Proceedings of the 19th World Congress

The International Federation of Automatic Control

Cape Town, South Africa. August 24-29, 2014

An energy perspective on modelling, supervision, and control of large-scale

industrial systems: Survey and framework
George van Schoor*, Kenneth R. Uren**
Michael A. van Wyk***, Pieter A. van Vuuren**, Carel P. du Rand**

*Unit for Energy Systems, North-West University, Potchefstroom,
South Africa (Tel: +2718 299 1962; e-mail: george.vanschoor@nwu.ac.za)
**School of Electrical, Electronic and Computer Engineering, North-West University, Potchefstroom,
South Africa (e-mail: kenny.uren@nwu.ac.za, pieter.vanvuuren@nwu.ac.za, charl.durand@nwu.ac.za)
*** School of Electrical and Information Engineering, University of the Witwatersrand,
Johannesburg, South Africa, (e-mail: anton.vanwyak@wits.ac.za)

Abstract: Energy is a universal concept that can be used across physical domains to describe complex
large-scale industrial systems. This brief survey and framework gives a perspective on energy as a
unifying domain for system modelling, supervision, and control. Traditionally, modelling and control
problems have been approached by adopting a signal-processing paradigm. However, this approach
becomes problematic when considering non-linear systems. A behavioural viewpoint, which incorporates
energy as basis for modelling and control, is considered a viable solution. Since energy is seen as a
unifying concept, its relationship to Euler-Lagrange equations, state space representation, and Lyapunov
functions is discussed. The connection between control and process supervision using passivity theory
coupled with a system energy balance is also established. To show that complex industrial systems
comprising multiple energy domains can be modelled by means of a single electric circuit, its application
to a large-scale thermo-hydraulic system is presented. Next, a simple non-linear transmission impedance
electric circuit is used to illustrate how energy can be used to not only describe a system, but also serve as
basis for system optimisation. An energy-based framework is proposed whereby energy is used as a
unifying domain to work in, to analyse, and to optimise large-scale industrial systems.
Keywords: Energy, modelling, supervision, control, multi-domain, large-scale, industrial systems,
equivalent electric circuit.

stability analysis and control design framework for large-
1. INTRODUCTION
scale non-linear interconnected dynamic systems. The
Large-scale industrial systems are characterised by a proposed framework stands on the legs of vector Lyapunov
multitude of sub-systems exchanging matter and energy to functions and passivity theory. In the analysis of large-scale
accomplish a common goal. The interactions of the sub- non-linear industrial systems, several Lyapunov functions
systems can be complex and take place across different arise naturally from the stability properties of each individual
physical domains such as thermal, chemical, fluid, subsystem. Furthermore, with many input, state, and output
mechanical, and electrical, to name but a few. Most physical properties related to the supply, storage, transport, and
systems also portray non-linear behaviour of varying dissipation of energy, extending classical dissipativity theory
complexity. Global system optimisation, meeting specific (Willems, 1972a, 1972b) to include storage and dissipation
performance objectives while maintaining a healthy energy on the subsystem level, leads to a natural energy-based
profile, is therefore a complex and multi-faceted problem. framework for large-scale industrial systems. Dissipativity
refers to the system characteristic where a fraction of the
Energy is seen as a unifying concept that can be used across
energy supplied to the system is transformed into heat or
physical domains to characterise and describe complex large- losses. Passivity is a special case of dissipativity, where the
scale industrial systems. The engineering challenge can be energy stored in the passive systems cannot exceed the
described as one of achieving some global plant objectives
energy supplied by the external environment. It provides a
through the effective manipulation and transformation of
fundamental framework for the analysis and design of control
energy. The notion to consider energy as a measure of system
using a state space formalism based on system energy related
stability is of course the basis of Lyapunov’s second stability considerations.
criterion (Shinners, 1998), where the sum of the system’s
kinetic and potential energy is considered as a function, and Linear or non-linear complex systems have long been
the time derivative of the function is taken. Haddad & modelled by means of equivalent electric circuits. This is due
Nersesov (2011) proposes the global optimisation of a to the striking similarity that exists between the differential
complex system from an energy perspective, with a unified equations that describe the behaviour of physical systems in a

