2. Mathematical Modeling of Dynamical Systems

Introduction and Context

Control engineering is concerned with influencing the behavior of physical systems so that they achieve specified objectives. Before a controller can be designed, however, the system must be represented mathematically. This representation is called a mathematical model. A model does not reproduce every physical detail; rather, it captures the relationships that are important for prediction, analysis, and control.

Mathematical modeling is therefore the bridge between a real plant and analytical control theory. A motor, generator, suspension system, thermal process, or power converter exists in the physical domain, whereas stability tests, frequency-response methods, state estimation, and controller synthesis are performed using equations. The quality of the model strongly influences the quality of the resulting controller. An oversimplified model may produce poor performance or instability, while an unnecessarily detailed model may make analysis computationally expensive and obscure the dominant dynamics.

This chapter develops models using differential equations, Laplace transforms, transfer functions, block diagrams, signal-flow graphs, and state variables. Electrical, mechanical, and electromechanical examples are used to show how physical laws become system equations. Particular attention is given to assumptions, initial conditions, linearization, model reduction, parameter interpretation, and the limitations of idealized models.

Learning Objectives

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

1. Explain the role and purpose of mathematical models in control engineering.

2. Derive differential-equation models from electrical, mechanical, and electromechanical laws.

3. Obtain transfer functions using Laplace transforms and zero initial conditions.

4. Interpret poles, zeros, gains, time constants, and system order physically.

5. Construct and simplify block diagrams and signal-flow graphs.

6. Formulate state-space models and convert between representations.

7. Linearize nonlinear dynamical systems about an operating point.

8. Select an appropriate model complexity and identify common modeling errors.

9. Apply modeling methods to a practical electromechanical control problem.

2.1 What a Dynamical-System Model Represents

A dynamical system is one whose output depends not only on present inputs but also on stored energy or past history. Capacitors store electrical energy, inductors store energy in magnetic fields, masses store kinetic energy, and thermal bodies store heat. These storage mechanisms produce differential equations and distinguish dynamic systems from purely algebraic systems.

A model normally identifies:

• input variables, which represent commanded, injected, or environmental quantities;
• output variables, which represent measured or controlled quantities;
• state variables, which summarize the stored energy and therefore the system's internal condition;
• parameters, such as resistance, mass, damping, inductance, or stiffness; and
• disturbances, uncertainties, and neglected effects.

The same physical plant can have several valid models. A motor may be represented by a nonlinear differential equation, a linear transfer function around a rated operating point, or a state-space model containing current and angular velocity as states. Model selection depends on the intended use, available measurements, operating range, and acceptable computational effort.

A useful model satisfies three requirements. It must be physically consistent, sufficiently accurate for the intended purpose, and mathematically manageable. These requirements often conflict. A model for preliminary controller design may neglect magnetic saturation, friction nonlinearities, and parasitic capacitance, whereas a model for high-fidelity simulation must include them.

2.2 From Physical Laws to Differential Equations

The first modeling step is generally the identification of conservation laws and constitutive relations. In electrical systems, Kirchhoff's voltage and current laws are combined with

\[

v_R=Ri, \qquad v_L=L\frac{di}{dt}, \qquad i_C=C\frac{dv_C}{dt}.

\]

In translational mechanics, Newton's law gives

\[

\sum F=M\frac{d^2x}{dt^2},

\]

while the constitutive relations for a spring and viscous damper are

\[

F_k=Kx, \qquad F_b=B\frac{dx}{dt}.

\]

For rotational mechanics,

\[

\sum T=J\frac{d^2\theta}{dt^2},

\]

with viscous friction torque \(B\dot\theta\) and torsional spring torque \(K\theta\).

The order of a differential equation is related to the number of independent energy-storage elements after algebraic constraints have been considered. A series RLC circuit is generally second order because it contains two independent storage elements, the inductor and capacitor. A mass-spring-damper system is also second order. The order is important because it indicates the number of independent initial conditions and, in a minimal realization, the number of system states.

Example 2.1: Series RLC Circuit

Consider a series circuit driven by voltage \(v_i(t)\), with resistor \(R\), inductor \(L\), and capacitor \(C\). If the output is the capacitor voltage \(v_o(t)=v_C(t)\), Kirchhoff's voltage law gives

\[

v_i(t)=Ri(t)+L\frac{di(t)}{dt}+v_C(t).

