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Metric on the factorizations factor-classes as the measure of robustness
Proceedings of the 7th IFAC Symposium on Robust Control Design The International Federation of Automatic Control Aalborg, Denmark, June 20-22, 2012
Metric on the factorizations factor-classes as the measure of robustness S. Iantchenko ∗ A. Ghulchak ∗∗ ∗
Department of Mathematics, Lund University, Box 118, 221 00 Lund, Sweden, e-mail: [email protected] ∗∗ Department of Mathematics, Lund University, Box 118, 221 00 Lund, Sweden, e-mail: [email protected] Abstract: The factor space of factorization is defined. A metric, called δF −metric, was introduced. It has been shown that δF −metric has close connection with the ν−gap metric. Furthermore, the δF -metric is more appropriate to robust stabilization of parametrically uncertain plants then ν−gap metric. 1. NOTATION By R (or C) we denote the field of real (or complex) numbers. The unit circle and the open unit disc in C are denoted by T respectively D T = {z ∈ C | |z| = 1} D = {z ∈ C | |z| < 1}. Let Y ⊂ Cn be any measurable set and 1 ≤ p < ∞. Denote by Lp (Y ) the standard Lebesgue space of functions f : T → Y equipped with the norm ∫ 1 ( |f (z)|p dm(z)) p , 1 ≤ p < +∞, ∥f ∥p = T ess sup |f (z)|, p = +∞ z∈T
where | · | denotes the usual 2-norm in Cn √ |f | = |f1 |2 + |f2 |2 + . . . + |fn |2 . The Hardy class Hp (Y ) consists of functions in Lp (Y ) that can be analytically continued inside the unit disk. A function θ analytic in D is called an inner function if θ ∈ H∞ and |θ(z)| = 1 for almost all z ∈ T. A function h analytic in D is called an outer function if there exists a real function g ∈ L1 and a complex number c of modulus 1 such that (∫ ) z+λ h(λ) = c exp g(z)dm(z) , λ ∈ D T z−λ If f ∈ Hp , then f admits the representation f = θh, where θ is an inner function and h is an outer function. Let Hp0 (Y ) denote the shifted Hp (Y ) Hp0 (Y ) = zHp (Y ) = {f ∈ Hp (Y )|f (0) = 0}. The disk algebra A(Y ) is by definition the subspace of H∞ that consists of analytic functions in D ⊂ Y that can be extended continuously to the closed unit disk. The brief notations A, H∞ etc. will be used if Y = Cn and the dimension of the space is clear from context. 978-3-902823-03-8/12/$20.00 © 2012 IFAC
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Two matrices N, M ∈ H∞ are right-coprime if they have equal number of columns and there exist X, Y ∈ H∞ such that XN + Y M = I. Let G be a matrix function. We say that the factorization G = N M −1 is a right coprime factorization over H∞ if N and M are right-coprime H∞ -matrices. The factorization G = N M −1 is called a normalised right coprime factorization if N ∗ N + M ∗ M = I. The matrix function V is called co-outer if V ⊤ is outer. η[G] is the number of open right half plane poles of G(s). Z[G] is the number of open right half plane zeros of G(s). Notation [P, C] refer to the standard feedback connection of a plant P with a controller C. The superscript ⊤ stands for transposition and † stands for pseudoinverse. The bar denotes the complex conjugate and ∗ denotes conjugate transpose. The prefix B denotes the unit ball in the corresponding space. 2. INTRODUCTION The problem of robust optimization have been considered by a number of authors. Much work has been devoted to problems stated in terms of convex optimization. In particular, it has been shown in Rantzer and Megretski [1994] that the robust stabilization problem under parametric uncertainties has a convex formulation if the characteristic polynomial depends linearly on the uncertain parameters (so called rank one problem). The convex parametrization gives great benefits both for theoretical analysis and for practical computations. However, when an optimization problem is infinite-dimensional, a reduction to finite-dimensional form (like Ritz projection, Galerkin finite element scheme, grid methods, etc.) is needed. Such an approximation introduces conservatism to the problem. The gap between the true optimal value and its finitedimensional counterpart can be arbitrarily large. To overcome this difficulty and to estimate the conservatism, convex duality has been used. The dual problem for systems with uncertainties of rank one was introduced in Ghulchak 10.3182/20120620-3-DK-2025.00074
7th IFAC Symposium on Robust Control Design Aalborg, Denmark, June 20-22, 2012
and Rantzer [2002]. It was obtained an analytical expression of the infinite-dimensional dual problem. It was shown that result can be used for numerical optimization of the robustness margin by primal-dual methods based on finitedimensional approximation. The dual factorization of all plant factors with destabilizing uncertainties was obtained in Iantchenko and Ghulchak [2007]. In this paper, we continue to study the duality principle in robust optimization problems of rank one. We show that the dual problem can be interpreted as a minimization of all destabilizing uncertainties in a certain metric that has interesting connections to the well-known ν-gap metric. 3. PRELIMINARIES The main problem of robust stabilization is stated as follows: Given a uncertain plant P ( ) ( ) y w =P , z u w = δ ⊤ z, where δ ∈ ν∆ for some convex compact ∆, with 0 ∈ ∆, the problem is to find a controller u = Ky that robustly stabilizes the plant for as large ν as possible. According to the Youla parametrization of internal stabilizing controllers we assume that the transfer function from w to z is given on the form z = (T1 + T2 Q)w, and T2 ∈ RH∞ where T1 ∈ m×n are fixed and parameter Q is any transfer matrix in RH∞ n×1 . The system becomes robustly stable if and only if RH∞ m×1
[1 − δ ⊤ (T1 + T2 Q)]−1 ∈ RH∞ for all |δ| ≤ ν, here | · | is some norm on Rm . The main problem is stated as follows ∞ Given T1 ∈ RH∞ m×1 , T2 ∈ RHm×n , find a convex ∞ parametrization of all Q ∈ RHn×1 such that [1 − δ ⊤ (T1 + T2 Q)]−1 ∈ RH∞ for all δ ∈ Rm with |δ| ≤ ν.
The problem was solved in Rantzer and Megretski [1994]. ∞ Theorem 1. Suppose T1 ∈ RH∞ m×1 , T2 ∈ RHm×n . Then the following two conditions on the rational matrix Q are equivalent: (1) Q ∈ RH∞ n×1 and for all δ ∈ Rm with |δ| ≤ ν [1 + δ ⊤ (T1 + T2 Q)]−1 ∈ RH∞ . (2) There exist α ∈ RH∞ and β ∈ RH∞ such that Q = β/α and ∀ω ∈ R ∪ {∞}, |Re[T1 α + T2 β](iω)|d < ν −1 Re α(iω). Here | · |d is the dual norm defined by |x|d = max{x⊤ y : |y| ≤ 1}. The convex parametrization of all controllers that stabilize the system for all possible combinations of parameters makes it possible to use convex optimization to find a robustly stabilizing controller.
