A Hyperbolic View on Robust Control ?

Preprints of the 19th World Congress The International Federation of Automatic Control Cape Town, South Africa. August 24-29, 2014

A hyperbolic view on robust control ?

Z. Szab´o ∗ J. Bokor ∗∗ T. V´amos ∗

∗ Institute for Computer Science and Control, Hungarian Academy of Sciences, Hungary, (Tel: +36-1-279-6171; e-mail: [email protected]). ∗∗ Institute for Computer Science and Control, Hungarian Academy of Sciences, Hungary, MTA-BME Control Engineering Research Group.

Abstract: The intention of the paper is to demonstrate the beauty of geometric interpretations in robust control. We emphasize Klein’s approach, i.e., the view in which geometry should be defined as the study of transformations and of the objects that transformations leave unchanged, or invariant. We demonstrate through the examples of the basic control tasks that a natural framework to formulate different control problems is the world that contains as points the equivalence classes determined by the stabilizable plants and whose natural motions are the Möbiustransforms. Transformations of certain hyperbolic spaces put light into the relations of the different approaches, provide a common background of robust control design techniques and suggest a unified strategy for problem solutions. Besides the educative value a merit of the presentation for control engineers might be a unified view on the robust control problems that reveals the main structure of the problem at hand and give a skeleton for the algorithmic development.

1. INTRODUCTION AND MOTIVATION on the relationship between the properties of analyticity and a hyperbolic metric on the disk. In these develop- Geometry is one of the richest areas for mathematical ments rational inner functions (finite Blaschke products) exploration. The visual aspects of the subject make explo- are central objects and, due to the time invariant setting, ration and experimentation natural and intuitive. At the a characterization of shift invariant subspaces in terms of same time, the abstractions developed to explain geomet- (rational) inner functions – a Krein space generalization ric patterns and connections make the approach extremely of the Beuerling–Lax theorem – was the main ingredient powerful and applicable to a wide variety of situations. of the approach, see, e.g., Helton [1978, 1987]. In the nineteenth century development of the Bolyai- All these topics involves an advanced mathematical ma- Lobachevsky geometry, as the first instance of non- chinery in which often the underlying geometrical ideas euclidean geometries, had a great impact on the evolution remain hidden. The aim of the paper is to highlight some of mathematical thinking. Non-Euclidean geometry has of these geometric governing principles that facilitate the turned out to be more than just a logical curiosity, and solution of these problems. We try to avoid the technical many of its basic features continue to play important roles details which can be found in the cited references. in several branches of mathematics and its applications. In many of Euclid’s theorems, he moves parts of figures The non-Euclidean world is something that escapes our on top of other figures. Felix Klein, in the late 1800s, everyday view and its rules and behaviors are only made developed an axiomatic basis for Euclidean geometry that accessible by some auxiliary tools, the so called models, started with the notion of an existing set of transforma- that help our imagination and understanding. It happens tions and he proposed that geometry should be defined that the main building blocks of these models has a certain as the study of transformations (symmetries) and of the relevance for systems and control theory as well, see, e.g., objects that transformations leave unchanged, or invari- Helton [1980, 1982], Khargonekar and Tannenbaum [1985], ant. This view has come to be known as the Erlanger Hassibi et al. [1999], Allen and Healy [2003]. Program. The set of symmetries of an object has a very nice algebraic structure: they form a group. By studying Early approaches of the H∞ theory were formulated in this algebraic structure, we can gain deeper insight into the frequency domain, and the solution of multivariate the geometry of the figures under consideration. Another interpolation problems played the central role. The con- advantage of Klein’s approach is that it allows us to relate struction amounts to finding solutions to Nevanlinna-Pick different geometries. In this paper we put an emphasize on interpolation problems, which is related, at a mathemati- this concept of the geometry and its direct applicability to cal level, to the Schwarz lemma. The later is a statement control problems.

? The research has been conducted as part of the project TAMOP-´ Section 2 lists some basic features of hyperbolic geometry 4.2.2.A-11/1/KONV-2012-0012: Basic research for the development relevant to our presentation. We are not going to elaborate of hybrid and electric vehicles. The Project is supported by the the hyperbolic geometry in details, just highlight some Hungarian Government and co-ﬁnanced by the European Social facts illustrated through the Poincar´edisc model of this ge- Fund.