\]

Since \(i(t)=C\,dv_C/dt\),

\[

LC\frac{d^2v_o}{dt^2}+RC\frac{dv_o}{dt}+v_o=v_i.

\]

This equation shows directly how resistance affects the first-derivative term, while inductance and capacitance determine the second-order energy storage. It also makes clear that the output definition matters: the model would be different if the output were resistor voltage or inductor voltage.

Example 2.2: Mass-Spring-Damper System

For a mass \(M\), damper \(B\), and spring \(K\) acted on by force \(F(t)\), with displacement \(x(t)\), force balance yields

\[

M\ddot{x}+B\dot{x}+Kx=F(t).

\]

The equation is mathematically analogous to that of an RLC circuit, but analogy does not imply identical physical behavior. The parameter correspondence depends on the chosen analogy. What matters for control analysis is the structure of the governing equation and the meaning of each coefficient.

2.3 Laplace-Domain Models and Transfer Functions

The Laplace transform converts differential equations into algebraic equations in the complex variable \(s\). For a function \(f(t)\),

\[

\mathcal{L}\{\dot f(t)\}=sF(s)-f(0^-),

\]

and

\[

\mathcal{L}\{\ddot f(t)\}=s^2F(s)-sf(0^-)-\dot f(0^-).

\]

In control-system transfer functions, initial conditions are normally taken as zero. The transfer function from input \(U(s)\) to output \(Y(s)\) is

\[

G(s)=\frac{Y(s)}{U(s)}\bigg|_{\text{zero initial conditions}}.

\]

The zero-initial-condition qualification is essential. A nonzero initial capacitor voltage or mechanical displacement contributes an additional response that is not caused by the input and is not included in the transfer function.

For the RLC circuit,

\[

G(s)=\frac{V_o(s)}{V_i(s)}=\frac{1}{LCs^2+RCs+1}.

\]

For the mass-spring-damper system,

\[

G(s)=\frac{X(s)}{F(s)}=\frac{1}{Ms^2+Bs+K}.

\]

A transfer function displays input-output behavior but does not uniquely describe internal behavior. Different internal realizations may have the same transfer function. Transfer functions are especially useful for linear time-invariant systems, while state-space models are more general for multivariable systems, nonlinear systems, and systems with nonzero initial conditions.

Poles, zeros, and physical interpretation

A transfer function may be written as

\[

G(s)=K\frac{(s-z_1)(s-z_2)\cdots}{(s-p_1)(s-p_2)\cdots}.

\]

The values \(p_i\) are poles and the values \(z_i\) are zeros. Poles are values of \(s\) at which the natural response is generated and the transfer function becomes unbounded in the ideal mathematical model. Zeros are values that suppress particular modes of the input-output response. For a continuous-time linear system, poles in the left half of the complex plane correspond to asymptotically decaying modes; poles on the imaginary axis indicate sustained modes under ideal conditions; right-half-plane poles indicate instability.

For the standard second-order denominator,

\[

s^2+2\zeta\omega_n s+\omega_n^2,

\]

\(\omega_n\) is the undamped natural frequency and \(\zeta\) is the damping ratio. Comparing the mass-spring-damper model gives

\[

\omega_n=\sqrt{K/M}, \qquad \zeta=\frac{B}{2\sqrt{KM}}.

\]

Thus increasing stiffness raises the natural frequency, increasing mass lowers it, and damping changes the transient character. Parameter interpretation is valuable because it allows design decisions to be made in physical terms rather than only through algebraic coefficients.

2.4 Block Diagrams and Signal-Flow Graphs

A block diagram represents functional relationships between variables. A block labeled \(G(s)\) means that the output is \(G(s)\) times the input. Summing points and takeoff points show how signals combine and branch.

For a cascade of blocks,

\[

G_{eq}=G_1G_2\cdots G_n.

\]

For parallel blocks with the same input,

\[

G_{eq}=G_1+G_2+\cdots+G_n.

\]

For a negative-feedback loop with forward path \(G(s)\) and feedback path \(H(s)\),

\[

\frac{Y(s)}{R(s)}=\frac{G(s)}{1+G(s)H(s)}.

\]

With positive feedback, the denominator becomes \(1-GH\). The sign must be determined from the summing junction, not guessed from the appearance of the diagram.

