High Gain and High Frequency Controller

1. Introduction and System Setup

1.1 System Model

The system dynamics are given by:

where:

is the state vector.

is an uncertain function representing system dynamics or disturbances.

is the control input.

1.2 Control Objective

The goal is to make the system state track a desired state . This is achieved by driving the error to zero:

1.3 Uncertainty Bound

The uncertainty is assumed to be bounded:

where is a known bounding function.

1.4 Error Dynamics

The derivative of the error is:

2. General Control Law Structure

A common control law is proposed as:

where:

is a positive gain constant.

is a proportional feedback term.

is adesired value for x.

is an auxiliary control term designed to compensate for the uncertainty .

3. Controller Types based on Auxiliary Control ()

3.1 Sliding Mode Controller (SMC)

Auxiliary Control ():

This is a signum-like function scaled by the uncertainty bound .

Problem: The discontinuous nature of leads to chattering (high-frequency oscillations of the control input). This can be a huge challenge for actuators.

3.2 Continuous Approximation Controllers (High-Gain / High-Frequency Inspired)

The main idea is to compensate for the uncertainty with an adequate (smoother) input to mitigate chattering.

3.2.1 Controller Option A:

This auxiliary control term is continuous:

where is a small constant. (The source material also shows this form, written as , where is equivalent to ).

Lyapunov Stability Analysis:

Choose the Lyapunov candidate function: .

The derivative is:

Using :

Case 1: If (i.e., ): Then . So, . This is Negative Definite (N.D.), implying stability.
Case 2: If : Then . .

Since : . Substituting : .

Solving the differential inequality : Let , where . The solution is . Thus, . Substituting back :

Result: The system is Globally Uniformly Ultimately Bounded (GUUB).

As , the error converges to a bound:

If , then .

Trade-off: The control input . If (for better error convergence), then if . This is a significant practical issue. It is a trade-off!

Since < 1 for > 0 (or equal to 0 if ), it follows that < . Therefore:

Substituting :

3.2.2 Controller Option B:

This auxiliary control term is also continuous and provides a saturation-like effect:

(This form is also presented as ). The graphical representation shows a similar structure (where could be ), illustrating how this function smoothly approximates the signum function as . This approach is associated with high frequency control ideas.

Lyapunov Stability Analysis:

Lyapunov candidate function: .

The derivative is:

Using :

Simplifying the term in the parenthesis:

So,

Result: This differential inequality is the same as in Controller Option A. The system is Globally Uniformly Ultimately Bounded (GUUB). As , the error converges to the same bound:

Advantage over Option A: Consider . As (for ):

This means remains bounded (by ) as , unlike which tends to infinity. This makes Controller Option B more practical.

4. Summary

Sliding Mode Control (SMC) offers robustness but suffers from chattering.

Continuous approximations, inspired by high-gain and high-frequency concepts, aim to reduce chattering.

leads to GUUB, but the control effort can become unbounded as .
also achieves GUUB with the same error bound, but crucially, the control effort remains bounded as , offering a better trade-off.

Both continuous controllers result in an ultimate error bound of , indicating a trade-off between the smallness of (for accuracy) and the gain .