Feedback and Control Systems: MATLAB Fundamentals, Laplace Transforms, and Transfer Functions

MATLAB Environment and Interface Fundamentals

• Overview of MATLAB:

  • MATLAB (Matrix Laboratory) is a high-level programming language and interactive environment tailored for numerical computation, data analysis, visualization, algorithm development, and application development.

  • It allows engineers and scientists to explore multiple computational approaches and solve mathematical problems significantly faster than using traditional spreadsheets or low-level programming languages such as C, C++, or Java.

• Core Applications:

  • Signal processing and communications

  • Image and video processing

  • Control systems engineering (calculating transfer functions, determining root locus, and analyzing time response)

  • Test and measurement systems

  • Computational finance

  • Computational biology

• Key Features:

  • High-level language for numerical computation and application development.

  • Interactive environment for iterative design, data exploration, and problem-solving.

  • Built-in mathematical functions covering linear algebra, statistics, Fourier analysis, filtering, optimization, numerical integration, and ordinary differential equations (ODEs).

  • Graphics tools for standard data visualization and custom plotting.

  • Development tools for code quality, debugging, performance optimization, and maintainability.

  • Graphical user interface (GUI) building tools.

  • Functions for integrating MATLAB algorithms with external applications written in C, Java, .NET, and Microsoft Excel.

• Simulink Companion Tool:

  • Simulink is a block-diagram environment integrated with MATLAB designed for multidomain simulation and model-based design.

  • In control systems, Simulink is utilized for simulating dynamical systems and graphing complex output responses.

• User Interface Layout (Five Core Areas):     

  

  • 1. Command Window: The primary text interface where commands are entered at the prompt marker >>. It acts similarly to the Windows Command Prompt (cmd) or Linux terminal. Commands like why or help why can be issued directly here.

  • 2. Command History: Logs all commands entered into the Command Window during previous and current sessions, allowing users to track and reuse past code.

  • 3. Current Folder: Works as an embedded file explorer to navigate working directories and manage files.

  • 4. Workspace: Displays all variables currently created or loaded into memory, showing variable names, current values, sizes, and data types/classes.

  • 5. Toolstrip: The top toolbar organized into tabs, similar to Microsoft Office ribbons:

    • Home Tab: Contains controls for file handling, variable management, workspace operations, layout customization, and environment preferences.

    • Plots Tab: Facilitates one-click plotting of variables selected in the Workspace.

    • Apps Tab: Houses specialized toolboxes for specific engineering disciplines (e.g., Image Processing, Digital Signal Processing).

  • Layout Modification: The interface configuration can be modified under the HOME tab by selecting Environment $\rightarrow$ Layout.

MATLAB Basics and Matrix Operations

• Variable Assignments:

  • Assigning a scalar value: entering a = 1 creates a variable storing a floating-point number (default type double).

  • Default variable ans: if a command returns a result without an explicit destination variable, MATLAB stores the result in ans (short for answer).

  • Suppressing output: placing a semicolon ; at the end of a command executes the statement but suppresses output display in the Command Window (e.g., e = a*b;).

• Array and Matrix Creation:

  • MATLAB operates natively on whole matrices and arrays. All variables are treated as multidimensional arrays.

  • Row Vector Creation: Separate elements with spaces or commas:         $\mathbf{a} = \begin{pmatrix} 1 & 2 & 3 & 4 \end{pmatrix}$         Command syntax: a = [1 2 3 4] or a = [1, 2, 3, 4].

  • Matrix Creation: Separate rows with semicolons:         $\mathbf{a} = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10 \end{pmatrix}$         Command syntax: a = [1 2 3; 4 5 6; 7 8 10].

  • Built-in Matrix Generators:

    • zeros(m, n) creates an $m \times n$ matrix of zeros (e.g., z = zeros(5,1) creates a $5 \times 1$ column vector of zeros).

    • ones(m, n) creates an $m \times n$ matrix of ones.

    • rand(m, n) creates an $m \times n$ matrix of uniformly distributed random numbers.

• Matrix and Element-Wise Arithmetic Operations:

  • Scalar Addition: a + 10 adds $10$ to every element in matrix a.

  • Function Application: sin(a) computes the sine of each individual element in radians.

  • Matrix Transpose: The single quote ' transposes a matrix (e.g., a').

  • Standard Matrix Multiplication: The * operator performs linear algebra matrix multiplication:         $\mathbf{p} = \mathbf{a} \times \mathbf{a}^{-1}$         Command syntax: p = a*inv(a).

  • Display Formats: MATLAB stores numbers as double-precision floating-point values:

    • format short: displays 4 decimal places (default).

    • format long: displays 15 decimal places.

    • Note: The format command alters display output only and does not impact internal precision or mathematical calculations.

  • Element-Wise Operations: Preceding an operator with a dot . forces element-by-element evaluation rather than linear algebra operations:

    • .* Element-wise multiplication (e.g., p = a.*a).

