_{1}

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Backstepping technique usually adopts back step design to construct the Lyapunov function gradually, and then to design the corresponding virtue controller. The backstepping technique based on error also adopts back step design process, but the design of virtue controllers depends on the corresponding errors which are designed to satisfy some expected behaviors. Six different error equations are deduced by chang
ing the results of the virtue controls arbitrarily while guaranteeing the system behaviors such as stability, and an example shows the effectiveness of these six versions. Simulated results illustrate that these
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six versions of backstepping technique based on error are effective.

Backstepping design methodology is among the most important nonlinear control design techniques with numerous applications. It was first presented by P. V. Kokotovic and his coauthors, see [_{G}V-backstepping. In reference [

In all the references mentioned above, the purpose of backstepping design methodology is the construction of various types of control Lyapunov functions: stable, adaptive, robust, etc. D. Swaroop et al. [

A new redesign backstepping technique, backstepping technique based on error is presented in this paper, which also adopts backstepping design process, but doesn’t construct the system, control Lyapunov function; the design of the virtue controller depends on the corresponding errors which are designed to satisfy some expected behaviors. The method is also more flexible than DSC. Based on some results of [

In this paper, we call backstepping technique based on control Lyapunov functions as conventional backstepping, and call backstepping technique based on error as error backstepping. Section 2 presents the design and six results of error backstepping, in Section 3 an example shows the effectiveness of these six versions, and Section 4 concludes this paper.

Consider a usual strictly feedback nonlinear system as follows:

(1)

where

The backstepping design process is as follows.

Step 1: Consider the control goal

Define the first tracking error

Error

Then the following equality corresponds to desired behaviour

Step 2: but we cannot just choose

Introduce

Define the second tracking error:

The error

Then the following equality corresponds to desired behavior.

Step i: Choose

Select its time derivative to satisfy the following behaviou

Introduce

Choose

Then has

Step n: Choose u to solve the

Its derivative satisfies.

In fact:

The last equality corresponds to what we are forcing.

The deduced error equation is:

Defining

Proposition 2.1: When choose parameters

where

Proposition 2.2: When

Proof: error Equation (19) is in short as

Introduce

Its time derivative is

If parameters

End of proof.

Proposition 2.3: Assume the expression of virtual control (14) is changed into:

where

Proof: when virtual control (24) takes place (14) in recursive procedure, the new error equation is changed into

or

It is the same as (2.27). Then all the errors converge to origin globally.

End of proof.

Proposition 3.4: Similarly assume the expression of virtual control (8) is changed into:

where

or

When choose the parameters

Proof:

Defining

where

(29)

Then

Introduce a positive definite quadratic function

It can be obtained

Then

End of proof.

Proposition 2.5: The virtual control can also be choose as

where

or

When choose parameters

Errors

Proof:

Define

(36)

Then it can be obtained

where

Introduce a positive definite function

Because

So

It is obvious that

Substituting (42) into (43), it can be deduced that

End of proof.

Proposition 2.6: The virtual control can also be choose as

where

or

When choose parameters

Errors

Proof: it is similar to the proof of Proposition 2.5.

Consider the following two-order system:

The control objective is to design a state feedback control to asymptotically stabilize the origin.

We adopt backstepping technique based on error to design control law. The calculated results are presented in

As it is shown in

Virtual control | Control law | error equation | |
---|---|---|---|

The backstepping technique based on error is the expansion of the backstepping technique, it adopts backstepping design process, but the design of the virtue controller depends on the corresponding errors which are designed to satisfy some expected behaviors. This method makes the design systematical and structural, and it can change the result of the virtue control law arbitrarily in six forms while guaranteeing the system stability. The method can be used for both stabilization control problems and tracking control problems. Subjects of future research include the discussions of systems that contain uncertain terms, unknown parameters or unmeasured signals.

Wang, L. (2016) Backstepping Technique Based on Error. Open Access Library Journal, 3: e3061. http://dx.doi.org/10.4236/oalib.1103061