Output-Feedback Contraction in Control Theory

• Output-feedback contraction is a control framework that uses contraction metrics to certify incremental exponential stability via output-feedback laws.
• It applies across continuous, discrete, and infinite-dimensional settings using LMIs, metric bounds, and separation principles for observer and controller synthesis.
• The approach enhances robust control with integral action and learned-perception architectures, accommodating multistability and complex system dynamics.

Searching arXiv for recent and foundational papers on output-feedback contraction and related contraction-based feedback design. Search query: "output-feedback contraction control contraction metrics observer contraction metric arXiv" Output-feedback contraction denotes a family of control-theoretic constructions in which an output-feedback law is designed or analyzed so that the resulting closed-loop system is contracting, incrementally exponentially stable, or, in generalized settings, $2$-contracting. In finite-dimensional nonlinear systems, the term is most closely associated with control contraction metrics (CCMs), dual observer contraction metrics, and separation principles for dynamic output-feedback controllers (Manchester et al., 2014). In discrete time, the same theme appears in contraction-based LMIs for state-feedback and observer synthesis on learned Structured State-space Models (SSMs) with bi-Lipschitz input and output scaffoldings (Zakwan et al., 8 Apr 2026). In infinite dimensions, contraction is formulated at the level of nonlinear contraction semigroups or contraction semigroups generated by closed-loop operators, and output-based integral action or static output feedback is used to obtain regulation and stabilization (Vanspranghe et al., 2022, Curtain et al., 2021). A broader usage also includes contraction certification for feedback interconnections, two-time-scale output-feedback loops, and learned-perception control architectures in which state estimates are produced from high-dimensional observations (Ofir et al., 2024, Cothren et al., 2023, Chou et al., 2022).

1. Differential and semigroup notions of contraction

In the continuous-time finite-dimensional setting, the basic plant is

$$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$

with differential dynamics

$$\dot\delta_x(t) = A(x,t)\delta_x(t)+B(t)\delta_u(t), \qquad \delta_y(t) = C(t)\delta_x(t),$$

where $A(x,t)=\partial f(x,t)/\partial x$. A control contraction metric is a Riemannian metric of the form

$$V(x,\delta_x,t)=\delta_x' M(x,t)\delta_x,\qquad \alpha_1I\le M(x,t)\le \alpha_2I,$$

such that uncontrolled differential directions are strictly contracting (Manchester et al., 2014). In discrete time, contraction is expressed through

$$(A_k + B_k K_k)^T M_{k+1} (A_k + B_k K_k) - (1-\rho) M_k \prec 0,$$

for some $0<\rho<1$, with $M_k\succ 0$; this is the discrete-time analogue of standard contraction inequalities and implies that squared differential distances measured in metric $M$ shrink by at least factor $1-\rho$ per step (Zakwan et al., 8 Apr 2026).

The semigroup formulation is the infinite-dimensional analogue. For a nonlinear evolution

$$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$0

on a Hilbert space $$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$1, strong monotonicity,

$$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$2

implies the exponential contraction estimate

$$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$3

where $$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$4 is the nonlinear strongly continuous contraction semigroup generated by $$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$5 (Vanspranghe et al., 2022). In the linear operator-theoretic setting, a closed-loop operator generates a contraction semigroup precisely when it is maximal dissipative, by the Lumer–Phillips theorem; this is the fundamental mechanism behind several PDE feedback constructions (Kurula et al., 2014).

These formulations share the same structural idea: contraction is not merely asymptotic stability of one trajectory, but a uniform decay of distances between trajectories in a chosen metric or norm. This suggests why output-feedback contraction is naturally tied to observer design, disturbance rejection, and trajectory tracking rather than only equilibrium stabilization.

