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:

Note: the friction forces f_v1 and f_v2 can be thought of as damping forces.

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 = find the A, B, C and D matrices

Question: Consider the transfer function:

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.

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:

                                 (1)

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

                             (2)

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

With a state vector given by:

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