Globally convergent algorithms for DC operating point analysis of nonlinear circuits

An important objective in the analysis of an electronic circuit is to find its quiescent or dc operating point. This is the starting point for performing other types of circuit analysis. The most common method for finding the dc operating point of a nonlinear electronic circuit is the Newton-Raphson method (NR), a gradient search technique. There are known convergence issues with this method. NR is sensitive to starting conditions. Hence, it is not globally convergent and can diverge or oscillate between solutions. Furthermore, NR can only find one solution of a set of equations at a time. This paper discusses and evaluates a new approach to dc operating-point analysis based on evolutionary computing. Evolutionary algorithms (EAs) are globally convergent and can find multiple solutions to a problem by using a parallel search. At the operating point(s) of a circuit, the equations describing the current at each node are consistent and the overall error has a minimum value. Therefore, we can use an EA to search the solution space to find these minima. We discuss the development of an analysis tool based on this approach. The principles of computer-aided circuit analysis are briefly discussed, together with the NR method and some of its variants. Various EAs are described. Several such algorithms have been implemented in a full circuit-analysis tool. The performance and accuracy of the EAs are compared with each other and with NR. EAs are shown to be robust and to have an accuracy comparable to that of NR. The performance is, at best, two orders of magnitude worse than NR, although it should be noted that time-consuming setting of initial conditions is avoided.     

Globally stable and exponentially convergent adaptive control

In this paper three new adaptive control strategies for linear discrete-time single-input/single-output plants are proposed. They enable globally stable and exponentially convergent adaptive controllers to be designed based on various approaches, such as pole placement and optimal LQ. The principle idea of the strategies proposed is to apply a fixed-parameter control law while the estimates of the plant parameters leave some appropriately chosen region. The persistency of excitation of some external signals is crucial for the stability analysis of an overall adaptive system.     

Globally convergent algorithms with local learning rates

A novel generalized theoretical result is presented that underpins the development of globally convergent first-order batch training algorithms which employ local learning rates. This result allows us to equip algorithms of this class with a strategy for adapting the overall direction of search to a descent one. In this way, a decrease of the batch-error measure at each training iteration is ensured, and convergence of the sequence of weight iterates to a local minimizer of the batch error function is obtained from remote initial weights. The effectiveness of the theoretical result is illustrated in three application examples by comparing two well-known training algorithms with local learning rates to their globally convergent modifications.     

Globally convergent decomposition methods for nonconvex optimization problems

In this paper we consider nonlinear optimization problems of a separable form with nonconvex objective and convex constraints. A convexification procedure preserving separability is given in order that primal-dual methods are applicable. A globally convergent algorithm observing computational aspects is given. This algorithm was applied to a real world problem with 1007 variables and 4030 constraints for controlling the heads of a hydroenergy power station.     

Globally convergent algorithms for robust pole assignment by statefeedback

It is observed that an algorithm proposed in 1985 by Kautsky, Nichols, and Van Dooren (KNV) amounts to maximizing, at each iteration, the determinant of the candidate closed-loop eigenvector matrix X with respect to one of its columns (with unit-length constraint), subject to the constraint that it remains an achievable closed-loop eigenvector matrix. This interpretation is used to prove convergence of the KNV algorithm. It is then shown that a more efficient algorithm is obtained if det (X) is concurrently maximized with respect to two columns of X, and such a scheme is easily extended to the case where the eigenvalues to be assigned include complex conjugate pairs. Variations exploiting the availability of multiple processors are suggested. Convergence properties of the proposed algorithms are established. Their superiority is demonstrated numerically     

Multicriterion optimization using interval analysis

Interval Analysis methods have been applied for obtaining the global optimum of the multimodal multivariable functions. We discuss here the multicriterion optimization problem, where several objective functions must be optimized in conflicting situations.     

Constrained optimization using interval analysis

An interval analysis method is described for finding the global maximum of a multimodal multivariable function subject to equality and/or inequality constraints. By discarding subregions where the global solution can not exist and applying the interval Newton method to solve the Lagrange equation, one can always find the solution with the rigorous error bound. Some numerical examples are given.     

Maximum likelihood autocalibration

This paper addresses the problem of autocalibration, which is a critical step in existing uncalibrated structure from motion algorithms that utilize an initialization to avoid the local minima in metric bundle adjustment. Currently, all known direct (not non-linear) solutions to the uncalibrated structure from motion problem solve for a projective reconstruction that is related to metric by some unknown homography, and hence a necessary step in obtaining a metric reconstruction is the subsequent estimation of the rectifying homography, known as autocalibration. Although autocalibration is a well-studied problem, previous approaches have relied upon heuristic objective functions, and have a reputation for instability. We propose a maximum likelihood objective and show that it can be implemented robustly and efficiently and often provides substantially greater accuracy, especially when there are fewer views or greater noise.     

