# New Algorithm Based on Sign Decomposition to Verify the Robust Stability Property for a Class of Linear Time-Delay Systems

Authors: Iván D. Díaz-Rodríguez; Gerardo Romero; Cesar Elizondo
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

The main focus of this paper is to analyze the
robust stability property for a class of time-delay systems when
parametric polynomic uncertainty is considered. The analysis
is made by replacing the time-delay part with an auxiliary
equation and then using the sign definite decomposition to
deal with the polynomic parametric uncertainty. Also, it is
shown that it is possible to verify the robust stability property
by first obtaining the Hurwitz matrix from the characteristic
equation for this class of systems and then checking the leading
principal minors positivity using the sign definite decomposition.
Finally, an algorithm codified in MATLAB is used to evaluate and
graphically show the robust stability property. This is shown by
a series of points that were calculated using the sign definite
decomposition.


---

## Body

## 1. Introduction

Time-delay systems arose as a result of inherent delays in system’s components and also due to the introduction of deliberated delay in the system for control purposes; see [1–3]. Over the years, time-delay systems interest and popularity have grown steadily. In particular, in the last 10 to 15 years there has been a surge in research and a proliferation of new techniques and results. Many of these new results include systems not only with time-delay analysis, but also with uncertainty in the system to be considered. For example, in [4–6] an analysis of robust stability for time-delay dynamical systems with parametric uncertainty in the mathematical model and in the time-delay is shown; in addition, the value set concept and the zero exclusion condition are used to verify the robust stability property of interval plants; see also [7, 8]. In [9, 10] the robust stabilization problem for a class of time-delay systems is considered where they involve parametric affine perturbations; in [11], new results to compute the time-delay of the hot-dip galvanizing control system are considered. In [12, 13] the authors present a robust model predictive control for systems represented by Takagi-Sugeno models and this technique was applied to the continuous stirred tank reactor (CSTR). They use Linear Matrix Inequalities (LMI) to solve the optimization problem. A technique based on a representation in the time domain of a class ofdifferential-difference systems is presented in [14, 15]. Here, the authors make an application to the active suspension systems with actuator delay using the aforementioned technique. In [16, 17] new robust stability results for LTI systems with parametric uncertainty using sign definite decomposition were developed. In [18] the robust stability problem for a polynomial family was considered whose coefficients are polynomial functions of the parameters of interest. They used the sign definite decomposition for the controllers design. In [19] the robust stability positivity of a real function f(x) is considered while the real vector x varies over a box. They determined Hurwitz robust stability for a polynomial family using the sign definite decomposition described in [16].In 1981, the characteristic polynomial including a time-delay, for a linear differential-difference system, considered replacing the terme-sτ by a regular polynomial (1-Ts)2/(1+Ts)2. After this, it was possible to verify the asymptotic stability property for a class of time-delay systems; see [20]. Previously, a different approach was made in [21] by replacing e-sτ by (1-Ts)/(1+Ts). But, it was found in [20] that the main problem with this substitution was that the two sets of image points were not identical for all s=jω,ω>0,T>0 and  τ>0. In the present paper we outline a new algorithm to verify the robust stability property for a class of linear time-delay systems including a special case of polynomic parametric uncertainty, which one has not been considered in systems involving a time-delay. This is by using sign definite decomposition theory to verify the robust stability of the system using the stability conditions of Hurwitz matrix. We illustrate this using a numerical example.This paper is organized as follows. In the preliminaries section, the Hurwitz stability criterion, a special polynomic parametric uncertainty case, and the sign definite decomposition are described. Then, the problem statement is presented. After that, the methodology and proposed algorithm are shown. An illustrative numerical example is presented to show the effectiveness of this approach. Finally, we discuss our results and future research.


## 2. Main Contribution

As it may be seen from the section above, some of the previous results use techniques based on a representation in the time domain ofdifferential-difference systems. Thus, to analyze and design them it is necessary to use the Lyapunov technique. It should be also mentioned that the uncertainty that they experiment should be represented by time functions. However, there are many applications where the uncertainty depends on variables other than time, such as resistors, capacitors, and inductors in an electrical circuit, which have parameters that are uncertain and whose uncertainty depends mainly on temperature and therefore could not be analyzed with these techniques. Also, in previous section some other results were mentioned that consider uncertainty structures like interval or linear affine and systems without delay. However, the main result of this paper is to obtain sufficient conditions to verify the robust stability property of a class of quasi-polynomials that represent the characteristic equation ofdifferential-difference dynamics systems. It considers polynomic parametric uncertainty structure in the coefficients of quasi-polynomials and also interval uncertainty in the time-delay. First of all, a transformation of the delay’s operator is performed in order to get a two-variable polynomial; after this, to obtain the robust stability property, a result based on the Hurwitz matrix is applied, and then checking the leading principal minors positivity using the sign definite decomposition.


