Sources and References

Primary textbook — G.F. Franklin, J.D. Powell & A. Emami-Naeini, Feedback Control of Dynamic Systems, 8th ed., Pearson, 2019. Supplementary — R.C. Dorf & R.H. Bishop, Modern Control Systems, 14th ed., Pearson; K. Ogata, Modern Control Engineering, 5th ed., Pearson. Online resources — [Karl Åström & Richard Murray, Feedback Systems (open access, Caltech); MIT OCW 6.302; MATLAB Control System Toolbox documentation]

---

Chapter 1: Introduction and Mathematical Foundations

1.1 What Is a Control System?

A control system is an arrangement of physical and computational components organized to regulate, command, or direct the behaviour of another system — the plant — so that its output follows a desired reference signal despite the presence of disturbances and model uncertainty. This deceptively simple statement encompasses the heating element in your kitchen oven, the attitude-control thrusters on a satellite, the cruise-control module in an automobile, and the insulin-pump algorithm that keeps a diabetic patient’s blood glucose within safe bounds.

From the Systems Design Engineering perspective that defines SYDE 352, control is inseparable from design. Every controller is an artefact embedded in a larger sociotechnical system, and the engineer’s job is not merely to stabilize a plant but to satisfy a constellation of objectives — speed of response, energy efficiency, robustness to manufacturing variation, cost, and safety — while acknowledging the limits of the mathematical model used during design. This dual perspective — rigorous mathematical analysis combined with engineering judgment about model fidelity — runs throughout the course and its associated laboratory program.

1.1.1 Open-Loop vs. Closed-Loop Control

The most fundamental architectural distinction in control is between open-loop and closed-loop (feedback) systems.

In an open-loop system the controller sends a command to the plant based entirely on the reference input, with no measurement of the actual plant output. An electric toaster is a canonical example: the heating element stays on for a preset time regardless of how brown the bread actually becomes. Open-loop control is simple and cheap but fails whenever the plant dynamics differ from the design model or whenever disturbances push the output away from the desired value.

In a closed-loop system a sensor measures the plant output \( y(t) \), a comparator forms the error

\[ e(t) = r(t) - y(t), \]

and a controller \( C \) processes \( e(t) \) to generate the control input \( u(t) \) that drives the plant \( P \). The loop of information flow — plant output feeds back to modify the plant input — is why closed-loop control is synonymous with feedback control.

1.1.2 Components of a Feedback Control System

A standard unity-feedback block diagram contains the following elements:

• Reference (setpoint) \( R(s) \): the desired value of the output.
• Error signal \( E(s) = R(s) - Y(s) \): the discrepancy between reference and output.
• Controller \( C(s) \): the compensator that processes \( E(s) \) to produce the control signal \( U(s) \).
• Plant \( G(s) \): the physical process to be controlled.
• Sensor/measurement \( H(s) \): transduces the plant output to a feedback signal (often \( H(s) = 1 \) in unity-feedback configurations).
• Disturbance \( D(s) \): an unwanted external input that enters the plant.
• Noise \( N(s) \): corruption of the sensor measurement.

The closed-loop transfer function from reference to output is

\[ T(s) = \frac{C(s)\,G(s)}{1 + C(s)\,G(s)}. \]

1.2 Mathematical Models of Physical Systems

Before designing a controller we need a mathematical description of the plant. For the lumped-parameter systems treated in this course such a description takes the form of an ordinary differential equation (ODE) relating the output \( y(t) \) to the input \( u(t) \).

1.2.1 Mechanical Translational Systems

For a mass-spring-damper system with mass \( m \), damping coefficient \( b \), and spring constant \( k \) driven by force \( u(t) \):

\[ m\ddot{y} + b\dot{y} + ky = u(t). \]

1.2.2 Electrical Circuits

A series RLC circuit with input voltage \( u(t) \) and output voltage across the capacitor \( y(t) = v_C(t) \) obeys

\[ LC\,\ddot{y} + RC\,\dot{y} + y = u(t). \]

Notice the structural isomorphism with the mechanical system: \( L \leftrightarrow m \), \( R \leftrightarrow b \), \( 1/C \leftrightarrow k \). This universality of the second-order ODE means that the analysis developed once applies across all physical domains.

1.2.3 Rotational Systems

For a rotating disc of moment of inertia \( J \) with viscous friction coefficient \( b \) and applied torque \( \tau(t) \):

\[ J\ddot{\theta} + b\dot{\theta} = \tau(t). \]

1.3 Laplace Transforms — A Review

The Laplace transform converts a time-domain ODE into an algebraic equation in the complex frequency variable \( s \), which can then be manipulated with the tools of algebra before being inverted back to the time domain.

Key transform pairs used throughout this course:

\( f(t) \)	\( F(s) \)
\( \delta(t) \)	\( 1 \)
\( u_s(t) \) (unit step)	\( 1/s \)
\( t \) (unit ramp)	\( 1/s^2 \)
\( e^{-at} \)	\( 1/(s+a) \)
\( \sin(\omega_0 t) \)	\( \omega_0/(s^2 + \omega_0^2) \)
\( \cos(\omega_0 t) \)	\( s/(s^2 + \omega_0^2) \)
\( t^n e^{-at} \)	\( n!/(s+a)^{n+1} \)

The key operational properties are differentiation (\( \mathcal{L}\{\dot{f}\} = sF(s) - f(0^-) \)) and integration (\( \mathcal{L}\{\int_0^t f\,d\tau\} = F(s)/s \)), which allow us to replace derivatives with multiplications by \( s \).