978-3-902823-62-5/2014 © IFAC 6692

19th IFAC World Congress
Cape Town, South Africa. August 24-29, 2014

variety of domains, ranging from electric circuits to paper aims to establish an apparent relation between the tasks
mechanical movement and thermodynamics. The above mentioned using energy as the unifying domain. The paper
mentioned pattern becomes even more evident when so- starts out in Section 2 with a historical development of
called ''through'' and ''across'' variables are used (Dorf & energy as the basis for system modelling, supervision, and
Bishop, 2011). Examples of "through" variables (flow control. Section 3 presents the relevance of energy as a
variables in Bond graph terminology) include current, force, unifying domain for the tasks mentioned. A perspective on
torque, fluid volumetric flow rate, and heat flow rate. realistic systems is then given in Section 4, portraying
Examples of "across" variables (effort variables in Bond equivalent electric circuits as representative of complex
graph terminology) include voltage, velocity difference, multi-domain industrial systems. Next, a non-linear electric
angular velocity difference, pressure difference, and circuit case study is examined in Section 5, demonstrating
temperature difference. The energy contained in through how energy can form the basis for global system
variables is stored inductively in the form of inductors optimisation. Section 6 describes an envisaged energy-based
(electric circuits), springs (mechanical systems), and fluid framework using energy formalisms as a unifying concept for
inertia. The energy contained in across variables is stored system modelling, supervision, and control. Some final
capacitively in the form of capacitors (electric circuits), mass thoughts and concluding remarks are presented in Section 7.
(mechanical systems), fluid capacitance, and thermal
2. HISTORICAL DEVELOPMENT
capacitance. Lastly, energy is also dissipated in a similar
manner across various domains by means of resistors Considering the general history of control theory, it is
(electric circuits), dampers (mechanical systems), and interesting to notice that the control system design process
thermal resistance. In addition, equivalent electric circuits has been traditionally approached from a signal theoretic
typically represent lumped parameter models of the examined perspective (R Ortega, van der Schaft, Mareels, & Maschke,
system. Combined with the use of either positive or negative 2001). This perspective can be traced back to the 1930s with
feedback, it is therefore possible to approximate most the development of the first feedback amplifiers. In this
systems (in a particular domain) by means of an equivalent paradigm, the sub-systems of the control system are viewed
electric circuit. as signal processing devices that transform input signals into
output signals. The design specifications are based on
The notion of using energy as basis for global optimisation is
minimising error signals and reducing the effect of
extended in this work to the more general concept of
disturbance signals in the presence of model uncertainties.
applying energy for the purpose of system representation.
Mathematically this translates to the assumption that the
The idea stems from two apparently unrelated works. The
disturbance signals and unmodelled dynamics are norm
first derives a generic procedure for state space model
bounded. This means that the control performance is
extraction of large-scale thermo-hydraulic systems, whereby
determined by the size of the operator gains that map the
the transparency of the system components is retained in the
various input signals to the output signals. The mathematical
final model (Uren & van Schoor, 2013a, 2013b). In the state
framework that supported the modelling, analysis, and
space model, each state represents stored energy associated
synthesis of control systems was based on input-output ideas
with that particular state, and therefore, an important link can
with Fourier and Laplace transforms being the dominant
be made between energy as basis for modelling and control.
mathematical tools. This paradigm worked particularly well
In the second work, a methodology is devised to extract
in the case of linear time-invariant control systems since
enthalpy and entropy fault signatures of a large-scale thermo-
filtering using frequency-domain considerations can be
hydraulic system for the purpose of fault detection and
implemented successfully.
diagnosis (FDD) (du Rand & van Schoor, 2012a, 2012b). A
connection is thus established between energy and process During the 1960s and 1970s, this paradigm took a new
supervision, and visualising the condition of the system using direction due to the introduction of the state-space formalism.
energy signatures. In a mathematical framework, mapping between the inputs
and outputs of the control system are based on the
Note that in this work, terminology in the field of process
transformation of the internal state of the control system.
supervision and FDD are adopted according to the IFAC
Moreover, in this approach, the mathematical tools turned
SAFEPROCESS Technical Committee (Isermann & Ballé,
towards ordinary differential equations. The celebrated
1997). Supervision constitutes a continuous task of
concepts of controllability, observability, and optimality were
determining the condition of the process (monitoring)
introduced that led to powerful controller design techniques.
whereby system anomalies are detected, diagnosed, and
corrected. The diagnosis task comprises fault isolation (type, From the 1990’s research is focused towards the development
location, and time) and fault identification (magnitude and of a paradigm that allows for the treatment of a more
time-variant behaviour). Also, an energy signature does not generalised class of systems, the goal typically being to
signify a simple “best-fit” line between specific calculated consider non-linear systems represented by
indices (Belussi & Danza, 2012; Yu & Chan, 2005), but aims
to optimally depict system knowledge based on energy x  f ( x, u), (1)
indicators.
y  h( x, u). (2)
This paper gives an energy perspective on modelling,
supervision, and control of large-scale industrial systems in This notion was inspired by (Willems, 1991, 2007) and is
the form of a brief survey and proposed framework. The generally referred to as the behavioural approach towards

6693
19th IFAC World Congress
Cape Town, South Africa. August 24-29, 2014

systems modelling and control. In this paradigm a  It has the ability of treating both finite-dimensional
mathematical model is viewed as a subset of a universum of and infinite-dimensional components.
possible descriptions of reality. That is, before a
mathematical model is derived of the real system, all
outcomes in the universum are in principle possible. After a
mathematical model is accepted as a convenient description
of reality, only then a certain subset of outcomes is possible.
This subset is called the behaviour of the mathematical
model. Proceeding from this perspective, one arrives at the
notion of a dynamical system as simply a subset of time-
trajectories. This paradigm is not as restrictive as the
input/output point of view. In fact, most physical systems do
not have a preferred signal flow direction, and it is important
to let the mathematical structures reflect this. The
behavioural paradigm therefore starts from a mathematical
model obtained from first principles resulting in a set of
differential and algebraic equations. Among the vector of
time trajectories satisfying these equations are components
that are available for interconnection. The process of
controller design reduces to defining an additional set of
equations for these interconnection variables to impose a
desired behaviour on the system.
This paradigm naturally supports the fundamental concept of
energy conservation. Therefore, complex dynamic systems
consisting of sub-systems and controllers are viewed as
energy-transforming devices that interconnect via power
Fig. 1. Paradigms for modelling and control of large-scale
conserving connections to achieve not only a desired
dynamic systems.
response, but also an optimal system response.
The port-Hamiltonian formalism is able to match the “old”
Considering the modelling and control of large-scale systems,
framework of port-based network modelling of multi-domain
the same kind of restrictions surface as with the signal
physical systems with the “new” framework of geometrical
theoretic paradigm. According to (Haddad & Nersesov,
dynamic systems and control theory. This allows for the
2011), the behavioural paradigm is a much more natural fit.
systematic approach of modelling, analysis, condition
It follows that energy-based modelling arises naturally in
monitoring/fault detection and control, via
large-scale dynamic systems. Fig. 1 illustrates the two
paradigms regarding the modelling and control of large-scale  separation of the interconnection structure of the
dynamic systems. Considering the behavioural paradigm, system from the constitutive relations of its
three modelling approaches can be followed using energy components;
concepts (Janschek, 2011):
 enforcing power conservative interconnections by
 Energy-based modelling employing scalar energy means of Dirac structures;
functions using either Euler-Lagrange or Hamiltonian  analysing the system making use of the
formalisms. interconnecton structure and component constitutive
 Multi-port modelling employing component-based relations;
system models with power-conserving rules utilising e.g.  the achievement of control by means of Casimir
Kirchhoff networks or Bond/linear graph approaches. generation, energy shaping, energy routing and port
 A combination of the Hamiltonian and port-based and impedance control.
modelling formalism called Port-Hamiltonian modelling. From a geometric perspective, the Dirac structure lies central
The port-Hamiltonian formalism is especially of great in describing port-Hamiltonian systems. The Dirac structure
importance regarding the modelling of complex, large-scale has a strong link with bond graphs, especially 0- and 1-
systems due to the following advantages (Duindam, juncionts are prime examples of the general concept of a
Macchelli, Stramigioli, & Bruyninckx, 2009): Dirac structure. Generalised flow and effort vectors (e, f )
are elements of abstract finite-dimensional linear spaces
 It is highly scalable, and therefore naturally allows
and respectively. The effort space is defined as the dual
the modelling of very large interconnected multi-
physics systems. space of , that is : * . The total space of flow and
effort variables is  * and is generally called the space of
 Due to a strong differential geometric base, it has port variables.
the ability to incorporate non-linearities while
retaining underlying conservation laws.

