history of computer based control systems

... Introduction to Computer Based Control Systems Introduction to Computer Based Control Systems 1.4.1 Centralized Computer Control The early digital computer based control ... to Computer Based Control Systems Introduction to Computer Based Control Systems 1.4.3 Hierarchical Computer Control Systems As the name implies, in this type of Systems ... Computer Based Control Systems Introduction to Computer Based Control Systems Control Level (Level-1) • Systems at... process Figure 1.4(b) is a block diagram of the computerized

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... discrete-time systems, ” IEEE e Transactions on Automatic Control, vol 41, no 6, pp 903–907, 1996 12 E N Chukwu, Stability and Time-Optimal Control of Hereditary Systems, vol 188 of Mathematics ... exponential stabilizability of discrete-time systems with time-delayed states Finally, as an application, we consider the global stabilization of the nonlinear control systems Preliminaries In this ... Analysis of Nonlinear Systems, vol 125 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1989 P Niamsup and V N Phat, “Asymptotic stability of nonlinear control

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... conjugate pair. Therefore we extract the quadratic factors that are the products of the pairs of 92 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES complex roots, and then complex arithmetic ... gives a pair of complex roots or a pair of real roots. Let us divide the given polynomial p n (x) by a quadratic factor x 2 + px + q, we obtain a quotient polynomial Q n–2 (x) of degree (n–2) ... polynomial. Another quadratic factor is of obtained using this default polynomial. If the initial approximation of p and q are not known then the last three terms of given polynomial a n – 2 x 2 +

Ngày tải lên: 04/07/2014, 15:20

... of x in the given range. However if the function f is not known, the value of y can be obtained, when a set of values of x is given. The method to find out such values is based on principle of ... independent variable of function y = f(x). To determine the values of function y = f(x) for given intermediate or argument values of x, three types of differences are useful: 104 CALCULUS OF FINITE DIFFERENCES ... roots and multiple roots of polynomials and also for determining the Eigen values of a matrix. An important feature of the method is that it gives approximate values of all the roots simultaneously

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... operator D of differential calculus. In finite differences, we deal with ratio of simultaneous increments of mutually dependent quantities where as in differential calculus, we find the limit of such ... displacement or translation operator. It shows the operation of increasing the argument value x by its interval of differencing h so that. CALCULUS OF FINITE DIFFERENCES 113 Ef (x) = f(x + h) or Ey x ... from each value of y except y 0 , the previous value of y, we get y 1 – y 0 , y 2 –y 1 , y 3 – y 2 , y n – y n–1 . These differences are called first backward differences of y and are denoted

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... ). Let one of the value of f(x) is missing. Say it f(i). To determine this missing value of f(x), assume that f(x) can be represented by a polynomial of degree (n – 1) since n values of f(x) are ... given a set of equidistant values of arguments and its corresponding value of f(x). Suppose for n + 1 equidistant argument values x = a, a + h, a + 2h, , a + nh, are given. CALCULUS OF FINITE DIFFERENCES ... 10. Assuming that the following values of y belong to a polynomial of degree 4, compute the next three values: x0 1 2 3 4567 y1-11 11−−−− 140 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES

Ngày tải lên: 04/07/2014, 15:20

... coefficients of different powers of x in order beginning with the coefficients of higher power of x. 2. Put 1 in the left hand side column and write zero below the coefficients of highest power of x ... which is to be written below 3. The remainder –10 is the value of D. 4. Add the terms of corresponding columns of (a) and get 2, –1 and 2 of (b) 5. Now again apply the steps (1) and (3): in this way ... coefficient of x 2 on both sides of (1), we get A = 2 Hence, the required polynomial in factorial notation will be f(x)= 2x (3) + 3x (2) + 2x (1) – 10 Again by the simple rule of differentiation,

Ngày tải lên: 04/07/2014, 15:20

... = 0.047875 190 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Example 4. Use Gauss’s forward formula to find a polynomial of degree four which takes the following values of the function ... 186 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Take the mean of equation (1) and (2) 22 (0) (1) (0) ( 1) (1) (1) () (0) 22 2!2 ... Apply Gauss forward formula to find the value of U 9 , if u(0) = 14, u(4) = 24, u(8) = 32, u (12) = 35, u(16) = 40 [Ans. 33.1162109] 192 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES 5.

