#
Model based MIN/MAX override control of centrifugal compressor systems

Rico Schulze1 and Hendrik Richter2

1 AviComp Controls GmbH, Ostwaldstr. 4,
D–04329 Leipzig, Germany
2 HTWK Leipzig University of Applied Sciences, Faculty of Engineering,
Postfach 30 11 66, D–04251 Leipzig, Germany

###### Abstract

We consider an application-oriented nonlinear control of centrifugal
compressors. Industrial applications require the compressor system to adjust
to variable process demands and to be restricted to the valid operation range
(e.g. surge limit). We modify a compressor model of Gravdahl and Egeland to
account for characteristic features of industrial compressors and combine the
framework of nonlinear output regulation via the internal model principle with
MIN/MAX-override control in order to implement trajectory tracking between
given state constraints. Furthermore the switching scheme as well as the
practical stability of the closed-loop MIMO system is analysed by the
corresponding switched and impulsive error system. The override control is
demonstrated by applying discharge pressure control, anti-surge control and
maximum discharge pressure limitation.

KEYWORDS: Compressor systems, Override control, Surge, Tracking

##  1 Introduction

Rotating compressors are an essential part of most plants of process industry.
They are used to generate a pressure difference between the suction and the
discharge side of the compressor, thus producing a continuous flow. Compressor
operation is bound to a limited working range. The most important limitation
is called surge limit and restricts operation to a certain minimum flow.
Crossing the surge limit causes strong aerodynamic instabilities. This leads
to a broad spectrum of negative consequences ranging from high vibrations to
complete breakdown. Hence, most of the existing literature concerning control
designs focuses on aerodynamic instabilities. Especially so-called active
control used for suppression of stall and surge by different design methods
has been intensively studied. Among the methods, Lyapunov-based designs (e.g.
via backstepping) are most popular, e.g. [13, 15, 22, 25]. But also adaptive
[2] and high-gain control [3] as well as bifurcation-based [26] and optimum
criteria-based methods [35] have been used. Moreover, research was done on
tracking fast setpoint changes for a coupled compressor/gas turbine system
using flatness-based feedforward control [30] and gain-scheduled decoupling
control [32]. An advantage of active control is the stabilization of unstable
open-loop working points. This leads to an extension of the stable operation
area in the closed-loop setup. However, an operation very close to the surge
line is explicitly allowed. This contradicts the safety and availability
demands which industrial applications of plant operations usually require.
Therefore, surge avoidance rather than active suppression is typically applied
in industrial applications. This corresponds to introducing state constraints
in the controller design. Another important but rather rarely discussed aspect
of industrial compressor operation is adjustment to variable process demands.
Both aspects are taken into account by model predictive control (MPC). Linear
MPC [1, 4, 5, 6] and nonlinear MPC [21, 37, 40] have been applied to
centrifugal compressor systems.

Another way for meeting the requirements of industrial control of centrifugal
compressors (tracking of constant as well as time-dependent trajectories
together with introducing state constraints) is override control. Override
control is based on independent controllers that are organized in an override
scheme and allows tracking of trajectories as long as no state constraint is
violated. If any violation occurs, the control law is switched in order to
return the state to the unrestricted domain. Despite the practical importance,
there is little research effort on it. The most comprehensive work was done by
Glattfelder and Schaufelberger [9, 10, 11]. In particular, they investigated
linear plant models in single-loop MIN/MAX override control. Another early
work was conducted by Foss [8] on a linear gas turbine model. A somewhat
different formulation of override control has been developed by Turner &
Postlethwaite [38]. Instead of a MIN/MAX structure switching between multiple
sub-controllers, a single output violation compensator considering soft bounds
for multivariable linear models has been designed. An extension of this
approach to feedback linearizable systems has been reported by the same
authors [39], while Herrmann et al. [17] introduced strict bounds on linear
multivariable plants. However, to the best of our knowledge only linear plant
models have been considered for override control of centrifugal compressors so
far. In this paper, a nonlinear model is studied.

A problem with override control of nonlinear compressor models is the need for
a proof of stability of the switched system. To tackle the stability issue, we
propose to combining MIN/MAX override control with the framework of internal
model based nonlinear output regulation. This lead to a multiple controller
formulation of MIN/MAX override control that provides practical stability as
well as a simple setup and intuitive understanding in industrial environments.

The paper is structured as follows. In Section 2 we give the centrifugal
compressor model used for MIN/MAX override control, see also Appendix A for
the full modelling procedure. It expands the compressor model of Gravdahl and
Egeland [12] to account for characteristic features of industrial compressors.
The flow through the impeller and the diffuser is modelled separately so that
the total effective passage length depends on the pressure ratio. In addition,
the modified model describes the jointed action of the adjustable positions of
the guide vane and the blow-off valve while compensating variable disturbances
introduced by the process valve. This is a typical scenario in process control
of industrial compressors. Section 3 discusses the model based override
control including a proof of practical stability. The application of the
control scheme to the compressor model using nonlinear output regulation is
demonstrated in Section 4. Section 5 gives conclusions and recommendations on
further work.

##  2 The centrifugal compressor model

Several models are available for describing stable and unstable compressor
operations. Well-known examples for axial compressors are the models of
Greitzer [14] and Moore & Greitzer [29]. Based on Greitzer’s lumped-parameter
approach, turbocharger and centrifugal compressor models have been developed
by Fink et al. [7] and Gravdahl & Egeland [12], respectively. We consider a
fixed-speed centrifugal compressor of industrial size. Since we focus on an
application-oriented perspective we will expand the model of Gravdahl &
Egeland [12] to higher differential pressures that are common for centrifugal
compressor stages of industrial size. This involves the flow through the
impeller, the diffuser as well as the flow through control valves.
Furthermore, we consider an application-oriented setup with a downstream
process valve (PV) and a blow-off valve (BOV) that is used for the MIN/MAX
anti-surge override control, see Figure 9 of Appendix A for a schematic
description.

To facilitate easy practical interpretability, the dynamics of the centrifugal
compressor is described by the physical dimensional quantities, see the
complete derivation of the model equations in Appendix A. In particular, we
take the impeller velocity $c_{2}$ representing the flow though the compressor
and the pressure ratio $\Pi=p_{4}/p_{1}$ accounting for the relation between
plenum pressure and ambient pressure, as well as the guide vane position
$r_{GV}$, the process valve position $r_{PV}$ and the blow-off valve position
$r_{BOV}$. The compressor can be controlled by adjusting the positions of the
guide vane $u_{GV}$ and the blow-off valve $u_{BOV}$. Adjusting the position
of the process valve $u_{PV}$ is considered to model an external disturbance
$z_{PV}:=u_{PV}$ to the compressor.

Using the Equations (33), (36), (40), (43) and (44) of Appendix A, and
defining the state vector $x=(c_{2},\Pi,r_{GV},r_{PV},r_{BOV})^{T}$, the
control input vector $u=(u_{GV},u_{BOV})^{T}$ and the disturbance input
$z=z_{PV}$, the compressor model becomes:

$\displaystyle\dot{x}_{1}$
$\displaystyle=\frac{1}{L(x_{2})}\left[Y_{C}(x_{1},x_{3})-k_{1}{\left(x_{2}^{r_{k}}-1\right)}\right],$
(1a) $\displaystyle\dot{x}_{2}$
$\displaystyle=k_{2}x_{2}^{r_{k}}\left[{x_{1}-\left[k_{PV}Y_{PV}(x_{4})+k_{BOV}Y_{BOV}(x_{5})\right]\sqrt{{2k_{1}\left(x_{2}^{r_{k}}-1\right)}/r_{k}}}\right],$
(1b) $\displaystyle\dot{x}_{3}$
$\displaystyle=\frac{1}{\tau_{GV}}\left[u_{1}-x_{3}\right],$ (1c)
$\displaystyle\dot{x}_{4}$
$\displaystyle=\frac{1}{\tau_{PV}}\left[z-x_{4}\right],$ (1d)
$\displaystyle\dot{x}_{5}$
$\displaystyle=\frac{1}{\tau_{BOV}}\left[u_{2}-x_{5}\right],$ (1e)

with the model parameters $k_{1}$, $k_{2}$, $r_{k}$ and the total effective
passage length through the compressor $L(x_{2})$:

$\displaystyle k_{1}$ $\displaystyle=R_{S}T_{1},\quad k_{2}=\kappa
A_{2}/V,\quad r_{k}=(\kappa-1)/\kappa,$ (2a) $\displaystyle L(x_{2})$
$\displaystyle=L_{12}+\frac{L_{23}}{x_{2}^{\frac{1}{\kappa}}\frac{A_{3}}{A_{2}}-1}\ln\left(x_{2}^{\frac{1}{\kappa}}\frac{A_{3}}{A_{2}}\right)+\frac{L_{34}}{x_{2}^{\frac{1}{\kappa}}\frac{A_{3}}{A_{2}}{\left(\frac{A_{4}}{A_{3}}-1\right)}}\ln\left(\frac{A_{4}}{A_{3}}\right).$
(2b)

The nonlinear static compressor map $Y_{C}(x_{1},x_{3})$ is modeled by cubic
polynomials according to a $\beta$-line interpolation along isolines, see
Equation (35) and Figure 11 of Appendix A. The PV characteristic
$Y_{PV}(x_{4})$, the BOV characteristic $Y_{BOV}(x_{5})$ and the coefficients
$k_{PV}$ and $k_{BOV}$ are given by (41).

