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Closed-loop stability of uncertain linear systems is studied under the state feedback realized by a linear quadratic regulator (LQR). Sufficient conditions are presented that ensure the closed-loop stability in the presence of uncertainty, initially for the case of a non-robust LQR designed for a nominal model not reflecting the system uncertainty. Since these conditions are usually violated for a large uncertainty, a procedure is offered to redesign such a non-robust LQR into a robust one that ensures closed-loop stability under a predefined level of uncertainty. The analysis of this paper largely relies on the concept of inverse optimal control to construct suitable performance measures for uncertain linear systems, which are non-quadratic in structure but yield optimal controls in the form of LQR. The relationship between robust LQR and zero-sum linear quadratic dynamic games is established.
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Nonlinear Feedback Control Design via NEOC
Quadratic performance indices associated with linear plants offer simplicity and lead to linear feedback control laws, but they may not adequately capture the complexity and flexibility required to address various practical control problems. One notable example is to improve, by using possibly nonlinear laws, on the trade-off between rise time and overshoot commonly observed in classical regulator problems with linear feedback control laws. To address these issues, non-quadratic terms can be introduced into the performance index, resulting in nonlinear control laws. In this study, we tackle the challenge of solving optimal control problems with non-quadratic performance indices using the closed-loop neighboring extremal optimal control (NEOC) approach and homotopy method. Building upon the foundation of the Linear Quadratic Regulator (LQR) framework, we introduce a parameter associated with the non-quadratic terms in the cost function, which is continuously adjusted from 0 to 1. We propose an iterative algorithm based on a closed-loop NEOC framework to handle each gradual adjustment. Additionally, we discuss and analyze the classical work of Bass and Webber, whose approach involves including additional non-quadratic terms in the performance index to render the resulting Hamilton-Jacobi equation analytically solvable. Our findings are supported by numerical examples.
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- Award ID(s):
- 2120430
- PAR ID:
- 10535362
- Publisher / Repository:
- IEEE Xplore
- Date Published:
- Journal Name:
- IEEE Control Systems Letters
- ISSN:
- 2475-1456
- Page Range / eLocation ID:
- 1 to 1
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1109/LCSYS.2024.3417891
