[Solved] 24677 Homework4-Canonical forms

Exercise 1. Canonical forms Consider the system given by:

Find the controllable canonical form state representation.

Exercise 2. Realization matrix form of realizable MIMO system

Find a state-space realization for

Exercise 3. Minimum Realizations

Are the two state equations

and

equivalent, i.e. do they have the same transfer function? Are they minimal realizations?

Exercise 4. Realization

Consider the following transfer function

• Determine the standard controllable realization.
• Determine the standard observable realization.
• Determine a minimal realization.

Exercise 5. Controllable decomposition

Reduce the state equation

to a controllable form. Is the reduced state equation observable?

Exercise 6. kalman decomposition

Decompose the state equation

to a form that is both controllable and observable.

Exercise 7. Controllable Canonical Form

Figure 1: An electromechanical system

The dynamic model of this system can be derived in three segments: a circuit model, electromechanical coupling, and a rotational mechanical model. For the circuit model, Kirchoffs voltage law yields a first order differential equation relating the armature current to the armature voltage; that is,

) (1)

Motor torque is modeled as being proportional to the armature current, so the electromechanical coupling equation is

(t) = k_Ti(t) (2)

where k_T is the motor torque constant. For the rotational mechanical model, Eulers rotational law results in the following second-order differential equation relating the motor shaft angle (t) to the input torque (t).

J(t) + b(t) = (t) (3)

Converting the ODEs into transfer functions and multiplying them together, we eliminate the intermediate variables to get the overall transfer function:

(4)

Write the controllable canonical form of this system.