6. Dynamic-System Representation and Identification

Introduction and Context

A mechanical system is understood not only by its physical components but also by the relationships among forces, motions, energy storage, energy dissipation, and external inputs. A shaft, suspension, robot joint, cutting tool, or vehicle can therefore be represented in several mathematically equivalent ways. The choice of representation determines how easily the system can be analysed, simulated, controlled, or identified from measurements.

Dynamic-system representation converts a physical system into a mathematical model. Common representations include differential equations, transfer functions, block diagrams, signal-flow graphs, and state-space models. System identification proceeds in the opposite direction: measured input–output data are used to estimate a model when the physical parameters are unknown, uncertain, or too complex to derive directly. These activities are central to machine dynamics and control because controller design, vibration prediction, fault diagnosis, and performance assessment all depend on an adequate model.

No model is universally best. A differential equation preserves physical meaning; a transfer function is convenient for frequency-domain analysis; a block diagram reveals functional interconnection; a signal-flow graph supports algebraic reduction; and a state-space model is well suited to multivariable, time-varying, and computational applications. The engineer must therefore understand both the mathematical forms and the assumptions under which they are valid.

Learning Objectives

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

1. derive dynamic equations for mechanical systems using force, torque, energy, and Newton–Euler methods;

2. formulate transfer-function, block-diagram, signal-flow, and state-space models;

3. distinguish between linear, nonlinear, time-invariant, and time-varying models;

4. linearize nonlinear mechanical equations about an operating point;

5. estimate model parameters from measured data using appropriate identification methods;

6. interpret frequency-response and modal-test data;

7. validate models using independent data and suitable error measures;

8. recognise practical sources of modelling error, uncertainty, and non-identifiability; and

9. select an appropriate model for simulation, control, diagnosis, or design.

From Physical Laws to Dynamic Equations

A dynamic model expresses how system variables change with time. For a translational mass–spring–damper system,

m x¨ + c x˙ + kx = f(t),

where m is mass, c is viscous damping, k is stiffness, x is displacement, and f(t) is the applied force. The terms represent inertia, energy dissipation, elastic restoring force, and excitation, respectively. For a rotational system,

J θ¨ + b θ˙ + k_θ θ = T(t),

where J is rotational inertia, b is rotational damping, k_θ is torsional stiffness, and T is applied torque.

These equations may be derived directly from Newton’s second law, from torque balance, or from an energy formulation. In complex machines, Lagrange’s equations are particularly useful. If q_i are generalised coordinates, the equations are

d/dt(∂L/∂q˙_i) − ∂L/∂q_i + ∂D/∂q˙_i = Q_i,

where L = T − V is the Lagrangian, T and V are kinetic and potential energies, D is a dissipation function, and Q_i represents non-conservative generalised forces. This approach reduces repeated force-balance calculations and is effective for coupled mechanisms and flexible structures.

A model should identify inputs, outputs, internal variables, parameters, initial conditions, and assumptions. For example, a suspension model may take road displacement as input and body acceleration as output. The same physical suspension can have different models depending on whether the purpose is ride comfort, actuator control, fatigue assessment, or fault detection.

Important Concepts and Terminology

A system is a set of interacting elements whose behaviour is studied through specified inputs and outputs. An input is an imposed excitation or command, while an output is a measured or calculated response. A state is the smallest set of variables that, together with future inputs, determines future system behaviour. Order is the highest derivative in a differential equation, or equivalently the number of independent energy-storage variables in a minimal model.

A system is linear if superposition and homogeneity apply. It is time invariant if its parameters do not change with time. A causal system does not respond before its input is applied. A stable system produces bounded output for every bounded input under the specified stability definition. Controllability concerns the ability to move the state using the input; observability concerns the ability to reconstruct the state from measured outputs.

A parameter is a quantity such as mass, stiffness, damping, or gain that characterises a model. A residual is the difference between measured and model-predicted output. Identifiability is the possibility of uniquely estimating parameters from available data. Validation evaluates whether a model is sufficiently accurate for its intended use, whereas verification checks whether equations and software have been implemented correctly.

Linear and Nonlinear Dynamic Models

Most analytical control techniques require linear time-invariant (LTI) models. A nonlinear mechanical equation can contain terms such as sin θ, x², x x˙, Coulomb friction, backlash, saturation, or contact switching. These effects may be essential at large amplitudes but negligible near a chosen operating condition.

