ISE Block for Nonlinear System ID

• The ISE Block is a module that generates excitation signals by balancing uniform regressor-space coverage with dynamic transitions for nonlinear system identification.
• It employs the IDS-FID strategy, solving local optimization problems to select input subsequences that offer a tunable balance between steady-state and dynamic probing.
• Empirical results indicate that adjusting trade-off parameters can yield 20-40% RMSE improvements compared to traditional excitation methods.

The Information-based Selective Excitation (ISE) Block is a principled module for designing excitation signals tailored to nonlinear dynamic system identification. Its primary objective is to generate input sequences that maximize information acquisition regarding system parameters or dynamic characteristics, employing a rigorous combination of space-filling and dynamic excitation principles. At its core, the ISE Block leverages the Incremental Dynamic Space-Filling Design (IDS-FID) strategy, which balances uniform regressor-space exploration with targeted dynamic transients by solving a sequence of local optimization problems over candidate signal segments. The result is an adaptive signal design capable of prioritizing steady-state coverage or dynamic exploration, tunable via explicit trade-off parameters and executable using only minimal knowledge of the underlying system.

1. Information-Theoretic Foundations

In the context of nonlinear system identification, the primary information-theoretic goal is to select excitation signals that maximize the information content relevant to all unknown model parameters. While direct maximization of the Fisher information matrix or mutual information requires full system knowledge, the ISE Block adopts two surrogate objectives: (1) achieving uniform coverage of the regressor space, thereby preventing undersampling of any region of the input–output characteristic, and (2) emphasizing dynamic transitions, which are necessary for identifying time constants or frequency-dependent behaviors.

The ISE Block operationalizes these surrogates via a maximin-style distance criterion in the proxy regressor space, ensuring that new input subsequences are placed in the largest existing gaps. This is paired with a length-penalty mechanism that favors rapid, short-lived transitions, thereby enabling selective emphasis on dynamic content. Unlike spectral methods that target frequency content in aggregate, the ISE Block constructs the signal in an incremental, information-driven manner (Herkersdorf et al., 2024).

2. Mathematical Structure and Signal Optimization

Let the nonlinear dynamic process be represented in NARX form as

$$y(k) = f( x(k) ) + e(k), \quad x(k) = [u(k-1), \ldots, u(k-m), y(k-1), \ldots, y(k-m)]^\top.$$

Since $y(k)$ is unknown during excitation design, the algorithm employs a simple linear proxy model $\hat y = G_{\text{proxy}}(q)u$ to approximate past values and construct a proxy regressor $\tilde x(k).$

The input $u$ is constructed in incremental piecewise-constant "chunks" (subsequences). At each iteration, the algorithm:

1. Selects the input channel with the fewest generated samples.
2. Generates a library of candidate subsequences, defined by pairs $(u_{\text{new},j}, L_{\text{new},j})$, where $u_{\text{new},j}$ is chosen from a finite grid and $L_{\text{new},j}$ is the segment length (bounded per estimated time constant).
3. For each candidate, computes the sequence quality:

$J = J_1 - \lambda J_2,$

where $J_1 = \sum d_{\text{NN}}(\tilde X, \tilde x(k))$ is the sum of nearest-neighbor distances in proxy regressor space (promoting maximin coverage), and $J_2 = F\cdot(L^n)_{\text{scaled}}$ is a penalty favoring shorter, more dynamic sequences. $F$ inversely measures the average step size in proxy space (dynamic content), and the exponent $n \geq 1$ shapes how strongly short sequences are preferred.

The optimal candidate is appended to the sequence. The process is repeated until the total desired sequence length is achieved. The trade-off parameter $\lambda \geq 0$ allows smooth interpolation between pure space-filling (steady-state emphasis) and strongly dynamic regimes, while $n$ controls the distribution of segment lengths.

3. Selective Emphasis and Trade-Off Control

The signal quality function $J = J_1 - \lambda J_2$ provides an explicit mechanism for targeting different identification objectives:

• For $\lambda = 0$, the algorithm prioritizes space-filling, yielding long, flat subsequences optimized for steady-state or slowly varying dynamics.
• For $\lambda \gg 0$, the length-penalty dominates, encouraging short, frequent transitions and increased frequency richness, even if this requires revisiting already-filled regions.

