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Welcome to the lecture titled Off-line Identification of Process Dynamics. Earlier in the our in
our earlier lectures we have seen relay control systems, inducing limit cycle outputs. And
when the relay is substituted by an equivalent gain, then the analysis of closed loop relay
control systems, become easy. In today’s lecture, we shall see how off-line identification
can be done using equivalent gain of relays.
An off-line identification structure can be given like this, where r 0 equal to 0, when
the relay is in action, whereas r 0 is not equal to 0, when the controller is in action.
That means when the controller is in action, the process is in operation, then the relay
test cannot be initiated, then when retuning of the controller is required, then we need
to set the setting value reference input value to 0 r to 0. And disconnect the controller
and put a relay in the feed forward path that is how off-line identification is done using
relay control systems. Now, for identifying specific transfer function
models for process dynamics, let us assume a first-order plus dead time process model
of the form G (s) is equal to K e to power minus D s upon T s plus minus 1. So, when
plus sign is used in the denominator at that time we shall get a stable process model whereas,
for the negative sign, we get unstable process model.
Now, in the process model, we have got three unknown parameters, the unknown parameters
are K, D and T; K is the steady state gain steady state gain of the process, and K is
also equal to G (0), that means when we take the limiting of s to 0 G (s) gives us the
steady state gain of a process. Then D is the time delay of the process model whereas,
T is the time constant of the process transfer function model. Let the controller be a PI
PID controller given by G c (s) is equal to K c times 1 plus 1 upon T I s plus T d s upon
beta T d s plus 1, where K c T i T d are the parameters of the PID controller and beta
is the filter constant, derivative filter constant.
So, we we will not concentrate on the dynamics of PID controller, because we are going for
off-line identification, where controller will be out of the loop, so we get no role
to play. Now to identify the dynamics of an unknown process, let the process be assumed
as the unknown, we do not know the dynamics of this process, rather the dynamics of this
unknown process will be modeled by a first-order plus dead time transfer function model.
Now, for that during off-line relay test G c (s) is replaced with a relay, then we get
limit cycle output, from the closed loop. So, the relay induces limit cycle output as
shown here, where the output signal y (t), output signal has got some parameters; since
we are getting sustained oscillatory output. Therefore, we can acquire or measure some
information from the limit cycle output. So, let A be the peak amplitude of the limit
cycle output signal, so the A is the peak amplitude of the limit cycle output, then
similarly, P u be the ultimate period of the limit cycle output. Ultimate period means
nothing but, the time period of the output signal, then the upper waveform gives us a
typical limit cycle output waveform for the relay control systems.
Whereas, the bottom one, the waveform shown below the upper one is nothing but, a typical
relay output waveform this is the relay output waveform of the same period of the output
limit cycle output; h is the peak amplitude of the relay output waveform. Where h is also
nothing but, the amplitude of relay amplitude of relay often known as relay height and is
the nothing but, a parameter for relay setting, h is we get from the relay setting.
So, when the autonomous relay control system is consider, we get some typical limit cycle
output of the unknown process of this form and some typical relay output of this form.
So, making measurement of some features of the limit cycle output, the features known
as A P u of the limit cycle output it is possible to establish some relationship with the unknown
transfer function model parameters, with the measurements. And then using the measurements
it is possible to estimate the unknown parameters of a transfer function model, first-order
plus dead time transfer function model.
So, the relay is substituted by it is describing function for the analysis, the relay is defined
by a function N (A) which is equal to 4 h by pi A, where h is as we know the relay amplitude;
and A is the peak amplitude of the relay input, I can say relay input signal. So, from an ideal, for an ideal relay the
describing function of the relay can be given as N A is equal to 4h by pi A. Next the critical
frequency is defined as omega c r is equal to 2 pi upon P u, where P u is the ultimate
period of the limit cycle output ultimate period of the limit cycle output.
