E
MBEDDING THE BASIC CONCEPTS OF CONTROL in the system of geometry and the in-terpretation (and re-inin-terpretation) of the results of mathematical system theory by using a geometric approach was initiated in the beginning of the 1970’s by Basile, Marro and Won-ham, see (Basile and Marro, 1969a; Wonham and Morse, 1970). By now, the approach has proved to be an effective means to the analysis and design of control systems and the idea gained some popularity that was followed by many authors succesfully. Good summaries of the subject can be found in the classical books of (Wonham, 1974; Wonham, 1979) and (Basile and Marro, 1991).The term geometric suggests several things. On the one hand it suggests that the setting is linear state space and the mathematics behind is primarily linear algebra (with a geometric flavor). On the other hand it suggests that the underlying methodology is geometric. The theory treats many important system concepts, for example the classical Kalman controllability theory, as geometric properties of the state space or its subspaces. These are the properties that are not affected by coordinate changes and always preserved under linear transformations, for example, the so-called invariant or controlled invariant subspaces. Using these concepts the geometric approach captures the essence of many analysis and synthesis problems and treat them in a coordinate-free fashion. By characterizing the solvability of a problem as a verifiable property of some constructible subspace, calculation of the solution law becomes much easier.
The computational aspects are considered independently of the theory and handled by means of the standard methods of matrix algebra, once a suitable coordinate system is defined. In many cases, the geometric approach can convert what is usually a difficult time varying or nonlinear problem into a more easier linear time invariant one.
The linear geometric systems theory was extended to nonlinear systems in the 1980’s, see e.g., (Isidori, 1985). In the nonlinear theory, the underlying fundamental concepts are almost the same, but the mathematics is different. For nonlinear systems the tools from differential geometry and Lie-theory are primarily used.
In the first part of this chapter some important concepts of geometric system theory are referred, basically those which will be used in the forthcoming discussions of this volume. In the rest of the chapter, we use some typical problems to illustrate the basic ideas of the geometric
approach to control and filtering with special emphasis to the problem of residual generation to fault detection and isolation where the residual is given a geometric interpretation.
The concepts of input observability and the design of input estimators have evident ap-plications in system supervision and fault detection which will be used in many times when discussing fault detectors in various contexts in the next chapters. Therefore, the problem of input reconstruction from the viewpoint of input observability is presented through the in-troduction of the unknown input observer problem. The unknown input observer is a very important idea as it leads out from the theory of classical state estimation methods by linking the theory of traditional observer-based residual generators to geometric ideas.
It will be shown that the solution of this problem, while firmly based on the geometric approach, has a tight relation with system inversion. This knowledge will serve useful in later parts when the concept of system inversion will be used more extensively.
2.1. INTRODUCTION
In this section we give a short summary on the basic notions and notations of linear geometric system theory. Two important concepts: the invariant subspace and the controlled invariant subspace, that will be used later on in the discussion, are introduced. For further details the interested reader is directed to the books of (Wonham, 1979; Basile and Marro, 1991).
2.1.1. Elementary invariant subspaces Consider then-dimensional linear system
˙
x=Ax (2.1)
withx∈Rn.
DEFINITION 2.1. The setΩ⊆Rnis called an invariant set of (2.1) if for any initial condition xo∈Ω, we havex(xo, t) =eAtxo∈Ω,for all t≥0.Trivial examples of invariant sets areRn
andx=0. ¤
In this volume, only a special class of invariant sets is considered, i.e., invariant subspaces which can be interpreted in the following way. Consider the conditions of a subspaceWto be invariant: Since by the Taylor series expansion
x(xo, t) =xo+tAxo+ t2
2A2xo+. . . ,
it is obvious that ifAixo ∈W foralli≥0,thenx(xo, t) ∈W, for allt≥ 0.Obviously again, this argument is true only ifWis a linear subspace. In other words, this condition implies that if a mapping fromRntoRn:w=Azis defined, then the image ofW⊆Rnis contained inW.
This property can be denoted asAW⊆W.
