Minimax Observers for Linear DAEs
11 Minimax Observers for Linear DAEs
Sergiy Zhuk and Mihaly Petreczky
Abstract —In this note we construct minimax observers for linear sta-tionary DAEs with bounded uncertain inputs, given noisy measurements.We prove a new duality principle and show that a finite (infinite) horizonminimax observer exists if and only if the DAE is (cid:96) -impulse observable( (cid:96) -detectable) . Remarkably, the regularity of the DAE is not required.
I. I
NTRODUCTION
Consider a linear Differential-Algebraic Equation (DAE): d p F x p t qq dt “ Ax p t q ` f p t q , y p t q “ Hx p t q ` η p t q , (1)where F, A P R m ˆ n and H P R p ˆ n are given matrices and f , η are deterministic noises. We will be interested in solutions for which p F x p q , f, η q are a priori unknown elements of a bounding set E : “ Ş t ą tp x , f, η q : ρ p x , f, η, t , Q , Q, R q ď u where ρ p x , f, η, t , Q , Q, R q : “ x J Q x `` ż t p f J p t q Qf p t q ` η J p t q Rη p t qq dt , (2)and Q , Q, R are symmetric positive definite matrices. Intuitively, p F x p q , f, η q P E implies that the initial state F x p q is boundedand the noise signals p f, η q have bounded energy, i.e. bounded L norm. The latter is a standard assumption in robust control [1].In this paper we extend the classical results on minimax observers(see [1]–[4]) to DAEs. Namely, we estimate a linear function t ÞÑ Lx p t q , L P R ˆ n of the state trajectory of (1), based on the output y . Since (1) contain uncertain terms f, η , it follows that, even if F was invertible, there could be various state trajectories x of (1), andhence several values of Lx p t q , which are consistent with y . For thecase of rectangular F , the DAE (1) may have several solutions. Thisintroduces an additional source of uncertainty. Hence, the best wecan do is to construct a function t ÞÑ x Lx p t q such that for all statetrajectories x of (1), which are consistent with y and correspond to p F x p q , f, η q P E , Lx p t q lies in the interval r x Lx p t q´ r, x Lx p t q` r s ,and r is as small as possible. A dynamical system whose output is x Lx and whose input is y is called a minimax observer . In whatfollows we take Lx “ (cid:96) J F x for some (cid:96) P R m . The reason for thisis that the state vector x of (1) may contain components which arefree, possibly discontinuous, exogenous variables. The estimation ofthe latter is problematic even for LTI systems [5], if the observer isrequired to be an LTI system. In contrast, F x is always continuousand it is subject to a differential equation.
1) Contribution:
Our main contribution is a new duality principle(Theorem 2). This result allows us (i) to find necessary and sufficientconditions for existence of a minimax observer x Lx for finite andinfinite time horizons (Theorems 1,2), and (ii) to show that x Lx canbe represented as an output of a time-varying linear system in thefinite horizon case (Theorem 3), and as an output of a stable LTIsystem in the case of infinite time horizon (Theorem 4). Finally weprove that for bounded f and η with possibly unbounded L norm,the estimation error remains bounded. S. Zhuk is with IBM Research - Ireland, Damastown, Dublin 15, Ireland, [email protected]
CNRS, Centrale Lille, UMR 9189 - CRIStAL-Centre de Rechercheen Informatique, Signal et Automatique de Lille, F-59000 Lille, France, [email protected]
2) Motivation:
The need for estimating the state of a general DAEarises in many applications, as it has been demonstrated by variousauthors, e.g., [6]–[13]. In this paper we extend the classical minimaxframework to DAEs of the form (1). One motivation is to providecontrol-theoretic grounds for solving an issue of approximation andstate estimation for Partial Differential Equations (PDEs), describedin [14]. This issue may be resolved by using our results to incorporatePDE’s approximation error into the estimation process, see Section IVfor an example.
3) Related work:
To the best of our knowledge, the results ofthis paper are new. Its preliminary version appeared in [15]. Withrespect to [15] the main differences are: (i) new (necessary and suf-ficient) conditions for existence of minimax observers (Theorem 2),(ii) detailed proofs. Duality principle for non-stationary DAEs wasintroduced in [16], provided
F x p q “ . It was then used to derivea sub-optimal observer. In contrast, our duality theorems hold forDAEs with uncertain F x p q “ x and the constructed observers areoptimal in that the worst-case estimation errors associated with theobservers are minimal (see Definitions 3-4). The algorithm of [17]constructing a finite horizon minimax observer by using projectors F F ` is a special case of the one of this paper. Minimax observersfor discrete time DAEs were considered in [18].We note that some papers (see [19] for an overview) deriveobservers for regular DAEs by converting it to the Weierstrasscanonical form. If DAE has index j , then the resulting ODE dependson the derivatives of the noise of order j ´ , and so the observerwould require bounds for them [20]. In contrast, our observer isindependent of DAE’s index and works for bounded f and η . Thepapers [8], [9] present design of H -observers which force H -normof the “error system” to be less than a given number. In contrast, (i)minimax observers of this paper minimize a different error measure,(ii) unlike [9] we allow for non-regular DAEs, and (iii) unlike [8]we present necessary and sufficient conditions for observer existence( (cid:96) -detectability, Theorem 2). Sufficient existence conditions (partialimpulse observability) for the asymptotic functional observer for (1)were proposed in [11] provided f “ , η “ . In contrast, we dealwith noisy systems and present necessary and sufficient conditions.Design of unknown input observers for DAEs was considered in [10].Since we deal with bounded unknown inputs and estimate just apart of DAE’s state, our results do not compare directly with [10].In this paper we consider deterministic noise, stochastic or fuzzyuncertainties have been addressed for example in [21], [22].
4) Outline of the paper:
The definitions of minimax observers areformulated in Section II. The main results are presented in Section III.The application of the observers to PDEs is given in Section IV. Theproofs are given in Section V.
