Unit Vector Control: Finite-Time Stability

• Unit Vector Control (UVC) is a normalized-direction feedback method that regulates system states using unit-vector signals, ensuring finite-time convergence even under polytopic uncertainties and saturation.
• The approach employs LMI-based synthesis for designing gain matrices, enabling explicit treatment of actuator saturation and guaranteeing stability through convex optimization.
• In multivariable extremum seeking, UVC uses normalized gradient estimates to drive the system towards optimal values, offering robustness against gain uncertainties with finite-time convergence in the averaged sense.

Unit Vector Control (UVC) denotes a direction-only, or “unit-vector,” feedback paradigm in which the control input is generated from a normalized signal rather than from its magnitude. In the formulations developed in 2025, the normalized signal is either the state itself, $\phi(\sigma)=\sigma/\|\sigma\|$, or an estimated gradient, $\phi(\hat G)=\hat G/\|\hat G\|$, and the resulting control law takes the form $u=K\phi(\cdot)$. The recent literature treats UVC in two technically distinct but structurally related settings: multivariable polytopic uncertain systems under saturating actuators, and multivariable extremum seeking with an unknown Hessian. In both cases, the core claims are finite-time stability, LMI-based gain synthesis, and explicit treatment of uncertainty via convex polytopic embeddings (Vitório et al., 9 Apr 2025, Silva et al., 9 Apr 2025).

1. Core definition and control-theoretic role

In the saturating-actuator setting, UVC is introduced as a direction-only feedback law

$$u = K\,\phi(\sigma), \qquad \phi(\sigma)=\frac{\sigma}{\|\sigma\|},$$

with the stated objectives of finite-time convergence of the state $\sigma(t)$ to the origin, robustness to polytopic uncertainty in the input matrix $B$, and explicit handling of actuator saturation. In the extremum-seeking setting, the same normalization principle is applied to a gradient estimate,

$$u(t)=K\,\phi(\hat G(t)), \qquad \phi(\hat G)=\frac{\hat G}{\|\hat G\|}.$$

Geometrically, $\phi(\hat G)$ is described as a discontinuous “sliding-mode” direction that always points in the gradient estimate direction but with unit norm. The gain matrix $K\in\mathbb{R}^{n\times n}$ scales that direction (Vitório et al., 9 Apr 2025, Silva et al., 9 Apr 2025).

A central feature of UVC, as stated for uncertain systems with saturation, is that it enforces bounded control by its construction and yields sliding-mode-like finite-time convergence with very simple implementation. This is contrasted directly with LQR or backstepping. In extremum seeking, the use of the normalized gradient estimate rather than the raw estimate is associated with robustness to gain uncertainties and finite-time convergence in the average sense to the optimum. The two formulations therefore share a common normalized-direction mechanism, but they target different objects: direct state regulation in one case and optimization through gradient-estimate feedback in the other (Vitório et al., 9 Apr 2025, Silva et al., 9 Apr 2025).

2. UVC for multivariable polytopic uncertain systems with saturating actuators

The saturating-actuator formulation considers the uncertain plant

$$\dot{\sigma}=B\,\operatorname{sat}(u), \qquad \sigma\in\mathbb{R}^n,\; u\in\mathbb{R}^m,$$

where $\phi(\hat G)=\hat G/\|\hat G\|$0 is the component-wise saturation at levels $\phi(\hat G)=\hat G/\|\hat G\|$1, and

$\phi(\hat G)=\hat G/\|\hat G\|$2

The chosen UVC law is

$\phi(\hat G)=\hat G/\|\hat G\|$3

To represent saturation explicitly, the paper defines the dead-zone nonlinearity

$\phi(\hat G)=\hat G/\|\hat G\|$4

so that the closed loop becomes

$\phi(\hat G)=\hat G/\|\hat G\|$5

This decomposition isolates the effect of actuator limits as an additive nonlinear term rather than absorbing it implicitly into a bounded-input assumption (Vitório et al., 9 Apr 2025).

For the finite-time analysis, the paper introduces the coordinate transformation

$\phi(\hat G)=\hat G/\|\hat G\|$6

and then works with the standard quadratic Lyapunov function

$\phi(\hat G)=\hat G/\|\hat G\|$7

This change of coordinates is used specifically to handle finite-time stability with quadratic tools. A plausible implication is that UVC’s discontinuous normalization is not treated directly in the original coordinates; instead, the transformed coordinates permit an LMI-compatible Lyapunov analysis while preserving the finite-time character of the closed-loop dynamics (Vitório et al., 9 Apr 2025).