DEFINITION 2.2. LetA:X→X. Then a subspaceW⊂Xis calledA-invariantif it has the
propertyAW⊆W. ¤
Extensions of invariance provide means for analysis and synthesis of linear systems such as controllability and observability. Controlled invariant subspaces are associated to the control-lability (reachability) properties of dynamical systems. In fact they relates input and state in the dynamics of the systems. More specifically, controlled invariant subspaces are subspaces such that, from any initial state of the linear system ˙x = Ax+Bubelonging to these subspaces, at least one state trajectory can be maintained on them by means of a suitable control action.
DEFINITION 2.3. We say that a subspace W ⊂ X is (A, B)-controlled invariant or simply (A, B)-invariant, if assumingA : X → X andB : U → X there exist a map F : X → Usuch that(A+BF)W⊆W. In other words, the subspaceWis called a controlled invariant subspace of the control system ˙x=Ax+Buif there exists a feedback controlu=Fxsuch thatWis an
invariant subspace of ˙x= (A+BF)x. ¤
By duality, an analysis similar to controllability can be carried out by relating the interaction of state and output: the dual concept of(A, B)-controlled invariance is (C, A)-invariance or (C, A)-conditioned invariance.3
DEFINITION 2.4. LetA : X → X and C : X → Y. A subspace W ⊆ Xis (C, A)-invariant or(C, A)-conditioned invariantif there exists a mapD:Y →X such that(A+DC)W ⊂W.
Equivalently,W⊆Xis(C, A)-invariant ifA(W∩kerC)⊆W. ¤ Note that any A-invariant is also an (A, B)-controlled invariant for any B and an (A, C)-conditioned invariant for anyC.
2.1.2. Self-contained controlled and conditioned invariants
Self-contained controlled and conditioned invariants are particular classes of invariants which have interesting properties the most important of which is to admit both a supremum and an infimum.
PROPERTY 2.5. The sum of any two (A, B)-controlled invariant subspaces is an (A, B)-controlled invariant subspace.
PROPERTY 2.6. The intersection of any two (A, C)-conditioned invariant subspaces is an (A, C)-conditioned invariant subspace.
An immediate consequence of Property 2.5 is that the set of(A, B)-controlled invariants con-tained in a given subspaceE⊆Xadmits a supremum, the maximal(A, B)-controlled invariant contained in E. Similarly, Property 2.6 implies that the set of all (A, C)-conditioned invari-ants containing a given subspace D ⊆X admits an infimum, the minimal(A, C)-conditioned invariant containingD.
Subspaces which are the supremum of all(A, B)-controlled invariants contained in a given subspaceEand the infimum of all(A, C)-conditioned invariants containing a given subspaceD are introduced through the following definitions, respectively.
3 The naming conventions of these invariant subspaces are due to Basile and Marro that have been independently adopted by Wonham, see (Basile and Marro, 1969a) and (Wonham and Morse, 1970).
DEFINITION 2.7. Consider a subspace L ⊆ X. We use the notation hA| Li to denote the family of all A-invariant subspaces bounded to (containing or contained in) the subspace L whereL = Im LandL is a linear mapL:Y→ X. The extremal subspaces from this set (i.e., the maximal or minimal) are denoted bysuphA| LiandinfhA| Li, respectively. ¤ DEFINITION 2.8. Given any subspace E ⊆X the family of(A, B)-controlled invariant sub-spaces contained inEis defined asV=h(A, B)| Ei. The supremal subspace from this set, which will be referred to frequently in this book, is denoted byV∗ =supVh(A, B)| Ei. ¤ DEFINITION 2.9. Analogously, if we have D ⊆ Xthe set of all (C, A)-invariant subspaces containing Dis denoted byW = h(A, C)| Di. In the following, the infimal nonzero subspace from this set will be referred to W∗ = infWh(A, C)| Di. The characterizing property of these subspaces is to provide the possibility to make a part of the state space unobservable, that is to
place it in kerC, by means of output injection. ¤
The most elementary class of controlled invariant subspaces are the reachability (controllabil-ity) subspaces. Controlled invariants are subspaces such that, from any initial state belonging to them, at least one state trajectory can be maintained on them by means of a suitable control action. Consider for instance the system ˙x= Ax+Bu. The reachable (controllable) subspace of this system is defined as
infhA|Im Bi=span{B, AB, . . . , An−1B}
i.e., the minimalA-invariant subspace that containsIm B.