    • ./ Element-wise division.

    • .^ Element-wise exponentiation (e.g., a.^3 raises each element of a to the third power).

• Array Concatenation:

  • Square brackets [] act as the concatenation operator.

  • Horizontal Concatenation: Joins matrices side-by-side using commas or spaces (requires matching row counts):         A = [a, a]

  • Vertical Concatenation: Stacks matrices vertically using semicolons (requires matching column counts):         A = [a; a]

• Array Indexing:

  • Two-Subscript Indexing: Reference elements using A(row, column). For example, A(4,2) accesses the element at row 4, column 2.

  • Linear Indexing: Uses a single subscript A(k) traversing down columns sequentially. For a $4 \times 4$ matrix, element A(8) corresponds to row 4, column 2.

  • Index Out of Bounds: Referencing an out-of-bounds element on the right side of an assignment generates an error. Referencing outside bounds on the left side (e.g., A(4,5) = 17) dynamically expands the array size, padding unassigned elements with zeros.

  • Colon Operator : Usage:

    • Sub-range selection: A(1:3, 2) selects rows 1 through 3 in column 2.

    • Full dimension selection: A(3, :) selects all columns in row 3.

    • Equally spaced vector generation: start:step:end (e.g., B = 0:10:100 yields values from $0$ to $100$ incremented by $10$). Omitting step defaults to an increment of $1$ (start:end).

• Character Strings:

  • Strings are sequences of characters enclosed in single quotes (e.g., myText = 'Hello, world').

  • Including a single quote within a string requires typing two consecutive single quotes: otherText = 'You''re right' yields You're right.

  • Data type/class is char. String variables are stored as character arrays.

  • Concatenating strings: longText = [myText, ' - ', otherText].

  • Conversion functions: num2str(x) converts numerical values to character strings; int2str(x) converts integers.

• Help Systems:

  • doc <function>: Opens detailed documentation in a standalone help window.

  • help <function>: Displays text documentation inside the Command Window (e.g., help mean, help sin, help max, help rand).

• 2D Plotting:

  • plot(x, y): Generates a basic line plot.

  • Adding labels and titles:

    • xlabel('x') adds horizontal axis label.

    • ylabel('sin(x)') adds vertical axis label.

    • title('Plot of the Sine Function') adds plot title.         

      

  • Line formatting specs: plot(x, y, 'r--') plots a red dashed line; 'g:*' plots a green dotted line with asterisk markers.

  • Overlaying plots: calling plot clears existing figure windows by default. To overlay graphs, use hold on. To release the overlay mode, use hold off. matlab x = 0:pi/100:2*pi; y = sin(x); plot(x,y) hold on y2 = cos(x); plot(x,y2,'r:') legend('sin','cos')                  

    

• Polynomial Handling:

  • Polynomials are represented as row vectors containing coefficients ordered by descending powers.

    • Example: $p(x) = x^3 - 2x - 5$ is represented as p = [1 0 -2 -5].

  • polyval(p, x): Evaluates polynomial p at scalar value x (e.g., polyval(p, 5) returns $110$).

  • polyvalm(p, X): Evaluates polynomial p in a matrix sense ($p(\mathbf{X}) = \mathbf{X}^3 - 2\mathbf{X} - 5\mathbf{I}$) for a square matrix X.

  • roots(p): Calculates the roots of polynomial p and returns them in a column vector.

  • poly(r): Constructs polynomial coefficients given a vector of roots r.

  • conv(p1, p2): Performs polynomial multiplication (algebraic convolution) (e.g., p3 = conv([1 2 3], [2 0 5])).

MATLAB Programming: Control Flow, M-Files, and Functions

• M-Files:

  • Text files containing MATLAB commands saved with a .m file extension.

  • Key utility commands used inside m-files:

    • clc: Clears the Command Window display.

    • clear: Erases all variables from the Workspace memory.

    • disp('string'): Prints text or variable contents to the Command Window.

• Control Flow Statements:

  • Conditional Statements:

    • if, elseif, else, end structure: matlab a = randi(100, 1); if a < 30 disp('small') elseif a < 80 disp('medium') else disp('large') end &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

    • switch, case, otherwise, end structure: used for discrete equality checks. matlab [dayNum, dayString] = weekday(date, 'long', 'en_US'); switch dayString case 'Monday' disp('Start of the work week') case 'Friday' disp('Last day of the work week') otherwise disp('Weekend!') end             

  • Loop Statements:

    • for loops: Execute a code block a specified number of times with an incrementing index variable. matlab x = ones(1,10); for n = 2:6 x(n) = 2 * x(n - 1); end             

    • while loops: Repeat execution as long as a evaluated conditional statement remains true. matlab n = 1; nFactorial = 1; while nFactorial < 1e100 n = n + 1; nFactorial = nFactorial * n; end &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

  • Loop Termination Control:

    • break: Terminates loop execution prematurely.

