State-Space and Generalized Network Models

Introduction

Classical network analysis commonly describes a circuit through a driving-point impedance, admittance, or transfer function. Such descriptions are effective for single-input–single-output, linear time-invariant networks under zero initial conditions, but they conceal internal energy storage and become cumbersome for multiport, switched, constrained, or high-order systems.

State-space modeling instead represents a network through a minimum set of internal variables whose instantaneous values, together with the future input, determine future behavior. Capacitor voltages and inductor currents are natural physical states because they determine stored electric and magnetic energy. Generalized network models extend this idea to descriptor systems, multiports, modified nodal analysis, nonlinear devices, discrete-time models, and parameter-varying systems.

These models support transient analysis, stability assessment, control design, estimation, model reduction, power-electronic analysis, and computer-aided network synthesis.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Formulate state-space equations from electrical networks.
2. Interpret state, input, output, and feedthrough matrices physically and dimensionally.
3. Convert between transfer-function and state-space representations.
4. analyze modes, stability, controllability, observability, and minimality.
5. formulate descriptor models using modified nodal analysis.
6. distinguish impedance, admittance, hybrid, transmission, and scattering descriptions.
7. apply state-space methods to a practical converter model.
8. recognize numerical, modeling, safety, and model-governance limitations.

1. State Variables and Dynamic Order

A state is a set of variables containing sufficient information to determine the future response of a system when the future input is specified. For an electrical network, typical state candidates are:

• independent capacitor voltages, measured in volts;
• independent inductor currents, measured in amperes;
• flux linkages or charges when these produce a more natural model;
• internal controller, machine, or distributed-element approximation states.

For a capacitor and an inductor,

where q is charge in coulombs, C is capacitance in farads, v_C is capacitor voltage, \phi is flux linkage in webers, L is inductance in henries, and i_L is inductor current.

The stored energy is

for linear capacitor and inductor matrices C and L that are symmetric positive semidefinite. If both are positive definite on the independent energy-storage coordinates, the energy is positive for every nonzero physical state.

The number of storage elements is not always equal to the dynamic order. Loops containing only capacitors and ideal voltage sources, or cutsets containing only inductors and ideal current sources, can impose algebraic constraints. Mutual coupling and dependent sources may introduce further dependence. Consequently, state selection must respect network topology.

2. Continuous-Time State-Space Representation

For a finite-dimensional linear time-invariant network, the standard model is

where:

The term Du(t) represents an output response that occurs without state integration. It is present, for example, when the measured output includes a resistor voltage directly determined by an applied current.

Matrix entries do not necessarily share one physical unit. If the states have different units, each entry of A must convert the corresponding state into the units of the relevant state derivative. Thus, assigning a single unit such as s⁻¹ to an entire heterogeneous-state matrix is generally incorrect.

Figure: A complete state-space realization. The integrator maps ẋ to x, while the D branch represents direct input-to-output feedthrough.

For an initial state x(t₀)=x₀, the solution is

This formula assumes that A and B are constant and the input is sufficiently regular for the integral to exist. The first term is the zero-input response; the second is the zero-state response.

3. Deriving a State Model from a Circuit

Worked Example 1: Series RLC Network

Consider a series circuit with R=20\ \Omega, L=0.1\ \text{H}, and C=100\ \mu\text{F}. The applied voltage u(t) is the input, and capacitor voltage v_C(t) is the output. Choose

where i_L is measured in amperes and v_C in volts. Kirchhoff’s voltage law gives

and the capacitor relation gives

Therefore,

The numerical state model is

Dimensional interpretation is essential. The entry −200 has units s⁻¹, whereas −10 converts volts to amperes per second and therefore has units A/(V·s). Similarly, 10000 has units V/(A·s). The heterogeneous matrix does not possess one common entry unit.

The characteristic polynomial is

and the poles are

The negative real parts imply asymptotic stability. The capacitor-voltage transfer function is

under zero initial conditions.

