1
Analysis toolbox
▶
1.1
Continuity toolbox
1.2
Limits superior
1.3
Dini derivatives
▶
1.3.1
Relationship to the classical derivative
1.3.2
Characterization by difference quotients
1.3.3
Linearity properties
1.3.4
The comparison step
1.4
Interval integrals
1.5
Positive definite functions
2
Ordinary differential equations
▶
2.1
Gronwall–Bellman inequality
2.2
Existence, uniqueness and continuous dependence
▶
2.2.1
The integral and differential formulations
2.2.2
Time invariance and local existence
2.2.3
Continuous dependence
2.3
Comparison lemma
2.4
Lie derivatives
2.5
\(\omega \)-limit sets
3
Systems
▶
3.1
Continuous time
3.2
Discrete time
4
Linear systems
▶
4.1
The linear system objects
▶
4.1.1
Continuous-time LTV systems
4.1.2
Continuous-time LTI systems
4.1.3
Discrete-time LTV systems
4.1.4
Discrete-time LTI systems
4.2
Peano–Baker series and the state transition matrix
▶
4.2.1
Uniqueness
4.2.2
Semigroup, invertibility and composition
4.2.3
Variation of constants
4.2.4
The forced response as a trajectory
4.2.5
The time-invariant case: the matrix exponential
4.2.6
The forced response of a time-invariant system
4.3
The matrix exponential and complexification
▶
4.3.1
Complexification
4.3.2
Spectral facts about the exponential
4.4
The discrete-time state transition matrix
▶
4.4.1
Discrete variation of constants
4.4.2
The discrete responses as trajectories
5
Comparison functions
▶
5.1
Class \(\mathcal{K}\) and \(\mathcal{K}_{\infty }\) functions
▶
5.1.1
Building class \(\mathcal{K}\) functions
5.1.2
Building class \(\mathcal{K}_{\infty }\) functions
5.2
Inverses of antitone functions
5.3
Class \(\mathcal{KL}\) functions and the Osgood construction
▶
5.3.1
Building class \(\mathcal{KL}\) functions
5.3.2
Smoothing a bound into a class \(\mathcal{KL}\) function
5.4
Class \(\mathcal{K}\) bounds for positive definite functions
5.5
The decay system and the comparison bound
6
Stability of continuous-time systems
▶
6.1
Definitions
6.2
Implications
6.3
Comparison-function characterizations
6.4
Lyapunov functions
6.5
Lyapunov’s direct method
6.6
LaSalle’s invariance principle
6.7
Instability
7
Stability of discrete-time systems
▶
7.1
Definitions
7.2
Implications
7.3
Lyapunov’s direct method
8
Stability of linear systems
▶
8.1
Linear time-varying systems
▶
8.1.1
Hespanha’s Definition 8.1
8.2
Linear time-invariant systems
▶
8.2.1
The Hurwitz condition and its rate-indexed refinement
8.2.2
Equivalent analytic conditions
8.2.3
Hespanha’s Theorem 8.1
8.2.4
The Lyapunov equation
8.2.5
The rate form
8.2.6
Theorem 8.2
8.2.7
Quadratic forms along a linear vector field
8.2.8
A quadratic certificate for an unstable eigenmode
8.3
Discrete-time linear systems
▶
8.3.1
Hespanha’s Definition 8.2
8.3.2
The Schur condition
8.3.3
Hespanha’s Theorem 8.3
8.3.4
The discrete Lyapunov equation
8.3.5
Hespanha’s Theorem 8.4
9
Lyapunov’s indirect method
▶
9.1
The linearization error
9.2
The stable branch
9.3
The unstable branch
10
Controllability and observability
▶
10.1
Observability and controllability matrices
▶
10.1.1
Observability matrix
10.1.2
Controllability matrix
10.1.3
Milestone: observability iff trivial kernel of \(\mathcal{O}\)
10.1.4
Rank characterizations
10.1.5
Hautus observability
10.1.6
Hautus controllability
10.2
Reachable subspace and Kalman decomposition
Dependency graph
LeanForControl
