Control System Toolbox: LTI Models

Control System Toolbox

• Series connections of LTI models use the series function.
• The * operator connects LTI models in series.
• By default, the feedback function assumes negative feedback.
• Positive feedback in the feedback function is applied using feedback(H1,H2,+1).
• Parallel connections of LTI models use the parallel function.
• Inputs of two models are combined using [H1, H2].
• [H1; H2] outputs one input and two outputs.
• The append function appends inputs and outputs of two models.
• tf(2,[1 3 0]) creates a transfer function model.
• The connect function returns a State-space or FRD model.
• FRD models have the highest precedence in model interconnections.
• The feedback function creates a closed-loop transfer function.
• By default feedback(H1, H2) results in negative feedback.
• feedback(H1, H2) is used for the closed-loop transfer from u to y.
• H2 = zpk([], -5, 5) gives a Zero/pole/gain model.
• The append function appends inputs and outputs.
• When models of different types are connected, the model with the highest precedence determines the result.
• The connect function builds MIMO models.
• The series function connects two models in series.
• sigma(H) results in a singular value plot.
• The grid command adds gridlines to plots.
• sumblk creates summation blocks.
• connect is primarily used for building general block diagrams.
• The connect function automatically creates a model conversion.
• step(T) produces a step response plot.
• class(H) returns the model type.
• The tf function creates transfer functions.
• PID models have the lowest precedence.
• feedback(G*C, S) computes a closed-loop transfer function.
• The precedence rule is FRD > SS > ZPK > TF > PID.

MATLAB Simulink Onramp and Simscape Onramp

• Simulink is primarily used for modeling and simulating dynamic systems.
• Simulink integrates with MATLAB.
• Simulink provides a graphical editor.
• Simulink allows export of simulation results to MATLAB.
• Physical component modeling is not a feature of Simulink.
• Simscape is used to model physical systems.
• Simscape integrates with Simulink.
• Simscape uses physical connections.
• Simulink's solver is responsible for solving mathematical models over time.
• Simulink does not support hardware manufacturing.
• The first step in accessing Simulink Onramp is accessing the online course.
• Simscape allows models to be built based on physical connections.
• MATLAB Academy hosts the Simulink and Simscape Onramp courses.
• Simulink Onramp is best suited for learning Simulink basics.
• A MATLAB online account is needed to access the Onramp courses.
• Simscape models are constructed using physical components.
• Simulink simulations can be analyzed further in MATLAB.
• Simscape Onramp focuses on physical system modeling.
• Integrated solvers enable Simulink to handle different types of systems.
• Simulink can incorporate algorithms from MATLAB.
• The main benefit of Simscape is rapid modeling of physical systems.
• Simulink's graphical editor is mainly used to create block diagrams.
• The output of a Simulink model consists of simulation results.
• Simulink integrates with Simscape to simulate physical systems.
• To access the Simulink Onramp, a USC email is required.
• Simulink supports exporting results to MATLAB.
• Simulink and Simscape are used together to simulate mechanical systems.
• Simulink blocks represent mathematical operations.
• The Simulink solver handles time integration.
• Simscape integrates directly with Simulink.

Transfer Functions and Time Response

• A transfer function represents the input-output relationship in the Laplace domain in a control system.
• The Laplace transform of the input signal in the transfer function equation is called R(s).
• In T(s) = Y(s)/R(s), Y(s) represents the output while R(s) represents input.
• The Laplace transform operation is used to derive the transfer function from a differential equation.
• The MATLAB command tf creates a transfer function model.
• "sys = tf(n, d)" in MATLAB creates a transfer function.
• In a second-order system, the denominator of the transfer function represents a characteristic polynomial.
• The inverse Laplace transform of Y(s) gives the time-domain response.
• Partial fraction decomposition simplifies inverse Laplace transforms.
• Rise time (Tr) is the time to reach 90% of the final value in system response.
• Peak time (Tp) is the time to reach peak value.
• Percent overshoot is the maximum deviation from the steady-state value.
• "step" function in MATLAB simulates step response.
• The "roots" command finds roots of a polynomial in MATLAB.
• An underdamped response type has complex conjugate poles with negative real parts.
• A response type with two real distinct poles is overdamped.
• Critically damped systems have a double pole.
• The natural response in an overdamped system decays exponentially without oscillation.
• The characteristic polynomial of s^2 + 4s + 100 is s^2 + 4s + 100.
• Poles of a transfer function are the roots of the denominator.
• Zeros of a transfer function are the roots of the numerator.
• An overdamped system has no oscillation in its repsonse.
• The ilaplace function calculates the inverse Laplace transform.
• The steady-state value of y(t) = 1 - e^(-2t) is 1.
• The plot function plots the system response in MATLAB.
• clc clears the command window in MATLAB.
• "clear all" clears the workspace in MATLAB.
• linspace is used to generate a time vector in MATLAB.
• Settling time indicates the time for the transient to settle.
• Overshoot indicates the maximum deviation from steady-state value.

