Decentralized supervisory based switching control for uncertain multivariable plants with variable input output pairing

ISA Transactions 51 (2012) 132–140
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ISA Transactions
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Decentralized supervisory based switching control for uncertain multivariable
plants with variable input–output pairing
Omid Namaki-Shoushtari∗
, Ali Khaki-Sedigh1
Faculty of Electrical and Computer Engineering, K. N. Toosi University of Technology, P.O. Box 16315-1355, Tehran 1431714191, Iran
a r t i c l e i n f o
Article history:
Received 4 April 2011
Received in revised form
9 August 2011
Accepted 28 August 2011
Available online 13 October 2011
Keywords:
Uncertain multivariable control system
Supervisory switching control
Decentralized control
Changing Control Configuration
QFT
Bumpless transfer
a b s t r a c t
In this paper, the design of decentralized switching control for uncertain multivariable plants is
considered. In the proposed strategy, the uncertainty region is divided into smaller regions with a
nominal model and specific control structure. The underlying design is based on the quantitative feedback
theory (QFT). It is assumed that a MIMO-QFT controller exists for robust stability and performance of
the individual uncertain sets. The proposed control structure is made up by these local decentralized
controllers, which commute among themselves in accordance with the decision of a high level decision
maker called the supervisor. The supervisor makes the decision by comparing the local models’ behaviors
with the one of the plant and selects the controller corresponding to the best fitted model. A hysteresis
switching logic is used to slow down the switching to guarantee the overall closed loop stability. It
is shown that this strategy provides a stable and robust adaptive controller to deal with complex
multivariable plants with input–output pairing changes during the plant operation, which can facilitate
the development of a reconfigurable decentralized control. Also, the multirealization technique is used to
implement a family of controllers to achieve bumpless transfer. Simulation results are employed to show
the effectiveness of the proposed method.
© 2011 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
In spite of the indisputable performance advantage of central-
ized multivariable controllers, many complex multivariable plants
employ decentralized controllers [1]. The main advantages that
have led to the widespread use of decentralized controllers are
the ease of implementation and maintenance, the design and tun-
ing simplicity, and fault tolerance. To effectively design such con-
trollers, input–output pairing prior to the commencement of the
design is a key step for the achievement of the desired closed loop
stability and performance. There are different approaches to in-
put–output pairing and RGA is the first and the most widely used
analytical tool for this problem [1–3].
In the face of large plant parameter variations, unknown or un-
certain multivariable plants, the input–output structure or the so
called control configuration of the plant may endure fundamental
changes, which will severely degrade the decentralized controller
performance and can easily lead to closed loop instability [3,4]. The
quadruple-tank process [5], the Wood and Berry distillation col-
umn [6], and the Shell heavy oil fractionator [7] are examples of
such plants. The well-known input–output pairing techniques are
∗ Corresponding author. Tel.: +98 21 8406 2161; fax: +98 21 8846 2066.
E-mail addresses: onamakis@dena.kntu.ac.ir, onamakis@ieee.org
(O. Namaki-Shoushtari), sedigh@kntu.ac.ir (A. Khaki-Sedigh).
1 Tel.: +98 21 8406 2317; fax: +98 21 8846 2066.
unable to analyze the effect of uncertainty on input–output pair-
ing and only recently, pairing methods are proposed for uncertain
multivariable plants [3,8]. These methods can only provide a suit-
able input–output pairing in the presence of model uncertainties
and are unable to tackle the problem of major structural changes in
the plant and are also unable to provide an online input–output se-
lection based on the present state of the plant [4]. A reconfigurable
structure for the design of the decentralized controller based on
the nonlinear adaptive control strategies is proposed in [4].
The conventional adaptive control has some inherent limita-
tions which have been well recognized in the literature. Most
notably, if unknown parameters enter the process model in com-
plicated ways, it may be very difficult to construct a continuously
parameterized family of candidate controllers [9]. To overcome the
above mentioned difficulties with the classical approach to adapta-
tion and control, researchers have focused on the switching control
systems during the past decade [9–12].
Finding a single controller which can deal with the entire
range of parameter variations may be impossible for a highly
uncertain process. To design a satisfactory control plant in the
presence of large modeling uncertainties, noise, and disturbances,
a hierarchical control structure could be used. The control
architecture consists of a bank of candidate controllers supervised
by a logic-based switching [9].
In each fixed predetermined region of uncertainty, the local
controller can achieve the desired performance. Switching is made
between the local controllers to support all ranges of uncertainties.
0019-0578/$ – see front matter © 2011 ISA. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.isatra.2011.08.008

