Description
Exercise 1. Canonical forms Consider the system given by:
Find the controllable canonical form state representation.
Exercise 2. Realization matrix form of realizable MIMO systemÂ
Find a state-space realization for
Exercise 3. Minimum RealizationsÂ
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 decompositionÂ
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, Kirchoff’s voltage law yields a first order differential equation relating the armature current to the armature voltage; that is,
)Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (1)
Motor torque is modeled as being proportional to the armature current, so the electromechanical coupling equation is
Ï„(t) = kTi(t)Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (2)
where kT is the motor torque constant. For the rotational mechanical model, Euler’s rotational law results in the following second-order differential equation relating the motor shaft angle θ(t) to the input torque τ(t).
Jθ¨(t) + bθ˙(t) = τ(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.