4. Transfer Functions and Realizations

Taking Laplace transforms with zero initial conditions gives

Hence,

for a single-input–single-output system. For multiple inputs and outputs, H(s) is a transfer matrix.

A transfer function does not uniquely determine the state coordinates. For a strictly proper monic transfer function

one controllable companion realization is

This realization is a mathematical construction; its states need not equal physical currents or voltages. For a proper transfer function with nonzero feedthrough D, first separate the constant term and realize the remaining strictly proper part.

Similarity Transformations

If x=Tz, where T is nonsingular, then

Thus,

Similarity transformations preserve the transfer function, eigenvalues, controllability, and observability, but change state coordinates and often their scaling. Poor scaling can lead to numerical errors even when the analytical model is correct.

5. Modes and Stability

The natural modes are associated with eigenvalues of A. For a continuous-time finite-dimensional LTI system:

• it is asymptotically stable if every eigenvalue satisfies \operatorname{Re}(\lambda_i)<0;
• it is unstable if any eigenvalue has positive real part;
• eigenvalues on the imaginary axis require examination of Jordan blocks for internal stability.

A repeated imaginary-axis eigenvalue with a nontrivial Jordan block produces unbounded polynomial growth. Therefore, poles alone must be interpreted with multiplicity and realization structure.

The Lyapunov theorem states that A is Hurwitz if and only if, for every symmetric Q>0, there exists a unique symmetric P>0 satisfying

The quadratic function V=x^TPx then decreases along every nonzero unforced trajectory.

External, or bounded-input bounded-output, stability is determined by transfer-function poles after pole-zero cancellations. Internal stability is stronger: hidden uncontrollable or unobservable unstable modes remain physically important even if absent from an input-output transfer function.

6. Controllability, Observability, and Minimality

The pair (A,B) is controllable if an input can transfer the state from any initial condition to any final condition in finite time. The controllability matrix is

The system is controllable if

The pair (C,A) is observable if the initial state can be uniquely determined from input-output data over a finite interval. Define

The system is observable when \operatorname{rank}(\mathcal{O})=n.

For large systems, the Popov–Belevitch–Hautus tests are often more numerically appropriate:

for every eigenvalue \lambda of A establishes controllability, while

establishes observability.

A realization of a rational transfer matrix is minimal if it has the smallest possible state dimension. A finite-dimensional LTI realization is minimal if and only if it is both controllable and observable.

7. Generalized or Descriptor State Models

Ideal network constraints frequently lead to differential-algebraic equations rather than explicit ordinary differential equations. A descriptor model is

where E may be singular. The matrix pencil sE-A must be regular, meaning

for a square pencil. Regularity ensures that the model has a well-defined finite and algebraic mode structure. Under zero initial conditions, its transfer matrix is

wherever the inverse exists. With initial state x₀, the transformed state equation contains Ex₀, not merely x₀. Initial values must also satisfy the network’s algebraic constraints.

Singular descriptor models can possess finite eigenvalues and infinite eigenvalues. Infinite eigenvalues represent algebraic constraints; high-index formulations may produce impulsive behavior or severe numerical sensitivity. Index reduction must preserve the physical constraints and should not be performed solely by deleting apparently redundant equations.

Modified Nodal Analysis

For a linear RLC network, let v be the vector of nonreference node voltages and i_L the inductor-current vector. One common modified nodal form is

Here G is the nodal conductance matrix, C_n is the assembled nodal capacitance matrix, L is the inductance matrix, and A_L is an inductor incidence matrix under a consistent branch orientation. Ideal voltage sources normally require additional current unknowns and algebraic equations. Modified nodal analysis is attractive because it can be generated directly from a circuit netlist.

8. Generalized Multiport Network Descriptions

A multiport can be represented in several equivalent forms when the required matrix inverses exist.

For equal, real, positive reference impedance Z₀, incident and reflected wave variables may be defined as

with current directed into the port. Both a and b have units of √W. Scattering models avoid impractical open- and short-circuit measurements at microwave frequencies.

