On Stability Equivalence between Dynamic Output Feedback and Static Output Feedback for a Class of Second Order Infinite-Dimensional Systems

Feng, Hongyinping; Guo, Bao‐Zhu

doi:10.1137/140962346

Cited by 13 publications

(12 citation statements)

References 27 publications

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“…So ( 27) holds. Owing to (27) and the exponential stabilities of e Ãt , e A 1 t and eG t , the operator  1 generates an exponentially stable C 0 −semigroup. The proof of Lemma 1 is complete.…”

Section: Lemmamentioning

confidence: 99%

“…The motivation of such construction is from Reference 27, where various of coupled systems were proposed by decoupling the coupled systems as the controlled plants with their dynamic feedbacks. We consider system (16) in the state space

ℋ = ℂ^{n} \times L^{2} (0, 1) \times ℂ^{m},

in which the inner product is equipped with

{⟨ (p_{1}, f_{1}, q_{1}), (p_{2}, f_{2}, q_{2}) ⟩}_{ℋ} = {⟨ p_{1}, p_{2} ⟩}_{ℂ^{n}} + {⟨ f_{1}, f_{2} ⟩}_{L^{2} (0, 1)} + {⟨ q_{1}, q_{2} ⟩}_{ℂ^{m}} .

So, system (16) can be written as

\frac{d}{d t} (\tilde{X}, {\tilde{w}}_{1}, \tilde{v}) = 𝒜_{1} (\tilde{X}, {\tilde{w}}_{1}, \tilde{v}),

where the operator

𝒜_{1}

is defined by

\{\begin{matrix}  \end{matrix}

…”

Section: Observer Designmentioning

confidence: 99%

See 1 more Smart Citation

Performance output tracking for cascaded heat partial differential equation‐ordinary differential equation systems subject to unmatched disturbance

Wen

Feng

2021

Intl J Robust & Nonlinear

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In this paper, we are concerned with the performance output tracking for two cascades of one-dimensional heat partial differential equation-ordinary differential equation systems with Dirichlet and Neumann interconnection, respectively, where both the external disturbances are non-collocated to the controller, and moreover, the disturbances come from exosystem. By means of the proper trajectory planning, the non-collocated configurations can be converted into the collocated ones, so that the difficulties caused by these non-collocateds are overcome. Exponential convergences of the regulation errors are proved. Finally, the numerical simulations validate the effectiveness of this method. K E Y W O R D Scascaded systems, exponential convergence, non-collocated boundary control, output tracking INTRODUCTIONThe output tracking for distributed parameter systems has received a lot of attention in recent years as practically motivated. Internal model principle (IMP), partial results of which have been generalized to the infinite-dimensional systems in References 1-7, has been applied to the output tracking systematically for ordinary differential equations (ODEs) in References 8-10. However, from a practical point of view, the IMP has some distance to solve partial differential equations (PDEs) because it is difficult to solve some related Sylvester operator equations, which may not be easily realized. 6 Feng and Guo 11 proposed a new active disturbance rejection control to stabilize an antistable wave equation, see also References 12,13, which method can also be applied to cope with output tracking in Reference 14. The output tracking problem for a general 2 × 2 system of first-order linear hyperbolic PDEs was considered in Reference 15, which no uncertainty and disturbance were taken into consideration. Recently, output tracking problem with non-collocated configuration was considered by means of adaptive control method in References 16,17, an early effort by adaptive control can also be seen in Reference 18, which rejected the harmonic disturbance only. It is noted that the reference signal is not specified due to disturbance in the above works, and moreover, the authors are only concerned with the performance output tracking for a single control system. Although there are some results about boundary control of PDE-ODE cascades, [19][20][21][22] how to deal with the output tracking problem of cascaded control system subject to unmatched disturbance is quite challenging. System control through actuator dynamics can usually be modeled as a cascaded control system. 23 In this paper, we consider the output tracking for an ODE with actuator dynamics dominated by a heat equation. To the best

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Section: Lemmamentioning

confidence: 99%

ℋ = ℂ^{n} \times L^{2} (0, 1) \times ℂ^{m},

in which the inner product is equipped with

{⟨ (p_{1}, f_{1}, q_{1}), (p_{2}, f_{2}, q_{2}) ⟩}_{ℋ} = {⟨ p_{1}, p_{2} ⟩}_{ℂ^{n}} + {⟨ f_{1}, f_{2} ⟩}_{L^{2} (0, 1)} + {⟨ q_{1}, q_{2} ⟩}_{ℂ^{m}} .