O. Namaki-Shoushtari, A. Khaki-Sedigh / ISA Transactions 51 (2012) 132–140 133
The overall control architecture consists of a bank of controllers
(multi-controllers), and a supervisor. At each time instant, a high
level decision maker, the so called supervisor, determines which
controller should be placed in the feedback loop. In other words,
when estimates of the plant are changed, a new controller may
be selected, which is similar to the idea of adaptive control. But
unlike the traditional adaptive control strategies, this adaptation
takes place in a discrete fashion.
One of the main advantages of the supervisory control is
its modularity. Designing multi-estimators, multi-controllers, and
switching logic can be done in a mutually independent manner.
This feature enables the designer, to use ‘‘off-the-shelf’’ robust
control laws [9]. The objective of this paper is to provide a
practical solution to the problem of decentralized control of
uncertain multivariable plants with variable control configuration.
Decentralized MIMO-QFT is used as the underlying linear robust
design strategy, because it is a powerful design methodology that
provides a transparent tradeoff between different, often conflicting
design specifications, and suggests a controller with minimum cost
of feedback that satisfies the set of performance specifications in
spite of the plant uncertainty [13]. However, as with any other
linear robust design methodology, it fails in the face of variable
control configuration. By combining and switching robust designs,
the designed controller optimizes the time response of the plant by
fast adaptation of the controller parameters during the transient
response based on the error amplitude and ensures closed loop
stability [14].
The problem of robust adaptive control via combining QFT
and switching supervisory control is introduced in [15] for single-
input–single-output (SISO) plants; the entire region of uncertainty
is partitioned into smaller regions. For each region, a QFT
controller is employed to attain robust stability and performance
despite uncertainties and disturbances. A supervisory architecture
orchestrates controller selection. This selection is based on the
values of monitoring signals. In this paper, the combined switching
QFT control is extended for multi-input–multi-output (MIMO)
plants. The key idea is to divide the original uncertainty set
into smaller subsets with certain control configuration as in [16].
Then, for each uncertainty subset, a decentralized MIMO QFT
control is worked out to satisfy the robust stability and tracking
requirements. Finally, a switching supervisory structure is used to
recognize to which uncertainty subset, the ‘‘real’’ uncertain plant
is matched with. Then, the supervisor places the corresponding
controller in the feedback loop.
This paper is organized as follows. In Section 2, switching
supervisory control is briefly reviewed. Fundamentals of MIMO-
QFT are given in Section 3. Our proposed design structure is
described in Section 4. Simulation results and conclusions are
made in Sections 5 and 6, respectively.
2. Switching supervisory control
In this section, the supervisory control framework for adap-
tively controlling plants with large uncertainty is briefly re-
viewed [11,12]. The basic idea is to divide the uncertainty set,
which is compact, into a finite number of subsets with nominal val-
ues and then employ a family of controllers, one for each nominal
value. Switching among the controllers is orchestrated by a super-
visor in such a way that closed-loop stability is assured, as is shown
in Fig. 1. The benefits gained by this approach include (i) simplicity
and modularity in design: controller design amounts to controller
design for linear time-invariant plants with smaller uncertainty,
for which various computationally efficient controller design tools
are available; (ii) ability to handle larger classes of plants than is
possible with other approaches (see [9] for more discussion).
Fig. 1. The supervisory control framework.
Consider an uncertain linear plant Mp parameterized by a
parameter p, and denote by p∗
the true but unknown parameter.
The corresponding realization is:
Mp∗ :

