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**Contents:**

In Mamdani fuzzy systems, the consequent of each rule is a fuzzy set. This is in contrast to Takagi—Sugeno T—S fuzzy systems see Chapter 6 , whose consequents are mathematical expressions. The fuzzification section converts crisp inputs into fuzzy sets. There are several strategies for fuzzification, but the one most commonly used, and the only strategy used in this book, is singleton fuzzification. This assumes the measured input is the true input i. Thus in fuzzification the input fuzzy sets are evaluated exactly at the measured inputs. The inference mechanism determines the extent to which each rule in the rule base applies in the present situation, and forms a corresponding implied fuzzy set for each rule.

If the rule premises contain a conjunction of several inputs, the degree of firing of each rule is calculated by taking a T-norm of the individual members of the conjunction. The inference mechanism also calculates an implied fuzzy set for each rule. Also discussed in this chapter are the input—output characteristic of the fuzzy system, and singleton fuzzy sets. Singleton fuzzy sets are often used on the output universe of discourse to simplify calculations. Singleton output fuzzy sets obviate the need for COG defuzzification, which usually gives comparable results and is much more computationally expensive to implement.

Use the Wind Chill fuzzy system of Section 3.

Use minimum T-norm and center of gravity defuzzification. For this wind speed, three fuzzy sets on the S universe are nonzero. Repeat the Wind Chill problem of Section 3.

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Use the triangular fuzzy sets of Figure 3. Use the same T-norm and defuzzification as Problem 3. Specify another shape for membership functions that is similar to triangular memberships i. Consider a fuzzy system for braking a car. Specify triangular memberships forming partitions of unity for all the above linguistic values. Give a complete rule base for a fuzzy system for braking a car. Use the fuzzy sets you defined in Problem 3. In the Wind Chill fuzzy system of Section 3. Higher relative humidity causes higher wind chill.

Prove that an n-input, m-output fuzzy system is equivalent to m n-input, single-output fuzzy systems. The output or states of the system are measured and fed back to the controller. On the basis of this information, the controller decides how to change the system input in order to improve the system performance. In some cases, however, these methods fail because a sufficiently accurate mathematical model of the system is not known.

In fact, one of the main uses for fuzzy systems is in closed-loop control of nonlinear systems whose mathematical models are unknown or poorly known. This kind of controller needs no mathematical model of the system, hence it is not model based. Its design is rather based on expert knowledge. Most plants to be controlled are continuous-time, therefore their inputs and outputs are piecewise-continuous functions of time.

If the control objective is tracking, the controller configuration usually involves unity feedback with a cascade compensator, as in Figure 1.

When the compensator is a fuzzy system, the configuration of Figure 4. The fuzzy compensator takes e t as its input and formulates a plant input to force the plant output to follow the reference input r t. In this chapter, the compensator will be designed from only expert knowledge about how to control the system. Therefore, no plant models will be needed. Because the compensator is a fuzzy system, it is virtually always implemented with a digital computer.

Therefore, time must be discretized with an appropriate sampling time depending on the speed of the analog signals. The input to the fuzzy compensator is a sampled version of e t , resulting in an output of the fuzzy compensator occurring at every sample time. This characteristic does not change with time, therefore the fuzzy compensator implements a time-invariant mapping from e t to u t with e t [hence u t ] changing at every time step.

Ball and Beam Plant Consider the ball and beam plant of Section 1. The system is depicted in Figure 4. The input to the ball and beam system is the voltage v supplied to the motor. In general, determination of the information necessary to accomplish a given control task is not trivial, but for many problems the determination can be made using common sense. These are characterized by the triangular memberships shown in Figures 4. Triangular fuzzy sets on e universe.

These are characterized by the memberships shown in Figure 4. A good choice of input and output fuzzy sets is crucial to the success of any fuzzy controller. The locations of the fuzzy sets in Figure 4. The fuzzy sets in Figure 4. The singleton fuzzy sets in Figure 4. Recall that in Chapter 3 we saw that using singleton output fuzzy sets gave comparable results to triangular, Gaussian, or other types of output fuzzy sets under certain conditions.

For some applications, especially classification [20], it may be necessary to be very meticulous about shaping output membership functions. However, for control applications, it is usually sufficient to use singleton output fuzzy sets because the controller can be otherwise adjusted by adjusting scaling gains or other parameters. Using singleton output memberships simplifies defuzzification as well, since no areas under memberships of implied fuzzy sets need to be calculated. Efficiency of calculation is crucial for control with a digital computer due to short sampling times available for the fuzzy controller to perform its calculations.

For these reasons, we will use only product T-norm and singleton output fuzzy sets in the remainder of this book. The rules should reflect our common sense about how to balance the ball motionless at the center of the beam. Thus, for instance, if the ball is in the position depicted in Figure 4. This is an example of heuristic expert knowledge; it is based on our past experience and common sense, not on any mathematical model of the ball and beam. Below we address several scenarios to indicate how the rule base is constructed. In this case, because the ball is moving quickly to the right away from the center of the beam, v should be large and negative i.

