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September
2019, 9(3): 517-539.
doi: 10.3934/mcrf.2019024
Discretized feedback control for systems of linearized hyperbolic balance laws
RWTH Aachen University, IGPM, Templergraben 55, 52062 Aachen, Germany |
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.
Keywords:
Systems of hyperbolic balance laws,
feedback stabilization,
Lyapunov function,
isothermal Euler equations,
shallow water equations.
Mathematics Subject Classification:
Primary: 37L45, 93B52, 35B35; Secondary: 35B30, 93B18, 93C73, 93D15, 93D05.
Citation:
Stephan Gerster, Michael Herty. Discretized feedback control for systems of linearized hyperbolic balance laws. Mathematical Control and Related Fields,
2019, 9
(3)
: 517-539.
doi: 10.3934/mcrf.2019024
![]()
References:
| [1] |
M. K. Banda and M. Herty,
Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws, Mathematical Control and Related Fields, 3 (2013), 121-142.
doi: 10.3934/mcrf.2013.3.121. |
| [2] |
M. K. Banda, M. Herty and A. Klar,
Gas flow in pipeline networks, Networks and Heterogeneous Media, 1 (2006), 41-56.
doi: 10.3934/nhm.2006.1.41. |
| [3] |
G. P. Barker, A. Berman and R. J. Plemmons,
Positive diagonal solutions to the Lyapunov equations, Linear and Multilinear Algebra, 5 (1978), 249-256.
doi: 10.1080/03081087808817203. |
| [4] |
G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-d Hyperbolic Systems, 1st edition, Progress in nonlinear differential equations and their applications, Birkhäuser, Switzerland, 2016.
doi: 10.1007/978-3-319-32062-5. |
| [5] |
J.-M. Coron,
Local controllability of a 1-d tank containing a fluid modeled by the shallow water equations, ESAIM: Control, Optim. and Calculus of Variations, 8 (2002), 513-554.
doi: 10.1051/cocv:2002050. |
| [6] |
J.-M. Coron, Control and Nonlinearity, vol. 136 of Mathematical surveys and monographs, Providence, RI, 2007. |
| [7] |
J.-M. Coron and G. Bastin,
Dissipative boundary conditions for one-dimensional quasilinear hyperbolic systems: Lyapunov stability for the $C^1$-norm, SIAM Journal on Control and Optimization, 53 (2015), 1464-1483.
doi: 10.1137/14097080X. |
| [8] |
J.-M. Coron, G. Bastin and B. d'Andréa-Novel,
A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws, IEEE Transactions on Automatic Control, 52 (2007), 2-11.
doi: 10.1109/TAC.2006.887903. |
| [9] |
J.-M. Coron, G. Bastin and B. d'Andréa-Novel, Boundary feedback control and Lyapunov stability analysis for physical networks of 2$\times$2 hyperbolic balance laws, Proceedings of the 47th IEEE Conference on decision and Control, (2008), 1454-1458. |
| [10] |
J.-M. Coron, G. Bastin and B. d'Andréa-Novel,
Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems, SIAM Journal on Control and Optimization, 47 (2008), 1460-1498.
doi: 10.1137/070706847. |
| [11] |
J.-M. Coron, G. Bastin and B. d'Andréa-Novel,
On Lyapunov stability of linearised Saint-Venant equations for a sloping channel, Networks and Heterog. Media, 4 (2009), 177-187.
doi: 10.3934/nhm.2009.4.177. |
| [12] |
J.-M. Coron, G. Bastin, B. d'Andréa-Novel and B. Haut,
Lyapunov stability analysis of networks of scalar conservation laws, Networks and Heterogeneous Media, 2 (2007), 751-759.
doi: 10.3934/nhm.2007.2.751. |
| [13] |
C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3rd edition, A series of comprehensive studies in mathematics, Springer, Providence, RI, 2010.
doi: 10.1007/978-3-642-04048-1. |
| [14] |
J. de Halleux, C. Prieur, J.-M. Coron, B. d'Andréa-Novel and G. Bastin,
Boundary feedback control in networks of open channels, Automatica, 39 (2003), 1365-1376.
doi: 10.1016/S0005-1098(03)00109-2. |
| [15] |
M. Dick, M. Gugat and G. Leugering,
Classical solutions and feedback stabilization for the gas flow in a sequence of pipes, Networks and Heterogeneous Media, 5 (2010), 691-709.
doi: 10.3934/nhm.2010.5.691. |
| [16] |
M. Gugat and M. Herty,
Existence of classical solutions and feedback stabilization for the flow in gas networks, ESAIM: Control, Optim. and Calculus of Variations, 17 (2011), 28-51.
