Dynamic Mode Adaptive Control (DMAC)

• Dynamic Mode Adaptive Control (DMAC) is a model-free, data-driven method that approximates local system dynamics using techniques like RLS and DMD.
• It integrates online identification with full-state and integral feedback to synthesize adaptive tracking controllers for nonlinear or high-dimensional systems.
• DMAC has been validated through benchmarks and engineering applications, demonstrating fast convergence, minimal overshoot, and robustness to parameter variations.

Dynamic Mode Adaptive Control (DMAC) is a model-free, data-driven adaptive control methodology designed for high-dimensional, nonlinear, or partially observed systems where traditional model-based control strategies are either inapplicable or computationally prohibitive. DMAC integrates online local dynamics approximation using matrix recursive least squares (RLS) or dynamic mode decomposition (DMD) with an adaptive tracking controller that employs full-state and integral feedback. The DMAC architecture is now well-established as a practical and robust alternative to conventional adaptive and robust control, especially for systems lacking explicit, compact models or facing substantial time-varying behavior (Oveissi et al., 17 May 2025, Oveissi et al., 4 Jan 2026, Oveissi et al., 4 Jan 2026).

1. Definition, Motivation, and Basic Structure

DMAC synthesizes output tracking controllers for sampled-data systems described by

$\dot{x}(t) = f(x(t),u(t)), \quad y(t) = h(x(t))$

where the measured signal $\xi_k = h(x(kT_s)) \in \mathbb{R}^{l_\xi}$ is available at discrete sampling times, and the control input $u_k\in \mathbb{R}^m$ is piecewise constant on $[kT_s,(k+1)T_s)$. DMAC is comprised of two key modules:

• Dynamics Approximation Module: Learns low-dimensional, time-varying linear approximations of the local system dynamics in state-space form:

  $\xi_{k+1} \approx A_k \xi_k + B_k u_k$

  using a matrix RLS algorithm with exponential forgetting and regularization.

• Controller Module: Synthesizes a full-state, integral-action feedback law exploiting the estimated $(A_k,B_k)$:

  $u_k = K_{\xi,k}\xi_k + K_{q,k}q_k + v_k$

  where $q_k$ accumulates the tracking error, and $v_k$ is a small stochastic signal to ensure persistence of excitation.

DMAC is motivated by the difficulty of constructing accurate and tractable models for nonlinear or high-dimensional plants—especially when only sparse or aggregate system measurements are available. It bypasses explicit modeling by continuously (re-)identifying a local linear system and linking this identification to online controller synthesis, enabling real-time closed-loop adaptation (Oveissi et al., 17 May 2025, Oveissi et al., 4 Jan 2026, Oveissi et al., 4 Jan 2026).

2. Dynamic Mode Approximation and Matrix Recursive Least Squares

At each sampling instant $k$, DMAC collects the regressor vector

$$\phi_k = \begin{bmatrix} \xi_k^\top & u_k^\top \end{bmatrix}^\top \in \mathbb{R}^{n}$$

with $n = l_\xi + m$, and the next measured state $\xi_{k+1}$. The goal is to compute

$$\Theta_k = [A_k \;\; B_k] \in \mathbb{R}^{l_\xi \times n}$$

by minimizing the regularized, exponentially weighted least-squares cost: $$J_k(\Theta) = \sum_{i=0}^{k}\lambda^{k-i}\|\xi_{i+1}-\Theta\phi_i\|_2^2 + \lambda^{k}\operatorname{Tr}(\Theta^\top R_\Theta\Theta)$$ where $0 < \lambda \le 1$ is the forgetting factor, and $R_\Theta \succ 0$ provides Tikhonov regularization.

Recursive updates proceed as follows:

• Covariance matrix update:

  $P_k = \lambda P_{k-1} + \phi_k\phi_k^\top$

• Inverse covariance update:

  $$P_k^{-1} = \lambda^{-1}P_{k-1}^{-1} - \lambda^{-1}P_{k-1}^{-1}\phi_k\gamma_k^{-1}\phi_k^\top P_{k-1}^{-1}$$

  where $$\gamma_k = \lambda + \phi_k^\top P_{k-1}^{-1}\phi_k$$.

