Persistent Excitation in Adaptive Systems

• Persistent excitation (PE) is a richness condition on input or measured signals that guarantees every parameter direction is sufficiently probed for adaptive estimation and control.
• PE underpins exponential convergence and uniform stability by ensuring the moving-window Gramian or information matrix remains uniformly positive definite.
• Modern generalizations extend PE to contexts such as RKHS, hybrid systems, and finite excitation, thereby broadening its applications in adaptive control and data-driven learning.

Searching arXiv for recent and foundational papers on persistent excitation across adaptive control, RKHS estimation, data-driven control, and related formulations. Persistent excitation (PE) is a richness condition on measured or applied signals that ensures parameter directions are sufficiently probed over time for identification, adaptive estimation, or learning to converge. In its classical form, PE requires a moving-window Gramian to remain uniformly positive definite; in contemporary work, the same idea appears in discrete time, hybrid systems, switched systems, reproducing kernel Hilbert spaces (RKHS), data-driven control, and distributed estimation, often with modified notions such as partial PE, local-PE, hybrid PE, interval excitation (IE), or finite excitation (FE). Across these settings, PE functions as a structural bridge between signal informativeness and strong convergence properties such as parameter consistency, uniform exponential stability, identifiability, or validity of Hankel-data representations (Guo et al., 2020).

1. Classical definition and geometric meaning

In finite-dimensional adaptive control, PE is imposed on a regressor vector so that every direction in parameter space is excited on every sliding time window. A standard continuous-time form recalled in RKHS-based adaptive estimation is

$$\int_t^{t+\Delta} v^T \Phi(x(\tau))\Phi^T(x(\tau))v\,d\tau \ge \gamma \|v\|^2,\qquad \forall t\ge T_0,\ \forall v\in\mathbb{R}^n,$$

and a standard discrete-time form is

$$\sum_{i=k+1}^{k+\Delta T} \phi_i \phi_i^\top \succeq \alpha I, \qquad \forall k\ge k_0.$$

Both formulations express the same moving-window requirement: the accumulated regressor information must be uniformly positive definite (Guo et al., 2020, Cui et al., 2021).

In online identification for nonlinear stochastic systems, the same principle is expressed asymptotically through the average information matrix. One formulation requires

$$\frac{1}{T}\sum_{t=1}^{T}\nabla_{\theta}h(\theta, x_{t}, u_{t})\nabla_{\theta}^{\top}h(\theta, x_{t}, u_{t})$$

to converge almost surely to a positive definite matrix. This is described as a “strong PE condition,” and it encodes uniform asymptotic richness of the gradient or regressor directions (Zhang et al., 3 Apr 2025).

For linear time-invariant systems, recent work separates PE from “sufficient richness” (SR). PE remains the positivity condition on a signal or stacked regressor, while SR is a property of an input relative to a system and initial condition: the input is SR if it guarantees that the generated regressor is PE. In this formulation, PE is a property of the resulting signal, whereas SR is a design property of the applied input (Borghesi et al., 6 Feb 2025).

Geometrically, the finite-dimensional interpretation is consistent across these papers: a trajectory or regressor must not remain confined to a lower-dimensional subspace that leaves some parameter directions unobserved. This geometric reading becomes more explicit in later generalizations.

2. Why PE matters for convergence, identifiability, and stability

The classical role of PE is to convert boundedness or tracking into parameter convergence. In the discrete-time parameter-estimation algorithm for time-varying parameters, PE yields a uniform lower bound on the information matrix,

$$\Omega_{PE} I \preceq \Omega_k \preceq \Omega_{\max} I,$$

and then on the adaptive gain,

$$\Gamma_{PE} I \preceq \Gamma_k \preceq \Gamma_{\max} I.$$

These bounds allow a Lyapunov analysis proving exponential decrease of the error measure outside a compact set whose size is proportional to the parameter-variation bound $\Delta^*$; in the constant-parameter case $\Delta^*=0$, exponential convergence to the origin follows under PE (Cui et al., 2021).

