Zeros. Carsten Scherer. Mathematisches Institut. Am Hubland. D-8700 Wurzburg

Transcription

1 H -Optimization without Assumptions on Finite or Innite Zeros arsten Scherer Mathematisches Institut Am Hubland D-87 Wurzburg Abstract We present explicit algebraic conditions for the suboptimality of some parameter in the H - optimization problem by output measurement control. Apart from two strict properness conditions, no articial assumptions restrict the underlying system. In particular, the plant may have zeros on the imaginary axis or at innity. These suboptimality characterizations are applied to show how to compute the optimal value by quadratically convergent algorithms and to solve the almost disturbance decoupling problem with internal stability. Keywords H -optimization, Invariant zeros, Riccati inequalities, Almost disturbance decoupling AMS(MOS) subject classication 9327, 935, 9335, 9345, 936, 93D5, Introduction After Zames introduced the basic motivations for H -optimization in [4], this problem has attracted a lot of research and several techniques for its solutions have been proposed. We only want to mention briey the approaches in the frequency domain using operator theory [8, 9], J-spectral factorization [2, ], polynomial methods [8], or an all-pass imbedding (including a treatment of the H -problem at optimality) [2]. The direct solutions of the H -problem in the state-space are primarily based on the ounded Real Lemma and on ideas from LQ-theory as well as dierential games [3, 4, 5, 4, 5, 6, 7, 24, 33, 35]. We refer the reader to [8, 9, 5] for rather comprehensive discussions of the historical development. The paper [5] may be viewed as the breakthrough for the state-space techniques to solve the H -problem. The suboptimality tests are formulated in terms of the solvability of two indenite Riccati equations and a coupling condition on their solutions. Some authors tried to remove several articial assumptions needed in the approach of [5]. In [4] one can nd generalized formulae if certain direct feedthrough matrices do not vanish. From a practical point of view, the main restriction results from the hypothesis that the plant cannot have zeros neither on the imaginary axis nor at innity. Using tools of the geometric theory, the zero assumption at innity was removed in [33] and the suboptimality criteria were given in terms of the solvability of two quadratic matrix inequalities and a coupling condition on their solutions. The solutions of these inequalities may be constructed by transforming the system and solving again two reduced order indenite Riccati equations. The general problem without any assumption on the zeros on the imaginary axis and at innity has not been solved up to now. supported by Deutscher Akademischer Austauschdienst

2 In [3] we explained the diculties resulting from the imaginary axis zeros in the regular H - problem by state-feedback: One cannot infer from the solvability of an algebraic Riccati inequality (ARI) the solvability of the corresponding algebraic Riccati equation (ARE). The same problems arise in the general H -problem by output feedback. Indeed, from the very beginning of the direct state-space approach to the H -problem [24] it became clear that Riccati inequalities could be viewed as a central tool for studying this problem. In fact, new algebraic tests for the solvability of general strict Riccati inequalities lead to a satisfactory solution of the regular state-feedback H -problem [3]. In this paper, these ideas are extended to attack the general H -optimization problem by output measurement and without assumptions on nite or innite zeros of the underlying plant. Our approach only refers to a strict version of the ounded Real Lemma and proceeds, by simple algebraic manipulations, completely in the state-space. The paper is organized as follows. In Section 3, we will be able to provide a very short but complete proof of a characterization of suboptimality in terms of the solvability of two Riccati inequalities and a coupling condition on their solutions where the ARIs still depend on an unknown feedback and injection matrix (Theorem ). We shortly discuss the results appearing in the literature. For a reformulation of this characterization into algebraic tests, one needs to transform the system data which is explained in Section 4. Then we show in Section 5 how to check the solvability of the feedback depended ARI by the solvability of a xed welldened reduced order ARI (Theorem 6). An abstract reformulation (Theorem ) displays how to express the suboptimality directly in terms of the original system matrices. These considerations lead to the solution of the general H -optimization problem by state-feedback. Section 6 is devoted to clarify how the coupling condition can be checked algebraically which culminates in the solution of the general H -problem by output measurement (Theorem 4). Section 7 contains the application of these results to the computation of the optimal value by quadratically convergent algorithms. In addition, we display certain special cases for which it may be determined by solving Hermitian eigenvalue problems. Finally, we derive in Section 8 the geometric solution of the almost disturbance decoupling problem with stability. We adopt all the notations from [3]. Moreover, we denote by A + the Moore-Penrose inverse of A 2 R nm. The symbol g is used for any nonempty subset of which is symmetric with respect to the real axis. 2 Problem Formulation The system is described by _x = Ax + u + Gd; x() = ; y = x + Dd; z = Hx + Ed; () where x 2 R n is the state, d 2 R is the external disturbance, u 2 R m is the control, z 2 R k is the controlled output and y 2 R is the measured output available for control. A compensator is any dynamic output feedback system This system is identied with r (s) = _w = Kw + Ly; w() = ; u = Mw + Ny: K? si M 2 L N (2) where, for clarity, the dimension r 2

3 N [ fg (K 2 R rr ) appears as an index. The closed-loop system is given A A + N? si M G + ND L K? si LD H + EN EM END x with state and initial value. Moreover, the compensator w r (s) is called stabilizing if it internally stabilizes the resulting closed-loop system in the sense of A + N L M K? : It is well-known that a stabilizing controller exists i (A? si ) is stabilizable and A? si is detectable: Apart from Section 4, these are the standing assumptions throughout this paper. For any stabilizing compensator r (s), let ( r (s)) denote the H -norm of the controlled closedloop system. The general H -problem by output measurement consists of minimizing ( r (s)) over all stabilizing controllers r (s). For the same reasons as in [3], we rather consider the equivalent problem: Maximize ( r (s)) := ( r (s)) 2 over all stabilizing r (s) under the usual convention = =. We dene the optimal value by opt := supf( r (s)) j r (s) is stabilizing, r 2 Ng: Now we can describe the subjects of the present paper: Give checkable (algebraic) conditions for some parameter > to be suboptimal in the sense of < opt. Find a fast algorithm for computing opt. Try to nd out when opt can be computed explicitly or characterize whether opt is innite. All these problems will nd rather satisfactory answers. The only restriction on the plant: There is no direct feedthrough from u to y (which is easy to handle) and from d to z (which is more dicult to deal with). For techniques to overcome these assumptions we refer the reader to [26]. A particular case has its own interest: The state-feedback H -problem which is dened by the specication = I and D = ; i.e., the whole state is available for control and it is not directly corrputed by the disturbances. In the frequency domain [8, 9], the plant (including weighting matrices) is described by the real rational proper matrix G(s) as z(s) G = (s) G 2 (s) d(s) : y(s) G 2 (s) G 22 (s) u(s) 3