In this case, because the ball is moving quickly to the left toward the center of the beam, no rotation of the beam is necessary. Therefore, v should be zero i. The parameter e is NL and c is PL. Therefore, v should be a small negative amount i. The above three scenarios reflect our common sense i. In this way, we can write all 25 rules in the rule base: These are given in tabular form Table 4. This situation corresponds to the ball being slightly to the right of center and traveling rapidly to the left. Referring to Figure 4. Similarly, referring to Figure 4. Ball and Beam Plant Since the output memberships are singletons see Section 3.

Note that the denominator in 4. These gains are used to tune the compensator to achieve better performance. The closed-loop system is shown in Figure 4. The ball moves toward zero, however, the response is very oscillatory. The fuzzy controller is doing its job, but must be tuned to quicken the response and get rid of the oscillations. This can be done by adjusting the gains g0, g1, and h. In some cases, the fuzzy controller originally designed with all scaling gains equaling unity may not produce a stable closed-loop system, even though the controller is correctly designed according to our best judgment.

In such cases, the problem is likely that the universes of discourse are incorrectly sized for the problem. Then it may be necessary to adjust the scaling gains initially to produce a stable closed-loop system before the gains are further adjusted to improve the response. This reduces the oscillations significantly, but the ball takes 6 s to settle motionless at the center of the beam, which is too slow.

The resulting response is shown in Figure 4. The time taken by the ball to reach the center of the beam is significantly decreased, however, the ball overshoots its target. When the output scaling gain h is increased to 7 while keeping g0 and g1 unchanged, the response of Figure 4. If a sufficiently powerful supply is not available, it may be necessary to accept a slower response. The control effort necessary to accomplish a control task is always a concern in the implementation of every practical control system.

The beam angle producing the response of Figure 4. This may be a problem: Such a quick movement may throw the ball off the beam! Note that, similar to triangular membership functions, adjacent Gaussians may cross wherever desired. The spreads of the Gaussians of Figures 4. Gaussian fuzzy sets on e universe. The rule base of Section 4. However, the inference calculation is different due to the different membership function shapes. Specifically, we have the following: However, the degrees of firing of most rules are very small.

To avoid calculating the degrees of firing for rules that are minimally fired, let us consider a quantity to be in a fuzzy set only if its degree of membership in that set is at least 0. That is, we consider a quantity to be in a fuzzy set only if it is in the 0. Once again, rules 9, 10, 14, and 15 are on, and the rest are not fired.

In fact, the closed-loop behavior of the plant with Gaussian controller is very close to that of the plant with triangular controller. This is not surprising, because the shape of Gaussian memberships is similar to that of triangular memberships. The input—output characteristic of the fuzzy controller with triangular memberships is shown in Figure 4. The two characteristics are essentially the same, except for the local waves and undulations in the characteristic of Figure 4.

These were discussed in Section 3. These local waves are due to the curvature of the Gaussians; they have nothing to do with the controller itself. Input—output characteristic of fuzzy controller with triangular input memberships, product T-norm, and singleton output fuzzy sets.

Input—output characteristic of fuzzy controller with Gaussian input memberships, product T-norm, and singleton output fuzzy sets. This makes it possible to derive a fuzzy controller from a PID controller. It is very useful to be able to do this because designing a fuzzy controller can be initially difficult as was seen in Sections 4. However, there exist well-known methods of designing PID controllers e. Even when these methods are not used, an expert can often manually adjust a PID controller to achieve stable performance.

Similarly, an effective strategy if the response is too oscillatory is often to increase the derivative gain. If the steady-state error is too large, it can be reduced by adjusting the integral gain. Using these and similar rules of thumb, an effective PID controller can be designed for a plant relatively easily see Section 4. If this PID controller can then be converted into a fuzzy controller, it may be easier to adjust the fuzzy controller in a nonlinear manner to enhance robustness. Assume that we have a nonfuzzy PD controller that gives satisfactory closed-loop behavior.

In order for the equivalent fuzzy controller to be exactly equal to the nonfuzzy PD, it is necessary that the system trajectory always remain within the linear part of the characteristic. Therefore, with the nonfuzzy PD in operation, measure the maximum control effort necessary to accomplish the control task. This is the maximum absolute value of u t that is output by the PD in controlling the plant.

After the equivalent fuzzy controller has been constructed, it can be easily altered to improve performance. For simulation purposes, its mathematical model is given by 2 9. Control effort [output of nonfuzzy PD controller 4. The performance of the inverted pendulum in closed loop with this fuzzy PD compensator is identical to Figures 4. If the cart is bumped with an impulsive force of 50 N, this disturbance is enough to cause the rod to fall with this controller. Therefore, let the new output singletons be as in Figure 4. The input—output characteristic of the redesigned nonlinear fuzzy PD controller is shown in Figure 4.

