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BacksteppingFrom Wikipedia, the free encyclopedia
In control theory, backstepping is a technique developed circa 1990 by Petar V. Kokotovic and others[1][2] 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.[3]
Contents
1 Recursive Control Design Overview
2 Integrator Backstepping
2.1 Single-integrator Equilibrium
2.2 Single-integrator Backstepping
2.3 Motivating Example: Two-integrator Backstepping
2.4 Many-integrator backstepping
3 Generic Backstepping
3.1 Single-step Procedure
3.2 Many-step Procedure
4 Flexible-joint manipulators (the Lozano-Brogliato's scheme)
5 See also
6 References
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 form[3]
where
with ,
are scalars,
u is a scalar input to the system,
vanish at the origin (i.e., ),
are nonzero over the domain of interest (i.e., for ).
Also assume that the subsystem
is stabilized to the origin (i.e., ) by some known control such that . It is also assumed that a Lyapunov function Vx 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 u has its most immediate stabilizing impact on state zn.
The state zn then acts like a stabilizing control on the state zn − 1 before it.
This process continues so that each state zi is stabilized by the fictitious "control" zi + 1.
The backstepping approach determines how to stabilize the subsystem using z1, and then proceeds with determining how to make the next state z2 drive z1 to the control
required to stabilize . Hence, the process "steps backward" from out of the strict-feedback form system until the ultimate control u is designed.
Recursive Control Design Overview
1. It is given that the smaller (i.e., lower-order) subsystem
is already stabilized to the origin by some control where . That is, choice of ux to stabilize this system must occur using some other method. It
is also assumed that a Lyapunov function Vx for this stable subsystem is known. Backstepping provides a way to extend the controlled stability of this subsystem to
the larger system.
2. A control is designed so that the system
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is stabilized so that z1 follows the desired ux control. The control design is based on the augmented Lyapunov function candidate
The control u1 can be picked to bound away from zero.
3. A control is designed so that the system
is stabilized so that z2 follows the desired u1 control. The control design is based on the augmented Lyapunov function candidate
The control u2 can be picked to bound away from zero.
4. This process continues until the actual u is known, and
The real control u stabilizes zk to fictitious control uk − 1.
The fictitious control uk − 1 stabilizes zk − 1 to fictitious control uk − 2.
The fictitious control uk − 2 stabilizes zk − 2 to fictitious control uk − 3.
…
The fictitious control u2 stabilizes z2 to fictitious control u1.
The fictitious control u1 stabilizes z1 to fictitious control ux.
The fictitious control ux stabilizes to the origin.
This process is known as backstepping because it starts with the requirements on some internal subsystem for stability and progressively steps back out of the system, maintaining
stability at each step. Because
fi vanish at the origin for ,
gi are nonzero for ,
the given control ux has ,
then the resulting system has an equilibrium at the origin (i.e., where , z1 = 0, z2 = 0, … , zk − 1 = 0, and zk = 0) that is globally asymptotically stable.
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 a system 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 system
where and z1 is a scalar. This system is a cascade connection of an integrator with the subsystem (i.e., the input u enters an integrator, and the integral z1 enters the
subsystem).
We assume that , and so if u1 = 0, and z1 = 0, then
So the origin is an equilibrium (i.e., a stationary point) 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 (1) 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
with has a Lyapunov function such that
where is a positive-definite function. That is, we assume that we have already shown that this existing simpler subsystem is stable (in the sense of
Lyapunov). Roughly speaking, this notion of stability means that:
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The function Vx 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.
Our task is to find a control u that makes our cascaded system also stable. So we must find a new Lyapunov function candidate for this new system. That
candidate will depend upon the control u, and by choosing the control properly, we can ensure that it is decaying everywhere as well.
Next, by adding and subtracting (i.e., we don't change the system in any way because we make no net effect) to the part of the larger system,
it becomes
which we can re-group to get
So our cascaded supersystem encapsulates the known-stable subsystem plus some error perturbation generated by the integrator.
We now can change variables from to by letting . So
Additionally, we let so that and
We seek to stabilize this error system by feedback through the new control v1. By stabilizing the system at e1 = 0, the state z1 will track the desired control ux which will
result in stabilizing the inner subsystem.
From our existing Lyapunov function Vx, we define the augmented Lyapunov function candidate
So
By distributing , we see that
To ensure that (i.e., to ensure stability of the supersystem), we pick the control law
with k1 > 0, and so
After distributing the e1 through,
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So our candidate Lyapunov function V1 is a true Lyapunov function, and our system is stable under this control law v1 (which corresponds the control law u1 because
). Using the variables from the original coordinate system, the equivalent Lyapunov function
As discussed below, this Lyapunov function will be used again when this procedure is applied iteratively to multiple-integrator problem.
