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Optimal control of a linear system subject to partially specified input noise

Alexander Erreygers (UGent) , Jasper De Bock (UGent) , Gert De Cooman (UGent) and Arthur Van Camp (UGent)
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Abstract
One of the most basic problems in control theory is that of controlling a discrete-time linear system subject to uncertain noise with the objective of minimising the expectation of a quadratic cost. If one assumes the noise to be white, then solving this problem is relatively straightforward. However, white noise is arguably unrealistic: noise is not necessarily independent and one does not always precisely know its expectation. We first recall the optimal control policy without assuming independence, and show that in this case computing the optimal control inputs becomes infeasible. In a next step, we assume only knowledge of lower and upper bounds on the conditional expectation of the noise, and prove that this approach leads to tight lower and upper bounds on the optimal control inputs. The analytical expressions that determine these bounds are strikingly similar to the usual expressions for the case of white noise.
Keywords
Linear system, quadratic cost, optimal control, partially specified noise

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MLA
Erreygers, Alexander, et al. “Optimal Control of a Linear System Subject to Partially Specified Input Noise.” INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, vol. 29, no. 12, Wiley, 2019, pp. 3892–914, doi:10.1002/rnc.4583.
APA
Erreygers, A., De Bock, J., De Cooman, G., & Van Camp, A. (2019). Optimal control of a linear system subject to partially specified input noise. INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, 29(12), 3892–3914. https://doi.org/10.1002/rnc.4583
Chicago author-date
Erreygers, Alexander, Jasper De Bock, Gert De Cooman, and Arthur Van Camp. 2019. “Optimal Control of a Linear System Subject to Partially Specified Input Noise.” INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL 29 (12): 3892–3914. https://doi.org/10.1002/rnc.4583.
Chicago author-date (all authors)
Erreygers, Alexander, Jasper De Bock, Gert De Cooman, and Arthur Van Camp. 2019. “Optimal Control of a Linear System Subject to Partially Specified Input Noise.” INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL 29 (12): 3892–3914. doi:10.1002/rnc.4583.
Vancouver
1.
Erreygers A, De Bock J, De Cooman G, Van Camp A. Optimal control of a linear system subject to partially specified input noise. INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL. 2019;29(12):3892–914.
IEEE
[1]
A. Erreygers, J. De Bock, G. De Cooman, and A. Van Camp, “Optimal control of a linear system subject to partially specified input noise,” INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, vol. 29, no. 12, pp. 3892–3914, 2019.
@article{8618849,
  abstract     = {One of the most basic problems in control theory is that of controlling a discrete-time linear system subject to uncertain noise with the objective of minimising the expectation of a quadratic cost. If one assumes the noise to be white, then solving this problem is relatively straightforward. However, white noise is arguably unrealistic: noise is not necessarily independent and one does not always precisely know its expectation. We first recall the optimal control policy without assuming independence, and show that in this case computing the optimal control inputs becomes infeasible. In a next step, we assume only knowledge of lower and upper bounds on the conditional expectation of the noise, and prove that this approach leads to tight lower and upper bounds on the optimal control inputs. The analytical expressions that determine these bounds are strikingly similar to the usual expressions for the case of white noise.},
  author       = {Erreygers, Alexander and De Bock, Jasper and De Cooman, Gert and Van Camp, Arthur},
  issn         = {1049-8923},
  journal      = {INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL},
  keywords     = {Linear system,quadratic cost,optimal control,partially specified noise},
  language     = {eng},
  number       = {12},
  pages        = {3892--3914},
  publisher    = {Wiley},
  title        = {Optimal control of a linear system subject to partially specified input noise},
  url          = {http://dx.doi.org/10.1002/rnc.4583},
  volume       = {29},
  year         = {2019},
}

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