## 3. Preliminaries

### 3.1. Hurwitz Stability Criterion

Theorem 1 (Hurwitz stabilty).
Given a real polynomialp(s)=a0+a1s+a2s2+⋯+ansn, the polynomial p(s) is stable; that is, all its roots lie in the open left half plane (LHP) of the complex plane, if and only if, all of the leading principal minors, defined by Δi, of the matrix H are positive; see [22]:(1)H=an-1an-3an-5⋯⋯0anan-2an-4⋯⋯00an-1an-3an-5⋯00anan-2an-4⋯0⋮⋮⋮⋮⋱00000⋯a0.


### 3.2. Uncertainty

There exists a case where the precise value of the parameters of the mathematical model is unknown; however, its lower and upper bounds are knownqi-,qi+, respectively. The collection of all l parameters involved in the mathematical model forms a vector of parameters q=q1,q2,…,qlT which is an element of a parametric uncertainty box Q:(2)Q=q=q1,q2,…,qlT∣qi∈qi-,qi+,i=1,2,…,l.For different lower and upper bound values, it is always possible to make a coordinate transformation of the physical parameters without losing their original properties. Such transformation can be ρi=qi-qi-/qi+-qi-, and in this case qi∈qi-,qi+ is taken in ρi=0,1, where, for simplicity, we can name q to the new coordinate qi=0,1. When we consider a parametric uncertainty, we have a polynomial family defined as(3)Ps,Q≜ps,q:q∈Q.There exist a class of polynomials with parametric uncertainty p(s,q). It is calledpolynomic uncertainty structure; that is, it has all of its coefficients where at least one parameter appears with power greater than one. For example, p(s,q)=q1+2q12q3s2+q1q23+q1s+2q2q3.Definition 2 (see [23]).
LetP be a positive convex cone in a vector space Rl, for all x,y∈Rl, it is said that x≥y(x>y) with respect to P if x-y∈P(x-y∈P0, the interior of P).From this point, we will considerQ⊂P and qi-≥0. This implies that q∈Q⊂P.Definition 3 (see [23]).
f
:
R
l
→
R being a continuous function and Q⊂P⊂Rl a convex subset, it is said that f(·) is a nondecreasing function in Q, if x≥y implies f(x)≥f(y),∀x,y∈Q.


