elements 1. A rule-base (a set of If-Then rules), which contains a fuzzy logic quantification of the expert's linguistic description of how to achi leve good control 2. An inference mechanism (also called an "inference engine"or "fuzzy inference"module), which emulates the expert's decision making in interpreting and applying knowledge about how best to control the plant 3. A fuzzification interface, which converts controller inputs into information that the inference mechanism can easily use to activate and apply rules. 4. A defuzzification interface, which converts the conclusions of the inference mechanism into actual inputs for the rocess We introduce each of the components of the fuzzy controller for a simple problem of balancing an inverted pendulum on a cart, as shown in Figure 2.2. Here, y denotes the angle that the pendulum makes with the vertical(in radians), I is the half-pendulum length(in meters), and u is the force input that moves the cart(in Newtons). We will use r to denote 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) Figure 2.2 Inverted pendulum on a cart 2.2. 1 Choosing fuzzy Controller Inputs and outputs How do we choose fuzzy controller inputs and outputs Consider a human-in-the-loop whose responsibility is to control the pendulum, as shown in Figure 2.3. The fuzzy controller is to be designed to automate how a human expert who is successful at this task would control the system First, the expert tells us(the designers of the fuzzy controller) what information she or he will use as inputs to the decision-making process Suppose that for the inverted pendulum, the expert (this could be you! says that she or he will use e(t)=r(t)-y(t)and e(n as the variables on which to base decisions. Certainly, there are many other choices (e. g the integral of the error e could also be used) but this choice makes good intuitive sense Next, we must identify the controlled variable. For the inverted pendulum, we are allowed to control only the force that moves the cart, so the choice here is simpleelements: 1. A rule-base (a set of If-Then rules), which contains a fuzzy logic quantification of the expert's linguistic description of how to achieve good control. 2. An inference mechanism (also called an "inference engine" or "fuzzy inference" module), which emulates the expert's decision making in interpreting and applying knowledge about how best to control the plant. 3. A fuzzification interface, which converts controller inputs into information that the inference mechanism can easily use to activate and apply rules. 4. A defuzzification interface, which converts the conclusions of the inference mechanism into actual inputs for the process. We introduce each of the components of the fuzzy controller for a simple problem of balancing an inverted pendulum on a cart, as shown in Figure 2.2. Here, y denotes the angle that the pendulum makes with the vertical (in radians), l is the half-pendulum length (in meters), and u is the force input that moves the cart (in Newtons). We will use r to denote 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). Figure 2.2 Inverted pendulum on a cart 2.2.1 Choosing Fuzzy Controller Inputs and Outputs How do we choose fuzzy controller inputs and outputs ? Consider a human-in-the-loop whose responsibility is to control the pendulum, as shown in Figure 2.3. The fuzzy controller is to be designed to automate how a human expert who is successful at this task would control the system. First, the expert tells us (the designers of the fuzzy controller) what information she or he will use as inputs to the decision-making process. Suppose that for the inverted pendulum, the expert (this could be you!) says that she or he will use et rt yt () () () = − and ( ) d e t dt as the variables on which to base decisions. Certainly, there are many other choices (e.g., the integral of the error e could also be used) but this choice makes good intuitive sense. Next, we must identify the controlled variable. For the inverted pendulum, we are allowed to control only the force that moves the cart, so the choice here is simple