Sampled-Data Adaptive Controller

• Sampled-data adaptive controllers are online control strategies that adapt to uncertainties in systems with discrete measurements and event-triggered sampling.
• They integrate observer designs and companion variables to maintain regularity in controller dynamics despite asynchronous, sample-and-hold operations.
• Their adaptive laws, based on Lyapunov stability, ensure bounded output errors and robust performance in strict-feedback nonlinear systems.

A sampled-data adaptive controller is an online control strategy tailored for systems where measurements and control updates occur at discrete, potentially asynchronous time instants. These controllers adapt to parametric uncertainties and nonlinearities using only sampled signals, and are fundamentally distinguished by their event-triggered architectures, adaptation laws, output feedback mechanisms, and provable stability properties. Advances in this area address challenges of non-differentiability, asynchronous sampling, and computation limitations in practical networked systems.

1. Fundamental System Model and Problem Setting

Sampled-data adaptive controllers are most frequently formulated for uncertain strict-feedback nonlinear plants of the form

$$\begin{aligned} \dot x_i &= x_{i+1} + \theta\,\psi_i(y),\quad i=1,\dots,n-1,\ \dot x_n &= u + \theta\,\psi_n(y),\ y &= x_1 \end{aligned}$$

where $x\in\mathbb{R}^n$ is the state, $y$ is the output, $u$ is the scalar control input, and $\theta$ is an unknown but bounded parameter ($|\theta|\le\bar\theta$). The nonlinearity $\psi_i(\cdot)$ is required to be $\mathcal{C}^1$ and Lipschitz on a known compact set. This class of systems is generic in adaptive control literature and emphasizes the separation between continuous plant dynamics and discrete controller operation.

2. Event-Triggered Sampling Architecture

Communication and computation constraints in networked systems motivate the use of event-triggered sampling. Two event detectors are introduced:

• Plant-side trigger (ED1):

Output $y$ is sampled and transmitted only when the difference

$e_y(t) = |y(t)-y(\bar t_k)| \ge \gamma_y$

exceeds a threshold $\gamma_y$. The transmission instants $\bar t_{k+1}$ are defined as the first time $t > \bar t_k$ when $e_y(t) \ge \gamma_y$. This configuration reduces communication load by transmitting only significant output variations.

• Controller-side trigger (ED2):

All internal controller states $\xi,\zeta,\hat\theta,\alpha_f$ are resampled simultaneously when at least one state changes by more than its threshold (e.g., $e_\xi(t)\ge\gamma_\xi$). This multi-condition trigger enables simultaneous state and output updates, ensuring computational efficiency and avoiding constant monitoring.

At each inter-event interval $[t_j, t_{j+1})$, all controller computations rely solely on piecewise-constant sampled data, facilitating closed-form trajectory predictions and algebraic computation of future event times.

3. Sampled-Data Controller Dynamics and Observer Construction

In contrast to traditional continuous observers, sampled-data adaptive controllers operate with sample-and-hold dynamics: $$\begin{aligned} \dot{\xi}(t) &= A_c\,\xi(t_j)+k\,\bar y(t_j)+b\,u(t_j), \ \dot{\zeta}(t) &= A_c\,\zeta(t_j)+\psi(\bar y(t_j)), \ \dot{\hat\theta}(t) &= \bar y(t_j)\left[\psi_1(\bar y(t_j))+\zeta_2(t_j)\right] - \delta\,\hat\theta(t_j), \ \dot{\alpha}_f(t) &= \varsigma(\alpha_f(t_j), \hat\theta(t_j), \hat x(t_j), \bar y(t_j)) \end{aligned}$$ with control

$$u(t) = \kappa(\alpha_f(t_j), \hat\theta(t_j), \hat x(t_j), \bar y(t_j)).$$

The observer structure splits state estimates into $\hat x = \xi + \theta\,\zeta$, where $A_c$ is Hurwitz with gains $k$ explicitly constructed. The estimation error $\varepsilon=x-(\xi+\theta\zeta)$ evolves under the influence of sampling-induced mismatch and bounded output differences, enabling simplified adaptive laws compared to conventional designs with extra parameter adaptation for observer gains.

