Carsten Scherer. Mechanical Engineering Systems and Control Group. Delft University of Technology. Mekelweg CD Delft

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1 Mixed H =H ontrol arsten Scherer Mechanical Engineering Systems and ontrol Group Delft University of Technology Mekelweg 68 D Delft The Netherlands Abstract In this article we provide a solution to the mixed H =H problem with reduced order controllers for time-varying systems in terms of the solvability of dierential linear matrix inequalities and rank conditions, including a detailed discussion of how to construct a controller. Immediate specializations lead to a solution of the full order problem and the mixed H =H problem for linear systems whose description depends on unknown but in real-time measurable time-varying parameters. As done in the literature for the H problem, we resolve the quadratic mixed H =H problem by reducing it to the solution of a nite number of algebraic linear matrix inequalities. Moreover, we point out directions how to overcome the conservatism caused by assuming a particular parameter dependence or by using constant solutions of the dierential matrix inequalities. For linear time-invariant systems, we reveal how to incorporate robust asymptotic tracking or disturbance rejection as an objective in the mixed H =H problem. Finally, we address the specializations to the fully general pure H or generalized H problem, and provide quadratically convergent algorithms to compute optimal values. Our techniques do not only lead to insights into the structure of the solution sets of the corresponding linear matrix inequalities, but they also allow to explicitly describe the inuence of various system zeros on the optimal values. Notation =? [ [ + is the complex plain partitioned into open half planes and the imaginary axis. R n is equipped with the Euclidean norm, and R nm with the corresponding induced In: A. Isidori, Ed., Trends in ontrol, A European Perspective, Springer-Verlag, erlin 995, pp.

2 norm, both denoted as k:k. (A) denotes the spectral radius of the matrix A, and A the unique positive semidenite square root of A if A. L p denotes the signal space L n p [; ) (for an appropriate n) and is equipped with the standard norm k:k p. Functions are tacitly assumed to be continuous and bounded, and smooth functions are, in addition, continuously dierentiable. Time functions are functions dened on [; ). For a symmetric valued function X dened on S, X is said to be strictly positive (X ) if there exists an > with X(s) I for all s S. For the system" or input # output mapping A A _x = Ax + u; x() = ; y = x + Du, we use the notation. If is D D a constant matrix, the system is called LTI, if it is a time function, the system is called LTV. The time function A is exponentially stable if the system _x = Ax has this property. Introduction Recently, linear systems which depend on time-varying a priorily unknown but on-line measurable parameters have gained a lot of interest [5, 38],[3]-[8]. These so-called linear parametrically-varying (LPV) systems appear in robustness problems, in gain-scheduling techniques for nonlinear systems, or in synthesis problems for nonlinear systems that can be described by a dierential inclusion []. For a detailed discussion we refer to the literature [5, 4]. Given an LPV system, the goal is to construct a controller which not only uses a measured output but, in addition, the on-line measured actual parameters as information in order to exponentially stabilize the system and to provide good performance properties. Until now, the performance objective was specied as an L disturbance attenuation problem with the standard interpretation such as guaranteeing robust stability or tracking [36],[3]-[8]. This is the so-called H problem for LPV systems. In principle, an LPV system can be viewed to be time-varying, and any design technique which is available for a specic choice of the performance measure can be used for controller construction. However, the actual parameter curve is not known a priorily and many existing synthesis techniques (as e.g. H and H control for LTV systems) instantaneously require the knowledge of the parameter values over the whole time interval of interest. For LPV systems and the H problem, it is not dicult to propose a (pretty conservative) way out of this dilemma: Assume that the parameters are contained in an a priorily given set. Then replace the time-varying solutions of dierential Riccati equations along the actual parameter curve by constant solutions of algebraic Riccati inequalities over the whole set of possible parameters. Only under additional hypotheses on the structure of the parameter set (polytopic) and on the dependence of the system on the parameters (ane and partly constant), the verication of the existence of a suboptimal controller and its construction can be reduced to solving a set of linear matrix inequalities [3]-[8]. The purpose of this article is to show that these ideas can be extended to the so-called mixed H =H problem (only whose LTI version has been addressed previously [9, 7, 5,

3 8, 9, 33, 4, 43, 5, 6]) such that not only robustness specications (in terms of an H constraint) but also performance specications (measured in H norm like criteria or upper bounds thereof) can be taken into account. In fact, we provide a full proof for our central result, a solution of the reduced order mixed H =H problem for LTV systems in terms of the solvability of dierential linear matrix inequalities and rank coupling conditions, and we give explicit formulas for a full order controller. Then we address various specializations of our main result. We point out how to recover results on the pure H problem for LTV systems [39, 55]. More importantly, we solve the mixed H =H problem for LPV systems not only in the spirit of previous work but also including possible renements to avoid conservatism. This encompasses a solution of the H problem for LPV systems. If the system is LTI, we obtain a new solution of the mixed H =H problem in terms of linear matrix inequalities where the underlying system is in no way restricted. Moreover, we show how to incorporate asymptotic tracking or disturbance rejection requirements by extending the system with a suitable internal model. This leads to a solution of the mixed H =H problem with robust regulation (for possibly large plant uncertainties) and extends [, ]. For the pure H problem, we generalize [47, 49] to systems having a nonzero direct feedthrough from the disturbances to the controlled outputs and provide a quadratically convergent algorithm to compute the optimal value. Finally, we reveal that such a computational scheme can be also obtained for the generalized H problem [4], and we provide an explicit formula for the optimal value of the genuine H problem, both for general LTI systems [5]. As auxiliary considerations, we investigate in detail the estimation inequality both for time-varying and for time-invariant data. In the latter case, we are not only able to gain insights into the structure of the solutions of this algebraic linear matrix inequality, but we can also explicitly display the inuence of various system zeros on the solvability. The article is organized as follows. In Section, we dene the mixed H =H objective for time-varying systems and address the related analysis tests, including the role of scalings, for reasons of space mainly without proofs. Section 3 contains our main result, a solution to the reduced order controller mixed H =H control problem in terms of dierential linear matrix inequalities, including explicit formulas for controllers. In Section 4, we show how the estimation dierential inequality can be reduced to an initial value problem for a perturbed Riccati dierential equation. For time-invariant data, we investigate the corresponding algebraic linear matrix inequality in full generality. Section 5 summarizes the consequences for our main result. In Section 6, we discuss linear parametricallyvarying systems and demonstrate a controller construction by solving a nite number of algebraic linear matrix inequalities. In Section 7, we address the mixed H =H problem with robust regulation for LTI systems, and the Sections 8 and 9 are devoted to the pure H and (generalized) H problems respectively. Short proofs which provide insights into construction schemes are included in the text whereas more technical proofs are collected in the appendix. 3

