Simulation Lab (EC-406) - Unit 3 Short Notes

EC-406: Simulation Lab - UNIT 3 SHORT NOTES

3.0 UNIT OVERVIEW & LEARNING OBJECTIVES

• Purpose: Move beyond basic modeling to advanced hierarchical system design, multi-domain analysis (time/frequency), and performance validation.

• Key Skills:

  • Building reusable, modular models with subsystems and buses.

  • Configuring solvers for accuracy vs. speed trade-offs.

  • Extracting linear models from complex simulations.

  • Simulating and analyzing control and communication systems.

  • Interpreting results using advanced visualization tools.

3.1 ADVANCED SIMULINK MODELING TECHNIQUES

3.1.1 Model Hierarchy & Subsystems

• Subsystem: A group of blocks represented as a single block. Promotes modularity and readability.

• Masked Subsystem: A subsystem with a custom icon and dialog box. Allows user to parameterize internal blocks without opening the subsystem.

  • Use Case: Creating a reusable "PID Controller" block where user sets Kp, Ki, Kd via the mask.

• Model Referencing: A top model references and runs a separate .slx file. Changes to the referenced model propagate automatically. Better for large projects than copying subsystems.

• Custom Library: A collection of frequently used masked subsystems/blocks. Drag-and-drop from library browser ensures consistency.

3.1.2 Signal Routing & Bus Management

• Bus Creator: Combines multiple signals into a bus object (a single composite signal).

• Bus Selector: Extracts specific signals from a bus.

• Bus Assignment: Modifies or adds elements to a bus signal.

• Mux/Demux: Combines/splits signals into vector signals (elements must be of the same data type).

• Best Practice: Use buses for heterogeneous signals (different types) and mux for homogeneous vectors. Name bus elements clearly for automatic Bus Selector population.

3.1.3 Solver Configuration & Simulation Settings

• Solver Types:

  • Variable-step: Adjusts step size for accuracy (e.g., ode45 - non-stiff, ode15s - stiff). Good for transient analysis.

  • Fixed-step: Constant step size. Required for real-time simulation and discrete systems.

• Key Parameters:

  • Start/Stop time: Simulation window.

  • Max step size: Upper bound on solver step (controls detail/speed).

  • Relative/absolute tolerance: Error tolerances for variable-step solvers.

• [!TIP] Common Pitfall: Using a variable-step solver for a model with discrete (sampled) components can cause missed sample hits. Use a fixed-step solver for discrete/hybrid systems.

3.2 TIME-DOMAIN ANALYSIS OF SYSTEMS

3.2.1 Transient Response Analysis

• Standard Inputs: Step (Step block), Impulse (Pulse Generator with very small width), Ramp.

• Key Performance Metrics (for a step response):

  • Rise Time ($$\displaystyle t_r $$): Time to go from 10% to 90% of final value.

  • Settling Time ($$\displaystyle t_s $$): Time to enter and remain within a tolerance band (e.g., ±2%) around steady-state.

  • Overshoot ($$\displaystyle M_p $$): Maximum peak value minus steady-state value, as a percentage.

  • Steady-State Error ($$\displaystyle e_{ss} $$): Final value minus desired input.

• Tools: To Workspace block (exports data as timeseries), Scope block (visualizes). Use MATLAB's stepinfo(sys) for automatic calculation.

3.2.2 Analysis of Linear & Nonlinear Systems

• Nonlinearities: Implement using blocks like Saturation, Dead Zone, Relay, Piecewise Linear.

• Approach: Simulate nonlinear model. For small-signal analysis around an operating point, use linearization (see 3.3.1).

• [!TIP] Exam Insight: Be prepared to compare the step response of a linear model (e.g., transfer function) with its nonlinear counterpart (e.g., motor with saturation) and explain differences (e.g., reduced overshoot due to saturation).

3.2.3 Discrete-Time System Simulation

• Implementation: Use Unit Delay block to implement difference equations: y[k] = a1*y[k-1] + b0*u[k] + b1*u[k-1].

• Zero-Order Hold (ZOH): Zero-Order Hold block holds the input constant between sampling instants. Models the effect of a DAC.

• Discretization: Convert continuous TF to discrete using c2d in MATLAB or Discrete Transfer Fcn block with specified Sample time.

3.3 FREQUENCY-DOMAIN ANALYSIS IN SIMULINK

3.3.1 Linearization of Models

• Purpose: Obtain a linear time-invariant (LTI) model (state-space (A,B,C,D) or transfer function) of a nonlinear or complex model at a specific operating point.

• Tools:

  • Simulink Control Design™: Linear Analysis Tool (right-click on model).

  • Functions: linmod (for continuous models), dlinmod (for models with discrete blocks).

• Operating Point: Must be a steady-state condition where derivatives are zero. Can be found using trim function or by simulating to steady-state.

3.3.2 Bode Plot & Frequency Response

• From Linear Model: Use MATLAB bode(sys) on the linearized sys.

• From Simulink Model: Use Bode Plot block (from Simulink Control Design) which linearizes the model internally at each frequency.

• Key Metrics from Bode:

  • Gain Crossover Frequency ($$\displaystyle \omega_{gc} $$): Freq. where |G(jω)| = 1 (0 dB).

  • Phase Crossover Frequency ($$\displaystyle \omega_{pc} $$): Freq. where ∠G(jω) = -180°.

  • Gain Margin (GM): $$\displaystyle GM = \frac{1}{|G(j\omega_{pc})|} $$ (in dB: $$\displaystyle 20\log_{10}(GM) $$).

  • Phase Margin (PM): $$\displaystyle PM = \angle G(j\omega_{gc}) + 180° $$.

  • Bandwidth: Freq. where |G(jω)| drops to -3 dB of DC gain.

