Flatness-Based Control: Theory & Applications

• Flatness-based control is a geometric and algebraic framework that uses differential flatness to express all system states and inputs in terms of flat outputs and their derivatives.
• It enables explicit open-loop trajectory planning and feedforward control for nonlinear systems and distributed parameter systems, including PDEs like the Euler–Bernoulli beam and Schrödinger equation.
• The method integrates advanced techniques such as series expansions, Gevrey class functions, and feedback stabilization to ensure rigorous controllability and accurate shape tracking.

Flatness-based control is a geometric and algebraic methodology for trajectory planning and feedback design in nonlinear and infinite-dimensional systems. At its core, it relies on the system's differential flatness property: there exists a set of "flat outputs" such that all system states and inputs can be expressed as (typically explicit) functions of these outputs and a finite number of their time derivatives. This framework offers a systematic route for constructing both open-loop and closed-loop controls for highly nonlinear or distributed parameter systems, under-actuated devices, and systems described by partial differential equations (PDEs) with boundary or in-domain actuation.

1. Theory of Differential Flatness and Model Parameterization

For a nonlinear system

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

flatness entails the existence of an output vector $y = \psi(x, u, \dot{u}, ..., u^{(r)})$ such that the full state $x$ and input $u$ can be parametrized by $y$ and its time derivatives up to order $r$: $$x = \Phi_x(y, \dot{y}, ..., y^{(r)}), \quad u = \Phi_u(y, \dot{y}, ..., y^{(r)}).$$ In infinite-dimensional settings such as PDEs, the flat output may parametrize the state and boundary or distributed inputs via infinite series expansions (generalized Birkhoff normal forms), e.g., as in

$w(x, t) = \sum_{n=0}^\infty H_n(x) y^{(2n)}(t)$

with $H_n(x)$ analytic functions and $y(t)$ a trajectory of the flat output.

Flatness is preserved under coordinate change and is stronger than classical controllability; all flat systems are controllable, but not all controllable systems are flat.

2. Application to Distributed Parameter Systems: Boundary and In-domain Actuation

Flatness-based control was extended to boundary- and in-domain-actuated PDEs, e.g., for the Euler–Bernoulli beam and Schrödinger/heat equations.

For an Euler–Bernoulli beam actuated at $N$ in-domain points: $$w_{tt}(x, t) + w_{xxxx}(x, t) = \sum_{j=1}^{N+1} \alpha_j(t) \delta(x - x_j),$$ the inhomogeneous PDE is mapped via a lifting transformation to a standard boundary-controlled analogue. The boundary actuation is synthesized by solving auxiliary ODEs for the lifting functions $H_j(x)$ that satisfy

$H_j^{(4)}(x) = \varphi(x - x_j),$

with smooth mollifier $\varphi \to \delta$ in the distribution limit. The original in-domain actuation is replaced by boundary inputs (equivalent in the steady-state), thereby leveraging the full machinery of flatness-based boundary control for PDEs (Badkoubeh et al., 2015).

Similarly, for the 1D Schrödinger equation with Dirichlet boundary control, the state and input are explicitly expressed in terms of the spatial derivative at $x = 0$: $$y(t) = \theta_x(t, 0), \qquad \theta(t, x) = \sum_{j=0}^\infty \frac{x^{2j+1}}{(2j+1)!} (-i)^j y^{(j)}(t).$$ This permits exact controllability via open-loop, series-based construction of the boundary input (Martin et al., 2017).

3. Flatness-based Motion Planning and Feedforward Control

For systems amenable to flat parameterizations, motion planning is transformed to the problem of designing trajectories for the flat output $y(t)$ that satisfy initial and final state consistency. Practically, one selects $y(t)$ as a trajectory in a high-regularity class (e.g., Gevrey class of order $1 < \sigma < 2$) so that all derivatives are bounded and the associated series converge.

In the motion planning phase:

• The desired deformation or final state is typically mapped to the amplitudes of the flat output via inversion of an input–output map (often involving Green’s functions, as in steady-state interpolation for a micro-beam).
• The state and input trajectories are synthesized by evaluating the explicit series or parameterizations for the entire time horizon, yielding open-loop controls that steer the system exactly along the prescribed path in the absence of uncertainty.