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Let us consider the family of perturbed plants Pδ =
N + G1 δ , δ ∈ ν∆. M + G2 δ
The problem is to find a controller K that robustly stabilizes Pδ for ν as large as possible. Denote by F = (N M ) ∈ An×1 the right coprime factorization of the plant P , by G = (G1 G2 ) ∈ An×m the weight matrix and the set ∆ is a convex compact set in Cm . According to a convex parametrization of all robustly stabilizing controllers (Theorem 1) we define the primal problem as the problem of finding a function h ∈ BH∞ such that the condition Re h(z)(F (z) + G(z)δ) > 0, ∀z ∈ T, ∀δ ∈ ν∆, (1) is satisfied. We would like to solve it for ν as large as possible, that is, for νopt = sup{ν|∃h ∈ BH∞ : Re h(F + Gδ) > 0 ∀δ ∈ ν∆}. (2) In Ghulchak and Rantzer [2002] the dual problem was introduced as follows: Theorem 2. Let F ∈ An×1 , G ∈ An×m and denote Φδ = F + Gδ. Then the optimal value νopt takes the following form νopt = min{νopt|c , νopt|s } with the regular part νopt|c = inf{ν|∃w ∈ L1 (R+ )\0, δ ∈ L∞ (ν∆) : Φδ w ∈ H10 } (3) and the singular part νopt|s = inf{ν|∃z ∈ T, δ ∈ ν∆ : Φδ (z) = 0}. (4)
We will consider the regular case, i.e. νopt|c < νopt|s . The canonical parametrization of the dual problem in the regular case was obtained in Iantchenko and Ghulchak [2007] as follows: We need to find the functions wopt ∈ L1 (R+ )\0 and δopt ∈ L∞ (ν∆) such that Φδ w ∈ H10 . The last condition means that there exists a vector function p ∈ H1 such that Φδ (z)w(z) = zp(z). Note that in the regular case we have |Φδ | ≥ ε, which implies, in particular, that log(w) ∈ L1 . A real positive function with this property can be factorized as w = f ∗ f , where f is a scalar outer function. Then Φδ can be written as p f p Φδ = z ∗ = z ∗ 2 . f f f f Furthermore, the function p can be factorized into inner and co-outer factors p = pi pco , where pi is scalar, and the function Φδ may be written as f pco (5) Φδ = zpi ∗ 2 = uψ, f f where u = zpi ff∗ is a scalar function and ψ ∈ H∞ is an coouter function. In the rational case, the rational function u collects all common zeros of the plant factors and gives a factorization of the dual problem via unstable zeropole cancelations. For non-rational systems the situation with the common zeros is more complicated. The nominal
7th IFAC Symposium on Robust Control Design Aalborg, Denmark, June 20-22, 2012
factors can contain a singular component and cannot be treated by unstable cancelations. It was shown that in the case when the uncertainty set is the ball ∆ = BCm , the winding number for the function u is well-defined. It plays the same role as the unstable cancelations in the rational case. We introduce the class U of all unimodular functions with the well-defined and finite winding numbers U = {u ∈ L∞ : |u| ≡ 1, wno u > 0}. Theorem 3. Let νopt|c < νopt|s . Then the plant numerator and denominator with the worst uncertainty Φδ = F + Gδ can be factorized in the following way Φδ = uψ, (6) ∞ where u ∈ U and ψ ∈ H is co-outer. 4. METRIC ON THE FACTOR-CLASS Consider the dual parametrization F + Gδ = uψ, u ∈ U, ψ ∈ H∞ . Recall that for the optimal δ we have wno u > 0, i.e. on the boundary of the stability region, we have a nonstabilizable plant, the plant that has an unstable pole/zero cancelation. This gives an idea how to define the classes of equivalence in the factor space. Then we can introduce a distance between them. Let U0 be the class of all unimodular functions with welldefined winding numbers given by U0 = {u ∈ L∞ : |u| ≡ 1, wno u = 0}. Suppose [ ] that we have two plants P1 and P2 . Let[F1 = ] N1 N2 be a coprime factorization of P1 and F2 = M1 M2 be a coprime factorization of P2 . We will call two factorizations equivalent F1 ∼ F2 if we can find Q such that Q ∈ U0 and F2 = F1 Q. We define the distance between equivalent classes as follows: δF (F1 , F2 ) = inf{∥δ∥∞ : F1 + Gδ ∈ class (F2 )}. Theorem 4. δF is a metric on the factor space of factorizations.
