By Torkel Glad, Lennart Ljung

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Let Ul(t) be orders for day t. Ordered goods is, however, not delivered until two days later. Let U2(t) be the sales for day t. 27), with one output and two inputs, n == 2, and with the coefficients Here u(t) is a column vector u(t) 1 = [ U2 Ul (t) (t) We then have y(t) - y(t 1) = [0 -1] u(t - 1) + [1 0] u(t - 2) The transfer function matrix for this system is G(z) = [ z2 1 _ Z -1 Z - 1 1 State Space FornI The discrete time variant of the state form is a system of first order difference equations, written as x(t + 1) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) with a n-dimensional state vector x(t).

In an €-environment around x* (t). This definition of stability of solutions to differential equations is also called Lyapunov-stability. 4 and Chapter 12 we shall return to stability of linear and nonlinear systems, respectively. 6) • defined signaillorms IIzll, system gain IIS//, and stability • shown the small gain theorem: that a closed loop system is stable if the product of the gains in the loop is less than 1. Literature Mathematical foundations. Control theory builds on a mathematical foundation and we shall use concepts and results from a variety of mathematical subfields.

Tx: I;. +TBu y = CT-Il;. 5) so that T AT- I is diagonal. i = Ti X (Ti is the ith row of T) which are certain linear combinations of the original state vector X, corresponding to A '8 eigenvectors. The mode is generally named for its corresponding eigenvalue ("the Inode Ai"). 2 Controllability and Observability 45 up in the output. They are also important for understanding what happens when factors are cancelled in the transfer operator as well as for the control synthesis with observer and state feedback.