2. Controller–observer synthesis in finite dimensions

The standard finite-dimensional realization is the CCM/observer-metric framework for nonlinear control-affine systems. A convex sufficient condition for universal exponential stabilizability by state feedback is the existence of $$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$6, $$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$7, and constants $$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$8 such that

$$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$9

and

$$\dot\delta_x(t) = A(x,t)\delta_x(t)+B(t)\delta_u(t), \qquad \delta_y(t) = C(t)\delta_x(t),$$0

Then the control contraction metric is $$\dot\delta_x(t) = A(x,t)\delta_x(t)+B(t)\delta_u(t), \qquad \delta_y(t) = C(t)\delta_x(t),$$1, and the differential gain is

$$\dot\delta_x(t) = A(x,t)\delta_x(t)+B(t)\delta_u(t), \qquad \delta_y(t) = C(t)\delta_x(t),$$2

used together with a geodesic integration formula

$$\dot\delta_x(t) = A(x,t)\delta_x(t)+B(t)\delta_u(t), \qquad \delta_y(t) = C(t)\delta_x(t),$$3

to track a reference trajectory $$\dot\delta_x(t) = A(x,t)\delta_x(t)+B(t)\delta_u(t), \qquad \delta_y(t) = C(t)\delta_x(t),$$4 (Manchester et al., 2014).

The dual observer problem replaces $$\dot\delta_x(t) = A(x,t)\delta_x(t)+B(t)\delta_u(t), \qquad \delta_y(t) = C(t)\delta_x(t),$$5 by $$\dot\delta_x(t) = A(x,t)\delta_x(t)+B(t)\delta_u(t), \qquad \delta_y(t) = C(t)\delta_x(t),$$6. An observer contraction metric $$\dot\delta_x(t) = A(x,t)\delta_x(t)+B(t)\delta_u(t), \qquad \delta_y(t) = C(t)\delta_x(t),$$7 satisfies

$$\dot\delta_x(t) = A(x,t)\delta_x(t)+B(t)\delta_u(t), \qquad \delta_y(t) = C(t)\delta_x(t),$$8

with the same type of metric bounds. In the constant-metric case this yields a Luenberger-type observer, while in the general case the observer uses a minimizing path from the estimate $$\dot\delta_x(t) = A(x,t)\delta_x(t)+B(t)\delta_u(t), \qquad \delta_y(t) = C(t)\delta_x(t),$$9 to the output-consistent set

$A(x,t)=\partial f(x,t)/\partial x$0

in the metric $A(x,t)=\partial f(x,t)/\partial x$1 (Manchester et al., 2014). The resulting theorem is a nonlinear separation principle: if a state-feedback CCM and a dual observer contraction metric both exist, then the dynamic output-feedback controller obtained by evaluating the CCM-based feedback at the state estimate is exponentially stable (Manchester et al., 2014).

A discrete-time variant appears in SSM-based indirect data-driven control. The nonlinear SSM model

$A(x,t)=\partial f(x,t)/\partial x$2

uses a linear recurrent unit together with bi-Lipschitz neural-network scaffoldings $A(x,t)=\partial f(x,t)/\partial x$3. Local controllability is inherited from controllability of $A(x,t)=\partial f(x,t)/\partial x$4 and bi-Lipschitzness of $A(x,t)=\partial f(x,t)/\partial x$5; local observability is inherited from observability of $A(x,t)=\partial f(x,t)/\partial x$6 and bi-Lipschitzness of $A(x,t)=\partial f(x,t)/\partial x$7 (Zakwan et al., 8 Apr 2026). State-feedback synthesis is performed through the LMI

$A(x,t)=\partial f(x,t)/\partial x$8

with $A(x,t)=\partial f(x,t)/\partial x$9, while observer synthesis is performed through

$$V(x,\delta_x,t)=\delta_x' M(x,t)\delta_x,\qquad \alpha_1I\le M(x,t)\le \alpha_2I,$$0

with $$V(x,\delta_x,t)=\delta_x' M(x,t)\delta_x,\qquad \alpha_1I\le M(x,t)\le \alpha_2I,$$1 (Zakwan et al., 8 Apr 2026). The paper then proves a discrete-time separation principle for the full output-feedback loop. A common misconception is that contraction-based output feedback is restricted to continuous time; these SSM results show that the same architecture extends to learned discrete-time surrogates.

