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MAE 280B

Linear Control Design – Spring 2020

Final Exam

Instructions:

• Due on 06/07/2020 by 11:59 PM on Canvas

• Use Matlab or Mathematica

• You get marks for clarity

• You lose marks for obscurantism

• This exam has 4 questions, 42 total points and 4 bonus points

Figure 1: Inverted pendulum on wheels

Questions

1. LQG Design.

The motion of an inverted pendulum on wheels as the one in Fig. 1 is described by the nonlinear

differential equations:

c

¨θ + b cos(θ) ˙ω = d sin(θ) + 2G

2rk(ω − ˙θ) −2Grs¯Vmaxu, (1)b cos(θ)¨θ + a ω˙ = b sin(θ)˙θ

2 − 2G2rk(ω − ˙θ) + 2Grs¯Vmaxu, (2)

where θ is the angle the pendulum makes with a vertical plane (θ = 0 points up), ω is the

angular speed of the wheels, u is the DC motor voltage,

a = 2Iw + (mb + 2mw)r2, b = mbr `, c = Ib + mb`2, d = mbg `,

and the constants

g = 9.8m/s2, ` = 0.036m, r = 0.034m,s¯ = 0.003Nm, Gr = 35.57, ωm = 1760rad/s, Vmax = 7.4V,mb = 0.263Kg, mw = .027Kg, Ib = 0.0004Kgm2, Im = 3.6 × 10−8Kgm2,and Iw = mwr2/2 + G2rIm, k = ¯s/ωm.

(a) (2 points) Convert the equations (1)-(2) into nonlinear state-space equations and calculate

all equilibrium points for which u = 0.

(b) (1 point) Linearize the state-space equations about the equilibrium point θ = u = 0. Is

the linearized system asymptotically stable?

(c) (5 points) It is easy to measure the angular speeds ˙θ and ω. Design an LQG controller

using measurements of ˙θ and ω that can stabilize the inverted pendulum about the equilibrium

point θ = u = 0. Consider a process noise that enters through the input voltage

u and is uncorrelated with the measurement noise.

(d) (2 points) Calculate the closed-loop transfer-function from the reference inputs ¯˙θ and ¯ω

to the outputs θ,

˙θ and ω. Where are the poles and zeros located?

(e) (1 point (bonus)) Measuring θ is much more complicated. A classmate argued that this

is not the case, and that he or she can easily estimate θ using an accelerometer attached

to the body of the pendulum. What do you think?

(f) (5 points) Can you stabilize the MIP using a single output measurement? If so, which one

would you use, θ,

˙θ or ω? If possible design an LQG controller and a controller using any

classic control design technique (e.g. rooto-locus, Nyquist, etc) and compare with your

answer to part (c).

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2. Robust Control.

In this question you will attempt to determine whether the controller you designed in Question 1

is capable of stabilizing the inverted pendulum beyond a small neighborhood of the equilibrium

point θ = u = 0. With that in mind consider the approximation:

sin(θ)

˙θ

2 ≈ 0 and sin(θ) ≈ θ. (3)

These approximations should hold in the range θ ∈ [−π/2, π/2]. Substitute (3) into (1)-(2) to

obtain the approximate nonlinear differential equations:

(6)

Comment on the quality of this approximation for θ ∈ [−π/2, π/2].

(b) (3 points) Use part (a) to show that the nonlinear equations (4)-(5) can be approximated

by the nonlinear state-space equations

β(θ) = 2Gr [c(α(θ) − 1) + bα(θ)] , γ(θ) = 2Gr [a(α(θ) − 1) + bα(θ)]

(c) (3 points) Use part (b) to construct an uncertain time-varying model of the form

x˙(t) = A(ξ(t))x(t) + Bu(ξ(t))u(t)

where

A(ξ) = (1 − ξ)A1 + ξA2, Bu(ξ) = (1 − ξ)Bu,1 + ξBu,2,

and

ξ(t) = b2 − acb2(1 − α(θ(t))) (7)

is the relationship between ξ and α(θ). Verify that when θ ∈ [−π/2, π/2] then ξ ∈ [0, 1].

(d) (1 point) Is any of the matrices A1 or A2 Hurwitz?

(e) (1 point (bonus)) If you did this question correctly so far, one of the pair of matrices

(Ai

, Bu,i) calculated in part (c) coincides with the linearized pair (A, Bu) calculated in

Question 1. Why?

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(f) (3 points) Use the above model and what you learned about robust stability in MAE280B

to determine if the closed-loop connection of the above uncertain time-varying model with

the LQG controller you designed in Question 1 is robustly stable for all ξ ∈ [0, 1].

Hint: use the notion of quadratic stability.

(g) (1 point (bonus)) Is robust stability as assessed in part (f) enough to guarantee asymptotic

stability of the closed-loop connection of the nonlinear model (4)-(5) with the LQG

controller you designed in Question 1? Explain.

3. Gain Scheduling Control.

(a) (5 points) Consider the uncertain time-varying model from Question 2 part (c) and solve

the following semidefinite program

to calculate a dynamic gain-scheduled LQG controller using the exact same settings you

employed in Question 1. The corresponding gain scheduled LQG controller is

x˙ c(t) = Ac(ξ(t)) xc(t) + Bc(ξ(t)) y(t), (8)

u(t) = Cc xc(t),

In the above formulas, U and V are any matrices such that Y X + V U = I.

4. Comparison and Simulation.

(a) (3 points) Use (6) and (7) to substitute ξ(t) for θ(t), which turns the gain scheduled

controller (8)-(9) into a nonlinear controller. Calculate a state-space realization for such

controller.

(b) (9 points) Simulate the closed-loop response of the nonlinear MIP model to test and compare

the performance of the LQG controllers you designed in Question 1 with the nonlinear

controller you calculated in part (a). Use graphics, cost functions, transfer-functions, or

whatever you think necessary to adequately compare the controllers.

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