A Novel Approach for Active Surge Control in Multistage Centrifugal
Compressor
YACINE ARIBI, RAZIKA ZAMMOUM BOUSHAKI
Signals and Systems Laboratory, Institute of Electrical and Electronics Engineering,
UniversitƩ M'hamed Bougara Boumerdes,
Avenue de lā€™indĆ©pendance, Boumerdes,
ALGERIA
Abstractā€” Compressors are used in the hydrocarbon industry from upstream to downstream to boost the
pressure required to transport gas into processing and transportation pipelines inside and toward processing
facilities. Turbo-centrifugal compressors are subject to repetitive damages resulting from the manifestation
phenomenon known as surge, observed as fluid flow reversal at low mass flows. Active surge control aims to
increase the compressor operating region by allowing the compressor to operate at low mass flow in the
naturally-unstable region of the compressor. Surge is avoided in the industry by forcing the operating point to
the stable region of the compressor curve. Active surge control aims to control the surge phenomenon by
extending the stable operating region. This work developed a novel model-based control approach for effective
compression system active surge control on an automatic coupled recycle valve as a primary actuator. The
feedback linearization technique is used to reduce the system complexity by providing a linear representation of
the highly nonlinear system. A detailed description of the feedback linearization method for centrifugal
compressors and a dual controller design was developed in this paper.
Keywords: - Centrifugal Compressor, Active Surge Control, Feedback Linearization, Nonlinear Systems, I/O
Linearization, PID Controller.
Received: November 24, 2021. Revised: October 22, 2022. Accepted: December 5, 2022. Published: January 26, 2023.
1 Introduction
In Centrifugal compressors, the potential rise of the
mass flow through the compressor to the nominal
values often results in a decrease of inlet fluid
density and hydrostatic pressure at the suction
plenum, thus, resulting in an even larger intake rate
from the gas storage vessels towards the centrifugal
compressor, [1]. Centrifugal compressors are
subject to a high oscillatory phenomenon that
emerges as an intense vibration and gas flow
reversal, this phenomenon is known as surge, which
is a dangerous unstable operating mode that triggers
inversion of the rotorā€™s axial push. Surge
prevention has been treated in the literature by anti-
pumping regulation systems to preserve the system
from reaching the surge line, in similar control
systems, the section flow is always larger than the
minimum flow that triggers the surge at any
compression rise. The surge prevention control is
based on a feed-backward loop that connects the
discharge and suction manifolds and is controlled
through a recycle valve that regulates the flow fed-
back, nevertheless, this method forces the system to
work at high feed flow which limits the gas
production and hence the efficiency of the
compressor. Since the compression system model is
nonlinear, the design of an active surge controller is
not a straightforward process because of the high
nonlinearities present in the system and the
undamped oscillatory behavior of the system.
Feedback linearization is a non-linear control
design method that has attracted lots of research in
recent years, [2], [3], [4], as it involves
transforming the nonlinear system into a
controllable linear system by using state feedback
and coordinate transformations. The input-output
decoupling method is used to transform the
nonlinear system into a new linear-equivalent form
where each output is independently and uniquely
controlled by only one of the newly defined inputs.
In this work, a combined approach aims to use a
speed regulator and an overall feedback
linearization on a centrifugal compression system
as a novel approach to tackle the active surge
control problem in linear form. A linear PID
controller, which is the gas production industryā€™s
first choice for the recycle actuators control, is used
to control the linearized system. Furthermore,
traditional PID design techniques are combined
with pole placement techniques to ensure the
WSEAS TRANSACTIONS on POWER SYSTEMS
DOI: 10.37394/232016.2023.18.1
Yacine Aribi, Razika Zammoum Boushaki
E-ISSN: 2224-350X
1
Volume 18, 2023
overall system stability for a linear controller meant
to control a complex and dangerous industrial
process.
2 Centrifugal Compressor Model
The mathematical model adapted in this paper and
used as a backbone to create the simulation model
is based on the model provided by [5] and [6] and
stated in equation (1), this model is widely accepted
as a solid description for centrifugal and axial
compressors and has found wide acceptance in
literature, [5], [6].
ī³•
ī³–
ī³–
ī³–
ī³”
ī³–
ī³–
ī³–
ī³“
īµŒó°‡—ī„³ļŒī“½ī„²ī„³
ī„“
ī“øī„³ļˆļ«ī“³īµ‚ļ†ī“³ļ…ī“³īµŽļÆ
īµŒó°‡—ī„“ļŒī“½ī„²ī„³
ī„“
ī“øī„“ļˆļ«ī“³ļ†ī˜ƒī“³īµŽļ†ī“³īµļÆ
ī“³ó°‡—ļŒī“£ī˜ƒ
ī“®ļˆļ«ó°©µó°‡›ī“³īŸ”ī·±ó°‡œļˆīµŒī„³ļ†īµŒī„“ļÆ
ī·±ó°‡—ļŒī„³ī“¬ļˆó°‡›ī·¬īµ€ļ†ī·¬ī“æ󰇜
(1)
The model is realizable as per the diagram in Fig. 1.
The state space model of equation (1), with four (4)
states is used to simulate the disturbance effect on
the system, to define the surge limit, and to test the
control system efficiency.
The compressor parameters, set points, and ambient
conditions used for simulation purposes are defined
in Table 1 in section 7.
To check the stability of the system in equation (1)
around the systemā€™s equilibrium point, direct
stability analysis shows that the uncontrolled
system is stable if and only if the system is
operating at ļ‡—ļˆ…
ļ‡—ļ‡Šļ‘ī·, thus the operating point is at
the negative slope, or at most, small positive slope
of the ī·–ļ†ī“³ characteristic, this conclusion is
derived by analyzing at a single constant Speedline
ī·±ļ„“ and by considering the non-controlled
compressor case, thus, setting the recycle flow ī“³ļ‡„
to zero, then linearizing equations (1) around the
equilibrium point and substituting the estimated
values into the original state-space equation.