### 3.3. Sign Definite Decomposition

Definition 4 (see [16, 17]).
f
:
R
l
→
R being a continuous function and Q⊂P⊂Rl a convex subset, it is said that f(·) has sign decomposition in Q if there exist two nondecreasing bounded functions fn(·)≥0, fp(·)≥0, such that f(q)=fp(q)-fn(q) for all q∈Q. One will call those functions the positive fp(·) and negative fn(·) parts of the function:(4)fq=fpq-fnq∀q∈Qfp·≜Positive  part  off·fn·≜Negative  part  off·.The negative and positive partsfn(·),fp(·) constitute a representation fn,fp of the function in R2 with a graphic representation in the plane fn(·),fp(·) according to Figure 1.Figure 1
Rectangle containing the functionf(q).Definition 5 (see [16, 17]).
It will be called minimum and maximum euclidean vertexvmin, vmax to the vectors elements of Q⊂P⊂Rl with the minimum and maximum Euclidean norm, respectively:(5)vmin2=minq∈Q⁡q2,vmax2=maxq∈Q⁡q2.Since the negativefn(q) and positive fp(q) parts are nondecreasing functions in a vector space, the graphic representation of f(q),∀q∈Q in the plane fn,fp is contained in Figure 1, where if the lower right vertex fnvmax,fpvmin is above the 45° line, then the function f(q)>0,∀q∈Q.Definition 6 (see [16, 17]).
f
p
(
q
) and fn(q) are the elements of a function f(q) with sign definite decomposition in Q. T being the linear transformation described such that there exists T-1, then it is called a representation of the function f(q) in α,β coordinates to the linear transformation α(q),β(q)=Tfn(q),fp(q) of the function:(6)T=11-11T-1=0.51-111αqβq=Tfnqfpqfnqfpq=T-1αqβqαq=fpq+fnqfpq=0.5αq+βqβq=fpq-fnqfnq=0.5αq-βq.In order to define the positivity or negativity of a function using theα,β representation, when a polynomial uncertainty set is included, we need to use the following theorem.Theorem 7 ((rectangle) [16, 17]).
f
:
R
l
→
R being a continuous function with sign definite decomposition in Q such that Q⊂P⊂Rl is a box with euclidean minimum and maximum vertex vmin,vmax then (a) the lower bound of f(q) is 0.5αvmin+βvmin-αvmax+βvmax and its upper bound is 0.5αvmax+βvmax-αvmin+βvmin, respectively; (b) the graphic representation of the function f(q),∀q∈Q in the plane α,β is contained in the rectangle with vertices: αizq=αvmin,  βizq=βvmin,  αder=αvmax,  βder=αvmax; αinf=0.5[α(vmin)+α(vmax)]-0.5[β(vmax)-β(vmin)],  βinf=0.5[β(vmin)+β(vmax)]-0.5[α(vmax)-α(vmin)],  αsup=0.5[α(vmin)+α(vmax)]+0.5[β(vmax)-β(vmin)],  βsup=0.5[β(vmin)+β(vmax)]+0.5[α(vmax)-α(vmin)]; (c) if the lower vertex αinf,βinf is above the α axis in the α,β plane, then the function f(q)<0,∀q∈Q.Theorem 8 ((box partition) [16, 17]).
f
:
R
l
→
R being a continuous function with sign definite decomposition in Q such that Q⊂P⊂Rl is a box with minimum and maximum euclidean vertices vmin,vmax then the function is positive (negative) in Q if and only if there exist a set of boxes Γ, such that Q=⋃j‍Γj and the lower bound ≥c>0 for every box Γj (upper bound ≤c<0 for every box Γj).The determinant of the matrixM is comprised of additions and subtractions of products of the elements of the matrix and if those are formed of polynomial type, the determinant det(M) has sign definite decomposition. The programming development in order to get the sign definite decomposition of the determinant in the representation fn,fp can be quite complicated; however, in the (α,β) representation there exists a less complicated way to do it.Definition 9 (see [16, 17]).
M
(
q
) being a square matrix with elements mi,j(q) with sign definite decomposition in Q with representation αi,j(q),βi,j(q), then it will be called Mα(q) to the matrix formed with the elements αi,j(q) and it will be called detαMα(q)=Mα(q)α to the function similar to the determinant of the matrix Mα(q) but without applying the sign rule (-1)i+j; it will be Mβ(q)=M(q) and detβMβ(q)=detM(q).Lemma 10 (see [16, 17]).
LetM(q) be a square matrix with elements mi,j(q) with sign definite decomposition in Q with αi,j(q),βi,j(q) representation. Mα(q) being the square matrix with αi,j(q) elements, then the α,β representation of the matrix determinant M(q) is given by(7)αq=det⁡αMαq,βq=det⁡Mq.


## 3.1. Hurwitz Stability Criterion

Theorem 1 (Hurwitz stabilty).
Given a real polynomialp(s)=a0+a1s+a2s2+⋯+ansn, the polynomial p(s) is stable; that is, all its roots lie in the open left half plane (LHP) of the complex plane, if and only if, all of the leading principal minors, defined by Δi, of the matrix H are positive; see [22]:(1)H=an-1an-3an-5⋯⋯0anan-2an-4⋯⋯00an-1an-3an-5⋯00anan-2an-4⋯0⋮⋮⋮⋮⋱00000⋯a0.