6694
19th IFAC World Congress
Cape Town, South Africa. August 24-29, 2014

On this total space of port variables, the power is defined by realised by monitoring this inequality for “fault” energy.
Since implicit FDD is only required for part of faults under
P  e| f ( f , e)   *
, (3) the passivity framework, certain conditions on the system
structure can be relaxed for implicit fault diagnosis (Gertler,
where e | f denotes the dual product, that is, the linear 1998). (W Chen, Ding, Khan, & Abid, 2010) extends the
functional e  *
acting on f  . A Dirac structure on passivity framework to a more inclusive energy based
framework taking into account the system’s dissipative
 *
is then a subspace   *
such that
properties. Therefore, unlike an energy inequality, an energy
 e | f  0, for all (e, f )  , balance is achieved which offers optimal FDD. For
unmeasurable system states, FDD is accomplished by an
 dim  dim . optimal approximation of the energy balance based on system
inputs and outputs. In (Wei Chen, 2011), the energy balance
This illustrates the notion that port-Hamiltonian system based FDD is further developed to accommodate passive
descriptions share common ground with geometric nonlinear non-linear systems. FDD design procedures are established
control theory and geometric mechanics. for two classes of passive non-linear systems namely input-
Traditional and advanced methods for process supervision affine and Lagrangian systems.
and FDD are well documented in the literature (Bokor & 3. ENERGY AS A UNIFYING CONCEPT
Szabó, 2009; Das, Maiti, & Banerjee, 2012; Hwang, Kim,
Kim, & Seah, 2010; R Isermann & Ballé, 1997; Rolf 3.1 Prelude to energy as a universal concept
Isermann, 1984; Qin, 2012; Venkatasubramanian,
Rengaswamy, Kavuri, & Yin, 2003a, 2003b, 2003c). Energy is a universal concept in systems and processes found
However, in the last few decades, FDD based on energy in all domains, and therefore, also in multi-domain systems.
formalisms (i.e. not energy based signal transformations) did In this work, the term system implies a closed environment
not develop to any great extent. The most notable that represents energy exchanges internal to the system, but
contributions relate to an energy balance or conservation also to and from the system. A simple example of a multi-
principle, which offers great possibilities due to its clear domain system demonstrating this concept is a permanent
physical meaning and easy implementation (Wei Chen, magnet DC motor, in which electrical energy is converted to
2011). The energy can be representative of the true physical mechanical energy and vice versa. Specifically, the armature
system energy, or an abstract energy function defined via current is converted to mechanical torque, which results in
Lyapunov theory. acceleration of the motor’s rotor. Conversely, the mechanical
angular speed of the motor results in a back electromotive
Model-based fault detection based on energy balance
force. This inherently feeds back to the electrical subsystem
calculations takes its origins from chemical process control in
of the motor, thereby controlling the magnitude of the rotor
the 1970s (Gertler, 1998 that refers to previous works of
current. Consequently, the amount of mechanical torque
Himmelblau, 1978; Vaclavek, 1974). Berton & Hodouin
produced is controlled. Another important point to be made is
(2003) introduced a conservation model obtained via linear
that the general theory of systems assumes that the examined
and bilinear state equations describing mass and energy
system is linear, and if not, that the system can be linearised
balances. Interactions between different conservation laws,
in the operating region of interest. An important salient
i.e. mass and energy, are evaluated using single bilinear FDD
feature of energy is that it applies equally to both linear and
residual vectors. (Sunde & Berg, 2003) successfully achieved
non-linear systems. Therefore, by considering a system from
the notion of fault detection by way of plant-wide mass and
the viewpoint of energy, the very limiting requirement of the
energy balances for a 3,300 MW t boiling water reactor
system to be linear proves to be superfluous and can thus be
turbine cycle. The balance equations were implemented as
disposed of.
constraints to a minimisation problem. (Theilliol, Noura,
Sauter, & Hamelin, 2006) exploits the energy balance of a 3.2 Euler-Lagrange equations and energy
SISO closed-loop system to generate residuals of the energies
involved without an input-output model. Very often, such The Euler-Lagrange equations used for deriving the
input-output models are almost impossible to obtain for differential equations that models a given problem forms part
complex large-scale industrial systems. In this case, the of the subject called Calculus of Variations. The basic
energy balance offers an intuitive way to perform FDD. problem herein is to infer a function x(.) (i.e. not a variable’s
Energy indices are used by (Tinaut, Melgar, Laget, & value) that minimises a specified definite integral. The
Domínguez, 2007) to investigate the interchange of energy integrand of the latter is a function of the original function as
between different engine components for the purpose of fault well as certain derivatives thereof. In its simplest form, the
detection. An energy model corresponding to the change in problem is that of determining a once-differentiable scalar
total kinetic energy of the moving parts facilitates a function x(t) of a single independent variable t, for which the
transparent and straightforward FDD approach. integral
Subsequent energy supported FDD works follow from
I [ x(t )]   F  t , x(t ), x(t )  dt
t1
passivity theory. (Yang, Cocquempot, & Jiang, 2008) (4)
t0
constructs a global passivity energy relation by an inequality
that comprises system states, inputs and outputs. FDD is

6695
19th IFAC World Congress
Cape Town, South Africa. August 24-29, 2014

is minimised. The function F in (4) is called the Lagrange are those represented by the product of two different state-
function. variables.
It follows readily (Hildebrand, 2012) that the function x(t) For an arbitrary n-th order driven system with m outputs, the
that minimises the integral (4) satisfies the Euler-Lagrange general form of the state space representation is given by the
equation state equation
d  F  F x(t )  f ( x(t ), t ), f : Rn 1  Rn (7)
dt  x   x  0. (5)

together with the output equation
Conversely, (5) is a necessary but not sufficient condition for
y(t )  h( x(t ), t ), h : Rn 1  Rm (8)
a function x(t) to minimise the integral (4). It is possible to
generalise the above to problems involving n dependent and
for all t ≥ t0 with x(t0) = x0.
m independent variables. The particular case of interest is
that of modelling a system using an ordinary differential If the functions f and h are independent of time, then the
equation of order n with t as the independent variable. In this system is said to be autonomous (i.e. not driven). A large
case, the set of Euler-Lagrange equations (Hildebrand, 2012) class of non-linear driven systems can be represented in the
in (5) needs to be solved. form

d  F  F x(t )  g ( x(t ))  Bu(t ), y(t )  Cx(t ) (9)

