System Identification in the Behavioral Setting

Abstract

System identification is a fast growing research area that encompasses a broad range of problems and solution methods. It is desirable to have a unifying setting and a few common principles that are sufficient to understand the currently existing identification methods. The behavioral approach to system and control, put forward in the mid 80’s, is such a unifying setting. Till recently, however, the behavioral approach lacked supporting numerical solution methods. In the last 10 years, the structured low-rank approximation setting was used to fulfill this gap. In this paper, we summarize recent progress on methods for system identification in the behavioral setting and pose some open problems. First, we show that errors-in-variables and output error system identification problems are equivalent to Hankel structured low-rank approximation. Then, we outline three generic solution approaches: (1) methods based on local optimization, (2) methods based on convex relaxations, and (3) subspace methods. A specific example of a subspace identification method—data-driven impulse response computation—is presented in full details. In order to achieve the desired unification, the classical ARMAX identification problem should also be formulated as a structured low-rank approximation problem. This is an outstanding open problem.

A Subspace Method for Impulse Response Estimation

Let $\mathscr {B}$ be a linear time-invariant system of order $\mathtt {n}$ with lag $\ell$ and let $w = (u,y)$ be an input-output partitioning of the variables. In [16], it is shown that, under the following conditions,

• the data $w_\mathrm{d}$ is exact, i.e., $w_\mathrm{d}\in \mathscr {B}$,

• $\mathscr {B}$ is controllable,

• $u_\mathrm{d}$ is persistently exciting, i.e., $\mathscr {H}_{\mathtt {n}+\ell +1}(u_\mathrm{d})$ is full rank,

the Hankel matrix $\mathscr {H}_{t}(w_\mathrm{d})$ with t block-rows, composed from $w_\mathrm{d}$, spans the space  of all t-samples long trajectories of the system $\mathscr {B}$, i.e.,

This implies that there exists a matrix G, such that

$$\begin{aligned} \mathscr {H}_{t}( {y_\mathrm{d}})G = H, \end{aligned}$$

where H is the vector of the first t samples of the impulse response of $\mathscr {B}$. The problem of computing the impulse response H from the data $w_\mathrm{d}$ reduces to the one of finding a particular G.

Define $U_\mathrm{p}$, $U_\mathrm{f}$, $Y_\mathrm{p}$, $Y_\mathrm{f}$ as follows

where

$$ \mathrm{row\,dim}(U_\mathrm{p}) = \mathrm{row\,dim}(Y_\mathrm{p}) = \ell $$

and

$$ \mathrm{row\,dim}(U_\mathrm{f}) = \mathrm{row\,dim}(Y_\mathrm{f}) = t. $$

Then if $w_\mathrm{d} = (u_\mathrm{d},{y_\mathrm{d}})$ is a trajectory of a controllable linear time-invariant system $\mathscr {B}$ of order $\mathtt {n}$ and lag $\ell$ and if $u_\mathrm{d}$ is persistently exciting of order $t+\ell +\mathtt {n}$, the system of equations

is solvable for , and for any particular solution G, the matrix $Y_{\mathrm{f}} G$ contains the first t samples of the impulse response of $\mathscr {B}$, i.e.,

$$\begin{aligned} Y_{\mathrm{f}} G = H. \end{aligned}$$

This gives Algorithm 1 for the computation of H.

Algorithm 1 computes the first t samples of the impulse response; however, the persistency of excitation condition imposes a limitation on how big t can be. This limitation can be avoided by a modification of the algorithm. L consecutive samples, where L is a user specified parameter that is small enough to allow the application of Algorithm 1, are computed iteratively. Then, provided the system is stable, by monitoring the decay of H in the course of the computations, gives a way to determine how many samples are needed to capture the transient behavior of the system.

In case of noisy data, the system of Eq. (**) on step 1 in Algorithm 1 has no exact solution. Using a least squares approximate solution instead, turns Algorithm 1 in a heuristic for approximate system identification. The algorithm is heuristic because the maximum likelihood estimator requires structured total least squares solution of (**). The structured total least squares problem, however, is nonlinear optimization problem [10].