Sequential SISO Loop-Shaping Procedure

• Sequential SISO loop-shaping is a method for decomposing control system dynamics into nested SISO loops to achieve enhanced stability, disturbance rejection, and bandwidth maximization.
• Extensions such as LPV scheduling, iterative Youla–Kučera synthesis, and fractional-order compensation refine classical designs to address position-dependent dynamics and multiband disturbances.
• Modern applications demonstrate its versatility through direct digital implementations and decentralized control for high-precision motion and distributed parameter systems.

Sequential single-input, single-output (SISO) loop-shaping is a fundamental methodology for control system synthesis that constructs high-performance feedback systems through a series of nested, frequency-domain loop-shaping operations. The procedure is widely applied in precision motion control, distributed parameter systems, digital control, robust multiband disturbance rejection, and modern stochastic synthesis. It targets stability, disturbance rejection, bandwidth maximization, and robustness by incrementally closing control loops on hierarchically decomposed system dynamics. Extensions include linear-parameter-varying (LPV) scheduling, iterative Youla-Kučera synthesis, LQG/LTR weighting augmentation, fractional-order compensator embedding, algebraic group-theoretic affine feedback synthesis, field-tested distributed control architectures, and direct digital filter design. Each approach refines classical sequential SISO loop-shaping for contemporary applications and advanced specification environments.

1. Classical Sequential SISO Loop-Closing Framework

Industry-standard sequential loop closure decomposes a MIMO motion system into nested SISO loops, exploiting rigid-body and actuator-level decoupling. Inner current or velocity loops are first closed at high bandwidth to linearize and decouple the actuator dynamics, then outer position loops shape rigid-body and flexible-mode behavior. Each SISO loop-shaping stage aims to:

• Achieve the target crossover frequency $\omega_c$, with loop gain $L(s)$ crossing $0$ dB at $\omega_c$ and appropriate phase margin $\phi_m$.
• Suppress low-frequency disturbances by maintaining sensitivity $S(s) = 1/(1+L(s))$ below $M_d$ for $\omega \lesssim \omega_d$.
• Guarantee robustness with minimum gain margin and phase margin constraints on $1+L(s)$.

Practical SISO controllers are constructed by cascading proportional gain, integral action, lead filters for phase boost, and notch filters suppressing resonant modes (Broens et al., 2022). Controller tuning is guided by frequency-domain trade-offs: disturbance rejection, noise attenuation ($T(s)$ bounds at high frequencies), and phase/gain margin requirements.

2. Extensions: LPV Loop-Shaping and Position-Dependent Dynamics

High-precision motion systems such as moving-magnet planar actuators display position-dependent resonance frequencies and mode shapes. LPV sequential loop-shaping generalizes the conventional SLC strategy by modeling the closed-loop plant as: $\dot{x} = A x + B(p)x + B(p)u, \quad y = C(p)x$ where $A$ is constant and $B(p)$, $C(p)$ vary with the scheduling variable $p = q(t)$ (position).

The LPV algorithm selects a grid of frozen positions, designs local SISO controllers for each, and fits polynomial scheduling maps to the notch filter coefficients. The full LPV controller has position-dependent notch and backbone elements, enabling real-time adaptation of controller gains to local resonance properties. This scheduling recovers bandwidth lost to position-dependent uncertainty, achieving much higher crossovers ($\omega_c \approx 175$ Hz vs. $95$ Hz) and dramatically reducing trajectory-tracking error (91.7% moving-average error reduction) (Broens et al., 2022).

3. Iterative Youla–Kučera Sequential Loop-Shaping for Multiband Disturbance Rejection

The iterative Youla–Kučera (YK) framework guarantees closed-loop stability by parameterizing all stabilizing controllers $C(s)$ via $Q(s)$: $C(s) = Q(s)/(1 - P(s) Q(s))$ with $Q(s)$ stable and proper. The sequential algorithm introduces shaping one frequency band at a time via band-pass weighting functions $W_i(s)$, enforcing $\|W_i(s) S_i(s)\|_\infty \leq \gamma_i$ per band. At each step, $Q_k(s)$ is updated by adding a notch term $\Delta Q_k(s)$ designed for band $k$, with order reduction and gain regularization to control numerical robustness and avoid waterbed amplification outside shaped bands.

Performance trade-offs are inherent: deeper, sharper notches (high $g_k$) cause sensitivity peaks elsewhere; balanced-truncation and spectral-factor reduction mitigate controller complexity and ill-conditioning (Hu et al., 19 Aug 2025). The approach provides systematic multi-frequency disturbance rejection, is modular, and guarantees stability via the YK architecture.