1.3.1 Initial and Final Value Theorems

---

Chapter 2: Transfer Functions and Block Diagrams

2.1 The Transfer Function

For a system governed by the \( n \)-th order ODE

\[ a_n y^{(n)} + a_{n-1} y^{(n-1)} + \cdots + a_0 y = b_m u^{(m)} + \cdots + b_0 u, \]

the transfer function is

\[ G(s) = \frac{b_m s^m + b_{m-1} s^{m-1} + \cdots + b_0}{a_n s^n + a_{n-1} s^{n-1} + \cdots + a_0}. \]

A transfer function is said to be proper if \( m \leq n \) and strictly proper if \( m < n \). Physical systems are generally strictly proper because real components cannot respond instantaneously.

2.2 Poles and Zeros

Poles dominate the transient behaviour of a system. A pole at \( s = -a \) (real, negative) contributes a decaying exponential \( e^{-at} \) to the output. A complex conjugate pole pair at \( s = -\sigma \pm j\omega_d \) contributes a damped sinusoid \( e^{-\sigma t}\sin(\omega_d t + \phi) \).

2.3 Block-Diagram Reduction

Complex control systems are represented by block diagrams — signal flow graphs in which each block represents a transfer function and arrows represent signal paths. Block-diagram reduction converts a multi-block diagram into a single equivalent transfer function.

The three elementary configurations are:

Series (cascade): \( G_{\text{eq}}(s) = G_1(s)\,G_2(s) \).

Parallel: \( G_{\text{eq}}(s) = G_1(s) + G_2(s) \).

Feedback loop: With forward-path transfer function \( G(s) \) and feedback transfer function \( H(s) \):

\[ T(s) = \frac{G(s)}{1 + G(s)H(s)}. \]

Additional rules include moving a summing junction past a block, moving a branch (pickoff) point past a block, and combining nested loops from the innermost outward.

2.4 First-Order System Step Response

The prototype first-order transfer function is

\[ G(s) = \frac{K}{\tau s + 1}, \]

where \( K \) is the DC gain and \( \tau \) is the time constant. For a unit step input the output is

\[ Y(s) = \frac{K}{s(\tau s + 1)}, \quad y(t) = K\left(1 - e^{-t/\tau}\right), \quad t \geq 0. \]

The output reaches \( 63.2\% \) of its final value at \( t = \tau \), and is within \( 2\% \) of final value at \( t \approx 4\tau \). The rise time (10% to 90% of final value) is \( t_r \approx 2.2\tau \).

---

Chapter 3: Time-Domain System Response

3.1 The Standard Second-Order System

The vast majority of control specifications are written in terms of the step response of a second-order system. Understanding this prototype is therefore central to the entire subject.

The poles are located at

\[ s_{1,2} = -\zeta\omega_n \pm \omega_n\sqrt{\zeta^2 - 1}. \]

Four regimes are distinguished by the value of \( \zeta \):

Regime	Condition	Pole type
Underdamped	\( 0 < \zeta < 1 \)	Complex conjugate, \( s = -\sigma \pm j\omega_d \)
Critically damped	\( \zeta = 1 \)	Repeated real, \( s = -\omega_n \)
Overdamped	\( \zeta > 1 \)	Two distinct real poles
Undamped	\( \zeta = 0 \)	Pure imaginary, \( s = \pm j\omega_n \)

In the underdamped case the damped natural frequency is \( \omega_d = \omega_n\sqrt{1-\zeta^2} \).

3.2 Step-Response Specifications

For a unit step input applied to the standard second-order system in the underdamped regime \( (0 < \zeta < 1) \):

\[ y(t) = 1 - \frac{e^{-\zeta\omega_n t}}{\sqrt{1-\zeta^2}}\sin\!\left(\omega_d t + \phi\right), \quad \phi = \arccos(\zeta). \]

The four canonical performance metrics are:

Rise time (0% to 100%):

\[ t_r \approx \frac{1.8}{\omega_n} \quad \text{(for } 0.3 \leq \zeta \leq 0.8 \text{)}. \]

Peak time (time of first overshoot peak):

\[ t_p = \frac{\pi}{\omega_d}. \]

Percent overshoot:

\[ \%OS = 100\,\exp\!\left(\frac{-\pi\zeta}{\sqrt{1-\zeta^2}}\right). \]

Settling time (2% criterion):

\[ t_s \approx \frac{4}{\zeta\omega_n}. \]

3.3 Higher-Order Systems and Dominant Poles

Real plants have more than two poles. The standard approach is to identify a dominant pole pair — the pair closest to the imaginary axis — that controls the transient behaviour, and to treat the remaining poles as negligible provided they are at least five times further into the left half-plane.

Zeros near the dominant poles can significantly affect overshoot — a zero in the left half-plane close to the dominant pair increases overshoot, while a right-half-plane (non-minimum phase) zero causes an initial undershoot before the response rises.