4 Mixed H =H Performance ounds onsider the LTV system z = T w = z z = T T w = 6 4 A G H F H F w: () We interpret w z as the robustness channel and w z as the performance channel. To be more specic, we assume that the uncertainty of the system is described by w = z where comprises the set of (possibly nonlinear) operators L L with incremental gain [3] not larger than =. If A is exponentially stable, the small-gain theorem implies that stability is preserved if kt k <, where kt k := sup kt wk wl ;kwk = denes the operator norm of T induced by signals in L. In the LTI case, there is a wellknown test on the state space matrices which characterizes stability of A and kt k <, the so-called ounded Real Lemma (RL) [57,, 6]. It is not dicult to prove the following generalization to LTV systems. Recall that time functions are bounded, and that a symmetric valued time function X is strictly positive (X ) if there exists an > with X(t) I for all t. Theorem (LTV Strict ounded Real Lemma) The time function A is exponentially " A G stable and # < i there exists a smooth time function X such that H F X ; X _ + A T X + XA XG H T G T X?I F T H F?I A : () Hence norm bounds can be characterized by the existence of strictly positive solutions to a strict dierential linear matrix inequality (DLMI). Note that asking for the existence of a constant solution is equivalent to the popular concept of quadratic H performance for an LTV system [3]-[8]. If T is LTI, it is no loss of generality to conne X in Theorem to be constant and the inequalities () become algebraic. The channel w z of () is used for describing performance specications. Indeed, we have in mind to generalize certain H norm like criteria for LTI systems [4] to the LTV system T. Among the several possibilities, we pay special attention to the deterministic criterion of assessing performance by the largest amplitude of z for all w of nite and bounded energy. This is particularly useful if (components of) z are interpreted as tracking errors. To quantify the gain of T mapping L into L, we dene the induced norm kt k := sup wl ;kwk = kt wk : 4

5 If A is exponentially stable, let Y denote the (bounded) solution of the initial value problem _Y = AY + Y A T + GG T ; Y () = : (3) If F =, one has kt k = sup t kh Y H T (t)k. This allows to prove the following analogue of Theorem. Theorem A is exponentially stable and a smooth Z with " A G H F # < i F = and there exists Z ; _Z + A T Z + ZA + ZGG T Z ; H Z? H T I: (4) If dening the size of the amplitude of z with the spatial norm max j jx j j, the squared gain of T equals sup t max j d j [H Y H T (t)], where d j (M) denotes the j-th diagonal element of the square matrix M, and Y solves (3). Theorem remains valid for this norm after replacing H Z? H T I by max j d j (H Z? H T ). An alternative measure arises with a stochastic interpretation. If w is white noise, we recall (due to x = and hence E(x x T) = ) that E(z z T) = H Y H T [3]. Then kt k := sup t E(z T z ) = sup trace[h Y H T (t)] t denes the maximal output variance and is a generalization of the genuine H norm to LTV systems. Theorem persists to hold for the H norm with H Z? H T I replaced by trace(h Z? H T ). Even for LTI systems, the synthesis problem of optimizing k:k or k:k of over all stabilizing controllers which keep a bound on the norm k:k for a dierent channel seems very hard [8, 4]. This motivates to replace the objective functional by an upper bound. Let us dene J(T ) := inff j 9 smooth time function X with () and H X? H Ig (5) (including, as usual, J(T ) = if no solution of () exists.) Then J(T ) < implies H X? H I for some solution X of (). Note that any such X satises _ X + A T X + XA + XGGT X. If F =, Theorem allows to infer kt k <. We conclude kt k J(T ) and J(T ) is indeed an upper bound of kt k =. If = ; H = ; F =, the solution sets of the DLMI in () and of the dierential Riccati inequality in (4) are clearly identical, what implies kt k = J(T ) and recovers kt k. Similar conclusions hold for the other norms of T. One can clearly dene J(T ) via (5) conning X to constant solutions of (). This generalizes the so-called quadratic H performance specication to the quadratic mixed H =H 5

6 specication and allows a specialization to the pure H case. If T is LTI, it is wellknown that this restriction to constant X causes no loss of generality, and we recover the denitions in [4, 4]. Let us nally comment on scalings. If one can take structural or other properties of the system uncertainty into account, the perturbation is restricted to a certain subset of all operators with incremental gain at most =. It might then be possible to identify a class of scalings containing pairs of matrix valued time functions (S; S ) with bounded inverses such that SS has incremental gain at most = for all : The existence of a pair of scalings (S; S ) in this class with ks? T S? k < is an obvious sucient condition for stability robustness against uncertainties in, and it is weaker than kt k <. A most prominent specic example of this concept is the -upper bound with constant scalings. This gives a systematic tool to incorporate in this channel not only stability robustness requirements against structured uncertainties, but also robust performance specications in the induced L -norm, as usually done for LTI systems [37]. One might as well specify a set of scalings S for weighting the performance output and modeling alternative performance specications [44]. The benet of incorporating scalings S for the channel z remains to be explored. Summarizing, if having xed a class of scalings S consisting of triples (S ; S ; S) of time functions with bounded inverse, we can alternatively dene J s (T ) to be the inmal for which there exists an (S ; S ; S) S and a solution to the RL inequality corresponding to ks? T S? k < such that S? H X? H T S?T I: 3 The Mixed H =H Synthesis Problem Suppose a specic control tasks (including the specication of weightings) leads to the generalized LTV plant 3 A G y z A = 6 D 7 u 6 4 H E F 7 (6) 5 w z H E F where A is of size n n. With the LTV controller " K L u = Ry = M N 6 # y; (7)

