithm for fast computation of sparse logistic PCA

Sparse logistic principal component analysis was proposed in Lee et al. (2010) for exploratory analysis of binary data. Relying on the joint estimation of multiple principal components, the algorithm therein is computationally too demanding to be useful when the data dimension is high. We develop a computationally fast algorithm using a combination of coordinate descent and majorization–minimization (MM) auxiliary optimization. Our new algorithm decouples the joint estimation of multiple components into separate estimations and consists of closed-form elementwise updating formulas for each sparse principal component. The performance of the proposed algorithm is tested using simulation and high-dimensional real-world datasets.

Norm Optimal Iterative Learning Control with Auxiliary Optimization: A Switching Approach

The paper describes a substantial extension of Norm Optimal Iterative Learning Control (NOILC) that permits tracking of a class of finite dimensional reference signals whilst simultaneously minimizing a quadratic cost function. Motivated by practical problems in automation and control, this enables point-to-point motion tasks to be performed whilst reducing effects such as payload spillage, vibration tendencies and actuator wear. Solutions combine feedforward and feedback actions, and are experimentally tested using a robotic arm.

On the robust control design for a class of nonlinearly affine control systems: The attractive ellipsoid approach

This paper is devoted to a problem of robust control design for a class of continuous-time dynamic systems with bounded uncertainties. We study a family of nonlinearly affine control systems and develop a computational extension of the conventional invariant ellipsoid techniques. The obtained method can be considered as a powerful numerical approach that makes it possible to design a concrete stabilizing control strategies for the resulting closed-loop systems. The design procedure for this feedback-type control is based on the classic Lyapunov-type stability analysis of invariant sets for the given dynamic system. We study the necessary theoretic basis and propose a computational algorithm that guarantee some minimality properties of the stability/attractivity regions for dynamic systems under consideration. The complete solution procedure contains an auxiliary LMI-constrained optimization problem. The effectiveness of the proposed robust control design is illustrated by a numerical example.

Dynamic Output Feedback Robust Model Predictive Control

This paper addresses the synthesis approach of dynamic output feedback robust model predictive control for systems with both polytopic uncertainty and bounded disturbance.The system is quasi-linear parameter varing,i.e.,the time-varing parameters of the system are exactly known at the current time but unknown in the future.The notion of quadratic boundedness is utilized to characterize the stability properties of the augmented closed-loop system.The primary contribution of this paper is to propose a method for the auxiliary optimization to provide reliable bounds of true state for the main optimization problem at the next sampling time.A continuous stirred tank reactor control system is given to illustrate the e?ectiveness of the approach.

Games with Incomplete Information in Continuous Time and for Continuous Types

We consider a two-player zero-sum game with integral payoff and with incomplete information on one side, where the payoff is chosen among a continuous set of possible payoffs. We prove that the value function of this game is solution of an auxiliary optimization problem over a set of measure-valued processes. Then we use this equivalent formulation to characterize the value function as the viscosity solution of a special type of a Hamilton–Jacobi equation. This paper generalizes the results of a previous work of the authors (Math. Oper. Res. 34(4), 769–794, 2009), where only a finite number of possible payoffs is considered.

New Formulation of Dynamic Output Feedback Robust Model Predictive Control with Guaranteed Quadratic Boundedness

AbstractThe dynamic output feedback robust model predictive controller for a system with both polytopic uncertainty and bounded disturbance is addressed in this paper. This controller utilizes a main optimization problem to find the control law and a simple auxiliary optimization problem to refresh the bounds on the true state. The main optimization problem, which is not necessarily solved at each sampling instant, achieves the near‐optimal solution. The auxiliary optimization, which is solved at each sampling instant, is followed with a simple criterion which determines whether or not to solve the main optimization problem at the next sampling time. By applying the proposed method, the augmented state of the closed‐loop system is guaranteed to converge to the neighborhood of the equilibrium point.