---

Chapter 4: Stability Analysis — Routh-Hurwitz and Steady-State Performance

4.1 Stability of LTI Systems

The characteristic equation of the closed-loop system is

\[ 1 + C(s)\,G(s) = 0, \]

or equivalently the denominator polynomial of \( T(s) \) set to zero. Stability depends entirely on the roots of this polynomial.

4.2 Routh-Hurwitz Criterion

Finding the exact roots of a high-degree polynomial is computationally intensive and, more importantly, does not give insight into how design parameters affect stability. The Routh-Hurwitz criterion determines stability without computing roots.

Constructing the Routh Array. For a fourth-order polynomial \( a_4 s^4 + a_3 s^3 + a_2 s^2 + a_1 s + a_0 \):

\[ \begin{array}{c|cc} s^4 & a_4 & a_2 & a_0 \\ s^3 & a_3 & a_1 & 0 \\ s^2 & b_1 & b_2 & 0 \\ s^1 & c_1 & 0 & \\ s^0 & d_1 & & \end{array} \]

where

\[ b_1 = \frac{a_3 a_2 - a_4 a_1}{a_3}, \quad b_2 = \frac{a_3 a_0 - a_4 \cdot 0}{a_3} = a_0, \]\[ c_1 = \frac{b_1 a_1 - a_3 b_2}{b_1}, \quad d_1 = a_0. \]

4.3 Steady-State Error and System Type

Even a stable closed-loop system may exhibit a permanent offset between the reference and the output. Quantifying this offset is the goal of steady-state error analysis.

For a unity-feedback system with open-loop transfer function \( G(s) \) and input \( R(s) \), the steady-state error is

\[ e_{ss} = \lim_{s \to 0} s \cdot E(s) = \lim_{s \to 0} s \cdot \frac{R(s)}{1 + G(s)}. \]

The error constants are defined as:

\[ K_p = \lim_{s\to 0} G(s) \quad \text{(position constant)}, \qquad K_v = \lim_{s\to 0} s\,G(s) \quad \text{(velocity constant)}, \qquad K_a = \lim_{s\to 0} s^2 G(s) \quad \text{(acceleration constant)}. \]

The complete table of steady-state errors:

Input	Type 0	Type 1	Type 2
Step	\( 1/(1+K_p) \)	0	0
Ramp	\( \infty \)	\( 1/K_v \)	0
Parabola	\( \infty \)	\( \infty \)	\( 1/K_a \)

4.4 Sensitivity and Disturbance Rejection

Two of the most important benefits of feedback — sensitivity reduction and disturbance rejection — can be quantified precisely.

Sensitivity to plant gain variations. Define the sensitivity function

\[ S(s) = \frac{1}{1 + C(s)G(s)}. \]

The fractional change in the closed-loop transfer function \( T \) relative to the fractional change in the plant \( G \) is

\[ S_G^T = \frac{\partial T/T}{\partial G/G} = S(s) = \frac{1}{1 + CG}. \]

For large loop gain \( |CG| \gg 1 \), \( |S| \ll 1 \): feedback dramatically reduces the sensitivity to plant variation — the core argument for using feedback.

Disturbance rejection. If a disturbance \( D(s) \) enters at the plant input, the output response to the disturbance alone is

\[ Y_D(s) = \frac{G(s)}{1 + C(s)G(s)}\,D(s) = S(s)\,G(s)\,D(s). \]

Again, large loop gain suppresses disturbance response.

---

Chapter 5: State-Space Representation

5.1 State Variables and the State-Space Model

Transfer functions describe systems from an input-output perspective and are sufficient for many classical design problems. However, they conceal the internal dynamics of a system, cannot directly represent multi-input multi-output (MIMO) plants, and do not immediately reveal properties such as controllability and observability that are critical for modern control. The state-space representation addresses all of these limitations.

The state \( \mathbf{x}(t) \) encodes all past history of the system needed to determine future outputs given future inputs. For a mechanical system the natural state variables are positions and velocities; for electrical circuits they are capacitor voltages and inductor currents.

5.2 Conversion Between Transfer Function and State Space

5.2.1 Controller Canonical Form

Given the transfer function

\[ G(s) = \frac{b_{n-1}s^{n-1} + \cdots + b_0}{s^n + a_{n-1}s^{n-1} + \cdots + a_0}, \]

the controller canonical form (companion form) is:

\[ A = \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & & & \ddots & \vdots \\ -a_0 & -a_1 & -a_2 & \cdots & -a_{n-1} \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 1 \end{bmatrix}, \]\[ C = \begin{bmatrix} b_0 & b_1 & \cdots & b_{n-1} \end{bmatrix}, \quad D = 0. \]

This representation is particularly useful for pole-placement design because the characteristic polynomial coefficients appear explicitly in the last row of \( A \).

5.3 Controllability and Observability

These two concepts — introduced by Rudolf Kalman in 1960 — determine whether a state-space model is internally adequate for control and estimation.