7 the closed loop system is described as where z = T (R)w = z z = T (R) T (R) 6 4 A G H F H F = 6 4 w = 6 4 A G H F H F A + N M G + ND L K LD H + E N E M F + E ND H + E N E M F + E ND w The controller R is called stabilizing if A is exponentially stable. We intend to minimize J(T (R)) over all LTV controllers R. Recall that J(T (R)) < automatically implies that R is stabilizing. It is standard to approach this problem via a suboptimality test: haracterize whether there exists an R with J(T (R)) < or, equivalently, whether there exists an R and a smooth time function X such that X ; X _ + A T X + X A X G H T G T X?I F T H F?I : A ; H X? H : (8) This not only guarantees robust stability (against perturbations with incremental gain at most =) but also a performance level. Recent approaches to the H problem for LTI systems are based on the following observations: The RL inequality in (8) is, for a xed X, linear in the controller parameters. Hence one can eliminate these parameters using the so-called Projection Lemma (Lemma 7), what leads to a suboptimality test in terms of linear inequalities in (parts) of X and X? [, 6, 49]. In the mixed problem (8), we have to fulll three inequalities what makes it impossible to eliminate all controller parameters from the nal characterization. Instead, we intend to keep as many (transformed) controller parameters as possible such that, still, matrix inequalities result which are linear in parts of X and X? and in the remaining (transformed) controller parameters. In fact, the key idea has its origins in [45, 49]: Eliminate K in (8) (which only aects the RL inequality), transform L and M, and keep N to achieve the desired structure. A central step in the proof is the following very simple explicit result for the solvability of a specially structured inequality. In fact, it will turn out in Section 4 that this is a version of the so-called `completion of the squares argument' which is most suited for our purposes. Lemma 3 Let Q be a symmetric (partitioned) time function and consider the inequality Q Q T Q T 3 + X T Q Q Q T 3 Q 3 + X Q 3 Q 3 7 A (9)

8 in the unstructured time function X. This inequality has a solution X i Q Q T Q Q T 3 and : () Q Q Q 3 Q 3 If (9) is solvable, one particular solution is given by X = Q 3 Q? Q? Q 3 : () Proof. If (9) has a solution then () just follow from (9) by canceling the rst or third block row/column. Now suppose that () holds what implies Q : The central trick is to cancel in (9) the block Q by a congruence transformation. To be specic, (9) is equivalent to which rewrites to I?Q T Q? I I A (9) I?Q? Q I I A Q? Q T Q? Q Q T 3? Q T Q? Q T 3 + X T Q Q T 3 Q 3? Q 3 Q? Q + X Q 3 Q 3 X dened in () is a solution since () implies Q? Q T Q? Q. Now we are ready to formulate and prove our main result. A : " # K L Theorem 4 There exists a controller R := with K of size k n which M N satises J(T (R)) < i there exist >, time functions X, Y, Z with X ; Y ; X? Y? = ZZ T ; Z of size n k; Z T Z ; () and time functions F, J, N such that rank F Y I N I X A = k + n, rank N Y I J I X = k + n; (3) and such that the following DLMIs are satised: _X + A T X + XA + J + (J) T XG + JD (H + E N) T (XG + JD) T?I (F + E ND) T H + E N F + E ND?I? Y _ + AY + Y A T + F + (F ) T G + ND (H Y + E F ) T (G + ND) T?I (F + E ND) T H Y + E F F + E ND?I 8 A ; (4) A ; (5)

9 ? I H Y + E F H + E N (H Y + E F ) T Y I (H + E N) T I X A : (6) Proof of necessity. Suppose, for some controller of size k n, there exists a smooth time function X with (8). We x X andview the RL inequality in (8) as an inequality in K. To be specic, dene I := I to get A = A + N M L Then the RL inequality reads as + I K I = ~ A + I T KI: X _ + A ~ T X + X A ~ + I T K T IX + X I T KI X G H T G T X?I F T H F?I A : (7) The matrix J := I clearly satises IJ =. Hence (7) implies J T [ X _ + ~A T X + X ~A]J J T X G J T H T G T X J?I F T H J F?I A : (8) Similarly, with we infer (IX )YJ = and thus Y := X? ; J T [? Y _ + Y A ~ T + AY]J ~ J T G J T YH T G T J?I F T H YJ F?I A : (9) Let us partition X, Y according to A into n and k rows/columns as X = X U T U ^X ; Y = Y V T V ^Y ; () and let us recall Y = [X? U ^X? U T ]??[X? U ^X? U T ]? U ^X?? ^X? U T [X? U ^X? U T ]? [ ^X? U T : () X? U]? We can assume without loss of generality that U (of dimension n k) satises U T U ; 9

10 if not true, just perturb X suitably without violating (8). The formula () reveals Y? = X? U ^X? U T. If holesky factorizing ^X = W T W such that W is smooth, bounded, and has a bounded inverse, we arrive at () for Z := UW?. A simple computation shows, with the identities F = NY + MV T and J = XN + UL; () that the left-hand sides of (8), (4) and (9), (5) are identical. We remark that we dene F and J via these equations for proving necessity, and we view () as equations in L and M for constructing a controller in the suciency proof. Under the hypothesis (), it is now not dicult to see that () are solvable as equations in L and M i the rank conditions (3) hold true. We clarify this for the rst equation: It is solvable i ker(v T ) ker(f? NY ). The (,) block of X Y = I implies XY + UV T = I which reveals ker(i? XY ) = ker(v T ) since U has full column rank. Hence, solvability is equivalent to ker(i? XY ) ker(f? NY ). y (), I? XY has rank k and, therefore, this inclusion is equivalent to the rst condition in (3) since F Y I N I X A I?Y I Finally, there is an > with H X? H T = F? NY N I I? XY X A : (? )I which is equivalent to (? )I H H T X : (3) With the (n + k) n function Z := Y I V T ; (4) the inequality (3) implies (? )I H Z Z T H T Z T X Z I X Recalling the denition of F and computing Z T X Z = Z T U T to (6). This proves necessity. : (5) = Y I I X leads X U onstructive proof of suciency. Dene U := Z, ^X := I, and X := U T I Due to X? UU T, X is smooth, bounded, and strictly positive, and the same holds for Y := X?. Again because of () and (), Y has Y as its left-upper block. If using the partitions (), we have U T U by hypothesis and, since V =?Y U from (), V T V as a consequence. Let us now dene the time functions M := (F? NY )V (V T V )? and L := (U T U)? U T (J? XN):.