In hybrid systems, the same logic is generalized from pure flow or pure jump dynamics to mixed flow-jump signals. Hybrid persistency of excitation (HPE) is defined by a positive-definite lower bound on a hybrid integral over windows $E^A_{t_o,j_o,K}$, and the central implication is

$$\text{HPE} \Rightarrow \text{HUO} \Rightarrow \text{UES and ISS},$$

where HUO denotes hybrid uniform observability, UES uniform exponential stability, and ISS input-to-state stability. This explicitly connects excitation to observability and stability for linear non-autonomous hybrid systems (Saoud et al., 2023).

In data-driven predictive control, PE is not merely sufficient for good performance but is a prerequisite for the validity of the trajectory representation itself. The rank-based definition requires the Hankel matrix $\mathcal{H}_L(u)$ to have rank $$\sum_{i=k+1}^{k+\Delta T} \phi_i \phi_i^\top \succeq \alpha I, \qquad \forall k\ge k_0.$$0, and for the Fundamental Lemma the offline input must be persistently exciting of order $$\sum_{i=k+1}^{k+\Delta T} \phi_i \phi_i^\top \succeq \alpha I, \qquad \forall k\ge k_0.$$1. If PE is lost, the Hankel matrix loses full row rank and the data no longer span the admissible system behavior required by the controller (Faro et al., 6 Apr 2025).

For discrete-time linear switched systems, PE is formulated as a property of the input signal relative to an input-output map, not to any specific identification algorithm. A finite input is PE if the resulting data determine the Markov parameters of the switched input-output map; asymptotic PE requires that longer truncations yield approximations converging to the Markov parameters. This shifts the emphasis from a fixed regressor Gramian to recoverability of mode-dependent Markov parameters $$\sum_{i=k+1}^{k+\Delta T} \phi_i \phi_i^\top \succeq \alpha I, \qquad \forall k\ge k_0.$$2 (Petreczky et al., 2011).

Taken together, these results indicate that PE is best viewed not as a single formula but as an information-accumulation condition tailored to the state space, signal domain, and representation used by the estimator or controller.

3. Infinite-dimensional and geometric generalizations

A major extension of PE arises in RKHS-based adaptive estimation, where the unknown object is a function $$\sum_{i=k+1}^{k+\Delta T} \phi_i \phi_i^\top \succeq \alpha I, \qquad \forall k\ge k_0.$$3 rather than a finite parameter vector. The evaluation functional

$$\sum_{i=k+1}^{k+\Delta T} \phi_i \phi_i^\top \succeq \alpha I, \qquad \forall k\ge k_0.$$4

replaces the Euclidean regressor, and the adaptive error dynamics evolve in $$\sum_{i=k+1}^{k+\Delta T} \phi_i \phi_i^\top \succeq \alpha I, \qquad \forall k\ge k_0.$$5. Because exciting the entire RKHS is usually too strong, the relevant notion becomes partial persistent excitation on a kernel-generated subspace $$\sum_{i=k+1}^{k+\Delta T} \phi_i \phi_i^\top \succeq \alpha I, \qquad \forall k\ge k_0.$$6 associated with a subset $$\sum_{i=k+1}^{k+\Delta T} \phi_i \phi_i^\top \succeq \alpha I, \qquad \forall k\ge k_0.$$7 (Guo et al., 2020).

The subspace is defined by

$$\sum_{i=k+1}^{k+\Delta T} \phi_i \phi_i^\top \succeq \alpha I, \qquad \forall k\ge k_0.$$8

This decomposition isolates the observable component of the function on $$\sum_{i=k+1}^{k+\Delta T} \phi_i \phi_i^\top \succeq \alpha I, \qquad \forall k\ge k_0.$$9. Two PE notions are introduced. The first is

$$\frac{1}{T}\sum_{t=1}^{T}\nabla_{\theta}h(\theta, x_{t}, u_{t})\nabla_{\theta}^{\top}h(\theta, x_{t}, u_{t})$$0

for some $$\frac{1}{T}\sum_{t=1}^{T}\nabla_{\theta}h(\theta, x_{t}, u_{t})\nabla_{\theta}^{\top}h(\theta, x_{t}, u_{t})$$1, and the second is the energy form

$$\frac{1}{T}\sum_{t=1}^{T}\nabla_{\theta}h(\theta, x_{t}, u_{t})\nabla_{\theta}^{\top}h(\theta, x_{t}, u_{t})$$2