5) Notation: I n denotes the n ˆ n identity matrix; for a matrix A , A ` denotes its Moore-Penrose pseudoinverse. Let I : “ r , t s , ă t P R or I : “ r , `8q . Denote by AC p I, R n q , L p I, R n q , L loc p I, R n q respectively the set of all absolutely continuous, the setof all square integrable, and the set of all locally square integrablefunctions of the form f : I Ñ R n . Recall that f : I Ñ R n is locallysquare integrable if its restriction to any compact interval I Ď I is square integrable. We write AC p I q , L p I q and L loc p I q referringto AC p I, R n q , L p I, R n q and L loc p I, R n q respectively, when R n is clear from the context. We say that f p t q “ g p t q holds almost a r X i v : . [ m a t h . O C ] F e b everywhere (a.e.) on I if f p t q ‰ g p t q only on a subset of I ofmeasure zero. We identify linear time invariant (LTI) systems x “ Ax ` Bu , y “ Cx ` Du , with the tuple of matrices p A, B, C, D q . Byconvention, the infimum of an empty set will be `8 . Let I “ r , t s , ă t P R . For any v P L p I, R k q , define δ t p v q P L p I, R k q by δ t p v qp s q “ v p t ´ s q a.e. For any τ ą , positive definitematrices S , S P R m ˆ m , S P R p ˆ p , vector g P R m , and functions g P L loc p I, R m q , h P L loc p I, R p q , where I contains r , τ s , define ρ p g , g, h, τ, S , S , S q : “ g J S g `` ż τ p g J p t q S g p t q ` h J p t q S h p t qq dt. (3)The quantity ρ p x , f, η, t , Q , Q, R q defined in (2) is a special caseof the notation defined in (3) applied to τ “ t , g “ x , f “ g , h “ η , Q “ S , Q “ S , R “ S .II. P ROBLEM STATEMENT
We begin with a precise definition of DAE’s solution on the interval I of the form I “ r , t s , ă t P R or I “ r , `8q . Definition 1: A solution of (1) on I is a tuple p x, f, y, η q P L loc p I, R n qˆ L loc p I, R m qˆ L loc p I, R p qˆ L loc p I, R p q such that: F x is absolutely continuous and
F x p t q “ F x p q ` ş t p Ax p s q ` f p s qq ds for all t P I and y p t q “ Hx p t q ` η p t q a.e. on I . Denote the set ofall solutions on I by B I p F, A, H q .Let us fix the symmetric positive definite matrices Q , Q, R . Definition 2: A solution p x, f, y, η q P B I p F, A, H q is said to beadmissible if ρ p F x p q , f, η, t , Q , Q, R q ď for all t P I . Denotethe set of all admissible solutions by E E p I q . Remark 1:
The map p x , f, η q ÞÑ ρ p x , f, η, t , Q , Q, R q is anorm of the space H t : “ R m ˆ L p I, R m q ˆ L p I, R p q and E p t q : “ tp x , f, η q : ρ p x , f, η, t , Q , Q, R q ď u is the unitball of H t induced by this norm.Next, we define finite horizon minimax observers. Fix a time ă t P R and set I “ r , t s . Fix (cid:96) P R m . To define an observer werely upon classical results [2]–[4] on minimax (worst-case) estimators(observers) which state that for general linear equations the minimaxobserver is linear in outputs y [3]. The latter suggests that an estimate { (cid:96) J F x p t q of the true value (cid:96) J F x p t q , which is computed by theminimax observer, should be linear in y . It is then natural to modelthe effect of applying minimax observers to outputs as continuouslinear functionals, generated by functions U P L p I, R p q as follows: O U p t , y q : “ ż t y J p t q U p t q dt , @ y P L p I, R p q . Hence, given y , we compute the estimate { (cid:96) J F x p t q of the true value (cid:96) J F x p t q by selecting a function U and evaluating O U p t , y q . Inorder to select U optimally we introduce a cost function σ measuringthe worst-case estimation error: σ p U, (cid:96), t q : “ sup p x,f,y,η qP EE p I q p (cid:96) J F x p t q ´ O U p t , y qq . (4) Definition 3:
We say that p U P L p I, R p q is a minimax observer on I “ r , t s , if ˆ σ : “ σ p p U, (cid:96), t q “ inf U P L p I, R p q σ p U, (cid:96), t q ă `8 .If σ p U, (cid:96), t q ă `8 , then the estimation error satisfies p (cid:96) J F x p t q ´ O U p y qp t qq ď σ p U, (cid:96), t q for any admissible solution p x, f, y, η q P E E p I q . If, in addition, U is a minimax observer on I , then the errorbound σ p U, (cid:96), t q is the smallest possible. We show below that if thereexists a function U P L p I, R p q with finite worst-case error, thenthere also exists the minimax observer p U , and it can be implementedas an output of a linear system.Consider now the infinite horizon case, i.e. I “ r , `8q . Similarlyto the finite horizon case, observers of interest are devices which mapoutputs y to estimates { (cid:96) J F x p t q of (cid:96) J F x p t q , and { (cid:96) J F x p t q is a linear and continuous function of y | r ,t q . To model such observers we define F – the set of all maps U : tp τ, s q | τ ą , s P r , τ su Ñ R p suchthat @ τ ą , the function U p τ, ¨q : r , τ s Q s ÞÑ U p τ, s q belongs to L pr , τ s , R p q . An element U P F maps output y P L loc p I, R p q toa function O U p y q defined as: O U p y q : I Q t ÞÑ O U p y qp t q : “ ż t y J p s q U p t, s q ds . Now, to construct { (cid:96) J F x p t q we take U P F and compute O U p y q . Toselect U optimally we define the worst-case estimation error: σ p U, (cid:96) q : “ lim sup t Ñ8 σ p U p t, ¨q , (cid:96), t q . (5) Definition 4:
We say that q U P F is an infinite horizon minimaxobserver , if σ p q U, (cid:96) q “ inf U P F σ p U, (cid:96) q ă `8 . (6)If σ p U, (cid:96) q ă `8 , then for any (cid:15) ą , for large enough t , theestimation error satisfies p (cid:96) J F x p t q ´ O U p y qp t qq ď σ p U, (cid:96) q ` (cid:15) for any admissible solution p x, f, y, η q P E E pr , `8qq . If, in addition, U is an infinite horizon minimax observer, then the error bound σ p U, (cid:96) q is the smallest possible. We will be interested in infinitehorizon minimax observers which can be represented by stable LTIsystems. Definition 5: U P F is said to be represented by an LTI system p A o , B o , C o , q , if for any y P L loc p I, R p q , where I “ r , t s , ă t P R or I “ r , `8q , it holds that @ t P I : O U p y qp t q “ C o s p t q ,where s p t q “ A o s p t q ` B o y p t q , s p q “ .Note that p A o , B o , C o , q represents an element U of F , if and onlyif U p t, s q “ B J o e A J o p t ´ s q C J o for all t ą , s P r , t s . Moreover, wewill prove that if there exist a U P F with finite worst-case estimationerror, then there exist an infinite horizon minimax observer q U , which,in addition, can be represented by a stable LTI system.III. M AIN RESULTS
A. Duality and existence of minimax observers
We show that the minimax observer is related to the solution ofan optimal control problem for the dual DAE: d p F J q p t qq dt “ A J q p t q ´ H J u p t q . (7)To this end, we need to define what we mean by a solution of (7). Definition 6:
Let I be an interval of the form I “ r , t s , ă t P R or I “ r , `8q . A pair p q, u q P L loc p I, R m q ˆ L loc p I, R p q is asolution of (7) on I , if F J q is absolutely continuous and F J q p t q “ F J q p q` ş t p A J q p s q´ H J u p s qq ds for all t P I . Denote the set of allsolutions on I by B I p F J , A J , H J q and define D (cid:96) p I q : “ tp q, u q P B I p F J , A J , H J q | F J q p q “ F J (cid:96) u . Definition 7 ( (cid:96) -impulse observability):