As we followed the basic modeling approach of Gravdahl & Egeland, Equations
(1a) and (1b) are similar to [12] for constant rotational speed of the
impeller. In contrast to Gravdahl & Egeland, we have modeled the flow through
the impeller and the diffuser separately. In combination with the isentropic
valve flow model (38) the total effective passage length $L$ now depends on
the pressure ratio $\Pi=x_{2}$. Hence, an increasing pressure ratio will lead
to a reduction of the total effective passage length $L$ and vice versa. This
accounts for the effect of changing dynamic responses of the system due to
varying mass through the compressor. This effect is especially important for
starting and stopping of the compressor as well as for compressor sections
with a large operating range. In addition, we have taken into account the
specific valve characteristics $Y_{PV}$ and $Y_{BOV}$. Hence, the model can be
adopted to the application-specific designs of the valves where we use linear
characteristic for the BOV and equal percentage for the PV which complies with
typical industrial setups.

Thus, from a modeling perspective, the model (1) studied in this paper uses
the physical variables flow and pressure ratio to describe the compressor
operation, which is similarly to the Gravdahl & Egeland model [12], but
additionally accounts for the effect initiated by the jointed action of the
adjustable positions of the guide vane and the blow-off valve. As the
combination of guide vanes and blow-off valve is frequently used in process
control of industrial compressors, the model is more realistic from an
application point of view.

##  3 MIN/MAX override control

###  3.1 Problem statement

Because active surge control is not applicable in an industrial environment we
adopt a surge avoidance strategy. The proposed structure of the MIN/MAX
override control with state feedback is shown in Figure 1. From an
application-oriented point of view this modular design is very advantageous as
each controller can be designed and tuned independently. Roughly speaking such
an controller design procedure corresponds to a divide-and-conquer approach.
We consider one main controller and several limitation controllers. The main
controller takes over the function of performance control, e.g. discharge
pressure control. If an operating limit is reached the limitation controllers
bound the main controller’s output, e.g. for achieving anti-surge or maximum
discharge pressure control. In total we have $K$ controllers which are indexed
by $k$, i.e. $k\in\Omega\mathrel{\mathop{:}}=\\{1,\ldots,K\\}$. The compressor
model (1) is autonomous and input affine. At any time instant $t$ the
compressor model (1) is described by smooth vector fields
$f,g_{j}:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}^{n}$:

$\dot{x}=f(x)+\sum_{j=1}^{m}g_{j}(x)u_{j},$ (3)

where $n$ is the dimension of the state space model, $m$ is the number of
inputs and $u_{j}$ is the $j$-th input of the system. Now, we introduce state
feedback controllers $p_{k}:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}^{m}$:

$p_{k}(x)=\begin{bmatrix}p_{k1}(x)\\\ \vdots\\\ p_{km}(x)\end{bmatrix}.$ (4)

Here, $p_{kj}$ is the $j$-th sub-controller of the $k$-th controller. However,
only one of the $K$ sub-controllers $p_{kj}$ can access the input $u_{j}$ at
any time instant $t$ via the switching law $I_{j}$. In case of MIN/MAX
override control each switching law $I_{j}:{\mathbb{R}}^{K}\rightarrow\Omega$
comprises of a combination of MIN/MAX selectors and generates a switching
signal $\sigma_{j}\in\Omega$ via:

$\sigma_{j}(t)=I_{j}(p_{1j},\ldots,p_{Kj}).$ (5)

This implies that controller outputs are bounding each other. An override
takes place if the controller output difference $\Delta
p_{k_{1}k_{2}}^{\scriptscriptstyle j}$ of two sub-controllers $p_{k_{1}j}(x)$
and $p_{k_{2}j}(x)$ ($k_{1}\neq k_{2}$) is zero, i.e.:

$\Delta p_{k_{1}k_{2}}^{\scriptscriptstyle
j}(t)=p_{k_{1}j}(t)-p_{k_{2}j}(t)=0.$ (6)

An override from $p_{k_{1}j}(x)$ to $p_{k_{2}j}(x)$ will be marked by
$p_{k_{1}j}\rightarrowtail p_{k_{2}j}$. We assume that the controllers are
designed in a way to ensure $\Delta p_{k_{1}k_{2}}^{\scriptscriptstyle j}(t)$
to be continuous. Otherwise, the state space cannot be divided into defined
domains uniquely associated with a specific controller. This ensures that only
one controller is active at a given time instant. When $\Delta
p_{k_{1}k_{2}}^{\scriptscriptstyle j}(t)$ is continuous the non-empty zero set
$\mathcal{P}_{k_{1}k_{2}}^{\scriptscriptstyle j}$ of (6) will be a continuous
surface in the state space. The union of all zero sets
$\mathcal{P}_{k_{1}k_{2}}^{\scriptscriptstyle j}$ will be denoted as
$\mathcal{P}$.

Figure 1: General structure of MIN/MAX override consisting of a main
controller and $K$ limitation controllers.

As a control objective of the MIN/MAX override control is trajectory tracking,
any controller $p_{k}$ must successfully achieve output regulation of system
(3). Therefore, the compressor model is expanded by the outputs (to be
tracked) $h:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}^{m}\times{\mathbb{R}}^{K}$
with $h(x)=[h_{1}(x),\ldots,h_{K}(x)]$ and the setpoints
$q:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}^{m}\times{\mathbb{R}}^{K}$ with
$q(x)=[q_{1}(x),\ldots,q_{K}(x)]$ where $h_{k}(x)$ and $q_{k}(x)$ are given
by:

$h_{k}(x)=\begin{bmatrix}h_{k1}(x)\\\ \vdots\\\ h_{km}(x)\end{bmatrix},\quad
q_{k}(x)=\begin{bmatrix}q_{k1}(x)\\\ \vdots\\\ q_{km}(x)\end{bmatrix}.$ (7)

The choice of $h_{k}(x)$ and $q_{k}(x)$ depends on the control objective in
the given application. Now, let us introduce the $k$-th controller tracking
error $e_{k}:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}^{m}$:

$e_{k}=\begin{bmatrix}h_{k1}(x)-q_{k1}(x)\\\ \vdots\\\
h_{km}(x)-q_{km}(x)\end{bmatrix}.$ (8)

It is assumed that the tracking error $e_{k}$ is continuous. Hence, the non-
empty zero set $\mathcal{E}_{k}$ of the tracking error (8) related to
controller $p_{k}(x)$ will be a continuous surface in the state space. The
union of all zero sets $\mathcal{E}_{k}$ will be denoted as $\mathcal{E}$.
Using these definitions, system (3) can be rewritten:

$\displaystyle\dot{x}$
$\displaystyle=f(x)+\sum_{j=1}^{m}g_{j}(x)\,p_{\sigma_{\\!j}j}(x),$ (9a)
$\displaystyle e$
$\displaystyle=\begin{bmatrix}h_{\sigma_{\\!1}1}(x)-q_{\sigma_{\\!1}1}(x)\\\
\vdots\\\ h_{\sigma_{\\!m}m}(x)-q_{\sigma_{\\!m}m}(x)\end{bmatrix},$ (9b)

where $e:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}^{m}$ is the tracking error of
the active sub-controller. With respect to the above-named assumptions, the
closed-loop system (9) is continuous despite the switching scheme. Thus, no
sliding modes can occur which is a key advantage of the override control
considered here. The steady state of system (9) is given by $e(t)=0$ and will
be denoted as $\bar{x}(t)$; the steady state controller output is
$p_{k}(x(t))\\!\mid_{\,e(t)=0\;}=c_{k}(\bar{x}(t))$.

From an application point of view it is desirable to have a prescribed control
dynamics. Therefore, we want the override control to satisfy the condition

$\|e_{\sigma_{\\!j}}(t)\|<\delta,\;t\in[t_{\delta,i},t_{i+1})$ (10)

after each switch, where $\delta\in\mathbb{R}_{+}$ is a sufficiently small
positive constant and $t_{\delta,i}$ is a time instant with
$t_{i}<t_{\delta,i}<t_{i+1}$. We are referring to condition (10) as the
perfect regulation condition as is requires all elements of
$e_{\sigma_{\\!j}}(t)$ to be sufficiently small within a certain time interval
after each switch. As we will see later this condition is sufficient for
guaranteeing stability of the switched system as well as successful output
regulation in a practical sense. Note that successful output regulation of any
limitation controller also satisfies the given state constraints.

