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# Communications and Control (Mathematical Model)

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1.6 LINEARISATION

1.1 Specify variable x and coefficients a, b and c in the differential equation (1.5) for
a) the mass-spring-damper system in Fig 1.1 described by (1.1), and
b) the RLC circuit in Fig. 1.2 described by (1.3).

1.2 Let x = uc, where uc is the voltage on the capacitor shown in Fig 1.6. Show that the dynamics of the RLC circuit can be described by a 2nd-order differential equation in form (1.5).

1.3 If time t in (1.3) is in seconds. Show that, to balance the equation (1.3), the unit of L/R has to be in seconds, and LC in square of seconds. Moreover, RC also has to be in seconds.

1.4 The liquid level in a tank as shown in Fig 1.7 is governed by the differential equation h = -0.01 rooth + 00u, where u is in the control inflow to the tank. Derive a linear model of the system by linearising he non-linear one around the operating point h0 = 0.1m.

1.1 MASS SPRING DAMPER SYSTEM

By Newton's second law of motion, the simple mechanical system of Fig. 1.1 is described by the 2nd-order linear ODE for t >= 0

m(d^2)(y)/(dt^2) + f dy/dt + ky = u

In (1.1) y is in the displacement and u is the force as external excitation, m is the mass, f the friction factor and k the spring coefficient. m, f and k are constants. Eg (1.1) is said to be linear since each term on the both sides of the equation is a linear function of y, dy/dt, (d^2)y/dt^2 or u.

1.2 R-L-C CIRCUIT SYSTEM

According to Kirchhoff's current law, the voltage c in the basic RLC circuit system of Fig 1.2 obeys the linear integro-differential equation for t >= 0.

v/R + C*dv/dt + 1/L*integral as c goes to t of vdt = u

In 1.2 v is the voltage of the capacitance C and u is teh current source as excitation R is the resistance and L is the inductance. The values of T, L and C are constants.

If choose the current of the inductance, i = 1/L integral as 0 goes to t of v ddt as the bvariable of interest, the integro-differential equation (1.2) becomes the 2nd-order ODE

v/R + C (d^2)v/dt^2 + v/L = du/dt

Clearly, as shown in (1.2), (1.3) and (1.4), a system may have several models depending on the form of expressions and/or the choice of variables of interest. Different choices of forms and/or variabes lead to various mathematical models. However, as may be expected, some essential properties of the systme should remain unchanged in the distinct models. This will further be discussed.

1.3 ANALOGY OF DYNAMIC SYSTEMS

Dynamic systems with individual physical nature could have analogous behaviors. Although the mass-spring-damper system and the electric circuit are physically totally different, they are mathematically analogous. This is because differential equations (1.1) and (1.3) can commonly be denoted by

a(d^2)x/dt^2 + b(dx/dt) + cx = u

##### Solution Summary

Solution for 1.2 includes applying Kirchhoff's voltage law and the relationship between i and v for a capacitor. Other solutions are included as well.

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1.2) Applying Kirchhoff's voltage law to the circuit in the illustration gives (see attached).

We must now use the relationship between i and vc for a capacitor. Subsequently, we find the derivative to replace di/dt with a term including ...

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