A convex processAfromRnto itself is a set-valued map satisfyingA.x/CA.y/A.xCy/
for all ; 0, or, equivalently, a set-valued map whose graph is a convex cone. A convex process is closed if its graph is closed and that it is strict if its domain is the whole space. With a strict closed convex processAone can associate the Cauchy problem for the differential inclusion:
P
x.t /2A.x.t //; x.0/D0, for details see Filipov (1960) and Aubin and Cellina (1984).
IfG Rn, let us denote byGCits (positive) polar cone defined by GCD fp 2 Rnj hp; xi 0; 8x 2Gg:
The transposeA of A is defined as the set-valued map defined by p 2 A.q/ , 8.x; y/ 2 Graph.A/; hp; xi hq; yi:For2Rthe eigenvectorsvofAare the nonzero solutions of the inclusionv2A.v/.
Motivated by the terminology used for linear systems we say thatAsatisfies therank conditionif the subspace spanned by the coneAk.0/is the whole space for some integerk1.
Theorem 2(Frankowska et al. (1986)): The following conditions are equivalent:
a) the differential inclusionx.t /P 2A.x.t //; x.0/D0is controllable, b) the differential inclusion is controllable at some timeT > 0,
c) the rank condition is satisfied andAhas no eigenvectors, d) for somek 1, one hasAk.0/D. A/k.0/ DRn.
Controllability of a linear control system is equivalent to the controllability of the differential inclusion defined byx.t /P 2 Ax.t /CU; x.0/ D 0;withU D co.B˝/is a closed convex cone of controls, whereco.S /denotes the closure of the convex hull of the setS, see Aubin and Cellina (1984). The adjoint inclusion is q.t /P 2ATq.t /; q.t /2UC, see Frankowska et al. (1986).
C Geometry of LTI systems
Let us consider the LTI control system P
x DAxCBu with the output
y DC x
It is assumed that columns of the matrixB 2 Rnm and the rows of the matrixC are linearly independent.
C.1 Brunovsky canonical form
The set of points that lies on the same trajectory with the origin is called the reachability (control-lability) subspace. Let us denote the controllability subspace of the pair.A; B/byR.A; B/:
The state space of the system is partitioned by the manifolds of typexCR.A; B/;wherex2X:
By definition, the points of two different manifolds cannot be joined by a trajectory.
Any controllable linear system can be effectively transformed to Brunovsky canonical form, Brunovsky (1970) by feedback and a change of state and input coordinates as asserted in the next result:
Theorem 3 (Generalized Brunovsky canonical form): For every pair . A; B /, there exists a unique sequence of integersk D. k1; k2; km/satisfying
k1 k2 km; k1Ck2C Ckm Dnc;
a linear transformationGand invertible linear transformationsF andH such that the pair .A;Q B /Q D. F .A BH 1G/F 1; FBH 1/
is in Brunovsky normal form, i.e.,
AQD
C Geometry of LTI systems
andAJ a block diagonal matrix with Jordan blocks.
The Generalized Brunovsky form reveals two kinds of complete invariants: the controllability indiceski of the controllable part of. A; B /and the invariant factors of the uncontrollable part of . A; B /shown inAj:
An LTI system with unconstrained inputs is not only controllable on its controllability subspace, but it can also be driven on any sufficiently smooth trajectory that lies in the controllability sub-space. However, if there is a constrain on the inputu, this property might not hold, as the following small example shows:
Considering the measured output y D C x then by using feedback, and output injection one can obtain the Morse canonical form, i.e.,
.A;Q B;Q C /Q D. F .ACBH CKC /F 1; FBH; GCF 1/
If there is a constraint onu, the feedback might not be implemented, so the system is equivalent through similarity with a canonical form that also contains coupling terms.
C.2 Controlled and conditioned invariance
In the absence of control action a subspace of the state space X is a locus of trajectories if and only if it is anA-invariant1set. The extension of this property to the case in which the control is
1For the details concerning the notions and propositions used in this section the interested reader is sent to Basile and Marro (2002) and Wonham (1985).
C Geometry of LTI systems
present and suitably used to steer the state along a convenient trajectory leads to the concept of .A; B/-controlled invariantsubspaceV defined as:
AV V CB; B DImB:
The dual of a controlled invariant subspace is an.A; C / conditioned invariantsubspaceS, which is defined as:
A.S \C/S; C DKerC:
The set of all.A; B/ controlled invariantsVE contained in a given subspaceE is an upper semi-lattice that admits a supremum, themaximal.A; B/ controlled invariantcontained inE, which will be denoted by VE D maxV .A; B;E/. Similarly the set of all .A; C / conditioned invariants SD containing a given subspace D is a lower semilattice that admits an infimum, the minimal .A; C / conditioned invariant containing D, which will be denoted by SD D minS.A; C;D/.
