A Guide to Design Disturbance Observer

1. INTRODUCTION

• A DOB can estimate disturbances precisely within its low-pass-filter bandwidth, but robustness and noise limit how freely that bandwidth can be increased.The introduction also notes that higher bandwidth is desired to estimate disturbances over a wider frequency range.Source: §1 · pp. 2–3
• The relationship between DOB robustness and the dynamics of its low-pass filter and nominal plant had not been clearly reported.The introduction identifies this as a gap in systematic analysis and design methods.Source: §1 · pp. 2–3
• RHP plant zeros and time delay limit DOB bandwidth, whereas RHP poles impose a lower bandwidth bound for good robustness.The paper uses Bode and Poisson integral formulas to derive these constraints analytically.Source: §1 · p. 3
• Higher-order DOBs improve system performance by using bandwidth more effectively, but their bandwidth constraints become more severe and robustness deteriorates.The introduction presents this as a performance–robustness trade-off.Source: §1 · p. 4
• The proposed design methods use the derived bandwidth constraints, an approximate minimum-phase nominal model for non-minimum-phase plants, and a performance controller for the conventional two-degree-of-freedom structure.The approximate model addresses the internal-stability problem associated with non-minimum-phase plant zeros.Source: §1 · p. 4

2. DISTURBANCE OBSERVER

• The diagram identifies the DOB low-pass filter and outer-loop controller, alongside reference, disturbance, and noise inputs.Estimated disturbances include external disturbances and system uncertainties.Source: §2 · p. 4
• The figure is used to derive the open-loop, sensitivity, and co-sensitivity transfer functions.The section names these functions L, S, and T, respectively.Source: §2 · pp. 4–5
• The figure depicts a two-degree-of-freedom DOB-based robust-control system.The diagram distinguishes uncertain and nominal plant models.Source: §2 · p. 4, Figure 1 · p. 5

3.1. Minimum‐phase Plant

• The inner loop is strictly robust when Δ > 0; when Δ < 0, robustness can be guaranteed over a wide range of DOB bandwidths.HODOB use and/or plant delay limits the bandwidth over which robustness is guaranteed, even if Δ > 0.Source: §3.1 · p. 7
• For HODOBs, Bode-integral analysis bounds the sensitivity-integral range under an assumed low-pass-filter decay condition.This analysis is used to derive robustness constraints.Source: §3.1 · pp. 8–9
• HODOB design inequalities for the low-pass filter are proposed to provide good robustness while meeting a predefined performance criterion.The section contrasts this with a DOB, which can be robust over a wide frequency range but has performance limited by its dynamics.Source: §3.1 · p. 10
• Increasing HODOB order or bandwidth raises the sensitivity peak, tightening the bandwidth tradeoff.The section also notes that increasing order narrows the frequency gap between the stated sensitivity bounds.Source: §3.1 · p. 11
• For a minimum-phase plant, the sensitivity bound is conservative because it assumes sectionally constant sensitivity; more realistic bounds can lessen this conservatism.Source: §3.1 · p. 11

3.2. Plant with Time‐Delay

• For a plant with time delay, the proposed design tool constrains DOB bandwidth to obtain good robustness and predefined performance, but its sensitivity bounds are conservative.The theorem’s design inequalities account for the bandwidth limitation due to delay; the authors note that the sensitivity bounds are not realistic.Source: §3.2 · pp. 12–13
• The sensitivity peak increases as DOB order increases in the time-delay case.Source: §3.2 · p. 13

3.3.1. Performance Limitations

• RHP zeros cause undershoot and RHP poles cause overshoot in closed-loop step responses, motivating performance constraints on bandwidth.The passage identifies the bandwidth and RHP zero and pole locations as relevant quantities for the stated performance inequalities.Source: §3.3.1 · p. 14

3.3.2. The Poisson's Integral Formulas

• Poisson integral formulas are used to derive DOB bandwidth constraints analytically when a plant has RHP poles and/or zeros.Source: §3.3.2 · p. 14

3.3.3 Plant with RHP Zero(s)

• For plants with RHP zeros, Theorem 3 gives DOB low-pass-filter constraints for good robustness and predefined performance under stated sensitivity and co-sensitivity bounds.The constraints are derived for a plant with an RHP zero using an approximate nominal plant model with a stable inverse.Source: §3.3.3 · p. 16
• For plants with RHP zeros, the proposed tools target good robustness and predefined performance, using co-sensitivity bounds to control high-frequency performance and improve bandwidth bounds.Source: §3.3.3 · p. 17

• RHP zeros impose an upper bound, while unstable plants with RHP poles impose a lower bound on DOB bandwidth for good robustness.Source: §3.3.3 · p. 17, §3.3.4 · p. 18
• Both design constraints remain conservative: the RHP-zero theorem uses unrealistic sensitivity and co-sensitivity bounds, and the RHP-pole constraint also suffers from conservatism.Source: §3.3.3 · p. 17, §3.3.4 · p. 18

3.3.4. Plant with RHP Pole(s)

• For plants with RHP poles, the design tool requires a stabilizing outer-loop controller because the inner loop is always unstable.Source: §3.3.4 · pp. 17–18

4.1. Minimum‐Phase Plant:

• For minimum-phase plants, increasing DOB order improves bandwidth use and noise suppression, but deteriorates robustness and makes bandwidth constraints more severe.The first-order design should use bandwidth below 100 rad/s for robust stability.Source: §4.1 · pp. 19–20

4.2 Plant with Time‐Delay

• For plants with a time delay, the practical DOB bandwidth bound is 70 rad/s, substantially below the derived bound because of conservatism.For the second-order case, the derived bound is 500 rad/s.Source: §4.2 · p. 22

4.3. Plant with a RHP Zero

• The bounds are 12 ≤ W_B ≤ 24 and ω_β ≤ 15 rad/s.The other stated bounds are 6 ≤ W_B ≤ 55 and ω_β ≤ 35 rad/s.Source: §4.3 · p. 23

4.4. Plant with a RHP Pole:

• For an RHP-pole plant, increasing DOB bandwidth improves robustness, while high-frequency disturbances are estimated imprecisely and weaken robustness.Bandwidth selection therefore trades off robustness against noise response.Source: §4.4 · p. 24

4.5. Comments on Conservatism:

• More realistic sensitivity bounds raise the time-delay example’s DOB bandwidth limit from 70 to 95 rad/s, reducing conservatism.The paper identifies the gap between the approximate bound and actual sensitivity response as a source of conservatism.Source: Figure 9 · p. 25, §4.5 · p. 26

• The Bode and Poisson integral formulas are used to derive DOB bandwidth constraints analytically, and the paper proposes analysis and design tools.Source: §5 · p. 26
• More realistic sensitivity/co-sensitivity bounds can reduce conservatism, but increase analysis complexity.Source: §5 · p. 26
• The authors report that conservatism is not severe in their experience and that the tools provide insight into DOB design constraints.Source: §5 · pp. 26–27

• Table 1 summarizes the design constraints of a DOB.Source: Table Caption List · p. 30

• The figure list includes diagrams of DOB-based control systems, including a two-degrees-of-freedom system and cases with delay or an unstable plant.Source: Figure 1 · p. 33, Figure 2 · p. 34, Figure 3 · p. 35
• Several figures show sensitivity and co-sensitivity frequency responses, including inner-loop responses and an outer-loop response.Source: Figure 4 · p. 36, Figure 5 · p. 37, Figure 6 · p. 38
• The captions include a step-response figure for an unstable plant controlled by the proposed robust controller, alongside a table summarizing DOB design constraints.Source: Table 1 · p. 32, Figure 9 · p. 41