# state-space representation in phase-variable forms

Figure in state-space where the output is OL(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: Jxd2Oxdt2+DxdOxdt+KxOx=JwdOzdt 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.

At the same time, the manipulator must provide sufficient force to perform the task. In order to develop a control system to regulate these forces, the robotic manipulator and the target environment must be modeled. Assuming the model in the Figure, represent in state-space the robotic manipulator and its environment under the following condition. a. the manipulator is not in contact with its target environment. b. the manipulator is in constant contact with its target environment. 22. In the past, Type-1 diabetes patient need to inject themselves with insulin three to four times a day.

New delayed-action insulin analogues such as insulin Glargine require a single daily dose. A similar procedure to the one described in Pharmaceutical drugs absorption case study of this chapter is used to find a model in concentration time evolution of plasma for insulin Glargine. For specific patient State space model matrices are given by: Where the state vector is given by a. Find the system’s transfer function. b. Verify the result using MATLAB. 23. A linear, time invariant model of the hypothalamic-pituitary-adrenal axis of the endocrine system with five state variables has been proposed as follows: 4. In this chapter, we described the state-space representation of single input single output systems. In general, systems can have multiple inputs and multiple outputs. An autopilot is to be designed for a submarine in the Figure to maintain a constant depth under severe wave disturbances. We will see that this system has two inputs and two outputs and thus the scaler u becomes a vector, u, and scaler y becomes a vector, y, in the state equations. 25. Experiments to identify precision grip dynamics between the index finger and the thumb have been performed using a ball-drop experiment.

A subject holds a device with a small receptacle into which an object is dropped, and the response is measured. Assuming a step input, it has been found that the response of the motor subsystem together with the sensory system is with the form: Convert this transfer function into a state-space representation. 26. State-space representation are, in general, not unique. One system can be represented in several possible ways. For example, consider the following systems: Show that these systems will result in the same transfer function.