Lect 06 Hand Out

Time Delays in Control and Estimation (038806)

lecture no. 6

Leonid Mirkin

Faculty of Mechanical EngineeringTechnionIIT

esh

Outline

Fiagbedzi-Pearson reduction

Smith controller revised

Modified Smith predictor and dead-time compensation

Modified Smith predictor vs. observer-predictor

Coprime factorization over H1 and Youla parametrization

Nontrivial complication

So far, we studied problems in which h has finite spectrum. Let now

h W Px.t/ D A0x.t/C Ahx.t h/C Bu.t/;

which might haveI infinite number of open-loop poles complicates matters

Workaround:I move only part of these eigenvalues by feedback,

namely, unstable ones (we know that there is only a finite number of them).

Yet another transformation

Consider

Qx.t/ Qx.t/CZ tth

eQA.th/QAhx./d

for some QA 2 RQnQn and Q 2 RQnn. Then,

PQx.t/ D Q Px.t/C QAZ tth

eQA.th/QAhx./d C e QAhQAhx.t/ QAhx.t h/

D QA0x.t/CQBu.t/C QAZ tth

eQA.th/QAhx./d C e QAhQAhx.t/

D QA Qx.t/CQBu.t/ QAQ QA0 e QAhQAhx.t/:If we can choose QA satisfying the left characteristic matrix equation

QAQ D QA0 C e QAhQAh; (l.c.m.e.)

h reduces toQ W PQx.t/ D QA Qx.t/CQBu.t/:

Properties of solutions of l.c.m.e. QAQ D QA0 C e QAhQAhLet Q 2 spec. QA/ and Q be the corresponding left eigenvector. Then

Q QQ D Q QAQ D QQA0 C Q e QAhQAh D QQ.A0 C eQhAh/:

In other words, QQ. QI A0 eQhAh/ D 0, so thatI whenever QQ 0, every eigenvalue of QA is a pole of h

(remember, those poles are the roots of h.s/ D det.sI A0 Ahesh/).

Closed-loop spectrum

Suppose we can solve (l.c.m.e.) in QA and Q. Then

u.t/ D QF Qx.t/ D QFQx.t/C

Z tth

eQA.th/QAhx./d

for some QF leads to the closed-loop system

sI A0 Ahesh B QFQ QF e QAh R h

0e.sI QA/dQAh I

X.s/

U.s/

D I.C. :

and the characteristic quasi-polynomial cl.s/ D det.s/, where

.s/

sI A0 Ahesh B QF QC e QAh R h

0e.sI QA/dQAh

I

:

Closed-loop spectrum (contd)

Next,

.sI QA/QC e QAh

Z h0

e.sI QA/d QAh

D sQ QAQC .eAh eshI /QAh D Q.sI A0 Ahesh/;

so that

QC e QAhZ h0

e.sI QA/dQAh D .sI QA/1Q.sI A0 Ahesh/:

Then

.s/ D

sI A0 Ahesh B QF .sI QA/1Q.sI A0 Ahesh/ I

and

cl.s/ D det.sI A0 Ahesh/ B QF .sI QA/1Q.sI A0 Ahesh/

D detI B QF .sI QA/1Qdet.sI A0 Ahesh/:

Closed-loop spectrum (contd)

Because QA QB QF I

D QACQB QF Q

B QF I1

;

we have that

detI QF .sI QA/1QB D det.sI QA QB QF /

det.sI QA/and thus

cl.s/ D det.sI A0 Ahesh/

det.sI QA/ det.sI QA QB QF /:

In other words,I spec.h;cl/ D

spec.h/ n spec. QA/

Sspec. Qcl/

(remember, spec. QA/ spec.h/).

Implications

Thus, if we canI solve (l.c.m.e.) so that QA contains all unstable modes of h,I find QF so that QACQB QF is Hurwitz (requires stabilizability of . QA;QB/),

the control law

u.t/ D QFQx.t/C

Z tth

eQA.th/QAhx./d

stabilizes h by moving all its unstable modesthose in spec. QA/to theeigenvalues of QACQB QF and keeping the other modes of h untouched.

