Real Time Computationally Efficient MIMO System Identification Algorithm

Abstract

MIMO system identification is a fundamental concern in a variety of applications. Various iterative and recursive MIMO system identification algorithms exist in the literature. The iterative algorithms suffer from high computational cost due to large matrix computations and recursive algorithms suffer from slow convergence speed. This paper proposes a fast recursive exact least squares algorithm for MIMO system identification with fast convergence speed and low computational cost. The recursions of the algorithm give rise to a lattice structure. The lattice structure is less sensitive to round-off errors, coefficients variations and can also be used for model order reduction. Although in this work we estimate an FIR (finite impulse response) model of the MIMO system, the framework can also be used for IIR (infinite impulse response) models, and for estimating linear periodically time varying (LPTV) and multivariable systems. The theory proposed is validated using simulation results.

Appendix: Formulae Used in the Algorithm

We call a signal v(n) to be in a pre-windowed form if v(n) = 0 for n < 0. A given discrete time signal, v(Ln − i), is defined as 1 × K (K is a fixed number) vector, with K ≫ Ln − i, :

$$ \begin{array}{@{}rcl@{}} \mathbf{v}(Ln-i) \equiv \left[\begin{array}{ccccccc} 0 & {\cdots} & 0 & v(-i) & v(L-i) & {\cdots} & v(Ln-i) \end{array}\right] \end{array} $$

and the set of vectors, {v(Ln − k)|i ≤ k ≤ i + p}, forms a (p + 1) × K matrix, denoted as $V_{p+1}^{Ln-i}$, and is given as follows:

$$ \begin{array}{@{}rcl@{}} {V}_{p+1}^{Ln-i}\equiv \left[\begin{array}{c} \mathbf{v}(Ln-i)\\ \mathbf{v}(Ln-i-1)\\ \vdots\\ \mathbf{v}(Ln-i-p) \end{array}\right] .\end{array} $$

(29)

Here the superscript denotes the top row vector used and the subscript p + 1 denotes number of rows. The span of row-space of ${V_{p+1}^{Ln-i}}$ is given as:

$$ \begin{array}{@{}rcl@{}} S=span \{\mathbf{v}(Ln-i),\mathbf{v}(Ln-i-1),\cdots, \mathbf{v}(Ln-i-p)\} \end{array} $$

If the rows vectors are linearly independent, the span S can be written as:

$$ \begin{array}{@{}rcl@{}} S=span \{\mathbf{v}(Ln-i),\mathbf{v}(Ln-i-1),\cdots, \mathbf{v}(Ln-i-p)P^{\perp}\left[V_{p}^{Ln-i}\right]\}. \end{array} $$

The projection on span of $V_{p+1}^{Ln-i}$ can be written as:

$$ \begin{array}{@{}rcl@{}} &P\left[V_{1+p}^{Ln-i}\right]=P\left[\begin{array}{c} V_{p}^{Ln-i}\\ \mathbf{v}(Ln-i-p)P^{\perp}\left[V_{p}^{Ln-i}\right] \end{array}\right]\\ &=P[V_{p}^{Ln-i}]+P\left[\mathbf{v}(Ln-i-p)P^{\perp}\left[V_{p}^{Ln-i}\right]\right] \end{array} $$

(30)

Since the projection operator is defined as P[X] = X^T[XX^T]^− 1X, we can re-write the above equation as:

$$ \begin{array}{@{}rcl@{}} &P\left[V_{1+p}^{Ln-i}\right]=P[V_{p}^{Ln-i}]+P^{\perp}\left[V_{p}^{Ln-i}\right]\mathbf{v}^{T}(Ln-i-p)[\mathbf{v}(Ln-i-p) \\ &\times P^{\perp}\left[V_{p}^{Ln-i}]\mathbf{v}^{T}(Ln-i-p)\right]^{-1}\mathbf{v}(Ln-i-p)P^{\perp}\left[V_{p}^{Ln-i}\right] \end{array} $$

(31)

and the projection onto the orthogonal complement space, $P^{\perp }\left [V_{1+p}^{Ln-i}\right ]$, can be written as

$$ \begin{array}{@{}rcl@{}} &P^{\perp}\left[V_{1+p}^{Ln-i}\right]=I-P\left[V_{1+p}^{n-1}\right]\\ &=P^{\perp}[V_{p}^{Ln-i}]-P^{\perp}\left[V_{p}^{Ln-i}\right]\mathbf{v}^{T}(Ln-i-p)[\mathbf{v}(Ln-i-p)\times\\ &P^{\perp}\left[V_{p}^{Ln-i}]\mathbf{v}^{T}(Ln-i-p)\right]^{-1}\mathbf{v}(Ln-i-p)P^{\perp}\left[V_{p}^{Ln-i}\right] \end{array} $$

(32)

Pre-multiply (32) with v and post-multiply with w^T to get:

$$ \begin{array}{@{}rcl@{}} &\mathbf{v}P^{\perp}\left[V_{1+p}^{Ln-i}\right]\mathbf{w}^{T} =\mathbf{v}P^{\perp}[V_{p}^{Ln-i}]\mathbf{w}^{T}-\mathbf{v}P^{\perp}\left[V_{p}^{Ln-i}\right]\mathbf{v}^{T}(Ln-i-p)\times\\ &\left[\mathbf{v}(Ln-i-p)P^{\perp}\left[V_{p}^{Ln-i}\right]\mathbf{v}^{T}(Ln-i-p)\right]^{-1}\times\\ &\mathbf{v}(Ln-i-p)P^{\perp}\left[V_{p}^{Ln-i}\right]\mathbf{w}^{T} \end{array} $$

(33)

The above equation is known as the inner-product update formula, for more detail refer [29]. We now briefly discuss the projection update formula from [30]. This update relation is used to compute recursions for (17). Let us define $K_{p+1}^{Ln-i}$, the pseudo-inverse of $X^{Ln-i}_{p+1}$ i.e. $X^{Ln-i}_{p+1}K_{p+1}^{Ln-i}=I$, as follows:

$$ \begin{array}{@{}rcl@{}} K_{p+1}^{Ln-i}=\left[X^{Ln-i}_{p+1}\right]^{T}\left[{X^{Ln-i}_{p+1}}\left[{X^{Ln-i}_{p+1}}\right]^{T}\right]^{-1} \end{array} $$

(34)

Pre-multiply (31) with z and post-multiply by $K_{p+1}^{Ln-i}$, we get:

$$ \begin{array}{@{}rcl@{}} &{\mathbf{z} K_{p+1}^{Ln-i}}=\left[\begin{array}{c}{\mathbf{z} K_{p}^{Ln-i}}\\ 0 \end{array}\right] +{\mathbf{z} { P}^{\perp}[ V_{1:p}] \mathbf{\nu}_{p+1}^{T}}{[\mathbf{\nu}_{p+1} { P}^{\perp}[V_{1:p}] \mathbf{\nu}_{p+1}^{T}]^{-1}}.\\ & \left[\begin{array}{c}-\mathbf{\nu}_{p+1} K_{p}^{Ln-i}\\ 1 \end{array}\right] . \end{array} $$

(35)

For more information and derivation of (35) refer [30].