Stability Analysis of Feedback Systems:
Earlier, we have examined the dynamic characteristics of the response of closed-loop systems and developed the closed-loop transfer functions that determine the dynamics of such systems.
It is important to emphasize again that the presence of measuring devices, controllers, and final control elements changes the dynamic characteristics of an uncontrolled process.
An important consequence of feedback control is that it can cause oscillatory responses. If the oscillation has a small amplitude and damps out quickly, then the control system performance is generally considered to be satisfactory. However, under certain circumstances, the oscillations may be undamped or even have an amplitude that increases with time until a physical limit is reached, such as a control valve being fully open or completely shut. In these situations, the closed-loop system is said to be unstable.
So, while designing a feedback control system (i.e, selecting its components, and tuning its controller), we are seriously concerned about its stability characteristics. Thus, here we will learn how to analyze the stability characteristics of closed-loop systems and present several useful criteria for determining whether a system will be stable. Additional stability criteria based on frequency response analysis are discussed later on.
The Notion of Stability:
How do we define a stable or unstable system? In the most basic terms, a system is said to be unstable if, after it has been disturbed by an input change, its output “takes off” and does not return to the initial state of rest.
Definition of Stability . An unconstrained linear system is said to be stable if the output response is bounded for all bounded inputs. Otherwise, it is said to be unstable.
(*unconstrained here refers to the ideal situation, where there are no physical limits to the input and output variables; i.e, can increase/decrease to any value without any limit)
“Bounded” is an input that always remains between an upper and a lower limit (e.g., sinusoidal, step, but not the ramp)
Unbounded outputs exist only in theory and not in practice because all physical quantities are limited. Therefore, the term “unbounded” means very large.
According to the definition above, a system with a response like those in the below figure (a) is stable, while (b) shows the response of unstable systems.
Characteristic Equation:
We know that the closed-loop response of the generalized feedback control system is given by:
(s) (G(s)/1 ol(s)).ysp(s) (Gd(s)/1 ol(s)).d(s)
y = +G + +G
Where, G(s) = Gp.Gf.Gc; ol(s) Gp.Gf.Gc.Gm
G =
Or equivalently,
(s) Gsp.ysp(s) Gload.d(s)
y = +
Where,
Gp.Gf.Gc)/(1 p.Gf.Gc.Gm) Gsp,
( +G =
.
d/(1 p.Gf.Gc.Gm) Gload
G +G =
The stability characteristics of the closed-loop response will be determined by the poles of the transfer function Gspand Gload. These poles are common for both transfer functions (because they have the same denominator) and are given by the roots of the equation:
p.Gf.Gc.Gm 0
1 +G =
which is called the “characteristic equation” for the generalized feedback system.
General Stability Criterion. The feedback control system is stable if and only if all roots of the characteristic equation are negative or have negative real parts. Otherwise, the system is unstable.
The figure below provides a graphical interpretation of this stability criterion. Note that all the roots of the characteristic equation must lie to the left of the imaginary axis in the complex plane for a stable system to exist.
The qualitative effects of these roots on the transient response of the closed-loop system are shown in the figures below. The left portion of each part of the figure below shows representative root locations in the complex plane. The corresponding figure on the right shows the contributions these poles make to the closed-loop response for a step-change in the set point. Similar responses would occur for a step-change in a disturbance. A system that has all negative real roots will have a stable, nonoscillatory response, as shown in (a). On the other hand, if one of the real roots is positive, then the response is unbounded, as shown in (b). A pair of complex conjugate roots results in oscillatory responses as shown (c) and (d). If the complex roots have negative real parts, the system is stable; otherwise, it is unstable. Recall that complex roots always occur as complex conjugate pairs.
Routh-Hurwitz Criterion for Stability:
The criterion of stability for closed-loop systems does not require calculation of the actual values of the roots of the characteristic equation. It only requires that we know if any root is to the right of the imaginary axis. The Routh-Hurwitz procedure allows us to just do that and thus reach quickly a conclusion as to the stability of the closed-loop system without computing the actual values of the roots.