11 Since U and V have full column rank, these are the unique solutions of the equations () if they are solvable at all. The latter, however, is assured by (3) as claried in the necessity proof. With L, M, N given so far, we can dene A, ~ G, H j, F j, j = ;. If we introduce Z as in (4), we infer that (6) is the same as (5). Since Z has full row rank and thus a right-inverse, we can get back to (3) which leads to the third inequality in (8). Moreover, (4) and (5) are identical to (8) and (9) respectively. Hence it remains to nd a time function K which satises (7) and, therefore, leads to the RL inequality in (8). Z We use X I T KI = K I k where we display the size of the identity blocks I k by using the index. Due to Z T Z, there exists a smooth time function Z e such that Z Z e has a smooth bounded inverse. With the rst k rows S and the last n? k rows S of this inverse, we conclude that S S has a smooth bounded inverse and S Z = S I k : With S := S S?S I k A we get S Z = I k I k A and S = I k I k A : Note that both S and S? are smooth time functions. We can left-multiply the rst row of (7) with S and right-multiply the rst column of (7) with S T. After this congruence transformation, (7) is equivalent to Q Q + K Q 3 Q + K T Q Q 3 Q 3 Q 3 Q 3 A (6) for some computable Q. Now we just note that (8), (9) are equivalent to Q Q 3 Q and Q 3. Lemma 3 then reveals that K := Q 3 Q 3 Q 3 Q 3 Q 3 Q? 3 Q 3? Q is indeed a time function which leads to (6) and hence to (7). The controller construction is complete. The following comments on Theorem 4 also apply, without explicitly mentioning, to all the other problems that will be considered in this article. Remark on the dual problem. onsider the LTV system y z z A = 6 4 A G G D D H E F F H E F F u w w A :

12 z = z w admits a smooth X with (8) and G T X G is characterized w The existence of a controller (7) such that the closed loop system described by 6 4 A G G H F F H F F exactly as in Theorem 4 if just adding (? )I (XG + JD ) T (G + ND ) T XG + JD X I G + ND I Y Then the solution of the dual problem is obtained by canceling (6). A : Remark on the controller construction. It is not dicult to see that any X satisfying (8) can be transformed by a coordinate change in the controller state (which amounts to a congruence transformation on X ) such that its right-lower block is identical to I. Hence this choice in the controller construction can be made without loss of generality. Remark on scalings. Since scalings only change the data matrices, we can directly derive the corresponding characterization for J s (T (R)) < without new proof. In Theorem 4, we just need to require, in addition, the existence of scalings (S ; S ; S) S and replace the (,) identity blocks in (4) and (5) by S T S, the (3,3) identity blocks by S S T, and the (,) identity block in (6) by S S T. Hence the DLMIs are linear in (S T S; S S T ; S S T ) for (S ; S ; S) S. It might be possible to reparametrize f(s T S; S S T ; S S T ) j (S ; S ; S) Sg and to transform the inequalities (4)-(6) such that the new parameters enter linearly [37, 38, 36, 3]. Generally, however, one has to turn to heuristic ways out [4]. Remark on other performance measures. Let e j denote the standard unit vector. learly, max j d j (H X? H T ) is equivalent to? e T j H H T e for all j and j X some >. Hence, if we replace the performance inequality in (8) by this alternative one, we conclude without new proof that Theorem 4 remains valid if replacing (6) with 8j :? e T j (H Y + E F ) e T j (H + E N) (H Y + E F ) T e j Y I (H + E N) T e j I X A : If H has l rows, the same holds for trace(h X? H T ) if replacing (6) with? (H Y + E F ) T e Y I (H + E N) T e I X.... (H Y + E F ) T e l Y I (H + E N) T e l I X A :

13 Note that one can easily extend all this to a system with more than one performance output and dierent performance specications on each of these outputs; one just needs to add the corresponding inequalities in Theorem 4. Remarks concerning direct feedthroughs. " Just by# setting N =, we can extract K L a suboptimality test for controllers of the form. Due to the H nature of the M performance specication, one might wish to include the requirement F + E ND = on N. This puts another linear restriction on N without destroying the structure of the DLMIs. Remark on special problems. The state-feedback problem is obtained by setting = I and D =. As immediate necessary conditions for suboptimality, there exist time functions Y (smooth) and F with? Y _ + AY + Y A T + F + (F ) T G G T?I F T H Y + E F F?I A ; I H Y + E F Y : If these two inequalities hold, it is easily seen directly that N := F Y? provides a static suboptimal feedback controller. An analoguous specialization tackles the full information problem = (I ) T, D = ( I) T. Similarly, we remark that the inequality (5) is related to an H estimation problem [34, 46]. Remark on quadratic performance. For solving the quadratic mixed H =H control problem, one needs to characterize the existence of a controller with (8) where X is restricted to be constant. Trivial simplications of our proof lead to the conclusion that Theorem 4 fully applies if just asking for constant X, Y, and Z. Hence, one ends up having to solve algebraic linear matrix inequalities (LMIs) in constant X, Y, Z and in time functions F; J; N along the parameter curve dened by the system data. Remark on LTI systems. If the system and the controllers are LTI, it causes no loss of generality to conne X in (8) to be constant. Hence Theorem 4 remains valid by specializing to constant X, Y, Z, and F, J, N. This is an LMI solution to the reduced order [9] or full order mixed H =H problem [7, 5, 8, 9, 33, 4, 43, 5, 6]. In contrast to all earlier papers, no technical assumptions on the system data are required. The design of full order controllers can be directly based on Theorem 5: Find the parameters X, Y and F, J, N by solving three coupled linear matrix inequalities [35, 4]. Then compute L, M, and K according to the formulas given after Theorem 5. Note that we do not require to include the a priori hypotheses that (A; ) is stabilizable and (A; ) is detectable (7) since they are obvious necessary conditions for the existence of positive denite solutions of (4) and (5). Remark on the pure H Problem. The whole discussion includes the pure H problem obtained with H =, E =, F =. y (), this just amounts to canceling 3

