Two-Mode Loop Shaping in Control Systems

• Two-mode loop shaping is a control design approach that partitions feedback loops into distinct dynamic modes to manage nonlinearities and switching behaviors.
• It employs separate methodologies to address first-harmonic linear effects and higher-harmonic disturbances, ensuring robust performance across diverse conditions.
• Applications span hybrid robotics, wind turbine regulation, and oscillatory systems where dual-mode analysis improves tracking and disturbance rejection.

Two-mode loop shaping refers to a suite of control design methodologies that explicitly recognize, model, and tune feedback loops for systems exhibiting distinct dynamic regimes or multiple operational objectives—often by partitioning the control synthesis or synthesis targets into two distinct dynamical “modes”. The two-mode concept is central in advanced precision control, reset control, hybrid/impedance robotics, wind energy, and oscillatory dynamics, where linear or single-mode loop-shaping is insufficient to guarantee robust performance and predict closed-loop behavior across different task phases or system nonlinearities.

1. Theoretical Foundations and Motivation

Traditional loop shaping in control engineering is based on linear, single-mode feedback design, targeting frequency-domain attenuation, robustness, and disturbance rejection via open-loop shaping and sensitivity analysis. However, many real-world systems are inherently multimodal: they switch between regimes (e.g., free motion versus contact, or power maximization versus power tracking), produce non-sinusoidal responses due to nonlinear or hybrid elements, or require frequency-selective shaping due to multi-band disturbance scenarios.

Two-mode loop shaping generalizes linear Bode and sensitivity-based design by:

• Identifying two salient operational or dynamic regimes (modes), e.g., first-harmonic ("quasi-linear") mode and higher-harmonic ("disturbance") mode in reset systems (Saikumar et al., 2020), or stiff (free motion) vs. compliant (contact) impedance modes in hybrid robotics (Ruderman, 2024).
• Providing explicit design and analysis tools for both modes, linking open-loop design (classical or generalized, e.g., describing functions, Youla-Kucera parameterization, or weighting filters) to closed-loop performance and robustness guarantees across all modes.

2. Formulations in Reset Control and Nonlinear Frequency-Domain Methods

In reset control, the need for two-mode loop shaping arises due to the fundamental generation of higher harmonics (via resets) that cannot be captured by standard single-harmonic describing function (DF) analysis. The higher-order sinusoidal-input describing functions (HOSIDFs), $H_n(\omega)$ (for odd $n\geq3$), extend the DF formalism to characterize the magnitude and phase of all harmonics generated by the reset element $\mathcal{R}$:

• The open-loop $n$th-harmonic loop gain is $L_n(\omega) = H_n(\omega)\,\mathcal{P}(jn\omega)$.
• Only the first-harmonic loop, $L_1(\omega)=H_1(\omega)\mathcal{P}(j\omega)$, behaves quasi-linearly, admitting classic loop-shaping (gain/phase margin, crossover, etc.) (Saikumar et al., 2020).
• Higher-harmonic loops demand a second mode of analysis, as they manifest as exogenous disturbances acting through the base-linear loop (with the reset disabled) and must be attenuated/addressed via design of both the reset law and the base linear gains.

The closed-loop sensitivities for reference/disturbance/noise incorporate both modes: $$\begin{aligned} S_1(\omega) &= 1/(1+L_1(\omega)), \ S_n(\omega) &= -L_n(\omega)\,S_{l,bl}(n\omega)\,|S_1(\omega)| \exp(j n \arg S_1(\omega)),\ n > 1. \end{aligned}$$ This establishes that higher harmonics (mode 2) are filtered through the base-linear loop, and their impact can be reduced via judicious choice of reset law parameters (minimizing $|H_n(\omega)|$) and ensuring adequate base-linear loop phase margin to suppress amplification.

3. Hybrid and Mode-Switching Control: Two-Mode Structures in Robotics and Energy Systems

A structurally distinct application appears in hybrid, mode-switching, or gain-scheduled control architectures, as exemplified by compliant robot control and wind turbine regulation.

• In hybrid motion control with contact transitions, loop shaping is performed separately in the free-motion (mode 1, "stiff", for precise reference tracking) and contact (mode 2, "compliant/impedance") regimes. The two mode-specific controllers $C_s$ (stiff) and $C_v$ or $C_{ve}$ (viscous/viscoelastic) are designed using classical loop-shaping and transferred via a detection/switching logic based on actuator output bounds (sensor-free strategy) (Ruderman, 2024).
  • The free-motion mode targets high bandwidth ($\sim$100 rad/s), high phase margin (≥60°), and disturbance rejection.
  • Upon contact (detected by deviation in actuator effort), control switches to an impedance-matched loop shaped to provide safe compliance—achieved via dynamic reconfiguration without force sensing.
  • Stability is maintained either by slow switching (dwell-time) and verified via Lyapunov-based hybrid proofs.
• In wind turbine power control, two-mode loop shaping involves $\mathcal{H}_\infty$ synthesis for both power-maximizing (region 2) and active power-tracking (region 3) operating regimes. Two separate shaped feedback loops, with independently tuned weighting filters and controllers $K_1(s)$ and $K_2(s)$, are switched according to operating point, with bumpless transfer strategies eliminating transients at transitions (Grapentin et al., 10 Mar 2025).

