Relaxed Stabilization Conditions for TS Fuzzy Systems With Optimal Upper Bounds for the Time Derivative of Fuzzy Lyapunov Functions
This paper initially proposes an optimization problem and after presents its optimal solution. Then, this result is applied to obtain relaxed conditions to design controllers for nonlinear plants described by Takagi-Sugeno (TS) models, based on fuzzy Lyapunov function (FLF) and Linear Matrix Inequal...
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IEEE
2021-01-01
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| Series: | IEEE Access |
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| Online Access: | https://ieeexplore.ieee.org/document/9416573/ |
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| author | Adalberto Z. N. Lazarini Marcelo C. M. Teixeira Jean M. De S. Ribeiro Edvaldo Assuncao Rodrigo Cardim Ariel S. Buzetti |
| author_facet | Adalberto Z. N. Lazarini Marcelo C. M. Teixeira Jean M. De S. Ribeiro Edvaldo Assuncao Rodrigo Cardim Ariel S. Buzetti |
| author_sort | Adalberto Z. N. Lazarini |
| collection | DOAJ |
| description | This paper initially proposes an optimization problem and after presents its optimal solution. Then, this result is applied to obtain relaxed conditions to design controllers for nonlinear plants described by Takagi-Sugeno (TS) models, based on fuzzy Lyapunov function (FLF) and Linear Matrix Inequalities (LMI). The FLF is given by <inline-formula> <tex-math notation="LaTeX">$V(x(t)) = x(t)^{T}P(\alpha (x(t)))x(t)$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$x(t)$ </tex-math></inline-formula> is the plant state vector, <inline-formula> <tex-math notation="LaTeX">$P(\alpha (x(t))) = \alpha _{1}(x(t))P_{1} + \alpha _{2}(x(t))P_{2} + \cdots + \alpha _{r}(x(t))P_{r}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$P_{i}=P_{i}^{T} > 0$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\alpha _{i}(x(t))$ </tex-math></inline-formula> is the weight related to the local model <inline-formula> <tex-math notation="LaTeX">$i$ </tex-math></inline-formula> in the representation of the plant by TS fuzzy models, for <inline-formula> <tex-math notation="LaTeX">$i=1,2,\cdots,r$ </tex-math></inline-formula>. When one calculates the time derivative of this <inline-formula> <tex-math notation="LaTeX">$V(x(t))$ </tex-math></inline-formula>, it appears the term <inline-formula> <tex-math notation="LaTeX">$x(t)^{T}\dot {P}(\alpha (x(t)))x(t)$ </tex-math></inline-formula>, that is usually handled using conservative upper bounds, supposing that the bounds of the time derivative of <inline-formula> <tex-math notation="LaTeX">$\alpha _{i}(x(t))$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$i=1,2,\cdots,r$ </tex-math></inline-formula>, are available. The main result of this paper is a procedure to obtain optimal upper bounds for the term <inline-formula> <tex-math notation="LaTeX">$x(t)^{T}\dot {P}(\alpha (x(t)))x(t)$ </tex-math></inline-formula>, such that they contemplate the maximum value and are always smaller than or equal to the maximum value. It is a relevant result on this subject, because these optimal upper bounds do not add any constraint. With these optimal upper bounds, a relaxed design method for stabilization of TS fuzzy models is proposed. Two numerical examples illustrate the effectiveness of this procedure. |
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| institution | Directory Open Access Journal |
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| spelling | doaj.art-4f51dce9c18241d1a5d49befaf7749752022-12-21T21:36:05ZengIEEEIEEE Access2169-35362021-01-019649456495710.1109/ACCESS.2021.30760309416573Relaxed Stabilization Conditions for TS Fuzzy Systems With Optimal Upper Bounds for the Time Derivative of Fuzzy Lyapunov FunctionsAdalberto Z. N. Lazarini0https://orcid.org/0000-0001-8380-5573Marcelo C. M. Teixeira1https://orcid.org/0000-0002-2996-2831Jean M. De S. Ribeiro2https://orcid.org/0000-0002-9197-2475Edvaldo Assuncao3Rodrigo Cardim4https://orcid.org/0000-0002-1072-3814Ariel S. Buzetti5Department of Electrical Engineering, Faculty of Engineering of Ilha Solteira, São Paulo State University (UNESP), Ilha Solteira, BrazilDepartment of Electrical Engineering, Faculty of Engineering of Ilha Solteira, São Paulo State University (UNESP), Ilha Solteira, BrazilDepartment of Electrical Engineering, Faculty of Engineering of Ilha Solteira, São Paulo State University (UNESP), Ilha Solteira, BrazilDepartment of Electrical Engineering, Faculty of Engineering of Ilha Solteira, São Paulo State University (UNESP), Ilha Solteira, BrazilDepartment of Electrical Engineering, Faculty of Engineering of Ilha Solteira, São Paulo State University (UNESP), Ilha Solteira, BrazilDepartment of Electrical Engineering, Faculty of Engineering of Ilha Solteira, São Paulo State University (UNESP), Ilha Solteira, BrazilThis paper initially proposes an optimization problem and after presents its optimal solution. Then, this result is applied to obtain relaxed conditions to design controllers for nonlinear plants described by Takagi-Sugeno (TS) models, based on fuzzy Lyapunov function (FLF) and Linear Matrix Inequalities (LMI). The FLF is given by <inline-formula> <tex-math notation="LaTeX">$V(x(t)) = x(t)^{T}P(\alpha (x(t)))x(t)$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$x(t)$ </tex-math></inline-formula> is the plant state vector, <inline-formula> <tex-math notation="LaTeX">$P(\alpha (x(t))) = \alpha _{1}(x(t))P_{1} + \alpha _{2}(x(t))P_{2} + \cdots + \alpha _{r}(x(t))P_{r}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$P_{i}=P_{i}^{T} > 0$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\alpha _{i}(x(t))$ </tex-math></inline-formula> is the weight related to the local model <inline-formula> <tex-math notation="LaTeX">$i$ </tex-math></inline-formula> in the representation of the plant by TS fuzzy models, for <inline-formula> <tex-math notation="LaTeX">$i=1,2,\cdots,r$ </tex-math></inline-formula>. When one calculates the time derivative of this <inline-formula> <tex-math notation="LaTeX">$V(x(t))$ </tex-math></inline-formula>, it appears the term <inline-formula> <tex-math notation="LaTeX">$x(t)^{T}\dot {P}(\alpha (x(t)))x(t)$ </tex-math></inline-formula>, that is usually handled using conservative upper bounds, supposing that the bounds of the time derivative of <inline-formula> <tex-math notation="LaTeX">$\alpha _{i}(x(t))$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$i=1,2,\cdots,r$ </tex-math></inline-formula>, are available. The main result of this paper is a procedure to obtain optimal upper bounds for the term <inline-formula> <tex-math notation="LaTeX">$x(t)^{T}\dot {P}(\alpha (x(t)))x(t)$ </tex-math></inline-formula>, such that they contemplate the maximum value and are always smaller than or equal to the maximum value. It is a relevant result on this subject, because these optimal upper bounds do not add any constraint. With these optimal upper bounds, a relaxed design method for stabilization of TS fuzzy models is proposed. Two numerical examples illustrate the effectiveness of this procedure.https://ieeexplore.ieee.org/document/9416573/Fuzzy Lyapunov function (FLF)Takagi-Sugeno (TS) fuzzy systemslinear matrix inequalities (LMIs)fuzzy controlstabilitystabilization |
| spellingShingle | Adalberto Z. N. Lazarini Marcelo C. M. Teixeira Jean M. De S. Ribeiro Edvaldo Assuncao Rodrigo Cardim Ariel S. Buzetti Relaxed Stabilization Conditions for TS Fuzzy Systems With Optimal Upper Bounds for the Time Derivative of Fuzzy Lyapunov Functions IEEE Access Fuzzy Lyapunov function (FLF) Takagi-Sugeno (TS) fuzzy systems linear matrix inequalities (LMIs) fuzzy control stability stabilization |
| title | Relaxed Stabilization Conditions for TS Fuzzy Systems With Optimal Upper Bounds for the Time Derivative of Fuzzy Lyapunov Functions |
| title_full | Relaxed Stabilization Conditions for TS Fuzzy Systems With Optimal Upper Bounds for the Time Derivative of Fuzzy Lyapunov Functions |
| title_fullStr | Relaxed Stabilization Conditions for TS Fuzzy Systems With Optimal Upper Bounds for the Time Derivative of Fuzzy Lyapunov Functions |
| title_full_unstemmed | Relaxed Stabilization Conditions for TS Fuzzy Systems With Optimal Upper Bounds for the Time Derivative of Fuzzy Lyapunov Functions |
| title_short | Relaxed Stabilization Conditions for TS Fuzzy Systems With Optimal Upper Bounds for the Time Derivative of Fuzzy Lyapunov Functions |
| title_sort | relaxed stabilization conditions for ts fuzzy systems with optimal upper bounds for the time derivative of fuzzy lyapunov functions |
| topic | Fuzzy Lyapunov function (FLF) Takagi-Sugeno (TS) fuzzy systems linear matrix inequalities (LMIs) fuzzy control stability stabilization |
| url | https://ieeexplore.ieee.org/document/9416573/ |
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