Our choice of control v1 ultimately depends on all of our original state variables. In particular, the actual feedback-stabilizing control law
The states and z1 and functions fx and gx come from the system. The function ux comes from our known-stable subsystem. The gain parameter k1 > 0affects the convergence rate or our system. Under this control law, our system is stable at the origin .
Recall that u1 in Equation (3) drives the input of an integrator that is connected to a subsystem that is feedback-stabilized by the control law ux. Not surprisingly, the control
u1 has a term that will be integrated to follow the stabilizing control law plus some offset. The other terms provide damping to remove that offset and any other
perturbation effects that would be magnified by the integrator.
So because this system is feedback stabilized by and has Lyapunov function with , it can be used as the upper subsystem
in another single-integrator cascade system.
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 system
where and z1 and z2 are scalars. This system is a cascade connection of the single-integrator system in Equation (1) with another integrator (i.e., the input u2 enters
through an integrator, and the output of that integrator enters the system in Equation (1) by its u1 input).
By letting
,
,
then the two-integrator system in Equation (4) becomes the single-integrator system
By the single-integrator procedure, the control law stabilizes the upper z2-to- subsystem using the Lyapunov function , and so Equation (5) is
a new single-integrator system that is structurally equivalent to the single-integrator system in Equation (1). So a stabilizing control u2 can be found using the same single-integrator
procedure that was used to find u1.
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
that has scalar input ux and output states . Assume that
so that the zero-input (i.e., ux = 0) 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 .
That is, if output states are fed back to the input ux by the control law , then the output states (and the Lyapunov function) return to the origin after a single
perturbation (e.g., after a nonzero initial condition or a sharp disturbance). This subsystem is stabilized by feedback control law ux.
Next, connect an integrator to input ux so that the augmented system has input u1 (to the integrator) and output states . The resulting augmented dynamical system is
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This "cascade" system matches the form in Equation (1), and so the single-integrator backstepping procedure leads to the stabilizing control law in Equation (3). That is, if we
feed back states z1 and to input u1 according to the control law
with gain k1 > 0, then the states z1 and will return to z1 = 0 and after a single perturbation. This subsystem is stabilized by feedback control law u1, and the
corresponding Lyapunov function from Equation (2) is
That is, under feedback control law u1, the Lyapunov function V1 decays to zero as the states return to the origin.
Connect a new integrator to input u1 so that the augmented system has input u2 and output states . The resulting augmented dynamical system is
which is equivalent to the single-integrator system
Using these definitions of , f1, and g1, this system can also be expressed as
This system matches the single-integrator structure of Equation (1), and so the single-integrator backstepping procedure can be applied again. That is, if we feed back states
z1, z2, and to input u2 according to the control law
with gain k2 > 0, then the states z1, z2, and will return to z1 = 0, z2 = 0, and after a single perturbation. This subsystem is stabilized by feedback control law
u2, and the corresponding Lyapunov function is
That is, under feedback control law u2, the Lyapunov function V2 decays to zero as the states return to the origin.
Connect an integrator to input u2 so that the augmented system has input u3 and output states . The resulting augmented dynamical system is
which can be re-grouped as the single-integrator system
By the definitions of , f1, and g1 from the previous step, this system is also represented by
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Further, using these definitions of , f2, and g2, this system can also be expressed as
So the re-grouped system has the single-integrator structure of Equation (1), and so the single-integrator backstepping procedure can be applied again. That is, if we feed
back states z1, z2, z3, and to input u3 according to the control law
with gain k3 > 0, then the states z1, z2, z3, and will return to z1 = 0, z2 = 0, z3 = 0, and after a single perturbation. This subsystem is stabilized by
feedback control law u3, and the corresponding Lyapunov function is
That is, under feedback control law u3, the Lyapunov function V3 decays to zero as the states return to the origin.
This process can continue for each integrator added to the system, and hence any system of the form
has the recursive structure
and can be feedback stabilized by finding the feedback-stabilizing control and Lyapunov function for the single-integrator subsystem (i.e., with input z2 and output
) and iterating out from that inner subsystem until the ultimate feedback-stabilizing control u is known. At iteration i, the equivalent system is
The corresponding feedback-stabilizing control law is
with gain ki > 0. The corresponding Lyapunov function is
By this construction, the ultimate control (i.e., ultimate control is found at final iteration i = k).
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Hence, any system in this special many-integrator strict-feedback form can be feedback stabilized using a straight-forward procedure that can even be automated (e.g., as part of
an adaptive control algorithm).
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 system
where
,
z1 and u1 are scalars,
For all and z1, .