3 Feedback Linearization
Feedback linearization is a widely used control
technique for nonlinear systems. This model is
based on developing a transformation envelope for
the nonlinear system in such a way the residual
system is a linear equivalence for the original
nonlinear system, this transformation is obtained
via a change of variables (inputs and/or outputs)
and a set of controls, both methods work to cancel
the nonlinearities in the system resulting in a fully
or partially linear closed-loop dynamics so that the
well-known linear techniques that we have
developed in linear control systems and
multivariable control system courses can be
applied.
Feedback linearization has two major types: input-
state linearization and input-output linearization
depending on whether full or partial linearization of
the system is required.
Input-state linearization is focused on developing
transformation for the state variables and a
linearizing input that linearizes the whole system.
On the other hand,
input-output linearization is a method focused on
establishing the relationship between the nonlinear
system inputs and outputs by defining the I/O map
and then linearizing this map. If the class of
nonlinear systems in equation (2) is considered a
model:
ļœīµ”ó°‡—ļŒīµ‚󰇛īµ”󰇜ļ…īµƒó°‡›īµ”󰇜īµ‘
īµ•ļŒīµ„󰇛īµ”󰇜
(2)
Where:īµ‚īŸ£ī“¦ī­œī¬¹ļ‡” ,īµƒīŸ£ī“¦ī­œī¬¹ļ‡”ļˆļ‡¤ all defined on
the domain ī“¦ī°æī¬¹ļ‡” . n is the systemā€™s degree, and
q is the number of outputs.
The transformationā€™s ultimate objective is to define
the control signal: īµ‘ļŒī·™ó°‡›īµ”󰇜ļ…ī·šó°‡›īµ”󰇜īµ’
Along with variables transformation īµ–ļŒī“¶ó°‡›īµ”󰇜 to
fully or partially transform the nonlinear system,
[7].
Fig. 1: Drawing for compressor with recycle line
WSEAS TRANSACTIONS on POWER SYSTEMS
DOI: 10.37394/232016.2023.18.1
Yacine Aribi, Razika Zammoum Boushaki
E-ISSN: 2224-350X
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Volume 18, 2023
3.1 Input-Output linearization
To execute an input-output linearization that sets a
relation between the output function īµ• and the input
īµ‘ (the control signal), a set of repetitive
differentiations is carried until the relationship is
visible, then the input īµ’, is designed to cancel the
system nonlinearities, [8].
ī˜ƒīµ•ó°‡›īµó°‡œ
īµ’󰇛īµó°‡œļŒī„³
īµļ‡„
(3)
Nevertheless, if the systemā€™s relative degree is
undefined, the controlling input īµ’ cannot be
designed to cancel the systemā€™s nonlinearities.
3.2 Well-defined Relative Degree
A single input single output (SISO) system of
degree īµŠ similar to the form in equation (2) is said
to have relative degree īµŽ in a region ī ³ī˜ƒī˜ƒī˜ƒif, īƊī˜ƒīµ”īƐ
ī ³ī˜ƒī˜ƒ, [9]. where: īµŽļ‘īµŠ.
ī“®ļ‡šī“®ļ‡™ļ‡œīµ„󰇛īµ”󰇜ļŒī„²ī˜ƒī˜ƒī˜ƒī˜ƒī˜ƒī˜ƒī˜ƒī˜ƒī˜ƒī˜ƒī˜ƒī˜ƒī˜ƒī˜ƒī˜ƒī˜ƒī˜ƒī˜ƒī˜ƒī„²ļ‘ī˜ƒīµ…
ļīµŽļ†ī„³
ī“®ļ‡šī“®ļ‡™ļ‡„ļ„æļ„µīµ„󰇛īµ”󰇜ļī„²
(4)
Then the linearizing input can be defined as:
īµ•ó°‡—ļŒīƏīµ„󰇛īµ‚ļ…īµƒīŸ¤īµ‘󰇜ļŒī“®ļ‡™īµ„󰇛īµ”󰇜ļ…ī“®ļ‡šīµ„󰇛īµ”󰇜īŸ¤īµ‘
īµ•ó°‡˜ļŒī“®ļ‡™ļ„¶īµ„󰇛īµ”󰇜ļ…ī“®ļ‡šī“®ļ‡™īµ„󰇛īµ”󰇜īŸ¤īµ‘
īµ•ó°‡›ļ‡œó°‡œļŒī“®ļ‡™ļ‡œīµ„󰇛īµ”󰇜ļ…ī“®ļ‡šī“®ļ‡™ļ‡œļ„æļ„µīµ„󰇛īµ”󰇜īŸ¤īµ‘
If the system has relative degree r, then we can
write:
īµ‘ļŒ ī„³
ī“®ļ‡šī“®ļ‡™ļ‡„ļ„æļ„µīµ„󰇛ļ†ī“®ļ‡™ļ‡„īµ„ļ…īµ’󰇜
(5)
Since:ī˜ƒī“®ļ‡šī“®ļ‡™ļ‡„ļ„æļ„µīµ„󰇛īµ”󰇜ļī„²
By replacing īµ‘ from equation (5) into īµ•ó°‡›ļ‡„󰇜 :
ī˜ƒīµ•ó°‡›ļ‡„󰇜ļŒīµ’
(6)
A nonlinear system is said to have an undefined
relative degree if at the operating point
ī“®ļ‡šī“®ļ‡™ļ‡„ļ„æļ„µīµ„ļŒī„² but nonzero at some arbitrary point
near the operating point.
