Application of a Generalized Budan-Fourier Algorithm to Stability Analysis of Time-Delay Systems

Transcription

1 Application of a Generalized Budan-Fourier Algorithm to Stability Analysis of Time-Delay Systems Bernard Hanzon 1 Joint Work With Oliver Mason 2 1 Edgeworth Centre for Financial Mathematics, Department of Mathematics, University College Cork, Ireland 2 Hamilton Institute, NUI Maynooth, Ireland 10 July / 32

2 Introduction Introduction Differential-delay systems have many applications in various fields including financial mathematics, control etc. Here we consider classical LTI differential-delay systems and investigate how to determine whether the system is stable, and if not, where the unstable poles are located (in the complex plane). 1 / 32

3 Introduction Introduction This problem has been around for a long time. We will follow an exposition of [Luc Habets 1992] of how one could approach this problem. However we combine the method suggested there with a relatively recent Generalized Budan Fourier (GBF) algorithm, which allows one to determine the zeros of any real Exponential-Polynomial-Trigonometric (EPT) function on any finite interval of the real axis with arbitrary precision under the assumption that we can determine the (-,0 or +) of an EPT function in any given real point. The GBF algorithm will also allow us to isolate all the (finitely many) unstable poles with arbitrary precision, in the complex plane, if there are any such poles. 2 / 32

4 Stability Analysis for Linear Differential-Delay Systems Stability Analysis for Linear Differential-Delay Systems Consider a linear differential-delay system with state equation: ẋ(t) = Fx(t) + k j=1 G jx(t τ j ) + Lu(t), x(t) R n, τ j > 0, j = 1, 2,..., k This system is stable if the poles of the transfer function all lie in the open left half plane (so all have negative real part). The poles are the zeros of the denominator of the transfer function, which is given by (assuming minimality of the state space realization) f (z) = det(zi F k j=1 G je zτ j ) This function is also the characteristic function of the relevant differential delay operator. Consider the homogeneous polynomial q(u, v, w 1,..., w k ) = det(ui vf k j=1 G jw j ) Then f (z) = q(z, 1, e zτ 1,..., e zτ k ) 3 / 32

5 Stability Analysis for Linear Differential-Delay Systems Stability Analysis for Linear Differential-Delay Systems We can also write this in the form f (z) = z n + n 1 h=0 p h(e τ1z,..., e τkz ).z h, z C, where the p h are polynomials in k variables. Habets[1992] shows that f (z) has only a finite number of zeros in the closed right half plane. Stability holds if this number is actually zero. As f (z) is an entire function (analytic on the whole complex plane) we can count the number of zeros inside any Jordan curve in the complex plane by counting the number of windings around the origin that f (z) makes when z traverses (once) along the Jordan curve in counterclockwise direction. 4 / 32

6 Stability Analysis for Linear Differential-Delay Systems Stability Analysis for Linear Differential-Delay Systems By taking a Jordan curve consisting of an interval on the imaginary axis and a large semi-circle in the right half plane, one can reduce the problem to counting the total increase in the argument of f (z) along a sufficiently large interval on the imaginary axis, as the contribution to the increase in the argument of moving along the semi-circle is known to be nπ if the semi-circle is sufficiently big. This leads to Theorem 2.7 of [Habets 1992] which says that the number of right half plane zeros is n 2 1 π.totarg(f, [0, ik]) where totarg(f, [0, ik]) denotes the net increase in argument of f (z) when z traverses along the imaginary axis from the origin to the number ik, for some sufficiently large K. 5 / 32

7 Stability Analysis for Linear Differential-Delay Systems Stability Analysis for Linear Differential-Delay Systems Finally the net increase in argument can actually be found by counting the number of crossings by f (z) of the real and of the imaginary axis, when z runs from 0 to ik. This in turn boils down to locating the zeros of Re[f (z)] and Im[f (z)], and determining the sign of the other at any location where one of these is zero. See the picture from [Habets,1992] 6 / 32

8 EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computing Science Memorandum CaSaR A Reliable Stability Test for Exponential Polynomials L.C.G.J.M. Habets Eindhoven, NoveJIlber 1992 The Netherlands

