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Yet Another Tutorial of Disturbance Observer: Robust Stabilization and Recovery of Nominal Performance
Hyungbo Shim, Gyunghoon Park, Youngjun Joo, Juhoon Back, Nam Hoon Jo
TL;DR
Plant uncertainty and external disturbances make it important to recover nominal behavior while maintaining robust stability. This tutorial reviews DOB from an output-feedback perspective and explains its large-Q-filter-bandwidth analysis. Under stated conditions, DOB can stabilize against bounded parametric uncertainty and recover nominal steady-state and transient performance.
Problem
Method
The tutorial interprets conventional DOB as output-feedback robust control and analyzes its operation when Q-filter bandwidth is large.
Source: §1 · p. 1Results
Under conditions including minimum phase and sufficiently large Q-filter bandwidth, DOB enables robust stabilization against bounded parametric uncertainty with known bounds and recovery of nominal steady-state and transient performance.
Source: §1 · p. 1Takeaways & Limitations
DOB's inner-loop compensation can let an outer-loop controller designed for the nominal plant handle nominal stability and performance.
Source: §1 · p. 1The large-bandwidth analysis has restrictions: mismatched relative degree from unmodelled dynamics may lead to instability, and the tutorial is not an exhaustive survey of DOB contributions.
Source: §8 · pp. 8–9
Abstract
from arXiv · showhide
This paper presents a tutorial-style review on the recent results about the disturbance observer (DOB) in view of robust stabilization and recovery of the nominal performance. The analysis is based on the case when the bandwidth of Q-filter is large, and it is explained in a pedagogical manner that, even in the presence of plant uncertainties and disturbances, the behavior of real uncertain plant can be made almost similar to that of disturbance-free nominal system both in the transient and in the steady-state. The conventional DOB is interpreted in a new perspective, and its restrictions and extensions are discussed.
1 Introduction
DOB is presented as an inner-loop controller that compensates for plant uncertainty and disturbances, allowing an outer-loop controller designed for the nominal plant to address nominal stability and performance. This tutorial reframes DOB as an output-feedback approach to robust stabilization and recovery of nominal steady-state and transient performance under stated conditions.
1 Introduction
- DOB compensates for plant uncertainty and external disturbances so the inner loop behaves like a disturbance-free nominal plant, while the outer loop handles nominal stability and performance.This modular division allows the outer-loop controller to be designed for the nominal plant without considering robustness.Source: §1 · p. 1
- DOB offers design simplicity despite theoretical analysis being non-simple, and it has been used in industrial applications.Source: §1 · p. 1
- The tutorial presents recent author findings in a less formal style and interprets DOB as output-feedback robust control.The paper states that it is not an exhaustive survey of DOB contributions.Source: §1 · p. 1
- Under conditions including minimum phase and large Q-filter bandwidth, DOB enables robust stabilization against bounded parametric uncertainty with known bounds and recovery of nominal steady-state and transient performance.The tutorial focuses on the principles underlying DOB with large Q-filter bandwidth.Source: §1 · p. 1
2 2 System Description for Analysis
The analysis considers uncertain single-input, single-output linear time-invariant systems with smooth, uniformly bounded disturbances. It uses a normal-form representation to distinguish the plant's relative-degree dynamics from its zero dynamics.
2 System Description for Analysis
- The systems analyzed are single-input, single-output linear time-invariant systems with uncertain state-space matrices and a smooth disturbance whose derivatives are uniformly bounded.The realization without the disturbance term is assumed minimal.Source: §2 · p. 2
- The transfer-function parameters are uncertain, but the high-frequency gain is nonzero and its sign is known.Source: §2 · p. 2
- The analysis writes the plant in normal form, splitting its state into relative-degree coordinates x and zero-dynamics coordinates z.The disturbance signals in the transformed system include linear combinations of the disturbance and its derivatives.Source: §2 · p. 2
- The eigenvalues of S are the plant zeros, and the plant is minimum phase exactly when S is Hurwitz.The normal form is used for analysis; it is not required for DOB design.Source: §2 · p. 2
3 Required Action for DOB
The required action is to make the uncertain, disturbed plant behave like its nominal model, including its output behavior. The paper develops a desired input and explains how the DOB estimates it, while stability of the resulting zero dynamics remains necessary.
