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Finite Nyquist and Finite Inclusions Theorems for disjoint stability regions
Systems & Control Letters 58 (2009) 804–809
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Finite Nyquist and Finite Inclusions Theorems for disjoint stability regions Yuri Dolgin a,∗ , Ezra Zeheb a,b a
Department of Electrical Engineering Technion, Israel Institute of Technology, Israel
b
Department of Electronics Engineering, Jerusalem College of Engineering, Jerusalem, Israel
article
info
Article history: Received 28 March 2009 Received in revised form 30 July 2009 Accepted 21 September 2009 Available online 17 October 2009
abstract Finite Nyquist Theorem is an important tool in stability analysis and design of linear systems. Currently, Finite Nyquist Theorem can treat only simply connected convex stability regions with some extensions to simply connected non-convex regions. In this paper, we consider the generalization of Finite Nyquist Theorem for the case of union of disjoint convex stability regions. Based on this result, the Finite Inclusions Theorem is also formulated for a union of disjoint convex stability regions. © 2009 Elsevier B.V. All rights reserved.
Keywords: Stability Disjoint stability regions Finite Nyquist Theorem Finite Inclusions Theorem Uncertain systems
1. Introduction The Finite Nyquist Theorem and its generalization for uncertain systems, i.e. the Finite Inclusions Theorem, [1], are important tools in stability analysis and design of linear systems [2–6]. The above theorems allow us to consider only a finite subset of frequencies in order to establish stability. This subject remains a topic of continuous research: useful extensions to this method were devised for example in [7]. Recently, the question of stability, where stability domain is given as a set of disjoint convex regions, got much attention [8–10]. Below we state a few examples of control problems requiring stability analysis with disjoint stability regions: Problem 1. Model reduction of unstable LTI SISO system. When the unstable poles of the reduced system should be kept close to unstable poles of the original system, the stability region for the reduced system becomes the union of:
• Regular stability region (for example the unit circle) • Small regions around the unstable poles of the original system. Problem 2. Handling of systems with separate dynamics, for example the systems for which one part of the system should react
∗
Corresponding author. E-mail addresses: [email protected] (Y. Dolgin), [email protected] (E. Zeheb). 0167-6911/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2009.09.001
fast, while the other part should react slow. Separate dynamics naturally leads to stability region being a union of disjoint regions. Designing a controller for such a system, determining maximal stability radius and other problems can benefit from the proposed method. Problem 3. Non-fragility analysis of a given system. More precisely, given some nominal controller, find the maximal deviation from the nominal controller, so that the poles of the closed loop transfer function will remain in some predefined vicinity of the poles of the closed loop transfer function with nominal controller. In this case, again, the stability region is the union of small regions around the poles of the closed loop transfer function. The purpose of this paper is to generalize the results of Finite Nyquist Theorem, stated in [1] only for a simply connected convex region, to the case of union of disjoint convex stability regions. Namely, given a polynomial, our purpose is to find a finite number of checks to be performed on the phase of this polynomial in order to ensure that its zeros are located in given disjoint convex regions. Based on this result, the Finite Inclusions Theorem is also formulated for the union of disjoint convex stability regions. Note that, as shown in Section 3, the natural naive extension of the theorems in [1] to the case of disjoint regions does not hold. The structure of the paper is as follows. In Section 2 we provide the notations used in the paper. In Section 3 we state and prove the new Finite Nyquist Theorem for disjoint stability regions. Based on the results in Section 3, the Finite Inclusions Theorem is formulated in Section 4. In Section 5 we provide an example. We conclude in Section 6.
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2. Definitions and notations The following notations will be used throughout the paper. Given the monic polynomial p(s) = (s − zn )(s − zn−1 ) . . . (s − z1 ), its phase, arg(p(s)), is defined as arg(p(s)) = arg(s − zn ) + · · · + arg(s − z1 ) where arg(s − zi ) ∈ [0, 2π ], 1 ≤ i ≤ n. The arg(p(s)) is a continuous function of s unless s passes through some zero zi . Note, that arg(p(s)) can take values in [0, 2π n]. A given closed Jordan curve Ψ ⊂ C separates the complex plane into two regions. By reg Ψ we denote one of these regions, namely, either interior or exterior (note, that Ψ is not included in either of them). Given closed Jordan curve Γ ⊂ C, its convex interior int Γ , if exists, is defined as the points lying to the left of Γ when traversing the Γ in counterclockwise direction as seen from all points in int Γ . Let us consider a closed Jordan curve Γ ⊂ C with convex int Γ such that Γ is fully contained in a given region reg Ψ (in other words Γ ∩ Ψ = ∅). Consider also two points on Γ , say s1 and s2 . Consider all possible placements of m zeros of polynomial p(s) outside reg Ψ . These placements result in different phase changes of p(s) observed when going from s1 to s2 in counterclockwise direction along Γ . We denote the maximal phase change of p(s), which can be obtained from m zeros located outside reg Ψ , when going from s1 to s2 along Γ , by ∆reg Ψ ,Γ ,m (s1 , s2 ). The notion ‘‘stability region’’ is used to denote a region in which the zeros of a polynomial should be located.
We see that the ‘‘x’’ marks (i.e. misinterpreted sampled phase) satisfy the requirements in (1), replacing n with x(1) or with x(2) , for x(1) = 1 and x(2) = 1 in both disks, but actually none of the zeros is inside these disks. Thus, the Theorem is erroneous in this case. Below we provide a correct non-trivial generalization of the Finite Nyquist Theorem for the case of disjoint convex stability regions.
3. Finite Nyquist Theorem for disjoint stability region
Theorem 1. Let p(s) =
Fig. 1. Missed phase change.
αj sj , where n ∈ Z, n > 0, and αj ∈ C. Let Γ ⊂ C, 1 ≤ i ≤ q be closed Jordan curves such that int Γ (i) are convex and Γ (i) ∩ Γ (j) = ∅ for i 6= j. Only one of the Γ (i) is allowed Pint q (i) to be unbounded. Let x(i) ∈ Z be given such that = n. i=1 x Then, p(s) has x(i) zeros in int Γ (i) for 1 ≤ i ≤ q if and only if for (i) every 1 ≤ i ≤ q there exist m(i) ≥ 1 angles θk ∈ R, a 0 ≤ δ (i) < π , (i) a counterclockwise sequence of points sk ∈ Γi , 1 ≤ k ≤ m(i) (i) and a region reg Ψ ⊂ C in the complex plane, such that for every 1 ≤ i ≤ q: Pn
j =0
(i)
The Finite Nyquist Theorem for simply connected single region [1] is cited below Pn for jconvenience. Let p(s) = j=0 αj s , where n ∈ Z, n ≥ 0, and αj ∈ C and let Γ ⊂ C be a closed Jordan curve such that int Γ is convex. Then, p(s) is of degree n and has all its zeros in int Γ if and only if there exist m ≥ 1 angles θk ∈ R and a counterclockwise sequence of points sk ∈ Γ , 1 ≤ k ≤ m, such that
∀1≤k
(1a)
|2π n + θ1 − θm | < π ∀1≤k≤m p(sk ) 6= 0
(1b)
∀1≤k≤m arg p(sk ) = θk + 2π vk ,
v k ∈ Z.
