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Optimal Steady-State Regulation of Nonlinear Systems: A Feedback Optimization Approach

Wei He, Lei Wang, Jingjing Fang, ZhengGuang Wu, Hongye Su

arXiv:2610.34360v1eess.SYcs.SY

TL;DR

The paper addresses nonlinear output regulation when the desired steady-state reference is an optimum rather than a prescribed signal. It combines a generalized regulator equation with gradient-feedback reference generation and backstepping control; under the stated assumptions, the closed loop converges globally asymptotically to the optimal steady-state manifold.

  • Problem

    For nonlinear systems, the regulated output may need to track an optimal steady-state trajectory that is not prescribed a priori but determined by optimization.

    Source: §I · p. 1

  • Method

    A generalized regulator equation supports a two-layer design: a steady-state generator with gradient feedback produces the reference, and a backstepping-based state-feedback controller tracks it.

    Source: §I · p. 1, A. Design of optimization layer · p. 2, §V · p. 5

  • Results

    Under Assumptions 1–4 and suitable gains, the closed-loop system globally asymptotically converges to the optimal steady-state manifold.

    Source: B. Design of control layer · p. 4

  • Takeaways & Limitations

    The framework provides optimal steady-state regulation for the considered class of nonlinear systems in lower-triangular form subject to disturbances generated by a neutrally stable nonlinear exo-system.

    Source: §V · p. 5

    The framework's stated scope is lower-triangular nonlinear systems with neutrally stable exo-systems; state and input constraints and output-feedback designs with unknown exo-systems are future directions.

    Source: §V · p. 5

Abstract

from arXiv · show

This paper studies the optimal steady-state regulation problem for a class of nonlinear systems. The regulated output is required to track a trajectory that minimizes a steady-state cost depending on both the output and the exo-system. To characterize the desired steady-state behavior, we introduce a generalized regulator equation and develop a two-layer framework. The optimization layer generates the optimal reference online through a steady-state generator combined with gradient-feedback optimization dynamics, while the control layer employs backstepping to stabilize the plant and achieve the desired optimal steady-state regulation. Under suitable immersion and minimum-phase assumptions, global asymptotic stability is established.

I. INTRODUCTION

The introduction places optimal steady-state regulation (OSSR) in the context of output regulation, where the reference may need to be determined by the system rather than prescribed. For nonlinear plants, the paper combines a generalized regulator equation with optimization and backstepping in a two-layer framework.

I. INTRODUCTION

  • Earlier nonlinear output-regulation work developed the internal-model method and regulator equation, while later work considered removing the immersion condition in a pre-processing, minimum-phase setting.The cited literature also extends internal-model methods to hybrid and networked systems and addresses unknown non-vanishing measurements using a washout filter.Source: §I · p. 1
  • OSSR addresses cases where the regulated output must follow an optimal steady-state trajectory that is not prescribed a priori but determined through optimization.Prior work cited here treated disturbed nonlinear systems with equality and inequality constraints and convex optimal steady-state control for disturbed linear systems.Source: §I · p. 1
  • Related work also studies low-gain integral control, timescale separation, power-system optimization, and distributed optimal steady-state control.The cited power-system work connects Automatic Generation Control with Economic Dispatch Control from an optimization viewpoint.Source: §I · p. 1
  • The proposed two-layer framework combines the regulator equation with an optimality condition, uses immersion for optimization-layer design, and uses backstepping in the control layer to ensure global asymptotic stability of the optimal steady-state manifold.The considered problem is convex OSSR for nonlinear systems in lower-triangular form, with a cost depending on the output and an exo-system signal whose gradient is measurable.Source: §I · p. 1

II. PROBLEM FORMULATION

The problem is to regulate the output of a lower-triangular nonlinear plant to a steady-state reference minimizing a cost that depends on the output and exogenous signal. The formulation relies on convexity, measurable gradient, a generalized regulator equation, and minimum-phase and immersion assumptions.

