System Dynamics and Controls

Learn how Laplace transforms model dynamic systems, convert differential equations to algebraic forms in the complex plane, and apply linearity in example 1.

Apply Laplace transform tables to break down complex expressions using linearity, unit steps, and standard forms for sine, cosine, and exponential terms in example 2.

Explore the shifting property of Laplace transforms by multiplying f(t) with exponential terms and obtaining F(s±α), as shown in example 3.

Apply the differentiation theorem of Laplace transforms to convert derivatives into algebraic terms. Use L{f'}=sF(s)-f(0) and L{f''}=s^2F(s)-s f(0)-f'(0) to solve initial value problems.

Explore how to compute Laplace transforms when the standard table lacks a match, using the differentiation theorem and the integration theorem, and apply them to initial value problems.

Explore inverse Laplace transforms by returning to the table and using partial fraction expansion. Solve 1/(s^2-5s+6) with distinct roots s=2 and s=3, yielding -1/(s-2)+1/(s-3).

Utilize partial fraction expansion for a fraction with repeated roots, factor the denominator as s(s+1)^2, determine a, b, c, and rewrite for inversion.

Decompose a complex fraction into partial fractions for inverse Laplace transforms. Solve for constants A, B, and C, handling imaginary roots to reconstruct the original fraction.

Learn to solve linear differential equations with initial value problems, including second-order cases, in system dynamics and controls, using the Laplace transform differentiation theorem and partial fractions to obtain x(t).

Apply the Laplace transform to an initial value problem, perform inverse Laplace transform using partial fractions to derive X(t) = cos(2t) + sin(2t), and verify X(0) and X'(0).

Apply Laplace transforms to initial value problem x¨ - 3x˙ + 2x = 4t - 6, use the differentiation theorem and initial conditions, and obtain X(t) = 2t + e^{t}.

Solve a pair of coupled first-order initial value problems with Laplace transforms, convert to matrix form, apply initial conditions, and use Cramer's rule to obtain X1(t) and X2(t).

Explains how to use Matlab's residue function to compute partial fractions of X1(s) and X2(s) from polynomial numerators and denominators, then inverse Laplace and plot the responses.

Use matlab to compute the inverse laplace with ilaplace, form X = A inverse B, and plot X1(t) and X2(t) from the symbolic toolbox; compare with the residues method.

Review si and English units, define mass, weight, and force using F = ma, and introduce mechanical elements, focusing on particles with position, velocity, acceleration, kinetic energy, and linear momentum.

Explore how rigid bodies in planar motion use axes perpendicular to the motion plane, compute the center of mass and moments of inertia, and apply the parallel axis theorem.

Apply the parallel axis theorem to compute the net moment of inertia about the x axis by subtracting the hollow cylinder's inertia from the solid cylinder's, using the given density.

Review rigid-body dynamics by deriving kinetic energy, linear momentum, and angular momentum for planar motion, focusing on velocity of the center of mass, moment of inertia, and angular velocity.

Analyze the rigid bar in example 12 by applying relative motion to find the center-of-mass velocity, acceleration, kinetic energy, and angular momentum about the center.

Explore how springs store energy and relate force to displacement via F = k x, where x is spring stretch, and torsional springs with M = k theta for rotation.

Master damping elements in system dynamics, using viscous dampers for translational and torsional motion, with F = C xdot and moment = K theta_dot for linear modeling.

Explore mechanical inputs, work, and power in linear and rotational systems. Learn to draw free body diagrams, include springs and dampers, and derive equations of motion.

Draw a free body diagram for a mass-spring system, apply Newton's second law, and derive the differential equation m y'' + k y = F with downward positive.

Derive the equation of motion for a mass-spring-damper under an applied force, using a free-body diagram to obtain M y'' + C y' + K y = F.

Analyze a two-mass system with springs and a damper using three free-body diagrams per mass to derive coupled equations of motion via Newton's second law.

Derive two coupled equations of motion for a horizontal two-mass system with viscous friction, using free body diagrams, springs, dampers, and applied force F on M1 to model interaction forces.

Model two elements connected without a mass by inserting a zero-mass point mass to separate the spring and damper, derive equations of motion, and simplify analysis.

