Analysis and Control of Linear Systems - Chapter 7 docx

We can also use, in order to calculate the control, a highly simplified non-linear model, the ary between these two situations being the calculation of a set of linear models fordifferen

Process Modeling

7.1 Introduction

Obtaining a model of the industrial system to automate is the first task of a controlengineer – and not the smallest one, as the quality of his work significantly depends

on the adequacy between the model and the procedure

To begin, we note that there are several points of view on a complex physical

pro-cedure By complex, we mean a procedure with many variables and/or a procedure in

which the phenomena involved and the interactions between the variables, are cated There is no universal model; its design depends entirely on the task for which itwill be used

compli-Certain models pertain to the representation of physical components of

installa-tion and to their connecinstalla-tions (structural models) They can be described by diagrams

called PI (piping-instrumentation) which define the complete diagram of installation.Universal graphic symbols are used in order to facilitate the interpretation of this rep-resentation Nowadays, a computer representation is also adopted: these models aregenerally a database and/or an object oriented representation and they are used, forexample, for the maintenance of the procedure, for safety analysis, for the implemen-tation of block diagrams, etc

Other models are used for the design and dimensioning of the installation and formanaging the various operation modes; they often refer to different functions that the

installation must fulfill (functional models) They describe the role of each subsystem

Chapter written by Alain BARRAUD, Suzanne LESECQand Sylviane GENTIL

195

in performing the roles of the procedure, in connection with a structure and behavior

of components They are used for the design of procedure monitoring, i.e its veryhigh level of control: start-up, stop, failure management, manual reboot procedures,structure changes, etc

Let us consider the example of heating a room The goal of the system is to heat upthe room To do this, we specify several functions: generation of energy, water supply,water circulation These functions are based on several components For example,for water circulation, we use a heater, a pump and a control valve; for the function

of generating energy, we use a boiler, a pump and a fuel tank To all the functionsenumerated above, corresponding to a normal operation, we can add a “draining”function which would correspond to taking the installation out of service

The goal of other models is to describe the behavior of the installation

(behav-ioral models); this refers to describing the evolution of physical units during all the

operation phases, be it from a static or dynamic point of view

The complete description of an installation requires the representation of ous phenomena (main process) and of discrete aspects (discontinuous actions duringchanges in the operation mode, security actions, etc.) To date these two representa-tion modes have been separated; for example, under a purely continuous angle, thesynthesis of regulation loops is described by supposing that any state space is accessi-ble and the production planning is represented in a purely discrete manner However,nowadays, there are attempts to characterize the set in a hybrid model (combination

continu-of two aspects, continuous and discrete), but this path still has difficulties and is stillthe subject of research

The behavioral model may have different objectives The two main objectives are:the simulation of the installation in order to test its behavior in different situationsoffline (different control laws that the engineer seeks to compare, research into itslimits, training of control operators, etc.) and the design of controls to implement It

is not necessarily the same model that is used in these two cases: the first one oftenrequires more precision than the second one In fact, for the majority of time, the con-trol is calculated on a linear approximation of the system around the nominal workingpoint because the majority of industrial systems work (in normal operating mode) in alimited range, corresponding to an optimal zone for the production We can also use,

in order to calculate the control, a highly simplified non-linear model, the ary between these two situations being the calculation of a set of linear models fordifferent working points or operating modes However, in order to design the automa-tion of an installation in order to optimize the working points or train the operators,the model must be the most robust possible

intermedi-The complex model is based on a precise knowledge of physical, chemical, ical or other laws, describing the material phenomena governing the processes imple-

biolog-mented in the procedure We often speak, in this case, of a knowledge model or a

model based on the first principles It is thus quite naturally described in the form of

non-linear differential equations in the dynamic case and/or in the form of algebraicequations in the static case These equations describe the main laws of the physicalworld, which are in general material or energy balances When we can reduce thedifferential equations to first degree equations (by possibly introducing intermediaryvariables), we obtain an algebraic differential state model

The complex model can be simplified under the hypothesis of linearity, in order

to obtain linear differential equations from which we can move either to a state sentation or to an input-output representation by transfer function Then, if we want,

repre-we can also use the traditional methods in order to discretize these linear models, inorder to directly calculate a discrete control The first section provides a few exam-ples, which are trivial in comparison with the exhaustive task of the engineer for anindustrial procedure, but which illustrate the methodology