4 In our case, the system is only restricted to be internally stabilizable and to have a representation with strictly proper matrices G (s) and G 22 (s). In the usual frequency domain approach, one rst parametrizes the set of transfer matrices of all internally stabilized closed-loop systems as ft (s) + T 2 (s)q(s)t 3 (s) j Q(s) real rational, proper, stableg with T 2 (s) = (H + EF )(si? A? F )? (for some F satisfying (A + F )? ), T 3 (s) = (si? A? J)? (G + JD) (for some J satisfying (A + J)? ) and some real rational proper stable T (s). This step changes the H -problem to a model-matching problem. If T 2 () has maximal column rank and T 3 () has maximal row rank for all 2 [ fg, it is possible to transform the model-matching problem into the so called four block Nehari problem [9]. The aim of this paper is to overcome these rank assumptions which implies that the transformation to the four block problem is not possible any more. 3 Suboptimality and Strict Algebraic Riccati Inequalities We start by characterizing the suboptimality of some positive by the solvability of Riccati inequalities which still depend on some unknown feedback and injection matrix. Theorem > is suboptimal i there exist F and J as well as P > and Q > which satisfy (A + F ) T P + P (A + F ) + P GG T P + (H + EF ) T (H + EF ) < ; (3) (A + J)Q + Q(A + J) T + QH T HQ + (G + JD)(G + JD) T < ; (4) (P Q) < : (5) If these conditions are satised, there exists a stabilizing controller n (s) of dimension n with < ( n (s)). This result has a very nice interpretation. Suppose that F is any matrix such that (3) has a positive denite solution. Then this feedback F yields (A + F )? and k(h + EF )(si? A? F )? Gk <. One the other hand, if F satises these last two properties, there exists a P > which solves (3). Therefore, there exist F and P > with (3) i is suboptimal for the H -problem by static state-feedback for the plant and the disturbance input matrix G. S(s) := A? si H Dually, the existence of J and Q > with (4) is equivalent to the suboptimality of for the H -estimation problem (by linear observers) for the plant T (s) := A? si and the to be estimated output Hx [23]. However, for < opt it does not suce that is suboptimal for these two problems separately but one has to require the additional coupling condition (5). This constraint can be understood if trying to build a compensator: The controller may be constructed as an estimator for the desired suboptimal control function F x (which is 4 E G D

5 not implementable) using only the measurements y. The conditions for this estimation to be possible result in (5) [5]. From a practical point of view, Theorem is not very useful: It is not clear how to test these conditions in an algebraic way without resorting to perturbation techniques which would involve an additional parameter. Moreover, it does not provide any insight how the -zeros of the plants S(s) or T (s) inuence the optimal value opt. ontrary to the considerations in [27] which stop at this point, this is only a preliminary step in our approach. Theorem is proved in [27] (and was derived independently for the regular problem in the rst version of this paper). Nevertheless, we would like to include a complete and most direct proof which is still much shorter than that in [27] since there will be no need to distinguish between proper and nonproper controllers. Moreover, we prove the suciency part by starting with a compensator which is only a slight modication of that used for the regular problem in [4, 5]; the modication can be easily motivated and the required computations are very simple. Proof of necessity Suppose that > is suboptimal. Then there exists a stabilizing controller r (s) with < ( r (s)). Let us dene the abbrevations A := A + N M L K ; := G + ND LD ; := H + EN EM ; D := END: y (A)? and k(si? A)? + Dk 2 <, we can apply the strict version of the ounded Real Lemma and infer the existence of some Y > with [27, 42] A T Y + YA + T Y + T D T Y + D T D T D? I < (6) It is very simple to see [27] that satises X := Y (7) AX + X A T + T X T + D T < : (8) X X Let us now partition X = 2 Y Y X T and Y = 2 2 X 2 Y T according to A. Note that 2 Y 2 both matrices in (6) and (8) are in fact partitioned into three block rows/columns. Now we just compute the (,) block of the matrix in (6) to A T Y +Y A+ T [N T T Y +L T Y T 2+N T E T H]+[Y N +Y 2 L+H T EN]+H T H+(EN) T EN which equals (A + J) T Y + Y (A + J) + H T H + (EN) T (EN) for J := N + Y? Y 2 L + Y? H T EN. The (,3) block may be written with the help of J as Y (G + JD) + (EN) T (END): y (EN) T (EN) (END) T (EN) (EN) T (END) (END) T (END) ; 5

6 we immediately infer from (6) the inequality (A + J) T Y + Y (A + J) + H T H Y (G + JD) (G + JD) T Y? I < : The Schur complement of the matrix on the left in this inequality with respect to the (2,2) block is negative denite. If we multiply it with we infer that Q := (Y )? > satises (4). Since (8) is dual to (6), there is no need for any computation in order to deduce that P := (X)? is a positive denite solution to (3) for F := N + MX T 2 X? + NDG T X?. y the well-known formula for the inverse of a block matrix, the left upper block of Y? is given by (Y? Y 2 Y? 2 Y T 2 )?. Therefore, (7) shows X Y? which clearly leads, by the denition of P and Q, to (P Q). If the inequality is not strict, we can perturb P or Q (e.g. to P? I or Q? I, > ) such that these perturbations still satisfy the ARIs (3) or (4) and, in addition, the strict coupling condition (5) becomes true. Proof of suciency and controller construction Let R F and R J denote the left-hand sides of the Riccati inequalities (3) and (4). We assume without restriction Q? R J Q? < R F : (9) Suppose that this is not valid. Obviously, there exists some matrix R < with Q? R J Q? < R and R F < R. Since the ARI (A + F ) T X + X(A + F ) + XGG T X + (H + EF ) T (H + EF )? R < has the solution X = P and since A + F is stable, there exists some P < P [3] with (A + F ) T P + P (A + F ) + P GG T P + (H + EF ) T (H + EF )? R =. The stability of A + F and R < imply P >. We can hence replace P by P such that all the hypotheses in the theorem persist to hold and (9) becomes true. Motivated by the results for the regular problem [4, 5], one can write down the strictly proper compensator _w = (A + GG T P + )w + u + ~ J(( + DG T P )w? y); w() = ; u = F w in observer form, where we dene ~J := (I? QP )? J; :=?(Q?? P )? h (F ) T P + (EF ) T (H + EF ) i : The only modication consists of the introduction of. The formula for will immediately become clear by the compuations shown below. As usual, we consider the dynamics of the error e := x? w which is clearly given by _e = (A + GG T P + ~ J( + DG T P ) + )e? ((G + ~ JD)G T (P ) + )x + (G + ~ JD)d: This leads to the closed-loop system (with state (x T e T ) T A + F? si?f G?(G + JD)G ~ T (P )? A + GG T P + J( ~ + DG T P ) +? si G + JD ~ H + EF?EF which is again denoted as A)? k <. A? si A. We have to show (A)? and k p (si? 6

