By Panos J. Antsaklis
In keeping with a streamlined presentation of the authors’ profitable paintings Linear structures, this textbook presents an advent to platforms conception with an emphasis on keep an eye on. preliminary chapters current beneficial mathematical heritage fabric for a basic realizing of the dynamical habit of structures. every one bankruptcy comprises valuable bankruptcy descriptions and guidance for the reader, in addition to summaries, notes, references, and workouts on the finish. The emphasis all through is on time-invariant platforms, either non-stop- and discrete-time.
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Extra info for A Linear Systems Primer
3. If F depends continuously on parameters (say, λ ∈ Rl ) and if x0 depends continuously on λ, the solution φ is continuous in λ as well. 1a) will have a unique solution that exists for t ∈ R. 8, φ(t, t0 , x0 ) characterizes the state of the system at time t. 1b). We usually assume that g ∈ C(R × Rn × Rm , Rp ) or that g ∈ C 1 (R × Rn × Rm , Rp ). 3b) where A ∈ C(R, Rn×n ), B ∈ C(R, Rn×m ), C ∈ C(R, Rp×n ), and D ∈ C(R, Rp×m ). 6. 3a) possesses a unique solution that exists for all t ∈ R and that is continuous in (t, t0 , x0 ).
Fn (·)), u(k)T = (u1 (k), . . , um (k)), y(k)T = (y1 (k), . . , yp (k)), and g(·)T = (g1 (·), . . 13a) y(k) = g(k, x(k), u(k)). 13b) Throughout this section we will assume that f : Z × Rn × Rm → Rn and g : Z × Rn × Rm → Rp . Since f is a function, for given k0 , x(k0 ) = x0 , and for given u(k), k = k0 , k0 + 1, . . 13a) possesses a unique solution x(k) that exists for all k = k0 , k0 + 1, . . Furthermore, under these conditions, y(k) is uniquely deﬁned for k = k0 , k0 + 1, . . 1)], k0 denotes initial time, k denotes time, u(k) denotes the system input (evaluated at time k), y(k) denotes the system output or system response (evaluated at time k), x(k) characterizes the state (evaluated at time k), xi (k), i = 1, .
Sound waves exert an external force f (t) on the moving plate. The output voltage vs , which appears across the resistor R, will reproduce electrically the sound-wave patterns that strike the moving plate. When f (t) ≡ 0 there is a charge q0 on the capacitor. This produces a force of attraction between the plates that stretches the spring. When sound waves exert a force on the moving plate, there will be a resulting motion displacement x that is measured from the equilibrium position. The distance between the plates will then be x0 − x, and the charge on the plates will be q0 + q.
A Linear Systems Primer by Panos J. Antsaklis