The use of series expansions is critically dependent on convergence, for which Gevrey-class transitions and careful truncation strategies (N-term selection) are used to assure feasibility and practical implementability (Badkoubeh et al., 2015, Scholz et al., 2023).

4. Feedback Design and Closed-loop Stabilization

Open-loop flatness-based feedforward is susceptible to disturbances and unmodeled dynamics. Closed-loop stabilization is commonly incorporated by:

• Dedicating an actuator (or boundary control channel) to feedback. For a beam, e.g., one actuator is reserved for stabilization:

$\alpha_{N+1}(t) = -k w_t(x_{N+1}, t), \quad k > 0.$

• The overall closed-loop system (in Hilbert space formulation) is mathematically treated as a dissipative system, and exponential stability of the regulation error

$e(x, t) = w(x, t) - \bar{w}^d(x)$

is established using semigroup theory and energy methods.

In the spatial domain, specific actuation location (e.g., rational, coprime fractions) may affect stabilization guarantees through spectrum considerations (Badkoubeh et al., 2015).

5. Well-posedness, Regularity, and Implementation Aspects

Rigorous analysis of infinite-dimensional flatness-based control includes:

• Well-posedness of the closed-loop PDE system, typically in weak/distributional sense due to Dirac delta or blob inputs.
• Existence and uniqueness of solutions established in function spaces such as $C([0,T];\Phi) \cap C^1([0,T];L^2(0,1))$, with regularity inherited from the flat output trajectory.
• Stability proofs leveraging Lyapunov/energy functionals, dissipativity of the generator, and semigroup theory.

Implementation entails truncating the open-loop series for practical computation and dealing with actuator limitations. Numerical simulations indicate that increasing the number of actuators improves shape tracking but at the expense of higher actuation effort. Trade-offs are managed by selecting an intermediate number of actuators that achieves desired tracking accuracy with moderate input energy.

6. Representative Results and Numerical Verification

• Numerical studies for micro-beam deformation (e.g., with 8, 12, 16 actuators) show that regulation error decays to zero and the beam deformation tracks complex shapes (such as Gaussian combinations) accurately.
• The design avoids finite-dimensional mode truncation (which can result in spillover or unmodeled dynamics), as the control operates directly on the full infinite-dimensional system.
• Comparative metrics reported in the simulations include regulation error, input magnitude, and transient energy in the controlled beam (Badkoubeh et al., 2015).

7. Extensions and Comparative Remarks

Flatness-based control of PDEs provides a unified planning and control architecture for distributed parameter systems, distinct from modal truncation-based feedback or adjoint-based indirect controllability proofs. Its explicit construction of input and state trajectories makes it particularly powerful for reference tracking and shape control in flexible structures, and for systems with noncollocated or in-domain actuation.

The approach is compatible with model-free or intelligent control designs for robustness (HEOL framework (Join et al., 21 Aug 2024)) and can be generalized or combined with optimization-based MPC for handling constraints (Neve et al., 2023). For real-world applications, the choice of flat outputs, regularity constraints, and spatial actuation strategies are crucial for successful implementation.

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Summary Table: Flatness-based Control for In-domain Actuated Euler–Bernoulli Beam

Step	Key Mathematical Construction	Practical Outcome
PDE modeling & lifting	Inhomogeneous PDE → boundary control form	Enables use of flatness/boundary control methods
Flatness-based planning	Series representation (e.g., Eqn. (16))	Explicit state/input trajectories from flat outputs
Feedforward design	Flat output chosen as Gevrey function	Convergent series/planned rest-to-rest transitions
Output amplitude mapping	Input–output inversion (Green’s function)	Exact steady-state shape matching/interpolation
Feedback stabilization	Extra actuator, velocity feedback at control pt.	Exponential error decay, robustness to disturbances
Well-posedness & regularity	C([0,T];Φ) ∩ C¹([0,T];L²(0,1)); semigroup proof	Mathematical guarantee for the closed-loop solution
Experimental validation	Micro-beam simulations with various actuators	Verified tracking, input–output effort trade-off

The methodology set forth in this framework and its rigorous verification represent the state-of-the-art for distributed parameter shape control using flatness-based techniques (Badkoubeh et al., 2015).