7 To nish the proof, it suces (by the ounded Real Lemma [42]) to construct a solution Y of the strict ARI A T Y + YA + Y T Y + T < : () Again motivated by the solution of the regular problem [5], we are lead to try Y := P Q?? P which is positive denite by (5). Let us introduce the abbrevation Z := Q?? P. Now we just compute the blocks of the left-hand side of (). The (,) block is immediately seen to be R F which is negative denite. The (2,) block is given by?(f ) T (P )? Z(G + ~ JD)G T (P )? Z + Z(G + ~ JD)G T (P )? (EF ) T (H + EF ) = =?(F ) T P? (EF ) T (H + EF )? (Q?? P ) which just vanishes, by the very denition of. compute the (2,2) block to We rst stress the Z ~ J = Q? J and then (A + GG T P ) T Z + Z(A + GG T P ) + ZGG T Z + (EF ) T (EF ) + T Z + Z + + Q? JDD T J T Q? + ( T + P GD T )J T Q? + ZGD T J T Q? + + Q? J( + DG T P ) + Q? JDG T Z which is rewritten, by P GG T Z + ZGG T P + ZGG T Z = Q? GG T Q?? (P )GG T (P ), to A T Z + ZA? (P )GG T (P ) + T Z + Z + (EF ) T (EF ) + T J T Q? + Q? J + + Q? GG T Q? + Q? JDG T Q? + Q? G(JD) T Q? + Q? JD(JD) T Q? : The denition of shows that T Z + Z + (EF ) T (EF ) is equal to?(f ) T (P )? (P )F? (H + EF ) T (H + EF ) + H T H: Therefore, the (2,2) block is given by A T Z + ZA? (P )GG T (P )? (F ) T (P )? (P )F? (H + EF ) T (H + EF ) + + (J) T Q? + Q? (J) + H T H + Q? (G + JD)(G + JD) T Q? which obviously equals Q? R J Q?? R F < : The modication of the standard suboptimal compensator [4, 5] is due to the generality of F and J: We do not need to specify particularly chosen matrices F and J and can dene a compensator based on certain solutions of the Riccati inequalities. The controller has a simple observer structure with a similar interpretation as discussed in [5]. In completely the same fashion we could also construct a compensator based on the minimal solutions of those Riccati equations that correspond to (3), (4) (which exist since A + F and A + J are stable). These solutions still satisfy (5) by minimality. The question arises how to test the various conditions of Theorem in an algebraic way. The following results are available in the literature under combinations of the assumptions 7

8 (a) E has maximal column rank and D has maximal row rank. (b) S(s) and T (s) have no invariant zeros in (see Section 4). If (b) holds, our characterization may be reformulated as follows [33]: There exist solutions P, Q of the two quadratic matrix inequalities which satisfy A T P + P A + P GG T P + H T H P + H T E T P + E T H E T E Q + DG T DD T AQ + QA T + QH T HQ + GG T Q T + GD T (P Q) < : plus rank conditions; plus rank conditions If both (a) and (b) hold true, the quadratic matrix inequalities can obviously be reduced (by taking the Schur complements with respect to the (2,2) blocks) to indenite AREs and one arrives at the results of [4, 5]. If (a) fails to hold, the solutions of the quadratic matrix inequalities may as well be determined by solving two reduced order indenite AREs for subsystems of certain transformed versions of S(s) and T (s) [33]. A referee draw our attention to the paper [2] where algebraic tests for suboptimality are derived (in the frequency domain) if (a) and (b) do not necessarily hold true. However, these results are restricted to the one block problem, i.e., T 2 (s), T 3 (s) as appearing in Section 2 have to be square and invertible as rational matrices. The aim of this paper is to remove both assumptions (a) and (b) for the general four block problem as described in Section 2. We will generalize the suboptimality criteria we derived in [3] for the regular state-feedback problem ( = I, D =, E has maximal column rank) which are formulated in terms of the Riccati map R(X; ) = AX + XA T + XH T HX + GG T? ( + XH T E)(E T E) + ( T + E T HX) and the associated map A(X) := A + XH T H? ( + XH T E)(E T E) + E T H; dened on S n R and S n respectively. Note that we performed in [3] a preliminary feedback with F :=?(E T E) + E T H () (motivated by E T (H + EF ) = ) and one easily veries that R(X; ) equals (A + F )X + X(A + F ) T + X(H + EF ) T (H + EF )X + GG T? (E T E) + T and A(X) is given by (A + F ) + X(H + EF ) T (H + EF ): We already indicate the generalization to the singular problem by using the Moore-Penrose inverse: The two representations of these maps still coincide even if E does not have maximal column rank The undetectable subspaces and the observability normal form appearing in [3] have their natural generalizations in the corresponding conditionally invariant subspaces and a modication of Morses canonical form which are thoroughly investigated in geometric control theory. 8

