Feedback Linearized Physical System

• Feedback linearization is a method that converts nonlinear system dynamics into a linear form using state or output transformations combined with nonlinear feedback laws.
• Modern techniques integrate data-driven sparse regression and Lie derivative analysis to identify system models while enforcing strict linearization conditions.
• This approach enhances control performance in robotics, mechatronics, and biomedical applications by facilitating robust and interpretable linear controller design.

A feedback linearized physical system is a controlled dynamical system, typically nonlinear, whose open-loop input-output behavior is transformed into a linear system by applying a nonlinear change of variables (state or output transformation) and a nonlinear, possibly state-dependent, feedback law. The fundamental objective is to facilitate the use of linear system analysis and control design for systems whose underlying physical laws are nonlinear, by leveraging feedback to exactly cancel nonlinearities or render the closed-loop dynamics linear—either in the full state space, or in a subset of state variables, or along certain output channels.

1. Fundamental Principle and Mathematical Construction

The basic construction for a generic (control-affine) physical system is

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

where $x \in \mathbb{R}^n$, $u \in \mathbb{R}^m$, and $y \in \mathbb{R}^p$. Feedback linearization seeks a transformation

$z = \phi(x), \quad u = \alpha(x) + \beta(x) v$

such that, in the new coordinates, the closed-loop system is linear, typically in canonical (Brunovsky) form: $\dot{z} = A z + B v, \quad y = C z.$ The construction proceeds either by state (full) or output (input-output) linearization. For input-output linearization, Lie derivatives are employed to calculate the relative degree $r$ of the output function; the system is feedback linearized if the decoupling matrix (formed with the highest-order Lie derivatives) is nonsingular and zero dynamics are stable.

For physical systems with specific structures, e.g., mechanical systems governed by second-order Euler–Lagrange equations, feedback linearization can be constructed in a manner that preserves the mechanical structure, known as mechanical feedback linearization (Nowicki et al., 2022).

2. Data-Driven and Sparse Regression Approaches

Modern identification and controller synthesis for feedback linearized physical systems increasingly leverage data-driven techniques. The methodology in (K. et al., 18 Aug 2025) combines sparse regression for dynamic identification and Lie derivative analysis for feedback linearization:

• Sparse regression is used to infer the most parsimonious model structure for $f(x)$, $g(x)$, and output functions from data by searching a large function dictionary for a compact, best-fit representation.
• Lie derivatives are applied to estimate the system's relative degree and to check conditions ensuring that the control input directly affects the desired output derivatives at a specific order.
• Feedback design is realized by imposing an augmented constraint (involving Lie derivatives and relative degree) in the regression problem, ensuring no internal dynamics remain.
• A stacked regression formulation solves for dynamics and output mappings jointly, imposing sparsity and the linearization constraint as a (bilinear) optimization constraint.

The net result is a procedure that, from time-series data only (states, inputs, and measured outputs), discovers both the underlying governing equations and a feedback law that renders the closed-loop system input-output linear with the prescribed relative degree and no unobservable or hidden modes.

3. Feedback Controller Synthesis via Lie Derivatives

After model identification, the design of the feedback law exploits Lie derivatives to cancel nonlinearities and assign the desired closed-loop poles. Specifically, for an output $y = c(x)$, its derivatives under the nominal dynamics are:

• $L_f^0 c(x) = c(x)$
• $L_f c(x) = \frac{\partial c}{\partial x} f(x)$
• $$L_f^2 c(x) = \frac{\partial (L_f c(x))}{\partial x} f(x)$$, etc.

One differentiates until the input $u$ appears explicitly after $r$ steps; for example,

$$\frac{d^r}{dt^r} y = L_f^r c(x) + L_g L_f^{r-1} c(x) u$$

Assuming $L_g L_f^{r-1} c(x) \neq 0$, the feedback law

$u = v - [L_f^r c(x)] / [L_g L_f^{r-1} c(x)]$

yields a linear relationship $y^{(r)} = v$. By assigning $v$ as a state feedback controller (e.g., with gains chosen for pole placement), the closed-loop exhibits linear tracking behavior.