4. Frequency-Selective and Multi-Band Two-Mode Shaping

Two-mode loop shaping also refers to frequency-selective disturbance rejection, specifically in systems with multi-band resonances or vibration sources. Using the iterative Youla-Kucera (YK) parameterization (Hu et al., 19 Aug 2025):

• The controller design is formulated as an iterative notch-filtering process, synthesizing each mode (frequency band) sequentially via low-order, numerically well-conditioned Q-filters.
• Each stage implements a band-limited sensitivity notch at the target frequency, with trade-offs between depths (attenuation), bandwidths (via filter root locus), and water-bed effects (Bode integral constraints).
• The two-mode case specializes the iterative approach to exactly two notches (two Q-filters, each second order); this maintains tractable controller orders and robust frequency response while allowing explicit tuning of trade-offs (attenuation at resonance vs. amplification elsewhere).

5. Oscillatory Two-Mode Systems: Mixed Feedback and Limit Cycle Tuning

Systems that intrinsically generate oscillations leverage two-mode loop shaping by interpolating between negative and positive feedback regimes. In "mixed feedback" frameworks, the relative strength of positive vs. negative feedback is parameterized (via balance $\beta$), and oscillation characteristics are controlled through explicit loop-phase and magnitude relations (Che et al., 2021):

• Harmonic-balance regime (mode 1): frequency set by the phase condition on the linearized open loop (with mostly negative feedback), amplitude by magnitude requirement, and robustness via dominance theory (p=2).
• Fast/slow relaxation regime (mode 2): strong positive feedback induces slow drift bounded by the reset or saturation nonlinearity (limit cycle mechanics), with period predicted via singular perturbation analysis.
• Explicit two-mode tuning protocols select $(\beta,\,k)$ to govern which oscillatory regime (harmonic vs. relaxation) dominates, supporting applications from pattern generation to mechanical resonance exploitation.

6. Design Guidelines, Performance Trade-offs, and Validation

Key guidelines and observed trade-offs across two-mode loop shaping illustrate both versatility and limitations:

• The base linear or first-harmonic mode remains amenable to standard linear loop-shaping logic; crossovers, bandwidth, and phase margins are shaped as in conventional Bode design.
• The higher-mode (harmonic, compliance, frequency-selective, etc.) loop requires additional attention to sensitivity peaks, dynamic reconfiguration logic (in hybrid/impedance systems), or water-bed effects (in iterative multi-band shaping). Overly aggressive shaping of one mode can degrade performance in the other (Bode integral/water-bed tradeoff).
• Stability in the presence of mode transitions is certified via nonlinear criteria (e.g., $H_\beta$ condition in reset systems (Saikumar et al., 2020), multiple Lyapunov functions in hybrid control (Ruderman, 2024)), dominance theory in oscillatory mixed feedback (Che et al., 2021), or robust coprime-factor bounds in $\mathcal{H}_\infty$ designs (Grapentin et al., 10 Mar 2025).
• Experimental results in precision positioning, compliant interaction, HDD servo, and wind turbine testbeds confirm the necessity and predictive accuracy of two-mode (or more generally, multi-mode) loop shaping. Higher-order or multimodal analyses (HOSIDF, iterative YK, multiple weighting filters) consistently outperform classical single-mode/TI (time-invariant) predictions by wide margins.

7. Impact, Applications, and Future Directions

Two-mode loop shaping now underpins a range of advanced control strategies:

• Nonlinear/Reset Control: enabling high-bandwidth motion control beyond classical PI/PID limits (Saikumar et al., 2020).
• Hybrid and Robot Impedance Control: delivering safe, high-precision interaction in sensor-free, dynamically switching environments (Ruderman, 2024).
• Power and Motion Systems: robust, mode-specific control with bumpless transition logic to address distinct operational needs (e.g., energy maximization vs. tracking/grid-support) (Grapentin et al., 10 Mar 2025).
• Frequency-Selective Rejection: supporting dual-mode vibration attenuation in precision mechatronics (Hu et al., 19 Aug 2025).
• Oscillatory Mechanics: facilitating robust CPG-like rhythms for locomotion and force generation (Che et al., 2021).

Current research advances focus on systematic generalization to N-mode architectures, improved theoretical guarantees for hybrid and switching stability, and tighter integration with learning-adaptive frameworks for real-time mode-detection and shape tuning.

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References:

• Loop-shaping for reset control systems -- A higher-order sinusoidal-input describing functions approach (Saikumar et al., 2020)
• Loop Shaping of Hybrid Motion Control with Contact Transition (Ruderman, 2024)
• Iterative Youla-Kucera Loop Shaping For Precision Motion Control (Hu et al., 19 Aug 2025)
• $\mathcal{H}_\infty$ Loop-shaping for Power Tracking Control of Wind Turbines (Grapentin et al., 10 Mar 2025)
• Shaping oscillations via mixed feedback (Che et al., 2021)