linear control theory - unibo.it · summer school on time delay equations and control theory...
TRANSCRIPT
Summer School
on Time Delay Equations and Control Theory
Dobbiaco, June 25–29 2001
Linear Control Theory
Giovanni MARRO∗, Domenico PRATTICHIZZO‡
∗DEIS, University of Bologna, Italy‡DII, University of Siena, Italy
References
WonhamLinear Multivariable Control – A Geometric Approach,3rd edition, Springer Verlag, 1985.
Basile and MarroControlled and Conditioned Invariants in Linear Sys-tem Theory, Prentice Hall, 1992
Trentelman, Stoorvogel and HautusControl Theory for Linear Systems, Springer Verlag,2001
Early References
Basile and MarroControlled and Conditioned Invariant Subspaces inLinear System Theory, Journal of Optimization The-ory and Applications, vol. 3, n. 5, 1969.
Wonham and MorseDecoupling and Pole Assignment in Linear Multivari-able Systems: a Geometric Approach, SIAM Journalon Control, vol. 8, n. 1, 1970.
Introduction to Control Problems
Consider the following figure that includes a controlledsystem (plant) Σ and a controller Σr, with a feedbackpart Σc and a feedforward part Σf .
+_
+
+
rp Σfr e Σc
d1 d2
u Σy2
y1
Σr
Fig. 1.1. A general block diagram for regulation.
• rp previewed reference
• r reference
• y1 controlled output
• y2 informative output
• e error variable
• u manipulated input
• d1 non-measurable disturbance
• d2 measurable disturbance
1
d
Σu y
Σrrp e
Fig. 1.2. A reduced block diagram.
In the above figure d := {d1, d2}, y := {y1, y2, d1}.All the symbols in the figure denote signals, repre-sentable by real vectors varying in time.
The plant Σ is given and the controller Σr is to bedesigned to (possibly) maintain e(·)=0.
Both the plant and the controller are assumed to belinear (zero state and superposition property).
The blocks represent oriented systems (inputs, out-puts), that are assumed to be causal.
In the classical control theory both continuous-timesystems and discrete-time systems are considered.
t k0 0
2
An example
+
_
++∫
e
K
1T
∫amplifier
va
tachometer
crω
r
motor
vc
PI controller
z
Fig. 1.3. The velocity control of a dc motor.
The PI controlled yields steady-state control with noerror.
This property is robust against parameter variations,provided asymptotic stability of the loop is achieved.
This is due to the presence of an internal model ofthe exosystem that reproduces a constant input sig-nal (an integrator).
Thus, a step signal r of any value is reproduced withno steady-state error and the disturbance cr is steady-state rejected. This is called a type 1 controller.
Similarly, a double integrator reproduces with nosteady-state error any linear combination of a stepand a ramp and rejects disturbances of the same typeThis is a type 2 controller.
3
+
_
e va
cr
ωrPI M
T
Fig. 1.4. The simplified block diagram.
w
Σu y
Σr
eΣe
Fig. 1.5. The reduced block diagram.
In Fig. 1.5 w accounts for both the reference andthe disturbance. The control purpose is to achieve a“minimal” error e in the response to w.
If w is assumed to be generated by an exosystem Σe
like in the previous example, the internal model en-sures zero stedy-state error.
This approach can easily be extended to the multi-variable case with geometric techniques.
Modern approaches consider, besides the internalmodel, the minimization of a norm (H2 or H∞) ofthe transfer function from w to e to guarantee a sat-isfactory transient.
4
A more complex example
+
rp
delay re
controller
motor
reductiongear
dgage
transducer
v0
Fig. 1.6. Rolling mill control.
This example fits the general control scheme given inFig. 1.1.
The gage control has an inherent transportation de-lay. If the aim of the control is to have given amountsof material (in meters) at a specified thickness, it isnecessary to have a preview of these amounts, that istaken into account with the delay.
Of course, this preview can be used with negligibleerror if the cilinder rotation is feedback controlled bymeasuring the amount of material with a type 2 con-troller.
Thus, robustness is achieved with feedback and makesfeedforward (preview control) possible.
There are cases in which preaction (action in advance)on the controlled system significantly improves track-ing of a reference signal. The block diagram shownin Fig. 1.1 also accounts for these cases.
5
Mathematical Models
Let us consider the velocity control of a motor shownin Fig. 1.3 and its reduced block diagram (Fig. 1.5):
w
Σu y
Σr
eΣe
Mathematical model of Σ:
va(t) = Ra ia(t) + Lad ia
dt(t) + vc(t) (1.1)
cm(t) = B ω(t) + Jdω
dt(t) + cr(t) (1.2)
In (1.1) va is the applied voltage, Ra and La the arma-ture resistance and inductance, ia and vc the armaturecurrent and counter emf, while in (1.2) cm is the mo-tor torque, B, J, and ω the viscous friction coefficient,the moment of inertia, and the angular velocity of theshaft, and cr the externally applied load torque.
Mathematical model of Σr:
d z
dt(t) =
1
Te(t) (1.3)
va(t) = K e(t) + z(t) (1.4)
where z denotes the output of the integrator in thePI controller.
6
Their state space representation is
x(t) = Ax(t) +B1 u(t) + B2 d(t)
y(t) = C x(t) +D1 u(t) +D2 d(t)(1.5)
where for Σ, x := [ia ω]T, u := va, d := cr, y := ω and
A =
[−Ra/La −k1/La
k2/J −B/J
]
B1 =
[1/La
0
]B2 =
[0
−1/J
]C =
[0 1
]D1 = 0 D2 = 0
while for Σr, xr := z, ur := e, yr := va and
Ar = 0 Br = 1/T Cr = 1 Dr = K
Mathematical model of Σe:
d r
dt= 0
d cr
dt= 0 (1.6)
This corresponds to an autonomous system (withoutinput) having xe = y := [r cr]T and
Ae =
[0 00 0
]Ce =
[1 00 1
]
7
The overall system (controlled system and controller)can be represented with a unique mathematical modelof the same type:
˙x(t) = A x(t) + B1 u(t) + B2 d(t)
y(t) = C x(t) + D1 u(t) + D2 d(t)(1.7)
where for x := [ia ω z]T, u := r, d := cr y := ω and
A =
−Ra/La −(k1 +K)/La 1/La
k2/J −B/J 00 −1/T 0
B1 =
K/La
01
B2 =
0−1/J0
C =[0 1 0
]D1 = 0 D2 = 0
The regulator design problem is: determine T and Ksuch that the system (1.7) is internally stable, i.e. theeigenvalues of A have stricly negative real parts andthis property is maintained in presence of admissibleparameter variations.
8
If only its behavior with respect to step inputs mustbe considered, the overall system in Fig. 1.3 can berepresented as the autonomous system
˙x(t) = A x(t)
y(t) = C x(t)(1.8)
where for x := [ia ω z r cr]T, y := ω and
A =
−Ra/La −(k1 +K)/La 1/La K/La 0k2/J −B/J 0 0 −1/J0 −1/T 0 1 00 0 0 0 00 0 0 0 0
C =[0 1 0 0 0
]The regulator design problem is: determine T and Ksuch that the autonomous system (A, C) is externallystable, i.e., limt→∞ y(t) = 0 for any initial state andthis property is maintained in presence of admissibleparameter variations.
9
State Space Models
Continuous-time systems:
x(t) = Ax(t) +B u(t)
y(t) = C x(t) +Du(t)(1.9)
with the state x∈X =Rn, the input u∈U=Rp, theoutput y∈Y=Rq and A, B C, D real matrices of suit-able dimensions. The system will be referred to as thequadruple (A,B,C,D) or the triple (A,B,C) if D = 0.Most of the theory will be derived referring to triplessince extension to quadruples is straightforward.
Discrete-time systems:
x(k+1) = Ad x(k) + Bd u(k)
y(k) = Cd x(k) +Dd u(k)(1.10)
Recall that a continuous-time system is internallyasymptotically stable iff all the eigenvalues of A be-long to C− (the open left half plane of the complexplane) and a discrete-time system is internally asymp-totically stable iff all the eigenvalues of Ad belong toC� (the open unit disk of the complex plane).
