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High Performance Computing of SDRE models with Discrete Kalman Filtering for Robust $H_{\infty}$-Controls
Yunfeng Cai, Tiexiang Li, Wen-Wei Lin, Junxin Zhang
TL;DR
SDRE control requires repeated online CARE solutions, while under-actuated and partially observed systems also need realizable disturbance and performance channels. This work projects those channels, combines repeated controller CARE solvers with discrete Kalman estimation, and tests the framework on aircraft and quadrotor models. With measured computation delays, SDA and Newton reduce aircraft tracking errors across all six paired runs.
Problem
SDRE requires repeated online CARE solutions, and under-actuated, partially observed systems need channel formulations compatible with physical actuator and sensor dimensions.
Source: §1 · p. 2Method
The framework projects disturbance and performance channels while retaining physical actuator and sensor dimensions, uses SDA or safeguarded warm-started Newton–Kleinman for the controller CAREs, and applies a discrete Kalman recursion.
Source: §1 · pp. 2–3Results
SDA and Newton reduce airspeed RMSE versus icare by 11.6–18.7% and 8.5–19.9%, respectively, and reduce pitch RMSE in all six paired aircraft runs.
Source: §6.1.2 · p. 18Takeaways & Limitations
The discrete recursion avoids an additional estimator CARE at each frozen-state update, leaving the two controller equations as the repeated Riccati solves.
Source: §4 · p. 12The synthesis conditions are stated for a frozen-state model satisfying assumptions (A1)–(A3).
Source: §2 · p. 4
Abstract
from arXiv · showhide
State-dependent Riccati equation (SDRE) control requires repeated online solution of continuous-time algebraic Riccati equations (CAREs). A projected-channel $H_\infty$ formulation preserves the physical actuator and sensor dimensions of under-actuated, partially observed systems. We derive a computable sufficient attenuation bound for positive-semidefinite stabilizing Riccati solutions and construct an update that enforces their spectral-radius coupling condition. A discrete Kalman recursion supplies the scheduling state estimates. The two CAREs arising in controller synthesis are solved by a structure-preserving doubling algorithm (SDA) or a warm-started Newton--Kleinman iteration equipped with a doubling Lyapunov solver. Numerical studies use a twelve-state F-16 model with complete aerodynamic forces, moments, and trim, together with quadrotor spiral tracking. With instantaneous command updates, all three CARE backends give essentially the same closed-loop response. In the tracking simulations, measured computation times determine when new control inputs are applied, and the previous inputs are held during computation. SDA and Newton reduce airspeed and pitch errors in all six paired aircraft tracking runs. In the faster quadrotor task with limited computational resources, SDA and Newton complete all paired runs, while runs using MATLAB \texttt{icare} terminate early.
1 Introduction
SDRE preserves state-dependent dynamics through repeated frozen-state CARE solves, making online computation a bottleneck; under-actuated, partially observed systems also require realizable channel dimensions. This work combines projected-channel H∞ synthesis, efficient repeated CARE solvers, and discrete Kalman estimation, with aircraft and quadrotor validation.
1 Introduction
- SDRE retains nonlinear state dependence in a Riccati-based feedback law but requires repeated online CARE solutions as the state evolves.Those repeated solves are identified as a major computational bottleneck.Source: §1 · pp. 1–2
- A warm-started Newton–Kleinman iteration with a stabilizing safeguard and doubling Lyapunov solver complements SDA for the two repeated controller CAREs.The iteration reuses preceding accepted Riccati solutions along the frozen-state trajectory.Source: §1 · pp. 2–3
- The projected-channel formulation preserves physical actuator and sensor dimensions while providing a sufficient attenuation bound and an update enforcing condition (C3).Disturbance and performance channels are projected onto the actuator matrix's column space and measurement matrix's row space, respectively.Source: §1 · p. 3
- A discrete Kalman recursion updates the state estimate, covariance, and gain, replacing the estimator CARE with recursive matrix operations.This leaves the two H∞ controller CAREs as the repeated algebraic Riccati solves.Source: §1 · p. 3
- With measured computation delays, SDA and Newton reduce airspeed and pitch errors in all six paired aircraft tracking runs.The studies also cover a twelve-state F-16 model and nominal and faster quadrotor spiral tracking; instantaneous updates yield essentially solver-independent accuracy.Source: §1 · p. 3
2 SDRE for H∞-control systems
The section sets out frozen-state measurement-feedback H∞ synthesis: a controller must stabilize the plant while meeting a prescribed attenuation level. Its construction relies on two stabilizing Riccati solutions and a spectral-radius coupling condition.
