Controlling a Dynamic System - Part 3

Since $a = e^h$ and $b = e^h-1$ , we obtain the relation $x(T+h) \; =\; a(1-b)x(T)$ and hence that $x(jh) \; = \; 10\big[a(1-b)\big]^j \hspace{2cm} j \ge 0$ Thus $x(jh)$ , and hence $E(jh)$ , will be a decreasing sequence provided that $0 < a(1-b) = 2e^h - e^{2h} = 1 - (1 - e^h)^2 < 1$ . This is certainly true for any $0 < h < \ln2$ .

A more normal (there are many others) method of creating a discrete version of a differential equation is to make the approximation (using the first two terms of the Taylor series for $x(T+h)$ ): $x(T + h) \; \approx x(T) + hx'(T)$ which leads to the equation $x(T+h) \; = \; (1+h)x(T) + hu(T)$ The method of this question is a little strange, since it makes the control term $u(T) = -ax(T)$ dependent on the size the discrete time interval $h$ , since $a = e^h$ .