    • continue: Skips the remainder of the current loop iteration and advances to the next iteration.

    • Ctrl+C: Halts infinite execution loops manually.

• Relational Operators:

  • Relational operators compare arrays element-by-element and return logical arrays containing ones (true) and zeros (false).

  • <: Less than

  • <=: Less than or equal to

  • >: Greater than

  • >=: Greater than or equal to

  • ==: Equal to

  • ~=: Not equal to

• Scripts vs. Functions:

  • Scripts: Sequential files that execute commands directly within the base Workspace. They take no input arguments and return no output arguments.

  • Functions: Programs that accept inputs and return outputs. Declared using the function keyword and stored in a file matching the function name (function_name.m). Functions operate inside their own isolated workspace.

    • Function declaration syntax: function a = triarea(b,h)matlab function a = triarea(b,h) a = 0.5*(b.* h);             

• Laboratory Exercises and Activities:

  • Activity 1a Requirements:

    • 1.a. Add matrices $\mathbf{A} = \begin{pmatrix} 2 & 1 & 1 \\ 0 & -3 & 4 \end{pmatrix}$ and $\mathbf{B} = \begin{pmatrix} 3 & -1 & 3 \\ 2 & 0 & 5 \end{pmatrix}$.

    • 1.b. Compute $3\mathbf{A} - 2\mathbf{B}$ for $\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 3 & 0 \end{pmatrix}$ and $\mathbf{B} = \begin{pmatrix} 1 & 3 \\ 0 & -4 \end{pmatrix}$.

    • 1.c. Compute $5\mathbf{A} - 2\mathbf{B}$ using matrices from item 1.a.

    • 2. Compute $\mathbf{C}(\mathbf{A} + \mathbf{B})$ where $\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 3 & 0 \end{pmatrix}$, $\mathbf{B} = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}$, and $\mathbf{C} = \begin{pmatrix} 2 & -2 \\ 1 & 3 \\ 4 & -1 \end{pmatrix}$.

    • 3. Compute $\mathbf{C}\mathbf{A} + \mathbf{C}\mathbf{B}$ using matrices from item 2.

    • 4. Plot $y = x^2 + 5x + 3$ (red solid line) and $y = x^3 + 4$ (blue dashed line) on a single plot.

    • 5. Plot the signals from item 4 on separate subplots using subplot.

    • 6. Find the roots of polynomials:

      • a. $s^7 + 32s^6 + 8s^5 + 85s^4 + 4s^3 + s^2 + 3s + 1 = 0$

      • b. $3s^5 - s^4 + 24s^3 + 9s^2 + 6s + 2 = 0$

      • c. $s^3 + 77s^2 + 11s + 1 = 0$

    • 7. Compute polynomial convolutions of item 6.a with 6.b, and 6.a with 6.c.

  • Activity 1b Requirements:

    • Fibonacci Generator: Function newFibonacci.m accepting two boundary integers and outputting all Fibonacci numbers within that closed interval.

    • Palindrome Checker: Function verifying whether a string input reads identically forward and backward (e.g., racecar, anna).

Laplace Transform Theory and Principles

• Role in Control Systems:

  • Developing mathematical models is a primary phase of control system design.

  • Dynamic systems are governed by linear time-invariant (LTI) differential equations that map input signals $r(t)$ to output signals $c(t)$.

  • Solving high-order differential equations directly in the time domain is mathematically cumbersome.

  • The Laplace transform simplifies dynamic system analysis by converting calculus operations (differentiation and integration) into simple algebraic operations in the complex frequency domain ($s$--domain).

• Definition of the Laplace Transform:

  • The unilateral Laplace transform of a time-domain signal $f(t)$ is defined by the integral:         $\mathcal{L}[f(t)] = F(s) = \int_{0^-}^{\infty} f(t) e^{-st}\,dt$         where $s = \sigma + j\omega$ is a complex frequency variable.

  • Initial Condition Evaluation: The lower limit $0^-$ allows system analysis to incorporate initial conditions present immediately prior to initial input disturbances or discontinuities at $t = 0$

• Fundamental Laplace Transform Derivations:

  • Ramp Function $f(t) = t\,u(t)$:         Using integration by parts (where $u = t$, $du = dt$, $dv = e^{-st}dt$, $v = -\frac{1}{s}e^{-st}$):         $F(s) = \int_{0}^{\infty} t e^{-st}\,dt = \left[ -\frac{1}{s} t e^{-st} \right]_{0}^{\infty} + \frac{1}{s} \int_{0}^{\infty} e^{-st}\,dt = 0 + \left[ -\frac{1}{s^2} e^{-st} \right]_{0}^{\infty} = \frac{1}{s^2}$