Analyze a rigid-body rotation with translation in a pulley-like disc, using free-body diagrams and moments to derive an equation in X by linking X double dot to theta double dot.

Apply the small angle approximation to linearize the system, replacing sin theta with theta, cos theta with one, and 1 minus cos theta with 1/2 theta^2, as shown in example 18.2.

Derives a small-angle model for slender rods connected by a rigid massless link, with spring and damper, using free-body diagrams and parallel-axis theorem to obtain the theta equation of motion.

Learn transfer functions as the output-input ratio in the s-domain with zero initial conditions. Derive G(s) from a differential equation and visualize the system with a block diagram.

Derive the transfer function x1(s)/f(s) for a mass-spring-damper system by building free body diagrams, formulating equations of motion, and applying the Laplace transform.

Model a car on a bumpy road with two masses, springs, and a damper to illustrate the suspension system and derive its transfer function x1 over F(s) using Laplace transforms.

learn the impedance method to derive system equations directly in Laplace transform form, without free body diagrams, using spring, damper, and mass impedances via transfer functions F(s)/X(s).

Apply the impedance method to derive equations of motion for a two-mass system, replacing free-body diagrams with impedance calculations to obtain transfer functions.

Explore rotational dynamics with torsional dampers and springs, using free body diagrams and angular variables to derive equations of motion, then apply the impedance method to obtain the transfer function.

This lecture derives a transfer function matrix for multiple inputs and outputs, Ka'bah, using the impedance method and Cramer's rule.

Explore state space representation for orbital dynamics, converting a second-order satellite motion into first-order equations with a state vector, using MATLAB or C++ and A, B, C matrices.

Define state vector x with x1 = y and x2 = y dot; derive x dot = A x + B u; choose C to output displacement, velocity, or both.

Derive a state-space model for a mass-spring-damper by choosing x1 as displacement and x2 as velocity and form x1dot = x2, x2dot = (F - K x1 - B x2)/m.

derive the equations of motion for a two-mass, three-spring system with two inputs, define four state variables, and build the state-space model with outputs as the two displacements.

Learn to numerically integrate state-space models in matlab using ode45. Convert a spring-mass-damper system to state variables, and plot displacement and velocity.

Show state-space modeling of a mass-spring system in MATLAB, deriving A and B from m x'' + k x = u(t) with x1 displacement and x2 velocity, and plot results.

Demonstrate a four-by-four state-space model for a two-mass system with a two-input b matrix in Matlab, solve with ode45, and plot displacements and velocities.

Convert a transfer function to a state-space model to enable numerical integration; define state variables from the output and its derivatives, and derive A and B matrices for the system.

Convert a transfer function to a state-space model by deriving the differential equations with zero initial conditions and forming the state vector, A and B matrices. Output equals first state.

Derive the transfer function from state-space form by applying the Laplace transform to x dot = A x + B u, yielding G(s) = C (sI - A)^{-1} B.

Extract the system matrices from the state-space representation, compute (sI − A)⁻¹, and form the transfer function J(s) = C (sI − A)⁻¹ B.

Explore the time response through transfer functions, distinguishing natural and forced responses, and learn how poles and zeros—roots of the denominator and numerator—shape system behavior.

analyzes a unit step response of a system with transfer function (s+2)/(s+5); identifies pole-zero locations and derives the output time response c(t) = (2/5) + (3/5) e^{-5t}.

Explore how input types shape system dynamics, including free response, impulsive response, step response, and ramp input. Use transfer functions and the final value theorem to analyze outputs.

Learn to build transfer functions in MATLAB using tf, and simulate impulse, step, and ramp responses to analyze system dynamics.

derive the transfer function for a first-order system from the differential equation T ẋ + x = f(t) with x(0)=0, preparing for input-response analysis.

Explain the free (natural) response of a first-order system with zero input, and impulsive response for a damped spring and damper under a 10 N·s impulse, yielding X(t)=0.005 e^{-0.05 t}.

Explore the step response of a first-order system in system dynamics and controls, derive the time constant, rise time, and settling time, and connect exponential decay to the final value.

Solve a step input in a tank system from the system dynamics and controls course to determine the steady-state liquid level, step response, and settling time.

Revisit the tank flow example in MATLAB and demonstrate plotting the transfer function response using the step command, inverse Laplace, and settling time to show the final value.

