Page 1

Checking if controllers are stabilizing using closed-loop data

Alexander Lanzon, Andrea Lecchini, Arvin Dehghani and Brian D. O. Anderson

Abstract—Suppose an unknown plant is stabilized by a

known controller. Suppose also that some knowledge of the

closed-loop system is available and on the basis of that knowl-

edge, the use of a new controller appears attractive, as may arise

in iterative control and identification algorithms, and multiple-

model adaptive control. The paper presents tests using a limited

amount of experimental data obtained with the existing known

controller for verifying that introduction of the new controller

will stabilize the plant.

I. INTRODUCTION

Let [P, C0] be a feedback control interconnection. The

symbols P and C0 denote respectively the plant and the

controller. The Multiple Input Multiple Output (MIMO) case

is considered here. The transfer function P(s) is not known

while the transfer function C0(s) is known. The closed-loop

interconnection [P, C0] is known to be internally stable and

is available for experiments. Let C1denote a new controller

which has been designed to replace C0in the loop. In this

paper, we develop tests to check whether C1 (instead of

C0) stabilizes the feedback loop. These tests are based on

the knowledge of C0(s) and C1(s) and on data obtained

from experiments on the closed-loop system [P, C0], but

not directly on P. The tests are based on gross properties

of the behaviour of the closed-loop, and so should exhibit

significant tolerance of noise.

It should be noticed that many iterative control design

methods have been developed to use closed-loop data ob-

tained from an existing closed loop system in order to update

the current controller with a controller with better perfor-

mance [7], [8], [10]. Iterative data based control methods are

mainly focused on the objective of performance improvement

which is typically an objective competing with the robust

stability of the designed closed loop [4], [11]. Therefore,

alongside data based iterative control design methods a num-

ber of stability tests have been developed to ascertain stability

of the new controller before implementing the controller in

the loop. Existing tests are based either on the identification

of a parametric ‘full order’ model of the current closed-loop

transfer function or on the estimation of frequency bounds

Corresponding author. Andrea Lecchini is with the Department of Engi-

neering, University of Leicester, LE17RH, UK, al394@cam.ac.uk

Alexander Lanzon, Arvin Dehghani and Brian D. O. Anderson are

with the Research School of Information Sciences and Engineering,

The Australian National University, Canberra ACT 0200, Australia and

the National ICT Australia Ltd., Locked Bag 8001, Canberra, ACT

2601,Australia,

Alexander.Lanzon, Arvin.Dehghani,

Brian.Anderson@anu.edu.au

This work was supported by an ARC Discovery-Projects Grant

(DP0342683) and National ICT Australia. National ICT Australia is funded

through the Australian Government’s Backing Australia’s Ability initiative,

in part through the Australian Research Council.

on the magnitude of the current closed-loop transfer function

[3], [6], [9], [17].

One may argue that a mismatch exists between the nature

of these tests and their usual application. Iterative methods

as [7], [8], [10], [17] are based on limited closed loop

experiments which are intended to obtain information for the

design of small controller changes, see also [1], [2], [5], [12].

The existing validation tests are based on the identification of

the full dynamics of the current closed-loop system. Hence

the amount of experimental effort required for validation

purposes, can apparently be much larger that the amount of

experimental effort required for the design of the controller

update. In contrast to this fact we will show in Section IV

that our validation test requires gathering of information only

on a limited known frequency region whose size depends on

the size of the controller change. Hence the experimental

effort is linked to the size of the controller update.

In this paper we put forward the use of phase information.

Our validation tests rely on estimating the phase of the

current closed-loop transfer functions. The use of the phase

information to ascertain closed-loop stability derives from

the Nyquist stability criterion and leads to validation tests

which assess necessary and sufficient stability conditions.

This is in contrast with methods based on magnitude bounds

from which only sufficient conditions can be derived. We

will show that our validation experiments have can reflect

the limitation on the size of the controller update imposed

by the closed loop experimental setting. In particular it will

be shown that if the controller change has limited size then

it is sufficient to obtain an estimate of the phase of the

current closed loop system only up to a certain known finite

frequency. This fact makes the validation tests practical from

the experimental point of view.