These subspaces can be determined by efficient algorithms in finite steps.
A trajectory of the pair .A; B/can be controlled onE if and only if its initial state belongs to a controlled invariant contained inE, hence in VE. In general, for any initial state belonging to a controlled invariantVE, it is possible not only to continuously maintain the state onVE by means of a suitable control action, but also to leaveVE with a trajectory onE and to pass to some other controlled invariant contained in E. On the other hand there exist controlled invariants that are closed with respect to the control, i.e., that cannot be exited by means of any trajectory onE: these will be called self-bounded with respect toE. An .A; B/ controlled invariant V contained in a subspaceE is said to be self-bounded with respect toE ifVE\B V.
The duals of the self-bounded controlled invariants are the self-hidden conditioned invariants:
an.A; C / conditioned invariantS containing a subspaceDis said to be self-hidden with respect toD ifS SD CC.
In general, however, 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. In other words, given a subspace E, by leaving the origin with trajectories belonging to E, hence to VE, (the maximal.A; B/ controlled invariant contained inE), it is not possible to reach any point of VE, but only a subspace of it, which is called the reachable set on E and denoted by RE. It can be proved thatRE DVE\S withS DminS.A;E;B/.
Let us denote by V D maxV.A; B;C/the maximal . A; B /-controlled invariant subspace contained inCand bySDminS.A; C;B/the minimal. A; C /-conditioned invariant subspace containingB:
Theorem 4(Four Map Theorem): Let us consider the state transformation DT 1x defined by T D
C Geometry of LTI systems
whereA23 D NA23C3 andA43 D NA43C3. Moreover by a suitable feedback A31 andA32 can be zeroed out.
C.3 Left and right invertibility
It is well known that the response of the triple.A; B; C /is related to initial statex.0/and control functionu.t /by
y.t / D .A;B;C /x0 uDC eAtx.0/CC Z t
0
eA.t /Bu. /d D .A;B;C /x.0/C˚.A;B;C /0 u:
The term system invertibility denotes the possibility of reconstructing the input from the output function; more precisely the term invertibility refers to unknown-input invertibility, i.e., to the invertibility of map˚.A;B;C /0 such thatu.t /D.˚.A;B;C /0 / 1˚.A;B;C /0 .u.t //.
When.A; C /is notobservable(reconstructable), the initial or final state can be determined modulo the subspace
Ker .A;B;C / DQ
whereQdenotes the maximalA invariant subspace contained inC, which is calledunobservability subspace(unreconstructability subspace). This means that the state canonical projection on X=Q can be determined from the output function. Q is the locus of the free motions corresponding to the output function identically zero. A dynamical system is completely unknown-input state observable by means of differentiators if it is possible to determine its statex when an arbitrary short output segmentyis given.
The subspace of unknown input state observability by means of differentiators is minS.AT;KerBT;ImCT/DmaxV?.A; B; C /
and the subspace of functional input observability isBT minS.AT;KerBT;ImCT/. The or-thogonal projection of the state on the subspaceV;? can be deduced from the output and from its derivatives, moreover this is the greatest subspace where the orthogonal projection of the state can be recognized solely from the output. If the state is known the orthogonal projection of the input can be determined on BTV;? and it cannot be recognized a greater subspace (it can be determined moduloB 1;TV).
Definition 6: Assume that B has maximal rank. The system .A; B; C / with x.0/ D 0 is said to be invertible (left-invertible) if, given any output function y.t / defined on Œ0; t1; t1 > 0 belonging to Im˚.A;B;C /0 , there exists a unique input function u.t / such that ˚.A;B;C /0 u.t / D y.t / holds, i.e., KerT.A;B;C /0 D0.
The triple.A; B; C /, with B having maximal rank, is unknown-state (zero-state) invertible if and only if it is unknown-state, unknown-input (zero-state, unknown-input) completely recon-structable.
C Geometry of LTI systems
A dynamic system exists which, connected to the system output and with initial state suitably set as a linear function of the system state (which is assumed to be known), provides tracking of the system state moduloS. This system is is not necessarily stable. The observer equations, expressed in the basis that corresponds to the transformationT DŒT1T2, withImT1 DS, can be written T 1x.0/a state estimate moduloSis derived.