Copyright © 2014 IFAC 718 19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014 ometry. For a more detailed elaboration of the subject, see, 1+r Observe that the map r 7→ 1−r sends the interval [0, 1) to e.g., Gans [1973], Kelly [1981], Anderson [2005]. Section [1, ∞). Since the points 0, z lie on a diameter, it’s omega 3 presents the Klein view through the projective matrix points are given by ± z . Thus space. The material is based on Schwarz and Zaks [1981, |z| 1985]. The relevance of this model to control problems is z z 1 + |z| d (0, z) = | log(%(0, − , z, ))| = log( ). detailed in Section 4. H |z| |z| 1 − |z|

z−z1 The hyperbolic translation takes z1 7→ 0 and z2 7→ 2. ELEMENTS OF HYPERBOLIC GEOMETRY 1−z¯1z z2−z1 . Then the general formula follows as: 1−z¯1z2 To visualize hyperbolic geometry, we have to resort to z − z d (z , z ) = 2 arctanh | 2 1 |. (1) a model. In the Poincarémodel the hyperbolic plane is H 1 2 1 − z¯ z the unit disk, and points are Euclidean points. Lines are 1 2 portions of circles intersecting the disk and meeting the We often prefer to use the pseudo-hyperbolic distance p (u, v) = | v−u | since it is algebraic whereas the expres- boundary at right angles. The angles for the model are H 1−uv¯ the same as Euclidean angles. A model with the property sion for dH is not. The main difference is that the pseudo- that angles are faithfully represented is called a conformal hyperbolic distance is not additive along geodesics. model. A hyperbolic circle is drawn as a Euclidean circle, Hyperbolic geometry has no preferred points, but in the but its center becomes lopsided toward the outer edge of Poincarédisk model, however, the origin has a very special the unit disk. role. Similarly for diameters, especially the one on the real Since the reciprocal has a problem at z = 0, the complex axis. Since hyperbolic distance is based on cross ratios, and plane is extended to the Riemann sphere (one-point com- cross ratios are invariant under Möbiustransformations, we can measure distances between points by moving their pactification) denoted by ˆ = + {∞}. Given a matrix C C lines into a special position using a hyperbolic isometry. a b M = with det(M) = ad − bc = 1 one can define In general: any geometric property of a configuration of c d points which is invariant under a hyperbolic isometry, may a special fractional-linear transformation of the Riemann az+b be reliably investigated after the data has been moved into sphere, the Möbiustransformation, as µM (z) = cz+d . a convenient position in the model. This is the idea that Every Möbiustransformation is a bijection of the Riemann will be used in the solution of basic control problems, too. sphere and they form a group, i.e., µM ◦ µN = µMN and −1 µM = µM −1 . 2.1 Extension to operator balls The geometrical meaning of the parameters a, b, c, d is not so easy to determine, even if we normalize them to For the Hilbert spaces K, H consider the open unit ball B ad − bc = 1. Define the cross ratio as of all linear bounded operators from K to H (L(K, H)). (p − q) (r − s) The map h : B 7→ H is holomorphic on B if the Fréchet %(p; q, r, s) = . (p − s) (r − q) derivative of h at x exists as a bounded complex linear map of K into H for each x ∈ B. Invertible holomorphic maps Then the Möbiustransformation can be written in the from B onto B are called biholomorphic automorphisms. cross ratio form µM (z) = %(z; z0, z1, z2) with z0 = −b/a, z1 = (d − b)/(a − c), z2 = −d/c. In this setting the operator valued Blaschke functions Three points determine the Möbiustransformation: given −1/2 ∗ −1 1/2 BA(X) = NA (A + X)(1 + A X) NA∗ , (z0, z1, z2) defining a cross ratio, the corresponding Möbius ∗ z−z0 with NA = I − AA are biholomorphic automorphisms transformation is µM (z) = m with −1 ∗ ∗ z−z2 of B (B = B−A). Then BA(X) = BA∗ (X ) and A m −mz z1 − z2 kB (X)k ≤ B (kXk). Moreover, every biholomor- M = 0 , m = . A kAk 1 −z2 z1 − z0 phic mapping h is of the form h = Bh(0)(UXV ) = The cross ratio is invariant under the Möbiustransforma- UB−h−1(0)(X)V , where U and V are unitary operators. tion: %(z0; z1, z2, z3) = %(w0; w1, w2, w3) if wi = µM (zi). The metric defined as 1 + kB−A(B)k Möbiustranformations that map the unit disc onto ρ(A, B) = ln = arctanh(kB (B)k) D B −A itself form a subgroup, the hyperbolic group. They can be 1 − k −A(B)k iθ is invariant with respect to biholomorphic automorphisms written as Vα,θ(z) = e Bα(z) where α ∈ D and Bα(z) = z−α is an elementary Blaschke function. Substitution and provides an extension of the Poincarédisk model of 1−αz¯ the hyperbolic geometry to the operator ball. For details reveals that B (α) = 0, B (α ) = 1 and B (α ) = ∞, α α 1 α ∞ see, e.g., Harris [1973], Khatskevich [1984], Ostrovskii et al. where α = 1+α and α = 1 . Note that α is the point 1 1+¯α ∞ α¯ ∞ [2009]. symmetric to α with respect to the unit circle. |Bα(z)| = 1 on the unit circle |z| = 1 and |z| < 1 ⇒ |B (z)| < 1 and α 3. THE KLEIN VIEW OF GEOMETRY |z| > 1 ⇒ |Bα(z)| > 1. Four points lie on the same circle (line) if and only if We have already described the hyperbolic geometry using their cross ratio is a real number. Thus, if the points the Poincarédisk model defined on {z ∈ C : |z| < 1} z1, z2 are inside the unit disk, and ω0, ω1 (omega points) and we can introduce a half plane model on {z ∈ C : are on on the unit circle, then the function dH (z1, z2) = Im(z) > 0} by using a Möbiustransformation, the Cayley | log(%(z1, ω0, z2, ω1))| is a candidate to measure the dis- z − i 1−r 1+r map c : Cˆ → Cˆ defined by c(z) = , to convert between tance. Indeed: %(0, −1, r, 1) = 1+r , i.e., dH (0, r) = log 1−r . z + i

