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Stabilization and pole placement
Interconnection and Control
H.L. Trentelman1
1University of Groningen, The Netherlands
DISC Course Mathematical Models of Systems
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Outline
1 Stabilization and pole placement
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
Recall: In general, design specifications are desired properties of themanifest controlled behavior (Pfull ∧c C)w .In the context of stabilization and pole placement:
Stability of (Pfull ∧c C)w : the stabilization problem.
Stability of (Pfull ∧c C)w with arbitrary transient settling timeand frequencies of oscillation: the pole placement problem.
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
Stability
B ∈ Lw is called stable if for all w ∈ B we have:
limt→∞
w(t) = 0.
Recall: if B is stable, then it is autonomous. Stability in terms ofrepresentations:
Proposition
Let B ∈ Lw. Let R ∈ Rw×w[ξ] be such that R( ddt )w = 0 is a
minimal kernel representation of B. Then B is stable if and onlyif R is Hurwitz, i.e., the polynomial det R has all its roots in C− ={λ ∈ C | Re(λ) < 0}.
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
Stabilization
Given Pfull ∈ Lw+c, the stabilization problem is to find a controllerC ∈ Lc such that
the manifest controlled behavior (Pfull ∧c C)w is stable,
the interconnection of Pfull and C is regular.
In other words: given Pfull ∈ Lw+c, find K ∈ Lw such that K isregularly implementable and stable.
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
Characteristic polynomial of an autonomous behavior
Let B ∈ Lw be autonomous. Then there exists R ∈ Rw×w[ξ],det(R) 6= 0, such that R( d
dt )w = 0 is a kernel representation ofB.Obviously, for any non-zero α ∈ R, αR also yields a kernel repre-sentation of B.Hence: we can choose R such that det(R) is a monic polynomial.This monic polynomial is denoted by χB, and is called the the char-acteristic polynomial of B. The polynomial χB only depends onB, and not on the polynomial matrix R we have used to define it:if R1 and R2 both represent B, then there exists a unimodular Usuch that R2 = UR1. Hence if det(R1) and det(R2) are monic, thendet(R1) = det(R2).
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
The pole placement problem
Given Pfull ∈ Lw+c, the pole placement problem is to find, for everymonic polynomial r ∈ R[ξ], a controller C ∈ Lc such that
the characteristic polynomial χ(Pfull∧cC)w of the controlledbehavior is equal to r ,
the interconnection of Pfull and C is regular.
In other words:Given Pfull ∈ Lw+c, for every monic polynomial r ∈ R[ξ] find K ∈ Lw
such that K is regularly implementable and χK = r .
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
Stabilizability
Recall that B ∈ Lw is called stabilizable if for all w ∈ B there existsw ′ ∈ B such that
w ′(t) = w(t) for t < 0,
limt→∞ w ′(t) = 0.
Proposition
Let B ∈ Lw, and let R ∈ R•×w[ξ] be such that R( ddt )w = 0 is a
minimal kernel representation of B. Then B is stabilizable if and
only if there exists R ′ ∈ R•×w[ξ] such that
(RR ′
)is Hurwitz.
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
Interpretation in terms of full interconnection
Let B be represented by R( ddt )w = 0. Let B′ be the system rep-
resented by R ′( ddt )w = 0. The full interconnection B ∩B′ is then
represented by (R( d
dt )
R ′( ddt )
)w = 0.
If R,R ′ have full row rank then
(RR ′
)is nonsingular if and only
if rank(R) + rank(R ′) = rank(
RR ′
), Equivalently:
p(B ∩B′) = p(B) + p(B′).
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
Thus we get the following characterization of stabilizability in termsof stabilization by regular full interconnection:
Proposition
Let B ∈ Lw. Then B is stabilizable if and only if there existsB′ ∈ Lw such that the full interconnection B ∩ B′ is stable andregular.
Note: the entire manifest variable w is used as a control variable.
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
Controllability
Recall the definition of controllability: B ∈ Lw is controllable if forall w ′,w ′′ ∈ B there exists w ∈ B and T ≥ 0 such that
w |(−∞,0) = w ′|(−∞,0)
w |[T ,∞) = w ′′|[T ,∞)
Proposition
Let B ∈ Lw, and let R ∈ R•×w[ξ] be such that R( ddt )w = 0 is a
minimal kernel representation of B. Then B is controllable if andonly if for every monic polynomial r ∈ R[ξ] there exists R ′ ∈ R•×w[ξ]such that
det
(RR ′
)= r .
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
This yields the following characterization of controllability in termsof pole placement by regular full interconnection:
Proposition
Let B ∈ Lw, B 6= 0. Then B is controllable if and only if for eachmonic polynomial r ∈ R[ξ] there exists B′ ∈ Lw such that the fullinterconnection B ∩B′ regular, autonomous, and χB∩B′ = r .
Note again: the entire manifest variable w is used as a controlvariable.
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
Recall: given Pfull ∈ Lw+c, the stabilization problem is to find C ∈Lc such that (Pfull ∧c C)w is stable and the interconnection of Pfull
and C is regular.Recall the notions of manifest plant behavior:
P := {w | there exists c such that (w , c) ∈ Pfull},
and hidden behavior:
N := {w | (w , 0) ∈ Pfull}.
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
Also recall: for any Pfull ∈ Lw+c and any controller C ∈ Lc we have
N ⊂ (Pfull ∧c C)w ⊂ P
Thus, a necessary condition for the existence of a stabilizing con-troller C is that the hidden behavior N is stable.
