Deterministic Linear Systems Overview

• Deterministic linear systems are mathematical models governed solely by linear equations where system dynamics depend exclusively on initial conditions and inputs.
• They underpin classical control theory, signal processing, and network coding through precise methods such as the deterministic Kaczmarz algorithm and spectral analysis.
• Recent research extends these systems to high-dimensional and networked contexts, enabling robust model recovery, stabilization, and causal inference.

Deterministic linear systems are mathematical models in which the evolution of system variables is governed by linear equations and the dynamics are entirely determined by initial conditions, system parameters, and inputs, without any stochastic or random influences in the model specification. These systems form the backbone of classical control theory, communications, signal processing, and form the baseline against which robustness, uncertainty, and stochasticity are subsequently studied. Recent research has substantially broadened the scope of deterministic linear systems, encompassing aspects of system identification, control, causal inference, coding, stability, and extensions to high-dimensional and networked contexts.

1. Foundational Structures and Representations

Deterministic linear systems are classically represented in continuous or discrete time as:

• Continuous time: $\dot{x}(t) = A x(t) + B u(t)$, $y(t) = C x(t) + D u(t)$,
• Discrete time: $x_{k+1} = A x_k + B u_k$, $y_k = C x_k + D u_k$,

where $x$ is the state vector, $u$ the deterministic input, $y$ the output, and $A,B,C,D$ system matrices. For infinite-dimensional spaces or systems with convolutive dynamics, the impulse response $ϕ(t)$ and corresponding Hankel integral operators $T_ϕ$ become central objects (2409.15826).

Transfer function representations via the Laplace or $Z$-transform encapsulate the input-output relationships:

$G(s) = C(sI - A)^{-1}B + D,$

translating time-domain behaviors into algebraic operations over a complex frequency variable.

Layered and Networked Structures

In networked communication or relay systems, the system may be modeled as a layered directed graph in which nodes perform deterministic linear operations on the transmitted signals (1101.2937). These structures underpin network coding theory and distributed control.

2. Algorithmic Methods and System Identification

Deterministic approaches feature in both direct solution of equations and in system characterization from data.

Solving Linear Systems

For consistent linear systems, deterministic iterative algorithms such as the Kaczmarz method—adapted to use reflections rather than projections—exploit system geometry by generating sequences of points on spheres centered around a solution. The averaging procedure over spheres, controlled by an eigengap parameter $\eta(A)$, quickly yields accurate solutions. The relation to the condition number $\kappa(A)$ captures sensitivity to conditioning due to system matrix properties (2105.07736).

Algorithm Example: Deterministic Kaczmarz with Reflections

Let $A \in \mathbb{R}^{m \times n}$:

alpha = 2 # Reflection x_k1 = x_k + 2 * (b[i_k] - A[i_k] @ x_k) / np.linalg.norm(A[i_k])**2 * A[i_k]

After sufficiently many cycles, averaging iterates as per theoretical prescription yields the solution.

System Identification and Noise Rejection

Deterministic system identification leverages frequency analysis. Observed signals—viewed as deterministic almost periodic functions—are expanded in convergent trigonometric series. Dynamic models are derived without ensemble averaging: by matching frequency components between input and output, the exact system response may be reconstructed, and noise suppressed (1301.6596).

Frequency-based Identification Workflow:

1. Obtain synchronized measurements of all inputs $x_i(t)$ and outputs $y(t)$.
2. Perform spectral (Fourier) analysis to extract discrete frequency sets for each signal.
3. Identify and remove communication/coupling frequencies to isolate linearly independent channels.
4. For each input-output pair, find coincidence of frequencies and compute transfer functions at those frequencies.
5. Formulate and solve algebraic equations (arising from Fourier-transformed differential equations) to estimate system order and coefficients.

Application: Achieving high-order model recovery (orders up to 9) for real-world systems such as aircraft during auto-landing, using only a single trajectory.

3. Control and Stabilization Strategies

Deterministic Multicast Coding and Network Control

In networked relay settings, deterministic polynomial-time multicast coding can be achieved by constructing “flows” at each layer to maintain linearly independent global coding vectors for every destination. By derandomizing assignment of coding coefficients (using combinatorial lemmas), capacity-achieving schemes are produced using only $\lceil \log(g+1)\rceil$ rounds, where $g$ is the number of destinations (1101.2937).

Key formula for coding coefficient selection (for new vector $\mathbf{x}_i(q)$):

$$\mathbf{x}_i(q) = \sigma \cdot \mathbf{w},\quad \sigma \neq \frac{\mathbf{y}_i(p_\ell)\cdot \gamma_\ell - 1}{\mathbf{w}\cdot \gamma_\ell},\quad \sigma \neq -\frac{1}{\mathbf{w}\cdot \gamma_\ell}$$

for all destinations, avoiding at most $g$ “forbidden” values.

Stabilization in Switched Linear Systems

Deterministic synthesis of stabilizing switching signals is achieved by representing the system as a weighted digraph. Cycles/circuits with negative cumulative weight (contractive circuits) are sought via linear programming and graph algorithms; their repetition yields globally asymptotically stable switched systems (1405.1857).