   0, for i  1, ,n (6)
dt  xi  xi
where u(t) represents the input vector of the system. In
(Hrusak, Stork, & Mayer, 2009) it is shown that for the case
For our purpose, the Lagrange function is selected to be the
where the function g is of the form g(x) = A(x)x, the
energy difference F (t , x, x)  T ( x, x)  V ( x). Here, T and V instantaneous value of the output power P(t) and the
represent the total kinetic and potential energy of the system corresponding average energy E(t) of the state, up to time t,
respectively, and are expressed in terms of the variable set are related by
{x1,…,xn} called generalised coordinates, and their
derivatives {ẋ1,…,ẋn}. Taking this set of generalised dE 2
  P(t )   y(t ) . (10)
coordinates to be the state variables then links the Euler- dt
Lagrange equations to state space modelling. Therefore, by
applying this method for the purpose of system modelling, a Then, for the zero input case, the energy present in the system
state space representation of the system can be obtained. at time instant t0 is
Refer to (Jeltsema & Scherpen, 2009) for a more 
E (t0 )  
2
comprehensive discussion of Euler-Lagrange equations. y(t ) dt. (11)
t0

3.3 Relationship between state space representation and A state space representation with the A matrix of the form
energy
 11 2 0 0 0 0 
Just as energy is a universal concept across various domains,    22  3 0 0 0 
it comes as no surprise that the state space representation is  2
also a universal concept, allowing a system’s dynamical  0  3  33 0 0 
A  (12)
behaviour to be expressed as a set of first order ordinary  0 0  n 1 0 
differential equations (refer to equation (7) below).  0 0 0  n 1  n 1, n 1 n 
 
For an arbitrary multi-domain system, a particular state space  0 0 0 0  n  n n 
representation, called the standard form, can be obtained by
choosing the output of each energy storage element to be a exists and is termed physically correct (Hrusak et al., 2009).
state variable of the system (Shinners, 1998). Here, the output By (Hrusak et al., 2009), a large class of non-linear systems,
of a storage element is the dependent variable associated with all for which g(x) = A(x)x, can be represented by a state space
the storage element as dictated by the given system’s representation for which (12) holds. For the non-linear case,
configuration (e.g. the current passing through a voltage-fed components of A in (12) depend on x.
inductor). For this particular state-variable assignment, the
time average of a state-variable squared is easily shown to be In (Hrusak et al., 2009) and references therein, it is shown
proportional to the energy stored in the associated energy that a necessary and sufficient condition for dissipativity of
storage element. This confirms our suspicion that there exists this class of non-linear systems is that α1 > 0, while a
a deep connection between energy flow in a system, and the necessary and sufficient condition for conservativity is that α1
state space representation of the system. Moreover, in more = 0. A necessary but not sufficient condition for asymptotic
elaborate paradigms (e.g. as encountered in optimal control), stability is that α1 > 0. Refer to (Hrusak et al., 2009) for more
even generalisations of power and energy namely cross- detail. In (Mayer, Hrusak, & Stork, 2013), this energy state
power and cross-energy are considered. Cross-power terms space approach is applied to gain insight into the mechanisms

6696
19th IFAC World Congress
Cape Town, South Africa. August 24-29, 2014

responsible for chaotic behaviour of two non-linear coupled t

H  x  t   H  x  0   d  t   u   s y  s  ds .
T
oscillators. (14)
0
stored energy dissipated
3.4 Lyapunov functions and how they relate to energy energy supplied energy

Another subtle connection is worthwhile to emphasise. In the A control action u(t )   ( x(t ))  v(t ) may be applied such
theory of autonomous ordinary differential equations (ODEs), that the closed-loop system is again a passive system, with
Lyapunov functions are scalar functions that enable stability energy function Hd (x(t)), with respect to v y , and such that
analyses of an equilibrium point of the ODE. For a system Hd (x(t)) has a global minimum at a desired point x*(t). If a
that can be represented by the state space representation (7), a function  ( x(t )) can be found for some function Ha (x(t))
once-differentiable function V : D × R→R, with D ⊂ Rn a such that
neighbourhood of the state space’s origin, is called a
t
Lyapunov test or candidate function if both V(x,t) > 0 for x ≠
   T ( x( s)) y( s)ds  H a ( x(t )), (15)
0, and 0
V(0,t) = 0 for all t ≥ t0 (Jordan & Smith, 2007). A Lyapunov
candidate function that has the property of a non-positive then the closed loop system will also be passive, with input
orbital derivative, i.e. v(t) and an energy function
V H d ( x(t ))  H ( x(t ))  H a ( x(t )). (16)
V ( x, t )   xV  f ( x, t )  ( x, t )  0, t  t0 , (13)
t
This methodology of assigning an energy function with a
is called a Lyapunov function for the system (7). The minimum at the desired values is generally referred to as
existence of a Lyapunov function then guarantees that the energy shaping. In some cases the natural dissipation term
state space’s origin is stable. By coordinate translation, may be replaced by some function dd(t) ≥ 0. This is called
considering a signal that comprises the difference between damping injection (Romeo Ortega, Schaft, Maschke, &
the state vector of a system and a non-zero equilibrium point Escobar, 2002).
of the same system, the power of this signal is an indication
The notion of passivity based control adopting the energy
of how far the signal travels from the equilibrium point and
balance is extended to process supervision as previously
hence, is a Lyapunov candidate function for the equilibrium
discussed (W Chen et al., 2010). The system in (13) is
point. In (Stork, Hrusak, & Mayer, 2005), this approach is
dissipative with respect to the supply rate S(u,y) = yTMu and
referred to as the energy-metric approach. For a certain class
storage function V(x) = xTPx/2 if
of systems, this choice of candidate function does comprise a
Lyapunov function for equilibrium points. By non-linear T
PA  A P  0, (17)
warping of the power as a function of the state variable x,
Lyapunov functions for a larger class of non-linear systems and
may be obtained (Guckenheimer & Holmes, 1997). Clearly,
for a large class of systems, Lyapunov functions for studying T
PB  C M . (18)
stability are intimately connected with the concept of power
and hence to energy, relative to an equilibrium point in state Assuming x(0) = 0, the energy balance is
space.
1 1