3. Interconnections, small-gain conditions, and generalized $$V(x,\delta_x,t)=\delta_x' M(x,t)\delta_x,\qquad \alpha_1I\le M(x,t)\le \alpha_2I,$$2-contraction

Output-feedback contraction is not restricted to $$V(x,\delta_x,t)=\delta_x' M(x,t)\delta_x,\qquad \alpha_1I\le M(x,t)\le \alpha_2I,$$3-contraction of a single plant–observer pair. For the feedback interconnection

$$V(x,\delta_x,t)=\delta_x' M(x,t)\delta_x,\qquad \alpha_1I\le M(x,t)\le \alpha_2I,$$4

with Jacobian

$$V(x,\delta_x,t)=\delta_x' M(x,t)\delta_x,\qquad \alpha_1I\le M(x,t)\le \alpha_2I,$$5

a sufficient condition for $$V(x,\delta_x,t)=\delta_x' M(x,t)\delta_x,\qquad \alpha_1I\le M(x,t)\le \alpha_2I,$$6-contraction is formulated in terms of a $$V(x,\delta_x,t)=\delta_x' M(x,t)\delta_x,\qquad \alpha_1I\le M(x,t)\le \alpha_2I,$$7 Metzler matrix $$V(x,\delta_x,t)=\delta_x' M(x,t)\delta_x,\qquad \alpha_1I\le M(x,t)\le \alpha_2I,$$8 built from the $$V(x,\delta_x,t)=\delta_x' M(x,t)\delta_x,\qquad \alpha_1I\le M(x,t)\le \alpha_2I,$$9-contraction rates of the subsystems, the $$(A_k + B_k K_k)^T M_{k+1} (A_k + B_k K_k) - (1-\rho) M_k \prec 0,$$0-contraction of the subsystems, and the cross-gains $$(A_k + B_k K_k)^T M_{k+1} (A_k + B_k K_k) - (1-\rho) M_k \prec 0,$$1 (Ofir et al., 2024). If there exists a matrix measure $$(A_k + B_k K_k)^T M_{k+1} (A_k + B_k K_k) - (1-\rho) M_k \prec 0,$$2 induced by a monotonic norm $$(A_k + B_k K_k)^T M_{k+1} (A_k + B_k K_k) - (1-\rho) M_k \prec 0,$$3 such that

$$(A_k + B_k K_k)^T M_{k+1} (A_k + B_k K_k) - (1-\rho) M_k \prec 0,$$4

then the feedback system is $$(A_k + B_k K_k)^T M_{k+1} (A_k + B_k K_k) - (1-\rho) M_k \prec 0,$$5-contracting with rate $$(A_k + B_k K_k)^T M_{k+1} (A_k + B_k K_k) - (1-\rho) M_k \prec 0,$$6 (Ofir et al., 2024). The associated small-gain reduction yields the scalar inequality

$$(A_k + B_k K_k)^T M_{k+1} (A_k + B_k K_k) - (1-\rho) M_k \prec 0,$$7

This generalized setting is significant because $$(A_k + B_k K_k)^T M_{k+1} (A_k + B_k K_k) - (1-\rho) M_k \prec 0,$$8-contraction does not imply a unique equilibrium. For time-invariant $$(A_k + B_k K_k)^T M_{k+1} (A_k + B_k K_k) - (1-\rho) M_k \prec 0,$$9-contracting systems with bounded trajectories, every bounded trajectory converges to the set of equilibria, yet multiple equilibria and multiple locally stable equilibria are allowed (Ofir et al., 2024). Accordingly, output-feedback contraction need not be synonymous with monostability. The paper’s FitzHugh–Nagumo network example makes this explicit: the network can be $0<\rho<1$0-contractive but not $0<\rho<1$1-contracting, and trajectories still converge to equilibria (Ofir et al., 2024).