3. LMI synthesis for finite-time stability

The finite-time stabilization theorem in the saturating-actuator paper seeks $\phi(\hat G)=\hat G/\|\hat G\|$8 and $\phi(\hat G)=\hat G/\|\hat G\|$9 such that the origin is finite-time stable for all $u=K\phi(\cdot)$0 and despite actuator saturation. The key auxiliary result is a dead-zone sector bound. Define

$u=K\phi(\cdot)$1

Then, for any diagonal $u=K\phi(\cdot)$2,

$u=K\phi(\cdot)$3

whenever $u=K\phi(\cdot)$4. This bound supplies a quadratic inequality that controls the negative effect of the dead-zone term inside a specified polyhedral region (Vitório et al., 9 Apr 2025).

The theorem fixes $u=K\phi(\cdot)$5 and requires the existence of $u=K\phi(\cdot)$6, diagonal $u=K\phi(\cdot)$7, and full matrices $u=K\phi(\cdot)$8 such that, for each $u=K\phi(\cdot)$9, a vertex-dependent block LMI is satisfied with

$$u = K\,\phi(\sigma), \qquad \phi(\sigma)=\frac{\sigma}{\|\sigma\|},$$0

together with actuator-channel LMIs

$$u = K\,\phi(\sigma), \qquad \phi(\sigma)=\frac{\sigma}{\|\sigma\|},$$1

When these conditions hold, the synthesis variables recover the controller and Lyapunov matrices through

$$u = K\,\phi(\sigma), \qquad \phi(\sigma)=\frac{\sigma}{\|\sigma\|},$$2

and the ellipsoid

$$u = K\,\phi(\sigma), \qquad \phi(\sigma)=\frac{\sigma}{\|\sigma\|},$$3

is a guaranteed estimate of the region of attraction. The proof sketch given in the summary uses Schur complements to obtain

$$u = K\,\phi(\sigma), \qquad \phi(\sigma)=\frac{\sigma}{\|\sigma\|},$$4

for all vertices and all $$u = K\,\phi(\sigma), \qquad \phi(\sigma)=\frac{\sigma}{\|\sigma\|},$$5, and then concludes the finite-time bound

$$u = K\,\phi(\sigma), \qquad \phi(\sigma)=\frac{\sigma}{\|\sigma\|},$$6

The technical role of the $$u = K\,\phi(\sigma), \qquad \phi(\sigma)=\frac{\sigma}{\|\sigma\|},$$7-term and the projection bounds involving $$u = K\,\phi(\sigma), \qquad \phi(\sigma)=\frac{\sigma}{\|\sigma\|},$$8 is to guarantee the quadratic dominated bound needed for the Lyapunov argument (Vitório et al., 9 Apr 2025).

4. Region-of-attraction enlargement and explicit saturation handling

The same framework is extended to enlarge the certified region of attraction. For a prescribed upper bound $$u = K\,\phi(\sigma), \qquad \phi(\sigma)=\frac{\sigma}{\|\sigma\|},$$9 on the reaching time, two additional LMIs are introduced: $\sigma(t)$0 and

$\sigma(t)$1

The resulting convex program is

$\sigma(t)$2

According to the summary, the optimizer yields the largest ellipsoidal $\sigma(t)$3 in which any initial $\sigma(t)$4, hence $\sigma(t)$5, converges in time $\sigma(t)$6 even if actuators saturate (Vitório et al., 9 Apr 2025).

The explicit treatment of saturation is not incidental. The dead-zone nonlinearity $\sigma(t)$7 is inserted directly into the closed-loop model, the sector-type lemma upper-bounds its adverse effect on $\sigma(t)$8, and constraint (7) guarantees $\sigma(t)$9. The summary states that in the LMI condition the term involving $B$0 is handled without any additional non-convexity. This sharply distinguishes the method from analyses that rely only on bounded-input intuition: bounded control by construction does not remove the need for a formal saturation model, and the paper’s contribution is precisely to integrate that model into a convex finite-time synthesis procedure (Vitório et al., 9 Apr 2025).

5. Multivariable extremum seeking with unit-vector control

The extremum-seeking formulation considers an unknown static map

$B$1

where $B$2, $B$3 is the unknown minimizer, $B$4 is the corresponding extremum value, and $B$5 is the unknown Hessian. The probe signal is decomposed as

$B$6

The dither signals are chosen component-wise as

$B$7

with distinct nonresonant $B$8. By trigonometric identities and filtering, one obtains $B$9 in the high-frequency limit (Silva et al., 9 Apr 2025).

