Checking if controllers are stabilizing using closed-loop data

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 knowledge, 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

Full text

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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

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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).
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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
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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
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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
00.51 1.522.533.54
−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
012345678910111213 1415
−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
3664