In the data-driven setting, this construction is embedded in the regression problem by enforcing the relevant Lie derivative constraints via bilinear (or augmented) equations involving the sparse regression coefficients (K. et al., 18 Aug 2025).

4. Stacked Regression and Integrated Linearization

Distinct from traditional serial approaches, the stacked regression method fuses identification (of both drift and input vector fields) and output mapping discovery into a unified optimization with the explicit constraint for feedback linearization. The stacked vector of parameters—combining dynamic coefficients and output mapping coefficients—is found by minimizing a composite loss: $$\min_{\tilde{\Xi}, \hat{\Xi}, \zeta} \|\dot{X} - \theta_f(X)\tilde{\Xi} - \theta_g(X,u)\hat{\Xi}\|_2^2 + \|Y - \Phi(X)\zeta\|_2^2 + \lambda_1(\|\tilde{\Xi}\|_1 + \|\hat{\Xi}\|_1) + \lambda_2 \|\zeta\|_1$$ subject to the Lie derivative constraint: $$\zeta^\top \left( \frac{\partial \Phi(x)}{\partial x}^\top \theta_g(X,u) \hat{\Xi} \right) = 0$$ This ensures that feedback linearizability is enforced in the very act of system identification.

Such integrated formulations guarantee that the designed feedback linearized system is robust (only few, physically relevant terms are retained), free of spurious internal dynamics, and fully compatible with observed data.

5. Mathematical Summary and Comparative Features

The following table encapsulates core elements of the feedback linearized physical system as presented:

Aspect	Equation/Formulation	Role
System dynamics	$\dot{x}(t) = f(x(t)) + g(x(t)) u(t)$, $y(t) = c(x(t))$	Underlying system equations
Sparse regression model	$$\dot{X} = \theta_f(X)\tilde{\Xi} + \theta_g(X,u)\hat{\Xi}$$	Identification from data
Output mapping	$Y = \Phi(X)\zeta$	Output reconstruction/definition
Lie derivative cond.	$$\zeta^\top [ (\partial\Phi(x)/\partial x)^\top \theta_g(X,u) \hat{\Xi} ] = 0$$	Enforces relative degree for linearization
Feedback law	$u = v - [L_f^r c(x)] / [L_g L_f^{r-1} c(x)]$ or as derived in regression under constraint	Nonlinear cancellation for linear closed loop
Stacked regression	$\min_{\tilde{\Xi}, \hat{\Xi}, \zeta}\, \text{cost}$ s.t. $\zeta^\top M(x) \hat{\Xi}_k = 0$	Simultaneous model and controller identification

As highlighted in (K. et al., 18 Aug 2025), this methodology stands apart from prior approaches by guaranteeing, in closed form and directly from the data, that the synthesized feedback controller admits the absence of internal dynamics and minimality (in the sense of sparsity) of the underlying predictive model.

6. Applications and Broader Significance

Feedback linearized physical systems, realized through sparse data-driven discovery and Lie derivative analysis, have implications across scientific and engineering domains:

• In robotics and mechatronics, they enable control of manipulators, aerial vehicles, and flexible systems where precise models are intractable.
• In biomedical and neuronal systems, the methodology facilitates system identification and control under severe nonlinearity and limited parametric knowledge.
• Their integration with adaptive, learning-based, and online updating algorithms paves the way toward real-time control in uncertain or drifting environments.

Moreover, by directly linking model identification to the requirements of exact feedback linearization (relative degree, internal dynamics), the stacked regression and Lie derivative approach offers a pathway to robust, interpretable, and high-performing controller design—even for systems whose governing equations were, until now, inaccessible or poorly understood.

This synthesis, based strictly on factual content from (K. et al., 18 Aug 2025), provides a rigorous and contemporary perspective on feedback linearized physical systems within modern identification and control theory.