5.4 System Modeling

5.4 System Modeling

Introduction to System Modeling

System modeling is the process of developing mathematical representations of physical systems to predict, analyze, and control their behavior. It is the foundation of modern control theory and engineering design. A model abstracts the essential dynamic characteristics of a system—be it mechanical, electrical, thermal, or a hybrid—using differential equations, transfer functions, or state-space representations. This unit introduces the core concepts of control systems, establishes the framework for modeling diverse physical domains, and demonstrates powerful analogies that allow engineers to translate understanding from one domain (e.g., electrical) to another (e.g., mechanical). Mastery of modeling is the first critical step towards designing automated, precise, and stable control systems for everything from robotic arms to chemical plants.

---

1. Control Systems: Definition and Classification

1.1 Definition

A control system is an interconnection of components forming a system configuration that will provide a desired system response. It manages, commands, directs, or regulates the behavior of other devices or systems.

1.2 Classification of Control Systems

Control systems are broadly classified based on several criteria:

1. Based on the Presence of Feedback:

   • Open-Loop Control Systems

   • Closed-Loop (Feedback) Control Systems

2. Based on the Nature of Signals:

   • Continuous-Time Systems: Signals are functions of a continuous time variable. Modeled by differential equations.

   • Discrete-Time Systems: Signals are defined only at discrete instants of time (sampled). Modeled by difference equations.

3. Based on System Parameters:

   • Linear vs. Non-Linear Systems:

     • Linear Systems: Obey the principles of superposition and homogeneity. Parameters are constant.

     • Non-Linear Systems: Do not obey superposition. Most real-world systems are non-linear but are often linearized about an operating point for analysis.

   • Time-Invariant vs. Time-Varying Systems:

     • Time-Invariant: Parameters do not change with time.

     • Time-Varying: Parameters change with time.

4. Based on the Number of Inputs/Outputs:

   • SISO (Single-Input, Single-Output): One input, one output.

   • MIMO (Multi-Input, Multi-Output): Multiple inputs and outputs.

---

2. Open-Loop vs. Closed-Loop (Feedback) Systems

2.1 Open-Loop Control Systems

1. Definition: A system where the control action is independent of the output. There is no feedback to compare the output with the desired input.

2. Block Diagram: Input → Controller → Actuator → Process → Output.

3. Examples: Washing machine (timed cycle), traffic light controller, toaster, basic electric fan.

4. Advantages: Simple construction, low cost, easy maintenance, stable.

5. Disadvantages:

   • Inaccurate (output is not corrected for disturbances or changes in system parameters).

   • Sensitive to disturbances (e.g., wind affecting a fan's speed).

   • Cannot correct its own errors.

2.2 Closed-Loop (Feedback) Control Systems

1. Definition: A system where the control action is dependent on the output. The output is measured (sensed), fed back, and compared with the reference input to generate an error signal, which drives the controller.

2. Block Diagram: Input → Comparator (∑) → Controller → Actuator → Process → Output → Sensor (→ Feedback path).

3. Key Elements:

   • Error Detector (Comparator): Generates error, $e(t) = r(t) - b(t)$.

   • Feedback Path: Contains the sensor which measures the output.

4. Examples: Automatic electric iron (thermostat), cruise control in a car, water level controller, missile guidance.

5. Advantages:

   • Accurate (minimizes error).

   • Less sensitive to disturbances and parameter variations.

   • Stable performance over a wide range.

6. Disadvantages:

   • More complex and expensive.

   • Risk of instability due to feedback.

   • May introduce oscillations.

---

3. Modeling of Physical Systems

Modeling involves applying fundamental physical laws (Newton's, Kirchhoff's, etc.) to derive equations of motion.

3.1 Mechanical Systems (Translational & Rotational)

• Fundamental Laws: Newton's Second Law for translation, and its rotational equivalent.

• Basic Elements:

  1. Mass (M, kg) / Moment of Inertia (J, kg·m²): Stores kinetic energy. $F = M \frac{d^2x}{dt^2}$, $T = J \frac{d^2\theta}{dt^2}$.

