Automatic control
Course:- Mechanical Engineering
Reference No.:- EM13251

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Question: The two DOF system below has two moving masses M1, M2 with an input force of f(t) applied to M . The characteristics of the system are as follows:

459_Automatic Control.png

Note: the friction forces fv1 and fv2 can be thought of as damping forces.

298_Automatic Control1.png

a) Derive the system's dynamics (the governing differential equations) using the Lagrangian method

b) For an output of x1(t) and a state vector of x = 894_Automatic Control2.pngfind the A, B, C and D matrices

Question: Consider the transfer function:

1307_Automatic Control3.png

a) Find the differential equation describing the system

b) Write the state and output equation's in Observer Canonical (Left Companion)

c) Draw the block diagram of the system (using the form from part b above); how many states are observable?

Question: In the case of an inverted pendulum shown below, the equations of motion for control are a function of the movement in the horizontal direction and the angular motion of the bob. The angle of the bob is assumed to be sufficiently low to provide a linear model of the system.

2390_Automatic Control4.png

The properties of the system are as follows:

• M- Mass of Cart (5 kg)
• m- Mass (the bob) at end of rod (0.1 kg)
• θ- angular displacement of rod from vertical
• y- displacement in horizontal direction
• l- length of rod (0.5 m)

The sum of the forces in the horizontal direction can be shown to be:

1465_Automatic Control5.png                                 (1)

By approximating the pendulum as a point mass, the sum of the moments about the pivot point is:

2044_Automatic Control6.png                             (2)

By rearranging equations (1) and (2) and simplifying by assuming M >> m then the state and input matrices are:

2232_Automatic Control7.png

1518_Automatic Control8.png

With a state vector given by:

985_Automatic Control9.png

a) What is the rank of the A matrix and what does this tell you about the states used in the state space model? Relate this to the dynamic equations

b) For an output of θ the minimal realisation of the system will have a 2x2 A matrix. For this minimal realisation find the A, B, C, D matrices and the state vector

c) Find the eigenvalues of the system from the state-space model derived in b)

d) Sketch the poles of the system on the complex plane

e) Is the system stable?

f) Using the state space model from b) find the transfer function

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