Explore the ramp response for first order systems, derive x(t) from the s-domain using partial fraction expansion, and relate the time-domain behavior to exponential terms.

Turn a damped mass system into a first-order velocity model and solve for v(t) under a step force, yielding v(t) = (F/C)[1 − exp(−(C/M)t)].

Explore a first-order cooling model of temperature change in a soup cooling scenario with T(t) = 5 + 95 e^{-0.05 t}. Visualize its first-order response in MATLAB plots.

Learn how second-order systems differ from first-order ones by examining four response types: underdamped, overdamped, critically damped, and undamped, through pole locations, transfer functions, and unit-step responses.

Derive the second order mass–spring–damper model with damping, obtain its equation of motion and transfer function, define zeta and omega_n, and analyze pole locations across damping regimes.

Derive underdamped second-order system behavior with complex conjugate poles when zeta^2 < 1, yielding an exponentially decaying sinusoid with omega_d = omega_n sqrt(1 - zeta^2) and two X(t) forms.

Explore the overdamped and critically damped second-order responses by analyzing transfer function poles, damping ratio, and exponential terms, revealing no oscillation and the fastest approach to one.

Analyze a spring-mass-damper with displacement to determine overshoot at t = 0.605 s, using underdamped motion from x'' + Bx' + Kx = 0 and natural frequency, damping ratio, omega_d.

Analyze the step response of underdamped second-order systems using the transfer function and unit step. Derive peak time, overshoot, settling time, and rise time with key formulas.

Relate pole location to system response by connecting imaginary components to peak time and real components to settling time and damping, and show how moving poles alters overshoot and speed.

Determine pole locations for an underdamped second-order system from a 12 percent overshoot and 0.6 s settling time; compute zeta and omega_n, then locate poles.

Analyze a suspension model with mass 500 kg, spring 3.2e5 N/m, and damper 1e4 N s/m. A 12.8 mm road dip yields an underdamped response with a value 0.0128 m.

Plot the hand-derived X(t) in MATLAB, compare it to the transfer function response using a step input of 0.0128, and show overlapping plots of oscillation and damping.

Learn block diagrams with a heating system, illustrating a closed-loop with thermostat, temperature sensor, and a summing junction, and apply cascade and parallel transfer functions.

Explore feedback form concepts as the basis of closed-loop control engineering: negative feedback reduces error to keep systems stable, with HVAC, autopilot, and cruise control as examples.

Learn to form equivalent diagrams by reducing subsystems to one block while preserving outputs; move blocks around summing junctions and apply 1/G(s) to maintain equivalence.

Demonstrates reducing a three-block system to an equivalent transfer function by converting to standard feedback form with negative feedback, applying cascade reduction, and combining gains to yield a single block.

Identify the single feedback loop in the three-block H system and convert to the standard form G/(1+GH). Use cascade and parallel reductions to derive the equivalent transfer function (G3*G1*G2)/(1+G1*H1).

Simplify a complex block diagram into a single transfer function by identifying feedback loops, arranging into standard and parallel form, and applying the unity feedback rule.

Evaluate stability using bounded input bounded output and pole location. Determine stable, unstable, and marginally stable behavior from pole positions and exponential terms, illustrated in example 40.

Apply the Ruths (Routh-Hurwitz) stability criterion to the characteristic equation of the closed-loop transfer function to determine if any roots lie in the right half-plane.

Apply the routh's criterion to a cubic characteristic equation, build the routh array, count sign changes, and conclude two right-half-plane poles imply instability.

Apply the Routh stability criterion to a cubic equation by forming its coefficient array, counting first-column sign changes, and noting patterns in even-exponent rows for shortcuts.

Covers Routh array special cases: replace a row's zero with epsilon (case one) and handle a full zero row (case two); uses sign changes to assess stability.

Explain special case two in the Routh array, including how a row of zeros yields an auxiliary (even) polynomial and how differentiating it fills the zero row for root analysis.

Handle special case two by forming the auxiliary polynomial from the row above when zeros appear, then differentiate with respect to s and count sign changes to assess stability.

Explore special case two in the Routh array: symmetry about the origin from the auxiliary polynomial with real, imaginary, or quadrant roots, and handling a row of zeros.