Ngày tải lên: 04/07/2014, 15:20

... the normal weights of babies during the first 12 months of life: Age in Months :02581012 Weight in lbs : 7.5 10.25 15 16 18 21 Find the weight of babies during 5 to 5.6 months of life. [Ans. 15.67] ... y corresponding to argument value of x (for equal and unequal spaced argument). On the other hand the process of Estimating the value of x for a entry value of y (which is not in the table) is ... 5 (correct to 3 places of decimal) We have to find cube root of 10. Since 10 lies between the value of y corresponding to x =2 and x = 3, therefore the required value of 3 10 is x = x 0 +

Ngày tải lên: 04/07/2014, 15:20

... (°F) on three days, 30 days apart round the pds. of summer and winter. Estimate the app. dates and values of max. and min. temperature. 254 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Summer ... x 0 ] 2. The n th divided differences of a polynomial of n th degree are constant. Let f(x) = A 0 x n + A 1 x n–1 + + A n–1 x + A n by a polynomial of degree n provided A 0 ≠ 0 and arguments ... in powers of (x – 1), we use synthetic division method as, −+11 1 3 8 103 11 0 3 11 11 11 1 4 1 12 x 3 – x 2 + 3x + 8 = (x – 1) 3 + 2 (x – 1) 2 + 4 (x – 1) + 11. Ans. 250 COMPUTER BASED NUMERICAL

Ngày tải lên: 04/07/2014, 15:20

... (x), as well as to express the powers of x in terms of the Chebyshev polynomials. Some of the Chebyshev polynomials and the expansion for powers of x in terms of T n (x) are given as follows: T ... polynomial. These polynomial are used in the theory of approximation of function. Chebyshev polynomials: Chebyshev polynomial T n (x) of the first kind of degree n over the interval [–1, 1] is defined ... or modification of the co-efficients a 1 , a 2 a n will produce little error for small x near zero. But probably substantial error near the ends of the interval 272 COMPUTER BASED NUMERICAL

Ngày tải lên: 04/07/2014, 15:20

... Economization of power series: To describe the process of economization, which is essential due to Lanczos, we first express the given function as a power series in x. Let power series expansion of x ... good uniform approximation to ()fx in which the number of terms retained depends on the given tolerance of ∈ . However for a large number of functions, an expansion as in (2), converges more rapidly ... we take f(x) up to 4 24 x with a tolerance of ∈ = 0.02 S.t. f(x)= e x = 1 + x + 234 2624 xxx ++ (1) Changing each power of x in (1) in terms of Chebyshev polynomials, we get e x = 01 2

Ngày tải lên: 04/07/2014, 15:20

... , x 1 , x 2 , x n . We want to find the vaule of () b a Ifxdx = ∫ . Let the interval of integration (a, b) be divided into n equal subintervals of width ba h n −  =   so that 01020 0 ... fx xx xx fx xx xx xx=+−∆+−−∆ +−−− 34 001230 ()()()()()() fx xx xx xx xx fx∆ +−−−−∆ (1) 308 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES x f (x) ()fx∆ 2 () fx∆ () 3 fx∆ () 4 fx∆ 5 () fx∆ ... 26224 yy y y uuu fx u y y h −−− −−   ∆+∆ ∆ +∆ −−  ′ =+∆+ +∆+         310 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES ej ej Using difference table, we have 23 0111