Under the first form, one obtains

$$\frac{1}{T}\sum_{t=1}^{T}\nabla_{\theta}h(\theta, x_{t}, u_{t})\nabla_{\theta}^{\top}h(\theta, x_{t}, u_{t})$$3

and, assuming uniform embedding $$\frac{1}{T}\sum_{t=1}^{T}\nabla_{\theta}h(\theta, x_{t}, u_{t})\nabla_{\theta}^{\top}h(\theta, x_{t}, u_{t})$$4, pointwise convergence on $$\frac{1}{T}\sum_{t=1}^{T}\nabla_{\theta}h(\theta, x_{t}, u_{t})\nabla_{\theta}^{\top}h(\theta, x_{t}, u_{t})$$5 (Guo et al., 2020).

The preceding RKHS literature places this in a broader geometric context. For a continuous semiflow with positive orbit $$\frac{1}{T}\sum_{t=1}^{T}\nabla_{\theta}h(\theta, x_{t}, u_{t})\nabla_{\theta}^{\top}h(\theta, x_{t}, u_{t})$$6 and positive limit set $$\frac{1}{T}\sum_{t=1}^{T}\nabla_{\theta}h(\theta, x_{t}, u_{t})\nabla_{\theta}^{\top}h(\theta, x_{t}, u_{t})$$7, if the RKHS contains a rich family of bump functions, then PE of $$\frac{1}{T}\sum_{t=1}^{T}\nabla_{\theta}h(\theta, x_{t}, u_{t})\nabla_{\theta}^{\top}h(\theta, x_{t}, u_{t})$$8 implies

$$\frac{1}{T}\sum_{t=1}^{T}\nabla_{\theta}h(\theta, x_{t}, u_{t})\nabla_{\theta}^{\top}h(\theta, x_{t}, u_{t})$$9

The proof idea is that a point not revisited asymptotically would admit a bump function supported in a neighborhood eventually avoided by the trajectory, contradicting the PE lower bound. Thus, in RKHS settings, PE localizes learning to the asymptotic support of the dynamics (Kurdila et al., 2019).

On smooth compact embedded manifolds $$\Omega_{PE} I \preceq \Omega_k \preceq \Omega_{\max} I,$$0, sufficient conditions for RKHS PE can be stated in terms of repeated visitation of geodesic neighborhoods of finitely many kernel centers $$\Omega_{PE} I \preceq \Omega_k \preceq \Omega_{\max} I,$$1. If the trajectory spends a uniformly positive amount of time near every center in every time window, then PE$$\Omega_{PE} I \preceq \Omega_k \preceq \Omega_{\max} I,$$2-2 holds, with explicit lower bound

$$\Omega_{PE} I \preceq \Omega_k \preceq \Omega_{\max} I,$$3

In finite-dimensional RKHS subspaces this yields exact function convergence; in infinite-dimensional spaces it yields ultimate boundedness proportional to the approximation error $$\Omega_{PE} I \preceq \Omega_k \preceq \Omega_{\max} I,$$4 (Paruchuri et al., 2020).

A plausible implication is that infinite-dimensional PE is inherently representation-dependent: unlike the Euclidean case, the kernel, the induced topology, and the subset repeatedly visited by the trajectory all enter the definition.