(i)
⊃ (int Γ
(i)
(3a)
(i)
∪Γ )
(1c)
reg Ψ
(1d)
∀1≤k≤m(i) ∆reg Ψ (i) ,Γ (i) ,n (s(ki+) 1 , s(ki) ) ≤ δ (i)
(3c)
i) ∀1≤k≤m(i) |θk(+ 1
(3d)
The simple paraphrase of the above theorem is: if we can find sufficiently dense sampling of points on the stability region boundary so that the phase change of p(s), traversing all the sampled points, will be 2π n, then all n zeros are inside the stability region. The naive extension of this theorem to the case of disjoint regions does not hold. Namely, the theorem cannot be used to guarantee exactly x(i) < n zeros in each of the disjoint regions by requiring (1) to hold (with (1b) replaced by 2π x(i) + θ1 − θm < π ). As a counter-example, consider the following polynomial: p(s) = s2 + (−0.6 − 2i)s + (−0.91 + 0.6i)
∀1≤j≤q, i6=j reg Ψ (i) ∩ reg Ψ (j) = ∅
(2)
which has both zeros at 0.3 + i. The stability region under consideration is the union of two disjoint disks: the unit disk and the disk centered at 0.75 + 1.45i with radius 0.6. Note, that the zeros of p(s) are located outside both regions. The stability region together with zero locations of p(s) is shown in left part of Fig. 1. The phase evaluated on the unit circle and on the second circle are shown in the right-top and right-bottom parts of this figure, respectively. The solid line shows the real phase, the rectangles indicate the (correctly) sampled phase, the ‘‘x’’ marks indicate the misinterpreted sampled phase (namely, missed jump of 2π when passing from a sample which is close to the two zeros, to its neighbor sample).
(3b)
(i)
− θk | < π − δ
(i)
∀1≤k≤m(i) p(s(ki) ) 6= 0
(3e)
∀1≤k≤m(i) arg p(s(ki) ) = θk(i) + 2π vk(i) , where
(i) (i) (i) sm+1 , s1 , θm+1 ,
vk(i) ∈ Z
(3f)
(i)
θ1 + 2π x(i) .
For illustration of the definitions used in the Theorem see Fig. 2. Note, that when there is only one stability region (q = 1), the above Theorem coincides with the original Finite Nyquist Theorem if we choose reg Ψ (1) = C and δ (1) = 0. Evidently, conditions (3a)– (3c) become vacuous and condition (3d) satisfies both conditions (1a) and (1b). (i) A simple method for choosing the sequences sk is briefly discussed in Appendix B. Proof. ‘‘If’’ part: First, we prove the ‘‘if’’ part. Let us focus on one of i values. Our purpose is to prove that when (3) holds, there are exactly x(i) zeros inside int Γ (i) . In the following proof we remove the . . .(i) suffix in order to avoid cumbersome notation, unless required for clarification. Namely, we denote Γ , Γ (i) , Ψ , Ψ (i) , x , x(i) .
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a
b
Fig. 2. Illustration of notions used in theorem. Solid lines are Γ (i) , dashed lines are (i ) Ψ (i) , dark gray regions are int Γ (i) , light gray regions are reg Ψ (i) , black dots are sk samples. (a) Shows several stability regions int Γ (i) with corresponding reg Ψ (i) . (b) Shows the enlarged int Γ (1) from (a), with corresponding sampling of Γ (1) .
To simplify notations, let us define a function γ (t ), t ∈ [0, 1] which parameterizes Γ in counterclockwise direction. As a first step, our purpose is to show that there should be at least x zeros inside reg Ψ in order to satisfy (3). If there are at least x zeros inside int Γ , the statement trivially holds. Assume, in contradiction, that there are only x − r , r ∈ Z, r > 0 zeros inside int Γ . Then, arg (p(γ (1))) − arg (p(γ (0))) = 2π (x − r )
(4)
which can be written in the form m X
(arg (p(sk+1 )) − arg (p(sk ))) = 2π (x − r ).
(5)
k =1
However, the sampled phase change is, by definition,
θm+1 − θ1 = 2π x
(6)
which can be written in the form m X
(θk+1 − θk ) = 2π x.
(7)
k =1
Combining (5) and (7): m X
(arg(p(sk+1 )) − arg(p(sk ))) −
k =1
m X
(θk+1 − θk ) = −2π r .
(8)
k=1
According to (3f), for each k
(arg(p(sk+1 )) − arg(p(sk ))) − (θk+1 − θk ) = 2π w,
w ∈ Z. (9)
In other words, real phase difference and sampled phase difference can differ only by multiples of 2π . Thus, it follows from (8) and (9) that there exist not more than r indices h1 , . . . , hy , y ≤ r so that y X
arg(p(shk +1 )) − arg(p(shk ))
In other words, the cumulative phase change of argp(γ (t )) in intervals [shg , shg +1 ], 1 ≤ g ≤ y along Γ is negative and is greater in magnitude than r (π + δ). Negative phase changes of argp(γ (t )) can stem only from zeros located outside int Γ . Due to convexity of int Γ , the negative phase change which stem from a zero outside int Γ is less than or equal to π . The phase change between consecutive samples shg and shg +1 from all zeros outside reg Ψ is less than or equal to δ (by condition (3c)). Thus, when r > 0, we must have more than r zeros inside reg Ψ and outside int Γ to satisfy (11). We have thus shown that either there are at least x zeros inside int Γ , or there are x − r, r > 0, zeros inside int Γ and more than r zeros outside int Γ but inside reg Ψ . In both cases this means that there are at least x zeros inside reg Ψ . Given the fact that the same conclusion is valid for all disjoint stability regions (i.e. that there are at leastP x(i) zeros inside reg Ψ (i) for all 1 ≤ i ≤ q), and from the q constraint i=1 x(i) = n, it now follows that there are exactly x(i)
zeros inside each reg Ψ (i) . Now we want to show that all the x(i) zeros inside reg Ψ (i) are actually inside int Γ (i) . As we have already shown, there are either x(i) zeros inside int Γ (i) or x(i) − r (i) zeros, r (i) > 0, zeros inside int Γ (i) and more than r (i) zeros outside int Γ (i) but inside reg Ψ (i) . The assumption r (i) > 0 contradicts the fact that there are exactly x(i) zeros inside int Ψ (i) . Hence, we conclude that all x(i) zeros are inside int Γ (i) . This concludes the proof of the ‘‘if’’ part, i.e. when constraints (3) hold, each region int Γ (i) contains exactly x(i) zeros, as required. ‘‘Only if’’ part: Assume that p(s) has x(i) zeros inside int Γ (i) for 1 ≤ i ≤ q. It particularly means that there are no zeros on Γ (i) and thus (3e) holds. Then, the phase of p(s) is continuous when going along Γ (i) . According to the argument principle, the phase change when going along Γ (i) is 2π x(i) , thus it is always possible to find a sampling of points on Γ (i) satisfying (3d), (3f) with the property that any denser sampling will also satisfy (3d), (3f). It is rather intuitive fact that a sufficiently dense sampling of points on Γ (i) will satisfy (3c) for a reg Ψ (i) satisfying (3a), (3b). For the sake of completeness we provide the rigorous proof of this statement in Appendix. Since Γ (i) ∩ Γ (j) = ∅ for i 6= j, there always exist reg Ψ (i) satisfying (3a), (3b). In order to obtain the finite number of samples, our only requirement is that for the only unbounded int Γ (i) , say int Γ (i0 ) , the exterior of Ψ (i0 ) should be the region which includes Γ (i0 ) , while the interior of Ψ (i0 ) should be bounded. This is always possible when there is only one unbounded int Γ (i) . Hence, we have shown that if p(s) has x(i) zeros inside int Γ (i) for 1 ≤ i ≤ q, then all conditions (3) are satisfied. This concludes the proof of the ‘‘only if’’ part, and thus concludes the proof of the theorem. 4. Finite Inclusions Theorem
k =1
−
y X
θhk +1 − θhk ≤ −2π r .
