By Panos J. Antsaklis

According to a streamlined presentation of the authors’ profitable paintings Linear structures, this textbook offers an advent to platforms concept with an emphasis on keep watch over. preliminary chapters current beneficial mathematical historical past fabric for a primary figuring out of the dynamical habit of platforms. every one bankruptcy comprises important bankruptcy descriptions and directions for the reader, in addition to summaries, notes, references, and routines on the finish. The emphasis all through is on time-invariant platforms, either non-stop- and discrete-time.

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**Example text**

101). Generate a phase portrait for each case given above. 9. 35) by choosing x1 = x and x2 = x˙ 1 . 6 for numerical methods for solving diﬀerential equations). 35) tend to is an example of a limit cycle. 10. Consider a system whose state-space description is given by √ x˙ = −k1 k2 x + k2 u(t), √ y = k1 x. Linearize this system about the nominal solution √ u0 ≡ 0, 2 x0 (t) = 2 k − k1 k2 t, where x0 (0) = k. 11. 36) consider the hard, linear, and soft spring models given by g(x) = k(1 + a2 x2 )x, g(x) = kx, g(x) = k(1 − a2 x2 )x, respectively, where k > 0 and a2 > 0.

If g(x) = k(1 − a2 x2 )x, where k > 0 and a2 > 0 are parameters, then g represents the restoring force of a soft spring. Finally, if g(x) = kx, then g represents the restoring force of a linear spring. 5. 6. 34), let f (x) ≡ 0 and g(x) = k sin x, where k > 0 is a parameter. 34) assumes the form x¨ + k sin x = 0. 7. 5 Existence, Uniqueness, and Dependence on Parameters of Solutions 17 gravitational acceleration g, and the mass. The symbol x denotes the angle of deﬂection measured from the vertical.

The symbol x denotes the angle of deﬂection measured from the vertical. The present model is called a simple pendulum. 7. 34) can be represented by the system of ﬁrst-order ordinary diﬀerential equations given by x˙ 1 = x2 , x˙ 2 = −f (x1 )x2 − g(x1 ). 38) are given by x1 (0) and x2 (0). 12). 5. 39). Hence, no solution exists for this initial-value problem (in the sense deﬁned in this chapter). 6. 40) where x ∈ R, has at least two solutions given by φ1 (t) = [ 23 (t − t0 )]3/2 and φ2 (t) = 0 for t ≥ t0 .