3 Required Action for DOB
- The desired input is constructed to replace uncertain plant terms with nominal terms and make the real plant's output dynamics match the nominal model.Because the control does not directly affect the z-subsystem, the construction introduces a controller state z_n with its own dynamics.Source: §3 · pp. 2–3
- With the desired input applied, the plant matches the nominal input-output behavior, but its z-subsystem becomes standalone and no longer affects the output.The state z is then unobservable from y, leaving no input freedom to control it; therefore S must be Hurwitz for z to remain stable under bounded x and d_z.Source: §3 · p. 3
- Implementation requires estimating the unmeasured state x and estimating the desired input, which depends on unknown quantities, then driving u toward that estimate.Source: §3 · p. 3
- The conventional DOB uses Q-filters Q_A and Q_B to make the plant-with-DOB block behave like the nominal plant for an outer-loop controller designed for that model.The filters are conventionally stable low-pass filters; their bandwidth increases as τ decreases, and their relative degree is chosen to make the inverse-model/filter block proper.Source: §3 · p. 3, Figure 1 · p. 3
- The design chooses filter parameters to robustly stabilize the plant against uncertainty and reject external-disturbance effects on the output.The DOB may be combined with C(s), forming a feedback controller with reference r and output y as inputs.Source: §3 · p. 3, Figure 1 · p. 3
4 A Closer Look at Conventional DOB
The conventional-DOB analysis uses a third-order, relative-degree-two example to unpack the controller’s internal roles. It interprets the inverse-filter block as both a nominal z-subsystem constructor and an x-state estimator, while QA’s convergence and stability depend on fast dynamics and suitable filter design.
4 A Closer Look at Conventional DOB
- The conventional-DOB analysis specializes to a third-order uncertain plant with relative degree two and organizes its discussion around two findings.Source: §4 · pp. 3–4
4.1 Robust observer is embedded in P −1
- As τ shrinks, q approaches x, revealing P_n^-1(s)Q_B(s) as both a high-gain robust observer for x and a constructor of the nominal z-subsystem.The observer estimates x, but not z.Source: §4.1 · p. 4
4.2 Estimation of udesired is performed by QA(s).
- With suitably small τ and chosen filter coefficients, the QA dynamics make u approach udesired.Source: §4.2 · p. 5
- The approximation improves as p-dynamics become faster, since they can track the time-varying input more closely; smaller τ accelerates those dynamics.The authors describe this as intuition and refer to singular-perturbation proofs for a more rigorous treatment.Source: §4.2 · p. 5
- For higher-order Q-filters, ensuring the stability polynomial remains Hurwitz under gain variation requires careful coefficient design; bounded gain limits permit a design for arbitrarily large variation.When gain variation is small, the polynomial is automatically Hurwitz if the Q-filter is stable.Source: §4.2 · pp. 5–6
5 Robust Stability of DOB-based Control System
For sufficiently large Q-filter bandwidth, robust stability requires stable zero dynamics, a nominally stabilized plant, and a stable gain-variation polynomial. The characteristic roots separate into groups that clarify these stability checks.
5 Robust Stability of DOB-based Control System
- As τ → 0, the characteristic roots split into groups approaching roots of the limiting polynomials, making robust stability analysis more tractable.One group describes behavior when u = udesired; the other governs convergence of u toward udesired.Source: §5 · p. 6
- Robust stability requires N(s) Hurwitz, C(s) internally stabilizing Pn(s), and pf(s) Hurwitz for all gain variations, with sufficiently large Q-filter bandwidth.The conditions are necessary and sufficient except when roots of the limiting polynomials have zero real part, where the root direction is unclear.Source: §5 · p. 6
6 Robust Transient Response
Robust transient response means keeping the real output close to the nominal output from the initial time despite disturbances and uncertainties. Fast convergence toward the desired input helps, but excessive control excursions can undermine that match; saturation can limit them.
6 Robust Transient Response
- Robust transient response requires y(t) to remain close to the nominal output from t = 0 under disturbances and uncertainties.The nominal output is that of the nominal closed-loop system with the same initial condition.Source: §6 · pp. 6–7
- Achieving the nominal output trajectory requires u(t) = udesired(t) from the initial time, so slow convergence can leave the outputs mismatched afterward.The mismatch can persist after u reaches udesired because the output trajectories have already diverged.Source: §6 · p. 7
- Faster convergence of u toward udesired can cause large overshoots or undershoots that make robust transient matching difficult.A brief mismatch in the output’s νth derivative may impair output matching during and after the brief interval.Source: §6 · p. 7
- Saturation limits large transient control excursions, but restricts global stability; a sufficiently wide inactive range can provide semi-global stabilization.The saturation should remain inactive at steady state and may activate briefly during the transient.Source: §6 · p. 7, Figure 2 · p. 7
7 Extensions
The paper describes extensions of DOB analysis to nonlinear and MIMO plants, modified Q-filters, and other controller and implementation settings. These extensions include exact disturbance rejection under a known disturbance model and special discrete-time filter designs for sampling zeros.