  2. Damper (B, N·s/m or N·m·s/rad): Dissipates energy as heat. Force/Torque is proportional to velocity. $F = B \frac{dx}{dt}$, $T = B \frac{d\theta}{dt}$.

  3. Spring (K, N/m or N·m/rad): Stores potential energy. Force/Torque is proportional to displacement. $F = K x$, $T = K \theta$.

• Procedure: Draw Free Body Diagram (FBD) → Apply Newton's Law → Write differential equation.

3.2 Electrical Systems

• Fundamental Laws: Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL).

• Basic Elements (Impedances):

  1. Resistor (R, Ω): $V_R = iR$.

  2. Inductor (L, H): $V_L = L \frac{di}{dt}$.

  3. Capacitor (C, F): $V_C = \frac{1}{C} \int i \, dt$.

• Procedure: Write loop/mesh equations using KVL or node equations using KCL → Obtain integro-differential equation.

3.3 Liquid Level Systems

• Fundamental Law: Based on conservation of mass.

• Key Variables: Head ($h$, m), Volumetric Flow Rate ($Q$, m³/s).

• Basic Elements:

  1. Capacitance (Fluid Storage): $C = \frac{A}{\rho g}$ for a tank of area $A$. Relates change in stored volume to head: $C \frac{dh}{dt} = Q_{in} - Q_{out}$.

  2. Resistance (Fluid Flow): For laminar flow, $Q = \frac{h}{R}$, where $R$ is flow resistance (e.g., for a pipe/valve).

• Procedure: Apply mass balance: Accumulation = Input - Output.

3.4 Thermal Systems

• Fundamental Law: Based on conservation of energy.

• Key Variables: Temperature ($T$, °C or K), Heat Flow Rate ($q$, W).

• Basic Elements:

  1. Thermal Capacitance: $C_{th} = mc$, where $m$ is mass, $c$ is specific heat. $C_{th} \frac{dT}{dt} = q_{in} - q_{out}$.

  2. Thermal Resistance: For conductive heat transfer, $q = \frac{\Delta T}{R_{th}}$, where $R_{th} = L/(kA)$ (L: length, k: conductivity, A: area).

• Procedure: Apply energy balance: Rate of change of internal energy = Net heat flow rate.

---

4. Force-Voltage and Force-Current Analogies

These analogies provide a direct mapping between mechanical and electrical systems, allowing analysis using familiar electrical circuit techniques.

4.1 Force-Voltage Analogy (Direct/Mobility Analogy)

• Basis: Newton's Law (ΣF=0) is analogous to KVL (ΣV=0).

• Mapping:

  • Force (F) ↔ Voltage (V)

  • Velocity (v) ↔ Current (i)

  • Mass (M) ↔ Inductance (L)

  • Damper (B) ↔ Resistance (R)

  • Spring (K) ↔ Inverse Capacitance (1/C)

  • Displacement (x) ↔ Charge (q)

• Series mechanical system becomes a series electrical circuit.

4.2 Force-Current Analogy (Inverse/Impedance Analogy)

• Basis: Newton's Law (ΣF=0) is analogous to KCL (Σi=0).

• Mapping:

  • Force (F) ↔ Current (i)

  • Velocity (v) ↔ Voltage (V)

  • Mass (M) ↔ Capacitance (C)

  • Damper (B) ↔ Conductance (1/R)

  • Spring (K) ↔ Inverse Inductance (1/L)

  • Displacement (x) ↔ Flux Linkage (λ)

• Series mechanical system becomes a parallel electrical circuit.

4.3 Choosing an Analogy

• Force-Voltage is often more intuitive as series/parallel configurations are preserved similarly.

• Force-Current can be more convenient for certain mechanical configurations (e.g., gears).

---

5. Modeling of Specific Components

5.1 Gears

• Purpose: Transform torque and rotational speed.

• Modeling Parameters: Gear Ratio $N = \frac{\theta_1}{\theta_2} = \frac{T_2}{T_1} = \frac{r_2}{r_1}$.

