A Linear Systems Primer by Panos J. Antsaklis PDF
By Panos J. Antsaklis
In line with a streamlined presentation of the authors’ winning paintings Linear platforms, this textbook presents an creation to structures conception with an emphasis on keep an eye on. preliminary chapters current precious mathematical heritage fabric for a primary realizing of the dynamical habit of structures. each one bankruptcy contains beneficial bankruptcy descriptions and instructions for the reader, in addition to summaries, notes, references, and workouts on the finish. The emphasis all through is on time-invariant structures, either non-stop- and discrete-time.
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We can specialize the preceding discussion to linear systems of equations x˙ = Ax. 80) assumes the form t s1 A(s1 ) t0 s2 A(s2 ) t0 sm−1 A(s3 ) . . t0 t = Am s1 s2 sm−1 ... t0 A(sm )dsm · · · ds1 t0 t0 t0 1dsm · · · ds1 = t0 Am (t − t0 )m , m! 78) for φm assumes now the form m Ak (t − t0 )k x0 . k! 83) on compact subsets of R. We have ∞ φ(t, t0 , x0 ) = I + k=1 Ak (t − t0 )k x0 k! 83). ] By k ∞ making the analogy with the scalar ea = 1 + k=1 ak! , usage of the notation ∞ A e =I+ k=1 Ak k! 85) should be clear.
1b) where x ∈ Rn , y ∈ Rp , u ∈ Rm , f : R×Rn ×Rm → Rn , and g : R×Rn ×Rm → Rp . Here t denotes time and u and y denote system input and system output, 48 2 Introduction to State-Space and Input–Output Descriptions of Systems respectively. 1b) constitute the state-space description of continuous-time ﬁnite-dimensional systems. , u : R × Rn → Rm ). 1a) as x˙ = F (t, x). 14, if F ∈ C(R × Rn , Rn ) and if for any compact subinterval J0 ⊂ R there is a constant LJ0 such that ˜ for all t ∈ J0 and for all x, x˜ ∈ Rn , F (t, x) − F (t, x˜) ≤ LJ0 x − x then the following statements are true: 1.
When no such continuation is possible, then φ is called noncontinuable. 10. 43) 1 deﬁned on J = (−1, 1). This solution is continuable has a solution φ(t) = 1−t to the left to −∞ and is not continuable to the right. 11. 44) where x ∈ R, has a solution ψ(t) ≡ 0 on J = (−∞, 0). This solution is continuable to the right in more than one way. 44) for t ≥ 0. 3) In the next result, ∂D denotes the boundary of a domain D and ∂J denotes the boundary of an interval J. 12. 11) on an open interval J, then φ can be continued to a maximal open interval J ∗ ⊃ J in such a way that (t, φ(t)) tends to ∂D as t → ∂J ∗ when ∂D is not empty and |t| + |φ(t)| → ∞ if ∂D is empty.
A Linear Systems Primer by Panos J. Antsaklis