A state-space multiport model permits frequency-dependent port behavior:

where port voltages, currents, or waves are selected as input and output variables. Not every selection is causally admissible; singular port relations may require a descriptor representation.

9. Passivity and Network Synthesis

A network is passive if it cannot deliver more energy than was previously stored or supplied to it. For port variables chosen so that instantaneous power is u^Ty, passivity requires a nonnegative storage function V(x) satisfying

For a linear model and quadratic storage V=x^TPx/2, a sufficient dissipativity condition is the Kalman–Yakubovich–Popov inequality

The supply-rate variables must be power conjugate and consistently scaled. Under standard minimality and regularity conditions, this result is closely related to the positive-real property of the transfer matrix. Passivity preservation is important in model reduction because a nonpassive approximation of a passive physical network may create artificial energy and unstable interconnections.

10. Discrete-Time Models and Numerical Implementation

For sampling interval T_s and a zero-order-held input, exact discretization of an LTI model gives

where

and

The frequently used expression B_d=A^-1(A_d-I)B is valid only when A is nonsingular. Matrix-exponential block methods are preferable because they also handle singular A.

Practical computation should avoid explicitly forming matrix inverses. Linear systems should instead be solved using stable factorizations. State scaling, sparse matrices, stiff integration methods, and error tolerances are especially important for networks containing time constants that differ by several orders of magnitude.

11. Detailed Application: Small-Signal Buck Converter

Background

This is an illustrative engineering case. A design team must assess the open-loop dynamics of a continuous-conduction-mode buck converter before designing its feedback controller. The converter has:

• input voltage V_g=48\ \text{V};
• inductance L=2\ \text{mH};
• inductor series resistance r_L=0.2\ \Omega;
• capacitance C=100\ \mu\text{F};
• load resistance R=10\ \Omega.

The duty-ratio perturbation \hat d is the input, and output-voltage perturbation \hat v is the output. Continuous conduction, ideal switching devices apart from r_L, a stiff input source, and small perturbations around an operating point are assumed.

Problem and Model

Choose the perturbation states

The averaged small-signal equations are

Thus,

and

Substitution gives

No common unit is attached to the entire A matrix. Specifically:

This entry-by-entry interpretation is required because current and voltage are dimensionally different states.

Analysis

The characteristic polynomial is

The eigenvalues are

Therefore, the open-loop averaged plant is asymptotically stable and underdamped. Its duty-to-output transfer function is

The DC gain is

This agrees with the steady-state circuit result

A duty perturbation of 0.01 therefore produces a final output perturbation of approximately 0.471 V within the validity range of the linearization. A rough 2% settling-time estimate based on the real modal part is 4/550\approx7.3\ \text{ms}.

Approach and Outcome

The team can use this model to select controller bandwidth, estimate transient behavior, and test controllability and observability. The model should not be used uncritically near discontinuous conduction, saturation, duty limits, switching-frequency dynamics, or strongly nonlinear loads. A switching model or hybrid model is required when ripple, subharmonic behavior, or individual switching events are significant.

Learning Points

1. Averaging converts a switched network into a tractable low-frequency state model.
2. Linearization applies only near the selected operating point.
3. Matrix entries must be checked individually for dimensional consistency.
4. Agreement between state-space DC gain and steady-state circuit analysis is a valuable validation test.
5. Stable plant poles do not eliminate the need for robust closed-loop design.

12. Nonlinear, Time-Varying, and Hybrid Extensions

A nonlinear network is represented by

Linearization around an equilibrium (x₀,u₀) gives

where

and analogous Jacobians define C and D. This local approximation is not a global stability guarantee.

Time-varying networks use A(t), B(t), C(t), and D(t). Switched converters, protection systems, and reconfigurable networks may require hybrid models that combine continuous states with discrete switching logic and reset maps.

13. Model Reduction and Contemporary Developments

Large extracted interconnect, electromagnetic, and power-grid models may contain millions of states. Reduction techniques include:

• moment matching and Krylov-subspace methods;
• balanced truncation;
• singular perturbation;
• passivity-preserving reduction;
• structure-preserving reduction for second-order or port-Hamiltonian systems.

