Visual State-Space Blocks for Converter Modeling

• Visual State-Space Blocks are a modular framework for modeling electrical converter systems by encapsulating subsystems into canonical state-space forms.
• They enable seamless visual assembly through standardized interconnections that mirror physical topology and facilitate error reduction.
• The approach supports direct extraction of control-oriented transfer functions, aiding design validation and iterative model updates.

Visual State-Space Blocks constitute the modular foundation of a rigorous, hierarchical modeling framework for electrical converter systems, especially DC–DC converters, via state-space theory. Each block implements an explicit state-space description for a fundamental subsystem (e.g., a converter, filter, load, or controller), supports standardized interconnection, and facilitates visual composition into complex system-level models. This approach enables the systematic representation, integration, and structural updating of converter models, allowing for the seamless derivation of control-oriented frequency-domain characteristics and facilitating robust analyses across a wide spectrum of configurations.

1. State-Space Block Fundamentals and Mathematical Structure

At the core of the visual state-space block paradigm is the encapsulation of converter subsystems in canonical state-space form. For example:

• Passive two-port subsystem:

$$\begin{aligned} \dot{x}(t) &= A x(t) + [B_1 \;\; B_2] \begin{bmatrix} v_{\mathrm{in}}(t) \ i_{\mathrm{out}}(t) \end{bmatrix} \ \begin{bmatrix} i_{\mathrm{in}}(t) \ v_{\mathrm{out}}(t) \end{bmatrix} &= \begin{bmatrix} C_1 \ C_2 \end{bmatrix} x(t) + \begin{bmatrix} D_{11} & D_{12} \ D_{21} & D_{22} \end{bmatrix} \begin{bmatrix} v_{\mathrm{in}}(t) \ i_{\mathrm{out}}(t) \end{bmatrix} \end{aligned}$$

• Controlled converter block (additional control input):

$$\begin{aligned} \dot{x}(t) &= A x(t) + [B_1 \;\; B_2 \;\; B_3] \begin{bmatrix} v_{\mathrm{in}}(t) \ i_{\mathrm{out}}(t) \ ctl(t) \end{bmatrix} \ \begin{bmatrix} i_{\mathrm{in}}(t) \ v_{\mathrm{out}}(t) \end{bmatrix} &= \begin{bmatrix} C_1 \ C_2 \end{bmatrix} x(t) + \begin{bmatrix} D_{11} & D_{12} & D_{13} \ D_{21} & D_{22} & D_{23} \end{bmatrix} \begin{bmatrix} v_{\mathrm{in}}(t) \ i_{\mathrm{out}}(t) \ ctl(t) \end{bmatrix} \end{aligned}$$

• General controller block:

$$\begin{aligned} \dot{x}_C(t) &= A_C x_C(t) + B_C e(t) \ u(t) &= C_C x_C(t) + D_C e(t) \end{aligned}$$

This precise interface—two electrical ports plus (optionally) a control input/output—ensures composability and clarity in subsystem interaction.

2. Visual Compositionality and Block Diagram Integration

All building blocks are visualized as standard two-port entities (with explicit inputs/outputs: voltages and currents, plus control signals). This uniform formalism enables:

• Direct visual composition—connecting input/output ports to mimic the physical topology.
• Traceable signal flow: power flow (via electrical ports) and control flow (via control signal feeds).
• Seamless augmentation or reconfiguration, such as the insertion of input filters, loads, or the addition/alteration of controllers.

Hierarchically, this allows system-level diagrams akin to electrical schematics, but underpinned by exact state-space algebra.

3. Standardized Interconnection and System Assembly

Sub-blocks are connected using standardized algebraic rules rigorously derived from system interconnection theory.

For series/terminal interconnection (e.g., connecting a filter to a converter):

• Terminal variables are equated as per continuity of voltage and conservation of current (e.g., $v^{S}_{\mathrm{out}} = v^{L}_{\mathrm{in}}$, $i^{S}_{\mathrm{out}} = -i^{L}_{\mathrm{in}}$).
• The combined block’s state-space matrices (denoted $A$, $B$, $C$, $D$) are then constructed using explicit blockwise formulas (see eqn:ConnectSystems_Model and eqn:ConnectSystems_ABCD), allowing push-button system assembly once individual blocks are available.

When closing control loops (connecting a controller to a converter), the combined open-loop model is formed by stacking their states and augmenting the input vector. Closed-loop formation employs an algebraic update:

$$A_{\mathrm{CL}} = A_{\mathrm{OL}} - B_{\mathrm{OL}} K$$

where $K$ acts as the feedback gain, selected according to the loop type (current or voltage).

4. Control-Oriented Characteristic Extraction

The universal block format facilitates immediate extraction of transfer functions, including:

• Control-to-output:

$G_{co}(s) = C_2 (sI - A)^{-1} B_3 + D_{23}$

• Input admittance and output impedance via respective output/input matrices.

Because the same $A$, $B$, $C$, $D$ descriptors are maintained throughout composition, transfer functions for subsystems (or the whole system) are always accessible at each assembly stage.

5. Modularity, Updatability, and Error Reduction

Key advantages of the visual state-space block methodology are:

• Modularity: Any block, such as an input filter, load, or controller, can be replaced or modified without requiring global re-derivation of the system equations.
• Locality of updates: Model changes (e.g., swapping a controller design) require only local block adjustments.
• Reduction of human error: Standard operation formulas and structured interconnections eliminate manual algebraic errors endemic to monolithic modeling.
• Reusability: Standard blocks (e.g., common LC filters, converter topologies, controller architectures) can be pre-defined and reused across designs.
• Visual traceability: The block diagram mirrors the physical/electrical circuit, making dependencies and couplings explicit.

6. Examples Demonstrating Block Utility

Case studies include:

• Buck converter with multiloop control: Blocks for the converter, current loop (Type 1/2), and voltage loop are composed, with controller feedback realized via block-structured interconnections. Analytical and simulated responses confirm the constructed system’s fidelity and the correctness of the derived transfer functions.
• Boost converter with input filter: The effect of filter insertion on frequency-domain characteristics (e.g., impedance plots) is captured through system assembly, utilizing the series connection operation.
• Cascaded converter stages: Connecting boost and buck converters, each with their own models (including filters and multi-loop control), is handled via repeated application of standardized connection rules. Derived frequency and time-domain responses confirm model validity.

7. Impact on Analysis, Synthesis, and Design of Converter Systems

The visual state-space block approach enables converter system engineers and researchers to:

• Rapidly prototype system-level models accommodating arbitrary combinations of converters, filters, controllers, and loads.
• Systematically adjust or extend models, such as for design iterations, what-if studies, or tolerance analyses.
• Conduct in-depth frequency-domain and time-domain analyses for design validation, controller synthesis, and stability assessment.
• Integrate and analyze nested or multiloop controllers, cascaded converters, or systems of arbitrary modular complexity, without manual re-derivation.

This approach establishes a solid foundation for modular, visual-algebraic modeling of converter systems, substantially streamlining both the modeling workflow and the transition from schematic to analytical state-space description (Herbst, 2019).