PaperlayerView exact revision on arXiv

Source-linked AI summary

Model-Based Disturbance Rejection via Sliding-Mode Control: Continuous-Time Formulation and Proper Implicit Discretization

Benedikt Andritsch, Stefan Koch, Jaime A. Moreno

arXiv:2610.02890v1eess.SYcs.SY

TL;DR

The paper addresses reference tracking under disturbances generated by LTI models with bounded input, including disturbances that may be coupled with the plant. It proposes a robust exact sliding-mode controller and a proper implicit sampled-data discretization; in the nominal case, control accuracy is of order T^(p+1) times the disturbance bound.

  • Problem

    The paper studies reference tracking when disturbances are generated by an LTI model with bounded input and may be coupled with the plant.

    Source: §1 · p. 2

  • Method

    The RESMC incorporates the disturbance model and uses a proper implicit discretization for sampled-data implementation.

    Source: §1 · p. 2

  • Results

    T^(p+1) times the disturbance bound is the order of the proposed discrete-time controller's nominal-case accuracy.

    Source: §6 · p. 11

  • Takeaways & Limitations

    The nominal-case accuracy relates to the best possible worst-case accuracy by a factor depending on disturbance-model order and not sampling time.

    Source: §8 · p. 12

    The higher-order-plant extension applies to relative-degree-one outputs, with internal plant states required to stay bounded during tracking.

    Source: §2 · p. 2

Abstract

from arXiv · show

We propose a robust exact sliding-mode controller for reference tracking with model-based disturbance rejection, where the disturbance is generated by a linear time-invariant model driven by a bounded input and may be bidirectionally coupled with the plant. The controller embeds the internal model principle and its error dynamics mirror those of the robust exact differentiator, ensuring finite-time convergence. We develop a proper implicit discretization for sampled-data implementation that preserves stability and accuracy. In the nominal case, which contains a disturbance generated by a $p$-th order integrator chain, the method achieves a control accuracy of $T^{p+1}\bar D$, with discretization time $T$ and disturbance bound $\bar D$. This accuracy is shown to relate to the fundamental lower bound by a factor depending only on the order of the disturbance model. Simulations demonstrate the efficacy of the proposed methods.

1 Introduction

Sliding-mode control rejects matched disturbances by driving a sliding variable to zero in finite time, while higher-order variants can change the disturbance class they reject. This paper extends model-based sliding-mode control to LTI-generated disturbances and develops an implicit sampled-data discretization to avoid discretization chattering.

1 Introduction

  • First-order SMC is robust against bounded disturbances, whereas the super-twisting algorithm is robust against Lipschitz-continuous disturbances.Higher-order sliding-mode control was introduced to mitigate chattering for sliding variables with relative degree one.Source: §1 · p. 1
  • Explicit Euler discretization of sliding-mode controllers can cause steady-state high-frequency oscillations, or discretization chattering, that deteriorate control performance.The paper develops a proper implicit discretization to avoid this effect.Source: §1 · p. 2
  • The proposed RESMC handles disturbances generated by an arbitrary LTI model driven by bounded input, including disturbances that may be unbounded and non-Lipschitz continuous.Its closed-loop tracking-error dynamics are equivalent to those of the arbitrary-order robust exact differentiator.Source: §1 · p. 2
  • The paper generalizes the internal model control principle in sliding-mode controller design for sliding variables with relative degree one.The disturbance model may be driven by the sliding variable itself.Source: §1 · p. 2
  • The work contributes a reference-tracking RESMC, a sample-wise exact nominal plant discretization, a proper implicit controller discretization, and a fundamental lower bound on worst-case control accuracy.The paper also reports simulations.Source: §1 · p. 2

2 Problem Description

The problem is to make a scalar output track a differentiable reference despite a disturbance described by a known LTI model with bounded unknown input. An observable-canonical-form transformation recasts tracking as driving its first transformed state to zero.

