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Low-Complexity Control Under Input Saturation and Performance Constraints: A Bidirectional Modification Scheme

Jinyu Ni, Xiucai Huang, George A. Rovithakis, Yamin Yan

arXiv:2609.00827v1eess.SY

TL;DR

The paper addresses tracking control for high-order uncertain highly coupled MIMO nonlinear systems subject to input saturation and performance constraints. It develops a single first-order bidirectional modification mechanism and a low-complexity control scheme, with stability analysis establishing modified-constraint satisfaction and bounded closed-loop signals under parameter conditions. Simulations support the methodology, including better reported steady-state performance than a comparison controller.

  • Problem

    Input saturation can prevent standard PPC from satisfying performance constraints and can trigger instability, while existing remedies may increase complexity or restrict the plant class.

  • Method

    The paper combines a single first-order bidirectional constraint-modification signal with a continuous low-complexity state-feedback control scheme and a novel stability analysis framework.

  • Results

    The analysis establishes satisfaction of modified constraints and boundedness of all closed-loop signals when the stated parameter selection conditions are met; simulations validate the methodology.

  • Takeaways & Limitations

    The mechanism relaxes constraints during saturation, restores original constraints afterward, and tightens them during sustained nonsaturation while avoiding high-order or multiple modification dynamics.

Abstract

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This article addresses the output tracking control problem for a class of high-order uncertain highly-coupled MIMO nonlinear systems subject to input saturation and performance constraints. To resolve the problem, a bidirectional modification mechanism is constructed, which is able to not only relax the constraints when saturation occurs to alleviate potential conflict, but also accelerate the recovery of original constraints after saturation ceases, and further tighten the constraints to enhance control performance if saturation remains inactive at the steadystate phase. Based on the mechanism, a model-, approximationand complexity-explosion-free control scheme is proposed. To bypass the obstacle in Lyapunov analysis, a novel stability analysis framework is developed, which, given that two parameter selection conditions are met, ensures satisfaction of modified constraints and boundedness of all closed-loop signals. Simulation results validate the effectiveness and superiority of the methodology.

I. INTRODUCTION

The paper targets low-complexity tracking control for uncertain, highly coupled MIMO nonlinear systems under input saturation and performance constraints. It introduces bidirectional constraint modification to address feasibility and conservatism limitations in prior approaches.

  • Performance constraints specify output-error overshoot, convergence speed, and steady-state accuracy, while practical controllers must also handle input saturation with low complexity.
  • PPC transforms constrained tracking into an unconstrained stability problem but may demand control effort unavailable under actuator saturation.Under insufficient actuator capacity, the output error can reach the performance boundary in finite time and trigger controller instability.
  • Existing constraint-modification methods can retain low complexity but may require high-order or multiple first-order modification dynamics, large saturation limits, or restrictive plant classes and assumptions.
  • The proposed bidirectional mechanism relaxes constraints during saturation, accelerates recovery afterward, and tightens constraints when saturation remains inactive at steady state.Its single first-order modification dynamics is intended to reduce controller complexity and increase design flexibility.
  • The article presents a continuous low-complexity input-saturated PPC solution and extends the conference version from second-order SISO systems to higher-order MIMO systems.The extended work also adds analysis, proofs, parameter guidance, and comparative simulation.

A. System Description

The system description defines saturated MIMO control inputs and uncertain, time-varying nonlinearities. The analysis assumes BIBS stability and bounded growth conditions for the nonlinear terms.

  • The control input vector is componentwise saturated, with each saturation level satisfying ¯uj > 0.
  • The unknown vector and matrix nonlinearities are locally Lipschitz in the subsystem states and piecewise continuous in time.
  • The plant is assumed BIBS stable, excluding systems that become unstable under input saturation.The paper identifies the unstable system ˙x = 2 + sat(u) with ¯u = 1 as an excluded example.
  • The nonlinearities satisfy norm bounds through continuous, not necessarily known, state-dependent functions ¯fi and ¯gi.These bounds ensure the nonlinearities cannot grow unbounded as time increases.

B. Control Objective

The control objective is to track desired trajectories while respecting performance constraints despite input saturation. The paper seeks a continuous, low-complexity state-feedback controller that preserves modified constraints and bounded closed-loop behavior.

  • The output tracking errors are constrained using user-designed performance functions that are positive, differentiable, decreasing, and bounded.
  • The performance-function parameters determine maximum overshoot, minimum convergence rate, and maximum steady-state accuracy.The initial performance bound must exceed both the initial tracking error and the steady-state bound.
  • Under saturation, standard PPC may fail because its demanded control effort cannot be delivered, causing boundary contact and controller instability.
  • The proposed controller must alleviate saturation–performance conflict while ensuring modified error constraints and boundedness of all closed-loop signals.
  • Desired trajectories are assumed differentiable and bounded, with piecewise-continuous, bounded, and not necessarily known first-order derivatives.

A. Bidirectional Constraint Modification Mechanism

A single first-order modification signal adjusts the performance functions in both directions according to saturation deficiency. It relaxes constraints during saturation, restores them afterward, and can tighten them during sustained nonsaturation.

