LinearTimeInvariant.md

Linear Time-Invariant System Model

Overview & Motivation

A Linear Time-Invariant (LTI) system model is the fundamental mathematical object shared across state-space control, estimation, and signal processing. It captures the dynamics of a physical plant — a motor, a pendulum, a vehicle — in a compact matrix representation that enables the systematic application of optimal control and filtering theory.

Representing an LTI model as a first-class value type makes it possible to pass a single plant description into algorithms such as the Kalman Filter, LQR, LQG, and MPC, eliminating the error-prone practice of supplying the same system matrices (A, B, C, D) separately to each component.

Mathematical Theory

Discrete-Time State-Space Representation

A discrete-time LTI system with state $x_k \in \mathbb{R}^n$, input $u_k \in \mathbb{R}^m$, and output $y_k \in \mathbb{R}^p$ is described by two equations:

State equation: $$x_{k+1} = A x_k + B u_k$$

Output equation: $$y_k = C x_k + D u_k$$

where:

• $A \in \mathbb{R}^{n \times n}$ is the state transition matrix
• $B \in \mathbb{R}^{n \times m}$ is the input matrix (sometimes called the control matrix)
• $C \in \mathbb{R}^{p \times n}$ is the output matrix (sometimes called the measurement matrix)
• $D \in \mathbb{R}^{p \times m}$ is the feedthrough matrix (often zero for strictly proper systems)

Special Cases

Full-state output ($C = I$, $D = 0$): When all states are directly observable (e.g., a simulation or a state-feedback controller with perfect sensing), the output equation collapses to $y_k = x_k$. This is the typical case for LQR.

Autonomous system ($B = 0$, $u_k = 0$): A system driven only by initial conditions and process noise. Used in Kalman smoothing and spectral analysis.

Discretisation

Continuous-time plants described by $\dot{x} = A_c x + B_c u$ must be discretised before use in digital controllers:

$$A = e^{A_c T_s}, \quad B = A_c^{-1}(A - I) B_c$$

where $T_s$ is the sampling period. For small $T_s$ and stable $A_c$, the forward-Euler approximation $A \approx I + A_c T_s$, $B \approx B_c T_s$ may be used.

Stability and Controllability

An LTI model is:

• Stable if all eigenvalues of $A$ lie strictly inside the unit disc ($|\lambda_i| < 1$ for discrete-time systems).
• Controllable if the controllability matrix $\mathcal{C} = [B ; AB ; A^2B ; \cdots ; A^{n-1}B]$ has full row rank.
• Observable if the observability matrix $\mathcal{O} = [C^T ; (CA)^T ; \cdots ; (CA^{n-1})^T]^T$ has full column rank.

Controllability and observability are prerequisites for LQR and Kalman Filter stability, respectively.

Complexity Analysis

Operation	Time	Space	Notes
Step (state)	$O(n^2 + nm)$	$O(n)$	One matrix-vector multiply per term
Output	$O(pn + pm)$	$O(p)$	One matrix-vector multiply per term
Construction	$O(1)$	$O(n^2 + nm + pn + pm)$	Value-type struct, no allocation

Numerical Considerations

• Condition number of A: Poorly conditioned $A$ amplifies numerical errors during iterative integration. Monitor the spectral radius of $A$ to ensure stability is not lost due to finite-precision arithmetic.
• Feedthrough matrix D: For most physical systems $D = 0$ (strictly proper). A nonzero $D$ requires care in closed-loop analysis since direct input-to-output coupling may cause algebraic loops.
• Fixed-point use: LTI matrices are suitable for Q15/Q31 when all matrix entries and intermediate products remain within $[-1, 1]$. For real physical systems with gains exceeding unity, scale the representation (e.g., normalise state and input ranges) before using fixed-point arithmetic.
• Identity construction: When building the $C = I$ case (full-state output), use $0.9999$ rather than $1.0$ for Q15 compatibility, since $1.0$ is not representable in the Q15 format.

Pitfalls & Edge Cases

• Time-varying systems: The LTI model is invalid for systems whose matrices $A$, $B$, $C$, $D$ depend on time or operating point. Use scheduling or re-linearisation for such plants.
• Nonlinear plants: State-space linearisation is valid only in a neighbourhood of the operating point. Large deviations from the linearisation point invalidate the model.
• Sampling rate: Discrete-time LTI models are sampling-rate-specific. Changing $T_s$ requires re-discretisation and possibly re-tuning of dependent controllers and filters.

