On convexity of the frequency response of a stable polynomial
aa r X i v : . [ m a t h . O C ] S e p On convexity of the frequency response ofa stable polynomial
Didier Henrion , October 25, 2018
Abstract
In the complex plane, the frequency response of a univariate polynomial is theset of values taken by the polynomial when evaluated along the imaginary axis. Thisis an algebraic curve partitioning the plane into several connected components. Inthis note it is shown that the component including the origin is exactly representableby a linear matrix inequality if and only if the polynomial is stable, in the sensethat all its roots have negative real parts.
Keywords
Polynomial, stability, convexity, linear matrix inequality, real algebraic geometry.
Let p ( s ) ∈ R [ s ] be a polynomial of the complex indeterminate s ∈ C . We say that p ( s ) isstable if all its roots lie in the open left half-plane. Define the frequency response P = { p ( jω ) : ω ∈ R } as the set of values taken by the polynomial when evaluated along the stability boundary,namely the imaginary axis. The frequency response plays a key role when deriving resultsof robust control theory such as Kharitonov’s theorem [2, 3, 5].In [9] it was observed that the frequency response of a stable polynomial features inter-esting convexity properties, see also [3, Chapter 18]. More specifically, given a polynomial LAAS-CNRS, University of Toulouse, France Faculty of Electrical Engineering, Czech Technical University in Prague, Czech Republic ( s ), an arc is defined as a subset of the frequency response for a given range of the inde-terminate, i.e. { p ( jω ) : ω ∈ Ω } with Ω a subset of the real line R . A proper arc is an arcthat does not pass through the origin and such that the net change in the argument of p ( jω ) does not exceed π as ω increases over Ω. The Arc Convexity Theorem of [9] statesthat all proper arcs of the value set of a stable polynomial are convex. Alternative proofscan be found in [11] and [8].The frequency response ∂ P is a curve that partitions the complex plane into severalconnected components. We denote by P the connected component including the origin.It is called the inner frequency response set in [9]. In the case of a stable polynomial, theboundary of P therefore consists of a finite union of proper arcs. Theorem 4.1 in [9] usesthe Arc Convexity Theorem to establish convexity of P .In this note we provide a more accurate description of the geometry of P and an alternativeproof of its convexity. We derive an explicit representation of this set as a two-dimensionallinear matrix inequality (LMI). Instrumental to this derivation are standard results fromreal algebraic geometry [1, 4, 7, 14] and a recent characterization of two-dimensionalconvex polynomial level sets obtained in [10]. The frequency response of polynomial p ( s ) = p + p s + · · · + p n s n can be expressed asthe parametric curve ∂ P = { p ( jω ) = x ( ω ) + jy ( ω ) : ω ∈ R } (1)where x ( ω ) = q x ( ω ) q z ( ω ) y ( ω ) = q y ( ω ) q z ( ω ) (2)and q x ( ω ) = p − p ω + p ω + · · · q y ( ω ) = p ω − p ω + p ω + · · · q z ( ω ) = 1are polynomials of the real indeterminate ω ∈ R .Curve ∂ P is rationally (here polynomially) parametrized, so it is an algebraic plane curveof genus zero [1]. In control theory terminology, ∂ P is sometimes called the Mikhailov plotof polynomial p ( s ) or the Nyquist plot of the rational (here polynomial) transfer function p ( s ), see [3] or [5].Equations (1-2) provide a parametric description of curve ∂ P . With the help of elimina-tion theory and resultants, we can derive an implicit description ∂ P = { x + jy : f ( x, y ) = 0 } (3)2here f ( x, y ) is an irreducible bivariate polynomial, see [6, Section 3.3]. Lemma 1
Given two univariate polynomials g ( ω ) = g + g ω + · · · + g n ω n ∈ R [ ω ] and h ( ω ) = h + h ω + · · · + h n ω n ∈ R [ ω ] , there exists a unique (up to sign) irreducible polyno-mial b ( g, h ) ∈ R [ g , g , . . . , g n , h , h , . . . , h n ] called the resultant which vanishes whenever g ( ω ) and h ( ω ) have a common zero. To address the implicitization problem, we make use of a particular resultant, the B´ezoutian,see e.g. [14, Theorem 4.1] or [7, Section 5.1.2]. Given two univariate polynomials g ( ω ) , h ( ω ) of the same degree n as in Lemma 1, build the following bivariate polyno-mial g ( ω ) h ( v ) − g ( v ) h ( ω ) ω − v = n − X k =0 n − X l =0 b kl ω k v l called the B´ezoutian of g and h , and the corresponding symmetric matrix B ( g, h ) of size n × n with entries b kl bilinear in coefficients of g and h . As shown e.g. in [14, Theorem4.3] or [7, Section 5.1.2], the determinant of the B´ezoutian matrix is the resultant: Lemma 2 det B ( g, h ) = b ( g, h ) . Now we can use the B´ezoutian to derive the implicit equation (3) of curve ∂ P from theexplicit equations (1-2). Lemma 3