(10)
k=1
Equivalently, taking into account (3d) y
X
k =1
X
θhk +1 − θhk − 2π r
k=1
< y(π − δ) − 2π r ≤ r (π − δ) − 2π r = −r (π + δ)
(11a)
or y X k =1
arg(p(shk +1 )) − arg(p(shk )) < −r (π + δ).
j Theorem 2. Let p(s, q) = j=0 αj (q)s , q ∈ Q where Q is an arbitrary set (of possibly complex parameters), n ∈ Z, n > 0, and αj : Q → C. Let Γ (i) ⊂ C, 1 ≤ i ≤ q be closed Jordan curves such that int Γ (i) are convex and Γ (i) ∩ Γ (j) = ∅ for i 6= j. Only one of the int Γ (i) is allowed to be unbounded. Let x(i) ∈ Z be given such that P q (i) = n. i=1 x Then, every member of the polynomial family p(s, q) has x(i) zeros in int Γ (i) for 1 ≤ i ≤ q if for every 1 ≤ i ≤ q there exist m(i) ≥ 1 (i) (i) intervals (ak , bk ) ⊂ R, a 0 ≤ δ (i) < π , a counterclockwise sequence
Pn
y
arg(p(shk +1 )) − arg(p(shk )) ≤
In this section we provide the Finite Inclusions Theorem for a union of disjoint convex stability regions, based on the results of the previous section.
(11b)
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807
(i)
of points sk ∈ Γi , 1 ≤ k ≤ m and a region reg Ψ (i) ⊂ C in the complex plane, such that for every 1 ≤ i ≤ q:
∀1≤j≤q, i6=j reg Ψ (i) ∩ reg Ψ (j) = ∅ reg Ψ
(i)
⊃ (int Γ
(i)
(12a)
(i)
∪Γ )
(12b)
∀1≤k≤m(i) ∆reg Ψ (i) ,Γ (i) ,n (s(ki+) 1 , s(ki) )
≤δ
(i)
(12c)
∀1≤k≤m(i) max{bk(i+) 1 − a(ki) , b(ki) − a(ki+) 1 } < π − δ (i) (i)
(i)
(i) jθ (i)
∀1≤k≤m(i) p(sk , q) ⊂ Sk = R e
|R
(i)
> 0,
θ
(12d) (i)
(i)
(i)
∈ (ak , bk ) (12e)
(i)
(i)
(i)
(i)
where am+1 , a1 + 2π x(i) , bm+1 , b1 + 2π x(i) . (i)
In other words, if, for sufficiently dense sampling sk (i.e. satisfying (i)
(12c)), we can find a sequence of sectors Sk ‘‘spaced’’ less than π − δ (i) radians apart and revolving a net 2π x(i) radians about the (i) origin, and if p(sk , q) is contained in the above sectors, then every member of the polynomial family p(s, q) has x(i) zeros in int Γ (i) for 1 ≤ i ≤ q. The proof of this theorem will be omitted since it follows the same lines as in [1], taking into account the results of (i) the previous section. These results ensure that if Sk are ‘‘spaced’’ (i) less than π − δ radians apart, then, for each member of p(s, q), (3d), (3f) will hold, and since R(i) > 0 the condition (3e) will also hold. (i) A simple method for choosing the sequences sk and the sectors (i)
Sk is briefly discussed in Appendix B. 5. Example We consider a practical example requiring the Finite Inclusions Theorem for disjoint regions. The poles of the nominal plant for this example are taken from [9], where the lateral dynamics of an aircraft was considered:
λi = {−0.5, −2 ± 2i, −3 ± 2i} ⇒ p(s) = s5 + 10.5s4 + 50s3 + 122.5s2 + 154s + 52.
(13)
Our aim is to analyze the robustness of the pole locations in the presence of an uncertainty. We consider an interval polynomial of a predefined form (with parameterizable gain k) around the nominal polynomial (13): p(s, q) =
α5 α4 α3 α2 α2 α0
5 X
Fig. 3. Example 1: The stability regions (gray), the stability boundary sampling sk (‘‘x’’), the Ψ (i) (dashed) and final uncertain polynomial roots (blue [or dark gray, when printed in grayscale]).
samples is good enough. At each sample sk we form 2 constraints defining together a sector constraint — we require the uncertain polynomial to lie in sectors defined by the original nominal polynomial (and the constraint (12d)). Additionally, since we work with uncertain polynomial, for each sector we require all 12 vertices of the uncertain polynomial to reside in this sector. Finally, we end up with 3840 linear constraints on k, ensuring that the roots of (14) will lie in the desired disjoint regions. Then, the optimization problem is solved to find maximal k satisfying the above constraints — the final k is 0.234. In the Fig. 3 we show the stability regions as gray disks, the dashed lines are Ψ (i) , the ‘‘x’’ marks are the sampling points on the stability boundary and the blue (or dark gray, when printed in grayscale) regions are the roots of the final uncertain polynomial (e.g. with k = 0.234). We can see that the roots of final uncertain polynomial indeed come close to the stability boundary, thus the constraints are not very conservative. The legitimate question now is whether the conditions of the classical Finite Nyquist/Inclusions Theorem are not sufficient here. Consider the following polynomial p(s) = z 5 + 10.66z 4 + 52.1724z 3 + 133.2323z 2
αj (q)s , j
+ 175.156z + 72.8291
where
j=0
∈ [1 · 0.95k, 1 · 1.05k], ∈ [10.5 · 0.98k, 10.5 · 1.02k] ∈ [50 · 0.993k, 50 · 1.007k], ∈ [122.5 · 0.998k, 122.5 · 1.002k] ∈ [154 · 0.999k, 154 · 1.001k], ∈ [52 · 0.99k, 52 · 1.01k].
(15)
which has the roots at (14)
We wish to find the maximum gain k so that the roots of the resulting uncertain polynomial (14) will lie inside the disks of radius 0.4 around the roots of the nominal polynomial (13). First of all, since the stability regions are symmetric with respect to the real axis and since the considered polynomial is real, we can consider only 3 disjoint regions instead of 5, namely the disks centered at −3 + 2i, −2 + 2i and −0.5 (the other two regions will automatically satisfy the conditions of Theorem 2). We consider a total of 160 samples sk on the stability boundaries. To satisfy the conditions of the Theorem 2 it is enough to take much less samples, but then the constraints are too conservative. Obviously, more than 160 samples will result in less conservative constraints, but as we show below, even such moderate number of
{−0.7, −2.48 ± 2i, −2.5 ± 2i}
(16)
i.e. all roots except the first are far outside the desired stability region. As we show in Fig. 4, there exists a sampling of the stability boundary, which satisfies the conditions of classical Finite Nyquist Theorem for each region (namely, that in each region there is one root), but obviously, the conclusion is incorrect. Thus, to obtain correct stability constraints, it is not enough to use classical Finite Nyquist/Inclusions Theorem — the Theorems 1 and 2 should be applied instead. 6. Conclusion In this paper we have stated and proved the Finite Nyquist Theorem for a union of disjoint convex stability regions. We have shown that a finite number of phase checks is sufficient to guarantee that all roots of a polynomial lie inside the stability region, even when the stability region is a union of convex regions. Based on this theorem, we also formulated the Finite Inclusions Theorem for a union of disjoint convex stability regions.
