Learning-Based Nonlinear Tracking Control

• The paper presents a modular approach fusing ISS-based robust feedback with model-free learning modules (MES and GP-UCB) to enhance trajectory tracking.
• It decouples robust state feedback using Lyapunov methods from adaptive parameter estimation, ensuring bounded errors despite model uncertainties.
• Simulation on robotic manipulators shows improved tracking performance with MES for local adaptation and GP-UCB for global exploration.

A learning-based nonlinear algorithm for tracking control combines modern machine learning methodologies with nonlinear control-theoretic design to enable adaptive tracking in systems exhibiting model uncertainties and structural nonlinearities. These frameworks fuse robust nonlinear feedback (often with formal Lyapunov or input-to-state stability guarantees) and model-free or data-driven machine learning modules, typically integrated in a modular structure. The goal is to achieve high-precision trajectory tracking for nonlinear plants in the presence of parametric and unstructured uncertainties, with rigorous guarantees on stability and convergence. Approaches in this field can incorporate domain-agnostic learning methods, such as multi-parametric extremum seeking, Gaussian Process-based Bayesian optimization, and neural network adaptation, and are demonstrated on benchmark systems such as robotic manipulators. The following sections detail the underlying principles, architecture, learning algorithms, stability analysis, practical performance, and generalizations as established in contemporary research (Benosman et al., 2015).

1. Modular Indirect Adaptive Control Architecture

The modular indirect adaptive control paradigm separates the controller design into two principal components:

1. Robust Nonlinear State Feedback: A model-based nonlinear controller is synthesized to guarantee input-to-state stability (ISS) of the output tracking error, with respect to the parameter estimation error. The nominal design employs input–output linearization:

$y^{(r)}(t) = b(\xi(t)) + A(\xi(t))u(t)$

The nominal control law is:

$$u_n = A^{-1}(\xi) [v_s(t,\xi) - b(\xi)],\qquad v_s = y_{\text{id}}^{(r)} - K_r(y^{(r-1)} - y_{\text{id}}^{(r-1)}) - \cdots - K_1(y - y_{\text{id}})$$

yielding error dynamics:

$e^{(r)} + K_r e^{(r-1)} + \cdots + K_1 e = 0$

Under structured uncertainty, a robust compensation term $u_r$ is added, resulting in the full feedback:

$u_f = u_n + u_r$

where the robust term is designed (for affine uncertainties) as

$$u_r = -A^{-1}(\xi) \left[ \widetilde{B}^T P z \|Q(\xi,t)\|^2 + \hat{E}(t) Q(\xi,t) \right]$$

Here, $z$ is the tracking error state, $P$ a positive-definite matrix from a Lyapunov equation, and $\hat{E}(t)$ the current parameter estimate. The ISS property is established via Lyapunov analysis:

$\dot{V} \leq -\|z\|^2 + \frac{1}{2}\|e_\Delta\|^2$

leading to the ISS bound

$$\|z(t)\| \leq \beta(\|z(0)\|, t) + \gamma\left( \sup_{0 \leq \tau \leq t} \|e_\Delta(\tau)\| \right)$$

This ensures that the tracking error remains bounded and shrinks as the parameter estimation error decreases.

1. Model-Free Learning Module: Augmenting the ISS controller, a data-driven parameter estimator improves model accuracy using only cost-based feedback. Both multi-parametric extremum seeking (MES) and Gaussian Process Upper Confidence Bound (GP-UCB) Bayesian optimization are implemented as model-free learning algorithms (see Section 3).

This architecture supports modular design: the feedback controller assures bounded error for arbitrary learning-induced parameter errors, while the learning algorithm improves closed-loop performance solely through measurement-driven updates.

2. Robust Nonlinear Controller and ISS Analysis

The feedback design targets tracking for nonlinear systems with parametric uncertainty additive in a known structure (e.g., robot manipulators with linearly parameterized gravity or Coriolis force terms). The robust correction, derived via a Lyapunov approach, leads to a closed-loop system whose tracking error exhibits input-to-state stability from the parameter estimation error input to the system state. For example, for a mechanical system: $$y^{(r)}(t) = b(\xi(t)) + A(\xi(t)) u_f + \Delta b(t, \xi(t))$$ with $\Delta b(t, \xi) = E Q(\xi, t)$ (unknown constant matrix $E$), the correction term is: $$u_r = -A^{-1}(\xi) [ \widetilde{B}^T P z \| Q(\xi, t) \|^2 + \hat{E}(t) Q(\xi, t) ]$$ Lyapunov analysis ensures: $\dot{V} \leq -\|z\|^2 + \frac{1}{2}\|e_\Delta\|^2$ with $e_\Delta = E - \hat{E}(t)$, so the asymptotic tracking error can be arbitrarily reduced as estimation improves.

