should be independent of one another, and only dependent on a linear combination of the input vector and the
The state variables represent values from inside the system, that can change over time. In an electric circuit,
for instance, the node voltages or the mesh currents can be state variables. In a mechanical system, the forces
applied by springs, gravity, and dashpots can be state variables.
We denote the input variables with u, the output variables with y, and the state variables with x. In essence,
we have the following relationship:
Where f(x, u) is our system. Also, the state variables can change with respect to the current state and the
Where x' is the rate of change of the state variables. We will define f(u, x) and g(u, x).
The state equations and the output equations of systems can be expressed in terms of matrices A, B, C, and D.
Because the form of these equations is always the same, we can use an ordered quadruplet to denote a system.
We can use the shorthand (A, B, C, D) to denote a complete state-space representation. Also, because the state
equation is very important for our later analyis, we can write an ordered pair (A, B) to refer to the state
Obtaining the State-Space Equations
The beauty of state equations, is that they can be used to transparently describe systems that are both
continuous and discrete in nature. Some texts will differentiate notation between discrete and continuous
cases, but this text will not make such a distinction. Instead we will opt to use the generic coefficient
matrices A, B, C and D for both continuous and discrete systems. Occasionally this book may employ the
subscript C to denote a continuous-time version of the matrix, and the subscript D to denote the discrete-time
version of the same matrix. Other texts may use the letters F, H, and G for continuous systems and Γ,
and Θ for use in discrete systems. However, if we keep track of our time-domain system, we don't need to
worry about such notations.