Discretized feedback control for systems of linearized hyperbolic balance laws

Stephan Gerster; Michael Herty

doi:10.3934/mcrf.2019024

Article Contents

2019, Volume 9, Issue 3: 517-539. Doi: 10.3934/mcrf.2019024

This issue Previous Article Determining the shape of a solid of revolution Next Article A non-exponential discounting time-inconsistent stochastic optimal control problem for jump-diffusion

Discretized feedback control for systems of linearized hyperbolic balance laws

Stephan Gerster and
Michael Herty

RWTH Aachen University, IGPM, Templergraben 55, 52062 Aachen, Germany

Received: June 30, 2017

Revised: July 31, 2018

Published: September 2019

This work is supported by DFG HE5386/13-15.

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Abstract

Physical systems such as water and gas networks are usually operated in a state of equilibrium and feedback control is employed to damp small perturbations over time. We consider flow problems on networks, described by hyperbolic balance laws, and analyze the stability of the linearized systems. Sufficient conditions for exponential stability in the continuous and discretized setting are presented. The analysis is extended to arbitrary Sobolev norms. Computational experiments illustrate the theoretical findings.

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Mathematics Subject Classification: Primary: 37L45, 93B52, 35B35; Secondary: 35B30, 93B18, 93C73, 93D15, 93D05.

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Figure 1. $ \log_2 $ of $ \hat{L}^1_x $-error for Riemann invariants

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Figure 2. Lyapunov functions for $ {G = \mathbb{1}} $ and $ {\hat{\mu} = 0} $

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Figure 3. Pipeline with two compressors and stability domain for the $ L^2 $-norm

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Figure 4. Lyapunov functions for the steady states $ {\bar{h}(x): = 3} $ and $ {\bar{h}(x): = 3+10^{-3} \cos(2 \pi x)} $

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Figure 5. Non-decaying Lyapunov functions for the steady state $ {\bar{h}(x): = 3+0.1 \cos(2 \pi x)} $

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Table 1. $\hat{L}^1_x$-error and EOS for Riemann invariants

	$\hat{L}^1_x$-error					EOC
△x	d=0	d=1	d=2	d=3	d=4	d=0	d=1	d=2	d=3	d=4
$2^{-4}$	0.52	3.61	24.11	146.27	953.65
$2^{-5}$	0.27	1.87	12.46	75.48	492.38	0.95	0.95	0.95	0.95	0.95
$2^{-6}$	0.13	0.94	6.27	37.97	247.28	0.99	0.99	0.99	0.99	0.99
$2^{-7}$	0.07	0.46	3.11	18.80	121.91	1.01	1.01	1.01	1.01	1.02
$2^{-8}$	0.03	0.23	1.51	9.12	58.63	1.04	1.04	1.04	1.04	1.06
$2^{-9}$	0.02	0.11	0.70	4.26	26.98	1.10	1.10	1.10	1.10	1.12

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Table 2. Error of Lyapunov function for Euler]{$L^1_t$-, $L^2_t$-, $L^\infty_t$-error and EOC for Lyapunov functions, units in $0.01$

	${L_t^1\text{-error}}$					EOC
$\Delta x$	d=0	d=1	d=2	d=3	d=4	d=0	d=1	d=2	d=3	d=4
$2^{-4}$	6.90	7.08	7.08	7.08	7.00
$2^{-5}$	3.61	3.70	3.70	3.70	3.63	0.93	0.93	0.93	0.93	0.95
$2^{-6}$	1.84	1.88	1.88	1.88	1.81	0.98	0.98	0.98	0.98	1.00
$2^{-7}$	0.91	0.94	0.94	0.94	0.87	1.01	1.01	1.01	1.01	1.06
$2^{-8}$	0.45	0.46	0.46	0.46	0.39	1.04	1.04	1.04	1.04	1.17
$2^{-9}$	0.21	0.21	0.21	0.21	0.14	1.10	1.10	1.10	1.10	1.42

	${L_t^2\text{-error}}$					EOC
	d=0	d=1	d=2	d=3	d=4	d=0	d=1	d=2	d=3	d=4
$2^{-4}$	7.90	8.04	8.04	8.04	7.96
$2^{-5}$	4.15	4.23	4.23	4.23	4.15	0.93	0.93	0.93	0.93	0.94
$2^{-6}$	2.12	2.16	2.16	2.16	2.08	0.97	0.97	0.97	0.97	1.00
$2^{-7}$	1.06	1.08	1.08	1.07	1.00	1.00	1.00	1.00	1.00	1.06
$2^{-8}$	0.51	0.52	0.52	0.52	0.44	1.04	1.04	1.04	1.04	1.16
$2^{-9}$	0.24	0.25	0.25	0.25	0.17	1.09	1.09	1.09	1.09	1.42

	${L_t^\infty\text{-error}}$					EOC
$\Delta x$	d=0	d=1	d=2	d=3	d=4	d=0	d=1	d=2	d=3	d=4
$2^{-4}$	13.90	13.92	13.92	13.91	13.78
$2^{-5}$	7.41	7.43	7.43	7.42	7.29	0.91	0.91	0.91	0.91	0.92
$2^{-6}$	3.79	3.79	3.79	3.79	3.66	0.97	0.97	0.97	0.97	0.99
$2^{-7}$	1.87	1.87	1.87	1.87	1.74	1.02	1.02	1.02	1.02	1.07
$2^{-8}$	0.89	0.90	0.90	0.90	0.77	1.06	1.06	1.06	1.06	1.19
$2^{-9}$	0.42	0.42	0.42	0.42	0.29	1.09	1.09	1.09	1.09	1.40

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Table 3. Estimated decay rate µ^e (top), guaranteed rate µ^g (middle) and observed rate µ^o (bottom) for $\mathsf{\hat{\mu }}$ : = 0.25 with constant and perturbed steady states

Estimated rate
	constant	p=4	p=3	p=2	p=1
d=0	0.2499	0.2498	0.2485	0.2254	-0.1563
d=1	0.2499	0.2495	0.2420	0.0374	-2.2991
d=2	0.2499	0.2458	0.1288	-1.3167	-15.8406
Guaranteed rate
	constant	p=4	p=3	p=2	p=1
d=0	0.2499	0.2498	0.2487	0.2279	-0.1154
d=1	0.2499	0.2495	0.2428	0.0591	-2.0424
d=2	0.2499	0.2463	0.1412	-1.1566	-14.2202
Observed rate
	constant	p=4	p=3	p=2	p=1
d=0	0.2572	0.2572	0.2572	0.2537	0.1343
d=1	0.3461	0.3459	0.3416	0.2126	-1.3244
d=2	0.2632	0.2615	0.2089	-0.6659	-10.7687

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