Page-by-Page Notes on Time-Domain Analysis, Stability, and Steady-State Error

Page 1

• Analysis focus: time-domain (transient behavior) vs frequency-domain representations (steady-state behavior) including Bode plots and root-locus concepts.

• Emphasis on second-order systems for stability and dynamic response.

• Key quantities in time-domain analysis: time constant, rise time, delay time, settling time, overshoot, and damping ratio.

• Stability and root location: system behavior is governed by the poles in the s-plane; dominant pole pairs control the transient response.

Concepts mentioned: time-domain analysis, frequency-domain plots (Bode), time constants, delays, settling, stability, dominant pole pairs, damping ratio, overshoot, order of the system, root-locus, state analysis, and BIBO stability.

• Damping ratio: ζζ

• Natural frequency: ωnωn

• Dominant poles for a 2nd-order system:

s1,2=−ζωn±jωn1−ζ2s1,2=−ζωn±jωn1−ζ2

• Core takeaway: Damping ratio (ζζ) and natural frequency (ωnωn) govern transient response; pole locations in the s-plane determine stability.r.

• Core takeaway: For most control problems, the focus is on how a 2nd order (and low-order) system behaves with respect to stability and transient response, which is largely dictated by damping ratio and natural frequency, and how the root locations relate to stability criteria.

Page 2

• Time-domain analysis approach:

  • We obtain the system transfer function $M(s)$, then analyze its time-domain response to chosen inputs to study transient behavior and time-domain performance.

  • For steady-state analysis, we often examine the frequency-domain response (e.g., Bode plots).

• Test inputs for time-domain and steady-state analyses:

  • a) Unit impulse: used for time-domain impulse response analysis.

  • b) Unit ramp: tests ramp response (useful for evaluating steady-state error to ramp inputs).

  • c) Unit step: widely used for both transient and steady-state analyses; often the primary test signal.

  • d) Unit parabolic: tests parabolic input response (for higher-order aspects).

  • e) Unit sine wave: used for steady-state response and frequency-domain insight.

• Relationship between total and steady-state response:

  • Total response = Transient response + Steady-state response.

  • The steady-state behavior is governed by the transfer function and input, while the transient portion is governed by the pole locations and initial conditions.

• Statement: In practice, unit impulse signals are hard to generate in a lab; unit step is commonly used to study transient behavior and approximate impulse effects for analysis.

Page 3

• Stability concept: A system is stable if, for every bounded input, the output remains bounded (BIBO stability).

• Decomposition of response:

  • Total response = Transient response + Steady-state response.

  • Transient response depends on the network parameters (poles).

  • Steady-state response is governed by the transfer function and input parameters.

• Stability criteria and language:

  • If the transient response dies out as time goes to infinity, the system is stable (BIBO stable).

  • If the transient response becomes unbounded, the system is unstable.

  • If some poles lie on the imaginary axis (jw-axis) and the rest are in the left-half plane, the system is marginally stable (often called critically stable).

  • If any pole lies in the right-half plane, the system is unstable.

• BIBO stability and pole locations:

  • LHS of the S-plane corresponds to stability region.

  • RHS of the S-plane corresponds to instability region.

• Transient behavior and damping:

  • The damping ratio and the proximity of poles to the jw-axis determine the transient behavior (overshoot, oscillations, settling).

  • Complex conjugate poles near the jw-axis yield oscillatory responses with larger overshoot and longer settling.

Page 4

• Time-domain response characterization for a decaying exponential:

  • If a response is of the form $c(t) = A e^{-\alpha t}$, the time constant is defined as the time taken to reach (roughly) the steady state when the initial decay rate is maintained.

  • For a single pole at $s=-\alpha$, the time constant is $\tau = 1/\alpha$, and the response decays exponentially with rate (e^{-t/\tau}).

• In control terms, a common first-order model is described by a transfer function with a single pole; the time constant governs the speed of response.

• For higher-order systems, the concept of a single time constant is replaced by natural frequency and damping, but the idea of exponential decay from dominant poles remains central.