doi: 10.1051/cocv/2009035. |
| [17] |
M. Gugat and G. Leugering,
Global boundary controllability of the de St. Venant equations between steady states, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 20 (2003), 1-11.
doi: 10.1016/S0294-1449(02)00004-5. |
| [18] |
M. Gugat and G. Leugering,
Global boundary controllability of the Saint-Venant system for sloped canals with friction, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 26 (2009), 257-270.
doi: 10.1016/j.anihpc.2008.01.002. |
| [19] |
M. Gugat, G. Leugering and G. Schmidt,
Global controllability between steady supercritical flows in channel networks, Mathematical Methods in the Applied Science, 27 (2004), 781-802.
doi: 10.1002/mma.471. |
| [20] |
M. Gugat, G. Leugering, S. Tamasoiu and K. Wang,
$H^2$-stabilization of the isothermal Euler equations: A Lyapunov function approach, Chin. Ann. Math., 33 (2012), 479-500.
doi: 10.1007/s11401-012-0727-y. |
| [21] |
M. Gugat, L. Rosier and V. Perrollaz,
Boundary stabilization of quasilinear hyperbolic systems of balance laws: exponential decay for small source terms, Journal of Evolution Equations, 18 (2018), 1471-1500.
doi: 10.1007/s00028-018-0449-z. |
| [22] |
H. K. Khalil, Nonlinear Control, Pearson Education, 2015. |
| [23] |
G. Leugering and G. Schmidt,
On the modelling and stabilization of flows in networks of open canals, SIAM Journal on Control and Optimization, 41 (2002), 164-180.
doi: 10.1137/S0363012900375664. |
| [24] |
R. J. Leveque, Finite Volume Methods for Hyperbolic Problems, 1st edition, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002.
doi: 10.1017/CBO9780511791253.
|
| [25] |
P. Schillen and S. Göttlich,
Numerical discretization of boundary control problems for systems of balance laws: Feedback stabilization, European Journal of Control, 35 (2017), 11-18.
doi: 10.1016/j.ejcon.2017.02.002. |
| [26] |
A. Zlotnik, M. Chertkov and S. Backhaus, Optimal control of transient flow in natural gas networks, 54th IEEE Conference on Decision and Control (CDC), (2015), 4563-4570.
doi: 10.1109/CDC.2015.7402932. |
show all references
References:
| [1] |
M. K. Banda and M. Herty,
Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws, Mathematical Control and Related Fields, 3 (2013), 121-142.
doi: 10.3934/mcrf.2013.3.121. |
| [2] |
M. K. Banda, M. Herty and A. Klar,
Gas flow in pipeline networks, Networks and Heterogeneous Media, 1 (2006), 41-56.
doi: 10.3934/nhm.2006.1.41. |
| [3] |
G. P. Barker, A. Berman and R. J. Plemmons,
Positive diagonal solutions to the Lyapunov equations, Linear and Multilinear Algebra, 5 (1978), 249-256.
doi: 10.1080/03081087808817203. |
| [4] |
G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-d Hyperbolic Systems, 1st edition, Progress in nonlinear differential equations and their applications, Birkhäuser, Switzerland, 2016.
doi: 10.1007/978-3-319-32062-5. |
| [5] |
J.-M. Coron,
Local controllability of a 1-d tank containing a fluid modeled by the shallow water equations, ESAIM: Control, Optim. and Calculus of Variations, 8 (2002), 513-554.
doi: 10.1051/cocv:2002050. |
| [6] |
J.-M. Coron, Control and Nonlinearity, vol. 136 of Mathematical surveys and monographs, Providence, RI, 2007. |
| [7] |
J.-M. Coron and G. Bastin,
Dissipative boundary conditions for one-dimensional quasilinear hyperbolic systems: Lyapunov stability for the $C^1$-norm, SIAM Journal on Control and Optimization, 53 (2015), 1464-1483.
doi: 10.1137/14097080X. |
| [8] |
J.-M. Coron, G. Bastin and B. d'Andréa-Novel,
A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws, IEEE Transactions on Automatic Control, 52 (2007), 2-11.
doi: 10.1109/TAC.2006.887903. |
| [9] |
J.-M. Coron, G. Bastin and B. d'Andréa-Novel, Boundary feedback control and Lyapunov stability analysis for physical networks of 2$\times$2 hyperbolic balance laws, Proceedings of the 47th IEEE Conference on decision and Control, (2008), 1454-1458. |
| [10] |
J.-M. Coron, G. Bastin and B. d'Andréa-Novel,
Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems, SIAM Journal on Control and Optimization, 47 (2008), 1460-1498.
