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contour. Therefore, we will assume the clockwise traversal of a contour to be positive and the area enclosed within the contour to be on the right. We consider the transfer function Y sAs Àz 1 Y AX3X2 where A is a constant and z 1 de®nes a zero of Y s, s z 1 Re jj Y where R, j are variables, s À z 1 Re jj X If the contour C in the s-plane encircles the zero z 1 , this is equivalent to a rotation of the (s±z 1 ) vector by 2p in the case when the corresponding contour D in the Y s-plane encircles the origin in a clockwise direction. If the contour C does not encircle the zero z 1 , the angle of s±z 1 is zero when the traversal is in a clockwise direction along the contour and the contour D does not encircle the origin (Fig. A.3.2). Now, we consider the transfer function Y s A s À p 1 Y AX3X3 Figure A.3.1 Contour C in s-plane. Figure A.3.2 Contour D in Ys-plane. A. Appendix 709 Control where p 1 de®nes a pole of Y s s p 1 Re jj Y Then Y s A R e Àjj X AX3X4 We assume that the contour C encircles the pole p 1 . Considering the vectors s as shown for a speci®c contour C (Fig. A.3.3) we can determine the angles as s traverses the contour. Clearly, as the traversal is in a clockwise direction along the contour, the traversal of D in the Y s-plane is in the opposite direction. When s traverses along C a full rotation of 2p rad, for the Y s-plane we will have an angle À2p rad (Fig. A.3.4). We can generalize these results. If a contour C in the s-plane encircles Z zeros and P poles of Y s as the traversal is in a clockwise direction along the contour, the corresponding contour D in the Y s-plane encircles the origin of the Y s-plane N Z À P times in a clockwise direction. The resultant angle of Y s will be Df 2pZ À 2pPX AX3X5 The case in which the contour C encircles Z 3 zeros and P 1 pole in the s-plane is presented in Figs. A.3.5 and A.3.6. The corresponding contour D in the Y s-plane encircles the origin two times in a clockwise direction. Figure A.3.3 Contour C in s-plane with a pole p 1 . Figure A.3.4 Contour D in Ys-plane. 710 Control Control One of the most interesting mapping contours in the s-plane is from the Nyquist contour. The contour passes along the j o-axis from ÀjI to jI and is completed by a semicircular path of radius r (Fig. A.3.7). We choose the transfer function Y 1 s as Y 1 s k ts 1 Y AX3X6 which has the pole p 1 À1at real and negative. In this case, the Nyquist contour does not encircle a pole. Figure A.3.5 Contour C for Z 3 and P 1. Figure A.3.6 Contour D for N Z and P 2. Figure A.3.7 (a) Nyquist contour and a pole p 1 À1at. (b) Mapping contour for Y 1 s kats 1. A. Appendix 711 Control From (A3.4) we have Y 1 s k tR e Àjy X AX3X7 When s traverses the semicircle C with r 3I from o I to o ÀI, the vector Y 1 s with the magnitude katR has an angle change from Àpa2topa2. When s traverses the positive imaginary axis, s jo 0 ` o ` I, the mapping is represented by Y s k 1 jot k 1 Àjot 1 o 2 t 2 Y AX3X8 which represents a semicircle with diameter k (Fig. A.3.7b, the solid line). The portion from o ÀIto o 0 À is mapped by the function Y 1 sj sÀj o Y 1 Àj ok 1 jot 1 o 2 t 2 X AX3X9 Thus, we obtain the complex conjugate of Y 1 j o, and the plot for the portion of the polar plot from o ÀIto o 0 À is symmetrical to the polar plot from o Ito o 0 (Fig. A.3.7b). A.4 The Signal Flow Diagram A signal ¯ow diagram is a representation of the relationship between the system variables. The signal ¯ow diagram consists of unidirectional opera- tional elements that are connected by the unidirectional path segments. The operational elements are integration, multiplication by a constant, multi- plication of two variables, summation of several variables, etc. (Fig. A.4.1). Figure A.4.1 Signal ¯ow diagram elements. 