719 19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014 the two models. Note that lines in the Poincarédisc model Using the Hermitian form defined as passing through 1 are in one-to-one correspondence with ∗ I I −r2I 0 the lines that are vertical rays in the upper half plane H (P, r) = H˜ (r) ,H (r) = , e P e P e 0 I model. for r > 0, we can introduce the Euclidean circle as Another model is the Hyperboloid (Minkowski) model I where the points lie on the upper sheet of a hiperboloid, γ (0, r) = P ∈ : P = i ,H (P, r) = 0 . 2 2 2 e Pf P e i.e., H = {(x, y, z) | z = 1 + x + y ≥ 1}. We can map 0 D to H by stereographic projection from Z = (0, 0, −1) In analogy to the one dimensional case we denote by D using the map given by the Euclidean unit disc, i.e., D = {P : d(O,P ) < 1} and 2x 2y 1 + x2 + y2 by D¯ its closure. Note that this is exactly the contractive m(x, y) = ( , , ), 1 − x2 − y2 1 − x2 − y2 1 − x2 − y2 matrix ball in Mn(C). x y −1 p 2 2 −I 0 m (x, y, z = 1 + x + y ) = ( , ). Let J = . Then, a non–Euclidean geometry can z z 0 I The lines of the model are the intersections of with H be defined by using the J -unitary matrices Π ∈ M ( ), planes through the origin. This model is related to special 2n C i.e., Π∗J Π = J . These matrices form a group. relativity and it is also of considerable significance in geometry. The group for the Minkowski model is isomorphic Setting to a subgroup of the projective group P(2) which shows − P1 ∗ ∗ that hyperbolic geometry is related to projective geometry. D = P ∈ Pf : P = i , −P1 P1 + P2 P2 < 0 , P2 3.1 Projective Matrix Space note that D− = D and every P ∈ D− has a representant that can be completed to a J –unitary matrix P˜. Considering the ring M ( ) of the complex matrices and n C − ˜ ˜ the set M n ( ) of the pairs P ,P ∈ M ( ) with If P and Q are in D with P and Q the corresponding (2n,n) C 1 2 n C R1 R3 P1 J –unitary matrices we can set R˜ = = P˜∗J Q.˜ rank = n, we can define the equivalence relation: R2 R4 P2 The pseudo–chordal distance is defined as P Q 1 ∼ 1 if and only if there exists R ∈ GL ( ) P = Q n C %(P ,Q) 2 2 Ψ(P ,Q) = , P Q [1 + %2(P ,Q)]1/2 such that 1 = 1 R. Then the equivalence classes P2 Q2 1 − where %(P ,Q) = kR2k, which is a metric on D . P are considered as the points of the projective space P. n − Accordingly we introduce the map i from M(2n,n)(C) to the The projectivity S maps D onto itself and keeps the pseudo–chordal distance invariant (non–Euclidean mo- P1 P1 −1 space P such that P = i and = i (P ). ˜ P2 P2 tions) if and only if the corresponding matrices S are of the form S˜ = λT˜, T˜∗J T˜ = J , where λ 6= 0. P1 −1 − P is called finite if, for any ∈ i (P ), det(P1) 6= 0 In analogy to the scalar case, for any pair of points in D , P2 we define the non–Euclidean distance Eh(P ,Q) by and Pf denotes the set of finite points. While the points of P correspond to linear subspaces of a fixed dimension, 1 1 + Ψ(P ,Q) I Eh(P ,Q) = log . finite points are related to graph subspaces of the 2 1 − Ψ(P ,Q) P linear operators P ∈ M ( ). P1 n n C For arbitrary ∈ M2n,n(C) we set P2 We consider the action of nonsingular matrices on P ∗ AB P1 ˜ P1 (projectivity), i.e., for S = ∈ GL ( ) and P ∈ Hh(P1,P2, r) = Hh(r) CD 2n C P P2 P2 2 S S S P1 − sinh (r)I 0 the map defined by P = i(P ) where P = S . The with H˜h(r) = , 0 < r < ∞. P2 0 cosh2(r)I projectivities of P form a group under composition. The non–Euclidean circle γh(0, r) and the corresponding Then the matrix Möbiustransformation non–Euclidean disk D−(0, r) are defined as: −1 h MS(P ) = (C + DP )(A + BP ) P1 is the restriction of the projectivity S to the finite points γh(0, r) = P : P = i ,Hh(P1,P2, r) = 0 , P2 of the space, i.e., MS operates on Pf . − P1 Dh (0, r) = P : P = i ,Hh(P1,P2, r) < 0 . On the set of the points the Euclidean distance can be P2 defined as d(P ,Q) = kP − Qk. Then the projectivities − that keep this distance invariant are Both lie in D . For re = tanh(rh), 0 < re < 1 we obtain γ (0, r ) = γ (0, r ) and D−(0, r ) = D−(0, r ). U 0 e e h h e e h h S = λ 1 , U P U 2 0 2 1 For n = 1 this is the classical pseudo–chordal distance, i.e., where P0 is an arbitrary matrix and U1,U2 are unitary. Ψ(z, w) = |w − z|/|1 − zw¯ |.