## 3.2. Uncertainty

There exists a case where the precise value of the parameters of the mathematical model is unknown; however, its lower and upper bounds are knownqi-,qi+, respectively. The collection of all l parameters involved in the mathematical model forms a vector of parameters q=q1,q2,…,qlT which is an element of a parametric uncertainty box Q:(2)Q=q=q1,q2,…,qlT∣qi∈qi-,qi+,i=1,2,…,l.For different lower and upper bound values, it is always possible to make a coordinate transformation of the physical parameters without losing their original properties. Such transformation can be ρi=qi-qi-/qi+-qi-, and in this case qi∈qi-,qi+ is taken in ρi=0,1, where, for simplicity, we can name q to the new coordinate qi=0,1. When we consider a parametric uncertainty, we have a polynomial family defined as(3)Ps,Q≜ps,q:q∈Q.There exist a class of polynomials with parametric uncertainty p(s,q). It is calledpolynomic uncertainty structure; that is, it has all of its coefficients where at least one parameter appears with power greater than one. For example, p(s,q)=q1+2q12q3s2+q1q23+q1s+2q2q3.Definition 2 (see [23]).
LetP be a positive convex cone in a vector space Rl, for all x,y∈Rl, it is said that x≥y(x>y) with respect to P if x-y∈P(x-y∈P0, the interior of P).From this point, we will considerQ⊂P and qi-≥0. This implies that q∈Q⊂P.Definition 3 (see [23]).
f
:
R
l
→
R being a continuous function and Q⊂P⊂Rl a convex subset, it is said that f(·) is a nondecreasing function in Q, if x≥y implies f(x)≥f(y),∀x,y∈Q.


## 3.3. Sign Definite Decomposition

Definition 4 (see [16, 17]).
f
:
R
l
→
R being a continuous function and Q⊂P⊂Rl a convex subset, it is said that f(·) has sign decomposition in Q if there exist two nondecreasing bounded functions fn(·)≥0, fp(·)≥0, such that f(q)=fp(q)-fn(q) for all q∈Q. One will call those functions the positive fp(·) and negative fn(·) parts of the function:(4)fq=fpq-fnq∀q∈Qfp·≜Positive  part  off·fn·≜Negative  part  off·.The negative and positive partsfn(·),fp(·) constitute a representation fn,fp of the function in R2 with a graphic representation in the plane fn(·),fp(·) according to Figure 1.Figure 1
Rectangle containing the functionf(q).Definition 5 (see [16, 17]).
It will be called minimum and maximum euclidean vertexvmin, vmax to the vectors elements of Q⊂P⊂Rl with the minimum and maximum Euclidean norm, respectively:(5)vmin2=minq∈Q⁡q2,vmax2=maxq∈Q⁡q2.Since the negativefn(q) and positive fp(q) parts are nondecreasing functions in a vector space, the graphic representation of f(q),∀q∈Q in the plane fn,fp is contained in Figure 1, where if the lower right vertex fnvmax,fpvmin is above the 45° line, then the function f(q)>0,∀q∈Q.Definition 6 (see [16, 17]).
f
p
(
q
) and fn(q) are the elements of a function f(q) with sign definite decomposition in Q. T being the linear transformation described such that there exists T-1, then it is called a representation of the function f(q) in α,β coordinates to the linear transformation α(q),β(q)=Tfn(q),fp(q) of the function:(6)T=11-11T-1=0.51-111αqβq=Tfnqfpqfnqfpq=T-1αqβqαq=fpq+fnqfpq=0.5αq+βqβq=fpq-fnqfnq=0.5αq-βq.In order to define the positivity or negativity of a function using theα,β representation, when a polynomial uncertainty set is included, we need to use the following theorem.Theorem 7 ((rectangle) [16, 17]).
f
:
R
l
→
R being a continuous function with sign definite decomposition in Q such that Q⊂P⊂Rl is a box with euclidean minimum and maximum vertex vmin,vmax then (a) the lower bound of f(q) is 0.5αvmin+βvmin-αvmax+βvmax and its upper bound is 0.5αvmax+βvmax-αvmin+βvmin, respectively; (b) the graphic representation of the function f(q),∀q∈Q in the plane α,β is contained in the rectangle with vertices: αizq=αvmin,  βizq=βvmin,  αder=αvmax,  βder=αvmax; αinf=0.5[α(vmin)+α(vmax)]-0.5[β(vmax)-β(vmin)],  βinf=0.5[β(vmin)+β(vmax)]-0.5[α(vmax)-α(vmin)],  αsup=0.5[α(vmin)+α(vmax)]+0.5[β(vmax)-β(vmin)],  βsup=0.5[β(vmin)+β(vmax)]+0.5[α(vmax)-α(vmin)]; (c) if the lower vertex αinf,βinf is above the α axis in the α,β plane, then the function f(q)<0,∀q∈Q.Theorem 8 ((box partition) [16, 17]).
f
:
R
l
→
R being a continuous function with sign definite decomposition in Q such that Q⊂P⊂Rl is a box with minimum and maximum euclidean vertices vmin,vmax then the function is positive (negative) in Q if and only if there exist a set of boxes Γ, such that Q=⋃j‍Γj and the lower bound ≥c>0 for every box Γj (upper bound ≤c<0 for every box Γj).The determinant of the matrixM is comprised of additions and subtractions of products of the elements of the matrix and if those are formed of polynomial type, the determinant det(M) has sign definite decomposition. The programming development in order to get the sign definite decomposition of the determinant in the representation fn,fp can be quite complicated; however, in the (α,β) representation there exists a less complicated way to do it.Definition 9 (see [16, 17]).
M
(
q
) being a square matrix with elements mi,j(q) with sign definite decomposition in Q with representation αi,j(q),βi,j(q), then it will be called Mα(q) to the matrix formed with the elements αi,j(q) and it will be called detαMα(q)=Mα(q)α to the function similar to the determinant of the matrix Mα(q) but without applying the sign rule (-1)i+j; it will be Mβ(q)=M(q) and detβMβ(q)=detM(q).Lemma 10 (see [16, 17]).
LetM(q) be a square matrix with elements mi,j(q) with sign definite decomposition in Q with αi,j(q),βi,j(q) representation. Mα(q) being the square matrix with αi,j(q) elements, then the α,β representation of the matrix determinant M(q) is given by(7)αq=det⁡αMαq,βq=det⁡Mq.


