Outline
- Execution of demo
Execution of demo would motivates you to learn how control
system operates and how it should be designed.
- Derivation of the equation of motion
Check if the equation of motion in demo model is correct.
- Modification of the equation of motion
Since the moment of inertia of pendulum is neglected in the
demo model, the equation of motion is modified so that the
moment of inertia of pendulum is considered.
So far, preparation is finished.
Next, controller will be constructed from scratch.
In this text, state feedback controller is adopted as the
structure of controller.
This controller is usually adopted when all states of
plant(control object) can be measurable, where the control input
(control signal inputted to plant) is generated by the state
vector of the plant multiplied by a given constant matrix as shown
below:
f = F z
where f is the control input, F is the constant matrix with
suitable size which is called state feedback gain, and
z is the state vector of plant.
- Construction of state feedback control system
In our plant of inverted pendulum, z is composed of 4 scalar
variables, i.e. cart position, pendulum angle, and their
derivative.
Since the plant in demo model has no output of the derivatives
for cart position and pendulum angle, so first the plant is
modified to output these two derivative signals.
Moreover, the block diagram of model is modified so that the
control input, force input applied to the cart, is given as z
multiplied the feedback gain F.
- Derivation and linearization for state equation
Then, only task remaining so far is to set the feedback gain F
properly so that the pendulum stands as fast as possible.
Since trial-and-error tuning might be difficult to find the
matrix F. more systematic way, pole placement technique, is
utilized to find the matrix F in this text.
To do that, we need a linear model of the plant, i.e matrices A
and B are needed when the plant is represented by
(d/dt) z = A z + B f.
Note that the above differential equation is linear on the
vector z. (There are no non-linear terms e.g. sin or cos.)
However, in the equation of motion of inverted pendulum, there
are some non-linear terms.
Therefore, we first have to linerize the equation of motion.
Determination of the state feedback gain via pole assignment method
After determining matrices A and B, then the feedback gain F is
automatically obtained by specifying the closed loop poles
(equivalently, closed loop eigenvalues) by using a tool.
Relation between closed-loop poles and time response
There is a intuitive relation between closed-loop poles and the
resultant time response.
In other words, the resultant time response can be roughly estimated
by considering the location of given closed-loop poles.
Tune the location of closed-loop poles so that the desired
time response would be achieved with some trial and error by
using the relation.