Compendium

Control Algorithms — Family Index

19 min read

Control Algorithms — Family Index

A taxonomic survey of feedback-control algorithms used on robots. Scope: continuous-time and discrete-time regulators acting on actuator commands. Estimators (Kalman family, particle filters, factor graphs) are covered separately in bayesian-estimation; planning (RRT, A*, CHOMP) is in path-planning. The line between control and learning is intentionally blurred at the bottom — diffusion policies and RL controllers are first-class citizens here.

1. At a glance — the six pillars

Every robot control stack picks from these families and stacks them in cascades (inner loops fast and simple, outer loops slow and clever).

  1. Classical — PID and its variants. >95 % of all industrial loops by count. Sufficient when the plant is well-damped, near-linear, slowly time-varying, and disturbance bandwidth is below the bandwidth you can afford.
  2. Model-based linear — pole placement, LQR, LQG, H∞, state-space servo. Requires a linearised plant model around an operating point. Closed-form gains; provable stability and robustness margins.
  3. Nonlinear / robust — sliding-mode (SMC), super-twisting, backstepping, feedback linearisation (computed-torque), passivity-based, internal-model control (IMC), repetitive control. Used when nonlinearity, uncertainty, or interaction physics dominate.
  4. Adaptive — MIT-rule, MRAC, L1 adaptive, self-tuning regulator, gain-scheduling, adaptive PID. Used when plant parameters drift (payload mass changes, fluid viscosity changes, vehicle dry-mass during fuel burn).
  5. Optimisation-based / predictive — linear MPC, nonlinear MPC (NMPC), real-time iteration MPC, iterative LQR (iLQR), DDP, iterative learning control (ILC), MPPI. Pays online compute for explicit constraint handling and multi-step lookahead.
  6. Learning-based — RL (PPO, SAC, TD3, MuZero, Dreamer), imitation learning (BC, DAgger, GAIL), diffusion policies, foundation-model VLAs (RT-2, OpenVLA, PaLM-E). Used when the cost-to-model is higher than the cost-to-data, or when behaviour cloning beats hand-engineering.

A seventh “layer” — the estimator (Kalman / EKF / UKF / particle / factor-graph) — sits beside the controller and is treated separately. See bayesian-estimation.

2. PID family — the workhorse

2.1 Topologies

  • Pure P — proportional only; nonzero steady-state error to step disturbance. Common in the velocity loop of a cascade because the outer position integrator handles the offset.
  • PI — adds integral; eliminates steady-state error. Default for motor current loops, flow loops, temperature loops.
  • PD — proportional + derivative. Mechanical-position loops where the load is double-integrator (mass) — D term provides damping. Always with derivative filter.
  • PID — the textbook form. ISA “ideal” form: u = Kp(e + (1/Ti) integral e dt + Td de/dt).
  • PI-D — derivative on measurement only (not on error). Avoids derivative kicks on setpoint changes.
  • I-PD — proportional and derivative on measurement only; integral on error. Used when setpoint kicks must be suppressed completely.

2.2 Anti-windup

  • Clamping / conditional integration — most common. Freeze the integrator when the actuator is saturated and the error has the same sign as the integrator.
  • Back-calculation — Hippe / Astrom-Wittenmark; the integrator is driven by the actuator-saturation error through a separate tracking-time-constant Tt. Smoother recovery from saturation than clamping.
  • Tracking anti-windup — generalisation of back-calculation; used when the loop has feed-forward terms.
  • Observer-based anti-windup — model the saturation as an additive disturbance and let an observer compensate.

2.3 Gain scheduling

Lookup-table interpolation of (Kp, Ki, Kd) against operating-point variables. Standard in aerospace flight control where gains schedule on Mach number, altitude, and angle-of-attack; in process control where they schedule on production rate; in cobots where they schedule on payload mass estimate.

2.4 PID variants

  • Cascade PID — inner velocity loop (fast, ~kHz) wrapped by outer position loop (slow, ~100 Hz). Inner loop linearises the plant for the outer.
  • Feed-forward + PID — model-based open-loop term computes nominal actuator command from desired trajectory; PID only corrects model mismatch.
  • 2-DoF PID — two filters, one on the setpoint path and one on the feedback path, weighted independently (Astrom 1995). Decouples tracking from disturbance rejection.

