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Problem Set 3 Solutions: 6.245: Multivariable Control Systems by A. Megretski

This document provides solutions to problems from the course 6.245: Multivariable Control Systems. Problem 3.1 involves finding the optimal dynamic feedback controller for a system with state q, control input v, and measurement g, in the presence of noise. Analytical expressions are derived for the Hamiltonian matrices, stabilizing solutions, and optimal controller and observer gains. Problem 3.2 involves designing a linear filter to estimate the derivative of a bandlimited white noise signal, given a noisy measurement. Filters optimized for different noise bandwidths are tested on signals with those bandwidths, showing better performance when the filter design matches the signal bandwidth.

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119 views4 pages

Problem Set 3 Solutions: 6.245: Multivariable Control Systems by A. Megretski

This document provides solutions to problems from the course 6.245: Multivariable Control Systems. Problem 3.1 involves finding the optimal dynamic feedback controller for a system with state q, control input v, and measurement g, in the presence of noise. Analytical expressions are derived for the Hamiltonian matrices, stabilizing solutions, and optimal controller and observer gains. Problem 3.2 involves designing a linear filter to estimate the derivative of a bandlimited white noise signal, given a noisy measurement. Filters optimized for different noise bandwidths are tested on signals with those bandwidths, showing better performance when the filter design matches the signal bandwidth.

Uploaded by

sergiovelasquezg
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Massachusetts Institute of Technology

Department of Electrical Engineering and Computer Science


6.245: MULTIVARIABLE CONTROL SYSTEMS
by A. Megretski
Problem Set 3 Solutions
1
Problem 3.1
Consider a control system described by
q(t)a
2
q(t)=v(t)+f
1
(t), g(t)=q(t)+f
2
(t),
where f = [f
1
;f
2
] is a normalized white noise, v is the control signal, g is
the sensor measurement, and a > 0 is a parameter. The objective is to
find a dynamic feedback controller (with input g and output v) which
stabilizes the system while using a minimum of control effort (defined as
the asymptotic variance of v(t) as t).
(a) FindthecoefficientsoftheauxiliaryabstractH2optimizationprob-
lems associated with the original task.
For

q f
1
x=
q
, w=
f
2
, u=v, y=g, z=v,
we have

0 1 0 0 0
A=
a
2
0
, B
1
=
1 0
, B
2
=
1
,

C
1
= 0 0 , C
2
= 1 0 , D
11
= 0 0 ,
1
Version of March 9, 2004

2

D
12
=1, D
21
= 0 1 , D
22
=0.
The full information control abstract H2 optimization has coecients
a =A, b =B
2
, c =C
1
, d =D
12
.
The state estimation abstract H2 optimization problem has coecients
a =A

, b =C
2
, c =B
1
, d =D
21
.
(b) Write analytically the associated Hamiltonian matrices, bases of
their stable invariant subspaces, stabilizing solutions of the Riccati
equations, and optimal controller and observer gains.
The full information control Hamiltonian is

0 1 0 0

2

a A B
2
B
2

0 0 1

H
fi
=
A

=
0

0 0 0 a
2

.
0 0 1 0
Its eigenvalues are a (double multiplicity each). Hence the stable invariant sub-
space of H
fi
is the kernel of (H
fi
+aI)
2
. A basis in this kernel is given by

1 1/a

0

1

.
2a
3
,

0

2a
2
0
Hence

1

2a
3
2a
3
0 1 1/a 2a
2
P
fi
=
2a
2
=
0 0 1 2a
2
2a
,
and the optimal state feedback gain is given by

2a
2
K =B
2

P
fi
= 2a .
Agoodsanitycheckhere: theclosedlooppoles(eigenvaluesofA+B
2
K)shouldbe
identical to the stable eigenvalues of the Hamiltonian.
The state estimation Hamiltonian is

0 a
2
1 0

C
2

C
2
=

1 0 0 0

. H
se
=
B
1
B
1

0 0 0 1

0 1 a
2
0

3
2 2
Itscharacteristicpolynomialiss
4
2a s +a
4
+1,andhenceitseigenvaluesssatisfy
2 2
s =a j. The stable eigenvalue s=x+jy such that s
2
=a
2
+j is given by

a
4
+1+a
2

a
4
+1a
2
x= , y= .
2

2
The corresponding eigenvector is

x+jy


h= .

yjx
Since H
se
has real coecients, the eigenvector corresponding to eigenvalue xjy
will be the complex conjugate of h. Hence real and imaginary parts of h form a
basis in the stable invariant subspace of H
se
:

y x

0

1

.

,

0

y x
Hence

1

P
se
=
1 0 y x 1/y x/y
=
x y 0 1 x/y x
2
/yy
,
and

C

1/y
L=P
se
2
= .
x/y
(c) Derive an analytical expression for the transfer function of the
optimal dynamic feedback controller, and verify it using numerical
calculations with h2syn.m.
The closed loop transfer function is given by
G(s)=K(sIAB
2
KLC
2
)
1
L.
Toverifytheformulae,MATLABfunctionps3 1.mcanbeused. Thisfunctionrelies
on SIMULINK design diagram ps3 1a.m.
4
Problem 3.2
Random signal q = q(t) is assumed to be a bandlimited white noise of a
given bandwidth B (i.e. the result of passing the true white noise v
1
(t)
throughanideallow-passfilterofbandwidthB rad/sec). Ahighquality
sensor is assumed to measure q(t) accurately, except for a white additive
noise, with the signal-to-noise ratio of 10.
(a) Use h2syn.m to design a 10-th order linear filter which inputs the
sensor output, and outputs an estimate of q(t) which makes the mean
square estimation error as small as possible.
M-functionps3 2a.mdoesthejob. Itusesa10-thorderButterworthlterW with
cut-ofrequencyB tomodelq asq=Ww,wherew isthenormalizedwhitenoise.
(b) Test your design by comparing the simulated performance of filters
you have designed for B = 10 rad/sec and B = 1 rad/sec on signals
q() of bandwidths of B = 10 and B = 1 rad/sec. (One expects that
the filter optimized for B = 10 rad/sec will be better on the q()
with bandwidth B = 10 rad/sec than the filter optimized for B = 1
rad/sec, and vice versa.) Use the generator of bandlimited white
noise supplied with the SIMULINK to perform the simulations.
TheSIMULINKdiagramfortestingisps3 2c.mdl. Youmustrunps3 2b.mbefore
you open it. For B = 1 rad/sec, the degradation of performance when a lter
designed for B = 10 is used is dramatic (at least a 10-fold increase of error). For
B =10, the degradation of performance when a lter designed for B =1 is used is
not as big, but still quite noticeable.

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