System Stability Analysis

• A pole represents a point where a transfer function goes to infinity.
• The numerator of a transfer function is called the zero.
• The denominator of a transfer function is called the characteristic polynomial.
• The characteristic equation determines system poles.
• In a stable system, all poles should be located in the left half-plane.
• A pole in the right half-plane makes the system unstable.
• Repeated poles on the imaginary axis indicate marginal stability in a system.
• The system order is determined by the number of poles.
• The roots function in MATLAB finds the roots of a polynomial.
• Distinct real poles indicate a system is overdamped.
• The impulse function in MATLAB plots the impulse response.
• Poles in the right-half plane make the system unstable.
• A pole-zero plot is represented by a complex plane with poles and zeros.
• A zero indicates a point where the transfer function equals zero.
• Repeated real poles indicate a critically damped system.
• The tf function in MATLAB creates transfer functions.
• To plot a pole-zero plot in MATLAB use zplane.
• Moving poles further to the left in the complex plane reduces response time.
• Poles in the right half-plane indicate an unstable system.
• An unstable system has poles with positive real parts.
• The MATLAB command "clc" clears the command window.
• The command that would display the transfer function in MATLAB is tf.
• A repeated pole on the imaginary axis indicates marginal stability.
• A second-order system has two poles.
• The roots of the numerator polynomial determine the number of zeros in a transfer function.
• The impulse function simulates impulse response.
• "clear all" clears variables in MATLAB.
• The X-axis represents the real part of the poles/zeros in a pole-zero plot.
• A stable system type returns to equilibrium after a disturbance.
• Repeated poles on the imaginary axis have marginal stabilty.

Performance Specifications

• The two main performance aspects considered when designing a control system are transient and steady-state performance.
• wn represents natural frequency in a second-order system.
• ξ (zeta) represents the damping ratio in a control system.
• Natural frequency affects the speed at which a system reaches steady-state.
• When the damping ratio ξ = 0, the response is pure oscillation.
• A type of response produced when ξ > 1 is the overdamped.
• tp represents peak time.
• Peak time in a second-order system calculated by tp=πwn1−ξ2t_p = \frac{\pi}{w_n \sqrt{1 - \xi^2}}
• Percent overshoot measures maximum amplitude beyond steady-state.
• Percent overshoot is calculated using Mp=e−ξπ1−ξ2×100M_p = e^{-\xi \pi / \sqrt{1 - \xi^2}} \times 100.
• Settling time (ts) is the time for oscillations to stay within 2% of steady-state.
• Settling time is calculated using ts=−ln⁡(0.02)ξwnt_s = \frac{-\ln(0.02)}{\xi w_n}.
• In MATLAB, the step function generates the step response.
• Increasing the damping ratio decreases oscillation of system response.
• If damping ratio ξ = 1 in a step response, the outcome will be critically damped results.
• Adding a pole far from the imaginary axis increases response time.
• The "hold on" command holds current plot to overlay new plots.
• As the damping ratio increases, the step response reduces overshoot.
• The dominant pole is the one closest to the origin.
• Increasing the natural frequency results a faster response.
• A system with a damping ratio ξ > 1 is overdamped.
• A system with poles only on the imaginary axis is marginally stable.
• Adding a zero near the origin decreases response time.
• "clc" clears the command window in MATLAB.
• Increasing wd increases oscillation frequency on the step response.
• Natural frequency directly affects oscillation frequency.
• Increasing damping reduces overshoot in a second-order system.
• The "pause" command delays execution
• When a pole is added closer to the origin, increases rise time.
• Increasing the damping ratio results in reduced overshoot.