Weighted Graph Construction:

• Vertices: subsystems (indexed by $i$).
• Edge weights: $w(i,j) = \ln \mu_{ij}$, vertex weights: $w(j) = \ln \lambda_j$.
• Stability: find a closed walk $W$ with aggregated contractivity $\Xi(W) < 1$.

4. Causality, Spectral Properties, and Inference

Deterministic linear relationships are leveraged for causal inference by exploiting spectral asymmetries rather than noise-based innovations. The Spectral Independence Criterion (SIC) posits that, under true causality ($X \rightarrow Y$ via LTI filter: $y = h * x$), the PSD of the cause and the squared magnitude of the transfer function are uncorrelated:

$$\langle S_{XX}(\nu)|\hat{h}(\nu)|^2\rangle = \langle S_{XX}(\nu)\rangle \langle|\hat{h}(\nu)|^2\rangle$$

Measurement of the spectral dependency ratio in both directions facilitates identification of causal structure in deterministic or nearly deterministic time series, outperforming standard Granger causality in noise-insensitive contexts (1503.01299).

Deterministic Structural Causal Models

Extension of linear SCMs to deterministic relations allows identifiability and recovery of propagation structures where variables are exact functions of parents and source mixtures. Recovery is possible via careful comparison of source components and application of combinatorial criteria such as the “marriage condition” for unique identifiability (2111.00341).

5. Stability, Robustness, and Survivability

Joint Spectral Radius and Lyapunov Exponents

The analysis of stability for deterministic switched systems invokes the joint spectral radius (JSR) in discrete time and Lyapunov exponents in continuous time. The deterministic JSR defines worst-case exponential growth/decay rates. Comparisons with probabilistic (randomly switched) analogues reveal cases where the worst-case (deterministic) and average-case (probabilistic) exponents differ, and characterize when equality is attainable—for example, when every cycle under the Markov switching reaches the maximal JSR (1812.08399, 2112.07005).

Survivability Beyond Asymptotics

Survivability quantifies the probability that, for a random initial condition, the entire trajectory remains within a desirable region, not just ultimately converging to a safe state. For deterministic linear $N$-dimensional systems:

$$S(t) = \frac{\mathrm{Vol}(X^S_t)}{\mathrm{Vol}(X^+)}$$

where $X^S_t$ is the set of safe initial conditions. Semi-analytic lower bounds for linear systems relate the survivability directly to eigenvectors of system matrices, revealing a dependence beyond eigenvalues (decay rates) (1506.01257). This provides actionable metrics for real-world safety-critical systems such as power grids and engineered infrastructures.

6. Extensions: High-Dimensional, Data-Driven, and Networked Contexts

High-Dimensional Deterministic Systems and Linear Response

Remarkably, high-dimensional deterministic systems may obey linear response theory for macroscopic observables—even though all their microscopic components individually violate it. This is attributed to the ensemble average effect (central limit theorem) as the number and heterogeneity of unresolved degrees of freedom increase, smoothing out non-differentiabilities and enabling classical response predictions (1801.09377).

Data-Driven Control in Deterministic Systems

In data-driven predictive control (DPC), the solution to control optimization formulated purely from measured input–output data matches, in both tractability and explicit law complexity, the equivalent solution from model-predictive control (MPC). Systematic elimination of redundant variables in the DPC optimization yields the same number of regions in the piecewise affine explicit solution, offering a direct equivalence in deterministic settings (2206.07025).

Deterministic Solvers for Network Laplacians

Algorithmic advances provide deterministic, nearly-linear time solvers for linear systems involving directed graph Laplacians. Key developments include the partial symmetrization of Laplacians, which allows for efficient deterministic sparsification and preconditioning, leading to deterministic algorithms for computing random walk stationary distributions, PageRank, and fundamental network quantities (2208.10959).

7. Spectral Curves, Hankel Operators, and Integrable Structures

The interplay between deterministic continuous-time linear systems ($-A, B, C$) and spectral theory is articulated through the association to Hankel operators $T_\phi$ with kernel $\phi(x+y)$ and Fredholm determinants $\det(I + T_\phi)$. Through the Faddeev-Dyson formula and relations to the tau function, these determinants link directly to the potential of associated Schrödinger operators and, under certain hierarchy-terminating conditions, to algebraic curves—realizing Burchnall-Chaundy’s commutative algebra of differential operators as functions on hyperelliptic curves (2409.15826). This connects deterministic system theory to algebraic geometry and random matrix theory, particularly through the role of Hankel determinants in describing eigenvalue distributions.

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Deterministic linear systems thus bridge foundational mathematical structures, algorithmic and control theory developments, data-driven and spectral inference paradigms, and deep connections to integrable systems and mathematical physics. Recent literature underscores the robustness, tractability, and broad applicability of deterministic analysis, while also continually expanding its frontier toward high-dimensionality, uncertainty, causality, and networked interactions.