4. Relaxations of PE and PE-free alternatives

Many papers treat classical PE as too strong for closed-loop, feedback-driven, or practically constrained systems. One common relaxation is finite excitation (FE), which requires excitation only on a single finite interval rather than every sliding window. In combined adaptive control for uncertain nonlinear systems, FE replaces a global-in-time PE condition by a memory-based identification stage that constructs $$\Omega_{PE} I \preceq \Omega_k \preceq \Omega_{\max} I,$$5 after the excitation interval. The result is uniform stability for all $$\Omega_{PE} I \preceq \Omega_k \preceq \Omega_{\max} I,$$6 and exponential stability for $$\Omega_{PE} I \preceq \Omega_k \preceq \Omega_{\max} I,$$7, with decay rate

$$\Omega_{PE} I \preceq \Omega_k \preceq \Omega_{\max} I,$$8

independent of the excitation level (Patel et al., 4 Feb 2025).

A related FE construction appears in a two-layer forgetting RLS scheme. There, a bounded regressor $$\Omega_{PE} I \preceq \Omega_k \preceq \Omega_{\max} I,$$9 satisfying FE over one finite interval is used to build an augmented matrix $$\Gamma_{PE} I \preceq \Gamma_k \preceq \Gamma_{\max} I.$$0 via directional forgetting, and this augmented regressor satisfies a PE condition: $$\Gamma_{PE} I \preceq \Gamma_k \preceq \Gamma_{\max} I.$$1 This FE-to-PE conversion yields bounded information matrix $$\Gamma_{PE} I \preceq \Gamma_k \preceq \Gamma_{\max} I.$$2 and global exponential parameter convergence $$\Gamma_{PE} I \preceq \Gamma_k \preceq \Gamma_{\max} I.$$3 (Tsuruhara et al., 28 Apr 2025).

Another relaxation is interval excitation (IE). In adaptive attitude control without PE, IE requires excitation only on one interval,

$$\Gamma_{PE} I \preceq \Gamma_k \preceq \Gamma_{\max} I.$$4

and a DREM-based constructive linear time-varying filter transforms the original regressor into a scalar regressor $$\Gamma_{PE} I \preceq \Gamma_k \preceq \Gamma_{\max} I.$$5 that becomes bounded away from zero after finite time. This suffices for exponential convergence of both tracking and parameter-estimation errors without PE of the original regressor (Shao et al., 2021).

Other approaches avoid PE entirely. In online learning for nonlinear stochastic dynamical systems, vanishing average regret

$$\Gamma_{PE} I \preceq \Gamma_k \preceq \Gamma_{\max} I.$$6

and, under bounded outputs, $$\Gamma_{PE} I \preceq \Gamma_k \preceq \Gamma_{\max} I.$$7 are proved without PE. Parameter convergence is then tied to a weaker information-growth condition,

$$\Gamma_{PE} I \preceq \Gamma_k \preceq \Gamma_{\max} I.$$8

rather than to a positive-definite asymptotic Gramian (Zhang et al., 3 Apr 2025).

Concurrent learning provides another PE-free route. In disturbance estimation, a conventional observer relies only on the most recent sample and can lose precision when the disturbance gain is near singular. A concurrent-learning observer replaces this with a time-variant history stack

$$\Gamma_{PE} I \preceq \Gamma_k \preceq \Gamma_{\max} I.$$9

and proves uniform ultimate boundedness of the estimation error under stack conditions, without assuming PE of the current signal (Zhang et al., 2023).

These relaxations suggest that the core requirement is often not classical moving-window PE as such, but some weaker mechanism ensuring that accumulated information does not degenerate.

5. Variants for networks, hybrid dynamics, switched systems, and data-driven control

In distributed parameter estimation with Gaussian observation noises, excitation may be insufficient at an individual sensor but sufficient in a neighborhood. This motivates local persistent excitation (Local-PE): $\Delta^*$0 Here $\Delta^*$1 is the scalar DREM regressor at sensor $\Delta^*$2. Under bounded regressors, Local-PE, and diminishing step sizes, every sensor’s estimate converges to the true parameter in mean square even when an individual sensor is not PE on its own (Yan et al., 2023).