Then since the characteristic equation of the relay control system, relay closed loop
control system is 1 plus N (A) G y omega c r is equal to 0, this is the characteristics
equation of the relay control system. Then using that we can write N G j omega c r is
equal to minus 1, the same expression can be obtained from the fact that, limit cycle
output is obtained, when the loop gain is equal to 1 and loop phase is equal to minus
pi. So, the same can be transformed into some analytical expression form which is given
as N G j omega c r is equal to minus 1. Now, substitution of N (A) which is nothing
but, 4 h upon pi A and G j omega c r gives us the equation 4 h upon pi A K times e to
the power minus j omega c r D upon j omega c r T plus minus 1 is equal to minus 1, because
N A is equal to 4 h by pi A as we have seen earlier at G (s) is equal to K e to the power
minus D s upon T s plus minus 1. So, in frequency domain that can be written as G omega c r
is equal to K e to the power minus j omega c r D upon j omega c r T plus minus 1. So,
substitution of this N A and G omega c r over here give us this equation.
Equating the magnitude of both sides of the equation, we get 4 h upon pi A times K upon
omega c r T square plus 1 root is equal to 1. So, for both the stable and unstable transfer
function model, the magnitude of the analytical expression will be same. So, this can further
be simplified as written in the form of omega c r T square plus 1 root is equal to 4 h K
upon pi A implies omega c r T square plus 1 is equal to 4 h K upon pi A square.
Then next, that can be also written as omega c r T omega c r T square is equal to 4 h K
upon pi A square minus 1, then taking the square root of that will give us omega c r
T is equal to 4 h K upon pi A square minus 1 root. So finally, we get the expression
T is equal to 4 h K upon pi A square minus 1 root upon omega c r. So, this is how we
get the expression for the time constant of the transfer function model, the first-order
transfer function model expressed as T is equal to 4 h K upon pi A square minus 1 root
upon omega c r, this expression can be used for both stable and unstable process dynamics.
Next, equating the phase angles of both side, again we get we know that the phase angle
of both sides means 4 K h upon pi A e to the power minus j omega c r D upon j omega c r
T plus 1. Let us consider the stable process model first, so this angle should be equal
to minus pi, because the loop phase is equal to minus pi, which can be written in the form
of minus omega c r D minus tan inverse omega c r T is equal to minus pi. So, making all
these minus to plus, now will enable us to write omega c r D is equal to pi minus tan
inverse omega c r T, which will ultimately give us d is equal to d is equal to pi minus
tan inverse omega c r T upon omega c r. So, this is how we get the expression for
the the second unknown parameter in the transfer function model of a stable process dynamics,
the D time delay as D is equal to pi minus tan inverse omega c r T upon omega c r. So,
extending the similar analysis for the unstable process, now we can get the expression for
D as D is equal to equal to pi plus tan inverse omega c r T upon omega c r.
So, this is the way, we can estimate the unknown model parameters T and D T and D of the first-order
plus dead time transfer function model, using the measurements of P u and A. Please keep
in mind D and T, if we see T is now, a function of A and omega c r and omega c r is a function
of P u. Therefore, making use of the measurements of omega c measurements of P u ultimate period
and peak amplitude A, it is possible to estimate T and it is possible to estimate the this
for stable and unstable processes. So, from the analysis, we have found that
with the measurements of peak amplitude and frequency of the limit cycle output signal,
it is possible to estimate two parameters of the unknown transfer function model, first-order
plus dead time model of a process dynamics.
Let us, go for some simulation study and see, how accurate is our method; so in the simulation
study one, let us consider the process to be process dynamics actual process dynamics
to be G (s) is equal to e to the power minus 2 s upon 10 s plus 1. So, the actual process
is assumed to have a time delay of 2 seconds, a time constant of 10 seconds and a steady
state gain of 1. If this is the actual process has assuming that the actual process dynamics
is not known, we shall put a relay in the feed forward path along with the process and
make one relay control system. So, using the relay control system and with the setting
of relay amplitude of h equal to 1 relay amplitude h equal to 1.
What we get we get a typical limit cycle output of this form, where this is x axis, this is
the time axis, time in seconds and y axis is the output limit cycle output. Now we can
make measurements the period of the limit cycle output, often measurement gives us P
u to be of 7.332 seconds; similarly, the peak amplitude is found to be of magnitude 0.181.
So, this is of value 0.181 and P u is of value 7.332 seconds.