2.5 Tuning

  • Ziegler-Nichols (Ziegler + Nichols 1942) — ultimate-gain or step-response. Aggressive; oscillatory.
  • Cohen-Coon (1953) — better for systems with significant dead time.
  • Lambda tuning (IMC tuning) — Rivera + Morari 1986; specify desired closed-loop time constant lambda. Conservative; predictable.
  • Astrom-Hagglund relay autotuning (1984) — relay-feedback test extracts ultimate gain + ultimate period without operator intervention. Common in industrial controllers (Honeywell, Yokogawa, ABB).
  • System-identification derived — fit ARX / state-space model, design PID by pole-placement.

2.6 Implementation

  • Discretisation — backward Euler (simple, biased), Tustin / bilinear (preserves stability), trapezoidal integration (standard).
  • Integer / fixed-point arithmetic — common on motor-drive MCUs running at 20-50 kHz current-loop.
  • Derivative filter — first-order low-pass on the D term, N typically 8-20, to prevent noise amplification.

3. State-space linear

3.1 Pole placement / Ackermann

SISO with full-state feedback. Ackermann’s formula (1972) gives K such that the closed-loop A - BK has the chosen eigenvalues. Multi-input case uses Bass-Gura or numerical place().

3.2 LQR — Linear Quadratic Regulator

Kalman 1960. Minimise J = integral (x^T Q x + u^T R u) dt subject to xdot = Ax + Bu. Solution: u = -Kx where K = R^-1 B^T P and P solves the continuous-time algebraic Riccati equation (CARE). Provable infinite gain margin and 60-degree phase margin in the SISO case. Tuning is shifted from “where do I want poles?” to “how do I weight state vs control effort?” — often more intuitive.

3.3 LQG — Linear-Quadratic-Gaussian

LQR + Kalman filter. Separation principle: design the regulator assuming full-state, design the Kalman filter assuming you want state estimates, combine them. Robustness margins of LQR are not preserved by LQG — LQG/LTR (loop-transfer recovery, Doyle + Stein 1979) recovers them.

3.4 DLQR

Discrete-time LQR; solves discrete algebraic Riccati equation (DARE). Practical implementation. MATLAB dlqr(), Python scipy.linalg.solve_discrete_are.

3.5 iLQR / DDP

Iterative LQR (Todorov + Tassa 2007); differential dynamic programming (Mayne 1966, Jacobson + Mayne 1970). Nonlinear trajectory optimisation via repeated quadratic approximation around a nominal trajectory. Covered under MPC for trajectory-optimisation usage.

4. Robust + nonlinear

4.1 H∞ and μ-synthesis

Doyle, Glover, Khargonekar, Francis 1989 — minimax synthesis bounding the worst-case norm of the closed-loop sensitivity transfer matrix under norm-bounded uncertainty. μ-synthesis (Doyle 1985) extends to structured uncertainty via D-K iteration. Used in aerospace flight-control (X-29, F-117), high-precision optics, wafer-stages.

4.2 Sliding-mode control (SMC)

Utkin 1977; Edwards + Spurgeon 1998 textbook. Discontinuous switching of the control law on a sliding surface s(x) = 0 chosen so that motion confined to the surface is stable. Theoretically invariant to matched uncertainty. Practical drawback: chattering. Boundary-layer modification (Slotine + Li 1991) trades robustness for smoothness. Super-twisting (Levant 1993) is a second-order SMC that drives both s and sdot to zero with continuous control — much less chattering. Higher-order SMC generalises further.

4.3 Backstepping

Krstic, Kanellakopoulos, Kokotovic 1992-1995 (“Nonlinear and Adaptive Control Design”). Recursive Lyapunov-based design for strict-feedback / pure-feedback systems. Each step picks a virtual control that stabilises one subsystem; final step gives the real control. Used in adaptive flight control, marine vehicles, induction motors.

4.4 Feedback linearisation (computed-torque)

Exact algebraic cancellation of nonlinearity to leave a linear residue, then close a linear loop around the residue. For robot manipulators: tau = M(q) qddot_d + C(q,qdot) qdot + g(q) + tau_PD. Inverse-dynamics torque control. Sensitive to model error in M, C, g — robustness recovered via sliding-mode or adaptive overlay.