˙x = Ap∗ x + Bp∗ u
y = Cp∗ x,
(1)
where x ∈ Rnx
is the state, u ∈ Rnu
is the input, and y ∈ Rny
is
the output. The parameter p∗
∈ Rnp
belongs to a known finite set
P := {p1, . . . , pm}, where m is the cardinality of P.
It is assumed that (Ap, Bp) is stabilizable and (Ap, Cp) is
detectable for every p ∈ P.
The architecture of supervisory adaptive control comprises: (1)
a family of controllers, and (2) a decision maker (supervisory unit).
The decision maker, consisting of a multi-estimator, a monitoring
signal generator and a switching logic, produces a switching signal
that indicates at every time the active controller.
• Multi-estimator: A multi-estimator is a collection of models, one
for each fixed parameter p ∈ P. The multi-estimator takes in
the input u and produces a bank of outputs yp, p ∈ P. The
multi-estimator should have the following matching property:
there is ˆp ∈ P such that
|yˆp(t) − y(t)| ≤ cee−λe(t−t0)
|yˆp(t0) − y(t0)| (2)
for all t ≥ t0, for all u, and for some ce ≥ 0 and λe > 0. One
such multi-estimator for (1) is constructed as follows with the
state xE = (ˆxT
1 , . . . , ˆxT
m)T
where xE ∈ Rm×nx
and, the dynamics
˙ˆxp = Ap ˆxp + Bpu + Lp(yp − y),
yp = Cp ˆxp,
(3)
for all p ∈ P, and the property (2) is satisfied with ˆp = p∗
. The
matrices Lp in (3) are such that Ap + LpCp are Hurwitz
∀p ∈ P. (since (3) with p = p∗
and (1)
imply that (d/dt)(xp∗ − x) = (Ap + LpCp) (xp∗ − x)
and y = Cp∗ x).
• Multi-controller: A family of candidate controllers {Gp} is
designed such that the closed loop plant meets the desired
robust stability and performance specifications for every p ∈ P .
Then the family of controllers is
Gp, p ∈ P. (4)
• Monitoring signals: Monitoring signals µp, p ∈ P are norms of
the output estimation errors, ep = yp − y. Here, the monitoring
signals are generated as
µp := ε0 +
∫ t
0
e−λ(t−s)
γ ‖yp(s) − y(s)‖2
ds (5)
for some γ , ε0 > 0 and λ ∈ (0, λ0), where ‖.‖ denotes the
vector norm. The numbers γ , ε0, and λ are design parameters,
and λ0 is related to the eigenvalues of the closed-loop plant (for

134 O. Namaki-Shoushtari, A. Khaki-Sedigh / ISA Transactions 51 (2012) 132–140
Fig. 2. The two-degrees of freedom structure of QFT.
details on λ0, see [11]). Note that the interactions between the
outputs of the multivariable plant are taken into account in the
estimation error vector.
• Switching logic: The switching signal, which nominates the
active controller at each time instant, is produced by the scale
independent hysteresis switching logic [17]:
σ(t) :=



argmin
q∈P
µq(t) if ∃q ∈ P such that
(1 + h)µq(t) ≤ µσ(t−)(t)
σ(t−
) else,
(6)
where h > 0 is called a hysteresis constant and is a design
parameter that conveniently prevents excessive switching. The
control signal applied to the plant is u(t) = uσ (t). (See Fig. 1.)
3. Decentralized quantitative feedback design
The decentralized MIMO-QFT design technique provides a
design procedure to synthesize a fixed diagonal controller transfer
function matrix G(s) and pre-filter F(s) to satisfy specifications on
the closed-loop plant shown in Fig. 2, where P(s) is the MIMO
uncertain plant.
In solving an n × n multivariable design problem, the
synthesis problem is converted into n equivalent single-loop
multi-input–single-output (MISO) problems, where parameter
uncertainties, external disturbances and performance tolerances
are derived from the original problem, and the coupling effects
between MISO subsystems are treated as disturbance inputs. The
objective of the design is to achieve set point tracking, while
minimizing the outputs due to the disturbance inputs (cross-
coupling effects) [13]. It is desired that the closed-loop system is
stable and:
|tij(jω)| ≤ bij(ω), i ̸= j,
0 ≤ aii(ω) ≤ |tii(jω)| ≤ bii(ω),
(7)
where T(s) = [tij(s)] is the control ratio matrix, (note that the
closed-loop plant control ratio tij = yi/rj relates the ith output to
the jth input.) A(s) = [aij(s)] and B(s) = [bij(s)] are the desired
lower and upper tracking bounds for the MIMO plant.
4. Switching supervisory QFT control
Combining switching and QFT for SISO plants was first
introduced in [14] to prioritize some specifications over others
according to the state of the plant at each time. Switching is used to
select the appropriate controller which is determined based on the
error amplitude. However, the proposed method in [14] requires
that the underlying controllers must have the same poles and this
constraint, limits controllers type and therefore the performance
of the plant. A more general approach for SISO plants is introduced
in [15] to overcome this limitation in which the quantitative
feedback design was used in the context of switching supervisory
adaptive control to achieve robust stability and performance for
highly uncertain SISO plants. In the following, the decentralized
switching-QFT method is given for MIMO plants based on the
SISO version of [15]. It is shown that the proposed method could
be implemented for complex uncertain plants with I/O pairing
Fig. 3. The supervisory based switching QFT control architecture.
or control configuration changes during the multivariable plant
operation.
In the proposed method, the idea is to divide the original
uncertainty set into smaller subsets with certain I/O pairing.
Then, for each uncertainty subset a decentralized MIMO QFT
control is worked out to satisfy the robust stability and tracking
requirements. Finally, a switching supervisory structure as shown
in Fig. 3 is used to match the proper uncertainty subset to the ‘‘real’’
uncertain plant. Then, the supervisor places the corresponding
controller in the feedback loop.
4.1. Problem formulation
Regarding highly uncertain MIMO plants, two distinct cases
may occur:
• A single decentralized QFT controller exists for the entire
uncertainty set. However, the closed loop performance may not
be improved further as desired.
• A single decentralized QFT controller does not exist to ensure
closed loop robust stability and performance.
The second case is more severe, especially in cases where
parameter uncertainty can lead to input–output pairing change
which is detrimental to the decentralized control of multivariable
plants. The goal is to design a stable closed-loop control system
which can track a predetermined set point in spite of large plant
uncertainty and/or disturbances.
4.2. Class of admissible plants
The plant is assumed to be modeled by a stabilizable and
detectable MIMO linear plant with control input u and measured
output y, perturbed by a bounded disturbance input d. It is
assumed that the plant transfer function belongs to a known class
of admissible transfer function matrices of the form:
M :=

