Wathanyoo Khaisongkram

On computing the worst-case norm of linear systems subject to inputs with magnitude bound and rate limit

In this paper, we propose a practical and effective approach to compute the worst-case norm of finite-dimensional convolution systems. System inputs are modelled to have bounded magnitude and rate limit. The computation of the worst-case norm is formulated as a fixed-terminal-time optimal control problem. Applying Pontryagins maximum principle with the generalized Karush–Kuhn–Tucker theorem, we obtain necessary conditions which are subsequently exploited to characterize the worst-case input. Furthermore, we develop a novel algorithm called successive pang interval search (SPIS) to construct the worst-case input for general finite-dimensional convolution systems. The algorithm is guaranteed to converge and give an accurate solution within a prescribed error bound. To verify the accuracy of the algorithm, we derive bounds on computational errors including the truncation error and the discretization error. Then, the bounds on the errors yielded by our algorithm are compared with those of a comparative discr...

ON COMPUTING THE WORST-CASE NORM OF CONVOLUTION SYSTEMS: A COMPARISON OF CONTINUOUS-TIME AND DISCRETE-TIME APPROACHES

Abstract This paper compares two approaches to compute the worst-case norm of finite-dimensional convolution systems. All admissible inputs are defined to have bounded magnitude and limited rate of change. Due to physical and mathematical reasons, the inputs are also specified to start from zero. The first approach is based on continuous-time optimal control formulation. Necessary conditions obtained via the Pontryagins maximum principle provide a systematic means to characterize and construct the worst-case input. The second approach is based on discretization of the norm-computation problem which results in a large-scale finite-dimensional linear programming. We also investigate computational errors including truncation errors and discretization errors. Although the second approach seems to be simpler, the first approach is deemed to yield better accuracy.

A Linear Programming Approach for the Worst-Case Norm of Uncertain Linear Systems Subject to Disturbances with Magnitude and Rate Bounds

This paper presents methods to compute the worst-case norm (WCN) of uncertain linear time-invariant systems. The system input is modelled by the magnitude and rate bounds, and the impulse response of uncertain linear systems lies inside response bounds. Since the computation of the exact WCN is an NP-hard problem, we develop two methods, namely, the simplicial method and the tetrahedral method, to compute upper bounds of the WCN. Computing these upper bounds is equivalent to solving linear programming (LP) problems. In the solving process, we take into account of sparsity structures in the LP problems, and apply a primal interior-point method in the software implementation. Numerical examples reveal that the tetrahedral method outperforms the simplicial method. In particular, the tetrahedral method gives a tighter bound of the WCN and uses fewer flops. Hence, the tetrahedral method is applicable and efficient to approximate the WCN

MATLAB based GUIs for linear controller design via convex optimization

Owing to the current evolution of computational tools, a complicated parameter optimization problem could be effectively solved by a computer. In this paper, a CAD tool for multi-objective controller design based on MATLAB program is developed. In addition, simple GUIs which provide a visual approach in specifying the contraints are also constructed using GUIDE tools within MATLAB. The linear controller design problem can be cast as the convex optimization subjected to time domain and frequency domain constraints. This optimization problem is efficiently solved within a finite dimensional subspace by a practical ellipsoid algorithm. In the design process, we include a model reduction of the resulting controller to speed up the computational efficiency. Finally, a numerical example shows the capability of the program to design multi-objective controller for a one-link flexible robot arm.

A Branch-and-Bound Algorithm to Compute the Worst-Case Norm of Uncertain Linear Systems under Inputs with Magnitude and Rate Constraints

This paper extends the worst-case norm (WCN) of linear systems subject to inputs with magnitude and rate bounds to the case of uncertain linear systems. While the WCN for linear systems can be accurately obtained by simply solving a sparse linear programming, the computation of the WCN for uncertain linear systems leads to an NP-hard problem. In this paper, a branch-and-bound algorithm is adopted to calculate the WCN in the presence of uncertainty. Numerical examples demonstrate that computation time of the proposed algorithm is reasonable within certain problem dimensions. An exhaustive search is employed to validate the branch-and-bound algorithm, which later indicates the positive outcome. Finally, we suggest a means to improve the WCN computation for problems with higher dimensions

A combined geometric-volumetric calibration of inclined cylindrical underground storage tanks using the regularized least-squares method

Due to some technical and economical factors, the conventional geometric calibration cannot be completely applied to inclined underground storage tanks (UST), nor can the regular volumetric calibration. We propose a novel calibration method called combined geometric-volumetric calibration. The method is based on the data fitting of a level-volume (LV) characteristic curve where LV data are acquired through partially volumetric calibration. The approach aims to find the inclination of UST in order to generate the LV table. The calibrating problem is formulated as a series of standard regularized least-squares (RLS) problem. A computer program with user-friendly GUIs has also been implemented, on the basis of MATLAB, for the ease of the calibrating operation. The numerical example verifies that the presented method could determine the amount of inclination, and return the LV curve that satisfactorily matched the LV data. The result suggests that the presented method as well as the program can be realized with actual applications in industrial sections.