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**Example text**

This permits us to rewrite Eqn. 10) We now write Kirchhoff's voltage law, proceeding from point (a) to point (c) to point (b), and then back to point (a), as follows: e(t)-ei(t)-e2«)-eL(t) = 0. 11) Note that we have marked point (a) as the reference point or the "ground point" in the circuit. 12) We next substitute the relations of Eqn. 7 into Eqn. 13) Now we invoke Kirchhoff's current law at point (b). Kirchhoff pointed out that since no electric charge can accumulate at a junction of conductors, the sum of all the currents flowing into the junction at any instant t must equal the sum of all the currents flowing out of the junction at /.

4. See if the equilibrium state x(t) = [0 0 0 0] T dictates that all four ±f(t) must be zero. 5. Make h < d < 0 in your equations and repeat step 3. Do you get results which are physically sensible? 17 Obtain a linear state-variable model for the ball-rack-pendulum system that is valid for small departures from the equilibrium state [6 = 0, 0 = 0, z = 0, z = 0]. Let the input to your model be T(t), and let the output be z(t). Express each element of the A, B, C, and D matrices in terms of the eight physical parameters of the system.

18) eout(t) = Ksxl(t); t0 < t < tF\ Xi(t0), x2(t0)9 and w(0 are known. Note the similarity between this model for the rate gyroscope and that for the mass-spring-damper system in Eqn. 9. Here we have added a sensor equation to our model. The sensor equation is an integral part of a state-variable model for a system that is to be controlled automatically. z(t) fit) CART M SUPPORT RAIL VISCOUS FRICTION SUPPORT 3 FRICTIONLESS PIVOT PENDULUM m, 7CM, / Fig. 6 Cart-pendulum system. 4 Pendulum with Moving Base A cart of mass M supported on a horizontal rail is propelled along that rail by an applied force/(/), as shown in Fig.