On the Modulating Function Method for Control Problems

Abstract: The modulating function method is an algebraic framework that, thus far, has been used for state and parameter estimation, as well as fault detection, of linear, fractional-order, distributed, and some nonlinear systems. At the core of the method lies the modulating function, which can either be selected directly or be obtained as a solution to an auxiliary system. By introducing the notion of dual modulating functions and dual modulations using auxiliary systems and duality, this paper shows that this framework is not only an estimation framework, but also a controller design framework for LTV systems. In particular, necessary and sufficient conditions for the existence of the associated control laws are introduced; the well-known state feedback law is obtained as a particular case of the dual modulation approach, along with output feedback, LTI sliding mode control, the reachability gramian, and the state-transition matrix; and a new fixed-time control law is proposed for both LTI and LTV systems, including an estimate of the transient behavior. Moreover, numerical simulations of the newly proposed control law are performed, indicating similar performance levels to a benchmark LQR even when handling unmatched disturbances.

• The paper introduces dual modulating functions for control, linking estimation and controller design via algebraic duality.
• It establishes necessary and sufficient conditions for fixed-time stabilization through explicit operator equations and a modulating function regulator.
• Numerical results show the method achieves robust transient shaping and reduced actuation effort, validating its competitiveness against LQR.

Detailed Summary of "On the Modulating Function Method for Control Problems" (2607.04544)

Introduction and Motivation

The paper investigates the modulating function method (MFM), an algebraic framework well established in state and parameter estimation and fault detection for various classes of dynamical systems, including linear, fractional-order, and distributed systems. Despite its history in estimation, the application of MFM in control synthesis, particularly for linear time-varying (LTV) and multiple-input multiple-output (MIMO) systems, remained largely unexplored. The core contribution of this work is the formalization and application of dual modulating functions—derived via duality and auxiliary systems—to extend MFM from estimation to control, enabling rigorous controller synthesis for a broad class of LTV systems.

Modulating Functions and Duality

A modulating function is a carefully designed, sufficiently smooth windowing function used to manipulate system equations (typically by multiplication and integration followed by integration by parts). Specific classes of modulating functions—total (TMF), left (LMF), and right (RMF)—are defined by vanishing boundary conditions to tailor estimation or control tasks.

The paper introduces the dual modulating function for SISO LTI systems as:

$$\gamma(\tau) = \varphi^{(n)}(\tau) + \sum_{i=0}^{n-1} a_i \varphi^{(i)}(\tau)$$

where the $\varphi$ functions are modulating functions and $a_i$ are system coefficients. This object captures the duality between the structure of system dynamics and the algebraic manipulations permitted by the MFM, laying the groundwork for linking modulation-based estimation and controller design.

For general LTV systems in state space, the modulation operator is defined as a convolution:

$$\langle \mathbf{g}, \mathbf{x} \rangle (t_1, t_0) = \int_{t_0}^{t_1} \mathbf{g}^\top(t_1 - \tau) \mathbf{x}(\tau) d\tau$$

This operator admits powerful algebraic properties (e.g., derivative shifting via integration by parts), which are extensively used in both estimation and control contexts.

Dual Modulations and Controller Synthesis

Central to the extension from estimation to control is the dual modulation kernel:

$\bm{\uplambda}(\tau) = \mathbf{g}'(\tau) - \mathbf{A}^*(\tau) \mathbf{g}(\tau)$

where $\mathbf{A}^*(\tau)$ is the adjoint of the system matrix, and the evolution of $\mathbf{g}(\tau)$ constitutes a dual (adjoint) auxiliary system. By careful selection of $\bm{\uplambda}$, desired properties (controllability, reachability) can be imposed on the solutions.