• Parameter update:

  $$\Theta_k = \Theta_{k-1} + [\xi_{k+1}-\Theta_{k-1}\phi_k]\phi_k^\top P_k^{-1}$$

This approach ensures efficient, numerically stable updates that maintain real-time feasibility, accommodate strongly time-varying dynamics, and prevent overfitting or ill-conditioning, especially when operated with small regularization $\alpha$ in $R_\Theta = \alpha I_n$ (Oveissi et al., 17 May 2025, Oveissi et al., 4 Jan 2026, Oveissi et al., 4 Jan 2026).

A DMD-style variant with windowed data or singular value decomposition may be employed for dimensionality reduction or to discard modes with low energy, especially in systems with large state dimensions or when working with empirical snapshot data (Oveissi et al., 4 Jan 2026, Oveissi et al., 4 Jan 2026).

3. Synthesis of Adaptive Tracking Controllers

Once the local linear model is available, DMAC constructs an augmented state for output tracking: $$\bar{x}_k = \begin{bmatrix}\xi_k \ q_k \end{bmatrix}, \quad q_k = \sum_{i=0}^k (r_i - y_i)$$ with $y_k = C\xi_k$. The discrete-time system is then: $$\bar{x}_{k+1} = \begin{bmatrix} A_k & 0 \ -C & I \end{bmatrix}\bar{x}_k + \begin{bmatrix} B_k \ 0 \end{bmatrix}u_k + \begin{bmatrix} 0 \ I \end{bmatrix}r_k$$

Controller gains $[K_{\xi,k}, K_{q,k}]$ are synthesized by minimizing the discrete-time linear-quadratic cost: $$J = \sum_{k=0}^\infty \bar{x}_k^\top Q \bar{x}_k + u_k^\top R u_k$$ via the algebraic Riccati equation, yielding full-state feedback with integral action. The tracking control law becomes: $u_k = K_{\xi,k}\xi_k + K_{q,k}q_k + v_k$ where $v_k \sim \mathcal{N}(0,\sigma_v^2 I)$ is stochastic probing noise. This noise, properly tuned, guarantees the probing of unexcited modes, facilitating the convergence of RLS without external test signals (Oveissi et al., 17 May 2025, Oveissi et al., 4 Jan 2026, Oveissi et al., 4 Jan 2026).

State-feedback designs may use LQR or LQI criteria, and for systems with measured outputs derived from nonlinear maps (such as neural-network surrogates for physical models), the measurement Jacobian can substitute for $C$. The modularity of the controller design allows for separation between system identification and controller synthesis, as well as rapid adaptation in the presence of changes in system parameters or reference inputs (Oveissi et al., 17 May 2025, Oveissi et al., 4 Jan 2026).

4. Robustness, Sensitivity, and Performance Guarantees

Systematic sensitivity studies demonstrate that DMAC is robust to a broad range of choices for the forgetting factor $\lambda$, regularization $\alpha$, LQR weights, and probing variance $\sigma_v^2$:

• $\lambda \in [0.9,0.9999]$ controls adaptation speed; smaller $\lambda$ improves transient tracking but increases noise sensitivity.
• $\alpha \in [10^2,10^6]$ regularizes the RLS parameter estimates; small $\alpha$ can induce ill-conditioning, while large $\alpha$ damps parameter fluctuations.
• $\sigma_v^2 \in [10^{-6},10^{-2}]$ ensures persistence of excitation; excessive probing degrades tracking performance.

DMAC consistently provides geometric error decay and bounded parameter estimates across wide variations in physical plant parameters—robust to 2–3 orders of magnitude variation in coefficients such as mass, spring constants, drag, and nonlinear parameters (e.g., Van der Pol parameter $\mu$). RLS convergence is typically achieved within 50–100 samples in canonical benchmarks. Closed-loop stability and convergence of tracking error to zero follow under standard Lyapunov arguments for adaptive control, contingent on sufficient probing excitation and slow variation of plant parameters (Oveissi et al., 17 May 2025, Oveissi et al., 4 Jan 2026, Oveissi et al., 4 Jan 2026).