4.5 Passivity-based control

Ortega + Spong, “Adaptive motion control of rigid robots” 1989; “Putting energy back in control” (Ortega et al. 2001). Energy-shaping and damping-injection. Foundation of impedance control. Guarantees stability against any passive environment without exact model.

4.6 Internal model control (IMC)

Garcia + Morari 1982. Controller contains an explicit plant model; performance and robustness tuned by a single filter time constant. Strong industrial adoption in process control; the Lambda-tuning approach to PID is an IMC interpretation.

4.7 Repetitive control

Hara, Yamamoto, Omata 1988 (Inoue 1981 original). Exploits internal-model principle for periodic disturbances: include a delay of one period in the controller and the loop will reject any disturbance at the fundamental + harmonics. Used in optical-disc tracking, semiconductor scanners, helicopter rotor vibration cancellation.

5. Adaptive

5.1 MIT rule (the original MRAC)

Whitaker 1958, MIT Instrumentation Lab. Gradient descent on the squared tracking-error with respect to controller parameters. Not Lyapunov-stable in general; instability was discovered in the X-15 hypersonic program in the 1960s, which catalysed the modern theory.

5.2 MRAC — Model Reference Adaptive Control

Lyapunov-redesign (Parks 1966) made adaptation provably stable. Modern reference: Narendra + Annaswamy “Stable Adaptive Systems” 1989; Ioannou + Sun “Robust Adaptive Control” 1996. Plant tracks a reference model; adaptation law is derived via Lyapunov + Barbalat’s lemma. Sigma-modification, e-modification, projection operator added for robustness.

5.3 L1 adaptive control

Cao + Hovakimyan 2010 (“L1 Adaptive Control Theory”). Decouples adaptation speed from robustness via a low-pass filter on the adaptive control signal. Allows very fast adaptation with bounded transient response and guaranteed margins. Flight-tested on NASA AirSTAR sub-scale aircraft, GTM, Calspan Learjet.

5.4 Self-tuning regulator (STR)

Astrom + Wittenmark 1973. Two-step online: (1) recursive system identification (RLS or similar); (2) recompute controller (pole-placement, minimum-variance, LQG). Indirect adaptive control. Industrial: Novatune (Astrom + Hagglund), Foxboro EXACT.

5.5 Adaptive PID

Practical hybrid: keep PID structure, schedule or adapt the gains either through (a) online ID + Lambda tuning, (b) fuzzy adaptation, (c) gradient on a performance index, or (d) neural-network gain adjustment.

6. Impedance and admittance

The robot-environment-interaction layer.

6.1 Impedance control

Hogan 1985 — three-paper series “Impedance Control: An Approach to Manipulation”. The robot is rendered as a virtual mass-spring-damper from the environment’s perspective. Causality: position-in, force-out. Realised with torque-controlled or backdrivable actuators. Foundation of safe human-robot contact.

6.2 Admittance control

Newman 1990s (DeSchutter, Whitney earlier). Opposite causality: measured force in, commanded position out. Used on position-controlled industrial arms with a wrist F/T sensor (FANUC, KUKA KR series). Less stiff coupling than impedance, harder to push around without instability.

6.3 Hybrid position/force

Mason 1981 (constraint frames); Raibert + Craig 1981 (selection matrix). Decompose the Cartesian directions into orthogonal subspaces — position-controlled in some, force-controlled in others. Classic application: peg-in-hole, surface following.

6.4 Variable impedance

Albu-Schäffer at DLR (DLR LWR, KUKA LBR iiwa). The stiffness and damping matrices are not fixed but selected per task or even per timestep. Variable-impedance learning (Calinon, Buchli, Khatib) ties this to imitation learning.

7. Optimisation-based and predictive

7.1 Linear MPC

Cutler + Ramaker 1979 (Shell, “Dynamic Matrix Control” DMC); Richalet 1978 (Adersa, IDCOM). Solve a QP at every control step over a receding finite horizon with explicit constraints on states and inputs. Industrial rollout 1980s-1990s: Honeywell RMPCT, AspenTech DMCplus, ABB Predict & Control / OptimizeIT. The bulk of refinery / chemicals advanced-process-control today.

7.2 Nonlinear MPC (NMPC)

Sequential quadratic programming over nonlinear dynamics. Frameworks: ACADO / acados (Diehl), CasADi (Andersson 2019), FORCES Pro, Pyomo / IPOPT. Covered in detail in mpc-for-robots.

