Brockett’s condition for smooth feedback stabilization
E615222
Brockett’s condition for smooth feedback stabilization is a fundamental result in nonlinear control theory that provides a necessary topological criterion for when a nonlinear system can be stabilized by a smooth state-feedback law.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in nonlinear control theory ⓘ |
| alsoKnownAs |
Brockett’s necessary condition for smooth feedback stabilization
NERFINISHED
ⓘ
Brockett’s topological obstruction NERFINISHED ⓘ |
| appliesTo |
control-affine systems
ⓘ
finite-dimensional dynamical systems ⓘ nonlinear control systems ⓘ |
| assumes |
continuous-time dynamical system
ⓘ
smoothness of the feedback map ⓘ |
| basedOn |
differential topology
ⓘ
properties of continuous maps ⓘ topology ⓘ |
| clarifies |
difference between linear and nonlinear stabilizability
ⓘ
limitations of smooth static state feedback ⓘ |
| concerns |
asymptotic stabilization of equilibria
ⓘ
smooth state-feedback stabilization ⓘ topological obstructions to stabilization ⓘ |
| contrastsWith |
results on discontinuous feedback stabilization
ⓘ
results on time-varying feedback stabilization ⓘ |
| doesNotGive | sufficient condition for smooth stabilizing feedback ⓘ |
| field |
applied mathematics
ⓘ
control theory ⓘ nonlinear control theory ⓘ |
| gives | necessary condition for existence of smooth stabilizing feedback ⓘ |
| hasImpactOn |
feedback design methodologies
ⓘ
theory of stabilization of nonholonomic systems ⓘ understanding of structural properties of control systems ⓘ |
| historicalPeriod | late 20th century ⓘ |
| implies | existence of topological obstructions to smooth stabilization ⓘ |
| involves |
continuous feedback law
ⓘ
continuous surjections ⓘ equilibrium point of the closed-loop system ⓘ image of vector field under feedback map ⓘ local asymptotic stability ⓘ smooth feedback law ⓘ |
| motivated |
research on discontinuous feedback stabilization
ⓘ
research on hybrid control strategies ⓘ research on time-varying feedback laws ⓘ |
| namedAfter | Roger W. Brockett NERFINISHED ⓘ |
| relatedTo |
Brockett integrator example
ⓘ
nonholonomic integrator ⓘ |
| statesThat | for certain nonlinear systems no smooth static state feedback can render the origin asymptotically stable ⓘ |
| typeOf | necessary condition ⓘ |
| usedIn |
analysis of nonholonomic systems
ⓘ
design of control laws for mobile robots ⓘ geometric control theory ⓘ study of driftless control systems ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.