3.3 Zero Dynamics
I/O feedback linearized system is composed of an
external segment (input-output) and an internal
segment, where the internal segment is
unobservable. The observable external part of the
system can be easily used to design a control signal
to control the input īµ’ via negative feedback to
obtain a desired output īµ• with behavior that
matches our expectations. Since the new input v
does not affect the internal dynamics, their behavior
must be taken into consideration. It is important to
note that to design an I/O controller, the internal
dynamics of the system have to be stable, [10].
By applying the notion of normal forms, when the
relative degree īµŽ is defined and īµŽļīµŠ, the
nonlinear system (equation (2)) can be transformed
into the normal form using the new states:
īµ•ī˜ƒīŸ”īµ•ó°‡—ī˜ƒīŸ”īŸ„īŸ”īµ•ó°‡›ļ‡„ļ„æļ„µó°‡œ.
In this case, and if we define:
ī·¤ļŒó°‡Ÿī·¤ļ„µī·¤ļ„¶ī˜ƒī˜ƒīŸ„ī˜ƒī˜ƒī˜ƒī˜ƒī˜ƒī˜ƒī˜ƒī·¤ļ‡„󰇠ļ‡= 󰇟īµ•ī˜ƒī˜ƒī˜ƒī˜ƒīµ•ó°‡—ī˜ƒī˜ƒīŸ„ī˜ƒī˜ƒī˜ƒī˜ƒī˜ƒī˜ƒī˜ƒīµ•ó°‡›ļ‡„ļ„æļ„µó°‡œó°‡ ļ‡
Then in a neighborhood of the operating point ī ³,
the normal form of the system is:
ī·¤ó°‡—ļŒļ¦ī·¤ļ„¶
īŸ„
īŸ„
ī“½ó°‡›ī·¤īŸ”ī ²ó°‡œļ…ī“¾ó°‡›ī·¤īŸ”ī ²ó°‡œīµ‘ļŖ
(7)
With the output defined as īµ•ļŒī·¤ļ„µīŸ¤
The last 󰇛īµŠļ†īµŽó°‡œ equations ī·–ó°‡—ļŒīµ“󰇛ī·¤īŸ”ī·–󰇜 of the
normal form defines the internal dynamics for the
I/O linearized form.
The zero dynamics are identified by placing the
restraint that the output īµ• and all its derivatives in
the multidimensional space of dimension ļŒī˜ƒó°‡›īµŠļ†
īµŽó°‡œ, are all equal to zero, i.e., īµ•ļŒī„², and īµ•ó°‡›ļ‡œó°‡œļŒī„²,
for i=0,ā€¦,(n-r).
The zero dynamics describes the systemā€™s internal
dynamics in a restricted 󰇛īµŠļ†īµŽó°‡œ-dimensional
smooth surface defined by ī·¤ļŒī„².
Calculating the zero dynamics allows us to build a
conclusion on the internal dynamicsā€™ stability.
From equation (5) the input must satisfy:
īµ‘ļŒ ļ†ī“®ļ‡™ļ‡„īµ„
ī“®ļ‡šī“®ļ‡™ļ‡„ļ„æļ„µīµ„
(8)
By assuming that the systemā€™s initial state is
located in the surface defined by ī·¤ļŒī„², i.e., that
ī·¤ó°‡›ī„²ó°‡œļŒī„², the normal form of the system dynamics
is:
WSEAS TRANSACTIONS on POWER SYSTEMS
DOI: 10.37394/232016.2023.18.1
Yacine Aribi, Razika Zammoum Boushaki
E-ISSN: 2224-350X
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Volume 18, 2023
ļœī·¤ó°‡—ļŒī„²
ī˜ƒī ²ó°‡—ļŒīµ“󰇛ī·¤īŸ”ī ²ó°‡œ
(9)
The part corresponding to the internal dynamics is:
ī“½ó°‡›ī·¤īŸ”ī ²ó°‡œļŒī“®ļ‡™ļ‡„īµ„󰇛īµ”󰇜, ī“¾ó°‡›ī·¤īŸ”ī ²ó°‡œļŒī“®ļ‡šī“®ļ‡™ļ‡„ļ„æļ„µīµ„󰇛īµ”󰇜 and
ī“®ļ‡šī ²ļ‡ļŒī„²ī˜ƒī˜ƒī˜ƒī˜ƒī˜ƒī˜ƒī„³ļ‘ī˜ƒīµ†ļ‘īµŠļ†īµŽ
By assuming that the system in equation (1) has
well defined relative degree, and the zero dynamics
are asymptotically stable around the operating
point, if we design a control input īµ’ that places the
roots of the new linearized system inside the stable
region, then the control law in equation (5) will
produce an asymptotically stable closed-loop
system around the operating point, [10].
For the system to obtain local asymptotic stability
in the vicinity of the operating point, it is enough to
ensure that the system has stable zero dynamics.
4 Feedback Linearization of the
Centrifugal Compressor
4.1 Compressor Flow Design
The system has one input which is the percentage
of opening the recycle valve and one main variable
which is the compressor flow, hence we could
rewrite our system by defining the recycle valve as
an additional state to the system, based on the mass
balance. In this case, the recycle flow ī“³ļ‡„ is
considerably taken into the calculations of mass and
energy balances at all plenums, this will result in
the complete design of equation (1) for the
controlled system.
