Understanding extreme value search control through three cases

Extremum search control is an adaptive control algorithm based on the search for the optimal extremum. Its core idea is to adjust the parameters of the controller by optimizing the performance index to gradually achieve adaptive regulation of the system. This algorithm is usually applied to nonlinear systems with uncertain parameters or that cannot be accurately modeled.

The extreme value search control method can be divided into two stages: the first stage is to establish a mathematical model of the system and determine the parameters of the initial controller by calculating the gradient information of the performance indicators; the second stage is based on an iterative optimization method, using the optimal extreme value search algorithm to continuously fine-tune the parameters of the controller to achieve adaptive adjustment of the control system.

In the second stage of extreme value search control, commonly used optimization algorithms include gradient descent method, Newton method, conjugate gradient method, etc. These algorithms can adjust the parameters of the controller according to the specified criteria to optimize the performance indicators of the system. Among them, the multi-parameter extreme value search method is a type of global extreme value search algorithm based on multi-parameter functions, which can ensure global convergence.

Extreme value search process

Verified by Simulink, this method can search for extreme points without building a model.

Extreme value search control

The figure below shows a block diagram of the extreme value search control applied to the control system. Since the ESC needs to track the minimum or maximum value, if the search signal is too smooth, it will affect the search process. Therefore, compared with the above extreme value search process, a high-pass filter is added between the control system and the extreme value search link to reduce the impact of low-frequency interference from the controlled object on the extreme value search process.

Extreme value search control is a data-driven control method. It is not necessary to describe the model establishment process like other control methods. Therefore, two specific examples will be used below to help readers understand.

Adaptive Gain Adjustment for Uncertain Linear Systems

The block diagram of the system using the extreme value search control method is as follows:

Build a Simulink simulation model.

The reference output and actual output are shown below

Anti-Lock Braking Using Extremum Search Control

The model for implementing anti-lock braking using extreme value search control is shown below.

Perform Simulink simulation, the controller is

function tau = fcn(v,w,lambda0,vdot,lambda)

m = 400;

B = 0.01;

R = 0.3;

I = 1;

c = 2;

if v<= 0.001

tau = 0;

end

tau = -c*I*v/R*(lambda-lambda0)-B*wI*w/v*vdot-m*R*vdot;

Simulation Results

From the above two simulation cases, we can conclude that extreme value search control is suitable for the following application scenarios:

• System optimization process: If the cost function is nonlinear or a nonlinear system, it is not easy to find the local optimum of the cost function using analytical methods.

• Complex systems: Searching state space to find optimal control strategies.

• Fuzzy system: used to find the optimal solution (note that it is not fuzzy control, but refers to a system with many unmodeled dynamics).

• Multivariable systems: Extreme value search methods can determine the optimal control strategy by considering the relationships between multiple variables.

The disadvantages of extreme value search control are also very obvious, including falling into local optimality and being unable to find the global optimal solution, slow search process, high dependence on initial values, and determination of parameters.