The paper is organized as follows. In Section II we

recall coprime factors representations and stability results

in this framework. In this work we adopt coprime factors

representations because they allow us to obtain very neat

statements and simple derivations. In Section III we present

the result which defines the experimental setting for a

stability test based on phase information. Some stability

falsification and validation tests are derived in Section IV.

Numerical illustrations and conclusions complete the paper.

II. COPRIME FACTOR REPRESENTATIONS AND STABILITY

We shall denote by H∞the space of functions bounded

and analytic in the open right-half complex plane, and the

same function spaces with prefix R their real-rational proper

subspaces. The plant is assumed to be a MIMO linear time-

invariant system with m inputs and p outputs. The transfer

Proceedings of the 45th IEEE Conference on Decision & Control

Manchester Grand Hyatt Hotel

San Diego, CA, USA, December 13-15, 2006

ThIP1.3

1-4244-0171-2/06/$20.00 ©2006 IEEE. 3660

Page 2

d

r

C

P

y

u

−

−

Fig. 1.Standard Feedback Configuration

function of the plant belongs to Rm×p,the set of real rational

transfer functions, and is denoted by P. The transfer function

of the controller is denoted by C. In this work we will

use coprime factor representations of P and C, and without

further comment we adopt as a standing assumption that the

plant and all controller transfer functions are always proper.

Hence, in this section we collect definitions and stability

results related to this representation.

Definition 1: The interconnection [P, C] (Fig. 1) is “well-

posed” if the transfer function matrix mapping?r

exists. Put another way, [P, C] is well-posed if (I−CP)−1∈

R. In this case, these four transfer functions can be written

as?y

u

I

Definition 2: The interconnection [P, C] is said to be

“internally stable” if it is well-posed and H(P,C) ∈ RH∞;

i.e., each of the four transfer functions in?r

to RH∞.

Definition 3: The ordered pair {N,M}, with M,N ∈

RH∞, is a right-coprime factorization (rcf) of P ∈ R if

M is invertible in R, P = NM−1, and N and M are right-

coprime over RH∞. Furthermore, the ordered pair {N,M}

is a normalized rcf of P if {N,M} is a rcf of P and

M∗M + N∗N = I.

Definition 4: The ordered pair {˜U,˜V }, with ˜U,˜V

RH∞, is a left-coprime factorization (lcf) of C ∈ R if˜V is

invertible in R, C =˜V−1˜U, and˜U and˜V are left-coprime

over RH∞. Furthermore, the ordered pair {˜U,˜V } is a nor-

malized lcf of C if {˜U,˜V } is a lcf and˜V˜V∗+˜U˜U∗= I. 2

Then, we define

d

?to?y

u

?

?

=

?P

?

(I − CP)−1?−CI??r

d

?

= H(P,C)

?r

d

?

.

d

??→?y

u

?belongs

2

∈

G :=

?N

M

?

,

(1)

˜K :=?−˜U

˜V?,

(2)

where G and will be referred to as the graph symbols of P,

and˜K will be referred to as the inverse graph symbol of C.

Then the following results hold.

Theorem 5: [16, Proposition 1.9] Let G and˜K be defined

as in (1) and (2). Then the following are equivalent:

a) [P, C] is internally stable;

b) (˜KG)−1∈ RH∞;

c) det(˜KG)(jω) ?= 0 ∀ω and wnodet(˜KG) = 0.

2

r

P

y

u

˜V−1

˜U

Fig. 2.Controller Implementation with C =˜V−1˜U

r

h

−˜

UI +˜

V

i

P

y

u

−

Fig. 3.Alternative Implementation of C =˜V−1˜U

In this work we will also refer to the “Observer-form

implementation” of the controller, see [16, Chapter 5]. In

this form the factor˜V−1of C is implemented in the feed-

forward path and the factor˜U of C is implemented in the

feedback path as depicted in Fig. 2 (in Figures 2, 3 and 4 we

omit the signal d because it is not relevant to the discussion).

This is typically done in order for the poles and zeros

of the controller not to impose restrictions on the response

from r to y. Simple manipulations show that the controller

equation can also be rewritten as:

u =

?

−˜UI +˜V

??y

u

?

− r

which is depicted in Fig. 3. This figure shows why this

configuration is referred to as the observer-form.

III. EXPERIMENTAL SETTING FOR THE STABILITY TESTS

The following theorem defines the experimental setting for

the stability tests proposed in this paper.