7 Extensions
- The analysis extends to SISO nonlinear plants with well-defined relative degree and to MIMO plants.The nonlinear case uses a normal-form representation like the one in the paper’s analysis.Source: §7 · p. 7
- A known disturbance-generating model permits exact disturbance rejection without relying on small τ, using the internal model principle.A sinusoidal disturbance with known frequency is given as a typical example.Source: §7 · p. 7
- Because QA(s) and QB(s) have different roles, they can be refined separately, including for exact rejection or greater measurement-noise reduction.The generating model can be embedded in QA(s) for exact rejection, while a higher-order QB(s) can reduce measurement-noise effects.Source: §7 · p. 8
- The DOB can be combined with state-feedback outer-loop control when the full state is treated as (x, zn), with zn supplied by the DOB.DOB-based control under input saturation has also been studied using an LMI to find controller and DOB parameters.Source: §7 · p. 8
- Fast sampling can introduce unstable sampling zeros when relative degree ν ≥ 3, so discrete-time Q-filters require special design to account for them.The cited design aims to handle unstable sampling zeros while maintaining continuous-time DOB properties.Source: §7 · p. 8
8 Restrictions
The tutorial identifies stability and noise constraints associated with large Q-filter bandwidth, while noting that useful bandwidths are system-dependent and need not always be very large. It also discusses alternative robustness approaches and clarifies that the tutorial is selective rather than exhaustive.
8 Restrictions
- Large Q-filter bandwidth can destabilize plants with relative-degree mismatch or non-minimum-phase dynamics; unknown high-frequency-gain sign and high-frequency measurement noise also constrain the design.The paper notes that remedies exist for some cases, but the non-minimum-phase problem remains open in general.Source: §8 · p. 8
- If Q-filter bandwidth is severely limited, the analysis no longer guarantees the desired steady-state or transient performance and robust stabilization.Suitable bandwidth varies by system and need not be too large in many practical cases; theoretical upper bounds for τ can be conservative, so τ is often tuned by simulation or experiment.Source: §8 · p. 8
- The small-gain theorem offers another robust-stability approach, but its sufficient condition can be conservative and severely limit allowable multiplicative uncertainty.For the typical case 0 ≤ ∥Q∥ ≤ 1, the cited condition restricts the size of uncertainty.Source: §8 · pp. 8–9
- Other approaches use state measurements or estimate disturbances through an exosystem, often alongside established controllers such as sliding-mode or model-predictive control.The paper points to for state-measurement approaches and for exosystem-based observers.Source: §8 · p. 9
- Despite these restrictions, studying DOBs with large-bandwidth Q-filters illustrates each component’s role, shows achievable ideal performance under arbitrarily large parametric uncertainty and disturbances, and derives constructive Q-filter design guides.Source: §8 · p. 9
- The tutorial presents the authors’ perspective rather than an exhaustive survey, so some important DOB contributions may be omitted.It uses a less formal style and directs readers to cited references for more rigorous theorem-proof arguments.Source: §8 · p. 9
Appendix
The appendix explains the transformation to the paper’s system representation and connects its matrix structure to the plant’s zeros. It also identifies the high-frequency gain within the transformed description.
Appendix
- For relative degree ν, the rows cA^i−1 are linearly independent, enabling selection of m additional independent rows and construction of Φ for the coordinate transformation.The appendix uses cA^i−1b = 0 for i = 1, …, ν−1 and cA^ν−1b ≠ 0 to establish the needed independence.Source: Appendix · pp. 10–11
- The appendix verifies the transformed coordinates by relating output derivatives to x and disturbance terms, then setting g = cA^ν−1b, φ = cA^νTa, and ψ = cA^νTb.It defines T−1 = [Ta, Tb] when describing the coordinate representation.Source: Appendix · p. 11
- With no disturbance, conversion from (1) to (3) is the similarity transformation by T, preserving the poles and zeros of P(s); eigenvalues of S are system zeros.Source: Appendix · p. 11
- The transformed high-frequency gain g equals βm, since s^νP(s) has βm as its constant high-frequency term.The appendix relates this to the appearance of βm in the νth derivative of the output.Source: Appendix · p. 11
- A value λ that makes the Rosenbrock system matrix lose rank is a system zero.The appendix identifies the left matrix in the displayed relation as the Rosenbrock system matrix of (3).Source: Appendix · p. 12