II. PROBLEM FORMULATION

  • The controller must keep closed-loop trajectories bounded and asymptotically steer the regulated output to the steady-state reference minimizing the cost in the presence of the exogenous signal.The exo-system is neutrally stable and evolves on an invariant compact set; unlike conventional regulation, the steady-state output is determined by the plant and optimization problem.Source: §II · p. 2
  • The stated results concern relative degree two, although the paper says the control design can be extended directly to higher relative degrees.The plant is described in lower-triangular form with an exogenous state, internal state, regulated output, and control input.Source: §II · pp. 1–2
  • The cost is assumed sufficiently smooth and strongly convex in the regulated output, uniformly over the exo-system state, with its output gradient measurable for controller design.The paper notes that the optimum is not directly solvable because the exogenous state is unknown, motivating online use of the measurable gradient.Source: §II · p. 2
  • The generalized regulator equation assumes smooth steady-state maps, and the internal error dynamics are required to be input-to-state stable with the regulated-output error as input.The immersion conditions constrain regulator maps; the paper says they facilitate constructive design, while the internal-dynamics assumption simplifies analysis and could be relaxed using extra measurement.Source: §II · p. 2

III. MAIN RESULTS

The proposed controller design framework has an optimization layer and a control layer.

III. MAIN RESULTS

  • The controller design framework is composed of two layers: an optimization layer and a control layer.The section identifies the two-layer architecture without describing the layers' design here.Source: §III · p. 2

A. Design of optimization layer

The optimization layer generates a reference using a steady-state generator, gradient feedback, and an output-error correction term. Under stated assumptions, its reference-state subsystem is input-to-state stable when the gain exceeds a threshold.

A. Design of optimization layer

  • The optimization dynamics combine a steady-state generator, gradient feedback, and output-error correction to generate a reference, enforce the cost optimality condition, and add stability margin.The generator term is motivated by the immersion condition; the correction term is introduced to guarantee the desired closed-loop stability property.Source: A. Design of optimization layer · p. 2
  • Under Assumptions 1, 2, and 4, the reference-state subsystem is input-to-state stable with input e when gain g_r exceeds a threshold.The lemma states that there exists g*_r such that the property holds for any g_r > g*_r and smooth κ(·).Source: A. Design of optimization layer · p. 3
  • The subsystem's ISS property is certified by a quadratic Lyapunov function with a positive-definite matrix and a class-K-infinity bound on the input.The supplied passage specifies V_o(ξ̃_r) = ξ̃_r^T Qξ̃_r and positive α_ξ and α_e terms.Source: A. Design of optimization layer · p. 3

B. Design of control layer

The control layer uses backstepping to stabilize the plant around the reference generated by the optimization layer. Under Assumptions 1–4 and suitable gain choices, its staged internal-model design yields global convergence to the optimal steady-state manifold.

B. Design of control layer

  • The control-layer design follows backstepping, treating x2 as the control input for the x1 subsystem.The design is intended to complete the previously specified optimization layer.Source: B. Design of control layer · p. 3
  • For g1 above its threshold, a smooth stabilizer makes the (ξ̃r, x̃0, η̄1, ϑ1) subsystem ISS with ϑ2 as input.This result holds under Assumptions 1–4.Source: B. Design of control layer · p. 3
  • A second pre-processing internal model introduces η2 and a residual input v2 for the x2 subsystem.The design then studies the η2 dynamics with the first subsystem and ϑ2 as inputs.Source: B. Design of control layer · pp. 3–4
  • With the prescribed earlier gains and g2 above its threshold, the closed loop globally converges to the optimal steady-state manifold yr = π1(w).The proof uses a small-gain argument to establish global asymptotic stability.Source: B. Design of control layer · p. 4

IV. SIMULATION

The simulation applies the proposed architecture to a nonlinear plant driven by a neutrally stable exo-system. The generated reference and regulated output converge to the optimal steady-state trajectory, while the measured gradient vanishes asymptotically.