This framework yields notable results:

• Full-state Feedback: By setting $\bm{\uplambda} = -\mathbf{K}^* \mathbf{v}$, where $\mathbf{K}$ is a feedback gain matrix, one recovers classic full-state feedback laws $\varphi$0 within the modulation framework.
• Output Feedback: Similar construction produces output feedback control via output injection matrices.
• Sliding Mode Control: With appropriate nonlinear maps in $\varphi$1, one directly obtains classic sliding-mode structures for affine systems.
• State-transition Representations: Choosing $\varphi$2 recovers the classical LTV state transition matrix solution.
• Reachability and Controllability: The sets of reachable and null-controllable states are characterized directly in terms of dual modulations and associated modulated integrals.

Finite-Time (Fixed-Time) Control and Feasibility Analysis

A major claim is the derivation of necessary and sufficient conditions for finite-time stabilization. The paper shows that the constructed control law steers the state to the origin in finite time $\varphi$3 if and only if a specific matrix $\varphi$4, defined as:

$\varphi$5

where $\varphi$6 is the system's state-transition matrix. This explicit criterion enables systematic construction of fixed-time stabilizers for general LTV MIMO systems.

To check whether solutions to the associated (often Fredholm or Volterra) integral equations exist for the control law, the paper introduces:

• The self-modulation operator $\varphi$7 (generalizing the reachability Gramian), whose full rank implies existence of solutions for arbitrary reachable states.
• The controllability bracket $\varphi$8, which provides a tractable operator equation for verifying whether the modulated control law is feasible and preserves controllability.

By directly synthesizing kernels using families of modulating functions (TMFs for off-diagonal and unitary RMFs for diagonal components), one can guarantee the boundary and rank conditions required for fixed-time convergence.

The Modulating Function Regulator (MFR)

This theoretical machinery culminates in a fixed-time control law—the modulating function regulator (MFR)—which works for LTV MIMO systems. The input is explicitly given in terms of system parameters, modulating function derivatives, and sampled states and inputs over a moving window of length $\varphi$9. The control law is optimal in the least-squares sense relative to $a_i$0 due to the use of the Moore-Penrose pseudoinverse. For LTI systems, the MFR gains can be pre-computed; for LTV systems, computation is done online.

Numerical Results and Comparative Analysis

The paper provides simulations for both unstable LTI and disturbed LTV systems (e.g., a DC motor with time-varying resistance), comparing the MFR to baseline linear-quadratic regulator (LQR) controllers:

• The MFR achieves closed-loop stabilization with similar transient performance and chattering characteristics to LQR.
• Notably, actuation effort is sometimes reduced under MFR compared to LQR, especially in disturbance scenarios.
• Fixed convergence time is observed—strongly supporting the finite-time property.
• The effect of the number of integration partitions and kernel choice on numerical error and transient shape is quantified.
• The flexibility in transient trajectory shaping through MF selection is emphasized, enabling both performance tuning and robustness assessment.

Theoretical and Practical Implications

The extension of MFM to controller synthesis via dual modulations establishes a unified algebraic approach to estimation and control for a broad class of linear (and possibly nonlinear) systems. The explicit conditions for existence and feasibility expand the theoretical tools available for system analysis, providing new operator-based tests for controllability and solvability in continuous time.

Practically, the MFR offers:

• Potential for embedded, model-based fixed-time stabilizers where classic time-domain eigenvalue placement is not applicable.
• Flexible transient shaping via modulating function selection.
• Direct robustness analysis with respect to parametric changes, as the controller structure is transparent in parameter dependence.

Future work is anticipated on:

• Embedded implementation and real-time feasibility.
• Extension to more general nonlinear or hybrid systems.
• Deeper exploration of modulating function families optimizing robustness, transient response, or actuation effort.
• Integration with tracking control objectives and multi-objective designs.

Conclusion

This work rigorously establishes dual modulating functions and dual modulations as a foundation for extending the MFM from estimation to control. The proposed fixed-time control law is applicable to LTV MIMO systems, with both theoretical guarantees and practical methods for verification and synthesis. Numerical evaluations validate its competitiveness with established optimal controllers, and the theoretical framework opens multiple avenues for both theoretical development and applied control design.