Continuous-time Laguerre-based subspace identification utilising nuclear norm minimisation

Online continuous time system identification in the presence of impulsive terms

“…Note that the proposed structure requires a single tuning parameter ℏ subject to the constraint (7), although alternative approaches to state variable filters, such as, for example, generalized Poisson moment functional or Laguerre filters in all-pass domain, can be also considered. 29,30 It is worthy of note that a sufficiently large dwell time ℏ allows for the cut-off frequency of h(s) to emphasize the system's frequency band of interest. Furthermore, noticing that the block Hankel matrix Ψ m,q k in ( 8) is generated from the sequence 𝜙 T j,k = (h * 𝜉 j 𝜑 T )(t k ), j ∈ {0, … , (q + 1)n}, a balanced matrix with centered frequencies can be obtained by letting j ∈ {− ⌊((q + 1)n∕2⌋ , … , ⌈(q + 1)n]∕2⌉}.…”

“…The computation of Ψ 11,2 k has been implemented with ℏ = 1 and a downsampling factor of 20 (N = 500 samples). An example of trajectories is depicted in Figure 4 (top) where 13 input values are randomly chosen within an interval ± [30,40], and at random time instants under the dwell time constraint ℏ D ≥ 1. A first transient phase is free of deviations for illustration.…”

“…noise in the data, Table 1 shows the results of a Monte Carlo study of 100 runs, in an EIV framework with different SNR ratios. Here, both LS and TLS schemes are compared, using at each experiment 15 input outliers with random values within the range ± [30,40], and chosen at random time instants with ℏ D ≥ 1. While the TLS estimator shows satisfactory results, the relative performances of the online LS scheme, with low SNR ratio, show that, in its present form, the projection operation in (58) is not capable of simultaneously tackling the problem of online disturbance annihilation and bias-compensation.…”

“…The following state variable filter $$$ h $$$, and complex exponential function $$$ \xi $$$ are adopted for the subsequent applications: $$$$ \kern0.5em \xi (t)={e}^{\iota \omega t},\kern1em \omega =2\pi /\hslash, \kern1em \left({\iota}^2=-1\right), $$$$ $$$$ \kern0.5em h(s)={\left(\frac{\gamma }{s+\gamma}\right)}^{\overline{\kappa}},\kern1em \gamma =4\overline{\kappa}/\hslash, \kern1em \overline{\kappa}=1+\underset{i}{\max}\left({\kappa}_i\right). $$$$ Note that the proposed structure requires a single tuning parameter $$$ \hslash $$$ subject to the constraint (), although alternative approaches to state variable filters, such as, for example, generalized Poisson moment functional or Laguerre filters in all‐pass domain, can be also considered 29,30 . It is worthy of note that a sufficiently large dwell time $$$ \hslash $$$ allows for the cut‐off frequency of $$$ h(s) $$$ to emphasize the system's frequency band of interest.…”

“…Simulated system (75) subject to unobservable input deviations within the range ±[30, 40]. Top: Input-output trajectories.Bottom: Parameters estimates using the regressor 𝜂 T t (blue lines), and the regressor 𝜙 T 0,t (black lines).…”