Systems and Control Theory Lecture Notes
Laura Giarr´ e
Ese1: Model Based Controller
Design a Model Based Controller for an inverted pendulum
A cart of mass M slides on a frictionless surface.
The cart is pulled by a force u (t).
On the cart a pendulum of mass m is attached via a frictionless hinge.
The pendulum’s center of mass is located at a distance l from either end.
The moment of inertia of the pendulum about its center of mass is denoted by I.
The position of the center of mass of the cart is at a distance s(t) from a reference point.
The angle θ(t) is the angle that the pendulum makes with respect to the vertical axis which is assumed to increase clockwise.
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Inverted pendulum
MODEL
Choosing the following state space:
x(t) =
⎡
⎢ ⎢
⎣
s(t) ˙s(t) θ(t) ˙θ(t)
⎤
⎥ ⎥
⎦
The nonlinear system is
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Linearized MODEL
The linearized system is (for M
t= M + m and
L = (I + ml
2)/ml )
Design
Let M = 2kg,m = 0.1kg,l = 0.5m, I = 0.025kgm
2and g = 9.8m/s
2.
Design an observer such that the eigenvalues are in
−4, −4, −4, −4
Design a feedback controller such that the eigenvalues are in
−1, −1, −3, −3
Plot the response of the linearized controlled system starting by initial conditions: s = 0,˙s = 0,θ = 0.2, ˙θ = 0 and input r = 0.
Plot the response also of the nonlinearized system.
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Procedure
Define the State Space Model
Check eigenvalues
Check reachability and oservability
Evaluate L and F
Build the overall system
Check the convergence of the state and the estimated one
Analyze the state response for the linearized system
Apply it to the nonlinear system