Consider a pendulum equation,

m l² θ¨ + c θ˙ + mgl sin θ = T.

For small angular displacement, sin θ ≈ θ, giving

m l² θ¨ + c θ˙ + mgl θ = T.

More generally, if a nonlinear model is written as

x˙ = f(x,u), y = h(x,u),

and the operating point is (x₀,u₀), define small perturbations δx = x − x₀ and δu = u − u₀. First-order Taylor expansion gives

δx˙ = Aδx + Bδu,

δy = Cδx + Dδu,

where A = ∂f/∂x, B = ∂f/∂u, C = ∂h/∂x, and D = ∂h/∂u, all evaluated at the operating point. Linearization is local: a model accurate around one equilibrium may be misleading at another or under large disturbances.

Neglecting nonlinearities must be justified rather than assumed. Friction, magnetic effects, geometric stiffness, bearing clearance, and actuator saturation can strongly influence a system even when the nominal plant appears linear. A useful practice is to compare the expected operating range with the range over which the linear approximation has been experimentally verified.

Transfer-Function Representation

For a linear time-invariant system with zero initial conditions, the transfer function is the ratio of the Laplace transform of the output to that of the input:

G(s) = Y(s)/U(s).

For the mass–spring–damper system,

G(s) = X(s)/F(s) = 1/(ms² + cs + k).

The denominator determines the poles and therefore the natural dynamic modes and stability properties. The numerator determines zeros and input–output transmission characteristics. The undamped natural frequency and damping ratio are

ω_n = √(k/m), ζ = c/(2√(km)).

The transfer function is compact and useful for pole–zero analysis, frequency response, classical control design, and convolution. However, it does not by itself reveal every internal variable, and pole–zero cancellation can conceal internally unstable or weakly observable dynamics. Transfer functions are most naturally applied to SISO LTI systems, although matrix transfer functions can represent multivariable systems.

The final-value and initial-value theorems may provide useful checks, but they require their mathematical conditions to be satisfied. For example, a static force applied to a stable mass–spring–damper system produces a final displacement F/k, while a pure mass without stiffness has no finite static displacement. Such physical checks often reveal modelling errors before numerical simulation begins.

Block Diagrams and Signal-Flow Graphs

A block diagram represents functional relationships using blocks, summing points, and take-off points. A block labelled G(s) transforms an input signal into an output signal. In a simple feedback loop with forward transfer function G(s) and feedback H(s), the closed-loop transfer function for negative feedback is

T(s) = G(s)/[1 + G(s)H(s)].

Block diagrams are valuable because they separate plant, sensor, actuator, controller, and feedback components. They also make design changes transparent. However, reduction rules must be used carefully when blocks are not in series or when summing and take-off points are displaced.

A signal-flow graph consists of nodes representing variables and directed branches representing gains. Mason’s gain formula gives the overall transfer function as

T = (Σ P_k Δ_k)/Δ,

where P_k are forward-path gains, Δ = 1 − (sum of individual loop gains) + (sum of products of non-touching loop gains) − …, and Δ_k excludes loops touching the kth forward path. Signal-flow graphs are particularly useful for coupled equations and multiple feedback paths, although state-space methods are usually more systematic for high-order systems.

State-Space Modelling

A state-space model has the form

x˙ = Ax + Bu,

y = Cx + Du.

The state vector may be selected as displacements and velocities. For the mass–spring–damper system, let x₁ = displacement and x₂ = velocity. Then

[ x˙₁ ] = [ 0 1 ] [x₁] + [ 0 ] f,

[ x˙₂ ] [−k/m −c/m] [x₂] [1/m]

and y = [1 0]x if displacement is measured.

State-space representation accommodates multiple inputs and outputs, initial conditions, time-varying matrices, nonlinear extensions, and numerical simulation. The eigenvalues of A are the system poles. A continuous-time pair (A,B) is controllable if the controllability matrix [B AB … Aⁿ⁻¹B] has full rank. The pair (A,C) is observable if [C; CA; …; CAⁿ⁻¹] has full rank. These rank tests are structural properties of the chosen model and measurement arrangement.

Different state choices can describe the same input–output behaviour. Similarity transformations change the coordinates but not the transfer function or eigenvalues. This distinction is important: physical states such as shaft angle and speed may be preferable for interpretation, while transformed or balanced states may be preferable for numerical reduction.

