Guillermo Fernández-Anaya and José-Job Flores-Godoy Departamento de Física y Matemáticas, Universidad Iberoamericana
México, D. F. 01490 MÉXICO Abstract-In this work it is presented a new class of operators
on polynomials which preserve realness of-roots and interlacing relationships; certain products of Hurwitz polynomials are preserved. In particular, sufficient conditions to preserve complex Hurwitz polynomials and strictly passivity are given. A new property of the operator derivative for Hurwitz polynomials and strictly passive rational functions is presented.
I. INTRODUCTION
As is pointed out in [1] and [2] the concept of positive realness of a transfer function plays a central role in Stability Theory. The definition of rational Positive Real functions (PR functions) arose in the context of Circuit Theory. In fact, the driving point impedance of a passive network is rational and positive real. If the network is dissipative (due to the presence of resistors), the driving point impedance of the network is a Strictly Positive Real transfer function (SPR function). Thus, positive real systems, also called passive systems, are systems that do not generate energy. The celebrated Kalman-Yakubovich-Popov (KYP) lemma [1], established the key role that strict positive realness plays in the design of Lyapunov functions associated to the stability analysis of Linear Time Invariant (LTI) systems with a single memory-less nonlinearity. For Hurwitz polynomials some work has been made based in the properties of “hyperbolic” polynomials i.e., polynomials with only real zeros, but almost nothing using results on preservation of properties in “hyperbolic”
polynomials. In the case of preservation of passivity based in these properties, as we know no work has been made.
In this paper based in a new class of operators on polynomials which preserve realness of roots and interlacing relationships, certain products of polynomials closed under Hurwitz property are preserved. In particular, sufficient conditions to preserve complex Hurwitz polynomials and strictly passivity are given. For instance, a new property of the operator derivative for Hurwitz polynomials and strictly passive rational functions is presented.
II. PRELIMINARIES
This section presents the notation and definitions which will be used throughout the paper.
Let ] and ]+ denote the integers and positive integers polynomials and ( )R s be the field of real rational functions.
Let H be the set of Hurwitz stable polynomials. The degree of a polynomial ( )p s is denoted by deg[ ( )]p s =n, with
n∈]+. Let \n be the vector space on \ with vectors of n components. Similarly, for the vector space \m.
Consider the rational function
0 to be a proper rational function.
A. Preliminary Definitions
In this subsection, we give a list of basic definitions used in this work.
Definition 1. An nth degree polynomial with real coefficients
1 1 0
( ) n n
p s =s +a s− + +" a is Hurwitz stable if all its roots, i.e., ( ) 0p s = have negative real part. Also a simple root is a root from the polynomial with multiplicity one. , Definition 2 ([3]). The Hadamard product of two polynomials ( ), ( )p s q s ∈R s[ ], is defined by little can be said about the setS. For certain polynomials
( )s are stable if and only if the set of polynomials
( )
{
p s p s( ) : ( )=λp s1( )+ −1 λ p s0( ),λ∈[0,1]}
T. Sobh et al. (eds.), Innovative Algorithms and Techniques in Automation, Industrial Electronics and Telecommunications, 155–157.
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is stable. Directions δ( )s =p s1( )−p s0( ) for which this proposition is true are called convex directions. ,
Now we recall some definitions of products which will be useful in the sequel.
Definition 4 ([3]). The circle-point product (:) of two standard real polynomials
0
is the polynomial
(
f :g s)
( )=∑
nk=0k f g s! k k k, In the following definition we introduce a new product of polynomials.
Definition 5. The Wagner product (⊗) of two real polynomials ( )f s and ( )g s is the polynomial Definition 7. Define the linear operator
(
( ) : [ ])
[ ]In this section the main results are given. The results are based on the definitions of products and operators of the previous section.
