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LINEAR SYSTEMS THEORY

Joao P. Hespanha

November 24, 2010

Comments and information about typos are very welcome.

Please contact the author at hespanha@ece.ucsb.edu.

Errata

1. In Figure 2.3, the angle 2 is incorrectly drawn, it should be drawn as follows:

1

2

1

2

m1

m2

Moreover, the matrix M q and the vector G q in Example 2.2 should be as follows:

M q

m222 2m212 cos2 m1 m2

21 m2

22 m212 cos2

m212 cos2 m222 m2

22

G

q

m2g2 cos 1 2 m1 m2 g1 cos1m2g2 cos 1 2

.

2. Equation (2.8) should be as follows:

x

0 I

0 0

x

0

I

v, y

I 0

x, x

q

q

R2k. (2.8)

(note the change in the B matrix)

3. The first equation in Section 2.4.4 should be as follows:

x1 x2

x2 M

1 x1

B x1,x2 x2 G x1 u

(note parenthesis in the equation for x2).

4. In Figure 2.5, the label y at the right should be replaced by

q

q

.

5. Condition 3. in Theorem 7.2 is incorrect. It should read as follows:

Theorem 7.2. For an n n matrix A, the following three conditions are equivalent:

http://hespanha@ece.ucsb.edu/http://hespanha@ece.ucsb.edu/http://hespanha@ece.ucsb.edu/

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(a) A is semisimple.

(b) A has n linearly independent eigenvectors.

(c) There is a nonzero polynomial without repeated roots that annihilates A; i.e., there is a nonzero polynomial

p s without repeated roots for which p A 0.

Note that condition 5c provides a simple procedure to check for diagonalizability. Since every polynomial

that annihilates A must have each eigenvalue of A as a root (perhaps with different multiplicities), one simply

needs to compute all the distinct eigenvalues 1, . . . , k (k n) of A and then check if the polynomial p s

s 1 s k annihilates A.

6. In the sidebar example in Section 4.3.2, the matrix N3 should be

N3

24 3

1 12

.

7. In the proof of Proposition 4.1, the second equation that relates the several Zk should be as follows

Zn 1

sZn 1, Zn 1

1

sZn 2, . . . , Z2

1

sZ1 Zk

1

sk 1Z1.

Note, in particular that Zn 1 1sZn 2 and not Zn 1 1sZn 1.

8. In Definition 8.1, the condition for exponential stability should be

x t ce t t0 x t0 , t 0.

Note the

sign in front of.

9. The second inequality in the Attention! box below Theorem 8.1 should be

x t eA t t0 x0 eA t t0

x0 c e

t t0

x0 , t R.

10. At the end of the proof of Theorem 8.2, the definition of should be

min Q

max P.

11. In equation (16.4a), the dimensions of the vectors xo and xu should read

xo x

o

xu x

u

Ao 0

A21 Au

xoxu

BoBu

u, xo Rn n, xu R

n, (16.4a)

where n denotes the dimension of the unobservable subspace UO of the original system.

Just below, the definition for detectability should read

Definition 16.1 (Detectable system). The pair A,C is detectable if it is algebraically equivalent to a system in

the standard form for unobservable systems (16.4) with n 0 (i.e., Au nonexistent) or with Au a stability matrix.

Note the condition n 0, instead of n n.

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Acknowledgements

I would like to thank several UCSB students, Prof. Bogdan Udrea, and Prof. Maurice Heemels for helping me to find

and correct several typos in the book.