Physical Processes and System Dynamics

Introduction and Context

A cyber-physical system (CPS) combines computation, communication, sensing, actuation, and a physical process. The physical process is not merely the environment surrounding a controller: it determines what can be measured, how rapidly the system responds, how much energy is required, and whether a control action is safe. A robot arm, autonomous vehicle, smart building, and automated manufacturing cell are all CPSs whose behaviour emerges from interactions between physical dynamics and digital decision-making.

This chapter develops models for four important engineering domains:

• mechanical systems, involving force, motion, inertia, damping, and stiffness;
• electrical systems, involving voltage, current, resistance, capacitance, and inductance;
• thermal systems, involving temperature, heat capacity, and heat transfer; and
• fluid systems, involving pressure, flow, resistance, compliance, and conservation of mass.

The central modelling task is to translate physical laws into differential equations, then express those equations in forms useful for analysis and control: state-space models, transfer functions, and energy-based representations. Every model is an approximation. A good engineer therefore states assumptions, checks units, identifies neglected effects, and validates the model against measured behaviour.

Learning Objectives

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

1. identify variables, parameters, inputs, outputs, and assumptions in a physical process;
2. derive first- and second-order differential equations from conservation laws and constitutive relations;
3. formulate continuous-time state-space models and interpret their states;
4. derive transfer functions using Laplace transforms and appropriate initial conditions;
5. relate mechanical, electrical, thermal, and fluid systems through analogous mathematical structures;
6. use energy and power balances to assess model consistency and passivity;
7. interpret time constants, poles, damping, stability, and bandwidth; and
8. select an appropriate model for simulation, control design, or embedded implementation.

1. Modelling Physical Processes

A dynamic system has variables that change with time. An input is an externally imposed quantity, an output is a quantity of interest, and a state is the minimum information required, together with future inputs, to predict future behaviour. Parameters describe physical properties such as mass, resistance, or thermal capacitance.

A model normally has four components:

1. balance laws, such as Newton’s second law, Kirchhoff’s laws, conservation of energy, or conservation of mass;
2. constitutive laws, such as Hooke’s law, Ohm’s law, Fourier’s law, or a valve characteristic;
3. boundary and initial conditions, which specify interactions with the surroundings and the starting condition; and
4. measurement relations, which connect internal variables to sensor outputs.

For a lumped-parameter model, spatial variation is neglected and each quantity is represented by a finite number of variables. This is appropriate when the physical dimensions are small relative to the relevant propagation distance or when internal equalisation is fast. Distributed models, described by partial differential equations, are required when temperature, pressure, or displacement varies significantly with position.

A model should be neither unnecessarily detailed nor too simple for its purpose. A low-order model may be ideal for controller design, while a high-fidelity finite-element or computational-fluid-dynamics model may be required for structural or thermal design.

2. Mechanical Dynamics

2.1 Translational systems

For one-dimensional translation, Newton’s law is

where ∑F is the resultant force in newtons (N), m is mass in kilograms (kg), x is displacement in metres (m), and ẍ is acceleration in m/s². A viscous damper produces a force proportional to velocity, and a linear spring produces a restoring force proportional to displacement:

Here b is the viscous damping coefficient in N·s/m and k is stiffness in N/m. For a mass–spring–damper system driven by an applied force F(t), the force balance is

The signs assume that damping and spring forces oppose positive displacement and velocity. The model is linear provided m, b, and k are constant and displacement is sufficiently small that the spring remains linear.

The undamped natural frequency and damping ratio are

where ω_n is in rad/s and ζ is dimensionless. The system is underdamped when 0 < ζ < 1, critically damped when ζ = 1, and overdamped when ζ > 1. These classifications apply to the free response of the ideal second-order model.

2.2 Rotational systems

The rotational equivalent of Newton’s law is

where τ is torque in N·m, J is rotational inertia in kg·m², and θ is angular displacement in radians. A rotational damper and torsional spring obey

where b_r has units N·m·s/rad and k_r has units N·m/rad. Thus, a rotational actuator coupled to a load may be represented by

Gear trains alter torque and speed. For an ideal gear pair with tooth counts N₁ and N₂, the angular speeds satisfy approximately ω₂/ω₁ = N₁/N₂, while torque is transformed inversely, neglecting losses. Real transmissions require an efficiency parameter and may introduce backlash, compliance, and friction.