###  3.2 Stability

The stability of system (9) is determined by the stability of its
corresponding error system. Hence, we shift an open-loop working point to the
origin by separating the state and the input using the error coordinates
$\tilde{x},\tilde{u}$ and steady state coordinates $\bar{x},\bar{u}$. Applying
the separation $x(t)=\tilde{x}(t)+\bar{x}(t)$ and
$u(t)=\tilde{u}(t)+\bar{u}(t)$ leads to the following switched impulsive
system with state-depended switching:

$\displaystyle\dot{\tilde{x}}(t)=\,$
$\displaystyle\tilde{f}_{\sigma}(\tilde{x})+\sum_{j=1}^{m}\tilde{g}_{\sigma_{\\!j}j}(\tilde{x})\bar{u}_{\sigma_{\\!j}j}+\sum_{j=1}^{m}g_{\sigma_{\\!j}j}(\tilde{x})\tilde{u}_{{\sigma_{\\!j}j}},\quad\tilde{x}(t)\notin\mathcal{S}(t),$
(11a) $\displaystyle\tilde{x}(t_{i}^{+})=\,$
$\displaystyle\tilde{x}(t_{i}^{-})+\Delta\bar{x}(t_{i}),\quad\tilde{x}(t)\in\mathcal{S}(t),$
(11b)

where $\Delta\bar{x}$ is the state reset and $\mathcal{S}(t)$ is the resetting
set which describes the set of all states in the error state space where a
switching (or override) takes place. Note, that $\mathcal{S}(t)$ is generated
from the unified zero set $\mathcal{P}$. This zero set is now time-varying and
exhibits impulsive changes during switchings. When
$\tilde{x}(t)\in\mathcal{S}(t)$ then the trajectory $\tilde{x}(t)$ will jump
from $\tilde{x}(t_{i}^{-})$ to $\tilde{x}(t_{i}^{+})$ at the time instant
$t_{i}$ which is given by the resetting law (11b). It is assumed that
$\tilde{x}(t_{i}^{+})\notin\mathcal{S}(t_{i}^{+})$. Hence, every switching
instant $t_{i}$ is unique. Because of the continuous flow of system (9) in the
error state space, we can evaluate the state reset as
$\Delta\bar{x}(t_{i})=\bar{x}(t^{-}_{i})-\bar{x}(t^{+}_{a})$, i.e. the state
reset is given by the difference of steady states in its original coordinates.
An inspection of the state reset reveals that in general $\exists t_{i}$ with
$\|\Delta\bar{x}(t_{i})\|>0$. Using this, we make the following assumption and
afterwards state the proposition.

Assumption 1: Each sub-controller $p_{k}(x)$ with $k=1,\ldots,m$ guarantees
asymptomatic stability of (11) in an open neighborhood $\tilde{U}_{A}$ around
$\tilde{x}=0$.

This assumption appears reasonable as asymptotic stability of each sub-
controller is a general requirement in industrial applications due to safety
and performance reasons. It can be ensured by an appropriate design procedure,
for instance as proposed by Huang [18], which has also been used in controller
design discussed in this paper, see Section 4.4.

Proposition 1: Suppose Assumption 1 holds. If $\exists t_{i}$ with
$\|\Delta\tilde{x}(t_{i})\|>0$ then there exists an open neighborhood
$\tilde{U}_{\epsilon}$ around $\tilde{x}=0$ so that
$\tilde{x}(t_{i}^{-})\in\tilde{U}_{\epsilon}\Rightarrow\tilde{x}(t_{i}^{+})\notin\tilde{U}_{\epsilon}$
follows $\forall\tilde{x}\in\tilde{U}_{\epsilon}$.

Proof: Define an open neighborhood
$\tilde{U}_{\epsilon}\mathrel{\mathop{:}}=\\{\tilde{x}\in\mathbb{R}^{n}\mid\|\tilde{x}\|<\epsilon\\}$
around $\tilde{x}=0$. From Assumption 1 it follows that there is some constant
$\epsilon>0$ and a time instant $t_{\epsilon}$ for which the error state
satisfies continuous motion $\|\tilde{x}(t)\|<\epsilon$ in the time interval
$(t_{\epsilon},t_{i})$. Now, choose
$0<\epsilon\leq\|\Delta\tilde{x}(t^{-}_{i})\|/2$. Then the error state
$\tilde{x}(t)$ will leave $\tilde{U}_{\epsilon}$ during qualitative change at
the time instant $t_{i}$ and the proposition follows.∎

Referring to the $\delta$-$\epsilon$ construction of Lyapunov stability, we
can deduce an important consequence from Proposition 1. It follows that
$\exists\epsilon>0$ so that $\forall\delta>0$,
$\exists\|\tilde{x}(t_{0})\|<\delta$ and $\exists t\geq t_{0}$ with
$\|\tilde{x}(t)\|\geq\epsilon$. Hence, the switched impulsive error system
(11) cannot be stable in the sense of Lyapunov. As only certain
$\varepsilon>0$ will satisfy the Lyapunov instability condition, we have to
refer to explicit bounds resulting in the concept of practical stability. We
will use the definition of [24]:

Definition 1: System (11) is said to be $(\lambda,A)$-practical stable if,
given explicit bounds $(\lambda,A)$ with $0<\lambda<A$, then
$\|\tilde{x}(t_{0})\|<\lambda$ implies $\|\tilde{x}(t)\|<A$, $t\geq t_{0}$ for
some $t_{0}\in\mathbb{R}_{+}$.

We are interested in uniform practical stability where Definition 1 holds
$\forall t_{0}\in\mathbb{R}_{+}$ [24]. For considering practical stability of
system (11) another assumption on the switching events is required.

Assumption 2: The sequence
$\Sigma_{\Delta}=\\{\|\Delta\bar{x}(t_{1})\|,\|\Delta\bar{x}(t_{2})\|,\|\Delta\bar{x}(t_{3})\|,\ldots\\}$
generated from the state resets at switching time instants $t_{i}$ is upper
bounded by a sufficiently small constant $L\in\mathbb{R}_{+}$, i.e.
$\|\Delta\bar{x}(t_{i})\|<L$ $\forall i$.

This assumption states that each state reset is bounded by an upper limit $L$.
This limit must be set in a way that the region of stability for the given
controller is not left after switching. We will assume that the controllers
are designed in a way that this condition is met. Note that Assumption 2 is
obsolete if the controllers are designed to provide global stability.

Now, a theorem on the practical stability of the closed-looped switched and
impulsive error system (11) can be stated. The well-known dwell time stability
approach as defined in [27] and [42] is applied.

Theorem 1: Suppose Assumption 1 and 2 hold. Consider the closed-loop error
system (11) with given initial conditions
$\tilde{x}(t_{0})\in\tilde{U}_{\lambda}$ and the open neighborhood
$\tilde{U}_{\lambda}$ around $\tilde{x}=0$. If $t_{i+1}-t_{i}>\tau_{i}$ with
sufficiently large dwell time $\tau_{i}$ holds for every override event, then
error system (11) is uniformly practically stable.

Proof: To show practical stability according to Definition 1 two conditions
must hold. The first condition is derived by examining the equation of
impulsive motion (11b). If the error state satisfies
$\tilde{x}(t_{i}^{-})\in\tilde{U}_{\lambda}$ at any switching time instant
$t_{i}$, then the state reset must be upper bounded in order to guarantee
$\tilde{x}(t_{i}^{+})\in\tilde{U}_{A}$. This is shown using the triangle
inequality deduced from the equation of impulsive motion (11b):

$\|\tilde{x}(t_{i}^{+})\|\leq\|\tilde{x}(t_{i}^{-})\|+\|\Delta\bar{x}(t_{i})\|<A.$
(12)

Now, if Assumption 2 holds with $\|\Delta\bar{x}(t_{i})\|<L$, then we have
$\|\tilde{x}(t_{i}^{-})\|<\lambda$ with the upper limit
$\lambda\in\mathbb{R}_{+}$ satisfying $\lambda<A-L$, i.e. Assumption 2
guarantees that the following condition holds for every switching event:

$\|\tilde{x}(t_{i}^{-})\|<\lambda\Rightarrow\|\tilde{x}(t_{i}^{+})\|<A.$ (13)

The second condition is derived by examining the equation of continuous motion
(11a). If $\tilde{x}(t)\notin\mathcal{S}(t)$, then Assumption 1 guarantees
that $\|\tilde{x}(t)\|$ will asymptotically decay to zero as time goes towards
infinity. The $i$-th active controller must be activated for a minimum dwell
time $\tau_{i}$ in order to reach $\tilde{U}_{\lambda}$ before the next
switching event occurs. Hence, if switching is slow enough then
$t_{i+1}-t_{i}>\tau_{i}$ holds and it follows that:

$\|\tilde{x}(t_{i}^{+})\|<A\Rightarrow\|\tilde{x}(t_{i+1}^{-})\|<\lambda.$
(14)

Combining both conditions (13) and (14) leads to:

$\|\tilde{x}(t_{1}^{-})\|<\lambda\Rightarrow\|\tilde{x}(t_{1}^{+})\|<A\Rightarrow\|\tilde{x}(t_{2}^{-})\|<\lambda\Rightarrow\|\tilde{x}(t_{2}^{+})\|<A\Rightarrow\ldots$
(15)

which completes the proof for practical stability as required by Definition
1.∎

Theorem 1 states that practical stability follows from (i.) asymptotic
stability of any controller within sufficiently large neighborhood of
$\tilde{x}=0$, (ii.) bounded state resets and (iii.) sufficiently slow
switching. Conversely, fast switching can cause instability which is a well-
known phenomenon in industrial compressor override control. Note that Theorem
1 also satisfies the perfect regulation condition (10) if $\lambda\leq\delta$,
i.e. practical stability guarantees successful output regulation of the main
controller and any limitation controller.