14 (6) in Theorem 4. In Theorem 5, one has to replace (8) by the coupling condition Y I : I X Remark on the pure H problem. As mentioned in Section, we can recover the pure generalized H problem by setting H =, E =, F =, =. In Theorem 4, this just amounts to canceling the third block rows/columns in (4) and (5). If allowing for nonproper controllers, one should include the linear constraint F + E ND = on N. To the best of our knowledge, this gives for the rst time a solution of the H problem in terms of DLMIs or, in the LTI case, in terms of LMIs. Let us now specialize Theorem 4 to the case without any a priori restriction on the controller size. In fact, a slight modication of the above proof (if the controller has size larger than n) leads to the following result. Theorem 5 There exists a controller R with J(T (R)) < i there exist smooth time functions X, Y and time functions F, J, N such that the DLMIs (4) and (5) and I H Y + E F H + E N (H Y + E F ) T Y I (H + E N) T I X are satised. If existing, the parameter K in R can be chosen of size n. A (8) Hence, suboptimality is characterized in terms of the solvability of two dierential inequalities and one algebraic inequality where all the unknowns X, Y, F, J, N enter linearly. This structural property will be essential for the discussions to follow. Remark on the construction of a full order controller. We would like to provide explicit formulas for how to construct a full order controller. Given X and Y, holesky factorize X? Y? = UU T such that U and U? are smooth and bounded. Motivated by XY + UV T = I, dene V = (I? Y X)U?T which is smooth and has a bounded inverse. Motivated by (), dene the controller parameters Finally, with M := (F? NY )V?T and L := U? (J? XN): (9) Q = _XY + _UV T + (A + N) T + X(A? N)Y + JY + XF; Q 3 Q 3 Q 3 = (G + ND) T (XG + JD) T?I (F + E ND) T H Y + E F H + E N F + E ND?I ; a suitable K is given by K = U? (Q T 3Q? 3 Q 3? Q )V?T : (This is shown literally as in the suciency proof of Theorem 4 by choosing S := Z T with Z dened in (4).) Note that we could as well start with U := X? Y? implying V =?Y or, dually, V := Y? X? implying U =?X. 4

15 4 Discussion of the Estimation DLMI In this section we would like gain some insight into the DLMI (4) which is related to an H estimation problem. We separate the time-varying case in Section 4. from the time-invariant situation in Section 4. since the latter leads to an algebraic linear matrix inequality. Most of the proofs are found in the appendix. 4. Time-Varying Data In the LTV case we proceed under the hypothesis DD T and E T E. If not true, these conditions can be enforced by (small) perturbations since the inequality (4) is strict. Then we can perform coordinate changes in the u- and y-space and orthogonal coordinate changes in the w- and z-space to obtain A G D H E F A = A G G I A = H E F F A G D H F H I F A = A G G I H F F H I F F A : (3) Under these hypotheses, the following result reduces the solvability of the DLMI to the test of whether a perturbed initial value problem has a bounded solution. We note that the proof provides, just by applying Lemma 3, a pretty quick derivation of the corresponding formulas. Theorem 6 If X satises the DLMI (4) then kf k ; kf k ; (3) and there exists an > such that the solution of the initial value problem Z() = I; _Z = (A? G )Z + Z(A? G ) T? Z T Z + I??I F? T G Z(H? F ) T F?I? G T (H? F )Z (3) exists on [; ) and remains bounded; for all small >, Z X?. onversely, (3) implies that?i F T N :=?F + F F?I? F yields kf + E NDk. If N is any time function with the latter property, and if the solution of (3) exists and is bounded on [; ) for some >, then Z, and (4) holds for X := Z? and J =?(XG + T ) + G T X H? F T?I F T?I F? F + E N (33) : (34) 5

16 Remark. The proof reveals that the set of all X satisfying the DLMI (4) (for some (J; N)) and the solution set of the dierential Riccati inequality are identical. _X + (A? G ) T X + X(A? G )? T??I F T? XG H T? T F T?I F? G T X H? F (35) 4. Time-Invariant Data If the data matrices are constant, we characterize in this section whether there exist constant X > ; J; N which satisfy (4); hence we discuss an algebraic linear matrix inequality. As a necessary condition for solvability, we infer that (A; ) is detectable; this is assumed from now on. For xed (X; N), we can view (4) as an LMI in J only. Dening this LMI reads as L(X; N) := L(X; N) + If we introduce a matrix I A T X + XA XG (H + E N) T G T X?I (F + E ND) T H + E N F + E ND?I A J D + T D T A ; A J T I < : K y whose columns form a basis of the kernel of D ; two obvious necessary conditions for the existence of a solution J are?i F + E ND (F + E ND) T?I < (36) and K T y L(X; N)K y <. The latter inequality is in fact independent of N since we have K T y L(X; N)K y = K T y L(X; )K y. The so-called Projection Lemma [, 6] reveals that these conditions are as well sucient for the existence of J. Lemma 7 (Projection Lemma) For arbitrary A,, Q = Q T, the LMI in the unstructered X has a solution i A T X + T X T A + Q < (37) Ax = or x = (x 6= ) imply x T Qx < : 6

17 We reproduce a proof since it provides a construction scheme for J. Moreover, it also reveals that, in suitable coordinates, Lemma 7 reduces to Lemma 3 if the kernels of A and together span the whole space. Proof. Necessity is trivial. For proving suciency, let S = (S S S 3 S 4 ) be a nonsingular matrix such that the columns of S span ker(a) \ ker(), those of (S S ) span ker(a), and those of (S S 3 ) span ker(). With ( A 3 A 4 ) := AS and ( 4 ) := S, we observe that (A 3 A 4 ) and ( 4 ) have full column rank. Hence, the equation A T 3 Z X A T 4 = 3 Z 34 has a solution X for each right-hand side. Therefore, S T (37)S is equivalent 4 Z 4 Z 44 to Q Q T Q T 3 + Z T 3 Q T 4 + Z T 4 Q Q Q T 3 Q T 4 Q 3 + Z 3 Q 3 Q 3 Q T 43 + Z 34 Q 4 + Z 4 Q 4 Q 43 + Z T 34 Q 4 + Z 44 + Z T 44 A < with free blocks Z ij. The hypotheses now just amount to () and, hence, we can nd a (constant) Z 3 such that the marked 3 3 block is negative denite. Since Z 44 is free, it can be chosen to render the whole matrix negative denite. We have fully proved the following result. Theorem 8 For xed X and N, the LMI (4) in J has a solution i (36) and hold. K T y A T X + XA XG H T G T X?I F T H F?I A K y < (38) We infer that N is only restricted by (36) and does not inuence (38). Moreover, for a xed N with (36), the set of all X satisfying (4) (for some J) is identical to the solution set of (38). This decouples the construction of X from the determination of (J; N) in (4). As for J, testing the existence of N with (36) and computing a solution just amounts to applying Lemma 7. Hence it remains to discuss how to test the existence of a positive denite solution of (38). If D has full row rank, one can reduce this LMI in exactly the same manner as in Section 4. to an algebraic Riccati inequality. In the general case, we need to transform the system data. For this purpose we look at A G D H F A S S S I A A G D H F A S? S 3 where S, S, S 3 are coordinate changes in the state-, y- and w-space (S 3 orthogonal) and S is an output-injection applied to (6). It is easily seen that (X; J; N) satises the LMI 7

