into a vector.
This is the system output value, and in the case of MIMO systems, we may have several. Output
variables should be independent of one another, and only dependent on a linear combination of the
input vector and the state vector.
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 system
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 equation:
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.