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Admissibility-Preserving Control for Multi-Input Systems with Joint Capacity Constraints
Saurabh Kumar, Lohitvel Gopikannan, Shashi Ranjan Kumar, Abhinav Sinha
TL;DR
Coupled actuator limits can make channel-wise constraint handling inadequate, and direction-blind filters may suppress commands tangent to the joint boundary. The paper proposes AJ-APIR, a geometry-aware input realization integrated with backstepping; under its compatibility condition, it preserves joint admissibility and yields exponential tracking convergence with bounded closed-loop signals.
Problem
Channel-wise constraint handling does not capture non-Cartesian joint input envelopes, while direction-blind filtering may attenuate commands tangent to the boundary.
Source: §I · p. 2Method
AJ-APIR uses a matrix-valued realization to attenuate boundary-normal input while preserving tangential input, and is integrated with recursive backstepping.
Source: §I · p. 2Results
Under the AJ-APC compatibility condition, tracking errors converge exponentially and all closed-loop signals remain uniformly bounded, while the joint admissible set is forward invariant.
Source: §I · p. 3, §III · p. 8, §V · p. 11Takeaways & Limitations
The reported comparison shows that satisfying individual actuator bounds alone does not ensure joint admissibility, whereas AJ-APIR maintains the shared capacity constraint.
Source: §IV · p. 9, §V · p. 11
Abstract
from arXiv · showhide
This paper addresses the control of multi-input strict-feedback nonlinear systems subject to a joint capacity constraint, in which the admissible input set is a coupled subset of the individual actuator limits. Unlike existing constraint-handling methods that enforce actuator bounds channel by channel and may unnecessarily suppress admissible control directions, we develop an Anisotropic Joint-Admissibility-Preserving Input Realization (AJ-APIR) framework that explicitly exploits the geometry of the joint constraint. The proposed realization constructs a state-dependent gain matrix whose spectral decomposition separates the commanded input into normal and tangential directions relative to the constraint boundary. The normal component is attenuated as the boundary is approached, while the tangential component is preserved, which allows the admissible control effort to be redistributed without loss of tracking authority. Integrated with a backstepping controller, the AJ-APIR framework guarantees forward invariance of the joint admissible set for all time. We establish exponential convergence of the tracking error to zero together with uniform boundedness of all closed-loop signals, and characterize the resulting command-demand behavior under the joint constraint. Simulation results for a representative second-order, two-input nonlinear system subject to a power-budget constraint demonstrate the efficacy of the proposed method to enforce the joint input constraint.
I. INTRODUCTION
Shared actuator resources create coupled input limits that channel-wise bounds cannot capture. The paper introduces AJ-APIR to exploit the joint constraint geometry, preserve feasible tangential control, and integrate constraint handling with backstepping.
I. INTRODUCTION
- Shared physical resources can make aggregate demand exceed the joint operating limit even when every actuator remains within its individual admissible range.The paper cites electrical buses, shared thermal paths, and marine-vessel prime movers as examples.Source: §I · p. 1
- Backstepping and barrier Lyapunov approaches constrain channels separately, so their enforced feasible sets are Cartesian products rather than coupled input envelopes.The cited backstepping work addresses asymmetric saturation and uncertain multi-input systems; standard barrier terms are also assigned independently by channel.Source: §I · p. 1
- CBF safety filters can represent smooth joint constraints but may attenuate tangent commands, while allocation feasibility and closed-loop stability require separate consideration.Reference governors and predictive architectures address feasibility at planning or reference generation, potentially at high computational cost for fast loops.Source: §I · p. 2
- Earlier APIR embeds admissibility in input dynamics but is channel-wise; the paper targets the missing dynamic realization for non-Cartesian joint envelopes that preserves boundary-tangent authority.The proposed AJ-APIR uses a matrix-valued realization to suppress the normal component near the boundary while retaining the tangential component.Source: §I · p. 2
- AJ-APIR is integrated with recursive backstepping, with a compact forward-invariant subset on which the realization channel remains uniformly nondegenerate.The stated compatibility condition yields exponential convergence of tracking and realization errors with uniformly bounded closed-loop signals.Source: §I · pp. 2–3
- In the reported studies, isotropic realization violates the shared capacity boundary despite respecting individual actuator bounds, whereas AJ-APIR maintains joint admissibility and tracking; a UAV path-following application is also reported.The first study compares both realizations under the same commanded input, and the second considers a nonlinear two-input strict-feedback system.Source: §I · p. 3
II. PROBLEM FORMULATION AND AJ-APIR
The section formulates tracking under individual actuator bounds and a shared capacity envelope, then contrasts isotropic input realization with a geometry-aware alternative. AJ-APIR attenuates boundary-normal commands while retaining tangential motion, under stated system and feasibility assumptions.
