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:
cθ ̈+bcos(θ)ω ̇ =dsin(θ)+2G2rk(ω−θ ̇)−2Grs ̄u, (1)
Vmax
bcos(θ)θ ̈+aω ̇ =bsin(θ)θ ̇2−2G2rk(ω−θ ̇)+2Grs ̄u, (2)
Vmax
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, and the constants
b=mbrl, c=Ib +mbl2, d=mbgl, r = 0.034m,
ωm = 1760rad/s, Vmax = 7.4V,
Ib = 0.0004Kgm2, Im = 3.6 × 10−8Kgm2,
(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 equi- librium 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).
g = 9.8m/s2,
s ̄ = 0.003Nm, mb = 0.263Kg,
l = 0.036m, Gr = 35.57,
mw = .027Kg, andIw =mwr2/2+G2rIm,k=s ̄/ωm.
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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:
cθ ̈+bcos(θ)ω ̇ =dθ+2G2rk(ω−θ ̇)−2Grs ̄u, (4)
Vmax bcos(θ)θ ̈+aω ̇ =−2G2rk(ω−θ ̇)+2Grs ̄u.
(5)
(6)
(b) (3 points) Use part (a) to show that the nonlinear equations (4)-(5) can be approximated by the nonlinear state-space equations
(a) (1 point (bonus)) Verify that
c bcos(θ)−1
α(θ) 1−α(θ)
≈c b,α(θ)=
ac
ac − b2 cos2(θ)
Vmax
b cos(θ) a
Comment on the quality of this approximation for θ ∈ [−π/2, π/2].
1−α(θ) α(θ) ba
θ ̈ −Grkβ(θ) Grkβ(θ) dα(θ) θ ̇ − s ̄ β(θ) bc bc c bcVmax
ω ̇ = Grkγ(θ) −Grkγ(θ) d(1−α(θ)) ω + s ̄ γ(θ) u ̇ ab ab b abVmax
θ100θ0
where
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
and
A(ξ) = (1 − ξ)A1 + ξA2, Bu(ξ) = (1 − ξ)Bu,1 + ξBu,2,
ξ(t) = b2 − ac (1 − α(θ(t))) (7) b2
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 asymp- totic 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 min trace(Z )
Z,X,Y,L,Fi,Qi
s.t. ATi +Qi
AiX+XATi +Bu,iL+LTBuT,i
Ai+QTi Bw,i
BwT,i
Z CzX+DzuL Cz
XCzT+LTDzTu X I≻0, i=1,…,N, CzT IY
YAi +ATi Y +FiCy +CyTFiT BwT,iY +DyTwFiT
YBw,i +FiDyw≺0 −I
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
where
x ̇ c(t) = Ac(ξ(t)) xc(t) + Bc(ξ(t)) y(t), u(t) = Cc xc(t),
(8) (9)
(10)
N
Ac(ξ(t)) Bc(ξ(t)) = ξi(t) Ac,i
i=1
Bc,i ,
and
Ac,i =V−1(Qi−YAiX−YBu,iL−FiCyX)U−1, Bc,i =V−1Fi,
Cc =LU−1. Intheaboveformulas,UandV areanymatricessuchthatYX+VU=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 com- pare 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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