p∈P
Mp
where p is a parameter taking values in some index set. Mp
is also a family of transfer functions ‘‘centered’’ around some
known nominal process model transfer function νp [18]. Allowable
unmodeled dynamics around the nominal process model transfer
functions νp could be specified as:
Mp :=

νp(1 + δm) + δa : ‖δm‖∞,λ ≤ ε, ‖δa‖∞,λ ≤ ε

,
where ε > 0 and λ ≥ 0 are two arbitrary numbers. Here, ‖.‖∞,λ
denotes eλt
-weighted H∞ norm of a transfer function matrix:

‖v‖∞,λ := supRe[s]≥0 ¯σ(v(s −λ)), ( ¯σ(.): the peak of the maximum
singular value) [18].
Throughout the paper, we will take P to be a compact subset of
a finite-dimensional normed linear vector space.
By this definition, the entire region of plant uncertainty is
partitioned into a set of smaller regions. Each smaller region is
presented by a parameter value p, and Mp is a model of the plant
in that small region.
Considering all permissible uncertainties and disturbances in
each smaller region, based on the appropriate I/O pairing, a
decentralized MIMO-QFT controller is designed to achieve robust
stability and robust set point tracking specifications.
4.3. Multi-estimator and multi-controller
A state-shared multi-estimator of the following form could be
utilized here.
˙xE = AE xE + LE y + BE u, yp = CpxE , ep = yp − y, (8)
where p ∈ P, and AE is an asymptotically stable matrix. This type
of structure is quite common in adaptive control [19]. One option
for AE , LE and BE is as follows (using the multi-estimators described
in (3)):
xE = (ˆxT
1 , . . . , ˆxT
m)T
, AE = diag[Ap + LpCp],
LE = block diag[−Lp], BE = block diag[Bp], p ∈ P.
For practical reasons, the bank of local controllers can be
efficiently implemented (multirealized) by means of a state-shared
parameter dependent feedback system. The method provided can
implement bumpless transfer, which is an effective way to improve
poor transient response of the switched systems [20,21].
4.4. Bumpless transfer using multirealization techniques
A major problem in the practical implementation of switching
based control strategies is the undesirable transient behavior of the
plant outputs during switching times. The key response property
that can guarantee a smooth transition during the switching times
is the bumpless transfer [20]. That is, control input u must be
continuous even in the time instants at which the supervisor
changes the controller. By bumpless transfer, the control signal is
smooth and there is no need to apply sudden changes to system
inputs by actuators, while unexpected changes in the control signal
can yield undesirable transient responses in switching times.
On the other hand, in a multicontroller architecture, only
one of the controllers at any instant of time is placed into the
feedback loop. Hence, at each time instant it is only necessary to
generate one control signal. This can be employed to simplify the
multicontroller architecture by generating all control signals by
a single controller. This concept is commonly identified as state
sharing [20]. State sharing can also be used in the multiestimation
part of switching multiple model control methods [19].
The state-shared implementation for a family of multivariable
LTI controllers with strictly proper transfer function matrices using
the right matrix function description (MFD) with denominator
matrix in the Popov form is presented in [21], in the form of
{A0 + B0Ki, B0, Ci} or its dual realization {A0 + LiC0, Bi, C0} (i =
1, 2, . . . , m). The multirealization form {A0 + LiC0, Bi, C0} is
preferred for the implementation of multicontroller structures
to achieve bumpless transfer. However, it is more convenient to
investigate the dual form {A0 + B0Ki, B0, Ci} because of the results
on invariant description of linear multivariable plants [22]. All
controller transfer function matrices can be written in the form of
a right MFD in which all the denominator matrices are in the Popov
form. Based on the results in [21], a modified and simpler algorithm
is presented in [23], which uses the Hermite form polynomial
matrix to achieve a multirealization in the form of {A0 +Ai, B0, Ci}.
In this multirealization form, A0 is a stable matrix and (A0, B0)