In the theorem we will refer to the unwrapped phase

of a transfer function which is the phase of the frequency

response when it is in the form of a continuous function of

the frequency [14].

Theorem 6: Let [P,C0] be internally stable. Let C0 =

˜V−1

0

˜U0and C1=˜V−1

1

˜U1be left coprime factorizations over

RH∞. Consider the configuration in Fig. 4 and define T to

be

T = [−˜U1

˜V1]

(I − C0P)−1

?P(I − C0P)−1

?

˜V−1

0

i.e. the mapping T : r → z in Fig. 4.

Let arg denote the unwrapped phase. Then the following

are equivalent:

a) [P, C1] is internally stable;

b) T−1∈ RH∞;

c) detT(jω) ?= 0 ∀ω and wnodet T = 0;

d) detT(jω) ?= 0 ∀ω and

argdetT(j∞) = argdetT(j0).

45th IEEE CDC, San Diego, USA, Dec. 13-15, 2006 ThIP1.3

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r

˜U1

˜V1

˜U0

˜V−1

0

y

u

P

z

−

Fig. 4.Experimental setting

Proof We have that T = (˜K1G)(˜K0G)−1. The proof is com-

pleted by noticing that: (b) and (Theorem 5 part b) are equiv-

alent since (˜K0G), (˜K0G)−1∈ RH∞; (c) and (Theorem

5 part c) are equivalent since {[P, C0] is internally stable

⇔ {det(˜K0G)(jω) ?= 0 ∀ω

and wnodet(T) = wnodet(˜K1G) − wnodet(˜K0G); (d)

and (c) are equivalent because T ∈ RH∞and is bi-proper

and therefore wnodet(T) = Z (T) =

argdetT(j0)] where Z (T) denotes the number of open

RHP zeros of T.

If the plant P is unknown, one cannot explicitly construct

the transfer function T in closed-form. However, the stable

mapping from r to z (resulting from T : r → z) can be

studied in a safe experiment, i.e. one where no instability

can occur, as shown in Fig. 4. Even though we do not

have an explicit characterization of T when P is unknown,

the reference signal r and the computed output signal z

(computed as a filtered version of the measured signals?y

via˜K1) can be used to infer the required properties of T.

In this work we adopt the observer form implementation of

the controller depicted in Fig. 2. If one is concerned in having

to split up the physical controller in two coprime factors

before injecting the reference signal, then the following

implementation will circumvent the concerns.

Let

?X

words, let P0= XY−1be some plant that stabilizes C0=

˜V−1

0

˜U0and satisfies the corresponding Bezout identity. Note

that P0 does not have to be an estimate of P). Then, it is

easy to see in Fig. 5 that

and wnodet(˜K0G) = 0}

1

π[argdetT(j∞) −

2

u

?

Y

?

be a right inverse of

?−˜U0

˜V0

?

(i.e., in other

?r1

r2

?

=

?X

Y

?

r

and

?y

u

?

= H(P,C0)

?r1

r2

?

.

Since H(P,C0) = G(˜K0G)−1˜K0, it easily follows that

?y

u

?

= G(˜K0G)−1r

as is the mapping from r to?y

that the requirement?−˜U0

?−˜U0

u

?in Fig. 2 and Fig. 3. Note

??X

˜V0

Y

?= I can be relaxed to

˜V0

??X

Y

?= Z where Z is a unit in RH∞since the

r1

r2

C0

P

y

u

−

−

X

Y

r

Fig. 5.Alternative experimental setting

transfer function from r to?y

?y

u

u

?then becomes

?

= G(˜K0G)−1Zr = G(ˆ˜ K0G)−1r

withˆ˜K = Z−1˜K0, i.e. only changing the particular coprime

factor representation of the controller.

An interesting observation is that there are several plants

P0 that stabilize C0 and furthermore there are several co-

prime factorizations of P0= Y−1X. This choice can be used

in the synthesis of X and Y to determine the frequency and

bandwidth characteristics of the physical reference signals r1

and r2. This facilitates the experiment by allowing the engi-

neer to control the excitation characteristics of the feedback

interconnection via alteration of the frequency characteristics

of the reference signals r1and r2.

IV. DATA-BASED STABILITY TESTS

In this section we develop data-based stability tests based

on the experimental setting defined in Section III. The tests

aim at verifying condition (d) in Theorem 6. We introduce

the following assumptions.