It can be applied only to systems whose characteristic equations are polynomials in s. Thus, the Routh stability criterion is not directly applicable to systems containing time delays, because an e^(-Өs) term appears in the characteristic equation where Ө is the time delay. Exact stability analysis of systems containing time delays can be performed by direct root-finding or by a frequency response analysis and the Bode or Nyquist stability criterion presented later on.
The Routh stability criterion is based on a characteristic equation that has the form:
s a s ... a s a 0 an n + n−1 n−1 + + 1 + 0 =
● We arbitrarily assume that an > 0. If an < 0, simply multiply the equation by -1 to generate a new equation that satisfies this condition.
● A necessary (but not sufficient) condition for stability is that all of the coefficients (ao, a1, ..., an) in the characteristic equation be positive. If any coefficient is negative or zero, then at least one root of the characteristic equation lies to the right of, or on, the imaginary axis, and the system is unstable.
● If all of the coefficients are positive, we next construct the following Routh array:
● The Routh array has n + 1 rows, where n is the order of the characteristic equation.
● The Routh array has a roughly triangular structure with only a single element in the last row. The first two rows are merely the coefficients in the characteristic equation, arranged according to odd and even powers of s.
● The elements in the remaining rows are calculated from the formulas:
● Having constructed the Routh array, we can now state the Routh stability criterion:
Routh Stability Criterion: A necessary and sufficient condition for all roots of the characteristic equation to have negative real parts is that all of the elements in the left column of the Routh array are positive.
Thus, it can be concluded that for R-H Stability Criteria:
Necessary Conditions (if not satisfied, no need to move forward, the system is unstable):
● No term should be missing in the characteristic equation.
● There should be no sign change in the given polynomial/characteristic equation.
Sufficient Conditions:
● If the necessary conditions are satisfied, then the system may or may not be unstable.
Then, sufficient conditions require us to form a Routh array.
● If there is no sign change in the first column of Routh array, then the closed-loop system will be stable.
● If there is a sign change, then the system will be unstable and the number of sign changes will indicate the number of roots that will lie on the right half of the complex plane.
Root Locus Diagrams/Analysis:
● In the previous section we have seen that the roots of the characteristic equation play a crucial role in determining system stability and the nature of the closed-loop responses.
● In the design and analysis of control systems, it is instructive to know how the roots of the characteristic equation change when a particular system parameter such as a controller gain, Kc changes.
● A root locus diagram provides a convenient graphical display of this information.
● Let us examine the construction of the root locus using a specific example.
● The complete root locus is given in figure 15.5 and since all its branches are located to the left of the imaginary axis, we conclude that the closed-loop system is stable for any value for Kc. Furthermore, we conclude that for Kc satisfying inequality (15.2) the response of the system to a step input is not oscillatory because the imaginary part of the two roots is zero. It becomes oscillatory for Kc satisfying the inequality (15.4).
The above example demonstrated that the root locus of a system not only provides information about the stability of a closed-loop system but also informs us about its general dynamic response characteristics as Kc changes. Therefore, the root locus analysis can be the basis of a feedback control loop design methodology, whereby the movement of closed-loop poles due to the change of the proportional controller gain can be clearly displayed.
Outline of the design problems:
In a general closed-loop system, when the load or the set-point change, the response of the process deviates and the controller tries to bring the output again close to the desired set point.
The below figure illustrates the response of a controlled process to a unit step change in the load when different types of controllers have been used. We notice that different controllers have different effects on the response of a controlled process.
Thus, the first question that arises is:
Q1: What type of feedback controller should be used to control a given process?
and since, we know that controller parameters such as Kc, τI, and τd have an important effect on the response of the controlled process. Thus, the second question that arises is:
Q2: How do we select the best values for the adjustable parameters of a feedback controller?