Distributed state / input delays

Leth W Px.t/ D

Z 0h

./x.t C /C ./u.t C /d:

Then transformation

Qx.t/ Qx.t/CZ tth

Z th

eQA.tC/Q

./x./C ./u./dd

with l.c.m.e.QAQ D

Z 0h

eQAQ./d (l.c.m.e.0)

yields reduced system

Q W PQx.t/ D QA Qx C QBu.t/; where QB Z 0h

eQAQ./d

andI spec.h;cl/ D

spec.h/ n spec. QA/

Sspec. Qcl/.

Is it that simple ?

Not quite, solving QAQ DZ 0h

eQAQ./d is highly nontrivial. Specifically,

I we have to find all troublesome modes of h(in most cases, have to rely on numerical approaches)

I solve (l.c.m.e.) / (l.c.m.e.0)(solution is non-unique and not especially elegant)

Only a handful of cases where the steps above can be solved analytically.One example is

./ D

2666664 0 ::::::: : :

::::::

0 0 0 0 0

3777775 ./CXi

26666640 0 0 ::::::: : :

::::::

0 0 0 0 0 0 0

3777775 . C hi /

in which case spec.h/ is finite and (l.c.m.e.0) is solvable with Q D I .

Example

Leth W Px.t/ D x.t/C x.t h/C u.t/;

whole characteristic quasi-polynomial is (here s D C j!)

h.s/ D s C 1 esh D C 1C j! ehej!h:

Solutions of h.s/ D 0 must satisfy the magnitude condition

. C 1/2 C !2 D e2h:

If > 0, this equation is unsolvable. If D 0, then ! D 0 is the only option.Indeed, s D 0 is a root. Then, by LHopitals rule,

lims!0

h.s/

sD 1C lim

s!01 esh

sD 1C lim

s!0h

1D 1C h;

which implies that s D 0 is a single root.

Example (contd)

Thus, we have only one unstable pole to shift and may pick Qn D 1, QA D 0.Eqn. (l.c.m.e.) then reads 0 D q C q, so we may pick q D 1. Then

Q W PQx.t/ D u.t/;

which is stabilized by u.t/ D k Qx.t/ for any k > 0. Thus

u.t/ D kx.t/C

Z tth

x./d

stabilizes h and renders its closed-loop characteristic polynomial

h;cl.s/ D s C ks

.s C 1 esh/:

In fact, the controller above has the transfer function

C.s/ D k1C 1 e

sh

s

2 H1:

Outline






Smith controller: preliminary conclusions

C.s/

reQeud

y- -Pr.s/e

sh QC.s/

Pr.s/.1 esh/

Remember, Smith controllerI works if Pr.s/ is stable

(stabilization problem reduces then to that for delay-free plant)

I does not necessarily work if Pr.s/ is unstable(might lead to unstable loop)

More rigorous analysis

wu

wyyu

Pr.s/esh

C.s/

System is said to be internally stable ifI transfer matrix from

wywu

toyu

,

Tcl 11 PrC esh

1 PreshC PrC esh

S TdTu T

2 H1:

LetsI analyze Smith controller from internal stability perspectives.

Loop shifting

QC.s/

Pr.s/wu

wyyu

Pr.s/esh

C.s/

Pr.s/.1 esh/

Pr.s/.1 esh/ -

Adding and subtracting block Pr.1 esh/ weI redistribute loop components w/o changing the whole system.

We end up with a new loop with the plant Pr and the controller

QC C1C CPr.1 esh/

so that C D

QC1 QCPr.1 esh/

Loop shifting: signal transformations

QC.s/

Pr.s/wu

wyyu

Pr.s/esh

C.s/

Pr.s/.1 esh/

Pr.s/.1 esh/ -

QC.s/

Pr.s/

wu

wy

./wu

yu

Pr.s/esh

C.s/

Pr.s/.1 esh/

Pr.s/.1 esh/ -

-

QC.s/

Pr.s/

wu QwyQy

yu

Pr.s/esh

C.s/

Pr.s/.1 esh/

Pr.s/.1 esh/ -

Loop shifting (contd)

wu

wyyu

Pr.s/esh

C.s/

,Qwu

QwyQyQu

Pr.s/

QC.s/

The new system is delay-free yet withI different signals.