By setting the following constants:
īµ‡ļ„µļŒī“½ļ„“ļ„µļ„¶
ī“øļ„µī˜ƒīŸ”īµ‡ļ„¶ļŒī“½ļ„“ļ„µļ„¶
ī“øļ„¶īŸ”īµ‡ļ„·ļŒī“£
ī“®īŸ”ī“­ļ„øļŒī„³ī“¬
The feed, throttle, and recycle flows are defined
below ī“³ļ‡™ļŒīµ‡ļ„µļ„¶ļˆļŽ„ī īµŒļ„“ļ„µļ†īµŒļ„µī 
ī“³ļ‡§ļŒīµ‡ļ„µļ„·ļˆļŽ„ī īµŒļ„¶ļ†īµŒļ„“ļ„µī 
ī“³ļ‡„ļŒīµ‘ļˆī“£ļ‡§ļˆļŽ„ī īµŒļ„¶ļ†īµŒļ„µī 
Where īµ‡ļ„µļ„¶ and īµ‡ļ„µļ„· are positive integers that
resemble the valve opening and īµ‘ī˜ƒis the recycle
valve opening control proportional to the valve
opening. The system in equation (9) can be written
as:
ī³•
ī³–
ī³”
ī³–
ī³“
īµŒļ„µó°‡— ļŒī˜ƒīµ‡ļ„µļˆó°‡›īµ‡ļ„µļ„¶ļˆļŽ„ī īµŒļ„“ļ„µļ†īµŒļ„µī ļ†ī“³ļ…īµ‘ļˆī“£ļ‡§ļˆļŽ„ī īµŒļ„¶ļ†īµŒļ„µī ó°‡œ
īµŒļ„¶ó°‡— ļŒī˜ƒīµ‡ļ„¶ļˆó°‡›ī“³ļ†īµ‘ļˆī“£ļ‡§ļˆļŽ„ī īµŒļ„¶ļ†īµŒļ„µī ļ†īµ‡ļ„µļ„·ļˆļŽ„ī īµŒļ„¶ļ†īµŒļ„“ļ„µī ó°‡œ
ī“³ó°‡—ļŒīµ‡ļ„·ļˆó°‡›ó°©µó°‡›ī“³īŸ”ī·±ó°‡œļˆīµŒļ„µļ†īµŒļ„¶ó°‡œ
ī·±ó°‡—ļŒīµ‡ī„¶ļˆó°‡›ī·¬ļ‡—ļ†īµ‡ī„³ī„¶ļˆī ī“³ī ļˆī˜ƒī·±ó°‡œ
(10)
Using a speed regulation module, the speed can be
considered constant while operating the compressor
on a constant Speedline, which allows the
assumption that the speed is constant. By setting:
īµ”ļ„µļŒīµŒļ„µīŸ”īµ”ļ„¶ļŒīµŒļ„¶īŸ”īµ”ļ„·ļŒī“³ and by substituting in
equation (10) and projecting to equation (1), we
get:
īµ‚󰇛īµ”󰇜ļŒļ¦īµ‡ļ„µļˆó°‡›īµ‡ļ„µļ„¶ļˆļŽ„ī īµŒļ„“ļ„µļ†īµ”ļ„µī ī˜ƒļ†īššļ„·ó°‡œ
īµ‡ļ„¶ļˆó°‡›īµ”ļ„·ļ†īµ‡ļ„µļ„·ļˆļŽ„ī īµ”ļ„¶ļ†īµŒļ„“ļ„µī ī˜ƒó°‡œ
īµ‡ļ„·ļˆó°‡›ó°©µó°‡›īµ”ļ„·ó°‡œļˆīµ”ļ„µļ†īµ”ļ„¶ó°‡œļŖ
īµƒó°‡›īµ”󰇜ļŒó°‡Æīµ‡ļ„µļˆī“£ļ‡§ļˆļŽ„īµ”ļ„¶ļ†īµ”ļ„µ
ļ†īµ‡ļ„¶ļˆī“£ļ‡§ļˆļŽ„īµ”ļ„¶ļ†īµ”ļ„µ
ī„²ó°‡°
īµ•ļŒīµ”ļ„·
(11)
by calculating the respective lie derivatives, we get:
ī»¦ļ‡šī»¦ļ‡™ļ„“īµ•ó°‡›īµ”󰇜ļŒī„²
ī»¦ļ‡šī»¦ļ‡™ļ„µīµ•ó°‡›īµ”󰇜ļī„²
Thus, the relative degree of the system is īµŽļŒī„“ļ
īµŠ and the system is Input-Output Linearizable.