Apply the Routh stability criterion to a closed-loop transfer function, building the denominator and auxillary polynomial to locate poles on the right, left, or imaginary axis in a fifth-order system.

Analyze a long closed-loop example by forming T(s)=G/(1+G), computing the closed-loop denominator, and using the auxiliary polynomial, symmetry, and sign-change analysis to assess stability and pole locations.

Using the Ruths criterion, this lecture shows how to choose k to keep the closed-loop system stable, ensuring all terms stay positive and k lies between minus one and nine.

Compute stability from state-space by finding eigenvalues of A using det(sI-A)=0, count poles on left vs right half-planes using the roots criterion; determine stability.

Explore root locus techniques to visualize how closed-loop poles move with the gain K, assess stability, and use Matlab to plot loci for higher-order systems.

Construct a root locus for a transfer function by identifying poles and zeros, setting k to zero, and tracing symmetric branches on the real axis to assess damping regimes.

Master root locus analysis of a transfer function with two poles and two zeros, tracing symmetric branches from poles to zeros and distinguishing underdamped, critically damped, and overdamped responses.

Compute poles and zeros from the transfer function and draw the root locus with symmetry about the real axis. Determine branches and asymptotes, applying step five when zeros are absent.

Apply root locus analysis to examples 53 and 54 by identifying poles and zeros, computing real-axis intercepts and asymptote angles, and determining K values to sketch locus and assess stability.

Explore how to compute root locus in Matlab using transfer functions, polynomials, and numerator and denominator arrays, then interpret poles, zeros, damping, overshoot, and frequency from the plots.

Explore steady state error by comparing input and output as time grows, using step, ramp, and parabolic inputs to assess position, velocity, and acceleration tracking and system stability.

Derive open-loop and closed-loop transfer functions for a unity-feedback system, then apply the final value theorem to express the steady-state error in terms of T(s) and G(s).

Derive the steady-state error equation using the open-loop transfer function G(s), apply the final value theorem, and analyze step, ramp, and parabolic inputs to assess the error.

Examine steady-state error for a unit step, showing zero error when the open-loop transfer function has at least one integration in the forward path; otherwise the error is finite.

Examine steady state error for ramp inputs using the open loop transfer function G(s). Identify zero, finite, and infinite error conditions for unit ramp inputs.

Analyze steady-state error for parabolic inputs (1/s^3) and identify when zero, finite, or infinite error occurs based on system order n. Illustrates testing with acceleration-like inputs.

In system dynamics and controls, evaluate steady-state errors for a given transfer function using step, ramp, and parabolic inputs from the Stansted Airport example 55.

Define error constants Kepi, KVI, and KA for step, ramp, and parabolic inputs; classify systems as type 0, 1, or 2 by exponent of s, with type reducing steady-state error.

analyze a unity feedback system with a ramp input to meet 1/6000 steady-state error; derive Kv as 60,000 and confirm a finite error requires a type-1 system.

Learn basic electrical circuit concepts using the impedance method, including current, voltage, and power, and identify passive components (resistor, capacitor, inductor) and active sources.

Apply Kirchoff's current and voltage laws to a circuit with a voltage source, resistors, and a capacitor, defining nodes and loops and using sign conventions to derive equations.

Apply Kirchhoff's voltage law to a three-loop circuit with six resistors and a voltage source, define loop currents i1, i2, i3, and compute currents and voltage drops.

Apply ohm's law to analyze current, voltage, and resistance in series and parallel circuits, then compute equivalent resistance, capacitance, and inductance to simplify circuits.

Reduce a three loop circuit to a single loop with an equivalent resistor, using series and parallel reductions, to obtain a single 12-volt source and a 5.38 ohm equivalent resistance.

Develop circuit models using sign conventions, apply Laplace transforms to compute impedances of passive elements, and derive transfer functions for a mixed resistor–capacitor–inductor circuit in example 59.

derive the time-domain math model for a three-loop circuit using kirchhoff's voltage law, establishing loop equations with capacitors, inductors, and resistors, and use the impedance method to obtain transfer functions.

Derive the circuit transfer function with impedance method and mesh analysis, solving three loops to obtain Vout/Vin for the given circuit.