Institutionen för ReglerteknikLunds Tekniska Högskola

Nonlinear Control FRT 075

Exam 2000-03-08

GradingSolutions should be clearly motivated. The exam consists of 25 points.

Grade 3: 12 points

Grade 4: 17 points

Grade 5: 22 points

Allowed materialSolutions to previous exams are not allowed. Otherwise, all material is allowed.

ResultsThe results will be posted outside the labs on the ground floor, M-building.

If you allow your result to be published on the course home page thenplease confirm this on the first page of your solutions.

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1. Which phase portrait in Figure 1 belongs to what system? No motivation isrequired.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x1

x2

A

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x1

x2

B

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x1

x2

C

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x1

x2

D

Figure 1 The phase portraits in Problem 1

(i) x1 = x2

x2 = −x2 − x1 − sign(x1)(ii) x1 = −x2

x2 = x1

(iii) x1 = x2

x2 = −x1 + x2x41

(iv) x1 = x2

x2 = sin(x1)

(2 p)

2

2. Consider the static nonlinearity f (u) in Figure 2. Draw a sketch of the

f

u

Figure 2 The nonlinearity f (u) in Problem 2

principal behavior of the describing function Nf (A). (2 p)

3. The following is a version of a model for the release of calcium in certaincells.

x1 = Kx3− x1

x2 = x1 − x2

x3 = x2 − x3

a. Determine the equilibria and the corresponding linearizations. (2 p)b. For which K > 0 is the system locally asymptotically stable at these equi-

libria? (1 p)

4. Construct an oscillator with approximate frequency 2 Hz using a staticnonlinearity of your choice, in feedback with a linear system, as in Figure 2.Motivate your construction, using for instance describing function analysisor Lyapunov theory. (3 p)

+

−G(s)

ϕ (⋅)

Figure 3 The block diagram for the oscillator

3

5. Consider the system

x =−x3

1 + u

x1

a. Is the system with u = 0 locally stable at x = 0? (Hint: You might find the

function E = x2 − 1x1

useful.) (2 p)

b. Use the Lyapunov function V (x) = x21 + x2

2 to find a control law

u = f (x1, x2)

that makes the origin x = 0 globally asymptotically stable. (2 p)

6. You are building a hydroaccoustic sender for creating “sounds” under water.The system uses a crude amplifier which has the static amplification shownin Figure 4. To make the amplifier behave linearly, you want to add aproportional feedback, as shown in Figure 5. The feedback loop also includesa time delay e−sT with unknown T ≥ 0. What is the largest K > 0 you canuse and still prove that the system is BIBO stable from the input r to theoutput y? Prove it.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−8

−6

−4

−2

0

2

4

6

y

u

Figure 4 The nonlinear amplifier in Problem 6. For large input signals the gain ap-proaches 5.

++

-

y u r

K e−sT

Nonlinear amplifier

Figure 5 The nonlinear amplifier with feedback in Problem 6.

(2 p)

4

7. You want to help a tired friend to win next year’s version of the famousSwedish ski race Vasaloppet. The problem is to minimize the finish time,but still only use exactly as much energy as one has available (includingintake of the traditional Blueberry-soup). You know that the height curve ofthe race looks approximately like in Figure 6. This implies that the heightcan be approximated by

f (x) = x − 0.04x2 + 300

where x is the distance from start in kilometers. The ski runner’s velocity

0 10 20 30 40 50 60 70 80 9050

100

150

200

250

300

350

Hei

ght[m]

Distance [km]Figure 6 The height of the track in Vasaloppet. The slope is the derivative of the height.

can be described by

v = −v− dfdx+ u

and the distance is just the velocity integrated

x = v, x(0) = 0.

The goal is to reach the goal x(t f ) = 89 km using the total available energy(approximated) ∫ tf

0u2(t)dt = 100

in as short time as possible. Write the problem as an optimal control prob-lem and show that the optimal control signal can be written as the outputof the linear system [

λ1

λ2

]=[

1 −1

−0.08 0

][λ1

λ2

]

withu(t) = Constant ⋅ λ1(t)

and λ1(t f ) = 0. You do not have to calculate any constants. (4 p)

5

8. The air throttel on laboration 3 can be modeled as in Figure 7. The non-

+1

s+d1s

−

x1x2

αβ

u

Figure 7 The throttel, with nonlinear springs

linearity f is odd and piece-wise linear with slope α > 0 for small angles,and slope β > 0 for larger angles, (i.e. f (ξ ) = αξ for 0 ≤ ξ ≤ α , andf (ξ ) = β(ξ − α ) + α for α ≤ ξ ), the damping d is positive. Show thatthe throttel, with u = 0, is globally asymptotically stable, so that x(t) → 0when t → ∞ for any initial condition x(0). (Hint: You might want to usethe Lyapunov function V (x) = ∫ x1

0 f (ξ ) dξ + x22/2). (3 p)

9. For which zero locations α is the linear system

G(s) = s+αs2 + 2ζ s+ 1

, ζ ≥ 0

passive? (2 p)

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