The new control signal is then defined based on
equation (6):
īµ’ļŒīµ•ó°‡›ļ‡„󰇜ļŒīµ”ļ„·ó°‡˜
(12)
Which gives: īµ’ļŒīµ”ī„µó°‡˜
Thus,
īµ’ļŒīµ‡ī„µļˆó°‡›ó°©µó°‡—󰇛īµ”ļ„·ó°‡œīŸ¤īµ”ļ„µ
ļ…󰩵󰇛īµ”ļ„·ó°‡œīŸ¤īµ‡ī„³īŸ¤ó°‡”īµ‡ī„³ī„“īŸ¤ļŽ„ī īµŒļ„“ļ„µļ†īµ”ļ„µī 
ļ…īµ‘īŸ¤ī“£ļ‡§īŸ¤ļŽ„ī īµ”ļ„¶ļ†īµ”ļ„µī ī˜ƒī˜ƒļ†īššļ„·ó°‡¢
ļ†īµ‡ī„“īŸ¤ó°‡”īµ”ļ„·ļ†īµ‡ī„³ī„µīŸ¤ļŽ„ī īµ”ļ„¶ļ†īµŒļ„“ļ„µī ī˜ƒ
ļ†īµ‘īŸ¤ī“£ļ‡§īŸ¤ļŽ„ī īµ”ļ„¶ļ†īµ”ļ„µī ó°‡¢
The linearizing input is then defined as:
īµ‘ļŒ
ļ„æ󰩵󰇛ļ‡«ļˆÆ󰇜
󰇗īŸ¤ļ‡«ļˆ­ļ„æ󰩵󰇛ļ‡«ļˆÆ󰇜ļ‡„ļˆ­ļ«ļ‡žļˆ­ļˆ®ļŽ„ī ļ‡£ļˆ¬ļˆ­ļ„æļ‡«ļˆ­ī ļ„æļ‡«ļˆÆļÆļ„¾ļ‡žļˆ®ļ«ļ‡«ļˆÆļ„æļ‡žļˆ­ļˆÆļŽ„ī ļ‡«ļˆ®ļ„æļ‡£ļˆ¬ļˆ­ī ļÆļ„¾ļˆ­
ļ‹–ļˆÆļ‡©
ļŽ„ī ļ‡«ļ„¶ļ„æļ‡«ļ„µī ļ†ŗļ‹Ÿó°‡›ļ‡žļ„µīƛ󰩵󰇛ļ‡«ļ„·ó°‡œļ„¾ļ‡žļ„¶ó°‡œ
(13)
WSEAS TRANSACTIONS on POWER SYSTEMS
DOI: 10.37394/232016.2023.18.1
Yacine Aribi, Razika Zammoum Boushaki
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Volume 18, 2023
Once the surge phenomenon is triggered, the mass
flow experiences an increasing oscillatory behavior,
depicting an instability in the mass flow and hence
all other system states. The whole system can be
stabilized by stabilizing the surge-triggering state,
which is the mass flow.
By setting the new state variables and control signal
as:
īŗ ļŒļ„īµ–ļ„µ
īµ–ļ„¶
īµ–ļ„·ļ©ļŒļ„īµ”ļ„·
īµ”ļ„·ó°‡—
īµ”ļ„·ó°‡˜ļ©
īµ’ļŒīµ”ļ„·ó°‡˜
(14)
The linearized system is written in Brunovsky
form, [11].
ļžīµ–ó°‡—ļŒļ„ī„² ī„³ ī„²
ī„² ī„² ī„³
ī„² ī„² ī„²ļ©īµ–ļ…ļ„ī„²
ī„²
ī„³ļ©īµ’
īµ•ļŒó°‡Ÿī„³ ī„² ī„²ó°‡ īµ–
(15)
This form is written for simplicity as
ļœīµ–ó°‡—ļŒī¹­īµ–ļ…ī¹®īµ’
īµ•ļŒī¹Æīµ–
(16)
The output deviation signal is defined as the
deviation of the actual output from the desired
output which is ideally equal to zero for perfect
control. The error is defined in the frequency
domain for the linearized system as:
ī¹±ó°‡›īµó°‡œļŒī¹¾ó°‡›īµó°‡œļ†īŗ…󰇛īµó°‡œ
ī¹±ó°‡›īµó°‡œļŒī¹¾ó°‡›īµó°‡œó°‡›ī„³ļ†ī¹Æ󰇛īµī¹µļ†ī¹­ó°‡œļ„æļ„µīŸ¤ī¹®ó°‡œ
Using the final value theorem and by assuming
īµó°‡›īµó°‡œļŒī„²ī˜ƒīµ‚īµ‹īµŽī˜ƒīµļī„², we can write the steady state
error as:
īšŽīš‹īš
ļ‡¦ī­œļ„“īµī¹±ó°‡›īµó°‡œļŒīšŽīš‹īš
ļ‡¦ī­œļ„“īµī¹¾ó°‡›īµó°‡œó°‡›ī„³ļ†ī¹Æ󰇛īµī¹µļ†ī¹­ó°‡œļ„æļ„µó°‡œī¹®
Applying the linearized compression system, we
get: ī¹±ó°‡›īµó°‡œļŒļ†ī˜ƒī„³īµīŸ¤ó°‡›ī„³
īµļ„·ī˜ƒļ†ī˜ƒī„³ó°‡œ
Then the steady-state error is: Error =
īšŽīš‹īš
ļ…±ī­œļ„“󰇛īµī¹±ó°‡›īµó°‡œó°‡œ= āˆž
Since the system is unstable, an increasing
oscillation with time is incepted for the non-
controlled system which leads to a theoretical
infinity blow-up of gases.
5 Pole Placement and Steady-State
Error Design
Taking the linearized form of equation (15), the
feedback-linearized system with the external linear
control is visualized in Fig. 2.
Fig. 2: Steady-state error control with PI.