Globally and superlinearly convergent algorithms for the solution of box-constrained optimization

Based on the identification technique of active constraints, we propose a Newton-like algorithm and a quasi-Newton algorithm for solving the box-constrained optimization problem. The two algorithms require only the solution of a lower-dimensional system of linear equations at each iteration. In the proposed quasi-Newton algorithm, we make use of an approximate direction derivative of the multiplier functions so that only first-order derivatives of the objective function are needed to evaluate. Under mild assumptions, global convergence of the two algorithms is established. In particular, locally quadratic convergence for the Newton-like algorithm and locally superlinear convergence for the quasi-Newton algorithm are obtained without assuming that the strict complementarity condition holds at the solution.     

Globally convergent adaptive pole assignment for a class of non-minimum-phase stochastic systems

A key difficulty of the explicit approach to self-tuning control—both theoretically and computationally—is the need to solve a polynomial identity to generate the required controller coefficients. For systems with uncorrelated output noise, however, the identity has a simple solution, and in this paper the implications of this phenomenon are discussed in relation to self-tuning regulation. A suitable explicit algorithm is introduced, and it is shown that, under certain conditions, global stability and system identifiability can be established without recourse to sophisticated estimator management techniques.     

Injectivity analysis using interval analysis: Application to structural identifiability

This paper presents a new numerical algorithm based on interval analysis able to prove that a differentiable function is injective. This algorithm also performs a partition of the domain A in subsets A_i where, for all , the cardinality of is constant. In the context of parameter estimation, we show how this algorithm provides an efficient and numerical method to study the structural identifiability of parametric models.     

Globally convergent multivariable adaptive decoupling controller and its application to a binary distillation column

This paper describes a new multivariable adaptive decoupling controller combining a decoupling compensator with a generalized minimum variance technique. The controller can completely decouple closed-loop systems both dynamically and in the steady state. It can control an unstable and/or non-minimum phase system and it can control the process with an arbitrary time delay structure. The proof of global convergence for the algorithm is also given. Successful experimental results of the adaptive decoupling control for a pilot-scale binary distillation column demonstrate the effectiveness of the proposed adaptive decoupling algorithm.     

Box particle filtering for nonlinear state estimation using interval analysis

In recent years particle filters have been applied to a variety of state estimation problems. A particle filter is a sequential Monte Carlo Bayesian estimator of the posterior density of the state using weighted particles. The efficiency and accuracy of the filter depend mostly on the number of particles used in the estimation and on the propagation function used to re-allocate weights to these particles at each iteration. If the imprecision, i.e. bias and noise, in the available information is high, the number of particles needs to be very large in order to obtain good performances. This may give rise to complexity problems for a real-time implementation. This kind of imprecision can easily be represented by interval data if the maximum error is known. Handling interval data is a new approach successfully applied to different real applications. In this paper, we propose an extension of the particle filter algorithm able to handle interval data and using interval analysis and constraint satisfaction techniques. In standard particle filtering, particles are punctual states associated with weights whose likelihoods are defined by a statistical model of the observation error. In the box particle filter, particles are boxes associated with weights whose likelihood is defined by a bounded model of the observation error. Experiments using actual data for global localization of a vehicle show the usefulness and the efficiency of the proposed approach.     

Finite time convergent control using terminal sliding mode

A method for terminal sliding mode control design is discussed. As we know, one of the strong points of terminal sliding mode control is its finite-time convergence to a given equilibrium of the system under consideration, which may be useful in specific applications. The proposed method, different from many existing terminal sliding model control design methods, is studied, and then feedback laws are designed for a class of nonlinear systems, along with illustrative examples.     

Global optimization using interval arithmetic

Interval arithmetic provides an efficient method of global optimization. With less efficiency all stationary points of a function can be found. A minimization method is described and applied to an econometric function. The results are compared with the method of simulated annealing on the same function.     

On global optimization using interval arithmetic

A method for finding all global minimizers of a real-valued objective function of several variables is presented. For this purpose a problem-oriented type of number is used: the set of real compact intervals. The range of the objective function over a rectangular set is estimated by natural interval extension of a suitable modelling function. An algorithm for interpolation and approximation in multidimensional spaces is developed. This optimization method can be applied successfully to conventionally, e.g. with real arithmetic, programmed functions.     

Volumetric data exploration using interval volume

A new type of geometric model called Interval volume for volumetric data exploration is presented. An interval volume represents a three dimensional subvolume for which the associate scalar values lie within a user specified interval, and provides one of the promising approaches to solid fitting, which is an extended concept of traditional surface fitting. A well known isosurfacing algorithm called Marching Cubes is extended to obtain a solid fitting algorithm, which extracts from a given volumetric data set a high resolution, polyhedral solid data structure of an interval volume. Branch-on-Need Octree is used as an auxiliary data structure to accelerate the extraction process. A variety of interval volume rendering methods and principal related operations, including measurements and focusing, are also presented. The effectiveness of measurement coupled visualization capabilities of the presented approach is demonstrated by application to visualizing a four dimensional simulated data from atomic collision research     

Stability analysis of dynamic interval systems

The stability analysis of interval matrices with certain stability margins is considered. By means of similarity transformation and Gershgorin's theorem, new criteria are developed and illustrated by examples. We also demonstrate a counter-example to a recent result by Argoun (1986).