Again,
Rearranging the above we obtain,
Hence,
Again
(IV.59)
Hence,
(IV.60)
Similarly,
(IV.61)
And,
(IV.62)
Now,
Rearranging the above we obtain,
(IV.63)
Hence,
(IV.64)
Using the above derivations,
(IV.65)
Using,
(IV.67)
Using the above expressions,
The optimal values for
and
are calculated by minimizing as follows,
(IV.68)
and
(IV.69)
There are two equations (IV.68-69) and two unknowns
. It is however left for the reader as an exercise to solve these equations and calculate the designed value of controller parameters. In a similar fashion, the controller parameters can also be designed by optimizing IAE and ITAE.
Use of simple criteria for controller design such as one-fourth decay ratio, minimum settling time or minimum overshoot are easy to use, however, they seldom produce multiple solutions and hence additional criteria are needed to break the multiplicity. Time integral performance criteria such as ISE, IAE or ITAE are most effective, however, they are heavily dependent on mathematical models. Real time application of such techniques is cumbersome. Use of semi-emperical techniques, on the other hand, is effective yet easy to use. Process Reaction Curve Method , developed by Cohen and Coon, is one of such empirical tuning methods for feedback controllers.
IV.4.5 Cohen-Coon technique of Controller Tuning
It is observed that the response of most of the processes under step change in input yields a sigmoidal shape (Fig. IV.13 ).
Fig. IV.13: Process Reaction Curve for Cohen Coon Method
Such sigmoidal shape can be adequately approximated by the response of a first order process with dead time.
(IV.70)
From the approximate response it is easy to estimate the parameters. The controllers are designed as given in Table IV.5.
Table IV.5: Controller settings using Cohen-Coon design method
IV.5 Stability of feedback control system
Frequency response analysis is an useful tool for designing feedback controllers. It enables the designer to study the stability characteristics of a closed loop system using Bode or Nyquist plots and also to select the appropriate design values of controller parameters.
IV.5.1 Bode Diagram of closed loop process
IV.5.1.1: Bode diagram of PID controler
It is also worth to study the Bode diagram of PID controller in the following figure.
Fig. IV.13(a): Bode Diagram of PID controller
The PID controllers have the following characteristics:
And
It is left to the reader to exercise how to arrive at above eqs. Note that the PID controllers ideally have three asymptotes.
• At
with a slope
• At
with a slope
• At
with a zero slope.
IV.5.1.2: Bode diagram of Processes in series
Let us have a control loop with two components, viz ., one PI controller and one first order plus dead time model.
The open loop transfer function is
The above open loop transfer function is a combination of four individual processes in series, viz., pure gain, pure dead time, first order system and PI controller. Two time constants are observed in the series that would yield the location of corner frequencies
viz .,
or
Fig. IV.13(b): Bode plot of processes in series given in example
The above figure shows the Bode plot of the open loop process indicated in this example. AR and phase shift for all individual transfer functions as well as the overall transfer function have been indicated along with the location of corner frequencies.
IV.5.2: Bode Stability Criterion
Consider a simple first order plus dead time process to be controlled by a proportional controller:
(a) Closed Loop
(b) Opened Loop
Fig. IV.14: Example of first order system for studying Bode stability criterion
The open-loop transfer function for this system is given by
(IV.71)
The Bode plot of the above open loop transfer function is given by the following figure.
Fig. IV.15: Bode plot
We are interested to know the frequency where the phase shift is
. Numerically it can be solved by the equation
(IV.72)
The frequency is
and the value of
at
is observed to be
which can also be found numerically by,
(IV.73)
The above exercise indicates that in order to obtain
at this frequency
one needs to set the value of
as,
(IV.74)
At this juncture, one needs to perform a thought experiment in order to understand the Bode stability criterion. Let us set the value of controller gain,
and let us “open up” the feedback loop as indicated in the figure before. Suppose, we vary the setpoint as a sinusoidal function
. As the loop is open, the error will be equal to the setpoint
and thereby yield an output,
(IV.75)
Now, suppose two events occur simultaneously
• The setpoint perturbation is stopped
• The feedback loop is reconnected
Then, the error signal will remain as
. In other words, the response of the system will continue to oscillate with constant amplitude even when the setpoint signal is withdrawn.
Alternatively, if we choose the value of controller gain less than
, (say
) then
(IV.76)
If we repeat the above thought experiment, the output signal will take the form
(IV.77)
Upon closing the loop and withdrawing the setpoint perturbation, the new value for the error for the next cycle will be
that will eventually yield an output response of
and so on. It is evident that the amplitude of the error signal would diminish at every cycle and eventually lead to zero.
In case we choose the value of controller gain greater than
, (say
) then
(IV.78)
The same thought experiment would lead to ever increasing error signal because the amplitude ratio is greater than 1.
Hence the above thought experiment indicates that we have been able to find a combination of frequency
and controller gain
such that the
of the process becomes 1 and phase shift becomes
simultaneously at that combination
. The output response shows a sustained oscillation with a time period
at this combination. Any frequency,
, will lead to oscillation with increasing amplitude and eventually will lead to instability. Hence, the frequency
is termed as the
crossover frequency , the gain value
is termed as
ultimate gain and
is called the ultimate period of oscillation of the closed loop system.
The conclusion drawn from the above thought experiment is the Bode Stability Criterion and can be stated as follows -A feedback control system is unstable if the amplitude ratio of the corresponding open loop transfer function is greater than one at the crossover frequency. The value of controller gain is the decisive factor in order to ensure its stability.
It is further understood from eq. (IV.72) that large dead time leads to smaller value crossover frequency. In other words, even a low frequency signal will be able to destabilize such process.