The Problem has been reduced to a linear SISO
system with three states. The control problem can
be tackled by either pole placement, PID controller,
adaptive PID controller, or simple Type I Fuzzy
controller, However, hybrid PID controllers and
pole placement methods can be more efficient
while designing the system for a preset transient
and steady-state performance. The system in Fig.2,
with a PI controller, extends the state space
description to:
ļīµ–ó°‡—ļŒī¹­īµ–ļ…ī¹®īµ’
ī˜ƒīµ‰ó°‡—ī˜ƒļŒļ†ī¹Æīµ–ļ…ī˜ƒīµŽ
īµ•ļŒī¹Æīµ–
(17)
The control signal from the modified PI controller
will be set then to:
īµ’ļŒīµ‡ļ‡£īƛīµ–󰇛īµó°‡œļ…īµ‡ļ‡‚īƛļŽ±īµó°‡›īµó°‡œīµ€īµ
ļ‡§
ļ‡§ļˆ¬
(18)
Substituting in the original system:
󰇣īµ–ó°‡—
īµ‰ó°‡—󰇤ļŒó°‡£ī¹­ ī„²
ļ†ī¹Æ ī„²ó°‡¤ó°‡£īµ–
īµ‰ó°‡¤ļ…󰇣ī¹®
ī„²ó°‡¤īµ’ļ…󰇣ļƒ™
ī„³ó°‡¤īµŽ
īµ•=󰇟ī¹Æ ī„²ó°‡ ó°‡£īµ–
īµ‰ó°‡¤
īµ’=ī¹·ļ‡£īµ–ļ…ī“­ļ‡‚īµ‰=󰇟ī¹·īŗ–ī“­ļ‡‚󰇠󰇣īµ–
īµ‰ó°‡¤
(19)
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Hence the following model:
󰇣īµ–ó°‡—
īµ‰ó°‡—󰇤ļŒļ¤ī¹­ļ…ī¹®ī¹·īŗ–ī¹®ī“­ļ‡‚
ļ†ī¹Æ ī„² ļØ󰇣īµ–
īµ‰ó°‡¤ļ…󰇣ļƒ™
ī„³ó°‡¤īµŽ
īµ•=󰇟ī¹Æ ī„²ó°‡ ó°‡£īµ–
īµ‰ó°‡¤
(20)
By defining desired performance parameters, and a
desired characteristic equation, ī¹·īŗ– and ī“­ļ‡‚ are used
to reach a zero steady-state error design for the
response and to optimize the oscillation damping
time.
5.1 Application to the Compressor Flow
To define a desired transient response as a frame of
reference, the desired settling time of 0.5seconds
and a percent overshoot of 5% will be adopted as
desired conditions, this requires setting up a system
with natural frequency = 11.59 and damping factor
= 8. we get the following characteristic equation:
ī“¦ó°‡›īµó°‡œļŒīµļ„¶ļ…ī„³ī„øīµļ…ī„³ī„µī„¶īŸ¤ī„¶
(21)
The third and fourth poles are set as double poles
for simplicity and are set to be insignificant
compared to the dominant pole. Based on this
design, the characteristic equation (22) is developed
to a 4th-order denominator:
ī“¦ó°‡›īµó°‡œļŒó°‡›īµļ„¶ļ…ī„³ī„øīµļ…ī„³ī„µī„¶īŸ¤ī„¶ó°‡œó°‡›īš•ļ…ī„“ī„²ó°‡œó°‡›īš•ļ…ī„“ī„²ó°‡œ
=īµļ„øļ…ī„·ī„øīµļ„·ļ…ī„³ī„³ī„¹ī„¶īµļ„¶ļ…ī„³ī„³ī„¹ī„ŗī„²īµļ…ī„·ī„µī„¹ī„·ī„²
(22)
ī¹·īŗ– is a three-element vector defined as
ī¹·ļ‡£ļŒó°‡Ÿī“­ļ‡£ļ„µ ī“­ļ‡£ļ„¶ ī“­ļ‡£ļ„·ó°‡ 
while ī“­ļ‡‚ is a scalar.
Two of the poles in equation (22) will be tuned by
the PI controller for zero steady-state error, and the
remaining poles can be placed using the pole
placement method to reduce their effect on the
transient performance.
Returning now to the original system, and
substituting the PI controls:
ļ¦īµ–ļ„µó°‡—
īµ–ļ„¶ó°‡—
īµ–ļ„·ó°‡—
īµ‰ó°‡—ļŖ=ļ¦ī„² ī„³ ī„² ī„²
ī„² ī„² ī„³ ī„²
ī“­ļ‡£ļ„µ ī“­ļ‡£ļ„¶ ī“­ļ‡£ļ„· ī“­ļ‡‚
ļ†ī„³ ī„² ī„² ī„²ļŖó°‡Æīµ–ļ„µ
īµ–ļ„¶
īµ–ļ„·
īµ‰ó°‡°+ļ¦ī„²
ī„²
ī„²
ī„³ļŖīµŽ
(23)
īµ•ļŒó°‡Ÿī„³ ī„² ī„² ī„²ó°‡ ī˜ƒó°‡Æīµ–ļ„µ
īµ–ļ„¶
īµ–ļ„·
īµ‰ó°‡°
The characteristic polynomial of this system is:
ī¦æ󰇛īµó°‡œļŒīš†īš‡īš–󰇛īµī¹µļ†ī¹­
ļŒ„󰇜
ī¦æ󰇛īµó°‡œ
ļŒīš†īš‡īš–ļ®ļ¦ īµ ļ†ī„³ ī„² ī„²
ī„² īµ ļ†ī„³ ī„²
ļ†ī“­ļ‡‰ļ„µ ļ†ī“­ļ‡‰ļ„¶ īµļ†ī“­ļ‡£ļ„· ļ†ī“­ļ‡‚
ī„³ ī„² ī„² īµ ļŖļ²
ī¦æ󰇛īµó°‡œļŒīµļ„øļ†ī“­ļ‡£ļ„·īŸ¤īµļ†ī“­ļ‡£ļ„¶īŸ¤īµļ†ī“­ļ‡£ļ„µīŸ¤īµļ…ī“­ļ‡‚
(24)
By identification we get:
ī“­ļ‡£ļ„·ļŒļ†ī„·ī„øīŸ”ī“­īµŒī„“ļŒļ†ī„³ī„³ī„¹ī„¶īŸ”ī“­īµŒī„³ļŒļ†ī„³ī„³ī„¹ī„ŗī„²īŸ”ī“­ī“«
ļŒī„·ī„µī„¹ī„·ī„²
Fig. 3: Designed step response
Fig. 4: Feed Flow Valve Opening (0 completely
closed- 1 Completely open) vs time.
