Balance car LQR high-level control

LQR (Linear Quadratic Regulator) is a commonly used high-order control method that can help the balancing car achieve stable movement.

LQR

LQR (Linear Quadratic Regulator) is a classic linear quadratic regulator used to design the optimal state feedback controller for continuous-time linear systems.

The performance of the system is optimized by minimizing a cost function, while considering the state and control input of the system.

LQR-Linear (Linear Model)

LQR-Linear refers to the use of the LQR algorithm for control design when the state space model of the system is linear.

The LQR-Linear algorithm can be used to calculate the optimal state feedback gain matrix K, and K can be adjusted to achieve stable control of the tilt angle and speed.

LQR-Quadratic (Quadratic Cost Function)

LQR-Quadratic refers to the use of a quadratic cost function to define the objective function, and the controller design is optimized by the LQR-Quadratic algorithm.

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The balance car can be adjusted to minimize the tilt angle deviation. We can add these factors to the quadratic cost function and use the LQR-Quadratic algorithm to design the corresponding optimal controller.

LQR-Regulator

LQR-Regulator can be used to adjust the state feedback matrix Q and the input feedback matrix R to achieve the regulation and stability of the system state.

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The response speed or stability of the balance car under different working conditions can be adjusted. The weight relationship between the controller's system state error and input power can be changed by adjusting the Q and R matrices, thereby achieving different regulation effects.

LQR advantages and disadvantages

LQR and PID

LQR (Linear Quadratic Regulator) and PID (Proportional-Integral-Derivative) are two commonly used control algorithms, which have some differences in design principles, adjustment methods and applicable scenarios.

LQR implementation ideas

LQR control block diagram: full-state feedback control system

A, B, C: state space equations

K: gain matrix

1. Establish a mathematical model of the balance car

Establish a dynamic model of the balance car, including the motion equations of the balance car, state space expressions, etc.

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State space equations can describe the dynamic characteristics of the system, including state variables, input variables, system matrices, and output matrices, etc.

2. Design state space expressions

Convert the dynamic model of the balance car into a state space form, that is, a set of linear equations that describe the relationship between the system state and input.

①: State equation

②: Output equation

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x is the system state vector, u is the control input vector

A, B, C, D are system parameter matrices

3. Design objective function

Based on the system performance indicators and requirements, design an objective function to evaluate the system performance.

③: Quadratic cost function

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Q and R are the state feedback matrix and input feedback matrix respectively

4. Design/implement LQR controller

Design LQR controller

Based on the state space expression and objective function of the system, use the LQR algorithm to solve the optimal controller gain matrix K.

Determine the optimal state feedback matrix Q and input feedback matrix R by minimizing the objective function.

Implement LQR controller

• Solve the optimal gain matrix K: Use the LQR algorithm to solve the optimal state feedback matrix Q and input feedback matrix R when the cost function is minimized.

④: Riccati equation → Solve the optimal control problem of the linear system

⑤: Solve the gain matrix K of the LQR controller

• Controller output: Control input 𝑢=−𝐾𝑥

Apply the optimal controller gain matrix K to the balancing car system, and adjust the tilt angle and speed in real time to maintain balance.

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The controller will adjust the control input in real time according to the current system state to achieve the best performance of the system

5. Simulation verification

Matlab can directly calculate K using the lqr function

Use Matlab to verify the designed LQR high-order controller to check whether the system can work stably and meet the performance index requirements.

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By continuously optimizing parameters and adjusting strategies, the system can achieve better stability, robustness and performance

6. Practical application

You need to use MATLAB to run the program parameter_LQR.m and modify the corresponding parameters in the program to adapt to your own car

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Product supporting materials source code path: Attachment → Source code summary → 6.LQR → Matlab

Use SIMULINK to simulate it:

Simulation results:

Program running results

You can modify the parameters obtained by running the program to the corresponding code, and then fine-tune the parameters to achieve the best results!

Code Implementation

When running this case code, the balancing car can only retain the standard package accessories, and other advanced accessories need to be removed; keep the balancing car vertical to the ground at 90° before starting.

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Product supporting materials source code path: Attachment → Source code summary → 6.LQR → STM32_code

Implementation code

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//LQR状态反馈系数
float K1= -62.1608, K2= -72.9295, K3=-356.0969, K4=-35.7579 , K5=15.8114, K6=15.8114;
float K5OLD=15.8114, K6OLD=15.8114;

L_accel=-(K1*x_pose+K2*(x_speed-Target_x_speed)+K3*(angle_x-Target_angle_x)+K4*gyro_x+K5*angle_z+K6*(gyro_z-Target_gyro_z));
R_accel=-(K1*x_pose+K2*(x_speed-Target_x_speed)+K3*(angle_x-Target_angle_x)+K4*gyro_x-K5*angle_z-K6*(gyro_z-Target_gyro_z));