When the objective is the development of a control on a linear model, it may besimpler to directly research this model This research can be done from specific exper-

imentations In that case we speak of identification, rather than modeling We obtain

a model of representation We know well the link between the transfer function and

the frequency response and it is thus easy to translate the latter into a cal model However, it is basically impossible to perform a harmonic analysis on anindustrial procedure – because it is incompatible with the production constraints – orwith the response time of the procedure Hence, faster means have been investigated inorder to obtain these models from time characteristic responses; the most widely used

mathemati-is of course the unit-step response because it corresponds to a change of the workingpoint of the installation, in other words to a current industrial practice Therefore, afew fast graphic constructions make it possible to obtain, for a minimal cost, a transferfunction close to the system The second section deals with this aspect

It was soon clear that, in order to make the model robust for the entire range ofoperation where linearization is valid, we should use input signals with a much largerspectrum than the step function, in order to excite all the modes of the system Assuch we use the identification on any input-output data (but that are full of informationregarding the behavior of the system); in this case, only the strong numerical methodsmake it possible to extract the information contained in these data sets These methodsare explained in the third section The method that will be the most developed can infact be used on non-linear representations and that is why we also use it in order toparameterize the knowledge methods mentioned above

7.2 Modeling

The behavioral modeling of a continuous procedure described in what follows isbased on a mathematical formalism: we search for a set of equations representingthe system in the largest possible operating range This is a task that may take severalmonths and pertains to the multi-disciplinary teams In fact, it requires a knowledge ofphysics, chemistry, biology, etc in order to be able to understand the phenomena thatthe model will describe, and knowing numerical analysis in order to write the modelequations in a form that is adapted to the numerical calculus It is also necessary tohave computer knowledge in order to be able to implement this calculations

A procedure is sufficiently complex in order to be able to describe straightawayits behavior by a system of equations In order to realize a global model, we need

to decompose the general system into simpler subsystems, through a descendingapproach, then recombine the various models into an ascending approach Thisdecomposition can be found in the methodology of software development: that iswhy we can use the same tools in order to manage these approaches (SADT, forexample) At the level of a basic subsystem, there is no optimal methodology: is itnecessary to start by writing the most complicated model possible – by calling uponthe description of detailed mechanisms – and later simplify it, either because wehave no knowledge regarding the coefficients present at this elementary level and nopossibility of estimating them in practice, or because this model is too complicated to

be used? Or is it necessary to start by writing a very rough model and not complicate

it unless the simulation results obtained are too inaccurate? It is obvious that themodel must be the result of a compromise between precision and simplicity When

it is established, we have to verify it: this means that we test it to make sure there is

no physical inconsistency between its behavior and the behavior of the system, due,for example, to numerical problems or to wrong initial hypotheses Then we have

to validate it; this means testing its adequacy with the set of tasks for which it was

The bond-graphs are a graphic representation tool for energy transfers in a physical

system, sometimes used as intermediaries between the physical description of a cedure and the writing of equations Through a formalism reuniting fields as various

pro-as mechanics, they describe electricity and hydraulics – simply because they are bpro-ased

on the description of power exchange between subsystems The graph consists of arcs

connecting the stress variables e or the stream variables f whose product represents

the power Forces, torques, tension and pressure are stress variables Speed, flow and

current are stream variables There are several main elements The resistances pate energy (electric resistances, viscous friction) The capacities store energy (electriccondenser, spring), as well as inertial elements (inductance, masses, moments of iner-

dissi-tia) The transforming elements preserve power e1f1 = e2f2while imposing a fixed

ratio between streams and input and output stresses (e1 = ne2, f1 = f2/n) Finally,the junctions are of two types (called 0 and 1) depending on whether they connectelements that preserve the stress and distribute the stream or the other way round Wewill not go into further detail on this method, which is dealt with in specific works(see [DAU 00], for example)

By admitting that we have conveniently traced the balance equations to write, theyare general in the form of non-linear differential equations They can be used as such

in the simulation fine model, but they will not be generally linearized in order to obtainthe control calculation model The linearization is operated as follows Let us assumethat the differential equation is:

We suppose that system [7.8] has a balance point Y0, E0for which the derivativesare zero, i.e which is defined by:

Now, we try to represent the trajectory of small variations x(t) and u(t) by carrying

[7.9] over [7.8] and by using a first order Taylor serial development, which leads to:

where the state matrix is the Jacobian of the non-linear relation vector f(y, e, t).