So, system (16) can be written as

\frac{d}{d t} (\tilde{X}, {\tilde{w}}_{1}, \tilde{v}) = 𝒜_{1} (\tilde{X}, {\tilde{w}}_{1}, \tilde{v}),

where the operator

𝒜_{1}

is defined by

\{\begin{matrix}  \end{matrix}

…”

Section: Observer Designmentioning

confidence: 99%

Performance output tracking for cascaded heat partial differential equation‐ordinary differential equation systems subject to unmatched disturbance

Wen

Feng

2021

Intl J Robust & Nonlinear

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“…In this paper we introduce new results for studying polynomial and the more general non-uniform stability for coupled passive abstract linear systems (1.1) and (1.2). Strong and exponential closed-loop stabilities of infinite-dimensional systems have been studied in the literature for passive one-dimensional boundary control systems [36,30], coupled systems with collocated inputs and outputs [15], and passive systems coupled with finite-dimensional systems [44]. Polynomial stability of coupled systems has been studied extensively in the context of coupled linear partial differential equations [3,1,6,2], and for abstract hyperbolic-parabolic systems [20].…”

Section: Introductionmentioning

confidence: 99%

Stability and Robust Regulation of Passive Linear Systems

Paunonen¹

2019

SIAM J. Control Optim.

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We study the stability of coupled impedance passive regular linear systems under power-preserving interconnections. We present new conditions for strong, exponential, and non-uniform stability of the closed-loop system. We apply the stability results to the construction of passive error feedback controllers for robust output tracking and disturbance rejection for strongly stabilizable passive systems. In the case of nonsmooth reference and disturbance signals we present conditions for non-uniform rational and logarithmic rates of convergence of the output. The results are illustrated with examples on designing controllers for linear wave and heat equations, and on studying the stability of a system of coupled partial differential equations.2010 Mathematics Subject Classification. 93C05, 47D06, 93D20, 93B52 (47A10, 35B35, 93D15).

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“…More recently, Wehbe studied the exact controllability of a wave equation with dynamical boundary control in [3]. In 2015, Feng and Guo systematically investigated stability equivalence between dynamic output feedback and static output feedback for a class of second-order systems in [22].…”

Section: Introductionmentioning

confidence: 99%

Spectrum and Stability of a 1‐d Heat‐Wave Coupled System with Dynamical Boundary Control

Jin

Zheng

2019

Mathematical Problems in Engineering

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In this paper, the negative proportional dynamic feedback is designed in the right boundary of the wave component of the 1heat-wave system coupled at the interface and the long-time behavior of the system is discussed. The system is formulated into an abstract Cauchy problem on the energy space. The energy of the system does not increase because the semigroup generated by the system operator is contracted. In the meanwhile, the asymptotic stability of the system is derived in light of the spectral configuration of the system operator. Furthermore, the spectral expansions of the system operator are precisely investigated and the asymptotical stability is not exponential and is shown in view of the spectral expansions.

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On Stability Equivalence between Dynamic Output Feedback and Static Output Feedback for a Class of Second Order Infinite-Dimensional Systems

Cited by 13 publications

References 27 publications

Performance output tracking for cascaded heat partial differential equation‐ordinary differential equation systems subject to unmatched disturbance

Performance output tracking for cascaded heat partial differential equation‐ordinary differential equation systems subject to unmatched disturbance

Stability and Robust Regulation of Passive Linear Systems

Spectrum and Stability of a 1‐d Heat‐Wave Coupled System with Dynamical Boundary Control

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