3. Model-Free Learning Schemes: MES and GP-UCB

The learning module employs cost-based, model-free gradient-free methods for online parameter estimation:

• Multi-Parametric Extremum Seeking (MES):

MES injects distinct dither (sinusoidal) signals into each parameter’s update, with amplitudes $a_i$ and frequencies $\omega_i$:

$$\dot{x}_i = a_i \sin(\omega_i t + \pi/2) J(\hat{\Delta}),\quad \hat{\Delta}_i = x_i + a_i \sin(\omega_i t - \pi/2)$$

where $J(\hat{\Delta})$ is an integral cost function over the tracking error and its derivative. Over time, MES locates a local minimum of $J$, thus driving parameter errors down and improving tracking.

• Gaussian Process Upper Confidence Bound (GP-UCB):

GP-UCB is a Bayesian optimizer that models the cost function $J(\Delta)$ as a sample from a GP with mean and variance estimates $\mu_t(\Delta), \sigma_t(\Delta)$ at iteration $t$. The next candidate parameter is chosen as:

$$\Delta_t \leftarrow \arg\min_{\Delta \in \mathcal{D}} [ \mu_{t-1}(\Delta) - \sqrt{\beta_t} \sigma_{t-1}(\Delta) ]$$

With an appropriate exploration parameter $\beta_t$, the algorithm satisfies theoretical regret bounds and is less likely than MES to be stuck in local minima. The learning proceeds using only cost function evaluations, not model gradients.

4. Integration: Modular Coupling and Closed-Loop Operation

The modular structure ensures that the feedback and learning are decoupled by design:

• The robust controller with feedback and correction laws guarantees the tracking error is bounded proportionally to the estimation error, regardless of learning progress.
• The model-free learner (MES or GP-UCB) operates asynchronously, updating parameter estimates to minimize the empirical cost function computed from the closed-loop system’s performance.

In deployment, the parameter estimate $\hat{E}(t)$ replaces the nominal parameters in the robust term. The learning module is supplied measurements of tracking performance, and its output feeds back into the correction term of the controller, establishing a closed adaptive loop.

Block-diagrammatically, the plant’s controlled output and tracking error feed into a cost function, which drives the update of parameter estimates via a learning module; the updated estimates feed back into the ISS controller correcting for uncertainties.

5. Application Example: Nonlinear Robot Manipulator Tracking

A two-link robot manipulator model demonstrates the approach: $H(q)\ddot{q} + C(q, \dot{q})\dot{q} + G(q) = \tau$ with uncertainty entering the acceleration equation. The nominal law is: $$\tau_n = [ C(q, \dot{q})\dot{q} + G(q) ] + H(q) [ \ddot{q}_d - K_d(\dot{q} - \dot{q}_d) - K_p(q - q_d) ]$$ with robust correction: $$\tau_r = -H(q)[ \widetilde{B}^T P z \| G(q) \|^2 + \hat{E} G(q) ]$$ Simulation shows (i) with MES, dither signals update the parameters, and after finite iterations, the estimates converge and tracking improves; (ii) with GP-UCB, similar or better convergence with reduced final oscillations due to omission of persistent dither.

Performance metrics reported include evolution of the tracking cost function, parameter estimate convergence, and tracking error decrease.

6. Properties, Generalizations, and Practical Limitations

Generalization

• The modular design decouples robust ISS controller synthesis from the learning-based estimation, allowing different learning algorithms to be swapped without redesigning the feedback.
• Applicability extends to uncertainties that are not linearly parameterized since the performance-based learning loop does not depend on the specific structure of the uncertainties.

Limitations

• MES may be trapped in local minima and exhibits residual oscillations determined by the choice of dither amplitude/frequency.
• GP-UCB generally achieves superior global exploration, but its computational cost increases with the dimension of the uncertainty and with the number of required cost evaluations; kernel hyperparameter tuning is critical.
• Practical use requires separation of the time scale of learning and the closed-loop dynamics—the learning process should converge sufficiently fast to impact tracking, but without destabilizing the system.

7. Summary Table: Comparison of MES and GP-UCB Modules

Feature	MES	GP-UCB
Gradient usage	Gradient-free, uses dither signals	Bayesian, gradient-free, uses GP inference
Local minima	Susceptible	Less susceptible
Oscillation	Persistent (dither-driven)	Typically reduced/absent
Computational	Low to moderate	Grows with data size, kernel complexity
Hyperparameters	Dither amplitude/frequency	Kernel hyperparameters, exploration coeff.
Exploration	Implicit via dither	Explicit via confidence bounds

8. Conclusions

Learning-based modular indirect adaptive tracking control for nonlinear systems, as formulated through ISS-feedback augmentation with model-free extremum-seeking or Bayesian optimization, achieves bounded tracking error even during transient learning phases. Simulation confirms improved tracking for nonlinear systems such as robot manipulators with both linearly and nonlinearly parameterized uncertainties. The modularity enables straightforward generalization to alternative learning modules, and theoretical guarantees from ISS and Lyapunov analysis extend to practical real-world deployments, conditioned on informed tuning of learning parameters and time-scale coordination between the feedback and learning components (Benosman et al., 2015).