2 SDRE for H∞-control systems
- At a fixed frozen state, synthesis seeks an internally stabilizing controller with closed-loop H∞ norm below the prescribed γ.The stated setup assumes stabilizability and detectability conditions and imposes additional feedthrough conditions.Source: §2 · pp. 4–5
- The normalized output-feedback existence conditions require two stabilizing positive-semidefinite Riccati solutions, X and Y, satisfying their respective CAREs.The central controller is obtained from the stable invariant subspace when conditions (C1)–(C3) hold.Source: §2 · pp. 6–7, (C3) The spectral radius of XY satisfies · p. 7
(C3) The spectral radius of XY satisfies
- The central H∞ controller is constructed from the two Riccati solutions, with its admissibility contingent on conditions (C1)–(C3).The classical stable-invariant-subspace construction supplies the controller for the frozen-state interconnection.Source: (C3) The spectral radius of XY satisfies · p. 7
3 Efficient computation of the H∞controller K
The section establishes CARE feasibility for projected output-feedback synthesis, then develops iterative solvers and an attenuation update for the coupled-controller condition.
3 Efficient computation of the H∞controller K
- The controller computation repeatedly solves the two CAREs and checks conditions (C1)–(C3) before recovering controller matrices.Source: §3 · pp. 7, 11
- For projected channels, both CAREs have stabilizing positive-semidefinite solutions above a sufficient attenuation threshold when the stated stabilizability and detectability conditions hold.The channel construction retains the physical actuator and sensor dimensions.Source: §3 · pp. 8–9
- The attenuation update uses Riccati monotonicity to raise γ until recomputing both CARE solutions gives χ(γ) < 1−τ.The result is stated in exact arithmetic.Source: §3 · pp. 9–10
- The structure-preserving doubling algorithm solves the CAREs through a Cayley-transformed symplectic pencil whose stable invariant subspace encodes the Riccati solution.Source: §3 · p. 10
- Newton–Kleinman iteration solves each CARE through Lyapunov equations, with the inner Lyapunov solve handled by a squared Smith doubling iteration.The implementation reuses accepted Riccati solutions as starting values and uses a stabilizing icare initialization when needed.Source: §3 · p. 11
4 Kalman state estimation
A discrete Kalman recursion supplies the state estimate used to schedule the state-dependent controller, while recursively updating its covariance and gain.
4 Kalman state estimation
- The posterior Kalman estimate determines the state-dependent coefficients used in controller synthesis.Source: §4 · p. 11
- A trapezoidal discretization of the frozen-state model uses the known effective input over each sampling interval to form the discrete estimation model.Source: §4 · p. 11
- Prediction and correction recursively update the estimate, covariance, and gain, avoiding an additional estimator CARE at each frozen-state update.Source: §4 · p. 12
5 Applications in aerial vehicle design
The applications use the framework for aircraft regulation and speed tracking, and for quadrotor position tracking.
