Integral Quadratic Constraint-Based Robustness Analysis of Launch Vehicles
Authors:   Technische Universität Dresden, Dresden, Germany, Biertümpfel Felix , Gkoutzos Dimitrios , Levi David , Valderrama Jorge , Pfifer Harald 
Published: January 2024

The certification process of space launchers is challenging due to the complexity of the overall system and the reduced means of full system tests. One of the most crucial mission segments is the atmospheric flight phase. During its ascent through the earth’s atmosphere the space launcher is exposed to rapidly changing environmental conditions and a multitude of disturbance. At the same time, the system dynamics are highly coupled, subject to a significant amount of uncertainty, and vary extremely fast over time along the ascent trajectory.

State-of-the-art approaches to assess the wind disturbances’ and parameter perturbations’ effects are simulation-based methods applied to the full nonlinear launcher model. These include Monte Carlo simulations and nonlinear optimization. Due to the complexity of the system and the variety of disturbances as well as perturbations, these methods are computationally expensive. Moreover, they can only provide a lower bound and some probabilistic information on the expected performance. Alternative methods include worst-case analysis methods using linear launcher representations. Common attempts in literature involve linear time-invariant worst-case methods based on the structured singular value (mu). Lately, linear parameter-varying (LPV) methods were applied to the launcher ascent problem. However, mu-based analyses are limited to frozen points in time along the trajectory. LPV-type analyses consider an infinite set of possible trajectories. Moreover, both approaches calculate the performance measure over an infinite analysis horizon. Due to this, they fail to adequately cover the launcher dynamics and the ascent/mission scenario – a prescribed finite trajectory connected to strictly time-varying dynamics. Moreover, both frameworks require the linear system to be stable at all times. To achieve load minimal conditions slight instability of the launcher for short periods of the flight is beneficial for its performance. However, classic linear analyses would deem these designs unfeasible. In summary, the calculated performance measures fail to provide meaningful (upper) bounds for the actual space launcher. This becomes obvious by an arbitrary validation gap comparing the simulation-based and the linear worst-case analysis results.

The paper addresses this short-coming by explicitly respecting the time varying dynamics of the launcher in a finite horizon linear time varying (LTV) worst case analysis of typical performance metrics (drift, aerodynamic loads, tracking). The analysis explicitly considers uncertainty in the dynamics and external wind disturbance in an easy to interpret fashion.

A gravity turn trajectory for an industry-sized nonlinear model of a light weight expendable launch vehicle (ELV) is calculated. It spans the time frame from the start of the gravity turn until first-sage burnout/ main engine cut off. The nonlinear dynamics can include the multi body dynamics of the space launcher and nozzle (moved by the thrust vectoring control (TVC)), tail-wag-dog effects, flexible modes, differential aerodynamic loads, sloshing effects, jet damping and aeroelastic couplings. The linear time-varying model of the space launcher is calculated about the reference trajectory. This model explicitly covers the launcher’s time-varying dynamics as well as the finite time horizon of the ascent.

The performance analysis is conducted inside the LTV integral quadratic constraints (IQC) framework. This framework allows to address a multitude of different perturbations, for example, nonlinearities (saturations), infinite dimensional systems (time delays), as well as dynamic and parametric uncertainties, in a single analytical analysis. Efficient worst-case gain optimization frameworks exist to solve the problem. The calculated performance bounds are guaranteed upper bounds for the nonlinear model. The finite horizon LTV framework also facilitates the analysis of unstable dynamics, which is beneficial for launch systems.

To include realistic wind disturbances which yield directly interpretable performance bounds inside the LTV IQC framework a tailored wind filter design is presented. This means the calculated upper bounds represent an actual physical value of e.g. the aerodynamic load or pitch error. The methodology allows to easily design wind filters dedicated to any provided reference wind profile, e.g., based on historical data from the launch side and/or weather balloon measurements.

The primary uncertainty will be implemented in a way to resemble disk margins. This type of uncertainty creates simultaneous phase and gain perturbations in the system. Hence, the LTV analysis directly correspond to phase and gain margin analysis typically required in certification processes.

The LTV performance analysis will be conducted for two representative control designs. One control design will provide a stable drift motion of the launcher. The alternative design will establish a load minimum condition, i.e. an unstable drift pole, in the maximum dynamic pressure region. A corresponding (mu-based) LTI analysis will be conducted representing the state-of-the-art approach in industry. This means the linear model will be analyzed on frozen points in time, over an infinite horizon.

To finish, a Monte Carlo simulation over a set of allowable uncertainties and wind profiles is conducted to validate that the LTV performance analysis is feasible and not overly conservative to calculate performance bounds for the nonlinear model. At the same time, the analysis will emphasize the limitations of the LTI approach and the gaps to the nonlinear simulation.