9 1m - Qll.is ""'!I""' Re ~ Qxis Figure 1: Jordan-curve JR Definition 2.3 Let R E R+. Then the half circle GR is defined as GR := {z EC Ilzl = RARe (z) ~ O}, the part IR of the imaginary axis as IR := {z Eel Re(z) = 0 A Izl < R}, and the Jordan-curve JR as JR:= GRUIR. This Jordan-curve is traversed in counter clockwise direction as depicted in Figure 1. Now suppose that the exponential polynomial f has no zeros on the imaginary axis. Let NJ denote the number of zeros of f in the open RHP. When R becomes large enough, JR will enclose all the NJ zeros of f. Hence NJ = lim _1_i f'(z) dz. R-+oo 21r~.7R f(z) (5) The computation of the integral (5) can be split into two parts: the integral over the half circle GR, and over the imaginary axis IR. The first term can be determined analytically, but before we can state this result we need some preliminary lemmas. Lemma 2.4 Let f(z) be an exponential polynomial of the form n-l f(z) = zn + ~ Pi(e-1"I Z,...,e-1""z)zi. i=o (6) Define A() d ( ( -1"IZ -1""Z)) Z := dz pn-l e,...,e. (7) Then for large values of Izl, such that Re (z) ~ 0 we have f'(z) = ~+ A(z) +O(~). f(z) z Z z2 (8) 3

10 1m -axis "Va. 'II. - '/a. 'r '/l... Ill. 'Iz..1/'1..'/ Figure 2: Counting the number of encirclements The only problem left is to track the curve r of I(f.W) for w E [0, K max ] in an accurate way that ascertains that all the intersections with the real axis (when n is odd) or the imaginary axis (when n is even) are detected. However, in an algorithm only a finite number of points on the curve r can be calculated. Therefore the tracking problem can be reformulated as the question to find a method for the selection of a finite number of points on the curve r in such a way that all the intersections of r with the real and imaginary axis can be detected from this finite set of points. For this purpose, linear search in the parameter space with constant step length I is until now most commonly used. Unfortunately, this method is not always reliable as will be illustrated by an example in the next section. Therefore we propose another, more reliable method to overcome this problem. The main idea is to make the step length variable depending on the curvature of the curve r. For a curve r = {(u(w),v(w)) IwE [O,K]} in a two-dimensional plane, parametrized by the variable w, the curvature in a point x(wo) = (u(wo), v(wo)) at w = wo, is given by u(wo)v(wo) - ij(wo)u(wo) (20) (u2(wo) + ij2(wo))3/2 (See for example [8, pp ]; the formula can be found literally in [2, p. 590]). The curvature in a point x(wo) on r is a measure for the rate of change of the tangent in x(wo), when proceeding along the curve. For example, a circle with radius R has in every point a curvature of *. It is easily seen that in a small neighborhood of the point x on the curve r, with curvature kx, the curve r itself behaves like a circle with radius Ik~ I. This implies that to track the curve accurately we have to take small steps along the curve when the curvature is large in absolute value, and we can take somewhat larger steps when the absolute value of the curvature is small. In this way we obtain the following rule: IcurvatureI X step length along the curve = constant (21) Note that the curve {(u(w), v(w)) I w E [0, K]} is parametrized by wand not by its length along the curve. The length of the curve between Wo and Wo + ~w is given by l wo +oo.w vu2(w) +ij2(w)dw. wo (22) For small values of ~w, this integral is approximately vu2(wo) + ij2(wo)~w. Substitution of this formula and (20) in (21) yields lu(wo)v(wo) - ij(wo)u(wo) I. ~ = C u2(wo) + ij2(wo) W. 8

11 Realization of Exponential-Polynomial-Trigonometric functions Realization of Exponential-Polynomial-Trigonometric functions So what s new? Firstly note that f (z) is an exponential-polynomial-trigonometric (EPT) function. Therefore it can be realized as the impulse response function of a linear system! So there exists a triple (A, b, c), A : N N, b : N 1, c : 1 N such that f (z) = ce Az b. Actually it is not too hard to show that an upper bound of the McMillan degree (i.e N in a minimal realization) is given by ( ) m + k + 1. k / 32