Block-diagram reduction is convenient for simple single-input, single-output systems, but careless movement of summing points or takeoff points can alter signal relationships. A block may be moved across a summing point only if the appropriate compensating factor is included. For complex interconnections, signal-flow graphs and Mason's gain formula provide a systematic alternative.

In a signal-flow graph, nodes represent variables and directed branches represent gains. Mason's formula is

\[

T=\frac{\sum_k P_k\Delta_k}{\Delta},

\]

where \(P_k\) is the gain of the \(k\)-th forward path, \(\Delta=1-\)(sum of individual loop gains) + (sum of products of non-touching loop gains) - ..., and \(\Delta_k\) is formed by excluding loops that touch path \(P_k\). The formula is particularly useful when several loops interact and ordinary block reduction becomes cumbersome.

2.5 State-Space Representation

A state-space model expresses a system using first-order equations:

\[

\dot{x}=f(x,u,t), \qquad y=g(x,u,t).

\]

For a linear time-invariant system,

\[

\dot{x}=Ax+Bu,

\qquad y=Cx+Du.

\]

The vector \(x\) contains state variables. A state is a minimum set of variables whose present values, together with future inputs, determine future system behavior. State variables are commonly selected as capacitor voltages, inductor currents, positions, and velocities.

For the mass-spring-damper system, choose \(x_1=x\) and \(x_2=\dot{x}\). Then

\[

\dot{x}_1=x_2,

\]

\[

\dot{x}_2=-\frac{K}{M}x_1-\frac{B}{M}x_2+\frac{1}{M}F,

\]

and, if the output is displacement,

\[

y=x_1.

\]

Thus

\[

A=\begin{bmatrix}0&1\\-K/M&-B/M\end{bmatrix},\quad

B=\begin{bmatrix}0\\1/M\end{bmatrix},\quad

C=\begin{bmatrix}1&0\end{bmatrix},\quad D=0.

\]

The transfer function associated with a state-space model is

\[

G(s)=C(sI-A)^{-1}B+D.

\]

State-space methods are central to modern control because they naturally accommodate multiple inputs and outputs, coupled subsystems, time-varying models, observers, optimal control, and numerical simulation. The choice of state coordinates is not unique. Similarity transformations can produce different matrices representing the same input-output dynamics.

2.6 Electromechanical Modeling

Electromechanical systems couple electrical and mechanical energy. A simplified armature-controlled DC motor illustrates the coupling. The armature circuit is modeled by

\[

v_a(t)=L_a\frac{di_a}{dt}+R_ai_a(t)+e_b(t),

\]

where the back electromotive force is

\[

e_b(t)=K_e\omega(t).

\]

The mechanical equation is

\[

J\frac{d\omega}{dt}+B\omega(t)=K_ti_a(t)-T_L(t),

\]

where \(K_ti_a\) is electromagnetic torque and \(T_L\) is load torque. For SI units and consistent motor conventions, the torque and back-emf constants are numerically related, although their physical units differ.

Ignoring load torque and taking angular velocity as output, the transfer function from armature voltage to speed is

\[

\frac{\Omega(s)}{V_a(s)}=

\frac{K_t}{(L_as+R_a)(Js+B)+K_tK_e}.

\]

If armature inductance is negligible over the bandwidth of interest, the electrical dynamics may be approximated algebraically, producing a lower-order model. This approximation is valid only when the neglected electrical time constant is significantly smaller than the dominant mechanical time constant and when fast electrical transients are not important to the application.

The motor example demonstrates why coupled variables must not be omitted. Current produces torque, speed produces back emf, and mechanical motion feeds back into the electrical subsystem. Treating the motor as a simple gain can be appropriate in a narrow operating range, but it hides the mechanism that determines transient response and stability.

2.7 Linearization of Nonlinear Systems

Many physical systems are nonlinear. Examples include friction with stiction, magnetic saturation, aerodynamic drag proportional to speed squared, diode conduction, and pendulum motion. A nonlinear model can be written as

\[

\dot{x}=f(x,u), \qquad y=g(x,u).