 x  T T Px  T      xT PA  AT P xdt  
T
Although not directly related to this paper’s focus, as a final 2 2 0 
note on Lyapunov functions, the reader is referred to the stored energy dissipated
energy (19)
development presented in (Malisoff & Mazenc, 2009). As
T
presented herein, Lyapunov functions are used to design 0
T
 y Mudt .
controllers to satisfy specific stability requirements for the supplied energy
closed-loop system.
For the faulty case, (19) becomes an inequality. The energy
3.5. Passivity and the energy balance balance for faults can therefore be expressed as
It is noteworthy to show that passivity theory and the energy 1
 f   x f T  Px f T 
T

balance can be used as a unifying energy paradigm to 2 

perform both system control and FDD. Passivity theory is an (20)
 
   xTf PA  AT P x f dt    yTf Mudt.
1
T T
established approach for stability analysis and control of non- 
 0 
 0
2
linear systems (Brogliato, Lozano, Maschke, & Egeland,
2006; R Ortega et al., 2001). Consider a system with states x Fault detection is realised for component, actuator, and
n m m
∈ R , inputs u ∈ R , and outputs y ∈ R . The mapping sensor faults by (20), (21), and (22) respectively.
u y is called passive if there exists a state function H(x),
T T
bounded from below, and a non-negative function d(t) ≥ 0  f   xTf PAx f dt   xTf PBudt (21)
0 0
such that

6697
19th IFAC World Congress
Cape Town, South Africa. August 24-29, 2014

T The PCU can be simplified by assuming that the heat

 f   xTf PBf a dt (22)
0 exchangers are all actively controlled, resulting in constant
outlet temperatures (Uren & van Schoor, 2013b). After
T
 f   f sT Mudt (23) simplification, the conceptual model of the PCU is given in
0 Fig. 3. In this figure, the helium injection and extraction
components are modelled by two externally controlled mass
In the equations, ∆A and ∆B are component faults, fa is an flow sources. The equivalent electric circuit for the hydraulic
actuator fault, and fs is a sensor fault. To perform fault domain is shown in Fig. 4. In this model, the turbines and
isolation, the type of energy change is first identified, i.e. compressors are modelled by non-linear frequency dependent
stored or dissipative. Next, the fault location is established by current sources, while pipe elements are modelled using RLC
writing (20) as a summation of energy-storing and energy- networks. A number of RC networks, voltage dividers, and
dissipating components, and checking a hypothesis involving amplifiers can be used to model each individual turbine in
the system states. For input-affine and Lagrangian non-linear more detail (see block diagram model of a generic turbine
systems, the energy balances can be written as (Wei Chen, (Dynamic models for fossil fueled steam units in power
2011) system studies, 1991)).
 V 
V  x     f ( x ) dt  0 y
T
udt , (24)
0
x Generator
stored energy supplied energy HPT LPT
Core Shaft
dissipated
energy

Recuperator
PT
and Shaft Shaft
Intercooler Pre-cooler
 R 
V    V  0    q
T
dt  0 q
T
udt . (25) High pressure Low pressure
0
q side of system HPC LPC
Bypass
valve
side of system
stored energy supplied energy
dissipated
energy

High pressure Low pressure

4. A PERSPECTIVE ON REALISTIC SYSTEMS extraction injection

Equivalent electric circuits is a suitable choice to describe Fig. 2. PCU of the PBMR.
large-scale industrial systems using energy:
 Energy and power are easily calculated in electric Generator

circuits. HPT LPT

 Equivalent electric circuits can be used to model systems

in a variety of domains. In fact, in the years predating Heat
PT

digital computers, differential equations could be solved source

by means of analogue computers (Howe, 2005). In an

analogue computer, operational amplifier circuits are
used to model phenomena and solve the resulting HPC LPC
differential equations. S2 S1

Constant inlet Constant inlet

 In general, there is no restriction that the components in temperature temperature

an equivalent electric circuit have to be linear. For

example, hysteresis can be modelled by means of non- Fig. 3. Conceptual model of the PCU.
linear inductors. Similar, amplifiers with saturation can
HPT LPT PT
be represented by voltage or current sources combined
with both normal or Zener diodes.
An example of a complex large-scale industrial system is the
power conversion unit (PCU) of the pebble bed modular
reactor (PBMR) concept (van Niekerk, Pritchard, van Schoor,
& van Wyk, 2006). The PBMR PCU entails a three-shaft
closed Brayton cycle and is depicted in Fig. 2. In addition to
piping and valves, the PCU consists of a pebble bed nuclear
reactor, high- and low-pressure turbines (HPT and LPT), a
HPC LPC
power turbine (PT), recuperator, pre-cooler, low-pressure S2 S1
compressor (LPC), intercooler, and a high-pressure
compressor (HPC).
Fig. 4. Equivalent electric circuit of the hydraulic sub-
systems of the PCU.

6698
19th IFAC World Congress
Cape Town, South Africa. August 24-29, 2014

Note that the equivalent circuit shown in Fig. 4 is limited to a In accordance with (Seshu & Reed, 1961), the power of the
single domain. In the case of the PBMR, four domains, i.e. individual components pi adds up to zero given by
electrical, mechanical, hydraulic, and thermal, are tightly knit N
together into a single system. It is difficult to describe energy p i  0. (27)
flows in such a system if the various domains are modelled i 1

by means of separate equivalent circuits. However,

equivalent circuits representing different domains can be In this case, the total number of components N equals six.
coupled by means of generalised transformers. Similar to the The components can be grouped into power sources, energy
operating principle of an electrical transformer, a generalised storage elements, dissipative components, and load
transformer can be used to model the coupling between components. Equation (27) therefore translates to
various energy domains (Cheng, Wang, & Arnold, 2007). In 7 1 2 2 1
terms of electric circuit components, a generalised p p pi si sti   pdi   pLi
(28)
transformer can be constructed from two current controlled i 1 si 1 sti 1 di 1 Li 1

voltage sources. As an example, Fig. 5 shows the coupling  Ps  Pst  Pd  PL  0

between the mechanical and electrical domain in a magnetic
energy harvesting device (Cheng et al., 2007). A single with psi, psti, pdi, and pLi representing the sources, storage,
equivalent electric circuit can be obtained from the model by dissipation, and load power components respectively. Ps, Pst,
reflecting the impedances of one domain over to the other Pd, and PL represent the total power associated with the
side (Cheng et al., 2007). respective component groups.

