Description
1: LQR design with linearized plant
A magnetic levitation system consists of an electromagnet and an optical sensor to detect
the vertical position of a steel ball. The objective is to establish a feedback loop to maintain
the ball floating in mid-air, between the sensor post and the electromagnet’s end surface.
A system schematic is shown in Fig. 1. Assume that we can control the current using an
Figure 1: Schematic of Maglev System
amplifier capable of delivering any requested current between -2.5 and 2.5 A. The upward
force exerted by the electromagnet on the ball is
F =
ki2
2x
2
where k = 6.5308 × 10−5 N-m2/A2
, i is the current on the coil and x is the gap between the
top of the ball and the electromagnet’s flat end. The mass of the ball is m = 0.068 kg and
the acceleration of gravity is g = 9.81 m/s2
.
1. Find the differential equation relating i (input) to x (output), accounting for gravity
and electromagnetic force.
2. Show that the current needed to maintain the ball in equlibrium at an air gap value
of xeq is
ieq =
r
2mg
k
xeq = kf fxeq (1)
3. Obtain a linearized differential equation model for operation near ieq and xeq. For this,
use a 2-variable Taylor expansion formula for mg − F at the equilibrium point.
4. Use the above to show that the linearized transfer function from ∆i to ∆x is
G(s) = ∆X(s)
∆I(s)
=
−2g/ieq
s
2 − 2g/xeq
(2)
5. Obtain a state-space representation of the linearized system at the equilibrium point.
6. Find ieq corresponding to xeq = 0.006 m from Eq. 1. Then obtain the numerical value
for the feedforward gain kf f .
Control Design by LQR
Use a state-space representation to design a linear state feedback controller for regulation to
a setpoint with the following specifications:
1. Zero steady-state error to step changes in target position.
2. Settling time to a step input of approx. 1 second
3. Overshoot as small as you can.
4. The range of motion is limited between x = 0 and x = 0.014 m. The current must be
kept within ±2.5 A.
5. Test for a stable response using a Simulink model. Use the linearized plant and a step
input of 1 mm. The settling time should be near 1 second, but the overshoot may be
high (100%).
6. Check that the current spike stays within ±1 A for a step input of 1mm.
7. Prepare a Simulink model to test the controller against the actual nonlinear plant, as
done in class (you have to account for offsets, in this case the bias current must be
injected in a feedforward fashion). Transfer the position from rest at the post (14mm)
to mid-air, at 6mm from the flat end of the electromagnet as fast as possible while
checking that the current remains within bounds.
2: Quadratic Lyapunov Function and Region of Attraction
The linearized system under the state feedback gain is a globally asymptotically stable
closed-loop system. The nonlinear system under the same controller has only local stability,
however. In this problem, a quadratic Lyapunov function will be used to obtain a crude
estimate of the region of attraction.
1. Use the closed-loop system matrix Ac = A − BK to obtain a quadratic Lyapunov
function V (x) = 1
2
x
TP x. For this, select any symmetric positive-definite matrix Q and
solve the Lyapunov equation for P:
A
′
CP + P
′Ac + Q = 0
In Matlab this is done with P=lyap(Ac’,Q)
2. Use symbolic and numeric processing to evaluate the derivative of V along the trajectories of the nonlinear closed-loop system. Then use a polar scan around the equlibrium
point to detect a failure of the V <˙ 0 condition. Essentially, you will be finding the
boundary defined by V˙ = 0.
3. The region determined above is conservative for 2 reasons: Q (in the Lyapunov equation) is selected arbitrarily and the Lyapunov method is only a sufficient condition
(quadratic functions are not necessarily the best to show stability of a given nonlinear
system). Show (by simulation example) that there are initial conditions inside the true
region of attraction that do not belong to the polar region you found.
4. Bonus points: iterate Q (manually or by any optmization method you feel comfortable
with) to obtain some improvement to the area of your polar region. You must report
the exact values of the area before and after optimization.