720 19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014

− Let M ∈ D and M be the corresponding J –unitary defined by ΞP is the plant P , it follows that the stabilizing U1 0 Q K matrix. Then S = M , with U1 and U2 arbitrary controllers are described by by ΠP ∼ , i.e.. 0 U2 I I unitary matrices, is the non–Euclidean motion that maps −1 Kstab = {K | K = (U + MQ)(V + NQ) ,Q stable} 0 into M. The circle γh(M, r) with radius 0 < r < ∞, − which is the well-known Youla parametrization and the disk Dh (M, r) are defined as maps of γh(0, r) and − 0 ∈ Dh (0, r) under this non–Euclidean motion. To emphasize this point we reformulate the standard stability result as follows: The Hermitian diameter l of D− is defined by h Proposition 4.1. The plant P has a double coprime factor- P l = P : P = i 3 ,P ∗P − P ∗P = 0 . ization if and only if there is a projectivity defined by an h P 4 3 3 4 4 outer Ξ with Ξ11 invertible such that MΞ(P ) = 0. Then The non–Euclidean straight lines are defined as the maps all stabilizing controllers are given by K = ML∗Ξ−1L(Q), of l under non–Euclidean motions. 0 I h Q stable, where L = . I 0 4. ROBUST CONTROL AND HYPERBOLIC GEOMETRY The proof is straightforward and it is omitted for brevity.

