By Bernard Brogliato
This moment version of Dissipative platforms research and keep watch over has been considerably reorganized to house new fabric and increase its pedagogical beneficial properties. It examines linear and nonlinear structures with examples of either in every one bankruptcy. additionally integrated are a few infinite-dimensional and nonsmooth examples. all through, emphasis is put on using the dissipative homes of a process for the layout of strong suggestions regulate laws.
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Extra resources for Dissipative Systems Analysis and Control: Theory and Applications
139) -(s) = exp [-2Ts] gL(s) a1 which is a time delay of 2T which is the wave propagation time from x = 0 to x =land back, in series with the load scattering function. It is seen that ~(s) is bounded real as it is the product of two bounded real functions. 137). We consider the following three important cases: 1. First consider impedance matching for the transmission line, which is achieved with ZL(s) = zo. 141) This shows that all the wave energy is absorbed in the load impedance. 2. Next assume that the load is a short circuit, which means that v2 = 0 and zL(s) = 0.
1 Dynamic model A dynamic model for a transmission line is obtained from Kirchhoff's voltage and current law for a length element dx of the transmission line. The transmission line is modelled by a distributed serial inductance Ldx and a parallel capacitance Cdx. The voltage along the line is denoted v(x, t) while the current is denoted i(x, t) where xis the coordinate along the line. 134) is the characteristic impedance of the transmission line. 135) CHAPTER 2. POSITIVE REAL SYSTEMS 44 We then find that the wave variables satisfy s 8a ox(x,s) = -ca(x,s) s 8b 'il(x,s) = -b(x,s) c ux which has solutions a(x,s) b(x,s) = a(O,s)exp [-~s] f- X ] = b(f,s)exp [ --c-s Inverse Laplace transformation then gives X a(x, t) = a(O, t - -) c f-x b(x, t) = b(f, t - - ) c The physical interpretation of this is that a is a wave moving in the positive x direction with velocity c, and b is a wave moving in the negative x direction with velocity c.
Then 1. The system is passive if and only if lg(jw)l ~ 1 for all w. 2. 87) = 1 - 4Re h jw jh(jw)+l It is seen that lg(jw)l ~ 1 if and only if Re(h(jw)] ~ 0. Result 1 then follows as the necessary and sufficient condition for the system to be passive is that Re(h(jw)] ~ 0 for all w. Concerning the second result we show the "iP' part. Assume that there is a 8 so that Re(h(jw)] ~ 8 > 0 and a 'Y so that lh (jw)l ~ 'Y for all w. ;) and the result follows with 0 < 'Y' < min ( 1, 2 ). Next assume that g(jwW ~ 1- 'Y' for all w.
Dissipative Systems Analysis and Control: Theory and Applications by Bernard Brogliato