712 Control Control These functions are often suf®cient to develop a simulation model of a system. For example, we consider a dynamic model described by the differential equation x a 1 x a 2 x uX AX4X1 Using the notations x x 1 x x 2 Y Eq. (A.4.1) can be rewritten as x 1 x 2 x 2 Àa 1 x 2 À a 2 x 1 uX AX4X2 The signal ¯ow diagram of (A.4.1) is presented in Fig. A.4.2. The diagram has two representations, one for time-domain variables and one for the Laplace transform representation. Figure A.4.2 Signal ¯ow diagrams (a) in the time domain, (b) in the Lap1ace variable domain. A. Appendix 713 Control References 1. R. J. Schilling, Fundamentals of RoboticsÐAnalysis and Control. Prentice Hall, Englewood Cliffs, NJ, 1990. 2. P. G. Darzin, Nonlinear Systems. Cambridge University Press, 1992. 3. I. E. Gibson, Nonlinear Automatic Control. McGraw-Hill, New York, 1963. 4. S. Ca Ï lin, Automatic Regulators. Ed. Did-Ped, Bucharest, 1967. 5. E. Kamen, Introduction to Signals and Systems. Macmillan, New York, 1990. 6. B. C. Kuo, Automatic Control Systems. Prentice Hall, Englewood Cliffs, NJ, 1990. 7. T. Yoshikawa, Foundation of Robotics. M.I.T. Press, Cambridge, MA, 1990. 8. G. I. Thaler, Automatic Control Systems. West Publishing, St. Paul, MN, 1990. 9. R. G. Dorf, Modern Control Systems. 6th ed. Addison-Wesley, Reading, MA, 1992. 10. B. W. Niebel, Modern Manufacturing Process Engineering. McGraw-Hill, New York, 1989. 11. S. C. Jacobsen, Control strategies for tendon driven manipulators. IEEE Control Systems, Vol. 10, Feb., 23±28 (1990). 12. I. I. E. Slotine and Li Weiping, Applied Nonlinear Control. Prentice-Hall International, New York, 1991. 13. M. Asada and I. I. E. Slotine, Robot Analysis and Control. Wiley-Interscience, New York, 1986. 14. H. BuÈhler, ReÂglage par mode de glissement. Presses Polytechniques Romandes, Lausanne, 1986. 15. M. Ivanescu and V. Stoian, A distributed sequential controller for a tentacle manipulator, in Computational Intelligence (Bernd Reusch, ed.), pp. 232±238. Springer Verlag, Berlin, 1996. 16. R. C. Rosenberg and D. C. Karnopp, Introduction to Physical System Design. McGraw-Hill, New York, 1986. 17. R. I. Smith and R. C. Dorf, Circuits, Devices and Systems, 5th ed. John Wiley & Sons, New York, 1991. 18. W. L. Brogan, Modern Control Theory. Prentice Hall, Englewood Cliffs, NJ, 1991. 19. R. E. Ziemer, Signals and Systems, 2nd ed. Macmillan, New York, 1989. 20. C. L. Phillps and R. D. Harbor, Feedback Control Systems. Prentice Hall, Springer Verlag, New York, 1988. 21. R. L. Wells, Control of a ¯exible robot arm. IEEE Control Systems, Vol. 10, Jan., 9±15 (1990). 714 Control Control Appendix Differential Equations and Systems of Differential Equations HORATIU BARBULESCU Department of Mechanical Engineering, Auburn University, Auburn, Alabama 36849 Inside 1. Differential Equations 716 1.1 Ordinary Differential Equations: Introduction 716 1.2 Integrable Types of Equations 726 1.3 On the Existence, Uniqueness, Continuous Dependence on a Parameter, and Differentiability of Solutions of Differential Equations 766 1.4 Linear Differential Equations 774 2. Systems of Differential Equations 816 2.1 Fundamentals 816 2.2 Integrating a System of Differential Equations by the Method of Elimination 819 2.3 Finding Integrable Combinations 823 2.4 Systems of Linear Differential Equations 825 2.5 Systems of Linear Differential Equations with Constant Coef®cients 835 References 845 715 1. Differential Equations 1.1 Ordinary Differential Equations: Introduction 1.1.1 BASIC CONCEPTS AND DEFINITIONS Operatorial Equation Let X , Y be arbitrary sets and f X X 3 Y a function de®ned on X with values in Y .Ify 0 P Y is given, and x P X must be found so that f xy 0 Y 1X1 then it is said that an operatorial equation must be solved. A solution of Eq. (1.1) is any element x P X that satis®es Eq. (1.1). The sets X , Y can have different algebraical and topological structures: linear spaces, metrical spaces, etc. If f is a linear function, that is, f ax 1 bx 2 af x 1 bf x 2 , and if X and Y are linear spaces, then Eq. (1.1) is called a linear equation.If Eq. (1.1) is a linear equation and y 0 y Y (the null element of space Y ), then Eq. (1.1) is called a linear homogeneous equation. Differential Equation An equation of the form (1.1) for which X and Y are sets of functions is called a functional equation. A functional equation in which is