Abstract
This paper presents a novel approach to the construction of explicit representation of robust model predictive feedback laws for constrained linear systems with parametric uncertainties. The approach is based on employing approximate dynamic programming where a large optimal control problem is split into a sequence of problems of smaller size. To be able to solve each such problem explicitly, we employ a bounded approximation of optimal value functions. Such a procedure is demonstrated to be superior, both in terms of construction time as well as in the complexity of the resulting controllers, to traditional approaches based on the one-shot approach. The proposed method is demonstrated on a simulation case study involving a continuous stirred tank reactor (CSTR) where a fast multi-component chemical reaction takes place.
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Abbreviations
- \(A\) :
-
State matrix
- \(B\) :
-
Input matrix
- \(c\) :
-
Concentration (\({\text{mol m}}^{ - 3}\))
- \(F\) :
-
Feed flow rate (\({\text{m}}^{3} {\text{s}}^{ - 1}\))
- \(I\) :
-
Identity matrix
- \(J\) :
-
Value function
- \(k\) :
-
Prediction step
- \(k_{1} ,k_{2} ,k_{3}\) :
-
Reaction rate constants (\({\text{m}}^{3} {\text{mol}}^{ - 1} {\text{s}}^{ - 1}\))
- \(m\) :
-
The number of control inputs
- \(n\) :
-
The number of system states
- \(N\) :
-
Prediction horizon
- \(N_{\text{R}}\) :
-
Number of critical regions
- \(P\) :
-
Terminal penalty matrix
- \(q\) :
-
Number of uncertain systems
- \(Q\) :
-
State penalty matrix
- \(R\) :
-
Input penalty matrix
- \(t\) :
-
Time (\(s\))
- \(u\) :
-
Input vector
- \(U\) :
-
Optimal control sequence
- \(v\) :
-
Molar feed rate (\({\text{mol s}}^{ - 1}\))
- \(V\) :
-
CSTR volume (\({\text{m}}^{3}\))
- \(x\) :
-
State vector
- \(\Delta\) :
-
Sampling time (\(s\))
- \(\theta\) :
-
Uncertainty parameter vector
- \(\varTheta\) :
-
Uncertainty parameter set
- \({\mathcal{P}}\) :
-
Critical region
- \({\mathcal{T}}\) :
-
Terminal state constraint set
- \({\mathcal{U}}\) :
-
Input constraint set
- \({\mathcal{X}}\) :
-
State constraint set
- \({\text{A}}\) :
-
Compound A
- \({\text{B}}\) :
-
Compound B
- \({\text{C}}\) :
-
Compound C
- feed:
-
Feed
- S:
-
Steady state
- \({ \top }\) :
-
Matrix/vector transpose
- \({ \star }\) :
-
Optimal value
- \((i)\) :
-
The \(i\)th realization of parametric uncertainty
- \(^{ - }\) (overline):
-
Nominal value
- \(^{ \sim }\) (tilde):
-
Approximation
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Acknowledgements
The Authors gratefully acknowledge the contribution of the Slovak Research and Development Agency under the project APVV 15-0007, the contribution of the Scientific Grant Agency of the Slovak Republic under the Grants 1/0112/16 and 1/0585/19, and the Research and Development Operational Programme for the project University Scientific Park STU in Bratislava, ITMS 26240220084, supported by the Research 7 Development Operational Programme funded by the ERDF.
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Bakaráč, P., Kvasnica, M. Approximate explicit robust model predictive control of a CSTR with fast reactions. Chem. Pap. 73, 611–618 (2019). https://doi.org/10.1007/s11696-018-0630-4
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DOI: https://doi.org/10.1007/s11696-018-0630-4