\]

Suppose the system operates near an equilibrium \((x_0,u_0)\), satisfying \(f(x_0,u_0)=0\). Define perturbation variables \(\delta x=x-x_0\), \(\delta u=u-u_0\), and \(\delta y=y-y_0\). A first-order Taylor expansion gives

\[

\delta\dot{x}=A\delta x+B\delta u,

\qquad \delta y=C\delta x+D\delta u,

\]

where

\[

A=\left.\frac{\partial f}{\partial x}\right|_{x_0,u_0},\quad

B=\left.\frac{\partial f}{\partial u}\right|_{x_0,u_0},

\]

\[

C=\left.\frac{\partial g}{\partial x}\right|_{x_0,u_0},\quad

D=\left.\frac{\partial g}{\partial u}\right|_{x_0,u_0}.

\]

Linearization is local. It describes small deviations near a specified operating point and may become inaccurate for large signals or when the operating point changes substantially. A linear model should therefore always be accompanied by its operating condition and range of validity.

For example, the pendulum equation

\[

J\ddot{\theta}+B\dot{\theta}+mgl\sin\theta=T

\]

becomes, near \(\theta=0\),

\[

J\delta\ddot{\theta}+B\delta\dot{\theta}+mgl\delta\theta=\delta T,

\]

because \(\sin\theta\approx\theta\) for small angles. Near the inverted position, the sign of the gravity term changes, leading to a different local model and different stability characteristics.

2.8 Model Reduction, Assumptions, and Validation

Model reduction is the controlled removal of dynamics that have little influence on the intended analysis. Common approaches include neglecting small time constants, retaining dominant poles, combining parameters, and using quasi-steady-state approximations. Reduction should be based on time-scale separation, frequency range, and performance requirements rather than algebraic convenience alone.

A useful workflow is:

1. Define the modeling purpose, inputs, outputs, operating range, and required accuracy.

2. Draw the physical interconnection and identify energy-storage elements.

3. State assumptions explicitly, including linearity, lumped parameters, negligible delays, and parameter constancy.

4. Apply conservation laws and constitutive relations.

5. Check dimensions, signs, limiting cases, and energy consistency.

6. Obtain a differential, transfer-function, or state-space representation.

7. Compare model predictions with measurements, simulation, or trusted reference data.

8. Refine or reduce the model according to the application.

Dimensional analysis is a simple but powerful error check. In \(M\ddot{x}+B\dot{x}+Kx=F\), all terms must have force units. Sign conventions must also be defined before equations are written. A reversed current or torque reference can create apparent negative resistance or negative damping in the derived equations.

Parameter uncertainty is unavoidable. Resistance changes with temperature, friction varies with lubrication and load, and motor constants may depend on saturation. A model should therefore distinguish nominal parameters from uncertain parameters. Sensitivity analysis can reveal which parameters most strongly affect poles, settling time, or steady-state gain.

Applied Case Study: Speed Control of a DC Motor

Background

Consider an illustrative industrial conveyor driven by a separately excited DC motor. The conveyor must maintain speed despite changes in product load. The available measurements are armature voltage and shaft speed; load torque is treated as a disturbance.

Situation and modeling problem

A controller designed from a static voltage-to-speed gain may appear adequate at steady state but fail to predict overshoot, current transients, or disturbance rejection. A dynamic model is required to determine the relevant time constants and to support controller design.

Analysis

The electrical and mechanical equations are

\[

v_a=L_a\dot{i}_a+R_ai_a+K_e\omega,

\]

\[

J\dot{\omega}=K_ti_a-B\omega-T_L.

\]

The states are selected as \(x_1=i_a\) and \(x_2=\omega\). Therefore,

\[

\dot{x}_1=-\frac{R_a}{L_a}x_1-\frac{K_e}{L_a}x_2+\frac{1}{L_a}v_a,

\]

\[

\dot{x}_2=\frac{K_t}{J}x_1-\frac{B}{J}x_2-\frac{1}{J}T_L.

\]

The load torque enters as a disturbance input. The equations reveal two important feedback effects: current generates acceleration through torque, while speed generates opposing back emf. They also show that electrical resistance and inductance govern current dynamics, whereas inertia and friction govern mechanical dynamics.

Approach

Suppose measurements and preliminary tests indicate that the electrical time constant is much smaller than the mechanical time constant. For a speed controller whose bandwidth is well below the electrical pole, the armature current may be approximated as quasi-steady:

\[

i_a\approx \frac{v_a-K_e\omega}{R_a}.