The response of the inverted pendulum under closed-loop control with the redesigned nonlinear fuzzy controller is shown in Figure 4. In this figure, we see that now the cart catches the rod before it falls, unlike Figure 4. Note that the fuzzy PD was easily redesigned using expert knowledge to handle the impulsive disturbance. The original nonfuzzy PD 4. The reader is reminded that all designs of fuzzy controllers done in this chapter were done using only expert knowledge, that is, our common sense and experience with the processes, not any control theory or knowledge of mathematical models of the plants, which were needed only for purposes of simulation.

So far, we have considered only position form control laws. Position form control laws prescribe what the control u t should be i. In incremental form control laws, the change in plant input, rather than the input itself, is prescribed by the controller. To formulate an incremental PID control law, we need [from 4. First, a nonfuzzy PD incremental controller is designed in an ad hoc manner using basic knowledge of PID controllers increasing proportional gain quickens the response, increasing derivative gain decreases oscillations, etc.

Fuzzy sets on d k universe for fuzzy incremental PD controller for ball and beam plant. The rule base of the fuzzy incremental PD controller is given in tabulated form in Table 4. We consider the most common interconnection for tracking: For the ball and beam controller Sections 4. These are based on our best common sense about the system, but scaling gains are included in the design to improve performance.

The equivalent fuzzy controller is exactly equal to the PID within its effective universe. This technique is useful because nonfuzzy PID controllers can be readily designed using well-known techniques. Once the equivalent fuzzy controller has been obtained, it can be easily adjusted in a nonlinear manner using heuristic knowledge to improve performance. In this example, a nonfuzzy PD controller for an inverted pendulum plant is converted to a fuzzy controller. This fuzzy controller is then adjusted in a nonlinear manner to increase robustness of the system to impulsive disturbances.

This controller approximately duplicates the closed-loop performance of the position-form controller derived in Section 4. Design a fuzzy controller for the ball and beam similar to the one in Section 4.

Derive a fuzzy controller equivalent to the PID controller 4. The maximum control effort from the PID is 3. Using simulation, find a fuzzy incremental controller that balances the rod for the inverted pendulum Hint: Initially designing a nonfuzzy incremental PD, then converting to fuzzy may help. Consider the robotic link of Section 1. This is one of the strengths of fuzzy control. However, more advanced fuzzy control methods, such as some types of parallel distributed compensation and fuzzy adaptive control, as well as fuzzy system identification, do require at least an assumption of some particular structure of the model.

Some methods assume continuous-time linear or nonlinear state-space model structures while others assume discrete-time state space or input—output difference equation model structures. We emphasize that a particular dynamic system can be modeled with any of these model structures, as will be demonstrated below. The particular structure used depends on the one required by the control or identification method. Therefore, we give a brief summary of several well-known model structures for dynamic systems. Of these, only the linear or nonlinear timeinvariant state-space descriptions are useful for fuzzy identification and control.

Now we give brief summaries of these model structures. This means that it can be linearized by an appropriately designed feedback law. Under certain assumptions, by differentiating the output it is possible to transform 5. The integer m is known as the relative degree of the system. It has a massless shaft of length L with a mass M concentrated at the end, and a coefficient of friction B at the attach point. This is a version of the motor-driven robotic link of Section 1. B L M Figure 5. This is not in the form of 5. Then, the model can be expressed in the form of 5. The resulting pendulum angle is shown in Figure 5.

If such a system is to be controlled by connecting it to a digital computer, which is virtually always the case with fuzzy control, the control input must eventually be represented in discretetime form, necessitating some type of method that approximates continuous-time signals by discrete-time signals. This could be done by calculating the continuoustime control signal from a continuous-time model in the form of 5.

In the former case, the approximation error is the result of discretizing a continuous-time control signal. In the latter case, the error is the result of approximating a continuous-time system by a discrete-time system. In this section, we give two forms of discrete-time models for linear systems. This is known as an autoregressive moving average ARMA model.

It is also possible to convert to linear discrete-time state-space form directly from linear continuous-time state space form 5. The behavior of the linearized discrete-time system is nearly identical to that of the original nonlinear system. The behavior of the system 5. It should be emphasized that the models 5. Each is inaccurate in its own way. The original nonlinear model 5. This is due to several reasons: This is not true for an actual pendulum. In an actual pendulum, the shaft is flexible to some extent.

This kind of controller needs no mathematical model of the system, hence it is not model based. Your display name should be at least 2 characters long. The reader is reminded that all designs of fuzzy controllers done in this chapter were done using only expert knowledge, that is, our common sense and experience with the processes, not any control theory or knowledge of mathematical models of the plants, which were needed only for purposes of simulation. Note that the denominator in 4. The initial parameter values are chosen randomly within the range of the training data. Of these, only the linear or nonlinear timeinvariant state-space descriptions are useful for fuzzy identification and control. The collection of rules as a whole is called a Rule base.

