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TECHNICAL BRIEFS

# A Reliable Algorithm to Compute the Spectral Set of a Polytope of Polynomials to Prescribed Accuracy

[+] Author and Article Information
Nataraj S. Paluri

Systems and Control Engineering Group, Room 114, ACRE Building, Indian Institute of Technology, Bombay, India 400076Nataraj@ee.iitb.ac.in

Jayesh J. Barve

Systems and Control Engineering Group, Room 114, ACRE Building, Indian Institute of Technology, Bombay, India 400076

J. Dyn. Sys., Meas., Control 128(2), 400-405 (Mar 28, 2005) (6 pages) doi:10.1115/1.2196416 History: Received April 01, 2003; Accepted March 28, 2005

## Abstract

We propose an algorithm to compute the spectral set of a polytope of polynomials. The proposed algorithm offers several key guarantees that are not available with existing techniques. It guarantees that the generated spectral set: (i) contains all the actual points, (ii) is computed to a prescribed accuracy, (iii) is computed reliably in face of all kinds of computational errors, and (iv) is computed in a finite number of algorithmic iterations. A further merit is that the computational complexity of the proposed algorithm is $O(n)$ in contrast to $O(n2)$ for existing techniques, where $n$ is the degree of the polynomial. The algorithm is demonstrated on a few examples.

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## Figures

Figure 1

Spectral set of Example 5.1 computed using the proposed algorithm. The domain and range accuracy tolerances are εz=[0.1,1deg], εp=0.1. Only the outer boundary boxes of the computed set are shown. For comparison, the spectral set of ∼105 fixed polynomials picked randomly from the polynomial family are computed using roots routine of MATLAB . These are the inner points shown as the lightly shaded area in the plot.

Figure 2

Spectral set of Example 5.2 computed using the proposed algorithm. The domain and range accuracy tolerances are εz=[0.01,1deg], εp=0.1. Only the outer boundary boxes of the computed set are shown. For comparison, the spectral set of ∼105 fixed polynomials picked randomly from the polynomial family are computed using roots routine of the MATLAB . These are the inner points shown as the lightly shaded area in the plot.

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