12 Realization of Exponential-Polynomial-Trigonometric functions Realization of Exponential-Polynomial-Trigonometric functions The real as well as the imaginary part of f (z) on the imaginary axis are again EPT functions and we can show that their McMillan degree is bounded by ( ) m + k (m + 1). k + 1 A realization for f (z) can be found in various ways. For instance one can use well-known realization formulas for the sum and the product of realizations of EPT functions (see e.g. [H.-Ober 2002] and references given there) and the well-known realizations of polynomials and scalar exponential functions. 8 / 32

13 It follows that the problem can be solved once we have a method to find the zeros of any given real EPT function on a finite real interval. This can be done using the GBF algorithm [H.-Holland 2009;2010] which was first used in the context of non-negativity of impulse-response functions of Finite Dimensional LTI systems and applications in financial matematics (non-negativity of Nelson-Siegel forward rate curves, and one-sided and two-sided EPT probability densities). As this method has not been used before in the context of locating roots in the complex plane we will give an overview of it here. 9 / 32

14 Consider the class of EPT functions k y(t) = Re p j (t)e µ j t, j=0 where p j (t) C[t] polynomials with complex coefficients, j = 0, 1, 2,..., k, µ j C, j = 0, 1, 2,..., k. This class contains the polynomials, the exponential functions and the scaled trigonometric polynomials (such as sin(νt), cos(νt), ν R but not tan(νt),..), and products and sums of such functions (the set of functions is a ring!). 10 / 32

15 A (real) EPT function can be written as y(t) = ce At b, (A, b, c) minimal and real. Distinguish three cases: (1) y(t) : polynomial iff σ(a) = {0}; then A nilpotent, A n = 0, e At = I + At A2 t (n 1)! An 1 t n 1 (2) If σ(a) R, real, then by bringing A in Jordan canonical form, one can show y(t) = ce At b = k h=0 p h(t)e λht, p h (t) R[t], real polynomial, λ 0 = 0 and λ h R, real eigenvalue of A, h=1,2,...,k. These will be called real exponential polynomials ( EP class). If deg(p h ) = 0, h = 0, 1, 2,..., K, we will speak of the exponential sums class ( E ) 11 / 32

16 (3) If σ(a) C the function will be called a function of the exponential-polynomial-trigonometric EPT class. In this case: { k } ce At b = Re h=0 p h(t)e λ ht, p h (t) C[t], λ h C, h = 0, 1, 2,..., k. This class/these classes go under many different names: quasi-exponential matrix-exponential exponential-polynomial; in systems theory they were sometimes called Bohl functions: this is now fading out. 12 / 32

17 Now consider the problem of determining the sign-changing zeros of such an EPT function on a finite interval (a, u). sign-changing zeros : zeros where the sign of the function changes from + to or vice versa. (EPT functions are real analytic, hence the zeros are isolated). CAN WE CONSTRUCT AN ALGORITHM THAT PRODUCES ALL SIGN-CHANGING ZEROS OF AN EPT FUNCTION ON A GIVEN FINITE INTERVAL? 13 / 32

18 For polynomials: Yes! Consider polynomial of degree n 1. Use Budan-Fourier sequence of higher order derivatives: p (0) (t) = p(t) = c.e At.b, σ(a) = {0} p (1) (t) = c.ae At.b, p (2) (t) = c.a 2.e At.b,. p (n) (t) = c.a n.e At.b 0, (A n = 0) We know k = n is the smallest integer value for which p (k) 0 so p (n 1) is a non-zero constant and p (n 2) is either strictly increasing (if p (n 1) > 0) or strictly decreasing (if p (n 1) < 0). 14 / 32

19 Let a < u. If p (n 2) (a) and p (n 2) (u) have the same (non-zero) sign then p (n 2) (t) 0, t [a, u]. If p (n 2) (a) and p (k 2) (u) have opposite (non-zero) signs then there is exactly one sign-changing zero of p (n 2) (t) on (a, u)]. This zero can be calculated using bisection, with arbitrary precision! One way of viewing this is as follows: Definition An open interval (a, u) R will be called simple for the function f : R R, f continuous, if f has at most one sign-changing zero on (a, u). 15 / 32

