Vincent Verdult

Subspace identification of multivariable linear parameter-varying systems

A subspace identification method is discussed that deals with multivariable linear parameter-varying state-space systems with affine parameter dependence. It is shown that a major problem with subspace methods for this kind of system is the enormous dimension of the data matrices involved. To overcome the curse of dimensionality, we suggest using only the most dominant rows of the data matrices in estimating the model. An efficient selection algorithm is discussed that does not require the formation of the complete data matrices, but processes them row by row.

Kernel methods for subspace identification of multivariable LPV and bilinear systems

Subspace identification methods for multivariable linear parameter-varying (LPV) and bilinear state-space systems perform computations with data matrices of which the number of rows grows exponentially with the order of the system. Even for relatively low-order systems with only a few inputs and outputs, the amount of memory required to store these data matrices exceeds the limits of what is currently available on the average desktop computer. This severely limits the applicability of the methods. In this paper, we present kernel methods for subspace identification performing computations with kernel matrices that have much smaller dimensions than the data matrices used in the original LPV and bilinear subspace identification methods. We also describe the integration of regularization in these kernel methods and show the relation with least-squares support vector machines. Regularization is an important tool to balance the bias and variance errors. We compare different regularization strategies in a simulation study.

Wiener model identification and predictive control for dual composition control of a distillation column

The benefits of using the Wiener model based identification and control methodology presented in this paper, compared to linear techniques, are demonstrated for dual composition control of a moderate–high purity distillation column simulation model. An identification experiment design is presented which enables one to identify both the low and high gain directions of the distillation column, properties which are important for control and hard to identify in a conventional identification experiment setup as is demonstrated in the paper. Data from the proposed experiment design is used for indirect closed-loop identification of both a linear and a Wiener model, which shows the ability of the Wiener model to approximate the nonlinearity of the distillation column much closer than the linear model can. The identified Wiener model is used in a MPC algorithm in which the nonlinearity of the Wiener model is transformed into a polytopic description. In this way a convex optimisation problem is retained while the effect of the nonlinearity on the input–output behaviour of the plant is still taken into account. The performance of the proposed Wiener MPC is compared with linear MPC based on the identified linear models, and with a Wiener MPC in which the nonlinearity of the Wiener model is removed from the control problem via an inversion, a popular way to handle Wiener models in a MPC framework. The simulations demonstrate that the proposed Wiener MPC outperforms the other MPC algorithms.

Subspace identification of piecewise linear systems

Subspace identification can be used to obtain models of piecewise linear state-space systems for which the switching is known. The models should not switch faster than the block size of the Hankel matrices used. The nonconsecutive parts of the input and output data that correspond to one of the local linear systems can be used to obtain the system matrices of that system up to a linear state transformation. The linear systems obtained in this way cannot be combined directly, because the state transformation is different for each of the local linear systems. The transitions between the local linear systems can be used to transform the models to the same state space basis. We show that the necessary transformations can be obtained from the data, if the data contains a sufficiently large number of transitions for which the states at the transition are linearly independent. An algorithm to determine the transformations is presented, and the sensitivity with respect to noise is investigated using a Monte-Carlo simulation.

Identification of linear parameter-varying state-space models with application to helicopter rotor dynamics

A considerable amount of work has been dedicated in the past to the problem of the system identification of helicopter flight dynamics, while much less activity has been oriented to the goal of developing suitable identification procedures for rotor dynamics, mainly because of the difficulties associated with the task. This paper shows that subspace and optimization based identification techniques can be used to determine discrete-time linear parameter-varying models that have the potential to provide accurate descriptions for the (intrinsically time-varying) dynamics of a rotor blade. The identification techniques are presented and applied to simulated data generated by a physical model that describes the out-of-plane bending dynamics of a helicopter rotor blade.

FAULT DETECTION AND IDENTIFICATION OF ACTUATOR FAULTS USING LINEAR PARAMETER VARYING MODELS

Abstract A method is proposed to detect and identify two common classes of actuator faults in nonlinear systems. The two fault classes are total and partial actuator faults. This is accomplished by representing the nonlinear system by a Linear Parameter Varying (LPV) model, which is derived from experimental input-output data. The LPV model is used in a Kalman filter to estimate augmented states, which are directly related to the faults. Decision logic has been developed to determine the fault class from the estimated augmented states. The proposed method has been validated on a nonlinear simulation model of a small commercial aircraft.

Identification of multivariable bilinear state space systems based on subspace techniques and separable least squares optimization

We discuss identification of discrete-time bilinear state space systems with multiple inputs and multiple outputs. Subspace identification methods for bilinear systems suffer from the curse of dimensionality. Already for relatively low order systems, the matrices involved become so large that the method cannot be used in practice. We have modified the subspace algorithm such that it reduces the dimension of the matrices involved. Only the rows that have the largest influence on the model are selected; the remaining rows are discarded. This obviously leads to an approximation error. The initial model that we get from the subspace method is optimized using the principle of separable least squares. According to this principle, we can first solve for the matrices that enter non-linearly in the output error criterion and then obtain the others by solving a linear least squares problem.

Efficient and systematic identification of MIMO bilinear state space models

We present a systematic way to identify multi-input, multioutput (MIMO) bilinear state space systems subject to white noise inputs, in the presence of process and measurement noise. The algorithm we present is based on a family of subspace identification algorithms for linear systems. It requires the linear pair (A, C) to be observable. We use subspace identification to determine the model order, to identify the linear part of the model, and to compute an initial estimate for the nonlinear part. The final estimate of the nonlinear part is computed by numerically solving a nonlinear optimization problem. A series of simulation experiments showed that the initial estimate is close to the optimum and allows convergence of the nonlinear optimization problem.

A switching detection method based on projected subspace classification

In this paper an innovative switching detection method for piecewise linear systems is presented. The principle used for switching detection is based on finding projected subspaces from batches of input-output data, which are taken from the full data set. The method runs off-line, incrementally over all the data and, at each time, a different batch is used to compute the projected subspace. In this way, the segmentation and classification of data are entirely based on the information retrieved from the projected subspace, i.e. the subspace dimension and basis. The output of the method is a matrix of weights that assigns each pair of input-output measured data to the respective local system. Simulation experiments show the effectiveness of the proposed approach.

Bilinear State Space Systems for Nonlinear Dynamical Modelling

We discuss the identification of multiple input, multiple output, discrete-time bilinear state space systems. We consider two identification problems. In the first case, the input to the system is a measurable white noise sequence. We show that it is possible to identify the system by solving a nonlinear optimization problem. The number of parameters in this optimization problem can be reduced by exploiting the principle of separable least squares. A subspace-based algorithm can be used to generate initial estimates for this nonlinear identification procedure. In the second case, the input to the system is not measurable. This makes it a much more difficult identification problem than the case with known inputs. At present, we can only solve this problem for a certain class of single input, single output bilinear state space systems, namely bilinear systems in phase variable form.