implied an unknown function and its derivatives of some order is called a differential equation. The maximum derivation order of the unknown function is called the order of the equation. When the unknown function depends on a single independent variable, the equation is termed an ordinary differential equation (or, more brie¯y; a differential equation). If the unknown function depends on more independent variables, the corresponding equation is called a partial differential equation. The general form of a differential equation of order n is F t Y x Y x H Y x HH Y FFFY x n 0Y 1X2 where t is the independent variable, x xt is the unknown function, and F is a function de®ned on a domain D R n2 (R is the set of real numbers). It is called a solution of Eq. (1.2) on the interval I aY b&R, a function j jt of C n I class [i.e., jt has continuous derivatives until n-order], which has the following properties: 1. tY jtY j H tY FFFY j n t P DY Vt P I 2. FtY jtY j H tY FFFY j n t 0Y Vt P I If the function F can be explicated with the last argument, it then yields x n f t Y x Y x H Y FFFY x nÀ1 Y 1X3 which is called the normal form of the n-order equation. 716 Appendix: Differential Equations and Systems of Differential Equations Differential Equations Cauchy's Problem Let G & R n1 be the de®nition domain of the function f from Eq. (1.3) and t 0 Y x 0 0 Y x 0 1 Y FFFY x 0 nÀ1 PG. If the solution of Eq. (1.3) satis®es the initial conditions xt 0 x 0 0 Y x H t 0 x 0 1 Y FFFY x nÀ1 t 0 x 0 nÀ1 Y 1X4 then the problem of ®nding that solution of Eq. (1.3) is called the Cauchy's problem for Eq. (1.3). The Cauchy's problem for Eq. (1.2) is formulated analogously. The general solution of Eq. (1.3) is a function family depending on the independent variable and on n arbitrary independent constants j jt Y c 1 Y c 2 Y FFFY c n , and which satis®es the conditions 1. jtY c 1 Y c 2 Y FFFY c n is a solution for Eq. (1.2) on an interval I c 2. For any initial conditions (1.4), there could be determined the values c 0 1 Y c 0 2 Y FFFY c 0 n of the constants c 1 Y c 2 Y FFFY c n so that jtY c 0 1 Y c 0 2 Y FFFY c 0 n is the solution for the Cauchy's problem with the conditions (1.4) A solution obtained from the general solution for particular constants c 1 Y c 2 Y FFFY c n is called a particular solution.Asingular solution is a solution that cannot be obtained from the general solution. EXAMPLE 1.1 Consider the following equation: x H2 x 2 À 1 0X 1X5 This is a differential equation of the ®rst order. Expressing it with x H ,we obtain x H 1 Àx 2 p and x H À 1 Àx 2 p Y 1X6 hence two equations of normal form. Let us consider the ®rst equation. This is of the form of Eq. (1.3) with n 1, the right-hand side function being f x 1 Àx 2 p and de®ned on G R ÂÀ1Y 1. In its expression, the independent variable t does not appear explicitly. The function jtsin t is derivable, and substituting it in the equation, we ®nd the equality cos t jcos tj. This is an identity on any interval I k of the form I k Àpa22kpY pa22kp, k P Z (Z is the set of integer numbers). Then, on each of these intervals the function jtsin t is the solution for equation x H 1 Àx 2 p . Now, consider the functions family jt Y c sint c, c P R. Let us set the interval I c Àpa2 ÀcY pa2c. For t P I c , t c P Àpa2Y pa2 and jtY csint c is the solution on I c for the equation x H 1 Àx 2 p .Ift 0 Y x 0 PG R ÂÀ1Y 1 settled from the condition jt 0 Y cx 0 , then sint 0 cx 0 and the value of c obtained from this condition and denoted by c 0 is c 0 arcsin x 0 À t 0 . The function jtY c 0 sint c 0 is a solution for the equation and satis®es the initial condition, so it is the general solution. m 1. Differential Equations 717 Differential Equations Remark 1.1 The constant functions j 1 t1 and j 2 tÀ1 are solutions on R for the equation x H 1 Àx 2 p , but they cannot be obtained from the general solution for any particular constant c, and hence there are singular solutions. m 1.1.2 SYSTEMS OF DIFFERENTIAL EQUATIONS A system of differential equations is constituted by two or more differential equations. A system with n differential