Switched Observer Design for a Class of Non-Linear Systems
Abstract
:1. Introduction
- A non-linear Luenberger-like switched observer with a guaranteed decay rate and cost for the estimation of error dynamics is designed in terms of LMI constraints, considering a class of non-linear systems locally satisfying a Lipschitz condition.
- , , , and represent, respectively, the set of non-negative integer numbers, real numbers, n-dimensional real vectors, and of real matrices;
- represents the subspace of square symmetric matrices;
- For , denotes the transpose of A;
- For , means that P is positive definite (negative definite), and represents the Hermitian operator applied to P;
- The set is the convex hull of ;
- is a vector of the canonical basis of ;
- The set of all vectors , such that , and is designated by ;
- The convex combination of a set of matrices is denoted by
- The subset of Metzler matrices denoted consists of all matrices with elements such that and
- is the Euclidean norm in , for every ;
- The set of trajectories with finite energy, i.e., , is denoted by .
- is locally Lipschitz continuous in x, i.e., there exists a constant scalar such thatholds for all .
- for all there exist functionsand constants and , such that ,and
2. System and Observer Description
2.1. Switched Luenberger-like Observer
2.2. LPV Formulation
2.3. Minimum-Type Switching Rule
3. Switched Observer Design
3.1. Guaranteed Decay Rate
3.2. Guaranteed Cost
3.3. Stability Regions
3.4. Limited Observer Gain Norm
Algorithm 1: Switched Observer Design with Guaranteed Stability Region |
|
4. Simulation Results
4.1. Optimized Initial Conditions
4.2. Observer Tests
5. Concluding Remarks
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
LPV | Linear parameter varying |
TS | Takagi–Sugeno |
DMVT | Differential mean value theorem |
SDP | Semi-definite programming |
LMIs | Linear matrix inequalities |
SOSs | Sum of squares |
LM | Lyapunov–Metzler |
BMIs | Bilinear matrix inequalities |
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Parameters | Numerical Values |
---|---|
kg m2 | |
kg m2 | |
h | m |
m | kg |
B | m |
Nm/V | |
k | 1 Nm/rad |
g | m/s2 |
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Yupanqui Tello, I.F.; Coutinho, D.; Mendoza Rabanal, R.M. Switched Observer Design for a Class of Non-Linear Systems. Appl. Syst. Innov. 2024, 7, 71. https://doi.org/10.3390/asi7040071
Yupanqui Tello IF, Coutinho D, Mendoza Rabanal RM. Switched Observer Design for a Class of Non-Linear Systems. Applied System Innovation. 2024; 7(4):71. https://doi.org/10.3390/asi7040071
Chicago/Turabian StyleYupanqui Tello, Ivan Francisco, Daniel Coutinho, and Renzo Martín Mendoza Rabanal. 2024. "Switched Observer Design for a Class of Non-Linear Systems" Applied System Innovation 7, no. 4: 71. https://doi.org/10.3390/asi7040071