The common tool in formulating robust feedback control 4.1 Linear relations and LFTs problems is to use system interconnections that can be P P described as linear fractional transforms (LFTs), as a If P is partitioned as P = zw zu then, provided that general framework to include the rational dependencies Pyw Pyu that occur. It is apparent that Möbiustransformations are the corresponding inverses exists, a lower and an upper −1 special LFTs. The set of stabilizing controllers for a given LFT is defined as Fl(P,K) = Pzw +PzuK(I−PyuK) Pyw −1 plant and the set of all suboptimal H∞ can be expressed and Fu(P, ∆) = Pyu + Pyw∆(I − Pzw∆) Pzu. P is called by using certain Möbiustransformations. In what follows the coefficient matrix of the LFT. the common background of these transformations and 0 their relations with the hyperbolic geometries will be Möbiustransformations Z = MΣ(Z) relate two graph highlighted. subspaces, GZ and GZ0 , through the projectivity Xi, i.e., GZ0 = ΞGZ and inherit the group structure of the linear

Even the system P is unbounded, through its coprime operators, i.e., MΞ2 ◦ MΞ1 = MΞ2Ξ1 . factorization P = NM −1, with M,N suitable bounded It turns out that LFTs can be obtained in the same M causal operators, the associated graph P = Πξ = ξ way as the Möbiustransformations, by performing some N interchange in the signal spaces and by considering linear is formulated in terms of bounded with ξ ∈ Hp (a suitable relations instead of the linear operators, Shmulyan [1976, Hilbert space) operators. We are not very restrictive if it is 1980]. assumed that there exists a double coprime factorization, i.e., P = NM −1 = M˜ −1N˜ and causal bounded U, V, U˜ If X and Y are two sets, a relation T ⊂ X × Y is defined and V˜ such that as a set of pairs (x, y) ∈ T , where x ∈ X, y ∈ Y . If ˜ ˜ X and Y are linear spaces (X ⊕ Y = X × Y ) a linear V −U MU I 0 relation T is a linear subspace of X ⊕ Y . If x ∈ dom(T ) ˜ ˜ = , (2) −N M NV 0 I then T (x) = {y ∈ Y :(x, y) ∈ T } and correspondingly if an assumption which is often made when setting the y ∈ ran(T ), then T −1(y) = {x ∈ X :(x, y) ∈ T }. stabilization problem, see, e.g., Vidyasagar [1985]. Recall Let T ⊂ X × Y and R ⊂ Y × Z be linear relations. Then M U that and are determined only up to outer, i.e., the product RT ⊂ X × Z is the linear relation defined by N V RT = {{x, z} ∈ X × Z : {x, y} ∈ T, {y, z} ∈ R} . stable with stable inverse, factors S and S0. An operator P : X 7→ Y is equivalent to a special relation, It is not hard to make the connection with the construction the graph subspace GP . For details see, e.g., Arens [1961]. presented in the previous section: plants P , represented by M With X = Xw ⊕ Xu and Y = Yz ⊕ Yy consider L = X ⊕ Y P = { }, are the finite points while the controllers N Pf and L˜ = (Yy ⊕ Xu) ⊕ (Xw ⊕ Yz). Observe that we have −1 ˜ K = UV in a feedback connection are described by the L = ΠlL with a permutation matrix Πl. Thus every linear U operator P : X 7→ Y induces a relation R ⊂ L˜ through inverse relation (inverse graph), i.e., K = { }. Well- P V its graph subspace, i.e., posedness (feedback stability) of the pair (P,K) means Pyw Pyu MU 0 Iu that the matrix ΞP = has a bounded causal RP = ΠlGP ∼   , (3) NV  Iw 0  inverse. Stable plants Q form a subset of the finite points, Pzw Pzu I i.e., those for which ∈ i−1(Q). It is obvious that the called scattering transformation. It turns out that evalu- Q ating this relation on the graph subspaces GK , i.e., on the zero plant is stabilized by the entire stable set, and only linear operators K : Yy 7→ Xu, we obtain a graph subspace by that set. Since the image of 0 under the projectivity GF = RT GK that corresponds to the linear operator