##  4 Output regulation based on MIN/MAX override control

###  4.1 Internal model based output regulation

We use the framework of the Byrnes-Isidori internal model principle for output
regulation [20] as a basis for designing the individual sub-controllers in the
given MIN/MAX override scheme. It will be shown later that this framework can
be applied to the compressor model (1) which can be described by system (9)
when MIN/MAX switching is considered. Following Isidori, system (9) is
expanded by the exo-state $v$:

$\displaystyle\dot{x}$
$\displaystyle=f(x,v)+\sum_{j=1}^{m}g_{j}(x,v)\,p_{\sigma_{\\!j}j}(x,v),$
(16a) $\displaystyle\dot{v}$ $\displaystyle=s(v),$ (16b) $\displaystyle e$
$\displaystyle=\begin{bmatrix}h_{\sigma_{\\!1}1}(x)-q_{\sigma_{\\!1}1}(v)\\\
\vdots\\\ h_{\sigma_{\\!m}m}(x)-q_{\sigma_{\\!m}m}(v)\end{bmatrix},$ (16c)

where the exo-state $v=[w,d]^{T}$ consists of the controller setpoint exo-
state $w$ and disturbance exo-state $d$. We assume that
$s:{\mathbb{R}}^{o}\rightarrow{\mathbb{R}}^{o}$ is smooth and marginally
stable as defined by [19]. With the combined state $[x,v]^{T}$, the
corresponding error system is again of the same structure as (11). Now, assume
that $e=e_{k}$ holds, i.e. the $k$th sub-controller $p_{k}(x,v)$ accesses all
control inputs of the system. An important feature of system (16) under
$e=e_{k}$ is the existence of a center manifold with the mapping
$x=\chi_{k}(v)$ for the $k$-th sub-controller. Following [19], this center
manifold is locally invariant and guarantees exponential stability for the
$k$-th sub-controller with
$\|x(t)-\chi_{k}(v(t))\|\leq\beta_{k}\exp(-\alpha_{k}t)\|x(t_{i})-\chi_{k}(v(t_{i}))\|$
($\alpha_{k},\beta_{k}\in\mathbb{R}_{+}$) in the time interval
$(t_{i},t_{i+1})$ if the initial tracking error satisfies
$\|e_{k}(t_{i}^{+})\|<A$ and the system is linear stable under $v(t)=0$, i.e.
in the absence of the exo-state. Hence, the combined state $[x,v]^{T}$
converges to the steady state that is given by the mapping $x=\chi_{k}(v)$ and
the objective of output regulation is reached for the $k$-th sub-controller
$p_{k}(x,v)$. Hence, Assumption 1 will be satisfied as the statements above
hold $\forall k\in\Omega$.

The approach just presented illustrates the main advantage of MIN/MAX override
control: each sub-controller can be designed independently of each other. This
is achieved by the specific MIN/MAX switching scheme together with the proof
of stability from the last section. Hence, control design can be done $\forall
k\in\Omega$ independent of the specific realization of the switching signal
$\sigma$. It follows that the design for the switched system (16) can be
reduced to $K$ controller designs of the non-switched system:

$\displaystyle\dot{x}$
$\displaystyle=f(x,w,d)+\sum_{j=1}^{m}g_{j}(x,w,d)\,p_{kj}(x,w,d),$ (17a)
$\displaystyle\dot{w}$ $\displaystyle=s_{w}(w),$ (17b) $\displaystyle\dot{d}$
$\displaystyle=s_{d}(d),$ (17c) $\displaystyle e_{k}$
$\displaystyle=\begin{bmatrix}h_{k1}(x)-q_{k1}(w,d)\\\ \vdots\\\
h_{km}(x)-q_{km}(w,d)\end{bmatrix},$ (17d)

where $v=[w,d]^{T}$ has been applied as mentioned before. System (17) is used
for control design. We will refer to system (17) as the combined compressor
system. Before starting control design for the combined compressor system,
some additional control requirements are discussed.

###  4.2 Compressor control requirements

We consider discharge pressure control using an inlet or outlet guide vane as
presented in Section 2. This is our main controller. The process valve (PV) is
modeling the flow demand by the downstream process. Repositioning the PV will
cause the compressor to change its working point. The main control objective
is to keep the desired discharge pressure despite the disturbance introduced
by the PV. Furthermore, compressor operation will be limited to a given
maximum pressure as well as the given surge control line (SCL) using the BOV,
where the BOV is opened to restrict compressor operation to the SCL during
stationary operation. The BOV is the most effective way to avoid surge in
industrial applications assuming that the handled gas can be vented to the
environment. As long as the compressor is operating away from its limits, the
BOV is closed. Hence, no gas is blown off at the discharge side and we have
$k_{BOV}(x_{5})=0$ and $u_{BOV}=0$. In this case the model is SISO. If the BOV
is open the system is MIMO. It follows that in addition to the main control,
two limiting controllers must be designed. The limiting controller will be
referred to as maximum pressure limiter and anti-surge controller (ASC). The
next step of control design is the trajectory generation for the setpoint and
disturbance exo-systems (17b) and (17c).

###  4.3 Trajectory generation

Industrial application of rotating compressor control mostly involve either
constant setpoints or ramp-like setpoint motion. Both can be generated by
using the concept of marginally stable exo-systems. However, the generation of
a ramp signal is not possible for a single marginally stable exo-system [19].
Therefore, we will approximate a ramp-like trajectory by using the suggested
method of MIN/MAX override. Figure 2 illustrates the basic idea behind this
approximation. A sinusoidal target trajectory, generated by the exo-system
$\dot{w}_{1}=\left[0,\omega w_{1,3},-\omega w_{1,2}\right]^{T}$, moves between
two constant limiting trajectories with $w_{2}$ and $w_{3}$ being their exo-
states. Before reaching $t_{i}$ the system is in a steady state given by
$w_{2}$ under control of $p_{2}(x,w_{2})$. When reaching $t_{i}$ the MIN/MAX
override structure selects $p_{1}(x,w_{1})$ and starts to ramp. After reaching
$t_{i+1}$ the MIN/MAX override structure selects $p_{3}(x,w_{3})$ and forces
the system to approach the steady state given by $w_{3}$. To prevent the
sinusoidal target trajectory from accessing the system again, $p_{1}(x,w_{1})$
must be switched off once $p_{3}(x,w_{3})$ is activated. Now, combining the
ramp generator according to Figure 2, we obtain the following exo-system:

$\displaystyle\dot{w}_{1}$ $\displaystyle=\left[0,\omega w_{1,3},-\omega
w_{1,2}\right]^{T},$ (18a) $\displaystyle\dot{w}_{2}$ $\displaystyle=0,$ (18b)
$\displaystyle\dot{w}_{3}$ $\displaystyle=0,$ (18c)

where the parameter $\omega$ determines the frequency of the sinusoidal main
trajectory and at the same time defines the slope of the approximated ramp.
Depending on the initialization of the corresponding exo-systems, the state
either performs a positive or negative ramp-like motion. Note that exo-system
(18) is marginally stable, which makes the presented approach a
straightforward method to approximate ramp-like motion using marginally stable
exo-systems.

Figure 2: Ramp generation by using MIN/MAX override of sinusoidal controller
$p_{1}$ and two limitation controllers $p_{2}$ and $p_{3}$. An exemplary
trajectory of $x_{1}(t)$ is plotted on the right hand side where the state
$x_{1}(t)$ exhibits ramp-like motion (solid black line).

The method is now applied to the combined compressor model. The setpoint for
the main controller (discharge pressure) and the ASC is provided by the above
ramp generator with a sinusoidal trajectory that will be bounded by two
constant trajectories with the exo-state variables $w_{1}$ to $w_{5}$. For the
maximum pressure limiter we use a constant trajectory with just the exo-state
variable $w_{6}$. This leads to the following setpoint exo-system
$\dot{w}=s_{w}(w)$ with $s_{w}(w)$ given by:

$s_{w}(w)=\left[0,\omega_{w}w_{3},-\omega_{w}w_{2},0,0,0\right]^{T}.$ (19)

Furthermore, we are modeling a sinusoidal motion for the disturbance that is
introduced by the PV. Hence, the same structure as exo-system (18) is used for
$\dot{d}=s_{d}(d)$ with $s_{d}(d)$ given by:

$s_{d}(d)=\left[0,\omega_{d}d_{3},-\omega_{d}d_{2}\right]^{T}.$ (20)

This completes the definition of the exo-system. The next step of control
design contains the compressor model analysis and the definition of the
control law.