Ngày tải lên: 04/07/2014, 15:20

... 326 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES [] 2(10) 7(0) 32(4) 12(7) 32(9) 7(12) 32(15) 12(14) 32(8) 7(3) 45 = +++++ + ++ = 708 Hence the required area of the cross-section of ... equation of motion, deflection of a beam etc., are represented by differ- ential equations. Hence, the solution of differential equation is a necessity in such studies. There are number of differential ... the hour, we go in for numerical solutions of differential equations. In researches, especially after the advent of computer, the numerical solutions of the differential equations have become

Ngày tải lên: 04/07/2014, 15:20

... 338 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Example 7. Find the solution of dy =1+ xy, y(0) = 1 dx which passess through (0, 1) in the interval (0, 0.5) such that the value of ... ∫ (2) Now, we solve (2) by the method of successive approximation to find out the solution of (1). The first approximate solution (approximation) 1 y of y is given by () 0 10 0 , x x yy ... approximation 1 y of y is given by 23 1 0 0 01 3 x xx ydx =+ = + ∫ NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATION 339 Similarly, the second approximation 2 y of y is given by

Ngày tải lên: 04/07/2014, 15:20

... truncation error in Milne’s method is also of O (h 5 ). The relative error of an approximate solution is the ratio of the total error to the exact value. It is of greater importance than the error ... SOLUTION OF ORDINARY DIFFERENTIAL EQUATION 371 Inherent stability is determined by the mathematical formulations of the problem and is dependent on the eigen values of Jacobian Matrix of the differential ... is a function of the error propagation in the numerical method. Three types of errors occur in the application of numerical integration methods: (a) Truncation error (b) Round-off error (c) Propagation

Ngày tải lên: 04/07/2014, 15:20

... the absolute values of the leading diagonal elements of the coefficient matrix A of the system AX = B are greater than the sum of absolute values of the other coefficients of that row. The condition ...    378 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES This iterative methods is not always successful to all systems of equations. If this method is to succeed, each equation of the system ... superiority of this method is that it gives the solution of a system, provided its determinant does not vanish in finite number of steps. 8.4 ILL-CONDITIONED SYSTEM OF EQUATIONS A system of equations

Ngày tải lên: 04/07/2014, 15:20

... Method of group averages III. Method of moments IV. Principle of least square. Out of above four methods, we will only discuss and study here principle of least square. 9.2 PRINCIPLE OF LEAST ... (5), we get the value of a, b and c. Putting the value of a, b and c in equation (1), we get the equation of the parabola of best fit. 9.2.3 Change of Scale When the magnitude of the variable in ... () 3 and () 4 , we get the value of a and b. Putting the value of a and b in equation () 1 , we get the equation of the line of best fit. 9.2.2 Fitting of Parabola Let a parabola 2 yabxcx=+

Ngày tải lên: 04/07/2014, 15:20

... nb=+ ∑∑ 2 1 y na b x x =+ ∑∑ Where n is the number of pair of values of x and y. 9.2.7 Fitting of the Curve 0 1 c y= +c x x Error of estimate for i th point () , ii xy is 0 1ii i i ... the method of least squares a parabola of the type 2 yabx=+ . Sol. Error of estimate for i th point (,) ii xy is () 2 ii i eyabx =−− By the principle of least squares, the values of a and ... The error of estimate for the ith point () , ii xy is () 2 [/] iii i eyaxbx =− − . By the principle of least square, the values of a and b are such so that the sum of the square of error S,

Ngày tải lên: 04/07/2014, 15:20

... development The educational component of the programme was based on the correct identification of images of healthy and unhealthy foods. The correct selection of the healthy food would be rewarded ... working on the computer. Where the level of verbal interaction is normal, even when children work on their own on the computer they often engage in “self- talk” which is talking to the computer explaining ... satisfaction of success and the enjoyment of participation. Liberman, (2006) conducted a study of children from age six to eleven and found that 49 of 50 preferred learning using interactive computer...

Ngày tải lên: 28/03/2014, 20:20