Page 5

• Block diagram/signal-flow perspective (SFG) for control systems:

  • Uses a block diagram representation with forward paths and feedback paths.

  • Mason’s gain formula can be used to compute overall transfer functions by accounting for all forward paths and loops.

  • The notes show a structured approach assigning intermediate gains (M1, M2, …) and feedback paths (D, H, etc.) to derive the closed-loop transfer function.

• General takeaway:

  • For complex interconnections, Mason’s rule provides a systematic way to compute the overall transfer function from the block diagram without explicitly solving large sets of equations.

• Stability considerations in SFG terms:

  • Stability analysis ultimately reduces to the locations of the closed-loop poles obtained from the characteristic equation, which are found via the denominator of the transfer function obtained from the block diagram.

Page 6

• Continued Mason’s gain framework with more blocks (M1, M2, M3, …, H1, H2, H3, H4, etc.).

• The process advances through constructing the overall transfer function by summing the contributions of the forward paths and loops, and applying Mason’s rule to determine the denominators that define the characteristic equation.

• The final expressions (D1, D2, etc.) correspond to the determinant-like components in the overall transfer function, whose zeros/poles determine stability.

Page 7

• Conditions for system stability (summary):

  • (i) The system must obey BIBO concepts (bounded input -> bounded output).

  • (ii) The transient response must die down to zero in the steady state (for a stable system).

  • (iii) The roots of the characteristic equation, given by $1 + G(s)H(s) = 0$, must lie in the left-half of the S-plane.

  • (iv) If any root lies on the right-hand side (or on the jw-axis in a way that does not settle), the system is unstable.

• Visual interpretation:

  • The S-plane divided into stable (LHS) and unstable (RHS) regions; the transient response is governed by dominant poles (those nearest the jw-axis).

  • A complex conjugate pair near the jw-axis dominates damping and oscillatory behavior.

• Summary of stability:

  • If all poles are strictly in the LHS, the system is stable.

  • If any pole lies on the jw-axis, the system is marginally or critically stable.

  • If any pole lies in the RHS, the system is unstable.

Page 8

• Standard second-order closed-loop transfer function:

  • Denominator form: $s^2 + 2\zeta\omega<em>n s + \omega</em>n^2 = 0$

  • Compare with the canonical second-order denominator $s^2 + 2\zeta\omega<em>n s + \omega</em>n^2$ to identify $\omega_n$ and $\zeta$ from circuit parameters.

• Example with denominator $s^2 + 20s + 100$:

  • Compare to $s^2 + 2\zeta\omega<em>n s + \omega</em>n^2$

  • So, $\omega<em>n^2 = 100 \Rightarrow \omega</em>n = 10$ and $2\zeta\omega_n = 20 \Rightarrow \zeta = \frac{20}{2\cdot 10} = 1.0$ (critically damped).

• Damping scenarios and ranges:

  • Underdamped: 0 < \zeta < 1

  • Critically damped: $\zeta = 1$

  • Overdamped: \zeta > 1

• Our goal is to observe how the damping ratio affects the natural response, overshoot, and settling in the standard second-order form.

Page 9

• Reiterating the standard second-order closed-loop form and its implications:

  • The natural frequency and damping ratio completely determine the transient response characteristics (overshoot, rise time, peak time, and settling time) for a given input.

  • The wavenumbers, damping, and natural frequency are linked to the denominator coefficients:$s^2 + 2\zeta\omega<em>n s + \omega</em>n^2 = 0$.

• For an undamped system (ζ = 0): persistent oscillations; for very high damping (ζ >> 1): very slow, non-oscillatory response; for moderate damping (0 < ζ < 1): oscillatory with overshoot.

• Practical advice from the notes: a damping ratio in the range roughly between 0.4 and 0.7 is often desirable for good trade-offs between speed and overshoot.

Page 10

• S-plane interpretation and dominant poles:

  • In the S-plane, the left half-plane (LHS) is stable; the right half-plane (RHS) is unstable.

  • Among the multiple poles, those closest to the jw-axis (the dominant poles) govern the damping ratio and transient response.

  • A complex conjugate pair closer to the jw-axis yields a damping ratio that determines the oscillatory nature and overshoot.