Therefore, we obtain a linear state representation of the non-linear system

outgoing flows of the second tank link this subsystem to its environment There are

no losses or intermediary transport element, hence we will write two matter balanceequations, one for each of the storage elements

Let Q e be the volume flow rate entering the first tank, Q s1the volume flow rate

leaving the tank, A1its section, N1the water level in the tank and K the restriction

coefficient of the output tank The outgoing flow is proportional to the square root of

the pressure difference ∆p at the edges of the tank, which is itself linked to the level

(law of turbulent flows) Hence, we have – if the atmospheric pressure is the referencepressure:

The same law describes the second tank, where, in order to simplify the notations,

we suppose that the tanks have the same coefficient K:

In general, we can assume that the levels are subjected to small variations with

respect to the balance given by the working points Q e0 , N10and N20 The balance isdefined by:

It should be noted, however, that this equation would be sufficient if we intended

to size up the system, i.e to choose the tanks (K1, K2coefficients) according to theaverage levels and flows wanted We write:

n1

N10



[7.21]

and similarly:

Q s2 = K2 N20



1 +12

n2

N20



[7.22]Equation [7.15] thus becomes:

Q e0 + Q0− K1 N10



1 + 12

R and L be the resistance and the inductance of the armature, u(t) the supply voltage,

i(t) the armature current, e(t) the back electromotive force, γ(t) the engine torque, J

and f the inertia and frictions of the tree rotating at a speed ω(t) The electric equation

of the armature is:

u(t) = Ri(t) + L di(t)



X +

0



[7.41]

which gives the state representation:



X +

01

Obviously, these two representations have the same transfer function

In order to have the control the question that arises is: can the value of all thesephysical parameters intervening in these knowledge models be obtained? We can usethe manufacturers’ documentation for small systems as the ones developed below Formore complex systems (like chemical or biotechnological systems) this task may bevery complicated That is why we determine, often directly from experimental record-ings, the parameters of transfer functions (the two time constants of the tanks, forexample) The following two sections describe this approach

7.3 Graphic identification approached

When the objective of modeling is the research of a (simple) linear model in view

of the control, we can use direct methods based on the use of experimental ings Two methods are available: the use of the harmonic response of the system orthe analysis of time responses with specific excitations The goal researched is, for aminimal cost, to obtain an input-output representation of the procedure in the form of

record-a continurecord-al trrecord-ansfer function F (p) Let us recrecord-all threcord-at F (p) models only the dynrecord-amic

part of the procedure The time expressions will entail the initial conditions

The first approach (harmonic response of the system) is rarely conceivable becauseits implementation is often incompatible with manufacturing requirements or, more so,because the response time of the procedure makes recording it particularly long andtedious

The second approach is based on the recording of the system’s response to thegiven excitations In particular, we use the recording of the unit-step response, whichcorresponds, from a practical point of view, to a change in the operating point Hence,from unique data, we identify the system by determining the coefficients of a standard-ized transfer function with a predefined structure It is important to point out that thesegraphic methods do not make it possible to estimate the precision of the parametersobtained In addition, the quality of the model depends on the operating mode (noiselevel, instrumentation, etc.) and on the operator (in particular during the use of graphs)

It is understood that the data must be collected in the absence of saturation and that it

is essential to verify the non-saturation at the level of internal regulation loops (whenthey exist) Finally, the graphic techniques based on the use of the unit-step responseand presented below suppose that the system to identify is asymptotically stable

The first criterion to consider for the choice of the method is whether or not the

final value, noted by s( ∞), which corresponds to a pseudo-periodical or a

period-ical response is exceeded In Table 7.1, there is a classification of various methodspresented in the remaining part of this section, as well as the standardized transferfunction and the parameters to identify

7.3.1 Pseudo-periodic unit-step response

From a balance position e(t −0), s(t−0), we apply at instant t0a step function ∆E.

Then we obtain a unit-step response in the form of the one in Figure 7.2 This response

presents a first exceedance A1, a final value s( ∞) Therefore, we use as a model the

of the model selected has the form:

with tan θ = 1− ζ2/ζ , for t
t0 The (relative) amplitude of the first exceedance,

noted by A1 %, depends only on damping ζ By using the graph given in Figure 7.3,

we obtain the numeric value of ζ The angular frequency ω nis linked to the period ofoscillations Its numeric value is obtained by using Figure 7.4

Figure 7.2 Pseudo-periodic unit-step response

Graphs to use

The two graphs to use are parameterized by damping ζ Their use is described

below, based on the unit-step response in Figure 7.2

Figure 7.3 First exceedance (percentage)

according to ζ

Figure 7.4 Response time at5%

according to damping ζ

Determining the transfer function F (p): method

The method is divided up as follows:

1) determining the static gain of procedure K = s(∞)−s(0 ∆E −);

2) determining the first relative exceedance (percentage) A1 %= 100× A1

s(∞)−s(0 −);

3) with the help of Figure 7.3, determining the numeric value of ζ;

4) obtaining the response time at 5%, noted by t r5 %;

5) by using Figure 7.4, determining the numeric value of ω n ·t r5 %

2π , then obtaining

the numeric value of the angular frequency ω n

Transfer function to identify Method

Necessary tools Coefficients

to identify Pseudo-periodic unit-step response

Table 7.1 Graphic methods

7.3.2 Aperiodic unit-step response

The method used for the identification of transfer function depends on the generalfeatures of the obtained step response If this response does not have a horizontal

tangent at instant t = t0, the tester will obviously choose a 1storder model Otherwise,the choice will have to be between three models: a delayed 1st order model (Brọdamethod), a 2ndorder model (Cadwell’s method), and a model (strictly) superior to 1

delayed or not (Strejc method) Here again, the know-how and a priori knowledge are

very important with respect to the “quality” of the model selected In order to simplify

the expressions, we will choose from now on t0 = 0 If this is not the case, a simplevariable change makes it possible to have this situation

7.3.2.1 First order model

When the unit-step response appears similar to that in Figure 7.5, the model usedis:

F (p) = S(p)

E(p) =

K

1 + τ p The unit-step response of the model selected has the form:

con-Figure 7.5 Unit-step response for a1storder system

Determining the transfer function F (p): method

The method is divided up as follows:

1) determining the static gain of procedure K = s(∞)−s(0 ∆E −);

2) obtaining the time constant τ Two approaches are conceivable depending on whether or not we know the output final value s( ∞) In both cases, we use the prop-

erties which resulted from the mathematical expression [7.44]:

a) determining τ using s( ∞) (see Figure 7.5): the tangent at the unit-step

response t = 0 intersects the horizontal line s( ∞) for t = τ, the response time at

5%verifies t r5 % = 3τand the rise time1of the system verifies t m = 2.2τ,

b) determining τ without using s( ∞) We choose two instants t1and t2= 2t1

We have the numeric values of s(t −0), s(t1 and s(t2 By using [7.44], we obtain:

ln(x) Expression [7.45] then enables us to calculate K.

Other graphic methods can be used, in particular that of the semi-logarithmic plane[LAR 77] However, since its use is not immediate, we will not present it here

7.3.2.2 Second order model

We will present a unit-step response given in Figure 7.6

1 Rise time is defined between instants t1 and t2 such ass(t1) = s(t −

0) + 0.1 K ∆E,

s(t2) = s(t − ) + 0.9 K ∆E.

Figure 7.6 Aperiodic unit-step response:

choice of second order model

When the selected model is:

τ1− τ2e

−( t τ2)

for t
0

Determining the transfer function F (p): Cadwell method

The Cadwell method is based on the fact that the unit-step response accepts a particular point P1(t1, s1 which does not depend on the ratio x = τ2/τ1and another

point P2 whose coordinates (t2, s2 strongly depend on this ratio The approach toadopt is the following:

1) determining the static gain of procedure K = s(∞)−s(0 ∆E −);

2) obtaining the instant t1= 1.32(τ1+ τ2 such that:

s1= s(0 − ) + 0.74

s(∞) − s(0 −)

3) inferring the numeric value of τ sum = τ1+ τ2;

4) obtaining on the curve the point (t2, s2 such that t2= 0.5τ sum Inferring the

Similar method

This method lies on an approach similar to the previous one; it was developed byStrejc The approach to adopt is summed up below:

1) determining the static gain of procedure K = s(∞)−s(0 ∆E −);

2) obtaining the instant t1= 1.2564 (τ1+ τ2 such that:

s1= s(0 − ) + 0.72

s(∞) − s(0 −)

3) obtaining the value of τ sum = τ1+ τ2;

4) obtaining on the curve the point (t2, s2 such that t2 = 0.3574 τ sum Inferring

the value of n%= s(t s(∞)−s(02)−s(0 − −));

5) with the help of the graph in Figure 7.8, determining the value of x = τ2/τ1

Calculating the numeric values of τ1and τ2via the formulae of [7.46]

Figure 7.7 Cadwell method, n%= f(x)

Figure 7.8 Strejc method, n%= f(x)