5.4 Solution of the State Equation

The solution of \( \dot{\mathbf{x}} = A\mathbf{x} + B\mathbf{u} \) with initial condition \( \mathbf{x}(0) \) is

\[ \mathbf{x}(t) = e^{At}\mathbf{x}(0) + \int_0^t e^{A(t-\tau)} B\mathbf{u}(\tau)\,d\tau, \]

where the matrix exponential \( e^{At} = I + At + \frac{(At)^2}{2!} + \cdots \) is the state transition matrix. The eigenvalues of \( A \) determine the natural frequencies of the state trajectories, mirroring the pole locations in the transfer-function framework.

---

Chapter 6: PID Control and Ziegler-Nichols Tuning

6.1 The PID Controller

The Proportional-Integral-Derivative (PID) controller is the workhorse of industrial process control. It is estimated that more than 90% of all control loops in manufacturing use PID. Its enduring popularity stems from its intuitive structure, ease of tuning, and adequate performance across a vast range of plants.

Alternative parameterizations use the integral time \( T_i = K_p/K_i \) and derivative time \( T_d = K_d/K_p \):

\[ C(s) = K_p\left(1 + \frac{1}{T_i s} + T_d s\right). \]

Proportional action reduces steady-state error but cannot eliminate it (except for Type 1+ plants). Increasing \( K_p \) speeds response but reduces stability margins.

Integral action adds a pole at the origin, raising system type by one and eliminating steady-state error to step inputs. The penalty is a tendency toward overshoot and oscillation — integral windup occurs when the integrator saturates while the plant is in saturation.

Derivative action provides phase lead, anticipating the error trend and damping oscillations. It amplifies high-frequency noise, which is why practical implementations use a filtered derivative: \( K_d s / (\tau_f s + 1) \).

6.2 Effects of Each Term on Response

Action	Rise time	Overshoot	Settling time	Steady-state error
Increase \( K_p \)	Decreases	Increases	Slight change	Decreases
Increase \( K_i \)	Decreases	Increases	Increases	Eliminated
Increase \( K_d \)	Slight change	Decreases	Decreases	No effect

6.3 Ziegler-Nichols Tuning Rules

Two classical tuning methods are due to Ziegler and Nichols (1942), based on characterising the plant by a small number of parameters obtained from a simple experiment.

6.3.1 Step-Response Method (Reaction Curve)

Apply a step input to the open-loop plant and fit a first-order plus dead-time (FOPDT) model:

\[ G(s) \approx \frac{K_{\text{dc}}}{Ts + 1}\,e^{-Ls}, \]

where \( L \) is the apparent dead time and \( T \) is the time constant obtained by drawing a tangent to the inflection point of the step response.

Ziegler-Nichols step-response tuning formulas:

Controller	\( K_p \)	\( T_i \)	\( T_d \)
P	\( T/(K_{\text{dc}}L) \)	—	—
PI	\( 0.9\,T/(K_{\text{dc}}L) \)	\( 3.33\,L \)	—
PID	\( 1.2\,T/(K_{\text{dc}}L) \)	\( 2L \)	\( 0.5L \)

6.3.2 Ultimate Gain Method

Close the loop with proportional control only and increase \( K_p \) until sustained oscillations occur. The value of gain at that point is the ultimate gain \( K_u \) and the period of oscillation is the ultimate period \( P_u = 2\pi/\omega_u \).

Controller	\( K_p \)	\( T_i \)	\( T_d \)
P	\( 0.5\,K_u \)	—	—
PI	\( 0.45\,K_u \)	\( 0.83\,P_u \)	—
PID	\( 0.6\,K_u \)	\( 0.5\,P_u \)	\( 0.125\,P_u \)

---

Chapter 7: Root-Locus Analysis and Design

7.1 Concept and Definition

The root-locus method, introduced by W.R. Evans in 1948, provides a graphical picture of how the closed-loop poles of a system move in the complex plane as a parameter (typically the loop gain \( K \)) varies from zero to infinity. This geometric insight is invaluable for design: instead of solving the characteristic equation for each value of \( K \), we sketch the entire trajectory in one operation.

For the standard closed-loop system with open-loop transfer function \( KG(s)H(s) \), the closed-loop characteristic equation is

\[ 1 + KG(s)H(s) = 0 \quad \Rightarrow \quad G(s)H(s) = -\frac{1}{K}. \]

A point \( s \) lies on the root locus (for \( K \geq 0 \)) if and only if:

\[ \angle G(s)H(s) = \pm 180°(2k+1), \quad k = 0, 1, 2, \ldots \quad \text{(angle condition)}, \]\[ K = \frac{1}{|G(s)H(s)|} \quad \text{(magnitude condition)}. \]

7.2 Rules for Constructing the Root Locus

7.3 Controller Design Using Root Locus

Given a desired dominant closed-loop pole location \( s_d \), we need a controller whose presence shifts the root locus so that it passes through \( s_d \).

Gain adjustment alone: If \( s_d \) already lies on the uncompensated locus, simply find the gain from the magnitude condition:

\[ K = \frac{1}{|G(s_d)|}. \]

Phase addition via a lead compensator: If \( s_d \) is not on the uncompensated locus, the angle deficiency \( \phi_{\text{def}} \) must be supplied by the compensator. A lead compensator \( C(s) = (s+z_c)/(s+p_c) \) with \( p_c > z_c \) adds positive phase; placement of its zero and pole is chosen to provide exactly \( \phi_{\text{def}} \) at \( s_d \).