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Fig. 4. Example 1: Example of erroneous conclusions by classical Finite Nyquist Theorem. The phase using ideal (continuous) sampling is shown by solid line and the misinterpreted phase using sparse sampling is shown by ‘‘x’’ marks.
Appendix A In this Appendix we augment the proof of the ‘‘Only If’’ part of Theorem 1. Namely, we show that a sufficiently dense sampling of points on Γ (i) will satisfy (3c) for a reg Ψ (i) satisfying (3a), (3b). In the following proof we again omit the . . .(i) suffix in order to avoid cumbersome notation. We consider first the bounded int Γ residing inside reg Ψ and we show that there exists finite sampling of this Γ , which satisfies (3c). To show this, we shortly sketch the construction procedure for such sampling. Start from some initial point A ∈ Γ . Our purpose is to show that we can find a point B ∈ Γ next to A in counterclockwise direction along Γ , so that the maximal angle seen between A and B from any point outside reg Ψ will be less than any predefined number . For our purposes we choose = nδ . The construction procedure is as follows. Find the maximal radius, say rA,0 , rA,0 > 0, of the circle centered at A and fully contained in reg Ψ . rA,0 is not zero since A must be inside (and not on the boundary of) reg Ψ . Consider another circle of radius rA,1 < rA,0 centered at A. Take a point B ∈ Γ on the latter circle of radius rA,1 , in counterclockwise direction from A. Consider triangle with vertices A, B and any point outside reg Ψ , say C . Denote the angle at vertex C (i.e. 6 ACB) by 6 C . By law of sines,
6 sin 6 C sin 6 A r A ,1 = ⇒ | sin 6 C | = AB sin A ≤ AB BC BC r −r . A,0 A ,1 (17) Thus, we can get C as small as desired by decreasing rA,1 . We need to show only that the constructed sampling is finite, namely, that rA,1 is bounded from below by some positive number. Since Γ ⊂ reg Ψ , the distance between closest points of Γ and Ψ is positive, say R0 > 0. Considering also (17): 6
| sin()| inf rA,0 = R0 > 0 ⇒ inf rA,1 ≥ R0 >0 A∈Γ A∈Γ 1 + | sin()| thus the number of samples on Γ is finite. Thus, we proved that a sufficiently dense sampling of Γ will satisfy (3c). Consider now the unbounded int Γ (i0 ) residing inside exterior of Ψ (i0 ) , where Ψ (i0 ) is with bounded interior. Consider any point
C inside interior of Ψ (i0 ) . Denote the maximal distance from C to a point on Ψ (i0 ) by R1 . Consider a circle CC , with center at C and 2R1 with radius R2 = sin(0.25 . Then, by law of sines, for any point A ) outside CC , the angle seen from C to A differs from the angle seen from any other point inside interior of Ψ (i0 ) to A by less than 0.25 . Now, we consider two parts of Γ (i0 ) : the part inside CC and the part outside CC . The sampling inside CC is constructed exactly in the same way as for bounded Γ (i) - since this part of Γ (i0 ) is finite, the number of samples is also finite. To construct the sampling outside CC we draw (finite number of) lines from C with angles 0, 0.5, , 1.5, . . . , 2π . The intersection of these lines with the part of Γ (i0 ) outside CC gives the desired points — the angle seen between neighbor points from any point outside reg Ψ (i0 ) is less than (note that the exterior of Ψ (i0 ) now plays the role of reg Ψ (i0 ) ). Appendix B In this appendix we briefly discuss one of the methods to build (i) (i) the sk sequences (and the Sk sectors) mentioned in Theorems 1 and 2. Here we again omit the . . .(i) suffix to avoid cumbersome notations. As in the proof of Theorem 1 we define a function γ (t ), t ∈ [0, 1] which parameterizes Γ (the stability boundary) in counterclockwise direction. Let D(s) be any polynomial of order n having all its zeros inside the stability regions. When the ‘‘nominal’’ or ‘‘center’’ polynomial is known - it should be chosen as D(s). Consider a counterclockwise sampling of D(s) for s on the stability region boundary, i.e. consider a sequence of samples: D(sk ),
sk ∈ Γ , k = 1, . . . , m.
(18)
Consider a sequence of sectors covering the above D(sk ), Sk = rk ei θk ,
θk ∈
h
6
rk > 0,
(D(sk−1 )) +
π 2
, 6 (D(sk+1 )) −
πi 2
(19)
where sm+1 , s1 + 2π N and s0 , sm − 2π N. These sectors become more and more wide with denser sampling sk , approaching sectors with angle π when sampling is infinitely dense.
Y. Dolgin, E. Zeheb / Systems & Control Letters 58 (2009) 804–809
Our aim is to find a dense enough sample sequence sk (and the corresponding sectors Sk ) so that the conditions of Theorem 1 (or Theorem 2) will be satisfied for D(s). Then, the same samples sk (and the same sectors Sk ) can be chosen for p(s) (or p(s, q)), providing sufficient conditions for p(s) (or p(s, q)) to have all its roots inside the desired stability regions. The simplest guideline to achieve the desired sample sequence sk is to try first a uniform step size in t, namely sk = γ (tk ),
tk+1 = tk + step, t0 = 0.
(20)
If the constraints are not satisfied for D(s) - halve the step size and try again. Obviously, this way the obtained sample sequence size will be far from the minimal possible, but the construction procedure is very simple and is guaranteed to converge. Note, that since D(s) can be freely chosen (e.g. independent from the polynomial p(s) (or p(s, q))), the above construction procedure can be performed off-line. To achieve smaller size of a sample sequence sk , the methods described in [2–5,1] can be applied, augmented with additional samples in order to satisfy (3a)–(3c). The latter additional samples can be constructed using the procedure described in Appendix A. Note also, that the conditions (3a)–(3c) do not depend on the specific polynomial, but only on the stability region under consideration and the order of the polynomial, thus the mentioned
809
construction procedure for additional samples can be performed off-line. References [1] R.D. Kaminsky, T.E. Djaferis, The finite inclusions theorem, IEEE Transactions on Automatic Control 40 (3) (1995) 549–551. [2] L. Bruyere, A. Tsourdos, B.A. White, Robust analysis for missile lateral acceleration control using finite inclusion theorem, AIAA Journal of Guidance, Control and Dynamics 28 (4) (2005) 679–685. [3] T.E. Djaferis, Robust Control Design: A Polynomial Approach, Kluwer Academic Publishers, 1995. [4] T.E. Djaferis, The finite inclusions theorem: A tool for robust design, Kybernetika 34 (1998) 625–634. [5] T.E. Djaferis, I.C. Schik, System Theory: Modeling, Analysis and Control, Springer, 1999. [6] A. Rantzer, A finite zero exclusion principle, robust control of linear systems and nonlinear control, in: Proceeding of the Int. Symp. MTNS-89, vol. II, 1989, pp. 239–245. [7] S. Mondié, J. Santos, V. Kharitonov, Robust stability of quasi-polynomials and the finite inclusions theorem, IEEE Transactions on Automaic Control 50 (2005) 1826–1831. [8] C.R. Ashokkumar, R.K. Yedavalli, Eigenstructure perturbation analysis in disjointed domains for linear uncertain systems, International Journal of Control 67 (1997) 887–899. [9] O. Bachelier, D. Henrion, B. Pradin, D. Mehdi, Robust matrix root-clustering of a matrix in intersections or unions of subregions, SIAM Journal of Control and Optimization 43 (2004) 1078–1093. [10] O. Bachelier, D. Mehdi, Robust DU -stability analysis, International Journal of Robust and Nonlinear Control 13 (2003) 533–558.
