LQR control learning -lqr control matlab official tutorial -lqr controller_ Modeling and analysis of state space system with matlab/simulink

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LQR control

Ben blog Main records LQR Linear quadratic modulation system , Study There are two tutorials B Teaching video of the station .

The state space 4-LQR control MATLAB The official tutorial

Video link ： [ Original translation ] The state space 4-LQR control MATLAB The official tutorial

https://player.bilibili.com/player.html?aid=98406477

[ Original translation ] The state space 4-LQR control MATLAB The official tutorial

The author uses straightforward examples to describe LQR Of cost function Q Penalty cost for state quantity 、R For thrusters ( Input ) The penalty cost , By building J Objective equation , So yes J The smallest purpose , solve K, Thus, the gain matrix at the lowest cost is obtained .

LQR-UFO experiment ：

matlab Official code Original video website

test1

We adjust Q matrix ,penalize angular error ( Angle error cost ) by 1,penalize angular rate by 0.01 ( Angular velocity cost ), The cost of angular error is much greater than that of angular velocity .R = 1 ( Actuator input cost ).

The results are shown below , stay R penalize angular error At a high cost , Use lqr,Angular Error Finally, it can converge , And there is no significant overshoot .

test2

Adjust the Q matrix ,penalize angular error ( Angle error cost ) by 1,penalize angular rate by 100 ( Angular velocity cost )

The results are shown below , You can see , The acceleration converges faster , And stabilize , But correspondingly Angular Error There is no guarantee of convergence .

test3

Adjust the R matrix Penalize thruster effort The cost of the actuator is 3

The results are shown below , There is obviously less fuel to burn

【Advanced Control theory 】8_LQR controller _ State space system Matlab/Simulink Modeling analysis

Video link ：【Advanced Control theory 】8_LQR controller _ State space system Matlab/Simulink Modeling analysis

【Advanced Control theory 】8

Let's start with the formula

Aim to solve the appropriate Acl Closed loop matrix

Let's start with the formula

test1 Open loop system

simulink Simulation model download

Set up K1 、K2 = 0, Use scope Check the open-loop system ,x1 Displacement ;x2 Speed ;u Negative feedback input

As follows , You can see , Under the open-loop system ,x1 x2 Tend to diverge

test2 Add negative feedback input , Build a closed-loop system ,lqr To solve the parameter K

It will be set by itself Q R, Solved K, Write in simlink Medium k1 k2

The results are shown below , It can be seen that after adding the closed-loop system ,x1 x2 All tend to converge

test3 modify R, Controlled experiment

The whole closed loop is encapsulated into a three output Subsysytem, Conduct a control experiment

take Q Set the matrix to unit matrix ,R( Actuator cost ) Set to the original 100 times ,R = 10, And will calculate K1、K2 Enter into , Observe scope Output

The comparison results are as follows , Yellow is the above test2 The experimental results of , Blue test3 The experimental results of , As shown in the figure below ,scope1 The convergence speed of the yellow line is obviously faster than that of the blue line , Such as scope3 Shown , When promoted R After the loss weight of , Input ( The blue line ) Be more stable . For both systems , The end result is the same , Finally, the state convergence is realized . For the yellow line , We pay more attention to the effect of convergence 、 Response speed , And for the blue line , We pay more attention to energy consumption .

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