Introduction to State Space

In a state space representation we end up with a first order matrix differential equation:

Good things:

• An nth order differential equation (or transfer function) yields n first order differential equations (i.e., x is nx1 and A is nxn)
• A state space representation can be developed for any system (we will add it to our system model/differential equation/transfer function set of tools fir solving systems).
• It is useful because it is compact and easy to work with (even for large systems).
• It is useful because computers are adept at approximating solutions for first order differential equations.
• The choice of state variables is not unique - and some choices may make solution easier.

Potentially problematic things

• A choice for state variables is not necessarily obvious (especially if the input to the original nth order differential equation has derivatives of the input).  We will develop two standard ways to deal with this

By defining a system in state space form we can change an nth order differential equation (or transfer function) into n first order differential equations of the form:

where x is an n element column vector, y is the output and u is the input.  The formulation above is for a single-input single-output (SISO) system.  The input u can be a vector, as can the output y (either one makes the element "d" a matrix).

We discussed various manipulations that we can do with State Space representations of systems.

• Starting with the transfer function I showed you one way to convert to a state space system.  Your book calls this method 2 (There is also, unsurprisingly, a method 1).  These methods can be used without thought to do the transformation from state space to transfer function (the book gives the general case).
• Starting with the state space representation I showed how you can get the transfer function.
If

then

•  You can change from one state space representation , to another .  Given a state variable, , with n elements and a square nxn matrix, ,

if

then

(the state variables and matrices have changed, but the input and output have not).

Note, your book does everything in terms of P, Matlab uses T, where T=P-1.