The penalty scaling factor $F$, based on the inverse of average step size in proxy regressor space, further modulates the trade-off by reducing penalty for sequences with more dynamic content. As the proxy regressor space becomes saturated, the influence naturally shifts from coverage to dynamic probing. The average segment length $\langle L \rangle$ thus evolves with the stage of the signal design process:

• In sparse regimes, $J_1$ dominates (space-filling).
• As coverage improves, $\lambda J_2$ becomes the main contributor (dynamic probing) (Herkersdorf et al., 2024).

4. Computational Aspects

Each subsequence selection step involves evaluating approximately $M_j \cdot L_{\max,j}$ candidates, with a brute-force nearest-neighbor distance computation for $J_1$ costing $O(L \cdot N_{\text{reg}})$. Here, $N_{\text{reg}}$ denotes the current size of the regressor dataset. Typical $M_j$ and $L_{\max, j}$ values ($\approx 10 – 20$) keep computational demands practical.

Overall, the worst-case complexity is $$O(N_{\text{total}} \cdot M \cdot L_{\max} \cdot N_{\text{total}})$$. Efficiency can be significantly improved by employing $k$-d trees or approximate nearest neighbor methods for distance calculations. As a result, real-time feasibility is achievable for both offline signal planning and online/adaptive signal updating by shortening the look-ahead horizon. The piecewise-constant nature of the resulting sequences aligns well with actuator constraints in practical systems.

5. Empirical Performance and Application Example

In a benchmark Hammerstein-type process

$$y(k) = 0.2 f(u(k-1)) + 0.8 y(k-1),\quad f(x) = \frac{\operatorname{atan}(8x-4)+\operatorname{atan}(4)}{2\, \operatorname{atan}(4)},\quad u \in [0,1],$$

the ISE Block (IDS-FID) was evaluated using training datasets of 300 samples and tested for (1) steady-state accuracy (ramp input), (2) transient capture (APRBS with short holding periods), (3) low- and high-frequency response (multisine inputs). Test sets targeted static, mid-frequency, and high-frequency dynamics.

Key performance findings include:

• $\lambda = 0$ (pure space-filling) achieves minimal RMSE on steady-state (ramp) tests but performs less well on high-frequency dynamics.
• Intermediate values ($\lambda \approx 0.02–0.5$) yield 20–30% lower mid- and high-frequency RMSE compared to OMNIPUS, APRBS, and multisine signals, with only slight losses in steady-state accuracy.
• $\lambda = 2$ confers best high-frequency performance (up to 40% RMSE improvement) at the cost of $\sim 15\%$ higher ramp RMSE.
• Across combined static-dynamic test sets, the ISE Block outperforms traditional PRBS, chirp, and pure space-filling approaches in terms of model accuracy (Herkersdorf et al., 2024).

6. Advantages, Limitations, and Extensions

The principal advantages of the ISE Block, as realized by the IDS-FID algorithm, are:

• Integration of space-filling and frequency-shaping objectives within a single information-theoretically motivated algorithm.
• Minimal system knowledge required; only rough proxy model time constants are necessary.
• Direct user control of the coverage/dynamic trade-off via the parameters $\lambda$ and $n$.
• Generation of actuator-friendly, piecewise-constant excitation sequences.

Key limitations include higher computational demand than fixed chirp or PRBS approaches, albeit manageable with approximate nearest neighbor search and feasible for offline or online deployment. The approach's coverage fidelity depends on the proxy model's accuracy, especially regarding time constants, but performance degradation from reasonable mismatches is modest.

Potential extensions include:

• Online/adaptive deployment with proxy model updates for sequential experiment design.
• Explicit frequency-domain weighting within $J_1$, supporting frequency-focused identification tasks.
• Multi-objective designs combining determinant-based (D-optimal) and space-filling criteria.
• Use in dual-control frameworks, enabling joint identification and control objectives.

In practical scenarios, the ISE Block can function as a standalone software module, returning custom excitation sequences based on user-specified priorities and physical constraints. If used in an online fashion, it can be updated recurrently as model certainty improves, thereby closing the gap between modern design of experiments (DoE) theory and practical nonlinear identification (Herkersdorf et al., 2024).