In this video, we are going to look at linear dynamics systems and the state based representation of
the systems, a system can be considered a collection of interrelated entities or in our case, differential
equations that can be considered as a whole.
If the different processes that make up the systems change of time, then it is considered as a dynamic
system.
The differential equations that make up the system are called the state equations of the dynamic system.
The state variables of the system are dependent variables of those state equations.
So if we consider the system of time varying first order differential equations as shown on slide here,
and we'll be using the Newton's dot notation.
So the dot notation just means that if the variable has a dot over the top, it means we just taking
the derivative of that variable with respect to time.
So I can say these here are first order differential equations.
They're a function of time of the different states x different inputs.
You and they calculate the state rates or the rate of change of the variable or state.
So on the slide here we have instead equations and each of these equations are going to be a time varying.
So you can see that they're a function of time and we have nine states.
So x1 x2 all the way up to extend all the different states of the equations.
And you can see for every state we have a corresponding state right equation that calculates the derivative
of that state.
We also have an input.
So you can see we have you want you to own up to um.
So each of these functions here are a function of the the state's inputs and time.
And you can see that all these are correlated with each other.
So each of these functions can be a function of the other states themselves.
So this is why it's a system of interrelated entities, because each function can be a function of the
different states.
So how all these evolve time is a function of how all the other states and inputs evolve with time.
So the system of equations in the last slide can be completely written in this form here.
So instead of writing out all the different functions individually, what we do is we change the representations
so that we write everything out in vector form.
So here we have a vector of the states and the vector of the inputs to get the vector of the derivatives.
So here exactly is a vector of the different states you have here is a vector of the different inputs
and then X of T here is going to be the derivative of the state vector with respect to time.
So this is normally called the state vector.
This is normally called the input vector.
This derivative here is usually called the state, right?
So it's a derivative of the states with respect to time and if function.
Here are the state equations.
Many of the processes that occur in the world can be considered as linear or nonlinear differential
equations.
We can describe these processes in state's best form, which then allows us to use different mathematical
tools to extract useful information and perform various analysis on the system.
So using state's best form, we've come up with a standard mathematical tool that allows us to analyze
systems in general so we can write something in the state's best form.
Then we can use these tools that we've developed for other systems on the system.
If we know the state of the system for the current time and all the current and future inputs to that
system, then we can predict values of the future states and the future outputs of that system and a
lot more.
So if we consider the solar system as a dynamic system, then we can use the same process on this.
Isaac Newton's first use of calculus was to describe the differential equations of the orbit of objects
in the solar system.
So if we consider the solar system as a dynamic system, if you know the position of the planets for
a given point of time, then you can predict where the planets will be at any point in the future or
in the past by using these states based analysis.
So this is the power of using state space allows us to use different mathematical tools and allows us
to completely predict or estimate different values in the future or the past of that system.