Step-by-Step Walkthrough

Consider a double integrator (position and velocity state, force input, position output) with sample period $T_s = 0.1,\text{s}$:

Continuous-time plant: $\dot{x} = \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} x + \begin{bmatrix} 0 \ 1 \end{bmatrix} u$, $y = \begin{bmatrix} 1 & 0 \end{bmatrix} x$

Step 1 — Discretise using forward-Euler ($A \approx I + A_c T_s$, $B \approx B_c T_s$):

$$A = \begin{bmatrix} 1 & 0.1 \ 0 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 0 \ 0.1 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 0 \end{bmatrix}, \quad D = \begin{bmatrix} 0 \end{bmatrix}$$

Step 2 — Simulate one step from $x_0 = [1, 0]^T$ with $u_0 = -2$:

$$x_1 = A x_0 + B u_0 = \begin{bmatrix} 1 & 0.1 \ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 \ 0 \end{bmatrix} + \begin{bmatrix} 0 \ 0.1 \end{bmatrix}(-2) = \begin{bmatrix} 1 \ -0.2 \end{bmatrix}$$

Step 3 — Compute output at $k=0$:

$$y_0 = C x_0 = \begin{bmatrix} 1 & 0 \end{bmatrix} \begin{bmatrix} 1 \ 0 \end{bmatrix} = 1$$

The position measurement is $1.0$ (exact, since $D = 0$ and noise is excluded here).

Variants & Generalizations

• Full-state output ($C = I$, $D = 0$): All states are directly available as outputs. Used in LQR with perfect sensing. Constructed via the WithFullStateOutput factory, which sets $C$ to the identity.
• Autonomous system ($B = 0$): No input; the system evolves freely from initial conditions. Used in Kalman smoothing and spectral analysis (e.g., auto-regressive models).
• Strictly proper systems ($D = 0$): Most physical plants have zero feedthrough; the $D$ matrix is omitted from the signal path. The LTI representation preserves $D$ for generality.
• Continuous-time counterpart: Described by $\dot{x} = A_c x + B_c u$, $y = C_c x + D_c u$. Requires discretisation before use in digital controllers.
• Multi-rate LTI: Different sensors and actuators may operate at different sample rates; a lifted LTI representation can handle periodic multi-rate systems.

Applications

• Digital control design: The LTI model is the canonical starting point for controller synthesis (LQR, MPC, pole placement). A linearised plant description is the single shared artefact that all design tools consume.
• Kalman Filter initialisation: The $(A, B, C)$ matrices are passed directly into the Kalman Filter to define the predict and update equations, ensuring consistency between plant and estimator.
• Simulation and validation: Monte Carlo simulations propagate an LTI model forward to verify closed-loop stability margins, noise sensitivity, and disturbance rejection before deployment on hardware.
• System identification: Identified discrete-time state-space models from input-output data (e.g., via subspace identification or prediction-error methods) are naturally expressed as LTI structs.
• Transfer function analysis: The $z$-domain transfer function $H(z) = C(zI - A)^{-1}B + D$ can be extracted from the LTI matrices for frequency-response computation and Bode plotting.

Connections to Other Algorithms

• LQR: Uses the $(A, B)$ pair to solve the DARE and compute the optimal state-feedback gain.
• Kalman Filter: Uses $(A, B, C)$ to predict the next state and relate states to measurements.
• LQG: Composes both — $(A, B)$ for the regulator and $(A, B, C)$ for the estimator.
• MPC: Uses $(A, B)$ to build the prediction model over a finite horizon.
• Kalman Smoother: Uses $(A, C)$ for the forward-backward pass in batch smoothing.
• DARE Solver: Receives $(A, B)$ as inputs to compute the Riccati solution.

References & Further Reading

• C.-T. Chen, Linear System Theory and Design, 4th ed., Oxford University Press, 2013. Chapters 4–5.
• R. E. Kalman, "Mathematical Description of Linear Dynamical Systems," SIAM Journal on Control, 1(2):152–192, 1963.
• G. F. Franklin, J. D. Powell, and A. Emami-Naeini, Feedback Control of Dynamic Systems, 8th ed., Pearson, 2019. Chapter 7.
• T. Kailath, Linear Systems, Prentice Hall, 1980. Comprehensive reference on state-space representations and transformations.
• K. J. Åström and B. Wittenmark, Computer-Controlled Systems: Theory and Design, 3rd ed., Dover, 2011. Chapter 3 (discretisation methods).