A related small-gain viewpoint appears in two-time-scale online feedback optimization. For the closed loop

$0<\rho<1$2

the plant is the fast contractive subsystem and the reduced gradient-flow controller is the slow contractive subsystem. The full shifted system is analyzed through a $0<\rho<1$3 Metzler gain matrix

$0<\rho<1$4

and if this matrix is Hurwitz the full output-feedback loop is strongly infinitesimally contractive in a composite norm (Cothren et al., 2023). This suggests that output-feedback contraction can be organized hierarchically: contraction margins of subsystems, interconnection gains, and time-scale separation jointly determine closed-loop incremental stability.

4. Infinite-dimensional formulations and contraction semigroups

In nonlinear infinite-dimensional systems, output-feedback contraction appears in a forwarding framework for robust output regulation. The plant is

$0<\rho<1$5

with $0<\rho<1$6 generating a nonlinear strongly continuous contraction semigroup on $0<\rho<1$7. After introducing an integral state $0<\rho<1$8,

$0<\rho<1$9

the design seeks a Lyapunov functional

$M_k\succ 0$0

based on a forwarding map $M_k\succ 0$1 satisfying

$M_k\succ 0$2

(Vanspranghe et al., 2022). The feedback law

$M_k\succ 0$3

then yields a transformed closed loop in coordinates $M_k\succ 0$4 whose operator is locally strongly monotone, hence locally exponentially contractive in a weighted norm on $M_k\succ 0$5 (Vanspranghe et al., 2022). Under additional coercivity conditions, global asymptotic stability is obtained, and the equilibrium satisfies $M_k\succ 0$6 (Vanspranghe et al., 2022). The paper explicitly notes that it does not construct an observer, so this is output-oriented integral action rather than pure output-only feedback.

A linear operator-theoretic counterpart is static output feedback for system nodes $M_k\succ 0$7 with generating triple $M_k\succ 0$8 and transfer function $M_k\succ 0$9. If there exists $M$0 such that $M$1, obtained by replacing $M$2 by $M$3, is impedance passive, and if

$M$4

with $M$5, then the closed-loop generator $M$6 generates a contraction semigroup on $M$7 (Curtain et al., 2021). If, moreover, the plant is approximately observable or approximately controllable in infinite time, then the closed-loop semigroup is weakly stable; if $M$8 is countable, then the closed-loop semigroup and its dual are both strongly stable (Curtain et al., 2021).

The feedback-theoretic semigroup construction makes the same point in a broader PDE context. A larger operator $M$9 is represented by a passive system node via the external Cayley system transform, and an internal maximal accretive operator $1-\rho$0 is mapped to a static output feedback operator

$1-\rho$1

If $1-\rho$2 is admissible, then the internal-loop operator $1-\rho$3 is the closed-loop main operator and generates a contraction semigroup on $1-\rho$4 (Kurula et al., 2014). This framework is used to show well-posedness of the heat equation, wave equations with viscous or structural damping, and degenerate parabolic equations (Kurula et al., 2014). In this literature, “output-feedback contraction” is therefore literally generation of a contraction semigroup by closing a passive system with a static output feedback law.

