This paper concerns the estimation of a dynamic model from two measured signals when it is not clear which signal should be used as input to the model. In this case, both a forward and an inverse model can be estimated. Here, a basic instrumental variable approach is used and it is shown that the forward and inverse model estimators give identical parameter estimates provided that corresponding model structures have been used. Furthermore, it is shown that this scenario occurs when properties of a quadcopter are estimated from accelerometer and gyro signals and, hence, that it does not matter which signal is used as input.

In this paper, an approach to estimate the mass of a quadcopter using only inertial measurements and pilot commands is presented. For this purpose, a lateral dynamic model describing the relation between the roll rate and the lateral acceleration is formulated. Due to the quadcopter’s inherent instability, a controller is used to stabilize the system and the data is collected in closed loop. Under the effect of feedback and disturbances, the inertial measurements used as input and output are correlated with the disturbances, which complicates the parameter estimation. The parameters of the model are estimated using several methods. The simulation and experimental results show that the instrumental-variable method has the best potential to estimate the mass of the quadcopter in this setup.

In this work, we estimate a model of the vertical dynamics of a quadcopter and explain how this model can be used for mass estimation and diagnosis of system changes. First, a standard thrust model describing the relation between the calculated control signals of the rotors and the thrust that is commonly used in literature is estimated. The estimation results are compared to those using a refined thrust model and it turns out that the refined model gives a significant improvement. The combination of a nonlinear model and closed-loop data poses some challenges and it is shown that an instrumental variables approach can be used to obtain accurate estimates. Furthermore, we show that the refined model opens up for fault detection of the quadcopter. More specifically, this model can be used for mass estimation and also for diagnosis of other parameters that might vary between and during missions.

A ship's roll dynamics is very sensitive to changes in the loading conditions and a worst-case scenario is the loss of stability.  This paper proposes an approach for online estimation of a ship's mass and center of mass. Instead of focusing on a sensor-rich environment where all possible signals on a ship can be measured and a complete model of the ship can be estimated, a minimal approach is adopted. A model of the roll dynamics is derived from a well-established model in literature and it is assumed that only motion measurements from an inertial measurement unit together with measurements of the rudder angle are available. Furthermore, identifiability properties and disturbance characteristics of the model are presented. Due to the properties of the model, the parameters are estimated with an iterative instrumental variable approach to mitigate the influence of the disturbances and it uses multiple datasets simultaneously to overcome identifiability issues. Finally, a simulation study is presented to investigate the sensitivity to the initial conditions and it is shown that there is a low sensitivity for the desired parameters.

A ship's roll dynamics is very sensitive to changes in the loading conditions and a worst-case scenario is the loss of stability.  This paper proposes an approach for online estimation of a ship's mass and center of mass. Instead of focusing on a sensor-rich environment where all possible signals on a ship can be measured and a complete model of the ship can be estimated, a minimal approach is adopted. A model of the roll dynamics is derived from a well-established model in literature and it is assumed that only motion measurements from an inertial measurement unit together with measurements of the rudder angle are available. Furthermore, identifiability properties and disturbance characteristics of the model are presented. Due to the properties of the model, the parameters are estimated with an iterative instrumental variable approach to mitigate the influence of the disturbances and it uses multiple datasets simultaneously to overcome identifiability issues. Finally, a simulation study is presented to investigate the sensitivity to the initial conditions and it is shown that the sensitivity is low for the desired parameters.

This paper concerns the estimation of approximate (linear) inverse models of block-oriented systems and the presented results give an improved understanding of these approximations. The estimated inverse is intended to be used as a pre- or postdistorter of the original system and a good inverse model would thus be one that, when used in series with the original system, produces a signal that resembles the original input. An inverse model of a nonlinear system can either be estimated in the standard way (from input to output) and then inverted, or directly (from output to input). This choice will affect the model. In the general case, the two modeling approaches will lead to different models, which will be shown for Hammerstein and Wiener systems. However, for a noise-free Hammerstein system with a white Gaussian input, the two approaches will result in the same model, up to a constant. When the two models are not equal, and the goal is to use the inverse as described above, it can be beneficial to estimate an approximate inverse directly. It will also be illustrated in an example how the inverse estimate can be used to get a nonparametric estimate of the nonlinearity in a block-oriented system. 

A common issue with many system identification problems is that the true input to the system is unknown. In this paper, a framework, based on indirect input measurements, is proposed to solve the problem when the input is partially or fully unknown, and cannot be measured directly. The approach relies on measurements that indirectly contain information about the unknown input. The resulting indirect model formulation, with both direct- and indirect input measurements as inputs, can be used to estimate the desired model of the original system. Due to the similarities with closed-loop system identification, an iterative instrumental variable method is proposed to estimate the indirect model. To show the applicability of the proposed method, it is applied to data from an inverted pendulum experiment with good results.

A common issue with many system identification problems is that the true input to the system is unknown. In this paper, a framework, based on indirect input measurements, is proposed to solve the problem when the input is partially or fully unknown, and cannot be measured directly. The approach relies on measurements that indirectly contain information about the unknown input. The resulting indirect model formulation, with both direct and indirect input measurements as inputs, can be used to estimate the desired model of the original system. Due to the similarities with closed-loop system identification, an iterative instrumental variable method is proposed to estimate the indirect model. To show the applicability of the proposed method, it is applied to data from an inverted pendulum experiment with good results. 

A ship's roll dynamics is sensitive to the mass and mass distribution. Changes in these physical properties might introduce unpredictable behavior of the ship and a {worst-case} scenario is that the ship will capsize. In this paper, a recently proposed approach for online estimation of mass and center of mass is validated using experimental data. The experiments were performed using a scale model of a ship in a wave basin. The data was collected in free run experiments where the rudder angle was recorded and the ship's motion was measured using an inertial measurement unit. The motion measurements are used in conjunction with a model of the roll dynamics to estimate the desired properties. The estimator uses the rudder angle measurements together with an instrumental variable method to mitigate the influence of disturbances. The experimental study shows that the properties can be estimated with quite good accuracy but that variance and robustness properties can be improved further.

A ship's roll dynamics is sensitive to the mass and mass distribution. Changes in these physical properties might introduce unpredictable behavior of the ship and a worst-case scenario is that the ship will capsize. In this paper, a recently proposed approach for online estimation of mass and center of mass is validated using experimental data. The experiments were performed using a scale model of a ship in a wave basin. The data were collected in free run experiments where the rudder angle was recorded and the ship's motion was measured using an inertial measurement unit. The motion measurements are used in conjunction with a model of the roll dynamics to estimate the desired properties. The estimator uses the rudder angle measurements together with an instrumental variable method to mitigate the influence of disturbances. The experimental study shows that the properties can be estimated with quite good accuracy but that variance and robustness properties can be improved further.