Pseudo-random multilevel sequences: Spectral properties and identification of Hammerstein systems

In system identification literature, pseudo-random sequences are commonly used as test inputs. Their ease of generation and persistent excitation properties makes them quite attractive for practical applications (Ljung, 1999; Soderstrom & Stoica, 1989). Pseudo-random binary sequences (PRBS) are known to be persistently exciting for linear systems, but not so for nonlinear systems. In this paper, we study the generation of pseudo-random multilevel sequences (PRMLS) suitable for identification of nonlinear systems. A closed form solution is provided for the optimal signal level selection, which is formulated as a nonlinear optimization problem to maximize the similarity to white Gaussian noise (WGN). Similarities of higher-order statistical properties of the generated PRMLS to white Gaussian noise signal are also established. Following that, we apply the results of the paper to identify an important class of nonlinear systems, those of the Hammerstein model structure. In that regard, we prove the necessity of p + 1 > N, and sufficiency of p > N and (pⁿ - 1)/(p - 1) > M for persistent excitation of a Hammerstein system with Nth-order nonlinearity and M shifts. We also show how to construct a persistently exciting PRMLS of length at most 4(M + 1)N² - 1 and number of signal levels at most 2N for the identification of the Hammerstein system. Finally, we derive an efficient Hammerstein system identification algorithm which involves only N × N matrix inversion and singular value decompositions. Hence, it is numerically reliable for the M ≫ N case, and has significant computational advantages.