\]

Substitution into the mechanical equation gives a reduced first-order speed model. The motor's back emf appears as an electrical contribution to effective damping. This reduced model can be used for initial controller tuning, while the full second-order model is retained for verification of current limits and fast transients.

Outcome

The resulting modeling strategy uses two models rather than forcing one model to serve every purpose. The reduced model supports transparent controller design and parameter interpretation. The full model checks whether the design produces excessive current, unacceptable electrical transients, or sensitivity to neglected inductance.

Learning points

The case illustrates that model order is a design choice, not merely a property copied from a data sheet. A lower-order model is useful only when its assumptions match the control bandwidth and operating conditions. Disturbances should be included explicitly, and a model should be validated against the phenomena that matter operationally.

Professional Practice, Risks, and Emerging Tools

In professional engineering, model documentation is part of technical governance. Assumptions, parameter sources, calibration procedures, units, sign conventions, operating ranges, and validation evidence should be recorded. An undocumented model can be misapplied after a system is modified or transferred to another team.

Common mistakes include confusing force with displacement models, omitting initial-condition qualifications, using a linear approximation over a wide operating range, canceling unstable pole-zero pairs without considering internal behavior, and ignoring sensor or actuator dynamics. Another error is treating identified numerical parameters as physically meaningful without checking identifiability or parameter correlation.

Simulation environments and computer algebra systems can generate state equations, linearize nonlinear models, and compare frequency responses. Digital twins and real-time simulators extend these capabilities for commissioning and predictive maintenance. Data-driven and machine-learning models can complement physics-based models when unmodeled effects are difficult to describe. Nevertheless, data-driven models require representative data and may fail outside the training domain. Hybrid physics-informed models are increasingly used to combine physical constraints with measured behavior.

Professional decisions must also consider safety and ethics. A model used to tune a high-energy converter, aircraft actuator, or industrial drive should not be trusted solely because a simulation is stable. Model uncertainty, sensor faults, saturation, delays, and worst-case operating conditions must be considered. Validation, independent review, and conservative testing are essential where incorrect control action can damage equipment or endanger people.

Best Practices for Mathematical Modeling

• Define the purpose and validity range before deriving equations.
• Select inputs, outputs, and state variables with physical meaning.
• Preserve units and sign conventions throughout the derivation.
• Separate nominal models from disturbances and uncertainties.
• Use transfer functions for suitable linear input-output analysis and state-space models for coupled, multivariable, or time-varying problems.
• Record every approximation and test its effect on the intended performance measure.
• Validate both transient and steady-state behavior.
• Retain a higher-fidelity model for verification even when a reduced model is used for design.

Chapter Summary

Mathematical modeling converts physical laws and energy-storage relationships into representations suitable for control analysis. Differential equations describe the system directly; Laplace transforms yield transfer functions under zero initial conditions; block diagrams and signal-flow graphs organize interconnections; and state-space equations represent internal dynamics and naturally support multivariable analysis. Poles, zeros, natural frequency, damping ratio, and time constants connect algebraic models with physical behavior.

Electrical, mechanical, and electromechanical systems can often be modeled using common structural ideas, although their parameters have different physical meanings. Nonlinear systems may be linearized locally around an equilibrium, while model reduction can remove fast or weakly relevant dynamics when justified by time-scale and bandwidth arguments. Reliable modeling requires explicit assumptions, dimensional and sign checks, validation, and awareness of uncertainty. In professional practice, the appropriate model is not the most detailed one in every situation, but the one whose accuracy, complexity, and validity match the decision or control task.

Key Terms

Dynamical system: A system whose behavior depends on stored energy and past history.

Mathematical model: An equation-based representation of a physical system.

State variable: A variable in a minimum set that determines future behavior with future inputs known.

Input: An externally applied variable that influences system behavior.

Output: A system variable selected for observation or control.

Transfer function: The zero-initial-condition ratio of output transform to input transform.

Pole: A root of the transfer-function denominator associated with a system mode.

Zero: A root of the transfer-function numerator that suppresses an input-output mode.

System order: The number of independent dynamic storage modes in a minimal representation.

Natural frequency: The characteristic oscillation frequency of an undamped second-order system.

Damping ratio: A dimensionless measure of damping relative to critical damping.

Block diagram: A graphical representation of functional signal relationships.

Signal-flow graph: A directed graph whose branches represent gains between system variables.

