Question

Consider a (continuous-time) linear system x=Ax + Bu. We introduce a time discretization tk-kAT, where ΔT = assume that the input u(t) is piecewise constant on the equidistant intervals tk, tk+1), , and N > 0, and N 1 a(t) = uk for t E [tk, tk+1). (a) Verify that the specific choice of input signals leads to a discretization of the continuous-time system x = Ax + Bu in terms of a discrete-time system with states x,-2(tr) and inputs uk = u(tk). (b) We say that a discrete-time system 2k+1 Fzk + Guk is controllable if for any zf R" there exists some NEN and an input sequence uo,ui,...,UN-1 such that the state of the discrete-time system is steered from 20-0 to zN Zf. Derive a controllability condition for the discrete-time system 2k+1 Fzk+Guk.

Answer 1

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We want the controllability matrix C of dimension n by N to have rank N when N <= n

When N > n, the system of equations CU = zf has multiple solutions for U (control signal). Then C should have rank n.

U = [u0 u1 .... uN-1]