A First Course in FUZZY and NEURAL CONTROL

2.7. PROPORTIONAL-INTEGRAL-DERIVATIVE CONTROL 71 integral gain so that the steady-state error is within a tolerable margin, typi ...

72 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL In standard international (SI) units, the armature constantKt,isequaltothe motor co ...

2.7. PROPORTIONAL-INTEGRAL-DERIVATIVE CONTROL 73 Design specifications Since the DC motor is being used as an actuator, we wish ...

74 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL Figure 2.40. Open-loop response The step response in Figure 2.41 is then obtained b ...

2.7. PROPORTIONAL-INTEGRAL-DERIVATIVE CONTROL 75 Gc(s)=KP+ KI s +KDs= KDs^2 +KPs+KI s The objective in the controller design is ...

76 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL Figure 2.44. System response Figure 2.44, where the system response is examined for ...

2.8. NONLINEAR CONTROL SYSTEMS 77 Figure 2.46. System response a satisfactory controller design. The basic steps in developing P ...

78 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL type controller is used in missile and spacecraft control systems. Other types of n ...

2.9. LINEARIZATION 79 higher thanfirst-order terms gives x ̇i(t)=fi[x 0 (t),u 0 (t)] + Xn j=1 ∂fi ∂xj Ø Ø Ø Ø Ø Ø x 0 ,u 0 (xj−x ...

80 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL Then Equation (2.46) gives x 1 (t)=−t+1 (2.49) Therefore, the nominal trajectory fo ...

2.10. EXERCISES AND PROJECTS 81 Each of the following is a linear time-invariant system. (i) ∑ x ̇ 1 x ̇ 2 ∏ = ∑ 10 22 ∏∑ x 1 ...

82 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL A system is described by the following differential equation d^2 x dt^2 +2 dx dt ...

2.10. EXERCISES AND PROJECTS 83 (b) If the inputu(t)is a unit step function, withx 1 (0) =x 2 (0) = 0,find y(t),t> 0. (c) If ...

84 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL Synthesize a set of PID parameters for different values ofαin the interval [0.1; 20 ...

Chapter 3 FUZZY LOGIC FOR CONTROL In this chapter, we set forth the basic mathematical ideas used in fuzzy control. These ideas ...

86 CHAPTER 3. FUZZY LOGIC FOR CONTROL If a strong wind is blowing right to left, then aim to the right of the goalposts. The use ...

3.2. FUZZY SETS IN CONTROL 87 0 0.2 0.4 0.6 0.8 10 20 30 40 50 Figure 3.2.A=[10,40] Definition 3.1Afuzzy subsetAof a setXis a fu ...

88 CHAPTER 3. FUZZY LOGIC FOR CONTROL Trianglesandtrapezoids, which are piecewise-linear functions, are often used in applicatio ...

3.2. FUZZY SETS IN CONTROL 89 TheGaussian functions, the familiar bell-shaped curve, are of the form A(x)=e− (x−c)^2 2 σ^2 These ...

90 CHAPTER 3. FUZZY LOGIC FOR CONTROL The values ofσdetermine either increasing or decreasing functions, while the parametermshi ...