  • The performance-function modification signal σ(t) is generated by a single first-order dynamics.
  • The saturation deficiency is defined from the difference between the control norm and a design threshold when the norm is below that threshold, and otherwise equals zero.
  • During input saturation, the deficiency term increases σ(t), thereby relaxing the performance constraints to alleviate potential conflict.
  • After saturation ceases, a non-positive term accelerates σ(t)'s decrease and speeds recovery of the original constraints.
  • If saturation remains inactive after the performance function enters its 2% steady-state band, the mechanism further tightens the constraints to enhance control performance.
  • Unlike methods requiring plant-order or separate per-constraint dynamics, this design uses one scalar first-order signal, reducing added complexity and increasing flexibility.An L2-norm implementation is possible but would increase computational burden through square-root and multiplication operations.

B. Low-Complexity Control Scheme Design

The control scheme combines PPC-style normalized and mapped errors with backstepping-like virtual-controller design and a single scalar modification signal. Parameterized constraint dynamics and selection conditions keep the transformed errors and modified constraints well-defined while limiting added complexity.

  • Control transformation: A differentiable mapping T transforms normalized errors into mapped errors while diverging at the constraint boundaries.The mapping satisfies T(0)=0, approaches ±∞ as its argument approaches ±1, and has a positive locally Lipschitz derivative.
  • Backstepping-like design: The design recursively constructs virtual controllers from first-order normalized and mapped errors through a backstepping-like program.The scheme defines ξ1 and ε1, designs α1 with gain k1>0, and repeats the construction for higher-order errors and virtual controllers.
  • Constraint functions: Each higher-order error constraint uses a strictly positive, differentiable, bounded performance function whose initial value exceeds the corresponding initial tracking mismatch.The functions have bounded, piecewise-continuous first derivatives and are selected with positive design parameters.
  • Well-posedness conditions: The initial conditions and parameter choices ensure normalized errors remain inside the mapping domain and prevent the modified performance-function denominator from reaching zero.The construction establishes |ξi,j(0)|<1 and requires ρi,j,∞+ηi,jσ>0 for all relevant indices and times.
  • Complexity and flexibility: Only a scalar first-order modification signal is added, reducing controller complexity relative to higher-order or multiple first-order modification dynamics.The parameters μ and ci,j also embed feasibility of modified constraints and an explicit trade-off between modification magnitude and allowable tightening.

IV. STABILITY ANALYSIS

The stability analysis establishes global existence, boundedness of closed-loop signals, and satisfaction of the modified performance constraints under the theorem’s assumptions and parameter conditions. A three-stage argument keeps the transformed states inside the admissible region and prevents finite escape time.

  • Theorem 1: Theorem 1 requires Assumptions 1–3 and parameter choices satisfying the stated inequality conditions to solve the control problem.Under these conditions, the control scheme addresses the input-saturated system while targeting modified performance constraints.
  • Proof framework: The proof analyzes z(t) in three stages: local existence, confinement to a compact subset with bounded signals, and exclusion of finite maximal time.The state domain is Ωz = (−1, 1)^(mn) × R, and the final stage extends the solution to all t ≥ 0.
  • Signal boundedness: The modification signal σ(t) remains finite, with its lower behavior controlled by construction and its upper unboundedness excluded by the stability argument.The proof establishes an unknown finite bound σ̄ such that |σ(t)| ≤ σ̄ over the considered interval.
  • State boundedness: The transformed errors remain strictly inside the barrier boundaries, with unknown bounds satisfying |ξi,j(t)| ≤ ξ̄i,j < 1.The argument excludes boundary-reaching behavior for intermediate and final transformed errors through continuity, bounded virtual controllers, and contradiction.
  • Conclusion: The resulting solution is global, all closed-loop signals are bounded, and the modified performance constraints are ensured.The continuation argument yields tmax = +∞ before concluding satisfaction of the modified constraints.

V. SIMULATIONS

The comparative simulation shows that both controllers complete the tracking task under input saturation and performance constraints, while this work reduces controller complexity and improves steady-state accuracy for highly-coupled MIMO systems.

  • The simulation uses equivalent settings to, including x1(0) = [−1, 1.6]⊤, x2,j(0) = 0, saturation levels ¯uj = 5, and yd(t) = [sin t, cos t]⊤.
  • Both control schemes complete the output tracking task under input saturation and performance constraints in the numerical example.
  • |x1,j(t)−yd,j(t)| < 0.005 in the proposed method, compared with |x1,j(t)−yd,j(t)| < 0.01 in during the steady-state phase.The proposed method's modification term η1,jσ(t) is approximately within [−0.007, −0.005] at steady state.
  • The proposed scheme applies to highly-coupled MIMO systems with square gain matrices, while the method in is limited to diagonal control gain matrices.

VI. CONCLUSION

The paper addresses input-saturated tracking for high-order uncertain highly-coupled MIMO nonlinear systems. It proposes bidirectional constraint modification, a low-complexity control scheme, and a stability framework that establishes modified-constraint satisfaction and bounded closed-loop signals.

  • The study targets input-saturated prescribed-performance control for high-order uncertain highly-coupled MIMO nonlinear systems.
  • The bidirectional modification mechanism alleviates conflict during saturation while reducing conservatism in control performance.
  • The proposed low-complexity control scheme ensures satisfaction of modified performance constraints and boundedness of all closed-loop signals.
  • A novel stability analysis framework bypasses the obstacle in Lyapunov analysis.
  • Future work includes extending the method to non-BIBS-stable or non-square MIMO nonlinear systems and removing parameter selection condition (23).
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