9. Fuzzy Control Systems

Transcription

1 9. Fuzzy Control Systems

2 Introduction A simple example of a control problem is a vehicle cruise control that provides the vehicle with the capability of regulating its own speed at a driver-specified set-point (e.g., 60 Km/hr). One solution to the automotive cruise control problem involves adding an electronic controller that can sense the speed of the vehicle via the speedometer and actuate the throttle position so as to regulate the vehicle speed as close as possible to the driver-specified value even if there are road grade changes, head winds, or variations in the number of passengers or amount of cargo in the vehicle.

3 Input Fuzzifier Inference Engine Fuzzy Knowledge base Defuzzifier Plant Output

4 u(t) = input: y(t) = output: r(t) = reference: d(t) = disturbance throttle position vehicle speed desired speed

5 Basically, while differential equations are the language of conventional control, heuristics and rules about how to control the plant are the language of fuzzy control. Design Constraints: Disturbance rejection properties (e.g., for the cruise control problem, that the control system will be able to dampen out the effects of winds or road grade variations). Insensitivity to plant parameter variations (the control system will be able to compensate for changes in the total mass of the vehicle) Stability Rise-time Overshoot Settling time Steady-state error Hany Selim 5

6 Fuzzy Control System Design What is the motivation for turning to fuzzy control? Basically, the difficult task of modeling and simulating complex real-world systems for control systems development represents usually a hurdle. It is for this reason that in practice conventional controllers are often developed via simple models of the plant behavior that satisfy the necessary assumptions, and via the ad hoc tuning of relatively simple linear or nonlinear controllers. Conventional control engineering approaches that use appropriate heuristics (Heuristics are rules of thumb, or common sense rules) to tune the design have been relatively successful. 6

7 ow how much of the success can be attributed to the use of the mathematical model and conventional control design approach, and how much should be attributed to the clever heuristic tuning that the control engineer uses upon implementation? And if we exploit the use of heuristic information throughout the entire design process, can we obtain higher performance control systems? Fuzzy control provides a formal methodology for representing, manipulating, and implementing a human s heuristic knowledge about how to control a system. 7

8 Fuzzy Controller Fuzzy controller embedded in a closed-loop control system 8

9 (1) The rule-base holds the knowledge, in the form of a set of rules, of how best to control the system. (2) The inference mechanism evaluates which control rules are relevant at the current time and then decides what the input to the plant should be. (3) The fuzzification interface simply modifies the inputs so that they can be interpreted and compared to the rules in the rule-base. (4) the defuzzification interface converts the conclusions reached by the inference mechanism into the inputs to the plant. Basically, you should view the fuzzy controller as an artificial decision maker that operates in a closed-loop system in real time. The rule-base is constructed so that it represents a human expert in-the-loop. 9

10 Fuzzy Controller Elements Fuzzy control system design essentially amounts to (1) choosing the fuzzy controller inputs and outputs. (2) choosing the preprocessing that is needed for the controller inputs and possibly postprocessing that is needed for the outputs. (3) designing each of the four components of the fuzzy controller shown. 10

11 For instance, one rule that a human driver may use is If the speed is lower than the set-point, then press down further on the accelerator pedal. A rule that would represent even more detailed information about how to regulate the speed would be If the speed is lower than the set-point AD the speed is approaching the set-point very fast, then release the accelerator pedal by a small amount. This second rule characterizes our knowledge about how to make sure that we do not overshoot our desired goal (the set-point speed). 11

12 Causes for concern when employlng a strategy of gathering heuristic control knowledge: Will the behaviors that are observed by a human expert and used to construct the fuzzy controller include all situations that can occur due to disturbances, noise, or plant parameter variations? Can the human expert realistically and reliably foresee problems that could arise from closed-loop system instabilities? Will the human expert be able to effectively incorporate stability criteria and performance objectives (e.g., rise-time, overshoot, and tracking specifications) into a rule-base to ensure that reliable operation can be obtained?