Assumption 7: The factors ˜V0 and ˜V1 are such that

˜V0(j∞) =˜V1(j∞) = I.

Assumption 8: The transfer functions PC0and PC1are

strictly proper.

Assumption 7 is without loss of generality and assumption

8 captures a typical situation. Notice that the transfer function

T can be written as

2

2

T =˜V1(I − C1P)(I − C0P)−1˜V−1

0

.

(3)

Hence under Assumptions 7 and 8 we have that

detT(j∞) =det˜V1(j∞)

det˜V0(j∞)

det(I − C1P)(j∞)

det(I − C0P)(j∞)= 1.

Therefore detT(j∞) is strictly positive and known and will

be used as a datum for the verification of condition (d) in

Theorem 6.

To start with we have the following falsification test based

on step responses.

Theorem 9: Let the suppositions of Theorem 6 and As-

sumptions 7 and 8 hold. Let ei denote a reference signal

where a step is applied at the i−th input while the other

inputs are kept at 0. Perform m experiments with reference

45th IEEE CDC, San Diego, USA, Dec. 13-15, 2006ThIP1.3

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signal r(t) = ei(t), i = 1,...,m up to steady state

conditions. Let ¯ zibe the steady state output of T recorded

in each experiment and define¯Z = [¯ z1,..., ¯ zm]. Then

[P,C1] is internally stable ⇒ det¯Z > 0.

Therefore if det¯Z ≤ 0, stability of [P,C1] is falsified.

Proof A necessary condition for condition (d) in Theorem 6

to hold true is that detT(j0) and detT(j∞) have the same

sign. By the final value Theorem we have

¯Z = [¯ z1¯ z2, ··· ¯ zm] = lim

s→0s

?

T(s)1

s

?

= T(j0).

Hence detT(j0) = det¯Z. The proof is completed by

noticing that if condition (d) in Theorem 6 holds true, then

detT(j0) must have the same sign of detT(j∞), which

was set to be 1 without loss of generality.

The experimental test devised above is quite simple to carry

out; it simply consists in recording the steady states of m

step responses. However such an experiment can only be

used to check a necessary stability condition.

Condition (d) in Theorem 6 can be verified in both its

necessary and sufficient parts by using more sophisticated

identification techniques. In principle, one could inject a

white noise signal r or a full sine sweep, measure the corre-

sponding output z and compute the full frequency response

for T. However, this is not practical and hence one needs to

determine an alternative, smarter, experiment. The key point

that has to be noticed in designing the experiment is that there

is no need to estimate the full frequency response of T but

what is instead needed is to measure its frequency response

up to a certain finite frequency ω0. The measurement can

tolerate significant error, as its purpose is simply to allow

computation of a certain phase change. A way to estimate ω0

can be worked out from the structure of the transfer function

T. We have the following result.

Lemma 10: Let the suppositions of Theorem 6 hold. Then

the transfer function T has the following expression.

2

T = I + T?

(4)

T?= [−(˜U1−˜U0)(˜V1−˜V0)]

?P(I − C0P)−1

(I − C0P)−1

?

˜V−1

0. (5)

Proof The expression for T?is derived as follows

T?=˜K1G(˜K0G)−1− I

= (˜K1−˜K0)G(˜K0G)−1

=

?

−(˜U1−˜U0) (˜V1−˜V0)

??P

I

?

(I − C0P)−1˜V−1

0

The last expression coincides with (5).

The expression for the transfer function T presented in

the lemma shows that T is the sum of a known term

(i.e. I) and a term which, under Assumptions 7 and 8, is

strictly proper. Hence it can be expected that measuring

the frequency response of T up to a frequency where the

response of T?has vanished is enough to characterize the

full frequency response of T. This fact is illustrated in the

following theorem specialized for the SISO case.

2

Theorem 11: Let the suppositions of Theorem 6 and As-

sumption 7 and 8 hold. Let P be a SISO transfer function.

Let ω0∈ [0,∞) be a frequency such that |T?(jω)| ≤ 1 ∀ω ≥

ω0i.e.

???˜V−1(jω)

Then the condition

???

?????

−(˜U1−˜U0)P + (˜V1−˜V0)

1 − C0P

(jω)

?????≤ 1 ∀ω ≥ ω0.