is controllable. Also, to achieve bumpless transfer, the duality
property can be employed to obtain a multirealization in the form
{A0 +Ai, Bi, C0}. In the following, the method based on the Hermite
form polynomial matrix to achieve a multirealization in the form
of {A0 + Ai, B0, Ci} is briefly reviewed.
A given n×n nonsingular polynomial matrix can be transformed
to a unique row Hermite form DH (s) by elementary column
operations, where DH (s) is an upper triangular polynomial matrix
with each diagonal element monic and of higher degree than any
other element in the same row, if a diagonal element is 1, all other
entries in that row will be zero [24]. With a unique denominator
DH (s) in the row Hermit form polynomial matrix, a unique and
canonical right MFD for H(s) = NH (s)D−1
H (s) is obtained and is
called the row Hermite form MFD.
In general, we can always write a given polynomial matrix D(s)
in the following form:
D(s) = SL(s)Dhr + ΨL(s)Dlr
where SL(s) = diag{sli , ı = 1, 2, . . . , n}, and li is the degree
of the ith row of D(s). Moreover, Dhr is the highest-row-degree
coefficient matrix of D(s) and the term ΨL(s)Dlr accounts for the
lower-row-degree terms of D(s), with Dlr a matrix of coefficients
and ΨL(s) = block diag{[sli−1
, sli−2
, . . . , 1], i = 1, 2, . . . , n} [24].
Theorem 1 ([23]). Consider a polynomial matrix DH (s) which is in
row Hermite form. Let li denote the degree of the ith row of DH (s)
and µi is the controllability index corresponding to the ith column of
a matrix B (µi independent vectors bi, Abi, . . . , Aµi−1
bi), where (A, B)
is a controllable pair that corresponds to a state realization of D−1
H (s).
Then,
µi = li.
Theorem 2 ([23]). Consider a set of ni-input no-output strictly
proper systems Hi(s) (i = 1, 2, . . . , m). There exist a set of realiza-
tions in the canonical controllability form {Ai, B0, Ci} where Ai is an
n×n matrix and {Ai, B0, Ci} are controllable realizations of Hi(s) (i =
1, 2, . . . , m) if and only if there exists a right row Hermite form MFD
for all Hi(s), with Hi(s) = NHi
(s)D−1
Hi
(s), and deg{DHi
(s)} = n for
i = 1, 2, . . . , m, provided that
lik = ljk, i, j = 1, 2, . . . , m and k = 1, 2, . . . , ni
where lik is the degree of the kth row of the DHi
(s).
Based on Theorem 2, the following algorithm is given to obtain
the multirealization in the form of {A0 + Ai, B0, Ci}.
The Multirealization algorithm:
Step 1. Write irreducible row Hermite form MFD descriptions for
all the MIMO systems Hi(s) = NHi
(s)D−1
Hi
(s).
Step 2. Let lmaxj = maxi lij, where lij denotes the degree of the jth
column in DHi
(s). Then, for all DHi
(s) matrices, multiply the
jth column by (s + a)lmaxj − lij for arbitrary chosen a > 0.
Step 3. Applying changes in the previous steps, it may be possible
that DHi
(s) is not in the row Hermite form. Convert all DHi
(s)
to the row Hermite form and apply appropriate changes on
NHi
(s) to have Hi(s) unchanged (Hi(s) = ˜NHi
(s)˜D−1
Hi
(s)).
Step 4. Using the controllable canonical form realization for
systems Hi(s) = ˜NHi
(s)˜D−1
Hi
(s), (i = 1, 2, . . . , m), the
multirealization in the {A0 + Ai, B0, Ci} form is obtained.
In the following section, the proposed algorithm is used
for practical implement of the bank of controllers. Instead of
switching between different controller transfer function matrices,
a multirealization of these controllers is employed. In the first

Table 1
Plant conditions used in Example 1.
No. γ11 γ22 γ12 γ21 δ11 δ22 δ12 δ21
1 1 2 0.5 1 1 2 2 3
2 1 2 0.5 1 0.5 1 1 2
3 1 2 0.5 1 0.2 0.4 0.5 1
4 1 2 4 5 2 3 1 2
5 1 2 4 5 1 2 0.5 1
6 1 2 4 5 0.5 1 0.2 0.4
7 10 8 2 4 1 2 2 3
8 10 8 2 4 0.5 1 1 2
9 10 8 2 4 0.2 0.4 0.5 1
10 5 8 16 20 2 3 1 2
11 5 8 16 20 1 2 0.5 1
12 5 8 16 20 0.5 1 0.2 0.4
step, the multirealization algorithm is applied to the transpose of
transfer function matrices, HT
i (s). Then, using the duality theorem,
the multirealization in the {A0 +Ai, Bi, C0} form is obtained. Finally,
the controller with the bumpless transfer property is implemented
using this multirealization technique.
5. Simulation results
Example 1. In this section, a numerical example is used to
illustrate the proposed design method. Consider the following 2×2
uncertain plant:
M(s) =




