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Disturbance rejection for classes of nonlinear systems
Saverio Messineo
TL;DR
The paper addresses non-adaptive global robust disturbance rejection for two nonlinear-system classes where disturbances and unavailable states complicate control design. It develops high-gain and bounded sliding-mode control for C1, then combines an open-loop observer with the C1 results for C2, obtaining boundedness and regulation toward arbitrarily small attractors.
Problem
Non-adaptive global robust disturbance rejection is sought for C1 systems with unavailable zero-dynamics states and for C2 systems with unknown forcing disturbances.
Method
The approach uses high-gain control with a bounded sliding-mode unit for C1, and an open-loop observer plus an adapted output-feedback dynamic controller for C2.
Results
For C1, the architecture achieves input-to-state stability and convergence to an arbitrarily small attractor; for C2, it ensures global uniform boundedness and asymptotic regulation to one.
Takeaways & Limitations
The two designs provide global robust disturbance rejection across the stated C1 and C2 system classes while allowing the residual attractor to be made arbitrarily small.
Takeaways & Limitations
The observer-error stability cannot be inferred from the observer-error dynamics alone when Γ(y) is not uniformly bounded.
Abstract
from arXiv · showhide
This paper addresses the problem of non-adaptive global robust disturbance rejection for two distinct classes of nonlinear systems. The first class, denoted by C1, consists of nonlinear systems in strict-feedback form, with linear and Hurwitz zero-dynamics (whose states are unavailable for feedback), and enhanced - within this work - by forcing, unmatched, additive disturbances. Nonlinear tools are herein employed to demonstrate that the proposed control architecture - based on the high-gain paradigm - achieves closed-loop input-to-state stability with respect to the forcing disturbances, along with global asymptotic convergence towards an attractor which can be rendered as small as desired. Then, owing to the established input-to-state stability property, a uniformly bounded control action is additionally embedded within the control architecture. The additional unit, designed following the sliding-mode paradigm, is aimed at improving the disturbance rejection task, by potentially lowering the required high-gain control expenditure. The second class of systems, denoted by C2, is constituted by minimum-phase, uncertain, nonlinear systems with relative degree greater than one, featuring possibly unbounded, with possibly unbounded derivatives, output-dependent nonlinearities, with matched additive forcing disturbances. To solve the problem of output-feedback, non-adaptive, global robust disturbance rejection for systems within C2, first, an open-loop observer is employed in lieu of a classic dynamic extension adopted in earlier works, as the latter is no longer implementable due to the presence of unknown forcing disturbances. Subsequently, the results derived for C1 are then adapted to C2, to yield an output-feedback dynamic controller providing closed-loop global uniform boundedness, along with asymptotic regulation towards an attractor which can be rendered as small as desired.
1 Introduction
The paper targets non-adaptive global robust disturbance rejection for two nonlinear-system classes, addressing gaps involving unavailable zero-dynamics states and unknown forcing disturbances.
- C1 motivation: For C1, the paper studies strict-feedback systems with linear Hurwitz zero-dynamics whose states are unavailable for feedback, and with unmatched additive disturbances.
- C1 motivation: Non-adaptive global robust disturbance rejection under partial-state feedback had remained unresolved for C1 systems.
- C1 contribution: A high-gain architecture establishes input-to-state stability and convergence toward an attractor whose size can be made arbitrarily small for C1.
- C1 contribution: A bounded sliding-mode unit supplements high-gain control to improve disturbance rejection and potentially reduce the required high-gain effort.
- C2 motivation: For C2, the paper addresses minimum-phase uncertain nonlinear systems with relative degree greater than one, output-dependent nonlinearities, and matched additive disturbances.
- C2 contribution: An open-loop observer replaces an infeasible dynamic extension, and the adapted output-feedback controller provides global uniform boundedness and regulation toward an arbitrarily small attractor.
2 Class C1 systems
Class C1 comprises strict-feedback systems with unmeasured linear Hurwitz zero-dynamics and bounded forcing disturbances, controlled through partial-state feedback.
- System definition: C1 models strict-feedback dynamics with unmeasured zero-dynamics state z and available feedback states ξ_i.
- System definition: The disturbances ¯d_i and Δ_m are unknown and uniformly bounded, and may depend on state variables without preserving strict-feedback structure.
- Assumptions: The zero-dynamics matrix F(¯µ) is Hurwitz, and each input coefficient b_i is uniformly lower-bounded by a positive constant.
- System definition: Unknown parameters are divided so that ¯µ remains constant to preserve Hurwitz stability, while bounded µ may vary with time or state.
- Preliminary result: The preliminary scalar system separates control into u_1 and a uniformly bounded disturbance-rejection input u_d, with the available feedback state y and unmeasured state z.
- Preliminary result: For any desired δ > 0, the design selects u_1 = −γ(y)y and a bounded u_d = ¯u_d(y).
(3) System (3)-(6) possesses an ultimate bound which
The stability analysis combines Lyapunov estimates with high-gain and bounded sliding-mode components to establish arbitrarily small ultimate bounds under bounded disturbances.
- The proof uses separate Lyapunov functions for the unmeasured dynamics and output state, then combines their inequalities through a composite Lyapunov function.
- Input-to-state stability with respect to the disturbance follows from the derived Lyapunov inequality and the Comparison Lemma.
- The closed-loop system possesses an ultimate bound that can be rendered as small as desired by choosing the design parameters appropriately.
- The bounded disturbance-rejection component remains admissible because its uniform boundedness preserves boundedness of the combined disturbance.