5. Applications: Benchmarks and Engineering Systems

Demonstrated applications span both canonical benchmarks and complex physics-based engineering systems:

• Benchmarks: Mass–spring–damper, 3-mass chain, Van der Pol oscillator, Burgers' PDE with sparse measurements. Consistent outcomes include settling times of 1–3 seconds, sub-2% overshoot, and robust tracking under wide parameter variations (Oveissi et al., 17 May 2025).
• Ramjet Thrust Regulation: DMAC regulates thrust in solid-fuel ramjets using only measured outputs and formalizes the architecture for high-fidelity CFD-in-the-loop settings. Reported metrics include settling time $\sim$50 samples (0.5 s), steady-state error under 1 N, and insensitivity to $\lambda$ and $R_\Theta$ tuning over $10^{0}-10^{4}$ (Oveissi et al., 4 Jan 2026, Oveissi et al., 4 Jan 2026).
• LPV–MAPS Extension: The Mode-Aware Probabilistic Scheduling framework applies DMAC-inspired ideas to LPV systems, using IMM estimators for probabilistic gain scheduling in DC-motor systems with time-varying friction, achieving hardware-in-the-loop validation and LMI-based exponential stability certificates (Kim et al., 16 Sep 2025).

A summary table of experimental results appears below:

Example	n	λ	α	Settling Time	Max Error	Overshoot
Mass–spring–damper	3	0.995	$10^4$	$\sim$2 s	0.02 N	$<$2%
3-mass chain	4	0.999	$10^4$	$\sim$3 s	0.01 N	$<$1.5%
Van der Pol	3	0.995	$10^4$	$\sim$2.5 s	0.03 N	$<$3%
Burgers PDE (7/100 nodes)	8	0.9995	$10^4$	$\sim$1 s	0.1–1	$<$5%

(Oveissi et al., 17 May 2025, Oveissi et al., 4 Jan 2026)

6. Design Guidelines and Comparative Analysis

Design of DMAC systems follows several empirically grounded recommendations:

• Initialize $\lambda\sim0.995$, $\alpha\sim10^3\text{–}10^4$, and $\sigma_v\sim10^{-4}$, then refine by sensitivity analysis.
• For fast-adapting plants, decrease $\lambda$; for noise suppression, increase $\lambda$.
• Use sparse probing only in directions unexcited by normal reference signals.
• Set LQR/LQI weights to balance tracking accuracy and control effort, with state/integral penalty blocks as appropriate.

DMAC does not require an explicit plant model; it adapts to the local operating regime, tolerates significant unmodeled dynamics, and can handle partially observed or overparameterized states, especially when combined with reduced-order modelling via DMD or SVD truncation (Oveissi et al., 17 May 2025, Oveissi et al., 4 Jan 2026, Oveissi et al., 4 Jan 2026). In contrast to MRAC/RCAC or classical robust control, DMAC decouples the identification and control synthesis steps, imposing no parametric structure on the system evolution.

7. Extensions and Theoretical Considerations

The DMAC paradigm generalizes to:

• Plants controlled through parameter-varying surfaces (variable-geometry inlets in ramjets) and systems exhibiting time-varying or uncertain parameters.
• Frameworks where local mode probabilities drive probabilistic scheduling of controller gains (as in the MAPS architecture) (Kim et al., 16 Sep 2025).
• Environments where only a subset of states or outputs is directly measured, leveraging DMD-based surrogate models or neural-network-based measurement Jacobians within the feedback loop (Oveissi et al., 4 Jan 2026, Oveissi et al., 4 Jan 2026).

Theoretical analysis establishes quadratic Lyapunov stability when persistence of excitation is enforced in the input channel. Under mild regularity and bounded parameter drift, DMAC ensures exponential convergence of both parameter error and state-tracking error. Monte Carlo investigations confirm empirical robustness across reference types, plant parameter distributions, and operational regimes (e.g., solid-fuel ramjet thrust, with 97.5% success rates over randomized trials) (Oveissi et al., 4 Jan 2026).

In summary, Dynamic Mode Adaptive Control provides a computationally efficient, generalizable, and rigorous adaptive control framework—combining data-driven local model identification and integral feedback regulation—demonstrated in both benchmark and high-fidelity engineering applications, with strong robustness to hyperparameter tuning and system-orientated uncertainties (Oveissi et al., 17 May 2025, Oveissi et al., 4 Jan 2026, Oveissi et al., 4 Jan 2026, Kim et al., 16 Sep 2025).