The tuple p F, A, H q is (cid:96) -impulse observable , if D (cid:96) pr , t sq ‰ ∅ for some ă t P R .We define now the following cost function for (7). Fix ă t P R and let I be an interval of the form I “ r , τ s for some t ď τ P R ,or I “ r , `8q . For any p q, u q P D (cid:96) p I q , define J p q, u, t q : “ ρ p q p t q , q, u, t , ¯ Q , Q ´ , R ´ q , (8)where ¯ Q : “ F p F J` ´ M opt q J Q ´ p F J` ´ M opt q F J with M opt “ P p P Q ´ P q ` P Q ´ F J` and P “ p I m ´ p F J q ` F J q and ρ is defined in (3) with τ “ t , g “ q p t q , g “ q , h “ u , S “ ¯ Q , S “ Q ´ , S “ R ´ . We can now state the followingtheorem relating existence of minimax observers to minimizing thecost function (8). Theorem 1:
The following statements are equivalent: (i) p F, A, H q is (cid:96) -impulse observable,(ii) there exists U P L pr , t s , R p q such that σ p U, (cid:96), t q ă `8 ,(iii) there exists a minimax observer p U on r , t s , such that p U “ δ t p u ˚ q and σ p p U, (cid:96), t q “ J ˚ t , where p q ˚ , u ˚ q P D (cid:96) pr , t sq isa solution of the dual control problem: J ˚ t : “ J p q ˚ , u ˚ , t q “ inf p q,u qP D (cid:96) p I q J p q, u, t q .Moreover, for any U P L pr , t s , R p q , σ p U, (cid:96), t q “ inf p q,u qP D (cid:96) p I q ,u “ δ t p U q J p q, u, t q . (9)In particular, the set tp q, u q P D (cid:96) p I q | u “ δ t p U qu is not empty, ifand only if σ p U, (cid:96), t q ă `8 .Next, we present the notion of (cid:96) -detectability and a new dualityprinciple for the infinite horizon case. Definition 8 ( (cid:96) -detectability):
The tuple p F, A, H q is (cid:96) -detectable ,if there exists p q, u q P D (cid:96) pr , `8qq such that lim t Ñ8 F J q p t q “ .Note that definition 8 implies that D (cid:96) pr , t sq ‰ ∅ for some ă t P R : indeed, for p q, u q from the definition 8 it follows that itsrestriction p q | r ,t s , u | r ,t s q onto r , t s belongs to D (cid:96) pr , t sq andhence for any t ą D (cid:96) pr , t sq ‰ H . Hence, (cid:96) -detectability implies (cid:96) -impulse observability by definition 7. Theorem 2:
The following statements are equivalent:(i) p F, A, H q is (cid:96) -detectable,(ii) there exists an infinite horizon minimax observer q U , such that q U p t , s q “ δ t p u ˚ qp s q , ď s ď t and σ p q U, (cid:96) q “ J ˚8 , where p q ˚ , u ˚ q P D (cid:96) pr , `8q satisfies J ˚8 : “ lim sup τ Ñ8 J p q ˚ , u ˚ , τ q“ lim sup τ Ñ8 inf p q,u qP D (cid:96) pr ,τ sq J p q, u, τ q ă `8 . (10)The assumption Q , Q, R ą is essential for Theorems 1 – 2. Remark 2:
Note that p F, A, H q is (cid:96) -impulse observable for all (cid:96) P R m , if and only if it is impulse controllable in the senseof [7]. The latter can be characterized in terms of p F, A, H q . If rank “ F J A J H J ‰ J “ n , then p F, A, H q is (cid:96) -impulse ob-servable for all (cid:96) P R m if and only if p F, A, H q is impulseobservable in the sense of [8], [12], [13]. The latter is equivalentto rank ” F A H F ı “ n ` rank F . We stress that the condition abovecharacterizes when it is true that for all (cid:96) , p F, A, H q is (cid:96) -impulseobservable. Hence, this condition does not involve (cid:96) . In contrast, theconditions which characterize (cid:96) -impulse observability of p F, A, H q for a given (cid:96) will depend on that vector (cid:96) . From [6, Lemma 4.3.7] itfollows that p F, A, H q is (cid:96) -detectable for all those (cid:96) P R m for whichit is (cid:96) -impulse observable ô @ λ P C , Re λ ě “ λF ´ AH ‰ “ max s P C rank “ sF ´ AH ‰ . B. Observer design
Below we present algorithms for computing minimax observers.Recall from [24] the notion of LTI systems associated with (7).
Definition 9:
Let A a P R ˆ n ˆ ˆ n , B a P R ˆ n ˆ k , C a P R p m ` p qˆ ˆ n , D a P R p m ` p qˆ k for some ˆ n, k ą . The LTI system S “p A a , B a , C a , D a q is said to be associated with (7), if the followingconditions hold:1) Either k “ and “ D a B a ‰ “ or D a is full column rank.2) Let C s and D s be the matrices formed by the first m rows of C a and D a respectively. Then F J D s “ and Rank F J C s “ ˆ n .3) Let I be an interval of the form I “ r , t s , ă t P R or I “ r , `8q . Then p q, u q P B I p F J , A J , H J q if and onlyif there exists p p, g q P AC p I, R ˆ n q ˆ L loc p I, R k q such that p “ A a p ` B a g a.e. and r qu s “ C a p ` D a g a.e.The matrix M : “ p F T C s q ` is called the state map of S . Associated LTI systems allow us to represent solutions of DAEs asoutputs of linear systems. Recall from [24, Theorem 2-3] that thereexists an LTI system S “ p A a , B a , C a , D a q associated with (7)and that all LTI systems which are associated with (7) are feedbackequivalent.The matrix M allows us to relate the state trajectories of (7)and the state trajectories of S : from [24, Theorem 1] it followsthat for any p q, u q P B I p F J , A J , H J q , for v : “ M F J q and g : “ D ` a pr qu s ´ C a v q , v “ A a v ` B a g a.e. and r qu s “ C a v ` D a g a.e. Consequently (see [24, Corollary 1]) it follows that p F, A, H q is (cid:96) -impulse observable if and only if F J (cid:96) P Im F J C s . An algorithmfor computing the associated LTI was discussed in [24, Proof ofTh.1] and it is summarized in Algorithm 1. This algorithm was im-plemented, the Matlab function DAE Lin implementing Algorithm1 is available in the file main T A .m of the supplemenatry materialof this report. Now we describe how to construct a minimax observer p U on the interval I “ r , t s , t ą . Let S “ p A a , B a , C a , D a q be an LTI system associated with (7). Consider the equation: P p t q“ A J a P p t q ` P p t q A a ´ K J p t qp D J a SD a q K p t q ` C J a SC a ,P p q “ p F J C s q J ¯ Q F J C s , S “ ” Q ´ , , R ´ ı ,K p t q“ " p D J a SD a q ´ p B J a P p t q ` D J a SC a q if D a ‰ , P R ˆ ˆ n if D a “ . (12)where C s is as in Definition 9. Note that (12) is well defined, aseither D a is full column rank, and so D J a SD a is invertible, or B a “ , D a “ , k “ . Let u ˚ P L p I, R p q , v ˚ P AC p I, R ˆ n q be suchthat u ˚ p s q “ p C u ´ D u K p t ´ s qq v p s q , v p s q “ p A a ´ B a K p t ´ s qq v p s q , v p q “ M p F J (cid:96) q , (13)where C u and D u are the matrices formed by the last p rows of C a and D a respectively, and K is as in (12). Theorem 3 (Minimax observer: finite horizon): If p F, A, H q is (cid:96) -impulse observable, then p U p s q “ u ˚ p t ´ s q is a finite hori-zon minimax observer on I “ r , t s . Moreover, σ p p U, (cid:96), t q “ (cid:96) J F M J P p t q M F J (cid:96) and O p U p t , y q “ (cid:96) J F M J r p t q , where r P AC p I, R ˆ n q , r p q “ and r p t q “ p A a ´ B a K p t qq J r p t q ` p C u ´ D u K p t qq J y p t q a.e. (14)Note that (cid:96) -impulse observability can easily be checked, see Algo-rithm 2 below. The Matlab function IsObservable implementingAlgorithm 2 is available in the file main T A .m of supplementarymaterial of this report. Algorithm 2
Checking (cid:96) -impulse observability
Input: p F, A, H, (cid:96) q Output:
Yes, if p F, A, H q is (cid:96) -impulse observable, No otherwise Use Algorithm 1 to compute an LTI system S “p A a , B a , C a , D a q associated with (7). Let C s be the matrix formed by the first m rows of C a .Check if (cid:96) belongs to im p F T C s q , for example, by checking if F T C s p F T C s q ` (cid:96) “ (cid:96) . If (cid:96) P im p F T C s q , return Yes, otherwise return No.Theorem 3 yields an algorithm for computing the finite horizonminimax estimate. This algorithm is presented in Algorithm 3 below. Algorithm 1
Computation of the associated LTI system [24, Proofof Th.1].