(known as the controller tuning problem)
To answer these two design questions, we need to have a quantitative measure in order to compare the alternatives and select the best type of controller and the best values of its parameters. Thus, the third design question arises:
Q3: What performance criterion should we use for the selection and the tuning of the controller?
There are a variety of performance criteria we could use, such as:
● Keep the error as small as possible
● Achieve short settling times
● Minimize the integral of the errors until the process has settled to its desired set-point, and so on.
A point to note here is that a different performance criterion leads to a different control design.
Simple Performance Criteria:
We start with the performance criteria since we need to establish some basis for the comparison of alternative controller designs.
Consider two different feedback control systems producing the two closed-loop responses as shown below:
Response A has reached the desired level of operation faster than response B. If our criterion for the design of the controller had been:
Return to the desired level of operation as soon as possible
Then, clearly we would select the controller which gives the closed-loop response of type A. But if our criterion had been:
Keep the maximum deviation as small as possible Or
Return to the desired level of operation and stay close to it in the shortest time
We would have selected the controller which gives the closed-loop response of type B. Similar dilemmas will be encountered quite often during the design of the controller.
Steady-State Performance Criteria:
The principal steady-state performance criterion usually is zero-error at steady-state. We have seen already that in most situations, the proportional controller cannot achieve zero steady-state error, while a PI controller can. Also, we know that for proportional control, the steady-state error(offset) tends to 0 as Kc tends to .∞
Dynamic Response Performance Criteria:
The evaluation of the dynamic performance of a closed-loop system is based on two types of commonly used criteria:
1. Criteria that uses only a few points of the response. They are simpler but only approximate. (Simple Performance Criteria)
2. Criteria that uses the entire closed-loop response from time t = 0 until t equals very large. These are more precise but also more cumbersome to use.
The simple performance criteria are based on some characteristic features of the closed-loop response of a system. The most often quoted are:
Overshoot
Rise time (time needed by the response to reach the desired value for the first time) Settling time (time needed by the response to settle within ± 5% of the desired value) Decay ratio
Frequency of oscillation of the transient
Every one of the characteristics above could be used by the designer as the basic criterion for selecting the controller and the values of its adjusted parameters.
From all the performance criteria above, the decay ratio has been the most popular among practicing engineers. Specifically, experience has shown that a decay ratio:
/A 1/4
C =
is a reasonable trade-off between fast rise time and reasonable settling time. This criterion is usually known as the one-quarter decay ratio criterion.
Example: Controller tuning with the one-quarter decay ratio criterion:
Time-Integral Performance Criteria:
The shape of the complete closed-loop response, from time t = 0 until steady-state has been reached, could be used for the formulation of a dynamic performance criterion. This criterion is based on the entire response of the process, unlike the simple criteria that use only isolated characteristics of the dynamic response. The most often used are:
The problem of designing the “best” controller can now be formulated as follows:
Select the type of the controller and the values of its adjusted parameters in such a way as to minimize the ISE, IAE, or ITAE of the system’s response.
Which one of the three criteria mentioned above we will use depends upon the characteristics of the system we want to control and some additional requirements we impose on the controlled response of the process. The following are some general guidelines:
We should remember the following two points:
1. Different criteria lead to different controller designs.
2. For the same time-integral criterion, different input change lead to different designs.
Let us analyze these two statements based on the following example:
Select the Type of Feedback Controller:
Which one of the three popular feedback controllers should be used? The question can be answered in a very systematic manner as follows:
1. Define an appropriate performance criterion (e.g., ISE, IAE, or ITAE).
2. Compute the value of the performance criterion using a P, or PI, or PID controller with the best settings for the adjusted parameters Kc, τI and τd.
3. Select the controller which gives the “best” value for the performance criterion.
The method, although mathematically rigorous, has several serious practical drawbacks:
● It is very tedious.
● It relies on models (transfer functions) for the process, sensor, and final control element, which may not be known exactly.
