Linear Input-Varying Systems (LIVs)

• Linear Input-Varying Systems (LIVs) are linear control systems whose matrices or coefficients vary with an external scheduling parameter, extending classical LTI and LTV models.
• They leverage state-space and input–output representations where the system matrices depend affinely or functionally on scheduling signals, facilitating precise system identification and robust controller synthesis.
• Applications span aerospace, process control, and smart grids, with advanced methods like generalized convolution and minimal realization techniques ensuring efficient observer and controller design.

Linear Input-Varying Systems (LIVs) are linear control systems whose defining matrices or coefficients vary as a function of an exogenous, time-dependent signal—typically referred to as a “scheduling parameter.” LIVs inherit and generalize properties of classical Linear Time-Varying (LTV) and Linear Parameter-Varying (LPV) systems, playing a central role in modeling, analysis, system identification, and robust controller synthesis for complex systems subject to external influences that alter linear dynamics. In state-space and input–output formulations, the linear structure is retained with respect to the state and input, but crucially, the system matrices depend affinely, polynomially, or functionally on a scheduling parameter, which itself is determined operationally by external signals or possibly the input itself.

1. Formal Definitions and State-Space Structures

A prototypical discrete-time LIV (or affine LPV) system is characterized by state and output equations with matrices affinely dependent on a scheduling signal $p(t)$,

$$x(t+1) = \sum_{q=1}^{D} [A_q x(t) + B_q u(t)] p_q(t) \ y(t) = \sum_{q=1}^{D} C_q x(t) p_q(t)$$

where $p(t) = [p_1(t), ..., p_D(t)]^\top$ typically spans a basis of $\mathbb{R}^D$. This models dynamic behavior whose structure changes according to an observed, time-varying parameter. In input–output form, LIVs may be described via equations where output and input coefficients are affine in the current and delayed scheduling parameter sequence,

$$y_k + \sum_{i=1}^{a} a_i(p_{k-i}) y_{k-i} = \sum_{i=0}^{b} b_i(p_{k-i}) u_{k-i}$$

By recasting the scheduling parameter as an input-varying signal, this structure generalizes the linear time-invariant and time-varying cases, allowing system coefficients to reflect external or operational variability.

2. Realization Theory, Generalized Convolution and Markov Parameters

Kalman-style realization theory for LIVs (specifically affine LPV systems) is rooted in generalized convolution representations and the construction of Markov parameters as functions of input–output maps over finite scheduling segments (Petreczky et al., 2012). For an input–output sequence $w = (p(0), u(0)), ..., (p(t), u(t))$,

$$f(w) = \sum_{k=0}^{t-1} \bigg\{ \sum_{v \in Q^+, |v|=t-k+1} S^f(v) \cdot (p(k)...p(t))^v \bigg\} u(k)$$

where $S^f(v)$ encapsulate system responses to impulse-like sequences under a particular scheduling path. The associated Hankel matrix $H_f$, constructed from $S^f(v)$, has finite rank if and only if a finite-dimensional affine realization exists, serving as the necessary and sufficient realization condition. Classical algorithms, such as the Kalman–Ho method, are extended to extract such realizations for LIVs.

3. Minimality, Reachability, Observability

Minimal realizations—those achieving the smallest state space consistent with external input–output behavior—require both reachability and observability in the affine dependence on the scheduling parameter (Petreczky et al., 2012). For $x(t)$,

• A realization is minimal iff the reachable subspace (under all admissible scheduling sequences and inputs) spans the entire state space, and the observability map distinguishes all states.
• Two minimal affine LPV realizations (and thus LIVs) generating identical input–output behavior are isomorphic via a parameter-independent similarity transformation. Minimality supports effective controller synthesis and system identification by ensuring no redundant internal structure.

4. Connections to Switched and Hybrid Systems

LIVs are closely related to linear switched systems (Petreczky et al., 2012). By constraining the scheduling parameter to the standard basis vectors, an affine LPV system becomes a switched linear system. This relation is formalized so that properties such as reachability, observability, and minimality are preserved through the transformation. The large body of technical results and realization theory for switched systems (e.g., via formal power series and Hankel matrices) is thus applicable to LIVs, providing algorithmic and theoretical tools for their analysis and synthesis.

5. Input–Output Equations and Algebraic Constraints

Beyond state-space realization, LIVs can be equivalently described via input–output equations involving affine polynomials in the scheduling parameter and shifted signals (Petreczky et al., 2012). A generic form for a single-output system is

$$E(P, Y, U) = \sum_{j=0}^{n} Q_j(P) Y_j + \sum_{i=1}^n \sum_{l=1}^m L_{i,l}(P) U_{i,l} = 0$$

where $Q_j(·)$ and $L_{i,l}(·)$ are polynomials in the scheduling parameters at specific delays. The existence of such equations is both necessary and sufficient for realization by an affine LPV (and thus LIV) system. This offers an alternative for identification and controller design, sometimes more tractable in handling noisy or complex data.

6. System Identification, Controller Synthesis, and Data-Driven Methods

Identification and synthesis for LIVs benefit from direct state-space realization results (Kon et al., 26 Feb 2025), which provide algorithms to realize input–output models without introducing dynamic or nonlinear scheduling dependency. Reachability is ensured under a coprimeness condition of the coefficients, and reconstructibility guarantees that non-observable directions decay in finite steps. Data-driven gain scheduling (Miller et al., 2022), dissipativity analysis (Verhoek et al., 2023), and optimization methods via quadratic matrix inequalities (QMIs) offer practical frameworks for robust control and performance certification. Vertex enumeration and polytopic representations are leveraged to avoid computational bilinearities and facilitate tractable optimization.

7. Applications and Extensions

LIVs are foundational to robust control of time-varying or parameter-dependent systems in aerospace, process control, automotive and energy systems, multi-agent networks, and smart grids. Their theoretical foundations underpin advanced methods such as set-valued $\mathcal{H}_\infty$ observers (Khajenejad et al., 2020), optimal filtering for unknown input estimation (Lu et al., 2016, Yong et al., 2016), and extensions to the Koopman operator framework, where lifted nonlinear models exhibit state-dependent input matrices in the LPV form (Iacob et al., 2022). Control strategies for LIVs have been shown effective under unknown time-varying input delays using parameterized Lyapunov functionals for stabilization (Zhou et al., 30 Apr 2025). Moreover, the modeling and stability analysis of “live systems” generalize LIVs to encompass systems with time-varying state dimension, leveraging input-to-state stability (ISS) and Lyapunov methods (Mironchenko, 27 Jan 2025).

8. Summary Table: Key Elements in LIV Theory and Practice

Feature	Technical Property	Significance
State-space realization	Affine in scheduling parameter	Enables efficient model extraction
Hankel finite-rank	Necessary and sufficient for realization	Guides system identification
Minimality (reach/observe)	Isomorphic realizations	Supports robust analysis and controller design
Switched system equivalence	Transformation via basis restriction	Facilitates use of switched systems theory
Input-output polynomials	Algebraic constraint equivalence	Alternative route to system realization
Data-driven/LMI/SDP	Vertex enumeration, QMI, finite LMIs	Addresses computational issues

The development of comprehensive realization theory for LIVs, together with associated identification, control, and observer synthesis methods, establishes a rigorous foundation for the analysis and control of systems subject to dynamic external variability. LIV theory extends the reach of classical linear systems to modern, scalable, and data-driven applications where simple time-invariant models are no longer sufficient.