(6)

T(jω) ?= 0 ∀ω and π

?argT(jω0)

π

?

= argT(j0)

(7)

where [·] denotes the closest integer, is equivalent to condi-

tion (d) in Theorem 6.

Proof The proof consists in showing that

argT(j∞) = π

?argT(jω0)

π

?

.

(8)

Lemma 10 shows that T(j∞) = I under Assumption 7 and

8. Hence, in the SISO case, the inequality

|T(jω) − 1| ≤ 1∀ω ≥ ω0

(9)

certainly implies (8). Inequality (9) is equivalent to (6).

Note that since T(j∞) = I, then argT(j∞) is an

integer multiple of 2π and hence both π[argT(jω0)/π] and

argT(j0) needs to be an integer multiple of 2π for condition

(7) to hold. The two theorems presented in this section

outline experimental tests to assess stability of [P, C1] before

inserting controller C1 in the loop. Theorem 9 holds for

the MIMO case and implies a very simple experiment

which consists in recording the steady state value of m step

responses. The outcome of the test can only be used to falsify

stability of [P, C1]. Theorem 11 holds for the SISO case

and implies the estimation of the frequency response of the

current closed loop system up to a certain frequency ω0.

The Theorem states a necessary and sufficient condition for

the stability of [P, C1]. For the application of Theorem 11

it is important to note that under Assumption 7 and 8 the

left hand side of inequality (6) tends to zero as ω tends to

infinity. In practice, it is reasonable to assume that one has

a rough estimate of the bandwidth of the current closed-

loop [P, C0] which can then be used be used to obtain a

possibly conservative estimate of ω0 by assuming that the

left hand side of inequality (6) remains below one over

some known high-frequency region. Notice that the left-hand

side of inequality (6) depends on the size of the controller

change. A small controller change certainly implies a smaller

frequency ω0 and hence reduced experimental effort. The

estimate of the frequency response of T up to frequency ω0

can be obtained using either parametric or non parametric

estimation methods [13][15]. The unwrapped phase can be

obtained with phase unwrapping techniques [14]. It seems

that Theorem 11 extends to the MIMO case quite readily and

the remaining question is how to easily device an experiment

and compute and interpret the corresponding MIMO results.

2

45th IEEE CDC, San Diego, USA, Dec. 13-15, 2006ThIP1.3

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V. SIMULATION EXAMPLES

In this section, we consider a MIMO system and a SISO

system to illustrate the advantages and effectiveness of the

stability tests proposed in Theorem 6 and Theorem 11.

Although the theorems do not assume that the plant is known,

for the sake of simulation the underlying unknown plants are

given.

A. Example 1: A MIMO System

Let the unknown plant, P ∈ R2×2, be given by

P =

1

s2+ 2s + 4

?−(s − 2)

−3

2(s + 0.5)

−(s − 2)

?

and let C0be a stabilizing controller, [P, C0] ∈ RH∞, given

by

C0=2(s + 2)(s2+ 2s + 4)

s(s + 1)(s2+ 2s + 7)

with a left coprime factorization, C0=˜V−1

?(s − 2)

−3

2(s + 0.5)

(s − 2)

?

0

˜U0,

˜

V0=

(s + 1)

(s2+ 3.89s + 3.8)(s2+ 1.94s + 2.58)(s2+ 2.03s + 4.07)

2

6

4

˜

V11

0

˜

V12

0

˜

V21

0

2

6

4

˜

˜

V22

0

˜

U12

0

3

7

5

˜

U0=

(s + 2)(s2+ 2s + 4)

(s2+ 3.89s + 3.8)(s2+ 1.94s + 2.58)(s2+ 2.03s + 4.07)

˜

U11

0

U21

0

˜

U22

0

3

7

5

˜V11

0

˜V12

0

˜V21

0

˜V22

0

˜U11

˜U12

˜U21

˜U22

= −0.22s(s2+ 4.72s + 6.01)(s2+ 2.24s + 4.51)

= 0.71s(s + 2.03)(s2+ 1.98s + 3.8)

= 0.27s(s − 3.12)(s + 2.04)(s2+ 2s + 3.9)

= −0.71s(s + 1.93)(s + 0.2)(s2+ 2.02s + 4.14)

0 = −0.437(s + 1.65)(s2+ 1.31s + 1.81)

0 = −0.872(s + 1.88)(s2+ 1.96s + 2.94)

0 = 0.545(s + 2.36)(s2+ 2.44s + 3.74)

0 = −0.341(s + 1.62)(s2+ 0.78s + 2.35) .