● It incorporates certain ambiguities as to which is the most appropriate criterion and what input changs to consider.
Fortunately, we can choose the most appropriate type of feedback controller using only general qualitative considerations stemming from earlier analysis. In summary, the conclusions were as follows:
1. Proportional Control
● Accelerates the response of a control process.
● Produces an offset (non zero steady-state error) for mostly all the processes.
2. Integral Control
● Eliminates any offset.
● The elimination of the offset usually comes at the expense of higher maximum deviations.
● Produces sluggish, long oscillating responses.
● On increasing Kc to produce a faster response, the system becomes more oscillatory and may lead to instability.
3. Derivative Control
● Anticipates future errors and introduces appropriate action.
● Introduces a stabilizing effect on the closed-loop response of a process.
It is clear from the above that a three-mode PID controller should be the best. This is true in the sense that it offers the highest flexibility to achieve the desired control response by having three adjustable parameters. On the other hand, it introduces a more complex tuning problem because we have to adjust three parameters.
To balance the quality of the desired response against the tuning difficulty, we can adopt the following rules in selecting the most appropriate controller.
1. If possible, use a simple proportional controller. The simple proportional controller can be used if (a) We can achieve acceptable offset with moderate values of Kc, or (b) The process has an integrating action(i.e., a term 1/s in its transfer function) for which the P-control does not exhibit offset. Therefore, for gas pressure or liquid-level control, we can use only P-controller.
2. If a simple P-controller is unacceptable, use a PI. A PI controller should be used when a P-controller alone cannot provide sufficiently small steady-state errors (offsets).
Therefore, PI will seldom be used in liquid-level or gas pressure control systems but very often (almost always) for flow control. The response of a flow system is rather fast.
Consequently, the speed of the closed-loop system remains satisfactory despite the slowdown caused by the integral control mode.
3. Use a PID controller to increase the speed of the closed-loop response and retain robustness. The PI eliminates the offset but reduces the speed of the closed-loop response. For a multi capacity process whose response is very sluggish, the addition of a PI controller makes it even more sluggish. In such cases, the addition of the derivative control action with its stabilizing effect allows the use of higher gains which produce faster responses without excessive oscillations. Therefore, derivative action is recommended for temperature and composition control where we have sluggish, multi capacity processes.
Controller Tuning:
After the type of feedback controller has been selected, we still have the problem of deciding what values to use for its adjusted parameters, known as the controller tuning problem. There are three general approaches we can use for tuning a controller:
1. Use simple criteria such as one-quarter decay ratio, minimum settling time, minimum largest error, and so on. Such an approach is simple and easily implementable on an actual process. Usually, it provides multiple solutions and additional specifications on the closed-loop performance is needed to break the multiplicity and select a single set of values for the adjusted parameters.
2. Use time integral performance criteria such as ISE, IAE, or ITAE. This approach is rather cumbersome and relies heavily on the mathematical model of the process. Applied experimentally on an actual process, it is time consuming.
3. Use semi-empirical rules which have been proven in practice.
As we have already discussed the first two approaches, now, we will discuss the most popular of the empirical tuning method known as process reaction curve method, developed by Cohen and Coon.
Consider the control system in the below figure, which has been “opened” by disconnecting the controller from the final control element.
Introduce a step change of magnitude A in the variable c which actuates the final control element. In the case of valve, c is the stem position. Record the value of the output with respect to time. The curve Ym(t) is called the process reaction curve. Between Ym and c we have the following transfer function:
This equation shows that the process reaction curve is affected not only by the dynamics of the main process but also by the dynamics of the measuring sensor and final control element.
Cohen and Coon observed that the response of most processing units to an input change, such as above had a sigmoidal shape (below figure (a)), which can be adequately approximated by the response of a first order process with dead time(dashed curve in (b)):
Which has three parameters: static gain K, dead time td, and time constant τ. From the approximate response of figure (b), it is easy to estimate the values of the three parameters.