20 Remark. Suppose f has simple interval (a, u). Then if lim Sign [f (a + ɛ)] = lim Sign [f (u + ɛ)] 0 (1) ɛ 0 ɛ 0 then there is no sign-changing zero of f in (a, u). If lim ɛ 0 Sign [f (a + ɛ)] lim ɛ 0 Sign [f (u + ɛ)] < 0, then bisection gives the unique sign-changing zero c (a, u) such that f (c) = 0. (Of course the case f 0 on (a, u) can be handled in a straightforward manner. ) 16 / 32

21 Definition A grid {a 0, a 1,..., a N }, a 0 = a, a N = u, a 0 < a 1 <... < a N, is called simple for f if each interval (a k 1, a k ), k = 1, 2,..., N is simple for f. Remark. Given a simple grid the sign-changing zeros of f on the interval (a, u) can be found all and with arbitrary precision, using a bisection algorithm. 17 / 32

22 Now note that h {1, 2,..., n} the zeros of p (h) (t), together with the boundary points a, u, form a simple grid for p (h 1) (t), where p (0) (t) p(t) is the original polynomial. Therefore the Budan-Fourier sequence gives a guaranteed method to find all the real zeros of the original polynomial p with arbitrary precision. 18 / 32

23 Simple grid properties and a Generalized Budan-Fourier sequence for EP functions Simple grids have some nice properties: (i) If {a = a 0, a 1,..., a N = u} is a simple grid for f, then also for f.g, where g(t) 0, t (a, u)\{a 0, a 1,..., a N } and g continuous on (a, u). (ii) If {a 0, a 1,..., a N } consists of the boundary points together with the sign-changing zeros of a function h, h continuous on (a, u); then the same grid is obtained if d is replaced by d.k for any continuous function d with d(t) 0, t (a, u). 19 / 32

24 We can now construct a GBF sequence for the EP class! Let g(t) = ce At b, σ(a) R, (A, b, c) minimal and real. Let λ 1, λ 2,..., λ n denote the (real) eigenvalues of A : n n. Let g (0) (t) := g(t) = c.e At.b, g (h) := c(a λ h.i )... (A λ 1.I ).e At.b, h = 1, 2,..., n 1, g (n) (t) := c(a λ n.i )... (A λ 1.I )e At b 0 (Cayley-Hamilton!) 20 / 32

25 Theorem This is a Generalized Budan-Fourier sequence, for any pair of boundary points {a, u}, a < u. 21 / 32

26 Proof Note that for every h {0, 1, 2,..., n 1}, g (h+1) is obtained from g (h) by applying the formula: g (h+1) = e λ h+1.t d [ ] e λh+1.t.g (h) (t) dt hence d [ ] e λh+1.t.g (h) (t) = e λh+1.t.g (h+1) (t) dt Using (i),(ii) and the fact that e λ h+1.t > 0, t R, it follows that the sign-changing zeros of g h+1) (t) (which are also the sign-changing zeros of e λ h+1.t g (h+1) (t)), together with the boundary points {a, u}, form a simple grid for e λ h+1.t g (h) (t), hence for g (h) (t). 22 / 32

27 As g (n) 0 because (A λ n I )(A λ n 1 I )... (A λ 1 I ) = 0, it follows that g (n 1) has constant sign (no zeros). The theorem now follows by induction. QED 23 / 32

28 Now consider a general EPT function g(t) = c.e At.b, where σ(a) can contain complex elements, but g(t) R, t R. Assuming (A, b, c) minimal, it follows that if λ σ(a) with multiplicity m(λ), then λ σ(a), with the same multiplicity m(ˆλ) = m(λ). Let us order the eigenvalues λ 1, λ 2,..., λ N such that the complex conjugates come in pairs λ h = θ h + iν h, λ h+1 = θ h iν h, with ν h > / 32