In the discrete-time case a significant linear modelis also the FIR (Finite Impulse Response) system,defined by the finite convolution sum
y(k) =∑N
l=0 W(l)u(k − l) (1.11)
where W(k) (k=0, . . . , N) is a q× p real matrix, re-ferred to as the gain of the FIR system, while N iscalled the window of the FIR system.
10
Transfer Matrix Models
By taking the Laplace transform of (1.9) or the Ztransform of (1.10) we obtain the transfer matrix rep-resentations
Y (s) = G(s)U(s) with
G(s) = C (sI − A)−1B+D(1.12)
and
Y (z) = Gd(z)U(z) with
Gd(z) = Cd (zI −Ad)−1Bd +Dd
(1.13)
respectively.
The H2 norm in the continuous-time case is
‖G‖2 =
(1
2πtr
(∫ ∞
−∞G(jω)G∗(jω) dω
))1/2(1.14)
=
(tr
(∫ ∞
0g(t) gT(t) dt
))1/2(1.15)
where g(t) denotes the impulse response of the system(the inverse Laplace transform of G(s)), and in thediscrete-time case it is
‖Gd‖2 =
(1
2πtr
(∫ π
−π
Gd(ejω)G∗d(e
jω) dω
))1/2(1.16)
=
(tr
( ∞∑k=0
gd(k) gTd (k) dt
))1/2(1.17)
where Gd(ejω) denotes the frequency response of thediscrete-time system for unit sampling time and gd(k)the impulse response of the system (the inverse Ztransform of Gd(z)).
11
Geometric Approach (GA)
Geometric Approach: is a control theory for multivari-able linear systems based on:
• linear transformations
• subspaces
(The alternative approach is the transfer function ap-proach)
The geometric approach consists of
• an algebraic part (theoretical)
• an algorithmic part (computational)
Most of the mathematical support is developed incoordinate-free form, to take advantage of simplerand more elegant results, which facilitate insight intothe actual meaning of statements and procedures; thecomputational aspects are considered independentlyof the theory and handled by means of the standardmethods of matrix algebra, once a suitable coordinatesystem is defined.
12
A Few Words on the Algorithmic Part
A subspace X is given through a basis matrix of max-imum rank X such that X =imX.
The operations on subspaces are all performedthrough an orthonormalization process (subroutineima.m in Matlab) that computes an orthonormal ba-sis of a set of vectors in Rn by using methods of theGauss–Jordan or Gram–Schmidt type.
Basic Operations
• sum: Z = X + Y• linear transformation: Y = AX• orthogonal complementation: Y = X⊥• intersection: Z = X ∩ Y• inverse linear transformation: X = A−1Y
Computational support with Matlab
Q = ima(A,p) Orthonormalization.
Q = ortco(A) Complementary orthogonalization.
Q = sums(A,B) Sum of subspaces.
Q = ints(A,B) Intersection of subspaces.
Q = invt(A,X) Inverse transform of a subspace.
Q = ker(A) Kernel of a matrix.
In program ima the flag p allows for permutations ofthe input column vectors.
13
Basic relations
X ∩ (Y + Z) ⊇ (X ∩ Y) + (X ∩ Z)X + (Y ∩ Z) ⊆ (X + Y) ∩ (X + Z)
(X⊥)⊥ = X(X + Y)⊥ = X⊥ ∩ Y⊥(X ∩ Y)⊥ = X⊥+ Y⊥A (X ∩ Y) ⊆ AX ∩AYA (X + Y) = AX + AY
A−1 (X ∩ Y) = A−1X ∩A−1YA−1 (X + Y) ⊇ A−1X +A−1Y
Remarks:
1. The first two relations hold with the equality signif one of the involved subspaces X , Y, Z is con-tained in any of the others.
2. The following relations are useful for computa-tional purposes:
AX ⊆ Y ⇔ ATY⊥ ⊆ X⊥
(A−1Y)⊥ = ATY⊥
where AT denotes the transpose of matrix A.
14
Invariant Subspaces
Definition 2.1 Given a linear map A : X →X , a sub-space J ⊆X is an A-invariant if
AJ ⊆ J
Property 2.1 Given the subspaces D, E contained inX and such that D⊆E, and a linear map A : X → X ,the set of all the A-invariants J satisfying D⊆J ⊆Eis a nondistributive lattice Φ0 with respect to ⊆,+,∩.
We denote with maxJ (A, E) the maximal A-invariantcontained in E (the sum of all the A-invariantscontained in E) and with minJ (A,D) the mini-mal A-invariant containing D (the intersection ofall the A-invariants containing D): the above lat-tice is non-empty if and only if D⊆maxJ (A, E) orminJ (A,D)⊆E.
{E
maxJ (A, E)
minJ (A,D)
D
Φ0
Fig. 2.1. The lattice Φ0.
15
The Algorithms
Algorithm 2.1 Computation of minJ (A,B)
Z1 = BZi = B+ AZi−1 (i= 2,3, . . .)
minJ (A,B) = B+ AminJ (A,B)(2.1)
Algorithm 2.2 Computation of maxJ (A, C)
Z1 = CZi = C ∩A−1Zi−1 (i = 2,3, . . .)
maxJ (A, C) = C ∩A−1maxJ (A, C)(2.2)
Property 2.2 Dualities
maxJ (A, C) = minJ (AT, C⊥)⊥
minJ (A,B) = maxJ (AT,B⊥)⊥
Computational support with Matlab
Q = mininv(A,B) Minimal A-invariant containingimB
Q = maxinv(A,C) Maximal A-invariant containedin imC
16
Internal and External Stability of an Invariant
The restriction of map A to the A-invariant subspaceJ is denoted by A|J ; J is said to be internally stable ifA|J is stable. Given two A-invariants J1 and J2 suchthat J1⊆J2, the map induced by A on the quotientspace J2/J1 is denoted by A|J2/J1. In particular, anA-invariant J is said to be externally stable if A|X/Jis stable.
Algorithm 2.3 Matrices P and Q representing A|Jand A|X/J up to an isomorphism, are derived as fol-lows. Let us consider the similarity transformationT := [J T2], with imJ=J (J is a basis matrix of J )and T2 such that T is nonsingular. In the new basisthe linear transformation A is expressed by
A′ = T−1AT =
[A′11 A′12O A′22
](2.3)
The requested matrices are defined as P :=A′11,Q :=A′22.
Complementability of an Invariant
An A-invariant J ⊆ X is said to be complementableif an A-invariant Jc exixts such that J ⊕Jc=X ; if so,Jc is called a complement of J .
Algorithm 2.4 Let us consider again the change ofbasis introduced in Algorithm 2.3. J is comple-mentable if and only if the Sylvester equation
A′11X −X A′22 = −A′12 (2.4)
admits a solution. If so, a basis matrix of Jc is givenby Jc := J X+T2.
17
Refer to the autonomous system
x(t) = Ax(t) x(0) = x0 (2.5)
or
x(k+ 1) = Ad x(k) x(0) = x0 (2.6)
The behavior of the trajectories in the state space withrespect to an invariant can be represented as follows.
x(0)
x(0)
J
Fig. 2.2. External and internal stability of aninvariant.
Computational support with Matlab
[P,Q] = stabi(A,X) Matrices for the internaland external stability ofthe A-invariant imX
18
Controllability and Observability
Consider a triple (A,B,C), i.e., refer to
x(t) = Ax(t) +B u(t)
y(t) = C x(t)(2.7)
Let B := imB. The reachability subspace of (A,B),i.e., the set of all the states that can be reachedfrom the origin in any finite time by means of controlactions, is R=minJ (A,B). If R=X , the pair (A,B)is said to be completely controllable.
Let C :=kerC. The unobservability subspace of (A,C),i.e., the set of all the initial states that cannot be rec-ognized from the output function, is Q=maxJ (A, C).If Q= {0}, (A,C) is said to be completely observable.