## 4. Problem Statement

The main interest of this research is to analyze the robust stability property of difference-differential dynamical systems which are characterized by polynomic parametric uncertainty and time-delay of the form:(8)x˙t=A0qxt+A1qxt-τ,where A0(q), A1(q)∈Rnxn are matrices with dependent parameters of qi∈Q and τ∈[0,τmax]; for example,(9)A0q=q12q2q34q1q2q12q22q32A1q=q32q12q23q3q15q2q3,q=q1q2q3qi∈qi-,qi+.The parameters qi represent the polynomic structure uncertainty and τ the uncertain time-delay. Then, system (8) is asymptotically stable if and only if the following condition is satisfied:(10)ps,q,e-τs=det⁡sI-A0q-A1qe-τs≠0.⁡One has ∀s∈C+, wWhere C+ represents the RHP of the complex plane. The quasi-polynomials that satisfy the last condition are called stable quasi-polynomials.


## 5. Results

The robust stability property is determined by the analysis of the characteristic equation (10). Such equation is called characteristic quasi-polynomial with polynomic parametric uncertainty. The following transformation is needed in order to determine the robust stability condition for this class of systems.Definition 11.
A polynomialp(s,T) associated with a quasi-polynomial p(s,e-τs) will be defined as follows:(11)ps,T=∑i=0npis1-Ts2i1+Ts2n-2ips,e-τs=∑i=0npise-iτs.The roots of this associated polynomial have an important relation with the roots of the quasi-polynomial. This relation is presented in the following theorem.Theorem 12 (see [20]).
Suppose thats0=jω0 for some value of ω0≥0; then s0=jω0 is a matrix of the characteristic equation p(s,e-τs) for some value of τ≥0 if and only if s0=jω0 is a root of p(s,T) for some value of T≥0.With this transformation we can get the relation betweenτ and T, which is valid in the imaginary axis jω; see [20]. The time-delay τ and T are related by the following equation:(12)T=tan⁡τωi/4ωi,where ωi∈W; the set of  W is defined as follows:(13)W≡0<ω<ω∗:pjω∗,T=0paraT>0,where p(jω,T) is a polynomial associated with p(s,e-τs) evaluated in the jω axis. Note that for each value of ωi, there exists a direct relation τ and  T; also, this relation is a continuous function and strictly increasing in the range T∈[0,∞). For this reason, for every fixed value of ωi, the interval τ∈[0,τmax] generates an interval T∈[0,Tmax]. Now, it is clear that for all values of ωi∈W there exists a relation between τ and T; we will define Tmax in the following relation:(14)Tmax⁡=min⁡TiTi=tan⁡τωi/4ωi∀ωi∈W.Definition 13.
The Hurwitz matrix is(15)Hps,q,T=h1,1q,Th1,2q,T⋯0h2,1q,Th2,2q,T⋯00h3,2q,T⋯00h4,2q,T⋯0⋮⋮⋱000⋯hn,nq,T,where every element of the matrix depends of the values of qi∈Q⊂P and T∈(0,Tmax].Definition 14.
The Hurwitz matrix being (15), then fpi,j(q,T) and fni,j(q,T) will be denoted to the positive and negative parts, respectively, for every element of the Hurwitz matrix such that(16)hi,jq,T=fpi,jq,T+fni,jq,T,where i=1,2,…,n,j=1,2,…,n.According to the(α,β) representation, we can express the following.Definition 15.
