Nonlinear Feedback Control

In nonlinear control problems, we have a system

Sometimes, the control impacts the state evolution in an affine manner.

Definition 52

A control affine system is given by the differential equation

When designing controllers, there is a wide variety of techniques we can use. Some simple techniques involve canceling out various types of nonlinearities in the system using the input. Here are some examples.

While these techniques can work, there are also more principled ways of designing controllers to satisfy different criteria, particularly for the case of control affine systems.

Control Lyapunov Functions

This result motivates the following definition.

Definition 53

Once we have a control lyapunov function, we can prove that it is possible to find a state feedback law that will make the origin globally asymptotically stable.

Theorem 27

Suppose that we have a control affine system, and we want to construct a control lyapunov function for the system.

Theorem 28

if

Definition 54

The small control property means that CLF will lead to a controller which has a small value that does not get too large when close to the equilibrium.

Hence, let

When the plant dynamics are naturally stabilizing, this controller exerts no control effort. When the plant dynamics are not naturally stabilizing, then the controller applies some control to stabilize the system. We can show that this is a minimum norm controller as it solves the optimization problem

Another type of controller is known as the Sontag controller.

Theorem 29

Feedback Linearization

SISO Case

Suppose we have a SISO control-affine system

Definition 55

If we differentiate again, then

Therefore, the relative degree of the system is essentially telling us which derivative of the output that we can control. By sequentially taking derivatives, we are essentially looking at the system

This change of coordinates allows us to put the system into a canonical form.

Definition 56

The normal form of a SISO control affine system is given by

With this parameterization, it is quite easy to see how we can make our system behave linearly. In particular, choose

Theorem 30

MIMO Case

Suppose instead we have a MIMO control affine system where

Definition 57

\label{thm:vector-relative-degree}

Definition 58

The normal form of a square MIMO system is given by

Theorem 31

Dynamic Extension

Sliding Mode Control

In sliding mode control, we design a controller

  1. Reaching Mode

  2. Sliding Mode

Backstepping

Definition 59

A system expressed in strict feedback form is given by

When systems are expressed in this way, we have a convenient method of designing controllers.

Theorem 32 (Backstepping Lemma)

the function

is a valid CLF and the control input

is a stabilizing controller.

If we apply Theorem 32 to a system expressed in strict feedback form, then we can recursively define controllers until we arrive at a controller for the full system.

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