2 Problem Description

  • The control goal is for the scalar output y(t) to track a reference r(t), with known plant parameters and the reference and its derivative given.The setup considers sampled output measurements rather than continuous access to y(t).Source: §2 · pp. 2–3
  • The paper focuses on a simple first-order setup, while noting that results can transfer to a relative-degree-one output of a higher-order plant.For that higher-order setting, internal plant states must remain bounded when the output tracks the reference.Source: §2 · p. 2
  • The disturbance is modeled by a known LTI system driven by a bounded, unknown input, and its output has relative degree p with respect to that input.The super-twisting controller is a special case with an integrator disturbance model; bounded p-th derivative disturbances also fit the model.Source: §2 · p. 2

2.1 Transformation to Observable Canonical Form

  • Because the disturbance output has relative degree p, the tracking error has relative degree p + 1 with respect to the unknown input, making the augmented system strongly observable.The system is transformed to observable canonical form for controller design.Source: §2.1 · p. 3
  • In the transformed coordinates, tracking is equivalent to driving ζ1(t) to zero.The closed-loop error dynamics are intended to match those of the robust exact differentiator.Source: §2.1 · p. 3

3 Continuous-Time Robust Exact Sliding-Mode Controller

The continuous-time RESMC compensates the transformed plant and disturbance dynamics while stabilizing tracking-error states. With suitable gains, it achieves finite-time tracking despite bounded disturbance input.

3 Continuous-Time Robust Exact Sliding-Mode Controller

  • The controller copies the structure of the transformed disturbance dynamics, using compensating functions for disturbance effects and stabilizing functions for error dynamics.The disturbance and disturbance-model state are unavailable to the controller; it uses the output error and reference information.Source: §3 · p. 4
  • With suitable gains, the RESMC makes the output converge to the reference in finite time and remain there despite bounded disturbance input.Its controller states converge to the corresponding transformed disturbance states.Source: §3 · p. 4
  • The closed-loop error dynamics match those of the robust exact differentiator, allowing gains to be selected for finite-time convergence under bounded disturbance.This relation motivates the RESMC name and supports the theorem's finite-time tracking result.Source: §3 · p. 4
  • Bounded disturbance-model states imply bounded controller states, but that disturbance-state boundedness condition does not hold generally, for example when S is not Hurwitz.The boundedness result is conditional on the disturbance-state assumption.Source: §3 · p. 4

4 Plant Discretization

The plant discretization is designed to preserve sampled output values and an implicit-Euler closed-loop structure. The nominal integrator-chain case admits a sample-wise exact representation, whereas the general LTI case uses an approximation.

4.1 Exact Plant Discretization in the Nominal Case

  • The nominal sampled-data problem is tracking an integrator-chain-disturbed output error, whose p-th derivative disturbance is bounded.The desired discrete representation preserves exact sampled output values, has implicit-Euler form, and retains the disturbance bound.Source: §4.1 · p. 5
  • The nominal case admits a discrete representation that is sample-wise exact and in implicit-Euler form, with a discrete disturbance bounded by the continuous-time disturbance bound.The construction defines discrete disturbance states and input to satisfy the required properties.Source: §4.1 · pp. 5–6
  • T^p D bounds the p-th forward difference of the nominal disturbance signal, yielding a discrete disturbance input with the same bound as the continuous-time disturbance.This bound supports consistency of controller tuning between continuous and discrete time.Source: §4.1 · pp. 5–6

4.2 Approximative Plant Discretization in the General Case

  • For a general transformed LTI system, a sample-wise exact representation is unavailable, so the method constructs an approximation with implicit-Euler structure and bounded disturbance.The approximation uses Lie–Trotter splitting: zero-order-hold discretization for one subsystem and backward Euler for the other.Source: §4.2 · pp. 6–7
  • The general-case approximation corrects ζ1,k from each available sampled error measurement, and its first-order splitting has per-step error O(T^2).Approximating the time-varying signals as piecewise constant is valid when they vary slowly enough, or the sampling time is small enough.Source: §4.2 · pp. 6–7

5 Proper Implicit Discretization of the RESMC

The proper implicit RESMC is reconstructed in discrete time from a discretization of the desired closed-loop error system, rather than by directly discretizing the controller. In the nominal integrator-chain case, it gives finite-time convergence to a disturbance-dependent neighborhood, with exact finite-time convergence when the disturbance is absent.