---

Chapter 8: Frequency-Domain Analysis — Bode and Nyquist

8.1 Frequency Response

For a stable LTI system with transfer function \( G(s) \), the frequency response \( G(j\omega) \) describes how the system responds to sinusoidal inputs \( u(t) = U_0\sin(\omega t) \). The steady-state output is

\[ y_{ss}(t) = U_0|G(j\omega)|\sin\!\left(\omega t + \angle G(j\omega)\right). \]

Thus \( |G(j\omega)| \) is the gain and \( \angle G(j\omega) \) is the phase shift at each frequency \( \omega \). Plotting these versus frequency is the Bode diagram.

8.2 Bode Diagrams

A Bode diagram consists of two plots on a logarithmic frequency axis:

1. Magnitude plot: \( 20\log_{10}|G(j\omega)| \) in decibels (dB) vs. \( \log_{10}(\omega) \).
2. Phase plot: \( \angle G(j\omega) \) in degrees vs. \( \log_{10}(\omega) \).

The power of the Bode diagram lies in the fact that a transfer function expressed as a product of elementary factors has a magnitude plot that is simply the sum of the individual magnitude plots (because \( 20\log_{10}|G_1 G_2| = 20\log_{10}|G_1| + 20\log_{10}|G_2| \)).

8.2.1 Elementary Bode Factors

Constant gain \( K \): Horizontal line at \( 20\log_{10}K \) dB; zero phase.

Integrator \( 1/s \): Slope \( -20 \) dB/decade passing through 0 dB at \( \omega = 1 \) rad/s; phase \( = -90° \).

Differentiator \( s \): Slope \( +20 \) dB/decade; phase \( = +90° \).

First-order factor \( 1/(1 + s/\omega_b) \): Asymptotic magnitude is 0 dB for \( \omega \ll \omega_b \) and \( -20 \) dB/decade for \( \omega \gg \omega_b \), with corner at \( \omega = \omega_b \) (actual: \( -3 \) dB at corner). Phase transitions from \( 0° \) to \( -90° \) over two decades centred at \( \omega_b \): \( \phi = -\arctan(\omega/\omega_b) \).

Second-order factor \( \omega_n^2/(s^2 + 2\zeta\omega_n s + \omega_n^2) \): For \( \omega \ll \omega_n \) the magnitude is 0 dB; for \( \omega \gg \omega_n \) the slope is \( -40 \) dB/decade. At \( \omega = \omega_n \) there is a resonance peak of \( 1/(2\zeta) \) (in linear scale) for small \( \zeta \). Phase goes from \( 0° \) to \( -180° \).

8.3 Gain and Phase Margins

Typical design targets for robust control: GM \( \geq 6 \) dB and PM \( \geq 45° \). These margins are not merely mathematical conveniences — they are engineering safety factors that account for unmodelled dynamics, parameter drift, and nonlinearities.

8.4 Resonant Peak and Bandwidth

For a second-order system the resonant frequency is

\[ \omega_r = \omega_n\sqrt{1 - 2\zeta^2}, \quad \zeta < \frac{1}{\sqrt{2}} \approx 0.707, \]

and the resonant peak magnification (peak value of \( |T(j\omega)| \)) is

\[ M_r = \frac{1}{2\zeta\sqrt{1-\zeta^2}}. \]

The bandwidth \( \omega_B \) (where \( |T(j\omega_B)| = 1/\sqrt{2} \) of the DC gain) is

\[ \omega_B = \omega_n\sqrt{1 - 2\zeta^2 + \sqrt{4\zeta^4 - 4\zeta^2 + 2}}. \]

For a first-order system with time constant \( \tau \), the bandwidth is simply \( \omega_B = 1/\tau \).

8.5 Polar Plots and the Nyquist Criterion

A polar plot (Nyquist plot) is the locus of \( G(j\omega)H(j\omega) \) in the complex plane as \( \omega \) varies from \( -\infty \) to \( +\infty \). Because real systems have conjugate-symmetric frequency responses, the full plot is the reflection of the \( \omega > 0 \) portion about the real axis.

8.6 Bode Stability Criterion

For non-minimum-phase plants (which have RHP zeros, e.g., from right-half-plane transmission zeros or time delays) the Bode criterion must be applied with care — the simple encirclement count may differ, and one must use the full Nyquist criterion.

---

Chapter 9: Phase-Lead and Phase-Lag Controller Design

9.1 Frequency-Domain Design Framework

In the frequency domain, controller design amounts to reshaping the Bode diagram of the open-loop transfer function \( L(j\omega) = C(j\omega)G(j\omega) \) so that the compensated system meets specifications on gain crossover frequency \( \omega_{gc} \) (related to bandwidth and speed), phase margin PM (related to damping), and DC gain (related to steady-state error).

Two fundamental compensator types accomplish this shaping: phase-lead (adds phase near the crossover frequency, improving PM) and phase-lag (boosts DC gain without significantly affecting high-frequency behaviour, improving steady-state error).