R
Fig. 2.3. The reachability subspace.
19
If R !=X , but R is externally stabilizable, (A,B) is saidto be stabilizable.
If Q != {0}, but Q is internally stabilizable, (A,C) issaid to be detectable.
Pole Assignment
+
+
v u y
x
u y
Σ Σ
F G
Fig. 2.4. State feedback and output injection
State feedback
x(t) = (A+BF ) x(t) + B v(t)
y(t) = C x(t)(2.8)
Output injection
x(t) = (A+GC)x(t) +B u(t)
y(t) = C x(t)(2.9)
The eigenvalues of A+BF are arbitrarily assignableby a suitable choiche of F iff the system is com-pletely controllable and those of A+GC are arbitrarilyassignable by a suitable choice of G iff the system iscompletely observable.
20
Complete Pole Assignment through an Observer
+
+
v u y u y
Σ Σ
F −G
Σc Σoxx
Fig. 2.5. Dynamic pre-compensator and observer
+
+
v u y
Σ
F
−G
Σox
Fig. 2.6. Pole assignment through an observer
The eigenvalues of the overall system are the unionof those of A+BF and those of A+GC, hence com-pletely assignable if the triple (A,B,C) is completelycontrollable and observable.
21
Controlled and Conditioned Invariants
Definition 2.2 Given a linear map A : X →X anda subspace B⊆X a subspace V ⊆X is an (A,B)-controlled invariant if
AV ⊆ V + B (2.10)
Let B and V be basis matrices of B and V respectively:the following statements are equivalent to (2.10):
- a matrix F exists such that (A+BF )V ⊆ V- matrices X and U exist such that AV = V X + BU
- V is a locus of trajectories of the pair (A,B)
V
Fig. 2.7. The controlled invariant as a locus oftrajectories.
22
The sum of any two controlled invariants is a con-trolled invariant, while the intersection is not; thusthe set of all the controlled invariants contained in agiven subspace E ⊆X is a semilattice with respect to⊆,+, hence admits a supremum, the maximal (A,B)-controlled invariant contained in E, that is denoted bymaxV(A,B, E) (or simply V∗(B,E)). We use the symbol V∗for maxV(A, imB, kerC), which is the most importantcontrolled invariant concerning the triple (A,B,C).
Referring to the pair (A,B), we denote with RV thereachable subspace from the origin by trajectories con-strained to belong to a generic (A,B)-controlled invari-ant V. Owing to the first property above, it is derivedas RV = minJ (A+BF,V ∩B) and, clearly being an(A+BF )-invariant, it also is an (A,B)-controlled in-variant.
A generic (A,B)-controlled invariant V is said to be in-ternally stabilizable or externally stabilizable if at leastone matrix F exists such that (A+BF )|V is stable orat least one matrix F exists such that (A+BF )|X/Vis stable. It is easily proven that the eigenstructureof (A+BF )|V/RV is independent of F ; it is called theinternal unassignable eigenstructure of V. V is bothinternally and externally stabilizable with the same Fif and only if its internal unassignable eigenstructureis stable and the A-invariant V +R = V+minJ (A,B)is externally stable. This latter is ensured by the sta-bilizability property of the pair(A,B).
23
Definition 2.3 Given a linear map A : X →X anda subspace C ⊆X a subspace S ⊆X is an (A, C)-conditioned invariant if
A (S ∩ C) ⊆ S (2.11)
Let C be a matrix such that C=kerC. The followingstatement is equivalent to (2.11):
- a matrix G exists such that (A+GC)S ⊆ S
The intersection of any two conditioned invariants isa conditioned invariant while the sum is not; thusthe set of all the conditioned invariants containinga given subspace D⊆X is a semilattice with respectto ⊆,∩, hence admits an infimum, the minimal (A, C)-conditioned invariant containing D, that is denotedby minS(A, C,D) (or simply S∗(C,D)). We use the simple
symbol S∗ for minS(A,kerC, imB), which is the mostimportant conditioned invariant concerning the triple(A,B,C).
Controlled and conditioned invariants are dual to eachother. Controlled invariants are used in control prob-lems, while conditioned invariants are used in obser-vation problems.
The orthogonal complement of an (A, C)-conditionedinvariant is an (AT, C⊥)-controlled invariant, hence theorthogonal complement of an (A, C)-conditioned in-variant containing a given subspace D is an (AT, C⊥)-controlled invariant contained in D⊥. External and in-ternal stabilizability of conditioned invariants are easilydefined by duality.
24
Self-bounded Controlled Invariants
Definition 2.4 Given a linear map A : X →X and twosubspaces B ⊆ X , E ⊆ X , a subspace V ⊆X is an(A,B)-controlled invariant self-bounded with respectto E if, besides (2.10), the following relations hold
V ⊆ V∗(B,E) (2.12)
V∗(B,E) ∩ B ⊆ V (2.13)
The set of all the (A,B)-controlled invariants self-bounded with respect to E is a nondistributive latticewith respect to ⊆,+,∩, whose supremum is V∗(B,E) andwhose infimum is RV∗
(B,E).
Given subspaces D, E contained in X and suchthat D⊆V∗, the infimum of the lattice of allthe (A,B)-controlled invariants self-bounded withrespect to E and containing D is the reach-able set on V∗ with forcing action B+D, i.e.,Vm :=minJ (A+BF,V∗(B,E)∩ (B+D)), with F such
that (A+BF )V∗(B,E)⊆V∗(B,E).
25
The infimum of the lattice of all the (A,B)-controlledinvariants self-bounded with respect to a given sub-space E can be expressed in terms of conditioned in-variants as follows.
Property 2.3 Let D⊆V∗(B,E). The infimum of the
lattice Φ of all the (A,B)-controlled invariants self-bounded with respect to E and containing D is ex-pressed by
Vm = V∗(B,E) ∩ S∗(E ,B+D) (2.14)
Note, in particular, that RV∗(B,E)
= V∗(B,E) ∩S∗(E ,B). The
dual of Property 2.3 is
Property 2.4 Let S∗(C,D)⊆E. The infimum of the lat-
tice Ψ of all the (A, C)-conditioned invariants self-hidden with respect to D and contained in E is ex-pressed by
SM = S∗(C,D) + V∗(D,C∩E) (2.15)
26
E
V∗(B,E)
Vm
D
E
SM
S∗(C,D)
D
Φ Ψ
Fig. 2.8. The lattices Φ and Ψ.
The main theorem and its dual
Theorem 2.1 Let D⊆V∗(B,E). There exists at least
one internally stabilizable (A,B)-controlled invariantV such that D⊆V ⊆E if and only if Vm is internallystabilizable.
Theorem 2.2 Let S∗(C,D)⊆E. There exists at least
one externally stabilizable (A, C)-conditioned invariantS such that D⊆S ⊆E if and only if SM is internallystabilizable.
27
The Algorithms
Algorithm 2.5 Computation of S∗=minS(A, C,B)
S1 = BSi = B+A (Si−1 ∩ C) (i = 2,3, . . .)
S∗ = B+A (S∗ ∩ C)(2.16)
Algorithm 2.6 Computation of V∗=maxV(A,B, C)
V1 = CVi = C ∩A−1 (Vi−1 + B) (i = 2,3, . . .)
V∗ = C ∩A−1(V∗+ B)(2.17)
Property 2.5 Dualities
maxV(A,B, C) = minS(AT,B⊥, C⊥)⊥
minS(A, C,B) = maxV(AT, C⊥,B⊥)⊥
Remark: Refer to the discrete-time triple (Ad,Bd, Cd),i.e., to equations (1.10) with Dd=0. Algorithm 2.5with A=Ad, B=imBd and C=kerCd at the generici-th step provides the set of all states reachable fromthe origin with trajectories having all the states butthe last one belonging to kerCd, hence invisible atthe output. Thus S∗ has a control meaning in thediscrete-time dynamics: it is the maximum subspaceof the state space reachable from the origin with thistype of trajectories in ρ steps, being ρ the number ofiterations required for (2.16) to converge to S∗.