  • Exponential Function $f(t) = e^{-at}u(t)$:         $F(s) = \int_{0}^{\infty} e^{-at} e^{-st}\,dt = \int_{0}^{\infty} e^{-(s+a)t}\,dt = \left[ -\frac{1}{s+a} e^{-(s+a)t} \right]_{0}^{\infty} = -\frac{1}{s+a}[0 - 1] = \frac{1}{s+a}$

  • Cosine Function $f(t) = \cos(\omega t)u(t)$:         Applying Euler's formula $\cos(\omega t) = \frac{e^{j\omega t} + e^{-j\omega t}}{2}$:         $F(s) = \int_{0}^{\infty} \left( \frac{e^{j\omega t} + e^{-j\omega t}}{2} \right) e^{-st}\,dt = \frac{1}{2} \int_{0}^{\infty} e^{-(s-j\omega)t}\,dt + \frac{1}{2} \int_{0}^{\infty} e^{-(s+j\omega)t}\,dt$         $F(s) = \frac{1}{2(s - j\omega)} + \frac{1}{2(s + j\omega)} = \frac{(s + j\omega) + (s - j\omega)}{2(s - j\omega)(s + j\omega)} = \frac{2s}{2(s^2 + \omega^2)} = \frac{s}{s^2 + \omega^2}$

• Standard Laplace Transform Pairs:     

  

  • 1. Impulse Function: $\delta(t) \longleftrightarrow 1$

  • 2. Unit Step Function: $u(t) \longleftrightarrow \frac{1}{s}$

  • 3. Ramp Function: $t\,u(t) \longleftrightarrow \frac{1}{s^2}$

  • 4. Polynomial Function: $t^n u(t) \longleftrightarrow \frac{n!}{s^{n+1}}$

  • 5. Exponential Decay: $e^{-at} u(t) \longleftrightarrow \frac{1}{s+a}$

  • 6. Sine Wave: $\sin(\omega t) u(t) \longleftrightarrow \frac{\omega}{s^2 + \omega^2}$

  • 7. Cosine Wave: $\cos(\omega t) u(t) \longleftrightarrow \frac{s}{s^2 + \omega^2}$

Laplace Transform Theorems & Properties

• Summary of Key Laplace Theorems:     

  

  • Definition Theorem:         $\mathcal{L}[f(t)] = F(s) = \int_{0^-}^{\infty} f(t) e^{-st}\,dt$

  • Linearity Theorems:         $\mathcal{L}[k\,f(t)] = k\,F(s)$         $\mathcal{L}[f_1(t) + f_2(t)] = F_1(s) + F_2(s)$

  • Frequency Shift Theorem:         $\mathcal{L}[e^{-at}f(t)] = F(s+a)$

  • Time Shift Theorem:         $\mathcal{L}[f(t-T)] = e^{-sT} F(s)$

  • Scaling Theorem:         $\mathcal{L}[f(at)] = \frac{1}{a} F\left(\frac{s}{a}\right)$

  • First Differentiation Theorem:         $\mathcal{L}\left[\frac{df}{dt}\right] = s\,F(s) - f(0^-)$

  • Second Differentiation Theorem:         $\mathcal{L}\left[\frac{d^2f}{dt^2}\right] = s^2 F(s) - s\,f(0^-) - f'(0^-)$

  • $n$-th Differentiation Theorem:         $\mathcal{L}\left[\frac{d^n f}{dt^n}\right] = s^n F(s) - \sum_{k=1}^{n} s^{n-k} f^{(k-1)}(0^-)$

  • Integration Theorem:         $\mathcal{L}\left[ \int_{0^-}^{t} f(\tau)\,d\tau \right] = \frac{F(s)}{s}$

  • Final Value Theorem:         $f(\infty) = \lim_{s \to 0} s\,F(s)$

  • Initial Value Theorem:         $f(0^+) = \lim_{s \to \infty} s\,F(s)$

• Applications of Theorems (Review Derivations):

  • Damped Sine Wave: Deriving $\mathcal{L}[e^{-at}\sin(\omega t)u(t)]$         Applying Frequency Shift Theorem to Pair 6 ($\frac{\omega}{s^2 + \omega^2}$):         $\mathcal{L}[e^{-at}\sin(\omega t)u(t)] = \frac{\omega}{(s+a)^2 + \omega^2}$

  • Damped Cosine Wave: Deriving $\mathcal{L}[e^{-at}\cos(\omega t)u(t)]$         Applying Frequency Shift Theorem to Pair 7 ($\frac{s}{s^2 + \omega^2}$):         $\mathcal{L}[e^{-at}\cos(\omega t)u(t)] = \frac{s+a}{(s+a)^2 + \omega^2}$

  • Cubic Power: Deriving $\mathcal{L}[t^3 u(t)]$         Applying Pair 4 for $n = 3$:         $\mathcal{L}[t^3 u(t)] = \frac{3!}{s^{3+1}} = \frac{6}{s^4}$

Inverse Laplace Transform and Partial Fraction Expansion

• Definition of the Inverse Laplace Transform:

  • The complex inversion integral reconstructs the time-domain signal $f(t)$ from $F(s)$:         $\mathcal{L}^{-1}[F(s)] = \frac{1}{2\pi j} \int_{\sigma - j\infty}^{\sigma + j\infty} F(s) e^{st}\,ds = f(t) u(t)$

  • Because direct evaluation of this complex surface integral is difficult, Partial Fraction Expansion (PFE) combined with transform tables is used instead.