In hybrid systems, HPE merges continuous-time and discrete-time excitation into a single hybrid-window condition,

$\Delta^*$3

with $\Delta^*$4 taking the value $\Delta^*$5 during flows and $\Delta^*$6 at jumps. HPE reduces to classical continuous-time PE or discrete-time PE in the corresponding limiting cases, but can also hold when neither purely continuous nor purely discrete excitation is adequate by itself (Saoud et al., 2023).

For discrete-time linear switched systems, PE must probe both continuous inputs and mode sequences. The sufficient asymptotic condition requires positive asymptotic frequencies $\Delta^*$7 for every mode word $\Delta^*$8 together with covariance limits such as

$\Delta^*$9

This formulation is algorithm-independent and is rooted in recoverability of Markov parameters rather than in a fixed Euclidean regressor (Petreczky et al., 2011).

In data-driven predictive control, PE can be actively maintained online. The nonexciting input set

$\Delta^*=0$0

is shown, in the relevant one-rank-deficient case, to be the hyperplane

$\Delta^*=0$1

Two disjoint linear inequalities,

$\Delta^*=0$2

therefore preserve PE by keeping the input away from the nonexciting hyperplane. These constraints can be implemented either as a one-binary mixed-integer formulation or as two quadratic programs solved in parallel (Faro et al., 6 Apr 2025).

A complementary line of work designs PE inputs a priori. For any controllable LTI system, a sparse sequence of discrete impulses is guaranteed to be PE: $\Delta^*=0$3 and $\Delta^*=0$4. Extensions guarantee PE of basis-function sequences in Hammerstein systems and collective PE for certain nonlinear systems via structured multi-experiment inputs (Alsalti et al., 2023).

6. Pathologies, structural refinements, and recurring misconceptions

Recent work has emphasized that classical PE, taken alone, is not always well behaved as a geometric notion. A counterexample uses a pulse-like switching signal $\Delta^*=0$5 to define

$\Delta^*=0$6

where both $\Delta^*=0$7 and $\Delta^*=0$8 are non-PE, yet $\Delta^*=0$9 is PE. In the two-dimensional example $E^A_{t_o,j_o,K}$0, the non-PE set is the union of the coordinate axes rather than a subspace. This motivates the notion of a regular regressor, defined by requiring the non-PE set $E^A_{t_o,j_o,K}$1 to be a subspace (Uzeda et al., 8 Jul 2025).

For regular regressors, excitation is confined to a canonical PE subspace

$E^A_{t_o,j_o,K}$2

and the signal admits a PE decomposition

$E^A_{t_o,j_o,K}$3

where the projected component $E^A_{t_o,j_o,K}$4 is PE and the complementary component is non-PE. This refinement makes precise the idea that only certain directions are genuinely informative, and it supplies an intrinsic notion of partial or semi-PE in finite dimensions (Uzeda et al., 8 Jul 2025).

A recurrent misconception is that PE is a purely signal-level condition independent of representation. The cited literature suggests otherwise. In RKHS estimation, PE depends on the kernel-induced topology and on the subspace generated by visited points (Guo et al., 2020). In switched systems, PE depends on mode words and Markov parameters rather than only on input covariance (Petreczky et al., 2011). In neural networks, PE can fail even with large datasets because intermediate-layer activations become low-dimensional during training; for cross-entropy, only support vectors continue to excite the parameters, and if the activated support-vector Gram matrices lose rank, the robustness-related PE bounds disappear (Nar et al., 2019).

Another misconception is that PE is always necessary in its classical form. The recent literature does not support that blanket statement. Some schemes still rely on PE for exponential parameter convergence, but others replace it with FE, IE, Local-PE, conditional expected richness, or information-matrix growth, and some obtain regret minimization or boundedness without PE altogether (Patel et al., 4 Feb 2025, Zhang et al., 3 Apr 2025).

Persistent excitation is therefore best understood as a family of nondegeneracy conditions whose precise form depends on the state representation, information structure, and convergence objective. What remains invariant across these variants is the core idea: successful adaptive inference requires that informative directions are revisited often enough, strongly enough, or cumulatively enough to preclude ambiguity.