Theorem 8. If ( )h s has all its roots in the interval (−∞, 0), it is simple-rooted, and ( )f s is a Hurwitz stable polynomial, then ⊗h
(
f s( ))
is a Hurwitz stable polynomial. , Corollary 10. For each fixed Hurwitz stable polynomial( )
f s . The linear operator Δf : [ ]\s →\[ ]s defined as
(
( ))
n 0 k ( )k ( )f g s k= λ f gk s
Δ
∑
preserves Hurwitz stable polynomials, for each fixed λ∈\+.,
The linear operators⊗h, :h, Δf, map Hurwitz real polynomials into Hurwitz real polynomials.
Theorem 11. If : [ ]φ \s →\[ ]s is a linear operator that preserves simple roots and negative real roots. Then the operator : [ ]φ \s →\[ ]s defined as
(
f s( )) ( )
fe( )
s2 s( )
fo( )
s2φ φ + φ
preserves Hurwitz stable polynomials and it is linear. , The linear operators ⊗h, :h, Δf , can be considered as particular cases of Theorem 11.
Defintion 12 ([5]). A rational function ( )q s of zero relative degree is strictly positive real (SPR0) if and only if ( )q s is analytic in Re[ ] 0s ≥ and Re[ (q jω)] 0> for all ω∈\. Theorem 13. If : [ ]φ \s →\[ ]s is a linear operator that preserves simple-roots and negative real roots. Then the operator : [ ]φ \s →\[ ]s defined as
(
f s( )) ( )
fe( )
s2 s( )
fo( )
s2φ φ + φ
1) Preserves complex Hurwitz stable polynomials i.e., if ( ) ( ) ( ) Notice that with linear operators preserving Hurwitz stable polynomials is possible to build new classes of products of Hurwitz stable polynomials. Two such examples are the following, let ( )f s and ( )g s be Hurwitz stable polynomials, and , : [ ]φ ϕ \s →\[ ]s linear operators preserving Hurwitz stable polynomials. Define the product Dφ ϕ, of Hurwitz stable polynomials as
( ) , ( ) ( ( )) ( ( )) f s Dφ ϕg s φ f s Dϕ g s
where D is the Hadamard product as seen in Definition 2. The product •φ ϕ, of Hurwitz stable polynomials as
FERNÁNDEZ-ANAYA AND FLORES-GODOY 156
( ) , ( ) ( ( )) ( ( )) f s •φ ϕg s φ f s ϕ g s
The products Dφ ϕ, and •φ ϕ, are linear in each component, associative and preserve Hurwitz stability, but they are not commutative. The last corollary is a clear consequence for convex directions.
Example 15. The Gauss-Lucas Theorem states that if a polynomial ( )p s has its zeros contained in some given convex set K, then its derivative dp sds( ) has all its zeros in K as well (unless ( )p s is constant). In particular, if all the zeros are real, then also all the zeros of the derivative are real. For instance, if all zeros of ( )p s are real and negative, then so are the zeros of the derivative (see [5]). In consequence, it is clear that the derivative operator
( )
dsd is a linear operator preserving simple-roots and negative real roots. In particular consider the following polynomial:4 3
Applying the derivative operator
( )
dsd to the Hurwitz stable polynomial ( )p s we obtainNow consider the polynomial
8 7 6 5 4 3
Applying the derivative operator
( )
dsd to the Hurwitz stable polynomial ( )q s we obtain function. Applying the operator( )
dsd in the numerator and the denominator, we obtain( )
In this paper we present new linear operators which preserve products of (real and complex) Hurwitz polynomials and strictly passive rational functions. We believe that these results provide an insight into the theory of stability and passive real functions preserved by these products. This can also be extended in several directions. A possible application of the results presented in this paper is in robust stability theory.
REFERENCE
[1] K. S. Narendra and A. M. Annaswamy, Stable Adaptive Systems. Prentice-Hall. Englewood Cliffs, NJ, USA, 1989.
[2] K. S. Narendra and J. H. Taylor, Frequency domain criteria for absolute stability. Academic Press. New York, NY, USA, 1973.
[3] J. Garloff and D. Wagner. Hadamard products of stable polynomials are stable. Journal of mathematical analysis and applications, 202:797-809, 1996.