• \boxed{PM > 0^\circ \text{ and } GM > 1 \text{ (0 dB)} \text{ indicates stability for minimum-phase systems.}}

3.3.3 Root Locus & Pole-Zero Maps

• Root Locus: Plot of closed-loop pole locations as a scalar gain K varies from 0 to ∞. Generated by rlocus(sys).

• Pole-Zero Map: Plot of open-loop poles (x) and zeros (o) on the complex s-plane. Use pzmap(sys).

• Design Insight: Poles in Left Half Plane (LHP) → stable. Poles on imaginary axis → marginally stable (oscillations). Poles in Right Half Plane (RHP) → unstable. Distance from imaginary axis relates to settling time.

3.4 APPLICATION-SPECIFIC SIMULATION CASE STUDIES

3.4.1 Control Systems

• PID Controller: Implement using PID Controller block (continuous) or Discrete PID Controller. Tuning via simulation (e.g., Ziegler-Nichols: find ultimate gain Ku and period Pu).

• Closed-Loop Analysis: Simulate with Step reference, add Disturbance (e.g., Step block at plant input) and Noise (e.g., Band-Limited White Noise) at sensor. Observe performance via Scope.

3.4.2 Communication Systems (Baseband & Passband)

• Typical Chain: Random Integer (Source) → Modulator (e.g., BPSK Modulator Baseband) → AWGN Channel → Demodulator → Error Rate Calculation.

• BER vs. Eb/N0: Sweep Eb/N0 (using For subsystem or MATLAB script), simulate many bits, compute BER using Error Rate Calculation block. Plot using semilogy.

• Theoretical BER for BPSK in AWGN: $$\displaystyle P_b = Q\left(\sqrt{\frac{2E_b}{N_0}}\right) $$ where $$\displaystyle Q(x) = \frac{1}{\sqrt{2\pi}}\int_x^\infty e^{-t^2/2} dt $$.

• [!TIP] Common Mistake: Forgetting to account for pulse shaping (e.g., raised cosine) in passband simulations, leading to incorrect bandwidth/ISI.

3.4.3 Signal Processing Systems

• Filter Design: Use Filter Designer app or fdesign/design functions to get coefficients. Implement with Discrete Transfer Fcn or DFilt block.

• FFT Analysis: Use FFT block (DSP System Toolbox) or Spectrum Analyzer block to visualize frequency content. Input: multi-tone signal or noisy signal.

• Key Test: Apply a square wave to a low-pass filter. Observe fundamental attenuation and harmonic suppression in time and frequency domains.

3.5 ADVANCED VISUALIZATION & RESULT INTERPRETATION

• Scope Block: Configure Number of axes, Time range, Y-limits. Enable Log data to workspace for post-processing.

• Spectrum Analyzer: Real-time spectral view. Configure FFT length, Window type (e.g., Hamming), Frequency scale (Hz/rad/s).

• MATLAB Workspace Plotting:

  • Data from To Workspace (as timeseries): plot(data.time, data.signals.values).

  • For frequency response: [mag, phase, w] = bode(sys); semilogx(w, 20*log10(squeeze(mag))).

• Programmatic Metrics: stepinfo(sys) returns RiseTime, SettlingTime, Overshoot, etc. bandwidth(sys) returns 3-dB bandwidth.

3.6 TROUBLESHOOTING & BEST PRACTICES

• Algebraic Loop: Occurs when a signal's value depends on itself directly (no unit delay). Fix: Introduce Unit Delay or Memory block.

• Solver Failure (Singularity): Often from uninitialized states or stiff systems. Fix: Check initial conditions, try a different solver (ode15s for stiff), or adjust tolerances.

• Diagnostic Viewer: Primary tool. Read warnings (e.g., "Signal dimension mismatch")—they often indicate root cause.

• Best Practices:

  • Use meaningful signal names (right-click signal → Properties).

  • Add annotations (Ctrl+I) to explain subsystems.

  • Disable logging for signals not needed for analysis to reduce memory/speed overhead.

  • Use model advisor (Analysis → Model Advisor) to check for common issues.

3.7 PRACTICAL LAB EXPERIMENT THEMES (Typical for Unit 3)

1. PID-Controlled DC Motor:

   • Model: Motor (TF K/(Js+b)) + PID + Step reference.

   • Task: Tune Kp, Ki, Kd to meet specs (e.g., $$\displaystyle t_s < 0.5s $$, $$\displaystyle M_p < 5\% $$). Compare with/without integral windup protection.

2. Linearization of Pendulum:

   • Model: Nonlinear pendulum (sin(θ)). Linearize around θ=0 (upright) and θ=π (inverted). Compare step responses.

3. BPSK over AWGN:

   • Model: Random bits → BPSK mod → AWGN (Eb/N0 sweep) → BPSK demod → Error Rate Calculation.

   • Task: Simulate for Eb/N0 = 0:2:10 dB. Plot simulated BER vs. theoretical curve.

4. Butterworth Low-Pass Filter:

   • Design: [b,a] = butter(4, 0.2) (4th order, fc=0.2*fs/2). Implement.

   • Task: Input square wave (fs=100 Hz). Observe output in Scope and Spectrum Analyzer. Note harmonic attenuation.

5. Hierarchical Digital Comm System:

   • Build top-level: Source → Channel Coder → Modulator → Channel → Demodulator → Channel Decoder → Sink.

   • Use masked subsystems for "Modulator" (select BPSK/QPSK) and "Channel" (set SNR). Reuse across experiments.

3.8 KEY MATLAB/SIMULINK FUNCTIONS & BLOCKS (Quick Reference)

Final Exam Reminder: Always state your assumptions (e.g., "Assuming linearization around zero operating point..."). For simulation results, include screenshots of the model and key outputs (Scope/Bode plot) in your answer.