Abstract
This paper investigates the problem of feedback control for a class of affine T–S fuzzy models using piece-wise Lyapunov functions. Although a large number of works on the issue have been published, several crucial problems still remain open. First, the paper shows what problems arise when using the affine T–S fuzzy model to design a controller, and in turn by employing the S-procedure, what kind of quadratic inequalities are required to help solve the resulting LMIs. It turns out that by partitioning the state space into certain cells based on the information of the antecedents of fuzzy rules, the required quadratic inequalities can be formularised. Taking advantage of the cell partition, a fuzzy controller is proposed using piece-wise Lyapunov functions, in which ensuing problems such as continuity functions used in the piece-wise Lyapunov functions and control input chattering also are addressed. Finally, examples are provided to illustrate the effectiveness of the proposed approach.
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Tanaka, K., Sugeno, M.: Stability analysis and design of fuzzy control systems. Fuzzy Sets Syst. 45(2), 135–156 (1992)
Kim, E., Kim, D.: Stability analysis and synthesis for an affine fuzzy control system via LMI and ILMI: discrete case. IEEE Trans. Syst. Man Cybern. 31(1), 132–140 (2001)
Kim, E., Lee, C.H., Cho, Y.W.: Analysis and design of an affine fuzzy systems via bilinear matrix inequality. IEEE Trans. Fuzzy Syst. 13(1), 115–123 (2005)
Qiu, Jianbin, Tian, Hui, Qiugang, Lu., Gao, Huijun: Nonsynchronized robust filtering design for continuous time T–S fuzzy affine dynamic systems based on piecewise Lyapunov functions. IEEE Trans. Cybern. 43(6), 1755–1766 (2013)
Qiu, J., Feng, G., Gao, H.: Static-output-feedback H\(_\infty\) control of continuous-time T–S fuzzy affine systems via piecewise Lyapunov function. IEEE Trans. Fuzzy Syst. 21(2), 245–261 (2013)
Ji, W., Qiu, J., Wu, L., Lam, H.: Fuzzy-affine-model-based output feedback dynamic sliding mode controller design of nonlinear systems. IEEE Trans. Syst. Man Cybern. Syst. 25(6), 1808–1823 (2016)
Wei, Y., Qiu, J., Lam, Hak-Keung., Ligang, W.: Approaches to T–S fuzzy-affine-model-based reliable output feedback control for nonlinear Ito stochastic systems. IEEE Trans. Fuzzy Syst. 25(3), 569–583 (2017)
Wei, Y., Qiu, J., Shi, P., Chadli, M.: Fixed-order piecewise-affine output feedback controller for fuzzy-affine-model-based nonlinear systems with time-varying delay. IEEE Trans. Circuits Syst. I 64(4), 945–958 (2017)
Wei, Y., Qiu, Ji., Shi, P., Lam, H.-K.: A new design of \(H\)-infinity piecewise filtering for discrete-time nonlinear time-varying delay systems via T–S fuzzy affine models. IEEE Trans. Syst. Man Cybern. Syst. 47(8), 2034–2047 (2017)
Wei, Y., Qiu, J., Lam, H.: A novel approach to reliable output feedback control of fuzzy-affine systems with time delays and sensor faults. IEEE Trans. Fuzzy Syst. 25(6), 1808–1823 (2017)
Wei, Y., Qiu, J., Reza Karimi, H.: Reliable output feedback control of discrete-time fuzzy affine systems with actuator faults. IEEE Trans. Circuits Syst. I 64(1), 170–181 (2017)
Wei, Y., Qiu, J., Shi, P., Ligang, W.: A piecewise-Markovian Lyapunov approach to reliable output feedback control for fuzzy-affine systems with time-delays and actuator faults. IEEE Trans. Cybern. 48(9), 2723–2735 (2018)
Ji, W., Zhang, H., Qiu, J.: Fuzzy affine model-based output feedback controller design for nonlinear impulsive systems. Commun. Nonlinear Sci. Numerical Simul. 79, 104894 (2019)
Wang, M., Qiu, J., Feng, G.: A novel piecewise affine filtering design for T-S fuzzy affine systems using past output measurements. IEEE Trans. Cybern. 50(4), 1509–1518 (2020)
Ji, W., Qiu, J., Lam, H.: Fuzzy-affine-model-based sliding-mode control for discrete-time nonlinear 2-d systems via output feedback. IEEE Trans. Cybern. 53(2), 979–987 (2023)