In reality, they are only approximately known, and B may change over time. If the difficulties 1—4 listed above were not present, that is, if an absolutely accurate model were used to describe the pendulum, it would be impossibly complex. This gives support for using the various models above, since all mathematical models are inaccurate to some extent. Below we give the methods that will be used in later chapters in conjunction with T—S fuzzy systems. The matrix C is called the controllability matrix. Thus the feedback control law that places the closed-loop eigenvalues at 0.

Then it has an input—output ARMA model as in 5. The resulting closed-loop system has bounded inputs and outputs provided that the poles of the inverse model i. This design procedure applies to n-step-ahead tracking problems with obvious modifications. The system can be rewritten as 5. Also, we are assured the tracking objective can be accomplished with a bounded control because the poles of the inverse model [i. The control law 5. The following development is a generalization of the tracking controller above. Let the plant be described by the input—output ARMA model 5.

A similar development applies for systems with delays greater than 1. This system is in the form of 5. Also, we are assured that the tracking objective can be accomplished with a bounded control because the poles of the inverse model i. Thus the nonlinear system has been linearized by the feedback 5. Then the closed-loop system is linear with transfer function: The block diagram of the closed-loop system is shown in Figure 5. All of these will be used in various fuzzy control strategies implemented with T—S fuzzy systems in the following chapters.

The presentation is brief, that is, some familiarity with controls is assumed. The model forms considered for continuous-time systems are the continuoustime nonlinear state equations 5. The model forms considered for discrete-time systems are the discrete-time linear state equations 5. Continuous-time nonlinear state equations are not used directly for fuzzy control, but are used to derive models in feedback-linearizeable form.

Feedbacklinearizable model forms are used in indirect adaptive control schemes for adaptive model following Section 5. Continuous- and discrete-time linear state equation model forms are used in parallel distributed compensators for pole placement Section 5. Input—output difference equation models are used in tracking and model reference controllers Sections 5. Put the inverted pendulum of Example 4. Consider the linear continuous-time state-space system 5. Consider the linear discrete-time system 5. Consider the system described by the linear continuous-time state-space equations given in Problem 5.

Find a linear state feedback law 5. Use the method of Example 5. For the system of Problem 5. In fact, it can be shown that Mamdani systems are special cases of T—S fuzzy systems. The T—S systems are important because they enable a kind of control called parallel distributed control, they facilitate fuzzy identification of dynamic systems and adaptive fuzzy control, and they enable stability proofs for certain closed-loop systems involving fuzzy controllers.

Their drawback is that they are less intuitive than Mamdani systems. In T—S systems, the consequents of the rules do not involve fuzzy sets as do Mamdani systems, but instead are mathematical expressions. The mathematical expressions can be any linear functions of any variables. In the former case, the T—S fuzzy system performs an interpolation between memoryless functions. In the latter case, the T—S system performs an interpolation between dynamic systems. The latter case is useful for fuzzy identification and control.

The output ycrisp is a nonlinear function of the inputs x1, … , xn. Inside the effective universe of discourse the fuzzy sets P11, P12 , P21, and P22 are characterized by the following membership functions: Given the crisp input x1, x2 , the crisp output of this system is [see 6.

These values yield the following fuzzy basis function values: Characteristic surface of the T—S fuzzy system with individual consequents plotted at appropriate corners. The input universes X1, X2, … , Xn are as before, with various fuzzy sets defined on them, but now the consequents are continuoustime linear constant-coefficient dynamic systems involving the states.

This fuzzy system does not have a single crisp output, rather it produces a time-varying system described by the differential equation: Matrices A t and b t contain functions of the system states, making the system 6. For an arbitrary input x1, x2 inside the effective universe of discourse i. These yield the following basis function values: As t changes, A t and b t change. The nonlinear nature of the system can be seen in this response. The Matlab simulation program producing Figure 6. Consider a T—S fuzzy system with R rules of the form: The input universes X1, X2, … , Xn are as before, with various fuzzy sets defined on them, but now the consequents are discrete-time linear constant-coefficient state-space systems.

This fuzzy system does not have a single crisp output, rather it produces a time-varying system described by the state-space difference equation: Therefore, the fuzzy basis functions and the matrices A k and b k vary with time. Matrices A k and b k contain functions of the system states, making the system 6.

As k changes, A k and b k change. Input and corresponding states for discrete-time system created by a four-rule T—S fuzzy system. This form of system model is particularly useful when system identification techniques are being used because an unknown system is often identified in the form of a discrete-time input—output difference equation rather than state-space equations as above. If statespace equations are desired e. This way of describing a discretetime system is commonly used in system identification techniques. The Matlab program producing Figure 6.

