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A local recursive least squares approach for discrete-time adaptive fuzzy control

Víctor Costa da Silva Campos, Mariella Maia Quadros

arXiv:2610.01859v1eess.SYcs.SY

TL;DR

Unknown nonlinearities and approximation error complicate adaptive control, while prior fuzzy-control results address a limited range of system structures. This paper combines local TS-model RLS estimation with LMI synthesis for matched, sector-bounded, and norm-bounded nonlinearities. Its examples illustrate bounded closed-loop responses and show the local adaptation law overcoming covariance windup relative to standard forgetting-factor RLS.

  • Problem

    Approximation error, disturbances, and unmodeled dynamics may contribute to parameter drift, while many prior adaptive fuzzy controllers focus on feedback-linearizable or strict-feedback systems.

    Source: §1 · pp. 3–4

  • Method

    The paper uses a membership-weighted local RLS law for TS approximations and presents LMI synthesis conditions for matched, sector-bounded, and norm-bounded unknown nonlinearities.

    Source: §1 · p. 5, §6 · pp. 41–42

  • Results

    The three examples illustrate the proposed cases; the BLDC motor example shows the local adaptation law overcoming covariance windup compared with standard RLS with a forgetting factor.

    Source: §6 · pp. 42–43

  • Takeaways & Limitations

    The proposed offline LMI gains, used with the adaptation law, guarantee ultimately bounded closed-loop responses for the considered nonlinearities.

    Source: §6 · p. 42

    The adaptation law changes rule consequent parameters but does not add or remove rules, limiting its flexibility relative to evolving TS models.

    Source: §6 · p. 43

Abstract

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This paper proposes a local recursive least squares (RLS) estimation strategy for discrete-time adaptive fuzzy control of nonlinear systems represented in quasi-Linear Parameter Varying (qLPV)/Takagi--Sugeno (TS) form. Unknown nonlinear terms are approximated by a constant-consequent TS fuzzy model, and the consequent parameters are updated by a membership-function-weighted RLS law with a forgetting factor. The proposed estimator keeps a different covariance for each rule, considerably reducing the memory footprint of the least-squares updates. The adaptation is also simplified since each rule is only adapted when it is active. From these properties, we are capable of showing that the adaptation law ensures bounded local adaptation errors. Building upon this adaptation law, Linear Matrix Inequality (LMI) synthesis conditions are presented for matched, sector-bounded and norm-bounded unknown nonlinearities, guaranteeing ultimate uniform boundedness of the adaptive control system in closed loop. Three numerical examples are presented to illustrate the adaptive control conditions in the three cases: a planar manipulator with unknown gravity direction, a two-tank system with unknown coupling, and a Brushless DC (BLDC) motor acting as thrust for an efficiency vehicle.

1. Introduction

The introduction motivates local RLS for fuzzy adaptive control by the scaling and covariance-windup challenges of standard RLS, alongside limitations in the system structures addressed by prior controllers. It proposes local estimation and LMI-based synthesis for several classes of unknown nonlinearities.

1. Introduction

  • Universal approximators can leave nonzero approximation error, and disturbances, unmodeled dynamics, and measurement noise may contribute to parameter drift without robustness.The introduction notes that least-squares objectives can reduce estimation sensitivity to these errors compared with gradient-based adaptation.Source: §1 · p. 3
  • Standard RLS with forgetting can suffer covariance windup without persistent excitation, while large estimation problems require covariance matrices whose size grows quadratically with parameter count.The proposed separate covariances for local rules yield linear rather than quadratic covariance growth.Source: §1 · p. 4
  • The proposed local RLS uses constant-consequent TS models and separate rule covariances, with parameters and covariance updated only when a rule is active.The introduction also reports an asymptotic bound on local adaptation errors.Source: §1 · pp. 4–5
  • The paper addresses limits in prior adaptive fuzzy control structures by using TS/LMI methods for a broader class of systems, including sector-bounded mismatched nonlinearities.For norm-bounded mismatched nonlinearities, it also proposes a feedforward term that approximately rejects the estimated nonlinearity's effect on a desired output.Source: §1 · pp. 4–5
  • For matched and sector-bounded nonlinearities, LMI conditions ensure ultimate uniform boundedness of the closed-loop system with respect to approximation error.For norm-bounded nonlinearities, the conditions also address adaptation error and can be paired with a feedforward term for approximate rejection at a desired output.Source: §1 · p. 5

2. Problem statement

The problem statement represents the discrete-time nonlinear system in qLPV form, with known state-dependent matrices and an unknown state function. A TS model and constant-consequent fuzzy approximator provide a local representation of the system and its unknown nonlinearity.

2. Problem statement

  • The system has state-dependent known matrices and an unknown function of the state, which the model seeks to estimate.The state, control input, and matrix dimensions are specified in the system description.Source: §2 · p. 6
  • The TS representation composes local submodels with membership functions describing the nonlinear system.The stated model can also be viewed as a TS model with nonlinear consequents, but the nonlinear function is unknown.Source: §2 · p. 6
  • A constant-consequent TS fuzzy model approximates the unknown nonlinearity, with rule parameters chosen to minimize maximum approximation error within the analysis region.The text describes Gaussian or bounded-support membership functions and identifies the approximation error explicitly.Source: §2 · pp. 6–7

3. Local Model Estimation

The local estimation strategy analyzes membership-weighted errors and applies a forgetting-factor RLS law to local TS models. Each rule's parameters and covariance remain unchanged while that rule is inactive, and the theorem establishes bounded local adaptation error.

