In ight system identi cation, ight test data is used to estimate model parameters. The aircraft is excited during the ight with a set of inputs (maneuvers) to generate the data for parameter estimation. Di erent approaches exist to design the inputs which result in an adequate information content in the acquired data during the ight tests. In most cases, inputs are applied to the aircraft which are designed based on a priori knowledge about the system. In this paper we investigate a method to update the model parameters during the ight and update the inputs onboard the aircraft. This method can increase the achieved parameter accuracy and is expected to reduce the required time for the ight test program, and therefore, also the program cost.
In ight vehicle system identi cation, a mathematical model is deducted from
recorded ight data using a series of test maneuvers. If a xed model structure is
assumed, the system identi cation task narrows down to a parameter estimation
problem. It has been shown that the type of inputs applied to an aircraft has a
signi cant in uence on the accuracy of the parameter estimates [
Besides ight testing for system identi cation and parameter estimation,
indirect adaptive control is also an important use case for onboard adaption of
optimal input design for parameter estimation. In such cases, the controller is
adjusted based on the change in the system parameters. Therefore, maneuvers
are pursued which lead to more accurate parameter estimates and therefore a
better performing controller. Many previous studies such as [
The problem discussed in this paper can be formulated as a Dual-Mode
Batch-to-Batch Optimization as presented in [
The problem in parameter estimation is to nd the parameter values in a xed model structure ^ = ~(Z; u) : (1)
In the above equation, ^ is the vector of the parameter estimates, ~ is the estimator and Z and u are the measured outputs and inputs. In this paper, we limit our study to linear systems of the following form (2) (3) (4) x_ = Ax + Bu; y = Cx + Du; where x is the vector of the system states, y is the output vector, u is the input vector and A, B, C and D are the system-, input-, output- and feed-through matrices. Moreover, v is a zero-mean Gaussian white noise sequence with the covariance matrix R.
It can be shown that the Fisher information matrix M provides an indication about the information content in the gathered ight test data during the experiment
The inverse of the Fisher information matrix, the Dispersion matrix D =
M 1 is the theoretical lower bound for the covariance matrix of the parameter
estimates if an unbiased estimator is used [
Since the Cramer-Rao bounds and the Dispersion matrix do not depend on the measurements, an initial model and initial values for parameters can be used to design inputs that minimize the remaining uncertainty in the parameters. In this study, we choose to minimize the trace of the Dispersion matrix which is the sum of the estimate for the parameter variances under consideration of practical state and input constraints. The optimal control problem (OCP) can be written as minimize
u subject to tr M
1 x_ = f (u; x; ) ; x 2 [xlb; xub] ; u 2 [ulb; uub] ;
= 0; t 2 [0; T ]; where T is the duration of the maneuver.
Since the estimation of the parameter values is the ultimate goal of the ight
test campaign, no precise values of the parameter estimates are available before
the ight tests. A common approach is to perform ight tests with suboptimal
inputs or optimal inputs based on inaccurate parameters, record the data and
perform o ine parameter estimation. The new parameter values are used to
design new inputs which are used for subsequent ight tests. This becomes a costly
iterative process. Robust optimal control methods have been used in [
The solution method applied in this study consists of an Extended Kalman Filter
(EKF) for the online parameter estimation. The approaches for designing the
optimal inputs are based on [
Di erent methods exist for online ight vehicle parameter estimation. These
methods allow to update the parameter values onboard the aircraft as new
measurements become available rather than recording the measurements over a time
period and analyzing them in batches. Online parameter estimation methods
can be formulated in time and frequency domain as described in [
: _ = 0:
The dynamics of the original system states is governed by (2). Since we assume the parameters to be constant, their dynamics is governed by
The governing equations for the augmented system become with the initial conditions and x_a = Aaxa + Bau;
y = Caxa + Du; z(ti) = y(ti) + v(ti); i = 1; 2; : : : ; N;
E[xa(0)] = xa;0; Ef[xa(0) xa;0][xa(0) xa;0]g = P a;0; (7) (8) (9) (10) (11) (12) (13) (14) (15) Aa =
A( ) 0 0 0 ; Ba =
B( ) 0 ; Ca = B( ) 0 ; xa;0 = x0 ; where xa;0 is the mean value of the augmented states at the initial time point and P a;0 is its respective covariance matrix.
The dynamic system described above is not linear anymore since the new augmented states which include the parameters are elements of the system matrix A and are multiplied with the state vector x. Therefore, the equations governing the nonlinear system (9) can be written as
x_a = f (xa; u) ;
y = g(xa) ; z(ti) = y(ti) + v(ti); i = 1; 2; : : : ; N: (16) (17) (18)
The parameters of the system (2) are now states of the augmented
system (16) and can be estimated via an EKF as described in [
As mentioned above, optimal inputs for parameter estimation are determined by
solving the optimal control problem described in (6). The cost function of the
mentioned problem is non-convex and previous studies in [
Onboard Adjustment of the Inputs As stated before, the optimal inputs computed o ine are based on an initial model with an initial set of parameters. As shown in Fig. 1 an optimal maneuver is designed o ine with the initial values of the parameters. This maneuver is injected during the ight. An EKF updates the parameter values during the optimization as new data becomes available.