Spring Damper Coil resistance

An optimal operating point from a power loss perspective
will be at maximum power efficiency. The power efficiency
Force Mass Load
can be written in terms of Pd and PL as follows
Coil inductance
PL
P  . (29)
PL  Pd
Mechanical domain Electrical domain
Given that Rx1 and Rx2 are non-linear functions of x1 and x2
Fig. 5. Equivalent circuits coupled by a generalised respectively, the total dissipation losses is
transformer. 2
x2 2
Pd   pdi  x12 Rx1 ( x1 )  , (30)
5. ELECTRIC CIRCUIT CASE STUDY di 1 Rx 2 ( x2 )
In the previous section, a connection was made between
with the load power
realistic systems and equivalent electric circuits. Therefore, to
illustrate the application of energy as unifying concept for x2 2
system optimisation, consider the simple non-linear electric PL  . (31)
RL
circuit in Fig. 6. Non-linearities are introduced to realistically
represent actual industrial systems. The circuit takes the form
of a typical transmission system with a source vs, a Substituting (29) and (30) into (28) therefore results in
transmission impedance represented by non-linear resistances x2 2
Rx1 and Rx2 associated with an inductance Lx1 and capacitance
Cx2 respectively, and a load RL. RL
P  . (32)
x2 2 x2 2
The state space formulation of the circuit is given by  x12 Rx1 ( x1 ) 
RL Rx 2 ( x2 )
  Rx1 1 
   1  For this case study, Rx1 and Rx2 are chosen to be non-linear
 x1   Lx1 Lx1   L v (26)
x   functions of x1 and x2, as portrayed in Figs. 7 and 8
1   
x1 s
 2  1 1  1
     0  respectively. Fig. 9 shows the mesh diagram of power
 Cx 2 Cx 2  Rx 2 RL  
efficiency as a function of x1 and x2.
with the state variables x1 and x2 denoting the current through The theoretic maximum efficiency operating point (x1m, x2m) is
Lx1 and the voltage across Cx2 respectively. given by
Rx1 Lx1 ( x1m , x2 m )  arg max P ( x1 , x2 ). (33)
x1 , x2 R

This maximum point will be one of the critical points of ηP,

vs Cx 2 Rx 2 RL
determined from the partial derivatives of ηP

P x1 ( x1 , x2 )  P ( x1 , x2 )  0 (34)
x1
Fig. 6. Simple electric circuit.

6699
19th IFAC World Congress
Cape Town, South Africa. August 24-29, 2014

and

P x 2 ( x1 , x2 )  P ( x1 , x2 )  0. (35)
x2

Applying (34) and (35) to (32) results in

d
2 Rx1  x1 Rx1  0 (36)
dx1

and
x23 d
2 x12 Rx1  Rx 2  0. (37)
Rx 2 2 dx2

Solving the expressions in (36) and (37) will give the maxima
and minima of  P , including the point ( x1m , x2m ) . The
simulation results in Fig. 9 produced a maximum power Fig. 9. Power efficiency vs. x1 and x2.
efficiency point of (2.98, 100.55), corresponding with an
In the context of global system optimisation through
efficiency of 98.74 %.
modelling, supervision, and control (the latter two are not
The concept of reachability (Ohta, Maeda, & Kodama, 1984) shown), this case study demonstrates how an equivalent
now becomes important to determine whether the point of electric circuit can serve to represent an energy model with
maximum efficiency is reachable within finite time using a associated state space that is suitable for dissipativity
specific input. The concept of least norm input for analyses. The simple circuit representation of Fig. 6 can be
reachability (Boyd, Ghaoul, Feron, & Balakrishnan, 1994) extended to a more general system representation constituting
can be used to realise the desired state transition with the sources (current and voltage), energy storage elements
least amount of energy input. (inductive and capacitive), dissipative elements, and loads of
which some elements can also be non-linear functions of the
50 system states. The development of generic topological
45 representations of such electric circuits is a topic for
40 continued research, and is therefore not included in this
35 paper.
30
6. AN ENERGY-BASED FRAMEWORK
Rx1 ( )

25

20
The fact that energy is a universal concept that holds across
15
different domains is the reason for us to focus on the use
10
thereof for large-scale industrial system analyses. It is
5
envisioned that unified energy formalisms can be established
0
0 2 4 6 8 10 12 to facilitate modelling, supervision, and control of these
x (A)
1 energy systems. Fig. 10 depicts the method implied by the
vision. The system block represents any large-scale industrial
Fig. 7. Non-linear function Rx1 ( x1 ) . system such as a power or petrochemical plant. In these
systems, the principles described will not only apply on a
1200 global systems level, but also on sub-system or even
component level.
1000
Reviewing Fig. 10, the examined system is firstly
800 transformed to an equivalent electric circuit. The process of
abstraction to obtain an energy signature implies obtaining a
Rx2 ( )

600
representation of the system based on an energy formalism.
400
This can for instance denote the system’s energy distribution
in terms of energy supplied, stored, transported, and
200 dissipated. The energy formalism is then used for feature
extraction to describe a reference energy signature. This
0
0 20 40 60 80 100 120 transformation will entail finding a global optimum in terms
x (V)
2
of some global system objectives, encapsulated in an optimal
Fig. 8. Non-linear function Rx 2 ( x2 ) . energy distribution profile. It is envisaged that such an
optimal energy distribution will be associated with an optimal
set of system states. Comparing an actual energy signature
with the reference case will serve to evaluate and supervise