IV.5.3 Gain Margin and Phase Margin
The Bode stability criterion states that the maximum value of the controller gain that can be chosen for stable closed loop response is called the
ultimate gain . In other words, the value of controller gain must always be less than
in order to ensure stability. The gain margin (GM) is a design parameter such that
(IV.79)
Gain margin should always be chosen as greater than one (GM>1) to ensure stability .
Gain margin acts as a safety factor for model uncertainty. Since process parameters such as gain, time constant and dead time can never be estimated exactly, a safety factor of magnitude more than one is necessary for stable operation. For relatively well modeled processes, a low safety factor will be acceptable whereas poorly modeled processes need higher safety factors. For an example, let us choose GM=2 for the process we have discussed above (eq. IV.71), the design value of the controller gain is
; suppose there exists a modeling error of 50% in estimating the dead time of the process and the true value of the dead time is 0.45 instead of 0.3, then the revised value of crossover frequency is
(IV.80)
or,
, and the corresponding
which is still higher than the designed value of
. The system is still stable despite the error by 50% we made in estimation of dead time of the process.
Phase margin is another safety factor which is used for controller design. Here we are interested to compute a frequency
that satisfies the following expression,
(IV.81)
is called phase margin (PM) and it is the extra phase lag needed to destabilize a system. For an example, let us choose
.
can be calculated from the following expression
(IV.82)
or,
. The gain is designed from the expression
(IV.83)
or,
Suppose there exists a modeling error of 50% in estimating the dead time of the process and the true value of the dead time is 0.45 instead of 0.3,then the phase lag encountered by the process would be
(IV.84)
which is 2° more than the safety limit for stability. Hence, the phase margin of
is not sufficient for handling 50% error in dead time estimation. It is left to the reader to verify that a phase margin of 45° will suffice for handling 50% error in dead time estimation.
IV.5.4 Ziegler Nichols Tuning technique
Unlike process reaction curve method which uses open loop response data, Ziegler Nichols tuning technique uses closed loop response data. The following settings are given by this technique for feedback controllers:
Table IV.6: Controller settings through Ziegler-Nichols tuning technique
Controller
P
PI
PID
IV.5.5 Nyquist Stability Criterion
The Bode stability criterion is valid for systems where amplitude ratio and phase shift decreases monotonically with Nyquist stability criterion does not have any such limitation and is applicable in more general sense. The criterion states that:
If open-loop Nyquist plot of a feedback system encircles the point (-1,0) as the frequency varies from to the closed loop response is unstable.
(a) Stable process
(b) Unstable process
Fig. IV.16: Example of systems for studying Nyquist stability criterion
IV.6 Problems with large dead time and/or inverse response
Feedback control systems with large dead time and/or inverse response cause immense problem for the controller designers. A suitable method to avoid such problem(s) is the aim of the following discussions.
IV.6.1 Processes with large dead time
Dead time in a system is caused by delay in transportation of fluids, delay and/or inefficiency of measuring device(s) and/or the actuating device(s), delay in decision making by “human” controller, etc .
A large dead time yields lower value of crossover frequency which in turn reduces ultimate gain. In order to avoid instability, as deadtime increases the value of controller gain should be lowered to ensure stability; however this leads to sluggish response.
IV.6.1.1 Pad é Approximation
Dead time transfer function of a process can be expanded with Taylor series and truncated with first/second order terms as follows
(IV.85)
as first order approximation or
(IV.86)
as second order approximation. This is known as Padé approximation. Nevertheless, it should also be noted that Padé approximation leads to zero in the system, which in turn could be problematic.
IV.6.1.2 Smith Predictor
A method of dead time compensation has been proposed by O.J.M. Smith, named
Smith Predictor . Let us assume that the process
is having a dead time
. This process can be expressed as
(IV.87)
Where
does not have any dead time component. Schematically this arrangement can be shown as in the following figure.
Fig. IV.17: Schematic of a process with dead time
If one introduces an additional loop as in the following figure .
Fig. IV.18: Schematic of a process with Smith Predictor
Then the resultant configuration appears to be the following:
Fig. IV.19: Schematic of a dead time compensated loop
The configuration indicates that the feedback carries only the information on "process minus delay" part and not the dead time part. The controller apparently takes a control action without any “delayed” information. It should be noted that by this compensation, one cannot discard the delay in the process, only its effect is nullified in the calculation of control action. It should also be noted that this arrangement is quite sensitive to the “correct” value of dead time. A Smith predictor based on an “incorrect” value of deadtime may actually worsen the situation.
IV.6.2 Processes with inverse response
Let us assume two first order processes
and
, where
, are arranged in the following manner
Fig. IV.20: Schematic of a process producing inverse response
As
the Process 2 will be faster than Process 1. Hence, the initial direction of the overall process response will be guided by the Process 2 in the negative direction. On the other hand,
indicates that Process 1 will guide the ultimate steady state in the positive direction by virtue of its higher gain than the Process 2. As a result the overall process response curve will initially transit towards in the opposite direction of its final steady state. However, the transition will change its direction after some time and finally settle at the steady state value. This type of response is termed as inverse response.
The open-loop response of the system is
(IV.88)
This system has a positive zero
(IV.89)
It is noted that any system having positive zero will demonstrate an inverse response.
IV.6.2.1 Compensation for inverse response
Similar to the Smith Predictor , compensation for inverse response is formulated as per the following schematic.
(a) Conventional
(b) With compensator
Fig. IV.21: Schematic of a process with inverse response
In order to have a non-positive zero for this system, the value of
, should be chosen in such a way that
(IV.90)