Approximation of LQR-Problems for Delay Systems: A Survey

Kappel, Franz

doi:10.1007/978-1-4612-0427-5_14

Cited by 7 publications

(12 citation statements)

References 71 publications

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“…In some cases one can relax the POES assumption and still obtain strong (and even norm) convergence of the Riccati operators. For example, the spline scheme developed for delay systems by Kappel and Salamon in [36] produced convergent Riccati operators even though POES was not satisfied (see [35] and [37]). Kappel and Salamon replaced the POES assumption with a uniform output and input-output stability condition and proved strong convergence of the Riccati operators.…”

Section: A Summary Of Approximation Results For Lqr Controlmentioning

confidence: 99%

“…The first method is based on constructing numerical schemes that are dual convergent. The excellent survey by Kappel [35] focuses on this approach. A second approach is based on constructing separate numerical schemes for the forward problem and the dual problem.…”

Section: A Summary Of Approximation Results For Lqr Controlmentioning

confidence: 99%

“…It is important to note that even standard numerical schemes may not preserve these important control system properties. However, for the delay systems considered below it is known that all of the schemes discussed in Kappel's survey [35] satisfy these conditions (see [15], [18], [19], and [28]). Moreover, as we see below, although these conditions are sufficient for the finite dimensional Newton iterates to converge, they do not guarantee that the limit of the Newton iterates X N ∞ converges to X ∞ as N → +∞.…”

Section: Convergence Of Approximating Riccati Operatorsmentioning

confidence: 99%

“…Observe that the linear operator A is not normal; this can cause problems when approximating the LQR control problem (see [8], [13], [16], [15], and [35]). …”

Section: Control Of Delay Systemsmentioning

confidence: 99%

“…Because this spline scheme failed to produce strongly convergent Riccati operators, several modifications were developed to overcome this issue. A nice summary of these schemes and their properties can be found in Kappel's survey paper [35]. We briefly describe the scheme below.…”

Section: 1mentioning

confidence: 99%

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Mesh Independence of Kleinman–Newton Iterations for Riccati Equations in Hilbert Space

Burns¹,

Sachs²,

Zietsman³

2008

SIAM J. Control Optim.

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Abstract. In this paper we consider the convergence of the infinite dimensional version of the Kleinman-Newton algorithm for solving the algebraic Riccati operator equation associated with the linear quadratic regulator problem in a Hilbert space. We establish mesh independence for this algorithm and apply the result to systems governed by delay equations. Numerical examples are presented to illustrate the results. There are two basic issues that need to be addressed in developing practical numerical approximations for control. First, it is essential that the approximation scheme leads to finite dimensional approximating Riccati equations that converge (under mesh refinement) to the solution of the infinite dimensional Riccati equation. This is a well-studied problem (see [7], [14], [26], [33], and [43]). It is now well known that to obtain norm convergence for the Riccati equation, the approximation scheme must satisfy some form of convergence, dual convergence, and uniform preservation of stabilizability and detectability under mesh refinement (see [7] and [33]). These concepts will be made more precise in section 7.1. The important point here is that many "standard" convergent approximation schemes do not satisfy all the conditions necessary for norm convergence of the Riccati operators (see [16]). If this issue is ignored when one develops an approximation scheme for control design and optimization, then the resulting numerical algorithm can fail to produce accurate and useful results (see the numerical examples in section 9). In this paper we show that these properties are also key ingredients in establishing mesh independence of Newton-type algorithms.

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Section: A Summary Of Approximation Results For Lqr Controlmentioning

confidence: 99%

Section: A Summary Of Approximation Results For Lqr Controlmentioning

confidence: 99%

Section: Convergence Of Approximating Riccati Operatorsmentioning

confidence: 99%

“…Observe that the linear operator A is not normal; this can cause problems when approximating the LQR control problem (see [8], [13], [16], [15], and [35]). …”

Section: Control Of Delay Systemsmentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

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