Then the critical frequency is calculated as omega c r is equal to 2 pi upon P u is
equal to 2 pi upon 7.332 is equal to 0.857 second sorry radian per second. Now, the steady
state gain is assume to be 1 as we know that using the describing function technique, we
can estimate at most two unknown parameters of a process model.
Therefore, we have to make use of some assumption, because the first-order plus dead time process
transfer function model has got three unknowns; the steady state gain, the time constant and
the time delay. So, the steady state gain is assume to be known a priory or it can be
obtained from some other tests, so assuming the steady state gain to be one K is equal
to 1. The T can be the time constant can be estimated
as T is equal to 4 h K upon pi A square minus 1 root upon omega c r is equal to 4 into 1
into 1 upon pi into 0.181 minus square minus 1 root upon 0.857 gives us T is equal to 8.1249
seconds. Similarly, D is calculated using the measurements as D is equal to pi minus
tan inverse omega c r T upon omega c r is equal to pi minus tan inverse 0.857 into 8.1249
upon 0.857 is equal to 1.9993 seconds. Thus the process model parameters time constant
is found to be of value 8.1249 seconds instead of the actual value of 10 seconds and the
time delay is of value 1.9993 seconds instead of or in place of 2 seconds. Those are the
actual plant parameters or the actual process has got the dynamics with time constant of
10 seconds and time delay of 2 seconds. So, the model parameter time constant is underestimated
by 18.75 percentage and the time delay is underestimated by 0.035 percentage.
So, we can tell that the time delay has been estimated accurately, whereas, the time constant
has not been estimated accurately. Since the relay has been assumed by it is equivalent
gain therefore, there is estimation error in the identification scheme. If the relay
is approximated by it is exact gain or if the relay is substituted by it is exact gain
or you making use of the exact analysis of relay control system. It is possible to estimate
the model transfer function model parameters accurately whereas, we have to scarifies some
accuracy in the case of relay control system using some describing function technique.
Let, us go to one more simulation study where the unknown plant dynamics is assume to be
G (s) is equal to 1 upon s plus 1 to the power 5. So, in this case what we are going to do
the actual plant is a plant all-pole plant with 5 repeated poles located at s equal to
minus 1. So, the actual plant is something else. Whereas, we are going to estimate the
plant dynamics by some first-order transfer function model.
What is that model, so we are going to estimate this dynamics by some equivalent model G m
s of the form K e to the power minus D s upon T s plus 1. So, it is possible to capture
the dynamics of actual plants by some transfer function models, not necessarily one has to
assume the form of the transfer function as same as the plant dynamics.
In that case a symmetrical relay with amplitude h is equal to 1 produces a sustained symmetrical
process output. And making measurement of the limit cycle output signal, we obtain the
peak amplitude peak amplitude to be of A is equal to 0.474. And the peak and the ultimate
period is of value 8.732 seconds, so the critical frequency is calculated as omega c r is equal
to 0.7196 second.
Again assuming the steady state gain to be 1 assuming the steady state gain to be one
T is calculated as T is equal to 4 h K upon pi A square minus 1 root upon omega c r is
equal to 3.4645 seconds. So, the time constant of the transfer function model is computed
or estimated as T is equal to 3.4645 seconds. Now, the time delay can be estimated using
the formula D is equal to pi minus tan inverse omega c r T upon omega c r, which is equal
to 2.713 seconds. So, the time delay of the first-order plus
dead time transfer function model is found to be of the value 2.713 seconds, then the
unknown plant dynamics is now represented by a transfer function model given as G m
s is equal to e to the power minus 2.713 s upon 3.4645 s plus 1. Whereas, the actual
plant dynamics is G (s) is equal to 1 upon s plus 1 to the power 5; so actual plant with
the dynamics given as this has been represented by a first-order plus dead time model, with
time delay of 2.713 seconds and time constant of 3.4645 seconds. Now, how to validate the
accuracy of identification, for that what we have done?