9 4 Transformation by Restricted oordinate hanges For some arbitrary (n + k) (n + m) system S(s) = A? si H E by [] (S(s)) the set of invariant zeros, and any g we denote V g (S(s)) the maximal element under all subspaces U R n for which there exists an F with (A + F )U U and U ker(h + EF ) such that the restriction of A + F to U has only eigenvalues in g. S g (S(s)) the minimal element under all subspaces U R n for which there exists a J with (A + JH)U U and im( + JE) U such that (A + JH) T ju? has only eigenvalues in g. We recall the important duality relation S g (S(s)) = V g (S(s) T )? : For g =, g =?, g = and g = +, we denote the corresponding spaces by replacing `g' with `', `?', `' and `+' respectively. If we choose some g with (S(s)) \ g = ;, we dene (independently of the choice of g ) R (S(s)) := V g (S(s)) and N (S(s)) := S g (S(s)): Motivated by the results in [3], we introduce for any 2 the complex subspace V (S(s)) := fx 2 n j 9u 2 m : S() x u = g as well as S (S(s)) := fx 2 n j 9u 2 n+m : x = S()ug and obtain the duality relation S (S(s)) = V (S(s) T )? : These various subspaces are displayed in the best way using the famous canonical form of Morse [22] which was given for not necessarily strictly proper systems in []. This canonical form is derived for the so-called full transformation group consisting of coordinate changes, `statefeedback' and `output-injection' transformations. The orbit of S(s) under this transformation group can be easily described as fls(s)r j L = U J V ; R = U? F W ; U 2 R nn ; V ; W square and nonsingularg: The transformation properties V g (LS(s)R) = UV g (S(s)) and S g (LS(s)R) = US g (S(s)) (where L and R are partitioned and denoted as above) are well-known and one easily veries the same relations for the spaces V (S(s)) and S (S(s)). It will be instrumental to restrict this transformation group to pure coordinate changes (F = and J = ) or, since we work with the Euclidean norm in the output space, even to 9

10 restricted coordinate changes: F =, J = ; V and W orthogonal. Note that the set of coordinate changes and restricted coordinate changes still form a group. The structure of the transformed system in the following result cannot be immediately extracted from [22, ], not even for general coordinate transformations. Theorem 2 A + F? si The system S F (s) := H + EF restricted coordinate changes to ~S(s) = ~A? si ~ ~H E ~ E with F dened by () can be transformed by A? si J H 2 2 F A 2? si 2 2 H H 2 A with the following properties: (a) is symmetric and nonsingular, (b) A 2? I 2 H 2 (c) S g ( ~ S(s)) = f (d) S ( ~ S(s)) = f has maximal row rank for all 2, x 2 R x n A j x 2 S? si g g for any 2 H g. x 2 x n A j x 2 S? si g for all 2. 2 H The proof of this transformation result is based on an interesting solvability criterion for a linear equation which has its origin in the solution of the regulator problem (combine Korollar 7.2 with Satz 7.4 in [9]). Lemma 3 Suppose that A? I H of the size (n + k) (n + m) has maximal row rank for all 2. Given any M 2 R ll, R 2 R nl and S 2 R kl (l 2 N), there exist X and Y with AX? XM + Y = R; HX = S: Proof of Proposition 2 The singular value decomposition yields orthogonal V, W and a nonsingular = T such that V EW equals. y the choice of F, we obtain I V S F (s) I W A? si H A :

11 We now transform the strictly proper system S(s) A? si := to Morses canonical H form [22]. If reversing the performed state-feedback and output-injection transformations and repartitioning suitably, the transformed system admits the shape A? si J H2 A 2 A2? si 2 H H2 A such that 2? I has maximal row rank for all 2. Since we can obviously reverse H 2 the coordinate change in the input-space without violating these properties, this transformation is possible by general coordinate changes in the state-space and output-space; we denote the corresponding matrices by U and V. We clearly have HU? = V? H H2 and can decompose V? as V ~ T T with some orthogonal V ~ to obtain T 2 T 3 ~V H U? = T H T 2H T 3H2 : Therefore, S(s) can be transformed by restricted coordinate changes to and, still, A 2? I 2 H A? si J H 2 A 2 A2? si 2 H H 2 H 2 A A has maximal row rank for all 2. This is the point to apply Lemma 3 in order to remove H 2 and to shape A2. There exist matrices X and F with A 2 X? X( A? J H 2 ) + A 2? 2 F = ; H 2 X + H 2 = : If we add the X-right multiple of the second column of (2) to the rst one and then the (?X)-left multiple of the rst row to the second one (which is just a coordinate change in the state-space), we cancel H 2 and change A 2 to 2 F as desired. Since the block A 2 is just transformed to A 2 := A2? XJ H 2, the condition (b) persists to hold. If S 2 (s) denotes the system appearing in (b) and n 2 is the dimension of A 2, the observations S g (S 2 (s)) = R n 2 (for any g ) and S (S 2 (s)) = n 2 (for any 2 ) nish the proof. The main reason for transforming to this special form is the property (b) in Theorem 2. As it is known from the theory of almost disturbance decoupling, such a right-invertible system without any zeros has the following very appealing property [39, 36]. Theorem 4 A? I If has maximal row rank for all 2, there exists a sequence of feedback H matrices F j such that A + F j is stable and lim kh(si? A? F j)? k = as well as lim j j Z khe (A+F j )t k 2 dt = (2)

12 hold true. As noted in Section 3, we will work with the Riccati map R(:; :) and with A(:) even if E does not have maximal column rank. It is important to know how these maps behave under restricted coordinate changes. Let U denote the state-transformation matrix in S(s) ~ S(s); i.e., U(A + F )U? = A. ~ Then we have to transform G as ~G = G G 2 := UG: We denote by ~ R(:; :) and ~ A(:) the maps for the transformed data ~ S(s) and ~ G which are given by ~R(X; ) = ~ AX + X ~ A T + X ~ H T ~ HX + ~ G ~ G T? ~ ( ~ E T ~ E) + ~ and ~ A(X) = ~ A + X ~ H T ~ H: The following result is veried by computation. The verication exhibits the reason for introducing the notion of restricted coordinate changes: One has to make essential use of the orthogonality of both transformations in the input- and the output-space. Lemma 5 For any X 2 S n and 2 R, the equations UR(X; )U T = ~ R(UXU T ; ) and UA(X)U? = ~A(UXU T ) hold true. 5 H -Optimization by Static State-Feedback Our aim is to provide reasonable and veriable characterization for the set P := fp > j9f : (A + F ) T P + P (A + F ) + P GG T P + (H + EF ) T (H + EF ) < g to be nonempty if is positive. The reader should recall the interpretation of the ARI (3) as explained in Section 3 which shows that all the results of this section have immediate consequences for the general H -optimization problem by static state-feedback. Let us denote its optimal value by := supfk(h + EF )(si? A? F )? Gk?2 j (A + F )? g: As an example, we mention the problem of maximizing the complex stability radius by feedback which is just our H -optimization problem for E = [3]. Remark From Theorem 6, we extract the following well-known conclusion for the state-feedback H - problem: In case of = I and D =, the optimal value opt for dynamic controllers coincides with. We dene S(s), G, R(:; :), A(:) and the transformed objects ~ S(s), ~ G, ~ R(:; :), ~ A(:) as well as the state transformation matrix U as in Section 4, where ~ S(s) has all the properties listed in Theorem 2. If any of the subspaces introduced in Section 4 is dened with respect to S(s), we drop the argument. Our rst suboptimality test is one of the central results of this paper. 2