Fig. 3 Shows the desired mass flow step response
with a reference gain of 53750, knowing that the
amplitude is an arbitrary set.
At t=5sec, the feed flow valve is chocked to 20%, if
this disturbance is applied to the throttle valve, an
improvement in the systemā€™s stability will be
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observed. Fig. 4 and Fig. 5 show the feed flow
valve closure pattern and the development of the
Feed Flow vs. time.
Fig. 5: Feed Flow (kg/sec) vs time (uncontrolled).
In the compressor characteristic, shown in Fig. 6,
and after applying the feed flow disturbance, the
operating point perfectly stabilized at the desired
flow without any observable flow reversal or
fluctuation at the Speedline corresponding to
30000rpm which is set as the desired speed.
In Fig. 6, while the feed flow rises into the
desirable suction value, the speed regulator pushes
the whole system to stabilize toward the 30000rpm
Speedline, when the system reached the desired
Speedline, the operating point stabilizes till the feed
flow is disturbed at t= 5sec.
Fig. 6: Controlled Compressor map
At the time t=2 seconds, the feed flow valve is
closed and reopened to ensure that the system is
stabilized at the required Speedline and the
controller is responding to the changes in the
system, [12], [13], this also meant to push the
compressor to the surge line limit through reducing
the mass flow and increasing the pressure rise to
preliminary assess the system behavior. The desired
mass flow through the compressor is set to 0.1 kg/s
once the compressor stabilizes to be able to test the
system without having a negative control signal that
results in a full valve closure.
In Fig. 7a and Fig. 7b, the controller is deactivated
by completely closing the recycle valve to assess
the effect of the controller on the stability of the
system. The mass flow in Fig. 7a and the pressure
ratio in Fig. 7b suffers from large oscillations when
the feed flow valve is closed with the controller
deactivated, the flow reversal (negative mass flow)
can be observed.
The Controlled System in Fig. 8 shows more stable
plenum pressures with no flow reversal when
operated near the surge area, denoting that the
linear controller can stabilize the system operation
near the surge line and dampen the sinusoidal
oscillations originally generated by the non-
controlled system. The peak in the transient reached
at the time t=7.2s may result in a perturbation to the
physical system as it presents a significant drop in
the mass flow and the plenum-1 pressures,
nevertheless, this perturbation is not significantly
visible in the plenum-2 pressure at the centrifugal
compressor outlet since it is partially dampened via
the recycle line.
From Fig. 8f, it is noticeable that as we tend to push
the operating point to the Speedlineā€™s most efficient
point, the control signal moves toward negative
values which means a total closure for the recycle
valve, which is physically impossible due to the
non-realistic sense of valve ā€œnegativeā€ closure, thus
the control signal is limited to a minimum value of
zero. For large speeds, the compressor can be
operated on a more efficient pattern with ī“³ļ‡„ļī“³.
Once the mass flow stabilizes in the normally-
unstable region at 0.1kg/sec, the control signal
decreases to a minimum that ensures zero steady-
state error.
020 40 60 80
1
1.2
1.4
1.6
1.8
2
2.2
X: 6.204
Y: 1.389
Corrected mass flow lb/min
Pressure ratio
Compressor charcterstic
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Fig. 7a: Feed Flow for under surge (uncontrolled)
Fig. 7b: Pressure ratio for under surge
(uncontrolled).
6 Conclusion
In this work, feedback linearization has been
presented along with the Moor-Greitzer compressor
model. A composite PI controller has been used on
a linearized compression system with the aim of
extending the compressorā€™s effective working area
and avoiding surge. The controller was designed to
allow the compressor to operate at maximum
efficiency while stabilizing the flow and the main
drive speed. The linearization of the system, when
the internal dynamics are stable, results in a system
that is both, less complex and easier to control.
This methodology can overcome significant
uncertainties in nominal models, [14], and can be
generalized to a considerable class of model-based
systems. The linear controllers designed using the
state space approach for both the drive speed
control and the mass flow control permits to
development of a new approach for centrifugal
compressorsā€™ active surge control.
The linearized model was simulated using the
polynomial approximation, this allowed the
operating point to be extended up to near-zero mass
flow while the system is operating under the
controller and the linearization blocks. The
combination of Feedback linearization, linear PID
controllers, and pole placement techniques in this
paper were able to extend the operating region to
cover up, theoretically, the whole right upper-hand
quarter of the characteristics plane allowing the
compressor to perfectly operate at the left of the
surge line while preventing the natural flow
reversal and the surge induced fluctuations.
The Results obtained in this work reduce the
system complexity and allow for the application of
other control techniques to reduce the transient time
and transient peak to allow for a stable and
homogeneous operating mode. Focusing on
damping the transient mode in the controlled
system through the controllerā€™s design enhancement
can result in smooth operational curves for sensitive
processes with large nonlinearities similar to the
centrifugal gas compression systems. In addition to
the above, simulating the system with a pulsation-
damping built-in pulsation-damping system may
result in smoother transient conditions.