• Standard 2nd-order system notation used:

  • Transfer function: $M(s) = \frac{V<em>o(s)}{V</em>i(s)} = \frac{?}{s^2 + ?s + ?}$ (the notes use a schematic form; use the canonical two-pole form below)

  • Canonical form for a standard 2nd-order closed-loop system: $M(s) = \frac{\omega<em>n^2}{s^2 + 2\zeta\omega</em>n s + \omega_n^2}$

• Key takeaway: The damping ratio and natural frequency determine the transient response, and the proximity of poles to jw-axis sets the damping characteristics.

Page 11

• Impulse response perspective for underdamped cases:

  • For analysis of transient behavior, impulse input is used to reveal the natural modes of the system.

  • For an underdamped system with poles at $s = -\alpha \pm j\omega<em>d$, the impulse response contains terms of the form $e^{-\alpha t} \cos(\omega</em>d t)$ and/or $e^{-\alpha t} \sin(\omega_d t)$.

• Relations among parameters:

  • $\alpha = \zeta \omega_n$

  • $\omega<em>d = \omega</em>n\sqrt{1-\zeta^2}$

• The resonant behavior and damping determine how quickly the system settles and how much overshoot occurs.

Page 12

• Characteristic equation and pole locations for a given pair of dominant poles:

  • For analysis, we consider a characteristic equation of the form $1 + G(s)H(s) = 0$ where poles are solutions of that equation.

  • A pair of complex conjugate poles on the left-half plane with real part near zero yields oscillatory response with slow decay; poles with large negative real parts yield quick damping and little overshoot.

• The notation used in the notes emphasizes the location of the roots of the characteristic equation and their influence on transient behavior.

• General takeaway: The dominant pair of poles largely determines the transient response; the rest of the poles have a lesser impact if they are far from the jw-axis.

Page 13

• Unit step response for a standard second-order system (underdamped case):

  • With unity feedback and a standard second-order denominator, the step response is typically written as:

  • $c(t) = 1 - \frac{e^{-\zeta \omega<em>n t}}{\sqrt{1-\zeta^2}} \sin\left(\omega</em>d t + \phi\right)$

  • where $\omega<em>d = \omega</em>n\sqrt{1-\zeta^2}$ and $\phi = \cos^{-1}(\zeta)$.

• An alternative form given in the notes introduces constants A and B related to the system parameters:

  • The impulse response and step response expressions can be written as combinations of terms involving $e^{-\alpha t}\sin(\omega t)$ and $e^{-\alpha t}\cos(\omega t)$ with appropriate coefficients determined by the input and the forward path gains.

• Note: The detailed algebra yields the same qualitative behavior: an exponentially decaying sinusoid approaching unity for the step input when the system is stable and underdamped.

Page 14

• Step-by-step expression forms and time-domain signals:

  • A typical form for the step response in the underdamped case involves exponential decay times sinusoidal components, with coefficients A and B determined by initial conditions and system gains.

  • The time-domain response can be written as a sum of terms of the type $e^{-\alpha t} \cos(\omega t)$ and $e^{-\alpha t} \sin(\omega t)$, plus the steady-state offset.

• Key derived quantities from step response:

  • The natural/undamped frequencies (and the damping) define the transient oscillation.

  • The ratio between the damped and undamped oscillations can be described via $\alpha = \zeta\omega<em>n$ and $\omega</em>d = \omega_n\sqrt{1-\zeta^2}$.

• For a unit step input and a standard second-order system, the steady-state value is typically 1 (unity feedback, unity DC gain) if the system is properly scaled. The transient part decays exponentially depending on the dominant poles.

Page 15

• Time constants and damping factors illustrated numerically:

  • Example values show how damping ratio affects time constants and settling behavior.

  • For a given ωn and damping ζ, the peak time and settling time depend on ωd and ζ respectively.

• Observations:

  • Critically damped (ζ = 1) yields the fastest non-oscillatory approach to the final value but can be relatively slower than a moderately underdamped system for some criteria.

