Backstepping
In control theory, backstepping is a technique developed circa 1990 by Petar V. Kokotovic and others for designing stabilizing controls for a special class of nonlinear dynamical systems. These systems are built from subsystems that radiate out from an irreducible subsystem that can be stabilized using some other method. Because of this recursive structure, the designer can start the design process at the known-stable system and "back out" new controllers that progressively stabilize each outer subsystem. The process terminates when the final external control is reached. Hence, this process is known as backstepping.
Backstepping approach
The backstepping approach provides a recursive method for stabilizing the origin of a system in strict-feedback form. That is, consider a system of the formwhere
- with,
- are scalars,
- is a scalar input to the system,
- vanish at the origin,
- are nonzero over the domain of interest.
is stabilized to the origin by some known control such that. It is also assumed that a Lyapunov function for this stable subsystem is known. That is, this subsystem is stabilized by some other method and backstepping extends its stability to the shell around it.
In systems of this strict-feedback form around a stable subsystem,
- The backstepping-designed control input has its most immediate stabilizing impact on state.
- The state then acts like a stabilizing control on the state before it.
- This process continues so that each state is stabilized by the fictitious "control".
Recursive Control Design Overview
- It is given that the smaller subsystem
- ::
- :is already stabilized to the origin by some control where. That is, choice of to stabilize this system must occur using some other method. It is also assumed that a Lyapunov function for this stable subsystem is known. Backstepping provides a way to extend the controlled stability of this subsystem to the larger system.
- A control is designed so that the system
- ::
- :is stabilized so that follows the desired control. The control design is based on the augmented Lyapunov function candidate
- ::
- :The control can be picked to bound away from zero.
- A control is designed so that the system
- ::
- :is stabilized so that follows the desired control. The control design is based on the augmented Lyapunov function candidate
- ::
- :The control can be picked to bound away from zero.
- This process continues until the actual is known, and
- * The real control stabilizes to fictitious control.
- * The fictitious control stabilizes to fictitious control.
- * The fictitious control stabilizes to fictitious control.
- *...
- * The fictitious control stabilizes to fictitious control.
- * The fictitious control stabilizes to fictitious control.
- * The fictitious control stabilizes to the origin.
- vanish at the origin for,
- are nonzero for,
- the given control has,
Integrator Backstepping
Before describing the backstepping procedure for general strict-feedback form dynamical systems, it is convenient to discuss the approach for a smaller class of strict-feedback form systems. These systems connect a series of integrators to the input of asystem with a known feedback-stabilizing control law, and so the stabilizing approach is known as integrator backstepping. With a small modification, the integrator backstepping approach can be extended to handle all strict-feedback form systems.
Single-integrator Equilibrium
Consider the dynamical systemwhere and is a scalar. This system is a cascade connection of an integrator with the subsystem.
We assume that, and so if, and, then
So the origin is an equilibrium of the system. If the system ever reaches the origin, it will remain there forever after.
Single-integrator Backstepping
In this example, backstepping is used to stabilize the single-integrator system in Equation around its equilibrium at the origin. To be less precise, we wish to design a control law that ensures that the states return to after the system is started from some arbitrary initial condition.- First, by assumption, the subsystem
- * The function is like a "generalized energy" of the subsystem. As the states of the system move away from the origin, the energy also grows.
- * By showing that over time, the energy decays to zero, then the states must decay toward. That is, the origin will be a stable equilibrium of the system – the states will continuously approach the origin as time increases.
- * Saying that is positive definite means that everywhere except for, and.
- * The statement that means that is bounded away from zero for all points except where. That is, so long as the system is not at its equilibrium at the origin, its "energy" will be decreasing.
- * Because the energy is always decaying, then the system must be stable; its trajectories must approach the origin.
- Next, by adding and subtracting to the part of the larger system, it becomes
- We now can change variables from to by letting. So
- From our existing Lyapunov function, we define the augmented Lyapunov function candidate
- Our choice of control ultimately depends on all of our original state variables. In particular, the actual feedback-stabilizing control law
Motivating Example: Two-integrator Backstepping
Before discussing the recursive procedure for the general multiple-integrator case, it is instructive to study the recursion present in the two-integrator case. That is, consider the dynamical systemwhere and and are scalars. This system is a cascade connection of the single-integrator system in Equation with another integrator.
By letting
- ,
- ,
By the single-integrator procedure, the control law stabilizes the upper -to- subsystem using the Lyapunov function, and so Equation is a new single-integrator system that is structurally equivalent to the single-integrator system in Equation . So a stabilizing control can be found using the same single-integrator procedure that was used to find.
Many-integrator backstepping
In the two-integrator case, the upper single-integrator subsystem was stabilized yielding a new single-integrator system that can be similarly stabilized. This recursive procedure can be extended to handle any finite number of integrators. This claim can be formally proved with mathematical induction. Here, a stabilized multiple-integrator system is built up from subsystems of already-stabilized multiple-integrator subsystems.- First, consider the dynamical system
- * so that the zero-input system is stationary at the origin. In this case, the origin is called an equilibrium of the system.
- *The feedback control law stabilizes the system at the equilibrium at the origin.
- *A Lyapunov function corresponding to this system is described by.
- Next, connect an integrator to input so that the augmented system has input and output states. The resulting augmented dynamical system is
- Connect a new integrator to input so that the augmented system has input and output states. The resulting augmented dynamical system is
- Connect an integrator to input so that the augmented system has input and output states. The resulting augmented dynamical system is
- This process can continue for each integrator added to the system, and hence any system of the form
Generic Backstepping
Systems in the special strict-feedback form have a recursive structure similar to the many-integrator system structure. Likewise, they are stabilized by stabilizing the smallest cascaded system and then backstepping to the next cascaded system and repeating the procedure. So it is critical to develop a single-step procedure; that procedure can be recursively applied to cover the many-step case. Fortunately, due to the requirements on the functions in the strict-feedback form, each single-step system can be rendered by feedback to a single-integrator system, and that single-integrator system can be stabilized using methods discussed above.Single-step Procedure
Consider the simple strict-feedback systemwhere
- ,
- and are scalars,
- For all and,.
which is possible because. So the system in Equation is
which simplifies to
This new -to- system matches the single-integrator cascade system in Equation . Assuming that a feedback-stabilizing control law and Lyapunov function for the upper subsystem is known, the feedback-stabilizing control law from Equation is
with gain. So the final feedback-stabilizing control law is
with gain. The corresponding Lyapunov function from Equation is
Because this strict-feedback system has a feedback-stabilizing control and a corresponding Lyapunov function, it can be cascaded as part of a larger strict-feedback system, and this procedure can be repeated to find the surrounding feedback-stabilizing control.
Many-step Procedure
As in many-integrator backstepping, the single-step procedure can be completed iteratively to stabilize an entire strict-feedback system. In each step,- The smallest "unstabilized" single-step strict-feedback system is isolated.
- Feedback is used to convert the system into a single-integrator system.
- The resulting single-integrator system is stabilized.
- The stabilized system is used as the upper system in the next step.
has the recursive structure
and can be feedback stabilized by finding the feedback-stabilizing control and Lyapunov function for the single-integrator subsystem and iterating out from that inner subsystem until the ultimate feedback-stabilizing control is known. At iteration, the equivalent system is
By Equation , the corresponding feedback-stabilizing control law is
with gain. By Equation , the corresponding Lyapunov function is
By this construction, the ultimate control .
Hence, any strict-feedback system can be feedback stabilized using a straightforward procedure that can even be automated.