Given polynomials q x , q y , q z of equations (1-2), the polynomial of equation (3)is given by f ( x, y ) = det F ( x, y ) where F ( x, y ) = ± ( B ( q x , q y ) − xB ( q y , q z ) − yB ( q x , q z )) (4) is a symmetric pencil, i.e. a polynomial matrix linear in x, y , of size n . Proof:
Rewrite the system of equations (2) as g ( ω ) = q x ( ω ) − xq z ( ω ) = 0 h ( ω ) = q y ( ω ) − yq z ( ω ) = 0and use the B´ezoutian resultant of Lemma 2 to eliminate indeterminate ω and obtainconditions for a point ( x, y ) to belong to the curve. The B´ezoutian matrix is B ( g, h ) = B ( q x − xq z , q y − yq z ) = B ( q x , q z ) − xB ( q y , q z ) − yB ( q x , q z ). Linearity in x, y follows frombilinearity of the B´ezoutian. Finally, note that the sign of F ( x, y ) = ± B ( g, h ) affects thesign of f , but not the implicit description f ( x, y ) = 0. (cid:3) Lemma 3 provides the implicit equation of curve ∂ P in symmetric linear determinantalform. 3 Convexity properties of the inner frequency re-sponse set
Curve ∂ P partitions the complex plane into several connected regions. We are interestedin the connected region containing the origin, denoted by P . In order to study thegeometry of this region, we need the following result. Lemma 4
The sign of pencil F ( x, y ) in (4) can be chosen such that F (0 ,
0) = B ( q y , q z ) is positive definite if and only if p is a stable polynomial. Proof:
The signature of the B´ezoutian matrix B ( q x , q y ), (the number of positive eigen-values minus the number of negative eigenvalues) is equal to the Cauchy index of therational function q x ( ω ) /q y ( ω ) (the number of jumps from −∞ to + ∞ minus the numberof jumps from + ∞ to −∞ ), see [4, Section 9.1.2]. The Cauchy index is maximum (resp.minimum) when B ( q x , q z ) is positive (resp. negative) definite. This occurs if and only ifpolynomials q x and q y satisfy the root interlacing condition, i.e. they must have only realroots and between two roots of q x ( ω ) there is only one root of q y ( ω ) and vice-versa. Since q x ( ω ) = Re p ( jω ) and q y ( ω ) = Im p ( jω ), this is equivalent to stability of p in virtue of theHermite-Biehler theorem, see [2, Section 8.1] or [5, Section 1.3]. (cid:3) The main result of this note can now be stated.
Theorem 1
The connected component including the origin and delimited by the frequencyresponse of polynomial p can be described by a linear matrix inequality (LMI) P = { x + jy : F ( x, y ) (cid:23) } if and only if p is stable. In the above description, F ( x, y ) is given by (4) and (cid:23) meanspositive semidefinite. Proof:
First we prove that p stable implies LMI representability of P . This set is theclosure of the connected component of the polynomial level set { x, y : f ( x, y ) > } thatcontains the origin, an algebraic interior in the terminology of [10]. Polynomial f ( x, y ) iscalled the defining polynomial. By continuity, the boundary of P consists of those points x + jy for which F ( x, y ) drops rank while staying positive semidefinite. Note that ingeneral this boundary is only a subset of the curve ∂ P .To prove the converse, namely that LMI representability of P implies stability of p , we usea result of [10] stating that a two-dimensional algebraic interior containing the origin hasan LMI representation if and only if it is rigidly convex. Geometrically, this means thata generic line passing through the origin must intersect the algebraic curve f ( x, y ) = 0a number of times equal to the degree of f ( x, y ). Rigid convexity implies that x ( ω )and y ( ω ), the respective real and imaginary parts of polynomial p ( jω ), satisfy the rootinterlacing property, and this implies stability of p by the Hermite-Biehler Theorem usedalready in the proof of Lemma 4. (cid:3) y Figure 1: Frequency response of a stable third degree polynomial. The shaded region isthe convex component containing the origin.