29 Theorem A GBF sequence with boundary points {0, u}, for g(t) = ce At b, (A, b, c) minimal, real, is given by: g (0) (t) = g(t), with boundary points {0, u}; if λ h R : g (h) = c(a λ h I )(A λ h 1 I )... (A λ 1 I ).e At.b with boundary points {0, u}; if λ h = θ h + [ iν h, λ h+1 = θ h iν h, ν h > 0 then] g (h) := Im e λ h t c(a λ h I )... (A λ 1 I )e At b with extended set of boundary points {0, π ν h, 2 π ν h,..., u π/ν h π ν h, u}; and g (h+1) := c(a λ h+1 I )(A λ h I )... (A λ 1 I )e At b, with (same) extended set of boundary points {0, π ν h, 2 π ν h,..., u π/ν h π ν h, u} 25 / 32

30 Note that (A λ h+1 I )(A λ h I ) = (A λ h )(A λ h ) is a real matrix. Notation: x = max{n N; n x}. 26 / 32

31 Proof. Consider g (h 1) (t) = c(a λ h 1 I )... (A λ 1 I )e At b, and consider an open interval J l = (l π ν h, (l + 1) π ν h ), l {0,..., u φ/ν h }. On J l the function sin(ν h t) has no zero, hence does not change sign. On I l (0, u) a simple grid for g (h 1) can be obtained by calculating the ( sign-changing ) zeros of sin 2 (ν h t) d g (h 1) (t)e θ h t dt = sin(ν h t) 27 / 32

32 sin(ν h t).( d dt g (h 1) (t))e θht + sin(ν h t).g (h 1) (t)( θ h )e θht + g (h 1) (t).e θht ν h cos(ν h t) = Im [ e θht (cos(ν h t) + i sin(ν h t)).( d dt g (h 1) (t) + g (h 1) (t)( θ h iν h )) ] = Im [ e θ ht+iν h t ( d dt λ h)(g (h 1) (t)) ] [ = ] Im e λ h t c.(a λ h I )(A λ h 1 I )... e At.b = g (h) (t). 28 / 32

33 Now, perhaps surprisingly, e θ h t d sin(ν h t) dt [ g (h) (t) ] = d2 dt 2 g (h 1) (t) 2θ h d dt g (h 1) (t) + (θ 2 + ν 2 h )g (h 1) (t) = ( d dt λ h I )( d dt λ h)g (h 1) (t) = c(a λ h I )(A λ h I )(A λ h 1 I )... (A λ 1 I )e At.b =: g (h+1) (t), where we use λ h+1 = λ h. Therefore the sign-changing zeros of g (h+1) (t), together with the (extended set of) boundary points {0, π ν h, 2 π ν h,..., u π/ν h π ν h, u} form a simple grid for g (h) (t) and the sign-changing zeros and the sign-changing zeros of g (h) (t), together with the (extended set of) boundary points {0, π ν h, 2 π ν h,..., u π/ν h π ν h, u} form a simple grid for g (h 1) (t). The theorem now follows by induction (in combination with the proof of the previous theorem). QED 29 / 32

34 Final remarks and conclusions Final remarks and conclusions Applying the GBF bisection method to the real and imaginary parts of the characteristic function of a differential-delay system one can determine stability. This provides an alternative to existing algebraic methods. A further application is to isolate the right-half plane zeros of the characteristic function. This can be done by using the fact that the increase of the argument over any Jordan curve that takes the form of a rectangle with horizontal and vertical sides, can be obtained again by determining the zero crossings of the real and imaginary parts of f (z) over the sides of such a rectangle. That will reveal the number of zeros inside the rectangle. By systematically cutting the rectangle into smaller ones one can isolate the complex zeros with arbitrary precision (as there only finitely many). We are not aware of other methods that can do this. 30 / 32

35 Final remarks and conclusions Final remarks and conclusions We are developing software that can handle arbitrary matrices F, G j, j = 1, 2,..., k as inputs. Presently we are considering possible generalizations to more general differential-delay systems (such as neutral systems) based on results of [H-Holland IWOTA 2010] concerning positivity of the tail of an EPT function. 31 / 32

36 Thank You Thank you! 1 For references and links to software contact us at b.hanzon@ucc.ie 1 Research support for B.Hanzon by Science Foundation Ireland, grant nrs RFP2007-MATF802 and 07/MI/ / 32