Polynomial Fuzzy Control Systems Stability Analysis

Polynomial fuzzy control systems merge the descriptive power of fuzzy logic with the analytical tractability of polynomial representations to govern nonlinear dynamical processes. Stability analysis in this context seeks to establish conditions under which the closed‐loop behaviour remains bounded and converges to a desired equilibrium despite inherent uncertainties and external perturbations. Central to recent advances is the development of Lyapunov‐based methods tailored to polynomial fuzzy models, often expressed through Takagi–Sugeno or more general conformable frameworks. By casting stability criteria into sum‐of‐squares (SOS) or linear matrix inequality (LMI) problems, researchers have achieved less conservative bounds and tractable computational procedures. Applications span from discrete‐time systems with time delays to large‐scale energy networks and autonomous vehicles, highlighting both theoretical depth and global practical importance. Techniques such as membership‐function partitioning, positivity constraints and co‐positive Lyapunov functions underpin modern approaches, enabling controllers to ensure robust performance under parameter variation, signal bounds and stochastic influences.

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Recent studies have refined stability criteria for discrete‐time polynomial fuzzy systems with intrinsic time delays and positivity requirements by constructing tailored Lyapunov functionals and translating positivity constraints into SOS conditions. These analyses utilise piecewise linear membership functions and boundary information of premise variables to reduce conservatism and guarantee positivity of state trajectories under bounded control inputs. Foundation for more general models has been laid by conformable polynomial fuzzy frameworks that extend beyond standard Takagi–Sugeno representations; here, delay‐dependent Lyapunov–Krasovskii functionals coupled with SOS optimisation deliver less conservative stability and stabilisation conditions via newly formulated polynomial fitting algorithms. A complementary line of work confronts non‐convexity in general polynomial control systems by proposing an iterative SOS‐based scheme that directly tackles non‐convex stability constraints without imposing restrictive assumptions on Lyapunov candidates. This iterative approach has demonstrated an enhanced ability to locate feasible solutions for asymptotic stability across a broad class of polynomial models, thereby broadening applicability to complex control tasks.

Polynomial Fuzzy Control Systems Stability Analysis publication trend

The graph below shows the total number of articles in polynomial fuzzy control systems stability analysis across all publications each year (not limited to Nature Index journals).

Technical terms

Polynomial fuzzy model: A representation of a nonlinear system combining fuzzy‐rule interpolation with polynomial equations to approximate dynamics.

Lyapunov stability: A method for assessing whether system trajectories remain near equilibrium using a scalar energy‐like function.

Sum‐of‐squares (SOS): An optimisation technique that certifies polynomial nonnegativity by expressing it as a sum of squared terms.

Linear matrix inequality (LMI): A convex constraint on matrix variables widely used to formulate control design and stability conditions.

Positivity constraint: A requirement that system states or control inputs remain nonnegative throughout operation.

Time delay: A finite interval between input application and system response, introducing additional complexity into stability analysis.

References

1. Model-Based Control and Stability Analysis of Discrete-Time Polynomial Fuzzy Systems With Time Delay and Positivity Constraints. IEEE Transactions on Fuzzy Systems (2019).
2. Relaxed stability conditions for polynomial‐fuzzy‐model‐based control system with membership function information. IET Control Theory and Applications (2017).
3. Stability analysis of discrete‐time positive polynomial‐fuzzy‐model‐based control systems through fuzzy co‐positive Lyapunov function with bounded control. IET Control Theory and Applications (2020).
4. Stability Analysis and Stabilization of General Conformable Polynomial Fuzzy Models with Time Delay. Symmetry (2024).
5. Iterative stability analysis for general polynomial control systems. Nonlinear Dynamics (2021).