Small gain methods consider stable perturbations. The homotopy methods give stronger results since the homotopy arguments are build on perturbations which are bounded on the imaginary axis, and on counting winding numbers. We introduce δF −metric techniques for studying robust stability and show that the δF −metric techniques gives the same strong results as the homotopy methods. 5.1 Additive uncertainty We consider the family of perturbed plans: {P + ∆ : ∆ ∈ L∞ , ∥W1 ∆W2 ∥∞ < 1.} Using the homotopy methods (see for exemple, Vinnicombe [2001]) we get the following theorem: Theorem 5. Given P, C real rational transfer functions, W1 , W1−1 , W2 , W2−1 ∈ RH∞ : The following are equivalent • [P1 , C] is stable for all P1 of the form {P1 = P + ∆ : ∆ ∈ H∞ , ∥W1 ∆W2 ∥∞ < 1}. • [P1 , C] is stable for all P1 of the form {P1 = P + ∆ : ∆ ∈ L∞ , ∥W1 ∆W2 ∥∞ < 1, η(P + ∆) = η(P )}. We will interpret the poles condition in term of δF −metric. Let (N, M ) be a coprime factorization of P. Then P + ∆ = N M −1 + ∆ = (N + ∆M )M −1 . [P1 , C] is stable for all] P1 =[P + ∆ ] if we can found Q ∈ U0 [ N1 N + M∆ =Q . Function Q will collect such that M M1 all common zeros of the plant factors and all poles. The poles of P are exactly the zeros of M. Furthermore the poles of N + M ∆ are the poles of ∆. Denote by Zc the common zeros of M and of N + M ∆. Since wno Q = 0, then Z(Q) − η(Q) = 0 ⇒ Zc − η(∆) = 0. We get following: η(P + ∆) = η(∆) + Z(M ) − Zc = Z(M ) = η(P ). We have shown that poles condition are equivalent with winding number condition on function Q in δF −metric. 5.2 Multiplicative uncertainty We allow the perturbed plans to take the form {(I + ∆)P : ∥W1 ∆W2 |∞ < 1.}
Proof. See Appendix A Now we can get an expression for our metric δF (F1 , F2 ). By the definition of δF (F1 , F2 ) we are looking for such a δ that F1 + Gδ ∈ class (F2 ). It means that there exists Q ∈ U0 such that F1 + Gδ = F2 Q. We have to take an infimum over all such Q in order to find the optimal δ, i.e. the distance between factor-classes. Then we get δF (F1 , F2 ) = inf ∥G† F1 − G† F2 Q∥∞ . (7) Q∈U0
5. COMPARISON WITH PREVIOUS RESULTS. We gather several problems in robust stability. There are two tools for the study of robust stability: • the small-gain theorem, • homotopy methods. 186
It was shown the similar result Theorem 6. Given P, C real rational transfer functions, W1 , W1−1 , W2 , W2−1 ∈ RH∞ : The following are equivalent • [P1 , C] is stable for all P1 of the form {P1 = (I + ∆)P : ∆ ∈ H∞ , ∥W1 ∆W2 ∥∞ < 1}. • [P1 , C] is stable for all P1 of the form {P1 = (I + ∆)P : ∆ ∈ L∞ , ∥W1 ∆W2 ∥∞ < 1, η((I + ∆)P ) = η(P )}. In the same way we interpret the poles condition in term of δF −metric. (1 + ∆)P = (1 + ∆)N M −1 , where (N, M ) is a coprime factorization of P. [P1 , C] is stable for [ all P1 =] (1 + [∆)P ]if we can found (1 + ∆)N N1 Q ∈ U0 such that =Q . We know that M M1
7th IFAC Symposium on Robust Control Design Aalborg, Denmark, June 20-22, 2012
the function Q will collect all common zeros of the plant factors and all poles. The poles of P are the zeros of M. The poles of (1 + ∆)N are the poles of ∆ minus possible cancelations with zeros of N. We denote the number of poles of (1 + ∆)N by ηc (∆). Zc is the common zeros of M and of (1 + ∆)N. Since wno Q = 0, then Zc − ηc (∆) = 0. We get the same result as in the previous case: η(1 + ∆)P = ηc (∆) + Z(M ) − Zc = Z(M ) = η(P ). Again the poles condition are equivalent with the winding number condition on function Q in δF −metric. 5.3 Inverse multiplicative uncertainty The inverse multiplicative uncertainty gives us the uncertainty model in the form: {(I − ∆)−1 P : ∥W1 ∆W2 ∥∞ < 1}. We have the following result: Theorem 7. Given P, P −1 , C real rational transfer functions, W1 , W1−1 , W2 , W2−1 ∈ RH∞ : The following are equivalent • [P1 , C] is stable for all P1 of the form {P1 = (I − ∆)−1 P : ∆ ∈ H∞ , ∥W1 ∆W2 ∥∞ < 1} and (I − ∆) is invertible. • [P1 , C] is stable for all P1 of the form {P1 = (I − ∆)−1 P : ∆ ∈ L∞ , ∥W1 ∆W2 ∥∞ < 1, Z((I − ∆)−1 P ) = Z(P )} and (I − ∆) is invertible. In the same way we interpret in this case the zeros condition in term of δF −metric. (1−∆)−1 P = N ((1−∆)M )−1 , where (N, M ) is a coprime factorization of P. −1 [P1 , C] is stable for [ all P1 = (1 ] − ∆)[ P ]if we can found N N1 Q ∈ U0 such that =Q . The function (1 − ∆)M M1 Q will collect all common zeros of the plant factors and all poles. The zeros of P are the zeros of N. We denote by Zc the common zeros of N and the zeros of (1 − ∆)M. Since wno Q = 0, then Zc = η((1 − ∆)M ). We get the following result: Z((I − ∆)−1 P ) = η((1 − ∆)M ) + Z(N ) − Zc = Z(N ) = Z(P ). The zeros condition are equivalent with the winding number condition on function Q in δF −metric. 5.4 Coprime factor uncertainty Coprime factor perturbations can be describe in the form
[ { ] }
∆N 1 −1
P1 = (N + ∆N )(M + ∆M ) : , < ∆ M ∞ γ
where P = N M −1 is normalized right coprime factorization of P and γ > 1. The following result was obtained: Theorem 8. Given P real rational transfer function and γ > 1, the following are equivalent • [P1 , C] is stable for all P1 of the[ form] ∆N P1 = (N + ∆N )(M + ∆M )−1 : ∈ H∞ , ∆M
[ ]
∆N 1
∆M ≤ γ and (M + ∆M ) is invertible. ∞ 187
• [P1 , C] is stable for all P1 of the[ form] ∆N P1 = (N + ∆N )(M + ∆M )−1 : ∈ L∞ , ∆M
[ ]
∆N 1
∆M ≤ γ , (M + ∆M ) is invertible and ∞ η(P1 ) = wno det(M + ∆M ). We analyze the winding number condition on the M +∆M with help of δF −metric. Let Zc be the number of common zeros of N + ∆N and M +∆M . Since wno Q = 0, then Zc −η(∆N )−η(∆M ) = 0. We have the following result η(P1 ) = η(∆N ) + Z(M + ∆M ) − Zc = η(∆N ) + Z(M + ∆M ) − η(∆N ) − η(∆M ) = Z(M + ∆M ) − η(M + ∆M ) = wno (M + ∆M ). We get the same result: the winding number condition on the M + ∆M are equivalent with the winding number condition on function Q in δF −metric. 5.5 δF −metric and ν−gap metric. In Vinnicombe [2001] it was considered the problem of minimizing