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Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle
Finite Nyquist and Finite Inclusions Theorems for disjoint stability regions Yuri Dolgin a,∗ , Ezra Zeheb a,b a
Department of Electrical Engineering Technion, Israel Institute of Technology, Israel
b
Department of Electronics Engineering, Jerusalem College of Engineering, Jerusalem, Israel
article
info
Article history: Received 28 March 2009 Received in revised form 30 July 2009 Accepted 21 September 2009 Available online 17 October 2009
abstract Finite Nyquist Theorem is an important tool in stability analysis and design of linear systems. Currently, Finite Nyquist Theorem can treat only simply connected convex stability regions with some extensions to simply connected non-convex regions. In this paper, we consider the generalization of Finite Nyquist Theorem for the case of union of disjoint convex stability regions. Based on this result, the Finite Inclusions Theorem is also formulated for a union of disjoint convex stability regions. © 2009 Elsevier B.V. All rights reserved.
Keywords: Stability Disjoint stability regions Finite Nyquist Theorem Finite Inclusions Theorem Uncertain systems
1. Introduction The Finite Nyquist Theorem and its generalization for uncertain systems, i.e. the Finite Inclusions Theorem, [1], are important tools in stability analysis and design of linear systems [2–6]. The above theorems allow us to consider only a finite subset of frequencies in order to establish stability. This subject remains a topic of continuous research: useful extensions to this method were devised for example in [7]. Recently, the question of stability, where stability domain is given as a set of disjoint convex regions, got much attention [8–10]. Below we state a few examples of control problems requiring stability analysis with disjoint stability regions: Problem 1. Model reduction of unstable LTI SISO system. When the unstable poles of the reduced system should be kept close to unstable poles of the original system, the stability region for the reduced system becomes the union of:
• Regular stability region (for example the unit circle) • Small regions around the unstable poles of the original system. Problem 2. Handling of systems with separate dynamics, for example the systems for which one part of the system should react
∗
Corresponding author. E-mail addresses: [email protected] (Y. Dolgin), [email protected] (E. Zeheb). 0167-6911/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2009.09.001
fast, while the other part should react slow. Separate dynamics naturally leads to stability region being a union of disjoint regions. Designing a controller for such a system, determining maximal stability radius and other problems can benefit from the proposed method. Problem 3. Non-fragility analysis of a given system. More precisely, given some nominal controller, find the maximal deviation from the nominal controller, so that the poles of the closed loop transfer function will remain in some predefined vicinity of the poles of the closed loop transfer function with nominal controller. In this case, again, the stability region is the union of small regions around the poles of the closed loop transfer function. The purpose of this paper is to generalize the results of Finite Nyquist Theorem, stated in [1] only for a simply connected convex region, to the case of union of disjoint convex stability regions. Namely, given a polynomial, our purpose is to find a finite number of checks to be performed on the phase of this polynomial in order to ensure that its zeros are located in given disjoint convex regions. Based on this result, the Finite Inclusions Theorem is also formulated for the union of disjoint convex stability regions. Note that, as shown in Section 3, the natural naive extension of the theorems in [1] to the case of disjoint regions does not hold. The structure of the paper is as follows. In Section 2 we provide the notations used in the paper. In Section 3 we state and prove the new Finite Nyquist Theorem for disjoint stability regions. Based on the results in Section 3, the Finite Inclusions Theorem is formulated in Section 4. In Section 5 we provide an example. We conclude in Section 6.
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2. Definitions and notations The following notations will be used throughout the paper. Given the monic polynomial p(s) = (s − zn )(s − zn−1 ) . . . (s − z1 ), its phase, arg(p(s)), is defined as arg(p(s)) = arg(s − zn ) + · · · + arg(s − z1 ) where arg(s − zi ) ∈ [0, 2π ], 1 ≤ i ≤ n. The arg(p(s)) is a continuous function of s unless s passes through some zero zi . Note, that arg(p(s)) can take values in [0, 2π n]. A given closed Jordan curve Ψ ⊂ C separates the complex plane into two regions. By reg Ψ we denote one of these regions, namely, either interior or exterior (note, that Ψ is not included in either of them). Given closed Jordan curve Γ ⊂ C, its convex interior int Γ , if exists, is defined as the points lying to the left of Γ when traversing the Γ in counterclockwise direction as seen from all points in int Γ . Let us consider a closed Jordan curve Γ ⊂ C with convex int Γ such that Γ is fully contained in a given region reg Ψ (in other words Γ ∩ Ψ = ∅). Consider also two points on Γ , say s1 and s2 . Consider all possible placements of m zeros of polynomial p(s) outside reg Ψ . These placements result in different phase changes of p(s) observed when going from s1 to s2 in counterclockwise direction along Γ . We denote the maximal phase change of p(s), which can be obtained from m zeros located outside reg Ψ , when going from s1 to s2 along Γ , by ∆reg Ψ ,Γ ,m (s1 , s2 ). The notion ‘‘stability region’’ is used to denote a region in which the zeros of a polynomial should be located.
We see that the ‘‘x’’ marks (i.e. misinterpreted sampled phase) satisfy the requirements in (1), replacing n with x(1) or with x(2) , for x(1) = 1 and x(2) = 1 in both disks, but actually none of the zeros is inside these disks. Thus, the Theorem is erroneous in this case. Below we provide a correct non-trivial generalization of the Finite Nyquist Theorem for the case of disjoint convex stability regions.
3. Finite Nyquist Theorem for disjoint stability region
Theorem 1. Let p(s) =
Fig. 1. Missed phase change.
αj sj , where n ∈ Z, n > 0, and αj ∈ C. Let Γ ⊂ C, 1 ≤ i ≤ q be closed Jordan curves such that int Γ (i) are convex and Γ (i) ∩ Γ (j) = ∅ for i 6= j. Only one of the Γ (i) is allowed Pint q (i) to be unbounded. Let x(i) ∈ Z be given such that = n. i=1 x Then, p(s) has x(i) zeros in int Γ (i) for 1 ≤ i ≤ q if and only if for (i) every 1 ≤ i ≤ q there exist m(i) ≥ 1 angles θk ∈ R, a 0 ≤ δ (i) < π , (i) a counterclockwise sequence of points sk ∈ Γi , 1 ≤ k ≤ m(i) (i) and a region reg Ψ ⊂ C in the complex plane, such that for every 1 ≤ i ≤ q: Pn
j =0
(i)
The Finite Nyquist Theorem for simply connected single region [1] is cited below Pn for jconvenience. Let p(s) = j=0 αj s , where n ∈ Z, n ≥ 0, and αj ∈ C and let Γ ⊂ C be a closed Jordan curve such that int Γ is convex. Then, p(s) is of degree n and has all its zeros in int Γ if and only if there exist m ≥ 1 angles θk ∈ R and a counterclockwise sequence of points sk ∈ Γ , 1 ≤ k ≤ m, such that
∀1≤k
(1a)
|2π n + θ1 − θm | < π ∀1≤k≤m p(sk ) 6= 0
(1b)
∀1≤k≤m arg p(sk ) = θk + 2π vk ,
v k ∈ Z.