18 (4) i (S XS ; (S J? S )S ; N) satises the same LMI for the transformed data. Hence solutions of (38) transform as X S?T XS? : As claried in [46, 49], one can transform in this manner A G D H F A to A r G r G s r A s G s s D r D s I H r H s F r F r F r A (39) with the crucial property that A s? I s G s has full column rank for all : (4) This transformation separates a regular subsystem 6 4 A G r r I D r I H r F r F r F r 3 " 7 5 from a singular part As G s s # (4) (what claries the indices s and r). For some more detailed explanations of this structure in the language of geometric control theory we refer to [49, 46]. It is important to observe that, on the basis of the structure algorithm, the regular subsystem can be computed in a numerically reliable manner [3]. For our purposes, the transformation is motivated by (4). It is well-known that left-invertible systems without any zeros have very nice properties. As an example, we mention a well-known fact from the theory of almost disturbance decoupling [58, 56]: if H r and F vanish, (38) has a positive denite solution for all >. This makes it plausible to expect that the singular part of the system can be factored out, and that the solvability of (38) can be characterized in terms of the regular subsystem only. The precise result reads as follows. Lemma 9 The inequality (38) has a (positive denite) solution i A T r X r + X r A r? (r T r + Dr T D r ) X r G r G T r X r?i Fr T H r? F r r? F r D r F r?i has a (positive denite) solution X r. A < (4) Let us introduce the abbreviations ~H r := H r? F r r? F r D r ; F := 8 I?F r I?F T r :

19 Then (4) is equivalent to F > and R(X r ) := A T r X r + X r A r? ( T r r + D T r D r ) + X r G r ~ H T r F? G T r X r ~H r < : Note that this is just the ARI which corresponds to (3) for the regular subsystem (4). If (?A r ; G r ) is stabilizable, a standard result implies that this strict ARI has a (positive denite) solution i the largest or antistabilizing solution of the corresponding algebraic Riccati equation (ARE) R(X r ) = exists (and is positive denite) [48, Theorem ]. If (A r ; G r ) has uncontrollable modes in? or, the validation test is more involved. Let us display the critical uncontrollable modes by choosing S in (39) (w.l.o.g.) such that where A r G r = A A A 3 G A A 3 A (?A ; G ) is stabilizable, (A ) ; (A 3 )? : This separates the uncontrollable modes of (A r ; G r ) in the open right-half plane (in (A ; G )) from those on the imaginary axis (in A ) and in the open left-half plane (in A 3 ). Let us partition the columns of r = 3 ; ~ Hr = H ~ H ~ H3 ~ D r according to those of A r. Finally, if A has the eigenvalues i j, let E j be a basis of the (complex) kernel of A? i j I. The following result provides a complete veriable characterization of whether (38) has a positive denite solution. Theorem The LMI (38) has a positive denite solution i F = the unique solution X of I?F r I?F T r > ; (43) A T X + XA? T + XG ~ H T F? G T X ~H = (44) with A + G F? G T X ~H + (45) exists and satises X >, and the unique solution Y of the linear equation A T Y + Y A + A T X? T + Y G ~ H T F? G T X ~H = (46) 9

20 satises E j " A T Y T + Y A? T + Y G ~ H T F? G T Y T ~H # E j < (47) for all j. Remark. Although these formulas look complicated, they have a very simple origin: After partitioning X r = X Y T Y Z W A ; R(X r) = R (X r ) R T (X r ) R (X r ) R (X r ) R 3 (X r ) A (48) according to A r, the block R (X r ) is indentical to the left-hand side of (44), R (X r ) is that of (46), and the left-hand side of (47) is nothing else than E j R (X r )E j. Let us now comment on how to test these properties. learly, (43) is just a matter of verication. Since (?A ; G ) is stabilizable, to test the existence of a solution X of (44) with (45) is a standard problem. We can apply the results in [8, Section 7.] to infer that X exists i the Hamiltonian matrix H := A + G F? T? ~ H T F? ~H ~H G F?? A? G F? G T T ~H X does not have eigenvalues on the imaginary axis. If true, and if the columns of X span the generalized eigenspace of H with respect to its eigenvalues in +, then X is square and nonsingular, and X = X X? satises (44)-(45). After all, one can easily verify X >. Due to (45) and (A ), we can solve the linear equation (46) for a unique Y, and (47) is, again, only a matter of verication. Remarks. (a) This theorem provides insight in which parts of the system are relevant for the solvability of (38). First, the singular part (4) does not play any role. Due to (4), it is easily seen that the zeros of A? si are the uncontrollable modes of (A r ; G r ), which are separated according to the partition + [ [? into the uncontrollable modes of (A ; G ) and the eigenvalues of A, A 3 respectively. Hence, zeros in the open left-half plane are irrelevant (A 3 does not appear) and the inuence of the zeros on the imaginary axis is explicitly displayed by (47). Similarly, the aect of the + zeros can be made explicit [46]. If G =, we note that (44) is a linear equation in X and (45) trivially holds; the resulting simplications for the following results are easy to extract [46, 49]. G D A (49)

