By Panos J. Antsaklis
In keeping with a streamlined presentation of the authors’ profitable paintings Linear structures, this textbook presents an advent to structures conception with an emphasis on keep an eye on. preliminary chapters current worthy mathematical heritage fabric for a basic realizing of the dynamical habit of platforms. every one bankruptcy contains important bankruptcy descriptions and guidance for the reader, in addition to summaries, notes, references, and routines on the finish. The emphasis all through is on time-invariant structures, either non-stop- and discrete-time.
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Additional resources for A Linear Systems Primer
5. Remark. 5, can be found in many of the standard texts on numerical analysis. 7. 15 to solve the initial-value problem x˙ = ax+ t, x(0) = x0 for t ≥ 0. Here a ∈ R. 8. 101) where x ∈ R and A ∈ R . , λ1 and λ2 are the roots of the equation det(A − λI) = 0, where det denotes determinant, λ is a scalar, and I denotes the 2 × 2 identity matrix. Make speciﬁc choices of A to obtain the following cases: 2 2×2 42 1. λ1 2. λ1 3. λ1 4. λ1 5. λ1 6. λ1 7. λ1 8. λ1 1 System Models, Diﬀerential Equations, and Initial-Value Problems > 0, λ2 > 0, and λ1 = λ2 < 0, λ2 < 0, and λ1 = λ2 = λ2 > 0 = λ2 < 0 > 0, λ2 < 0 √ = α + iβ, λ2 = α − iβ, i = −1, α > 0 = α + iβ, λ2 = α − iβ, α < 0 = iβ, λ2 = −iβ Using t as a parameter, plot φ2 (t, 0, x0 ) vs.
77). 86), we ﬁrst let t = t0 . 87), we have φ(t0 , t0 , x0 ) = x0 . 86). 86). 87) is called the variation of constants formula, which is discussed further in Chapter 3. 87), using a change of variables. 87) reduces to φ(t, t0 , x0 ) φh (t) = Φ(t, t0 )x0 . 86) may be viewed as consisting of a component that is due to the initial conditions (t0 , x0 ) and another component that is due to the forcing term g(t). This type of separation is in general possible only in linear systems of diﬀerential equations.
12). Linearization Given is x˙ = f (t, x) and a solution φ. The Jacobian matrix is ⎤ ⎡ ∂f1 ∂f1 ∂x1 (t, x) · · · ∂xn (t, x) ∂f ⎥ ⎢ .. 54) ⎦. . ∂x ∂fn ∂fn ∂x1 (t, x) · · · ∂xn (t, x) For A(t) = ∂f ∂x (t, φ(t)), z˙ = A(t)z is the linearized equation about the solution φ. 57) References • • 33 Existence and uniqueness of solutions of x˙ = A(t)x + g(t). 20. 80) t0 the Peano–Baker series. In the time-invariant case x˙ = Ax, ∞ φ(t, t0 , x0 ) = I + k=1 Ak (t − t0 )k x0 k! 84) = eA(t−t0 ) x0 , where ∞ eA = I + k=1 Ak .
A Linear Systems Primer by Panos J. Antsaklis