7.3 Real-time iteration (RTI)

Diehl 2005 — single Newton-type iteration per sampling step on a warm-started QP. Trades suboptimality for guaranteed completion within the control period. Used on quadrotor MPC, autonomous-driving MPC.

7.4 iLQR / DDP

Mayne 1966; Jacobson + Mayne 1970; Tassa et al. 2012 (MuJoCo demos). Locally optimal trajectories via backward Riccati-like sweep + forward rollout. Underlies CMU’s Aerobat work, MIT Cheetah’s whole-body planner, Boston Dynamics’ offline trajectory generation (publicly described in their conference talks).

7.5 Iterative Learning Control (ILC)

Arimoto, Kawamura, Miyazaki 1984. For tasks repeated identically; the controller updates the feed-forward command between iterations based on the previous iteration’s error. Provable monotonic error reduction under mild conditions. Used in semiconductor wafer-scan, robotic welding, batch processes.

7.6 MPPI — Model Predictive Path Integral

Williams et al. 2017 (Georgia Tech). Sampling-based MPC for highly nonlinear or non-differentiable dynamics. Used in autonomous-rally car control (AutoRally), aggressive drone flight, off-road wheeled vehicles.

8. Force and interaction

  • Direct force control — measured force versus reference; closed-loop force tracking. Used when contact is guaranteed and stable. Common loop bandwidth 50-200 Hz limited by F/T sensor noise + contact dynamics.
  • Indirect force control via impedance — more common in practice. The robot enforces a relationship between force and motion rather than tracking a force trajectory.
  • Series-elastic actuator (SEA) torque control — Pratt + Williamson 1995 (MIT Leg Lab). A spring between motor and load turns force control into a position-control problem (Hooke’s law: F = k delta x). Used on Baxter, Sawyer, ATRIAS, Cassie, Digit.
  • Cartesian impedance with null-space stiffness — on 7-DoF redundant arms (KUKA LBR iiwa, Franka Panda) the Cartesian task can be commanded with stiffness while the null-space (elbow swivel) is independently posture-regulated. Standard formulation: Khatib’s operational-space framework (Khatib 1987).

9. Reinforcement learning for control

Robot RL has moved from research curiosity (2015) to production capability (2024-2026). The chart of important algorithms:

  • Tabular Q-learning / SARSA (Watkins 1989; Sutton 1988) — legacy; discrete-state-action only.
  • DQN — Deep Q-Network (Mnih et al., DeepMind, 2013 + Nature 2015) — value-based; convolutional Q-net; experience replay + target network. Atari benchmark.
  • TRPO + PPO — Trust-Region / Proximal Policy Optimisation (Schulman et al. 2015 / 2017) — on-policy policy-gradient; clipped surrogate objective. The workhorse for continuous-action robot RL (locomotion, manipulation). Used by OpenAI Dactyl, ANYmal, MIT Mini-Cheetah, sim-to-real quadruped pipelines.
  • DDPG (Lillicrap et al. 2015) — deterministic policy-gradient; off-policy; actor-critic. First to scale to continuous robot control.
  • TD3 — Twin Delayed DDPG (Fujimoto et al. 2018) — addresses DDPG’s overestimation bias via twin critics + target-policy smoothing + delayed policy updates.
  • SAC — Soft Actor-Critic (Haarnoja et al. 2018) — maximum-entropy RL; off-policy; sample-efficient. Strong real-robot success (Berkeley quadruped + arm).
  • MuZero (Schrittwieser et al., DeepMind 2019) — model-based + MCTS tree search; learns its own dynamics model. Generalisation of AlphaZero.
  • Dreamer / DreamerV2 / DreamerV3 (Hafner et al. 2020 / 2021 / 2023) — model-based RL with a learned world model in latent space; train policy by imagined rollouts. DreamerV3 (2023) solved Atari, DMControl, Crafter with the same hyperparameters and scaled to Minecraft diamond.
  • Decision Transformer (Chen, Lu et al. 2021) — RL as conditional sequence modelling; transformer maps (return-to-go, state) → action. Offline-RL friendly.
  • Diffusion Policy (Chi et al. 2023, TRI + Columbia) — denoising-diffusion model used as a policy. Multi-modal action distributions, strong on bimanual manipulation. Massive 2023-2026 adoption.
  • Action Chunking with Transformers (ACT) (Zhao et al. 2023, Stanford ALOHA) — chunked action prediction; complement to diffusion for low-cost bimanual.