7 Nomenclature
Table 1. Nomenclature and Abbreviations
Parameter
Definition
Value
(UM)
ī“²ī“«ī“¦
Proportional-Integral-Derivative
--
ī“µī“«ī“µī“±
Single-Input Single-Output
--
ī“«ī €ī“±
Input/Output
--
ī“²ī“«
Proportional-Integral
--
ī“²ļ„“ļ„µ
Atmospheric Pressure
101325 (ī“²ī“½)
ī“²ļ„µ
Plenum-1 Pressure
ī“²ī“½
ī“øļ„µ
Plenums-1 Volumes
0.07 (īµ‰ļ„·)
ī“²ļ„¶
Plenum-2 Pressure
ī“²ī“½
ī“øļ„¶
Plenum-2 Volumes
0.12 (īµ‰ļ„·)
ī“³
Mass flow
īµ‡īµƒī €īµī˜ƒ
ī“£
Duct Cross-section Area
0.08 (īµ‰ļ„¶)
ī“®
Duct Length
2.93 (īµ‰)
ī·–ļ‡–
Pressure rise
--
ī·±
Rotation velocity
rī“½īµ€ī €īµ
ī“³ļ‡™īŸ”ī“³ļ‡§īŸ”ī“³ļ‡„
Feed, Throttle, and Recycle flows
īµ‡īµƒī €īµ
īµŽļ„µ
Inducer Radius
0.0393 (īµ‰)
īµŽļ„¶
Impeller Radius
0.0567 (īµ‰)
ī·¬ļ‡—īŸ”ī·¬ļ‡–
Drive and Compressor torques
ī“°ļˆīµ‰
ī“¬
Impeller inertia
īµ‡īµƒļˆīµ‰ļ„¶
ī“½ļ‡£
Speed of sound
343 (īµ‰ī €īµ)
ī·¤
Energy transfer coefficient
0.99
ī·™
Incidence loss cst
--
ī“„ļ‡£
Flow coefficient
--
ī“¶ļ„“ļ„µ
Atmospheric temperature
25 (ī¬Ø)
īµ‡ļ‡§īŸ”īµ‡ļ‡„
Throttle and Recycle valves gains
--
īµ‡ļ‡™
Friction constant
--
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References:
[1] Hamedi, H., Rashidi, F. and Khamehchi, E.,
A novel approach to the gas-lift allocation
optimization problem, Petroleum Science and
Technology, 29(4), 2011, pp.418-427.
[2] Isidori, Alberto, ed, Nonlinear control
systems: an introduction, Berlin, Heidelberg:
Springer Berlin Heidelberg, 1985.
[3] Aranda-Bricaire, Eduardo, Claude H. Moog,
and J-B. Pomet, A linear algebraic
framework for dynamic feedback
linearization, IEEE Transactions on
Automatic Control 40, no. 1 (1995), pp.127-
132.
[4] Vidyasagar, Mathukumalli, Nonlinear
systems analysis. Society for Industrial and
Applied Mathematics, 2002.
[5] Greitzer, Edward M, Surge and rotating stall
in axial flow compressorsā€”Part I:
Theoretical compression system model,
(1976), pp.190-198.
[6] Gravdahl, Jan Tommy, and Olav
Egeland, Compressor surge and rotating
stall: Modeling and control, Springer Science
& Business Media, 2012.
[7] CortƩs, Jorge, Lecture 3: Feedback
linearization of MIMO systems, MAE 281B:
Nonlinear Control, 2008, pp.1-11.
[8] Marquez, Horacio J, Nonlinear control
systems: analysis and design, Hoboken^ eN.
JNJ: John Wiley, Vol.161, 2003.
[9] Khalil, Hassan K, Nonlinear systems third
edition, Patience Hall, Vol.115, 2002.
[10] Slotine, Jean-Jacques E., and Weiping
Li, Applied nonlinear control, Englewood
Cliffs, NJ: Prentice hall, Vol. 199, no. 1,
1991.
[11] Pomet, J.B., Moog, C.H. and Aranda, E., A
non-exact Brunovsky form and dynamic
feedback linearization, Proceedings of the
31st IEEE Conference on Decision and
Control IEEE, 1992, pp. 2012-2017.
[12] Chetate, B., Zamoum, R., Fegriche, A. and
Boumdin, M., PID and novel approach of PI
fuzzy logic controllers for active surge in
centrifugal compressor, Arabian Journal for
Science and Engineering, Vol.38, No.6,
2013, pp.1405-1414.
[13] Boushaki, R.Z., Chetate, B. and Zamoum, Y.,
Artificial neural network control of the
recycle compression system, Studies in
Informatics and Control, Vol. 23, No,1,
pp.65-76.
[14] Wang, Y., Wang, L. and Brdys, M.A.,
Designing robust feedback linearisation
controllers using imperfect dynamic models
and sensor feedback, Cogent
Engineering, Vol.3, No.1, p.1173529.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
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(Attribution 4.0 International, CC BY 4.0)
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_US
Fig. 8a: Mass Flow (kg/sec) vs time
Fig. 8d: Plenum2 pressure development vs, time
Fig. 8b: Feed flow for the controlled system vs time
Fig. 8e: Recycle flow vs. time
Fig. 8c: Plenum1 pressure development vs, time
Fig. 8f: Control signal ā€œuā€ vs. time.
Fig. 8: Controlled compressor parameters
0 1 2 3 4 5 6 7 8 9 10
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
X: 8.285
Y: 0.1003
Mass flow
t [sec]
kg/s
0 1 2 3 4 5 6 7 8 9 10
1.01
1.02
1.03
1.04
1.05
1.06
1.07
1.08
1.09 x 105Pressures
t [sec]
p1/p01
plenuum2 pressure
ambient pressure
0 1 2 3 4 5 6 7 8 9 10
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
feed flow
t [sec]
kg/s
0 1 2 3 4 5 6 7 8 9 10
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
recycle flow
t [sec]
kg/s
0 1 2 3 4 5 6 7 8 9 10
6.5
7
7.5
8
8.5
9
9.5
10
10.5 x 104Pressures
t [sec]
p1/p01
plenuum1 pressure
ambient pressure
0 1 2 3 4 5 6 7 8 9 10
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
control input
t [sec]
u
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