2.3 Example: suspension actuator

Consider a linear actuator applying force to a 40 kg platform. Let k = 2,000 N/m and b = 240 N·s/m. The natural frequency is

The damping ratio is

Since ζ < 1, the platform is underdamped and will oscillate after a disturbance, although damping causes the oscillation to decay. The model helps a control engineer select feedback gains and helps a mechanical engineer determine whether additional damping is needed.

3. Electrical Dynamics

3.1 Fundamental relations

For a resistor, capacitor, and inductor, respectively,

R is resistance in ohms (Ω), C is capacitance in farads (F), and L is inductance in henrys (H). Kirchhoff’s current law states that the algebraic sum of currents at a node is zero; Kirchhoff’s voltage law states that the algebraic sum of voltages around a closed loop is zero.

For a series RLC circuit driven by voltage v(t), the charge q and current i satisfy i = dq/dt. Applying Kirchhoff’s voltage law gives

or, after differentiating when the input is sufficiently smooth,

The capacitor voltage and inductor current are natural state variables because they represent stored electrical energy.

3.2 RC and RL first-order systems

For a series resistor–capacitor circuit with output capacitor voltage and input voltage,

The time constant is

with units of seconds. For a resistor–inductor circuit with output current,

and the time constant is τ = L/R. A first-order system reaches approximately 63.2% of its final step change after one time constant and approximately 95% after three time constants, assuming a stable, isolated first-order response.

4. Thermal Dynamics

Thermal systems are modelled using energy conservation. For a lumped body with uniform temperature T,

where C_th = mc_p is thermal capacitance in J/K, m is mass in kg, c_p is specific heat capacity in J/(kg·K), and P is heat-transfer rate in watts (J/s).

For convection to an ambient temperature T_a,

where G_th = hA is thermal conductance in W/K, h is the convection coefficient in W/(m²·K), and A is exposed area in m². With an applied heating power P_h,

The thermal time constant is

and the steady-state temperature rise for constant heating is ∆T = P_h/G_th. This model assumes uniform temperature, constant properties, and approximately linear heat transfer. Radiation, contact resistance, phase change, and temperature-dependent properties may invalidate the approximation.

Worked example: heated enclosure

A small enclosure has thermal capacitance 12,000 J/K and thermal conductance 60 W/K to its surroundings. A heater supplies 180 W continuously. The time constant is

The steady-state temperature rise is

If the initial temperature equals ambient temperature, the ideal response is

At t = 200 s, the rise is 3(1 − e⁻¹) ≈ 1.90 K. The calculation shows why a heater can produce a modest final temperature rise while still requiring several minutes to approach equilibrium.

5. Fluid Dynamics Using Lumped Models

Fluid systems can often be approximated by pressure, flow, resistance, and compliance. Volumetric flow rate Q has units m³/s, pressure p has units pascals (Pa), and fluid capacitance or compliance C_f has units m³/Pa.

For laminar flow through a hydraulic resistance R_f,

This linear relation is an approximation. Turbulent flow often follows a nonlinear relation closer to ∆p ∝ Q². A compliant chamber satisfies

because flow changes the stored fluid volume. Conservation of flow at a compliant node gives

The fluid time constant is τ_f = R_f C_f. This equation has the same mathematical form as an RC electrical circuit and a thermal lumped model.

Fluid inertance can also be included. If fluid density is ρ and an effective passage has inertance I_f, then the pressure drop associated with changing flow is

A hydraulic line may therefore be represented by resistance, compliance, and inertance, analogous to R, C, and L in an electrical network. Pump curves, valve saturation, leakage, cavitation, compressibility, and fluid viscosity must be considered when the model is used beyond a narrow operating range.

6. State-Space Representation

A continuous-time state-space model has the form

where x ∈ ℝⁿ is the state vector, u ∈ ℝᵐ is the input vector, y ∈ ℝᵖ is the output vector, A is the system matrix, B is the input matrix, C is the output matrix, and D is the direct-feedthrough matrix. The symbols A, B, C, and D are matrices of compatible dimensions.