doi: 10.1137/070706847. |
| [11] |
J.-M. Coron, G. Bastin and B. d'Andréa-Novel,
On Lyapunov stability of linearised Saint-Venant equations for a sloping channel, Networks and Heterog. Media, 4 (2009), 177-187.
doi: 10.3934/nhm.2009.4.177. |
| [12] |
J.-M. Coron, G. Bastin, B. d'Andréa-Novel and B. Haut,
Lyapunov stability analysis of networks of scalar conservation laws, Networks and Heterogeneous Media, 2 (2007), 751-759.
doi: 10.3934/nhm.2007.2.751. |
| [13] |
C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3rd edition, A series of comprehensive studies in mathematics, Springer, Providence, RI, 2010.
doi: 10.1007/978-3-642-04048-1. |
| [14] |
J. de Halleux, C. Prieur, J.-M. Coron, B. d'Andréa-Novel and G. Bastin,
Boundary feedback control in networks of open channels, Automatica, 39 (2003), 1365-1376.
doi: 10.1016/S0005-1098(03)00109-2. |
| [15] |
M. Dick, M. Gugat and G. Leugering,
Classical solutions and feedback stabilization for the gas flow in a sequence of pipes, Networks and Heterogeneous Media, 5 (2010), 691-709.
doi: 10.3934/nhm.2010.5.691. |
| [16] |
M. Gugat and M. Herty,
Existence of classical solutions and feedback stabilization for the flow in gas networks, ESAIM: Control, Optim. and Calculus of Variations, 17 (2011), 28-51.
doi: 10.1051/cocv/2009035. |
| [17] |
M. Gugat and G. Leugering,
Global boundary controllability of the de St. Venant equations between steady states, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 20 (2003), 1-11.
doi: 10.1016/S0294-1449(02)00004-5. |
| [18] |
M. Gugat and G. Leugering,
Global boundary controllability of the Saint-Venant system for sloped canals with friction, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 26 (2009), 257-270.
doi: 10.1016/j.anihpc.2008.01.002. |
| [19] |
M. Gugat, G. Leugering and G. Schmidt,
Global controllability between steady supercritical flows in channel networks, Mathematical Methods in the Applied Science, 27 (2004), 781-802.
doi: 10.1002/mma.471. |
| [20] |
M. Gugat, G. Leugering, S. Tamasoiu and K. Wang,
$H^2$-stabilization of the isothermal Euler equations: A Lyapunov function approach, Chin. Ann. Math., 33 (2012), 479-500.
doi: 10.1007/s11401-012-0727-y. |
| [21] |
M. Gugat, L. Rosier and V. Perrollaz,
Boundary stabilization of quasilinear hyperbolic systems of balance laws: exponential decay for small source terms, Journal of Evolution Equations, 18 (2018), 1471-1500.
doi: 10.1007/s00028-018-0449-z. |
| [22] |
H. K. Khalil, Nonlinear Control, Pearson Education, 2015. |
| [23] |
G. Leugering and G. Schmidt,
On the modelling and stabilization of flows in networks of open canals, SIAM Journal on Control and Optimization, 41 (2002), 164-180.
doi: 10.1137/S0363012900375664. |
| [24] |
R. J. Leveque, Finite Volume Methods for Hyperbolic Problems, 1st edition, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002.
doi: 10.1017/CBO9780511791253.
|
| [25] |
P. Schillen and S. Göttlich,
Numerical discretization of boundary control problems for systems of balance laws: Feedback stabilization, European Journal of Control, 35 (2017), 11-18.
doi: 10.1016/j.ejcon.2017.02.002. |
| [26] |
A. Zlotnik, M. Chertkov and S. Backhaus, Optimal control of transient flow in natural gas networks, 54th IEEE Conference on Decision and Control (CDC), (2015), 4563-4570.
doi: 10.1109/CDC.2015.7402932. |
Figure 3.
Pipeline with two compressors and stability domain for the $ L^2 $ -norm
Figure 4.
Lyapunov functions for the steady states $ {\bar{h}(x): = 3} $ and $ {\bar{h}(x): = 3+10^{-3} \cos(2 \pi x)} $
Figure 5.
Non-decaying Lyapunov functions for the steady state $ {\bar{h}(x): = 3+0.1 \cos(2 \pi x)} $
Table 1.
$\hat{L}^1_x$ -error and EOS for Riemann invariants
| 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 | ||
| 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 | ||
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 | ||
| ${L_t^1\text{-error}}$ | 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 | ||
Table 3.
Estimated decay rate µe (top), guaranteed rate µg (middle) and observed rate µo (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 |
| 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 |
| [1] |
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