[ ]
P
−1
I (I − CP ) [ −C I ] , ∞ which is the transfer function from v1 and v2 , to u and y, in the standard feedback configuration, P is the nominal plant, C is the controller. Define [ ]
−1
P (I − CP )−1 [ −C I ] , [P, C] stable
bP,C := I ∞ 0, otherwise, and bopt (P ) = sup bP,C . C
The problem of finding bopt can be considered as primal problem. This primal problem is quite similar to the our primal problem. In order to measure uncertainty it was introduced the ν− gap metric as follows: Let P1 and P2 be the two plants such that η[P2 , −P1∗ ] = η[P1 , −P1∗ ]. Then
[ ]
−1 √
P2
∗ ∗ −1 2
1 − δν (P1 , P2 ) = (I + P1 P2 ) [ P1 I ]
. I ∞ The most fundamental property is that the stability margin bP,C and the uncertainty measure δν (P1 , P2 ) are related by the inequality arcsin bP2 ,C ≥ arcsin bP1 ,C − arcsin δν (P1 , P2 ). It means that if a compensator C performs well with the plant P1 and δν (P1 , P2 ) is sufficiently small, then C is guaranteed to stabilizes P2 with a certain level of performance. The measure δν (P1 , P2 ) can be considered as dual to the bP,C (and for optimal controller, as dual to the bopt (P ).) It was shown the following relation: Given a nominal plant P1 , a perturbed plant P2 and and a number β < bopt (P1 ) then:
7th IFAC Symposium on Robust Control Design Aalborg, Denmark, June 20-22, 2012
[P2 , C] is stable for all compensators C, satisfying bP,C > β if, and only if, δν (P1 , P2 ) ≤ β. The ν−gap metric can be calculated by using the following formula: δν (P1 , P2 ) =
inf ∥G1 − G2 Q∥∞ , Q, Q ∈ L∞ wno det(Q) = 0 −1
(8)
where G1 and G2 are the normalized coprime factorizations of P1 respectively P2 . We compare the factor-class metric, δF -metric (7), with ν−gap metric (8). Let G = I and F1 = G1 is the normalized coprime factorization of the nominal plant P1 . Then according to (7) δF (F1 , F2 ) = inf ∥F1 − F2 Q∥∞ . Q∈U0
The calculations formulaes for δF −metric and ν−gap metric are quite similar. Since U0 ⊂ {Q : Q, Q−1 ∈ L∞ , wno det(Q) = 0}, then δ ≥ δν . But the radius of the largest stabilizable ball in the metric δν is the same as in the δF −metric, and is precisely the distance in the metric to the nearest non-stabilizable plant. Indeed, in Vinnicombe [2001] it was proved the following theorem: Theorem 9. Given a plant P0 , then there exists a sequence of plants, Pε , with the properties that limε→0 δν (P0 , Pε ) = bopt (P0 ), and given any compensator C, there exists an ε0 such that C destabilizes Pε ∀ ε < ε0 . We know from theorem (3) that in the case of worst uncertainty, δopt , Φδ = F +Gδ can be factorize as Φδ = uψ, where wno u > 0 ⇒ in the boundary there is a plant that has an unstable pole/zero cancelation. We have a nonstabilizable plant on the boundary of the stability region in ν−gap metric as well as in the δF metric. In all other cases, when G ̸= I or F1 is not normalized coprime factorization, the ν-gap metric does not generally provide the optimal stability margin, whereas the δF metric does. The most advantage of the δF -metric is that we consider the structured uncertainty. In this sense, the δF -metric is more appropriate to robust stabilization of parametrically uncertain plants. 6. CONCLUSION In this paper we studied the duality principle in the robust optimization problem of rank one. We introduced a metric in the factor space of factorizations of the plant as a distance between the equivalent classes. It has been shown that this metric has close connection with the ν-gap metric. Appendix A. PROOF OF THEOREM 4 Proof. Function δF (·, ·) is a metric on the space of factorization if, for all factorizations F1 , F2 and F3 , we have (1) δF (F1 , F2 ) ≥ 0, with δF (F1 , F2 ) = 0 if and only if F1 ∼ F2 ; (2) δF (F1 , F2 ) = δF (F2 , F1 ); 188
(3) δF (F1 , F2 ) ≤ δF (F1 , F3 ) + δF (F3 , F2 ). 1. It is clear that δF (F1 , F2 ) = inf ∥δ∥∞ ≥ 0. Next, if F1 ∼ F2 then F1 ∈ class (F2 ) and δF (F1 , F2 ) = 0. If now δF (F1 , F2 ) = 0 then we can find a sequence δn such that there are Qn ∈ U0 satisfying F1 + Gδn = F2 Qn , and ∥δn ∥∞ → 0. There is Q0 ∈ U0 such that ∥Qn − Q0 ∥∞ → 0. Wi will show that F1 = F2 Q0 . Indeed, ∥F2 Q0 − F1 ∥∞ ≤ ∥F2 Q0 − F2 Qn ∥∞ + ∥F2 Qn − F1 ∥∞ ≤ ∥F2 ∥∞ ∥Q0 − Qn ∥∞ + ∥F2 Qn − F1 ∥∞ ≤ ε, ⇒ F1 = F2 Q0 . It means that F1 ∈ class (F2 ) and F1 ∼ F2 . To prove 2. we take δ such that F1 + Gδ = F2 Q. Then we multiply both parts with Q∗ and we get (F1 + Gδ)Q∗ = F2 QQ∗ = F2 and F2 − GδQ∗ = F1 Q∗ ⇒ F2 − GδQ∗ ∈ class (F1 ), since wno Q∗ = −wno Q = 0. Finely, δF (F1 , F2 ) = inf ∥δ∥∞ = inf ∥ − δQ∗ ∥∞ = δF (F2 , F1 ). To prove 3. we take δ13 such that we can find Q1 : F1 + Gδ13 = F3 Q1 . Take δ23 such that we can find Q2 : F3 + Gδ23 = F2 Q2 . Then by the simple calculation we get F1 + Gδ13 = F2 Q2 Q1 − Gδ23 Q1 F1 + G(δ13 + δ23 Q1 ) = F2 Q2 Q1 , where wno Q2 Q1 = wno Q2 + wno Q1 = 0. It means that F1 + G(δ13 + δ23 Q1 ) ∈ class (F2 ). Then δF (F1 , F2 ) ≤ ∥δ13 + δ23 Q∥∞ ≤ (∥δ13 ∥∞ + ∥δ23 Q∥∞ ) ⇒ δF (F1 , F2 ) ≤ inf inf (∥δ13 ∥∞ + ∥δ23 Q∥∞ ) = δ13 δ23
δF (F1 , F3 ) + δF (F3 , F2 ).
REFERENCES A. Ghulchak and A. Rantzer. Duality in H∞ cone optimization. SIAM J.Control Optim., 2002. S. Iantchenko and A. Ghulchak. Canonical Parametrization of the Dual Problem in Robust Optimization: NonRational Case. European Control Conference, Kos, Greece, 2007. A. Rantzer and A. Megretski. A convex parametrization of robustly stabilizing controllers. IEEE Transaction on Automatic Control, 39(9):1802–1808, 1994. G. Vinnicombe. Uncertainty And Feedback. Imperial College Press, 2001.