13 Theorem 6 Suppose >. Then < i the ARI A X + XA T + XH T H X + G G T? (J? )(J? ) T < (3) has a positive denite solution. Therefore, < can be tested by looking whether a certain reduced order algebraic Riccati inequality has a positive denite solution. The ARI is written down in a suggestive way. If we introduce := (J? ); (4) the ARI (3) has a positive denite solution i > is suboptimal for the regular H -problem in which the underlying system is given by S (s) A? si H I A (5) and G is the disturbance input matrix. This shows the only dierence if comparing regular and singular problems: For regular problems, the suboptimality can be tested by looking at an ARI dened in terms of the original system matrices. For singular problems, one has to transform the system and to test the suboptimality for a certain (regular) subsystem. The reduction to the regular subsystem may be viewed as factoring out the `unproblematic' part of the system which is constituted by the the nite but assignable zeros and by the innite zero structure (the S -part of the system). The subsystem (5) is determined by the `relevant' part of the plant which is determined by the zeros of S(s) (and in particular by those in and + ) as well as the part of S(s) `outside N ' which is related to the `observable part' of to the nontriviality of H [3, Section 4]. A? si and due H Proof y the very denition of restriced coordinate changes, we can work without restriction with the transformed data ~ S(s), ~ G. Suppose < and choose some F and X > with ( A ~ + F ~ )X + X( A ~ + F ~ ) T + X( H ~ + EF ~ ) T ( H ~ + EF ~ )X + G ~ G ~ T < : (6) X We now partition X = X 2 X T according to A 2 X ~ and F X as according to the 2 ~F column partition of ~ and the row partition of ~ A. Then we compute the (,) block of the ARI (6) to obtain (recalling = T ) A X + X A T + J H 2 X T 2 + (J H 2 X T 2) T + ~ F + ( ~ F ) T + An obvious completion of the squares delivers + X H T H X + X 2 H T 2 H 2 X T 2 + ~ F T 2 ~ F + G G T < : A X + X A T + X H T H X + G G T? J J T??2 + + (J + X 2 H T 2 )(J + X 2 H T 2 ) T + (?2 T + ~ F ) T 2 (?2 T + ~ F ) < : This already proves the necessity part. 3

14 For the proof of suciency it is enough to construct some ~ F with and such that the H -norm of is smaller than = p. ( ~ A + ~ ~ F )? (7) ( ~ H + ~ E ~ F )(si? ~ A? ~ ~ F )? ~G (8) Now suppose that the ARI (3) has the positive dene solution X. Then there exists some F (e.g. F :=? T X? ) [3] such that A + F is stable and the inequality holds for H(s) := (A + F? si)?. Let us partition F = partitioning of in (4). H F H(s)G < p (9) F J according to the column F Exploiting the properties of the structure at innity, it is possible to approximate the transfer matrix in (9) by (8) in the H -norm, if F ~ is a suitably chosen feedback matrix with (7). If the error is small enough, the proof is nished. An obvious feedback transformation of S(s) ~ leads A +? F? si J H 2 G A 2? si 2 2 G 2 H H 2 F According to Lemma 3, there exist X and Y with A 2 X? X(A + F )? 2 Y = and H 2 X = F J : A : (2) We now add the X-right multiple of the second column of (2) to the rst one and the (?X)-left multiple of the rst row of the resulting pencil to its second row (which amounts to a coordinate change for (2) in the state-space). After a suitable feedback to eliminate the (2,) block, we obtain A + F? si J H 2 G A2 ~? si 2 ~ 2 G2 ~ F J H 2 A F with A2 ~ := A 2? XJ H 2, ~ 2 := 2? X and G2 ~ ~A := G 2? XG. Therefore, 2? I 2 H 2 still has maximal rank for all 2. y Theorem 4, we can hence construct a sequence F 2 (j) such that A2 ~ + 2 F 2 (j) is stable and lim kh 2H j (s)k 2 = (2) j 4

15 holds for H j (s) := ( ~ A F 2 (j)?si)?. A last feedback nally leads to the closed-loop A + F? si J H 2 G A2 ~ + 2 F 2 (j)? si G2 ~ H F J H 2 F This system is internally stable. Moreover, its transfer matrix H F J H 2 F A H(s)?H(s)J H 2 H j (s) H j (s) A : G ~G 2 by the formula for the inverse of a block matrix. Hence we can infer from (9) and (2) that the H -norm of (22) is less than = p for some j which is suciently large. The proof contains an explicit construction of stabilizing suboptimal feedback matrices which yield a closed-loop system with a block triangular structure. Since we make no special choices for the -suboptimal feedbacks for the regular H -subproblem (called F ) and the sequence of feedbacks on the innite zero structure used for the approximation (denoted as F 2 (j)), it is possible to take additional freedom (e.g. pole placement requirements) or simplications for the construction of these matrices into account [3, 36]. If we combine Theorem 6 with the results in [3], we conclude: The optimal value of the general state-feedback H -problem for S(s) and G can be computed by a quadratically convergent algorithm. The innite zeros of S(s) only cause additional algebraic eort for transforming S(s) into ~ S(s). Another technique to construct suboptimal static feedback matrices is to perturb H and E such that the perturbation of E has maximal colunm rank. Since we want to allow for rather general perturbation schemes, we introduce the following notion. Denition 7 The family (; ) 3 (H E ) 2 R (k+p)(n+m), p 2 N, is called an admissible perturbation of (H E) 2 R k(n+m) if the following conditions are satised: (a) lim & (H E ) = H E 2 R (k+p)(n+m), (b) (H E) T (H E) (H E ) T (H E ) for >, (c) E has maximal column rank for >. Theorem 8 Suppose that P > saties A T P + P A + H T H + P GG T P? (P + H T E )(E T E )? (E T H + T P ) < (23) for some >. If dening F by F :=?(E T E )? (E T H + T P ); P satises the ARI (3) and is thus an element of P. If P is contained in P, there exists an > such that P solves (23). 5 (22)

