Course Control and System Theory of Rational Systems Motivated by
the Life Sciences
Homeworkset 7
Date issued: 18 October 2018.
Date due: 25 October 2018.
1. Controllability. Please check the controllability of the following structured linear system
which system representation is slightly different from that considered in Chapter 7.
dx(t)
dt
= Ax(t) + Bu(t), x(0) = x0,
A =




a11 1 0 1
a21 0 1 1
a31 0 0 1
0 0 0 a44



 , B =




0
0
1
0



 ,
nx = 4, nu = 1, a11, a21, a31, a44 ∈ R\{0}.
2. Observability. Consider a time-invariant linear systems. Call the observability map injective
if the next displayed map is injective,
x0 → y(∗; 0, x0) = {y(t; 0, x0) ∈ Y, ∀ t ∈ [0, ∞)}.
Recall that a map h : X → Y is called injective if ∀ x1, x2 ∈ X, h(x1) = h(x2) implies that
x1 = x2. Prove equivalence of the following two statements:
(a) The linear system is observable according to the characterization of Th. 7.3.11 of the
lecture notes.
(b) The observability map of the system is injective as defined above.
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3. Observable canonical form. Consider the specific form of a time-invariant linear system of
state-space dimension 3 and output dimension 2.
dx(t)
dt
= Ax(t), x(0) = x0,
y(t) = Cx(t),
A =


0 1 0
a21 a22 0
a31 a32 a33

 , C =
1 0 0
0 0 1
,
= nx = 3, ny = 2, a21, a22, a31, a32, a33 ∈ R.
Prove by the following three steps that this particular form is a canonical form.
(a) Prove that any system of this form is observable.
(b) If there are two systems of the above form possibly with different values for the parameters
ai,j which are similar as defined in Def. 7.4.1 of the lecture notes, then the system
matrices A and C of the considered two systems have to be identical.
(c) Prove that any linear system with observability indices equal to nobs = (2, 1) can be
transformed to the above defined special form.
It follows from the three properties that the above defined special form is an observable
canonical form.
Reading advice for Lecture 7
Please read of the lecture notes the Sections 7.7 - 7.6. The reading of the proof of Section 7.5 is not
urgent.
Reading advice for the future Lecture 8
On Thursday 23 October, Lecture 8 will be presented. Please read of the lecture notes the Sections
7.7 - 7.12. Several of these sections are not yet entered in the lecture notes. As mentioned before,
this advice is a recommendation only.
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