Frequency Response and Experimental Modal Analysis

When a structure is excited by a sinusoidal force at frequency ω, the steady-state response is described by the frequency response function (FRF),

H(jω) = Y(jω)/U(jω).

For a lightly damped mode, amplitude rises near resonance and phase changes rapidly. A single-degree-of-freedom receptance is

H(ω) = 1/[k − mω² + j cω].

Different output choices produce different FRFs: displacement/force is receptance, velocity/force is mobility, and acceleration/force is accelerance. Their units and resonance shapes differ, so the measurement definition must always be recorded.

Experimental modal analysis estimates natural frequencies, damping ratios, mode shapes, and modal constants from measured responses. A typical test includes sensor calibration, excitation selection, data acquisition, frequency-response estimation, curve fitting, and validation. Excitation may be supplied by an impact hammer, electrodynamic shaker, rotating imbalance, or operational environment. Accelerometers are common, but their mass, mounting, orientation, and frequency range influence results.

Coherence is a useful quality indicator for an FRF estimate. Low coherence may indicate noise, nonlinear behaviour, leakage, inadequate excitation, poor sensor attachment, or an unmeasured input. Windowing, averaging, frequency resolution, and anti-alias filtering must be selected with the dynamic objective in mind. A short record may miss low-frequency behaviour, whereas insufficient sampling frequency may alias high-frequency content into the measured band.

Modal testing is not identical to system identification. Modal analysis focuses on dynamic modes and parameters, while system identification may estimate a predictive input–output model for control or simulation. The two activities overlap and often use the same experimental data.

System Identification from Measured Data

System identification is the process of constructing a mathematical model from observations of inputs, outputs, and sometimes internal signals. It is especially valuable when parameters are uncertain, physical derivation is impractical, or the system includes unmodelled joints, flexibility, and actuator dynamics.

A disciplined identification study begins with a purpose. A low-order model may be sufficient for controller tuning but inadequate for fatigue prediction. The model structure should then be selected: static or dynamic, continuous or discrete, parametric or non-parametric, linear or nonlinear, and SISO or MIMO.

Experimental design and data preparation

The input must sufficiently excite the dynamics of interest. Step, impulse-like, swept-sine, pseudorandom binary sequence, multisine, and ambient excitation each have different advantages. A step reveals transient characteristics but may be unsuitable for systems with severe saturation. A swept sine provides clear frequency information but may take considerable time. Random or multisine signals can excite several frequencies efficiently.

Measured data require removal of offsets, synchronization of channels, calibration, outlier inspection, and appropriate filtering. Filtering should not distort the dynamics being estimated. The data should be divided into estimation and validation sets; using the same record for both can produce an overly optimistic assessment.

Parametric estimation

For a discrete-time model, an autoregressive model with exogenous input can be written as

A(q)y(k) = B(q)u(k−n_k) + e(k),

where q⁻¹ is the delay operator and e(k) is an error term. Least squares estimates parameters by minimizing the sum of squared prediction errors. In matrix form, if y = Φθ + e,

θ̂ = (ΦᵀΦ)⁻¹Φᵀy,

provided ΦᵀΦ is nonsingular. Ill-conditioning indicates inadequate excitation, correlated regressors, or excessive model order.

More general structures include output-error models, Box–Jenkins models, state-space subspace identification, and prediction-error methods. Prediction-error estimation adjusts parameters to minimise a selected criterion, often

V(θ) = (1/N) Σ [y(k) − ŷ(k|k−1,θ)]².

The criterion may be weighted to reflect sensor uncertainty or the importance of particular frequency ranges. Regularisation, Bayesian estimation, and sparsity-promoting methods can improve robustness when data are limited, but they introduce assumptions that must be documented.

Non-parametric identification

Non-parametric methods estimate impulse responses, correlation functions, or frequency-response functions without initially imposing a low-order model. They are useful for revealing resonances, anti-resonances, delays, and frequency-dependent uncertainty. A parametric model can subsequently be fitted to the estimated response.

Model order and parameter confidence

Increasing model order usually improves fit over the estimation data but may capture noise rather than physical dynamics. Order selection should combine fit, residual behaviour, stability, parsimony, and intended application. Confidence intervals can be estimated from noise assumptions, repeated experiments, bootstrap resampling, or sensitivity analysis. A parameter with a large uncertainty should not be presented as a precise physical property.

