Daniel J. Simon

Biogeography-Based Optimization

Biogeography is the study of the geographical distribution of biological organisms. Mathematical equations that govern the distribution of organisms were first discovered and developed during the 1960s. The mindset of the engineer is that we can learn from nature. This motivates the application of biogeography to optimization problems. Just as the mathematics of biological genetics inspired the development of genetic algorithms (GAs), and the mathematics of biological neurons inspired the development of artificial neural networks, this paper considers the mathematics of biogeography as the basis for the development of a new field: biogeography-based optimization (BBO). We discuss natural biogeography and its mathematics, and then discuss how it can be used to solve optimization problems. We see that BBO has features in common with other biology-based optimization methods, such as GAs and particle swarm optimization (PSO). This makes BBO applicable to many of the same types of problems that GAs and PSO are used for, namely, high-dimension problems with multiple local optima. However, BBO also has some features that are unique among biology-based optimization methods. We demonstrate the performance of BBO on a set of 14 standard benchmarks and compare it with seven other biology-based optimization algorithms. We also demonstrate BBO on a real-world sensor selection problem for aircraft engine health estimation.

Optimal State Estimation

This is a list of errors in the book Optimal State Estimation, John Wiley & Sons, 2006. The main web site for the book is at http://academic.csuohio.edu/simond/estimation. My email address is listed on my home page at http://academic.csuohio.edu/simond. I enthusiastically welcome feedback, comments, suggestions for improvements, and corrections. I also gratefully acknowledge those who have pointed out many of the errata that are documented here: Ali Javadi, Juan Luque, Ibrahim Abdel Hameed, Rick Rarick, Memo Ergezer, David Schwartz, Jeff Gove, Kevin Sharp, Stephan Busch, Yeoh WeeSoon, Michael Haralambous, Ville Kyrki, George Dontas, Felix Monasterio-Huelin, Cagdas Ozgenc, Cheng Zhong, Ning Lei, Roberto Rigamonti, Vincent Sircoulomb, Sami Fadali, Max Medvetsky, Jonathan How, Ismar Masic, Arthur Menikoff, Nedzad Arnautovic, Gabriel Zigelboim, Antje Westenberger, Laurent de Vito, Warner Losh, HunCheol Im, Martin Grossman, Christof Voemel, Bill Jordan, Bruno Stratmann, Philipp Warode, John McFarland, and James Tursa.In 1960, R.E. Kalman published his famous paper describing a recursive solution to the discrete-data linear filtering problem. Since that time, due in large part to advances in digital computing, the Kalman filter has been the subject of extensive research and application, particularly in the area of autonomous or assisted navigation. The Kalman filter is a set of mathematical equations that provides an efficient computational (recursive) means to estimate the state of a process, in a way that minimizes the mean of the squared error. The filter is very powerful in several aspects: it supports estimations of past, present, and even future states, and it can do so even when the precise nature of the modeled system is unknown. The purpose of this paper is to provide a practical introduction to the discrete Kalman filter. This introduction includes a description and some discussion of the basic discrete Kalman filter, a derivation, description and some discussion of the extended Kalman filter, and a relatively simple (tangible) example with real numbers & results.

Kalman filtering with state equality constraints

Kalman filters are commonly used to estimate the states of a dynamic system. However, in the application of Kalman filters there is often known model or signal information that is either ignored or dealt with heuristically. For instance, constraints on state values (which may be based on physical considerations) are often neglected because they do not fit easily into the structure of the Kalman filter. A rigorous analytic method of incorporating state equality constraints in the Kalman filter is developed. The constraints may be time varying. At each time step the unconstrained Kalman filter solution is projected onto the state constraint surface. This significantly improves the prediction accuracy of the filter. The use of this algorithm is demonstrated on a simple nonlinear vehicle tracking problem.

Blended biogeography-based optimization for constrained optimization

Biogeography-based optimization (BBO) is a new evolutionary optimization method that is based on the science of biogeography. We propose two extensions to BBO. First, we propose a blended migration operator. Benchmark results show that blended BBO outperforms standard BBO. Second, we employ blended BBO to solve constrained optimization problems. Constraints are handled by modifying the BBO immigration and emigration procedures. The approach that we use does not require any additional tuning parameters beyond those that are required for unconstrained problems. The constrained blended BBO algorithm is compared with solutions based on a stud genetic algorithm (SGA) and standard particle swarm optimization 2007 (SPSO 07). The numerical results demonstrate that constrained blended BBO outperforms SGA and performs similarly to SPSO 07 for constrained single-objective optimization problems.

