Design switching function/sliding mode surface S(x)

Linear: $\text{[math]}$

Sliding surface parameter determination method: Hurwitz

The polynomial is guaranteed $\text{[math]}$ to be Hurwitz, i.e. the eigenvalues are in the left half plane.

It is mainly suitable for nonlinear systems that do not require high speed and accuracy

Nonlinear "Review of the Theory of Sliding Mode Variable Structure Control"

Terminal sliding mode control: reach the equilibrium point in a limited time, introduce nonlinear part $\text{[math]}$ , improve convergence speed

Integral sliding mode control: increasing the integral term of the state variable can weaken chattering and reduce errors, but when the initial state is too large, it will cause overshoot or saturation

Piecewise linear sliding mode control: divided into multi-segment linear sliding mode surfaces

Time-varying sliding mode surface: The time-varying sliding mode surface can change with the state or time of the system, so that the system always runs in the sliding mode state

Find the control law u

Design the reaching law and combine it with $\text{[math]}$ the equation to obtain the control law

Using the Lyapunov criterion, use $\text{[math]}$ , inversely deduce the control law that satisfies the conditions

In actual controller design, in order to ensure stability, the Lyapunov criterion cannot be implemented for each output, so the approach law method is often used.

common reaching law

Constant speed approach law: $\text{[math]}$ $\text{[math]}$

Exponential reaching law: $\text{[math]}$ $\text{[math]}$

Power reaching law: $\text{[math]}$ $\text{[math]}$

General reaching law: $\text{[math]}$ $\text{[math]}$

Lyapunov function design

There is no general method to construct Lyapunov function, each method has its own limitations and is tricky.

The commonly used Lyapunov function of linear sliding mode surface is $\text{[math]}$ , and its derivative is $\text{[math]}$

Design Steps for Sliding Mode Control

According to the system equation of state

design sliding surface

Find the expression of the controller by designing the reaching law

Use the Lyapunov function to prove stability and ensure that s=0 is reachable

If the control law is obtained by the back-calculation method, the Lyapunov function is designed first, and the control law is designed after derivation to meet the stability conditions.

Chattering

When the system reaches the sliding surface, the velocity is not 0, so it will pass through the sliding surface and cause chattering.

The root cause is a control discontinuity caused by the switching action of the switch.

Change the switching function (here is not the meaning of sliding mode surface) to reduce chattering:

Symbolic function: sgn(s)

saturation function $\text{[math]}$

Hyperbolic tangent function: $\text{[math]}$

change reaching law

Filtering method: Eliminate high-frequency chattering

Observer method: use the observer to observe external disturbances and uncertain items, and design the switching gain according to the observed values

Intelligent control method: fuzzy control, neural network control, optimization method

The difference between sliding mode controller and sliding mode observer

The sliding mode observer is a kind of dynamic system, which is a kind of dynamic system that obtains the estimated value of the state variable according to the measured value of the external variable (input and output) of the system. It is similar to the principle of sliding mode control, but has different uses.

Sliding mode observer design steps:

Build the equation of state of the system

Add the sliding mode control amount on the basis of the equation

Use MATLAB to test the appropriate parameter value

equivalent sliding mode control

Using reverse deduction to find the control law is similar to the equivalent sliding mode control, in the equivalent sliding mode control: $\text{[math]}$ . First, by $\text{[math]}$ obtaining the equivalent control $\text{[math]}$ , and then by $\text{[math]}$ , the control law in this formula is the whole u, and then obtain the switching robust term $\text{[math]}$