DEFINITION 2.10. If we consider the feedback lawu=Fx+Gv, the corresponding closed-loop system ˙x= (A+BF)x+BGvwill have the reachable (controllable) subspace
R=suphA+BF|Im(BG)i. (2.2)
ThusRis precisely the reachable (controllable) subspace of the pair(A+BF, BG). ¤ The controllable subspace of the pair ((A+BF, BG)is called a controllability subspace of the original system (A, B). The significance of the reachability (controllability) subspace derives from the fact that by the restriction ofA+BFto an(A+BF)-controlled invariant subspace, an arbitrary spectrum can be assigned by suitable choice of the feedbackF.
DEFINITION 2.11. A subspace R is called a reachability (controllability) subspace of ˙x = Ax+Buif there existFandGwith the feedback lawu= Fx+Gvsuch that (2.2) holds. The
reachability subspaceRis(A, B)-controlled invariant. ¤
In general, it is not possible to reach any point of a controlled invariant from any other point (in particular, from the origin) by a trajectory completely belonging to it. For this property consider the following definition:
DEFINITION 2.12. Given a subspaceE⊆X, by leaving the origin with trajectories belonging toE(i.e., to the maximal(A, B)-controlled invariant subspace contained inE, which is denoted asV∗), it is not possible to reach all the points ofV∗, but only a subspace ofV∗, which is called the reachable set onE(or onV∗) and denoted byRE(orRV∗). ¤
THEOREM 2.13. (Basile and Marro, 1991).The reachable set onE(or onV∗), coincides with the maximal(A, B)-controlled invariant contained inE,i.e.,
RE,RV∗=suphA+BF| V∗∩Im Bi. ¤ Note that the reachable set onEcan be defined independently of the feedback matrixF. This definition is more practical becauseFis not unique.
DEFINITION 2.14. The reachable set onEis defined as
RE,RV∗ =V∗∩S∗2 with S∗2,infhE, A|Im Bi.
The duality relation, which defines the reachable set on Eas the intersection of the maximal (A, B)-controlled invariant contained inEwith the minimal(E, A)-conditioned invariant con-taining Im Bwas first derived by (Morse, 1973). This observation led to the construction of important computational algorithms which will be discussed in Section 2.2. ¤
DEFINITION 2.15. Dual to the properties of reachability is the observability. The pair of maps(C, A)is observable if
n
\
i=1
ker(CAi−1) =0. (2.3)
¤ DEFINITION 2.16. (Unobservable subspace). Definition 2.15 with condition (2.3) suggests that the subspaceS⊆Xdefined as
S=
n
\
i=1
ker(CAi−1) (2.4)
is the unobservable subspace of the pair(C, A). SinceAS⊂S, in fact,Sis the largest in family A-invariant conditioned subspace contained in kerC,i.e.,
S∗ =suphA|kerCi.
¤
DEFINITION 2.17. (Unobservability subspace). We say a subspaceS ⊆ Xis a (C, A) unob-servability subspace ifS= hA+DC| kerCi for some output injection mapD:Y→ X,i.e.,S is the unobservable subspace of the pair¡
C,(A+DC)¢
. The significance of the unobservability subspace derives from the fact that by the restriction ofA+DCto a(C, A+DC)-conditioned invariant subspace can be assigned an arbitrary spectrum by suitable choice of the output
injec-tion matrixD. ¤
Clearly, from the orthogonality property, the annihilator ofSisS⊥=hAT+CTDT|Im Ciand S⊥is an(AT, CT)controllability subspace,cf. (Wonham, 1979).