Thus,
(a) Process reaction curve; (b) its approximation with a first order plus dead-time system.
Cohen and Coon used the approximate model of first order with time delay and estimated the values of the parameters K, td and as indicated above.τ
Then they derived expressions for the “best” controller settings using load changes and various performance criteria, such as:
One-quarter decay ratio Minimum offset
Minimum integral square error (ISE)
From the above equations which give the value of the proportional gain Kc for the three controllers, we notice the following:
1. The gain of the PI controller is lower than that of the P-controller. This is due to the fact that the integral control mode makes the system more sensitive (which may lead to instability) and thus the gain value needs to be more conservative.
2. The stabilizing effect of the derivative control mode allows the use of higher gains in the PID controller (higher than gain for P or PI controllers).
On-Line Controller Tuning:
The control systems for modern industrial plants typically include thousands of individual control loops. During control system design, preliminary controller settings are specified based on process knowledge, control objectives, and prior experience.
After a controller is installed, the preliminary settings often prove to be satisfactory. But for critical control loops, the preliminary settings may have to be adjusted in order to achieve satisfactory control. This on-site adjustment is referred to by a variety of names: on-line tuning, field tuning, or controller tuning.
Because on-line controller tuning involves plant testing, often on a trial-and-error basis, the tuning can be quite tedious and time-consuming. Consequently, good initial controller settings are very desirable to reduce the required time and effort. Ideally, the preliminary settings from the control system design can be used as the initial field settings. If the preliminary settings are not satisfactory, alternative settings can be obtained from simple experimental tests. If necessary, the settings can be fine-tuned by a modest amount of trial and error.
Next, we make a few general observations:
1. Controller tuning inevitably involves a tradeoff between performance and robustness. The performance goals of excellent set-point tracking and disturbance rejection should be balanced against the robustness goal of stable operation over a wide range of conditions.
2. Controller settings do not have to be precisely determined. In general, a small change in a controller setting from its best value (for example, ±10%) has little effect on closed-loop responses.
3. For most plants, it is not feasible to manually tune each controller. Tuning is usually done by a
control specialist (engineer or technician) or by a plant operator. Because each person is typically responsible for 300 to 1,000 control loops, it is not feasible to tune every controller.
Instead, only the control loops that are perceived to be the most important or the most troublesome receive detailed attention. The other controllers typically operate using the preliminary settings from the control system design.
Continuous Cycling Method (Ziegler-Nichols Tuning Method):
Over 60 years ago, Ziegler and Nichols (1942) published a classic paper that introduced the continuous cycling method for controller tuning. It is based on the following trial-and-error procedure:
Step 1. After the process has reached a steady-state (at least approximately), eliminate the integral and derivative control action by setting τd to zero and τI to the largest possible value.
Step 2. Set Kc equal to a small value (e.g., 0.5) and place the controller in the automatic mode.
Step 3. Introduce a small, momentary set-point change so that the controlled variable moves away from the set point. Gradually increase Kc in small increments until continuous cycling occurs. The term continuous cycling refers to a sustained oscillation with a constant amplitude.
The numerical value of Kc that produces continuous cycling (for proportional only control) is called the ultimate gain, Kcu. The period of the corresponding sustained oscillation is referred to as the ultimate period, Pu·
Step 4. Calculate the PID controller settings using the Ziegler-Nichols (Z-N) tuning relations as mentioned in the below table.
Step 5. Evaluate the Z-N controller settings by introducing a small set-point change and observing the closed-loop response. Fine-tune the settings, if necessary.
The tuning relations reported by Ziegler and Nichols (1942) were determined empirically to provide closed-loop responses that have a 1/4 decay ratio. For proportional-only control, the Z-N settings in the above table provide a safety margin of two for Kc, because it is equal to one-half of the stability limit, Kcu. When integral action is added to form a PI controller, Kc is reduced from 0.5Kcu to 0.45Kcu. The stabilizing effect of derivative action allows Kc to be increased to 0.6 Kcu for PID control.