IV. SIMULATION

  • The example uses a nonlinear plant driven by the neutrally stable oscillator ẇ1 = −0.5w2, ẇ2 = 0.5w1.The cost function is specified for this example.Source: §IV · p. 4
  • The implementation uses the two-layer architecture, prime-form internal models, and selected gains for its optimization and control layers.The optimization layer uses gradient measurements with κ ≡ 0.Source: §IV · p. 4
  • The generated reference and regulated output converge to the optimal steady-state trajectory, and the measured gradient vanishes asymptotically.Fig. 1 presents trajectories of regulation errors and real-time gradients.Source: §IV · p. 5, Figure 1 · p. 5

V. CONCLUSION

The paper formulates optimal steady-state regulation for lower-triangular nonlinear systems subject to disturbances from a neutrally stable nonlinear exo-system. It combines the generalized regulator equation with a gradient-feedback reference generator and backstepping-based tracking control, and identifies constraints, output feedback, unknown exo-systems, and large-scale applications as future directions.

V. CONCLUSION

  • The studied OSSR setting covers lower-triangular nonlinear systems subject to disturbances generated by a neutrally stable nonlinear exo-system.Source: §V · p. 5
  • The generalized regulator equation incorporates the steady-state optimality condition into a constructive nonlinear regulation framework.A steady-state generator with gradient-feedback dynamics generates the optimal reference, while a backstepping-based state-feedback controller ensures tracking.Source: §V · p. 5
  • Future work will extend the framework toward state and input constraints, output-feedback designs with unknown exo-systems, and large-scale applications.Source: §V · p. 5

A. Proof of Lemma 2

The proof constructs an observer-like error subsystem whose stability depends on a gain choice. A Hurwitz matrix design and a lower bound on gr establish an ISS property for that subsystem.

A. Proof of Lemma 2

  • Assumption 4 supplies global Lipschitz continuity of ϕ1, with Lϕ1 as its Lipschitz constant.Source: A. Proof of Lemma 2 · p. 5
  • Choosing G0 makes Mr(t) = Ar − ω(w(t))G0Cr Hurwitz.The construction uses the stated lemma for the time-varying matrix.Source: A. Proof of Lemma 2 · p. 5
  • For gr above the stated threshold, the ξ̄r subsystem is ISS with e as input and ξ̄r as state.The threshold is bounded below by (2 + 2Lϕ1|P|)/λ0.Source: A. Proof of Lemma 2 · p. 5

B. Proof of Lemma 3

The proof analyzes interconnected error and exosystem dynamics using Lyapunov functions and gain conditions. It establishes input-to-state stability for subsystem components and uses a small-gain argument to complete the stability analysis.

B. Proof of Lemma 3

  • The proof treats (˜ξr, ˜x0, ¯η1) as zero dynamics and examines its interconnection with the ϑ1 dynamics under assumption 3.The analysis also considers bounded class K∞ functions αϑ1 and αξr.Source: B. Proof of Lemma 3 · p. 5
  • The proof constructs a Lyapunov function for ϑ1 and scales ¯η1 by Dg1 to analyze its stability.It uses Vϑ1 = 1/2ϑ1^2 and defines ˆη1 = Dg1^-1 ¯η1.Source: B. Proof of Lemma 3 · p. 6
  • For sufficiently large g1, the ¯η1 dynamics are ISS with respect to (˜ξr, ˜x0, ϑ1, ϑ2).The proof establishes a threshold g∗1 and requires g1 > g∗1.Source: B. Proof of Lemma 3 · p. 6
  • The (˜z1, ϑ1) subsystem is ISS with input ϑ2 by the small-gain theorem.The feedback v1 = −ρ1(ϑ1) is used, with ρ1 designed to meet a gain condition.Source: B. Proof of Lemma 3 · p. 6
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