28
Algorithm 2.7 Computation of matrix F such that(A+BF )V ⊆ V. Let V be a basis matrix of the (A,B)-controlled invariant V. First, compute[
XU
]= [V B]+AV
where the symbol + denotes the pseudoinverse. Then,compute
F := −U (V T V )−1 V T
Algorithm 2.8 Computation of the internal unas-signable eigenstructure of an (A,B)-controlled invari-ant. A matrix P representing the map (A+BF )|V/RV
up to an isomorphism, is derived as follows. Let usconsider the similarity transformation T := [T1 T2 T3],with imT1=RV, imT2=V and T3 such that T is non-singular. In the new basis matrix A+BF is expressedby
(A+BF )′ = T−1(A+BF )T =
A′11 A′12 A′13
O A′22 A′23O O A′33
The requested matrix is P := A′22.
29
Computational support with Matlab
Q = mainco(A,B,X) Maximal (A, imB)-controlledinvariant contained in imX
Q = miinco(A,C,X) Minimal (A, imC)-conditionedinvariant containing imX
F = effe(A,B,X) State feedback matrix such that(A+BF ) imX ⊆ imX
[P,Q] = stabv(A,B,X) Matrices for the internal andexternal stability of the (A, imB)-controlledinvariant imX
F = effest(A,B,X,ei,ee) Stabilizing feedback matrixsetting the assignable eigenvalues as ei andthe assignable external eigenvalues as ee
30
The Geometric Characterization
of Some Properties of Linear Systems
Consider the standard continuous-time system – triple(A,B,C)
x(t) = Ax(t) +B u(t)
y(t) = C x(t)(3.1)
or the standard discrete-time system – triple(Ad,Bd, Cd)
x(k+1) = Ad x(k) +Bd u(k)
y(k) = Cd x(k)(3.2)
(we consider triples since they provide a better insightand extension to quadruples is straightforward – ob-tainable with a suitable state extension)
Systems (3.1) and (3.2) with x(0)=0 define linearmaps Tf : Uf → Yf from the space Uf of the admissi-ble input functions to the functional space Yf of thezero-state responses. These maps are defined by theconvolution integral and the convolution summation
y(t) = C
∫ t
0eA (t−τ)B u(τ) dτ (3.3)
y(k) = Cd
k−1∑h=0
A(k−h−1)d Bd u(h) (3.4)
The admissible input functions are:- piecewise continuous and bounded functions of timet for (3.3);- bounded functions of the discrete time k for (3.4).
31
Left and Right Invertibility
Definition 3.1 Assume that B has maximal rank.System (3.1) is said to be invertible (left-invertible)if, given any output function y(t), t∈ [0, t1] t1>0 be-longing to imTf , there exists a unique input functionu(t), t∈ [0, t1), such that (3.3) holds.
Definition 3.2 Assume that Bd has maximal rank.System (3.2) is said to be invertible (left-invertible)if, given any output function y(k), k∈ [0, k1], k1≥nbelonging to imTf there exists a unique input functionu(k), k∈ [0, k1−1] such that (3.4) holds.
Definition 3.3 Assume that C has maximal rank.System (3.1) is said to be functionally controllable(right-invertible) if there exists an integer ρ≥1 suchthat, given any output function y(t), t∈ [0, t1], t1>0with ρ-th derivative piecewise continuous and suchthat y(0)=0, . . . y(ρ)(0)=0, there exists at least oneinput function u(t), t∈ [0, t1) such that (3.3) holds.The minimum value of ρ satisfying the above state-ment is called the relative degree of the system.
Definition 3.4 Assume that Cd has maximal rank.System (3.2) is said to be functionally controllable(right-invertible) if there exists an integer ρ≥1 suchthat, given an output function y(k), k∈ [0, k1], k1≥ ρsuch that y(k)=0, k∈ [0, ρ−1], there exists at leastone input function u(k), k∈ [0, k1−1] such that (3.4)holds. The minimum value of ρ satisfying the abovestatement is called the relative degree of the system.
32
Let V∗ :=maxV(A, imB, kerC) and S∗ :=minV(A,kerC, imB)
Theorem 3.1 System (3.1) or (3.2) is invertible ifand only if
V∗ ∩ S∗ = {0} (3.5)
Theorem 3.2 System (3.1) or (3.2) is functionallycontrollable if and only if
V∗+ S∗ = X (3.6)
Note the duality: if system (A,B,C) or (Ad,Bd, Cd) isinvertible (functionally controllable), the adjont sys-tem (AT, CT, BT) or (AT
d , CTd , BT
d ) is functionally con-trollable (invertible).
Relative Degree
Property 3.1 Assume that (3.6) holds and considerthe conditioned invariant computational sequence Si(i=1,2, . . .). The relative degree is the least integerρ such that
V∗+ Sρ = X
Computational support with Matlab
r = reldeg(A,B,C,[D]) Relative degree of (A,B,C)or (A,B,C,D)
33
+
_
+
_
Σi Σ
Σf
e
Σ Σi
Σf
e
Fig. 3.1. Connections for right and left inversion
In Fig. 3.1 Σf denotes a suitable relative-degree filterin the continuous-time case or a relative degree delayin the discrete-time case. The inverse system Σi is tobe designed to null the error e.
If the system is nonminimum phase, i.e, has some un-stable zeros, the inverse system is internally unstable,so that the time interval considered for the systeminversion must be finite.
34
Invariant Zeros
Roughly speaking, an invariant zero corresponds toa mode that, if suitably injected at the input of adynamic system, can be nulled at the output by asuitable choice of the initial state.
Definition 3.5 The invariant zeros of (A,B,C) arethe internal unassignable eigenvalues of V∗. Theinvariant zero structure of (A,B,C) is the internalunassignable eigenstructure of V∗.
Recall that RV∗=V∗ ∩S∗. The invariant zeros are theeigenvalues of the map (A+BF )|V∗/RV∗, where F de-notes any matrix such that (A+BF )V∗ ⊆ V∗.
V∗
RV∗
unstablezero
stablezero
Fig. 3.2. Decomposition of the map (A+BF )|V∗
35
Property 3.2 Let W be a real m×m matrix havingthe invariant zero structure of (A,B,C) as eigenstruc-ture. A real p×m matrix L and a real n×m matrix Xexist, with (W,X) observable, such that by applyingto (A,B,C) the input function
u(t) = LeWt v0 (3.7)
where v0∈Rm denotes an arbitrary column vector, andstarting from the initial state x0=X v0, the output y(·)is identically zero, while the state evolution (on kerC)is described by
x(t) = X eWt v0 (3.8)
v0 x0 = X v0
Σe L Σv u y
Fig. 3.3. The meaning of Property 3.2
Remark. In the discrete-time case equations(3.7) and (3.8) are replaced by u(k)=LWk v0 andx(k)=XWk v0, respectively.
Computational support with Matlab
z = gazero(A,B,C,[D]) Invariant zeros of (A,B,C)or (A,B,C,D)
36
Extension to Quadruples
Extension to quadruples of the above definitions andproperties can be obtained through a simple con-trivance.
vΣd
uΣ
y
integratorsor delays
uΣ
yΣd
z
integratorsor delays
Fig. 3.4. Artifices to reduce a quadruple to a triple
Refer to the first figure: system Σd is modeled by
u(t) = v(t)
and the overall system by
˙x(t) = A x(t) + B v(t)y(t) = C x(t)
with
x :=
[xu
]A :=
[A B0 0
]
B :=
[0Ip
]C :=
[C D
]
37
The addition of integrators at inputs or outputs doesnot affect the system right and left invertibility, whilethe relative degree of (A, B, C) must be simply reducedby 1 to be referred to (A,B,C,D)
In the discrete-time case Σd is described by
u(k+1) = v(k)
and the overall system by
x(k+1) = Ad x(k) + Bb v(k)y(k) = Cd y(t)
with the extended matrices Ad, Bd, Cd defined like inthe continuous-time case in terms of Ad, Bd, Cd, Dd.