9.2 Phase-Lead Compensator

Design procedure for phase-lead:

1. From the specifications, determine the required PM and the desired gain crossover frequency \( \omega_{gc}^* \).
2. Evaluate the uncompensated phase margin \( PM_{\text{uncomp}} = 180° + \angle G(j\omega_{gc}^*) \).
3. The phase to be added is \( \phi_{\text{add}} = PM_{\text{desired}} - PM_{\text{uncomp}} + \epsilon \) (where \( \epsilon \approx 5°\text{–}10° \) accounts for the shift in crossover frequency).
4. Compute \( \alpha = (1 - \sin\phi_{\text{add}})/(1 + \sin\phi_{\text{add}}) \).
5. Set \( \omega_{\max} = \omega_{gc}^* \). Then \( T = 1/(\omega_{gc}^*\sqrt{\alpha}) \).
6. Adjust \( K_c \) so that \( |C_{\text{lead}}(j\omega_{gc}^*)G(j\omega_{gc}^*)| = 1 \). Since the lead compensator adds magnitude \( 1/\sqrt{\alpha} \) at \( \omega_{\max} \), we typically set \( K_c = 1/\alpha \) (to keep the crossover frequency unchanged) and then fine-tune.

9.3 Phase-Lag Compensator

Design procedure for phase-lag:

1. Find the gain crossover frequency \( \omega_{gc}^* \) such that \( \angle G(j\omega_{gc}^*) = -180° + PM_{\text{desired}} + 5°\text{–}12° \) (the extra angle is a buffer against the phase lag added by the compensator at crossover).
2. Set the high-frequency attenuation: \( \beta = |G(j\omega_{gc}^*)| \) (so that compensated gain crossover occurs at \( \omega_{gc}^* \)).
3. Place the lag zero one decade below crossover: \( 1/T = \omega_{gc}^*/10 \).
4. The lag pole is then at \( 1/(\beta T) \).

Phase-lag design is preferred when the uncompensated system already has adequate phase margin at a sufficiently low frequency, and the only problem is that the gain is too high (causing the crossover to occur where there is insufficient phase). Lag compensation does not add phase; it repositions the crossover to a frequency with better phase.

9.4 Comparison of Lead and Lag Compensation

Property	Phase-Lead	Phase-Lag
Primary effect	Adds phase at crossover	Attenuates gain at high \( \omega \)
Bandwidth	Increases	Decreases
Transient response	Faster, better damping	Slower
Steady-state error	Slight improvement	Significant improvement
Noise sensitivity	Higher (boosts high frequencies)	Lower
Typical use	Speed up slow system	Fix steady-state error

---

Chapter 10: Laboratory Methods — System Identification and Controller Validation

10.1 The Role of the Laboratory in SYDE 352

The five laboratories in SYDE 352L form an integrated sequence that mirrors the complete control engineering design cycle: model the plant, design a controller using the model, implement the controller, and validate its performance against specifications. Two physical plants are used: a rotating disc (Labs 1–2) and an inverted pendulum (Labs 3–4), preceded by a MATLAB/Simulink orientation (Lab 0).

This sequence is not merely an illustration of lecture material — it is where the engineering judgment that transforms mathematical techniques into working solutions is developed. Discrepancies between the model prediction and the physical response are not failures; they are opportunities to understand model fidelity, sensor noise, actuator nonlinearity, and computational constraints.

10.2 Lab 0: MATLAB and Simulink Orientation

The orientation acquaints students with the MATLAB Control System Toolbox and Simulink. Key functions encountered throughout the labs include:

• tf(num, den) — create a transfer function model.
• step(sys), impulse(sys) — time-domain response plots.
• bode(sys) — Bode diagram.
• nyquist(sys) — Nyquist plot.
• rlocus(sys) — root locus.
• margin(sys) — gain and phase margins.
• pole(sys), zero(sys) — poles and zeros.
• feedback(G, H) — close a feedback loop.
• pid(Kp, Ki, Kd) — create a PID controller object.
• sisotool or controlSystemDesigner — interactive design GUI.

In Simulink, students build block diagrams of the feedback loop, add saturation blocks to model actuator limits, and use the Real-Time Workshop interface to download controllers to the lab hardware.

10.3 Lab 1: System Identification — Rotating Disc Plant

10.3.1 The Plant Model

The rotating disc is a motor-driven inertial disc with a tachometer providing angular velocity feedback. Neglecting coulomb friction, the governing equation is

\[ J\ddot{\theta} + b\dot{\theta} = K_m u(t), \]

where \( K_m \) is the motor gain constant. Taking the angular velocity \( \omega_{\text{disc}} = \dot{\theta} \) as output and the input voltage \( u \) as input:

\[ G_{\text{disc}}(s) = \frac{\Omega(s)}{U(s)} = \frac{K_m/J}{s + b/J} = \frac{K}{s + a}, \]

a first-order plant with DC gain \( K/a = K_m/b \) and time constant \( \tau = J/b \).

10.3.2 Step-Response Identification

Apply a known step voltage \( u_0 \) to the motor and record \( \omega(t) \). The response is

\[ \omega(t) = \frac{K_m u_0}{b}\left(1 - e^{-(b/J)t}\right). \]

From the data:

• DC gain: \( K/a = \omega_{\text{final}}/u_0 \).
• Time constant: \( \tau = J/b \) is the time to reach \( 63.2\% \) of final value.