LetΔi[αinf],i=1,2,…,n be the matrices with elements αi,jinf obtained from the leading principal minors Δi of the Hurwitz matrix H[p(s,q,T)]:(17)Δ1αinf⁡=α1,1inf⁡Δ2αinf⁡=α1,1inf⁡α1,2inf⁡α2,1inf⁡α2,2inf⁡⋮Δnαinf⁡=α1,1inf⁡α1,2inf⁡α1,3inf⁡0⋯0α2,1inf⁡α2,2inf⁡α2,3inf⁡0⋯00α3,2inf⁡α3,3inf⁡α3,4inf⁡⋯00α4,2inf⁡α4,3inf⁡α4,4inf⁡⋯0⋮⋮⋮⋮⋱00000⋯αn,ninf⁡,where every element of the matrices αi,jinf⁡=fp(vmin)+fn(vmax) is formed by the addition of the positive and negative parts taken from the corresponding element of the Hurwitz matrix H[p(s,q,T)].Definition 16.
LetΔi[βinf⁡],i=1,2,…,n, be the matrices with elements βi,jinf⁡ obtained from the leading principal minors Δi of the Hurwitz matrix H[p(s,q,T)]:(18)Δ1βinf⁡=β1,1inf⁡Δ2βinf⁡=β1,1inf⁡β1,2inf⁡β2,1inf⁡β2,2inf⁡⋮Δnβinf⁡=β1,1inf⁡β1,2inf⁡β1,3inf⁡0⋯0β2,1inf⁡β2,2inf⁡β2,3inf⁡0⋯00β3,2inf⁡β3,3inf⁡β3,4inf⁡⋯00β4,2inf⁡β4,3inf⁡β4,4inf⁡⋯0⋮⋮⋮⋮⋱00000⋯βn,ninf⁡,where every element of the matrices βi,jinf=fp(vmin)-fn(vmax) is formed by the subtraction of the positive and negative parts taken from the corresponding element of the Hurwitz matrix H[p(s,q,T)].Definition 17.
LetΔi[αder],i=1,2,…,n, be the matrices with elements αi,jder obtained from the leading principal minors Δi of the Hurwitz matrix H[p(s,q,T)]:(19)Δ1αder=α1,1derΔ2αder=α1,1derα1,2derα2,1derα2,2der⋮Δnαder=α1,1derα1,2derα1,3der0⋯0α2,1derα2,2derα2,3der0⋯00α3,2derα3,3derα3,4der⋯00α4,2derα4,3derα4,4der⋯0⋮⋮⋮⋮⋱00000⋯αn,nder,where every element of the matrices αi,jder=fp(vmax)+fn(vmax) is formed by the addition of the positive and negative parts from the corresponding element of the Hurwitz matrix H[p(s,q,T)].Definition 18.
LetΔi[βder],i=1,2,…,n, be the matrices with elements βi,jder obtained from the leading principal minors Δi of the Hurwitz matrix H[p(s,q,T)]:(20)Δ1βder=β1,1derΔ2βder=β1,1derβ1,2derβ2,1derβ2,2der⋮Δnβder=β1,1derβ1,2derβ1,3der0⋯0β2,1derβ2,2derβ2,3der0⋯00β3,2derβ3,3derβ3,4der⋯00β4,2derβ4,3derβ4,4der⋯0⋮⋮⋮⋮⋱00000⋯βn,nder,where every element of the matrices βi,jder=fp(vmax)-fn(vmax) is formed by the subtraction of the positive and negative parts taken from the corresponding element of the Hurwitz matrix H[p(s,q,T)].Definition 19.
LetΔi[αizq],i=1,2,…,n, be the matrices with elements αi,jizq obtained from the leading principal minors Δi of the Hurwitz matrix H[p(s,q,T)]:(21)Δ1αizq=α1,1izqΔ2αizq=α1,1izqα1,2izqα2,1izqα2,2izq⋮Δnαizq=α1,1izqα1,2izqα1,3izq0⋯0α2,1izqα2,2izqα2,3izq0⋯00α3,2izqα3,3izqα3,4izq⋯00α4,2izqα4,3izqα4,4izq⋯0⋮⋮⋮⋮⋱00000⋯αn,nizq,where every element of the matrices αi,jizq=fp(vmin)+fn(vmin) is formed by the addition of the positive and negative parts from the corresponding element of the Hurwitz matrix H[p(s,q,T)].Definition 20.
LetΔi[βizq],i=1,2,…,n, be the matrices with elements βi,jizq obtained from the leading principal minors Δi of the Hurwitz matrix H[p(s,q,T)]:(22)Δ1βizq=β1,1izqΔ2βizq=β1,1izqβ1,2izqβ2,1izqβ2,2izq⋮Δnβizq=β1,1izqβ1,2izqβ1,3izq0⋯0β2,1izqβ2,2izqβ2,3izq0⋯00β3,2izqβ3,3izqβ3,4izq⋯00β4,2izqβ4,3izqβ4,4izq⋯0⋮⋮⋮⋮⋱00000⋯βn,nizq,where every element of the matrices βi,jizq=fp(vmin)-fn(vmin) is formed by the subtraction of the positive and negative parts from the corresponding element of the Hurwitz matrix H[p(s,q,T)]. Now, the main result is presented. This presents the robust stability condition for the class of systems described in (1).