Metric on the factorizations factor-classes as the measure of robustness S. Iantchenko ∗ A. Ghulchak ∗∗ ∗
Department of Mathematics, Lund University, Box 118, 221 00 Lund, Sweden, e-mail: [email protected] ∗∗ Department of Mathematics, Lund University, Box 118, 221 00 Lund, Sweden, e-mail: [email protected] Abstract: The factor space of factorization is defined. A metric, called δF −metric, was introduced. It has been shown that δF −metric has close connection with the ν−gap metric. Furthermore, the δF -metric is more appropriate to robust stabilization of parametrically uncertain plants then ν−gap metric. 1. NOTATION By R (or C) we denote the field of real (or complex) numbers. The unit circle and the open unit disc in C are denoted by T respectively D T = {z ∈ C | |z| = 1} D = {z ∈ C | |z| < 1}. Let Y ⊂ Cn be any measurable set and 1 ≤ p < ∞. Denote by Lp (Y ) the standard Lebesgue space of functions f : T → Y equipped with the norm ∫ 1 ( |f (z)|p dm(z)) p , 1 ≤ p < +∞, ∥f ∥p = T ess sup |f (z)|, p = +∞ z∈T
where | · | denotes the usual 2-norm in Cn √ |f | = |f1 |2 + |f2 |2 + . . . + |fn |2 . The Hardy class Hp (Y ) consists of functions in Lp (Y ) that can be analytically continued inside the unit disk. A function θ analytic in D is called an inner function if θ ∈ H∞ and |θ(z)| = 1 for almost all z ∈ T. A function h analytic in D is called an outer function if there exists a real function g ∈ L1 and a complex number c of modulus 1 such that (∫ ) z+λ h(λ) = c exp g(z)dm(z) , λ ∈ D T z−λ If f ∈ Hp , then f admits the representation f = θh, where θ is an inner function and h is an outer function. Let Hp0 (Y ) denote the shifted Hp (Y ) Hp0 (Y ) = zHp (Y ) = {f ∈ Hp (Y )|f (0) = 0}. The disk algebra A(Y ) is by definition the subspace of H∞ that consists of analytic functions in D ⊂ Y that can be extended continuously to the closed unit disk. The brief notations A, H∞ etc. will be used if Y = Cn and the dimension of the space is clear from context. 978-3-902823-03-8/12/$20.00 © 2012 IFAC
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Two matrices N, M ∈ H∞ are right-coprime if they have equal number of columns and there exist X, Y ∈ H∞ such that XN + Y M = I. Let G be a matrix function. We say that the factorization G = N M −1 is a right coprime factorization over H∞ if N and M are right-coprime H∞ -matrices. The factorization G = N M −1 is called a normalised right coprime factorization if N ∗ N + M ∗ M = I. The matrix function V is called co-outer if V ⊤ is outer. η[G] is the number of open right half plane poles of G(s). Z[G] is the number of open right half plane zeros of G(s). Notation [P, C] refer to the standard feedback connection of a plant P with a controller C. The superscript ⊤ stands for transposition and † stands for pseudoinverse. The bar denotes the complex conjugate and ∗ denotes conjugate transpose. The prefix B denotes the unit ball in the corresponding space. 2. INTRODUCTION The problem of robust optimization have been considered by a number of authors. Much work has been devoted to problems stated in terms of convex optimization. In particular, it has been shown in Rantzer and Megretski [1994] that the robust stabilization problem under parametric uncertainties has a convex formulation if the characteristic polynomial depends linearly on the uncertain parameters (so called rank one problem). The convex parametrization gives great benefits both for theoretical analysis and for practical computations. However, when an optimization problem is infinite-dimensional, a reduction to finite-dimensional form (like Ritz projection, Galerkin finite element scheme, grid methods, etc.) is needed. Such an approximation introduces conservatism to the problem. The gap between the true optimal value and its finitedimensional counterpart can be arbitrarily large. To overcome this difficulty and to estimate the conservatism, convex duality has been used. The dual problem for systems with uncertainties of rank one was introduced in Ghulchak 10.3182/20120620-3-DK-2025.00074
7th IFAC Symposium on Robust Control Design Aalborg, Denmark, June 20-22, 2012
and Rantzer [2002]. It was obtained an analytical expression of the infinite-dimensional dual problem. It was shown that result can be used for numerical optimization of the robustness margin by primal-dual methods based on finitedimensional approximation. The dual factorization of all plant factors with destabilizing uncertainties was obtained in Iantchenko and Ghulchak [2007]. In this paper, we continue to study the duality principle in robust optimization problems of rank one. We show that the dual problem can be interpreted as a minimization of all destabilizing uncertainties in a certain metric that has interesting connections to the well-known ν-gap metric. 3. PRELIMINARIES The main problem of robust stabilization is stated as follows: Given a uncertain plant P ( ) ( ) y w =P , z u w = δ ⊤ z, where δ ∈ ν∆ for some convex compact ∆, with 0 ∈ ∆, the problem is to find a controller u = Ky that robustly stabilizes the plant for as large ν as possible. According to the Youla parametrization of internal stabilizing controllers we assume that the transfer function from w to z is given on the form z = (T1 + T2 Q)w, and T2 ∈ RH∞ where T1 ∈ m×n are fixed and parameter Q is any transfer matrix in RH∞ n×1 . The system becomes robustly stable if and only if RH∞ m×1
[1 − δ ⊤ (T1 + T2 Q)]−1 ∈ RH∞ for all |δ| ≤ ν, here | · | is some norm on Rm . The main problem is stated as follows ∞ Given T1 ∈ RH∞ m×1 , T2 ∈ RHm×n , find a convex ∞ parametrization of all Q ∈ RHn×1 such that [1 − δ ⊤ (T1 + T2 Q)]−1 ∈ RH∞ for all δ ∈ Rm with |δ| ≤ ν.
The problem was solved in Rantzer and Megretski [1994]. ∞ Theorem 1. Suppose T1 ∈ RH∞ m×1 , T2 ∈ RHm×n . Then the following two conditions on the rational matrix Q are equivalent: (1) Q ∈ RH∞ n×1 and for all δ ∈ Rm with |δ| ≤ ν [1 + δ ⊤ (T1 + T2 Q)]−1 ∈ RH∞ . (2) There exist α ∈ RH∞ and β ∈ RH∞ such that Q = β/α and ∀ω ∈ R ∪ {∞}, |Re[T1 α + T2 β](iω)|d < ν −1 Re α(iω). Here | · |d is the dual norm defined by |x|d = max{x⊤ y : |y| ≤ 1}. The convex parametrization of all controllers that stabilize the system for all possible combinations of parameters makes it possible to use convex optimization to find a robustly stabilizing controller.