###  4.4 Application to the compressor model

Following Section 4.2, several sub-controllers need to be designed for the
SISO and MIMO domain. There is one main controller consisting of the sub-
controllers $C_{1}$ (sine), $C_{2}$ (sine lower bound) and $C_{3}$ (sine upper
bound). The ASC consists of the sub-controllers $C_{5}$ (sine), $C_{6}$ (sine
lower bound) and $C_{7}$ (sine upper bound). The maximum pressure limitation
controller consists of $C_{4}$ and $C_{8}$ both representing the maximum
allowed pressure. Figure 3 illustrates the control structure for the MIMO
case.

Figure 3: Structure for internal model based output regulation controllers
applied to compressor system model (1).

Now, the control law is defined. We will use
$p_{kj}(x,w,d)=\bar{u}_{kj}(w,d)+G_{kj}(w,d)(x-\chi_{k}(w,d))$ for each sub-
controller with the gain $G_{kj}(w,d)$ and the steady state setpoint
$\bar{u}_{kj}(w,d)=p_{kj}(\chi_{k}(w,d),w,d)$. The state-feedback controller
$p_{kj}(x,w,d)$ is a standard form that can solve the problem of output
regulation and must be designed $\forall k\in\Omega$. Table 1 contains the
measurement equations and the setpoints used for control design, where
$Y_{ASC}$ is the measurement equation for the ASC. We will assume a bivariate
polynomial depending of the compressor flow $x_{1}$ and the pressure ratio
$x_{2}$:

$Y_{ASC}(x_{1},x_{2})=\sum_{n_{1}=0}^{N_{1}}\sum_{n_{2}=0}^{N_{2}}b_{n_{1},n_{2}}x_{1}^{n_{1}}x_{2}^{n_{2}}$
(21)

with $N_{1}>0$. Furthermore, the following switching laws $I_{1}$ and $I_{2}$
according to Equation (5) are used:

$\displaystyle I_{1}$
$\displaystyle=\max(\min(\max(p_{11},p_{21}),p_{31},p_{41}),\min(\max(p_{51},p_{61}),p_{71},p_{81})),$
(22a) $\displaystyle I_{2}$
$\displaystyle=\max(p_{12},p_{22},p_{32},p_{42},\min(\max(p_{52},p_{62}),p_{72},p_{82})).$
(22b)

Before we can solve the output regulation problem for the proposed MIN/MAX
override structure, it is verified whether the combined compressor system (17)
satisfies the necessary conditions for the control design presented by Huang
[18]. Therefore, the relative vector degree and the zero dynamics are examined
for the SISO and MIMO domain.

Trajectory | SISO domain | MIMO domain | MIMO domain
---|---|---|---
|  | $j=1$ | $j=2$
Sine | $h_{1}=x_{2}$ | $h_{1}=x_{2}$ | $h_{2}=Y_{ASC}(x_{1},x_{2})$
$q_{w,1}=w_{1}+w_{2}$ | $q_{w,1}=w_{1}+w_{2}$ | $q_{w,2}=Y_{ASC}(w_{1},w_{2})$
Sine lower limit | $h_{1}=x_{2}$ | $h_{1}=x_{2}$ | $h_{2}=Y_{ASC}(x_{1},x_{2})$
$q_{w,1}=w_{4}$ | $q_{w,1}=w_{4}$ | $q_{w,2}=Y_{ASC}(w_{4})$
Sine upper limit | $h_{1}=x_{2}$ | $h_{1}=x_{2}$ | $h_{2}=Y_{ASC}(x_{1},x_{2})$
$q_{w,1}=w_{5}$ | $q_{w,1}=w_{5}$ | $q_{w,2}=Y_{ASC}(w_{5})$
Constant | $h_{1}=x_{2}$ | $h_{1}=x_{2}$ | $h_{2}=Y_{ASC}(x_{1},x_{2})$
$q_{w,1}=w_{6}$ | $q_{w,1}=w_{6}$ | $q_{w,2}=Y_{ASC}(w_{6})$
Table 1: Measurement equations and setpoint definition.

The relative vector degree for the combined compressor system (17) is
determined by using the combined notation:

$\displaystyle\dot{x}_{e}=\;$ $\displaystyle f_{e}(x_{e})+g_{e}(x_{e}),$ (23a)
$\displaystyle e=\;$ $\displaystyle h_{e}(x_{e}),$ (23b)

with the combined state $x_{e}=[x,w,d]^{T}$ and the vector fields
$f_{e}(x_{e})=[f(x,d),s_{W}(w),s_{D}(d)]^{T}$,
$g_{e}(x_{e})=[g_{1}(x),0_{m_{W}+m_{D}\times m_{E}}]^{T}$ and
$h_{e}(x_{e})=[h(x)-q_{W}(w)]$. For the SISO domain with $n=4$, $m_{W}=6$ and
$m_{D}=3$ the relative degree is $r=3$. For the MIMO domain with $n=5$,
$m_{W}=6$ and $m_{D}=3$. the relative vector degree is $r=\\{2,2\\}$. In both
cases the combined internal dynamics is given by:

$\begin{bmatrix}\dot{x}_{4}\\\ \dot{w}\\\
\dot{d}\end{bmatrix}=\begin{bmatrix}f_{4}(x_{4},d)\\\ s_{W}(w)\\\
s_{D}(d)\end{bmatrix}=\begin{bmatrix}(q_{D}(d)-x_{4})/\tau_{PV}\\\ s_{W}(w)\\\
s_{D}(d)\end{bmatrix}.$ (24)

The combined compressor system (17) can be separated with
$x^{1}=[x_{1},x_{2},x_{3}]^{T}$ and $x^{2}=[x_{4}]^{T}$ in the SISO domain as
well as $x^{1}=[x_{1},x_{2},x_{3},x_{5}]^{T}$ and $x^{2}=[x_{4}]^{T}$ in the
MIMO domain. The combined internal dynamics (24) does neither depend on
$x^{1}$ nor on the inputs $u_{1}$ and $u_{2}$. Hence, the combined internal
dynamics (24) is also the combined zero dynamics of the combined compressor
system. The output regulation problem can be solved if the subsystem specified
by the $\dot{x}_{4}=f_{4}(x_{4},d)$ of the combined zero dynamics in Equation
(24) is linearly stable [18]. As the subsystem specified by
$\dot{x}_{4}=f_{4}(x_{4},d)$ is a first order low pass filter system, its
characteristic polynomial is given by $\lambda\tau_{PV}+1$. Assuming
$\tau_{PV}>0$, exponential stability is given for
$\dot{x}_{4}=f_{4}(x_{4},d)$. Hence, the combined compressor system (17) is
minimum phase and we can apply the control design algorithm of Huang [18] to
solve the center manifold $\chi_{k}(w,d)$ $\forall k\in\Omega$ and the
corresponding steady state setpoint
$\bar{u}_{kj}(w,d)=p_{kj}(\chi_{k}(w,d),w,d)$. Requirements on the control
accuracy or on the tracking speed can be achieved by adjusting the control
gains $G_{kj}(w,d)$. Each controller can be adjusted separately. As allready
mentioned, this is considered an advantage of the presented modular design of
a MIN/MAX override control.

###  4.5 Simulation results

In this section we show the results of two simulation scenarios of the
combined compressor system (17) under MIN/MAX override control as presented in
the last chapter. Thus, working principles of the control as well as
successful output regulations are demonstrated. The combined compressor system
(17) uses the compressor model (1) as plant description. The parameters of the
model are given in Table 3, see Appendix A. In both scenarios the main control
objective is to regulate the discharge pressure while simultaneously rejecting
process disturbance introduced by actions of the process valve (PV). Thus, the
main controller trajectory is designed to increase the pressure ratio from
$x_{2}=1.7$ to $x_{2}=2.0$. Scenario 1 is depicted in Figures 4(a), 6(a),
6(c), 8(a) and 8(c), while for scenario 2, see Figures 4(b), 6(b), 6(d), 8(b)
and 8(d). The main difference between scenario 1 and scenario 2 is the
severity of the process disturbance. While in scenario 1 the discharge
pressure limit is set to $x_{2,max}=1.9$, in scenario 2 it is set to
$x_{2,max}=2.05$, i.e. the discharge pressure limitation is virtually switched
off in scenario 2. Although in both scenarios a process disturbance is
introduced by the PV; see Figures 6(c) and 6(d), the disturbance in scenario 2
is designed to be larger to drive the compressor towards the surge limit.
Thus, the blow-off valve (BOV) becomes active.

According to the setup described above, the compressor will reach the maximum
pressure limit in the scenario 1 (see Figure 4(a)) while reaching the surge
control line (SCL) in scenario 2 (see Figure 4(b)). Simulation results show
that the compressor system first follows the ramp-like main trajectory as
shown in Figures 6(a) and 6(b) before switching to the corresponding limiting
controller.

(a) Simulation results for scenario 1.

(b) Simulation results for scenario 2.

Figure 4: Compressor working point motion displayed in the
$x_{1}$-$x_{2}$-plane (solid black) together with surge control line and
maximum pressure limit (both dashed gray).