Input: p F, A, H q Output:
LTI system S “ p A a , B a , C a , D a q associated with thedual system (7) and the state map M . Compute the SVD decomposition F T “ U Σ V T of F T andassume that Σ “ diag p σ , . . . , σ r , , . . . , q P R n ˆ m , r “ rank F , U P R n ˆ n , V P R m ˆ m , U T U “ I n , V T V “ I m . S “ diag p ? σ , . . . , ? σ r , , . . . , looomooon n ´ r q U T ,T “ V diag p ? σ , . . . , ? σ r , , . . . , looomooon m ´ r q . Note that SF T T “ „ I r
00 0 . Let r A P R r ˆ r , B P R r ˆ p , G P R r ˆ κ , r D P R p n ´ r qˆ κ , κ “ m ´ r ` p , r C P R p n ´ r qˆ r , be such that SA T T “ „ r A A A A , SH T “ ´ „ B B G “ “ A , B ‰ , r D “ “ A , B ‰ and r C “ A . Consider the linear system p “ r Ap ` Gq, z “ r Cp ` r Dq. (11)Using the algorithms discussed in [25, Section 4.5, page 241],compute a full row rank matrix V and a matrix r F P R κ ˆ r , where κ is the number of columns of G , such that ‚ im V is the largest output nulling controlled invariant sub-space of (11), and ‚ p r A ` G r F qp im V q Ď im V .For instance, p V , r F q can be computed by applying the Matlabfunction vstar from [26] to p r A, G, r C, r D q . Let L P R κ ˆ k be such that im L “ ker r D X G ´ p V q , and either k “ Rank L , or L “ p , . . . , q T P R κ .For instance, using Matlab functions invt , ints , ker of [26], thefunction call ints p invt p G, V q , ker p r D qq will return L . Define ¯ C “ „ T I n „ I r r F and ¯ D “ „ T I n „ L ,A a “ P V p r A ` G r F q V , B a “ P V GL, C a “ ¯ C V ,D a “ ¯ D, M “ p F J C s q ` , where P V “ V ` , and C s is the matrix formed by the first m rows of C a . Algorithm 3
Finite horizon minimax state estimate O p U p t , y q Input: p F, A, H, (cid:96) q and the observed output signal y P L pr , t s , R p q Output:
Minimax estimate O p U p t , y q Use Algorithm 1 to compute the linear system S “p A a , B a , C a , D a q associated with (7) and the state map M . Compute P and K by solving (12) and find r p t q by solving (14).The numerical approximations of P and r may be done by usingM¨obius time integrators [27]. Return O p U p t , y q “ (cid:96) J F M J r p t q .Next, we present an algorithm for computing an infinite horizonminimax observer p U . To this end, recall from [24] the notion ofa stabilizable LTI system associated with the DAE (7). Let S “ p A a , B a , C a , D a q be an LTI system associated with (7) and considerthe stabilizability subspace V g of S . From [28] it then follows that V g is A a -invariant and im B a Ď V g . Hence, there exists a basistransformation T such that T p V g q “ im “ I l
00 0 ‰ , l “ dim V g andin this new basis, T A a T ´ “ “ A g ‹ ‹ ‰ , T B a “ “ B g ‰ , C a T ´ “ ” C J g ‹ ı J , where A g P R l ˆ l , B g P R l ˆ k , C g P R p n ` m qˆ l . Definition 10:
The LTI system S g “ p A g , B g , C g , D g q with D g “ D a is called a stabilizable LTI system associated with (7). The matrix M g “ r I l s T M is called the associated state map.The LTI system S g is a restriction of S to the subspace V g . It followsthat S g is stabilizable. Moreover, since all associated LTI systems of(7) are feedback equivalent, all stabilizable associated LTI systemsof (7) are feedback equivalent. From the discussion after Lemma 1 of[24] it follows that the LTI S g and the map M g can be computedfrom the matrices F, A, H . For the convenience of the reader, wepresent the algorithm in Algorithm 4 below. The Matlab function
Lin StabLin implementing Algorithm 4 can be found in the file main T A .m in supplementary material of this report. Algorithm 4
Stabilizable associated linear system [24, Lem.1 anddiscussion after].
Input: p F, A, H q Output:
Stabilizable LTI system S g “ p A g , B g , C g , D g q associatedwith (7) and the state map M g . Use Algorithm 1 to compute the LTI system S “p A a , B a , C a , D a q associated with (7) and the state map M . Compute the controllability matrix R of S , Compute a matrix V such that im V is largest A a -invariantsubspace such that the restriction of A a to im V is stable. Ifwe use the Matlab tool [26], then V can be taken as the firstcomponent of the output of the subsplit p A a q . Let V g be any matrix whose columns form a basis of im “ R, V ‰ . Then V is full column rank, and from [29,Theorem 4.30, page 96] it follows that im V g equals the largeststabilizable subspace V g of S .In Matlab, V g can be taken as the output of orth p “ R, V ‰ q . A g “ P V g A a V g , B g “ P V g B a , C a “ C g V g , M g “ P V g M , where P V g “ V ` g .Note that by [24], p F, A, H q is (cid:96) -detectable if and only if F T (cid:96) P im p F T C s q and M p F T (cid:96) q belongs to the stabilizability subspace V g of S . This remark yields the following algorithm, presented inAlgorithm 5, for checking (cid:96) -detectability. Algorithm 5
Checking (cid:96) -detectability
Input: p F, A, H, (cid:96) q Output:
Yes, if p F, A, H q is (cid:96) -detectable, No otherwise Use Algorithm 3 to compute an LTI system S “p A a , B a , C a , D a q associated with (7) and the state-map M . Use Steps 2-4 of Algorithm 4 to compute a full column rankmatrix V g such that im V g is the largest stabilizable subspace V g of S . Like in Step 2 of Algorithm 2, check if (cid:96) P im F T C s , where C s is the matrix formed by the first m rows of C a . If (cid:96) R im F T C s ,then quit the algorithm and return No. If (cid:96) P im F T C s , then check if M p F T (cid:96) q P im V g . The lattercan be done, for example, by checking if V g V ` g p M p F T (cid:96) qq “ M p F T (cid:96) q . If M p F T (cid:96) q P im V g , then return Yes, otherwise return No.The Matlab function IsDetectable implementing Algorithm 5 canbe found in the file main T A .m of the supplementary material of this report.Let S g “ p A g , B g , C g , D g q and M g be as in Definition 10.Consider the following algebraic Riccati equation: “ P A g ` A J g P ´ K J p D J g SD g q K ` C J g SC g ,K “ " p D J g SD g q ´ p B J g P ` D J g SC g q if D g ‰ , P R ˆ l if D g “ ,S “ ” Q ´ R ´ ı . (15)Note that either D g “ D a is full column rank, and so, by positivedefiniteness of Q, R , D J g SD g is invertible, or B g “ , D g “ , k “ . If P “ P T ą is a solution of (15), then let u ˚ P L loc pr , `8q , R p q , v ˚ P AC pr , `8q , R l q be such that u ˚ p t q “ p ˆ C u ´ ˆ D u K q v ˚ p t q , v ˚ p t q “ p A g ´ B g K q v ˚ p t q , v ˚ p q “ M g p F J (cid:96) q , (16)where ˆ C u and ˆ D u are the matrices formed by the last p rows of C g and D g respectively. Define A o “ p A g ´ B g K q J , B o “ p ˆ C u ´ ˆ D u K q J , C o “ (cid:96) J F M Tg . Theorem 4 (Minimax observer: infinite horizon):
