State-Space and System Realization

Introduction

Transfer functions are powerful descriptions of linear time-invariant (LTI) systems, but they do not directly expose internal variables, initial conditions, interconnections, or multivariable structure. State-space modelling addresses these limitations by representing a system through a set of first-order equations involving internal state variables. This representation is central to modern control, estimation, signal processing, circuit simulation, robotics, aerospace systems, and embedded implementation.

A state-space model separates a system into three conceptual elements: the state, which summarizes the relevant memory of the system; the input, which excites it; and the output, which is measured or otherwise of interest. System realization is the systematic construction of such an internal model from an external description such as a transfer function or impulse response.

This chapter develops continuous-time and discrete-time state-space models, derives their relation to transfer functions, examines controllability and observability, explains canonical and minimal realizations, and discusses practical realization issues for modern engineering systems.

Learning objectives

By the end of this chapter, the reader should be able to:

1. Define state, state vector, state-space model, and realization.
2. Derive a transfer function from a state-space model.
3. Construct state-space realizations from transfer functions.
4. Test controllability and observability.
5. Distinguish minimal, nonminimal, canonical, and balanced realizations.
6. Analyse the role of coordinate transformations and initial conditions.
7. Apply realization concepts to a practical engineering problem.
8. Recognize numerical, modelling, and implementation limitations.

1. State-space representation

For a continuous-time LTI system with input vector u(t) ∈ ℝᵐ, state vector x(t) ∈ ℝⁿ, and output vector y(t) ∈ ℝᵖ, the general state-space equations are

Here, A ∈ ℝⁿˣⁿ is the state matrix, B ∈ ℝⁿˣᵐ is the input matrix, C ∈ ℝᵖˣⁿ is the output matrix, and D ∈ ℝᵖˣᵐ is the direct-transmission matrix. The dimensions must be compatible: Ax and Bu must both be n-dimensional, while Cx and Du must both be p-dimensional.

The corresponding discrete-time model, with sampling interval T_s, is

The matrices with subscript d describe the sampled system. In a discrete model, k is an integer sample index rather than continuous time.

1.1 Meaning of the state

A state is a minimum set of variables whose present values, together with future input, determine future state and output. For a system of order n, a state vector generally has n independent components, although many different choices of state coordinates are possible.

For an RLC circuit, capacitor voltages and inductor currents are natural state variables because they represent stored electrical energy. In a mechanical system, position and velocity are common states. In a digital filter, delayed samples or internal delay-register values may serve as states.

A state is not necessarily a directly measurable physical quantity. It may instead be a mathematical coordinate selected to obtain a convenient realization.

1.2 State trajectories and the state-transition matrix

For the continuous-time model, the solution from initial state x(t₀) is

The matrix exponential e^At is called the state-transition matrix. It describes autonomous evolution when the input is zero. The output follows from substitution into the output equation:

For a zero-order-held discrete input, the sampled matrices are often obtained from

These relations assume that the continuous input remains constant over each sampling interval. If A is nonsingular, an equivalent expression is B_d = A^-1(A_d−I)B, although the integral form is more general and numerically safer.

2. From state-space equations to transfer functions

Assume zero initial conditions and apply the Laplace transform to the continuous-time equations:

Thus,

and

Therefore, the transfer-function matrix is

where I is the n×n identity matrix. For a single-input single-output system, G(s) is a scalar rational function; for a multi-input multi-output system, it is a p×m matrix of transfer functions.

The poles of the transfer function are related to the eigenvalues of A, but cancellation can occur. Consequently, an eigenvalue of A need not appear as a pole of G(s) if the associated mode is uncontrollable, unobservable, or both.

For the discrete-time model, the analogous expression is

provided the z-transform uses zero initial conditions.

3. Constructing a realization from a transfer function

Consider a strictly proper SISO transfer function

Strictly proper means that the numerator degree is less than the denominator degree, so there is no direct feedthrough term. One controllable companion realization is

The notation in the matrix for A indicates the usual superdiagonal entries equal to one. This realization is controllable when the denominator is monic and the stated order is used.

If the transfer function is proper but not strictly proper, perform polynomial division:

For a physically realizable finite-dimensional causal LTI system, Q(s) is normally a constant D; a polynomial of degree greater than zero represents ideal differentiation and is not a standard proper state-space transfer function.

Worked example 1: realization and verification

Let

A controllable companion realization is

To verify it, calculate

Its determinant is s(s+4)+5 = s²+4s+5. The inverse is

Therefore,

which gives

as required. The state dimension is two, matching the denominator order.