(i)
⊃ (int Γ
(i)
(3a)
(i)
∪Γ )
(1c)
reg Ψ
(1d)
∀1≤k≤m(i) ∆reg Ψ (i) ,Γ (i) ,n (s(ki+) 1 , s(ki) ) ≤ δ (i)
(3c)
i) ∀1≤k≤m(i) |θk(+ 1
(3d)
The simple paraphrase of the above theorem is: if we can find sufficiently dense sampling of points on the stability region boundary so that the phase change of p(s), traversing all the sampled points, will be 2π n, then all n zeros are inside the stability region. The naive extension of this theorem to the case of disjoint regions does not hold. Namely, the theorem cannot be used to guarantee exactly x(i) < n zeros in each of the disjoint regions by requiring (1) to hold (with (1b) replaced by 2π x(i) + θ1 − θm < π ). As a counter-example, consider the following polynomial: p(s) = s2 + (−0.6 − 2i)s + (−0.91 + 0.6i)
∀1≤j≤q, i6=j reg Ψ (i) ∩ reg Ψ (j) = ∅
(2)
which has both zeros at 0.3 + i. The stability region under consideration is the union of two disjoint disks: the unit disk and the disk centered at 0.75 + 1.45i with radius 0.6. Note, that the zeros of p(s) are located outside both regions. The stability region together with zero locations of p(s) is shown in left part of Fig. 1. The phase evaluated on the unit circle and on the second circle are shown in the right-top and right-bottom parts of this figure, respectively. The solid line shows the real phase, the rectangles indicate the (correctly) sampled phase, the ‘‘x’’ marks indicate the misinterpreted sampled phase (namely, missed jump of 2π when passing from a sample which is close to the two zeros, to its neighbor sample).
(3b)
(i)
− θk | < π − δ
(i)
∀1≤k≤m(i) p(s(ki) ) 6= 0
(3e)
∀1≤k≤m(i) arg p(s(ki) ) = θk(i) + 2π vk(i) , where
(i) (i) (i) sm+1 , s1 , θm+1 ,
vk(i) ∈ Z
(3f)
(i)
θ1 + 2π x(i) .
For illustration of the definitions used in the Theorem see Fig. 2. Note, that when there is only one stability region (q = 1), the above Theorem coincides with the original Finite Nyquist Theorem if we choose reg Ψ (1) = C and δ (1) = 0. Evidently, conditions (3a)– (3c) become vacuous and condition (3d) satisfies both conditions (1a) and (1b). (i) A simple method for choosing the sequences sk is briefly discussed in Appendix B. Proof. ‘‘If’’ part: First, we prove the ‘‘if’’ part. Let us focus on one of i values. Our purpose is to prove that when (3) holds, there are exactly x(i) zeros inside int Γ (i) . In the following proof we remove the . . .(i) suffix in order to avoid cumbersome notation, unless required for clarification. Namely, we denote Γ , Γ (i) , Ψ , Ψ (i) , x , x(i) .
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a
b
Fig. 2. Illustration of notions used in theorem. Solid lines are Γ (i) , dashed lines are (i ) Ψ (i) , dark gray regions are int Γ (i) , light gray regions are reg Ψ (i) , black dots are sk samples. (a) Shows several stability regions int Γ (i) with corresponding reg Ψ (i) . (b) Shows the enlarged int Γ (1) from (a), with corresponding sampling of Γ (1) .
To simplify notations, let us define a function γ (t ), t ∈ [0, 1] which parameterizes Γ in counterclockwise direction. As a first step, our purpose is to show that there should be at least x zeros inside reg Ψ in order to satisfy (3). If there are at least x zeros inside int Γ , the statement trivially holds. Assume, in contradiction, that there are only x − r , r ∈ Z, r > 0 zeros inside int Γ . Then, arg (p(γ (1))) − arg (p(γ (0))) = 2π (x − r )
(4)
which can be written in the form m X
(arg (p(sk+1 )) − arg (p(sk ))) = 2π (x − r ).
(5)
k =1
However, the sampled phase change is, by definition,
θm+1 − θ1 = 2π x
(6)
which can be written in the form m X
(θk+1 − θk ) = 2π x.
(7)
k =1
Combining (5) and (7): m X
(arg(p(sk+1 )) − arg(p(sk ))) −
k =1
m X
(θk+1 − θk ) = −2π r .
(8)
k=1
According to (3f), for each k
(arg(p(sk+1 )) − arg(p(sk ))) − (θk+1 − θk ) = 2π w,
w ∈ Z. (9)
In other words, real phase difference and sampled phase difference can differ only by multiples of 2π . Thus, it follows from (8) and (9) that there exist not more than r indices h1 , . . . , hy , y ≤ r so that y X
arg(p(shk +1 )) − arg(p(shk ))
In other words, the cumulative phase change of argp(γ (t )) in intervals [shg , shg +1 ], 1 ≤ g ≤ y along Γ is negative and is greater in magnitude than r (π + δ). Negative phase changes of argp(γ (t )) can stem only from zeros located outside int Γ . Due to convexity of int Γ , the negative phase change which stem from a zero outside int Γ is less than or equal to π . The phase change between consecutive samples shg and shg +1 from all zeros outside reg Ψ is less than or equal to δ (by condition (3c)). Thus, when r > 0, we must have more than r zeros inside reg Ψ and outside int Γ to satisfy (11). We have thus shown that either there are at least x zeros inside int Γ , or there are x − r, r > 0, zeros inside int Γ and more than r zeros outside int Γ but inside reg Ψ . In both cases this means that there are at least x zeros inside reg Ψ . Given the fact that the same conclusion is valid for all disjoint stability regions (i.e. that there are at leastP x(i) zeros inside reg Ψ (i) for all 1 ≤ i ≤ q), and from the q constraint i=1 x(i) = n, it now follows that there are exactly x(i)
zeros inside each reg Ψ (i) . Now we want to show that all the x(i) zeros inside reg Ψ (i) are actually inside int Γ (i) . As we have already shown, there are either x(i) zeros inside int Γ (i) or x(i) − r (i) zeros, r (i) > 0, zeros inside int Γ (i) and more than r (i) zeros outside int Γ (i) but inside reg Ψ (i) . The assumption r (i) > 0 contradicts the fact that there are exactly x(i) zeros inside int Ψ (i) . Hence, we conclude that all x(i) zeros are inside int Γ (i) . This concludes the proof of the ‘‘if’’ part, i.e. when constraints (3) hold, each region int Γ (i) contains exactly x(i) zeros, as required. ‘‘Only if’’ part: Assume that p(s) has x(i) zeros inside int Γ (i) for 1 ≤ i ≤ q. It particularly means that there are no zeros on Γ (i) and thus (3e) holds. Then, the phase of p(s) is continuous when going along Γ (i) . According to the argument principle, the phase change when going along Γ (i) is 2π x(i) , thus it is always possible to find a sampling of points on Γ (i) satisfying (3d), (3f) with the property that any denser sampling will also satisfy (3d), (3f). It is rather intuitive fact that a sufficiently dense sampling of points on Γ (i) will satisfy (3c) for a reg Ψ (i) satisfying (3a), (3b). For the sake of completeness we provide the rigorous proof of this statement in Appendix. Since Γ (i) ∩ Γ (j) = ∅ for i 6= j, there always exist reg Ψ (i) satisfying (3a), (3b). In order to obtain the finite number of samples, our only requirement is that for the only unbounded int Γ (i) , say int Γ (i0 ) , the exterior of Ψ (i0 ) should be the region which includes Γ (i0 ) , while the interior of Ψ (i0 ) should be bounded. This is always possible when there is only one unbounded int Γ (i) . Hence, we have shown that if p(s) has x(i) zeros inside int Γ (i) for 1 ≤ i ≤ q, then all conditions (3) are satisfied. This concludes the proof of the ‘‘only if’’ part, and thus concludes the proof of the theorem. 4. Finite Inclusions Theorem
k =1
−
y X
θhk +1 − θhk ≤ −2π r .