10. Imitation learning

  • Behavioural cloning (BC) — supervised learning of pi(a|s) from demonstrations. Suffers from compounding error / distribution shift.
  • DAgger — Dataset Aggregation (Ross, Gordon, Bagnell 2011). Iterative: roll out current policy, get expert correction on visited states, retrain. Addresses BC’s covariate shift.
  • GAIL — Generative Adversarial Imitation Learning (Ho + Ermon 2016). Discriminator distinguishes expert vs policy trajectories; policy minimises a learned reward via GAN-style adversarial training.
  • AIRL — Adversarial Inverse RL (Fu et al. 2018) — recovers a reward function rather than just a policy.
  • IRL — Inverse Reinforcement Learning (Ng + Russell 2000; Abbeel + Ng 2004) — infer the reward function from demonstrations. Maximum-entropy IRL (Ziebart 2008) is the dominant variant.

11. Foundation-model robotics (VLA)

  • RT-1 — Brohan et al., Google Brain 2022 — 35M-param transformer trained on 130k episodes from 13 robots.
  • RT-2 — Brohan et al., Google DeepMind 2023 — vision-language model (PaLI-X / PaLM-E) co-finetuned on robot trajectories; tokenises actions; demonstrates semantic generalisation.
  • RT-X / Open-X-Embodiment — RT-2-X 2023 / 2024 — cross-embodiment dataset of 22 robot platforms, ~1M trajectories.
  • PaLM-E — Driess et al. 2023 — embodied multimodal LLM.
  • OpenVLA — Kim et al., Stanford 2024 — 7B-param open-source VLA built on Llama-2 + DinoV2 + SigLIP.
  • OctoPi-Tactile — multi-sensor foundation including tactile (CMU + others).
  • Tesla Optimus stack — public talks 2024-2025 describe an end-to-end neural-net policy from camera + IMU + joint encoders to joint torques.
  • Pi-0 / Pi-0.5 — Physical Intelligence 2024-2025 — VLA with flow-matching action head.

These are replacing the upper layers of classical control stacks (task and skill levels) rather than the low-level torque controllers, which remain PID / impedance.

12. Specialty and niche

  • Fuzzy logic control — Mamdani 1975; deployed Sendai subway 1987 (Hitachi). Smooth nonlinear gain mapping by linguistic rules. Rare in modern systems but still found in consumer appliances and process loops.
  • Neural-network feedback — historical Narendra + Parthasarathy 1990; modern Adam-trained MLP for friction / cogging compensation overlaid on PID.
  • ADRC — Active Disturbance Rejection Control — Han 1998-2009 (Cleveland State University). Extended-state observer (ESO) estimates all uncertainty as a lumped disturbance; controller cancels it. Strong industrial adoption in motion control (Galil, Texas Instruments controlSUITE).
  • Time-delay control (TDC) — Youcef-Toumi + Ito 1990. Uses recent past control + state to cancel unknown dynamics. Suits large-mass robots with slow sampling.
  • Active vibration control + AMD (active mass damper) — covered in vibration-damping-arms.
  • Resonant control / proportional-resonant — for tracking and rejecting pure sinusoidal references (grid-tied inverters, gimbals).

13. Robot-specific control architectures

How the families combine in practice:

  • Industrial 6-DoF arm (FANUC, ABB, KUKA, Yaskawa) — joint-level PD or PI + computed-torque feed-forward (model-based). Cartesian impedance / admittance overlay when force interaction is required.
  • Cobot (Universal Robots, Franka Emika, KUKA iiwa, Doosan) — joint torque control (motor-current-based for UR; strain-gauged harmonic-drive flexspline for Franka and iiwa) + Cartesian impedance + safety monitoring (ISO 10218-1/-2, ISO/TS 15066 PFL).
  • Mobile base (warehouse AMR, AGV) — DWA (Fox 1997) or TEB (Rosmann 2012) local planner producing (v, omega); low-level wheel PID closes around motor encoders.
  • Drone — cascaded P (position) → P (velocity) → PID (attitude) → PID (rate) → PWM. Modern Pixhawk firmware (PX4, ArduPilot) supports LQR + INDI (incremental nonlinear dynamic inversion) + NMPC variants.
  • Quadruped (MIT Cheetah 3, ANYmal, Boston Dynamics Spot, Unitree Go2) — convex MPC on simplified single-rigid-body dynamics (Di Carlo, Wensing et al. 2018) generating ground-reaction forces, distributed to joint-level torque controllers. Modern stacks (2024+) replace the MPC with PPO/Dreamer-trained neural policy.
  • Humanoid (Atlas, Sanctuary Phoenix, Figure 02, Tesla Optimus, Apptronik Apollo) — whole-body QP at 500-1000 Hz (Wensing + Orin; Sentis + Khatib operational-space); centroidal dynamics + contact constraints; foot-trajectory optimisation. Increasing fraction of upper-layer behaviour from neural policies.
  • Surgical robot (Intuitive da Vinci, CMR Versius, Medtronic Hugo) — bilateral teleoperation with Cartesian impedance on patient side; cable-stretch compensation; haptic feedback on master.
  • Precision lithography stage (ASML) — state-space + LQG + ILC + feed-forward; nanometer servo error.
  • Semiconductor wafer-scan — ILC for repetitive scan trajectories; settling time dominates throughput.
  • Aerospace fly-by-wire — gain-scheduled PID + complementary filter on INS + air-data; redundancy-managed across 3-4 channels; certification under DO-178C / ARP4754A.
  • Autonomous-vehicle lateral control — kinematic-bicycle MPC at 20-50 Hz; Stanley controller as fallback on Stanford Stanley / Junior.
  • Autonomous-vehicle longitudinal control — PID + model-based feed-forward (grade, mass).
  • Biped balance — LIP-MPC (linear inverted pendulum MPC, Kajita 2003) + whole-body QP.
  • Manipulation imitation — diffusion policy + force-fallback / safety monitor.
  • Race car — MPPI or NMPC at 50-200 Hz; tyre-model identification online.
  • Valve process flow — PID + IMC; lambda-tuned.
  • Nuclear control rod — digital control with redundancy; IEC 61511 SIS architecture; separate plant-protection system independent of advanced control.

14. Comparison table

AlgorithmLinear?Model required?Tuning effortRobustnessTypical use
Pyesnotriviallowinner velocity loop
PIyesnolowmediummotor current, flow, temperature
PDyesnolowmediumservo position
PIDyesnolowmediumgeneral SISO
2-DoF PIDyesnomediummediumtracking + disturbance separately
Cascade PIDyesweakmediummedium-highservo, drives, process
Pole placementyesfull state-spacelowlowtextbook SISO
LQRyesfull state-space + Q,Rmediumhigh (60 deg PM)multivariable servo
LQGyes+ noise covariancesmediumLTR-recoverablestochastic multivariable
H∞yes+ uncertainty boundshighhighflight control, optics
SMCnoroughmediumvery highelectric drives, missiles
Super-twistingnoroughmediumvery highlow-chatter SMC
Backsteppingnofull nonlinearhighhighstrict-feedback systems
Feedback linearisationnoexacthighlow w/o overlayrobot manipulators
Passivity-basednopartialmediumhighimpedance, marine
MRACyes/noparameterisationmediummediumaerospace, motors
L1 adaptiveyes/noparameterisationmediumhighflight control
Self-tuningyesonline IDhighmediumprocess control
Linear MPCyes+ constraintsmediumconstraint-awarerefinery, AV
NMPCno+ constraintshighconstraint-awaredrones, quadrupeds
iLQR / DDPnononlinearhighnone guaranteedoffline trajectory
ILCyes/noweaklowvery high (in batch)repetitive tasks
MPPInononlinear + samplermediumempiricalaggressive driving
Impedancen/ainertia + Jacobianmediumpassivity-stableHRC
RL (PPO/SAC)nonone (model-free)high (reward)empiricallocomotion, dexterous
Imitation (BC, DAgger)nononemedium (data)empiricalmanipulation skills
Diffusion policynononemedium (data)empiricalbimanual manipulation
VLA foundationnononelow (pretrained)empiricalsemantic manipulation

15. Selection heuristics

By task:

  • Motor servo loop → cascaded PI current + PI velocity + P position; tune in that order from inside out.
  • AGV wheel speed → PI + feed-forward from commanded acceleration.
  • Drone attitude → PID + derivative filter at 500-1000 Hz; consider INDI when the model is well-known.
  • Legged robot stance → convex MPC over single-rigid-body + low-level joint torque tracking; or end-to-end RL.
  • Cobot HRC contact → joint torque control + Cartesian impedance + ISO 15066 PFL safety monitor.
  • Pharma cobot insertion task → admittance with VR teach-pendant for demonstrations; behaviour cloning or diffusion policy thereafter.
  • Precision lithography stage → state-space + LQG with feed-forward.
  • Semiconductor wafer-scan → ILC for repetitive scan; feed-forward dominates feedback in tracking error.
  • Aerospace fly-by-wire → gain-scheduled PID over operating envelope; complementary INS + air-data; H∞ for stability augmentation if envelope is wide.
  • Auto AV lateral → MPC at 20-50 Hz over kinematic bicycle.
  • Auto AV longitudinal → PID with model-based feed-forward (grade, mass, drag).
  • Biped balance → LIP-MPC + whole-body QP at 500-1000 Hz.
  • Manipulation imitation → diffusion policy or ACT trained on teleop demos + force-fallback.
  • Race car → MPPI or NMPC; online tyre-stiffness ID.
  • Valve process flow → PID + IMC; lambda tuned.
  • Nuclear control rod → digital control with redundancy; separate IEC 61511 SIS layer.

By symptom:

  • Steady-state error → add integral or feed-forward.
  • Overshoot → reduce Kp / increase derivative damping / set-point weighting.
  • Noise amplification → derivative filter / observer-based D.
  • Saturation oscillation → anti-windup.
  • Performance drift over operating envelope → gain scheduling or adaptive.
  • Periodic disturbance → repetitive control / resonant control.
  • Repeated task with persistent error → ILC.
  • Contact-rich task → impedance + force control.
  • Hard input/state constraints → MPC.
  • No model available, lots of data → RL or imitation.

16. Cross-references

17. Citations

  • Ogata, K. — “Modern Control Engineering”, 5th ed., Prentice Hall, 2009.
  • Astrom, K. J., Murray, R. M. — “Feedback Systems: An Introduction for Scientists and Engineers”, 2nd ed., Princeton University Press, 2020.
  • Khalil, H. K. — “Nonlinear Systems”, 3rd ed., Prentice Hall, 2001.
  • Slotine, J.-J. E., Li, W. — “Applied Nonlinear Control”, Prentice Hall, 1991.
  • Camacho, E. F., Bordons, C. — “Model Predictive Control”, 2nd ed., Springer, 2007.
  • Sutton, R. S., Barto, A. G. — “Reinforcement Learning: An Introduction”, 2nd ed., MIT Press, 2018.
  • Schulman, J. et al. — “Proximal Policy Optimization Algorithms”, arXiv:1707.06347, 2017.
  • Haarnoja, T. et al. — “Soft Actor-Critic: Off-Policy Maximum Entropy Deep RL”, ICML 2018.
  • Chi, C. et al. — “Diffusion Policy: Visuomotor Policy Learning via Action Diffusion”, RSS 2023.
  • Hogan, N. — “Impedance Control: An Approach to Manipulation”, ASME JDSMC, 1985 (three-paper series).
  • Krstic, M., Kanellakopoulos, I., Kokotovic, P. — “Nonlinear and Adaptive Control Design”, Wiley, 1995.
  • Narendra, K. S., Annaswamy, A. M. — “Stable Adaptive Systems”, Prentice Hall, 1989.
  • Cao, C., Hovakimyan, N. — “L1 Adaptive Control Theory”, SIAM, 2010.
  • Doyle, J. C., Glover, K., Khargonekar, P. P., Francis, B. A. — “State-space solutions to standard H2 and H∞ control problems”, IEEE TAC, 1989.
  • Kalman, R. E. — “Contributions to the theory of optimal control”, Bol. Soc. Mat. Mexicana, 1960.
  • Utkin, V. I. — “Variable structure systems with sliding modes”, IEEE TAC, 1977.
  • Hafner, D. et al. — “Mastering Diverse Domains through World Models” (DreamerV3), arXiv:2301.04104, 2023.
  • Brohan, A. et al. — “RT-2: Vision-Language-Action Models”, arXiv:2307.15818, 2023.

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