## Review Of Characterization And Identification Means For Linear Systems

Because neural system identification and systems-level modeling are very broad subjects in which many workers have developed a variety of approaches, one cannot hope to review them all in this chapter. However, those classical approaches that have had some impact on neural systems will be examined.

Many schemes have been devised by systems and control engineers for the characterization of single-input/single-output (SISO) linear systems, including those exhibiting nonminimum phase transfer functions, or having transport lags. Linear systems can be described in several ways. The most fundamental method is by a set of first-order, linear, time-invariant, ordinary differential equations (ODEs) to which the parameters (gains, natural frequencies, or A matrix) are known. Such state equations are generally the result of the analysis and modeling of the physics and biochemistry of the system.

When a linear system is investigated experimentally, its unique response to an impulse input yields its weighting function or impulse response. The impulse response can also be obtained by solution of the state ODEs for zero initial conditions and an impulse input. If a steady-state sinusoidal input is applied to the linear system, the output will, in general, be sinusoidal with the same frequency but having a different phase and amplitude from the input. The sinusoidal frequency response function of the system is defined as the vector:

Where Yo is the peak amplitude of the output sinusoid at frequency ra rad/s, Xo is the peak amplitude of the input sinusoid, and 0(ra) is the phase angle by which the output lags (or leads) the phase of the input. Often system frequency responses are presented as Bode plots for convenience. A Bode plot consists of two parts, a magnitude and a phase plot: 20 log10[Yo/Xo] vs. ra (log scale), and 0(ra) vs. ra (log scale). Note that the Fourier transform of the linear system impulse response function, h(t), is the frequency response function, H(jra). It is generally difficult to go from h(t) or H(jra) to the state ODEs. One must estimate the natural frequencies (poles and zeros) of the system graphically, which is not an accurate process; hence, the actual order of the system may be underestimated. 