For the mechanical system, choose x₁ = x and x₂ = ẋ. Then

Therefore,

If displacement is measured and there is no direct force-to-displacement path,

The state vector is not unique. Different coordinate choices can describe the same input–output behaviour, but physically meaningful states are valuable because they expose stored energy and support sensor and actuator placement decisions.

For a linear time-invariant system, the solution is

The first term is the zero-input response and the second is the forced response. The matrix exponential e^(At) describes the natural dynamics.

Stability, controllability, and observability

A continuous-time linear system is asymptotically stable when all eigenvalues of A have strictly negative real parts. Stability means that natural disturbances decay toward the equilibrium for the linear model.

The pair (A, B) is controllable if the controllability matrix

has rank n. The pair (A, C) is observable if

has rank n. Controllability concerns whether inputs can move the internal state; observability concerns whether the state can be reconstructed from outputs. These are algebraic properties of a model and do not guarantee that an actuator or sensor is physically practical.

7. Transfer Functions

For a linear time-invariant system with zero initial conditions, the transfer function is

where s is the Laplace variable. Taking the Laplace transform of the mass–spring–damper equation gives

and hence

The zero-initial-condition assumption is essential: nonzero initial energy produces an additional natural-response term that is not represented by the ordinary input/output ratio alone.

For a state-space model,

where I is the identity matrix of the same dimension as A. Poles are roots of the denominator and indicate natural modes. Zeros are values of s that make the transfer response vanish, subject to the usual conditions of the transfer-function representation.

Transfer functions are especially useful for frequency-response analysis, block-diagram algebra, and classical controller design. State-space models are generally more convenient for multiple inputs and outputs, nonzero initial conditions, state estimation, and implementation of modern control algorithms.

8. Energy-Based Modelling

Energy provides a powerful consistency check. Kinetic and potential energy in a translational mechanical system are

The total stored energy is E = ½mẋ² + ½kx². For the unforced mass–spring–damper system,

Thus the damper can only remove energy in this ideal model. If a derived equation predicts that a passive damper continuously creates energy, a sign or modelling error is likely.

Electrical storage energies are

Thermal energy relative to a reference temperature can be represented approximately by E_th = C_th(T − T_ref), while compressible fluid storage is associated with pressure and volume. Energy-based methods reveal analogies between domains and support bond-graph and port-Hamiltonian modelling, in which power flows between components and interconnections preserve conservation laws.

The power entering a mechanical force–velocity port is P = Fv, and the power entering an electrical voltage–current port is P = vi. Consistent units are a practical diagnostic: both expressions have units of watts.

9. Analogies and Model Selection

Several physical domains share a first-order balance structure.

The analogy is mathematical, not necessarily a statement that the physical mechanisms are identical. It allows engineers to transfer intuition about poles, time constants, filtering, and feedback between disciplines.

Model selection should follow the intended use:

• use a static model when transient behaviour is irrelevant;
• use a first-order model when one dominant storage mechanism controls the response;
• use a second-order model when inertia and compliance interact;
• use nonlinear models when saturation, Coulomb friction, turbulent flow, radiation, dead zones, or geometric effects are important; and
• use sampled-data models when the controller executes digitally.

When converting a continuous model for a digital controller, the sampling period must be sufficiently short relative to the fastest relevant dynamics. A zero-order-hold discretisation of a state model is

where A_d = e^(AT_s) and B_d = ∫₀^T_s e^(Aλ)B dλ, with sampling period T_s. Aliasing, computation delay, quantisation, and sensor filtering can materially alter the closed-loop behaviour.

10. Illustrative Case: Temperature Control of a Smart Battery Enclosure

Background

A battery enclosure in an automated vehicle contains cells, a fan, temperature sensors, and a microcontroller. Cell temperature affects efficiency, ageing, and safety. The controller must regulate temperature despite changing electrical load and ambient conditions.

Problem

During high-power operation, heat generation rises rapidly. A detailed computational model is available, but it is too slow for real-time control design. Engineers need a transparent model for selecting a fan command and predicting thermal transients.