To complete1 the picture, we have to calculate them: QyQuDy C Pr.1 esh/u

u

DI Pr.1 esh/0 I

y

u

and Qwy

QwuDwy Pr.1 esh/wu

wu

DI Pr.1 esh/0 I

wywu

:

1Well, we also have to guarantee that internal QC loop is well posed.

Loop shifting (contd)

Thus,

Tcl DI Pr.1 esh/0 I

QTclI Pr.1 esh/0 I

:

1. if Pr.1 esh/ 2 H1, then QTcl 2 H1 implies Tcl 2 H12. if Pr.1 esh/ 62 H1, then QTcl 2 H1 not necessarily implies Tcl 2 H1

(in fact, never; this can be verified by making use of explicit form of QTcl)

Key question:I when does Pr.1 esh/ 2 H1 ?

Obviously, this is true if Pr 2 H1. Yet this also true ifI Pr.s/ proper and its only unstable poles are single poles at j2h k, k 2 Z,

which are simple zeros of 1 esh.

Outline






Loop shifting: idea

QC.s/

Pr.s/wu

wyyu

Pr.s/esh

C.s/

Pr.s/.1 esh/

Pr.s/.1 esh/ -

Driving idea here is toI cancel (compensate) delay via Pr.s/esh C Pr.s/.1 esh/ D Pr.s/.

The question:I can we do it with stable compensation element ?

Dead-time compensation question

QC.s/

QP .s/wu

wyyu

Pr.s/esh

C.s/

.s/

.s/ -

Technicaly speaking, we are looking for .s/ 2 H1 such that

Pr.s/esh C.s/ D QP .s/

for some proper and rational QP .s/.

Dead-time compensation: aspirations

If we obtained required .s/ 2 H1, we would transformwu

wyyu

Pr.s/esh

C.s/

,Qwu

QwyQyQu

QP .s/

QC.s/

for rational (delay-free) QP .s/. Resulting C.s/,

C.s/ D QC.s/I .s/ QC.s/1 D yu QC.s/.s/

;

called dead-time compensator (DTC) andI QC.s/ internally stabilizes QP .s/ iff C.s/ internally stabilizes Pr.s/esh as

Tcl.s/ DI .s/0 I

QTcl.s/

I .s/0 I

(provided DTC loop well posed).

DTC question: stable Pr.s/

Choice of .s/ apparent and non-unique:I .s/ D QP .s/ Pr.s/esh for any QP .s/ 2 H1 does the job.

Some standard choices:I QP .s/ D Pr.s/ results in Smith predictorI QP .s/ D 0 results in internal model controller (IMC)

DTC question: unstable Pr.s/ in Example of Lect 5

With Pr.s/ D 1=s, the Smith predictor

.s/ D 1 esh

sDZ h0

esd 2 H1

indeed (integrand analytic and bounded in C0 and integration path finite).In this case

QP .s/ D Pr.s/ D 1s

can be interpreted as unstable rational part of partial fraction expansion2 of

Pr.s/esh D Res

eshsI 0

sC.s/

for some entire .s/ (Pr.s/ has only one pole, that at the origin).

2Here Res.f .s/I c/ lims!c.s c/f .s/ stands for residue of f .s/ at c.

DTC question: another unstable Pr.s/

Let Pr.s/ D 1=.s 1/. Were looking for stable QP .s/ such that

.s/ D QP .s/ 1s 1 e

sh 2 H1:

As Pr.s/esh has one singularity, pole at s D 1, we have that

1

s 1 esh D Res

1s1 e

shI 1s 1 C.s/;

for some entire .s/. This suggest choice

QP .s/ D Res1s1 e

shI 1s 1 D

eh

s 1;

for which .s/ is distributed-delay system

.s/ D .s/ D eh eshs 1 D e

hZ h0

e.s1/d 2 H1:

DTC question: first-order unstable Pr.s/

Likewise,

1

s a esh D Res

1sa e

shI as a C.s/ D

eah

s a eah eshs a ;

so that if Pr.s/ D 1sa , the choice

QP .s/ D eah

s agives us required

.s/ D eah

s a esh

s a D eah

Z h0

e.sa/d 2 H1

again.