So, to ascertain the modeling accuracy to investigate the modeling accuracy, using the
frequency response plots of both the actual and identified plant, we have obtained the
Nyquist plot of the nominal and identified plants. So, the Nyquist plots of the original
plant and the identified plants are plotted and shown over here; so there is no significant
differences between the two plots then, one can conclude that the identified model truly
represents the dynamics of the actual process. So, basically we concern we have concerned
about the part of the Nyquist plots in this range of frequencies and as we see there is
no much difference between the two plots. Therefore, we can tell that we can assume
that the identified transfer function model, represents the dynamics of the actual model
actual plant successfully.
Now, for the transfer function model having some generality and given as G (s) is equal
to K minus T 0 s plus 1 times e to the power minus theta s upon T 1 s plus 1 to the power
P; also we can make use of the describing function analysis and estimate the unknown
parameters of the transfer function model. Now, here the the transfer function model
involves a lot of parameters, the steady state gain K, time delay theta, here theta is the
time delay of the transfer function model, T 0 is the time constant of the transfer function
model. And T 1 is also one time constant of the transfer function model and P is the pole
multiplicity a transfer function model as then for various values of P we get different
type of transfer function models. And similarly, for a various values of theta and T 0 it is
possible to get transfer function model with time delay and transfer function model with
non-minimum phase characteristics. Now, when lambda is assumed as lambda is equal
to minus 1 upon T 1, when P is equal to 1 for the case T 0 is greater than 0 and theta
equal to 0 theta equal to 0 means, we basically get the non-minimum transfer function non-minimum
phase transfer function models. So, for those models one can establish equation or relations
using describing function for the relay control system and the set of equations can be obtained
as the peak amplitude can be given as a is equal to 4 K h upon pi root of 1 plus omega
c r T 0 square upon 1 plus omega c r upon lambda square.
Similarly, this we get from the loop gain condition and similarly, the loop phase condition
enables us to obtain an expression of the form pi minus tan inverse omega c r T 0 plus
tan inverse omega c r upon lambda is equal to 0. So, basically we are able to obtain
two relations, two equations using the describing function analysis.
Similarly, when T 0 is equal to 0 and theta is non 0, that means when the transfer function
model is a transfer function model with time delay for that case a is given as 4 K h upon
pi cos pi minus omega c r theta. And the phase condition of the relay control system will
give pi minus omega c r theta plus tan inverse omega c r upon pi is equal to 0. So, in any
case what we have found, we are able to get two equations using the describing function
analysis.
So, for higher values of P also we one can easily obtain similar expressions, when P
is equal to 2, when the transfer function model is of second-order. In that case for
the case of T 0 is greater than 0 and theta is equal to 0 the set of equations are obtained
as A is this much and the phase condition gives us this equation. Now for T is equal
to 0 and theta is not 0 for transfer function models, which are known as time delay models.
So, time delay models, then the two equations are A is equal to 4 K h upon pi cos square
pi minus omega c r theta upon 2. And the phase condition will give us pi minus omega c r
theta plus 2 tan inverse omega c r upon lambda is equal to 0. So, these equations can easily
be obtained using describing function analysis and it is not difficult to obtain similar
expressions for higher values of P. So, for P from 3 to onwards 3, 4, 5 onwards also,
we will get very simple analytical expression of this these forms.
Now, drawbacks of an off-line relay test, what are the drawbacks an off-line relay test
has, using the describing function the off-line relay test is found to give us only two analytical
expressions; therefore, two unknowns of the transfer function model can be estimated.
So, basically we make use of the measurements and the analysis to estimate the time constant
and time delay of a transfer function model. In that case it is required to estimate the
steady state gain by a separate relay test or using some other analysis, because we cannot
estimate three unknowns using the two equations, we developed for the relay control system.
So, one has to assume either assume the steady state gain to be known a priory or make use
of other tests to estimate or obtain the steady state gain value. So, that is one of the major
drawbacks associated with an off-line relay test.
Now, second problem with the off-line relay test is that, it is sensitive to the presence
of static load disturbance; how that is show we shall see by going back to some earlier
slide, where we have the structure for relay control system.