Anand Gokhale Francesco Bullo
University of California, Santa Barbara
1
Analysis toolbox
1.1
Continuity toolbox
1.2
Limits superior
1.3
Dini derivatives
1.3.1
Relationship to the classical derivative
1.3.2
Characterization by difference quotients
1.3.3
Linearity properties
1.3.4
The comparison step
1.4
Interval integrals
1.5
Positive definite functions
2
Ordinary differential equations
2.1
Gronwall–Bellman inequality
2.2
Existence, uniqueness and continuous dependence
2.2.1
The integral and differential formulations
2.2.2
Time invariance and local existence
2.2.3
Continuous dependence
2.3
Comparison lemma
2.4
Lie derivatives
2.5
\(\omega \)-limit sets
3
Systems
3.1
Continuous time
3.2
Discrete time
4
Linear systems
4.1
The linear system objects
4.1.1
Continuous-time LTV systems
4.1.2
Continuous-time LTI systems
4.1.3
Discrete-time LTV systems
4.1.4
Discrete-time LTI systems
4.2
Peano–Baker series and the state transition matrix
4.2.1
Uniqueness
4.2.2
Semigroup, invertibility and composition
4.2.3
Variation of constants
4.2.4
The forced response as a trajectory
4.2.5
The time-invariant case: the matrix exponential
4.2.6
The forced response of a time-invariant system
4.3
The matrix exponential and complexification
4.3.1
Complexification
4.3.2
Spectral facts about the exponential
4.4
The discrete-time state transition matrix
4.4.1
Discrete variation of constants
4.4.2
The discrete responses as trajectories
5
Comparison functions
5.1
Class \(\mathcal{K}\) and \(\mathcal{K}_{\infty }\) functions
5.1.1
Building class \(\mathcal{K}\) functions
5.1.2
Building class \(\mathcal{K}_{\infty }\) functions
5.2
Inverses of antitone functions
5.3
Class \(\mathcal{KL}\) functions and the Osgood construction
5.3.1
Building class \(\mathcal{KL}\) functions
5.3.2
Smoothing a bound into a class \(\mathcal{KL}\) function
5.4
Class \(\mathcal{K}\) bounds for positive definite functions
5.5
The decay system and the comparison bound
6
Stability of continuous-time systems
6.1
Definitions
6.2
Implications
6.3
Comparison-function characterizations
6.4
Lyapunov functions
6.5
Lyapunov’s direct method
6.6
LaSalle’s invariance principle
6.7
Instability
7
Stability of discrete-time systems
7.1
Definitions
7.2
Implications
7.3
Lyapunov’s direct method
8
Stability of linear systems
8.1
Linear time-varying systems
8.1.1
Hespanha’s Definition 8.1
8.2
Linear time-invariant systems
8.2.1
The Hurwitz condition and its rate-indexed refinement
8.2.2
Equivalent analytic conditions
8.2.3
Hespanha’s Theorem 8.1
8.2.4
The Lyapunov equation
8.2.5
The rate form
8.2.6
Theorem 8.2
8.2.7
Quadratic forms along a linear vector field
8.2.8
A quadratic certificate for an unstable eigenmode
8.3
Discrete-time linear systems
8.3.1
Hespanha’s Definition 8.2
8.3.2
The Schur condition
8.3.3
Hespanha’s Theorem 8.3
8.3.4
The discrete Lyapunov equation
8.3.5
Hespanha’s Theorem 8.4
9
Lyapunov’s indirect method
9.1
The linearization error
9.2
The stable branch
9.3
The unstable branch
10
Controllability and observability
10.1
Observability and controllability matrices
10.1.1
Observability matrix
10.1.2
Controllability matrix
10.1.3
Milestone: observability iff trivial kernel of \(\mathcal{O}\)
10.1.4
Rank characterizations
10.1.5
Hautus observability
10.1.6
Hautus controllability
10.2
Reachable subspace and Kalman decomposition