  • Underdamped (ζ < 1) yields overshoot and oscillations, with the amount of overshoot decreasing as ζ increases toward 1.

• Summary guidance from the notes: with a 0.4 ≤ ζ ≤ 0.7 range, one can often achieve a good compromise between rise time, overshoot, and settling.

Page 16

• Transient-domain indices for a second-order response to a unit step:

  • Damping ratio: $\zeta$

  • Overshoot: percent overshoot MP = 100 e^{-\frac{\zeta \pi}{\sqrt{1-\zeta^2}}} \%

  • Rise time: $t<em>r$ (approximate expression depends on ζ and ωn; commonly approximated as the time to go from 0 to 100% or 10% to 90% of the final value for underdamped systems)

  • Peak time: $t<em>p = \frac{\pi}{\omega</em>d} = \frac{\pi}{\omega_n \sqrt{1-\zeta^2}}$

  • Delay time: often defined as the time for the response to reach 50% of its final value in certain approximations (definitions vary by course); commonly linked to the system's phase delay.

  • Settling time: $T_s$ (time to remain within a specified tolerance around the final value, e.g., ±2% or ±5%).

  • Transient period vs steady-state period distinction: the transient period ends when the response stays within the settling band around the final value.

Page 17

• Mathematical relationship for tpeak via ωd and damping:

  • For a standard second-order response, the peak occurs at a time where the derivative of the response is zero, yielding:

  • $t<em>{ ext{peak}} = \frac{\pi}{\omega</em>d} = \frac{\pi}{\omega_n \sqrt{1-\zeta^2}}$.

• Overshoot in terms of ζ:

  • The overshoot is governed by the damping ratio and is given by:

  • $\text{Overshoot} = e^{ -\frac{\zeta \pi}{\sqrt{1-\zeta^2}} } \times 100\%.$

• Relationship between ω and damping to the transient response characteristics:

  • As ζ increases (more damping), overshoot decreases and settling time generally decreases (up to a point); excessive damping leads to slower response.

Page 18

• Overshoot vs damping ratio table (illustrative):

  • With a fixed ω_n = 10 rad/s, varying ζ yields a different percentage overshoot, e.g.:

  • ζ = 0.1 → large overshoot but still decays

  • ζ around 0.4–0.7 often yields a moderate overshoot and acceptable settling time

  • ζ = 0.9 or higher yields tiny overshoot but longer settling time

• Practical takeaway: designers often target a damping ratio in the ~0.4–0.7 range to balance speed and overshoot.

• Settling time estimates for 2% tolerance with ω_n = 10 rad/s:

  • For each ζ, the approximate settling time can be computed via $T<em>s \approx \frac{4}{\zeta \omega</em>n}$ (2% criterion) or similar approximations for other tolerances.

Page 19

• Example numerical tabulation for time-domain indices (illustrative):

  • For certain cases, the following times are listed (numbers are example values corresponding to specific ζ, ω_n, and tolerance bands):

  • tr (rise time), ts (settling time), t_d (delay), etc., with a rough pattern showing how increasing damping reduces overshoot but can increase rise/settling times.

  • Settling time and delay time examples show that higher damping often reduces overshoot but increases the time to settle.

• Definition recap:

  • Settling time: time to remain within the specified tolerance around the final value (often ±2% or ±5%).

  • Delay time: time to reach a specified fraction (often 50%) of the final value.

  • t_rise: time to go from a lower to a higher percentage of the final value (e.g., from 10% to 90%).

Page 20

• Practical approximate formulas for indices (curve-fitting style):

  • Rise time, delay time, and rise/fall approximations can be obtained via simple fitted expressions in terms of ω_n and ζ, such as:

  • Delay time, tdelay ≈ 1/(0.5) × 1/4 ωn (illustrative pattern from notes; exact coefficients vary by method)

  • Rise time and 10–90% rise: approximate forms depending on ω_n and ζ.

• An explicit example: For a given M(s) and K, one can compute the related time-domain indices once G(s) is known; a table of K values demonstrates how the root locations move in the S-plane with gain changes and how stability is affected.