Model Validation and Model-Form Error

Validation asks whether the model reproduces behaviour not used in its construction. Useful measures include root-mean-square error,

RMSE = √[(1/N)Σ(y_k − ŷ_k)²],

normalised fit measures, maximum error, frequency-domain magnitude and phase error, and prediction-error autocorrelation. A good residual should have approximately zero mean, little correlation with past inputs, and no obvious frequency-dependent structure. If residuals remain correlated with the input, the model has not captured all relevant dynamics.

Validation must be linked to use. A model may predict displacement accurately but fail to predict acceleration near a resonance. It may be adequate for a narrow control bandwidth but unsuitable for shock loads. Validation should therefore consider operating points, amplitudes, temperatures, payloads, and boundary conditions relevant to deployment.

Model-form error arises from omitted flexible modes, nonlinear friction, unmeasured disturbances, sensor dynamics, time delay, parameter variation, and incorrect boundary conditions. It cannot always be removed by estimating more parameters. In some cases, uncertainty models, grey-box formulations, gain scheduling, nonlinear models, or experimentally derived correction factors are more appropriate.

Applied Illustrative Case: Identification of a Flexible Servo Axis

Background

Consider an electrically driven positioning axis used in a precision manufacturing machine. The motor drives a carriage through a compliant coupling. A position encoder is mounted on the motor shaft, while a second encoder measures carriage position. The control engineer needs a model for tuning a position loop and predicting vibration.

Situation and problem

A rigid-body model predicts acceptable low-frequency behaviour but the carriage exhibits oscillation near 42 Hz. The motor current command, motor position, and carriage position are available at a fixed sampling rate. The objective is to estimate a model that captures rigid motion and the dominant flexible mode without fitting measurement noise.

Analysis and approach

A multisine current command is selected with energy from 1 to 150 Hz, while amplitude limits prevent actuator saturation. Before testing, the encoders are calibrated and their time channels synchronised. Several records are collected; some are reserved for validation. The frequency-response estimate from current to carriage position shows a low-frequency rigid-body trend, a resonance near 42 Hz, and an anti-resonance caused by the two-inertia compliance.

A grey-box model is formulated with motor inertia J_m, carriage equivalent mass m_c, coupling stiffness k_c, damping c_c, and an actuator gain. The equations include motor and carriage coordinates. Initial parameter estimates are obtained from known mass and geometry, then refined by prediction-error minimisation. A lower-order rigid model and a two-inertia model are compared.

The rigid model has a smaller parameter count but produces large phase error around the flexible mode. The two-inertia model predicts both encoders simultaneously and gives physically interpretable estimates. Residual analysis shows little correlation below 100 Hz, although high-frequency residuals remain because of neglected structural modes and sensor dynamics.

Outcome

The two-inertia model is selected for controller design. The position-loop bandwidth is kept sufficiently below the flexible resonance, and a notch filter is evaluated using the measured resonance frequency. Independent validation records confirm the predicted resonance location over the tested operating range. The model is not used to predict behaviour during collisions because the linear compliance and damping assumptions do not represent contact nonlinearities.

Learning points

The case demonstrates that model purpose determines required fidelity. Independent measurements of motor and carriage motion improve observability of the flexible mode. Excitation must cover the frequency range of interest, and validation must include conditions distinct from estimation. A physically interpretable grey-box model can be more useful than a numerically excellent but opaque high-order fit.

Practical Applications and Professional Decisions

Dynamic representations support controller synthesis, vibration isolation, machine-tool stability, vehicle suspension design, rotor balancing, robotics, condition monitoring, and digital-twin development. In industry, model selection also affects certification, maintainability, computational cost, and communication between mechanical, electrical, and software teams.

For early conceptual design, a low-order physical model often provides the best insight. During controller implementation, a discrete state-space model may be preferred. For test correlation, FRFs and modal models are natural. For fault diagnosis, a model must distinguish normal variation from fault signatures and must include sensor and actuator behaviour. For real-time deployment, model reduction may be necessary, but removed modes should be documented and bounded by uncertainty estimates.

Digital engineering environments increasingly combine finite-element models, measured FRFs, reduced-order models, and operational data. Machine-learning methods can identify nonlinear or time-varying patterns, but purely data-driven models may extrapolate poorly and may not preserve physical constraints. Hybrid physics-informed models, online parameter estimation, and uncertainty-aware digital twins are therefore important developments. Their use requires traceability of data, version control, cybersecurity, and clear limits on automated decisions.