13 Examples of Application Areas Aircraft/spacecraft: Flight control, engine control, avionic systems, failure diagnosis, navigation, and satellite attitude control. Automobiles: Brakes, transmission, suspension, and engine control. Manufacturing systems: Scheduling and deposition process control. Power industry: Motor control, power control/distribution, and load estimation. Process control: Temperature, pressure, and level control, failure diagnosis, distillation column control, and desalination processes. Robotics: Position control and path planning. Home automation. 13

14 Fuzzy Controller Design consider a simple problem of balancing an inverted pendulum on a cart, as shown in Figure 1. This is a very simple and academic nonlinear control problem, and many good techniques already exist for its solution. Indeed, for this standard configuration, a simple PID controller works well even in implementation. Fig.1

15 y l F r denotes the angle that the pendulum makes with the vertical (in radians), is the half-pendulum length (in meters), is the force input that moves the cart (in ewtons) denotes the desired angular position of the pendulum. The goal is to balance the pendulum in the upright position (i.e., r = 0) when it initially starts with some nonzero angle off the vertical (i.e., y 0). Fig.1 (again!) 15

16 I. Choosing Fuzzy Controller Inputs and Outputs Consider a human-in-the-loop whose responsibility is to control the pendulum. The fuzzy controller is to be designed to automate how a human expert who is successful at this task would control the system. Suppose that the expert needs to use e(t) = r(t) y(t) d/dt(e(t)),and as the variables on which to base decisions. ext, we must identify the controlled variable. In our case we are allowed to control only the force that moves the cart. Fig. 2 Human in the loop Hany Selim 16

17 The designer may implement some filtering or other processing of the plant outputs. The fuzzy control system for our system is shown in Figure3. Fig. 3 17

18 Putting Control Knowledge into Rule-Base Suppose that the human expert provides a description of how best to control the plant in natural language. We seek to take this linguistic description and load it into the fuzzy controller, as indicated by the arrow in Figure. The linguistic variables that describe each of the time varying fuzzy controller inputs and outputs are given by: error change-in-error force describes e(t) describes d/dt (e(t)) describes F(t) 18

19 The linguistic variables assume linguistic values. That is, the values that linguistic variables take on over time change dynamically. Suppose for our case that error, change-inerror, and force take on the following values: negative large negative small zero positive small positive large represented represented represented represented represented by by by by by L S Z PS PL 19

20 Recall that for the inverted pendulum r=0 and e=r y so that e = y and d/dt (e) = d/dt (y) since d/dt (r) = 0. 20

21 For the inverted pendulum each of the following statements quantifies a different configuration of the pendulum (refer to Figure1): The statement error is PL can represent the situation where the pendulum is at a significant angle to the left of the vertical. The statement error is S can represent the situation where the pendulum is just slightly to the right of the vertical, but not too close to the vertical to justify quantifying it as zero and not too far away to justify quantifying it as L. Fig. 1 (another time!) 21

22 The statement error is zero can represent the situation where the pendulum is very near the vertical position (a linguistic quantification is not precise, hence we are willing to accept any value of the error around e(t) = 0 as being quantified linguistically by zero since this can be considered a better quantification than PS or S ). The statement error is PL and change-in-error is PS can represent the situation where the pendulum is to the left of the vertical and, since d/dt(y) < 0, the pendulum is moving away from the upright position (note that in this case the pendulum is moving counterclockwise).!! 22

23 The statement error is S and change-in-error is PS can represent the situation where the pendulum is slightly to the right of the vertical and, since d/dt (y) < 0, the pendulum is moving toward the upright position (note that in this case the pendulum is also moving counterclockwise).!!! 23

24 II. Rule-Base The rule-base captures the expert s knowledge about how to control the plant. In particular, for the three positions shown in Figure 4, we have the following rules Fig. 4 Inverted Pendulum in various positions 24

25 Rule 1 If error is L and change-in-error is L Then force is PL This rule quantifies the situation in Figure 4a where the pendulum has a large positive angle and is moving clockwise; hence it is clear that we should apply a strong positive force (to the right) so that we can try to start the pendulum moving in the proper direction. 25

26 Rule 2 If error is zero and change-in-error is PS Then force is S This rule quantifies the situation in Figure 4b where the pendulum has nearly a zero angle with the vertical (a linguistic quantification of zero does not imply that e(t) = 0 exactly) and is moving counterclockwise; hence we should apply a small negative force (to the left) to counteract the movement so that it moves toward zero (a positive force could result in the pendulum overshooting the desired position). 26