### 5.1. Robust Stability Analysis Algorithm

In this subsection we describe the steps to follow to analyze the robust stability property of the system including time-delay and polynomic parametric uncertainty. The algorithm is the following.Step 1.
Consider the time-delay system described in (8) and the characteristic equation in (10).Step 2.
Determine the characteristic equationp(s,q,T) using (11).Step 3.
Define the Hurwitz matrixH[p(s,q,T)] given by (15).Step 4.
For each element in the Hurwitz matrix, perform a separation in positive and negative part. This is given in (16).Step 5.
With respect to the(α,β) representation given in the preliminaries, define the leading principal minors from the (α,β) representation taking the Hurwitz matrix H[p(s,q,T)]. That is, Δi[αiinf], Δi[βiinf], Δi[αider], Δi[βider], Δi[αiizq] and Δi[βiizq] given in (17), (18), (19), (20), (21), and (22), respectively.Step 6.
Determine the points(αiinf,βiinf), (αider,βider), and (αiizq,βiizq) in the (α,β) plane obtained by αiinf=det(Δi[αiinf]), βiinf=det(Δi[βiinf]), αider=det(Δi[αider]), βider=det(Δi[βider]), αiizq=det(Δi[αiizq]), and βiizq=det(Δi[βiizq]).Step
7. Verify the following:(i)
If the points(αiinf,βiinf), (αider,βider), and (αiizq,βiizq) are located above of the α axis, then the system described in (8) is robustly stable. Stop the algorithm.(ii)
If the points(αiinf,βiinf), (αider,βider), and (αiizq,βiizq) are located below the α axis, then the system described in (8) is not robustly stable. Stop the algorithm.(iii)
If we have at least one point(αider,βider) or (αiizq,βiizq) located below the α axis, the analysis is concluded with the knowledge that the system described in (8) is not robustly stable. Stop the algorithm.(iv)
If the points(αiizq,βiizq), (αider,βider) are located above the α axis but the points (αiinf,βiinf) are located below the α axis, then we can make a partition in a subset of boxes Γi as given in Theorem 8. Go to Step 5.