2.2. SOME SPECIFIC COMPUTATIONAL ALGORITHMS
Subspaces S∗ = infSh(C, A)| Di and V∗ = supVh(A, B)| Ei, which are respectively the in-fimum of the semilattice of all (C, A)-conditioned invariants containing a given subspace D and the supremum of the semilattice of all (A, B)-controlled invariants contained in a given subspace E, are frequently used in our detection theory. They can be determined with com-putational algorithms. The basic algorithms which will be referred to in the later parts of this work are the following.
ALGORITHM 2.18. (Minimal (C, A)-conditioned invariant containing the subspaceD). The subspaceS∗=inf Sh(C, A)| Dican be obtained in a recursive procedure
Wo=D
Wi=D+A(Wi−1∩kerC), i=1, . . . , k
where the terminatingk≤n−1is determined by the conditionWk+1=Wk,i.e.,W∗=limWk. We will refer to this algorithm as the (C, A)-invariant subspace algorithm (CAISA) in the following.
ALGORITHM 2.19. (Unobservability subspace). The unobservability subspace, which, by Definition 2.17 is, in fact a(C, A+DC)-conditioned invariant subspace can be given by the recursive procedure
Wo=X
Wi=S∗+ (A−1Wi−1) ∩ kerC, i=1, . . . , k
for k = 1, 2, . . . , n−1. The infimal element W∗ is obtained as limWk where the subspace S∗ = inf Sh(C, A)| Di can be precalculated by the CAISA above. The algorithm is called the unobservability subspace algorithm (UOSA) in the literature. ¤ From Property 2.5 and from
E⊇V ⇔ E⊥⊆V⊥ (2.5)
the orthogonality relation of subspaces
supVh(A, B)| Ei=¡
infSh(AT, B⊥)| E⊥i¢⊥
(2.6) can be easily checked which relates the determination of the subspaceV∗ = supVh(A, B)| Ei to that ofS∗ = infSh(A, C)| Di. Relation 2.6 makes the dualization of the (C, A)-conditioned invariant algorithm (Algorithm 2.18) possible in the following way:
ALGORITHM 2.20. (Maximal (A, B)-controlled invariant contained in the subspace E). A recursive procedure which provides the subspace V∗ = supVh(A, B)| Ei in the last recursion can be given as
Wo=E
Wi=E ∩ A−1(Wi−1+Im B), i=1, . . . , k
where the terminatingk ≤n−1is determined by the conditionWk+1 = Wk. It can be seen that the recursion converges to the orthogonal complement ofS∗ = infS(AT, B⊥,E⊥)which in fact equals toV∗ =supV(A, B,E)by (2.6).
ALGORITHM 2.21. (Reachability (controllability) subspace). For an arbitrary, fixed subspace R⊂Xdefine the sequence of recursion
Wo=0
Wi=R ∩ (AWi−1+Im B), i=1, . . . , k
fork=1, 2, . . . , n−1, whereW∗ =limWk. The algorithm is called the controllability subspace algorithm (CSA) in the following.
ALGORITHM 2.22. (Supremal reachability (controllability) subspace). Due to Definition 2.12, every subspaceE⊂Xcontains a unique supremal controllability subspaceR∗ =RV∗. The com-putation of this subspace requiresa priori knowledge ofV∗=V suph(A, B)| Ei. A method that can be used for precomputing the subspaceV∗ was presented by Algorithm 2.20. For the com-putation of this supremal subspaceRV∗ consider the recursive sequence
Wo=0
Wi=V∗ ∩ (AWi−1+Im B), i=1, . . . , k fork=1, 2, . . . , n−1. Then,RV∗ is obtained in limWk.