Rather than designing feedback-stabilizing control u1 directly, introduce a new control ua1 (to be designed later) and use control law
which is possible because . So the system in Equation (6) is
which simplifies to
This new ua1-to- system matches the single-integrator cascade system in Equation (1). Assuming that a feedback-stabilizing control law and Lyapunov function
for the upper subsystem is known, the feedback-stabilizing control law from Equation (3) is
with gain k1 > 0. So the final feedback-stabilizing control law is
with gain k1 > 0. The corresponding Lyapunov function from Equation (2) 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,
1. The smallest "unstabilized" single-step strict-feedback system is isolated.
2. Feedback is used to convert the system into a single-integrator system.
3. The resulting single-integrator system is stabilized.
4. The stabilized system is used as the upper system in the next step.
That is, any strict-feedback system
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has the recursive structure
and can be feedback stabilized by finding the feedback-stabilizing control and Lyapunov function for the single-integrator subsystem (i.e., with input z2 and output ) and
iterating out from that inner subsystem until the ultimate feedback-stabilizing control u is known. At iteration i, the equivalent system is
By Equation (7), the corresponding feedback-stabilizing control law is
with gain ki > 0. By Equation (8), the corresponding Lyapunov function is
By this construction, the ultimate control (i.e., ultimate control is found at final iteration i = k). Hence, any strict-feedback system can be
feedback stabilized using a straight-forward procedure that can even be automated (e.g., as part of an adaptive control algorithm).
Flexible-joint manipulators (the Lozano-Brogliato's scheme)
Almost simultaneously with Kanellakopoulos-Kokotovic-Morse (whose paper appeared in the 1991 November issue of IEEE Transactions on Automatic Control), Lozano and
Brogliato published a paper (see reference [2] below) in the 1992 February issue of the same journal, which proposed a solution for the adaptive tracking control of a class of
underactuated mechanical systems that possess a triangular form: flexible-joint manipulators. This controller, that belongs to the class of what has been named afterwards passivity-
based control, is in essence designed from a backstepping method, though this was not called this way by the authors (and has remained a largely ignored fact in the Control
literature). This is obvious looking at equation (6) and the derivations of Appendix B in [2]: the signal q2d in (6) plays the role of the "fictitious" input of the backstepping procedure.
This control algorithm may be seen also as the extension towards flexible-joint systems, of the celebrated Slotine-Li's scheme for rigid manipulators. It is noteworthy that this
adaptive scheme remains the only available solution for the global tracking control of such mechanical systems, without the a priori knowledge of the inertial parameters and of the
stiffness matrix K. This last point is quite important because the stiffness matrix is precisely the term which pre-multiplies the "fictitious" input in the backstepping process. Relaxing
its a priori knowledge is therefore a hard task. Even in the fixed-parameter case the proposed controller has a strong interest due to its intrinsic robustness properties. Experiments
and comparisons with other control methods have been presented in.[4][5] In [6] it is shown that the fixed-parameter Lozano-Brogliato's scheme has the advantage over other
backstepping-based schemes, that the Lyapunov function of the closed-loop system is fixed in advance (it looks like the total mechanical energy of the system), and allows one to
avoid undesirable behaviors that a "blind" backstepping design may have (like high-gain in the loop that may create undesirable oscillations which may hamper the practical
applications to work well). Further informations may be found in the monograph.[7]
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See also
Nonlinear control
Strict-feedback form
Robust control
Adaptive control
References
1. ^ Kokotovic, P.V. (1992). "The joy of feedback: nonlinear and adaptive" (http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=165507) . Control Systems Magazine, IEEE 12 (3): 7–
17. doi:10.1109/37.165507 (http://dx.doi.org/10.1109%2F37.165507) . http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=165507. Retrieved 2008-04-13.
2. ^ R. Lozano, B. Brogliato (1992). "Adaptive control of robot manipulators with flexible joints". IEEE Transactions on Automatic Control, 37 (2): 174–181.
3. ̂a b Khalil, H.K. (2002). &onlinear Systems (http://www.egr.msu.edu/~khalil/NonlinearSystems/) (3rd ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-067389-7.
http://www.egr.msu.edu/~khalil/NonlinearSystems/.
4. ^ B., Brogliato et al (1998). "Experimental comparison of nonlinear controllers for flexible joint manipulators". Int. Journal of Robotics Research 17 (3): 260–281.
5. ^ B. Brogliato, D. Rey (1998). "Further experimental results on nonlinear control of flexible joint manipulators". Proc. of the American Control Conference 4: 2209–2211.
6. ^ B., Brogliato et al (1995). "Global tracking controllers for flexible-joint manipulators: a comparative study". Automatica 31 (7): 941–956.
7. ^ B. Brogliato, et al (2007). "Dissipative Systems Analysis and Control (2nd Edition)". Springer Verlag, Communications and Control Engineering.
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Categories: Control theory | Nonlinear control
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