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Let us consider the family of perturbed plants Pδ =
N + G1 δ , δ ∈ ν∆. M + G2 δ
The problem is to find a controller K that robustly stabilizes Pδ for ν as large as possible. Denote by F = (N M ) ∈ An×1 the right coprime factorization of the plant P , by G = (G1 G2 ) ∈ An×m the weight matrix and the set ∆ is a convex compact set in Cm . According to a convex parametrization of all robustly stabilizing controllers (Theorem 1) we define the primal problem as the problem of finding a function h ∈ BH∞ such that the condition Re h(z)(F (z) + G(z)δ) > 0, ∀z ∈ T, ∀δ ∈ ν∆, (1) is satisfied. We would like to solve it for ν as large as possible, that is, for νopt = sup{ν|∃h ∈ BH∞ : Re h(F + Gδ) > 0 ∀δ ∈ ν∆}. (2) In Ghulchak and Rantzer [2002] the dual problem was introduced as follows: Theorem 2. Let F ∈ An×1 , G ∈ An×m and denote Φδ = F + Gδ. Then the optimal value νopt takes the following form νopt = min{νopt|c , νopt|s } with the regular part νopt|c = inf{ν|∃w ∈ L1 (R+ )\0, δ ∈ L∞ (ν∆) : Φδ w ∈ H10 } (3) and the singular part νopt|s = inf{ν|∃z ∈ T, δ ∈ ν∆ : Φδ (z) = 0}. (4)
We will consider the regular case, i.e. νopt|c < νopt|s . The canonical parametrization of the dual problem in the regular case was obtained in Iantchenko and Ghulchak [2007] as follows: We need to find the functions wopt ∈ L1 (R+ )\0 and δopt ∈ L∞ (ν∆) such that Φδ w ∈ H10 . The last condition means that there exists a vector function p ∈ H1 such that Φδ (z)w(z) = zp(z). Note that in the regular case we have |Φδ | ≥ ε, which implies, in particular, that log(w) ∈ L1 . A real positive function with this property can be factorized as w = f ∗ f , where f is a scalar outer function. Then Φδ can be written as p f p Φδ = z ∗ = z ∗ 2 . f f f f Furthermore, the function p can be factorized into inner and co-outer factors p = pi pco , where pi is scalar, and the function Φδ may be written as f pco (5) Φδ = zpi ∗ 2 = uψ, f f where u = zpi ff∗ is a scalar function and ψ ∈ H∞ is an coouter function. In the rational case, the rational function u collects all common zeros of the plant factors and gives a factorization of the dual problem via unstable zeropole cancelations. For non-rational systems the situation with the common zeros is more complicated. The nominal
7th IFAC Symposium on Robust Control Design Aalborg, Denmark, June 20-22, 2012
factors can contain a singular component and cannot be treated by unstable cancelations. It was shown that in the case when the uncertainty set is the ball ∆ = BCm , the winding number for the function u is well-defined. It plays the same role as the unstable cancelations in the rational case. We introduce the class U of all unimodular functions with the well-defined and finite winding numbers U = {u ∈ L∞ : |u| ≡ 1, wno u > 0}. Theorem 3. Let νopt|c < νopt|s . Then the plant numerator and denominator with the worst uncertainty Φδ = F + Gδ can be factorized in the following way Φδ = uψ, (6) ∞ where u ∈ U and ψ ∈ H is co-outer. 4. METRIC ON THE FACTOR-CLASS Consider the dual parametrization F + Gδ = uψ, u ∈ U, ψ ∈ H∞ . Recall that for the optimal δ we have wno u > 0, i.e. on the boundary of the stability region, we have a nonstabilizable plant, the plant that has an unstable pole/zero cancelation. This gives an idea how to define the classes of equivalence in the factor space. Then we can introduce a distance between them. Let U0 be the class of all unimodular functions with welldefined winding numbers given by U0 = {u ∈ L∞ : |u| ≡ 1, wno u = 0}. Suppose [ ] that we have two plants P1 and P2 . Let[F1 = ] N1 N2 be a coprime factorization of P1 and F2 = M1 M2 be a coprime factorization of P2 . We will call two factorizations equivalent F1 ∼ F2 if we can find Q such that Q ∈ U0 and F2 = F1 Q. We define the distance between equivalent classes as follows: δF (F1 , F2 ) = inf{∥δ∥∞ : F1 + Gδ ∈ class (F2 )}. Theorem 4. δF is a metric on the factor space of factorizations.
