Controlled invariance for nonlinear differential–algebraic systems

Berger, Toby

doi:10.1016/j.automatica.2015.11.024

Cited by 22 publications

(20 citation statements)

References 15 publications

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“…The following characterization of local controlled invariance, under the constant rank assumption (CR) satisfied for M , was given as theorem 9 in Reference 13 for DACSs whose

\ker E (x)

is an involutive distribution. The DACSs in Reference 13 is of the form

\frac{d e (x (t))}{dt} = f (x (t)) + g (x (t)) u (t)

.…”

Section: Preliminaries On Solutions Of Differential‐algebraic Control Systemsmentioning

confidence: 99%

“…The authors of Reference 11 offered a nonlinear generalization of the Kronecker canonical form using an algebraic inversion algorithm for differential‐algebraic equations DAEs of the general form

F (\dot{x}, x, t) = 0

, while we intend to find normal forms for nonlinear DACSs using geometric methods. A zero dynamics form for DACSs with outputs was proposed in Reference 12 using the notion of maximal output zeroing submanifold introduced in Reference 13. Note that our system

Ξ^{u}

is different in two ways from the DACSs studied in References 12,13.…”

Section: Introductionmentioning

confidence: 99%

“…A zero dynamics form for DACSs with outputs was proposed in Reference 12 using the notion of maximal output zeroing submanifold introduced in Reference 13. Note that our system

Ξ^{u}

is different in two ways from the DACSs studied in References 12,13. First, in References 12,13, the distribution

\ker E (x)

is assumed to be involutive while we consider any E ( x ).…”

Section: Introductionmentioning

confidence: 99%

“…Note that our system

Ξ^{u}

is different in two ways from the DACSs studied in References 12,13. First, in References 12,13, the distribution

\ker E (x)

is assumed to be involutive while we consider any E ( x ). Second, systems in References 12,13 are equipped with outputs.…”

Section: Introductionmentioning

confidence: 99%

“…First, in References 12,13, the distribution

\ker E (x)

is assumed to be involutive while we consider any E ( x ). Second, systems in References 12,13 are equipped with outputs. Calculating the zero dynamics of a DACS

Ξ^{u}

with zero output y = h ( x ) = 0 can be seen as studying an extended DACS

Ξ_{e x t}^{u} : E (x) \dot{x} = F (x) + G (x) u, 0 = h (x)

, because the maximal output zeroing submanifold of

Ξ^{u}

(with the output y = h ( x )) coincides with the maximal controlled invariant submanifold of

Ξ_{e x t}^{u}

.…”

Section: Introductionmentioning

confidence: 99%

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Normal forms and internal regularization of nonlinear differential‐algebraic control systems

Chen

Trenn

Respondek

2021

Intl J Robust & Nonlinear

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In this article, we propose two normal forms for nonlinear differential‐algebraic control systems (DACSs) under external feedback equivalence, using a notion called maximal controlled invariant submanifold. The two normal forms simplify the system structures and facilitate understanding the various roles of variables for nonlinear DACSs. Moreover, we study when a given nonlinear DACS is internally regularizable, that is, when there exists a state feedback transforming the DACS into a differential‐algebraic equation (DAE) with internal regularity, the latter notion is closely related to the existence and uniqueness of solutions of DAEs. We also revise a commonly used method in DAE solution theory, called the geometric reduction method. We apply this method to DACSs and formulate it as an algorithm, which is used to construct maximal controlled invariant submanifolds and to find internal regularization feedbacks. Two examples of mechanical systems are used to illustrate the proposed normal forms and to show how to internally regularize DACSs.

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“…The following characterization of local controlled invariance, under the constant rank assumption (CR) satisfied for M , was given as theorem 9 in Reference 13 for DACSs whose

\ker E (x)

is an involutive distribution. The DACSs in Reference 13 is of the form

\frac{d e (x (t))}{dt} = f (x (t)) + g (x (t)) u (t)

.…”

Section: Preliminaries On Solutions Of Differential‐algebraic Control Systemsmentioning

confidence: 99%

F (\dot{x}, x, t) = 0

Ξ^{u}

is different in two ways from the DACSs studied in References 12,13.…”

Section: Introductionmentioning

confidence: 99%

“…A zero dynamics form for DACSs with outputs was proposed in Reference 12 using the notion of maximal output zeroing submanifold introduced in Reference 13. Note that our system

Ξ^{u}

is different in two ways from the DACSs studied in References 12,13. First, in References 12,13, the distribution

\ker E (x)

is assumed to be involutive while we consider any E ( x ).…”

Section: Introductionmentioning

confidence: 99%

“…Note that our system

Ξ^{u}

is different in two ways from the DACSs studied in References 12,13. First, in References 12,13, the distribution

\ker E (x)

is assumed to be involutive while we consider any E ( x ). Second, systems in References 12,13 are equipped with outputs.…”

Section: Introductionmentioning

confidence: 99%

“…First, in References 12,13, the distribution

\ker E (x)

is assumed to be involutive while we consider any E ( x ). Second, systems in References 12,13 are equipped with outputs. Calculating the zero dynamics of a DACS

Ξ^{u}

with zero output y = h ( x ) = 0 can be seen as studying an extended DACS

Ξ_{e x t}^{u} : E (x) \dot{x} = F (x) + G (x) u, 0 = h (x)

, because the maximal output zeroing submanifold of

Ξ^{u}

(with the output y = h ( x )) coincides with the maximal controlled invariant submanifold of

Ξ_{e x t}^{u}

.…”

Section: Introductionmentioning

confidence: 99%

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Normal forms and internal regularization of nonlinear differential‐algebraic control systems

Chen

Trenn

Respondek

2021

Intl J Robust & Nonlinear

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Controlled invariance for DAEs

Berger

2015

Proc Appl Math and Mech

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We study the concept of locally controlled invariant submanifolds for nonlinear descriptor systems. In contrast to classical approaches, we define controlled invariance as the property of solution trajectories to evolve in a given submanifold whenever they start in it. It is then shown that this concept is equivalent to the existence of a feedback which renders the closed‐loop vector field invariant in the descriptor sense. This result is motivated by a preliminary consideration of the linear case. Local controlled invariance leads to the concept of output zeroing submanifolds. We show that the outcome of the differential‐algebraic version of the zero dynamics algorithm yields a locally maximal output zeroing submanifold. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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Feedback linearization of nonlinear differential‐algebraic control systems

Chen

Respondek²

2021

Intl J Robust & Nonlinear

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In this article, we study feedback linearization problems for nonlinear differential-algebraic control systems (DACSs). We consider two kinds of feedback equivalences, namely, external feedback equivalence, which is defined (locally) on the whole generalized state space, and internal feedback equivalence, which is defined on the locally maximal controlled invariant submanifold (i.e., on the set where solutions exist). We define a notion called explicitation with driving variables, which is a class of ordinary differential equation control systems (ODECSs) attaching to a given DACS. Then we give necessary and sufficient conditions for both internal and external feedback linearization problems of the DACS. We show that the feedback linearizability of the DACS is closely related to the involutivity of the linearizability distributions of the explicitation systems. Finally, we illustrate the results of the by an academic example and a constrained mechanical system.

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Controlled invariance for nonlinear differential–algebraic systems

Cited by 22 publications

References 15 publications

Normal forms and internal regularization of nonlinear differential‐algebraic control systems

Normal forms and internal regularization of nonlinear differential‐algebraic control systems

Controlled invariance for DAEs

Feedback linearization of nonlinear differential‐algebraic control systems

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