6700
19th IFAC World Congress
Cape Town, South Africa. August 24-29, 2014

system performance. The process of compiling an actual abstraction of usable energy formalisms and the subsequent
energy signature may include some measurement inference or transformation to energy signatures. This warrants further
estimation of system variables. Analogous to normal research to explore the viability of the energy-based
feedback control, the comparison of energy signatures yields framework.
a set of residuals. If this energy mismatch violates certain
REFERENCES
bounds, a fault is detected. The diagnosis task then
establishes the fault location and type based on the energy
fault signature. A control action finally aims to restore or Belussi, L., & Danza, L. (2012). Method for the prediction of
optimise system performance (may also include system malfunctions of buildings through real energy
maintenance or conditioning). consumption analysis: Holistic and multidisciplinary
approach of Energy Signature. Energy and Buildings,
Actual 55(0), 715–720.
Measurements
System energy
signature doi:http://dx.doi.org/10.1016/j.enbuild.2012.09.003
Feature Berton, A., & Hodouin, D. (2003). Linear and bilinear fault
Modelling extraction detection and diagnosis based on mass and energy
Equivalent
electric
balance equations. Control Engineering Practice,
circuit 11(1), 103–113. doi:http://dx.doi.org/10.1016/S0967-
Abstraction 0661(02)00116-8
Bokor, J., & Szabó, Z. (2009). Fault detection and isolation in
Energy nonlinear systems. Annual Reviews in Control, 33(2),
formalism 113–123.
doi:http://dx.doi.org/10.1016/j.arcontrol.2009.09.001
Boyd, S., Ghaoul, L. E., Feron, E., & Balakrishnan, V.
Reference + (1994). Linear Matrix Inequalities in System and
energy Σ Control Theory. Society for Industrial and Applied
signature Feature
extraction Residuals Mathematics.
Control/Optimisation
Brogliato, B., Lozano, R., Maschke, B., & Egeland, O.
(2006). Dissipative Systems Analysis and Control:
Energy
Error Theory and Applications (2nd ed.). Springer.
FDD fault
minimiser
signature Chen, W. (2011). Fault detection and isolation in nonlinear
Supervision systems: observer and energy-balance based
approaches. Dissertation, Faculty of Eng. Automatic
Fig. 10. Energy-based framework for modelling, supervision, Control and Complex Systems, Duisubug-Essen
and control of large-scale industrial systems. University.
Chen, W., Ding, S. X., Khan, A. Q., & Abid, M. (2010).
Implementing the framework in Fig. 10 presents some unique
Energy based fault detection for dissipative systems. In
challenges. The abstraction of an energy formalism that is
Control and Fault-Tolerant Systems (SysTol), 2010
usable from a global optimisation point of view is seen as the
greatest challenge. Here the use of graph theory may provide Conference on (pp. 517–521).
a suitable platform to capture the energy attributes of the Cheng, S., Wang, N., & Arnold, D. P. (2007). Modeling of
multi-domain equivalent electrical circuit in a structured magnetic vibrational energy harvesters using equivalent
manner. circuit representations. Journal of Micromechanics and
Microengineering, 17(11), 2328.
7. CONCLUSION Das, A., Maiti, J., & Banerjee, R. N. (2012). Process
From this brief survey the tendency to use energy as a basis monitoring and fault detection strategies: a review.
for system analysis and optimisation seems prevalent. International Journal of Quality & Reliability
Especially when considering a larger class of system such as Management, 29(7), 720–752.
non-linear systems, traditional signal-processing concepts Dorf, R. C., & Bishop, R. H. (2011). Modern Control
that relate to modelling and control are limited. Considering Systems (12th ed.). Pearson.
the current research, a shift towards the development of a Du Rand, C. P., & van Schoor, G. (2012a). Fault diagnosis of
more general theory to describe and analyse non-linear generation IV nuclear HTGR components – Part I: The
systems is deduced. The root towards achieving this goal error enthalpy–entropy graph approach. Annals of
tends strongly to the adoption of a behavioural paradigm with Nuclear Energy, 40(1), 14–24.
energy as the core concept. The link between complex multi- doi:http://dx.doi.org/10.1016/j.anucene.2011.09.013
domain large-scale industrial systems and equivalent electric Du Rand, C. P., & van Schoor, G. (2012b). Fault diagnosis of
circuits provide a unified representation of multi-domain generation IV nuclear HTGR components – Part II: The
systems. The framework proposed, forms a direct analogy area error enthalpy–entropy graph approach. Annals of
with conventional control theory, using energy as a basis for Nuclear Energy, 41(0), 79–86.
global system performance optimisation. The greatest doi:http://dx.doi.org/10.1016/j.anucene.2011.11.009
challenge in terms of the proposed framework is the

