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{{one source|date=July 2011}} | |||
In [[control theory]], [[dynamical systems theory|dynamical systems]] are in '''strict-feedback form''' when they can be expressed as | |||
:<math>\begin{cases} \dot{\mathbf{x}} = f_0(\mathbf{x}) + g_0(\mathbf{x}) z_1\\ | |||
\dot{z}_1 = f_1(\mathbf{x},z_1) + g_1(\mathbf{x},z_1) z_2\\ | |||
\dot{z}_2 = f_2(\mathbf{x},z_1,z_2) + g_2(\mathbf{x},z_1,z_2) z_3\\ | |||
\vdots\\ | |||
\dot{z}_i = f_i(\mathbf{x},z_1, z_2, \ldots, z_{i-1}, z_i) + g_i(\mathbf{x},z_1, z_2, \ldots, z_{i-1}, z_i) z_{i+1} \quad \text{ for } 1 \leq i < k-1\\ | |||
\vdots\\ | |||
\dot{z}_{k-1} = f_{k-1}(\mathbf{x},z_1, z_2, \ldots, z_{k-1}) + g_{k-1}(\mathbf{x},z_1, z_2, \ldots, z_{k-1}) z_k\\ | |||
\dot{z}_k = f_k(\mathbf{x},z_1, z_2, \ldots, z_{k-1}, z_k) + g_k(\mathbf{x},z_1, z_2, \dots, z_{k-1}, z_k) u\end{cases}</math> | |||
where | |||
* <math>\mathbf{x} \in \mathbb{R}^n</math> with <math>n \geq 1</math>, | |||
* <math>z_1, z_2, \ldots, z_i, \ldots, z_{k-1}, z_k</math> are [[scalar (mathematics)|scalar]]s, | |||
* <math>u</math> is a [[scalar (mathematics)|scalar]] input to the system, | |||
* <math>f_0, f_1, f_2, \ldots, f_i, \ldots, f_{k-1}, f_k</math> [[vanish (mathematics)|vanish]] at the [[origin (mathematics)|origin]] (i.e., <math>f_i(0,0,\dots,0) = 0</math>), | |||
* <math>g_1, g_2, \ldots, g_i, \ldots, g_{k-1}, g_k</math> are nonzero over the domain of interest (i.e., <math>g_i(\mathbf{x},z_1,\ldots,z_k) \neq 0</math> for <math>1 \leq i \leq k</math>). | |||
Here, ''strict feedback'' refers to the fact that the [[nonlinear system|nonlinear]] functions <math>f_i</math> and <math>g_i</math> in the <math>\dot{z}_i</math> equation only depend on states <math>x, z_1, \ldots, z_i</math> that are ''fed back'' to that subsystem.<ref name="Khalil">{{cite book | |||
| last = Khalil | |||
| first = Hassan K. | |||
| authorlink = Hassan K. Khalil | |||
| year = 2002 | |||
| edition = 3rd | |||
| url = http://www.egr.msu.edu/~khalil/NonlinearSystems/ | |||
| isbn = 0-13-067389-7 | |||
| title = Nonlinear Systems | |||
| publisher = [[Prentice Hall]] | |||
| location = Upper Saddle River, NJ}}</ref> That is, the system has a kind of [[triangular matrix|lower triangular]] form. | |||
==Stabilization== | |||
:{{main|Backstepping}} | |||
Systems in strict-feedback form can be [[Lyapunov stability|stabilized]] by recursive application of [[backstepping]].<ref name="Khalil"/> That is, | |||
# It is given that the system | |||
#::<math>\dot{\mathbf{x}} = f_0(\mathbf{x}) + g_0(\mathbf{x}) u_x(\mathbf{x})</math> | |||
#:is already stabilized to the origin by some control <math>u_x(\mathbf{x})</math> where <math>u_x(\mathbf{0}) = 0</math>. That is, choice of <math>u_x</math> to stabilize this system must occur using some other method. It is also assumed that a [[Lyapunov function]] <math>V_x</math> for this stable subsystem is known. | |||
# A control <math>u_1(\mathbf{x},z_1)</math> is designed so that the system | |||
#::<math>\dot{z}_1 = f_1(\mathbf{x},z_1) + g_1(\mathbf{x},z_1) u_1(\mathbf{x},z_1)</math> | |||
#:is stabilized so that <math>z_1</math> follows the desired <math>u_x</math> control. The control design is based on the augmented Lyapunov function candidate | |||
#::<math>V_1(\mathbf{x},z_1) = V_x(\mathbf{x}) + \frac{1}{2}( z_1 - u_x(\mathbf{x}) )^2</math> | |||