• General takeaway: The gain K affects the pole locations, which in turn change the transient indices like tr, tp, and overshoot.

Page 21

• Numerical extraction of time-domain indices from a particular transfer function example (M(s)):

  • Specific coefficients and gains (K) yield different pole locations and thus different tp, tr, and overshoot values.

  • Example results show how choosing K changes the damping and dominance of poles.

• Note on c(t) expressions:

  • The step response c(t) can be written in a form that includes exponential decay and sinusoidal terms (as shown in previous pages), with constants determined by the system gains and input type.

• Practical outcome:

  • By adjusting K, one can place poles to achieve desired transient performance (overshoot, rise time, settling time) while keeping stability.

Page 22

• Root locus and pole-location analysis (continued):

  • Root locus plots illustrate how closed-loop poles move in the S-plane as a gain K is varied.

  • Examples show several K values (e.g., K = 1, 4, 16, 24, 48, 96) and the corresponding poles.

  • Observations: as K increases, poles move, potentially crossing the jw-axis indicating a transition from stability to marginal stability or instability.

• Example findings:

  • For certain K values, the system remains stable with poles in the LHS.

  • At a specific K (e.g., K around 8 in the notes), complex conjugate poles cross onto the jw-axis, indicating critical stability.

  • Further increases in K push poles into the RHS, yielding instability.

Page 23

• More root-locus data and pole positions for varying K:

  • A set of pole locations is listed for K values such as K = 1, 4, 16, 24, 48, 96, etc.

  • The table shows the real and imaginary parts of the poles (e.g., -0.284 ± j1.87 for K=1, etc.).

• Practical interpretation:

  • By inspecting pole locations, one can determine stability and approximate transient behavior without computing full time-domain simulations.

  • The behavior near jw-axis indicates potential for sustained oscillations if poles are too close to jw-axis or on jw-axis.

Page 24

• Gain Margin (GM) concept and calculation:

  • GM is defined as the ratio of the critical gain to the actual gain that yields the boundary of stability:

  • $GM = \frac{K<em>{critical}}{K</em>{actual}}$

  • In decibels: $GM<em>{dB} = 20 \log</em>{10}(GM)$

• Interpretation of GM:

  • A positive GM (GM > 1, GM_{dB} > 0) indicates a buffer above the current gain before the system becomes unstable.

  • GM = 0 dB corresponds to the boundary of stability (critical stability). If GM < 1 (negative dB), the system is unstable for the current gain.

• Example data from notes:

  • If Kad = 1 yields GM ≈ 18.06 dB; Kad = 4 yields GM ≈ 6.02 dB; Kad = 8 yields GM ≈ 0 dB (borderline stability); Kad = 16 yields GM ≈ -6.02 dB (unstable beyond this gain).

• Practical implication:

  • GM provides a quantitative measure of robustness against gain variations; higher GM means more robust stability margins.

Page 25

• Summary insight on stability margins:

  • The s-plane view shows that as gain increases, some complex conjugate poles approach the jw-axis and eventually cross into the right half-plane, making the system unstable.

  • GM indicates how much margin remains before instability occurs.

  • A system with complex conjugate poles on the jw-axis is critically stable; beyond this gain, transient behavior becomes unbounded.

• Qualitative takeaway on stability regions:

  • There exists a stable gain region where all closed-loop poles lie in the LHS; outside this region, the system is unstable.

  • The GM and root-locus together give a practical picture of how gain changes affect stability and transient behavior.

Page 26

• Steady-state error concepts and input types:

  • The steady-state error ess depends on the type of input signal (step, ramp, parabolic, etc.) and the system type (number of integrators in the forward path G(s)).

  • Common input types:

  • Step input (positional): R(s) = 1/s

  • Ramp input (velocity): R(s) = 1/s^2

  • Parabolic input (acceleration): R(s) = 1/s^3

• Steady-state error (ess) basics:

  • ess is the final difference between the input and the scaled output as t → ∞ for a given input.

  • ess depends on the type of control system and the presence of integral action.

• Steady-state error control constants (for unity feedback):

  • Kp: steady-state error constant for positional input (step input).