Challenges, Risks, and Common Mistakes

Several errors recur in modelling and identification:

• confusing an input command with the actual physical input after actuator dynamics and saturation;
• using an incorrect sign convention or inconsistent units;
• fitting a high-order model to noise and calling the result a physical model;
• estimating parameters from data that do not sufficiently excite the system;
• ignoring sensor dynamics, time delay, filtering, or sample-and-hold effects;
• validating only on the data used for estimation;
• interpreting a local linear model as globally valid;
• neglecting boundary-condition changes, preload, temperature, or payload;
• treating low coherence or correlated residuals as acceptable without investigation; and
• reporting point estimates without uncertainty or applicability limits.

Numerical conditioning is also important. State variables with very different magnitudes can produce poorly scaled matrices. Continuous-to-discrete conversion must account for sampling time and input assumptions. Aliasing can create apparently plausible but false dynamics. Regular checks using dimensional analysis, limiting cases, energy balance, and independent measurements are essential.

Ethical, Safety, and Governance Considerations

A model used in a safety-critical machine must be accompanied by assumptions, validation evidence, configuration records, and known limitations. Experimental testing should respect mechanical guarding, stored-energy hazards, overspeed limits, and emergency-stop provisions. Data integrity matters: altered, selectively discarded, or poorly documented measurements can lead to unsafe design decisions.

Engineers should distinguish measured facts, estimated quantities, and assumptions. Automated identification tools should not be treated as authoritative when the training data exclude important operating conditions. For industrial systems, access controls and cybersecurity are relevant because manipulated sensor or command data can corrupt both identification and control. Professional judgement remains necessary when a model is used to approve a design, set operating limits, or diagnose a dangerous fault.

Best Practices for a Reproducible Identification Study

A defensible workflow generally includes:

1. define the intended use, outputs, accuracy, and operating envelope;

2. draw the physical system boundary and identify inputs, disturbances, sensors, and states;

3. derive a baseline physical model and record assumptions;

4. design an excitation that covers the relevant dynamics without unsafe operation;

5. calibrate, synchronise, and inspect the measurements;

6. estimate candidate models with justified structures and orders;

7. compare physical plausibility, fit, residuals, stability, and uncertainty;

8. validate on independent data and, when possible, another operating condition;

9. document model version, data, parameter units, sampling details, and limitations; and

10. monitor model performance after deployment and update it under controlled procedures.

Chapter Summary

Dynamic-system representation translates mechanical behaviour into differential equations, transfer functions, block diagrams, signal-flow graphs, and state-space models. Differential equations preserve physical relationships, transfer functions support input–output and frequency-domain analysis, while state-space models provide a flexible framework for multivariable systems, initial conditions, controllability, observability, and computation. Nonlinear systems can often be linearized locally about an operating point, but the resulting model has a limited validity range.

System identification estimates model structure and parameters from measured data. Reliable identification depends on purposeful excitation, accurate and synchronised measurements, appropriate preprocessing, suitable estimation methods, and independent validation. Experimental modal analysis uses frequency-response data to estimate resonances, damping, and mode shapes. Model quality must be judged against its intended application, not merely by numerical fit. Physical plausibility, uncertainty, residual analysis, safety, data integrity, and documented limitations are essential features of professional modelling practice.

Key Terms

System: A set of interacting elements studied through specified inputs and outputs.

Dynamic model: A mathematical description of how system variables evolve with time.

State: A minimal set of variables that determines future behaviour when input is known.

Transfer function: The zero-initial-condition ratio of output transform to input transform for an LTI system.

Pole: A root of a transfer-function denominator or an eigenvalue governing natural response.

Zero: A value of the complex variable at which a transfer-function numerator vanishes.

State-space model: First-order vector equations relating state, input, and output variables.

Linearization: Approximation of a nonlinear model by its first-order Taylor expansion near an operating point.

Controllability: The ability to drive system states using available inputs.

Observability: The ability to infer internal states from measured outputs.

Frequency-response function: The complex steady-state output-to-input ratio as a function of frequency.

Experimental modal analysis: Measurement-based estimation of modal frequencies, damping, and shapes.

System identification: Estimation of a dynamic model from measured input–output data.

Parametric model: A model with a specified finite-dimensional structure and unknown parameters.

Non-parametric model: A response description estimated without imposing a fixed low-order parameter structure.

Residual: The difference between measured and model-predicted output.

Identifiability: The possibility of uniquely estimating model parameters from available data.