This contrivance can also be used in most of thesynthesis problems considered in the sequel.
38
Disturbance Decoupling
The disturbance decoupling problem is one of the ear-liest (1969) applications of the geometric approach.
u
de
x
Σ
F
Fig. 3.5. Disturbance decouplingwith state feedback
Let us consider the system
x(t) = Ax(t) +B u(t) +Dd(t)e(t) = E x(t)
(3.9)
where u denotes the manipulable input, d the distur-bance input. Let B := imB, D := imD, E :=kerE.
The disturbance decoupling problem is: determine, ifpossible, a state feedback matrix F such that distur-bance d has no influence on output e.
The system with state feedback is described by
x(t) = (A+B F )x(t) +Dd(t)e(t) = E x(t)
(3.10)
It behaves as requested if and only if its reachable setby d, i.e., the minimum (A+BF )-invariant containingD, is contained in E.
39
Let V∗(B,E) :=maxV(A,B, E). Since any (A+BF )-
invariant is an (A,B)-controlled invariant, the inacces-sible disturbance decoupling problem has a solution ifand only if
D ⊆ V∗(B,E) (3.11)
Equation (3.11) is a structural condition and does notensure internal stability. If stability is requested, wehave the disturbance decoupling problem with stabil-ity. Stability is easily handled by using self-boundedcontrolled invariants. Assume that (A,B) is stabiliz-able (i.e., that R=minJ (A,B) is externally stable)and let
Vm := V∗(B,E) ∩ S∗(E ,B+D) (3.12)
This subspace has already been defined in Property2.3. The following result, providing both the struc-tural and the stability condition, is a direct conse-quence of Theorem 2.1.
Corollary 3.1 The disturbance decoupling problemwith stability admits a solution if and only if
D ⊆ V∗(B,E)Vm is internally stabilizable
(3.13)
If conditions (3.13) are satisfied, a solution is providedby a state feedback matrix such that (A+BF )Vm ⊆Vm and σ(A+BF ) is stable.
If the state is not accessible, disturbance decouplingmay be achieved through a dynamic unit similar to astate observer. This is called disturbance decouplingproblem with dynamic measurement feedback, andwill be considered later.
40
Feedforward
Decoupling of Measurable Signals
Consider now the system
x(t) = Ax(t) +B u(t) +H h(t)e(t) = E x(t)
(3.14)
The triple (A,B,E) is assumed to be stable. This issimilar to (3.9), but with a different symbol for thenon-manipulable input, to denote that it is accessiblefor measurement. Let H := imH. Signals d1 and rp inthe general block diagram in Fig. 1.1 are of this type.
h
u Σ
Σc
e
Fig. 3.6. Measurable signal decoupling
The measurable signal decoupling problem is: deter-mine, if possible, a feedforward compensator Σc suchthat the input h has no inflence on the output e. Con-ditions for this problem to be solvable with stabilityare similar to those of disturbance decoupling prob-lem, but state feedback is not required (a feedforwardsolution with a pre-compensator of the type shown inFig. 2.5 is possible). Define
Vm := V∗(B,E) ∩ S∗(E ,B+H) (3.15)
41
The solvability conditions once again are consequenceof Theorem 2.1.
Corollary 3.2 The measurable signal decouplingproblem with stability admits a solution if and onlyif
H ⊆ V∗(B,E) + BVm is internally stabilizable
(3.16)
The feedforward unit Σc has state dimension equalto the dimension of Vm and includes a state feedbackmatrix F such that (A+BF )|Vm
is stable. It is notnecessary to reproduce (A+BF )|X/Vm
in Σc since itis not influenced by input h. The assumption thatΣ is stable is not restrictive. It can be relaxed to Σbeing stabilizable and detectable, so that the stabi-lizing feedback connection shown in Fig. 2.5 can beused. This does not influence conditions (3.16) sinceinput v in Fig 2.5 clearly overrides the feedback signalthrough F .
Note that internal stabilizability of Vm is ensured if theplant is minimum phase (with all the invariant zerosstable), since the internal unassignable eigenvalues ofVm are a part of those of V∗(B,E), that are invariant zerosof the plant.
It is possible to include feedthrough terms in (3.14)by using the extensions to quadruples previously de-scribed. In this case addition of a dynamic unit withrelative degree one at the output achieves our aim.
42
The Dual Problem: Unknown-Input Observation
Consider the system
x(t) = Ax(t) +Dd(t)y(t) = C x(t)e(t) = E x(t)
(3.17)
Triple (A,D,C) is assumed to be stable. Output edenotes a linear function of the state to be estimated(possible the whole state).
++
de
y
−e
Σ
Σo
ε
Fig. 3.7. Unknown-input observation
The unknown-input observation problem is: deter-mine, if possible, an observer Σo such that the inputu has no inflence on the output ε. Conditions for thisproblem to be solvable with stability are dual to thoseof the measurable signal decoupling problem. Theproblem can be solved by duality. Define
SM = S∗(C,D) + V∗(D,C ∩E) (3.18)
like in (2.15). The solvability conditions are conse-quence of Theorem 2.2.
Corollary 3.3 The unknown-input observation prob-lem with stability admits a solution if and only if
S∗(C,D) ∩ C ⊆ ESM is externally stabilizable
(3.19)
43
Decoupling of Previewed Signals (Discrete-Time)
The role of controlled and conditioned invariants isvery clearly pointed out by the previewed signal de-coupling problem in the discrete-time case. Consideragain signal decoupling, but suppose that there issome preview (knowledge in advance) of the signalh to be decoupled. To take into account preview,replace the block diagram in Fig. 3.6 with that inFig. 3.8.
hpdelay
hu Σ
Σc
e
Fig. 3.8. Previewed signal decoupling
a) relative-degree preview
If a relative-degree preview is available, the structuralcondition in Corollary 3.2 is relaxed as follows.
Corollary 3.4 The relative-degree previewed signaldecoupling problem with stability admits a solutionif and only if
H ⊆ V∗(B,E) + S∗(E ,B)
Vm is internally stabilizable(3.20)
where Vm is defined again by (3.15).
Note that the first condition in (3.20) is satisfied ifΣ is right invertible and the second is satisfied if it isminimum-phase.
44
a) large preview
A large preview time enables to overcome the stabilitycondition, thus making it possible to obtain signal de-coupling also in the nonminimum-phase case. “Large”means significantly greater than the time constant ofthe unstable zero closest to the unit circle.
Property 3.3 The “largely” previewed signal decou-pling problem with stability admits a solution if andonly if
H ⊆ V∗(B,E) + S∗(E ,B) (3.21)
Suppose that an impulse is scheduled at input h attime ρ. It can be decoupled with an input signal u ofthe type shown in the following figure with preactionconcerning unstable zeros and postaction stable zeros.
−ka 0 ρ
preaction
dead-beat
postaction
Fig. 3.9. Input sequence for decouplingan impulse at time ρ.
Localization of a previewed generic signal h(·) isachievable through a FIR system having such typeof functions as gain.
45
Two different strategies are outlined according towhether condition 2 in Corollary 3.4 is satisfied ornot. The basic idea is synthesized as follows.
Denote by ρ the least integer such that H⊆V∗(B,E) +Sρ.Let us recall that Vm is a locus of initial states in Ecorresponding to trajectories controllable indefinitelyin E, while (Sρ) is the maximum set of states that canbe reached from the origin in ρ steps with all the statesin E except the last one. Suppose that an impulse isapplied at input h at the time instant ρ, producing aninitial state xh∈H, decomposable as xh=xh,s+ xh,v,with xh,s ∈Sρ and xh,v ∈Vm. Let us apply the controlsequence that drives the state from the origin to −xh,s
along a trajectory in Sρ, thus nulling the first compo-nent. The second component can be maintained onVm by a suitable control action in the time intervalρ≤ k<∞ while avoiding divergence of the state if allthe internal unassignale modes of Vm are stable orstabilizable. If not, it can be further decomposed asxh,v= x′h,v+x′′h,v, with x′h,v belonging to the subspaceof the stable or stabilizable internal modes of Vm andx′′h,v to that of the unstable modes. The former com-ponent can be maintained on Vm as before, while thelatter can be nulled by reaching −x′′h,v with a controlaction in the time interval −∞<k≤ ρ−1 correspond-ing to a trajectory in Vm from the origin.