## 5.1. Robust Stability Analysis Algorithm

In this subsection we describe the steps to follow to analyze the robust stability property of the system including time-delay and polynomic parametric uncertainty. The algorithm is the following.Step 1.
Consider the time-delay system described in (8) and the characteristic equation in (10).Step 2.
Determine the characteristic equationp(s,q,T) using (11).Step 3.
Define the Hurwitz matrixH[p(s,q,T)] given by (15).Step 4.
For each element in the Hurwitz matrix, perform a separation in positive and negative part. This is given in (16).Step 5.
With respect to the(α,β) representation given in the preliminaries, define the leading principal minors from the (α,β) representation taking the Hurwitz matrix H[p(s,q,T)]. That is, Δi[αiinf], Δi[βiinf], Δi[αider], Δi[βider], Δi[αiizq] and Δi[βiizq] given in (17), (18), (19), (20), (21), and (22), respectively.Step 6.
Determine the points(αiinf,βiinf), (αider,βider), and (αiizq,βiizq) in the (α,β) plane obtained by αiinf=det(Δi[αiinf]), βiinf=det(Δi[βiinf]), αider=det(Δi[αider]), βider=det(Δi[βider]), αiizq=det(Δi[αiizq]), and βiizq=det(Δi[βiizq]).Step
7. Verify the following:(i)
If the points(αiinf,βiinf), (αider,βider), and (αiizq,βiizq) are located above of the α axis, then the system described in (8) is robustly stable. Stop the algorithm.(ii)
If the points(αiinf,βiinf), (αider,βider), and (αiizq,βiizq) are located below the α axis, then the system described in (8) is not robustly stable. Stop the algorithm.(iii)
If we have at least one point(αider,βider) or (αiizq,βiizq) located below the α axis, the analysis is concluded with the knowledge that the system described in (8) is not robustly stable. Stop the algorithm.(iv)
If the points(αiizq,βiizq), (αider,βider) are located above the α axis but the points (αiinf,βiinf) are located below the α axis, then we can make a partition in a subset of boxes Γi as given in Theorem 8. Go to Step 5.


## 6. Example

Consider a first order system:(23)x˙t=-xt-2xt-τ,τ>0.Using Definition 11, we have that the system has the following auxiliary equation:(24)s+1+21-Ts1+Ts2=0.For this polynomial, we can find out that it has roots in the LHP if and only if T=1/3 and they are located in s=±j3. We can conclude that Tmax=1/3 and by (12) the maximum time-delay is τmax=2π/33. This means that this is the maximum value that can be taken by the time-delay before the system becomes unstable. Now if we consider polynomial parametric uncertainty involved in the parameters of the model we have(25)s+1q1q22+21-Ts1+Ts2=0.From (23) we have(26)ps,q,T=T2s3+2T+T2q1q22+2T2s2+2Tq1q22-4T+1s+2+q1q22,where the Hurwitz matrix H[p(s,q,T)] is(27)2T+T2q1q22+2T22+q1q220T22Tq1q22-4T+1002T+T2q1q22+2T22+q1q22.We applied the definitions presented before to analyze the positivity of the leading principal minors. Running the algorithm, we took the values of qi∈[0,1] and T∈[0,0.2]; Figure 2 was obtained.Figure 2
Sign definite decomposition of the Hurwitz matrix for the example.We can see in Figure2 that the (αiinf,βiinf) points symbolized with “+” and (αider,βider), (αiizq,βiizq) symbolized with “∗” are located above the α axis, where according toRobust Stability Analysis Algorithm, the system is robustly stable.If we analyze a small variation in the time-delay with values ofT∈[0,0.4], we have Figure 3. We can see that the points (αiinf,βiinf) symbolized with “+” and (αider,βider), (αiizq,βiizq) symbolized with “∗” take negative values below the axis α, which—in relation to theRobust Stability Analysis Algorithm—means that the conditions that guarantee the robust stability of the system are not satisfied.Figure 3
Sign definite decomposition of the determinant of the Hurwitz matrix for the example with a little variation in the time-delay.


## 7. Conclusions

In this research it was shown that the robust stability property of linear dynamical systems, which have polynomic uncertain parameters and time-delay, can be verified by the application of an algorithm based on the method of sign definite decomposition. The positivity of the determinant of the Hurwitz matrix is verified by checking the positivity of all leading principal minors of the matrix in terms of(α,β) representation. This Hurwitz matrix contains the elements of the polynomial from the transformation of the characteristic quasi-polynomial in an auxiliary equation in terms of T. For future research, the next step is to optimize the algorithm to perform faster computation time and new methods to identify the positivity of the determinants of the Hurwitz matrix.



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*Source: 217904-2015-05-26.xml*