Input u t and output y t 1 0. Output of T—S fuzzy system that interpolates four linear constant-coefficient discrete-time dynamic systems to produce a nonlinear discrete-time dynamic system. The consequents of T—S systems are general mathematical functions rather than fuzzy sets. T—S fuzzy systems make identification of nonlinear systems Chapter 9 possible, as well as adaptive fuzzy control Chapter They also enable stability proofs for closed-loop systems controlled by T—S fuzzy controllers Chapter 7.

The three types of T—S fuzzy systems considered in this book are those with consequents that are 1 affine functions, 2 normal form state-space models, or 3 input—output difference equation models. If the consequents are affine functions, the T—S fuzzy system realizes a static nonlinear function, as in Example 6. It was seen that this type of fuzzy system interpolates the hyperplanes in the individual consequents inside the effective universe of discourse.

This type of fuzzy system will be utilized in Chapters 9 and 10 to identify nonlinear plants in the form of 5. If the consequents are normal form state-space models, the T—S fuzzy system realizes a nonlinear time-varying state-space system, as in Sections 6. This type of fuzzy system will be utilized in Chapter 7 to design parallel distributed state-feedback controllers for nonlinear systems. This type of fuzzy system is utilized in Chapter 7 for parallel distributed tracking and model reference control, and again in Chapter 10 for adaptive versions of these.

Plot the characteristic input—output surface of the system of Problem 6. Consider the T—S fuzzy system with rule base 1. Let the input fuzzy sets be characterized by the memberships in Figure 6. If x1 is 2. If x1 is 3. If x1 is 4. If x1 is 5. If x1 is 6. If x1 is 7. If x1 is 8. If x1 is 9. The basic philosophy of PDC is to create a controller fuzzy system with rule premises identical to those of the plant fuzzy system.

The overall control law is thus a weighted average of the individual control laws, just as the overall nonlinear system is a weighted average of the linear systems in each consequent of the plant fuzzy system. Parallel distributed control is important because it constitutes a method of controlling nonlinear systems. Furthermore, as will be seen in Chapter 10, a nonlinear system can sometimes be identified online as a T—S fuzzy system.

If a parallel distributed controller can then be designed based on this identification, a type of real-time control known as adaptive control can be effected for nonlinear systems. The control methods utilized in this book in parallel distributed control schemes are the ones contained in Chapter 5, that is pole placement, tracking, and model reference.

Pole placement via parallel distributed control has the distinct advantage that there exist mathematical tools to prove closed-loop stability, as discussed in Theorems 7. Such tools are lacking for control via Mamdani fuzzy systems. Not every PDC controller produces a stable closed-loop system, despite the fact that each consequent system in the plant fuzzy system is stabilized by the control law in the corresponding consequent of the controller fuzzy system. Nevertheless, we have the following stability result for pole placement via parallel distributed control. Therefore, this T—S fuzzy system is nonlinear and unstable.

The parallel distributed controller is given by 1. Let the fuzzy sets P11 , P12 , P21 , and P22 be characterized by the membership functions shown in Figures 6. The trajectories increase unboundedly because the open-loop system is unstable. With the same initial conditions and parallel distributed pole placement feedback control law 7. The Matlab program producing Figures 7.

The stability result for discrete-time parallel distributed control is as follows. Therefore, this T—S fuzzy system is nonlinear and unstable, because the eigenvalues of A1, A2, and A4 have magnitudes that are not less than unity. A parallel distributed-type controller can be designed to force the output of a nonlinear system modeled as a T—S fuzzy system to track a desired reference signal.

The controller design is explained in Section 5. The parallel distributed tracking strategy is to apply this design procedure to every consequent plant in the plant fuzzy system. For tracking, the consequents in the plant fuzzy system are assumed to be in the form of 5. The parallel distributed one-step-ahead tracking controller for this system is another fuzzy system with R rules of the form: Tracking performance y t and r t of parallel distributed one-step-ahead tracking controller.

The tracking would be perfect if each individual control law were applied to each individual plant. This is not the case, however, with parallel distributed-type controllers. Instead, we only have a weighted average of controllers controlling a weighted average of plants. Therefore, the overall control is only an approximation of the controller needed to effect perfect tracking for the overall plant, which is in fact nonlinear. The Matlab code that produced Figures 7. A parallel distributed-type controller can be designed to control a nonlinear plant modeled as a T—S fuzzy system to match a desired model.

For model reference control, the consequents in the plant fuzzy system are assumed to be in the form of 5. Let the plant fuzzy system rule base consist of R rules of the form 7. Assume, without loss of generality, that the reference model is given by 5. Then the parallel distributed model reference controller for this system is another fuzzy system with R rules of the form: Model matching performance y t and ym t of parallel distributed model reference controller. The model following is not perfect; the absolute value of the error between y t and ym t for this example always remains in the range of 0.