27 Rule 3 If error is PL and change-in-error is S Then force is S This rule quantifies the situation in Figure 4c where the pendulum is far to the left of the vertical and is moving clockwise; hence we should apply a small negative force (to the left) to assist the movement, but not a big one since the pendulum is already moving in the proper direction. 27

28 Each of the three rules listed above is a linguistic rule since it is formed solely from linguistic variables and values. Since linguistic values are not precise representations of the underlying quantities that they describe, linguistic rules are not precise either. They are simply abstract ideas about how to achieve good control that could mean somewhat different things to different people. They are, however, at a level of abstraction that humans are often comfortable with in terms of specifying how to control a process. 28

29 The general form of the linguistic rules listed above is If antecedent Then consequent As you can see from the three rules listed above, the antecedents are associated with the fuzzy controller inputs and the consequents (sometimes called actions ) are associated with the fuzzy controller outputs. otice that each antecedent (or consequent) can be composed of the conjunction of several terms (e.g., rule 3 above). The number of fuzzy controller inputs and outputs places an upper limit on the number of elements in the antecedents and consequents. ote that there does not need to be a antecedent (consequent) term for each input (output) in each rule, although often there is. 29

30 Using the above approach, we could continue to write down rules for the pendulum problem for all possible cases. ote that for the pendulum problem, with two inputs and five linguistic values for each of these, there are at most 52 = 25 possible rules A tabular representation of one possible set of rules for the inverted pendulum is shown in Table1. Table1 Rule Table for the inverted pendulum 30

31 III. Fuzzy Quantification of Knowledge (Membership Functions): Figure 5 is a plot of a function μ versus e(t) that takes on special meaning. The function μ quantifies the certainty that e(t) can be classified linguistically as PS. Fig.5 31

32 For some applications if we are absolutely certain that any value of e(t) near π/ 4 is still PS and only when you get sufficiently far from π/ 4 do we lose our confidence that it is PS. One way to characterize this understanding of the meaning of PS is via the trapezoid-shaped membership function in Figure 6a. For other applications you may think of membership in the set of PS values as being dictated by the Gaussianshaped membership function (not to be confused with the Gaussian probability density function) shown in Figure 6b. Fig.6a,b 32

33 For still other applications values far away from π/ 4 are not accepted as being PS, so you may use the membership function in Figure 6c to represent this. Finally, while we often think of symmetric characterizations of the meaning of linguistic values, we are not restricted to these symmetric representations. For instance, in Figure 6d we represent that we believe that as e(t) moves to the left of π/ 4 we are very quick to reduce our confidence that it is PS, but if we move to the right of π/ 4 our confidence that e(t) is PS, diminishes at a slower rate. Fig.6c,d 33

34 For the output F, the membership functions at the outermost edges cannot be saturated for the fuzzy system to be properly defined. The basic reason for this is that we want to indicate to a process actuator, any value of F bigger than, say, 30, is not acceptable. Fig.8 Membership functions for an inverted pendulum over a cart 34

35 ote that e(t) changes its value over time. For instance, as e(t) changes from π/2 to π/2 the various membership functions will change their values. For example, at e(t) = π/2 we are certain that the error is L, and as the value of e(t) moves toward π/4 we become less certain that it is L and more certain that it is S. 35

36 IV. Fuzzification It is actually the case that for most fuzzy controllers the fuzzification block on slide 7 can be simplified and under certain conditions virtually ignored. The fuzzification process is the act of obtaining a value of an input variable and finding the numeric values of the membership function(s) that are defined for that variable. For example, If e(t) = 0 and d/dt( e(t)) = 3π/16, the fuzzification process amounts to finding the values of the input membership functions for these. In this case μz(e(t)) = 1 ; with all other values of membership functions=0 and µz (d/dt(e(t) ) = 0.25 and µps (d/dt(e(t) ) =

37 This information is then used in the fuzzy inference process that starts with matching. : The antecedent of all the rules are compared to the controller inputs to determine which rules apply to the current situation. Fig.9 Input membership functions with input values. 37

