Assessing 3-D Uncertain System Stability by Using MATLAB Convex Hull Functions

This paper is dealing with the robust stability of an uncertain three dimensional (3-D) system using existence MATLAB convex hull functions. Hence, the uncertain model of plant will be simulated by INTLAB Toolbox; furthermore, the root loci of the characteristic polynomials of the convex hull are obtained to judge whether the uncertain system is stable or not. A design third order example for uncertain parameters is given to validate the proposed approach.


INTRODUCTION
Dealing with higher order system can be considered a challenge and a difficult problem, therefore the contribution of this paper is to the utilization of existence built-in MATLAB Convex Hull algorithm and functions to handle such control problems with less time consuming as will be illustrated throughout this research paper.

A. Motivation and objectives
This paper is dealing with the robust stability of an interval or uncertain system.Developing an algorithm that checks robust stability of third order uncertain system such systems will be an efficient and helpful tool for control systems engineers.

B. Literature Review
The problem of an interval matrices was first presented in 1966 by Ramon E. Moore, who defined an interval number to be an ordered pair of real numbers [a,b], with a ≤ b [1][2].This research is an extension and contribution of previous publications and ongoing research of the author [3]- [9].

C. Paper approach
Three dimension (3-D) convex hull approaches is utilized within MATLAB novel codes that is developed to assess 3-D uncertain system stability, and the algorithm associated is discussed and presented in this paper.

II. UNCERTAIN SYSTEMS AND ROBUST STABILITY
Due to the changes in system parameters due to many reasons, such as aging of main components and environmental changes, this present an uncertain threat to the system, therefore such a system need special type of control system called Robust to grantee the stability to the perturbed parameters.For instances , in recent research robust stability and stabilization of linear switched systems with dwell time [10], as well stability of unfalsified adaptive switching control in noisy environments [11] were discussed.

A. Robust D-stability
Letting D(p,q) denote the uncertain denominator polynomial, then the roots of D(p,q) lie in a region D as shown in Fig. 1 , then we can say that the system has a certain robust D-stability properly.A family of polynomials P={p(.,q):qQ} is said to be robustly D-stable, if all qQ, p(.,q) is D-stable, i.e. all roots of p(.,q)lie in D region.For special case when D is the open unit disc, P is said to be robustly schur stable.

B. Edge Theorem
A polytope of a polynomial with invariant degree p(s, q) is robustly D-stable if and only if all the polynomials lying along the edges of the polytopic type are D-stable, the edge theorem gives an elegant solution to the problem of determining the root space of polytopic system [12], [13].
It establishes the fundamental property that the root space boundary of a polytypic family polynomial is contained in the root locus evaluated along the exposed edges, so after we www.ijacsa.thesai.orggenerate the set of all segments of polynomials we obtain the root locus for all the segments as a direct location for the edge theorem.

C. Uncertain 3x3 systems
Third order uncertain systems can take the following general form:  9) elements which mean 2 9 possible combination of matrix family if all elements were uncertain.Generally, we have 2 n possible combinations of an uncertain system where n is number of uncertain elements in the system.Characteristic equations for a general 3x3 matrix can be calculated as shown below in equation ( 1 The aim of this paper is to calculate the family of all possible combinations for a 3x3 uncertain matrix and so family of possible characteristic equations can be calculated.Then, using convex hull algorithm we will find exposed edges of calculated polynomials and so the roots of exposed edges to determine region of eigenvalues space for studied system.

D. Computing the Convex Hull of the Vertices
The convex-hull problem is one of the most important problems from computational geometry.For a set S of points in space the task is to find the smallest convex polygon containing all points [14].Definition 1: A set S is convex if whenever two points P and Q are inside S, then the whole line segment PQ is also in S. Definition 2: A set S is convex if it is exactly equal to the intersection of all the half planes containing it.Definition 3: The convex hull of a finite point set S = {P} is the smallest 2D polygon that contains S.

III. METHODOLOGY AND ALGORITHM
The main goal of this research is to provide a simple and efficient algorithm to determine the bounds of an interval matrix that represent three dimensional problems, hence assess the stability of such an uncertain system by generating a MATLAB Algorithm for three by three interval matrix.Therefore the methodology and algorithm associated with will be discussed and presented in the following sections.

A. Input data and program call
The developed program takes the nine elements of 3x3 uncertain matrix in a vector form and is called in MATLAB command, and these elements can be either real number for specific elements or interval for uncertain matrix entries.

B. Calculating family of possible matrices
To obtain the family of all possible matrices then the following steps are performed within the function <afamilynew4.m>:  Check for size of each input element to find position of uncertain elements. Declare input vector of 18   Check for repeated matrices and delete it and make sure that remaining matrices are the 2 ss unique matrices, AA.
 Calculate 2 ss possible characteristic polynomials, polypointsn, according to equation 2.1.Note that polypointsn is a matrix and is expected to have the size of (2 ss ,3.

C. Find the 3-D convex hull of polynomials
For this purpose, the existence QuickerHull algorithm for convex hulls is utilized and incorporated in the main MATLAB Program [15], [6].
This algorithm has the advantage of being quicker than convhulln, the built-in code in MATLAB, as illustrated in Fig. 2.

Figure 2. Comparison of processing time in 3D between normal and quicker hull
Then, 3D convex hull of system under study is plotted according to the MATLAB QuickerHull code shown below.

C. Program output
Our proposed program is supposed to show all of family of possible matrices in addition to roots values of characteristic equations.
Four figures are generated while processing our algorithm, Fig. 4 shows 3D convex hull of polynomials.Fig. 5 shows roots locations on s-plane and encircles them by a convex hull.While Fig. 6 shows encircling only imaginary parts of roots, i.e. an identical polygons are generated on and below real axis and other roots are shown also.Fig. 7 focuses on identical polygons encircle imaginary parts of roots.In this paper, the stability of uncertain system using convex hull algorithm was tested.And we use MATLAB and INTLAB toolbox to write program that can plot the 3Dconvex hull, root loci, step response, and frequency response for any uncertain system.This paper tested the robust stability of an interval 3x3 matrix by the implementation of Printer Belt-Drive System.An efficient and enhanced algorithm was introduced and improved for this purpose.This algorithm can be easily extended to deal with higher order matrices (n-dimensional system) without a very large increase of processing time.

Figure 1 .
Figure 1.D-region Definition 1: (D-stability) Let D  C and take P(s) to be a fixed polynomial, then P(s) is said to be D-stable if and only if all its roots lie in the region D. Definition 2: (Robust-D-stability) comparison in 3D Space Normal Quicker www.ijacsa.thesai.org

Figure 7 .
Figure 7. Convex hull of imaginary roots in focus V. CONCLUSION AND FUTURE WORK ):