• Partial Fraction Expansion (PFE) Principles:

  • For a rational function $F(s) = \frac{N(s)}{D(s)}$, the order of the numerator $N(s)$ must be strictly less than the order of the denominator $D(s)$.

  • If the degree of $N(s) \ge$ degree of $D(s)$, polynomial long division must be performed first until the fraction displays a remainder numerator of lower degree than its denominator:         $F(s) = \frac{s^3 + 2s^2 + 6s + 7}{s^2 + s + 5} = s + 1 + \frac{s + 2}{s^2 + s + 5}$         $f(t) = \frac{d\delta(t)}{dt} + \delta(t) + \mathcal{L}^{-1}\left[ \frac{s + 2}{s^2 + s + 5} \right]$

• PFE Case 1: Real and Distinct Roots:

  • Given $F(s) = \frac{2}{(s+1)(s+2)}$, expand into distinct linear fractions:         $F(s) = \frac{K_1}{s+1} + \frac{K_2}{s+2}$

  • Calculate residue $K_1$ by multiplying by $(s+1)$ and setting $s \to -1$:         $K_1 = \left. (s+1) F(s) \right|_{s \to -1} = \left. \frac{2}{s+2} \right|_{s \to -1} = \frac{2}{-1+2} = 2$

  • Calculate residue $K_2$ by multiplying by $(s+2)$ and setting $s \to -2$:         $K_2 = \left. (s+2) F(s) \right|_{s \to -2} = \left. \frac{2}{s+1} \right|_{s \to -2} = \frac{2}{-2+1} = -2$

  • Inverse transform result:         $f(t) = \mathcal{L}^{-1}\left[ \frac{2}{s+1} - \frac{2}{s+2} \right] = (2e^{-t} - 2e^{-2t})u(t)$

• PFE Case 2: Real and Repeated Roots:

  • Given $F(s) = \frac{2}{(s+1)(s+2)^2}$, expand accounting for root multiplicity:         $F(s) = \frac{K_1}{s+1} + \frac{K_2}{(s+2)^2} + \frac{K_3}{s+2}$

  • Calculate distinct residue $K_1$:         $K_1 = \left. (s+1) F(s) \right|_{s \to -1} = \left. \frac{2}{(s+2)^2} \right|_{s \to -1} = 2$

  • Calculate highest-power repeated residue $K_2$:         $K_2 = \left. (s+2)^2 F(s) \right|_{s \to -2} = \left. \frac{2}{s+1} \right|_{s \to -2} = -2$

  • Calculate lower-power repeated residue $K_3$ using differentiation:         $K_3 = \left. \frac{d}{ds} \left[ (s+2)^2 F(s) \right] \right|_{s \to -2} = \left. \frac{d}{ds} \left( \frac{2}{s+1} \right) \right|_{s \to -2} = \left. \frac{-2}{(s+1)^2} \right|_{s \to -2} = -2$

  • Inverse transform result:         $f(t) = (2e^{-t} - 2t e^{-2t} - 2e^{-2t})u(t)$

• PFE Case 3: Complex Conjugate Roots:

  • Given $F(s) = \frac{3}{s(s^2 + 2s + 5)}$, write quadratic factor directly:         $F(s) = \frac{K_1}{s} + \frac{K_2 s + K_3}{s^2 + 2s + 5}$

  • Solve for $K_1$:         $K_1 = \left. s F(s) \right|_{s \to 0} = \left. \frac{3}{s^2 + 2s + 5} \right|_{s \to 0} = \frac{3}{5}$

  • Equate numerator coefficients ($3 = K_1(s^2 + 2s + 5) + (K_2 s + K_3)s$):         $3 = (K_1 + K_2)s^2 + (2K_1 + K_3)s + 5K_1$

    • $s^2 \text{ term:} \quad K_1 + K_2 = 0 \implies K_2 = -K_1 = -\frac{3}{5}$

    • $s^1 \text{ term:} \quad 2K_1 + K_3 = 0 \implies K_3 = -2K_1 = -\frac{6}{5}$

  • Resulting expansion:         $F(s) = \frac{3/5}{s} - \frac{\frac{3}{5}s + \frac{6}{5}}{s^2 + 2s + 5} = \frac{3}{5}\left(\frac{1}{s}\right) - \frac{3}{5}\left[ \frac{s + 2}{(s+1)^2 + 2^2} \right]$

• Solving Differential Equations using Laplace Transforms:

  • Problem Statement: Solve for $y(t)$ given $\frac{d^2 y(t)}{dt^2} + 12\frac{dy(t)}{dt} + 32y(t) = 32u(t)$ with zero initial conditions.