[4] S. P. Battacharyya, H. Chapellat, and L. H. Kell. Robust Control: The Parametric Approach. Prentice-Hall, New York, NY, USA, 1995.
[5] A. Aleman, D. Beliaev, and H. Hedenmalm. Real zero polynomials and Pólya-Schur type theorems. J. Anal. Math., 94:49-60, 2004.
OPERATORS PRESERVING PRODUCTS OF HURWITZ POLYNOMIALS AND POSSIVITY 157
Abstract— In this paper, we extend our modeling approach by introducing verification methods so that the design process will be carried correctly. The Model of Component (MoC) and the Model of Function (MoF) are the basic models of our modeling approach. Thus, we formalize some properties of these models and we present the corresponding verification methods.
A computer aided tool for specification and verification is developed to illustrate our approach.
I. INTRODUCTION
Operating modes of Automated Production Systems (APS) are characterized by coherence and safety constraints. The main difficulty is to insure mode changing while guaranteeing the system coherence. The solution to this problem relate to modeling methods as well as verification means. Several approaches are proposed in the literature [6].
These approaches focus on modeling, without dealing with verification of the proposed models. That is why we extend in this paper our modeling approach of mode handling [3][5] by developing formal verification tools so that this design process will be performed correctly. We introduce some properties of the obtained models and we propose some verification methods using graph theory. The aim is to handle the verification stage of the design process.
In the following section we present briefly the main characteristics of our modeling approach for mode handling.
We extend this process by integrating appropriate verification methods. Section 3 presents some properties of the MoC and the MoF and the corresponding verification methods. In section 4, we illustrate our propositions through an application example.
II. A MODELING APPROACH FOR MODE HANDLING We proposed in [3] a modeling method for mode handling of APS. The system is modeled using a Functional Graph (FG) obtained according to an analysis approach based on production goals. We should determine the main goals for which the system was designed and the sub goals that allow to reach them; each sub goal can be decomposed in the same manner until obtaining the initial goals. These are the leaves of the graph; they are related to the resources which perform the initial goals. The behavior of the resources and the functions is then specified. For each function or resource, several concurrent families of modes are determined according to a multipoint of view method. The obtained
models representing the behavior of the resources and the functions are respectively called Model of Component (MoC) and Model of Function (MoF). They are the basic models of this design approach. The behavior represented by these models is characterized by a set of concurrent state-transition graphs (the set is called a family of modes and the graphs are called modes). The incompatibilities and constraints are taken into account in the design process by the addition of some specifications to respect them; called mechanisms.
1) Specification of the modes: This specification process is based on the point of view concept, which allows characterizing a resource according to the observer’s criteria.
For example from the exploitation point of view, the system is characterized according to two points of view: production and maintenance. These points of view can be characterized by other points of view. The method determines a set of families of modes and generic modes representing the behavior of a production system [3].
2) Specification of the Models of Component: The specification of the MoC (Fig. 1) follows three steps:
1st step- specification of the static part: This step consists in listing the states that can take the system. The modes which characterize an elementary component are the same ones as those which characterize a production system.
2nd step- specification of the dynamic part: This step consists in determining at first the initial states of the model then the change-of-state conditions. A matrix form called Matrix of the Change-of-state Conditions (MCC) is used to represent these conditions within a mode.
3rd step- study of the coherence of the modes: This study begins with the development of the Matrix of Coherence of the Modes (MCM) (also called Matrix of Compatibility) which characterizes the incompatibilities of the states. When two states are incompatible their simultaneous activation is not possible and this should be taken into account in the specifications. We distinguish forbidden states and transient states. The forbidden states correspond to situations that should not occur. So any switching to these states must be forbidden. The transient states are states from which we can only go through without staying (the switching is considered as instantaneous). The activation of such states causes then a switching to compatible states. In order to guarantee the coherence of the MoC, a specification solution called mechanism is proposed for each case. Then it is necessary to