II. PROBLEM FORMULATION AND AJ-APIR
- The system is a multi-input strict-feedback plant with a single measurable output, and its admissible inputs satisfy individual asymmetric bounds together with a shared envelope defined by φ(u).The objective is to keep the realized input in the joint admissible set while tracking the prescribed output trajectory.Source: §II · p. 3
- Isotropic APIR applies scalar admissibility factors independently by channel, attenuating commands uniformly even when they point tangent to the joint boundary.Tangential commands do not change φ(u), so isotropic attenuation increases commanded-input demand and can degrade tracking performance.Source: §II · p. 3
- AJ-APIR replaces the diagonal gain with a matrix-valued gain that separates normal and tangential directions relative to the joint boundary.Its normal gain is required to vanish near the joint boundary, while the tangential gain may remain bounded away from zero.Source: §II · pp. 3–4
- At a boundary point where box constraints are slack, isotropic realization can retain a boundary-crossing component, whereas AJ-APIR annihilates the normal component and leaves tangent motion available.The figure describes spherical versus boundary-flattened realizable-velocity sets for the two realizations.Source: §II · p. 4
- The anisotropic gain has eigenvalue G⊥ along the boundary normal and G∥ on its orthogonal complement, and reduces to the channel-wise form when the joint constraint is absent.This reduction is presented as generalizing the existing single-input framework to multiple inputs.Source: §II · p. 4
- The assumptions require a smooth convex joint constraint with a regular boundary and origin in the interior, plus an initial input inside the set and a reference separated from its boundary.The reference-separation condition is a feasibility requirement: perfect tracking must in principle be achievable within the joint admissible set.Source: §II · pp. 4–5
- The relative placement of the box and shared envelope determines which boundary portions are active; anisotropy matters where the envelope is active while box constraints remain slack.The stated results cover all three described configurations, including containment cases where anisotropy offers no advantage over the channel-wise form.Source: §II · p. 5
- The control objective requires joint admissibility for all time, output tracking convergence, and uniformly bounded closed-loop signals.The realized input is required to remain strictly inside the joint admissible set.Source: §II · p. 5
III. PARAMETRIC AJ-APIR DESIGN
This section develops the AJ-APIR gain hierarchy and its integration with backstepping under an AJ-APC compatibility condition, addressing admissibility preservation and tracking convergence. It also states a regional, design-dependent exponential convergence result and describes denominator regularization that preserves forward invariance but yields ultimate boundedness instead of (42).
III. PARAMETRIC AJ-APIR DESIGN
- The product gain structure makes both gains vanish at any box bound, while near the joint boundary it suppresses the normal gain and preserves positive tangential gain.Away from box boundaries, the ratio G∥/G⊥ tends to infinity as the joint boundary is approached.Source: §III · p. 5
- For every bounded piecewise-continuous command, Theorem 1 confines the realized input to a compact subset of Ujoint and guarantees forward completeness and invariance.The invariant set Kξ,0 contains u(0).Source: §III · pp. 5–6
- Backstepping introduces a realization error because AJ-APIR gives the effective input its own dynamics, then selects a command to stabilize that error while preserving joint admissibility.For multiple inputs, the residual command freedom is resolved by a minimum-energy choice in the metric induced by G; the resulting Lyapunov derivative is negative definite where the command map is well defined.Source: §III · p. 7
- In the multi-input design, the quadratic form g_n⊤Gg_n, rather than G alone, determines realization-channel conditioning and must remain bounded away from zero.On Kξ,0, the lower bound follows from positive minima of the gains and the assumed lower bound on ∥g_n∥.Source: §III · p. 7
- AJ-APC compatibility couples the initial-error sublevel and reference demand to finite actuator authority; bounded recursive quantities and zero-error trackability ensure compatible small sublevels and permit offline verification.The small-sublevel result holds for some c⋆ > 0 with the prescribed command bound ξ0.Source: §III · p. 8
- For an AJ-APC-compatible level c, the closed loop preserves u(t) in Kξ,0 ⊂ Ujoint, keeps all signals bounded, and achieves exponential tracking and realization-error convergence.The commanded input also remains bounded by ξ for all t ≥ 0.Source: §III · pp. 8–9
- A continuation argument extends solutions to all t ≥ 0; non-increasing V keeps z and ϱ in Ωc and, under the stated assumptions, the states and input remain bounded.The input bound follows from compactness of Kξ,0 and boundedness of uc.Source: §III · p. 9
- Using ϵc > 0 in the regularized denominator retains forward invariance, but incurs a realization deficit of order ϵc in (37) and ultimate boundedness instead of (42).Source: §III · p. 9
IV. NUMERICAL SIMULATION
The numerical studies compare AJ-APIR with isotropic realization and evaluate AJ-APIR in tracking and path-following settings under individual and joint constraints.
IV. NUMERICAL SIMULATION
- The isotropic realization repeatedly violates the joint constraint while AJ-APIR remains admissible; both keep every input channel within its individual box limits.Fig. 3 shows that staying within individual actuator bounds does not ensure joint admissibility.Source: §IV · p. 9, Figure 3 · p. 10
- The output converges to its reference after an approximately 3 s transient, with commanded demand decreasing substantially after the tracking error is nullified.The study uses a second-order, two-input strict-feedback system.Source: §IV · p. 9
- During output tracking, realized inputs remain within individual bounds and the joint constraint stays strictly satisfied, even near its boundary during peak demand.The input trajectory remains inside the joint admissible set throughout the simulation.Source: §IV · p. 10
- In the path-following application, AJ-APIR keeps realized angular rates within individual and joint limits while the vehicle converges to and follows the desired path.The joint constraint represents shared maneuvering authority for the pitch and yaw rate channels.Source: §IV · p. 10
V. CONCLUSIONS
AJ-APIR uses joint-constraint geometry to preserve tangential control while attenuating the normal component near the boundary. Integrated with backstepping, it preserves admissibility and supports exponential tracking convergence with uniformly bounded closed-loop signals.
V. CONCLUSIONS
- AJ-APIR separates normal and tangential directions relative to the joint boundary, attenuating the normal component while keeping the tangential component active.This avoids unnecessarily restricting admissible control redistribution.Source: §V · pp. 10–11
- With the AJ-APC compatibility condition, tracking errors converge exponentially while all closed-loop signals and the commanded input remain uniformly bounded.The backstepping design demonstrates forward invariance and derives a compact invariant set where the realization channel remains uniformly nondegenerate.Source: §V · p. 11
- The simulations show that individual actuator bounds alone do not ensure joint admissibility, whereas AJ-APIR preserves the shared capacity constraint during tracking.Source: §V · p. 11