Let p ( s ) = s + s + 4 s + 1. Then q x ( ω ) = 1 − ω and q y ( ω ) = 4 ω − ω in parametrization(2). With the help of the Control System Toolbox for Matlab, a visual representation ofcurve ∂ P can be obtained as follows: >> p = [1 1 4 1]; % polynomial in Matlab format>> nyquist(tf(p,1)) % frequency response>> axis([-5 2 -7 7]) % zoom around the origin see Figure 1. The B´ezoutian matrices of Lemma 3 can be computed with the followingMaple 10 instructions: > with(LinearAlgebra): qx:=1-w^2:qy:=4*w-w^3:qz:=1:> Bxy:=BezoutMatrix(qx,qy,w,method=symmetric);[-1 0 1][ ]Bxy := [ 0 -3 0][ ][ 1 0 -4]> Byz:=BezoutMatrix(qy,qz,w,method=symmetric);[ 0 0 1][ ]Byz := [ 0 1 0][ ][ 1 0 -4]> Bxz:=subs(e=0,BezoutMatrix(qx,qz+e*w^3,w,method=symmetric));[ 0 0 0][ ]Bxz := [ 0 0 -1][ ][ 0 -1 0] Note the use of the subs instruction to ensure that the last B´ezoutian matrix has appro-priate dimension 3. Matrix B ( q x , q z ) is negative definite, so a sign change is required tobuild F ( x, y ) = − x x − y − x − y − x and we obtain the following determinantal polynomial: > F:=-(Bxy-x*Byz-y*Bxz);> f:=Determinant(F); 2 2 3f := 9 - 3 x - 5 x - y - x describing algebraic plane curve ∂ P implicitly. The curve can be studied with the algcurves package of Maple: > with(algcurves):> genus(f,x,y); 0> plot_real_curve(f,x,y);
400 −300 −200 −100 0 100 200 300 400−150−100−50050100150 x y Figure 2: Frequency response of a stable eighth degree polynomial. The shaded region isrigidly convex.
A more complicated example is the eighth degree stable polynomial p ( s ) = 336 + 198 s +496 s + 117 s + 183 s + 20 s + 24 s + s + s whose frequency response is represented onFigure 2. The rigidly convex region around the origin has the LMI description −
20 0 117 0 − y −
66 0 298 y −
336 + x −
20 0 414 0 − y x − y −
66 0 1150 y − x − y − x
117 0 − y x − y − − x y y − x − y − x y − x − y x − y − − x y x − yy −
336 + x − y − x y − x − y − x (cid:23) .3 Unstable fourth degree polynomial This example is taken from [11]. Let p ( s ) = s − s −
1. The implicit equation of ∂ P is f ( x, y ) = det y y y y x = − − x + y = 0 . The component P including the origin is convex, see Figure 3, but it is not rigidly convexsince a generic line passing through the origin cuts the quartic ∂ P only twice. Hencethis region does not admit an LMI representation, and by Theorem 1, polynomial p ( s ) isunstable.Note however that P can be represented as the projection of an LMI set: P = { x + jy : ∃ z : (cid:20) x zz (cid:21) (cid:23) , (cid:20) z yy (cid:21) (cid:23) } by introducing a lifting variable z , but such constructions are out of the scope of thisnote. Convexity of the connected component containing the origin and delimited by the fre-quency response of a stable polynomial was already established in [9]. In this note wegive an alternative proof of this result based on B´ezoutians and we give a more accuratecharacterization of the geometry of this region. Namely, the region is rigidly convex inthe sense of [10], a property which is stronger than convexity, and which is equivalent tothe existence of an LMI representation of the set.In the terminology of convex analysis, the polynomial f ( x, y ) defining implicitly the fre-quency response in (3) is hyperbolic with respect to the origin. Equivalently, f ( x, y ) canbe expressed as the determinant of a symmetric pencil which is positive definite at theorigin. See [13] for a tutorial on hyperbolic polynomials and [12] for connections with theresults of [10]. This note therefore unveils a link between polynomial hyperbolicity andstability. Acknowledgments
This work benefited from discussions with Bernard Mourrain.8 y Figure 3: Frequency response of an unstable fourth degree polynomial. The shaded regionis convex, but not rigidly convex. 9 eferenceseferences