State-space model: A first-order vector differential equation with an algebraic output equation.

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

Operating point: A steady condition about which a system may be analyzed or linearized.

Model reduction: Simplification of a model by removing dynamics judged insignificant for a specified purpose.

Disturbance: An input not normally manipulated for control but affecting system behavior.

Time constant: A parameter indicating the characteristic response time of a first-order mode.

Review Questions

Short Answer Questions

1. Why are zero initial conditions specified when defining a transfer function?

2. What physical meaning do poles have in a dynamical-system model?

3. Distinguish between an input, an output, and a state variable.

4. State the standard negative-feedback closed-loop transfer function.

5. Why is linearization considered a local approximation?

Long Answer Questions

1. Derive the transfer function of a series RLC circuit when the capacitor voltage is the output.

2. Explain how Newton's law leads to the mass-spring-damper model and interpret each parameter.

3. Compare transfer-function and state-space representations, including their relative applications.

4. Explain the construction and use of a signal-flow graph and Mason's gain formula.

5. Derive the coupled electrical and mechanical equations of an armature-controlled DC motor.

Analytical / Application Questions

1. A mechanical system has \(M=2\), \(B=4\), and \(K=8\) in consistent units. Determine its natural frequency and damping ratio, and classify its transient character.

2. For a DC motor whose electrical time constant is one-tenth of its mechanical time constant, discuss when neglecting armature inductance is justified and what behavior may be lost.

3. A nonlinear actuator is linearized at two different equilibrium points. Explain why the resulting models may have different poles and describe how you would validate which model is appropriate for controller design.

22. Multiple Choice Questions with Answers

Question 1

Which statement correctly defines a transfer function?

A. The ratio of output to input for arbitrary initial conditions

B. The zero-initial-condition ratio of output transform to input transform

C. The ratio of stored energy to dissipated energy

D. The ratio of poles to zeros

Correct Answer: B. The zero-initial-condition ratio of output transform to input transform

Question 2

For a mass-spring-damper system, which parameter primarily determines the coefficient of the second derivative of displacement?

A. Mass

B. Damping coefficient

C. Spring stiffness

D. Applied force

Correct Answer: A. Mass

Question 3

A continuous-time linear system with all poles strictly in the left half of the complex plane has

A. An asymptotically decaying natural response

B. A sustained natural response

C. An unstable natural response

D. No dynamic response

Correct Answer: A. An asymptotically decaying natural response

Question 4

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

A. G(s)H(s)

B. G(s)/(1-G(s)H(s))

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

D. 1/(G(s)+H(s))

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

Question 5

Which pair is normally selected as state variables for a basic series RLC circuit?

A. Resistor voltage and resistor current only

B. Inductor current and capacitor voltage

C. Source voltage and resistor resistance

D. Circuit power and frequency

Correct Answer: B. Inductor current and capacitor voltage

Question 6

The main purpose of linearizing a nonlinear model about an equilibrium point is to

A. Represent large-signal behavior exactly

B. Obtain a local first-order approximation

C. Remove all uncertainty from the system

D. Eliminate the need for physical parameters

Correct Answer: B. Obtain a local first-order approximation

Question 7

In a DC motor, back emf is important because it

A. Creates a mechanical spring directly

B. Provides a speed-dependent electrical feedback effect

C. Eliminates electromagnetic torque

D. Makes armature resistance zero

Correct Answer: B. Provides a speed-dependent electrical feedback effect

Question 8

If a fast pole is neglected in a reduced-order model, the approximation is most defensible when

A. The control bandwidth is well below the fast pole frequency

B. The fast pole is unstable

C. The fast pole is the only pole in the system

D. The system is operated far outside the modeled range

Correct Answer: A. The control bandwidth is well below the fast pole frequency

Question 9

For the mass-spring-damper model, increasing the mass while holding stiffness constant generally

A. Raises the natural frequency

B. Lowers the natural frequency

C. Removes damping

D. Creates a zero at the origin

Correct Answer: B. Lowers the natural frequency

Question 10

Which practice is most appropriate when developing a professional control model?

A. Hide approximations to keep the model simple

B. Validate only the steady-state gain

C. Document assumptions, units, validity range, and validation evidence

D. Use the highest-order model without regard to purpose

Correct Answer: C. Document assumptions, units, validity range, and validation evidence