16 Proof According to the dierent possible representations of the Riccati map as presented in Section 4, the ARI (23) can be rearranged to (A + F ) T P + P (A + F ) + P GG T P + (H + E F ) T (H + E F ) < : (24) The inequality (b) implies (H +EF ) T (H +EF ) (H +E F ) T (H +E F ) which yields P 2 P. The convergence property (a) shows that there exists for P 2 P some F and some > for which (24) holds. Since E has maximal column rank by (c), a standard completion of the squares argument leads to the ARI (23). The union of all positive denite solutions of (23) over all > therefore conicides with P. This results allows to construct a suboptimal feedback matrix by nding a suitable >, in fact directly in terms of the system data without any preliminary transformation. In practice, however, the choice of > remains as a problem. From a computational point of view, we arrive at a satisfactory solution of the state-feedback H -problem. However, one wishes as well to have an algebraic characterization (without referring to perturbations) of suboptimality for which no preliminary transformation of the plant is required and which is directly formulated in terms of the original data S(s) and G. Indeed, it is possible to generalize the criterion we derived in [3]. We rst dene the set T () (apart from an additional invariance requirement) literally as in [3, Section 4]. Denition 9 For any 2 R, T () is the set of all matrices Z 2 S n such that S is A(Z)-invariant, the restriction of A(Z) T to S? + has all its eigenvalues in +, x T R(Z; )y vanishes for x 2 S? + and y 2 S? + + S?. We stress that A(Z)S S implies A(Z) T S? + S? + such that the denition makes sense. Now we can formulate an abstract but algebraically veriable suboptimality test in terms of the original data matrices. Theorem Suppose >. Then < i there exists some Z 2 T () such that Z is positive on S? + and R(Z; ) is negative on [ 2 S? n S? +: (25) How do the matrices in T () look like and how should these conditions be understood? The answer is most simple and motivated by Theorem 6. According to Denition 9, we introduce the set T () for the Riccati map R (X; ) := A X + XA T + XH T H X + G G T? T where the underlying system is S (s) as dened in (5). y A S g (S (s)) = S? si g H for any g or g = 2, this is just the linear manifold we discussed intensively in [3]. The following result displays the simple relation of T () and T (). Moreover, the combination of Theorem 6 and [3, Theorem?] with the properties of the quadratic forms appearing in the Lemma proves our abstract suboptimality criterion. 6 (26)

17 Lemma T () = U? f Z Z 2 Z T 2 S n j Z 2 Z 2 T (); Z 2 satises J + Z 2 H T 2 = gu?t : 2 Moreover, if choosing Z 2 T () and Z 2 T (), then hold for any 2. fx Zx j x 2 S? +g = fx Z x j x 2 S + (S (s))? g; fx R(Z; )x j x 2 S? g = fx R (Z ; )x j x 2 S (S (s))? g Proof Lemma 5 and the transformation properties of S g (:) for g = +; ; 2 show that we can prove the lemma without restriction for the system S(s) ~ and G. ~ If partitioning any Z 2 S n, A(Z) ~ and ~R(Z; ) as A, ~ one easily computes ~A(Z) = A + Z H T H J H 2 + Z 2 H T H 2 2 and the blocks ~ R (Z; ), ~ R2 (Z; ) and ~ R2 (Z; ) of ~ R(Z; ) which are given by A Z + Z A T + Z H T H Z + G G T??2 T? J J T + (J + Z 2 H T 2 )(J + Z 2 H T 2 ) T, (A + Z H T H )Z 2 + Z 2 A T 2 + Z F T T 2 + (J + Z 2 H T 2 )H 2Z 2 + G G T 2??2 T 2, A 2 Z 2 + Z 2 A T 2 + Z 2 H T 2 H 2 Z 2? 2?2 T 2 + G 2 G T 2 + Z T 2H T H Z F Z 2 + Z T 2F T T 2 respectively. The combination of Theorem 2 with (26) implies S? g ( ~ S(s)) = fx = x j x 2 S g (S (s))? g for all g or g = 2. We infer that ~ A(Z) leaves S ( ~ S(s)) invariant i J H 2 +Z 2 H T 2 H 2 =, or, since H 2 has maximal row rank, J + Z 2 H T 2 =. Now suppose that J + Z 2 H T 2 vanishes. Then ~ A(Z) T is antistable on S + ( ~ S(s))? i (A + Z H T H ) T is antistable on S + (S (s))?. Moreover, we obviously have ~ R (Z; ) = R (Z ; ). If we choose x = (x ) 2 S? + ( ~ S(s)) and y = (y ) 2 S? + ( ~ S(s)) + S? ( ~ S(s)), we infer x T R(Z; )y = x T R (Z ; )y. This already proves the relation of T () and T (). Now take x = (x ) 2 S +(? S(s)) ~ and y = (y ) 2 S? ( S(s)) ~ for some 2. We infer x Zx = x Z x and y T R(Z; )y = y T R (Z ; )y and just have to note that neither x Z x nor y T R (Z ; )y depend on the particular choice of Z 2 T () in order to nish the proof. Remark Lemma shows that T (), if nonempty, is a linear manifold in S n. Moreover, the positivity/negativity conditions (25) hold for one Z 2 T () i they hold for all Z 2 T (). Therefore, it suces to pick out one element Z 2 T () and to verify (25) for this particular Z. This can be done explicitly as explained in [3]. 7