  • Step 1: Transform equation into $s$-domain:         $s^2 Y(s) + 12s Y(s) + 32Y(s) = \frac{32}{s}$

  • Step 2: Isolate Output $Y(s)$:         $Y(s)(s^2 + 12s + 32) = \frac{32}{s} \implies Y(s) = \frac{32}{s(s^2 + 12s + 32)} = \frac{32}{s(s+4)(s+8)}$

  • Step 3: Perform Partial Fraction Expansion:         $Y(s) = \frac{K_1}{s} + \frac{K_2}{s+4} + \frac{K_3}{s+8}$

    • $K_1 = \left. \frac{32}{(s+4)(s+8)} \right|_{s \to 0} = \frac{32}{32} = 1$

    • $K_2 = \left. \frac{32}{s(s+8)} \right|_{s \to -4} = \frac{32}{-16} = -2$

    • $K_3 = \left. \frac{32}{s(s+4)} \right|_{s \to -8} = \frac{32}{32} = 1$

  • Step 4: Take Inverse Laplace Transform:         $Y(s) = \frac{1}{s} - \frac{2}{s+4} + \frac{1}{s+8}$         $y(t) = (1 - 2e^{-4t} + e^{-8t})u(t)$

The Transfer Function

• Concept and Definition:     

  

  • A transfer function is an algebraic model representing the ratio of the Laplace transform of the output variable to the Laplace transform of the input variable, assuming zero initial conditions.

  • Mathematical definition:         $G(s) = \frac{\text{Output}(s)}{\text{Input}(s)} = \frac{C(s)}{R(s)}$

• General Derivation for an $n$-th Order LTI Differential Equation:

  • Given an $n$-th order differential equation with constant coefficients:         $a_n \frac{d^n c(t)}{dt^n} + a_{n-1} \frac{d^{n-1} c(t)}{dt^{n-1}} + \dots + a_0 c(t) = b_m \frac{d^m r(t)}{dt^m} + b_{m-1} \frac{d^{m-1} r(t)}{dt^{m-1}} + \dots + b_0 r(t)$

  • Taking the Laplace transform of both sides under zero initial conditions:         $(a_n s^n + a_{n-1} s^{n-1} + \dots + a_0) C(s) = (b_m s^m + b_{m-1} s^{m-1} + \dots + b_0) R(s)$

  • Forming the output-to-input transfer ratio:         $G(s) = \frac{C(s)}{R(s)} = \frac{b_m s^m + b_{m-1} s^{m-1} + \dots + b_0}{a_n s^n + a_{n-1} s^{n-1} + \dots + a_0}$

• Transfer Function Worked Examples:

  • Example 1 (First-Order Equation):         Find $G(s) = \frac{C(s)}{R(s)}$ for $\frac{dc(t)}{dt} + 2c(t) = r(t)$.         $s C(s) + 2C(s) = R(s) \implies (s + 2)C(s) = R(s) \implies G(s) = \frac{1}{s + 2}$

  • Practice Question 1 (First-Order System):         Find $G(s) = \frac{C(s)}{R(s)}$ for $3\frac{dc(t)}{dt} + c(t) = 5r(t)$.         $(3s + 1)C(s) = 5R(s) \implies G(s) = \frac{5}{3s + 1}$

  • Practice Question 2 (Second-Order Mechanical System):         Find $G(s) = \frac{C(s)}{R(s)}$ for $\frac{d^2 c(t)}{dt^2} + 4\frac{dc(t)}{dt} + 6c(t) = 2r(t)$.         $(s^2 + 4s + 6)C(s) = 2R(s) \implies G(s) = \frac{2}{s^2 + 4s + 6}$

  • Practice Question 3 (Derivatives on Input Side):         Find $G(s) = \frac{C(s)}{R(s)}$ for $2\frac{dc(t)}{dt} + 8c(t) = 3\frac{dr(t)}{dt} + r(t)$.         $(2s + 8)C(s) = (3s + 1)R(s) \implies G(s) = \frac{3s + 1}{2s + 8}$

Electrical Network Transfer Functions

• Passive Component Relationships:     

  

  • Capacitor ($C$ in Farads $F$):

    • Voltage-Current relation: $v(t) = \frac{1}{C} \int_{0}^{t} i(\tau)\,d\tau$

    • Current-Voltage relation: $i(t) = C \frac{dv(t)}{dt}$

    • Voltage-Charge relation: $v(t) = \frac{1}{C} q(t)$

    • $s$-Domain Impedance: $Z(s) = \frac{V(s)}{I(s)} = \frac{1}{Cs}$

    • $s$-Domain Admittance: $Y(s) = \frac{I(s)}{V(s)} = Cs$

  • Resistor ($R$ in Ohms $\Omega$):