For an asymptotically stable minimal realization, balanced truncation ranks states using controllability and observability Gramians. If \sigma_i are Hankel singular values and the model is reduced from order n to r, the standard error bound is

This bound applies under the assumptions of continuous-time stability and balanced truncation; it should not be transferred automatically to unstable or descriptor models.

Data-driven and machine-learning methods can identify reduced state models from measurements. However, unconstrained models may violate stability, causality, reciprocity, or passivity. Physics-informed identification, uncertainty quantification, and independent validation are therefore preferred for safety-critical electrical systems.

14. Engineering Limitations, Governance, and Best Practice

State-space accuracy is limited by parameter uncertainty, parasitic elements, sensor dynamics, neglected delays, switching effects, and changing operating conditions. A mathematically elegant model can still be unsuitable for design if its bandwidth or operating range is not documented.

Recommended practice includes:

1. Define state, input, output, reference directions, and units explicitly.
2. Verify Kirchhoff laws and dimensional consistency equation by equation.
3. Compare DC, transient, and frequency responses with independent calculations.
4. Test controllability, observability, stability, and algebraic consistency.
5. Document linearization points and validity ranges.
6. Preserve passivity and physical structure when interconnecting reduced models.
7. Retain model versions, parameter sources, tolerances, and validation evidence.
8. Use conservative safety margins where model errors affect insulation, thermal limits, protection, or converter stability.

Regulated power, transport, medical, and grid applications require traceable verification rather than reliance on an undocumented simulation or AI-generated model. Automated model generation should be treated as an engineering aid, with final responsibility remaining with qualified professionals.

Chapter Summary

State-space models expose the internal dynamics of electrical networks through first-order differential equations. Capacitor voltages and inductor currents provide natural physical states, although network topology can impose dependencies. Transfer functions follow from C(sI-A)^-1B+D, while different realizations related by similarity transformations have identical external behavior.

Eigenvalues describe natural modes, but complete analysis also requires controllability, observability, minimality, and internal stability. Descriptor models accommodate algebraic constraints arising from ideal elements and modified nodal analysis. Generalized multiport descriptions include impedance, admittance, hybrid, transmission, and scattering models.

Passivity provides an energy-based connection between physical networks and positive-real synthesis. Accurate numerical implementation requires appropriate scaling, discretization, sparse computation, and constraint handling. The buck-converter application demonstrated physically based state selection, entry-wise unit verification, transfer-function derivation, and model validation against steady-state behavior.

Key Terms

State: A minimum set of variables sufficient to determine future system behavior with a specified input.

State vector: An ordered collection of state variables.

Realization: A set of state-space matrices representing a transfer function or dynamic system.

State matrix: The matrix A governing unforced state evolution.

Direct feedthrough: The instantaneous input-to-output term Du.

State transition matrix: The matrix exponential governing homogeneous state evolution.

Mode: A natural dynamic component associated with an eigenvalue and its structure.

Controllability: Ability of the input to move the system throughout its state space.

Observability: Ability to reconstruct the state from input-output observations.

Minimal realization: A realization that is both controllable and observable and has minimum order.

Similarity transformation: A nonsingular change of state coordinates preserving input-output behavior.

Descriptor system: A model of the form Eẋ = Ax + Bu, possibly with singular E.

Regular pencil: A square matrix pencil sE − A whose determinant is not identically zero.

Modified nodal analysis: A systematic formulation using node voltages and selected branch currents.

Multiport network: A network described through voltage-current or wave variables at multiple ports.

Scattering matrix: A matrix relating reflected waves to incident waves.

Passivity: Inability of a system to generate net energy beyond stored and supplied energy.

Positive-real function: An immittance function satisfying analytic and nonnegative-real-part conditions associated with passive realizability.

Balanced truncation: Model reduction based on joint controllability and observability measures.

Hybrid model: A model combining continuous dynamics with discrete switching or event logic.

