 KHAIRUNNISA BINTI ABD AZIZ  MOHD SHAHRIL BIN IBRAHIM  MOHD SUFI B AB MAJID  MOHD SUPIAN B MOHD NAZIP  MOKTAR BIN YAAKOP

PID CONTROLLERS :  Easy to understand, easy to explain to others, and easy to implement  Moreover, they are often available at little extra cost since they are often incorporated into the programmable logic controllers (PLCs) that are used to control many industrial processes.  Unfortunately, many of the PID loops that are in operation are in continual need of monitoring and adjustment since they can easily become improperly tuned  PID controller is placed in a unity feedback control system : i. the input to the plant is u, ii. the plant output is y, iii. the reference input is r, iv. the error input to the PID controller is e = r −y  The basic form for the PID controller is :

where ; i. Kp is the proportional gain ii. Ki is the integral gain iii. Kd is the derivative gain  Because PID controllers are often not properly tuned,there is a significant need to develop methods for the automatic tuning of PID controllers.  While there exist many conventional methods for PID auto-tuning, here we will strictly focus on providing the basic ideas on how you would construct a fuzzy PID auto-tuner.  The figure in next slide showing how a fuzzy PID auto-tuner may work.  There, the “behavior recognizer” seeks to characterize the current behavior of the plant in a way that will be useful to the PID designer.  The whole (upper-level) supervisor may be implemented in a similar fashion to an indirect adaptive controller.

 Alternatively, simple tuning rules may be used where the premises of the rules form part of the behavior recognizer and the consequents form the PID designer.  In this way, a single fuzzy system is used to implement the entire supervisory control level.  Some candidate rules for such a fuzzy system may include the following:

1.If steady-state error is large Then increase the proportional gain. 2.If the response is oscillatory Then increase the derivative gain. 3.If the response is sluggish Then increase the proportional gain. 4.If the steady-state error is too big Then adjust the integral gain. 5.If the overshoot is too big Then decrease the proportional gain.  The Zeigler-Nichols PID tuning rules can used but represent them with a fuzzy system.  For some applications, it is best to have a tuning phase and an on-line operation phase where tuning does not occur.  In the tuning phase, the system takes over and generates a set of reference inputs (e.g., a step) and observes and analyzes the step response to determine the values of the parameters of the premises of the above rules.  Then the rule-base is applied to these, the new PID gains are found, and the closed-loop control system is put into operation with these new gains.

 Clearly, however, it may be unacceptable to expect the system to be taken off- line for such tests and retuning.  On the other hand, for some applications (e.g., in process control) it is possible to follow such a retuning scenario.  Overall, we would emphasize that fuzzy PID auto-tuners tend to be very application dependent and it is difficult to present a general approach to on-line fuzzy PID auto-tuning that will work for a wide variety of applications.  At this point, however, the reader who understands how a fuzzy system operates and has a specific application in mind, can probably quickly write down a set of rules that quantifies how to keep the PID controller properly tuned.  This is often the primary advantage of the fuzzy systems approach to PID auto- tuning over conventional methods.

 Conventional gain scheduling involves using extra information from the plant, environment, or users to tune (via “schedules”) the gains of a controller.  There are many applications where gain scheduling is used.  For instance, in aircraft control you often perform linear control designs about a whole range of operating points (e.g., for certain machs and altitudes).

 These control designs provide a set of gains for the controller at each operating condition over the entire flight envelope.  A gain schedule is simply an interpolator that takes as inputs the operating condition and provides values of the gains as its outputs.  One way to construct this interpolator is to view the data associations between operating conditions and controller gains as forming a large table.  Then, when we encounter an operating condition that is not in the table, we simply interpolate between the ones in the table to find an interpolated set of gains.  In this way, as the aircraft moves about its flight envelope, the gain scheduler will keep updating the gains of the controller with the ones that we would have picked based on a linear design about an operating point.  This general gain scheduling approach is widely used in the aircraft industry (e.g., also for engine control).  The fuzzy system methodology also offers three different methodologies for gain schedule construction to conventional gain scheduling approaches.

 In this case the design of a set of linear controllers for various operating conditions may not be necessary as it may be possible to simply specify, in a heuristic fashion, how the gains should change.  The use of rules may be particularly useful here. For example, the tank level control application.  We could use a rule-base to schedule the gain of a linear proportional controller based on our knowledge that for higher liquid levels we will have bigger crossectional areas for the tank.

 In this case the rules may represent that we want to have a lower proportional gain for lower liquid levels since for lower levels the tank will fill faster with smaller amounts of liquid but for high levels the controller will have to pump more liquid to achieve the same change in liquid level.  We see that some practical knowledge of the tank’s shape, which is not normally used in the design of a linear controller, can possibly be effectively used to schedule the gain of a linear controller for the tank.  It is clear that the knowledge used to construct the gain schedule can be highly application-specific; hence, it is difficult to provide a general methodology for the heuristic construction of fuzzy gain schedulers beyond what has already been presented for techniques to design standard fuzzy controllers.

 In the case where we have a set of controller designs (based on linear or nonlinear theory) for each set of operating conditions that we can measure  We can use the methods for function approximation to identify the function that the data inherently represents.  This function that interpolates between the data is the gain schedule, and the methods of Chapter 5 can be used for its construction.  So, consider how we would schedule just one of the (possibly) many gains of the controller.  To do this we view the operating condition–gain associations produced by the design of the controllers as data pairs (x i, y i ) ∈ G where x i = i th operating point y i = the gain that the controller design procedure produced at this operating point. G = the set of all such data pairs—that is, operating condition–controller associations.

 Here we consider the case where the plant to be controlled is a nonlinear interpolation of R linear models  Similar rules could be created with consequents that use a discrete-time statespace representation or transfer functions, either discrete or continuous.  Here we will simply use the continuous-time linear state- space models as consequents for convenience, but the same basic concepts hold for other forms.

 2 ways that the rules can be constructed to represent the plant.  First, for some nonlinear systems the mathematical differential equations describing the plant can be transformed into which is the model resulting from the above rules.  A second approach would be to use system identification to construct the rules (e.g., use the least squares, gradient, or clustering with optimal output predefuzzification methods.  While it is possible to modify the identification methods so that they apply directly to continuous-time systems.

 Recall controller for plant used the rules  where Ki, i = 1, 2,..., R, are 1 × n vectors of control gains, and the premises of the rules are identical to the premises of the plant rules.  Under certain conditions this controller can be used to stabilize the plant  Many view parallel distributed compensation as a form of gain scheduling that we can achieve stability by designing the Kj, j = 1, 2,..., R then using Equation to interpolate between these gains.

 Regardless of which of the above methodologies is used to construct the gain schedule, these approaches also automatically provide for the fuzzy system implementation of the gain scheduler (interpolator), which can at times be an advantage over conventional methods.  It is important to note that computational complexity is of concern in the implementation of a gain scheduler.  Some may think that the fuzzy system methodologies could offer an approach to reduce the memory storage requirements since the rules will tend to provide an interpolation that may make it possible to omit some data.  The conventional methods used for gain scheduling use interpolation techniques that are basically quite similar (sometimes identical) to what is implemented by a fuzzy system.  If the original designers found that the extra data were needed to make the mapping that the gain scheduler implements more accurate, then these extra data are probably needed for the fuzzy system also.