We call this the plant fuzzy system. The rules of the plant fuzzy system have consequents that are dynamic linear systems in continuous- or discrete-time normal state space form, or linear input—output difference equation form. The parallel distributed control PDC strategy is to construct a controller fuzzy system whose rule premises are identical to those of the plant fuzzy system, and whose consequents are controllers designed for the corresponding plant in the plant fuzzy system.

This type of PDC results in nonlinear state regulation. Regulation can be guaranteed if the feedback law satisfies a series of linear matrix inequalities LMI. Such guarantees are rare for closed-loop systems containing fuzzy controllers due to the complexity of fuzzy controllers. If the plant fuzzy system consists of plants described by input—output difference equations, the controller fuzzy system consequents can be either tracking control laws or model reference control laws.

Remember that in the case of parallel distributed output tracking or parallel distributed model reference control, there is EXERCISES no formal proof that the control objective i. Nevertheless, these two approaches will be shown in Chapter 10 to be very useful for adaptive fuzzy control. Prove that the controller designed in Problem 7.

the subjects of fuzzy identification and control, culminating in the creation of a graduate cipal uses for fuzzy logic: identification and control. This book was. This book gives an introduction to basic fuzzy logic and Mamdani and Takagi- Sugeno fuzzy systems. The text shows how these can be used to.

Simulate the closed-loop system designed in Problem 7. Use fourth-order Runge—Kutta integration with a step size of 0. Consider the system of Example 7. Design a parallel distributed controller that places the closed-loop poles at 0. Consider the following T—S system: The nonlinear characteristic function of these fuzzy systems is a result of the rules, T-norm, membership functions, and defuzzification method chosen by the designer. In these systems, the input and output membership functions are fixed a priori.

It is also possible, given a particular nonlinear function, to find a fuzzy system whose input—output characteristic matches it. In the fuzzy system, the output and in some cases input membership functions are not fixed a priori, but are adjusted so that the input—output characteristic most closely matches the nonlinear function in some sense.

The determination of a fuzzy system to approximate a given nonlinear function is done utilizing well-known results from estimation theory. The estimation methods introduced in this chapter are the least-squares and gradient approaches. These are introduced because they have direct applications to control. It will be seen in Chapter 9 that the ability to model static nonlinear functions as fuzzy systems makes it possible to model dynamic nonlinear systems as fuzzy systems.

The motivation for modeling a dynamic nonlinear system as a fuzzy system is that with such a model, the parallel distributed control methods of the previous chapter can be employed to control the system. This enables a very powerful type of control known as adaptive fuzzy control, introduced in Chapter With adaptive fuzzy control, tracking and model reference controllers can be designed for unknown nonlinear systems. We first introduce the batch least-squares approach, which finds the minimum based on a finite number of stored input—output measurements from the plant.

This process is then made recursive resulting in an algorithm, called recursive least squares, that updates the parameter estimate at each time step as new measurements are obtained from the plant. This is sometimes called training data. It is always a requirement on the regressor frequency content in all types of identification and estimation methods, fuzzy or not. Hence, in this case no matrix inversion is required, significantly reducing the computational load.

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This can be derived from the batch least-squares estimate 8. Since the output membership function locations bi are unknown, the fuzzy system 8. Assume that we do this with a four-rule adjustable Mamdani fuzzy system. The centers are chosen to cover the input universe of discourse [0 6] evenly. The spreads are chosen so that adjacent memberships cross at 0. The input memberships are shown in Figure 8. These are plotted in Figure 8. Since there are four rules, there are four unknown parameters i. We will estimate these with batch least squares.

The input portion of the training data should sufficiently cover the domain over which we desire a good estimate of g. Even with so few fuzzy sets and data points, the fit is still quite good. The input Gaussians Fig. This example illustrates the importance of choosing good input membership functions when using least squares.

The Matlab code producing the results for this example is found in the Appendix. Let us do this with an adjustable Mamdani fuzzy system with two inputs x1 and x2. This results in a 2 input, 1 output fuzzy system whose crisp output f x1, x2 closely matches g x1, x2. The Matlab program used for Example 8. The resulting fuzzy system can again be described by 8. Then the TRLS algorithm proceeds as in 8. Again, assume that we do this with an adjustable Mamdani fuzzy system with four rules, four input fuzzy sets characterized by Gaussian membership functions, and four adjustable output fuzzy sets characterized by singleton membership functions.

As above, we fix the input membership function centers and spreads.

Then the crisp output of the fuzzy system is given by 8. These final parameter values are not unique, however. They depend on the input x k , which is a random sequence. Therefore, the final parameter values will be different every time the algorithm is run. The fit is quite good; however it could be better if a more complex fuzzy system i. To summarize, the fuzzy system with the above characteristic that approximates g x is completely specified as follows: If x is A1, then f 2. If x is A2, then f 3. If x is A3, then f 4. If x is A4, then f where [b1 b2 is is is is b1.