4. Coordinate transformations and equivalent realizations

Let T be any nonsingular n×n matrix and define a new state coordinate by

Then the transformed realization is

Hence,

This transformation changes the coordinates but not the external input-output behaviour. The matrices A and A_z are similar and therefore have identical eigenvalues. Physical interpretation may change: one realization may use energy variables, another may use modal coordinates, and a third may be optimized for numerical conditioning.

5. Controllability and reachability

A state is controllable if a suitable input can move the system from any initial state to any desired final state in finite time. For the continuous-time pair (A,B), the controllability matrix is

The system is controllable if and only if

For discrete-time systems the same rank test applies. In numerical work, rank should be assessed with a tolerance based on singular values rather than exact arithmetic.

A more general test is the Popov–Belevitch–Hautus (PBH) criterion. The pair (A,B) is controllable if and only if

for every eigenvalue λ of A.

For stable continuous-time A, the controllability Gramian is

It satisfies the Lyapunov equation

and is positive definite precisely when the realization is controllable, assuming A is asymptotically stable.

6. Observability

A state is observable if it can be uniquely reconstructed from the input and output over a finite interval. For the pair (A,C), the observability matrix is

The realization is observable if and only if

The PBH observability test is

for every eigenvalue λ of A.

For stable A, the observability Gramian is

and satisfies

It is positive definite exactly when the realization is observable.

Controllability concerns the ability to influence internal modes; observability concerns the ability to infer them. These properties are independent. A mode may be controllable but unobservable, observable but uncontrollable, both, or neither.

7. Minimal realizations and canonical forms

A realization is minimal if no realization of the same input-output system has a lower state dimension. For finite-dimensional LTI systems, a realization is minimal if and only if it is both controllable and observable. Its order equals the McMillan degree of the transfer-function matrix.

A nonminimal realization may contain hidden states. Such states can arise through pole-zero cancellation, redundant modelling, or the inclusion of physically present but externally inaccessible dynamics. Removing uncontrollable and unobservable modes produces a minimal realization, although the procedure must be performed carefully, especially in numerical computation.

Common canonical forms include:

Jordan form is mathematically informative but often ill-conditioned. In practical computation, real Schur forms, staircase algorithms, and singular-value-based methods are generally preferred.

8. Stability and realization structure

For continuous-time systems, internal asymptotic stability requires every eigenvalue of [[M:A]] to have negative real part. For discrete-time systems, internal asymptotic stability requires every eigenvalue to lie strictly inside the unit circle:

A transfer function can appear stable after pole-zero cancellation even when a nonminimal realization contains an unstable hidden mode. Thus, input-output stability and internal stability must be distinguished. In safety-critical systems, relying only on a cancelled transfer function is unacceptable; the internal realization must also be examined.

For a stable continuous-time SISO system, the impulse response is

where δ(t) is the Dirac delta distribution associated with direct feedthrough. The D term produces an instantaneous output response and must be considered in high-frequency behaviour and implementation.

9. Balanced realizations and model reduction

A balanced realization scales the state coordinates so that the controllability and observability Gramians are equal and diagonal:

The nonnegative values σ_i are the Hankel singular values. Large values indicate states that are both easy to excite and important to observe; small values indicate weak input-output contribution.

Balanced truncation retains the states associated with the largest Hankel singular values. For a stable continuous-time system, if states after index r are removed, a standard error bound is

Here G_r is the reduced transfer function and ‖·‖_∞ is the H-infinity norm. The bound assumes a stable minimal realization and uses consistent normalization. Balanced truncation preserves stability under its standard conditions, but it may not preserve structural properties such as passivity or positivity unless specialized methods are used.

10. Practical realization issues

10.1 Scaling and conditioning

Poorly scaled state variables can produce matrices with very different numerical magnitudes. This can make rank tests, eigenvalue calculations, and simulation inaccurate. State scaling, orthogonal transformations, balancing, and appropriate solver tolerances are therefore important.

10.2 Continuous-to-discrete conversion

Sampling too slowly can distort dynamics and cause aliasing. The sampling frequency should be chosen in relation to the fastest relevant dynamics, while also considering sensor bandwidth, computational delay, quantization, and noise. Zero-order-hold conversion is appropriate when actuators hold inputs between samples; other discretization methods are needed for different intersample assumptions.

10.3 Initial conditions

Transfer functions conventionally assume zero initial conditions. State-space models do not. Initial stored energy can generate an output even when the external input is zero. This makes state-space methods essential for transient analysis and switching systems.