38 ow take the Rule: If e(t) is Z and d/dt(e(t)) is Z then F is Z If we are not very certain about the truth of one statement, how can we be any more certain about the truth of that statement and the other statement? For this reason, we use the minimum of both degrees of membership i.e. μantecedent= min (1,0.25) = 0.25 Table 2 Rule Table with Rules That Are On Highlighted 38

39 V. Inference Step: Determining Consequences: We now consider the rules that are on or match: 1st Rule: If error is Z and change-in-error is Z Then force is Z µ1 = min {1,0.25} = 0.25 so that we are 0.25 certain that this rule applies to the current situation. Fig.10 Implied fuzzy set with membership function μ1(f) for Rule 1 Hany Selim 39

40 2nd Rule: If error is Z and change-in-error is PS Then force is S µ2 = min {1,0.75} = 0.75 Fig. 11 Implied fuzzy set with membership function μ2(f) for Rule 2 40

41 VI. Converting Decisions into Actions (Defuzzification) ext, we consider the defuzzification operation, which is the final component of the fuzzy controller. Defuzzification operates on the implied fuzzy sets produced by the inference mechanism and combines their effects to provide the most certain controller output (plant input). In other words we want to find the one output, which we denote by Fcrisp,that best represents the conclusions of the fuzzy controller that are represented with the implied fuzzy sets. 41

42 Fig.12 Implied fuzzy sets 42

43 In fig.8 it was stated that the Force F cannot be saturated, otherwise we would have got an output membership functions that have infinite area. ow we use the center of gravity (COG) or Centroid defuzzification method for combining the recommendations represented by the implied fuzzy sets from all the rules. Using simple geometry, we get for the area under a triangle chopped off at a height of h and with base width w: A = w(h h2/2) and in our case Fcrisp =[ (0).(20( (0.25)2) + (-10).(20( (0.75)2)]/[(20( (0.25)2) + (20( (0.75)2) Fcrisp = /[ ] =

44 It is interesting to note that the method of Center of gravity limits the range of force between -20 and+20 (see fig. 13). Practically speaking, this ability to limit the range of inputs to the plant is useful; it may be the case that applying a force of greater than 20 is impossible for this plant. Fig. 13 Output membership functions 44

45 Summing up: Fig.14 Graphical representation of fuzzy controller operations Hany Selim 45

46 Fuzzy Systems Are Universal Approximators Fuzzy systems have very strong functional capabilities. That is, if properly constructed, they can perform very complex operations. There always exists a way to define the fuzzy system f(u) by picking the membership function parameters so that the achieved error is arbitrarily small. But this does not say how to find the fuzzy system. Furthermore, for arbitrary accuracy you may need an arbitrarily large number of rules. 46

47 For control, practically speaking, it means that there is great flexibility in tuning the nonlinear function implemented by the fuzzy controller. Generally, however, there are no guarantees that you will be able to meet your stability and performance specifications by properly tuning a given fuzzy controller. You also have to choose the appropriate controller inputs and outputs, and there will be fundamental limitations imposed by the plant that may prohibit achieving certain control objectives no matter how you tune the fuzzy controller. 47

48 VII. Simulation results for the inverted pendulum consider the initial conditions: e(0) = 0.1 radians (= 5.73 ) and, e (0) = 0, and the initial condition for the actuator state F(t) = 0. The results are shown in Fig.15, where we see in the upper plot that the output appropriately moves toward the inverted position, and the force input in the lower plot that moves back and forth to achieve this. fig.15 First design of the Fuzzy Controller 48

49 To speed up the settling of the pendulum we may use standard ideas from control engineering to conclude that we ought to try to tune the derivative gain. To do this we introduce gains on the proportional and derivative terms, as shown in Fig.16. fig.16 Fuzzy controller for inverted pendulum with scaling gains 49

50 (a) fig.17 Fuzzy controller balancing with (b) (a) g0=1, g1=0.1 and h=1 (b) g0=2, g1=0.1 and h=1 50