The input fuzzy sets are shown in Fig. Input fuzzy sets fixed. This may be the case, for example, if we are unsure of what the input fuzzy sets should be. If input memberships are to be adjusted, leastsquares methods cannot be used because it is no longer possible to express the system as linear in the parameters. The gradient method [33,34] can be used to adapt both input and output fuzzy sets although other methods may be used as well [35]. The basic idea of the gradient method is to move the parameter estimate in a direction that is opposite that of the gradient or slope of the error surface in parameter space.

This insures the error always decreases with each new parameter update. One problem with the gradient method is that the error surface is not convex so it has many local minima. This means that the converged parameter vector depends on the initial parameter guesses and may not be the one that produces the absolute minimum error. Therefore, if the resulting fuzzy system does not approximate the nonlinear function to sufficient accuracy, it may be beneficial to try different initial parameter guesses.

Assume an n-input Mamdani fuzzy system with R rules, nR input fuzzy sets characterized by Gaussian membership functions with adjustable centers cij and 8. Let the rule base of the system be given by: This is in contrast to rule bases seen previously in this book, in which a relatively small number of fuzzy sets are defined on each input universe and a rule is written for every possible combination of these.

Arrangement of inputs and rules corresponding to rule base 8. Rule bases with rules in the form of 8. The form seen in 8. On the other hand, rule bases of the form 8. For the rule base of 8. In the previous notation, the superscript denoted a particular fuzzy set on a universe [see 3. In this new notation, the superscript denotes the rule number. Using product T-norm, the premise value of Rule i of rule base 8. In this way, it is insured that e k decreases or at worst does not increase at every time step. Note that in this book Gaussian membership functions are always used when gradient parameter update is used, because it is much more straightforward to take derivatives of Gaussians with respect to their parameters than it is to take derivatives of triangulars with respect to theirs.

The process is similar to calculation of weight updates via backpropagation for neural networks. Suppressing the time increment k for notational simplicity, we have from 8. The gradients in the update laws depend on the training data pairs x k , y k taken from the system. If several pairs of training data are available, which is usually the case, the above algorithm could be run over and over for the first pair of data until convergence occurs, then iterate again using the second pair of data with the converged parameters of the first set as initial conditions, and so on.

Alternatively, we could run through the entire set of training data once for each pair, then iterate over the entire set repeatedly until convergence is obtained. The latter method is used in this text for test data consisting of a finite number of pairs. In adaptive identifiers and controllers, the new data is fed to the algorithm as it occurs.

Let there be five rules and five Gaussian input fuzzy sets, one for each rule. Therefore, we have 15 parameters to adjust: The initial parameter values are chosen randomly within the range of the training data. Note that if we have some idea of the correct parameter values, we should use these as initial conditions for the gradient algorithm.

The characteristic of the converged fuzzy system after training steps i. The fit is not good after so few training steps. The evolution of the parameter estimates is shown in Figures 8. From these figures, it is evident that convergence has not occurred after training steps. If we perform training steps i. Output membership functions determined by gradient method. Thus, the rule base of the fuzzy system approximating g x in 8. This is due to the fact that the error function for the gradient method does not have one unique minimum, but rather many local minima.

For most reasonable initial parameter guesses and training data, the gradient algorithm will eventually converge to a fuzzy system whose input—output characteristic closely matches g x. Each fuzzy system is different, depending on the initial parameter guesses and training data used. A good fit to g x is not guaranteed, however.

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Let the rule base be given by 1. These are to be adjusted using the gradient method. We note from 8. Then, the parameter update laws for T—S fuzzy systems with affine consequents are given by 8. To do this, input—output data must be available from the function. Although there are many methods that can be used to create fuzzy systems that aproximate functions, we concentrate on two: This is because these methods are recursive, hence can be used for adaptive fuzzy control Chapter In the case of least-squares estimation whether batch or recursive , the fuzzy system must be linear in the unknown parameters.

Therefore, the input membership functions must be fixed a priori and only the output parameters adjusted.

Because the error function is quadratic, it has a unique minimum in parameter space. Therefore, the parameter estimates converge to unique values regardless of the initial guess. The recursive least-squares algorithm converges more quickly than some other methods, specifically the gradient algorithm. Because least squares is limited to adjusting only the output fuzzy sets, it is less flexible in constructing fuzzy systems to estimate nonlinear functions than methods that adjust both input and output fuzzy sets, specifically the gradient algorithm. This enables one to define intuitively a relatively small number of fuzzy sets on each input universe and write rules whose premises are various combinations of these.

This produces a rule base that is transparent to human understanding. In the case of gradient adaptation, there is no requirement for the fuzzy system to be linear in the parameters, so all input and output fuzzy sets can adjusted. Because of this, each rule must have its own separate input fuzzy sets that are not shared with any other rules. This necessitates a premise fuzzy set arrangement as in 8.

