interconnection and control

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



Outline

1 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.





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}.





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.





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).





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 .





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.





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′).





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.





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 .





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.





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}.





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.





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.





(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)





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.





(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).





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.





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 .





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.





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.





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.





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 .





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.





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.




Interconnection

Example

Applying this procedure to Pfull represented by ddt x = Ax +Bu, y =

Cx , with w = (x , u, y) and c = (u, y) yields the well-known condi-tions on A,B and C .



interconnection and control

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