Ji, W., Qiu, J., Ligang, W., Lam, H.-K.: Fuzzy-affine-model-based output feedback dynamic sliding mode controller design of nonlinear systems. IEEE Trans Syst. Man Cybern. Syst. 51(3), 1652–1661 (2021)
Qiu, J., Ji, W., Rudas, I.J., Gao, H.: Asynchronous sampled-data filtering design for fuzzy-affine-model-based stochastic nonlinear systems. IEEE Trans. Cybern. 51(8), 3964–3974 (2021)
Ji, W., Qiu, J., Lam, H.-K.: A new sampled-data output-feedback controller design of nonlinear systems via fuzzy affine models. IEEE Trans. Cybern. 52(3), 1681–1690 (2022)
Ji, W., Qiu, J.: Observer-based output feedback control of nonlinear 2-d systems via fuzzy-affine models. IEEE Trans. Instrum. Meas. 71, 1–10 (2022)
Ji, W., Qiu, J., Song, C., Yili, F.: New results on nonsynchronous-observer-based output-feedback control of fuzzy-affine-model-based discrete-time nonlinear systems. IEEE Trans. Fuzzy Syst. 31(8), 2836–2847 (2023)
Wang, M., Lam, H.-K., Qiu, J., Yan, H., Li, Z.: Fuzzy-affine-model-based filtering design for continuous-time Roesser-type 2-d nonlinear systems. IEEE Trans. Cybern. 53(5), 3220–3230 (2023)
Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philandelphia (1994)
Mou, S., Han, H.: Quadratic inequalities over affine T–S fuzzy models. In: Proceeding of the 7th International Conference on Smart Computing and Artificial Intelligence (IIAI AAI 2021) (2021)
Mou, S., Han, H.: A proposal of fuzzy rule partition and its application to controller design based on affine T–S fuzzy model. IEEJ Trans. EIS 17(10), 74–84 (2022)
O’Kane, C., Han, H.: Approach for affine T–S fuzzy models with uncertainty. In: Proceedings of Joint 12th International Conference on Soft Computing and Intelligent Systems and 23rd International Symposium on Advanced Intelligent Systems (SCIS &ISIS2022) (2022)
O’Kane, C., Han, H.: A controller based on a class of affine T–S fuzzy models. In: Proceedings of The 20th World Congress of the International Fuzzy Systems Association (IFSA 2023) (2023)
Lin, H., Antsaklis, P.J.: Stability and stabilizability of switched linear systems: a survey of recent results. IEEE Trans. Autom. Control 54(2), 308–322 (2009)
Leth, J., Wisniewski, R.: On formalism and stability of switched systems. J. Control Theory Appl. 10, 05 (2012)
Iervolino, R., Trenn, S., Vasca, F.: Stability of piecewise affine systems through discontinuous piecewise quadratic Lyapunov functions. In: 2017 IEEE 56th Annual Conference on Decision and Control (CDC), pp. 5894–5899 (2017)
Iervolino, R., Trenn, S., Vasca, F.: Asymptotic stability of piecewise affine systems with filippov solutions via discontinuous piecewise Lyapunov functions. IEEE Trans. Autom. Control 66(4), 1513–1528 (2021)
Poonawala, H.A.: Stability analysis via refinement of piece-wise linear Lyapunov functions. In: 2019 IEEE 58th Conference on Decision and Control (CDC), pp. 1442–1447 (2019)
Dehghan, M.M.Jr.: Stability of switched linear systems under dwell time switching with piece-wise quadratic functions. In: 2014 13th International Conference on Control Automation Robotics and Vision, ICARCV 2014 (2014)
Johansson, M., Rantzer, A., Arzen, K.E.: Piecewise quadratic stability of fuzzy systems. IEEE Trans. Fuzzy Syst. 7(6), 713–722 (1999)
Johansson, M.: Piecewise Linear Control Systems. Ph.D. dissertation, Detp. Automat. Contr., Lund Inst. Technol., Lund (1999)
Berna, M., Huseko, P.: Piecewise quadratic stability of affine Takagi–Sugeno fuzzy control systems. IFAC Proc. 37, 157–162 (2004)
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Appendix
Appendix
1.1 Continuity Matrix
The way to construct the continuity matrices is based on the existing works works [33, 34]. Without loss of generality, let us consider there are two antecedent variables \(x_1\) and \(x_2\), and cell \({{{\mathcal {S}}}}_k\) is corresponding to the i-th partition on \(x_1\) and j-th partition on \(x_2\).