    • Voltage-Current relation: $v(t) = R\,i(t)$

    • Current-Voltage relation: $i(t) = \frac{1}{R} v(t) = G\,v(t)$

    • Voltage-Charge relation: $v(t) = R \frac{dq(t)}{dt}$

    • $s$-Domain Impedance: $Z(s) = R$

    • $s$-Domain Admittance: $Y(s) = \frac{1}{R} = G$

  • Inductor ($L$ in Henries $H$):

    • Voltage-Current relation: $v(t) = L \frac{di(t)}{dt}$

    • Current-Voltage relation: $i(t) = \frac{1}{L} \int_{0}^{t} v(\tau)\,d\tau$

    • Voltage-Charge relation: $v(t) = L \frac{d^2q(t)}{dt^2}$

    • $s$-Domain Impedance: $Z(s) = Ls$

    • $s$-Domain Admittance: $Y(s) = \frac{1}{Ls}$

• Single-Loop RLC Circuit Analysis:

  • Apply Kirchhoff's Voltage Law (KVL) around a series RLC mesh containing input voltage source $v(t)$, inductor $L$, resistor $R$, and capacitor $C$:         $L \frac{di(t)}{dt} + R i(t) + \frac{1}{C} \int_{0}^{t} i(\tau)\,d\tau = v(t)$

  • Express current in terms of output capacitor voltage $v_C(t)$ using $i(t) = C \frac{dv_C(t)}{dt}$:         $L C \frac{d^2 v_C(t)}{dt^2} + R C \frac{dv_C(t)}{dt} + v_C(t) = v(t)$

  • Transform to $s$-domain assuming zero initial conditions:         $(L C s^2 + R C s + 1) V_C(s) = V(s)$

  • Form the capacitor voltage transfer function $G(s) = \frac{V_C(s)}{V(s)}$:         $G(s) = \frac{V_C(s)}{V(s)} = \frac{1}{LCs^2 + RCs + 1} = \frac{1/LC}{s^2 + \frac{R}{L}s + \frac{1}{LC}}$

• Multi-Loop Network Analysis via Mesh Equations & Cramer's Rule:

  • Procedure for Complex Networks:

    • 1. Replace element values with their $s$-domain impedances $Z(s)$.

    • 2. Replace time-domain sources and variables with their Laplace transforms.

    • 3. Assign loop currents $I_1(s), I_2(s), \dots, I_N(s)$ and establish loop directions.

    • 4. Write KVL mesh equations in matrix form:             $\begin{bmatrix} \text{Sum of } Z \text{ in Mesh 1} \end{bmatrix} I_1(s) - \begin{bmatrix} \text{Common } Z \text{ Mesh 1-2} \end{bmatrix} I_2(s) = \begin{bmatrix} \text{Applied } V \text{ Mesh 1} \end{bmatrix}$             $-\begin{bmatrix} \text{Common } Z \text{ Mesh 1-2} \end{bmatrix} I_1(s) + \begin{bmatrix} \text{Sum of } Z \text{ in Mesh 2} \end{bmatrix} I_2(s) = \begin{bmatrix} \text{Applied } V \text{ Mesh 2} \end{bmatrix}$

    • 5. Solve the system of simultaneous equations using Cramer's rule for the desired output variable.

    • 6. Construct the output-over-input transfer function.

  • Two-Mesh Example Derivation:

    • Network configuration: Mesh 1 contains source $V(s)$, series resistor $R_1$, and shared inductor $L$. Mesh 2 contains shared inductor $L$, series resistor $R_2$, and capacitor $C$

    • Mesh KVL Equations:             $(R_1 + Ls) I_1(s) - Ls I_2(s) = V(s)$             $-Ls I_1(s) + \left( Ls + R_2 + \frac{1}{Cs} \right) I_2(s) = 0$

    • Solve for Mesh 2 current $I_2(s)$ using Cramer's Rule:             $I_2(s) = \frac{\begin{vmatrix} (R_1 + Ls) & V(s) \\ -Ls & 0 \end{vmatrix}}{\Delta} = \frac{Ls V(s)}{\Delta}$             where system determinant $\Delta$ is:             $\Delta = \begin{vmatrix} (R_1 + Ls) & -Ls \\ -Ls & \left( Ls + R_2 + \frac{1}{Cs} \right) \end{vmatrix} = (R_1 + Ls)\left( Ls + R_2 + \frac{1}{Cs} \right) - (Ls)^2$