(10)
k=1
Equivalently, taking into account (3d) y
X
k =1
X
θhk +1 − θhk − 2π r
k=1
< y(π − δ) − 2π r ≤ r (π − δ) − 2π r = −r (π + δ)
(11a)
or y X k =1
arg(p(shk +1 )) − arg(p(shk )) < −r (π + δ).
j Theorem 2. Let p(s, q) = j=0 αj (q)s , q ∈ Q where Q is an arbitrary set (of possibly complex parameters), n ∈ Z, n > 0, and αj : Q → C. Let Γ (i) ⊂ C, 1 ≤ i ≤ q be closed Jordan curves such that int Γ (i) are convex and Γ (i) ∩ Γ (j) = ∅ for i 6= j. Only one of the int Γ (i) is allowed to be unbounded. Let x(i) ∈ Z be given such that P q (i) = n. i=1 x Then, every member of the polynomial family p(s, q) has x(i) zeros in int Γ (i) for 1 ≤ i ≤ q if for every 1 ≤ i ≤ q there exist m(i) ≥ 1 (i) (i) intervals (ak , bk ) ⊂ R, a 0 ≤ δ (i) < π , a counterclockwise sequence
Pn
y
arg(p(shk +1 )) − arg(p(shk )) ≤
In this section we provide the Finite Inclusions Theorem for a union of disjoint convex stability regions, based on the results of the previous section.
(11b)
Y. Dolgin, E. Zeheb / Systems & Control Letters 58 (2009) 804–809
807
(i)
of points sk ∈ Γi , 1 ≤ k ≤ m and a region reg Ψ (i) ⊂ C in the complex plane, such that for every 1 ≤ i ≤ q:
∀1≤j≤q, i6=j reg Ψ (i) ∩ reg Ψ (j) = ∅ reg Ψ
(i)
⊃ (int Γ
(i)
(12a)
(i)
∪Γ )
(12b)
∀1≤k≤m(i) ∆reg Ψ (i) ,Γ (i) ,n (s(ki+) 1 , s(ki) )
≤δ
(i)
(12c)
∀1≤k≤m(i) max{bk(i+) 1 − a(ki) , b(ki) − a(ki+) 1 } < π − δ (i) (i)
(i)
(i) jθ (i)
∀1≤k≤m(i) p(sk , q) ⊂ Sk = R e
|R
(i)
> 0,
θ
(12d) (i)
(i)
(i)
∈ (ak , bk ) (12e)
(i)
(i)
(i)
(i)
where am+1 , a1 + 2π x(i) , bm+1 , b1 + 2π x(i) . (i)
In other words, if, for sufficiently dense sampling sk (i.e. satisfying (i)
(12c)), we can find a sequence of sectors Sk ‘‘spaced’’ less than π − δ (i) radians apart and revolving a net 2π x(i) radians about the (i) origin, and if p(sk , q) is contained in the above sectors, then every member of the polynomial family p(s, q) has x(i) zeros in int Γ (i) for 1 ≤ i ≤ q. The proof of this theorem will be omitted since it follows the same lines as in [1], taking into account the results of (i) the previous section. These results ensure that if Sk are ‘‘spaced’’ (i) less than π − δ radians apart, then, for each member of p(s, q), (3d), (3f) will hold, and since R(i) > 0 the condition (3e) will also hold. (i) A simple method for choosing the sequences sk and the sectors (i)
Sk is briefly discussed in Appendix B. 5. Example We consider a practical example requiring the Finite Inclusions Theorem for disjoint regions. The poles of the nominal plant for this example are taken from [9], where the lateral dynamics of an aircraft was considered:
λi = {−0.5, −2 ± 2i, −3 ± 2i} ⇒ p(s) = s5 + 10.5s4 + 50s3 + 122.5s2 + 154s + 52.
(13)
Our aim is to analyze the robustness of the pole locations in the presence of an uncertainty. We consider an interval polynomial of a predefined form (with parameterizable gain k) around the nominal polynomial (13): p(s, q) =
α5 α4 α3 α2 α2 α0
5 X
Fig. 3. Example 1: The stability regions (gray), the stability boundary sampling sk (‘‘x’’), the Ψ (i) (dashed) and final uncertain polynomial roots (blue [or dark gray, when printed in grayscale]).
samples is good enough. At each sample sk we form 2 constraints defining together a sector constraint — we require the uncertain polynomial to lie in sectors defined by the original nominal polynomial (and the constraint (12d)). Additionally, since we work with uncertain polynomial, for each sector we require all 12 vertices of the uncertain polynomial to reside in this sector. Finally, we end up with 3840 linear constraints on k, ensuring that the roots of (14) will lie in the desired disjoint regions. Then, the optimization problem is solved to find maximal k satisfying the above constraints — the final k is 0.234. In the Fig. 3 we show the stability regions as gray disks, the dashed lines are Ψ (i) , the ‘‘x’’ marks are the sampling points on the stability boundary and the blue (or dark gray, when printed in grayscale) regions are the roots of the final uncertain polynomial (e.g. with k = 0.234). We can see that the roots of final uncertain polynomial indeed come close to the stability boundary, thus the constraints are not very conservative. The legitimate question now is whether the conditions of the classical Finite Nyquist/Inclusions Theorem are not sufficient here. Consider the following polynomial p(s) = z 5 + 10.66z 4 + 52.1724z 3 + 133.2323z 2
αj (q)s , j
+ 175.156z + 72.8291
where
j=0
∈ [1 · 0.95k, 1 · 1.05k], ∈ [10.5 · 0.98k, 10.5 · 1.02k] ∈ [50 · 0.993k, 50 · 1.007k], ∈ [122.5 · 0.998k, 122.5 · 1.002k] ∈ [154 · 0.999k, 154 · 1.001k], ∈ [52 · 0.99k, 52 · 1.01k].
(15)
which has the roots at (14)
We wish to find the maximum gain k so that the roots of the resulting uncertain polynomial (14) will lie inside the disks of radius 0.4 around the roots of the nominal polynomial (13). First of all, since the stability regions are symmetric with respect to the real axis and since the considered polynomial is real, we can consider only 3 disjoint regions instead of 5, namely the disks centered at −3 + 2i, −2 + 2i and −0.5 (the other two regions will automatically satisfy the conditions of Theorem 2). We consider a total of 160 samples sk on the stability boundaries. To satisfy the conditions of the Theorem 2 it is enough to take much less samples, but then the constraints are too conservative. Obviously, more than 160 samples will result in less conservative constraints, but as we show below, even such moderate number of
{−0.7, −2.48 ± 2i, −2.5 ± 2i}
(16)
i.e. all roots except the first are far outside the desired stability region. As we show in Fig. 4, there exists a sampling of the stability boundary, which satisfies the conditions of classical Finite Nyquist Theorem for each region (namely, that in each region there is one root), but obviously, the conclusion is incorrect. Thus, to obtain correct stability constraints, it is not enough to use classical Finite Nyquist/Inclusions Theorem — the Theorems 1 and 2 should be applied instead. 6. Conclusion In this paper we have stated and proved the Finite Nyquist Theorem for a union of disjoint convex stability regions. We have shown that a finite number of phase checks is sufficient to guarantee that all roots of a polynomial lie inside the stability region, even when the stability region is a union of convex regions. Based on this theorem, we also formulated the Finite Inclusions Theorem for a union of disjoint convex stability regions.