Typical results for the trial-and-error determination of Kcu are shown in the below figure. For, Kc < Kcu, the closed-loop response y(t) is usually overdamped or slightly oscillatory. For the ideal case where Kc = Kcu, continuous cycling occurs (b). For Kc > Kcu, the closed-loop system is unstable and will theoretically have an unbounded response (c). But in practice, controller saturation prevents the response from becoming unbounded and produces continuous cycling instead (d).
The continuous cycling method, or a modified version of it, is frequently recommended by control system vendors. Even so, the continuous cycling method has several major disadvantages:
1. It can be quite time-consuming if several trials are required and the process dynamics are slow. The long experimental tests may result in reduced production or poor product quality.
2. In many applications, continuous cycling is objectionable, because the process is pushed to the stability limits. Consequently, if external disturbances or process changes occur during the test, unstable operation or a hazardous situation could result (e.g., a "runaway" chemical reaction).
3. This tuning procedure is not applicable to integrating or open-loop unstable processes, because their control loops typically are unstable at both high and low values of Kc, while being stable for intermediate values.
4. For first-order and second-order models without time delays, the ultimate gain does not exist, because the closed-loop system is stable for all values of Kc, providing that its sign is correct.
However, in practice, it is unusual for a control loop not to have an ultimate gain.
There are two other tuning methods namely, relay auto-tuning method and the step test method which can eliminate or avoid the disadvantages with the Z-N tuning method.
Troubleshooting Control Loops:
If a control loop is not performing satisfactorily, then troubleshooting is necessary to identify the source of the problem. Ideally, it would be desirable to evaluate the control loop over the full range of process operating conditions during the commissioning of the plant. In practice, this is seldom feasible.
Furthermore, the process characteristics can vary with time for a variety of reasons, including changes in equipment and instrumentation, different operating conditions, new feedstocks or products, and large disturbances.
Surveys of thousands of control loops have confirmed that a large fraction of industrial control loops perform poorly. For example, surveys have reported that about one-third of the industrial control loops were in the manual mode, and another one-third actually increased process variability over manual control, a result of poor controller tuning. Clearly, controller tuning and control loop troubleshooting are important activities.
Here, we provide a brief introduction to the basic principles and strategies that are useful in troubleshooting control loops.
Systems Approach: An important consideration for troubleshooting activities is to be aware that the control loop consists of a number of individual components: sensor/transmitter, controller, a final control element, instrument lines, computer- process interface (for digital control), as well as the process itself. Serious control problems can result from a malfunction of any single component. On the other hand, even if each individual component is functioning properly, there is no guarantee that the overall system will perform properly. Thus, a systems approach is required.
The starting point for troubleshooting is to obtain enough background information to clearly define the problem. Many questions need to be answered:
1. What is the process being controlled?
2. What is the controlled variable?
3. What are the control objectives?
4. Are closed-loop response data available?
5. Is the controller in the manual or automatic mode? Is it reverse- or direct-acting?
6. If the process is cycling, what is the cycling frequency?
7. What control algorithm is used? What are the controller settings?
8. Is the process open-loop stable?
9. What additional documentation is available, such as control loop summary sheets, piping and instrumentation diagrams, etc.?
After acquiring this background information, the next step is to check out each component in the control loop. In particular, one should determine that the process, measurement device (sensor), and control valve are all in proper working condition. Typically, sensors and control valves that are located in the field require more maintenance than control equipment located in the central control room. Any recent change to the equipment or instrumentation could very well be the source of the problem. For example, cleaning heat exchanger tubes, using a new shipment of catalyst, or changing a transmitter span could cause control-loop performance to change.
Finally, after checking each component (sensors and control valves, mainly) and making sure they are functioning properly, controller re-tuning may be necessary if the control loop exhibits undesirable oscillations or excessively sluggish responses.