2.3. THE CONCEPT OF UNKNOWN INPUT OBSERVER FOR LINEAR SYSTEMS Different types of modeling uncertainties such as nonlinearities, parameter variations and other unmeasurable external disturbances can conveniently be represented as unknown inputs which, in general, are also termed disturbances. Consider the system
˙
x(t) =Ax(t) +Bu(t) +Ld(t) (2.7)
y(t) =Cx(t)
where u(t) denotes the manipulable known input, d(t) is the unknown input, which at the moment is assumed to be completely unaccessible for measurement, and inspect the problem of realizing, if possible, a state feedback of the type shown in Fig. 2.1 such that, starting at the known initial condition, y(t)=0 results for all admissible d(t). This problem can be identified as the unaccessible disturbance localization problem in the literature, seee.g., (Basile and Marro, 1969a; Basile and Marro, 1969b). The system with state feedback is described by
˙
x(t) = (A+BF)x(t) +Ld(t) (2.8)
y(t) =Cx(t),
F
C
x=Ax+Bu+Ld
-¾
u(t) x(t)
d(t) y(t)
•
Figure 2.1. The unaccessible disturbance localization problem with state feedback.
and presents the requested behavior if and only if its reachable set by d(t), i.e., the minimal (A+BF)-invariant containingIm L, is contained in kerC. Since any(A+BF)-invariant subspace is an (A, B)-controlled invariant, the unaccessible disturbance localization problem admits a solution if and only if the following structural condition holds
L⊆V∗ (2.9)
whereV∗=supVhA,B |kerCiandL=Im B.
It is worth noting two things here. First, the assumption of the possibility of the application of full state feedback shown in Fig. 2.1 is not feasible in practice since, in most cases, state is not completely accessible for measurement; second, the assumption of stability,i.e., the property of the system that the matrixF, besides disturbance localization, achieves stability of the overall closed-loop systemA+BFis to be taken into consideration especially from practical points of view. These issues are considered in the following part of this section.
Introducing some subtle differences in the setup of the previous problem we shall now con-sider the asymptotic estimation of the state (or a linear function of the state, possibly the whole state) in the presence of the unaccessible disturbance input. This problem is referred to the disturbance decoupled estimation problem (DDEP) in the literature. If the problem of simulta-neous observation of states and the estimation of unknown inputs is investigated the problem is also known as the unknown input observer problem (UIOP). As one of the first solutions to UIOP, Wanget al. proposed a minimal-order observer for the system (2.7) without making any assumptions on the properties of unknown inputs (Wang et al., 1975). This approach was fol-lowed by many authors presenting different unknown input observer design ideas (Willems and Commault, 1981; Hou and M¨uller, 1988; Hou and M¨uller, 1992). The geometric approach of the problem was first introduced by (Bhattacharyya, 1978). Different solutions of UIOP, from many obvious reasons, have a number of implications to fault detection and isolation, which have been investigated in a series of papers, see e.g., (Frank and W¨unnenberg, 1989; Frank, 1990; Hou and M¨uller, 1994).
The mathematical problem of obtaining the state from input and output measurements when some of the inputs are unknown has solvability conditions more extended than the prob-lem of estimating the state by using a dynamic observer. For the cases when the input function is (at least partially) unknown the concept of unknown-input observability or unknown-input
˙^
x= (A+DC)^x−Dy C
˙
x=Ax+Bu+Ld
--
-- ^x(t) ε(t)
d(t) y(t)
u(t) •
Figure 2.2. Estimation in the presence of unaccessible disturbancei.e.,the concept of the unknown input observer problem.
reconstructability4 is presented by showing that unknown-input reconstructability is closely related to invertibility.
Consider system (2.7) and the behavior of the observer designed for estimating the states (see the discussion in the previous chapter with Fig. 1.6) when, as represented in (2.7), the observed system has, besides the accessible inputu(t), the unaccessible inputd(t)
˙^
x(t) = (A+DC)^x(t) +Bu(t) −Dy(t). (2.10) Subtracting (2.10) from (2.7) the error dynamicsε(t) = ^x(t) −x(t)is obtained in the form
˙
ε(t) = (A+DC)ε(t) −Ld(t), (2.11) which shows that the estimation error does not converge asymptotically to zero, even ifA+DC is stable, but converges asymptotically to a subspace infhA+DC| Li, which, in fact, equals with the reachable setRof the system (2.11) (cf. Definition 2.11).
