On the transformation of time-variable systems to the phase-variable canonical form

Ramaswami, B.; Ramar, K.

doi:10.1109/tac.1969.1099215

Cited by 16 publications

(6 citation statements)

References 2 publications

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“…Under more rigorous conditions on the coefficient functions, the concept of controllability from linear system theory allows for a characterization for special canonical forms, which occur in the state space representation of CARMA processes (A is in companion matrix form). The following results summarize the key aspects of this characterization, which is mainly based on [36], but also [5,29,28]. Definition 4.7 ([32,Chapter 9 and 10]).…”

Section: Remark 44 If A(s) and A(t) Commute Ie [A(s) A(t)]mentioning

confidence: 99%

Continuous-time locally stationary time series models

Bitter¹,

Stelzer²,

Ströh³

2021

Preprint

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We adapt the classical definition of locally stationary processes in discrete-time (see e.g. [14]) to the continuous-time setting and obtain equivalent representations in the time and frequency domain. From this, a unique time-varying spectral density is derived using the Wigner-Ville spectrum. As an example, we investigate time-varying Lévy-driven state space processes, including the class of time-varying Lévy-driven CARMA processes. First, the connection between these two classes of processes is examined. Considering a sequence of time-varying Lévy-driven state space processes, we then give sufficient conditions on the coefficient functions that ensure local stationarity with respect to the given definition.

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Section: Remark 44 If A(s) and A(t) Commute Ie [A(s) A(t)]mentioning

confidence: 99%

Continuous-time locally stationary time series models

Bitter¹,

Stelzer²,

Ströh³

2021

Preprint

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“…Let us consider that the system as a uniformly controllable . The following phase‐variable transformation is based on the procedure given in , consider the nonlinear transformation

y (t) = T (t) x (t),

considering equation and the derivative of , we obtain

\overset{y}{overset} (t) = ((), T (t) A (t) + \overset{T}{overset}) T {(t)}^{- 1} y (t) + T (t) B (t) u (t)

where T ( t ) is such that

A_{c} = ((), T ((), t) A (t) + \overset{T}{overset} (t)) T {((), t)}^{- 1}

and B c = T ( t ) B ( t ) are given by

A_{c} = ((), \begin{matrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 1 \\ β_{1} (t) & β_{2} (t) & β_{3} (t) & \dots & β_{n} (t) \end{matrix}), B_{c} = ((), \begin{matrix} 0 \\ 0 \\ ⋮ \\ 1 \end{matrix})

which transform the system into the canonical form:

\overset{y}{overset} (t) = A_{c} (t) y (t) + B_{c} u (t)

in its turn, the cost function is transformed into …”

Section: Higher Order Singular Optimization For Nominal Linear Time Vmentioning

confidence: 99%

“…As is shown in , the necessary and sufficient condition for the unique existence of the transformation matrix T ( t ) is the uniform controllability of the system . In what follows, we outline the procedure to find T ( t ).…”

Section: Higher Order Singular Optimization For Nominal Linear Time Vmentioning

confidence: 99%

“…Now, proceed to find the transformation matrix that put the system into the phase variable form, so then y ( t ) = T ( t ) δ x ; the time derivative of y is given by

\overset{y}{true} (t) = (\overset{T}{true} + T A) T^{- 1} y + T B (t) δ u (t)

. The matrix T ( t ) is founded following :

T (t) = (\begin{matrix} T_{11} (t) & T_{12} (t) & T_{13} (t) \\ T_{21} (t) & T_{22} (t) & T_{23} (t) \\ T_{31} (t) & T_{32} (t) & T_{33} (t) \end{matrix})

for the first row,

\begin{matrix} T_{11} = \frac{H M ([], V^{2} + (H g_{l} - V μ) e^{((), \frac{V t}{H})})}{V^{4} + {((), H g_{l} - V μ)}^{2} e^{((), \frac{2 V t}{H})} + ((), 2 H V^{2} g_{l} - 3 V^{3} μ) e^{((), \frac{V t}{H})}} \\ T_{12} = \frac{H^{2} M μ e^{((), \frac{V t}{H})}}{V^{4} + ((), H g_{l} - ...)} \end{matrix}

…”

Section: Numerical Examplementioning

confidence: 99%

See 1 more Smart Citation

Robust singular optimization for linear time varying systems by integral high‐order sliding mode

Jimenez-Lizarraga

Ibarra

2016

Optim Control Appl Methods

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Summary This paper presents an approach to solve a singular quadratic optimization problem for linear time varying systems based on the so‐called integral high‐order sliding mode control. The plant which is time varying is affected for some bounded disturbances, and the criterion to minimize is degenerated, in the sense that the weighting matrix can possess any rank. It is shown the natural connection between the order of singularity of the time varying quadratic criterion, which is connected to the rank of the cost matrix and the order of the sliding mode. An integral high‐order sliding mode is proposed to control the behavior of the transit phase before arriving toward the corresponding higher order singular time varying optimal manifold in prescribed time. The transformation to the phase‐variable form for the Linear Time Variant Systems (LTVS) becomes the key step to solve the problem, and the proposed solution provides the insensitivity of trajectory w.r.t. matched bounded uncertainties. Such a design is applied to a probe landing problem to illustrate the effectiveness of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.

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“…In the literature there exist works dealing with the transformation of time varying systems to canonical forms, as in (1), see, e.g [20]. and[26].…”

mentioning

confidence: 99%

Continuous and discrete state estimation for switched LPV systems using parameter identification

Ríos

Mincarelli²,

Efimov

et al. 2015

Automatica

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In this paper the problem of discrete and continuous state estimation for a class of uncertain switched LPV systems is addressed. Parameter identification techniques are applied to realize an approximate identification of the scheduled parameters of a switched LPV system with certain uncertainties and/or disturbances. A discrete state estimation is achieved using the parameter identification. A Luenberger-like hybrid observer, based on discrete state information and LMIs approach, is used for the continuous state estimation. The simplicity of the proposed method is one of the main advantages of this paper. The feasibility of the proposed method is illustrated by simulations.

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On the transformation of time-variable systems to the phase-variable canonical form

Cited by 16 publications

References 2 publications

Continuous-time locally stationary time series models

Continuous-time locally stationary time series models

Robust singular optimization for linear time varying systems by integral high‐order sliding mode

Continuous and discrete state estimation for switched LPV systems using parameter identification

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