721 19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014

F : Xy 7→ Yz, provided that (I − PyuK) is boundedly in- nzw nzu required by the proposition, with MΞs (P ) = . vertible. This map is exactly the lower LFT F = Fl(P,K). n˜yw 0 Analogously, by considering another permutation Π , one r Actually one has Fl(P,K) = nzw + nzuqn˜yw, where q is can obtain the expression of the upper LFT. the Youla parameter of K. The details are simple algebraic This construction extends the linearization trick already computations and are left out. Note that P is either encountered for the Möbiustransforms to the LFTs: on stabilized by any of the stabilizing controllers of the inner the level of equivalence classes, the map is linear while loop Pyu or it is not stabilizable at all. on the level of the representants the map is rational (Möbius, LFT). Moreover, the group structure on the 4.2 The H∞ problem representants is also present, however, the familiar matrix product should be changed to the more complex Redheffer In this section we consider the suboptimal normalized H∞ (star) product, see, e.g., Zhou and Doyle [1999], that problem, i.e., we seek all controllers K that internally reflects the composition of the relations. stabilize the loop and achieve kFl(P,K)k < 1. While the conclusions of the section remain valid for much larger Note, that in this construction Πl is a projectivity that classes, including infinite dimensional LTI or LTV systems, sends a finite point P to a possible not finite R, cor- we consider here only the finite dimensional LTI case, i.e., responding to the subspace RP . If the image is also a systems having a state space description. finite point, then the representant can be obtained by the We have already seen that it is enough to consider stable Möbiustransform Pˆ = M (P ) which is the Potapov- Πl n n Ginsbourg transformation. P is in the domain of this generalized plants of type P = zw zu . There are n˜ 0 P P yw transformation if yw yu is invertible. 0 I basically two type of strategies, relevant for this paper, u to solve the problem. Both methods assume either left

The relation Fl(P,K) = MPˆ (K), if it exists, between an or right invertibility of P . The first uses the scattering LFT and a Möbiustransformations has the advantage to approach by augmenting the plant, if necessary, to obtain use a more accessible operation (matrix product) instead a well defined Potapov-Ginsburg transform Pˆ, Ball et al. of the star product for the factorizations that possibly [1991], Kimura [1997]. Then a J-inner outer factorization simplify a given problem. This fact was widely exploited Pˆ = Θˆ aRˆ, with a block tridiagonal structure of the in the solution of the robust control problems, see, e.g., outer factor that corresponds to the structure of the the factorization approach of Ball et al. [1991] or in the so augmentation, solves the problem. The controllers are called chain scattering-approach of Kimura [1997]. given by MRˆ−1 (Ha) with We conclude this section with an example that shows how 0 0 H = , kHk < 1, suitable projectivities can reveal the basic structure of a 0 H a control problem. We are to apply Proposition 4.1 for while the closed loop is given by M ˆ (Ha). Recall that Θa the stabilization problem related to LFTs, i.e., to find all Θa is an inner function, thus internally stabilizing controllers K that makes Fl(P,K) F (P,K) = M (H ) = F (Θ ,H ) < 1. (4) bounded. Even this is a known result, its geometric inter- l Θˆ a a l a a pretation remains hidden. The novelty here is to show the For the details on J-inner and J-lossless functions see Dym role of the projectivity in the solution of the problem. [1989] and Kimura [1997]. Proposition 4.2. The stabilization problem is solvable if The second approach, see Green et al. [1990], Green [1992], and only if there is an outer Ξ with Ξ11 invertible such that instead of augmentation uses two J-spectral factorizations. 0 0 0 0 MΞ(P ) = S is stable and = ML∗Ξ−1L( ), Q We briefly sketch the geometric content of the method: let 0 K 0 Q us start from the scattering form, i.e., stable. Then S can be chosen to be a model matching type, y n˜yw 0 w Iw 0 Szw Szu = ξ = Pˆyuξ, = ξ = Pˆwzξ, i.e., S = . u 0 Iu z nzw nzu Syw 0 under the constraints z = Fl(P, q)w = Fqw and u = qy. The proof is by construction: under the stabilizabil- −Ip 0 ity assumption, i.e., there is a stabilizing controller We generically use the notation Jpq for . Then, 0 0 0 Iq for P , the double coprime factorization, with 0 K if we have the J-spectral factorizations ∗ ∗ −1 ∗ ∗ ˜−1˜ −1 Pˆ JwzPˆwz = V JwuVr, PˆyuV Jwu(?) = WrJyuW , K = kuu kuy = kuykyy , has the special structure wz r r r     ˆ −1 Iw 0 0 0 Iw 0 0 0 with Vr,Wr outer, i.e., Φr = PwzVr is J-lossless and 0 k˜ 0 −k˜ Kn m 0 k −1 ˆ −1  uu uy   yw uu uy  = I. Ψr = W PyuVr is co-J-lossless, respectively. The first  −n −n˜ I n˜ K   nzw nzu Iz 0  ˜ ˜ zw zu z zu observation is that from (?)JwzΦrξ = (?)Jwuξ < 0 with −n˜yw −n˜yu 0m ˜ yy nyw nyu 0 kyy ˜ ξ = V ξ and using (−q Iu) Pˆyuξ = (−qn˜yw Iu) ξ = 0 ˜⊥ ˆ −1 Compare this general fact with Green [1992]. Thus the one has ξ Jwu(?) > 0, i.e., (−q Iu) PyuVr Jwu(?) >  Iw 0 0 0  0. Then we have that (−q Iu) WrJyu(?) > 0 which is ˜ ˜  0 kuu 0 −kuy  −1 Iy projectivity Ξs = has the property equivalent to (?)JyuWr < 0, which provides the  0 0 Iz 0  q 0 −n˜yu 0m ˜ yy desired controllers. The notation (?) stands for the adjoint