5 Applications in aerial vehicle design
- The applications cover F-16 recovery to steady flight and speed tracking, alongside quadrotor position tracking.Source: §5 · pp. 12–13
5.1 F-16 model for regulation and speed tracking
- The F-16 example uses a 12-state aircraft model with thrust and three control-surface inputs, tracking steady-flight trim or a varying-speed reference.The aerodynamic force and moment model is computed using F16Model.jl.Source: §5.1 · pp. 13–14
5.2 Quadrotor model for position tracking
- The quadrotor example tracks a three-dimensional spiral with a rigid-body model; its controller coefficients are evaluated at the Kalman posterior state.The performance output includes nonzero position-error weights.Source: §5.2 · pp. 15, 17
6 Numerical experiments
The experiments compare icare, SDA, and Newton on F-16 and quadrotor tasks, with instantaneous updates for regulation and nominal tracking and measured computation delays in selected tracking tests. SDA and Newton reduce aircraft tracking errors across all six pairs and complete every limited-computation quadrotor run, while icare stops early in that task.
6.1.1 Recovery to straight-level flight
- During F-16 recovery, solver trajectories overlap at plotting resolution; initial thrust saturation precedes approach to the moving straight-level reference.Table 1 summarizes the physical response.Source: §6.1.1 · p. 17, Figure 3 · p. 18, Table 1 · p. 18
6.1.2 Speed tracking with measured computation delays
- Across six paired runs, SDA and Newton reduce airspeed RMSE by 11.6–18.7% and 8.5–19.9%, respectively, and lower pitch RMSE in every pair.All methods complete all six runs.Source: Table 2 · p. 18, §6.1.2 · p. 18
6.2.1 Nominal spiral tracking
- All three solvers achieve 0.0482 m position RMSE; dynamic H∞ feedback records 0.0482 m versus 254.4354 m for feedforward only.The feedforward-only comparison uses the same model, reference, initial state, and disturbance samples as the controlled run.Source: §6.2.1 · pp. 19–20
6.2.2 Faster spiral tracking under limited computational resources
- Across six paired faster-spiral runs, SDA and Newton complete every run, while icare stops early.Table 3 summarizes the limited-computation comparisons.Source: §6.2.2 · p. 20, Table 3 · p. 21
6.3 Riccati computation time
- SDA and Newton require substantially less CPU time than icare for both vehicle models; timing covers both CARE solves per update.Newton reuses preceding accepted solutions as initial guesses, while SDA applies structure-preserving doubling to each CARE.Source: Table 4 · p. 21, §6.3 · p. 21
7 Conclusions
The paper concludes that projected disturbance and performance channels, an attenuation update, and repeated SDA or safeguarded Newton–Kleinman solves form an SDRE workflow with Kalman state estimation. Its experiments report solver-independent accuracy with instantaneous updates, reduced aircraft tracking errors with measured delays, and complete SDA/Newton runs in the faster limited-computation quadrotor task.
7 Conclusions
- Projected channels, explicit large-γ feasibility, Riccati-monotonicity updates, and SDA or safeguarded Newton–Kleinman solves form the reported SDRE workflow.The stable invariant-subspace construction connects the projected CAREs to the central H∞ controller.Source: §7 · p. 21
- With instantaneous command updates, aircraft recovery and nominal quadrotor experiments give essentially solver-independent accuracy.With measured computation delays, SDA and Newton reduce airspeed and pitch errors in all six paired aircraft tracking runs; in faster limited-computation quadrotor tracking, both complete all six paired runs while icare runs terminate early.Source: §7 · p. 21
A Parameters and coefficient matrices for aerial-vehicle models
- The appendix specifies filter covariance propagation from a local linearization and independent physical disturbance samples held on the sensor grid.The cited model passage gives disturbance standard deviations of 10 N for each body-force component and 1 N m for each moment component.Source: A Parameters and coefficient matrices… · p. 23
- The appendix reports m1 = 15, m2 = 4, and p2 = 9, while the performance dimension is p1 = 16 for the F-16 and p1 = 14 for the quadrotor.The quadrotor measured components are [ϕ, θ, ψ, p, q, r, x, y, z].Source: A Parameters and coefficient matrices… · pp. 24–25