Hello Everyone,

I have been studying control systems for sometime now. I came across state space variable analysis of a LTI system. Frankly i am just not able to figure out what it means. It just feels so stupid and senseless. I am so bad at it that i cant even come up with a question to ask in this thread. I know nothing about the topic. Please somebody pleaasee help me out by putting this topic into simpler words and most importantly giving an example to show how it works. Its so frustrating

 

The state variables are system variables that fully describe the system's condition in time at any given time. A good example is a pendulum.

To describe a pendulum taking into account gravity and friction, for small deflection angles, you need to know its angle (position) and its angular speed.

If you knew only its position at a given time, you couldn't tell if it's swinging left or right and how far it would reach.

If you knew only its speed at a given time, you couldn't tell if the system was at the beginning or at the end of its swing.

But knowing both angle and angular speed at a given time, you can tell exactly where it will be at any time in the future.

I hope this helps.
Keep in mind that system state variables don't always correlate with real measurable quantities as in our example.

 

Hello Georacer,

Your example has given me a good understanding of state space.However how do you practically put it to use using matrices and stuff...??

Regards

 

That is mathematical formality from now on, nothing too hard. For example, let's say that we have a car that accelerates as fast an the input U tells it to (acceleration=U). We want to know its position in any given time.

Let's set up our equations:
We will annotate position with X1 and velocity as X2.
\(
\left\ \begin{array}{c}
\frac{dX1}{dt}=X2\\
\frac{dX2}{dt}=U
\end{array} \right} \Leftrightarrow\\
\left\ \begin{array}{c}
\frac{dX1}{dt}=0*X1+1*X2+0*U\\
\frac{dX2}{dt}=0*X1+0*X2+1*U
\end{array} \right} \Leftrightarrow\\

\left[ \begin{array}{c} \dot{X1} \\ \dot{X2}\\ \end{array} \right] =\left[ \begin{array}{c} 0 & 1 \\ 0 & 0 \end{array} \right] \cdot \left[ \begin{array}{c} X1 \\ X2 \end{array} \right] +\left[ \begin{array}{c} 0 \\ 1 \end{array} \right] \cdot U \Leftrightarrow\\
\dot{X}= \left[ \begin{array}{c} 0 & 1 \\ 0 & 0 \end{array} \right] \cdot X+ \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] \cdot U
\)

Are you comfortable with how I set up and reformed my equation set? The selection of the state variables is the trickiest part.

 

@Georacer:

Thanks a tonne bro...u have just saved my life...i understood a lot of things from your posts. I wish the author of the book i am reading would have done the same. So basically you should have the knowledge of all the inputs given to the system and the state of the system at that point to determine a single parameter in time. The matrices have been introduced to make calculations a little easier in case of many inputs. In the example you have chosen inputs in such a way that only the position can be an output.However if we have many inputs and more than one output then my modeling of the system could be different than yours.

Regards

 

I didn't say anything about the system's output. It could be anything, from the position to any linear combination of position, speed and acceleration. That is matrix's C and D to define.

Matrices A and B describe only the internal system dynamics and its relation with its inputs.

Yes, the matrices only add an extreme ease to make the calculations, other than that it's the basic differential equations.