Robust Controller Design: Minimizing Peak-to-Peak Gain

Dahleh, Munther A.

doi:10.21236/ada460049

Cited by 1 publication

(4 citation statements)

References 54 publications

(116 reference statements)

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“…Standalone computation of the peak‐gain norm with high precision has been addressed in the literature 1,2,4–6,42,51 . For optimization, due to non‐smoothness of both norms in (8), we need to supply subgradients of closed‐loop integral functionals

ϕ_{i j} : K \to \int_{0}^{\infty} | c_{i} (K) e^{A (K) t} b_{j} (K) | d t

, those for the

H_{\infty}

‐norm being well‐known 13 .…”

Section: Methodsmentioning

confidence: 99%

“…It is well known 1,36,41,42,51 that for real‐rational systems G the peak‐gain or peak‐to‐peak norm is

‖ G ‖_{{pk}_{-} gn} = \max_{i = 1, \dots, m} \sum_{j = 1}^{p} (| g_{i j}^{0} |_{1} + | d_{i j} |),

where

g_{i j} (t) = c_{i} e^{A t} b_{j} + d_{i j} δ (t) = g_{i j}^{0} (t) + d_{i j} δ (t)

with

g_{i j}^{0} \in L_{1}

. A special case is the well‐known expression

| | | A | | |_{\infty} = \max_{i = 1, \dots, m} \sum_{k = 1}^{p} | a_{i k} | = \max_{i = 1, \dots, m} | {row}_{i} (A) |_{1}

of the maximum row‐sum‐norm of

A \in ℝ^{m \times p}

…”

Section: Peak‐to‐peak Normmentioning

confidence: 99%

“…Estimate (2) is mentioned in Reference 1 with non‐specified constants, and the SISO case

p = m = 1

is proved in Reference 51 (Theorem) for discrete SISO systems, and in Reference 54 (pp. 11–12) for continuous SISO systems, where in the latter reference the idea of proof is attributed to I. Gohberg.…”

Section: Peak‐to‐peak Normmentioning

confidence: 99%

“…In the SISO case, it is also known as the system

L_{1}

‐norm. As opposed to the more standard

H_{2}

‐ or

H_{\infty}

‐norms, computation or optimization of

‖ G ‖_{{pk}_{-} gn}

has found only mild attention in the control literature, even though its importance, for example, for the rejection of persistent perturbations was recognized 1–7 . One of the reasons of this disesteem is probably the link of

‖ \cdot ‖_{{pk}_{-} gn}

with the

H_{\infty}

‐norm

‖ \cdot ‖_{\infty}

, where in the chain

m^{- 1 / 2} ‖ G ‖_{\infty} \leq ‖ G ‖_{{pk}_{-} gn} \leq (2 n + 1) p^{1 / 2} ‖ G ‖_{\infty}

the right‐hand estimate holds for real‐rational systems G with n poles and p inputs, while the left‐hand estimate is valid even for infinite dimensional well‐posed BIBO‐stable systems with m outputs.…”

Section: Introductionmentioning

confidence: 99%

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Mixed ‐synthesis for ‐stability

Apkarian

Noll

2021

Intl J Robust & Nonlinear

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We consider stabilization and performance optimization of nonlinear controlled systems, where the nonlinearity satisfies a sector constraint asymptotically. This leads to optimization of the closed loop peak‐to‐peak system norm subject to H∞‐performance constraints. Nonlinear controlled systems tuned successfully by this novel approach are locally exponentially stable and globally BIBO‐stable.

show abstract

ϕ_{i j} : K \to \int_{0}^{\infty} | c_{i} (K) e^{A (K) t} b_{j} (K) | d t

, those for the

H_{\infty}

‐norm being well‐known 13 .…”

Section: Methodsmentioning

confidence: 99%

“…It is well known 1,36,41,42,51 that for real‐rational systems G the peak‐gain or peak‐to‐peak norm is

‖ G ‖_{{pk}_{-} gn} = \max_{i = 1, \dots, m} \sum_{j = 1}^{p} (| g_{i j}^{0} |_{1} + | d_{i j} |),

where

g_{i j} (t) = c_{i} e^{A t} b_{j} + d_{i j} δ (t) = g_{i j}^{0} (t) + d_{i j} δ (t)

with

g_{i j}^{0} \in L_{1}

. A special case is the well‐known expression

| | | A | | |_{\infty} = \max_{i = 1, \dots, m} \sum_{k = 1}^{p} | a_{i k} | = \max_{i = 1, \dots, m} | {row}_{i} (A) |_{1}

of the maximum row‐sum‐norm of

A \in ℝ^{m \times p}

…”

Section: Peak‐to‐peak Normmentioning

confidence: 99%

“…Estimate (2) is mentioned in Reference 1 with non‐specified constants, and the SISO case

p = m = 1

Section: Peak‐to‐peak Normmentioning

confidence: 99%

“…In the SISO case, it is also known as the system

L_{1}

‐norm. As opposed to the more standard

H_{2}

‐ or

H_{\infty}

‐norms, computation or optimization of

‖ G ‖_{{pk}_{-} gn}

‖ \cdot ‖_{{pk}_{-} gn}

with the

H_{\infty}

‐norm

‖ \cdot ‖_{\infty}

, where in the chain

m^{- 1 / 2} ‖ G ‖_{\infty} \leq ‖ G ‖_{{pk}_{-} gn} \leq (2 n + 1) p^{1 / 2} ‖ G ‖_{\infty}

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Mixed ‐synthesis for ‐stability

Apkarian

Noll

2021

Intl J Robust & Nonlinear

View full text Add to dashboard Cite

show abstract

Robust Controller Design: Minimizing Peak-to-Peak Gain

Cited by 1 publication

References 54 publications

Mixed ‐synthesis for ‐stability

Mixed ‐synthesis for ‐stability

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