Formal Definion

Knowledge of the state variables defines the initial, current and future response of a system. An n -dimensional state-space is defined as the space generated by the smallest set of linearly independent variables x(t₀ )…..x_n (t₀ ) at t=t₀ ( n states of the system), which, together with the m number of given inputs u₁ (t)….u_m (t) for t=t₀ determine the states at any future time t>t₀ . The standard form of state-space equations is expressed as

	(24.1)

where, x represents the n -dimensional state vector, y the output vector, u the control input, A the system matrix, B the actuator influence matrix, C provides the sensor influence matrix and D the direct transmission matrix. Usually, D is considered as zero except for feed-forward systems. The example in the next slide shows how to obtain the state-space representation of a multiple degrees of freedom system.