Assume that p F, A, H q is (cid:96) -detectable. Then the following holds:(i) A o is a Hurwitz matrix and (15) has a unique symmetric positivedefinite solution P ,(ii) q U P F , where q U p t, s q “ u ˚ p t ´ s q , t ě s ě , is an infinitehorizon minimax observer.(iii) the minimax error is σ p q U, (cid:96) q “ (cid:96) J F M J g P M g F J (cid:96) ,(iv) q U can be represented by the LTI system p A o , B o , C o , q , i.e. O q U p y qp t q “ C o r p t q where r P AC pr , `8q , R l q solves r p t q “ A o r p t q ` B o y p t q and r p q “ , a.e. (17) Remark 3:
We stress that one does not need to solve (17) severaltimes in order to compute the minimax observer O q U p y qp t q for various (cid:96) i , i “ . . . s . Indeed, A o and B o do not depend on (cid:96) , sincethey depend only on the matrices A g , B g , C g , D g of the associatedstabilizable LTI S g and on the weight matrices Q and R . In turn, S g depends only on the matrices F, A, H , see Algorithm 4. Hence,it is sufficient to define C o “ LF M Tg where L “ ” (cid:96) ...(cid:96) s ı , provided p F, A, H q is (cid:96) i -detectable, i “ . . . s . Lemma 1:
Assume p F, A, H q is (cid:96) -detectable and let q U be the infi-nite horizon minimax observer from Theorem 4. For any p x, f, y, η q P E E pr , `8qq , the estimation error e p t q “ (cid:96) J F x p t q ´ O q U p y qp t q satisfies d r rdt “ p A g ´ B g K q J r r p t q ` p C g ´ D g K q J ” f p t q´ η p t q ı , r r p q “ p C g ´ D g K q J “ Fx p q ‰ ,e p t q : “ (cid:96) J F x p t q ´ O q U p y qp t q “ (cid:96) J F M J g r r p t q . (18)In particular, if f “ , η “ , then lim t Ñ8 e p t q “ . If f, η are bounded, i.e. sup t ą t f J p t q f p t q , η J p t q η p t qu ă `8 , then theestimation error e p t q is bounded, i.e. sup t ą e J p t q e p t q ă `8 . If f P L pr , `8q , R n q , ν P L pr , `8q , R p q , then e P L pr , `8q , R q .This lemma reveals the following important points: (i) the infinitehorizon minimax observer from Theorem 4 behaves like an asymp-totic observer in the absence of noises; (ii) if the noise signalsare of bounded energy (i.e. they are in L pr , `8q ), then theestimation error is of bounded energy; (iii) if the noise signals arejust bounded, then the estimation error is bounded too, i.e., theupper bound of the error is proportional to the upper bound of thenoises. The discussion above allows us to formulate an algorithm for computing an infinite horizon state estimate (see Algorithm 6). TheMatlab function DAEInfHorionsObs in the file main T A .m in the supplementary material of this report implements Steps 1 – 3Algorithm 6. Algorithm 6
Infinite horizon minimax state estimate O q U p y qpq Input: p F, A, H, (cid:96) q and the observed output y P L loc pr , `8q , R p q Output:
Infinite horizon minimax state estimate O q U p y qpq . Use Algorithm 4 to compute the stabilizable LTI system S g “p A g , B g , C g , D g q associated with (7) and the state map M g . Compute the solution P of (15) and compute the feed-back matrix K from (15). In Matlab, the command care p A g , B g , C g SC Tg , C g SD Tg , D Tg SD g q can be used to cal-culate P and K . Set A o “ p A g ´ B g K q J , B o “ p ˆ C u ´ ˆ D u K q J , C o “ (cid:96) J F M Tg . Solve the differential equation (17) to find r p t q . Return O q U p y qpq “ C o r .IV. N UMERICAL EXAMPLE
Consider the following infinite-dimensional system with outputs,determined by the partial differential equation (PDE) of heat transfer: dV p t q dt “ A V p t q ` u p t q , y “ C V ` w p t q , (19)where V p t q and u p t q take values in the space H : “ L pr´ , s , R q ,i.e., V p t q is itself a function from r´ , s to R , and A v “ ´ c d dυ v for all v P H such that A v P H . Moreover, w P L loc pr , `8q , R p q , C : H Ñ R p is given by: C φ “ ş ´ g p υ q φ p υ q dυ for some g P L pr´ , s , R p q , φ P H . We say that a pair p V, u, y, w q solves (19)if V : r , `8q Ñ H is a Frechet differentiable function such that A V p t q is defined, and (19) holds a.e., and V p t qp´ q “ V p t qp q “ .Physically, V p t qp z q represents the temperature at time t P r , `8q atpoint z P r´ , s . The constant c in the definition of the operator A is the thermal diffusivity, in this example it is taken to be c “ . ms . From the control perspective, V is a state, u, w are noises and y isa measured output. We take g p υ q “ p sin p πn υ q , sin p πn υ qqq T and p “ , for other choices of g and p the discussion is similar. Based on y and assuming that u and w satisfy ş } w p t q} R ` } u p t q} H dt ď , } . } H denotes the standard norm on H , we would like to estimate V p t q from y .We recast the problem of estimating V as the minimax estimationproblem for DAEs as follows (see [14] for the details): consider thebasis t φ k u k “ of H , where φ k “ P k ` ´ P k , k ě , with P i denoting the i th Legendre polynomial, i ě . Fix an integer N ą and define @ ψ P H : P N p ψ q “ “ x ψ, φ y , . . . , x ψ, φ N y ‰ T @ z “ p z , . . . , z N q T P R N : P ` N p z q “ N ÿ k “ z k φ k ˆ M N “ px φ i , φ j yq Ni,j “ , A N “ px A φ i , φ j yq Ni,j “ M N “ Λ N px φ i , φ j yq Ni,j “ , Λ N “ p i ` δ i,j q Ni,j “ C N “ “ P N p sin p πn υ qq , P N p sin p πn υ qq ‰ T , where x¨ , ¨y is the standard inner product in H and δ i,j is theKronecker delta symbol for all i, j ě . Consider the DAE (1) with F “ »– M N ,
00 00 0 fifl , A “ »– Λ N A N , Λ N I N “ I N u ‰fifl , H “ „ C TN T . Assume p V, u, y, w q solve (19). From [14] it follows that the tuple p x, f, y, ν q is a solution of (1), provided a “ ˆ M ´ N P N V , e m “ P N A e , e “ V ´ P ` N P N V , and x “ p a T , e Tm q T , and ν “ C e ` w , where f “ Λ N P N u , f “ “ f T , f T , f T ‰ T for some f P L loc pr , `8q , R N q , f P L loc pr , `8q , R N u q . Moreover, if V andits partial derivatives are bounded (which is the case for physicallymeaningful solutions), then by [14], there exist Q ą , R ą such that ş f T p t q Qf p t q ` ν T p t q Rν p t q dt ď . The intuition behindthis is as follows: the component a p t q of x p t q represents the first N coordinates of V p t q w.r.t the basis t φ k u k “ , and e m models theimpact of e , the error of approximating V by its projection onto thefirst N basis functions t φ k u Nk “ , onto a p t q . For this reason, e m isincluded into the state of DAE. The latter allows to incorporate theprojection error e into the estimation process. The smaller e m is, thecloser the behavior of (1) is to that of (19).The algebraic equations a ` f “ and r I Nu s e m ` f “ together with ş f T Qf p t q ` ν T p t q Rν p t q dt imply that the L normof a and of the first N u components of e m is uniformly bounded,i.e. these functions live in an ellipsoid. Note that the expressions for Q, R provided by [14, Eq. (3.12) and (3.13)] are very conservative,and, in practice,