5 Proper Implicit Discretization of the RESMC

  • The controller is reconstructed from a discretization of the desired closed-loop error system, not by directly discretizing the continuous-time controller.The approach avoids approximating next-sample states by neglecting unknown information.Source: §5 · p. 7
  • The discrete-time controller compensates known terms and copies the disturbance-model structure while leaving unknown disturbance states and input uncompensated.The sampled error ζ1,k is available from the measured output and reference.Source: §5 · pp. 7–8
  • The design transforms the error states so the desired implicit error dynamics can be recovered while the next transformed state is known at the current sampling step.This makes the resulting controller implementable despite unknown future error states.Source: §5 · p. 8
  • For the nominal integrator-chain case, sampled tracking error is bounded by T^(p+1)D after finite time, and reaches zero in finite time when D=0.The theorem assumes |Δ(t)| ≤ D and suitable controller gains.Source: §5 · pp. 9–10
  • When |ζ1,k| ≤ T^(p+1)D, the proper implicit RESMC is deadbeat: its closed-loop error dynamics are linear in the disturbance input and have only zero eigenvalues.This is the stated condition and characterization for the discrete-time system.Source: §5 · p. 10

6 Control Accuracy in the Nominal Case

This section establishes a fundamental lower bound on worst-case control accuracy for nominal output dynamics disturbed by an integrator chain with unknown input. It compares that bound with the proposed discrete-time controller’s accuracy and explains when the bound is tight.

6 Control Accuracy in the Nominal Case

  • The lower bound applies to nominal integrator-chain disturbances with bounded unknown input when only sampled measurements are available.The proposition concerns accuracy attainable by any causal control sequence under these information constraints.Source: §6 · pp. 10–11
  • Two admissible disturbance signals yield identical sampled measurements through K but force an error of at least M0 at the next sample for any causal control choice.The proof selects between the signals according to the control choice at sample K, producing a next-sample value at least M0 or at most −M0.Source: §6 · p. 11
  • Table 1 lists rounded lower bounds M0 for disturbance-model orders p = 0 through 6.The values are rounded to two decimal places.Source: §6 · p. 11, Table 1 · p. 11
  • The proposed controller’s accuracy is order T^(p+1) and scales with D; the lower bound is optimal for p ≤ 1, while smaller accuracy bands may exist for p > 1.This is the condition identified for a proper implicit discretization.Source: §6 · p. 11

7 Simulation Examples

The simulations compare proper implicit RESMC with an explicit RESMC discretization in a nominal integrator-chain case and a rotating-inertias system. The implicit method avoids the significant discretization chattering seen with the explicit method, while sampled errors remain near or within the proved nominal bounds.

Discretization comparison

  • The explicit comparison discretizes the Jζ subsystem with forward Euler, whereas the proper implicit scheme uses backward Euler.Both approaches use the same approximate solution for the Γ subsystem and compensate nonnominal dynamics similarly.Source: §7 · p. 11

Nominal system

  • The nominal example uses a p = 6 integrator chain, T = 0.1, and a rectangular disturbance that makes the disturbance and required control diverge.The plant starts at x1(0) = 1 and the reference is zero.Source: §7 · pp. 11–12
  • With the rectangular disturbance in the nominal p = 6 case, explicit RESMC has a remaining control-error amplitude 103 times larger than implicit RESMC.Implicit RESMC sampled errors stay within the Theorem 12 bounds; both controllers show significant inter-sample control deviation under the diverging disturbance.Source: §7 · p. 12, Figure 2 · p. 12

Rotating inertias

  • In the rotating-inertias system, explicit RESMC exhibits significant discretization chattering, while implicit RESMC has little chattering and error almost within the nominal bounds.Even with zero disturbance input, the coupled disturbance model and nonzero reference produce a nonzero disturbance.Source: §7 · p. 12, Figure 3 · p. 12, Figure 4 · p. 12

8 Conclusion

The paper presents a robust exact sliding-mode controller and proper implicit discretization for rejecting disturbances generated by an LTI model with bounded input. In the nominal case, its accuracy relates to the best possible worst-case accuracy by a factor depending only on disturbance-model order, and simulations support the theory.

Conclusion

  • The controller rejects disturbances generated by an LTI model with bounded input, and the paper provides a proper implicit discretization.The discretization reproduces the implicit-Euler closed-loop error dynamics of the continuous-time design in transformed coordinates.Source: §8 · p. 12
  • In the nominal case, discretization accuracy relates to the best possible worst-case accuracy by a factor depending only on disturbance-model order, independent of sampling time.Source: §8 · p. 12
  • Very small discretization times are not strictly necessary when sliding-mode control uses advanced control concepts and implementation schemes.Source: §8 · p. 13
Loading 2610.02890v1…