Now, this structure shows that when the relay is connected with the process unknown process,
when some static load disturbance is presents present. In that case what will happen u the
process input will be a symmetrical, as we know when L is equal to 0 what will be the
form of the u; u will assume some symmetrical output waveform of the form some square or
rectangular pulses. So, this is what we have T and this is our
u T, this is what we get when L is equal to 0 when the static load disturbance is equal
to 0. Suppose L is of some magnitude and the relay amplitude is 1 therefore, magnitude
of this signal is equal to 1, now when L is equal to 0.5, then this 0 point this 0.5 will
get superimposed with this signal and ultimately it will get elevated. Therefore, we will have
some asymmetrical relay output signal of the form shown over here.
So, this will continue this is the time and this the u T, so for the for this case for
this situation what will happen when the input signal to the process is asymmetrical the
output the limit cycle output also will be asymmetrical. Definitely it will be asymmetrical
but, of the same period as well; so we shall get some asymmetrical limit cycle output asymmetrical
limit cycle output from the system. So, although the relay is symmetrical the
relay is symmetrical whereas, the limit cycle output signal is asymmetrical. Now how to
make use of those techniques the analysis we have made earlier all those will fail because
we cannot get correct measurement of the peak amplitude. Now we have got two peaks one positive
peak and one negative peak of different magnitudes; similarly, the period we can measure but,
when we reset to half period there will be a lot of difficulty. So, again measurement
of period will be there but, we will not get the correct period, what is required for the
symmetrical relay. So, when the asymmetrical output limit cycle
output is obtained, we do not know where from the asymmetricity is coming. So, is it due
to asymmetrical relay or is it due to disturbances that will actually create problem we may not
get correct information about the process dynamics. So, that is one major difficulties
associated with off-line relay test, so the off-line relay test is subjected to limitations
and inaccuracies, when the static load disturbances are present.
And in practical real time systems it is very difficult to avoid static load disturbances
or some form of the disturbances will always be there therefore, the off-line relay test
may give us in inaccurate estimation of transfer function model parameters.
Now, third point is often it is dangerous and also expensive to break the control loop
for the purpose of tuning only, for the purpose of PID controller tuning. When the loop is
broken then the control operation is disrupted; the normal operation of the system gets disrupted,
which is not desirable for many real time systems. What is desirable rather that often,
it is desirable to tune controller under tight continuous closed loop operation.
So desirable is to tune controller under tight closed loop operation and this is what we
are not going to get from off-line tuning schemes.
Next, we shall consider some on-line identification structure to overcome the limitations associated
with off-line identification schemes; we shall consider an on-line identification structure.
Where this r T is not necessarily has to be 0, so I can write it can be non 0 also this
is your r T or reference input r simply. So, this gives us the structure of an on-line
identification scheme. What we see what is the basic difference between
the earlier structure and this structure a relay has been connected in parallel with
the controller. A relay has been connected in parallel with the controller and the controller
remains in actions throughout the operation of the real time system real time process.
Whereas, whenever there is requirement for retuning the parameters of the controller,
what we can do the process information can be acquired by setting some relay amplitudes.
So, when the relay amplitudes h is equal to 0 when the relay amplitudes h are 0 at that
time, the controller is in action and the output of the plant can be of the form like
this, we do not get limit cycle output when relay is not in action.
Whereas, when the relay is in action now when the relay amplitudes are not equal to 0 at
that time, we will get the output of the form limit cycle output, superimposed over the
steady state output of the process this the beauty. So, during this time we make measurements
of the output limit cycle output signal and estimate the plant dynamics model parameters.
So, plant model parameters and based on the model parameters the parameters of a controller
are set, the parameters of a controller are set and this is how on-line identification
and tuning is done in real time.
Now, we can represent the on-line identification structure by some equivalent diagram; if you
look carefully, then in that case again r is not equal to 0 then the G (s) is now, subjected
to some inner feedback with a controller G c s. The earlier feed forward path controller
G c (s) now appears in the feedback path; so G (s) is now having some inner loop controller
G c s. And relay basically sees or experiences the process connected with an inner controller,
so the relay is experienced or experiences or sees a I can say controlled process a process
with a controller in the loop. Now, in contrast to the off-line control scheme
or the conventional relay control scheme, the relay is subjected to a closed loop system.