An algebraic reconstructor with differentiators provides as output a state estimatez1moduloV and works if neither the initial state nor the input function is known, while the dynamic tracking device provides as z2 a state estimate moduloS, but requires the initial state to be known. A state estimate moduloV\Sis obtained as a linear function of the outputs of both devices, i.e., z D M z1CN z2. This state reconstructor provides the maximal information on the system state when the input function is unknown and the initial state known, by observing the output in any nonzero time interval.
The term functional controllability denotes the possibility of imposing any sufficiently smooth (piecewise differentiable at least n times) output function by a suitable input function, starting at the zero state. Starting from the identityy.t / D ˚.A;B;C /0 .˚.A;B;C /0 / 1.y.t //it is also called right invertibility.
For multi input single output (MISO) systems a formal definition can be given as:
Definition 7: Assume that C has maximal rank. The system .A; B; C /is said to be functionally con-trollable (right-invertible) if there exists an integer 1such that, given any output functiony with th derivative piecewise continuous and such thaty.0/ D 0; y./.0/ D 0, there exists at least one input functionusuch that˚.A;B;C /0 u Dy holds. The minimum value ofsatisfying the above statement is called the relative degree of the system.
In order to define therelative degreefor MIMO systems in geometric terms the following extension of functional output controllability is introduced:
Definition 8(Constrained Functional Output Controllability): A subspaceY.h/ is said to be a func-tional output controllability subspace with respect to thehth derivative if the output of the triple.A; B; C / can be driven along any trajectoryysuch thaty 2Y.h/ with thehth derivative piecewise continuous.
This is possible exactly when there exist an.A; B/ controlled invariant subspaceV such that Y.h/ D CV. Let us considerE DC 1Y.h/ andVE.h/, the maximal.A; B/ controlled invariant subspace contained inE such that the output can be driven onCVE.h/along any trajectoryywith piecewise continuoushth derivative for all the initial statesx.0/2 VE.h/.
Definition 9(Multivariable Relative Degree): The relative degreei of outputyi is defined asi Dh (if exists), whereY.h/ DCVE.h/ assuming thatY.h/ D fyjyk D0; k¤ig.
C Geometry of LTI systems
The functional controller is realizable in exactly the same way as the (left)inverse system, i.e., by a state reconstructor completed with a further differentiator stage and an algebraic part. Its dynamic part is asymptotically stable if and only if all the invariant zeros of.A; B; C /are stable. In this case, however, the inputucorresponding to the desired output is not unique, in general. The difference between any two admissible input corresponds to a zero-state motion onRV D V\S which does not affect the output, so that the functional controller can be realized to provide any one of the admissible inputs, for instance by setting to zero input components which, expressed in a suitable basis, correspond to forcing actions belonging toV\I mB.
Left and right invertibility can be characterized in geometric terms as follows:
Theorem 5: Triple. A; B; C /is left-invertible if and only if V\B D0:
Condition of left-invertibility is equivalent toV\S D0.
Theorem 6: LetC WDKerC:Triple. A; B; C /is right-invertible if and only if SCC DX:
Condition of right-invertibility is equivalent toVCSDX.
D Invariant distributions and codistributions
Let be a distribution defined on an open set U. We are interested in finding the smallest distribution, which is invariant under given vector fields (1; : : : ; q) and which is denoted by the symbolh1; : : : ; qji. Given a distributionand a set1; : : : ; q of vector fields we define the nondecreasing sequence of distributions:
0D k Dk 1C
Xq
iD1
Œi; k 1; (D.1)
i.e., for allkone has thatk h1; : : : ; kj i. If there exists an integerksuch thatk D kC1thenk D h1; : : : ; kji.
Let˝be a codistribution defined on an open setU and we are interested in finding the smallest codistribution, which is invariant under the given vector fields (1; : : : ; q) and which is denoted by the symbolh1; : : : ; qj˝i. Given a codistribution˝ and a set1; : : : ; q of vector fields we define the dual version of (D.1), i.e.,
˝0 D˝
˝k D˝k 1C Xq
iD1
Li˝k 1: (D.2)
Then for allk one has˝k h1; : : : ; kj˝iwhile˝k D h1; : : : ; kj˝iprovided that there exists an integerksuch that˝k D˝kC1.
Example 3: In the special case of LTI systems the algorithm (D.1) ends up with the well-known controllable subspace of the system:
n 1.x/ DImŒB AB : : : An 1B; x 2 Rn
Considering the dual case let ˝0 be the codistribution spanned by the row vectors c1; : : : ; cp of C, the
D Invariant distributions and codistributions
algorithm (D.2) ends up with the subspace:
˝n 1.x/ Dspanfc1; : : : ; cp; c1A; : : : ; cpA; : : : ; c1An 1; : : : ; cpAn 1g D
By duality˝n 1? .x/ is the largest distribution invariant under the vector field fA and contained in the distribution˝0?.x/. Moreover, by construction, at eachx2 Rn,
˝0?.x/DKerC where the algorithm was initialized at a constant distribution.