#:The control <math>u_1</math> can be picked to bound <math>\dot{V}_1</math> away from zero. | |||
# A control <math>u_2(\mathbf{x},z_1,z_2)</math> is designed so that the system | |||
#::<math>\dot{z}_2 = f_2(\mathbf{x},z_1,z_2) + g_2(\mathbf{x},z_1,z_2) u_2(\mathbf{x},z_1,z_2)</math> | |||
#:is stabilized so that <math>z_2</math> follows the desired <math>u_1</math> control. The control design is based on the augmented Lyapunov function candidate | |||
#::<math>V_2(\mathbf{x},z_1,z_2) = V_1(\mathbf{x},z_1) + \frac{1}{2}( z_2 - u_1(\mathbf{x},z_1) )^2</math> | |||
#:The control <math>u_2</math> can be picked to bound <math>\dot{V}_2</math> away from zero. | |||
# This process continues until the actual <math>u</math> is known, and | |||
#* The ''real'' control <math>u</math> stabilizes <math>z_k</math> to ''fictitious'' control <math>u_{k-1}</math>. | |||
#* The ''fictitious'' control <math>u_{k-1}</math> stabilizes <math>z_{k-1}</math> to ''fictitious'' control <math>u_{k-2}</math>. | |||
#* The ''fictitious'' control <math>u_{k-2}</math> stabilizes <math>z_{k-2}</math> to ''fictitious'' control <math>u_{k-3}</math>. | |||
#* ... | |||
#* The ''fictitious'' control <math>u_2</math> stabilizes <math>z_2</math> to ''fictitious'' control <math>u_1</math>. | |||
#* The ''fictitious'' control <math>u_1</math> stabilizes <math>z_1</math> to ''fictitious'' control <math>u_x</math>. | |||
#* The ''fictitious'' control <math>u_x</math> stabilizes <math>\mathbf{x}</math> 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 | |||
* <math>f_i</math> vanish at the origin for <math>0 \leq i \leq k</math>, | |||
* <math>g_i</math> are nonzero for <math>1 \leq i \leq k</math>, | |||
* the given control <math>u_x</math> has <math>u_x(\mathbf{0}) = 0</math>, | |||
then the resulting system has an equilibrium at the '''origin''' (i.e., where <math> \mathbf{x}=\mathbf{0}\,</math>, <math>z_1=0</math>, <math>z_2=0</math>, ... , <math>z_{k-1}=0</math>, and <math>z_k=0</math>) that is [[Lyapunov function#Globally_asymptotically_stable_equilibrium|globally asymptotically stable]]. | |||
==See also== | |||
* [[Nonlinear control]] | |||
* [[Backstepping]] | |||
==References== | |||
{{reflist}} | |||
[[Category:Nonlinear control]] |
Revision as of 07:04, 20 January 2014
Template:One source In control theory, dynamical systems are in strict-feedback form when they can be expressed as
where
- with ,
- are scalars,
- is a scalar input to the system,
- vanish at the origin (i.e., ),
- are nonzero over the domain of interest (i.e., for ).
Here, strict feedback refers to the fact that the nonlinear functions and in the equation only depend on states that are fed back to that subsystem.[1] That is, the system has a kind of lower triangular form.
Stabilization
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Systems in strict-feedback form can be stabilized by recursive application of backstepping.[1] That is,
- It is given that the system
- 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.
- A control is designed so that the system
- A control is designed so that the system
- 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.
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
then the resulting system has an equilibrium at the origin (i.e., where , , , ... , , and ) that is globally asymptotically stable.
See also
References
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