State-Space Averaging Model

• State-space averaging is a technique that replaces detailed switching dynamics with a single averaged LTI model, enabling simplified analysis in power electronics.
• It relies on high-frequency and small-ripple assumptions and employs the Baker–Campbell–Hausdorff formula to connect the Poincaré map with continuous dynamics.
• Widely used in DC-DC converter design, it facilitates control linearization and simulation, while noting limitations in multi-interval switching and high-order effects.

State-space averaging (SSA) is a model order reduction technique applied to periodically switched, piecewise-linear dynamical systems, particularly in power electronics. SSA is utilized to replace the detailed, time-varying dynamics of a system with a single equivalent continuous-time linear time-invariant (LTI) model, greatly simplifying analysis and control design. Classical SSA arises as the leading-order truncation in an operator-theoretic reconstruction that relates the exact period-mapped baseline (Poincaré map) to continuous-time dynamics, valid under high-frequency and small-ripple assumptions (Yang et al., 20 Dec 2025, Nwachukwu, 12 Jul 2025). This framework is foundational in modern power converter analysis, especially for applications such as DC-DC conversion, where high-frequency switching and small ripples make the approximation accurate.

1. Mathematical Formulation and Operator-Theoretic Foundations

SSA formalizes the replacement of a piecewise-linear, periodically switched system over period $T$ by its averaged counterpart. Consider a period divided into $m$ subintervals with durations $d_k$, $\sum_{k=1}^m d_k = T$. The system evolves under

$$\dot{x}(t) = A_k x(t) + B_k u(t), \quad t \in [d_1+\cdots+d_{k-1},\,d_1+\cdots+d_k].$$

The exact one-period evolution is captured by the state transition (Poincaré) map:

$\Phi = \prod_{k=1}^m e^{A_k d_k}$

A continuous-time surrogate $A_{\mathrm{eq}}$ must satisfy

$$e^{A_{\mathrm{eq}} T} = \Phi \implies A_{\mathrm{eq}} = \frac{1}{T} \log\left(\Phi\right)$$

To bridge to classical SSA, the matrix logarithm is expanded using the Baker–Campbell–Hausdorff (BCH) formula. Writing $X_k \equiv A_k d_k$,

$$\log\left(\prod_{k=1}^m e^{X_k}\right) = \sum_{k=1}^m X_k + \frac{1}{2} \sum_{i<j} [X_j, X_i] + \mathcal{O}(\|X\|^3)$$

where $[X_j, X_i]=X_j X_i - X_i X_j$ is the commutator. Dividing by $T$, the exact reconstruction is

$$A_{\mathrm{eq}} \approx \frac{1}{T} \sum_{k=1}^m A_k d_k + \frac{1}{2T} \sum_{i<j} [A_j d_j, A_i d_i] + \mathcal{O}(T^2)$$

Classical SSA keeps only the first (“averaging”) term:

$A_{\mathrm{avg}} = \sum_{k=1}^m \frac{d_k}{T} A_k$

The neglected (ripple) corrections are higher-order in $T$ and depend on commutators among the $A_k$.

2. High-Frequency and Small-Ripple Regimes: Justification and Breakdown

The validity of SSA is predicated on two principal conditions:

• High switching frequency ($T$ small): All $X_k = A_k d_k = \mathcal{O}(T)$ are small, making commutator and higher-order terms $\mathcal{O}(T^2)$ and negligible compared to $A_{\mathrm{avg}}$.
• Small ripple: The true periodic orbit $X^*$ differs from the SSA steady state $X^*_{\mathrm{avg}}$ by terms of order (ripple). When this difference is small, duty-cycle injection is linearizable and well-approximated in the averaged form [(A.2) in (Yang et al., 20 Dec 2025)].

In systems with more than two subintervals ($m\geq3$), combinatorially many commutators emerge, their contributions are both pattern-dependent and less likely to cancel. In such cases, higher-order terms in the BCH expansion may introduce bias into the SSA model, leading not only to quantitative but also to qualitative errors.

3. SSA Model Construction for Switched Power Converters

SSA is especially prevalent in power electronics for simplifying nonlinear, switched dynamics into tractable LTI models. The canonical application is the DC-DC buck converter (Nwachukwu, 12 Jul 2025). In continuous-conduction mode (CCM), the model comprises “ON” and “OFF” states:

State variable: $x(t) = [i_L(t), v_C(t)]^T$, input $u(t) = V_g$, output $y = Cx = v_C$.

ON-state:

$\dot{x} = A_{\mathrm{on}} x + B_{\mathrm{on}} V_g$

with

$$A_{\mathrm{on}} = \begin{bmatrix} - R_L/L & -1/L\ 1/C & -1/(RC) \end{bmatrix} ,\quad B_{\mathrm{on}} = \begin{bmatrix} 1/L\ 0 \end{bmatrix}$$

OFF-state: $A_{\mathrm{off}} = A_{\mathrm{on}}$, $B_{\mathrm{off}} = [0; 0]$.