Oppositional biogeography-based optimization

We propose a novel variation to biogeography-based optimization (BBO), which is an evolutionary algorithm (EA) developed for global optimization. The new algorithm employs opposition-based learning (OBL) alongside BBOs migration rates to create oppositional BBO (OB O). Additionally, a new opposition method named quasi-reflection is introduced. Quasi-reflection is based on opposite numbers theory and we mathematically prove that it has the highest expected probability of being closer to the problem solution among all OBL methods. The oppositional algorithm is further revised by the addition of dynamic domain scaling and weighted reflection. Simulations have been performed to validate the performance of quasi-opposition as well as a mathematical analysis for a single-dimensional problem. Empirical results demonstrate that with the assistance of quasi-reflection, OB O significantly outperforms BBO in terms of success rate and the number of fitness function evaluations required to find an optimal solution.

Aircraft Turbofan Engine Health Estimation Using Constrained Kalman Filtering

ABSTRACTKalman ¯lters are often used to estimate the state variablesof a dynamic system. However, in the application of Kalman¯lters some known signal information is often either ignored ordealt with heuristically. For instance, state variable constraints(which may be based on physical considerations) are often ne-glected because they do not ¯t easily into the structure of theKalman ¯lter. This paper develops an analytic method of in-corporating state variable inequality constraints in the Kalman¯lter. The resultant ¯lter is a combination of a standard Kalman¯lter and a quadratic programming problem. The incorporationof state variable constraints increases the computational e®ort ofthe ¯lter but signi¯cantly improves its estimation accuracy. Theimprovement is proven theoretically and shown via simulationresults obtained from application to a turbofan engine model.This model contains 16 state variables, 12 measurements, and 8component health parameters. It is shown that the new algo-rithms provide improved performance in this example over un-constrained Kalman ¯ltering.INTRODUCTION

Training Radial Basis Neural Networks with the Extended Kalman Filter

Abstract Radial basis function (RBF) neural networks provide attractive possibilities for solving signal processing and pattern classification problems. Several algorithms have been proposed for choosing the RBF prototypes and training the network. The selection of the RBF prototypes and the network weights can be viewed as a system identification problem. As such, this paper proposes the use of the extended Kalman filter for the learning procedure. After the user chooses how many prototypes to include in the network, the Kalman filter simultaneously solves for the prototype vectors and the weight matrix. A decoupled extended Kalman filter is then proposed in order to decrease the computational effort of the training algorithm. Simulation results are presented on reformulated radial basis neural networks as applied to the Iris classification problem. It is shown that the use of the Kalman filter results in better learning than conventional RBF networks and faster learning than gradient descent.

Training fuzzy systems with the extended Kalman filter

The generation of membership functions for fuzzy systems is a challenging problem. We show that for Mamdani-type fuzzy systems with correlation-product inference, centroid defuzzification, and triangular membership functions, optimizing the membership functions can be viewed as an identification problem for a nonlinear dynamic system. This identification problem can be solved with an extended Kalman filter. We describe the algorithm and compare it with gradient descent and with adaptive neuro-fuzzy inference system (ANFIS) based optimization of fuzzy membership functions. The methods discussed in this paper are illustrated on a fuzzy filter for motor winding current estimation, and are compared with Butterworth filtering. We demonstrate that the Kalman filter can be an effective tool for improving the performance of a fuzzy system.

Biogeography-based optimization combined with evolutionary strategy and immigration refusal

Biogeography-based optimization (BBO) is a recently developed heuristic algorithm which has shown impressive performance on many well known benchmarks. In order to improve BBO, this paper incorporates distinctive features from other successful heuristic algorithms into BBO. In this paper, features from evolutionary strategy (ES) are used for BBO modification. Also, a new immigration refusal approach is added to BBO. After the modification of BBO, F-tests and T-tests are used to demonstrate the differences between different implementations of BBOs.

Markov Models for Biogeography-Based Optimization

Biogeography-based optimization (BBO) is a population-based evolutionary algorithm that is based on the mathematics of biogeography. Biogeography is the science and study of the geographical distribution of biological organisms. In BBO, problem solutions are analogous to islands, and the sharing of features between solutions is analogous to the migration of species. This paper derives Markov models for BBO with selection, migration, and mutation operators. Our models give the theoretically exact limiting probabilities for each possible population distribution for a given problem. We provide simulation results to confirm the Markov models.