γ11
1 + sδ11
γ12
1 + sδ12
γ21
1 + sδ21
γ22
1 + sδ22



 .
The corresponding twelve different parameter cases are given
in Table 1.
The complexity of this example lies in the fact that the
appropriate input–output pair changes as the parameters vary
and a single robust decentralized controller (regardless of the
underlying design methodology) cannot effectively control the
plant.
Closed loop specifications (for all plants)
(1)—The tracking specifications, Tlij ≤ |T(jω)|ij ≤ Tuij (i, j =
1, 2.) are basically non-interacting, and are enforced to ω ≤ ωh =
10 rad/s.
On-diagonal:
Tuii(ω) =




25
s2 + 6s + 25




s=jω
and
Tlii(ω) =




4
s2 + 4.4s + 4




s=jω
.
Off-diagonal:
Tuij(ω) = 0.1, and Tlij = 0.
(2)—Stability margin: |1(1 + Li)| ≤ 3.5 dB for all plants.
Where Li corresponds to the ith loop gain and this would
indicate a gain margin and phase margin of 9.6 dB and 39 deg
respectively [13].
The relative gain array (RGA) was introduced by Bristol [2]
as a measure of interaction in multivariable control plants and
is a practical tool for control configuration selection in many
multivariable plants. The RGA is defined as
Λ = M(0)∗
M−T
(0),
where the asterisk denotes the Schur product and −T is the inverse
transpose.
The elements of the RGA for the uncertain process M(s) is as
follows:
Λ =
[
λ 1 − λ
1 − λ λ
]
, λ =
γ11γ22
γ11γ22 − γ12γ21
.
Fig. 4. Variations of the plant parameters.
Fig. 5. Switching signal, note that the change in RGA matrix is recognized by
supervisor.
For cases 1–3, λ = 1.33; therefore, pairing along the diagonal is
proposed. In cases 4–6, the structural change in the multivariable
plant clearly leads to λ = −0.11 and the off diagonal input–output
pairing is proposed.
If a fixed structure decentralized controller is employed, the
parameter variation leads to closed-loop instability. However, in
the case of a switching supervisory control design methodology
shown in Fig. 3, the change in the RGA matrix resulting from
the plant parameter variations, shown in Fig. 4, is recognized
by the supervisor as shown in Fig. 5. Hence, when the change
has occurred, a new input–output pairing is chosen. Fig. 6 shows
that the decentralized switching-QFT controller can easily handle
the new control configuration. The corresponding control effort is
shown in Fig. 7.
For cases 7–9, λ = 1.11 where diagonal pairing is the appropri-
ate input–output pair.
Finally, in cases 10–12, λ = −0.14 which leads to the off
diagonal input–output pairing.
In order to design the multiple-model based decentralized
switching-QFT architecture, the region of uncertainty is divided
into the following smaller regions:
Cases 1, 2 and 3
Cases 4, 5 and 6
Cases 7, 8 and 9
Cases 10, 11 and 12.

Fig. 6. Plant outputs in the proposed switching structure. (The family of controllers
are implemented using multirealization technique to achieve bumpless transfer.)
Fig. 7. Control inputs in the proposed switching structure. (The family of
controllers are implemented using multirealization technique to achieve bumpless
transfer.)
Tank 3
Pump 1
1ν
Pump 1
2ν
1
y
2
y
Tank 1
Tank 4
Tank 2
Fig. 8. Schematic diagram of the quadruple-tank process.
Now, for each region, a decentralized MIMO-QFT design is
employed to attain robust stability and performance in spite of
uncertainties and disturbances.
The decentralized controllers and pre-filters are designed as
follows:
G1(s) =




185(s + 1.1)
s(s + 13.5)
0
0
10(s + 1.4)
s(s + 5.3)



 ,
F1(s) =



6
s + 6
0
0
5.8
s + 5.8



G2(s) =




0
57.9(s + 2.9)
s(s + 21.7)
38.4(s + 1.7)
s(s + 72.2)
0



 ,
F2(s) =



0
2.7
s + 2.7
6.5
s + 6.5
0



G3(s) =




23.1(s + 1)
s(s + 16.3)
0
0
18.6(s + 5.9)
s(s + 84.8)



