Backstepping Control

This example shows the modeling and simulation of a non-linear control system from [1] using

QFIRE Studio

 to implement a linear backstepping control law.

Introduction

Considering the system to be controlled:

x1˙=x1sinx1+x2x˙2=u\begin{align*}\dot{x_{1}}& = x_{1}\sin x_{1}+x_{2} \\ \dot{x}_{2}& = u\end{align*}

This system is represented by the diagram in Figure 1.

Figure 1 - Non-linear plant model

Simulation

The simple linear control law in Figure 2 is able to stabilize the non-linear plant.

u=k2(x2+k1x1)u = -k_{2}(x_{2}+k_{1}x_{1})

Figure 2 - Controller diagram

Figure 3 - Backstepping Control System

Figure 4 shows the response of the control system to the initial condition of the state vector

x1=20x_{1}=20
and
x2=0x_{2}=0
.

Figure 4 - State 1 (blue) and state 2 (orange)

The corresponding control signal is shown in Figure 5.

Figure 5 - Control signal computed by the control law in Figure 2

Bibliographic References

[1] Ola Härkegårds,  

BACKSTEPPING AND CONTROL ALLOCATION WITH APPLICATIONS TO FLIGHT CONTROL

,  2003.

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