  • Kv: steady-state error constant for velocity input (ramp input).

  • Ka: steady-state error constant for acceleration input (parabolic input).

• The note indicates that ess can be evaluated with final-value methods or via constants Kp, Kv, Ka.

Page 27

• Type and pole structure related to input response:

  • Type-0 systems (no integrators in forward path) can track step inputs with finite error but fail to track ramp or parabolic inputs (ess → ∞ for ramp/parabolic).

  • Type-1 systems (one integrator) track step with finite error, ramp with finite error, but fail to track parabolic inputs (ess → ∞ for parabolic).

  • Type-2 systems (two integrators) track step and ramp with finite error, and parabolic with finite error.

  • Type-3 systems (three integrators or more) generally become unstable during transient periods for certain inputs; Type-3 is usually not useful for stable tracking.

• How to interpret the table:

  • The number of pure integrators (poles at s=0) in the forward path determines the system type.

  • Higher type improves steady-state accuracy for higher-order inputs but may affect robustness and dynamic performance.

Page 28

• How to calculate ess (two methods):

  • Method A (reference input approach):

  • ess(t) = controlled output of the steady state when the input is a step, i.e., ess(t) = c(∞) for step input.

  • In the Laplace domain: ess = lim_{s→0} E(s) where E(s) = R(s) - H(s)C(s).

  • Method B (final value theorem approach):

  • Use the final value theorem on the error signal E(s) to compute ess for a given input type.

• Final Value Theorem approach:

  • ess = lim_{s→0} s E(s)

  • For unity feedback with forward path G(s), E(s) = R(s) - G(s)C(s) with C(s) = G(s)R(s)/(1+G(s)); this reduces to ess = lim_{s→0} R(s)/(1+G(s)) when R(s) is a step input (R(s) = 1/s).

Page 29

• Summary of steady-state error constants (method for computing ess):

  • For a unit step input (positional input):

  • Kp = lim_{s→0} G(s) (the forward path gain at s=0).

  • ess = 1/(1 + Kp) for unity feedback;

  • For a unit ramp input (velocity input):

  • Kv = lim_{s→0} s G(s).

  • ess = 1/Kv for unity feedback when the type is >= 1 (i.e., there is at least one integrator in G(s)).

  • For a unit parabolic input (acceleration input):

  • Ka = lim_{s→0} s^2 G(s).

  • ess = 1/Ka for type >= 2 systems.

• Remarks:

  • If the corresponding constant is finite, the system can track the input with finite steady-state error; if the constant is infinite, the ess is zero (perfect tracking within the specified input’s capability).

Page 30

• Explicit definitions of the steady-state error constants for various inputs (positional, velocity, acceleration) in the unity-feedback setting:

  • R(s) = 1 (step/positional input):

  • ess = 1/(1 + Kp) with $K</em>p = \lim_{s\to 0} G(s).$

  • R(s) = 1/s (ramp/velocity input):

  • ess = 1/Kv with $K</em>v = \lim_{s\to 0} s G(s).$

  • R(s) = 1/s^2 (parabolic/acceleration input):

  • ess = 1/Ka with $K</em>a = \lim_{s\to 0} s^2 G(s).$

• The constants Kp, Kv, Ka are also referred to as steady-state error constants for positional, velocity, and acceleration inputs respectively.

• The equations for ess can be rearranged in terms of R(s) and the closed-loop transfer function, but the essential idea remains: the low-frequency gain (and the presence of integrators) determines the steady-state error.

Page 31

• Summary of steady-state error constants (closing remark):

  • Kp, Kv, Ka quantify the system’s ability to track step, ramp, and parabola inputs, respectively.

  • The relationship between ess and these constants is central to design trade-offs in control systems: higher integrator action generally reduces steady-state error for low-order inputs but can affect stability and transient performance.

• Final note on how to apply these concepts practically:

  • Identify the system type (number of integrators in G(s)).

  • Compute the low-frequency gains Kp, Kv, Ka as appropriate for your input type.

  • Use ess formulas to determine steady-state error and adjust controller gains to meet design specifications while preserving stability.