Small gain methods consider stable perturbations. The homotopy methods give stronger results since the homotopy arguments are build on perturbations which are bounded on the imaginary axis, and on counting winding numbers. We introduce δF −metric techniques for studying robust stability and show that the δF −metric techniques gives the same strong results as the homotopy methods. 5.1 Additive uncertainty We consider the family of perturbed plans: {P + ∆ : ∆ ∈ L∞ , ∥W1 ∆W2 ∥∞ < 1.} Using the homotopy methods (see for exemple, Vinnicombe [2001]) we get the following theorem: Theorem 5. Given P, C real rational transfer functions, W1 , W1−1 , W2 , W2−1 ∈ RH∞ : The following are equivalent • [P1 , C] is stable for all P1 of the form {P1 = P + ∆ : ∆ ∈ H∞ , ∥W1 ∆W2 ∥∞ < 1}. • [P1 , C] is stable for all P1 of the form {P1 = P + ∆ : ∆ ∈ L∞ , ∥W1 ∆W2 ∥∞ < 1, η(P + ∆) = η(P )}. We will interpret the poles condition in term of δF −metric. Let (N, M ) be a coprime factorization of P. Then P + ∆ = N M −1 + ∆ = (N + ∆M )M −1 . [P1 , C] is stable for all] P1 =[P + ∆ ] if we can found Q ∈ U0 [ N1 N + M∆ =Q . Function Q will collect such that M M1 all common zeros of the plant factors and all poles. The poles of P are exactly the zeros of M. Furthermore the poles of N + M ∆ are the poles of ∆. Denote by Zc the common zeros of M and of N + M ∆. Since wno Q = 0, then Z(Q) − η(Q) = 0 ⇒ Zc − η(∆) = 0. We get following: η(P + ∆) = η(∆) + Z(M ) − Zc = Z(M ) = η(P ). We have shown that poles condition are equivalent with winding number condition on function Q in δF −metric. 5.2 Multiplicative uncertainty We allow the perturbed plans to take the form {(I + ∆)P : ∥W1 ∆W2 |∞ < 1.}
Proof. See Appendix A Now we can get an expression for our metric δF (F1 , F2 ). By the definition of δF (F1 , F2 ) we are looking for such a δ that F1 + Gδ ∈ class (F2 ). It means that there exists Q ∈ U0 such that F1 + Gδ = F2 Q. We have to take an infimum over all such Q in order to find the optimal δ, i.e. the distance between factor-classes. Then we get δF (F1 , F2 ) = inf ∥G† F1 − G† F2 Q∥∞ . (7) Q∈U0
5. COMPARISON WITH PREVIOUS RESULTS. We gather several problems in robust stability. There are two tools for the study of robust stability: • the small-gain theorem, • homotopy methods. 186
It was shown the similar result Theorem 6. Given P, C real rational transfer functions, W1 , W1−1 , W2 , W2−1 ∈ RH∞ : The following are equivalent • [P1 , C] is stable for all P1 of the form {P1 = (I + ∆)P : ∆ ∈ H∞ , ∥W1 ∆W2 ∥∞ < 1}. • [P1 , C] is stable for all P1 of the form {P1 = (I + ∆)P : ∆ ∈ L∞ , ∥W1 ∆W2 ∥∞ < 1, η((I + ∆)P ) = η(P )}. In the same way we interpret the poles condition in term of δF −metric. (1 + ∆)P = (1 + ∆)N M −1 , where (N, M ) is a coprime factorization of P. [P1 , C] is stable for [ all P1 =] (1 + [∆)P ]if we can found (1 + ∆)N N1 Q ∈ U0 such that =Q . We know that M M1
7th IFAC Symposium on Robust Control Design Aalborg, Denmark, June 20-22, 2012
the function Q will collect all common zeros of the plant factors and all poles. The poles of P are the zeros of M. The poles of (1 + ∆)N are the poles of ∆ minus possible cancelations with zeros of N. We denote the number of poles of (1 + ∆)N by ηc (∆). Zc is the common zeros of M and of (1 + ∆)N. Since wno Q = 0, then Zc − ηc (∆) = 0. We get the same result as in the previous case: η(1 + ∆)P = ηc (∆) + Z(M ) − Zc = Z(M ) = η(P ). Again the poles condition are equivalent with the winding number condition on function Q in δF −metric. 5.3 Inverse multiplicative uncertainty The inverse multiplicative uncertainty gives us the uncertainty model in the form: {(I − ∆)−1 P : ∥W1 ∆W2 ∥∞ < 1}. We have the following result: Theorem 7. Given P, P −1 , C real rational transfer functions, W1 , W1−1 , W2 , W2−1 ∈ RH∞ : The following are equivalent • [P1 , C] is stable for all P1 of the form {P1 = (I − ∆)−1 P : ∆ ∈ H∞ , ∥W1 ∆W2 ∥∞ < 1} and (I − ∆) is invertible. • [P1 , C] is stable for all P1 of the form {P1 = (I − ∆)−1 P : ∆ ∈ L∞ , ∥W1 ∆W2 ∥∞ < 1, Z((I − ∆)−1 P ) = Z(P )} and (I − ∆) is invertible. In the same way we interpret in this case the zeros condition in term of δF −metric. (1−∆)−1 P = N ((1−∆)M )−1 , where (N, M ) is a coprime factorization of P. −1 [P1 , C] is stable for [ all P1 = (1 ] − ∆)[ P ]if we can found N N1 Q ∈ U0 such that =Q . The function (1 − ∆)M M1 Q will collect all common zeros of the plant factors and all poles. The zeros of P are the zeros of N. We denote by Zc the common zeros of N and the zeros of (1 − ∆)M. Since wno Q = 0, then Zc = η((1 − ∆)M ). We get the following result: Z((I − ∆)−1 P ) = η((1 − ∆)M ) + Z(N ) − Zc = Z(N ) = Z(P ). The zeros condition are equivalent with the winding number condition on function Q in δF −metric. 5.4 Coprime factor uncertainty Coprime factor perturbations can be describe in the form
[ { ] }
∆N 1 −1
P1 = (N + ∆N )(M + ∆M ) : , < ∆ M ∞ γ
where P = N M −1 is normalized right coprime factorization of P and γ > 1. The following result was obtained: Theorem 8. Given P real rational transfer function and γ > 1, the following are equivalent • [P1 , C] is stable for all P1 of the[ form] ∆N P1 = (N + ∆N )(M + ∆M )−1 : ∈ H∞ , ∆M
[ ]
∆N 1
∆M ≤ γ and (M + ∆M ) is invertible. ∞ 187
• [P1 , C] is stable for all P1 of the[ form] ∆N P1 = (N + ∆N )(M + ∆M )−1 : ∈ L∞ , ∆M
[ ]
∆N 1
∆M ≤ γ , (M + ∆M ) is invertible and ∞ η(P1 ) = wno det(M + ∆M ). We analyze the winding number condition on the M +∆M with help of δF −metric. Let Zc be the number of common zeros of N + ∆N and M +∆M . Since wno Q = 0, then Zc −η(∆N )−η(∆M ) = 0. We have the following result η(P1 ) = η(∆N ) + Z(M + ∆M ) − Zc = η(∆N ) + Z(M + ∆M ) − η(∆N ) − η(∆M ) = Z(M + ∆M ) − η(M + ∆M ) = wno (M + ∆M ). We get the same result: the winding number condition on the M + ∆M are equivalent with the winding number condition on function Q in δF −metric. 5.5 δF −metric and ν−gap metric. In Vinnicombe [2001] it was considered the problem of minimizing