Validation: Assessment of model adequacy using evidence, commonly independent data.

Coherence: A frequency-dependent measure of the consistency between measured input and output signals.

Model-form error: Error caused by an inadequate model structure or omitted physical effects.

Review Questions

Short Answer Questions

1. What information is represented by the poles of a transfer function?

2. Distinguish between controllability and observability.

3. Why is linearization considered a local approximation?

4. What is the purpose of separating estimation data from validation data?

5. State two causes of low coherence in an experimental frequency-response measurement.

Long Answer Questions

1. Derive the transfer-function and state-space models of a mass–spring–damper system and explain the physical meaning of each state.

2. Explain how a nonlinear mechanical model is linearized about an operating point, including the roles of the Jacobian matrices.

3. Describe a complete experimental modal-analysis procedure from sensor selection to modal validation.

4. Compare transfer-function, block-diagram, signal-flow, and state-space representations in terms of strengths and limitations.

5. Explain the stages of a reliable system-identification study, including excitation, preprocessing, estimation, order selection, and validation.

Analytical / Application Questions

1. A measured FRF shows a resonance at a frequency that is absent from a rigid-body model. Propose a diagnostic and modelling procedure to determine whether the cause is structural flexibility, sensor dynamics, or measurement error.

2. A high-order identified model fits the estimation record almost perfectly but performs poorly on a second record. Analyse the likely causes and recommend a revised identification strategy.

3. A servo axis operates at two substantially different payloads. Decide whether a single linear model, gain scheduling, online parameter estimation, or a nonlinear model is most appropriate, and justify the decision using model validity and control requirements.

22. Multiple Choice Questions with Answers

Question 1

For a mass–spring–damper system with force input and displacement output, which transfer function is correct?

A. 1/(ms+c+k)

B. 1/(ms²+cs+k)

C. (ms²+cs+k)

D. 1/(m+c+k)s²

Correct Answer: B. 1/(ms²+cs+k)

Question 2

What does a full-rank controllability matrix indicate for a state-space model?

A. All states can be inferred from outputs

B. The system has no disturbances

C. The states can be driven through the input

D. The system is necessarily stable

Correct Answer: C. The states can be driven through the input

Question 3

A linearized model obtained using a Taylor expansion is generally valid:

A. For all amplitudes and operating points

B. Only near the selected operating point

C. Only for unstable systems

D. Only when there is no damping

Correct Answer: B. Only near the selected operating point

Question 4

Which observation most strongly suggests that an identified model has omitted relevant dynamics?

A. Residuals have zero mean

B. The validation error is small

C. Residuals remain correlated with past inputs

D. The model has physically meaningful parameters

Correct Answer: C. Residuals remain correlated with past inputs

Question 5

In experimental modal analysis, a rapid phase change and amplitude peak commonly indicate:

A. A resonance

B. Sensor zero offset only

C. Aliasing at zero frequency

D. A static equilibrium

Correct Answer: A. A resonance

Question 6

Why should identification and validation data usually be different records?

A. To increase the sampling frequency

B. To test performance on data not used for fitting

C. To eliminate the need for calibration

D. To guarantee a linear system

Correct Answer: B. To test performance on data not used for fitting

Question 7

Which excitation is most suitable when several frequencies must be excited efficiently within safe amplitude limits?

A. A carefully designed multisine

B. A constant input only

C. An unmeasured disturbance

D. A signal with zero bandwidth

Correct Answer: A. A carefully designed multisine

Question 8

In a feedback loop with forward transfer function G(s) and feedback H(s), the negative-feedback closed-loop transfer function is:

A. G(s)/[1−G(s)H(s)]

B. H(s)/[1+G(s)H(s)]

C. G(s)/[1+G(s)H(s)]

D. G(s)H(s)

Correct Answer: C. G(s)/[1+G(s)H(s)]

Question 9

Which feature is characteristic of a grey-box identification model?

A. It ignores all physical knowledge

B. It combines a physically motivated structure with estimated parameters

C. It contains no parameters

D. It can only be non-parametric

Correct Answer: B. It combines a physically motivated structure with estimated parameters

Question 10

A low FRF coherence near a resonance may result from:

A. Perfect synchronisation and no noise

B. Inadequate excitation, noise, leakage, or nonlinear behaviour

C. A guaranteed stable pole

D. A correct static stiffness measurement

Correct Answer: B. Inadequate excitation, noise, leakage, or nonlinear behaviour