18 One should recall the following intuitive interpretation of T (): T () is nonempty i a certain reduced order Riccati equation has a (unique) antistabilizing solution. A parametrization of T () is computed by solving this ARE and two (always solvable) linear equations. Positivity of T () on S? + is related to the positive deniteness of the ARE solution. The negativity of R(T (); ) on S? n S? + for all 2 is related to the solvability of a reduced order Lyapunov inequality. Let us introduce the following critical parameters max : < max () T () 6= ;; pos : < pos () < max ; T () is positive on S? ; + neg : < neg () < max ; R(T (); ) is negative on [ 2 S? n S? + ; which obviously leads to = minf pos ; neg g: We can again refer to [3] for the fast computation of all these parameters since they coincide with those for T (). Moreover, coincides with pos if S(s) has no invariant zeros on the imaginary axis. Due to the -zeros of S(s), the optimal value may jump from pos to minf pos ; neg g. In fact, neg has its own system theoretic signicance. Remark Suppose that (A? si ) is stabilizable with respect to. Along the lines of our approach to the H -problem and exploiting [3, Theorem 3 (a)], one can prove for any > the following equivalences: 9F : (A + F ) \ = ;; < k(h + EF )(si? A? F )? Gk?2 () 9P 2 S n : (A + F ) T P + P (A + F ) + P GG T P + (H + EF ) T (H + EF ) < () < neg : This implies that neg is the optimal value of the L -optimization problem supfk(h + EF )(si? A? F )? Gk?2 j (A + F ) \ = ;g: Let us conclude this section with an algebraic characterization of max =, pos =, and neg =. y [3, Theorem 3], the following result needs no proof any more. Theorem 2 (a) max = i im(g) N (S(s)). In the case of max =, the parameters pos and neg can be computed by solving Hermitian eigenvalue problems. (b) pos = i im(g) S +. (c) neg = i im(g) \ 2 S. 8

19 6 The Solution of the General H-Optimization Problem We have discussed in Section 5 how to test algebraically whether P is nonempty. If we introduce Q := fq > j9j : (A + J)Q + Q(A + J) T + QH T HQ + (G + JD)(G + JD) T < g; it is possible, by dualization, to check Q 6= ; in the same way. We have even shown how to compute the critical parameters and such that P, Q are nonempty i <, <. Therefore, it remains to verify the coupling condition (5) for < minf ; g. If we tried to test this condition directly, we would be confronted with the diculty to nd suitable P 2 P and Q 2 Q but it is not clear how to make a concrete choice. At this point, the introduction of lower limit as discussed in [3] for general Riccati inequalities proves to be very fruitful. Let P () denote the strict lower limit point of the set of inverses of all positive denite solutions of (3). We stress that it is simple to compute P () by solving one well-dened reduced order Riccati equation [3, Theorem 9]. Theorem 3 For <, the set P has a strict lower limit point P () which may be computed by P () = U T P () Dually, the set Q has a computable strict lower limit point Q() for <. This solves our problem: Fix < minf ; g. Suppose that P 2 P and Q 2 Q satisfy (5). y P () < P and Q() < Q, we obtain U: (P ()Q()) < : (27) Now assume (27). There exist sequences P j 2 P and Q j 2 Q with P j P () and Q j Q() for j. y (27), there exists a suciently large j with (P j Q j ) <. We arrive at the central result of this paper. Theorem 4 The positive parameter satises < opt i < ; < ; and (P ()Q()) < : As shown in Section 5, the inequalities < and < can be tested algebraically. Moreover, we recall the detailed investigation of the inuence of the various zeros of S(s) and T (s) on both values and in [3, Section 4]. From the explicit formula for P () and, dually, for Q() it is obvious that these matrices are not inuenced by the -zeros of S(s) and T (s) which may be expressed as follows: The -zeros of S(s) and T (s) do not cause additional coupling constraints. 9

20 It remains to prove Theorem 3 which is again based on the nice properties of the subsystem of ~ S(s) appearing in Theorem 2 (b). In particular, we exhibit in Proposition 5 the striking exibility of the Riccati map dened for a right invertible system without any zeros. Its proof is given in the appendix. Proposition 5 A? I Let S() = H R 2 S n there exist an > and a X 2 S n with have maximal row rank for all 2. Then for all X 2 S n and X > X and AX + XA T + XH T HX? T 2 < R : Proof of Theorem 3 We can again assume without restriction S(s) = ~ S(s) with F = and G = ~ G: hoose P 2 P. Take any F such that X := P? satises (6). We proved that the (,) block X of X solves the ARI (3) and infer by the denition of P () the inequality P () < X?. Lemma 4 in [3] shows one part of the claim: P () := P () Let us now dene the admissible perturbation ~H H H 2 and, for notational simplicity, the Riccati map A and ~ E := < I A ~R (X; ) := ~ AX + X ~ A T + X ~ H T ~H X + ~ G ~ G T? ~ ( ~ E T ~ E )? ~ T : According to Theorem 8, it suces to construct sequences j > and X(j) > which satisy ~R j (X(j); ) < and X(j)? P () for j. We exploit the obvious relation ~R (X; ) = ~ R(X; ) +? 2 2 T 2 and note that we already computed the blocks of R(X; ~ ) in the proof of Lemma. y the denition of P (), we can nd a sequence X (j) > of solutions of the ARI (3) with X (j)? P () for j. Moreover, we dene X 2 (j) X 2 where X 2 satises J + X 2 H T 2 =. If one has a look on the structure of R2 ~ (:; )? 2 2 T 2, one can deduce from Proposition 5 X the existence of sequences j > and X 2 (j) such that X(j) := (j) X 2 X T is positive 2 X 2 (j) denite, X 2 (j)? converges to for j and Rj ~ (X(j); ) is negative denite. Again by [3, Lemma 4], we infer X(j)? P () for j. 2