21 (b) The proofs of Theorems 9 and provide explicit insights into the structure of the solution set of (38). In particular, one can extract which blocks of the solutions can be freely chosen or which ones can be made arbitrarily large. (c) Suppose that D has full row rank and that GD T =. Then we can assume w.l.o.g. D = ( I) and conclude A G A D A = r G r D r I A : (5) H F H r F r F r Hence (39) does not have a singular part and R(X r ) < is an algebraic Riccati inequality for the original matrices. Theorem then characterizes the solvability of this general ARI without restrictions on the uncontrollable modes of (A; G). If GD T 6=, one just needs to replace A by A? GD T (see Section 4.). (d) If (49) has no zeros in, A is empty (such that (46), (47) have to be cancelled) and one only needs to test the existence and positivity of X with (44)-(45). If (49) has no zeros in [?, an ARE in terms of the regular subsystem (4) results. (e) Suppose D = ( I) and GD T =. If (49) has no zeros in? [, then X >, (44)-(45) are conditions in the original data matrices. With P = X?, they can be rewritten as P and AP + P A T? P T P + G r P (H T? DF T ) F? A? P T + G r P (H T? DF T ) F? G T r (H? F D T )P? : H? F D T X If (49) has no zeros, the same holds with P =? S T S (in the partition of A r ). This transforms the conditions (44)-(45) in special coordinates back to the original data matrices, and reveals the relation to the indenite Riccati equations as appearing in [4, 5]. Again, if GD T 6=, replace A by A? GD T. Our main interest lies in the quick computation of the critical parameter = ; c := inff > j The LMI (38) has a positive denite solution.g (5) For this purpose we rst clarify that one can give an explicit formula for the interval of those values for which (43) holds and a symmetric X with (44)-(45) exists. One needs to solve a standard LQ Riccati equation which is, due to the stabilizability of (?A ; G ), always possible. Theorem Let X satisfy A T X + X A? T + X G G T X = ; (A + G G T X ) + :

22 Then (43) holds and there exists a symmetric X with (44) and (45) i > e := kf r? ( ~ H + F r G T X )(si + A + G G T X )? G k : Remark. Due to this formula, e can be computed by quadratically convergent algorithms []. Here is now the central trick to determine c : View the unique solutions of (44)-(46) as functions X and Y of on the interval ( e ; ). Dening U =?diag j E j 4 A T Y T we can abbreviate Theorem as follows. + Y A? T + GT Y T T F ~H? 3 G T Y T 5 Ej ; (5) ~H orollary The LMI (38) has a positive denite solution i > c i > e, X >, U >. Due to (45), the implicit function theorem implies that X and, hence, also Y, U are analytic functions on ( e ; ). y dierentiating (44) with respect to, it is not dicult to see that X is nonincreasing and concave. A perturbation trick allows to show that U shares these properties with X. Lemma 3 X and U are analytic functions on ( e ; ) which are positive denite for some large, and which are nondecreasing (the rst derivative is positive semidenie) and concave (the second derivative is negative semidenite). With f := diag(x ; U ); orollary clearly implies c = inff > e j f > g. Since f > for some large, we infer c <. It might happen that f, although nondecreasing, remains positive if decreases to e, what implies c = e : In the other case, there exists some > e with f 6>. Due to the properties in Lemma 3, there is exactly one ( ; ) for which f is positive semidenite and singular, and this parameter in fact coincides with c [47]. (It cannot happen that f is positive semidenite and singular for more than one values of.) This implies c and can be taken as the starting point for the following Newton algorithm. Theorem 4 Suppose f has the properties as in Lemma 3. If e < j c, there exists a unique j+ such that f j + f j ( j+? j ) is positive semidenite and singular, and j+ satises j j+ c. The inductively dened sequence ( j ) converges montonically and quadratically to c.

23 Hence, given j, one just needs to solve a symmetric generalized eigenvalue problem in order to nd a better approximation j+ of c. For a proof we refer to [47]. As a nal result, we provide an important property of X := fx? j X > satises (38)g which is identical to the set of X? if X varies over all positive denite solutions of (4) (for some (J; N)). We show that there exists a P which is a strict lower bound and, at the same time, a limit point of X. For brevity, P is called a strict lower limit point. As easily seen, any such strict lower limit point is necessarily unique. Hence, existence is the nontrivial and crucial point in the following result. Theorem 5Let X be nonempty. With X > satisfying (44)-(45), dene P r () := X? A in the partition of A P r and P := S T r () S in the partition of A. Then P is the unique strict lower limit point of X. P is analytic on ( c ; ) and satises P, P. Remark. If D has full row rank and (49) has no zeros in, P just coincides with the solution of the estimation Riccati equation in [4, 5] (see Remark (e) after Theorem ). In this section we have generalized the results from [48, 49, ] to arbitrary data and we considerably simplied some of the earlier proofs. 5 onsequences for the Mixed H =H Problem We just observe that, for xed (Y; F; N), the left-hand side of (8) is monotone in X. If combining Theorem 5 and 6, we arrive at the following corollary for LTV systems. orollary 6 There exists a controller R with J(T (R)) < i there exists an > such that the solution of (3) exists on [; ) and remains bounded, and there exist time functions Y (smooth), F, N satisfying (5) and I H Y + E F H + E N (H Y + E F ) T Y I (H + E N) T I Z? A : This decouples the construction of (J; X) from the construction of (Y; F; N) in Theorem 5. In addition, it is interesting to observe that Z can be computed on-line which will become important for linear parametrically-varying systems as discussed in Section 6. 3