6701
19th IFAC World Congress
Cape Town, South Africa. August 24-29, 2014

Duindam, V., Macchelli, A., Stramigioli, S., & Bruyninckx, decomposition. Computing, 95(1), 723–749.
H. (2009). Modeling and Control of Complex Physical doi:10.1007/s00607-012-0280-2
Systems: The Port-Hamiltonian Approach. Springer. Ohta, Y., Maeda, H., & Kodama, S. (1984). Reachability,
Dynamic models for fossil fueled steam units in power Observability, and Realizability of Continuous-Time
system studies. (1991). Power Systems, IEEE Positive Systems. SIAM Journal on Control and
Transactions on, 6(2), 753–761. Optimization, 22(2), 171–180. doi:10.1137/0322013
Gertler, J. (1998). Fault Detection and Diagnosis in Ortega, R., Schaft, A. Van Der, Maschke, B., & Escobar, G.
Engineering Systems. New York, USA: Taylor & (2002). Interconnection and damping assignment
Francis. passivity-based control of port-controlled Hamiltonian
Guckenheimer, J., & Holmes, P. (1997). Nonlinear systems. Automatica, 38(4), 585–596.
Oscillations, Dynamical Systems, and Bifurcations of doi:10.1016/S0005-1098(01)00278-3
Vector Fields. Springer-Verlag. Ortega, R., van der Schaft, A., Mareels, I., & Maschke, B.
Haddad, W. M., & Nersesov, S. G. (2011). Stability and (2001). Putting energy back in control. Control
Control of Large-Scale Dynamical Systems: A Vector Systems, IEEE, (April), 18–33.
Dissipative Systems Approach. Princeton University Qin, S. J. (2012). Survey on data-driven industrial process
Press. monitoring and diagnosis. Annual Reviews in Control,
Hildebrand, F. B. (2012). Methods of Applied Mathematics. 36(2), 220–234.
Dover Publications. doi:http://dx.doi.org/10.1016/j.arcontrol.2012.09.004
Himmelblau, D. M. (1978). Fault detection and diagnosis in Seshu, S., & Reed, M. B. (1961). Linear Graphs and
chemical and petrochemical processes. Elsevier Electrical Networks. Addison-Wesley.
Scientific Pub. Co. Shinners, S. M. (1998). Modern Control System Theory and
Howe, R. M. (2005). Fundamentals of the analog computer: Design. Wiley.
circuits, technology, and simulation. Control Systems, Stork, M., Hrusak, J., & Mayer, D. (2005). continuous and
IEEE, 25(3), 29–36. digital nonlinear chaotic systems: energy-metric
Hrusak, J., Stork, M., & Mayer, D. (2009). Physical approach, simulation and implementation. In
correctness of systems, state space representations, Proceedings of the 9th International Conference on
minimality and dissipativity. In Proceedings of the 13th Circuits (pp. 56:1–56:6). Stevens Point, Wisconsin,
WSEAS international conference on Systems (pp. 202– USA: World Scientific and Engineering Academy and
209). Stevens Point, Wisconsin, USA: World Scientific Society (WSEAS).
and Engineering Academy and Society (WSEAS). Sunde, S., & Berg, Ø. (2003). Data reconciliation and fault
Hwang, I., Kim, S., Kim, Y., & Seah, C. E. (2010). A Survey detection by means of plant-wide mass and energy
of Fault Detection, Isolation, and Reconfiguration balances. Progress in Nuclear Energy, 43(1–4), 97–
Methods. Control Systems Technology, IEEE 104. doi:http://dx.doi.org/10.1016/S0149-
Transactions on, 18(3), 636–653. 1970(03)00015-5
doi:10.1109/TCST.2009.2026285 Theilliol, D., Noura, H., Sauter, D., & Hamelin, F. (2006).
Isermann, R. (1984). Process fault detection based on Sensor fault diagnosis based on energy balance
modeling and estimation methods—A survey. evaluation: Application to a metal processing. ISA
Automatica, 20(4), 387–404. Transactions, 45(4), 603–610.
doi:http://dx.doi.org/10.1016/0005-1098(84)90098-0 Tinaut, F. V, Melgar, A., Laget, H., & Domínguez, J. I.
Isermann, R., & Ballé, P. (1997). Trends in the application of (2007). Misfire and compression fault detection
model-based fault detection and diagnosis of technical through the energy model. Mechanical Systems and
processes. Control Engineering Practice, 5(5), 709– Signal Processing, 21(3), 1521–1535.
719. doi:http://dx.doi.org/10.1016/S0967- Uren, K. R., & van Schoor, G. (2013a). State space model
0661(97)00053-1 extraction of thermohydraulic systems – Part I: A linear
Janschek, K. (2011). Mechatronic Systems Design: Methods, graph approach. Energy, 61, 368–380.
Models, Concepts. Springer. doi:10.1016/j.energy.2013.06.043
Jeltsema, D., & Scherpen, J. (2009). Multidomain modeling Uren, K. R., & van Schoor, G. (2013b). State space model
of nonlinear networks and systems. Control Systems, extraction of thermohydraulic systems – Part II: A
IEEE, (August). linear graph approach applied to a Brayton cycle-based
Jordan, D., & Smith, P. (2007). Nonlinear Ordinary power conversion unit. Energy, 61, 381–396.
Differential Equations: Problems and Solutions: A doi:10.1016/j.energy.2013.08.052
Sourcebook for Scientists and Engineers. OUP Oxford. Vaclavek, V. (1974). Gross systematic errors or biases in the
Malisoff, M., & Mazenc, F. (2009). Constructions of Strict balance calculations. Prague Institute of Technology,
Lyapunov Functions. Springer. Prague, Czechoslovakia.
Mayer, D., Hrusak, J., & Stork, M. (2013). On state-space Van Niekerk, C. R., Pritchard, J. F., van Schoor, G., & van
energy based generalization of Brayton – Moser Wyk, M. A. (2006). Linear model of a closed three-
topological approach to electrical network shaft brayton cycle. SAIEE Africa Research Journal,
97(March), 50–56.

6702
19th IFAC World Congress
Cape Town, South Africa. August 24-29, 2014

Venkatasubramanian, V., Rengaswamy, R., & Kavuri, S. N.

(2003). A review of process fault detection and
diagnosis: Part II: Qualitative models and search
strategies. Computers & Chemical Engineering, 27(3),
313–326. doi:http://dx.doi.org/10.1016/S0098-
1354(02)00161-8
Venkatasubramanian, V., Rengaswamy, R., Kavuri, S. N., &
Yin, K. (2003). A review of process fault detection and
diagnosis: Part III: Process history based methods.
Computers & Chemical Engineering, 27(3), 327–346.
doi:http://dx.doi.org/10.1016/S0098-1354(02)00162-X
Venkatasubramanian, V., Rengaswamy, R., Yin, K., &
Kavuri, S. N. (2003). A review of process fault
detection and diagnosis: Part I: Quantitative model-
based methods. Computers & Chemical Engineering,
27(3), 293–311. doi:http://dx.doi.org/10.1016/S0098-
1354(02)00160-6
Willems, J. C. (1972a). Dissipative dynamical systems. Part
I: General theory. Arch. Rational Mech. Anal., 45, 321–
351.
Willems, J. C. (1972b). Dissipative dynamical systems. Part
II: Linear systems with quadratic supply rates. Arch.
Rational Mech. Anal, 45, 352–393.
Willems, J. C. (1991). Paradigms and Puzzles in the Theory
of Dynamical Systems. IEEE Transactions on
Automatic Control, 36(3), 259–294.
Willems, J. C. (2007). The Behavioral Approach to Open and
Interconnected Systems. IEEE Control Systems,
(December), 46–99.
Yang, H., Cocquempot, V., & Jiang, B. (2008). Fault
Tolerance Analysis for Switched Systems Via Global
Passivity. Circuits and Systems II: Express Briefs,
IEEE Transactions on, 55(12), 1279–1283.
doi:10.1109/TCSII.2008.2008059
Yu, F. W., & Chan, K. T. (2005). Energy signatures for
assessing the energy performance of chillers. Energy
and Buildings, 37(7), 739–746.
doi:http://dx.doi.org/10.1016/j.enbuild.2004.10.004

6703