21 7 The omputation of the Optimal Value We have intensively discussed [3, 3] how to compute the critical parameters pos, neg and max for (S(s) G) and those for (T (s) T H T ) denoted as pos, neg and max. In Section 6, we introduced P (:) on (?; ) and Q(:) on (?; ) and proved that we only have to nd in addition in order to determine opt. cou := supf < minf ; g j (P ()Q()) < g If we combine Theorem 3 with [3, Theorem 9], there exist nonsingular constant matrices S, T with and P () = S T X()? Q() = T T Z()? S; X() > ; for < T; Z() > ; for < : Let us denote the left upper block of ST T (in an obvious partition) as V. Then we extract from [3, Section 7] for 2 (?; minf ; g) the equivalence If we dene the analytic function we obtain (P ()Q()) < () V T Z()? V < X(): (?; minf ; g) 3 F () := X()? V T Z()? V; cou = supf j < < minf ; g; F () > g: Since F () is positive denite and F () as well as F () are both negative semidenite for < minf ; g (because both X(:) and Y (:) have these properties), we can literally apply to F (:) the algorithm formulated in [3, Section 7] in order to compute cou and then opt. As explained in [3, Section 5], one can combine the computation of cou with that of, which implies that there exists a quadratically convergent algorithm to determine opt. If one of the values max or max is innite, pos, neg or pos, neg can be found by solving linear equations and Hermitian eigenvalue problems [3, 3], and it may happen, depending in particular on V, that F (:) also has simple structural properties which allow the explicit computation of opt. If both values are innite, this can be assured and we summarize the consequences in the following result. Theorem 6 max = i im(g) N (S(s)) and, dually, max = i R (T (s)) ker(h). If both values are innite, there exists a computable square polynomial matrix P cou () such that the optimal value is given by minf ; g if P cou () has no zeros in (; minf ; g) or, otherwise, by opt = minf 2 (; minf ; g) j det(p cou ()) = g: 2

22 Proof We only have to consider the characterization of opt in the case of max = and max =. Then X(:) and Y (:) are both ane and, therefore, F (:) is a real rational function without poles in (?; minf ; g). Let d(:) denote the least common multiple of the denominators of all the elements of F (:) and dene the polynomial matrix P cou () := d()f (). We obtain for any 2 (?; minf ; g): det(f ()) = () det(p cou ()) = : The distinction between the following cases nishes the proof: P cou () has no zeros in (; minf ; g): Then det(f ()) does not vanish in this interval. This implies F () > for all 2 (; minf ; g) and thus opt = minf ; g: There exists a minimal zero of P cou () in (; minf ; g): Then F () is positive denite for 2 (; ). Hence F ( ) is positive semidenite and singular. y [3, Proposition 6], F () cannot be positive denite for > and we infer opt =. The reader should note the additional simplications if pos or/and pos is/are innite since the functions P (:) or/and Q(:) is/are constant in this case. There is no need to discuss all the details. We just give an interpretation of these results for the model-matching problem introduced in Section 2. If one of the matrices T 2 (s) or T 3 (s) T has full row rank over R(s) (the two block problem), either max or max is innite. If both matrices have full row rank (the one block problem), the comupation of opt amounts to the solution of an eigenvalue problem. Theorem 6, however, shows that this may even happen if none of these conditions holds true, i.e., for certain four block problems. 8 Almost Disturbance Decoupling with Stability Any geometric characterization of opt = solves the almost disturbance decoupling problem with? -stability by output measurement (ADDP). Up to now, the ADDP has been treated for closed stability sets [37] or the given conditions for the MIMO case are deduced in a rather ad hoc way from SISO results without geometric interpretations [2]. Our approach allows to derive the following new geometric solution of the ADDP in a straighforward manner. Theorem 7 opt is innite i V + (T (s)) + X \ im(g) S + (S(s)) \ S (S(s)); (28) 2 V (T (s)) ker(h); (29) 2 V + (T (s)) S + (S(s)): (3) Proof If we assume opt =, we obtain (28) from =, i.e., pos = and neg = and Theorem 2. The dual inclusion (29) follows from =. y pos = and pos =, the functions P (:) and Q(:) are constant. Therefore, (P ()Q()) < holds for all 2 R i P ()Q() = or im(q()) ker(p ()) are valid for one/all 2 R. Theorem 3 together with [3, Theorem 9] imply that the kernel of P () is given by S + (S(s)) and, dually, the image of Q() equals V + (T (s)). This yields (3). 22

23 The argmuents can be reversed to obtain opt = from (28), (29), and (3). One should compare this result with the solution of the ADDP for the closed stability set? [. The rst two conditions dier due to -zeros of S(s) and T (s) but the third condition is the same reecting the earlier statement that the -zeros do not cause additional coupling constraints. The solution of the state-feedback ADDP ( = I and D = ) is contained in the above result: (29) and (3) are automatically satised. As it was expected in [2], the relevant space on the right in (28) lies between S + (S(s)) \ S (S(s)) = V? (S(s)) + S (S(s)) and S + (S(s)) = V? (S(s)) + V (S(s)) + S (S(s)). We stress a simple consequence of our results which cannot be obtained from those in [2]: If =, it is possible to construct a sequence of static stabilizing feedback matrices in order to approach. We nally mention an application of the solution to the ADDP in robust stabilization: As discussed in [6], it is possible to test algebraically whether complete quadratic stabilization is realizable. A Proof of Proposition 5 Since one can decompose any R 2 S n as JJ T? GG T, it suces to prove the result for R =?GG T. Thus we have to show for an arbitrary real G 2 R n and any X 2 S n : 9 > 9X > X : AX + XA T + XH T HX? T + GG T < : y Theorem 4, there exist P > and F with (A + F ) T P + P (A + F ) + P GG T P + H T H <. H Since (H E ) = denes and admissible perturbation of (H E), we infer from I Theorem 8 the existence of some X > and with AX +XA T +XH T HX? T +GG T <. From now on, is always taken out of [ ; ) if not specied otherwise. The set X () := fx > j AX + XA T + XH T HX? T + GG T < g is nonempty. According to [3, Theorem 9], X ()? has a strict lower limit point which is denoted as P and is, by X () X () for <, nonincreasing. Therefore, the limit lim P exists. If we show that this limit vanishes, we have proved the proposition. [3, Theorem 9] also contains an explicit formula for P. We transform with some nonsingular S as such that SAS? = A? si H equal to S T X? A ; S = ; SG = G ; HS? = K 2 H A 2 2 G 2 only has unobservable modes in + S where X > is the unique matrix which satises H ; and (A 2 ) [?. Then P is A X + X A T + X H T H X + G G T? T = ; (A + X H T H ) + : Note that X is nondecreasing. Let us dene := X? X and ~ A := A + X H T H. Then it simple to verify [32] that satises ~A + ~ A T + H T H? (? ) T = ; ( ~ A T + H T H ) + : 23