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Fig. 4. Example 1: Example of erroneous conclusions by classical Finite Nyquist Theorem. The phase using ideal (continuous) sampling is shown by solid line and the misinterpreted phase using sparse sampling is shown by ‘‘x’’ marks.
Appendix A In this Appendix we augment the proof of the ‘‘Only If’’ part of Theorem 1. Namely, we show that a sufficiently dense sampling of points on Γ (i) will satisfy (3c) for a reg Ψ (i) satisfying (3a), (3b). In the following proof we again omit the . . .(i) suffix in order to avoid cumbersome notation. We consider first the bounded int Γ residing inside reg Ψ and we show that there exists finite sampling of this Γ , which satisfies (3c). To show this, we shortly sketch the construction procedure for such sampling. Start from some initial point A ∈ Γ . Our purpose is to show that we can find a point B ∈ Γ next to A in counterclockwise direction along Γ , so that the maximal angle seen between A and B from any point outside reg Ψ will be less than any predefined number . For our purposes we choose = nδ . The construction procedure is as follows. Find the maximal radius, say rA,0 , rA,0 > 0, of the circle centered at A and fully contained in reg Ψ . rA,0 is not zero since A must be inside (and not on the boundary of) reg Ψ . Consider another circle of radius rA,1 < rA,0 centered at A. Take a point B ∈ Γ on the latter circle of radius rA,1 , in counterclockwise direction from A. Consider triangle with vertices A, B and any point outside reg Ψ , say C . Denote the angle at vertex C (i.e. 6 ACB) by 6 C . By law of sines,
6 sin 6 C sin 6 A r A ,1 = ⇒ | sin 6 C | = AB sin A ≤ AB BC BC r −r . A,0 A ,1 (17) Thus, we can get C as small as desired by decreasing rA,1 . We need to show only that the constructed sampling is finite, namely, that rA,1 is bounded from below by some positive number. Since Γ ⊂ reg Ψ , the distance between closest points of Γ and Ψ is positive, say R0 > 0. Considering also (17): 6
| sin()| inf rA,0 = R0 > 0 ⇒ inf rA,1 ≥ R0 >0 A∈Γ A∈Γ 1 + | sin()| thus the number of samples on Γ is finite. Thus, we proved that a sufficiently dense sampling of Γ will satisfy (3c). Consider now the unbounded int Γ (i0 ) residing inside exterior of Ψ (i0 ) , where Ψ (i0 ) is with bounded interior. Consider any point
C inside interior of Ψ (i0 ) . Denote the maximal distance from C to a point on Ψ (i0 ) by R1 . Consider a circle CC , with center at C and 2R1 with radius R2 = sin(0.25 . Then, by law of sines, for any point A ) outside CC , the angle seen from C to A differs from the angle seen from any other point inside interior of Ψ (i0 ) to A by less than 0.25 . Now, we consider two parts of Γ (i0 ) : the part inside CC and the part outside CC . The sampling inside CC is constructed exactly in the same way as for bounded Γ (i) - since this part of Γ (i0 ) is finite, the number of samples is also finite. To construct the sampling outside CC we draw (finite number of) lines from C with angles 0, 0.5, , 1.5, . . . , 2π . The intersection of these lines with the part of Γ (i0 ) outside CC gives the desired points — the angle seen between neighbor points from any point outside reg Ψ (i0 ) is less than (note that the exterior of Ψ (i0 ) now plays the role of reg Ψ (i0 ) ). Appendix B In this appendix we briefly discuss one of the methods to build (i) (i) the sk sequences (and the Sk sectors) mentioned in Theorems 1 and 2. Here we again omit the . . .(i) suffix to avoid cumbersome notations. As in the proof of Theorem 1 we define a function γ (t ), t ∈ [0, 1] which parameterizes Γ (the stability boundary) in counterclockwise direction. Let D(s) be any polynomial of order n having all its zeros inside the stability regions. When the ‘‘nominal’’ or ‘‘center’’ polynomial is known - it should be chosen as D(s). Consider a counterclockwise sampling of D(s) for s on the stability region boundary, i.e. consider a sequence of samples: D(sk ),
sk ∈ Γ , k = 1, . . . , m.
(18)
Consider a sequence of sectors covering the above D(sk ), Sk = rk ei θk ,
θk ∈
h
6
rk > 0,
(D(sk−1 )) +
π 2
, 6 (D(sk+1 )) −
πi 2
(19)
where sm+1 , s1 + 2π N and s0 , sm − 2π N. These sectors become more and more wide with denser sampling sk , approaching sectors with angle π when sampling is infinitely dense.
Y. Dolgin, E. Zeheb / Systems & Control Letters 58 (2009) 804–809
Our aim is to find a dense enough sample sequence sk (and the corresponding sectors Sk ) so that the conditions of Theorem 1 (or Theorem 2) will be satisfied for D(s). Then, the same samples sk (and the same sectors Sk ) can be chosen for p(s) (or p(s, q)), providing sufficient conditions for p(s) (or p(s, q)) to have all its roots inside the desired stability regions. The simplest guideline to achieve the desired sample sequence sk is to try first a uniform step size in t, namely sk = γ (tk ),
tk+1 = tk + step, t0 = 0.
(20)
If the constraints are not satisfied for D(s) - halve the step size and try again. Obviously, this way the obtained sample sequence size will be far from the minimal possible, but the construction procedure is very simple and is guaranteed to converge. Note, that since D(s) can be freely chosen (e.g. independent from the polynomial p(s) (or p(s, q))), the above construction procedure can be performed off-line. To achieve smaller size of a sample sequence sk , the methods described in [2–5,1] can be applied, augmented with additional samples in order to satisfy (3a)–(3c). The latter additional samples can be constructed using the procedure described in Appendix A. Note also, that the conditions (3a)–(3c) do not depend on the specific polynomial, but only on the stability region under consideration and the order of the polynomial, thus the mentioned
809
construction procedure for additional samples can be performed off-line. References [1] R.D. Kaminsky, T.E. Djaferis, The finite inclusions theorem, IEEE Transactions on Automatic Control 40 (3) (1995) 549–551. [2] L. Bruyere, A. Tsourdos, B.A. White, Robust analysis for missile lateral acceleration control using finite inclusion theorem, AIAA Journal of Guidance, Control and Dynamics 28 (4) (2005) 679–685. [3] T.E. Djaferis, Robust Control Design: A Polynomial Approach, Kluwer Academic Publishers, 1995. [4] T.E. Djaferis, The finite inclusions theorem: A tool for robust design, Kybernetika 34 (1998) 625–634. [5] T.E. Djaferis, I.C. Schik, System Theory: Modeling, Analysis and Control, Springer, 1999. [6] A. Rantzer, A finite zero exclusion principle, robust control of linear systems and nonlinear control, in: Proceeding of the Int. Symp. MTNS-89, vol. II, 1989, pp. 239–245. [7] S. Mondié, J. Santos, V. Kharitonov, Robust stability of quasi-polynomials and the finite inclusions theorem, IEEE Transactions on Automaic Control 50 (2005) 1826–1831. [8] C.R. Ashokkumar, R.K. Yedavalli, Eigenstructure perturbation analysis in disjointed domains for linear uncertain systems, International Journal of Control 67 (1997) 887–899. [9] O. Bachelier, D. Henrion, B. Pradin, D. Mehdi, Robust matrix root-clustering of a matrix in intersections or unions of subregions, SIAM Journal of Control and Optimization 43 (2004) 1078–1093. [10] O. Bachelier, D. Mehdi, Robust DU -stability analysis, International Journal of Robust and Nonlinear Control 13 (2003) 533–558.