51 fig.18 Fuzzy controller balancing with g0=2, g1=0.1 and h=5 51

52 We see that the change in the scaling gains at the input and output of the fuzzy controller can have a significant impact on the performance of the resulting fuzzy control system, and hence they are often convenient parameters for tuning. The effect of the input scaling gain g1=0.1 results in scaling the horizontal axis of the membership functions by 1/g1=10 as shown in fig.19. fig.19 effect of g1=0.1 on the membership function 52

53 Tuning of Membership Functions It is important to realize that the scaling gains are not the only parameters that can be tuned to improve the performance of the fuzzy control system. Indeed, sometimes it is the case that for a given rule-base and membership functions you cannot achieve the desired performance by tuning only the scaling gains. Often, what is needed is a more careful consideration of how to specify additional rules or better membership functions. The problem with this is that there are often too many parameters to tune (e.g., membership function shapes, positioning, and number and type of rules) and often there is not a clear connection between the design objectives (e.g., better rise-time) and a rationale and method that should be used to tune these parameters. 53

54 Output Membership Function Tuning In the following one of the methods that are very useful for the tuning by real implementations of fuzzy control systems will be examined: In fig.13 the centers of the membership functions were at -20, -10, 0, 10, 20 that is Center = 10. i and if we scale by h then Center = 10. h. i We see that a linear relationship in the equation produces a linear (uniform) spacing of the membership functions. Suppose that we instead choose: Center = 5. h. sign(i ). i2 or Center = 5. h. i3 54

55 Both equations will have the effect of making the output membership function centers near the origin be more closely spaced than the membership functions farther out on the horizontal axis. The effect of this is to make the gain of the fuzzy controller smaller when the signals are small and larger as the signals grow larger. Hence, the use of the equations above for the centers indicates that if the error and change-in-error for the pendulum are near where they should be, then do not make the force input to the plant too big, but if the error and change-in-error are large, then the force input should be much bigger so that it quickly returns the pendulum to near the balanced position. 55

56 Ultimately, the goal of tuning is to shape the nonlinearity that is implemented by the fuzzy controller. This nonlinearity, sometimes called the control surface, is affected by all the main fuzzy controller parameters. Fig.20 Control surface of the fuzzy controller for g0 = 2.0, g1 = 0.1, and h = 5. Hany Selim 56

57 When we tune the fuzzy controller, it changes the shape of the control surface, which in turn affects the behavior of the closed-loop control system. ote, that the slope of the surface is greater for larger signals in Fig.21 than in Fig.20. This further illustrates the effect of the choice of the nonlinear spacing for the output membership function centers. Fig.21 Control surface of the fuzzy controller for g0 = 2.0, g1 = 0.1, and h = 10 and center = 5.h.sign(i ).i2 Hany Selim 57

58 Basic Design Guidelines 1.Be careful to choose the proper inputs to the fuzzy controller. Carefully assess whether you need proportional, integral, and derivative inputs (using standard control engineering ideas). Consider processing plant data into a form that you believe would be most useful for you to control the system if you were actually a human-in-the-loop. Specify your best guess at as simple a fuzzy controller as possible (do not add inputs, rules, or membership functions until you know you need them). 2.Try tuning the fuzzy controller using the scaling gains, as we discussed in the previous section. 3.Try adding or modifying rules and membership functions so that you more accurately characterize the best way to control the plant (this can sometimes require significant insight into the physics of the plant). 58

59 4. Try to incorporate higher-level ideas about how best to control the plant. For instance, try to shape the nonlinear control surface using a nonlinear function of the linguistic-numeric values, as explained in the previous section. 5. If there is unsmooth or chattering behavior, you may have a gain set too high on an input to the fuzzy controller (or perhaps the output gain is too high). Setting the input gain too high makes it so that the membership functions saturate for very low values, which can result in oscillations (i.e., limit cycles). 6. Sometimes the addition of more membership functions and rules can help. These can provide for a finer (or higher-granularity ) control, which can sometimes reduce chattering or oscillations. 59

60 Example: Air Conditioner Fuzzy Control 1a. Specify the problem: Air-conditioning involves the delivery of air which can be warmed or cooled and have its humidity raised or lowered. An air-conditioner is an apparatus for controlling, especially lowering, the temperature and humidity of an enclosed space. An air-conditioner typically has a fan which blows/cools/circulates fresh air and has cooler and the cooler is under thermostatic control. Generally, the amount of air being compressed is proportional to the ambient temperature. 60

