1 System Identification and Forward Modeling

Figure 2.7: Block diagram of the general adaptive system identification (forward modeling) problem.

System identification is an essential stage in control systems design. The goal of this stage is to establish a model of the physical system (plant) to be controlled. The control system is then designed to meet the design criteria based on the plant model. In many cases, the system to be controlled is slowly time varying and it is therefore of little use to design a control system based on the plant model. Instead, an adaptive control system is used and in most cases, the system identification is also performed on-line using an adaptive modeling or adaptive system identification as shown in Fig. 2.7. The controller in this case is referred to as an adaptive controller or a self-tuning regulator. Another common application in which adaptive system identification is widely used is active noise and vibration control (ANVC), which usually employ adaptive controllers. All ANVC adaptive algorithms rely on an adaptive model for the secondary path between the control actuators and the error sensors. ANVC system might perform their system identification step in a stage prior to operation or update the secondary path model during operation depends on how fast the changes in the physical secondary path occur. The adaptive model is obtained by exciting the physical system to be modeled and the adaptive filter by a spectrally rich and persistent input signal $x(n)$ as shown in Fig. 2.7. An adaptive algorithm is then used to minimize the difference $e(n)$ between the physical system output $d(n)$ and the adaptive filter output $y(n)$. On convergence, and in ideal cases, the error signal is reduced to zero and $y(n)$ approaches $d(n)$. This in turn means that the filter impulse response approaches the physical systems' response. It is important to note that successful adaptive system identification start with correctly choosing the adaptive filter structure. When the plant response is oscillatory in nature, an infinite impulse response adaptive filter updated using the equation error algorithm (section 6.2) or the output error algorithm (Section 6.3) might be used. When the system response is short, an FIR transversal filter is usually preferred for stability reasons. An FIR adaptive model might be updated using the least mean squares or its normalized version (Section 4.9 and 4.11, respectively). For longer FIR models, frequency domain adaptive algorithms (such as BFDAF and PBFDAF, discussed in Sections 4.2 and 4.12, respectively) give superior performance and huge computational saving. Lattice adaptive filters have also been successfully and advantageously used in some applications such as modeling of the earth layers in seismic explorations. Adaptive lattice filters might be updated using the LMS-lattice or RLS-lattice algorithms (Sections 5.4 and 5.5, respectively). The Adaptive Signal Processing Toolbox includes many system identification examples for different physical systems and using different adaptive algorithms. Examples of those examples are model_arlmsnewt (Section 10.21), model_eqerr (Section 10.22), model_lmslattice (Section 10.23), model_mvsslms (Section 10.24), model_outerr (Section 10.25), model_rlslattice (Section 10.26), model_sharf (Section 10.27), model_tdlms (Section 10.28), and model_vsslms (Section 10.29).