After each maneuver, the gradient based optimization is repeated with the updated parameter values to adapt the optimal inputs. Thereby, the optimal input generated with the last set of parameters and the corresponding system states are used to initialize the optimization with the direct method for optimal control. Similarly, each EKF run for online parameter estimation is initialized by previous estimates for the parameters, their covariance matrix and the state parameter correlations. In the rst parameter estimation run, we initialize the covariance matrix of the parameters by a diagonal matrix with a large value Pre-Flight Initial Model
Dynamic Programming Direct Method for Optimal
Control
In-Flight Online Parameter
Estimation Maneuver Injection
Not Met
Onboard Maneuver Adaption Check Iteration Criteria
Met
End
In the above equation, v(ti) is the i-th realization of a zero mean white Gaussian random process with the following covariance matrix (here 105) on its diagonal. No correlation is assumed between the parameters and the states of the system in the initial covariance matrix of the augmented states (for the rst maneuver). The initial state covariance matrix is considered as a tuning parameter and should be chosen considering the accuracy of the aircraft's navigation system and controller. 4
A short period model of an aircraft is used to illustrate the proposed method in
this study. The linear model was featured in multiple previous studies such as
[
The measurement equations of the system are: yymm12 ((ii)) = 01 10 y1(ti) + v1(ti) ; i = 1; 2; : : : ; N; y2(ti) v2(ti) _ Z q_ = M 1 Mq
Z q + M
; yy12((tt)) = 01 10
(t) q(t) : v(i) = v1(ti) : v2(ti) 2:0 0:0 R = 0:0 1:0 : (19) (20) (21) (22)
The model parameter values are provided in Table 1. The parameters M , Mq, and M are unknown. We design the inputs such that the information content with respect to these parameter is maximized (see (6)). Other parameters are considered to be known in this example. Signi cantly di erent initial parameter values (provided in Table 1) have been used to design the rst optimal input signal (prior to the experiment). Four parameter estimation and input design iterations (described in Section 3.3) are performed during the test run-time. After each maneuver optimization, new parameter values are provided by the EKF based on virtual measurement data generated from the model described in (19){(23). A sampling rate of 100 Hz is assumed and the run-time of each maneuver is 4 s. The discrete time step for the solution of the optimal control problem (6) with the direct method for optimal control is set to 0:05 s. We perform 4 iterations of parameter estimation and input optimization during the ight. The following input and state bounds are considered to design the inputs (24) (25) (26) (27) 2 [ 12:5 ; 12:5 ] ;
It can be seen in Fig. 2 that the structure of the optimal inputs remain roughly consistent as the parameter values get updated. This is in agreement with using the optimal inputs from the previous set of parameters as an initial guess for the maneuver optimization in the next step. Figure 3 shows the outputs of the dynamic system corresponding to the inputs of Fig. 2 at each iteration. The change of the parameters and the respective estimate for their standard deviations after each iteration are shown in Fig. 4. It can be seen that the nal values of the standard deviations in Fig. 4 are consistently lower than the Cramer-Rao bounds in Table 1. This is due to the fact that the Kalman lter uses the available data in all of the ve maneuvers, whereas only one maneuver run has been considered in Table 1. It is also worth mentioning that no process noise has been considered in this study. The initial state covariance matrix for each EKF run is set to
P 0 = 0:2 0:0 0:0 0:1 :
The achieved parameter Cramer-Rao bounds are consistently lower than the
values determined for the same model in previous studies such as [
2:5 1:5
2 t [s] 2:5 3 3 3:5 3:5 4 4 In this study we show that adaption of optimal inputs for parameter estimation during the experiment run-time signi cantly improves the results of a subsequent parameter estimation. However, this is a very complex process and can be di cult to successfully implement and apply to a real aircraft. First and foremost, the application of the proposed method in real-world systems is limited by the existence of process noise and de ciencies regarding the model structure, such as unmodeled nonlinearities. Among the implementation aspects, the low computing power and memory capabilities of common ight control computers can be considered challenges associated with this approach.
Furthermore, since the parameter values used to design the optimal inputs are not accurate, state bounds are not guaranteed to be satis ed during the ight.
0 0:2 M 0:4 q M
1 0:6
0 0:5 1:5 1:6 1:8
2 2:2 0 0 M 1 1 1 2 2 2 iter 3 3 3 4 4 4 5 5 5
0:2
True Parameter Value
Fig. 5. Absolute value of the correlation matrices.
To address this issue, more conservative state bounds can be considered for the rst maneuvers and the bounds can be expanded as more accurate parameter estimates become available.
The additional e orts required to implement this method can however be advantageous in use cases where ight test time is very expensive and no good initial guess for the system model exists. In the case of linear system identi cation, changing the trim point of the aircraft during the ight also changes the values of the linear system's matrix coe cients. This method can be applied to adapt the optimal inputs of the aircraft to the new trim point during the ight.
It might be possible, especially in the case of nonlinear systems, that signi
cantly di erent parameter values result in signi cantly di erent maneuvers and
therefore cause convergence problems in the short time between two maneuvers.
This can be solved by updating the maneuvers online, as new parameter values
become available. Online methods for optimal input design are currently being
investigated by authors. Furthermore, it is planned to test the algorithm and
its capabilities in a real-world experiment using an unmanned aircraft.
Utilization of other online parameter estimation algorithms (e.g. the frequency domain
method as discussed in [
This work is supported by the German Federal Ministry for Economic A airs and Energy as part of the LuFo program (grant-ID 20Q1719D).