P. S. V. Nataraj

A novel approach to multiparametric quadratic programming

Multiparametric (mp) programming pre-computes optimal solutions offline which are functions of parameters whose values become apparent online. This makes it particularly well suited for applications that need a rapid solution of online optimization problems. In this work, we propose a novel approach to multiparametric programming problems based on an enumeration of active sets and use it to obtain a parametric solution for a convex quadratic program (QP). To avoid the combinatorial explosion of the enumeration procedure, an active set pruning criterion is presented that makes the enumeration implicit. The method guarantees that all regions of the partition are critical regions without any artificial cuts, and further that no region of the parameter space is left unexplored.

Technical Communique: Template generation for continuous transfer functions using interval analysis

This paper presents a template generation algorithm for transfer functions that are continuous in the uncertain system parameters. The algorithm is developed using interval mathematics. The generated templates are of arbitrary accuracy, safe, and reliable. Further, if the magnitude and phase are continuously differentiable functions of the parameters, then the algorithm can be speeded up considerably with the help of an additional step based on the interval Gauss-Seidel method. The convergence, finite termination, safety, and reliability properties of the algorithm are proven, and an example is provided to demonstrate its versatility.

Global optimization of mixed-integer nonlinear (polynomial) programming problems: the Bernstein polynomial approach

In this paper, we propose an algorithm for constrained global optimization of mixed-integer nonlinear programming (MINLP) problems. The proposed algorithm uses the Bernstein polynomial form in a branch-and-bound framework. Ingredients such as continuous relaxation, branching for integer decision variables, and fathoming for each subproblem in the branch-and-bound tree are used. The performance of the proposed algorithm is tested and compared with several state-of-the-art MINLP solvers, on two sets of test problems. The results of the tests show the superiority of the proposed algorithm over the state-of-the-art solvers in terms of the chosen performance metrics.

Brief Computation of QFT bounds for robust tracking specifications

An algorithm is proposed for generation of QFT controller bounds to achieve robust tracking specifications. The proposed algorithm uses quadratic constraints and interval plant templates to compute the bounds, and presents several improvements over existing QFT tracking bound generation algorithms. The proposed algorithm (1) guarantees robustness against template inaccuracies, (2) guarantees robustness against phase discretization, (3) provides a posteriori error estimates, (4) is computationally efficient, achieving a reduction in flops and execution time, typically by 1-2 orders of magnitude. The algorithm is demonstrated on an aircraft example having five uncertain parameters.

Constrained global optimization of multivariate polynomials using Bernstein branch and prune algorithm

We propose an algorithm for constrained global optimization to tackle non-convex nonlinear multivariate polynomial programming problems. The proposed Bernstein branch and prune algorithm is based on the Bernstein polynomial approach. We introduce several new features in this proposed algorithm to make the algorithm more efficient. We first present the Bernstein box consistency and Bernstein hull consistency algorithms to prune the search regions. We then give Bernstein contraction algorithm to avoid the computation of Bernstein coefficients after the pruning operation. We also include a new Bernstein cut-off test based on the vertex property of the Bernstein coefficients. The performance of the proposed algorithm is numerically tested on 13 benchmark problems. The results of the tests show the proposed algorithm to be overall considerably superior to existing method in terms of the chosen performance metrics.

Computation of QFT bounds for robust sensitivity and gain-phase margin specifications

Algorithms are presented for generation of QFT controller bounds to achieve robust sensitivity reduction and gain-phase margin specifications. The proposed algorithms use quadratic constraints and interval plant templates to derive the bounds, and present several improvements over existing QFT bound generation algorithms: (1) The bounds can be generated over interval controller phases, as opposed to discrete controller phases in existing QFT algorithms. This feature essentially solves the safety problems associated with phase discretization process in QFT bound generation; (2) The generated bounds are guaranteed to be safe and reliable, even for very coarse interval templates (of poor accuracy), and despite all kinds of computational errors; (3) Inner as well as outer enclosures of the exact bound values can be obtained using the proposed algorithms. Such enclosures directly provide upper bounds on the error in the generated results for any given interval template; (4) A very significant reduction in computational effort-typically, reduction in flops by 2-3 orders of magnitude is achieved; (5) The vertical line (or rectangle) nature of plant templates exhibited in the low and high frequency ranges can be readily exploited to obtain bounds with very little effort; (6) The number of flops required to generate the bounds for any given template can be estimated closely a priori; (7) The (entire) algorithms can be programmed using vectorized operations, resulting in small execution times.

An efficient algorithm for range computation of polynomials using the Bernstein form

We present a novel optimization algorithm for computing the ranges of multivariate polynomials using the Bernstein polynomial approach. The proposed algorithm incorporates four accelerating devices, namely the cut-off test, the simplified vertex test, the monotonicity test, and the concavity test, and also possess many new features, such as, the generalized matrix method for Bernstein coefficient computation, a new subdivision direction selection rule and a new subdivision point selection rule. The features and capabilities of the proposed algorithm are compared with those of other optimization techniques: interval global optimization, the filled function method, a global optimization method for imprecise problems, and a hybrid approach combining simulated annealing, tabu search and a descent method. The superiority of the proposed method over the latter methods is illustrated by numerical experiments and qualitative comparisons.

Global Optimization with Higher Order Inclusion Function Forms Part 1: A Combined Taylor-Bernstein Form

Recently, two versions of the so-called Taylor-Bernstein (TB) form having the property of higher order convergence were proposed in [5] and [10]. However, in many application problems, both these TB forms encounter difficulties in computing the range enclosures for some domain widths, due to excessive memory and/or time requirements. In this paper, we present a combined TB form that is more successful in computing the range enclosures as the domain shrinks from large to small widths. We test and compare the performance of the proposed form with those of the existing TB forms, the Taylor model of Berz et al. [1], and the simple natural inclusion function form. For the testing, we consider six benchmark examples with dimensions varying from 1 to 6. The results of the tests show that the proposed combined TB form is indeed more effective than the existing TB forms and the Taylor model, over the entire range of domain widths considered.

Automated Synthesis of Fixed Structure QFT Controller using Interval Constraint Satisfaction Techniques

Abstract Robust controller synthesis is of great practical interest and its automation is a key concern in control system design. Automatic controller synthesis is still a open problem. In this paper a new, efficient method has been proposed for automated synthesis of a fixed structure quantitative feedback theory (QFT) controller by solving QFT quadratic inequalities of robust stability and performance specifications. The controller synthesis problem is posed as interval constraint satisfying problem (ICSP) and solved with interval constraint solver. The method is guaranteed to find all feasible controllers of given structure in the search domain. The proposed method is tested on two benchmark problems, and simple, low order controllers are successfully obtained in quick time.

A New Subdivision Strategy for Range Computations

We present a new subdivision strategy in interval analysis for computing the ranges of functions. We show through several real-world examples that the proposed subdivision strategy is more efficient than the widely used uniform and adaptive subdivision strategies of Moore (Methods and Applications of Interval Analysis, SIAM, Philadelphia, 1979).