 ,
F3(s) =



2.1
s + 2.1
0
0
1.9
s + 1.9



G4(s) =




0
95.6(s + 2.5)
s(s + 83.3)
24.7(s + 1.2)
s(s + 99.8)
0



 ,
F4(s) =



0
1.7
s + 1.74
1.8
s + 1.8
0


 .
Finally, a supervisory architecture determines the active
decentralized controller which should be placed in the feedback
loop. This selection is based on the values of the monitoring
signal. A schematic diagram of the overall multiple-model based
switching structure is depicted in Fig. 3.
Example 2 (Quadruple-Tank Process). The quadruple-tank process
shown in Fig. 8 is considered to illustrate the effectiveness
of the proposed design procedure. The tank consists of four
interconnected water tanks and two pumps. The input signals
are the voltages v1 and v2 applied to the two pumps and the
outputs are y1 and y2 representing the water level in tanks 1 and 2,
respectively. There are two valves that facilitate flows to the tanks.
One of the features of the linear dynamic model for the process
is the variable appropriate input–output pairing depending on
the user adjustable valve settings. This provides an interesting
challenge for linear decentralized design paradigms [5].
The control objective is the control of two lower tanks levels,
h1 and h2, using the two pumps. Using the mass balances and
Bernoulli’s law, the equations describing the plant are
dh1
dt
= −
a1
A1

2gh1 +
a3
A1

2gh3 +
γ1k1
A1
v1
dh2
dt
= −
a2
A2

2gh2 +
a4
A2

2gh4 +
γ2k2
A2
v2

Fig. 9. Results of switching PI control for the MP setting. The plots show simulations with the nonlinear physical model.
dh3
dt
= −
a3
A3

2gh3 +
(1 − γ2)k2
A3
v2
dh4
dt
= −
a4
A4

2gh4 +
(1 − γ1)k1
A4
v1
where hi, Ai and ai for i = 1, . . . , 4 denote the level, cross-section
and outlet hole cross sections of the ith tank, respectively. Also, g
denotes the gravity acceleration. In the above equations, kivi is the
corresponding flow to the pump voltage, vi. Moreover, γ1, γ2 ∈
(0,1) indicate the division ratio of flow using the valves. And, the
measured level signals are y1 = kc h1 and y2 = kc h2, where kc is a
measurement constant.
The linearized dynamics at a given stationary operating point is
determined from the state space
dx
dt
=











−
1
T1
0
A3
A1T3
0
0 −
1
T2
0
A4
A2T4
0 0 −
1
T3
0
0 0 0 −
1
T4











x
+











γ1k1
A1
0
0
γ2k2
A2
0
(1 − γ2)k2
A3
(1 − γ1)k1
A4
0











u
y =
[
kc 0 0 0
0 kc 0 0
]
with the variables xi := hi − ho
i and ui := vi − vo
i and the time
constants are given by Ti = Ai
ai

2 ho
i
g
, i = 1, . . . , 4. At a particular
ho
i , the stationary control signal is obtained by solving the following
equations
a1
A1

2gho
1 =
γ1k1
A1
vo
1 +
(1 − γ2)k2
A1
vo
2
a2
A2

2gho
2 =
(1 − γ1)k1
A2
vo
1 +
γ2k2
A2
vo
2.
Finally, the transfer function matrix corresponding to the
stationary operating point is
M(s) =




γ1T1k1kc
A1(1 + sT1)
(1 − γ2)T1k2kc
A1(1 + sT1)(1 + sT3)
(1 − γ1)T2k1kc
A2(1 + sT2)(1 + sT4)
γ2T2k2kc
A2(1 + sT2)



 .
The corresponding RGA is
Λ =
[
λ 1 − λ
1 − λ λ
]
where λ = γ1γ2
γ1+γ2−1
. If λ > 0 or 1 < γ1 +γ2 < 2 then (y1 −u1/y2 −
u2) is the appropriate input–output pair for decentralized control.
However, for 0 < γ1 + γ2 < 1, the closed-loop performance will
be better if y1 and y2 are permuted in the control structure and
the decentralized control is considered with pairings (y1, u2) and
(y2, u1).
It is also shown that the system is minimum phase (MP) for
1 < γ1 + γ2 < 2, and non-minimum phase (NMP) for 0 <
γ1 + γ2 < 1 that is inherently more difficult to control. Note that,
the aforementioned permutation, however, does not change the
location of the right half-plane zero, and experiments have shown
that the settling times are still much larger than the minimum-
phase setting [5]. The chosen operating points and the parameter
values corresponding to the MP and NMP settings are given in
Table 2.
In this section, decentralized PI control is applied to the
nonlinear process model. A decentralized PI controller as a

