Scope and generalization of the theory of linearly constrained linear regulator

Paolo Alessandro; Elena de Santis

Kybernetika (1999)

  • Volume: 35, Issue: 6, page [707]-720
  • ISSN: 0023-5954

Abstract

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A previous paper by the same authors presented a general theory solving (finite horizon) feasibility and optimization problems for linear dynamic discrete-time systems with polyhedral constraints. We derived necessary and sufficient conditions for the existence of solutions without assuming any restrictive hypothesis. For the solvable cases we also provided the inequative feedback dynamic system, that generates by forward recursion all and nothing but the feasible (or optimal, according to the cases) solutions. This is what we call a dynamic (or automatic) solution. The crucial tool for the development of the theory was the conical approach to linear programming, illustrated in detail in a recent book by the first author. Here we extend this theory in two different directions. The first consists in generalizations for more complex constraint structures. We carry out two cases of mixed input state constraints, yielding the dynamic solution for both of them. The second case is particularly interesting because it appears at first sight hopeless, but, again, resort to the conical approach provides the key to overcome the difficulty. The second direction consists in evaluating the possibility of obtaining at least one solution to problems in the present class, by means of linear, instead of inequative, feedback. We illustrate three mechanisms that exclude any linear solution. In the first the linear feedback cannot handle cases where the origin is in the constraining set for the state. In the second the linear feedback lacks the initial condition independence of the inequative solution. In the third the linear feedback cannot control the geometric multiplicity of eigenvalues of the system, and this prevents stabilization, when the constraint structure is such that we cannot allow the state to converge to the origin. These results clearly strengthen the significance and relevance of the theory of linear (optimal) regulator.

How to cite

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Alessandro, Paolo, and de Santis, Elena. "Scope and generalization of the theory of linearly constrained linear regulator." Kybernetika 35.6 (1999): [707]-720. <http://eudml.org/doc/33457>.

@article{Alessandro1999,
abstract = {A previous paper by the same authors presented a general theory solving (finite horizon) feasibility and optimization problems for linear dynamic discrete-time systems with polyhedral constraints. We derived necessary and sufficient conditions for the existence of solutions without assuming any restrictive hypothesis. For the solvable cases we also provided the inequative feedback dynamic system, that generates by forward recursion all and nothing but the feasible (or optimal, according to the cases) solutions. This is what we call a dynamic (or automatic) solution. The crucial tool for the development of the theory was the conical approach to linear programming, illustrated in detail in a recent book by the first author. Here we extend this theory in two different directions. The first consists in generalizations for more complex constraint structures. We carry out two cases of mixed input state constraints, yielding the dynamic solution for both of them. The second case is particularly interesting because it appears at first sight hopeless, but, again, resort to the conical approach provides the key to overcome the difficulty. The second direction consists in evaluating the possibility of obtaining at least one solution to problems in the present class, by means of linear, instead of inequative, feedback. We illustrate three mechanisms that exclude any linear solution. In the first the linear feedback cannot handle cases where the origin is in the constraining set for the state. In the second the linear feedback lacks the initial condition independence of the inequative solution. In the third the linear feedback cannot control the geometric multiplicity of eigenvalues of the system, and this prevents stabilization, when the constraint structure is such that we cannot allow the state to converge to the origin. These results clearly strengthen the significance and relevance of the theory of linear (optimal) regulator.},
author = {Alessandro, Paolo, de Santis, Elena},
journal = {Kybernetika},
keywords = {discrete-time system; feedback dynamic system; polyhedral constraints; conical approach; linear feedback; discrete-time system; feedback dynamic system; polyhedral constraints; conical approach; linear feedback},
language = {eng},
number = {6},
pages = {[707]-720},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Scope and generalization of the theory of linearly constrained linear regulator},
url = {http://eudml.org/doc/33457},
volume = {35},
year = {1999},
}

TY - JOUR
AU - Alessandro, Paolo
AU - de Santis, Elena
TI - Scope and generalization of the theory of linearly constrained linear regulator
JO - Kybernetika
PY - 1999
PB - Institute of Information Theory and Automation AS CR
VL - 35
IS - 6
SP - [707]
EP - 720
AB - A previous paper by the same authors presented a general theory solving (finite horizon) feasibility and optimization problems for linear dynamic discrete-time systems with polyhedral constraints. We derived necessary and sufficient conditions for the existence of solutions without assuming any restrictive hypothesis. For the solvable cases we also provided the inequative feedback dynamic system, that generates by forward recursion all and nothing but the feasible (or optimal, according to the cases) solutions. This is what we call a dynamic (or automatic) solution. The crucial tool for the development of the theory was the conical approach to linear programming, illustrated in detail in a recent book by the first author. Here we extend this theory in two different directions. The first consists in generalizations for more complex constraint structures. We carry out two cases of mixed input state constraints, yielding the dynamic solution for both of them. The second case is particularly interesting because it appears at first sight hopeless, but, again, resort to the conical approach provides the key to overcome the difficulty. The second direction consists in evaluating the possibility of obtaining at least one solution to problems in the present class, by means of linear, instead of inequative, feedback. We illustrate three mechanisms that exclude any linear solution. In the first the linear feedback cannot handle cases where the origin is in the constraining set for the state. In the second the linear feedback lacks the initial condition independence of the inequative solution. In the third the linear feedback cannot control the geometric multiplicity of eigenvalues of the system, and this prevents stabilization, when the constraint structure is such that we cannot allow the state to converge to the origin. These results clearly strengthen the significance and relevance of the theory of linear (optimal) regulator.
LA - eng
KW - discrete-time system; feedback dynamic system; polyhedral constraints; conical approach; linear feedback; discrete-time system; feedback dynamic system; polyhedral constraints; conical approach; linear feedback
UR - http://eudml.org/doc/33457
ER -

References

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  1. Blanchini F., 10.1109/9.272351, IEEE Trans. Automat. Control AC-39 (1994), 428–433 (1994) Zbl0800.93754MR1265438DOI10.1109/9.272351
  2. d’Alessandro P., Santis E. De, General closed loop optimal solutions for linear dynamic systems with linear constraints, J. Math. Systems, Estimation and Control 6 (1996), 2, 1–14 (1996) Zbl0844.93054MR1649944
  3. d’Alessandro P., A Conical Approach to Linear Programming, Scalar and Vector Optimization Problems, Gordon and Breach Science Publishers, 1997 Zbl0912.90216MR1475216
  4. d’Alessandro P., Santis E. De, Controlled Invariance and Feedback Laws, Research Report no. R.99-31, Dept. of Electrical Engineering, University of L’Aquila, 1999 (submitted) (1999) 
  5. Hennet J. C., Dorea C. E. T., Invariant regulators for linear systems under combined input and state constraints, In: Proc. of 33rd IEEE Conference on Decision and Control, Lake Buena Vista 1994, pp. 1030–1035 (1994) 

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