The switching can be seen in Figures 8(a) and 8(b). There are two switching
events in scenario 1. The first event corresponds to switching between the
constant and sinusoidal main trajectory that have been introduced in Section
4.3 and 4.4 to approximate a ramp-like setpoint motion; i.e.
$C_{2}\rightarrowtail C_{1}$. The second event corresponds to
$C_{1}\rightarrowtail C_{4}$ which is the override by the maximum pressure
discharge limiter. During the remaining simulation the system is limited to
the maximum value of $x_{2,max}=1.9$ while rejecting the process disturbance
introduced by the PV. As illustrated by Figure 6(c) the disturbance is
rejected by adjusting the guide vane (GV) position only. Hence, the BOV is
kept close and the compressor system is subjected to the controllers designed
for the SISO domain only. As the BOV does not become active in scenario 1, the
BOV position $x_{5}(t)=0$ for the entire simulation shown in Figure 6(c).

(a) Pressure ratio (solid black) together with defined trajectories (dashed
gray) for scenario 1.

(b) Pressure ratio (solid black) together with defined trajectories (dashed
gray) for scenario 2.

(c) Guide vane (GV) position $x_{3}$ (solid black), process valve (PV)
position $x_{4}$ (dashed light grey) and blow-off valve (BOV) position $x_{5}$
(solid dark grey) for scenario 1.

(d) Guide vane(GV) position $x_{3}$ (solid black), process valve (PV) position
$x_{4}$ (dashed light grey) and blow-off valve (BOV) position $x_{5}$ (solid
dark grey) for scenario 2.

Figure 6: Simulation results of the pressure ratio $x_{2}$ and actuator
positions $x_{3}$, $x_{4}$, $x_{5}$.

Keeping the BOV closed corresponds to a very energy efficient way of maximum
pressure limitation. In common practice, the BOV is used for maximum pressure
limitation due to its very fast response time. However, the BOV becoming
active implies venting of already compressed gas and therefore a rather energy
inefficient operation. Furthermore, Figure 8(c) shows that the system is
stable as defined in Section 3.2. Both switching events (indicated by black
circles) take place in the close neighbourhood of $\tilde{x}=0$. Hence, the
closed-loop system satisfies the regulation condition (10) in this simulation.

In scenario 2 there is one more switching event; see Figure 8(b). Similar to
scenario 1 the simulation starts by following the ramp-like main trajectory
motion; see Figure 4(b). As the PV is throttled much more, the system now
reaches the SCL. Hence, there are overrides $C_{1}\rightarrowtail C_{5}$ and
$C_{5}\rightarrowtail C_{6}$ by the anti-surge sub-controllers. These
controllers now stabilize the system on the SCL while rejecting the process
disturbance introduced by the PV. As the ASC is acting on the BOV, the system
is now subjected to the sub-controllers designed for the MIMO domain. Finally,
it can be seen from Figure 8(d) that the closed-loop system is again practical
stable and satisfies the regulation condition (10) as desired during the
controller design procedure.

(a) Switching signal for guide vane controllers (solid black) and BOV
controllers (grey) for scenario 1.

(b) Switching signal for guide vane controllers (solid black) and BOV
controllers (grey) for scenario 2.

(c) Compressor working point motion displayed in
$x_{\Delta,1}$-$x_{\Delta,2}$-plane (solid black) for scenario 1.

(d) Compressor working point motion displayed in
$x_{\Delta,1}$-$x_{\Delta,2}$-plane (solid black) for scenario 2.

Figure 8: Simulation results of the switching signal $\sigma$ and compressor
working point motion displayed in error coordinates.

##  5 Conclusions

We considered an application-oriented nonlinear control of centrifugal
compressors. The control is based on a modification of a model of Gravdahl and
Egeland [12]. The modification accounts for characteristic features of
industrial compressors and models the flow through the impeller and the
diffuser separately. Thus, the total effective passage length depends on the
pressure ratio and facilitates for higher differential pressures. Furthermore,
the modified model describes a typical scenario in process control of
industrial compressors with control signals simultaneously and independently
adjusting the positions of the guide vane and the blow-off valve while
rejecting disturbances introduced by the process valve.

In the paper, we combined the framework of nonlinear output regulation via the
internal model principle with MIN/MAX-override control. The overall design
meets the requirements of industrial applications of centrifugal compressors,
as the control is capable of tracking constant and time-dependent
trajectories, as well as generating state constraints. A setpoint generator
for time-varying setpoints has been designed using the proposed method of
MIN/MAX-override control. This allows approximating a ramp-like setpoint
motion by suitable overrides of the system responses generated from the
corresponding marginally stable exo-systems. In industrial compressor
operations we typically find surge avoidance rather than active suppression,
which corresponds to the state constraints in the controller design, while
tracking trajectories matches adjustment to variable process demands. Override
control implies a switching scheme and we have given a proof of practical
stability for the overall system. The analytical and simulation results show
that the override control can be applied to discharge pressure control, anti-
surge control and maximum discharge pressure limitation.

Future work can go in several directions. Particularly relevant from an
application point of view is the implementation of the presented MIN/MAX-
override concept in a real centrifugal air compressor system. In this context,
further aspects should be investigated. These could include robustness against
model and parameter uncertainties as well as using output feedback based on
measurement. In this case, an essential aspect is knowledge about disturbance
inputs acting on the system. Since the proposed concept also takes these
disturbances into account, it must be clarified how these disturbances can be
measured or modelled in a given application.

Further work could be done in extending the compressor model to incorporate
additional fluid dynamical phenomena. This may include the following aspects:
(1) considering changes in rotational speed, (2) accounting for different
input and output conditions which would allow modeling of multi-stage
compressors, and (3) improving the model of the compressor map, e.g. by using
pseudo-invariant characteristics or applying efficiency maps which would also
account for the compressor behaviour under variable gas conditions.

Finally, another further research direction is to augment control concepts as
described in this paper by applications of computational intelligence methods
for compressors [16, 31, 36, 41]. Thus, the amount of information extracted
from the compressor operation and available for analysis and subsequent
utilization would be enhanced considerably which is a promising way for an
advanced control performance.

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## Appendix

##  Appendix A Centrifugal compressor modelling

We propose a model which uses relevant physical dimensional parameters rather
than dimensionless parameters, as they can be easily interpreted by
practitioners. The modeling is done by presenting component models for each
part of the system: suction line, impeller, diffuser, plenum, process valve
(PV) and blow-off valve (BOV); see Figure 9 and Table 2 for details.
Comprising all component models will result in the complete model which will
be considered for control design.

Figure 9: Coordinate system for the centrifugal compressor model. Index $1$
refers to ambient conditions.

Symbol | Description
---|---
$s_{i}$ | Axial coordinate
$c_{i}$ | Velocity of fluid at $s_{i}$
$p_{i}$ | Static pressure of fluid at $s_{i}$
$T_{i}$ | Absolute temperature of fluid at $s_{i}$
$\rho_{i}$ | Density of fluid at $s_{i}$
$A_{i}$ | Cross section of component at $s_{i}$
$L_{ij}$ | Covered distance by fluid particle from $s_{i}$ to $s_{j}$
$\Delta h_{ij}$ | Specific enthalpy difference from $s_{i}$ to $s_{j}$
$\Delta u_{ij}$ | Specific inner energy difference from $s_{i}$ to $s_{j}$
$\dot{m}_{C}$ | Compressor mass flow
$\dot{m}_{PV}$ | Process valve mass flow
$\dot{m}_{BOV}$ | Blow-off valve mass flow

Table 2: Notation for the compressor model.

The suction line is described as one-dimensional, transient and threadlike
incompressible flow without friction and heat transfer across piping
boundaries. Therefore, we have $c_{1}=c_{2}$. Now, integrating the
corresponding transient Bernoulli-equation

$\frac{1}{\rho}\int_{p_{1}}^{p_{2}}dp=-\int_{s_{1}}^{s_{2}}\frac{\partial
c}{\partial t}ds$ (25)

gives:

$\frac{p_{2}-p_{1}}{\rho_{1}}=-L_{12}\frac{\partial c_{2}}{\partial t}$ (26)

with $L_{12}$ being the effective length of the suction line. The centrifugal
impeller does work on the system. Despite the complex flow field, the
component model is based on a one-dimensional, steady-state and threadlike
compressible flow that is derived from the first law of thermodynamics for
open and stationary flow systems. The specific enthalpy difference $\Delta
h_{23}$ is given by:

$\Delta h_{23}=\Delta h_{I}+\frac{c^{2}_{3}-c^{2}_{2}}{2}$ (27)

where $\Delta h_{I}$ is the specific enthalpy difference caused by the
impeller. To account for the transient response of the compressor, we apply a
deviation function that is proportional to the rate of change of the fluid’s
speed:

$\Delta h_{23}=\Delta
h_{I}-\frac{c_{3}^{2}-c_{2}^{2}}{2}-\int_{s_{2}}^{s_{3}}\frac{\partial
c}{\partial t}ds.$ (28)