The averaged model for duty ratio $d$ is:

$\dot{x} = A x + d\,B_{\mathrm{on}} V_g$

This LTI-in-$d$ structure underpins linearization and control design.

4. Small-Signal Model and Frequency Response

For control purposes, SSA enables small-signal linearization about a steady-state operating point. Decomposing all variables into steady and small-signal parts:

$$x(t) = X + \tilde{x}(t),\quad d(t) = D + \tilde{d}(t)$$

The averaged incremental model is

$$\Delta\dot{x} = A\,\Delta x + V_{g0} B_{\mathrm{on}}\,\Delta d$$

$\Delta y = C\,\Delta x$

Laplace transforming and solving yields the small-signal transfer function from duty cycle to output voltage:

$$G_{vd}(s) = \frac{V_{g0}}{L C} \cdot \frac{1}{s^2 + \left(\frac{R_L}{L} + \frac{1}{RC}\right) s + \left(\frac{R_L}{LRC} + \frac{1}{LC}\right)}$$

In the lossless limit ($R_L \approx 0$):

$$G_{vd}(s) = \frac{V_{g0}}{LC} \cdot \frac{1}{s^2 + \frac{1}{RC}\,s + \frac{1}{LC}}$$

This result underlines the tractability of control-oriented analysis enabled by SSA (Nwachukwu, 12 Jul 2025).

5. Limitations: Multi-Interval Fragility and Validity

SSA’s simplifying assumptions inherently introduce limitations:

• Loss of high-frequency detail: SSA filters out the explicit time-varying nature of switching, rendering it incapable of predicting subharmonic instabilities, EMI, or other carrier-frequency effects.
• Mode constraints and ripple: SSA is accurate in CCM with small ripple but does not capture discontinuous-conduction mode (DCM) or nonlinear behavior manifesting under large duty-ratio steps or light load.
• Multi-subinterval complexity: For converters with $m \ge 3$ subintervals, commutator-induced bias becomes significant, and leading-order averaging no longer dominates; model error can become both structural and quantitative (Yang et al., 20 Dec 2025).

A plausible implication is that advanced reconstruction—using higher-order BCH expansion or direct operator techniques—may be required in topologies with complex or non-binary switching schemes.

6. Low-Complexity SSA-Flavored Implementation

Implementation of exact or near-exact SSA-type models can be achieved without explicit matrix logarithm or eigendecomposition, through algebraic invariants and minimal real-lift constructions, especially for $2\times 2$ or sign-symmetric system matrices:

• Eigenvalue computation via trace and determinant:

$\lambda_{1,2} = \frac{\tr(\Phi) \pm \sqrt{[\tr(\Phi)]^2 - 4\det(\Phi)}}{2}$

• Exponentials and logarithms for $2\times2$ matrices:

$$e^{\Omega} = e^{\mu}\left(\cosh\Delta \, I + \frac{\sinh\Delta}{\Delta}(\Omega - \mu I)\right)$$

where $\mu = \tr(\Omega)/2$, $\Delta^2 = \mu^2 - \det(\Omega)$.

• Minimal real-lift: For matrices lacking a real logarithm due to negative eigenvalues, embedding into a higher-dimensional real matrix ($3\times3$ “lift”) recovers a continuous-time surrogate preserving exactness on the original state (Yang et al., 20 Dec 2025).

This strategy maintains SSA’s practical low algebraic complexity while remaining formally connected to the exact underlying sampled-data dynamics.

7. Applications and Significance in Power Electronics Control

SSA models are a fundamental tool in power electronics, facilitating classical control methodologies—such as Bode plot analysis and PI/lead-lag compensator design—by recasting piecewise-linear switched converter operation as a singular LTI problem (Nwachukwu, 12 Jul 2025). This supports robust closed-loop regulation, simplified tuning, and transparent performance trade-offs for converters under high-frequency, small-ripple operation.

The approach supports rapid simulation and analysis, as explicit time-stepping through each switching event is unnecessary. However, these advantages are contingent on remaining within the model’s regime of validity. For systems or regimes lying outside SSA’s assumptions, higher-order operator-theoretic reconstruction, time-domain simulation, or hybrid modeling become necessary.

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In summary, state-space averaging constitutes the leading-order formalism for high-switch-frequency, small-ripple, and binary-switching systems and can be rigorously embedded within operator-theoretic exact reconstruction frameworks. Its efficiency and analytic tractability have made it the default for converter control design, though recent advances clarify its limitations and avenues for refinement via higher-order models (Yang et al., 20 Dec 2025, Nwachukwu, 12 Jul 2025).