A more robust estimate uses least-squares fitting: parametrise the model output as \( \hat{\omega}(t; K, a) = (K/a)(1 - e^{-at}) \cdot u_0 \) and minimise \( \sum_k (\omega_{\text{meas}}(t_k) - \hat{\omega}(t_k))^2 \) over \( K \) and \( a \).

10.3.3 Frequency-Domain Identification

Alternatively, apply a chirp signal (a sinusoid whose frequency sweeps from low to high) and estimate \( |G(j\omega)| \) and \( \angle G(j\omega) \) at each frequency. The MATLAB command frd creates a frequency response data object, and tfest or invfreqs can fit a rational transfer function to it. This approach captures resonances and non-minimum-phase behaviour that a step-response fit may miss.

10.4 Lab 2: PID Controller Design for the Rotating Disc

Having identified the transfer function \( G_{\text{disc}}(s) = K/(s + a) \), the closed-loop with unity feedback and PID controller is

\[ T(s) = \frac{C_{\text{PID}}(s) G_{\text{disc}}(s)}{1 + C_{\text{PID}}(s) G_{\text{disc}}(s)}. \]

10.4.1 Design Workflow

1. Specify requirements: e.g., rise time \( t_r < 0.3 \) s, overshoot \( < 10\% \), zero steady-state error.
2. Translate to frequency domain: PM target \( \approx 50°\text{–}60° \), bandwidth \( \omega_B \approx 1.8/t_r \approx 6 \) rad/s.
3. Initial PID tuning: Use Ziegler-Nichols (Section 6.3) for a starting point.
4. Refinement in MATLAB: Use pidtune(G, 'PID') or the Control System Designer to iterate toward specifications.
5. Robustness check: Verify GM and PM in Bode; check pole locations in root locus.
6. Simulink validation: Simulate the closed-loop step response; add actuator saturation to check for integrator windup and enable anti-windup if needed.
7. Hardware implementation: Download controller to the lab interface; collect step-response data; compare with simulation.

10.4.2 Anti-Windup

When the actuator saturates (voltage limited), the integrator continues to accumulate error, causing large overshoot on recovery. Anti-windup schemes (back-calculation or conditional integration) stop the integrator from winding up beyond the actuator limits:

\[ u = \text{sat}(u_{\text{PID}}) + \frac{1}{T_t}(u_{\text{sat}} - u_{\text{PID}}), \]

where \( T_t \) is the tracking time constant and \( u_{\text{sat}} = \text{sat}(u_{\text{PID}}) \).

10.5 Lab 3: System Identification — Inverted Pendulum

10.5.1 Physics of the Inverted Pendulum

The inverted pendulum is inherently unstable: an upright rigid rod of mass \( m \), length \( L \), hinged at its base, with a cart or motor providing a horizontal force \( u \). Linearizing about the upright equilibrium (\( \theta \approx 0 \)):

\[ \ddot{\theta} = \frac{g}{l}\,\theta + \frac{1}{ml^2}\,u, \]

where \( l = L/2 \) is the distance to the centre of mass. This is an unstable second-order system with a pole in the right half-plane at \( s = +\sqrt{g/l} \).

The transfer function (angle output, force input) is

\[ G_{\text{pend}}(s) = \frac{1/(ml^2)}{s^2 - g/l} = \frac{K_p}{s^2 - \omega_p^2}, \]

where \( \omega_p = \sqrt{g/l} \approx \sqrt{9.81/0.25} \approx 6.26 \) rad/s for \( l = 0.25 \) m.

10.5.2 Identification Strategy

Because the pendulum is open-loop unstable it cannot be subjected to a step input without a stabilizing controller already active. The identification is therefore performed with a provisional stabilizing controller (e.g., a simple proportional controller designed from the nominal model), and system identification is then done on the closed-loop data using closed-loop system identification techniques (prediction error methods or correlation analysis). MATLAB’s System Identification Toolbox tfest function handles this workflow.

10.6 Lab 4: PID Controller Design for the Inverted Pendulum

Stabilizing and controlling an inverted pendulum with PID requires careful attention because:

1. The RHP pole at \( +\omega_p \) means the open-loop Nyquist plot encircles \( -1 \) when the gain is too high, leading to instability from both sides of the gain range. There is a finite gain margin — the system is stable only within a range \( K_{\min} < K < K_{\max} \).
2. Root-locus analysis is essential: the two open-loop poles at \( \pm\omega_p \) (one in RHP) must be pulled into the LHP by the PID controller. The root locus departs from the RHP pole and must be brought across the imaginary axis for stability.
3. Derivative action is almost always necessary because it provides the phase lead needed to stabilize the RHP pole — pure P or PI control generally cannot achieve stability for the inverted pendulum.

10.6.1 Root-Locus Analysis for the Unstable Plant

The characteristic equation with proportional control \( C(s) = K \) is

\[ s^2 - \omega_p^2 + K K_p = 0 \quad \Rightarrow \quad s^2 = \omega_p^2 - K K_p. \]

For \( K K_p > \omega_p^2 \), the roots are imaginary: the proportional controller can only reach the imaginary axis, not achieve damped response. Adding derivative action \( C(s) = K_p(1 + T_d s) \) gives

\[ s^2 + K K_p T_d s + (K K_p - \omega_p^2) = 0. \]

For \( K K_p > \omega_p^2 \) and \( K K_p T_d > 0 \) (both conditions easily satisfied), the roots have negative real parts — the PD controller stabilizes the pendulum and places both poles in the LHP.