61 1b. Define linguistic variables Consider an air conditioner which has five control switches: COLD, COOL, PLEASAT, WARM and HOT. The corresponding speeds of the motor controlling the fan on the air-conditioner has the graduations: MIIMAL, SLOW, MEDIUM, FAST and BLAST. Thus we have: Ambient Temperature Air-conditioner Fan Speed 61

62 2. Determine Fuzzy Sets: Temperature Temp (0C). 0< (T)<1 (T)=0 COLD COOL PLEASAT WARM HOT 0 Y* 5 Y Y 10 Y 12.5 Y* 15 Y 17.5 Y* 20 Y 22.5 Y* 25 Y 27.5 Y 30 Y* (T)=1 62

63 and corresponding fuzzy membership functions Temperature Fuzzy Sets Sets Temperature Fuzzy 1 1 Cold Cold Cool Cool Pleasent Pleasent Warm Warm Hot Hot Truth Value Truth Value Temperature Temperature Degrees Degrees CC 63

64 Fuzzy Sets: Fan Speed Rev/sec (RPM) MIIMAL SLOW MEDIUM FAST BLAST 0 Y* 10 Y 20 Y Y 30 Y* 40 Y 50 Y* 60 Y 70 Y* 80 Y Y 90 Y 100 Y* 64

65 and corresponding fuzzy membership functions Speed Fuzzy Sets Truth Value MIIMAL SLOW MEDIUM FAST BLAST Speed 65

66 3. construct fuzzy rules RULE 1: RULE 2: RULE 3: RULE 4: RULE 5: IF IF IF IF IF temp is temp is temp is temp is temp is cold THE cool THE pleasant THE warm THE hot THE speed is minimal speed is slow speed is medium speed is fast speed is blast 66

67 4. Encode into an Expert System... Consider now a temperature of 16oC, use the system to compute the optimal fan speed. Operation of a Fuzzy Expert System Fuzzification Inference Composition Defuzzification 67

68 Fuzzification Affected fuzzy sets: COOL and PLEASAT COOL(T) = T / PLST(T) = T /2.5-6 = 16 /2.5-6 = 0.4 = 16 / = 0.3 Temp=16 COLD COOL PLEASAT 0.4 WARM HOT

69 Inference RULE 1: RULE 2: RULE 3: RULE 4: RULE 5: IF IF IF IF IF temp is temp is temp is temp is temp is cold THE cool THE pleasant THE warm THE hot THE speed is minimal speed is slow speed is medium speed is fast speed is blast 69

70 Inference RULE 2: IF temp is cool (0.3) THE speed is slow (0.3) RULE 3: IF temp is pleasant (0.4) THE speed is medium (0.4) 70

71 Composition speed is slow (0.3) + speed is medium (0.4) 71

72 Composition speed is slow (0.3) + speed is medium (0.4) 72

73 Defuzzification COG = 0.125(12.5) (15) + 0.3( ) + 0.4( ) (57.5) (11) + 0.4(5) = 45.54rpm 73

74 Another Fuzzy ControlExample We want to conduct a simulation of the final descent and landing approach of an aircraft. The desired profile is shown in Fig. The two state variables for this simulation will be the height above ground, h, and the vertical velocity of the aircraft, v. The control output will be a force that, when applied to the aircraft, will alter its height, h, and velocity, v. The differential control equations are derived as follows.. Mass m moving with velocity v has momentum p = mv. If a force f is applied over a time interval Δt, a change in velocity of Δv = f.δt/m will result. If we let Δt = 1.0 (s) and m = 1.0 (lb s2/ft), we obtain Δv = f (lb), or the change in velocity is proportional to the applied force. In difference notation we get vi+1 = vi + fi hi+1 = hi + vi Δt Hany Selim 74

75 Define the membership functions as: 75

76 Control force (output) membership function And given the Rules 76

77 If: Initial height, h0: 1000 ft Initial Vertical velocity, v0: 20 ft/s We calculate the defuzzified force and apply it for which: vi+1 = vi + fi hi+1 = hi + vi we then conduct a simulation for four cycles. Δt =1s ; after 77