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

A local recursive least squares approach for discrete-time adaptive fuzzy control proposes a membership-weighted RLS law with a forgetting factor to approximate unknown nonlinearities in quasi-Linear Parameter Varying/Takagi–Sugeno (qLPV/TS) systems, providing LMI synthesis conditions that guarantee ultimate uniform boundedness of the closed loop adaptive system.

The gist: A membership-weighted local recursive least squares with forgetting factor approach is proposed for fuzzy TS models approximating unknown nonlinearities in nonlinear systems.

Proposed Estimation Strategy

The paper introduces a membership-weighted local recursive least squares with forgetting factor approach to estimate the consequent parameters of a constant-consequent TS fuzzy model. This strategy keeps a different covariance for each rule, considerably reducing the memory footprint of the least-squares updates. Furthermore, adaptation is simplified because each rule is only adapted when it is active. The adaptation law is defined by:

((8) pi(k+1) = pi ik / (1 - hikα + hikpik)

((9) θˆ i(k+1) = θˆ ik + pi(k+1)(G T(xk)G(xk))−1G T(xk)hike(i)

The covariance dynamics are also detailed, showing that whenever hik != 0, wik converges monotonically to zero, which in turn implies that, whenever hik != 0, p ik converges monotonically to 1/α.

Error Analysis and Boundedness

The local estimation error is analyzed by focusing on the errors weighted by their respective membership functions separately: hi(xk)e(i)k+1 = hi(xk)G(xk)θ˜ Tik + ε i(xk), where ε i(xk) is the local approximation error. The analysis of the Lyapunov function Vθik shows that for a chosen parameter γ ∈ (0, 1), the one-step difference ∆Vθi is bounded by:

((12) ∆Vθi ≤ hik - γa Vθik + p bar(1 - γ)α ε Tikεik)

This leads to the conclusion that whenever hik != 0, it monotonically converges such that 0 ≤ Vθik ≤ p bar(1 - γ)α squared. Theorem 1 formally states that the proposed law ensures the adaptation error is bounded and, if hik != 0, converges so that θ˜ ik squared ≤ 4δ.

LMI Synthesis Conditions for Different Nonlinearity Types

The paper presents three sets of Linear Matrix Inequality (LMI) synthesis conditions based on the nature of the unknown nonlinearity:

1. For matched nonlinearities (Assumption 1), Theorem 2 provides conditions ensuring ultimate uniform boundedness with bound (13). The control law involves a Parallel Distributed Compensation (PDC) structure where K(xk) = Xrm i=1 µi(xk)Ki.

2. For sector-bounded nonlinearities (Assumption 2), Theorem 3 presents LMIs involving diagonal positive definite matrices Ω̄ and matrices M, K¯ i, F¯ i. The adaptation law is modified with a projection modification to respect the bounds: hikϕ T 1jxk ≤ hikˆθjik ≤ hikϕ T 2jxk.

3. For norm-bounded nonlinearities (Assumption 3), Theorem 4 and Theorem 5 are used. Lemma 2 provides conditions ensuring boundedness with bound (25). The control law uses a feedforward term F(xk) = K(xk)H(xk) − U(xk).

Numerical Validation

Three numerical examples illustrate the proposed adaptive control conditions:

1. A planar manipulator with unknown gravity direction, demonstrating that the adaptive controller achieves a tighter ultimate bound than a controller that is robust against the nonlinearity.

2. A two-tank system with unknown coupling, showing that the adaptive control law can accommodate a larger uncertainty than a purely robust controller.

3. A Brushless DC (BLDC) motor acting as thrust for an efficiency vehicle, where the local RLS approach is shown to overcome the covariance wind-up problem compared to a standard recursive least squares with a forgetting factor.

Future Directions

The authors suggest that future research should focus on adapting these conditions to the output feedback case by employing state observers, as the current methods require full state information in order to work. Additionally, there is interest in adapting the approach to employ evolving TS models instead of regular ones for greater flexibility. The paper concludes by emphasizing that offline design of LMI conditions is a viable strategy for systems where only certain unknown functions are considered.

References

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As a fastidious researcher, I have analyzed this paper on A local recursive least squares approach for discrete-time adaptive fuzzy control. This work focuses on developing robust, adaptive control laws for nonlinear systems represented in Takagi–Sugeno (TS) fuzzy form by utilizing a novel local recursive least squares (RLS) estimation strategy.

Here are the specific improvements that can be implemented in AI systems using the methodologies described in this paper:

Primary Improvements and Capabilities:

1. The proposed framework allows AI systems to achieve high-precision tracking and regulation of nonlinear dynamics while maintaining robustness against unmodeled uncertainties, disturbances, and unknown nonlinearities.

2. The system can be designed to handle a wide spectrum of nonlinearity types—matched, sector-bounded, and norm-bounded—by selecting the appropriate LMI synthesis conditions (Theorem 2 for matched/sector-bounded cases; Theorems 4 & 5 for norm-bounded cases).

Specific Technical Improvements:

1. The AI system can utilize a control structure that is inherently adaptive to unknown nonlinear dynamics approximated by a constant-consequent TS fuzzy model.

2. The parameter estimation mechanism employs a membership-function weighted RLS law with a forgetting factor, which significantly reduces the memory footprint compared to standard RLS, especially when dealing with many rules (local models).

3. The proposed local estimation strategy ensures that parameters and covariances for each rule are only updated when its corresponding membership function is active, leading to guaranteed bounded local adaptation errors (Theorem 1).

4. For sector-bounded nonlinearities, the system can employ a projection modification to the adaptation law (Equation 19) to ensure that estimated parameters remain within known bounds, guaranteeing stability and convergence towards the desired dynamics.

5. For norm-bounded nonlinearities, a feedforward term can be designed (Theorem 5) that approximates and rejects the effect of the nonlinearity on a desired output, ensuring ultimate boundedness even when full compensation is not possible.

What the Improved AI System Can Do:

The improved AI system, equipped with this control framework, can perform complex tasks in dynamic environments where system dynamics are unknown or time-varying nonlinearities exist. Specifically:

1. The system can effectively control a robotic manipulator (like the planar two-link example) even when its gravitational effects (unknown nonlinearity) change unpredictably during operation, achieving tighter tracking than purely robust controllers.

2. It can regulate complex fluid systems (like the two-tank system with unknown coupling and resistance) around desired setpoints despite nonlinear hydraulic resistances that have only known bounds.

3. It can control high-speed physical actuators (like a BLDC motor in an efficiency vehicle) to maintain precise angular velocity regulation, successfully estimating and compensating for unknown load torques and friction terms online, avoiding the covariance wind-up issues common in standard recursive estimation methods.

4. The system can be designed to regulate specific outputs (e.g., tank levels or vehicle speed) by using the feedforward term (Theorem 5), making it highly effective for output regulation tasks rather than just state regulation.

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.