Uncertainty in the Hessian is handled through the polytopic embedding

$$u(t)=K\,\phi(\hat G(t)), \qquad \phi(\hat G)=\frac{\hat G}{\|\hat G\|}.$$0

where each $$u(t)=K\,\phi(\hat G(t)), \qquad \phi(\hat G)=\frac{\hat G}{\|\hat G\|}.$$1 is a known positive-definite vertex matrix. After the time-scale change $$u(t)=K\,\phi(\hat G(t)), \qquad \phi(\hat G)=\frac{\hat G}{\|\hat G\|}.$$2 and averaging, the gradient-estimate dynamics become

$$u(t)=K\,\phi(\hat G(t)), \qquad \phi(\hat G)=\frac{\hat G}{\|\hat G\|}.$$3

The paper then introduces

$$u(t)=K\,\phi(\hat G(t)), \qquad \phi(\hat G)=\frac{\hat G}{\|\hat G\|}.$$4

leading to

$$u(t)=K\,\phi(\hat G(t)), \qquad \phi(\hat G)=\frac{\hat G}{\|\hat G\|}.$$5

With $$u(t)=K\,\phi(\hat G(t)), \qquad \phi(\hat G)=\frac{\hat G}{\|\hat G\|}.$$6, the theorem gives a constructive LMI test: if there exist symmetric $$u(t)=K\,\phi(\hat G(t)), \qquad \phi(\hat G)=\frac{\hat G}{\|\hat G\|}.$$7, $$u(t)=K\,\phi(\hat G(t)), \qquad \phi(\hat G)=\frac{\hat G}{\|\hat G\|}.$$8, and full $$u(t)=K\,\phi(\hat G(t)), \qquad \phi(\hat G)=\frac{\hat G}{\|\hat G\|}.$$9 satisfying a vertex-wise block LMI for every $\phi(\hat G)$0, then

$\phi(\hat G)$1

render the origin of the average closed-loop system finite-time stable. The proof outline shows

$\phi(\hat G)$2

and then

$\phi(\hat G)$3

An important feature of this formulation is that $\phi(\hat G)$4 need not be diagonal: off-diagonal entries are permitted, so the algorithm can mix gradient components, which the summary states often reduces conservatism and enlarges the region of feasible $\phi(\hat G)$5 within the polytope (Silva et al., 9 Apr 2025).

For the actual, non-averaged closed loop, the right-hand side is discontinuous because of $\phi(\hat G)$6. The analysis therefore invokes the averaging theorem for differential inclusions (Plotnikov 1979). For sufficiently large $\phi(\hat G)$7, the actual trajectories remain within $\phi(\hat G)$8 of the average trajectories over $\phi(\hat G)$9 time intervals. The summary states that $K\in\mathbb{R}^{n\times n}$0 in finite time in the average sense and that

$K\in\mathbb{R}^{n\times n}$1

while the output satisfies

$K\in\mathbb{R}^{n\times n}$2

This establishes a distinction that is sometimes blurred in informal discussions: the finite-time result is for the average closed-loop error system, whereas the actual closed-loop system is shown to converge to a neighborhood of the unknown extremum point (Silva et al., 9 Apr 2025).

6. Numerical realizations and technical implications

The saturating-actuator paper reports two examples. In a planar kinematic manipulator with $K\in\mathbb{R}^{n\times n}$3, $K\in\mathbb{R}^{n\times n}$4, $K\in\mathbb{R}^{n\times n}$5 is a rotation by uncertain angle $K\in\mathbb{R}^{n\times n}$6, represented polytopically with $K\in\mathbb{R}^{n\times n}$7 vertices. With saturation levels $K\in\mathbb{R}^{n\times n}$8, $K\in\mathbb{R}^{n\times n}$9, and $$\dot{\sigma}=B\,\operatorname{sat}(u), \qquad \sigma\in\mathbb{R}^n,\; u\in\mathbb{R}^m,$$0, the LMI solution gives

$$\dot{\sigma}=B\,\operatorname{sat}(u), \qquad \sigma\in\mathbb{R}^n,\; u\in\mathbb{R}^m,$$1

