2. Represent the electrical network shown in the Figure in state space, where iR(t) is the output.

3. Find the state-space representation of the network shown in the Figure if the output is vo(t).

4. Represent the system shown in the Figure in state space where the output is x3(t).

5. Represent the translational mechanical system shown in the Figure in state-space where x1(t) is the output.

6. Represent the rotational mechanical system shown in the Figure in state-space, where Ɵ1(t) is the output.

7. Represent the system shown in the Figure in state-space where the output is ƟL(t).

8. Show that the system in the previous Figure in the text yields a fourth-order transfer function if we relate the displacement of either mass to the applied force, and a third-order one if we relate the velocity of either mass to the applied force.

9. Find the state-space representation in phase-variable form for each of the system shown in the Figure.

10. For each system shown in the Figure, write the state equations and the output equation for the phase-variable representation.

11. Represent the following transfer function in state space. Give your answer in vector-matrix form.

12. Find the transfer function G(s)=Y(s)/R(s) for each of the following systems represented in state space.

13. Use MATLAB to find the transfer function, G(s)=Y(s)/R(s), for each of the following systems represented in state space.

14. Repeat problem 13 using MATLAB, the Symbolic Math Toolbox, and Eq. (3.73).

15. Gyros are used on space vehicles, aircraft, and ships for inertial navigation. The gyro shown in the Figure is a rate gyro restrained by springs connected between the inner gimbal and the outer gimbal (frame) as shown. A rotational rate about the z-axis causes the rotating disk to precess about the x-axis. Hence, the input is a rotational rate about the z-axis, and the output is an angular displacement about the x-axis. Since the outer gimbal is secured to the vehicle, the displacement about the x-axis is a measure of the vehicle’s angular rate about the z-axis. The equation of motion is:

Jxd2Ɵxdt2+DxdƟxdt+KxƟx=JwdƟzdt

Represent the gyro in state space.

16. A missile in flight as shown in the Figure, is subject to several forces, thrust, lift, drag, and gravity. The missile flies at an angle of attack, a, from its longitudinal axis, creating lift. For steering, the body angle from vertical 0 is controlled by the rotating engine at the tail. The transfer function relating the body angle to the angular displacement of the engine is of the form:

Represent the missile steering control in state space.

17.Given the dc servo motor and load shown in the Figure, represent the system in state-space, where the state variable is the armature current Ia and the load displacement ƟL and angular velocity wL. Assume that the output is the angular displacement of the armature. Do not neglect armature inductance.

Solution:

* dIq/dt = Vla

* d0L/dt = wL

* d^20L/dt^2 =wL/dt = ?

* for VLa by kirchoff’s voltage law

Ea(t) = RaIa(t) + Vla

18. Consider the mechanical system of the Figure. If the spring is non-linear and the force Fs required to stretch the spring is Fs = 2x21 , represent the system in state-space linearized about x1 = 1 if the output is x2....