Adding integral action then addresses the steady-state error. The complete PID design follows the same workflow as Lab 2 but with additional attention to the stability boundaries on all three gains.

---

Appendix: Key Formulas and Reference Tables

A.1 Standard Laplace Transform Pairs

Time function \( f(t) \)	Transform \( F(s) \)	ROC
\( \delta(t) \)	1	All \( s \)
\( u_s(t) \)	\( 1/s \)	\( \text{Re}(s) > 0 \)
\( t^n u_s(t) \)	\( n!/s^{n+1} \)	\( \text{Re}(s) > 0 \)
\( e^{-at}u_s(t) \)	\( 1/(s+a) \)	\( \text{Re}(s) > -a \)
\( t e^{-at}u_s(t) \)	\( 1/(s+a)^2 \)	\( \text{Re}(s) > -a \)
\( \sin(\omega_0 t) u_s(t) \)	\( \omega_0/(s^2+\omega_0^2) \)	\( \text{Re}(s) > 0 \)
\( \cos(\omega_0 t) u_s(t) \)	\( s/(s^2+\omega_0^2) \)	\( \text{Re}(s) > 0 \)
\( e^{-at}\sin(\omega_0 t) \)	\( \omega_0/[(s+a)^2+\omega_0^2] \)	\( \text{Re}(s) > -a \)
\( e^{-at}\cos(\omega_0 t) \)	\( (s+a)/[(s+a)^2+\omega_0^2] \)	\( \text{Re}(s) > -a \)

A.2 Second-Order System Summary

For \( T(s) = \omega_n^2/(s^2 + 2\zeta\omega_n s + \omega_n^2) \), unit step input:

\[ \%OS = 100\,e^{-\pi\zeta/\sqrt{1-\zeta^2}}, \quad t_p = \pi/\omega_d, \quad t_s \approx 4/(\zeta\omega_n), \quad \omega_d = \omega_n\sqrt{1-\zeta^2}. \]

Inverse relation: from %OS specification,

\[ \zeta = \frac{-\ln(\%OS/100)}{\sqrt{\pi^2 + \ln^2(\%OS/100)}}. \]

A.3 Routh Array Special Cases

Zero in first column: Replace the zero with a small \( \varepsilon > 0 \) and evaluate stability as \( \varepsilon \to 0^+ \). A sign change of \( \varepsilon \) indicates a root in the RHP.

Row of all zeros: An entire row of zeros indicates roots symmetrically placed about the origin (on the imaginary axis or in symmetric RHP/LHP pairs). Form the auxiliary polynomial from the row above the zero row; its roots are the symmetric root locations. Replace the zero row with the derivative of the auxiliary polynomial and continue.

A.4 Gain and Phase Margin from Bode

\[ GM\,[\text{dB}] = 0 - 20\log_{10}|L(j\omega_{pc})|, \quad PM = 180° + \angle L(j\omega_{gc}). \]

Stable minimum-phase systems: GM \( > 0 \) dB and PM \( > 0° \).

A.5 State-Space Quick Reference

Property	Test	Tool
Stability	All eigenvalues of \( A \) in open LHP	\( \text{eig}(A) \)
Controllability	\( \text{rank}(\mathcal{C}) = n \)	\( \text{rank}([B,AB,\ldots]) \)
Observability	\( \text{rank}(\mathcal{O}) = n \)	\( \text{rank}([C;CA;\ldots]) \)
Transfer function	\( C(sI - A)^{-1}B + D \)	ss2tf in MATLAB

A.6 PID Ziegler-Nichols Quick Reference

Step-response method (parameters: \( K_{\text{dc}}, T, L \)):

\[ K_p^{\text{PID}} = \frac{1.2T}{K_{\text{dc}}L}, \quad T_i = 2L, \quad T_d = 0.5L. \]

Ultimate gain method (parameters: \( K_u, P_u \)):

\[ K_p^{\text{PID}} = 0.6K_u, \quad T_i = 0.5P_u, \quad T_d = 0.125P_u. \]

A.7 Root-Locus Angle and Magnitude Conditions

For \( s_d \) on the root locus:

\[ \sum_j \angle(s_d - z_j) - \sum_i \angle(s_d - p_i) = \pm 180°(2k+1), \]\[ K = \frac{\prod_i |s_d - p_i|}{\prod_j |s_d - z_j|}. \]

A.8 Phase-Lead Design Equations

\[ \phi_{\max} = \arcsin\!\left(\frac{1-\alpha}{1+\alpha}\right), \quad \alpha = \frac{1 - \sin\phi_{\max}}{1 + \sin\phi_{\max}}, \quad \omega_{\max} = \frac{1}{T\sqrt{\alpha}}. \]

Magnitude at \( \omega_{\max} \): \( |C_{\text{lead}}(j\omega_{\max})| = 1/\sqrt{\alpha} \).

---

Additional Worked Examples