The Strict Bounded Real Lemma for Linear Time-Varying Systems

Chen, Wanyi; Tu, Fengsheng

doi:10.1006/jmaa.1999.6693

Cited by 9 publications

(6 citation statements)

References 9 publications

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“…The next theorem states an equivalence between a bound on the quadratic cost J and the existence of a solution to a Riccati Differential Equation (RDE) or Riccati Differential Inequality (RDI). The theorem is expressed in terms of strict inequalities and generalizes existing results for the induced L 2 gain of LTV systems [29,24,10,4].…”

Section: Strict Bounded Real Lemmasupporting

confidence: 61%

“…These conditions can be equivalently re-written with a Riccati Differential Equation (RDE). This equivalence is based on a variation of the strict bounded real lemma (Theorem 1 in Section 2) which generalizes existing results in [29,24,10,4]. The algorithm to compute finite horizon robustness metrics (Section 4) leverages both forms of these conditions.…”

Section: Introductionmentioning

confidence: 88%

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Finite horizon robustness analysis of LTV systems using integral quadratic constraints

et al. 2019

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Section: Strict Bounded Real Lemmasupporting

confidence: 61%

Section: Introductionmentioning

confidence: 88%

Finite horizon robustness analysis of LTV systems using integral quadratic constraints

et al. 2019

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“…Proof. Since the equivalence between statements 2), 3) and 4) has been proved in [45], we here only prove the equivalence between 1) and 2). For a given storage function V (x(t)) = x(t) T P(t)x(t), we calculate…”

Section: Dissipation Propertiesmentioning

confidence: 91%

Fault-tolerant Coherent H-infinity Control for Linear Quantum Systems

Liu¹,

Dong²,

Petersen³

et al. 2020

Preprint

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Robustness and reliability are two key requirements for developing practical quantum control systems. The purpose of this paper is to design a coherent feedback controller for a class of linear quantum systems suffering from Markovian jumping faults so that the closed-loop quantum system has both fault tolerance and H ∞ disturbance attenuation performance. This paper first extends the physical realization conditions from the time-invariant case to the time-varying case for linear stochastic quantum systems. By relating the fault tolerant H ∞ control problem to the dissipation properties and the solutions of Riccati differential equations, an H ∞ controller for the quantum system is then designed by solving a set of linear matrix inequalities (LMIs). In particular, an algorithm is employed to introduce additional noises and to construct the corresponding input matrices to ensure the physical realizability of the quantum controller. For real applications of the developed fault-tolerant control strategy, we present a linear quantum system example from quantum optics, where the amplitude of the pumping field randomly jumps among different values. It is demonstrated that a quantum H ∞ controller can be designed and implemented using some basic optical components to achieve the desired control goal.

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“…This follows from the Schur complement lemma 34 and R(t, 𝜆) < 0. * The Bounded Real Lemma for LTV systems [17][18][19][20][21] states that there exists a differentiable function P satisfying P(T) ≽ 0 and ( 19) if and only if Condition 3 holds. The precise version of the LTV Bounded Real Lemma used here is Theorem 1 in References 10 and 11.…”

Section: There Exists a Differentiable Function Ymentioning

confidence: 99%

“…This step relies on a connection to Riccati differential equations (RDEs) using the LTV Bounded Real Lemma. [17][18][19][20][21] This connection can also be used to compute subgradients with minimal computational cost (Section 4.2). This builds on related prior work by As a result the ellipsoid algorithm can be used to solve the optimization to within a desired accuracy (Section 4.3).…”

Section: Introductionmentioning

confidence: 99%

Trajectory‐based robustness analysis for nonlinear systems

Seiler,

Venkataraman

2023

Intl J Robust & Nonlinear

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This article considers the robustness of an uncertain nonlinear system along a finite‐horizon trajectory. The uncertain system is modeled as a connection of a nonlinear system and a perturbation. The analysis relies on three ingredients. First, the nonlinear system is approximated by a linear time‐varying (LTV) system via linearization along a trajectory. This linearization introduces an additional forcing input due to the nominal trajectory. Second, the input/output behavior of the perturbation is described by time‐domain, integral quadratic constraints (IQCs). Third, a dissipation inequality is formulated to bound the worst‐case deviation of an output signal due to the uncertainty. These steps yield a differential linear matrix inequality (DLMI) condition to bound the worst‐case performance. The robustness condition is then converted to an equivalent condition in terms of a Riccati differential equation. This yields a computational method that avoids heuristics often used to solve DLMIs, for example, time gridding. The approach is demonstrated by a two‐link robotic arm example.

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The Strict Bounded Real Lemma for Linear Time-Varying Systems

Cited by 9 publications

References 9 publications

Finite horizon robustness analysis of LTV systems using integral quadratic constraints

Finite horizon robustness analysis of LTV systems using integral quadratic constraints

Fault-tolerant Coherent H-infinity Control for Linear Quantum Systems

Trajectory‐based robustness analysis for nonlinear systems

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