Review Questions

Short Answer Questions

1. Why can the dynamic order of a network be smaller than its number of storage elements?
2. What physical behavior is represented by the D matrix?
3. State the rank condition for observability.
4. What does regularity of the pencil sE − A mean?
5. Why should a heterogeneous-state A matrix not be assigned one common physical unit?

Long Answer Questions

1. Derive the state-space model of a series RLC network and obtain its transfer function.
2. Explain controllability, observability, internal stability, and minimality, including their relationships.
3. Develop the modified nodal descriptor formulation of a general linear RLC network.
4. Compare impedance, admittance, hybrid, transmission, and scattering network representations.
5. Explain passivity in state-space form and discuss its importance in network synthesis and model reduction.

Analytical / Application Questions

1. For the RLC example, determine the response to a 10 V step input with zero initial conditions and classify the damping.
2. Recalculate the buck-converter poles and DC gain if the load changes from 10 Ω to 5 Ω, keeping all other parameters unchanged.
3. Given a proposed reduced-order network model, formulate a validation plan covering stability, passivity, dimensional consistency, frequency-domain accuracy, and operating-range limitations.

Multiple Choice Questions with Answers

Question 1

In the model y=Cx+Du, what does the term Du represent?

A. Stored magnetic energy

B. State integration

C. Instantaneous input-to-output feedthrough

D. Unobservable state dynamics

Correct Answer: C. Instantaneous input-to-output feedthrough

Question 2

The poles of the series RLC example are:

A. -100\pm j300\ \text{s}^-1

B. 100\pm j300\ \text{s}^-1

C. -200\pm j100\ \text{s}^-1

D. -100\pm j100000\ \text{s}^-1

Correct Answer: A. -100\pm j300\ \text{s}^-1

Question 3

Which property is preserved by a nonsingular similarity transformation?

A. The numerical values of all state variables

B. The eigenvalues and transfer function

C. The physical units of every transformed state

D. The individual entries of the A matrix

Correct Answer: B. The eigenvalues and transfer function

Question 4

For an n-state LTI system, which condition establishes controllability?

A. \operatorname{rank}(C)=n

B. \det(B)\neq0

C. \operatorname{rank}[B\ AB\ \cdots\ A^n-1B]=n

D. \operatorname{rank}(A)=n

Correct Answer: C. \operatorname{rank}[B\ AB\ \cdots\ A^n-1B]=n

Question 5

In the buck-converter model, the unit of A₁₂=-1/L is:

A. s⁻¹

B. V/(A·s)

C. A/(V·s)

D. V/s

Correct Answer: C. A/(V·s)

Question 6

For the zero-initial-condition descriptor model E\dot{x}=Ax+Bu, the transfer matrix is:

A. C(sI-A)^-1B

B. C(sE-A)^-1B+D

C. E(sA-I)^-1B+D

D. C(E-A)^-1B+sD

Correct Answer: B. C(sE-A)^-1B+D

Question 7

A square descriptor pencil sE-A is regular when:

A. E is necessarily nonsingular

B. A is symmetric

C. \det(sE-A) is not identically zero

D. All entries of E have units of seconds

Correct Answer: C. \det(sE-A) is not identically zero

Question 8

What is the calculated DC duty-to-output gain of the illustrative buck converter?

A. 4.706 V per unit duty ratio

B. 47.06 V per unit duty ratio

C. 48.98 V per unit duty ratio

D. 24000 V per unit duty ratio

Correct Answer: B. 47.06 V per unit duty ratio

Question 9

Under exact zero-order-hold discretization, the discrete state matrix is:

A. A_d=I+AT_s for every sampling interval

B. A_d=A/T_s

C. A_d=e^AT_s

D. A_d=(sI-A)^-1

Correct Answer: C. A_d=e^AT_s

Question 10

A finite-dimensional LTI realization is minimal if and only if it is:

A. Stable and passive

B. Controllable and observable

C. Symmetric and reciprocal

D. Strictly proper and stable

Correct Answer: B. Controllable and observable