Fig. 10. Results of switching PI control for the NMP setting. The plots show simulations with the nonlinear physical model.
Table 2
Nominal operating conditions and parameter values of the quadruple-tank process.
Symbol State/parameters Value (MP setting) Value (NMP setting)
ho
1, ho
2, ho
3, ho
4 Nominal levels (cm) 12.4, 12.7, 1.8, 1.4 12.6, 13.0, 4.8, 4.9
vo
1, vo
2 Nominal pomp voltages (V) 3.00, 3.00 3.15, 3.15
A1, A2, A3, A4 Areas of the tanks (cm2
) 28, 32, 28, 32 28, 32, 28, 32
a1, a2, a3, a4 Areas of the drain in the tank (cm2
) 0.071, 0.057, 0.071, 0.057 0.071, 0.057, 0.071, 0.057
γ1 Ratio of flow in tank 1 to flow in tank 4 0.70 0.43
γ2 Ratio of flow in tank 2 to flow in tank 3 0.60 0.34
k1, k2 Pump constants (cm3
/Vs) 3.33, 3.35 3.14, 3.29
T1, T2, T3, T4 Time constants in the linearized model (s) 62, 90, 23, 30 63, 91, 39, 56
kc Measurement constant (V/cm) 0.50 0.50
G Gravitation constant (cm/s2
) 981 981
combination of two SISO PI controllers is considered. PI controllers
of the form
gl(s) = Kl

1 +
1
τils

, l = 1, 2
are tuned based on the linear physical model which is accurate and
agree with the response of the real process. Then, PI controllers
are incorporated into the proposed switching control structure to
deal with the variable input–output pairing of the multivariable
process.
For the MP setting, it is easy to find controller parameters
which yield good performance. The controller settings are adopted
from [5] as (K1, τi1) = (3.0, 30) and (K2, τi2) = (2.7, 40). Due to
the fact that the non-minimum phase process is normally harder
to control than the minimum phase one, it is difficult to find the
controller parameters which give good closed-loop performance
for the NMP setting. The controller parameters (K1, τi1) =
(0.94, 187.3) and (K2, τi2) = (0.99, 197.3) which are borrowed
from [25] stabilize the process and give reasonable performance.
Thus, the decentralized controllers are as follows:
G1(s) =




3.0

1 +
1
30s

0
0 2.7

1 +
1
40s




 ,
G2(s) =




0 0.94

1 +
1
187.3s

0.99

1 +
1
197.3s

0



 .
If a diagonal decentralized controller is employed, the appro-
priate input–output pairing corresponding to NMP setting leads
to much slower responses than in the minimum-phase case. The
settling time for the step responses, which is an important char-
acteristic of the system, is approximately ten times longer for the
nonminimum-phase setting [5]. However, in the case of a switch-
ing supervisory control design, the appropriate input–output pair
is recognized by the supervisor. Figs. 9 and 10 show that the decen-
tralized switching-PI controller can easily handle the two MP and
NMP settings.
To proceed with simulation, assume that the process is working
in MP settings. In addition, the hysteresis constant h in the
switching law (6) is set to be h = 0.1. Suppose that the second
candidate controller is connected into the loop initially, that is,
σ(0) = 2. Fig. 9 depicts the closed-loop input–output trajectories.
These exhibit satisfactory closed-loop performance. The switching
signal identifies the ‘‘right’’ controllers via 1 switch. Furthermore,
we set the first controller initially, i.e., σ(0) = 1, and carry out the
simulation again for the NMP settings. Simulation results are given
in Fig. 10. The ‘‘right’’ controller corresponding to the appropriate

input–output pair is identified by the supervisor and the closed
loop performance is good enough.
6. Conclusion
This paper presents the application of multivariable switching
multiple model based adaptive control for complex multivariable
plants. Large plant uncertainties lead to different input–output
pairings in multivariable plants. The proposed control structure
consists of a bank of candidate controllers and a supervisor. The
underlying design strategy is decentralized MIMO QFT. Each of
the candidate controllers is designed in order to achieve the
demanded performance in a subregion of the plant uncertainty.
The supervisor consists of a multi-estimator, a performance
signal generator and a hysteresis switching logic scheme. The
supervisor selects the active controller corresponding to the local
model which best fits the plant input–output data. Simulation
results are presented to show the effectiveness of the proposed
methodology in the face of varying control configuration in
complex multivariable plants.
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