First-order unstable Pr.s/: time-domain interpretation

Impulse responses of P.s/ Pr.s/esh D 1saesh and QP .s/ D 1saeah are

p.t/ D(0 it t < h

ea.th/ if t h and Qp.t/ D(0 it t < 0

ea.th/ if t 0

respectively. Thus,p.t/ Qp.t/; 8t h:

Impulse response of .s/ D QP .s/ Pr.s/esh then

.t/ D Qp.t/ p.t/ D(0 it t < 0 or t hea.th/ if 0 t < h

i.e., .s/ is FIR (finite impulse response).

Completion operator

Let G.s/ DA B

C 0

. Impulse response of Gh.s/ G.s/esh is

gh.t/ D(0 it t < h

C eA.th/B if t h

We are looking for rational QG.s/ such that Qg.t/ gh.t/, 8t h. Obviously,

Qg.t/ D(0 it t < 0

C eAheAtB if t 0 so thatQG.s/ D

A B

C eAh 0

:

LTI system

hG.s/esh

QG.s/ G.s/esh A BC eAh 0

A B

C 0

esh

completes, in a sense, G.s/esh to rational QG.s/, so we call it completion ofG.s/esh.

Completion operator (contd)

For any rational G.s/, hG.s/esh

is FIR with (bounded) impulse response

h.t/ D(C eA.th/B if 0 t < h0 otherwise

Hence, transfer function

hG.s/esh

D Z h0

h.t/estdt D

Z h0

C eA.th/Bestdt

is bounded in C0 and, as it also entire, belongs to H1. Thus,I h

G.s/esh

stable for every proper rational G.s/.

In time domain, y D hG.s/esh

u writes

y.t/ D CZ tth

eA.th/Bu./d D CZ h0

eA.h/Bu.t /d:

DTC question: general Pr.s/

Let

Pr.s/ DA B

C 0

:

We can always choose .s/ D hPr.s/esh

2 H1, for whichQP .s/ D Pr.s/esh C h

Pr.s/e

sh D A BC eAh 0

is indeed rational.

Modified Smith predictor

yu QC.s/

hPr.s/esh

Transfer function

C.s/ D QC.s/I hPr.s/esh QC.s/1Then,I if Pr.s/ is strictly proper, internal loop is well-posed for every proper QC ,I C.s/ stabilizes Pr.s/es iff QC stabilizes QP .s/

Example 1

C.s/

reQeud

y 1s1e

sh2eh

eh eshs1

- -

This QC stabilizes QP .s/ D ehs1 and then

Tu.s/ D 2eh.s 1/s C 1 ; T .s/ D

2eh

s C 1 esh;

and

Td .s/ D 1s C 1

1C 2.1 e

.s1/h/s 1

esh

are all stable, as expected.

Outline






Observer-predictor revised

Let Pr.s/ DA B

C 0

and consider observer-predictor control law

Pxo.t/ D .AC LC/xo.t/C Bu.t h/ Ly.t/

u.t/ D FeAhxo.t/C

Z tth

eA.t/Bu./d

where F and L are matrices making ACBF and ACLC Hurwitz. Denote

.t/ eAhxo.t/CZ tth

eA.t/Bu./d

(which is an h-prediction of xo for y 0). Then

P.t/ D eAh.AC LC/xo.t/C Bu.t h/ Ly.t/C Bu.t/ eAhBu.t h/ A Z t

theA.t/Bu./d

D A.t/C Bu.t/ eAhLy.t/ Cxo.t/:

Observer-predictor revised (contd)

In other words, observer-predictor control law writes as(P.t/ D A.t/C Bu.t/ eAhLy.t/ Cxo.t/u.t/ D F.t/

Substituting xo.t/ D eAh.t/ Z tth

eA.th/Bu./d , we get P.t/ D .AC eAhLC eAh/.t/C Bu.t/ eAhLC

Z tth

eA.th/Bu./d eAhLy.t/u.t/ D F.t/

Observer-predictor revised (contd)

Thus, we end up with control lawP D .AC BF C eAhLC eAh/ eAhL

y C C

Z tth

eA.th/Bu./d

u D F

Now, noting that the last term above is output of hPr.s/esh

when u is

its input, control law above is actually

yu QC.s/

hPr.s/esh

for

QC.s/ DAC BF C eAhLC eAh eAhL

F 0

:

Connections

yu QC.s/

hPr.s/esh

This is clearly MSP with primary controller,

QC.s/ DAC BF C eAhLC eAh eAhL

F 0

;

which is observer-based controller for

QP .s/ D

A B

C eAh 0

(note that AC eAhLC eAh D eAh.AC LC/eAh is Hurwitz). Thus,I observer-predictor is MSP when primary controller QC is observer-based

controller for QP

Outline






Coprime factorization over H1

We say that transfer function P.s/ has (strongly) coprime factorization overH1 if there are transfer functions

M.s/;N.s/; QM.s/; QN.s/; X.s/; Y.s/; QX.s/; QY .s/ 2 H1

such thatP.s/ D N.s/M1.s/ D QM1.s/ QN.s/

and X.s/ Y.s/

QN.s/ QM.s/ M.s/ QY .s/N.s/ QX.s/

DI 0

0 I

:

Coprime factorization and stabilizability

wu

wyyu

P.s/

C.s/

TheoremThere is controller C.s/ internally stabilizing this system iff 3P.s/ has strongcoprime factorization over H1. In this case all stabilizing controllers can beparametrized as (Youla parametrization)

C.s/ D QY .s/CM.s/Q.s/ QX.s/CN.s/Q.s/1D X.s/CQ.s/ QN.s/1Y.s/CQ.s/ QM.s/

for some Q.s/ 2 H1 but otherwise arbitrary.

3Most nontrivial part here, only if, was proved by Malcolm C. Smith (1989).

Coprime factorization and stabilizability (contd)

wu

wyyu

ses

C.s/

This is example of system, thatI cannot be internally stabilized.

Plant can be (weakly) coprime factorized over H1, i.e.,

ses D ses

s C 1

1

s C 11

yet there is no strongly coprime factorization.

Coprime factorization for rational systems

Let P.s/ DA B

C D

with .A;B/ stabilizable and .C;A/ detectable. Then

X.s/ Y.s/

QN.s/ QM.s/D24AC LC B C LD LF I 0C D I

35and

M.s/ QY .s/N.s/ QX.s/

D24 AC BF B LF I 0C CDF D I

35 ;where F and L are any matrices such that AC BF and AC LC Hurwitz.

Reduction to rational factorization

Let P.s/ be (not necessarily rational) proper transfer function such that

P.s/ D Pa.s/ .s/

for some .s/ 2 H1 and rational Pa.s/ with coprime factorizationXa.s/ Ya.s/

QNa.s/ QMa.s/ Ma.s/ QYa.s/Na.s/ QXa.s/

DI 0

0 I

:

Were looking forI strongly coprime factorization of P.s/ in terms of that of Pa.s/.

Reduction to rational factorization (contd)

LemmaP.s/ D Pa.s/ .s/ has strongly coprime factorization

X.s/ Y.s/

QN.s/ QM.s/DXa.s/ Ya.s/

QNa.s/ QMa.s/

I 0

.s/ I

2 H1

and M.s/ QY .s/N.s/ QX.s/

D

I 0

.s/ I Ma.s/ QYa.s/Na.s/ QXa.s/

2 H1

Proof.By direct substitution.

Resulting stabilizing controllers

Youla parametrization in this case is

C D . QY CMQ/. QX CNQ/1D . QYa CMaQ/

QXa CNaQ . QYa CMaQ/1Hence,

C. QXa CNaQ/ C. QYa CMaQ/ D QYa CMaQor, equivalently,

C. QXa CNaQ/ D .I C C/. QYa CMaQ/:

Thus, denoting Ca . QYa CMaQ/. QXa CNaQ/1, we end up with

.I C C/1C D Ca C D Ca.I Ca/1 Dyu

Ca.s/

.s/

Hmm, looks familiar. . .

Dead-time systems

If

P.s/ D Pr.s/esh DA B

C 0

esh

we already know how to present it in form

P.s/ D QP .s/ hP.s/

D A BC eAh 0

h

A B

C 0

esh

Thus, any stabilizing controller for P.s/ is of the form

yu QC.s/

hPr.s/esh

where QC.s/ is stabilizing controller for QP .s/. Thus, we end up withI modified Smith predictor yet again.

Fiagbedzi-Pearson reductionSmith controller revisedModified Smith predictor and dead-time compensationModified Smith predictor vs. observer-predictorCoprime factorization over H and Youla parametrization

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Lect 06 Hand Out