The ellipsoid $$\dot{\sigma}=B\,\operatorname{sat}(u), \qquad \sigma\in\mathbb{R}^n,\; u\in\mathbb{R}^m,$$2 and the surfaces defining $$\dot{\sigma}=B\,\operatorname{sat}(u), \qquad \sigma\in\mathbb{R}^n,\; u\in\mathbb{R}^m,$$3 are plotted, with $$\dot{\sigma}=B\,\operatorname{sat}(u), \qquad \sigma\in\mathbb{R}^n,\; u\in\mathbb{R}^m,$$4, and sample trajectories starting on $$\dot{\sigma}=B\,\operatorname{sat}(u), \qquad \sigma\in\mathbb{R}^n,\; u\in\mathbb{R}^m,$$5 converge in $$\dot{\sigma}=B\,\operatorname{sat}(u), \qquad \sigma\in\mathbb{R}^n,\; u\in\mathbb{R}^m,$$6 even though $$\dot{\sigma}=B\,\operatorname{sat}(u), \qquad \sigma\in\mathbb{R}^n,\; u\in\mathbb{R}^m,$$7 saturates. In an underwater ROV example with $$\dot{\sigma}=B\,\operatorname{sat}(u), \qquad \sigma\in\mathbb{R}^n,\; u\in\mathbb{R}^m,$$8, $$\dot{\sigma}=B\,\operatorname{sat}(u), \qquad \sigma\in\mathbb{R}^n,\; u\in\mathbb{R}^m,$$9,

$\phi(\hat G)=\hat G/\|\hat G\|$00

the saturation level is $\phi(\hat G)=\hat G/\|\hat G\|$01 on each channel, $\phi(\hat G)=\hat G/\|\hat G\|$02, and $\phi(\hat G)=\hat G/\|\hat G\|$03. The synthesized $\phi(\hat G)=\hat G/\|\hat G\|$04 gain satisfies $\phi(\hat G)=\hat G/\|\hat G\|$05, yet any $\phi(\hat G)=\hat G/\|\hat G\|$06 converges in $\phi(\hat G)=\hat G/\|\hat G\|$07 despite extended saturation on channel 4. The paper presents these examples as evidence that the proposed UVC design systematically accounts for actuator limits, handles polytopic uncertainty by convex LMIs, and provides an explicit estimate of initial-condition sets for guaranteed finite-time convergence (Vitório et al., 9 Apr 2025).

The extremum-seeking paper gives a two-dimensional example with

$\phi(\hat G)=\hat G/\|\hat G\|$08

and a $\phi(\hat G)=\hat G/\|\hat G\|$09 polytopic uncertainty

$\phi(\hat G)=\hat G/\|\hat G\|$10

Solving the LMI-based optimization, including additional LMIs to minimize the bound on $\phi(\hat G)=\hat G/\|\hat G\|$11, with $\phi(\hat G)=\hat G/\|\hat G\|$12, $\phi(\hat G)=\hat G/\|\hat G\|$13, and $\phi(\hat G)=\hat G/\|\hat G\|$14, yields

$\phi(\hat G)=\hat G/\|\hat G\|$15

The dither amplitudes are $\phi(\hat G)=\hat G/\|\hat G\|$16, the frequencies are $\phi(\hat G)=\hat G/\|\hat G\|$17 and $\phi(\hat G)=\hat G/\|\hat G\|$18, and the initial condition is $\phi(\hat G)=\hat G/\|\hat G\|$19. The figures reported in the summary show $\phi(\hat G)=\hat G/\|\hat G\|$20 switching along the estimated gradient, $\phi(\hat G)=\hat G/\|\hat G\|$21 converging to $\phi(\hat G)=\hat G/\|\hat G\|$22 in finite time, and $\phi(\hat G)=\hat G/\|\hat G\|$23 converging to $\phi(\hat G)=\hat G/\|\hat G\|$24, with robustness illustrated by choosing $\phi(\hat G)=\hat G/\|\hat G\|$25 randomly inside the polytope (Silva et al., 9 Apr 2025).

Taken together, these formulations suggest a unified contemporary view of UVC as a normalized-direction control architecture combined with finite-time Lyapunov analysis and convex synthesis under structured uncertainty. The two applications are nevertheless technically distinct. In state regulation under saturating actuators, the decisive issue is certification of finite-time stability and a region of attraction in the presence of a dead-zone nonlinearity. In multivariable extremum seeking, the decisive issue is certification of finite-time stability for the averaged error dynamics and neighborhood convergence for the true discontinuous system. The commonality lies in the use of unit-vector feedback and LMI machinery; the difference lies in what is being normalized, what uncertainty is embedded polytopically, and what notion of convergence is formally established.