4. Virtual Input Construction and Companion Variables

Virtual feedback inputs, often derived via backstepping in strict-feedback systems, become piecewise constant and non-differentiable under sampled operation. To circumvent the breakdown of differentiability, companion variables $\hat\alpha_i$ are introduced to form intermediate errors $\hat\upsilon_i = \alpha_{i,f} - \hat\alpha_{i-1}$. This device preserves the regularity required for Lyapunov-based analysis: $$\begin{aligned} \alpha_1 &= -c_1\,\bar y -\hat\theta\,\left[\psi_1(\bar y)+\zeta_2\right], \ \hat\alpha_1 &= -c_1\,y -\hat\theta\,\left[\psi_1(y)+\zeta_2\right], \end{aligned}$$ enabling the calculation of $\dot{\hat\alpha}_1$'s evolution despite piecewise continuity. This generalizes to deeper integrator steps, facilitating recursive Lyapunov construction under non-differentiable controller architectures.

5. Output Error Bound for Asynchronous Sampling

The sampled controller and observer architectures may asynchronously read plant output, introducing a discrepancy between $\bar y(t_j)$ (last output sample held at controller) and the true output $y(t)$. Lemma 1 guarantees that the output error remains bounded: $$|y(t) - \bar y(t_j)| \le \tilde\gamma_y = \gamma_y + \gamma_{\bar y}$$ for $t \in [t_j, t_{j+1})$, with all subsequent stability and Lyapunov estimates inheriting sampling thresholds as additive constants, inherently quantifying the sampling-induced robustness.

6. Adaptive Law and Lyapunov-Based Stability

Adaptation of the unknown parameter $\theta$ proceeds via a single-parameter update law: $$\dot{\hat\theta} = \bar y(t_j)\left[\psi_1(\bar y(t_j))+\zeta_2(t_j)\right] - \delta\,\hat\theta(t_j)$$ where $\delta>0$ is a design rate. The Lyapunov function combines observer error, companion variable error, and adaptation error: $$V_1(z_1,\hat\upsilon_2,\tilde\theta) = \frac{1}{2}(z_1^2+\hat\upsilon_2^2+\tilde\theta^2)+V_\varepsilon+V_\zeta$$ and similar terms at deeper steps. The derivative satisfies

$\dot V \le -c(\beta) V + d(\beta)$

for gain-dependent $c(\beta)>0$, $d(\beta)\geq0$ determined by sampling thresholds. Thus, all internal signals are semiglobally bounded, and the output error $|y(t)|$ is practically stabilized to the origin within an $O(d/c)$ neighborhood.

7. Practical Implementation Aspects and Main Results

Key implementation features include:

• Explicit computation of inter-event times via algebraic solution, avoiding real-time polling.
• All controller components reside at the controller side, eliminating the need for plant-side computation.
• The architecture excludes Zeno behavior by strictly lower-bounding inter-event intervals using system Lipschitz constants.
• Performance metrics demonstrate practical stabilization, with output error confined to a quantifiable neighborhood.
• The approach integrates event-triggered sampling, sample-and-hold operation, backstepping modifications with companion variables, and formal output error bounding.

This synthesis provides technical foundations for extending sampled-data adaptive control to strict-feedback nonlinear systems under communication and computational constraints, with guaranteed robustness against asynchrony and sampling gaps.

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Concretely, this class of sampled-data adaptive controllers is distinguished by unified event-triggered architectures, sample-and-hold controller dynamics, companion-variable regularization of backstepping constructs, a single-parameter adaptation law without observer-gain adaptation, and rigorous Lyapunov-based stabilization with explicit error bounds; these elements address major practical and theoretical obstacles in real-world implementation (Zuo et al., 16 Jul 2024).