It follows that, in order to obtain the state estimate in presence of unknown inputs, it is convenient to choose D to make this subspace of minimal dimension: since it is an (A, C)-conditioned invariant, the best choice of D corresponds to transforming into an (A+ DC)-invariant subspace:i.e., the minimal(A, C)-conditioned invariant which containsIm L.
2.3.1. System invertibility and reconstructability
In the following part let us briefly characterize the property of unknown-input observability.
From a strictly mathematical viewpoint, unknown-input observability can be introduced and characterized as follows. It is well known that, collecting the effects of the inputsuanddin a common term, the response of system (2.7) given by the triple(A, B, C)can be related to the initial statex(0)and control functionu(t)by
y(t) =CeAtx(0) +C Zt
0
eA(t−τ)Bu(τ)dτ. (2.12)
4 For continuous time linear systems observability and reconstructability is equivalent.
Equation (2.12) consists of two terms, the free and the forced responses of the system denoted by the operatorsγ1(x(0), t) andγ2(u, t, τ), respectively. By using this notation, Eq. (2.12) for the finite time interval[0, T]can be rewritten as
y(t)|[0,T]=γ¡
x(0), u|[0,T]¢
=γ1¡ x(0)¢
+γ2¡
u(t)|[0,T]¢
. (2.13)
The initial (and the final) state of the system can be calculated from input and output obser-vations: this requires the pair (A, C) to be observable (reconstructable). Recall that(A, C) is observable, ifγ1is invertible,i.e., kerγ1=0, see (Wonham, 1979).
For the cases when the input function is unknown the concept of reconstructability can be extended by relating it to the concept of system invertibility, see (Basile and Marro, 1991). The term system invertibility denotes the possibility of reconstructing the input from the output function. For the sake of precision it is possible to define both the state, unknown-input invertibility and the zero-state, unknown-unknown-input invertibility. Consider the following defi-nitions.
DEFINITION 2.23. The triple (A, B, C) is said to be unknown-state, unknown-input re-constructable (or unknown-state, unknown-input invertible) if, in (2.13) γ is invertible,i.e.,
kerγ=0. ¤
DEFINITION 2.24. The triple (A, B, C)is said to be zero-state, unknown-input reconstruct-able (or zero-state, unknown-input invertible) if, in (2.13)γ2is invertible,i.e., kerγ2=0. ¤ When(A, C)is not observable or reconstructable
kerγ1=S∗ ,suphA|kerCi, (2.14)
that follows from the definition of the unobservability subspace (cf. Definition 2.19). This means that the state canonical projection on the factor spaceX/S∗ can be determined from the output functiony(t).
Unknown-input reconstructability can be approached in a similar way: by linearity, when reconstructability is not complete, only the canonical projection of the final state onX/S1 or X/S2 can be determined, where S1 is the unknown-state, unknown-input unreconstructabil-ity subspace andS2 is the zero-state unknown-input unreconstructability subspace. From the mentioned properties it follows thatS2⊆S1.
The geometric characterization of these subspaces can be given by the following properties (Basile and Marro, 1991). The unknown-state, unknown-input unreconstructability subspace is
S1=V∗,suph(A, B)|kerCi, (2.15) i.e., the maximal(A, B)-controlled invariant subspace contained in kerC. The zero-state, unknown-input unreconstructability subspace is
S2=R=V∗∩S∗ with S∗ =infh(A, C)|Im Bi (2.16) i.e., the minimal(C, A)-conditioned invariant subspace containing inIm B.
Although both state, input invertibility and the zero-state, unknown-input invertibility was given by Definitions (2.23) and (2.24), in the following discussions the term ‘invertibility’ will be referred to the latter, i.e., zero-state, unknown-input invertibility which is related to the invertibility of the operatorγ2in (2.13). Furthermore, invertibility can
Although both state, input invertibility and the zero-state, unknown-input invertibility was given by Definitions (2.23) and (2.24), in the following discussions the term ‘invertibility’ will be referred to the latter, i.e., zero-state, unknown-input invertibility which is related to the invertibility of the operatorγ2in (2.13). Furthermore, invertibility can