10.4 Noise and estimation

In measured systems, outputs may be represented as

where v_k is measurement noise. Process disturbances may be included as x_k+1=A_dx_k+B_du_k+w_k. These models support observers and Kalman filters, provided stochastic assumptions and covariance matrices are specified correctly. State realization is therefore foundational to estimation, fault diagnosis, and sensor fusion.

11. Applied case: thermal monitoring of a two-zone building

This is an illustrative engineering case.

Background

A building has two thermally coupled zones. Each zone has a temperature state, while heaters provide controllable inputs. Temperature sensors measure both zones.

Model

Let x=[T₁,T₂]^T represent temperature deviations from ambient, and u=[q₁,q₂]^T represent heater powers. A simplified model is

The coefficients have units consistent with inverse time for A and temperature change per unit heater input per unit time for B. The positive off-diagonal terms represent thermal coupling, while negative diagonal terms represent heat loss.

Analysis

Because C=I, both temperatures are directly measured, so the state is immediately observable in this idealized model. Controllability is assessed using

The diagonal input matrix is nonsingular, so each zone can be directly heated. In realistic buildings, however, actuator saturation, uncertain coupling, sensor bias, and unmeasured disturbances may reduce effective control authority.

Approach and outcome

A discrete model can be generated using the selected sampling interval and zero-order-hold assumptions. An estimator can filter noisy temperature readings, while a predictive controller uses the state model to coordinate heaters and reduce energy consumption. If the two thermal modes have widely separated Hankel singular values, balanced truncation may justify a lower-order model for embedded implementation.

Learning points

1. Physical energy-storage variables provide an interpretable state choice.
2. Direct measurement does not eliminate the need to test actuator controllability.
3. Coupling makes independent single-zone models potentially misleading.
4. Sampling, uncertainty, and saturation must be considered in deployment.
5. A mathematically minimal model is not always the best model for diagnosis or physical interpretation.

12. Limitations, governance, and contemporary developments

State-space models are usually linear approximations. Nonlinearities such as saturation, hysteresis, dead zones, friction, and temperature-dependent parameters may invalidate a fixed LTI model outside its operating region. Linear parameter-varying, switched, nonlinear, and data-driven state-space models are used when these effects are material.

Data-driven realization methods, including subspace identification, estimate state-space matrices directly from measured input-output data. Their reliability depends on excitation richness, noise treatment, model order selection, and validation on independent data. Neural state-space models can represent complex dynamics, but interpretability, stability guarantees, extrapolation, and safety certification remain important concerns.

Engineering governance requires traceability of model assumptions, version control, validation against physical tests, uncertainty reporting, and monitoring for model drift. In safety-critical applications, an automatically learned realization should not be deployed merely because it fits historical data; stability, robustness, failure modes, and human oversight must be addressed.

Best-practice checklist

• Define state variables according to system memory, not merely available measurements.
• Check matrix dimensions and physical units.
• Verify a realization by recomputing its transfer function or impulse response.
• Test controllability and observability using numerically robust methods.
• Distinguish internal stability from input-output stability.
• Examine scaling and conditioning before numerical rank decisions.
• Include initial conditions, disturbances, delays, and saturation where relevant.
• Validate reduced or identified models against independent operating conditions.
• Preserve safety, traceability, and interpretability in automated modelling.

Chapter Summary

State-space representation describes an LTI system through first-order state and output equations. The matrices A, B, C, and D encode internal dynamics, input action, measurement, and direct feedthrough. The transfer function follows from G(s)=C(sI−A)^-1B+D, but it does not generally reveal initial-condition effects or hidden internal modes.

Realization theory explains how external input-output behaviour can be represented internally. Coordinate transformations create equivalent realizations, while controllability and observability determine whether states can be influenced and inferred. A realization is minimal exactly when it is both controllable and observable. Canonical forms support analysis, and balanced realizations support model reduction through Hankel singular values. Practical work additionally requires attention to sampling, numerical conditioning, noise, uncertainty, stability, and governance.

Key Terms

State: Minimum information required, with future input, to determine future system behaviour.

State vector: Vector containing the selected state variables.

State-space model: First-order differential or difference equations describing state and output.

Realization: A state-space representation having a specified input-output behaviour.

State matrix: Matrix A governing autonomous state evolution.

Input matrix: Matrix B describing how inputs affect states.

Output matrix: Matrix C mapping states to outputs.

Direct-transmission matrix: Matrix D representing instantaneous input-output transmission.

State-transition matrix: Matrix exponential e^At for continuous-time systems.

Controllability: Ability to drive states to desired values using inputs.

Observability: Ability to reconstruct states from measured outputs.

