Method for Real-Time Frequency Response and Uncertainty Estimation

F REQUENCY responses describe the steady-state response of a dynamic system to harmonic excitation in terms of relative magnitude and phase angle. Frequency responses facilitate gaining physical insight into a system, analyzing stability properties, and synthesizing control laws. Acquiring this information as quickly as possiblewould save time andmoney, and/or increase safety in several aeronautical applications. For example, updating stability margins in flight would expedite flight envelope verification and expansion. The ability to monitor frequency responses and their uncertainties in real time could shorten wind-tunnel and flight tests. Estimating timevarying dynamics could aid in accident investigation, adjudicate flight control laws quickly, evaluate adaptive control laws with simple metrics, estimate pilot models for human factors analysis and loss-of-control situations, or assist in reconfiguring control algorithms to handle changes such as fuel burn, mass redistribution, structural damage, and ice formation.Moreover, frequency responses are nonparametric and can describe new systems for which there is little or no prior dynamic information. Traditionally, aircraft frequency responses are computed after a flight test usingWelch’smethod [1–4], which is the industry standard for nonparametric modeling. Engineering judgment and experience are required to choose window number, size, shape, and overlap to obtain smooth and accurate spectral estimates, although several commercial software packages are available to aid in this process [2,3]. More recently, the “quasi-real-time” approach expedites data transmission and processing time to provide solutions soon after the maneuver has completed [5–9]. Conventional frequency sweeps, requiring 90–120 s of data per input to accurately model rigid-body dynamics [3], lead to relatively long test times; however, orthogonal phase-optimized multisines were developed that uniquely and simultaneously excite all inputs and reduce required record lengths [10–14]. Recently, multisine inputs and a recursive least-squares estimator in the time domain were used to compute frequency responses in real time [15]. Alternatively, a two-step method employingwavelets and amaximum likelihood estimatorwas used to estimate time-varying frequency responses of pilots [9]. A real-time parametric method was also developed that uses a recursive Fourier transform and equation-error parameter estimation in the frequency domain [12,16,17]. This work presents a new method for estimating potentially timevarying frequency responses and their uncertainties in real time. Orthogonal phase-optimized multisine excitations are employed to excite all the aircraft inputs simultaneously and within a specified frequency band. A Fourier transform is used to then convert input and output measurements, at those frequencies, into the frequency domain, where the relative magnitudes, phase angles, and uncertainties are extracted. A recursive formulation of this process using a data forgetting algorithm gives the method a real-time and adaptive capability. This method provides an accurate and computationally efficient method of estimating potentially time-varying frequency responses and their uncertainties in real time using relatively short data records. The following section details the input design, frequency response estimation, and uncertainty propagation. Section III describes the F-16 nonlinear simulation model and the T-2 test aircraft used to demonstrate the method. In Sec. IV, time-invariant and time-varying dynamics are estimated using F-16 simulation data and T-2 flight data, followed by further discussion and conclusions in Sec. V. Software for the experiment design, additional system identification techniques, and aircraft simulations used in this work are in a MATLAB® toolbox called System IDentification Programs for AirCraft, or SIDPAC [10]. The SIDPAC software toolbox was developed at NASA Langley Research Center, and it is continually expanded and improved.