Theorem
Let Pfull ∈ Lw+c. There exists C ∈ Lc such that (Pfull ∧c C)w isstable and the interconnection of Pfull and C is regular if and onlyif
1 N is stable,
2 P is stabilizable.
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
Proof
Minimal kernel representation of Pfull: R1( ddt )w + R2( d
dt )c = 0.By suitable unimodular premultiplication of (R1 R2), Pfull is repre-sented by (
R11( ddt ) R12( d
dt )
R21( ddt ) 0
)(wc
)= 0.
with R12 full row rank.Manifest plant behavior P: eliminate c ⇒ R21( d
dt )w = 0,
Hidden behavior N : set c equal to zero ⇒(
R11( ddt )
R21( ddt )
)w = 0.
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
(only if) Let C be a stabilizing controller. Since N ⊂ (Pfull ∧c C)w
this implies N stable.Let C have minimal kernel representation C ( d
dt )c = 0.Minimal kernel representation of Pfull ∧ C: R11( d
dt ) R12( ddt )
R21( ddt ) 0
0 C ( ddt )
( wc
).
Latent variable representation of (Pfull ∧c C)w : R11( ddt )
R21( ddt )
0
w = −
R12( ddt )
0C ( d
dt )
c
(latent variable c)
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
Note: (Pfull ∧c C)w = P ∩ P ′, with P ′ ∈ Lw represented by(R11( d
dt )0
)w = −
(R12( d
dt )
C ( ddt )
)c
(latent variable c).Interconnection of Pfull and C regular ⇒ Interconnection P ∩ P ′regular.P ∩ P ′ stable ⇒ P stabilizable.
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
(if) N is represented by
(R11( d
dt )
R21( ddt )
)w = 0,
P is represented by R21( ddt )w = 0.
N stable⇒(
R11
R21
)=
(R ′11
R ′21
)G , with
(R ′11(λ)R ′21(λ)
)full column
rank for all λ ∈ C, and G Hurwitz.Hence: Pfull has a representation of the form(
G ( ddt ) R ′12( d
dt )
0 R ′22( ddt )
)(wc
)= 0.
Note: (w , c) ∈ Pfull ⇒ G ( ddt )w = −R ′12( d
dt )c
(reconstruction of G ( ddt )w using c).
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
P stabilizable and R21 = R ′21G , G Hurwitz ⇒ R ′21(λ) full row rank
for λ ∈ C+ ⇒ there exists C0 ∈ R•×w[ξ] such that
(R ′21
C0
)is
Hurwitz.
Hence
(R21
C0G
)is Hurwitz.
Define now K := P ∩ P ′, with P ′ repr. by C0( ddt )G ( d
dt )w = 0.
Then K is stable.Since G ( d
dt )w = −R ′12( ddt )c for (w , c) ∈ Pfull, K is regularly imple-
mented by the controller C ∈ Lc represented by C0( ddt )R ′12( d
dt )c =0.
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
stable hidden behavior ⇔ detectability
Note: N is stable if and only if
(w , 0) ∈ Pfull ⇒ limt→∞
w(t) = 0.
By linearity, this is equivalent with:
(w1, c), (w2, c) ∈ Pfull ⇒ limt→∞
(w1(t)− w2(t)) = 0.
Conclusion: N is stable ⇔ in Pfull, w is detectable from c .
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
Reformulation of the theorem
Let Pfull ∈ Lw+c. There exists C ∈ Lc such that (Pfull ∧c C)w isstable and the interconnection is regular if and only if
1 in Pfull, w is detectable from c ,
2 the manifest plant behavior P of Pfull is stabilizable.
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
Example
Consider the full plant behavior Pfull with to-be-controlled variable(w1,w2) and interconnection variable (c1, c2), represented by
w1 + w2 + c1 + c2 = 0,
w2 + c1 + c2 = 0,
c1 + c1 + c2 + c2 = 0.
A stabilizing regular controller is given by C = {(c1, c2) | c2 +2c1 + c2 = 0}. Indeed, by eliminating c from the full controlledbehavior Pfull∧c C we find that (Pfull∧c C)w = ker(R), with R(ξ) =(
0 ξ + 1−1 2
), which is Hurwitz.
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
Solution of the pole placement problem
Recall: given Pfull ∈ Lw+c, the pole placement problem is to find,for every monic polynomial r ∈ R[ξ], a controller C ∈ Lc such thatχ(Pfull∧cC)w = r and the interconnection is regular.
Theorem
Let Pfull ∈ Lw+c. For every r ∈ R[ξ] there exists a controller C ∈ Lc
such that χ(Pfull∧cC)w = r if and only if
1 N = 0,
2 P is controllable and P 6= 0.
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
Zero hidden behavior ⇔ observability
Note: N = 0 if and only if
(w , 0) ∈ Pfull ⇒ w = 0.
By linearity, this is equivalent with:
(w1, c), (w2, c) ∈ Pfull ⇒ w1 = w2.
Conclusion: N = 0 ⇔ in Pfull, w is observable from c .
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
Reformulation of the theorem
Let Pfull ∈ Lw+c. For every r ∈ R[ξ] there exists a controller C ∈ Lc
such that χ(Pfull∧cC)w = r if and only if
1 in Pfull, w is observable from c ,
2 the manifest plant behavior P of Pfull is controllable and notequal to the zero behavior.
H.L. Trentelman University of Groningen
Interconnection and Control
Stabilization and pole placement
Stabilization and Pole Placement
From the general result to particular representations
Statement of the problems and of their solutions do not use repre-sentations of Pfull, C, N and P. Hence: applicable to any particularrepresentation of the full plant Pfull. Procedure:
for a given representation of Pfull, compute representations ofits hidden behavior N and its manifest plant behavior P.
Next: express the representation-free conditions of the mainresult in terms of the parameters of these representations.
Use the general construction of the stabilizing (pole placing)controller to set up algorithms in terms of the parameters ofthese representations.
H.L. Trentelman University of Groningen
Interconnection and Control