Q, R are found by tuning.The minimax observer for (1) can be used to estimate the state of(19) as follows. Assume that (1) is (cid:96) -detectable for (cid:96)
P t (cid:96) i u N u i “ ,where e i is the i th standard basis vector in R N ` N u . Let q U i denote the minimax observer corresponding to (cid:96) “ (cid:96) i according toTheorem 4, and let (cid:126) O N u p y q “ p O q U p y q , . . . , O q U Nu p y qq T . Define ˆ (cid:126)V N u p y qp t q “ P ` N u M ´ N u (cid:126) O N u p y qp t q . If p V, u, y, w q is a solution of(19), then, as noted above, the tuple p x, f, y, ν q solves (1), and P ` N u p (cid:96) T F x p t q , . . . , (cid:96) TN u F x p t qq “ P ` N u P N u V p t q , the projection of V to the first N u basis functions. By noting that O q U i p y q is theminimax estimate of (cid:96) Ti F x , we derive that ˆ (cid:126)V N u p y qp t q is the minimaxestimate of the vector P ` N u P N u V p t q . This latter estimate is “good”if the minimax error σ p q U i , (cid:96) i q is “small”.For the simulations we chose N “ , n “ , n “ , N u “ . Note that the entries of ˆ M N , and A N can be computedanalytically (see [30, Example 7.2, p. 121]) so that A, F, H can becomputed numerically. The numerical values of the matrices
F, A, H can be found in the file
DAE matrices.mat (Matlab format) or inthe files
F matrix.txt , A matrix.txt , H matrix.txt (csv textformal) of the supplementary material of this report. If we check rank “ F T A T H T ‰ T , then it follows that its rank is less thanits number of columns. Hence, by Remark 2, there exists (cid:96) suchthat p F, A, H q is not (cid:96) -impulse observable for this (cid:96) . Thus, theobserver proposed in [8] is inapplicable as it requires to solve a linearequation [8, eq. (5c)] which has no solution if there exists an (cid:96) forwhich p F, A, H q is not (cid:96) -impulse observable . Note that [9] is notapplicable either: by [9, Remark 1], existence of a filter in the senseof [9] implies that the DAE at hand is regular, which is not the case.However, using Algorithm 2 - 5, it can be checked numerically that p F, A, H q is (cid:96) -impulse observable and (cid:96) -detectable (Def.7-8) onlyfor (cid:96) “ (cid:96) i “ e i , i “ , . . . , N u , i.e we can estimate (with finiteworst-case error!) only the first N u components of F x .To generate the outputs we took an input u true p t qp υ q “ u p t q sin p πn υ q ` u p t q sin p πn υ q with u p t q “ cos p t q and u p t q “ sin p t q , initial condition V true p qp υ q “ . p πn υ q` . p πn υ q , and noise w true p t q “ . e ´ . π ct cos p t q .Then we computed the exact solution p V true , u true , y true , w true q of (19) by using eigen-basis of the operator A , i.e., V true p t, υ q “ z p t q sin p πn υ q ` z p t q sin p πn υ q and y true p t q “ p z p t q , z p t qq T ` w true p t q , z i “ ´ cn π z i ` u i p t q , i “ , . As noted, p V true , u true , y true , w true q yields a solution of (1), namely p x true , f true , y true , µ true q and F x true p t q “ ř i “ z i p t q Λ N P N p sin p πn i υ qq .We applied Algorithm 5 to check that the DAE at handis (cid:96) -detectable for all standard unit vectors of R N u . We usedthe Matlab function IsDetectable from the file main T A .m in the supplementary material of this report to check (cid:96) -detectability. We then applied Algorithm 6 to compute infi-nite horizon minimax observer (cid:126) O N u p y true qp t q for (1) to es-timate F x true p t q on r , s . To this end we set Q “ diag p ´ I N u , ´ I N ´ N u , ´ I N , . I N u q , R “ I p . Weused the Matlab function DAEInfiniteHorizonObs from thefile main T A .m of the supplementary material to calculate thematrices A o , B o , C o of the minimax observer. These matrices can befound in the files Ao matrix.txt, Bo matrix.txt, Co matrix.txt (csv text format) and
Observer.mat (Matlab data file format)included in the supplementary material of this report. The re-sults of the intermediate steps of the computation can also befound in the supplementary material. More precisely the matri-ces of the LTI S “ p A a , B a , C a , D a q associated with thedual system (7) and the state map M can be found in thecsv text files Aa matrix.txt , Ba matrix.txt , Ca matrix.txt , Da matrix.txt , LMAP matrix.txt of the supplementary mate-rial of this report. The matrices of the stabilizable associated LTI S g “ p A g , B g , C g , D g q and the state map M g can be found inthe csv text files Ag matrix.txt , Bg matrix.txt , Cg matrix.txt , Dg matrix.txt , LMAP g matrix.txt of the supplementary mate-rial of this report. The solution P of (15) and the gain K can befound in the csv text files P matrix.txt and
K matrix.txt of thesupplementary material.It turns out that (cid:126) O N u p y true qp t q is quite close to F x true p t q component-wise. Specifically, Figure 1 shows that the minimax esti-mate O q U i p y true qp t q tracks the components (cid:96) Ti F x true p t q , i “ , of DAE’s state vector quite well. Note that F x true p t q representsthe coordinates of the true solution V true with respect to the basis t φ k u k “ . Hence, it does not itself represent a physical quantity andtherefore it has no measurement unit.We also compared ˆ (cid:126)V N u p y qp t q , the minimax estimate of the vector P ` N u P N u V true p t q against V true p t q , and they were in a good agree-ment. In particular, on Figure 1 we present the plots of t ÞÑ V p t qp z q versus t ÞÑ ˆ (cid:126)V N u p y qp t qp z q for z “ ´ . . Note that ˆ (cid:126)V N u p y qp t qp z q represents an estimate of a physical quantity, namely, the temperatureat time t at point z . V. P ROOFS
Proof of Theorem 1:
Let us prove that the set tp q, u q P D (cid:96) p I q | u “ δ t p U qu is not empty, if and only if σ p U, (cid:96), t q ă `8 . This willalso prove p i q ô p ii q . Take w P L p I q such that F J w is absolutelycontinuous and F J w p t q “ F J (cid:96) . Define b p w, U qp t q : “ dF J w p t q dt ` A J w p t q´ H J U p t q . Clearly, b p w, U q P L p I q . Now, let p x, f, y, η q P E E p I q . Then, integrating by parts (see [16, F.