Theorem 6 puts forward a solution to the problem of

checking in advance using collected closed-loop data if the

controller C1given here by

C1=

2(s2+ 2s + 4)(s − 2)

(s2+ s + 1)(s2+ 2s + 7)

with a left coprime factorization, C1=˜V−1

?

(s − 2)

−3(s − 0.33)

2(s + 0.5)

(s − 2)

?

1

˜U1,

˜

V1=

(s2+ s + 1)(s2+ 2s + 7)

(s + 3.15)(s + 2.04)(s + 1.85)(s + 0.34)(s2+ 2.04s + 4.09)

2

6

4

˜

V11

1

˜

V12

1

˜

V21

1

2

6

4

˜

˜

V22

1

˜

U12

1

3

7

5

˜

U1=

(s − 2)(s2+ 2s + 4)

(s + 3.15)(s + 2.04)(s + 1.85)(s + 0.34)(s2+ 2.04s + 4.09)

˜

U11

1

U21

1

˜

U22

1

3

7

5

˜V11

1

˜V12

1

˜V21

1

˜V22

1

˜U11

1 = −0.43(s − 1.015)(s + 1.8)(s + 0.42)

˜U12

1 = −0.87(s + 2.01)(s + 3.54)(s + 0.27)

˜U21

1 = 0.88(s − 3.6)(s + 1.85)(s + 0.33)

˜U22

= −0.22(s + 2.95)(s + 2.06)

= −0.13(s + 1.93)

= −0.03(s + 7.43)(s + 1.97)

= −0.16(s + 1.83)(s + 0.64)

1 = −0.43(s + 0.13)(s2+ 2.92s + 2.17) .

−1

0.6

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

00.511.522.533.54

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Time (sec)

Amplitude

Time (sec)

Amplitude

0123456789101112131415

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

(a) r(t) = step(t) · e1

(b) r(t) = step(t) · e2

Fig. 6.Step Responses

We setup the experimental configuration of Fig. 4 and

perform two experiments with reference signals r(t) =

step(t) · e1and r(t) = step(t) · e2. The step responses are

shown in Fig. 6 and the steady state of T : r → z are given

in

¯Z =

−0.391

?−0.750.476

1.27

?

with det(¯Z) = −0.7664 < 0 and hence the stability of

[P, C1] is falsified. Indeed, computing H(P, C1) shows that

it has three RHP poles which conforms with the results.

B. Example 2: A SISO system

This example demonstrates the effectiveness of the stabil-

ity tests proposed in Theorem 11 when the results of The-

orem 6 stops shorts of unfalsifying the proposed controller

C1.

Let the unknown SISO plant be given by

P =−186.66(s − 5)(s + 4.5)

(s + 10)2(s + 7)(s + 6)

and let C0be a stabilizing controller, [P, C0] ∈ RH∞, given

by

C0=0.021(s + 10.92)(s + 8.87)(s + 7.31)(s + 5.93)

(s2+ 8.6s + 19.84)(s2− 0.603s + 5.34)

with a left coprime factorization, C0=˜V−1

0

˜U0,

˜V0=(s2+ 8.603s + 19.84)(s2− 0.602s + 5.34)

(s2+ 8.64s + 19.97)(s2+ 1.83s + 6.96)

satisfying Assumption 7,˜V0(j∞) = 1, and

˜U0=0.021(s + 10.92)(s + 8.87)(s + 7.31)(s + 5.93)

(s2+ 8.64s + 19.97)(s2+ 1.83s + 6.96)

.

Suppose that the data collected from the closed-loop

suggests the use of a new controller C1given by

C1=0.33(s + 0.586)(s + 2.99)(s + 3.416)

(s + 2)(s2+ 2.26s + 3.52)

with a left coprime factorization, C1=˜V−1

1

˜U1,

˜V1=

(s + 2)(s2+ 2.26s + 3.52)

(s + 1.87)(s2+ 2.81s + 3.712)

45th IEEE CDC, San Diego, USA, Dec. 13-15, 2006ThIP1.3

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