722 19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014 of the corresponding left (right) hand side of the J- Gans, D. (1973). An Introduction to Non-Euclidean Geometry. product. It turns out that for the considered class existence Academic Press. of the desired factorization is actually equivalent to the Green, G. M., Glover, K., Limebeer, D. J. N., and Doyle, J. C. solvability of the problem. (1990) A J-spectral factorization approach to H∞ control. SIAM J. Control and Opt., 28(3):1350-1371. In the original presentation an important part of the result Green, M. (1992). H∞ controller synthesis by J-lossless coprime is missing, namely that the closed loop is provided by factorization. SIAM J. Control and Opt., 30(3):522-547. an inner function Θ, whose scattered representation is Harris, L. A. (1973). Bounded symmetric homogeneous domains in infinite dimensional spaces, in Lecture Notes in Mathematics No. Ψ r . The existence of this function follows from the fact 364. Springer- Verlag, New York. Φr Hassibi, B., Sayed, A., and Kailath, T. (1999). Indefinite Quadratic ˜ ˜ that (?)JwzΦrξ < 0 ⇔ (?)JyuΨrξ < 0 implies equality Estimation and Control: A Unified Approach to H2 and H∞ of the two indefinite quadratic form, see Bognár[1974], Theories. SIAM Studies in Applied Mathematics. Shmulyan [1978]. It follows that kwk2 − kzk2 = ky0k2 − Helton, J. W. (1978). Orbit structure of the Möbiustransformation ku0k2, i.e., kwk2 + ku0k2 = kzk2 + ky0k2. semigroup action on H∞ (broadband matching). Adv. in Math. Suppl. Stud., 3:129–197. −1 Helton, J. W. (1980). The distance from a function to H∞ in the Thus the projectivity Ξh = Πldiag(Wr ,I) maps the Poincarémetric; electrical power transfer. J. Functional Analysis, original plant P to Θ, moreover Fl(P, q) = Fl(Θ, h), with q = M (h) on khk < 1. Thus, we can continue the series 38(2):273–314. Wr Helton, J. W. (1982). Non–Euclidean functional analysis and of prototype configurations related to significant control electronics. Bull. AMS, 7(1):1–64. problems that can be achieved with suitable projectivities: Helton, J. W. (1987). Operator theory, analytic functions, matrices Proposition 4.3. The suboptimal H∞ problem is solv- and electrical engineering, volume 68 of CBMS Lecture Notes. able if and only if there is a projectivity Ξh such that Amer. Math. Soc., Providence, RI. Kelly, P. (1981). The Non-Euclidean Hyperbolic Plane. Springer- MΞ (P ) = Θ is an inner function. h Verlag, New York. To conclude this section we return to formula (4). Since Khargonekar, P. and Tannenbaum, A. (1985). Non–Euclidian metrics the closed loop system is parametrized naturally by con- and the robust stabilization of systems with parameter uncer- tainty. IEEE Trans. on Automatic Control, 30(10):1005–1013. tractions Ha through a hyperbolic motion it is natural to consider a hyperbolic distance, introduced as in subsection Khatskevich, V. A. (1984). Generalized Poincaré metric on the operator ball. Mathematical notes of the Academy of Sciences 2.1, that relates directly controllers with the closed loop of the USSR, 17(4):321–323. plant. Kimura, H. (1997). Chain-scattering approach to H∞ control. Birkhauser. 