System Dynamics

Abstract

Considering physical systems as “black boxes”, one can characterize some as having a single scalar input port and a single scalar output port. Such systems are referred to as single input, single output. Frequently, the dynamic behavior of such a physical system can be adequately described, (modelled) in terms of linear, finite dimensional, time invariant, differential equations. A very useful “time domain” dynamic system model is the state space representation, which is given by two equations. One is the state equation a first order matrix differential equation, and the other is the output equation:

$$\dot x(t) = Ax(t) + Bu(t), y(t) = Cx(t)$$

((2.1))

The vector x is n-dimensional, u and y are scalars, A is a constant n × n matrix, B a constant n × 1 matrix, and C a constant 1 × n matrix. The vector x(t) is the state of the system, u(t) is the input and y(t) is the output. The initial condition of this differential equation is frequently taken to be the state at time zero, x(0). In many cases, when laws of physics are applied to system models, the resulting dynamic description is in terms of a scalar, n ^th order ordinary differential equation with constant coefficients. Such an equation can be very easily transformed to one in state space form.