3. Local Model Estimation

  • The estimator focuses on local approximation errors weighted by their membership functions rather than applying RLS directly to the global approximation error.The passage motivates this choice by the potentially large covariance matrices of a direct global estimator.Source: §3 · p. 8
  • The membership-function-weighted local RLS uses a forgetting factor and the TS model's local structure to simplify adaptation.The membership functions naturally restrict each local model to a subset of the state space.Source: §3 · pp. 8–9
  • When a rule's membership is zero, its parameter and covariance stop changing, so adaptation occurs only for active rules.The covariance dynamics are presented with the local update law.Source: §3 · p. 9
  • Theorem 1 establishes bounded adaptation error and convergence of the local error when the rule is active.The proof bounds the covariance and the local Lyapunov function during active updates.Source: §3 · pp. 11–12

4. LMI Controller Synthesis

The section develops LMI-based adaptive control conditions for matched, sector-bounded, and norm-bounded nonlinearities, using Lyapunov arguments to establish ultimate uniform boundedness under the stated assumptions. For norm-bounded nonlinearities, it also considers output tracking with an approximate guarantee for the ideal approximator and its use alongside Theorem 4.

4. LMI Controller Synthesis

  • A Lyapunov condition on the state dynamics yields ultimate uniform boundedness for the closed loop with the local adaptation law.The proof bounds the Lyapunov change outside a region and establishes the state bound using γ = 0.5.Source: §4 · pp. 12–13

4.1. Matched Nonlinearity

  • The adaptive control law (15), with K(xk) given by (16), Ki = ¯KiM−1, and F(xk) satisfying (14), together with adaptation law (8) and (9), ensures that closed-loop states xk are ultimately uniformly bounded with bound (13).Source: §4.1 · p. 17

4.2. Sector-bounded Nonlinearities

  • Under adaptive control law (21), with Ki = K̄iM⁻¹ and Fi = F̄iΩ̄⁻¹, and adaptation laws (8), (18), and (19), closed-loop states xk are ultimately uniformly bounded with bound (13).When the original Θ parameters being estimated are within the projection region, projection decreases adaptation error and Theorem 1 remains valid. Bounds are considered only when the corresponding membership function is active.Source: §4.2 · pp. 18, 21

4.3. Norm-bounded Nonlinearity

  • Because a ψ-related term remains, Lemma 1 cannot be used directly and is adapted; under Theorem 4’s conditions, the adaptive control law is ultimately uniformly bounded regardless of gains F(xk).Source: §4.3 · pp. 22, 24
  • Unlike the previous cases, the uncertainty cannot be completely compensated, so the design focuses on compensating a desired output and rejecting the nonlinearity rather than the states.Source: §4.3 · p. 25
  • Theorem 5’s feedforward term approximately preserves the desired output against the ideal approximator φ*(x_k) = Θ^T h(x_k).The passage frames this as rejecting the unknown nonlinearity in the output, rather than compensating the states.Source: §4.3 · p. 25
  • Theorem 5 is exact only with ideal parameters and when H̄(x_{k+1}) − H̄(x_k) = 0; the authors therefore recommend using it with Theorem 4 as the safest approach.The required zero change can be hard to ensure, even when H is state-independent, because the membership functions would need very slow dynamics.Source: §4.3 · pp. 26–27

5. Numerical examples

The examples compare adaptive and robust control for a rotated-plane manipulator, test sector-bounded control in a two-tank system, and evaluate local RLS for a norm-bounded BLDC motor load.

5.1. Matched Nonlinearity

  • The adaptive controller drives the manipulator considerably closer to the origin than the robust controller, while closely approximating gravity’s effect.The robust controller does not drive the system to the origin in this example.Source: §5.1 · pp. 30–31, Figure 2 · p. 30, Figure 3 · p. 31

5.2. Sector-bounded nonlinearities

  • For Rmax=600 s/m4, adaptive synthesis is feasible at Rmin ≥45, versus ≥71 for robust control.In closed loop, the two-tank controller follows setpoints and local RLS correctly estimates the unknown flow between tanks.Source: §5.2 · pp. 34–35, Figure 5 · p. 34, Figure 6 · p. 35

5.3. Norm-bounded nonlinearities

  • Standard RLS diverges around 13.5 s from covariance wind-up; local RLS keeps closed-loop signals bounded without persistent excitation, while robust control regulates angular velocity worse.The proposed controller trades estimator precision for robustness.Source: §5.3 · pp. 39–41, Figure 8 · p. 40, Figure 9 · p. 41

6. Conclusion and future works

The paper proposes local RLS adaptation and LMI synthesis for discrete-time adaptive fuzzy control of qLPV/TS systems. Three examples illustrate controller behavior across different nonlinearities, while the discussion identifies limits and future directions.

6. Conclusion and future works

  • For qLPV/TS systems, the paper proposes local RLS adaptation that updates a rule’s parameters and covariance only while active, with LMI conditions for matched, sector-bounded, and norm-bounded nonlinearities.The conditions yield ultimately bounded closed-loop responses when used with the proposed adaptation law.Source: §6 · pp. 41–42
  • Across three examples, the controller tightens the planar manipulator’s ultimate bound, accommodates larger two-tank uncertainty, and local RLS overcomes covariance wind-up versus forgetting-factor RLS in BLDC thrust.Figure 10 compares local and global RLS load estimates against the exact load.Source: §6 · pp. 42–43, Figure 10 · p. 42
  • The adaptation changes only consequent parameters; evolving TS models could add or remove rules, while the paper’s controller gains are designed with offline LMIs.The authors identify adapting the approach to evolving TS models as future work.Source: §6 · p. 43
  • The proposed controllers and adaptation laws require full-state information; adapting the conditions to output feedback with state observers remains future work.Source: §6 · p. 43
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