equations of the ®rst order in normal form is a system of the form x H 1 f 1 tY x 1 Y x 2 Y FFFY x n x H 2 f 2 tY x 1 Y x 2 Y FFFY x n ÁÁÁ x H n f n tY x 1 Y x 2 Y FFFY x n Y V b b ` b b X 1X7 f i X I ÂD & R Â R n .Asolution of the system of equations (1.7) on the interval J I is an assembly of n functions j 1 tY j 2 tY FFFY j n t derivable on J and that, when substituted with the unknowns x 1 Y x 2 Y FFFY x n , satis®es Eqs. (1.7) in any t P J . Initial Conditions Consider that t 0 P J and x 0 1 Y x 0 2 Y FFFY x 0 n PD are settled. The conditions x 1 t 0 x 0 1 Y x 2 t 0 x 0 2 Y FFFY x n t 0 x 0 n 1X8 are called initial conditions for the system of equations (1.7). Vectorial Writing of the System of Equations (1.7) If the column vector X t is written as X t x 1 t x 2 t ÁÁÁ x n t V b b ` b b X W b b a b b Y Y with f tY X f 1 tY x 1 Y x 2 Y FFFY x n f 2 tY x 1 Y x 2 Y FFFY x n ÁÁÁ f n tY x 1 Y x 2 Y FFFY x n V b b ` b b X W b b a b b Y Y then the system of equations (1.7) can be written in the form X H tf tY X 1X9 and the initial conditions (1.8) are X t 0 X 0 Y 1X10 718 Appendix: Differential Equations and Systems of Differential Equations Differential Equations [...]... À U1 m APPLICATION 1.5 The Emptying of a Vessel Study the law of leakage of water from a vessel that has the shape of a rotation surface about a vertical axis, with a hole A in the bottom part Study the following particular cases: (a) The vessel has a hemisphere shape of radius R (b) The vessel has a truncated cone shape with the small base as bottom, the radii R1 , R2 , and height H (c) The vessel... x1
t is a solution for Eq (1.12) Conversely, if u
t is a solution of Eq (1.12), then W V b x1
t b b b a ` x2
t X
t b ÁÁÁ b b b Y X xn
t with x1 Y x2 Y F F F Y xn given by Eq (1 .13) is a solution of the system of equations (1.15) In summary, ®nding the solution of an n-order differential equation is equivalent to ®nding the solution of an equivalent system with n equations of... capacitor C, having a voltage U The laws of electricity yield the differential equation LI HH
t RI H
t 1 I
t f
t Y C
1X31 where I
t is the intensity and f
t U H
t Equation of a Mechanical Oscillator The equation of motion of a material point with mass m, which moves on the Ox axis under the action of an elastic force F Ào2 x , is m x o2 x 0X
1X32 Taking into account... multiple of numbers n1 and n2 Finally, we will obtain a rational function m Remark 1.3 If the equation x b1 bx b2 0 has as solutions x k, then these are singular solutions of the initial equation Particular Cases
dx ln jxj
x dx xb1 1 b 0, b1 T 1, b x 1 b1 1
dx ln jx bj b1 1, b2 0, x b
dx 1 x b1 2, b2 0, b b 0, 2 p arctan p x b b b p
x À Àb... Equation of Order n Consider the equation of order n u
n f
t Y uY u H Y F F F Y u
nÀ1 Y
1X12 with the initial condition 0 u
t0 u0 Y 0 0 u H
t0 u1 Y F F F Y u
nÀ1
t0 unÀ1 X
1X13 Using the notation u x1 Y u H x2 Y F F F Y u
nÀ2 xnÀ1 Y u
nÀ1 xn Y by derivation we ®nd the system V H b x1 x2 b H b b ` x2 x3 ÁÁÁ b H bx b nÀ1 xn b X H xn f
t Y x1 Y x2 Y F F... integrating factor From the de®nitionof the integrating factor, d d dm dm dh2 dh1 À
mh1
mh2 , or h1 À h2 m Hence, dt dx dx dt dx dt h1 d ln m d ln m dh2 dh1 À h2 À Y dx dt dt dx
1X59 and a partial differential equation has been obtained The integrating factor is relatively easy to ®nd in the following cases 1 If m m
t, then dm 0, and Eq (1.59) will take the form dx dh1 dh2 À d ln m . Control Control Appendix Differential Equations and Systems of Differential Equations HORATIU BARBULESCU Department of Mechanical Engineering, Auburn University, Auburn, Alabama 36849 Inside 1. Differential Equations. the conditions (1.4) A solution obtained from the general solution for particular constants c 1 Y c 2 Y FFFY c n is called a particular solution.Asingular solution is a solution that cannot be obtained. A.4.2 Signal ¯ow diagrams (a) in the time domain, (b) in the Lap1ace variable domain. A. Appendix 713 Control References 1. R. J. Schilling, Fundamentals of RoboticsÐAnalysis and Control. Prentice Hall,
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