Comparable approaches have been proposed and successfully used by Moore [28],
Moore & Greitzer [29] and Spakovszky [34]. Using enthalpy’s definition and
assuming a linear change of the cross sectional area-density ratio
$A(s)/\rho(s)$ along $s$ as proposed by Spakovszky [34], the component model
reads:

$\frac{p_{3}}{\rho_{3}}-\frac{p_{2}}{\rho_{2}}=\Delta h_{I}-\Delta
u_{23}-\frac{c_{3}^{2}-c_{2}^{2}}{2}-L^{*}_{23}\frac{dc_{2}}{dt}ds$ (29)

where the effective impeller passage length is given by
$L^{*}_{23}\mathrel{\mathop{:}}=L_{23}\ln(r_{I})/(r_{I}-1)$ with
$r_{I}\mathrel{\mathop{:}}=\rho_{3}A_{3}/\rho_{2}A_{2}$. The term $\Delta
h_{I}-\Delta u_{23}-(c_{3}^{2}-c_{2}^{2})/2$ can be interpreted as the steady
state compressor map $Y_{C}$ of the centrifugal compressor leading to:

$\frac{p_{3}}{\rho_{3}}-\frac{p_{2}}{\rho_{2}}=Y_{C}-L^{*}_{23}\frac{dc_{2}}{dt}ds.$
(30)

The diffuser collects all gas streams originating from impeller blade channels
and transports them to the compressor outlet. The motion is described as one-
dimensional, transient and threadlike incompressible flow:

$\frac{1}{\rho}\int^{s_{4}}_{s_{3}}dp=-\frac{c^{2}_{4}-c^{2}_{3}}{2}-\int^{s_{4}}_{s_{3}}\frac{dc}{dt}ds.$
(31)

No work is done on the flow and no heat is exchanged. The diffuser converts
dynamic pressure into static pressure by increasing the cross section. Hence,
the fluid velocity is decreased. We assume a linear change of the cross
section along $s$. After performing the integration in (31) this leads to the
component model of the diffuser:

$\frac{p_{4}-p_{3}}{\rho_{4}}=-\frac{c^{2}_{4}-c^{2}_{3}}{2}-L^{*}_{34}\frac{dc_{2}}{dt}$
(32)

where the effective impeller passage length is given by
$L^{*}_{34}\mathrel{\mathop{:}}=L_{34}\ln(r_{D})/(r_{I}r_{D}-r_{I})$ with
$r_{D}\mathrel{\mathop{:}}=A_{4}/A_{3}$. The summation of Equations (26), (30)
and (32) now gives the momentum equation from $s_{1}$ to $s_{4}$:

$\frac{p_{4}}{\rho_{4}}-\frac{p_{1}}{\rho_{1}}=Y_{C}-L\frac{dc_{2}}{dt}$ (33)

with total effective passage length

$L\mathrel{\mathop{:}}=L_{12}+L^{*}_{23}+L^{*}_{34}.$ (34)

The common procedure is to replace $Y_{C}$ in Equation (33) by the measured
static compressor map describing the pressure increase depending on the
compressor inlet velocity, i.e. $Y_{C}=Y_{C}(c_{2})$. As we are considering
guide vane controlled centrifugal compressors, the guide vane position
$r_{GV}$ will be integrated, which is leading to the measured compressor map
$Y_{C}(c_{2},r_{GV})$. We are using an isoline approach for modeling the
compressor map. The first type of isolines corresponds to the characteristic
curve along the constant guide vane position $r_{GV}$. The second type of
isolines corresponds to the so-called $\beta$-lines. The basic idea for the
introduction of the $\beta$-lines is to create a bijective coordinate system
[23]. This allows unique identification of the compressor state by the guide
vane position $r_{GV}$ and the $\beta$-value. Furthermore, if the
$\beta$-lines are chosen appropriately, each of these isolines can be assigned
a physical interpretation, see Figure 11(a). For example, the $\beta_{max}$
line marks the transition between aerodynamically stable and aerodynamically
unstable operation, i.e. the surge limit. The $\beta_{min}$ line marks the
choke limit. Along the $\beta$-lines, the compressor exhibits similar
aerodynamic behavior. The stable characteristic for constant guide vane
position $r_{GV}$ will be modeled by cubic polynomials with continuous first
derivative along $c_{2}$. Hence, the stable characteristic is $C^{1}$ for
constant $r_{GV}$:

$Y_{C}(c_{2},r_{GV})=\begin{cases}a_{3,1}c_{2}^{3}+a_{2,1}c_{2}^{2}+a_{1,1}c_{2}+a_{0,1}&c_{2,1}\leq
c_{2}<c_{2,2},\\\
a_{3,2}c_{2}^{3}+a_{2,2}c_{2}^{2}+a_{1,2}c_{2}+a_{0,2}&c_{2,2}\leq
c_{2}<c_{2,3},\\\ \qquad\qquad\qquad\vdots&\qquad\quad\vdots\\\
a_{3,n}c_{2}^{3}+a_{2,n}c_{2}^{2}+a_{1,n}c_{2}+a_{0,n}&c_{2,n}\leq
c_{2}<c_{2,n+1}.\end{cases}$ (35)

where the index $n$ is the number of polynomials used to model the stable
characteristic for a given $r_{GV}$, see Figure 11(b). The coefficients
$a_{i,j}$ and interval boundaries $c_{i,j}$ are specific for a given $r_{GV}$,
i.e. $a_{i,j}(r_{GV})$ and $c_{i,j}(r_{GV})$. The index $n=1$ is reserved for
the surge limit of the compressor for a given $r_{GV}$.

(a) Interpolation of the stable compressor map $Y_{C}(c_{2},r_{GV})$ using
some guide vane positions $r_{GV}$.

(b) Modeling the stable part of the compressor characteristic at constant
guide vane positions $r_{GV}$.

Figure 11: Static compressor map modeling using $\beta$-line interpolation.

Now, the dynamic behavior of the guide vane adjustment is considered. The
guide vane motion is assumed to have first order low pass behavior. Hence, the
dynamics of the guide vane position $r_{GV}$ is modeled by:

$\frac{dr_{GV}}{dt}=\frac{1}{\tau_{GV}}\left[u_{GV}-r_{GV}\right].$ (36)

with the motion’s time constant $\tau_{GV}$ and the control input $u_{GV}$.

Figure 12: Example map for an air compressor with outlet guide vane. The
stable characteristics (black lines) are fitted to measured static compressor
working points (gray dots). The transition points for polynomial interpolation
are shown by gray dots.

For simulations the measured compressor map from a real centrifugal air
compressor with outlet guide vanes (OGV) has been used for parameterization.
An example of the resulting map of a centrifugal air compressor can be seen in
Figure 12.

The modeling is completed by deriving models of the plenum and the valves.
Using the conservation of mass and assuming an isentropic process, the plenum
is described by:

$\frac{dp_{4}}{dt}=\frac{\kappa
p_{4}}{V\rho_{4}}\left(\dot{m}_{C}-\dot{m}_{PV}-\dot{m}_{BOV}\right)$ (37)

where $V$ is the plenum volume and $\kappa$ is the isentropic exponent. The
flows through the PV and the BOV are usually modeled using a quadratic
approach, see for example [12]. However, its validity refers to subcritical
and incompressible valve flows only. Since centrifugal compressors can reach
high pressure ratios, the energy equation for ideal, frictionless and
compressible gases can be used to derive Bendemann’s equation. This approach
models the isentropic valve flow from a pressure vessel through a nozzle [33]:

$\displaystyle\dot{m}_{PV}$
$\displaystyle=A_{PV}\Psi\sqrt{\frac{2\kappa}{\kappa-1}\rho_{4}p_{4}},$ (38a)
$\displaystyle\dot{m}_{BOV}$
$\displaystyle=A_{BOV}\Psi\sqrt{\frac{2\kappa}{\kappa-1}\rho_{4}p_{4}},$ (38b)

with the cross section $A_{PV}$ and $A_{BOV}$ for process valve and blow-off
valve as well as the Nusselt outlet function $\Psi$:

$\Psi=\sqrt{\left(\frac{p_{5}}{p_{4}}\right)^{\frac{2}{\kappa}}-\;\left(\frac{p_{5}}{p_{4}}\right)^{\frac{\kappa+1}{\kappa}}}.$
(39)

Adjusting the PV and the BOV positions means changing the cross-sectional
areas $A_{PV}$ and $A_{BOV}$. The relative stroke $h_{PV}:=A_{PV}/A_{PV,max}$
for the PV and the relative stroke $h_{BOV}:=A_{BOV}/A_{BOV,max}$ for the BOV
are introduced with the maximum cross-sectional areas $A_{PV,max}$ and
$A_{BOV,max}$. However, when the valve positions $r_{PV}$ and $r_{BOV}$ are
changed, the relative strokes $h_{PV}$ and $h_{BOV}$ are adjusted depending on
specific valve designs and the resulting characteristics $Y_{PV}$ and
$Y_{BOV}$. Thus, $h_{PV}=Y_{PV}(r_{PV})$ and $h_{BOV}=Y_{BOV}(r_{BOV})$ with
$Y_{PV}(0)=A_{PV,min}$ and $Y_{PV}(1)=A_{PV,max}$ for the PV as well as
$Y_{BOV}(0)=A_{BOV,min}$ and $Y_{BOV}(1)=A_{BOV,max}$ for the BOV. Now,
considering the outlet function (39) and the valve characteristics $Y_{PV}$ as
well as $Y_{BOV}$, the plenum model (37) can be rewritten as:

$\frac{dp_{4}}{dt}=\frac{\kappa
A_{2}p_{4}\rho_{2}}{V\rho_{4}}\left[c_{2}-\left[k_{PV}Y_{PV}(r_{PV})+k_{BOV}Y_{BOV}(r_{BOV})\right]\Psi\sqrt{\frac{2\kappa}{\kappa-1}\frac{\rho_{4}p_{4}}{\rho_{2}^{2}}}\right].$
(40)

with:

$\displaystyle k_{PV}(x_{2})$ $\displaystyle=\begin{cases}{\scriptstyle
K_{PV}}\dfrac{{\scriptstyle A_{PV,\text{max}}}}{{\scriptstyle
A_{2}}}&{\scriptstyle x_{2}\leq}\left(\dfrac{{\scriptstyle
2}}{{\scriptstyle\kappa+1}}\right)^{-\frac{{\scriptstyle 1}}{{\scriptstyle
r_{k}}}},\\\ {\scriptstyle K_{PV}}\dfrac{{\scriptstyle
A_{PV,\text{max}}}}{{\scriptstyle\sqrt{2}A_{2}}}\left(\dfrac{{\scriptstyle
2x_{2}^{r_{k}}}}{{\scriptstyle\kappa+1}}\right)^{\frac{{\scriptstyle\kappa+1}}{{\scriptstyle
2\kappa-2}}}\left(\dfrac{{\scriptstyle\kappa-1}}{{\scriptstyle
x_{2}^{r_{k}}-1}}\right)^{\frac{{\scriptstyle 1}}{{\scriptstyle
2}}}&{\scriptstyle x_{2}>}\left(\dfrac{{\scriptstyle
2}}{{\scriptstyle\kappa+1}}\right)^{-\frac{{\scriptstyle 1}}{{\scriptstyle
r_{k}}}},\end{cases}$ (41a) $\displaystyle k_{BOV}(x_{2})$
$\displaystyle=\begin{cases}{\scriptstyle K_{BOV}}\dfrac{{\scriptstyle
A_{BOV,\text{max}}}}{{\scriptstyle A_{2}}}&{\scriptstyle
x_{2}\leq}\left(\dfrac{{\scriptstyle
2}}{{\scriptstyle\kappa+1}}\right)^{-\frac{{\scriptstyle 1}}{{\scriptstyle
r_{k}}}},\\\ {\scriptstyle K_{BOV}}\dfrac{{\scriptstyle
A_{BOV,\text{max}}}}{{\scriptstyle\sqrt{2}A_{2}}}\left(\dfrac{{\scriptstyle
2x_{2}^{r_{k}}}}{{\scriptstyle\kappa+1}}\right)^{\frac{{\scriptstyle\kappa+1}}{{\scriptstyle
2\kappa-2}}}\left(\dfrac{{\scriptstyle\kappa-1}}{{\scriptstyle
x_{2}^{r_{k}}-1}}\right)^{\frac{1}{2}}&{\scriptstyle
x_{2}>}\left(\dfrac{{\scriptstyle
2}}{{\scriptstyle\kappa+1}}\right)^{-\frac{{\scriptstyle 1}}{{\scriptstyle
r_{k}}}},\end{cases}$ (41b)

where $k_{PV}=K_{PV}A_{PV,max}/A_{2}$ and $k_{BOV}=K_{BOV}A_{BOV,max}/A_{2}$
are for subcritical valve flow. $K_{PV}$ and $K_{BOV}$ are correction factors
valid for a speciﬁc valve. In the simulations we use an equal percentage
characteristics for the PV with $Y_{PV}$ given by:

$Y_{PV}(r_{PV})=k_{PV,0}\left(\dfrac{1}{k_{PV,0}}\right)^{r_{PV}}$ (42)

with $k_{PV,0}$ being the dimensionless $Kv_{0}$ design parameter that can be
extracted from the corresponding data sheet. For the BOV we use a linear
characteristic.

The process valve position $r_{PV}$ and the blow-off valve position $r_{BOV}$
is controlled by the inputs $u_{PV}$ and $u_{BOV}$. The valve motion is
assumed to have first order low pass behavior. Hence, the dynamics of the PV
position $r_{GV}$ and the dynamics of the BOV position $r_{BOV}$ are modeled
by:

$\displaystyle\frac{dr_{PV}}{dt}$
$\displaystyle=\frac{1}{\tau_{PV}}\left[u_{PV}-r_{PV}\right],$ (43)
$\displaystyle\frac{dr_{BOV}}{dt}$
$\displaystyle=\frac{1}{\tau_{BOV}}\left[u_{BOV}-r_{BOV}\right].$ (44)

Equations (33), (36), (40), (43) and (44) constitute the complete compressor
system model. Now, these equations are rearranged to contain only the density
ratio $\rho_{4}/\rho_{2}$, the pressure ratios $p_{4}/p_{2}$ and $p_{5}/p_{4}$
as well as the pressure-density ratios $p_{1}/\rho_{1}$ and $p_{2}/\rho_{2}$.
These ratios are subsequently replaced by known quantities. If the compressor
works on ambient upstream and downstream conditions, then $p_{5}=p_{1}$. If
friction is neglected across the suction piping then $p_{1}=p_{2}$. This
reduces the number of required ratios. The model now depends only on the
density ratio $\rho_{4}/\rho_{1}$, the pressure-density ratio $p_{1}/\rho_{1}$
and the pressure ratio $\Pi\mathrel{\mathop{:}}=p_{4}/p_{1}$. The pressure-
density ratio $p_{1}/\rho_{1}$ can be described by the equation of state (EOS)
for ideal gases $p_{1}/\rho_{1}=R_{S}T_{1}$, where $R_{S}$ is the speciﬁc gas
constant and $T_{1}$ is the ambient temperature. Both are assumed to be known.
Now, we use the pressure ratio $\Pi=p_{4}/p_{1}$ describing the relation
between plenum pressure and ambient pressure as a further state variable and
replace the density ratio and the pressure-density ratio $\rho_{4}/\rho_{1}$.
For this we assume an isentropic process for perfect gas during impeller and
diffuser passage:

$\frac{\rho_{4}}{\rho_{1}}=\left(\frac{p_{4}}{p_{1}}\right)^{\frac{1}{\kappa}}=\Pi^{\frac{1}{\kappa}}.$
(45)

Thus, the total effective passage length $L$ in Equation (34) becomes:

$L(x_{2})=L_{12}+\frac{L_{23}}{\Pi^{\frac{1}{\kappa}}\frac{A_{3}}{A_{2}}-1}\ln\left(\Pi^{\frac{1}{\kappa}}\frac{A_{3}}{A_{2}}\right)+\frac{L_{34}}{\Pi^{\frac{1}{\kappa}}\frac{A_{3}}{A_{2}}{\left(\frac{A_{4}}{A_{3}}-1\right)}}\ln\left(\frac{A_{4}}{A_{3}}\right),$
(46)

which completes the modeling. The model can now be reformulated in its final
version using the state vector $x=(c_{2},\Pi,r_{GV},r_{PV},r_{BOV})^{T}$; see
Equation (1). In the numerical simulations presented in Section 4.5, the model
parameters given in Table 3 are used.

Description | Symbol | Value
---|---|---
Specific gas constant [$JK^{-1}mol^{-1}$] | $R_{S}$ | 287.0
Ambient temperature [K] | $T_{1}$ | 295.4
Isentropic exponent [-] | $\kappa$ | 1.4
Effective length of suction line [m] | $L_{12}$ | 13.0
Effective length of impeller [m] | $L_{23}$ | 2.5
Effective length of diffuser [m] | $L_{34}$ | 1.0
Cross section of impeller [$m^{2}$] | $A_{2}$ | 0.44
Cross section of diffuser inlet [$m^{2}$] | $A_{3}$ | 0.11
Cross section of diffuser outlet [$m^{2}$] | $A_{4}$ | 0.22
Plenum volume [$m^{3}$] | $V$ | 32.0
Process valve correction factor [-] | $K_{PV}$ | 1.0
Process valve maximum cross-sectional area [$m^{2}$] | $A_{PV,max}$ | 0.196
Process valve minimum flow parameter [-] | $Kv_{0}$ | 0.03
Blow-off valve correction factor [-] | $K_{BOV}$ | 1.0
Blow-off valve maximum cross-sectional area [$m^{2}$] | $A_{BOV,max}$ | 0.196
Time constant of guide vane motion [s] | $\tau_{GV}$ | 0.50
Time constant of process valve motion [s] | $\tau_{PV}$ | 0.35
Time constant of blow-off valve motion [s] | $\tau_{BOV}$ | 0.35

Table 3: Parameter of the compressor model.