5. CONCLUSIONS Ostrovskii, M. I., Shulman, V. S., and Turowska, L. (2009). Unitariz- able representations and fixed points of groups of biholomorphic The paper emphasizes Klein’s approach to geometry and transformations of operator balls. Journal of Functional Analysis, demonstrates that a natural framework to formulate dif- 257(8):2476–2496. ferent control problems is the world that contains as points Schwarz, B. and Zaks, A. (1981). Matrix Möbiustransformations. equivalence classes determined by stabilizable plants and Communications in Algebra, 9(19):1913-1968. whose natural motions are the Möbiustransforms. The Schwarz, B. and Zaks, A. (1985). Geometries of the projective matrix space. Journal of Algebra, 95(1):263-307. fact that any geometric property of a configuration, which Shmulyan, Y. L. (1976). The theory of linear relations and spaces is invariant under a hyperbolic motion, may be reliably in- with an indefinite metric. Funkts. Anal. Prilozh, 10(1):67–72. vestigated after the data has been moved into a convenient Shmulyan, Y. L. (1978). General linear-fractional transformations of position in the model, facilitates considerably the solution operator balls. Siberian Mathematical Journal, 19(2):293-298. of the problems. This method provides a common back- Shmulyan, Y. L. (1980). Transformers of linear relations in J-spaces. ground of robust control design techniques and suggests a Funkts. Anal. Prilozh., 14(2):39–43. unified strategy for problem solutions. Soumelidis, A., Bokor, J., and Schipp, F. (2011). On hyperbolic wavelets. In 18th World Congress of the International Federation Besides the educative value a merit of the presentation for of Automatic Control, Milano, pages 2309–2314. control engineers might be a unified view on the robust Soumelidis, A., Schipp, F., and Bokor, J. (2011). Pole structure control problems that reveals the main structure of the estimation from Laguerre representations using hyperbolic metrics problem at hand and give a skeleton for the algorithmic on the unit disc. In Proceedings of the 50th IEEE Conference on development. Decision and Control and European Control Conference, CDC- ECC 2011, Orlando, FL, USA, pages 2136–2141. REFERENCES Vidyasagar, M. (1985). Control System Synthesis: A Factorization Approach. MIT Press, Cambridge. Allen, J. C. and Healy, D. M. (2003). Hyperbolic geometry, Nehari’s Zhou, K. and Doyle, J. C. (1999). Essentials of Robust Control. theorem, electric circuits, and analog signal processing. Modern Prentice Hall. Signal Processing, 46:1–62. Anderson, J. (2005). Hyperbolic geometry. Springer-Verlag, London. Arens, R. (1961). Operational calculus of linear relations. Pacific Journal of Mathematics, 11(1):9–23. Ball, J. A., Helton, J. W., and Verma, M. (1991). A factorization principle for stabilization of linear control systems. Journal of Nonlinear and Robust Control, 1:229–294. Bognár,J. (1974). Indefinite inner product spaces. Springer–Verlag, Berlin. Dym, H. (1989). J–contractive matrix functions, reproducing kernel Hilbert spaces and interpolation. AMS, Providence, R.I.

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