46
Unknown-input Delayed Observation
++
de
y
ed
−ed
delay
Σo
Σ ε
Fig. 3.10. Unknown-input delayed observation
The dual problem is the unknown-input observationof a linear function of the state with relative degreedelay if Σ is minimum phase or “large” delay if not.The duals of Corollary 3.4 and Property 3.3 are statedas follows.
Corollary 3.5 The unknown-input observation prob-lem of a linear function of the state with relative de-gree delay and stability admits a solution if and onlyif
V∗(D,C) ∩ S∗(C,D) ⊆ E
SM is externally stabilizable(3.22)
where SM is defined again by (3.18).
Note that the unknown-input observation of any linearfunction of the state (possibly the whole state) withrelative degree delay is achievable if Σ is left-invertibleand minimum phase.
Property 3.4 The unknown-input observation prob-lem of a linear function of the state with “large” delayand stability admits a solution if and only if
V∗(D,C) ∩ S∗(C,D) ⊆ E (3.23)
47
Feedforward Model Following
The feedforward model following problem reduces todecoupling of measured signals, as the following figureshows.
+
_h
u y
ym
eΣΣc
Σm
ΣFig. 3.11. Feedforward model following
Assume that system Σ is described by the triple(A,B,C) and model Σm by the triple (Am,Bm,Cm).The overall sistem Σ is described by
A :=
[A 00 Am
]B :=
[B0
]
H :=
[0Bm
]E :=
[C −Cm
] (3.24)
Both system and model are assumed to be stable,square, left and right invertible. The structural con-dition expressed by the former of (3.16) is satisfied ifand only if the relative degree of Σm is at least equalto that of Σ.
48
It can be shown that the internal eigenvalues of Vm
are the union of the invariant zeros of Σ and theeigenvalues of Am, so that in general model followingwith stability is not achievable if Σ is nonminimum-phase. If, on the other hand, the model Σm consistsof q independent single-input single-output systems allhaving as zeros some invariant zeros of Σ, these arecanceled as internal eigenvalues of Vm. This makes itpossible to achieve both input-output decoupling andinternal stability, but restricts the model choice.
Note that the right inversion layout shown in Fig. 3.1is achievable with a model consisting of q independentrelative-degree filters in the continuous-time case or qindependent relative-degree delays in the discrete-timecase.
The dual problem of model following is model follow-ing by output feedforward correction, that reduces tothe left inversion layout shown in Fig. 3.1 if a modelconsisting of p independent relative-degree filters inthe continuous-time case or p independent relative-degree delays in the discrete-time case is adopted.
49
Feedback
Disturbance Decoupling by Dynamic Output Feedback
d
Σu y
Σc
e
Fig. 4.1. Disturbance decouplingby dynamic output feedback
Model of Σ:
x(t) = Ax(t) +B u(t) +Dd(t)
y(t) = C x(t)
e(t) = E x(t)
(4.1)
The inputs u and d are the manipulable input and thedisturbance input, respectively, while outputs y and eare the measured output and the controlled output,respectively.
Model of Σc:
z(t) = N z(t) +M y(t)
u(t) = Lz(t) +K y(t)(4.2)
The disturbance decoupling problem by dynamic out-put feedback is stated as follows: determine, if pos-sible, a dynamic compensator (N,M,L,K) such thatthe disturbance d has no influence on the regulatedoutput e and the overall system is internally stable.
50
It has been shown that output dynamic feedback ofthe type shown in Fig. 4.1 enables stabilization ofthe overall system provided that (A,B) is stabilizableand (A,C) detectable. Since overall system stabilityis required, these conditions on (A,B) and (A,C) arestill necessary.
The overall system is described by
˙x(t) = A x(t) + D d(t)
e = E x(t)(4.3)
with
x :=
[xz
]A :=
[A+BKC BL
MC N
]
D :=
[D0
]E :=
[E 0
] (4.4)
i.e., it can de described by a unique triple (A, B, C).
dΣ
e
Fig. 4.2. The overall system
Output e is decoupled from input d if and only ifminJ (A, imD) (the reachable subpace of the pair(A, D)) is contained in kerE or, equivalently, imD iscontained in maxJ (A,kerE). Furthermore, in orderthe stability requirement to be satisfied, A must bea stable matrix or minJ (A, imD) and maxJ (A,kerE)must be both internally and externally stable.
51
Stated in very simple terms, disturbance decouplingis achieved if and only if the overall system (A, D, E)exibits at least one A-invariant W such that
D ⊆ W ⊆ EW is internally and externally stable
(4.5)
Necessary and sufficient conditions for solvability ofour problem are stated in the following theorem.
Theorem 4.1 The dynamic measurement feedbackdisturbance decoupling problem with stability admitsat least one solution if and only if there exist an(A,B)-controlled invariant V and an (A, C)-conditionedinvariant S such that:
D ⊆ S ⊆ V ⊆ ES is externally stabilizableV is internally stabilizable
(4.6)
A short outline of the “only if” part of the proof.Define the following operations on subspaces of theextended state space x:
projection:
P (W) =
{x :
[xz
]∈ W
}(4.7)
intersection:
I(W) =
{x :
[x0
]∈ W
}(4.8)
52
Clearly, I(W)⊆P (W), D = I(D) = P (D), E = P (E) =I(E). The “only if” part of the proof of Theorem 4.1follows from (4.5) and the following lemmas.
Lemma 4.1 Subspace W is an internally and/or ex-ternally stable A-invariant only if P (W) is an internallyand/or externally stabilizable (A,B)-controlled invari-ant.
Lemma 4.2 Subspace W is an internally and/or ex-ternally stable A-invariant only if I(W) is an inter-nally and/or externally stabilizable (A, C)-conditionedinvariant.
The “if” part of the proof is constructive, i.e., if aresolvent pair (S,V) is given, directly provides a com-pensator (N,M,L,K) satisfying all the requirementsin the statement of the problem. This consists of aspecial type of state observer fed by the measuredoutput y plus a special feedback connection from theobserver state to the manipulable input u.
53
A more constructive set of necessary and sufficientconditions, based on the dual lattice structures af self-bounded controlled invariants and their duals, provid-ing a convenient set of resolvent pair, is stated in thefollowing theorem.
Theorem 4.2 Consider the subspaces Vm and SM de-fined in (2.14) and (2.15). The dynamic measurementfeedback disturbance decoupling problem with stabil-ity admits at least one solution if and only if
S∗(C,D) ⊆ V∗(B,E)
SM is externally stabilizableVM :=Vm + SM is internally stabilizable
(4.9)
If Theorem 4.2 holds, (SM,VM) is a convenient resol-vent pair. Similarly, define Sm :=Vm ∩SM . It can easilybe proven that (Sm,Vm) is also a convenient resolventpair.
Note that conditions (4.9) consist of a structural con-dition ensuring feasibility of disturbance decouplingwithout internal stability and two stabilizability condi-tions ensuring internal stability of the overall system.
54
The layout of the possible resolvent pairs in the duallattice structure is shown in the following figure, thatalso points out the correspondences between any self-bounded controlled invariant belonging to the firstlattice and an element of the second and viceversa.This enables to derive other resolvent pairs satisfyingTheorem 4.1.
V∗(B,E)
VM
Vm
SM
Sm
S∗(C,D)
+Vm
∩SM
Fig. 4.3. The resolvents with minimum fixed poles
55
The Autonomous Regulator Problem
Consider the block diagram shown in the followingfigure.
+r e u y
_Σ
rΣ
Σe2
p
d
Σe1
Fig. 4.4. The closed-loop control scheme.
The regulator Σr achieves:
(i) closed-loop asymptotic stability or, more generally,pole assignability;
(ii) asymptotic (robust) tracking of reference r andasymptotic (robust) rejection of disturbance d.