Gramian: Matrix measure of accumulated controllability or observability.

Minimal realization: Lowest-order realization of a given input-output system.

Canonical form: Structured realization emphasizing a particular property.

Modal form: Representation organized around system modes or eigenvalues.

Balanced realization: Realization with equal diagonal controllability and observability Gramians.

Hankel singular value: Measure of joint input-output importance of a state direction.

McMillan degree: Minimum state dimension of a finite-dimensional transfer-function realization.

Internal stability: Stability of all modes in the state dynamics.

Model reduction: Construction of a lower-order model that approximates the original system.

Review Questions

Short Answer Questions

1. What is meant by the state of a dynamical system?
2. State the continuous-time state-space equations and define their matrices.
3. What is the difference between controllability and observability?
4. Why can a transfer function fail to reveal an internal unstable mode?
5. What is a minimal realization?

Long Answer Questions

1. Derive the transfer-function matrix G(s)=C(sI−A)^-1B+D from the continuous-time state equations.
2. Explain controllability and observability, including their rank tests and PBH criteria.
3. Construct a controllable companion realization for a general strictly proper SISO transfer function.
4. Discuss coordinate transformations and explain why they preserve input-output behaviour.
5. Explain balanced realization and balanced truncation, including the standard error bound and its assumptions.

Analytical / Application Questions

1. For a given state-space model, calculate the controllability and observability matrices, determine their ranks, and identify whether the realization is minimal.
2. A sampled thermal system has a slow mode and a fast mode. Explain how sampling interval, state scaling, and model reduction could affect its implementation.
3. A transfer function exhibits pole-zero cancellation at an unstable pole. Analyse why the transfer function alone is insufficient for safety assessment and specify the additional state-space tests required.

Multiple Choice Questions with Answers

Question 1

For a continuous-time state-space model, which expression gives the transfer-function matrix under zero initial conditions?

A. G(s)=C(sI−A)^-1B+D

B. G(s)=A(sI−C)^-1B+D

C. G(s)=B(sI−A)^-1C+D

D. G(s)=C(sI+A)^-1B−D

Correct Answer: A. G(s)=C(sI−A)^-1B+D

Question 2

A continuous-time realization of order n is controllable when which condition holds?

A. rank(C)=n

B. rank([B,AB,…,Aⁿ⁻¹B])=n

C. det(A)=1

D. All eigenvalues of A are positive

Correct Answer: B. rank([B,AB,…,Aⁿ⁻¹B])=n

Question 3

Which rank condition tests observability?

A. rank([B,AB,…,Aⁿ⁻¹B])=n

B. rank(A)=n

C. rank([[C],[CA],[⋮],[CAⁿ⁻¹)=n]]

D. rank(D)=n

Correct Answer: C. rank([[C],[CA],[⋮],[CAⁿ⁻¹)=n]]

Question 4

A realization is minimal if and only if it is

A. stable and diagonal

B. controllable and observable

C. proper and strictly causal

D. symmetric and positive definite

Correct Answer: B. controllable and observable

Question 5

For a discrete-time internally asymptotically stable system, every eigenvalue of the state matrix must satisfy

A. |λ_i|>1

B. Re(λ_i)<0

C. |λ_i|=1

D. |λ_i|<1

Correct Answer: D. |λ_i|<1

Question 6

Under a nonsingular coordinate transformation x=Tz, the transformed state matrix is

A. TAT^-1

B. T^-1AT

C. T+A

D. A−T

Correct Answer: B. T^-1AT

Question 7

In a state-space model, the matrix D represents

A. the state-transition matrix

B. the direct input-to-output transmission

C. the observability Gramian

D. the denominator polynomial

Correct Answer: B. the direct input-to-output transmission

Question 8

For a stable continuous-time system, the controllability Gramian satisfies

A. AW_c+W_cA^T+BB^T=0

B. AW_c−W_cA^T−BB^T=0

C. A^TW_c+W_cB+C^TC=0

D. W_c=A+B

Correct Answer: A. AW_c+W_cA^T+BB^T=0

Question 9

What is the principal purpose of balanced truncation?

A. To increase the number of states

B. To remove all unstable poles without analysis

C. To reduce model order while retaining important input-output behaviour

D. To convert every system into a transfer function with no poles

Correct Answer: C. To reduce model order while retaining important input-output behaviour

Question 10

For the transfer function G(s)=(2s+3)/(s²+4s+5), the denominator poles are the roots of

A. s²+2s+3

B. s²+4s+5

C. 2s+3

D. s²+3s+4

Correct Answer: B. s²+4s+5