Controllability of linear systems on lie groups

Jouan, Philippe

doi:10.1007/s10883-011-9131-2

Cited by 46 publications

(45 citation statements)

References 13 publications

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“…The proof of items 1. to 3. can be found in [6], Proposition 2. The items 4. and 5. in [4] Proposition 2.13.…”

Section: E ∈ Int a If And Only If A Is Openmentioning

confidence: 93%

“…On the other hand, in [1] the authors give a first example of a local controllable linear system from e where the reachable set A(e) = A is not a semigroup. Later in [6] Jouan proves that A is a semigroup if and only if A = G. All this facts show that it is hard to understand the class of linear systems from the controllability point of view. For example, controllability of invariant control system, i.e., when the drift below is also an invariant vector field, is a local property for connected groups.…”

Section: Introductionmentioning

confidence: 99%

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Controllability of Linear Systems on Lie Groups with Finite Semisimple Center

Ayala¹,

Silva²

2017

SIAM J. Control Optim.

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Abstract. A linear system on a connected Lie group G with Lie algebra g is determined by the family of differential equationṡwhere the drift vector field X is a linear vector field induced by a g-derivation D, the vector fields X j are right invariant and u ∈ U ⊂ L ∞ (R, Ω ⊂ R m ) with 0 ∈ int Ω. Assume that any semisimple Lie subgroup of G has finite center and e ∈ int A τ0 , for some τ 0 > 0. Then, we prove that the system is controllable if the Lyapunov spectrum of D reduces to zero. The same sufficient algebraic controllability conditions were applied with success when G is a solvable Lie group, [4].

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“…The proof of items 1. to 3. can be found in [6], Proposition 2. The items 4. and 5. in [4] Proposition 2.13.…”

Section: E ∈ Int a If And Only If A Is Openmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Controllability of Linear Systems on Lie Groups with Finite Semisimple Center

Ayala¹,

Silva²

2017

SIAM J. Control Optim.

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“…However the above system is controllable which implies, in particular, that A is open (for the details see Example 5 of [8]). …”

Section: Examplementioning

confidence: 99%

“…Controllability of such systems, with unrestricted control functions, was studied in [3], [2] and [8] and null controllability in [1].…”

Section: Introductionmentioning

confidence: 99%

Controllability of Linear Systems on Solvable Lie Groups

Silva¹

2016

SIAM J. Control Optim.

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Abstract. Linear systems on Lie groups are a natural generalization of linear system on Euclidian spaces. For such systems, this paper studies controllability by taking in consideration the eigenvalues of an associated derivation D. When the state space is a solvable connected Lie group, controllability of the system is guaranteed if the reachable set of the neutral element is open and the derivation D has only pure imaginary eigenvalues. For bounded systems on nilpotent Lie groups such conditions are also necessary.

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“…To express the differential equation (18) in log coordinates we differentiate e with respect to time, making use of Lemma 7.1 and Proposition 3.1…”

Section: A Differential Equation On Gl (N R)mentioning

confidence: 99%

Local Observers on Linear Lie Groups With Linear Estimation Error Dynamics

Koldychev

Nielsen

2014

IEEE Trans. Automat. Contr.

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This paper proposes local exponential observers for systems on linear Lie groups. We study two different classes of systems. In the first class, the full state of the system evolves on a linear Lie group and is available for measurement. In the second class, only part of the system's state evolves on a linear Lie group and this portion of the state is available for measurement. In each case, we propose two different observer designs. We show that, depending on the observer chosen, local exponential stability of one of the two observation error dynamics, left or right invariant error dynamics, is obtained. For the first class of systems these results are developed by showing that the estimation error dynamics are differentially equivalent to a stable linear differential equation on a vector space. For the second class of system, the estimation error dynamics are almost linear. We illustrate these observer designs on an attitude estimation problem.

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Controllability of linear systems on lie groups

Cited by 46 publications

References 13 publications

Controllability of Linear Systems on Lie Groups with Finite Semisimple Center

Controllability of Linear Systems on Lie Groups with Finite Semisimple Center

Controllability of Linear Systems on Solvable Lie Groups

Local Observers on Linear Lie Groups With Linear Estimation Error Dynamics

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