• Reflected Load: When modeling a motor with a gear train, the load inertia ($J_L$) and damping ($B_L$) on the secondary side are "reflected" to the primary (motor) side.

  • Reflected Inertia: $J_{eq} = J_m + \frac{J_L}{N^2}$

  • Reflected Damping: $B_{eq} = B_m + \frac{B_L}{N^2}$

5.2 Transformers (Electrical)

• Purpose: Voltage/current transformation and isolation.

• Ideal Model: $\frac{V_1}{V_2} = \frac{N_1}{N_2} = a$, $\frac{I_1}{I_2} = \frac{1}{a}$.

• Reflected Impedance: An impedance $Z_2$ in the secondary is reflected to the primary as $Z_1 = a^2 Z_2$.

5.3 Sensors and Encoders

• Purpose: Measure system outputs (position, speed, etc.) for feedback.

• Modeling: Often represented as a simple gain (K_s) in the feedback path. For ideal sensors, $K_s = 1$ (with appropriate unit conversion).

  • Potentiometer (Position): $V_{out} = K_p \cdot \theta$.

  • Tachogenerator (Speed): $V_{out} = K_t \cdot \omega$.

  • Encoder (Digital Position/Speed): Generates a pulse train. Modeled by a gain (pulses/radian) and possibly a sampler.

5.4 Generators (DC & AC)

• Purpose: Convert mechanical energy to electrical energy. Often used as sensors (tachogenerator) or power sources.

• DC Generator Model: Generated voltage, $E_g = K_g \phi \omega_m$, where $K_g$ is a constant, $\phi$ is flux, $\omega_m$ is mechanical speed.

• Armature Circuit: $V_t = E_g - I_a R_a$, where $V_t$ is terminal voltage, $I_a$ is armature current, $R_a$ is armature resistance.

---

6. Modeling of Electromechanical and Mixed Systems

Many practical systems (actuators, motors) involve coupling between electrical and mechanical domains.

6.1 DC Motor (A Prime Example)

A DC motor converts electrical input (armature voltage $V_a$) to mechanical output (shaft speed $\omega_m$ and position $\theta_m$).

1. Electrical Subsystem (Armature Circuit):

   • $V_a(t) = R_a i_a(t) + L_a \frac{di_a(t)}{dt} + e_b(t)$

   • Back EMF: $e_b(t) = K_b \omega_m(t)$, where $K_b$ is the back EMF constant.

2. Mechanical Subsystem (Rotor):

   • $J_m \frac{d\omega_m(t)}{dt} + B_m \omega_m(t) = T_m(t)$

   • Motor Torque: $T_m(t) = K_t i_a(t)$, where $K_t$ is the torque constant. (For SI units, $K_t = K_b$).

3. Combined Model (Transfer Function):

   • Often, the armature inductance $L_a$ is neglected for small motors.

   • Taking Laplace transforms and combining equations yields the transfer function from input voltage to output speed: $\frac{\Omega_m(s)}{V_a(s)} = \frac{K_t}{(R_a J_m)s + (R_a B_m + K_t K_b)}$

   • For position output: $\frac{\Theta_m(s)}{V_a(s)} = \frac{1}{s} \cdot \frac{\Omega_m(s)}{V_a(s)}$

6.2 General Procedure for Mixed Systems

1. Identify the energy domains involved (electrical, mechanical, thermal, fluidic).

2. Identify the energy conversion elements (e.g., motor, solenoid, pump).

3. Write the governing equations for each subsystem separately using domain-specific laws.

4. Write the coupling equations that link the subsystems (e.g., motor torque, back EMF).

5. Combine the equations algebraically to eliminate intermediate variables and obtain a relationship between the system's input and output.

6. Express the final model in the desired form: differential equation, transfer function, or state-space representation.

Conclusion: System modeling is the essential language of control engineering. By abstracting a physical system into a mathematical model—whether it's a simple mass-spring-damper or a complex electromechanical actuator—engineers gain the power to simulate behavior, predict performance, and design effective controllers before building costly prototypes. The analogies between different physical domains are particularly powerful, providing a unified framework for understanding and designing complex, interconnected systems.