Both the reference and disturbance inputs are steps,ramps, sinusoids, that can be generated by the exosys-tems Σe1 and Σe2. The eigenvalues of the exosystemsare assumed to belong to the closed rigth half-placeof the complex plane.
The overall system considered, included the exosys-tems, is described by a linear homogeneous set ofdifferential equations, whose initial state is the onlyvariable affecting evolution in time.
56
The plant and the exosystems are modelled as aunique regulated system which is not completely con-trollable or stabilizable (the exosystem is not control-lable). The corresponding equations are
x(t) = Ax(t) +B u(t)
e(t) = E x(t)(4.10)
with
x :=
[x1x2
]A :=
[A1 A3
0 A2
]
B :=
[B1
0
]E :=
[E1 E2
]In (4.10) the plant corresponds to the triple(A1, B1, E1). Note that the exosystem state x2 in-fluences both the plant through matrix A3 and theerror e through matrix E2. (A1, B1) is assumed to bestabilizable and (A,E) detectable.
The regulator is modelled like in the disturbance de-couplig problem by measurement feedback, i.e.
z(t) = N z(t) +M e(t)
u(t) = Lz(t) +K e(t)(4.11)
57
exosystem
plant
regulator
u eregulated system
x 2
Σ
regulator
u e
Σ
Σr
Σe
Σp
Σr
regulated system
a) b)
Fig. 4.5. Regulated system and regulator connection
The overall system is referred to as the autonomousextended system
˙x(t) = A x(t)
e(t) = E x(t)(4.12)
with
x :=
x1
x2z
A :=
A1 +B1KE1 A3 + B1KE2 B1L
O A2 OME1 ME2 N
E :=[E1 E2 O
]58
Let x1∈Rn1, x2∈Rn2, z ∈Rm. If the internal modelprinciple is used to design the regulator, the au-tonomous extended system is characterized by an un-observability subspace containing these modes, thatare all not strictly stable by assumption. In geometricterms, an A-invariant W ⊆ kerE having dimension n2exists, that is internally not strictly stable.
Since the eigenvalues of A are clearly those of A2 plusthose of the regulation loop, that are strictly stable,W is externally strictly stable. Hence A|W has theeigenstructure of A2 (n2 eigenvalues) and AX/W that
of the control loop (n1+n2 eigenvalues).
The existence of this A-invariant W ⊆kerE is pre-served under parameter changes.
The autonumous regulator problem is stated as fol-lows: derive, if possible, a regulator (N,M,L,K) suchthat the closed-loop system with the exosystem dis-connected is stable and limt→∞ e(t)=0 for all the ini-tial states of the autonomous extended system.
In geometric terms it is stated as follows: refer to theextended system (A, E) and let E :=kerE. Given themathematical model of the plant and the exosystem,determine, if possible, a regulator (N,M,L,K) suchthat an A-invariant W exists satisfying
W ⊆ Eσ(A|X/L) ⊆ C−
(4.13)
59
In the extended state space X with dimensionn1+n2+m, define the A-invariant extended plant Pas
P := { x : x2 = 0} = im
In1 O
O OO Im
(4.14)
By a dimensionality argument, the A invariant W,besides (4.13), must satisfy
W ⊕ P = X (4.15)
The main theorem on asymptotic regulation simplytranslates the extended state space conditions (4.13)and (4.15) into the plant plus exosystem state spacewhere matrices A, B and E are defined. Define theA-invariant plant P through
P := { x : x2 = 0} = im
[In1
O
](4.16)
Theorem 4.3 Let E :=kerE. The autonomous regu-lator problem admits a solution if and only if an (A,B)-controlled invariant V exists such that
V ⊆ EV ⊕ P = X
(4.17)
The “only if” part of the proof derives from (4.13)and (4.15), while the “if” part provides a quadruple(N,M,L,K) that solves the problem.
60
Unfortunately the necessary and sufficient conditionsstated in Theorem 4.3 are nonconstructive. The fol-lowing theorem provides constructive sufficient and al-most necessary∗ conditions in terms of the invariantzeros of the plant.
Theorem 4.4 Let us define V∗ :=maxV(A,B, E).The autonomous regulator problem admits a solutionif
V∗+ P = XZ(A1, B1, E1) ∩ σ(A2) = ∅
(4.18)
Remark:We have again a structural condition and a stabilitycondition in terms of invariant zeros. However, thestability condition is very mild in this case since it isonly required that the plant has no invariant zerosequal to eigenvalues of the exosystem. Hence the au-tonomous regulator problem may be also solvable ifthe plant is nonminimum phase. In other words, min-imality of phase is only required for perfect tracking,non for asymptotic tracking.
Corollary 4.1 (Uniqueness of the resolvent) If theplant is invertible and conditions (4.18) are satisfied,a unique (A,B)-controlled invariant V satisfying con-ditions (4.17) exists.
∗The conditions become necessary if the boundednessof the control variable u is required. This is possiblealso when the output y is unbounded if a part of theinternal model is contained in the plant.
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Proof of Theorem 4.4:
Let F be a matrix such that (A+BF )V∗ ⊆ V∗. In-troduce the similarity transformation T := [T1 T2 T3],with imT1=V∗ ∩P, im[T1 T2]=V∗ and T3 such thatim[T1 T3]=P.
In the new basis the linear transformation A+BF hasthe structure
A′ = T−1(A+BF )T =
A′11 A′12 A′13
O A′22 OO O A′33
(4.19)
Recall that P is an A-invariant and note that, owingto the particular structure of B, it is also an (A+BF )-invariant for any F .
By a dimensionality argument the eigenvalues of theexosystem are those of A′22, while the invariant zerosof (A1, B1, E1) are a subset of σ(A′11) since RV∗ iscontained in V∗ ∩P. All the other elements of σ(A′11)are arbitrarily assignable with F . Hence, owing to(4.18), the Sylvester equation
A′11X −X A′22 = −A′12 (4.20)
admits a unique solution.
The matrix
V :=T1X+T2
is a basis matrix of an (A,B)-controlled invariant Vsatisfying the solvability conditions (4.17).
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Remarks:
• The proof of Theorem 4.4 provides the computa-tional framework to derive a resolvent when thesufficient conditions stated (that are also neces-sary if the boundedness of the plant input is re-quired) are satisfied.
• Relations (4.18) are respectively a structural con-dition and a spectral condition; they are easilycheckable by means of the algorithms previouslydescribed.
• When a resolvent has been determined by meansof the computational procedure described in theproof of Theorem 4.4, it can be used to derive aregulator with the procedure outlined in the “if”part of the proof of Theorem 4.3.
• The order of the obtained regulator is n (thatof the plant plus that of the exosystem) withthe corresponding 2n1+n2 closed-loop eigenval-ues completely assignable under the assumptionthat (A1, B1) is controllable and (E,A) observ-able.
• The internal model principle is satisfied since thefrom the proof of the “if” part of Theorem 4.3it follows that the eigenstructure of the regulatorsystem matrix N contains that of A2.
• It is necessary to repeat an exosystem for everyregulated output to achieve independent steady-state regulation (different internal models are ob-tained in the regulator).
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Feedback Model Following
The reference block diagram for feedback model fol-lowing is shown in Fig. 4.6. Like in the feedforwardcase, both Σ and Σm are assumed to be stable andΣm to have at least the same relative degree as Σ.
+
−
+
−Σ
Σm
Σcr h u
e
y
ym
Fig. 4.6. Feedback model following
Replacing the feedback connection with that shownin Fig. 4.7 does not affect the structural propertiesof the system. However, it may affect stability. Thenew block diagram represents a feedforward modelfollowing problem.
+
−
+
−Σ
Σm
Σcr h u
e
Fig. 4.7. A structurally equivalent connection
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In fact, note that h is obtained as the difference ofr (applied to the input of the model) and ym (theoutput of the model). This corresponds to the parallelconnection of Σm and the opposite of the identitymatrix, that is invertible, having zero relative degree.Its inverse is Σm with a feedback connection throughthe identity matrix, as shown in Fig. 4.8.