Analysis

An initial lumped model uses cell-and-enclosure thermal capacitance C_th, effective conductance G_th, ambient temperature T_a, and heat-generation input P_b:

Here P_f is the effective cooling power produced by the fan, approximated over a limited operating region as proportional to fan command. The temperature sensor output is

where v represents measurement noise. A state vector containing temperature is sufficient for the first model. If the enclosure wall and cells have noticeably different temperatures, a two-state model is preferable:

The two temperatures are cell temperature T_b and wall temperature T_w; C_b and C_w are thermal capacitances, while G_{bw} and G_{wa} are thermal conductances.

Approach and solution

Engineers first identify parameters from step tests: apply a known change in heater power or fan command, record temperatures, and estimate time constants and steady-state gains. They then validate the model using a different load profile. A feedback controller uses the measured temperature, but safety logic independently limits maximum temperature, fan current, and allowable rate of temperature rise.

Outcome

The one-state model is adequate for slow supervisory control, while the two-state model predicts short-term cell–wall differences more accurately. The resulting controller can be tested in simulation before deployment. However, the model must be recalibrated as airflow paths become blocked, ambient temperature changes, or cell ageing alters heat generation.

Learning points

1. A low-order model can be highly valuable when its assumptions match the control objective.
2. Parameter identification and independent validation are essential.
3. Sensor noise and actuator limits belong in the CPS model, not only in the physical equations.
4. Safety monitoring should not rely exclusively on a possibly inaccurate control model.

11. Limitations, Safety, and Contemporary Practice

Physical models have structural and parametric uncertainty. Structural uncertainty arises when an effect is omitted; parametric uncertainty arises when a coefficient is unknown or variable. Engineers should document operating ranges, estimate uncertainty, and use sensitivity analysis to determine which parameters most affect performance.

Nonlinearities are common. Examples include Coulomb friction, motor saturation, backlash, valve dead zones, temperature-dependent resistance, turbulent pressure loss, and radiation proportional approximately to the fourth power of absolute temperature. Linearisation about an operating point is useful for local analysis, but conclusions may fail far from that point.

In CPSs, modelling errors can become safety hazards. A controller may command excessive force, overheat a battery, or destabilise a fluid actuator if sensor units, sign conventions, or sampling assumptions are wrong. Good practice includes unit checking, independent software tests, limit monitoring, fault detection, conservative safe states, traceable requirements, and hardware-in-the-loop testing. Relevant engineering projects may also be governed by functional-safety, machinery-safety, vehicle, medical-device, or industrial-control standards; the applicable regulation depends on the application and jurisdiction.

Machine learning can assist parameter estimation, anomaly detection, and surrogate modelling, but a learned model should not automatically replace conservation-law models in safety-critical functions. Hybrid models combine known physics with data-driven correction terms. Their validation must include operating conditions not used during training, sensor failures, distribution shifts, and physically impossible predictions.

Chapter Summary

Physical-process modelling begins with conservation laws, constitutive relations, boundary conditions, and measurement equations. Mechanical systems are governed by force and torque balances; electrical systems by Kirchhoff laws and component relations; thermal systems by energy balances; and fluid systems by mass conservation, pressure-flow relations, and storage effects.

Differential equations describe time evolution. State-space models expose internal energy-storage variables and are well suited to multi-input, multi-output systems, estimation, and digital implementation. Transfer functions describe zero-initial-condition input–output behaviour and support pole, zero, and frequency-response analysis. Energy methods provide physical insight and reveal modelling errors through power and passivity checks.

Time constants indicate response speed, while poles determine natural modes and stability. Model complexity should match the intended use, and assumptions must be validated experimentally. In a CPS, the physical model must be considered together with sensors, actuators, computation, sampling, uncertainty, safety limits, and communication delays.

Key Terms

State: Minimum information required to predict future system behaviour given future inputs.

Input: Externally imposed quantity that influences a system.

Output: Measured or specified quantity produced by a system.

Parameter: Physical coefficient, such as mass, resistance, or capacitance.

Lumped-parameter model: Model that represents spatially distributed behaviour with a finite set of variables.

Differential equation: Equation relating variables and their time derivatives.

State-space model: First-order vector equations describing state evolution and output generation.

Transfer function: Laplace-domain ratio of output to input under zero initial conditions.

Pole: Root of a transfer-function denominator or eigenvalue associated with a natural mode.

Time constant: Characteristic time governing a first-order exponential response.

