Introduction to modern control theory + understanding

I think you have learned the theory of automatic control （ What is written at the back of this article is very casual ）

The state space

We can consider controlling the vehicle （ We assume that they are all linear systems ）, Our input has a steering wheel , throttle , Brake, etc . Our output has speed 、 The acceleration 、 angle 、 Angular velocity and other information . If it is automatic control theory , We build transfer functions for each input and output , Then analysis . It's troublesome , And not systematic .

Before entering modern control theory , A very important point is , We need to take Clarify the relationship between output and state . The speed we mentioned earlier 、 The acceleration 、 angle 、 Angular velocity is actually the state of the system , What we want to control is the state . Output is only a representation . It's like adding acceleration and speed together to output , Then reduce the speed and speed to output , It doesn't make any sense . But it can also restore the state of the system . The output just uses the sensor to show the state . State is something more essential than output .

therefore , We need to be more systematic 、 More essentially, it reflects The whole picture of the system , Modern control theory is coming .

What is state space ：
A car moving in a straight line ,$x=[r,\dot{r},\ddot{r}{]}^T$ The three states are position , Speed , The acceleration . These three quantities are enough to indicate the complete state of the car . And the possible assignment of all States , The space formed is the state space , If you add the direction of state change, it will form site , This is another story .
The following is what you will see when you first learn to control , Let me help you understand ：

$x$ Is the current state ,$u$ It's input ,$\dot{x}$ Is the rate of change of state , The three constitute a differential equation . In discrete form, it will become $x(n)=Ax(n-1)+Bx(n-1)$, Is it easier to understand , Namely Adding input to the current state will make the next state look like .
The second line $y$ Indicative output , The expression is in the current state and input , What results will you output （ What value will your sensor detect ）, Generally speaking D=0, Output is the reaction of state .

It's important ： State space representation is not unique , This is actually obvious , You can take $x=[x1,x2{]}^T$ become $x=[x1-x2,x1+x2{]}^T$, Through the previous state variables, we can calculate the following , The back one can also calculate the front one . The concrete manifestation is like this .

The next step will generally be how to write a state space expression according to the transfer function , Or vice versa , I won't even talk about it .

State space solution

Too lazy to write. . Let's learn how to solve it by ourselves

stability

Self control is BIBO（ Bounded input , Bounded output ）. The current control is different , Is to add a disturbance to a certain state , Can return to this state （ More specifically, you can take a look at this Lyapunov stability judgment ）

Too lazy to write. , later .