where \(k=(i-1)\times n_{x_2}+j\), \(p=\sum _{i=1}^2\left( n_{x_i}+1\right) +n\),
and the last row may be removed if the resulting \({{\bar{F}}}_k\) are of full column rank, and subsequently \(p=\sum _{i=1}^2\left( n_{x_i}+1\right)\) in this case. In the following, constructing \(\bar{\textbf{F}}_i=[ \textbf{F}_i~~ \textbf{f}_i]\) is given, while \(\bar{\textbf{F}}_j\) can be obtained in the same manner.
Let \(v=[v_1, \ldots , v_{n_{x_1}+1}]\) be corner points on \(x_1\), which means there are \(n_{x_1}\) partitions on \(x_i\), and \({{{\mathcal {S}}}}_i=[v_i, v_{i+1}]\) (\(i=1, \ldots , {n_{x_1}}\)).
-
Step 1:
Let \(\bar{\textbf{F}}_i\) be a \((n_{x_1}+1)\)-by-2 zero matrix, and
$$\begin{aligned} {{{\mathcal {E}}}}=\begin{bmatrix} v_i&{}v_{i+1}\\ 1&{}1 \end{bmatrix}; \end{aligned}$$ -
Step 2:
replace i-th and \((i+1)\)-th rows of \(\bar{\textbf{F}}_i\) by \({{{\mathcal {E}}}}^{-1}\).
However, \(\textbf{f}_i\) in \(\bar{\textbf{F}}_i\) cannot be guaranteed to be zero for \({{{\mathcal {S}}}}_i\) for \(i\in {{{\mathcal {I}}}}_0\). Therefore, we modify \(\bar{\textbf{F}}_i\) for \(i\in {{{\mathcal {I}}}}_0\), and subsequently others related to the modification. Let \(\bar{\textbf{F}}_i(j)\), and \(\bar{\textbf{F}}_i(j,k)\) be the j-th row, and the element in row j, column k of \(\bar{\textbf{F}}_i\), respectively.
-
Step 1:
Calculate:
$$\begin{aligned} r&=\left( \bar{\textbf{F}}_{i-1}(i,1)\cdot v_i+\bar{\textbf{F}}_{i-1}(i, 2)\right) /v_i=1/v_i,\\ l&=\left( \bar{\textbf{F}}_{i+1}(i+1,1)\cdot v_{i+1}+\bar{\textbf{F}}_{i+1}(i+1, 2)\right) /v_{i+1}\\&=1/v_{i+1}; \end{aligned}$$ -
Step 2:
Update \(\bar{\textbf{F}}_i\):
$$\begin{aligned} \textbf{F}_i(i)=[r~~ 0], \quad \textbf{F}_i(i+1)=[l~~0]; \end{aligned}$$ -
Step 3:
Update \(\bar{\textbf{F}}_1\sim \bar{\textbf{F}}_{i-1}\):
$$\begin{aligned} \bar{\textbf{F}}_{j}(i+1)=[l~~0], \quad \text {for} j=1\sim i-1; \end{aligned}$$ -
Step 4:
Update \(\bar{\textbf{F}}_{i+1}\sim \bar{\textbf{F}}_{n_{x_1}}\):
$$\begin{aligned} \bar{\textbf{F}}_{j}(i)=[r~~ 0], \text { for} j=i+1\sim n_{x_1}. \end{aligned}$$
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O’Kane, C., Han, H. A Controller Based on a Class of Affine T–S Fuzzy Models Using Piece-Wise Lyapunov Functions. Int. J. Fuzzy Syst. 26, 1030–1045 (2024). https://doi.org/10.1007/s40815-023-01651-6
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DOI: https://doi.org/10.1007/s40815-023-01651-6