[ ]
P
−1
I (I − CP ) [ −C I ] , ∞ which is the transfer function from v1 and v2 , to u and y, in the standard feedback configuration, P is the nominal plant, C is the controller. Define [ ]
−1
P (I − CP )−1 [ −C I ] , [P, C] stable
bP,C := I ∞ 0, otherwise, and bopt (P ) = sup bP,C . C
The problem of finding bopt can be considered as primal problem. This primal problem is quite similar to the our primal problem. In order to measure uncertainty it was introduced the ν− gap metric as follows: Let P1 and P2 be the two plants such that η[P2 , −P1∗ ] = η[P1 , −P1∗ ]. Then
[ ]
−1 √
P2
∗ ∗ −1 2
1 − δν (P1 , P2 ) = (I + P1 P2 ) [ P1 I ]
. I ∞ The most fundamental property is that the stability margin bP,C and the uncertainty measure δν (P1 , P2 ) are related by the inequality arcsin bP2 ,C ≥ arcsin bP1 ,C − arcsin δν (P1 , P2 ). It means that if a compensator C performs well with the plant P1 and δν (P1 , P2 ) is sufficiently small, then C is guaranteed to stabilizes P2 with a certain level of performance. The measure δν (P1 , P2 ) can be considered as dual to the bP,C (and for optimal controller, as dual to the bopt (P ).) It was shown the following relation: Given a nominal plant P1 , a perturbed plant P2 and and a number β < bopt (P1 ) then:
7th IFAC Symposium on Robust Control Design Aalborg, Denmark, June 20-22, 2012
[P2 , C] is stable for all compensators C, satisfying bP,C > β if, and only if, δν (P1 , P2 ) ≤ β. The ν−gap metric can be calculated by using the following formula: δν (P1 , P2 ) =
inf ∥G1 − G2 Q∥∞ , Q, Q ∈ L∞ wno det(Q) = 0 −1
(8)
where G1 and G2 are the normalized coprime factorizations of P1 respectively P2 . We compare the factor-class metric, δF -metric (7), with ν−gap metric (8). Let G = I and F1 = G1 is the normalized coprime factorization of the nominal plant P1 . Then according to (7) δF (F1 , F2 ) = inf ∥F1 − F2 Q∥∞ . Q∈U0
The calculations formulaes for δF −metric and ν−gap metric are quite similar. Since U0 ⊂ {Q : Q, Q−1 ∈ L∞ , wno det(Q) = 0}, then δ ≥ δν . But the radius of the largest stabilizable ball in the metric δν is the same as in the δF −metric, and is precisely the distance in the metric to the nearest non-stabilizable plant. Indeed, in Vinnicombe [2001] it was proved the following theorem: Theorem 9. Given a plant P0 , then there exists a sequence of plants, Pε , with the properties that limε→0 δν (P0 , Pε ) = bopt (P0 ), and given any compensator C, there exists an ε0 such that C destabilizes Pε ∀ ε < ε0 . We know from theorem (3) that in the case of worst uncertainty, δopt , Φδ = F +Gδ can be factorize as Φδ = uψ, where wno u > 0 ⇒ in the boundary there is a plant that has an unstable pole/zero cancelation. We have a nonstabilizable plant on the boundary of the stability region in ν−gap metric as well as in the δF metric. In all other cases, when G ̸= I or F1 is not normalized coprime factorization, the ν-gap metric does not generally provide the optimal stability margin, whereas the δF metric does. The most advantage of the δF -metric is that we consider the structured uncertainty. In this sense, the δF -metric is more appropriate to robust stabilization of parametrically uncertain plants. 6. CONCLUSION In this paper we studied the duality principle in the robust optimization problem of rank one. We introduced a metric in the factor space of factorizations of the plant as a distance between the equivalent classes. It has been shown that this metric has close connection with the ν-gap metric. Appendix A. PROOF OF THEOREM 4 Proof. Function δF (·, ·) is a metric on the space of factorization if, for all factorizations F1 , F2 and F3 , we have (1) δF (F1 , F2 ) ≥ 0, with δF (F1 , F2 ) = 0 if and only if F1 ∼ F2 ; (2) δF (F1 , F2 ) = δF (F2 , F1 ); 188
(3) δF (F1 , F2 ) ≤ δF (F1 , F3 ) + δF (F3 , F2 ). 1. It is clear that δF (F1 , F2 ) = inf ∥δ∥∞ ≥ 0. Next, if F1 ∼ F2 then F1 ∈ class (F2 ) and δF (F1 , F2 ) = 0. If now δF (F1 , F2 ) = 0 then we can find a sequence δn such that there are Qn ∈ U0 satisfying F1 + Gδn = F2 Qn , and ∥δn ∥∞ → 0. There is Q0 ∈ U0 such that ∥Qn − Q0 ∥∞ → 0. Wi will show that F1 = F2 Q0 . Indeed, ∥F2 Q0 − F1 ∥∞ ≤ ∥F2 Q0 − F2 Qn ∥∞ + ∥F2 Qn − F1 ∥∞ ≤ ∥F2 ∥∞ ∥Q0 − Qn ∥∞ + ∥F2 Qn − F1 ∥∞ ≤ ε, ⇒ F1 = F2 Q0 . It means that F1 ∈ class (F2 ) and F1 ∼ F2 . To prove 2. we take δ such that F1 + Gδ = F2 Q. Then we multiply both parts with Q∗ and we get (F1 + Gδ)Q∗ = F2 QQ∗ = F2 and F2 − GδQ∗ = F1 Q∗ ⇒ F2 − GδQ∗ ∈ class (F1 ), since wno Q∗ = −wno Q = 0. Finely, δF (F1 , F2 ) = inf ∥δ∥∞ = inf ∥ − δQ∗ ∥∞ = δF (F2 , F1 ). To prove 3. we take δ13 such that we can find Q1 : F1 + Gδ13 = F3 Q1 . Take δ23 such that we can find Q2 : F3 + Gδ23 = F2 Q2 . Then by the simple calculation we get F1 + Gδ13 = F2 Q2 Q1 − Gδ23 Q1 F1 + G(δ13 + δ23 Q1 ) = F2 Q2 Q1 , where wno Q2 Q1 = wno Q2 + wno Q1 = 0. It means that F1 + G(δ13 + δ23 Q1 ) ∈ class (F2 ). Then δF (F1 , F2 ) ≤ ∥δ13 + δ23 Q∥∞ ≤ (∥δ13 ∥∞ + ∥δ23 Q∥∞ ) ⇒ δF (F1 , F2 ) ≤ inf inf (∥δ13 ∥∞ + ∥δ23 Q∥∞ ) = δ13 δ23
δF (F1 , F3 ) + δF (F3 , F2 ).
REFERENCES A. Ghulchak and A. Rantzer. Duality in H∞ cone optimization. SIAM J.Control Optim., 2002. S. Iantchenko and A. Ghulchak. Canonical Parametrization of the Dual Problem in Robust Optimization: NonRational Case. European Control Conference, Kos, Greece, 2007. A. Rantzer and A. Megretski. A convex parametrization of robustly stabilizing controllers. IEEE Transaction on Automatic Control, 39(9):1802–1808, 1994. G. Vinnicombe. Uncertainty And Feedback. Imperial College Press, 2001.