Thus gradient-adapted rules are not as easily understood as least-squares-adapted rules. Gradient estimation takes longer to converge and may not converge at all in a finite time interval. However, gradient estimation tends to be more flexible in its estimation capabilities, due to the fact that the input memberships are adapted along with the output parameters. Derive the recursive least-squares algorithm 8. Let the rules be 1. Assume a fuzzy system with rules of the form Ri: Write the gradient update equations for a0i and a1i.

The determination of a model for an unknown system is known as identification. Usually, the object of doing so is to utilize the model in a parallel distributed control scheme see Chapter 7. Design of a parallel distributed controller is relatively straightforward because the consequents in the fuzzy model are linear systems. This enables parallel distributed control of the nonlinear system when derivation of the controller directly from the nonlinear equations of motion may he impossible. If the controller is to be capable of adjusting itself in real time as in adaptive fuzzy control , the method used for identification must be recursive.

In Chapter 8, we give two methods of recursive identification: Therefore, we will concentrate on these methods in this chapter. The basic idea of modeling it with a T—S fuzzy system is to express it as a series of linear dynamic systems, each one the consequent of one of the rules. Functions zi are determined such that nonlinear terms in the plant model can be expressed as zi x, where x is a plant state.

Each of these functions zi becomes an input to the fuzzy system. Two fuzzy sets forming a partition of unity are defined on each zi universe, centered at the maximum and minimum of zi. Assume a scalar nonlinear function z x. These are shown in Figure 9. Then the nonlinear system is exactly modeled on X by the T—S fuzzy system: Now that we have a T—S fuzzy model of 9. Then the required controller fuzzy system has the rule base 1. However, it can be verified via simulation though this is not a proof! The input sequence u k should have sufficient frequency content to identify the plant.

In general, the more complex the plant, the more different frequencies should be contained in u. It is generally not known what types of inputs provide sufficient excitation for nonlinear systems, but inputs consisting of many frequencies have the best chance of success. If the plant is unknown, we can attempt to estimate it as an R-rule T—S fuzzy system with rules in the form: We note that 9.

The number of past inputs and outputs in the RHS regressions could be different from each other, and the delay of the model, which equals 1 in 9. Therefore, the fuzzy basis functions are also known. The system is simulated with a fourth-order Runge—Kutta integration algorithm with a step size of 0. The fuzzy sets on each input universe are shown in Figure 9.

Input fuzzy sets for T—S fuzzy system approximating motor-driven robotic link. To test the approximating T—S fuzzy system called model validation , its behavior is compared with that of the true robotic link for a variety of inputs. These inputs should be significantly different from the input used to obtain the training data. This shows good agreement between the actual link and the T—S fuzzy model. The agreement using a variety of other inputs is also good, therefore the T—S fuzzy system is deemed to approximate the motor-driven robotic link plant to sufficient accuracy. True and estimated outputs 1.

Although it is not the subject of this discussion, we point out that now that we have an accurate T—S model of the link, it is possible to use the methods of 5. The input function chosen for the identification is a Gaussian pseudorandom sequence with zero mean and a standard deviation of Thus there are three rules and six unknown parameters to be determined via RLS.

The three input fuzzy sets are chosen as in Figure 9. This rule base together with the input memberships in Figure 9. The output of the fuzzy approximator closely matches that of the true system for a variety of inputs, hence the approximator is deemed valid. Once again, we point out that with the above fuzzy approximation it is now possible to design tracking and model reference controllers for this plant. Therefore the input fuzzy sets, as well as the output parameters, can be adjusted. Let the fuzzy approximator have R rules of the form: The fuzzy system is described by 9.

By a process similar to that of Sections 8. The input signal chosen for the identification consists of six frequencies covering the operating frequency range of the gantry: Thus there are unknown parameters to be determined via gradient: After iterations through the gradient algorithm, the parameters have converged to constant values. The resulting rule T—S fuzzy system produces close agreement with the truth model 9. An example is shown in Figure 9.

Comparison of gantry output and gradient-trained T—S fuzzy model output for test input 9. Although signal differentiation is undesirable in practice, it is unavoidable in this case. For the relative degree, one needs some idea of the system model. In most cases, m is treated as a design parameter and determined by trial and error.

This chapter discusses two model forms for the estimated system: Introduction to Optimum Design. Advances in Robust Fractional Control. Operator-Based Nonlinear Control Systems. Control and Estimation Methods over Communication Networks. Diagnosis and Fault-Tolerant Control. Applied and Computational Control, Signals, and Circuits. Intelligent Big Multimedia Databases. Explaining Algorithms Using Metaphors. The Regularized Fast Hartley Transform. Speech Processing in Embedded Systems. Computer Aided Design and Design Automation. How to write a great review. The review must be at least 50 characters long.

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