Natural frequency: Frequency associated with an undamped second-order mode.

Damping ratio: Dimensionless measure of damping relative to critical damping.

Controllability: Ability of inputs to drive a system’s states.

Observability: Ability to reconstruct states from measured outputs.

Constitutive law: Relation describing the behaviour of a component or material.

Thermal capacitance: Heat required to raise a body’s temperature by one kelvin.

Compliance: Change in stored volume or displacement per unit pressure or force.

Passivity: Property that a system cannot generate net energy without an external source.

Linearisation: Approximation of a nonlinear model near an operating point.

Model validation: Comparison of model predictions with independent physical measurements.

Review Questions

Short Answer Questions

1. What is the difference between a state variable and an output variable?
2. State the assumptions behind the linear mass–spring–damper model.
3. What physical quantities are naturally selected as states in an RC circuit?
4. Define the thermal time constant and state its units.
5. Why are transfer functions normally defined using zero initial conditions?

Long Answer Questions

1. Derive the differential equation and state-space model of a translational mass–spring–damper system, clearly defining all variables.
2. Explain the analogy between an RC circuit, a lumped thermal system, and a fluid resistance–compliance system.
3. Compare state-space and transfer-function representations in terms of initial conditions, multiple variables, and control design.
4. Explain how energy and power balances can be used to check the correctness of a physical model.
5. Discuss the limitations of linear models and explain when nonlinear or distributed models are required.

Analytical / Application Questions

1. A 5 kg mass is connected to a spring of stiffness 500 N/m and a damper of 20 N·s/m. Calculate its natural frequency and damping ratio, classify its response, and write its force-to-displacement transfer function.
2. A thermal device has C_th = 8,000 J/K, G_th = 40 W/K, and constant input power 120 W. Determine its time constant and steady-state temperature rise above ambient. State the assumptions used.
3. Design a low-order model for a fluid tank whose level is measured, whose inlet flow is manipulated, and whose outlet flow changes approximately linearly with level. Identify the state, input, output, balance equation, and conditions under which the model would fail.

Multiple Choice Questions with Answers

Question 1

For a mass–spring–damper system, which equation represents the force balance?

A. mẍ + bẋ + kx = F(t)

B. mẋ + bx + kẍ = F(t)

C. mẍ + kẋ + bx = 0

D. mẋ + bẍ + kx = F(t)

Correct Answer: A. mẍ + bẋ + kx = F(t)

Question 2

What are the units of a thermal conductance?

A. J/K

B. W/K

C. K/W

D. W·s

Correct Answer: B. W/K

Question 3

For a stable continuous-time linear system, the eigenvalues of A must have:

A. Positive real parts

B. Zero imaginary parts only

C. Strictly negative real parts

D. Unit magnitude

Correct Answer: C. Strictly negative real parts

Question 4

The transfer function is conventionally defined under which condition?

A. Infinite input energy

B. Zero initial conditions

C. Zero system output

D. Constant acceleration

Correct Answer: B. Zero initial conditions

Question 5

A first-order thermal system has C_th = 10,000 J/K and G_th = 50 W/K. Its time constant is:

A. 20 s

B. 50 s

C. 200 s

D. 500,000 s

Correct Answer: C. 200 s

Question 6

Which quantity is stored in an ideal capacitor?

A. Mechanical momentum

B. Magnetic flux only

C. Electrical energy associated with voltage

D. Thermal resistance

Correct Answer: C. Electrical energy associated with voltage

Question 7

For a translational mechanical port, instantaneous power is:

A. P = F/v

B. P = Fv

C. P = mF

D. P = kx

Correct Answer: B. P = Fv

Question 8

A system with damping ratio ζ = 0.5 is classified as:

A. Underdamped

B. Critically damped

C. Overdamped

D. Unstable by definition

Correct Answer: A. Underdamped

Question 9

Which property means that inputs can move all states of an n-state model?

A. Observability

B. Passivity

C. Controllability

D. Linearity

Correct Answer: C. Controllability

Question 10

For a compliant fluid chamber, which relation is appropriate in a lumped model?

A. Q = C_f dp/dt

B. Q = p/C_f²

C. p = C_f dQ/dt

D. Q = R_f p²

Correct Answer: A. Q = C_f dp/dt