+
−+
+
Σ
Σm
Σch u
e
Σ′m
r
Fig. 4.8. A structurally equivalent block diagram
Let the model consist of q independent single-inputsingle-output systems all having as zeros the unstableinvariant zeros of Σ. Since the invariant zeros of asystem are preserved under any feedback connection,a feedforward model following compensator designedwith reference to the block diagram in Fig. 4.8 doesnot include them as poles.
It is also possible to include multiple internal modelsin the feedback connection shown in the figure (this iswell known in the single input/output case), that arerepeated in the compensator, so that both Σ′m and thecompensator may be unstable systems. In fact, zerooutput in the modified system may be obtained asthe difference of diverging signals. However, stabilityis recovered when going back to the original feedbackconnection represented in Fig. 4.6.
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Geometric Approach to LQR Problems
Consider again the disturbance decoupling problem bystate feedback, corresponding to the state equations
x(t) = Ax(t) +B u(t) +Dd(t)e(t) = E x(t)
(5.1)
in the continuous-time case and to the equations
x(k+1) = Ad x(k) + Bd u(k) +Dd d(k)e(k) = Ed x(k)
(5.2)
in the discrete-time case. The corresponding blockdiagram is represented in Fig. 5.1.
u
de
x
Σ
F
Fig. 5.1. Disturbance decouplingby state feedback
Assume that the necessary and sufficient conditionsfor its solvability with internal stability
D ⊆ V∗(B,E)Vm is internally stabilizable
(5.3)
are not satisfied. In this case a convenient resort is tominimize the H2 norm of the matrix transfer functionfrom input d to output e, defined by equation (1.14)or (1.15) in the continuous-time case and equation(1.16) or (1.17) in the discrete-time case.
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The continuous-time case
Consider the following problem:
Problem 5.1 Referring to system (5.1), determine astate feedback matrix F such that A+BF is stableand the corresponding state trajectory for any initialstate x(0) minimizes the performance index
J =
∫ ∞
0e(t)Te(t) dt =
∫ ∞
0x(t)TETE x(t) dt (5.4)
This problem is the so-called “cheap version” ofthe classical Kalman regulator problem or LinearQuadratic Regulator (LQR) problem. In the Kalmanproblem the performance index is
J =
∫ ∞
0x(t)TQx(t) + u(t)TRu(t) dt
=
∫ ∞
0x(t)TCTC x(t) + u(t)TDTDu(t) dt
where matrices Q and R are symmetric positivesemidefinite and positive definite respectively, hencefactorizable as shown. It can be proven that the cheapversion is the more general, since the input to outputfeedthrough term u(t)TDTDu(t) can be accounted forwith a suitable state extension.
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Problem 5.1 is solvable with the geometric tools. Ac-cording to the classical optimal control approach, con-sider the Hamiltonian function
H(t) := x(t)TETE x(t) + p(t)T (Ax(t) +B u(t))
and derive the state, costate equations and stationarycondition as
x(t) =
(∂H(t)
∂p(t)
)T= Ax(t) +B u(t)
p(t) =
(∂H(t)
∂x(t)
)T= −2ETE x(t)− ATp(t)
0 =
(∂H(t)
∂u(t)
)T= BTp(t)
This overall Hamiltonian system can also be writtenas
˙x(t) = A x(t) + B u(t)
0 = E x(t)(5.5)
with
x =
[xp
]A =
[A 0
−2ETE −AT
]
B =
[B0
]E =
[0 BT
] (5.6)
Problem 5.1 admits a solution if and only if thereexixts an internally stable (A, B)-controlled invariant ofthe overall Hamiltonian system contained in E whoseprojection on the state space of system (5.1), definedas in (4.7), contains the initial state x(0).
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It can be proven that the internal unassignable eigen-values of V∗ := maxV(A, B, E) are stable-unstable bypairs. Hence a solution of Problem 5.1 is obtained asfollows:
1. compute V∗;2. compute a matrix F such that (A+BF )V∗ ⊆ V∗
and the assignable eigenvalues (those internal toRV∗) are stable;
3. compute Vs, the maximum internally stable(A+BF)-invariant contained in V∗;
4. if x(0)∈P (Vs) the problem admits a solution F ,that is easily computable as a function of Vs andF ; if not, the problem has no solution.
Refer to Fig. 5.1. The above procedure also providesa state feedback matrix F corresponding to the mini-mum H2 norm from d to e. This immediately followsfrom expression (1.15) of the H2 norm in terms ofthe impulse response. In fact, the impulse responsecorresponds to the set of initial states defined by thecolumn vectors of matrix D. Thus, the problem ofminimizing the H2 norm from d to e has a solution ifand only if
D⊆P (Vs)
Thus, the minimum H2 norm disturbance almost de-coupling problem has no solution if the above condi-tion is not satisfied. The discrete-time case is partic-ularly interesting since a solution always exist. Thereason for this will be pointed out below.
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The discrete-time case
The discrete-time cheap LQR problem is stated asfollows.
Problem 5.2 Referring to system (5.2), determine astate feedback matrix Fd such that Ad+BdFd is stableand the corresponding state trajectory for any initialstate x(0) minimizes the performance index
J =∞∑
k=0
e(k)Te(k) =∞∑
k=0
x(k)TETd Ed x(k) (5.7)
In this case the Hamiltonian function is
H(k) := x(k)TETd Ed x(k) + p(k)T (Ad x(k) +Bd u(k))
and the state, costate equations and stationary con-dition are
x(k+1) =
(∂H(k)
∂p(k+1)
)T= Ad x(k) +Bd u(k)
p(k) =
(∂H(k)
∂x(k)
)T= 2ET
d Ed x(k) +ATd p(k+ 1)
0 =
(∂H(k)
∂u(k)
)T= BTp(k+1)
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Like in the continuous-time case, it is convenient tostate the overall Hamiltonian system in compact form:
x(k+ 1) = Ad x(k) + Bd u(k)
0 = Ed x(k)(5.8)
with
x =
[xp
]Ad =
[Ad 0
−2A−Td ETd Ed −A−Td
]
Bd =
[Bd
0
]Ed =
[−2BT
d A−Td ETd Ed BT
d A−Td
](5.9)
A solution of Problem 5.2 is obtained again with ageometric procedure, but, unlike the continuous-timecase, in this case a dead-beat like motion is also feasi-ble and P (Vs) covers the whole state space of system(5.2). Hence both Problem 5.2 and the problem ofminimizing the H2 norm from d to e are always solvablein the discrete-time case.
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0 ρ
dead-beat
postaction
Fig. 5.2. Cheap H2 optimal control.
A typical control sequence is shown in Fig. 5.2: asthe sampling time approaches zero, the dead beatsegment tend to a distribution, which is not obtainablewith state feedback. For this reason solvability of theH2 optimal decoupling problem is more restricted inthe continuous-time case.
If the signal to be optimally decoupled is measurableand the system considered is stable, state feedbackcan be used in an auxiliary feedforward unit of thetype shown in Fig. 3.6, while the dual layout shown inFig 3.7 realizes the H2-optimal observation of a linearfunction of the state or possibly of the whole state(Kalman filter).
However, if the signal is not measurable and stateis not accessible, the problem of H2-optimal decou-pling with dynamic output feedback can be stated andsolvability conditions derived by using geometric tech-niques again.
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Conclusions
The three types of input signals:
• disturbance (eliminable only with feedback)
• measurable
• previewed
The seven characterizing properties of systems:
• (internal) stability
• controllability
• observability
• invertibility
• functional controllability
• relative degree
• minimality of phase
In general, the necessary and sufficient conditions forsolvability of control problems consist of
• a structural condition
• s stability condition
When a tracking or disturbance rejection problem isnot perfectly solvable with internal stability, it is pos-sible to resort to H2 optimal solutions that can alsobe obtained through the standard geometric tools andalgorithms.
http://www.deis.unibo.it/Staff/FullProf/GiovanniMarro/geometric.htm
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