Starting from the constant codistribution˝0 D QImC DImCT, one has
˝k D˝k 1C Xm
iD0
˝k 1Ai:
Let p1.x/; : : : ; pd.x/ be a set of smooth vector fields defined on an open set U, set P D spanfp1; : : : ; pdgand consider the nondecreasing sequence of distributions defined as follows:
S0 DP Sk DSk 1C
Xm
iD0
Œgi; Sk 1\Ker dhg whereS denotes the involutive closure ofS.
Suppose there exists an integerksuch thatSkC1 DSkand set˙P DSk. Then˙P is the minimal conditioned invariant and involutive distribution containingP. This algorithm is called termed asconditioned invariant distribution algorithm.
D Invariant distributions and codistributions
Example 5: By setting
g0.x/DAx; g1.x/DB; h.x/DC x one has
Œg0; Sk 1\KerC .x/DA.Sk 1 \KerC /;
thus one can obtain the well-known (C; A)-invariant subspace algorithm for LTI systems:
S0DP
Sk DSk 1CA.Sk 1 \KerC /:
Example 6: For bilinear systems, i.e.,gi.x/DAix it follows that:
S0 DP Sk DSk 1C
Xm
iD0
Ai.Sk 1\KerC /:
Example 7: Using the augmented state space DŒt; xT one can obtain the algorithm S0./DP
Sk./DSk 1./C @
@t A..t //
.Sk 1./\KerC /;
for a linear time varying dynamics, a readout map with constantC matrix and a constant distributionP. The dual is thecontrolled invariant distribution algorithmwhich is defined via codistributions:
˝0Dspan dh
˝k D˝k 1C Xm
iD0
Lgi.˝k 1\G?/: (D.3)
Suppose there exists an integerk such that˝kC1 D ˝k. Then˝k D ˝k, for allk > k and if ˝k \ G? and ˝k? are smooth, then ˝k? is the maximal controlled invariant smooth distribution contained inKer dh.
Example 8: Considering LTI systems, the algorithm
˝0 D QIm C DImCT
˝k D˝k 1C.˝k 1 \KerBT/A;
ends up in the minimal (BT; AT)-invariant subspace over ImCT so its dual is the maximal (A; B)-invariant subspace inKerC.
D Invariant distributions and codistributions
Remark 17: The derivation of the time-dependent form (i.e., in the augmented state space) of the controlled invariant distribution algorithm (D.3) will end up in
˝QkC1./Dspanfdhg C.˝Qk\B?/A./;
provided that there existsksuch that˝kC1D˝k. Then˝Qk? will be the maximal controlled invariant distribution inKerfdhgwhich containsG D spanfg1; : : : ; gmg. Considering constant codistributions in each step we get the dual form of (7.7):
˝QkC1 Dspanfdhg C XN
iD0
.˝Qk \B?/Ai:
Letbe a fixed codistribution and define the nondecreasing sequence of codistributions as:
Q0D\span dh QkC1 D\
Xm
iD0
LgiQkCspan dh
!
: (D.4)
Suppose that all the codistributions of this sequence are nonsingular, i.e., there exists an integer kn 1such thatQk DQk for allk > k, set˝DQk and use the notation:
˝Do.c.a../
where o.c.a. stands forobservability codistribution algorithm. Then Q0 D˝\span dhg
QkC1 D˝\ Xm
iD0
LgiQk Cspan dh
! :
provided that all the codistributions generated by the observability codistribution algorithm are nonsingular. As a consequence o.c.a..˝/ D ˝ and if is conditioned invariant, so is the codistribution˝.
˝ is said to be aobservability codistributionif fulfills the relations:
Lgi˝ ˝Cspan dh; i D0; 1; : : : ; m o.c.a..˝/D˝:
The distributionis calledunobservability distributionif its annihilator˝ D?is an observability codistribution. If the algorithm (D.4) is initialized at.˙P/?, then o.c.a.(.˙P/?) is an observability codistribution contained inP?. Moreover, it is the largest codistribution having this property.
D Invariant distributions and codistributions
Example 9: Let us consider the nonlinear system
P
xDA0xC Xm
iD1
uiAixCl.x/mC Xd
iD1
pi.x/wi
y DC x with the assumption that
P Dspanfp1; : : : ; pdg
is independent of x. Then the observability codistribution algorithm will be read as:
Q0 D\ImQC QkC1 D\
Xm
iD0
QkAi CImQC
!
: (D.5)
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