On the discretization of single-input single-output bilinear systems

Generalised tangential interpolation for model reduction of discrete-time MIMO bilinear systems

“…While in the linear case one can make use of the Tustin transform to create a discrete-time system which additionally preserves the stability properties of the continuous system, the situation becomes more complicated for bilinear models. For a more detailed overview on this topic, the reader is referred to Dunoyer, Balmer, Burnham, and James (1997) and Schwarz (1987). Since the main focus of this article is directed to the problem of model order reduction, we will be content with a semi-implicit Euler discretisation of the above system, i.e.…”

Control of nonlinear state-dependent systems using a receding horizon strategy

“…To see this, recall that in the design algorithm presented in the last section, the optimal control law (10) must be solved during each sampling instant. In equation (10), the matrix V NN and the vector C N are defined as respectively. Hence, either the input matrix B(x t , u t ) = 0 or the output matrix C(x t , u t ) = 0 will result in V NN being diagonal; consequently, the optimal control signal u t can be obtained from the first line of the optimality condition (9), without solving the optimal control law (10).…”

“…Proof. The first part of the proposition can be proved by collecting the results up to (10). Then, by assuming u t + N = u t + N À1 , it is seen that both V NN and C N can be calculated by recursively computing the states from the system dynamics and the input vector.…”

“…This state-dependent representation of nonlinear systems is very general that may even include chaotic processes and bilinear systems. 10,11 More importantly, systematic identification methods have been developed by Young and colleagues [12][13][14] in a series of papers.…”

A bilinear controller with PID structure

“…In (Dunoyer et al, 1997 a) it is shown that the discrete form of a first order SISO bilinear system is given by where the subscript d denotes that the parameters aod, bod and qod correspond to the discrete model and k denotes the discrete time index. The discrete model parameters are obtained from in which T is the sampling interval in seconds and h2 is an input dependent correction factor, and approximated by (Dunoyer et a1 1997 a) It is interesting to note that the scalar h2 of equation (7) is a function of the input U which is fixed over the sampling interval T. As a consequence, an invariant continuous first order bilinear system corresponds to an input dependent discrete time first order bilinear system.…”