- A sliding-mode control choice yields a uniformly bounded control action and an ultimate bound that can also be made arbitrarily small.
(1) The positive definite function
The analysis states that a positive-definite-function estimate contains a tunable term whose magnitude can be made arbitrarily small.
- The quantity ¯δ in the positive-definite-function estimate can be rendered as small as desired.
(2) System (28)-(33) is input-to-state stable with respect
The supplied passage is an incomplete fragment mentioning ¯, ∆, ∆1, and d2, so it does not state a complete input-to-state stability result.
- The passage contains only the fragment “to ¯ ∆, ∆1 and d2.”
- No complete stability property, system condition, or conclusion is stated in the supplied text.
- The symbols ¯, ∆, ∆1, and d2 are mentioned without definitions or a supported relation.
(3) System (28)-(33) possesses an ultimate bound which
System (28)-(33) possesses an ultimate bound that can be rendered as small as desired.
- The ultimate bound of system (28)-(33) can be rendered as small as desired.The proof obtains this by choosing the relevant design parameters sufficiently small while preserving the required stability properties.
- The result applies when p is selected large enough and η is selected as small as desired.
- The proof establishes input-to-state stability with respect to ¯∆, ∆1, and d2 before deriving the ultimate-bound property.
2.2 Closed-loop stability analysis
The closed-loop stability analysis constructs a smooth feedback law whose trajectories are input-to-state stable and possess an ultimate bound that can be made arbitrarily small.
- A smooth feedback law yields input-to-state stable closed-loop trajectories with an ultimate bound that can be made as small as desired.The design uses smooth functions and parameters p and η selected to support the stated properties.
- The theorem is proved by induction on the system's relative degree r, beginning with r = 1 and extending the result to r + 1.
- For r = 2, the system is rewritten in the form of system (28), allowing Lemma 2 to establish the theorem's points.
- The induction step shows that the hypotheses and results of Lemma 2 continue to hold when the relative degree increases from r to r + 1.
3 Class C2 systems
Class C2 covers uncertain, minimum-phase nonlinear systems with relative degree greater than one, output-dependent nonlinearities, and matched forcing disturbances. An open-loop observer replaces an infeasible dynamic extension, enabling an output-feedback controller with global boundedness and arbitrarily small asymptotic regulation error.
- Class definition: C2 systems have uncertain parameters, minimum-phase dynamics, relative degree r ≥ 2, output-dependent nonlinearities, and matched uniformly bounded disturbances.Only the output y is measurable; the state x is unavailable for feedback.
- Observer design: Unknown forcing disturbances make the classic dynamic extension unusable, motivating an open-loop observer as its proxy.The observer replaces the unknown disturbance term with a constant-gain term, avoiding potentially unbounded forcing in the observer-error dynamics.
- System transformation: The disturbance-dependent nonlinearity is decomposed so the transformed system has a constant input gain and a uniformly bounded disturbance contribution.The change uses Γ̄(y)=Γ(y)−Γ(0), with Γ(0) retained in the input channel and the remaining term absorbed into the nonlinear coefficient.
- Observer design: The observer trajectories are uniformly bounded, and its gains can be selected so the relevant observer state approaches any prescribed tolerance after a finite time.This bounded observer error enters the transformed plant as a uniformly bounded forcing term.
- Closed-loop stability: The resulting dynamic controller yields globally uniformly bounded closed-loop trajectories with an ultimate bound that can be made as small as desired.The proof applies the C1 result after treating the bounded observer error as an additional bounded disturbance.
4 Examples
The examples simulate representative C1 and C2 systems under uncertain parameters and forcing disturbances. In both cases, the stabilizer alone and the stabilizer plus additional module reach bounded attractors, while the combined strategy improves steady-state performance.
- C1 example: The C1 comparison disables the additional module on the left and enables both stabilizer and additional module on the right.Plots (c) and (d) show the full x2 history, while plots (e) and (f) zoom into its steady-state evolution.
- C1 example: In the C1 simulation, both strategies steer trajectories to a bounded attractor despite forcing disturbances and unknown parameters.The combined strategy improves steady-state performance for states x1 and x2, while no noticeable difference is reported for another state.
- C2 example: The C2 example uses output-only feedback, unmeasured internal states, uncertain parameters, and a uniformly bounded time-varying disturbance.The constructed system is verified to satisfy the defining C2 properties, including exponentially stable zero dynamics and relative degree r=3.
- C2 example: In the C2 simulation, both strategies reach a bounded attractor despite forcing disturbances and unknown parameters.The strategy with both modules enabled shows superior steady-state performance for all system states.
5 Conclusions
The paper establishes non-adaptive global robust disturbance rejection for two nonlinear-system classes using high-gain control, a bounded supplementary unit, and an observer-based extension. The resulting controllers provide stability or boundedness and regulation toward attractors that can be made arbitrarily small.
- Contributions: For C1, high-gain control achieves input-to-state stability and global asymptotic convergence to an arbitrarily small attractor.A supplementary sliding-mode unit is embedded to improve disturbance rejection while potentially reducing high-gain expenditure.
- Contributions: For C2, an open-loop observer replaces the infeasible dynamic extension, and the adapted output-feedback controller ensures global uniform boundedness and arbitrarily small asymptotic regulation error.The observer substitution is required because the forcing disturbances are unknown.
- Proof strategy: The stability proofs use an inductive structure intended to circumvent the curse of dimensionality in strict-feedback designs.