(2.1)]), we find: (cid:96) J F x p t q ´ O U p t , y q “ p F x p qq J p F ` q J F J w p q`` ż t p w J p t q f p t q ´ U J p t q η p t q ` b J p w, U qp t q x p t qq dt . (20)Let L denote a differential operator associated to (1): p Lx qp t q “ ` F x p q , d p Fx q dt ´ Ax p t q ˘ with domain D p L q : “ t x P L p I q | F x P AC p I q , dFxdt P L p I qu . Define the adjoint of L , namely Fig. 1. Comparing { (cid:96) Ti F x p t q : “ O q U i p y true qp t q versus (cid:96) Ti F x true p t q , i “ , and V true p t qp z q versus ˆ (cid:126)V N u p y qp t qp z q for z “ ´ . . L p g , g qp t q “ ´ dF J gdt ´ A J g , p g , g q P D p L q , where the domain D p L q of L is given by: D p L q : “ tp g , g q P R m ˆ L p I q | F J g P AC p I q , dF J gdt P L p I q ,F J g p t q “ , g “ F `J F J g p q ` d, F J d “ u . Now we claim that σ p U, (cid:96), t q ă `8 ô L p g , g q “ b p w, U q for some p g , g q P D p L q . Note that equality L p g , g q “ b p w, U q implies that, by definition of L and w , q : “ w ` g veri-fies d p F J q q dt “ ´ A J q ` H J U and F J q p t q “ F J (cid:96) , so that p δ t p q q , δ t p U qq P D (cid:96) p I q , and vice versa, if p δ t p q q , δ t p U qq P D (cid:96) p I q then g : “ q ´ w , g “ F `J F J g p q verifies L p g , g q “ b p w, U q . Thus, it suffices to prove the above claim. To thisend, recall the definition of E p t q (Remark 1). We note that sup p Fx p q ,f,η qP E p t q p F x p qq J p F ` q J F J w p q ` ş t w J p t q f p t q dt ă`8 by Cauchy-Swartz inequality in L p I q . Thus, by (4) and (20),we have that σ p U, (cid:96), t q ă `8 ô s p b, U q ă `8 , where s p b, U q : “ sup tp x,η q : p Lx,η qP E p t qu ż t p b J p w, U qp t q x p t q´ U J p t q η p t qq dt . To compute s p b, U q we recall from convex analysis [31] that thesupport function s E p x q : “ sup v P E x J v of the ellipsoid E : “ t v : v J Q v ď u equals to p x J Q ´ x q . Analogously, the support func-tional s E p t q p g , g, v q : “ sup p x ,f,η qP E p t q g J x ` ş t p g J f ` v J η q dt of the ellipsoid E p t q equals to p ρ p g , g, v, t , Q ´ , Q ´ , R ´ qq .Now we note that s p b, U q equals to the support function of the pre-image of E p t q with respect to the operator p x, η q ÞÑ p Lx, η q . Thelatter can be computed by using Young-Fenhel conjugate (see [16,F.(2.5),Lemma 2.2]): s p b, U q “ inf g ,g,v t ρ p g , g, v, t , Q ´ , Q ´ , R ´ q | p g , g q P D p L q ,v P L p I q , “ L p g ,g q v ‰ “ “ b p w,U q U ‰ u . This equality implies that s p b, U q ă `8 ô L p g , g q “ b p w, U q for some p g , g q P D p L q .Let us now prove (9). If σ p U, (cid:96), t q “ `8 , then from the dis-cussion above both sides of (9) are `8 . Assume σ p U, (cid:96), t q ă `8 . Take p p, u q P D (cid:96) p I q such that U “ δ t p u q . Set w : “ δ t p p q . Then b p w, U q “ in (20) and so we get: p σ p U, (cid:96), t qq “ s E p t q p w , w, U q : “ sup p x ,f,η qP E p I q t w J x ` ż t p w J p t q f p t q ´ U J p t q η p t qq dt u , (21)where w : “ p F ` q J F J w p q , E p t q is the set of all p x , f, η q P E p t q such that Lx “ p x , f q for some x P D p L q , and s E p t q isthe support function of E p t q . Thus, to compute σ p U, (cid:96), t q it sufficesto compute right hand side of (21). From [16, F.(2.6),Lemma 2.2]: s E p t q p w , w, U q “ inf p g ,g qP ker L ρ p w ` g , w ` g, U, t , Q ´ , Q ´ , R ´ q , (22)where ker L “ tp g , g q P D p L q : L p g , g q “ u . We claim that theright-hand side of (22) equals to inf p q,v qP D (cid:96) p I q ,v “ δ t p U q J p q, v, t q .Indeed, since g “ F `J F J g p q ` P ˜ d , ˜ d P R m , P “ p I m ´p F J q ` F J q it follows that we can fix g and eliminate ˜ d by mini-mizing the 1st term of ρ w.r.t. ˜ d . We get: inf t g |p g ,g qP ker L u } Q ´ pp F ` q J F J w p q ` g q} ““ inf ˜ d P R m } Q ´ pp F ` q J F J p w p q ` g p qq ´ P ˜ d q} “ } Q ´ pp F ` q J F J p w p q ` g p qq ´ P ˆ d q} ““ p w p q ` g p qq J ¯ Q p w p q ` g p qq , where ˆ d : “ p P Q ´ P q ` P Q ´ p F ` q J F J p w p q ` g p qq . Now, toprove the above claim we note that tp q, v q | v “ u, p q, v q P D (cid:96) p I qu “ tp δ t p g ` w q , u q | p g , g q P ker L u . In other words,if p q, u q P D (cid:96) p I q then q is a sum of p “ δ t p w q and δ t p g q provided L p g , g q “ . This allows us to write: inf t g |p ˆ g ,g qP ker L u rp w p q ` g p qq J ¯ Q p w p q ` g p qq` ż t p w p t q ` g p t qq J Q ´ p w p t q ` g p t qq dt s “ inf p q,u qP D (cid:96) p I q ,u “ δ t p U q rp q p t qq J ¯ Q q p t q ` ż t q J p t q Q ´ q p t q dt s , where ˆ g “ F `J F J g p q ` P ˆ d . Thus, the right-hand sideof (22) equals inf p q,v qP D (cid:96) p I q ,v “ δ t p U q J p q, v, t q . The latter and (21)prove (9). To prove p ii q ô p iii q we note that, by Definition 3, p iii q implies that σ p p U, (cid:96), t q ă `8 and then p ii q follows. Now, assumethat p ii q holds and p q ˚ , u ˚ q P D (cid:96) p I q denotes a minimizer of J overthe affine set D (cid:96) p I q . Then, by (9), p U “ δ t p u ˚ q is a minimizer for σ p U, (cid:96), t q and so p iii q holds by Definition 3. Proof of Theorem 2:
To prove p i q ñ p ii q we assume that p F, A, H q is (cid:96) -detectable. Then (7) is behaviorally stabilizable from (cid:96) (according to the terminology of [24]) and vice-versa. Then by [24,Theorem 6] there exists p q ˚ , u ˚ q P D (cid:96) pr , `8qq such that (10)holds. Let us prove that q U p t, s q “ u ˚ p t ´ s q verifies (6). To thisend, from (9) it follows that for any u P L pr , τ s , R p q : σ p u, (cid:96), τ q ě inf p q,v qP D (cid:96) pr ,τ sq J p q, v, τ q . (23)Take any U P F . Then by using (23) and (10) and (5) we get: σ p U, (cid:96) q ě lim sup τ Ñ8 inf p q,v qP D (cid:96) pr ,τ sq J p q, v, τ q “ J ˚8 . (24)Apply (9) to q U p τ, ¨q . Then σ p q U p τ, ¨q , (cid:96), τ q “ inf p q,v qP D (cid:96) pr ,τ sq ,v “ u ˚ | r ,τ s J p q, v, τ q ď J p q ˚ , u ˚ , τ q , so that σ p q U, (cid:96) q ď lim sup τ Ñ8 J p q ˚ , u ˚ , τ q “ J ˚8 . This and (24) implies that σ p q U, (cid:96) q ď J ˚8 ď σ p U, (cid:96) q . Since J ˚8 ă `8 by (10) it followsthat σ p q U, (cid:96) q ă `8 and so q U verifies (6).To prove p ii q ñ p i q let q U P F be an infinite hori-zon minimax observer. Then, by Definition 3, σ p q U, (cid:96) q : “ lim sup t Ñ8 σ p q U p t, ¨q , (cid:96), t q ă `8 . Hence, there exists ˜ t ą such that for all t ą ˜ t : σ p q U p t, ¨q , (cid:96), t q ă `8 .Now, by (9) of Theorem 1 we get that σ p q U p t, ¨q , (cid:96), t q “ inf p q,u qP D (cid:96) pr ,t sq ,u “ δ t p q U p t, ¨qq J p q, u, t q for all t ą ˜ t , and so lim sup t Ñ8 inf p q,u qP D (cid:96) pr ,t sq ,u “ δ t p q U p t, ¨qq J p q, u, t q ă `8 . The lat-ter and [24, Theorem 6] implies that (7) is behaviorally stabilizablefrom (cid:96) that, as noted above, implies p i q . Proof of Theorem 3:
Assume that v, u ˚ are defined by (13).Set q ˚ p s q “ p C s ` D s K p t ´ s qq v p s q , s P I . From [24, Theorem5] it then follows that J p q ˚ , u ˚ , t q “ inf p q,u qP D (cid:96) p I q J p q, u, t q and J p q ˚ , u ˚ , τ q “ (cid:96) J F M J P p t q M F J (cid:96) . From Theorem 1 it followsthat p U is indeed the minimax observer on I and σ p p U, (cid:96), t q “ J p q ˚ , u ˚ , t q “ (cid:96) J F M J P p t q M F J (cid:96). Finally, let r be the solutionof (14). Then ddt p r J p t q v p t ´ t qq “ y J p t q u ˚ p t ´ t q “ y J p t q p U p t q ,and by integrating both sides, O p U p t , y q “ (cid:96) J F M J r p t q . Proof Theorem 4: If p F, A, H q is (cid:96) -detectable, then (7) isbehaviorally stabilizable from (cid:96) (according to the terminology of [24])and vice-versa. Hence, the statement of p i q follows from [24, Lemma2]. From [24, Theorem 6] it follows that if v ˚ P AC pr , `8q , R l q is a solution of v ˚ “ p A g ´ B g K q v ˚ , v ˚ p q “ M g p F J (cid:96) q and p q ˚ , u ˚ q P L loc pr , `8q , R m q ˆ L loc pr , `8q , R p q are such that ” q ˚ u ˚ ı “ p C g ´ D g K q v ˚ , then p q ˚ , u ˚ q P D (cid:96) pr , `8qq and J ˚8 “ lim sup t Ñ8 J p q ˚ , u ˚ , t q “ lim sup t Ñ8 inf p q,u qP D (cid:96) pr ,t sq J p q, u, t q and J ˚8 “ (cid:96) J F M J g P M g F J (cid:96) . From p ii q , Theorem 2 it then followsthat p ii q , p iii q hold true. Now, to prove p iv q we note that from theabove discussion u ˚ p t q “ p C u ´ D u K q e p A g ´ B g K q t M g F J (cid:96) andthus p A o , B o , C o q represents q U . Proof of Lemma 1:
Let p q ˚ , u ˚ q P D (cid:96) pr , `8qq be suchthat p q ˚J , u ˚J q J “ p C g ´ D g K q p , p “ p A g ´ B g K q p , p p q “ M g p F J (cid:96) q . Define g : r , t s ÞÑ p F x q J p s qp F J q ` F J q ˚ p t ´ s q “ x J p s q F J q ˚ p t q . Since q ˚ and F x are both absolutely continuous, g is absolutely continuous. Note that Ax p s q ` f p s q is in the range of F a.e. and so F F ` p Ax p s q ` f p s qq “ Ax p s q ` f p s q a.e. Hence, dg p s q ds “ f J p s q q ˚ p t ´ s q ` p y p s q ´ η p s qq J u ˚ p t ´ s q a.e. Integrate both sides of the above equality and use q U p t, s q “ u ˚ p t ´ s q : g p t q´ O q U p y qp t q “ g p q` ż t r q ˚J p t ´ s q u ˚ p t ´ s q s ” f p s q´ η p s q ı ds . (25)Note that g p t q “ (cid:96) J F x p t q , and hence the left-hand side of (25) isthe estimation error e p t q . It is easy to see that the right-hand side of(25) equals (cid:96) J F M J g r r p t q where r r solves (18).VI. C ONCLUSIONS
We have presented an original solution of the minimax observerdesign problem for linear DAEs. It is “application friendly” as (i)it relies upon well developed tools (e.g. Riccati equations), (ii)it imposes easy to check necessary and sufficient conditions ( (cid:96) -detectability) on p F, A, H q and (iii) it is optimal for L inputs andprovides bounded estimation error for merely bounded inputs.R EFERENCES [1] M. Green and D. Limebeer,
Linear robust control . Pearson EducationInc., 1995.[2] A. J. Krener, “Kalman-Bucy and minimax filtering,”
IEEE Trans. onAutom. Control , vol. 25, no. 2, pp. 291–292, Apr 1980.[3] M. Milanese and R. Tempo, “Optimal algorithms theory for robustestimation and prediction,”
IEEE Trans. Automat. Control , vol. 30, no. 8,pp. 730–738, 1985. [4] A. Nakonechny, “A minimax estimate for functionals of the solutionsof operator equations,”
Arch. Math. , vol. 14, no. 1, pp. 55–59, 1978.[5] M. L. J. Hautus, “Strong detectability and observers,”
Linear Algebraand Its Applications , vol. 50, pp. 353–368, 1983.[6] T. Berger, “On Differential-Algebraic Control Systems,” Ph.D. disserta-tion, Technische Universit¨at Ilmenau, 2013.[7] T. Berger and T. Reis, “Controllability of linear differential-algebraicsystems: a survey,” in
Surveys in Differential-Algebraic Equations I ,A. Ilchmann and T. Reis, Eds. Springer, 2013, pp. 1–61.[8] M. Darouach, “ H unbiased filtering for linear descriptor systems viaLMI,” IEEE Trans. on Autom. Control , vol. 54, no. 8, pp. 1966–1972,2009.[9] S. Xu and J. Lam, “Reduced-order H filtering for singular systems,” System & Control Letters , vol. 56, no. 1, pp. 48–57, 2007.[10] F. J. Bejarano, T. Floquet, W. Perruquetti, and G. Zheng, “Observabilityand Detectability of Singular Linear Systems with Unknown Inputs,”
Automatica , vol. 49, no. 2, pp. 792–800, 2013.[11] M. Darouach, “On the functional observers for linear descriptor sys-tems,”
Systems & Control Letters , vol. 61, no. 3, pp. 427 – 434, 2012.[12] M. Hou and P. Muller, “Observer design for descriptor systems,”
IEEETrans. on Autom. Control , vol. 44, no. 1, pp. 164–169, Jan 1999.[13] J. Ishihara and M. Terra, “Impulse controllability and observability ofrectangular descriptor systems,”
IEEE Trans. on Autom. Control , vol. 46,no. 6, pp. 991–994, 2001.[14] S. Zhuk, J. Frank, I. Herlin, and R. Shorten, “Data assimilation for linearparabolic equations: minimax projection method,”
SIAM J. Sci. Comp. ,vol. 37, no. 3, p. A1174A1196, 2015.[15] S. Zhuk and M. Petreczky, “Infinite horizon control and minimaxobserver design for linear DAEs,” in
Proc. of 52nd IEEE Conf. onDecision and Control , 2013, extended version at arXiv:1309.1235.[16] S. Zhuk, “Kalman duality principle for a class of ill-posed minimaxcontrol problems with linear differential-algebraic constraints,”
AppliedMathematics & Optimization , vol. 68, no. 2, pp. 289–309, 2013.[17] ——, “Minimax state estimation for linear stationary differential-algebraic equations,” in
Proc. of 16th IFAC Symposium on SystemIdentification, SYSID 2012 , 2012.[18] ——, “Minimax state estimation for linear discrete-time differential-algebraic equations,”
Automatica , vol. 46, no. 11, pp. 1785–1789, 2010.[19] G.-R. Duan, “Observer design,” in
Analysis and Design of DescriptorLinear Systems , ser. Advances in Mechanics and Mathematics. SpringerNew York, 2010, vol. 23, pp. 389–426.[20] M. Gerdin, T. B. Sch¨on, T. Glad, F. Gustafsson, and L. Ljung, “On pa-rameter and state estimation for linear differential-algebraic equations,”
Automatica , vol. 43, no. 3, pp. 416–425, 2007.[21] R. Alikhani, F. Bahrami, and T. Gnana Bhaskar, “Existence and unique-ness results for fuzzy linear differential-algebraic equations,”
Fuzzy Setsand Systems , vol. 245, pp. 30–42, 2014.[22] R. Pulch, “Stochastic collocation and stochastic galerkin methods forlinear differential algebraic equations,”
Journal of Computational andApplied Mathematics , vol. 262, pp. 281–291, 2014.[23] L. Dai,
Singular Control Systems . Springer-Verlag New York, Inc.,1989.[24] M. Petreczky and S. Zhuk, “Solutions of differential-algebraic equationsas outputs of LTI systems: application to LQ control problems,”
Arxive1312.7547, a version is conditionally accepted to Automatica. , 2015.[25] G. Basile and G. Marro,
Controlled and conditioned invariants in linearsystems theory
Proc. IEEE Conference on Decision and Control ,2014.[28] F. Callier and C. Desoer,
Linear System Theory . Springer-Verlag, 1991.[29] H. Trentelman, A. A. Stoorvogel, and M. Hautus,
Control theory oflinear systems . Springer, 2005.[30] J. S. Hesthaven, S. Gottlieb, and D. Gottlieb,
Spectral Methods for Time-Dependent Problems . Cambridge University Press, 2007.[31] R. Rockafellar,