Earlier the relay was subjected to the process dynamics only whereas; in the on-line identification
scheme the relay is subjected to a process with a controller in the loop. Therefore,
it is possible to estimate more than two parameters from this scheme also with proper use of the
or with proper measurements of the limit cycle output signal, it is possible to make more
measurements from the limit cycle output or the output of the relay control system. And
it is possible to estimate more than two unknown parameters of a transfer function model.
So, basically in on-line identification scheme has got advantageous over the off-line identification
scheme, in the sense that it the planned operation is not disrupted. So, we will have normal
operation of the plant and the relay heights are to be or the relay amplitudes are to be
chosen suitably such that the plant operation is not affected at all. So, one can put some
conditions depending on the magnitude of the error signal e e is the error signal.
So, whenever the error signal is greater than some threshold value epsilon that time only
relay will come into picture will be in action and induce limit cycle output. And when the
error signal is less than some threshold value there is no need for retuning the PID controller
parameters. So, this is the beauty of the on-line identification
scheme, now using the on-line identification scheme a suitable controllers also can be
designed. The form of the controllers also can be decided often; so those issues will
be discussed in the next lecture.
Next, we shall go to the summary now, what we have seen in this lecture off-line process
identification is presented. So, an off-line identification scheme is presented and it
is found that it is possible to estimate two unknown parameters of a transfer function
model using the describing function gain or using the describing function analysis of
a relay. Now first-order plus dead time transfer function
models can be estimated or obtained using the measurements made on the limit cycle output,
not only first-order plus dead time models. But, also we can estimate we can obtain transfer
function models of various forms and those are nothing but, non-minimum phase non-minimum
phase transfer function models and higher order time delayed transfer function models.
In that case what we will have second-order plus dead time transfer function model, third-order
plus dead time transfer function model and so on.
So, it is possible to estimate two unknowns parameters of non-minimum phase transfer function
models or say first-order plus dead time transfer function models or second-order plus dead
time transfer function models or third-order plus dead time transfer function models using
describing function. Next now, it is not possible to estimate accurate
values for the steady study gains using off-line relay identification scheme. So, estimation
of steady state gain has to be done by some separate test or by some other techniques;
so this is one crucial issue and there is limitation with the off-line relay test mainly
in estimating the steady state gains. Now, coming to the next point as we have discussed
often it is dangerous and expensive to break the closed loop operation for the sake of
tuning of controllers and to overcome that we have introduced on-line identification
schemes. So, on-line identification schemes can be of different types one such on-line
identification scheme has been introduced. We can have other type of on-line identification
scheme as well where the scheme can be given in the form of suppose a relay outside the
closed loop, then we will have the controller the process, inner feedback and then, we can
have the feedback negative feedback to relay. So, this can make a probable scheme for on-line
identification also here, in this case what is happening, when the relay amplitudes are
0, we have got the normal operation of the plant or process with a controller. And for
nonzero heights of the relay or when the relay amplitudes are not 0, at that time the output
will have some limit cycle waveform. And this also can make a probable candidate as I have
told for the on-line identification scheme.
Some points to ponder, how difficult is it to identify a varieties of transfer function
models, describing function based methods enables one to estimate two unknown parameters
of a transfer function model. As we have seen the model can be of a general form or of any
form, but only two unknowns of the transfer function model can be estimated using the
describing function analysis. So, some other tests or multiple relay tests
might be used, might be made to identify a varieties of transfer function models. The
second point how is the identification scheme sensitive to external disturbances? As we
have seen, when we have got a static load disturbance then the limit cycle output become
asymmetrical, in spite of using asymmetrical relay in the relay control system. And asymmetrical
relay output may not give correct information of the process dynamics; to avoid that, what
is to be done as far as possible, we should try to get symmetrical limit cycle output
and we should try to take measurements from the symmetrical limit cycle output.
And for that, one has to make use of some other scheme and one has to go for on-line
identification scheme, also it is possible to make use of another analysis technique
known as Dual Input Describing Function analysis - DIDF, DIDF analysis for the relay control
system, which is subjected to external static load disturbances. So, in in this case, it
will be possible to estimate more number of parameters, and also it will be possible to
it might be possible to estimate accurate values for the transfer function model parameters
that is all in this lecture, thank you.