4. Sequential SISO Loop-Shaping in LQG/LTR Synthesis via Weighting Augmentation

The LQG/LTR technique enables simultaneous shaping of sensitivity $S(s)$ and controller-noise sensitivity $T_n(s) = K(s)S(s)$ by augmenting the plant with two frequency-shaping filters $W_1(s)$, $W_2(s)$, designed to meet analytic performance bounds at designated low and high-frequency points. The design steps are:

1. Specify magnitude bounds on $|S(i\omega_k)|$ (low-frequency) and $|T_n(i\omega_k)|$ (high-frequency).
2. Encode bounds as equations for weighting filter poles/zeros.
3. Construct augmented plant, solve standard LQG synthesis, obtain dynamic controller.
4. Adjust LQR cost ratio $\rho$ to perform loop-transfer-recovery: LQG open loop converges to Kalman filter open loop, guaranteeing specified shape is achieved.

A detailed example on DC-motor torque control demonstrates the translation of bounds into filter equations, solution for weighting parameters, loop shape recovery, and iterative retuning to optimize sensitivity peaks and robustness margins (Mahinzaeim et al., 20 Jul 2025). The procedure is analytic, fully systematic, and requires no heuristics or ad-hoc guesswork.

5. Fractional-Order Partial Cancellation Embedded in Sequential Loop-Shaping

Non-minimum phase zeros and unstable poles restrict classical SISO loop-shaping by imposing harsh phase deficits. Fractional-order compensators split the integer-order zero/pole into a product of fractional-order pseudo zeros/poles with chosen orders $\alpha_k$ summing to unity: $(s - z) = \prod_{k=1}^N (s - z_k)^{\alpha_k}$ Magnitude and phase response of each pseudo term is explicitly computable. The design procedure selects splitting order ($\alpha$ or $N$) to control phase margin impact at crossover, embeds the compensator in SISO loop-shaping, retunes backbone controller, and verifies robustness.

Numerical examples show that fractional-order compensation reduces step-response undershoot and steepens open-loop magnitude at crossover, thereby increasing achievable phase margin. Extensions to complex zeros and implicit FO compensators are included (Voß et al., 2022). This methodological augmentation provides a rigorous route to overcoming fundamental loop-shaping restrictions dictated by right-half-plane zeros/poles.

6. Sequential SISO Loop-Shaping for Distributed and Decentralized Systems

In spatially distributed systems—e.g., multi-pool irrigation channels—the sequential SISO loop-shaping paradigm applies with adaptations for distributed parameter models and recursive feasibility. Each controller is designed for its local SISO channel by:

1. Modeling local and interconnection dynamics via transfer functions derived from linearized PDEs.
2. Defining local error signals (e.g., weighted difference of downstream pool levels).
3. Sequentially closing SISO loops following a chosen permutation order, at each step forming the partially closed-loop transfer function incorporating prior closures.
4. Designing PI-type compensators per SISO plant, tuning crossover and phase margin.

Inductive arguments guarantee recursive feasibility (each plant retains a single integrator at each step), stability (Nyquist/phase margin verified for each sub-loop), and disturbance rejection (steady-state error vanishes for constant disturbances). Implementation on large-scale irrigation channels demonstrated robustness in both simulation and multi-kilometer field trials, with inter-pool level errors constrained below 5 cm over days of supply manipulation (Strecker et al., 24 Dec 2025).

7. Direct Digital Loop-Shaping in Sampled Systems

Digital sequential SISO loop-shaping employs general linear models (GLMs) for lag/lead compensator synthesis in the $z$-domain. Weighted least-squares estimation computes IIR filter coefficients for polynomial or sinusoidal basis functions, enabling precise shaping of low-frequency or arbitrary-frequency regions.

The complete design procedure: identify plant pulse-transfer function, choose lag/lead, select basis and order, compute Gram matrix, projection vectors, and synthesis vector, assemble $H(z)$, extract difference equation, insert into digital controller, and tune for gain/delay margins. Polynomial basis filters shape the near-zero region, sinusoidal bases target exact frequencies (e.g., resonances), and the approach is more robust to noise and delay than PID or linear state-space controllers (Kennedy, 2014).

Frequency-domain metrics confirm that GLM-based IIR compensators provide superior roll-off, phase boost, and noise/disturbance rejection compared to conventional PI/PID, with empirical and simulated step responses closely matching analytic predictions.

• LPV sequential loop closing for high-precision motion systems (Broens et al., 2022)
• Iterative Youla-Kucera Loop Shaping For Precision Motion Control (Hu et al., 19 Aug 2025)
• An approach to the LQG/LTR design problem with specifications for finite-dimensional SISO control systems (Mahinzaeim et al., 20 Jul 2025)
• Fractional-Order Partial Cancellation of Integer-Order Poles and Zeros (Voß et al., 2022)
• On Structural Non-commutativity in Affine Feedback of SISO Nonlinear Systems (S., 2024)
• Decentralized water-level balancing for irrigation channels in storage critical operations (Strecker et al., 24 Dec 2025)
• Direct Digital Design of Loop-Shaping Filters for Sampled Control Systems (Kennedy, 2014)