    • Expand and simplify $\Delta$:             $\Delta = R_1 Ls + R_1 R_2 + \frac{R_1}{Cs} + L^2 s^2 + L R_2 s + \frac{L}{C} - L^2 s^2 = (R_1 + R_2)Ls + (R_1 R_2 + \frac{L}{C}) + \frac{R_1}{Cs}$             $\Delta = \frac{(R_1 + R_2)LCs^2 + (R_1 R_2 C + L)s + R_1}{Cs}$

    • Form final transfer function $G(s) = \frac{I_2(s)}{V(s)}$:             $G(s) = \frac{I_2(s)}{V(s)} = \frac{Ls}{\Delta} = \frac{LCs^2}{(R_1 + R_2)LCs^2 + (R_1 R_2 C + L)s + R_1}$

Operational Amplifier Transfer Functions

• Ideal Operational Amplifier Properties:

  • Differential Input: $v_o(t) = A [v_2(t) - v_1(t)]$

  • High Input Impedance: $Z_i = \infty$ (ideal, resulting in zero terminal input currents: $I_a(s) = 0$)

  • Low Output Impedance: $Z_o = 0$ (ideal)

  • High Constant Voltage Gain: $A = \infty$ (ideal, leading to virtual short circuit between inputs: $v_1(t) \approx v_2(t)$)

• Inverting Operational Amplifier Circuit:     

  

  • Non-inverting input (+$) is grounded (v_2(t) = 0$), creating a virtual ground at the inverting input ($v_1(t) \approx 0).\n * Input current equals negative feedback current due to infinite input impedance (I_1(s) = -I_2(s)).\n * Current expressions: I_1(s) = \frac{V_i(s)}{Z_1(s)}$and$-I_2(s) = -\frac{V_o(s)}{Z_2(s)}.\n * Inverting Transfer Function Equation:\n        \frac{V_o(s)}{V_i(s)} = -\frac{Z_2(s)}{Z_1(s)}\n\n * *Inverting Op-Amp Numerical Example:*\n * Input impedance Z_1(s)$composed of parallel$R_1 = 360\,\text{k}\Omega$and$C_1 = 5.6\,\mu\text{F}:\n            Z_1(s) = \frac{1}{C_1 s + \frac{1}{R_1}} = \frac{1}{5.6 \times 10^{-6} s + \frac{1}{360 \times 10^3}} = \frac{360 \times 10^3}{2.016s + 1}\n * Feedback impedance Z_2(s)$composed of series$R_2 = 220\,\text{k}\Omega$and$C_2 = 0.1\,\mu\text{F}:\n            Z_2(s) = R_2 + \frac{1}{C_2 s} = 220 \times 10^3 + \frac{10^7}{s}\n * Substitute into inverting equation:\n            \frac{V_o(s)}{V_i(s)} = -\frac{Z_2(s)}{Z_1(s)} = -\frac{220 \times 10^3 + \frac{10^7}{s}}{\frac{360 \times 10^3}{2.016s + 1}} = -1.232 \frac{s^2 + 45.95s + 22.55}{s}\n\n* **Noninverting Operational Amplifier Circuit:**\n    \n * Input signal V_i(s)$is applied directly to the non-inverting terminal ($+$).

  • Output voltage equation: $V_o(s) = A [V_i(s) - V_1(s)]$

  • Feedback voltage divider equation at inverting terminal ($-$):         $V_1(s) = \frac{Z_1(s)}{Z_1(s) + Z_2(s)} V_o(s)$

  • Substitute $V_1(s)$ into output equation:         $V_o(s) = A \left[ V_i(s) - \frac{Z_1(s)}{Z_1(s) + Z_2(s)} V_o(s) \right]$         $\frac{V_o(s)}{V_i(s)} = \frac{A}{1 + A \frac{Z_1(s)}{Z_1(s) + Z_2(s)}}$

  • For ideal/infinite gain $A \to \infty$, unity in the denominator is neglected:         $\frac{V_o(s)}{V_i(s)} = \frac{Z_1(s) + Z_2(s)}{Z_1(s)} = 1 + \frac{Z_2(s)}{Z_1(s)}$

  • Noninverting Op-Amp Impedance Example:

    • Impedance $Z_1(s)$ in series branch to ground: $Z_1(s) = R_1 + \frac{1}{C_1 s}$

    • Impedance $Z_2(s)$ in parallel feedback branch: $Z_2(s) = \frac{R_2 (1/C_2 s)}{R_2 + (1/C_2 s)} = \frac{R_2}{R_2 C_2 s + 1}$

    • Substitute $Z_1(s)$ and $Z_2(s)$ into noninverting transfer function:             $\frac{V_o(s)}{V_i(s)} = \frac{C_2 C_1 R_2 R_1 s^2 + (C_2 R_2 + C_1 R_2 + C_1 R_1)s + 1}{C_2 C_1 R_2 R_1 s^2 + (C_2 R_2 + C_1 R_1)s + 1}$