Backstepping control for nonlinear systems
Pasanen, Jussi (2024)
2024
Teknis-luonnontieteellinen DI-ohjelma - Master's Programme in Science and Engineering
Tekniikan ja luonnontieteiden tiedekunta - Faculty of Engineering and Natural Sciences
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Hyväksymispäivämäärä
2024-07-30
Julkaisun pysyvä osoite on
https://urn.fi/URN:NBN:fi:tuni-202406297443
https://urn.fi/URN:NBN:fi:tuni-202406297443
Tiivistelmä
In systems theory a nonlinear system is a mathematical model that describes the behaviour of a state in a space with the use of differential equations. The state describes a point in the space and the differential equations describe the movement of the state in the space at a given time. Systems can also have some controllable outside inputs present in it, which are called control inputs. Controlling a system is a process in which the control input within a system is used to affect the behaviour of the state of a system.
The goal of this thesis is to familiarize the reader with the concept of stability and describe one nonlinear control method and give examples along the way to give better understanding of the method described. We introduce the backstepping method for two dimensional systems and see that the recursive application of this process can be used to stabilize larger systems.
Backstepping is a nonlinear control method, in which systems with a cascading structure are recursively stabilized starting from a one dimensional subsystem. We show that if a subsystem can be stabilized using one state variable as a "pseudo-controller" the controller can add an integrator to this "pseudo-controller" and use it to stabilize the next subsystem within a system. This process can be repeated until the whole system is stabilized and the process terminates when the actual stabilizing controller is designed using all of the controllers from previous steps.
We see that backstepping is especially useful when stabilizing systems in a strict feedback form, because the stability reached with backstepping for this class of systems is global and asymptotic. Other systems can also be stabilized with backstepping, but then the stability is contained within some neighbourhood around the origin.
We also show that backstepping can be implemented to stabilize systems with uncertainties, when the uncertainties fall under certain assumptions. These uncertainties can be any undesired forces affecting the system, which the controller wants to terminate as they affect the behaviour of the system. This process is called adaptive backstepping and we see that by overpowering or recursively estimating these uncertainties we can cancel them out with the final control input, so they do not affect the stability of a system.
The goal of this thesis is to familiarize the reader with the concept of stability and describe one nonlinear control method and give examples along the way to give better understanding of the method described. We introduce the backstepping method for two dimensional systems and see that the recursive application of this process can be used to stabilize larger systems.
Backstepping is a nonlinear control method, in which systems with a cascading structure are recursively stabilized starting from a one dimensional subsystem. We show that if a subsystem can be stabilized using one state variable as a "pseudo-controller" the controller can add an integrator to this "pseudo-controller" and use it to stabilize the next subsystem within a system. This process can be repeated until the whole system is stabilized and the process terminates when the actual stabilizing controller is designed using all of the controllers from previous steps.
We see that backstepping is especially useful when stabilizing systems in a strict feedback form, because the stability reached with backstepping for this class of systems is global and asymptotic. Other systems can also be stabilized with backstepping, but then the stability is contained within some neighbourhood around the origin.
We also show that backstepping can be implemented to stabilize systems with uncertainties, when the uncertainties fall under certain assumptions. These uncertainties can be any undesired forces affecting the system, which the controller wants to terminate as they affect the behaviour of the system. This process is called adaptive backstepping and we see that by overpowering or recursively estimating these uncertainties we can cancel them out with the final control input, so they do not affect the stability of a system.