A First Course in FUZZY and NEURAL CONTROL

2.2. STATE VARIABLES AND LINEAR SYSTEMS 31 Example 2.1 (Motion of an automobile)A classical example of a simpli- fied control sy ...

32 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL 2.3 Controllability and observability An importantfirst step in solving control pro ...

2.3. CONTROLLABILITY AND OBSERVABILITY 33 that a system that exhibits state controllability will also exhibit output con- trolla ...

34 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL Once again the property of observability is also a black and white issue. A system ...

2.4. STABILITY 35 Figure 2.10. Disturbance in system In general, a dynamical system can have several equilibrium states. Also, t ...

36 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL perturbations, from the equilibrium 0 at some timet 0 , the system remains close to ...

2.4. STABILITY 37 Figure 2.13. (a) Underdamped response (b) Undamped response the initial response is to oscillate beforeachievi ...

38 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL where eAt= X∞ k=0 tk k! Ak withA^0 the identityn◊nmatrix andx 0 =x(0).Now eAt= Xm k ...

2.4. STABILITY 39 2.4.3 Stability of nonlinear systems................ Stability analysis for nonlinear systems is more complica ...

40 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL are simultaneously satisfied. Also, sinceV ̇(x)≤ 0 for allxin the neighborhood kxk& ...

2.4. STABILITY 41 2.4.4 Robuststability The problem of robust stability in control of linear systems is to ensure system stabili ...

42 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL by the superscripts of the lower and upper bounds of the intervals. Example 2.4Take ...

2.5. CONTROLLER DESIGN 43 Figure 2.14. Set-point control The most commonly used configuration is shown in Figure 2.14 (a) in whi ...

44 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL for values ofsfor which this improper integral converges. Taking Laplace trans- for ...

2.5. CONTROLLER DESIGN 45 Figure 2.15. Closed-loop feedback system The error between the input valueE(s)and the outputC(s)of th ...

46 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL Figure 2.16. Controller in forward or feedback path controller either in the forwar ...

2.5. CONTROLLER DESIGN 47 position to choose any desired set of roots that yield satisfactory performance. Suppose we choose the ...

48 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL 2.6 State-variablefeedbackcontrol Here we address a controller design technique kno ...

2.6. STATE-VARIABLE FEEDBACK CONTROL 49 The characteristic polynomial therefore is(λ+1)(λ−2) = 0. Clearly this system is unstabl ...

50 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL or λ^2 +3λ+2=0 Comparing the coefficients, we get(− 1 −k 1 −k 2 )=3and(− 2. 0 −k 1 ...