5. Data-driven, learned, and perception-mediated realizations

A data-driven realization is given by unknown nonlinear systems of the form

$1-\rho$5

where $1-\rho$6 is expressible through a known dictionary $1-\rho$7. With data matrices $1-\rho$8, the controller

$1-\rho$9

is synthesized by semidefinite programs whose feasibility implies a constant-metric contraction inequality for the closed-loop Jacobian (Hu et al., 2024). In the noise-free case, the SDP uses the linear constraints

$$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$00

together with an LMI involving $$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$01, $$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$02, and a Jacobian bound on $$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$03 (Hu et al., 2024). For disturbances that are linear combinations of sinusoids of known frequencies and constants, an added linear constraint

$$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$04

yields contraction conditions that do not depend on the magnitude of the disturbances (Hu et al., 2024). The same machinery is extended to an integral controller

$$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$05

which makes the plant-plus-integrator extended state contracting and thereby achieves constant reference tracking and constant disturbance rejection (Hu et al., 2024). Although this framework assumes the full state is measured, it is directly relevant to output-feedback contraction because the integral state is driven by output error and the extended closed loop is the object made contractive.

A complementary learned-perception architecture is developed for uncertain control-affine systems with high-dimensional outputs,

$$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$06

A learned inverse perception module $$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$07 approximates a reduced observable state $$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$08, and contraction theory is used both to design a stabilizing state feedback controller and a convergent dynamic state observer (Chou et al., 2022). The observer uses the learned perception system through

$$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$09

and the coupled tracking and estimation errors satisfy a Metzler differential inequality in the Riemannian distances $$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$10 and $$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$11 (Chou et al., 2022). These bounds are incorporated into the CORRT planner so that the tracking tube $$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$12 remains in the safe set and in the trusted domain where both the CCM/OCM conditions and the perception error bounds are valid (Chou et al., 2022). This is a direct realization of output-feedback contraction from images: learned output inversion, contraction-based observer and controller, and set-based planning are combined in a single safety-critical loop.

6. Scope, limitations, and related directions

The literature uses the phrase “output-feedback contraction” in several nonequivalent senses. In the CCM tradition, it means a dynamic controller obtained by combining a state-feedback contraction design with a contracting observer, typically under a nonlinear separation principle (Manchester et al., 2014, Zakwan et al., 8 Apr 2026). In semigroup and PDE settings, it often means that static output feedback produces a maximal dissipative closed-loop generator and therefore a contraction semigroup (Curtain et al., 2021, Kurula et al., 2014). In regulation problems with integral action, the feedback is output-oriented because the integrator is driven by $$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$13, even when the plant state is still assumed available (Vanspranghe et al., 2022, Hu et al., 2024). A common misconception is that the term always denotes pure static laws of the form $$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$14; the cited work shows that it may also denote observer-based dynamic feedback, integral action on an extended state, or static output feedback in operator-theoretic form.

A second misconception is that contraction always implies a unique equilibrium. Classical $$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$15-contraction does imply that all trajectories converge exponentially to one another and, in the time-invariant case with a forward invariant convex set, a unique globally exponentially stable equilibrium (Ofir et al., 2024). By contrast, $$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$16-contraction permits multistationarity: every bounded trajectory converges to the set of equilibria, yet multiple locally stable equilibria may coexist (Ofir et al., 2024). This suggests that generalized output-feedback contraction can be used to certify global asymptotic structure without excluding multistability.

The boundary of the term is illustrated by open-loop contraction design. For systems in normal form with scalar output $$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$17, the existence of an output variable that ensures contraction of the inverse system facilitates the design of a contracting input perturbation (Lee et al., 2022). The paper explicitly studies open-loop perturbations $$\dot x(t) = f(x(t),t) +B(t)u(t), \qquad y(t) = C(t)x(t),$$18 rather than feedback laws, but its structural message is closely related: a contracting inverse system makes output-based trajectory design tractable (Lee et al., 2022). This suggests a plausible implication that output-feedback contraction and output-programming for contraction are adjacent rather than identical notions.

Across these strands, the recurring invariants are metric selection, incremental dissipation, small-gain compatibility of interconnections, and certification of the closed-loop geometry of trajectories. Output-feedback contraction is therefore best understood not as one theorem or one algorithm, but as a unifying viewpoint in which outputs, observers, integral states, or passive interconnections are organized so that the closed loop inherits contraction or a generalized contraction property from its constituent parts.