24 If the system is LTI, we let all the unknowns in Theorem 5 be constant as well. Moreover, we note that (8) is equivalent to X > and I H Y + E F? H + E N H X? + E N Y I I T > : (53) Now we exploit Theorem 5 to infer that the set of all X? in Theorem 5 has the lower limit point P. It is easily seen that we can replace X? in (53) by P. orollary 7 If the system is LTI, there exists an LTI controller R with J(T (R)) < i > c (as dened in (5)), and there exist a solution (Y; F; N) of (5) with I? (H + E N)P (H + E N) T > : (54) (H [Y? P ] + E [F? NP ]) T Y? P Indeed, (53) leads to (54) by?p >?X?, and (54) implies (53) since X can be chosen close to P. Note that (54) is equivalent (Schur complement) to the linear matrix inequality I W T (H + E N) T (H + E N)W I (H [Y? P ] + E [F? NP ]) T Y? P A > : in (Y; F; N) if decomposing P = W W T. Hence, with the reformulation in orollary 7, we have eliminated the variables (X; J) in the suboptimality test. Note that [4] contains a separation result which converts, under technical hypotheses, the output feedback problem to a state-feedback problem. orollary 7 is the full generalization to arbitrary LTI systems. 6 Linear Parametrically Varying Systems A practically relevant specic time dependence in (6) arises if specifying (continuous bounded) functions A(p) (p) G(p) (p) D(p) H (p) E (p) F (p) A H (p) E (p) dened on some set P R m, and letting the system be described by some unknown parameter curve p(t) P. The available a priori information consists of the set P and the on-line information is the actual parameter value p(t) at the time instant t. This structure comprises linear systems with time-varying measurable parameters and gain scheduling structures for possibly nonlinear systems. 4

25 Suppose we can nd constant X, Y and continuous functions F; J; N dened on P such that (4), (5), and (8) hold for all p P. As described earlier, one can construct a (full order) controller as a function of p and a constant X a such that, for the corresponding closed-loop matrices, (8) holds on P. As a consequence, if the controller is scheduled along a specic parameter curve (which is measured on-line), it solves the mixed H =H control problem with bounds and. Hence the original dynamic problem is reduced to a static problem of solving linear matrix inequalities over the parameter set P. Due to the (possibly) nonlinear dependence of the data on p and due to the unspecied structure of P, solving these inequalities results in a nonlinear problem. If the parameter space has a moderate dimension, simple discretization methods reduce this nonlinear problem to the convex problem of testing the feasibility of a nite number of LMIs. Let us briey describe one possible paradigm. ollect the parameters as a = (A; G; H ; H ; F ), b = (; E ; E ; ; D) and the unknowns as c = (F; J; N), x = (X; Y ). Dene the block diagonal function g(a; b; c; x) by putting the left-hand sides of (4) and (5) (without derivatives) and the negative of the left-hand side of (8) on the diagonal. For constant x, the three inequalities (4), (5), (8) can then be written as g(a(p(:)); b(p(:)); c(p(:)); x). The generally quadratic function g has the following properties: For xed system data (a; b) it is linear in (c; x), and for xed (b; x) it is linear in (a; c). In our case, a(p); b(p) are continuous functions on P. The problem is to nd a continuous function c(p) and a constant x such that g(a(p); b(p); c(p); x) on P : (55) Let us now choose a nite number of points p j P. If c(p) and x with (55) exist, one can nd j > such that, with c j := c(p j ), the inequalities are satised. g(a(p j ); b(p j ); c j ; x)? j I (56) For a specic choice of p j and j, let us now assume that (56) hold for some c j, x, and j. In addition, suppose that P is open, that the data functions (a(:); b(:)) are smooth, and that their derivatives are bounded on P. Let g (p; c; x) := k p kg(a(p); b(p); c; x)k k denote the vector of the norms of the partial derivatives of g(a(p); b(p); c; x) with respect to the components of p. Finally, let P j denote (large) open convex subsets P j of P with p j P j and supfkg (p; c j ; x)kkp? p j k j p P j g < j : (57) Then the mean value theorem implies g(a(p); b(p); c j ; x) on P j. If the sets P j cover P, one can easily nd a function c(p) which `interpolates' the controller parameters c j such that (55) is veried. Just choose a smooth partition of unity j which is subordinated 5

26 to [ j P j, and dene c(p) := j c j j (p) [3, Section 6.]. (Indeed, there exists an > with g(a(p); b(p); c j ; x)?i for all p P j and all j. If p meets P jk but no other sets, we infer P k jk (p) = and thus g(a(p); b(p); c(p); x) = P k jk (p)g(a(p); b(p); c jk ; x)? P k jk (p)i =?I.) If the sets P j do not cover P, one has to include more points from P in the list p j. If (55) is solvable, it is not dicult to see that there always exists a choice of (suciently dense) p j and (suciently small) j such that the corresponding P j cover P : Suppose x and the (bounded) function c(p) satisfy g(a(p); b(p); c(p); x)?i for p P. Let us introduce the bounds ^x and ^c by kxk < ^x and supfkc(p)k j p P g < ^c, and dene s := supfkg (p; c; x)k j p P ; kck < ^c; kxk < ^xg: For an arbitrary choice of p j P and with j :=, the LMIs (56) with the constraints kck < ^c, kxk < ^x have solutions c j and x. Then kp? p j k < =(s + ) implies kg (p; c j ; x)kkp? p j k < s=(s + ) < and thus P j satisfying (57) can be chosen to contain at least the ball fp P : kp? p j k < =(s + )g. Since the radius of this ball does not depend on the choice of p j, it is clear that a suciently dense set of points p j indeed leads to P j covering P. We have proved that we can indeed test whether (55) is solvable by some bounded continuous function c(p) and by some x: For p j P, j >, and some bounds ^c, ^x, test the feasibility of (56) with the constraints kck < ^c, kxk < ^x. If not feasible, decrease j >, increase the bounds ^c, ^x, and repeat. If (56) is never feasible then stop. Otherwise, construct P j (as large as possible). If the sets P j cover P then a c(p) can be computed. Otherwise, rene the choice of p j and return to the rst step. It is important to note that one does not need to start with a uniform partition of the parameter set but one can systematically vary the the density of the points p j according to the sizes of P j, which takes the rate of variation (derivatives) of the system data with respect to the parameter into account. In practice, this might allow to keep the number of points (and hence of LMIs in (56)) reasonably small. Under certain additional hypothesis on the system data and on the parameter set, one can exploit convexity. In the spirit of [], let us just describe how to proceed if the (possibly nonlinear) system can be written as _x y z z X A = j j (t) A j j G j j D j (H ) j (E ) j (F ) j (H ) j (E ) j (F ) j A where the nitely many (constant) vertices (a j ; b j ) are a priorily available and the continuous convex combination coecients j (t) with j (t) and P j j (t) = can be measured on-line. With this parameter dependence, a necessary condition for the solvability of (55) is the existence of c j and x satisfying g(a j ; b j ; c j ; x) < 6 x u w A