Sources and References

Primary reference — Chris Nielsen, ECE 380 Course Notes, University of Waterloo (public course notes).

Supplementary — Richard C. Dorf & Robert H. Bishop, Modern Control Systems, 14th ed., Pearson, 2022.

Online resources — MIT OCW 6.302 (Feedback System Design); Karl J. Åström & Richard M. Murray, Feedback Systems: An Introduction for Scientists and Engineers, 2nd ed., Princeton University Press (open-access at fbsbook.org); Katsuhiko Ogata, Modern Control Engineering, 5th ed., Prentice Hall; Gene F. Franklin, J. David Powell & Abbas Emami-Naeini, Feedback Control of Dynamic Systems, 8th ed., Pearson.

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Chapter 1: Introduction to Feedback Control

1.1 Why Control?

Every engineered system exists in a world that resists our intentions. A car’s engine must maintain a desired speed despite hills, wind, and varying payloads. A chemical reactor must hold a precise temperature despite fluctuating feed concentrations. A quadrotor drone must stay level despite gusts that knock it sideways. The discipline of control engineering is concerned with designing systems — usually involving sensors, actuators, and a computational element — that force physical plants to behave in ways they would not naturally behave on their own.

The word “control” in everyday speech means many things, but in engineering it refers to a specific technical activity: choosing inputs to a dynamical system so that its outputs track a desired reference signal, reject disturbances, and remain stable. Control theory provides the mathematical tools to analyse whether a given design achieves these goals and to synthesise new designs when it does not.

1.2 Open-Loop vs. Closed-Loop Control

The most fundamental architectural choice in control system design is whether to use feedback.

1.2.1 Open-Loop Control

In an open-loop (feedforward) architecture, the controller computes its output solely from the reference input, without measuring the actual plant output. A simple example is a toaster: the user sets a timer, and the heating element runs for that duration regardless of whether the bread is actually toasted. Another example is a stepper motor driven by a fixed pulse sequence in a 3D printer — there is no encoder measuring actual position.

The strength of open-loop control is its simplicity. The weakness is that it cannot correct for disturbances, model uncertainty, or parameter variations. If the plant changes (the bread is thicker, the motor skips a step), the open-loop controller has no way to know.

1.2.2 Closed-Loop (Feedback) Control

In a closed-loop architecture, the controller measures the plant output, compares it to the desired reference, forms an error signal, and computes a corrective input. This is feedback: information flows from the plant output back to the controller input.

The canonical feedback loop is:

\[ e(t) = r(t) - y(t), \qquad u(t) = C(e(t)), \qquad y(t) = P(u(t)) \]

where \( r(t) \) is the reference (setpoint), \( y(t) \) is the measured output, \( e(t) \) is the tracking error, \( u(t) \) is the control input (actuator signal), \( C \) denotes the controller, and \( P \) denotes the plant.

1.2.3 Advantages of Feedback

Feedback confers four principal advantages that open-loop control cannot provide:

1. Disturbance rejection. If an external disturbance pushes the output away from the reference, the feedback loop detects the resulting error and applies a corrective input.

2. Robustness to model uncertainty. Even if the plant’s parameters are poorly known or slowly time-varying, the closed-loop system may still perform acceptably because the controller continuously corrects for deviations.

3. Stabilisation of unstable plants. Some plants are inherently unstable — an inverted pendulum, for example, will fall without active control. Feedback can stabilise such systems.

4. Improved transient response. Feedback can be designed to make the output respond more quickly or with less overshoot than the plant would exhibit naturally.

These advantages come at a cost: feedback can also destabilise a system if poorly designed, and sensors introduce noise into the loop. The art of control engineering lies in capturing the benefits while managing the costs.

1.3 Motivating Examples

1.3.1 Cruise Control

A vehicle’s cruise control system measures vehicle speed \( v \) via a wheel-speed sensor, compares it to the driver’s desired speed \( v_d \), and commands the throttle (and sometimes the brakes) accordingly. The plant is the vehicle’s longitudinal dynamics, which includes engine torque, aerodynamic drag, and road slope. A simple model is:

\[ m\dot{v} = F_{\text{engine}}(u) - F_{\text{drag}}(v) - mg\sin\theta \]

where \( m \) is vehicle mass, \( u \) is the throttle command, and \( \theta \) is the road angle. Cruise control must reject disturbances (hills, headwinds) and handle model uncertainty (passenger load, tyre pressure). A purely open-loop system would require knowing \( \theta \) in advance and modelling the engine perfectly — impractical. A feedback controller solves both problems naturally.

1.3.2 Thermostat

A home thermostat is perhaps the oldest and most ubiquitous feedback controller. It measures room temperature \( T \), compares it to the setpoint \( T_{\text{set}} \), and switches the furnace or air conditioner on or off. This is an example of bang-bang (on/off) control — a nonlinear control law — rather than proportional control. The result is that room temperature oscillates around the setpoint (hysteresis), but the average tracking error is small.

1.3.3 Autonomous Driving

A self-driving vehicle must simultaneously control lateral position (steering), longitudinal velocity (throttle/brakes), and heading. Each of these is a closed-loop control problem. For lane keeping, the controller measures the vehicle’s lateral deviation from the lane centre using cameras or lidar, and steers to reduce that deviation. The difficulties are that the plant dynamics (tyre–road interaction) are nonlinear, the environment changes rapidly, and the system must operate safely across a vast range of conditions. Modern autonomous driving systems use a hierarchy of control loops operating at different timescales.

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Chapter 2: Mathematical Modelling of Dynamical Systems

2.1 Nonlinear State-Space Models

Physical systems are typically described by ordinary differential equations (ODEs) derived from first principles — Newton’s laws, Kirchhoff’s laws, conservation of energy, and so forth. The standard form for an \( n \)-dimensional dynamical system is:

\[ \dot{\mathbf{x}}(t) = \mathbf{f}(\mathbf{x}(t),\, u(t)) \]\[ y(t) = \mathbf{h}(\mathbf{x}(t),\, u(t)) \]

where \( \mathbf{x}(t) \in \mathbb{R}^n \) is the state vector, \( u(t) \in \mathbb{R} \) is the scalar control input, \( y(t) \in \mathbb{R} \) is the scalar output, \( \mathbf{f}: \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}^n \) is the state dynamics function, and \( \mathbf{h}: \mathbb{R}^n \times \mathbb{R} \to \mathbb{R} \) is the output map.

2.2 Equilibrium Points and Linearisation

Most practical control systems operate near a desired operating condition. Classical control theory is built on the assumption of linearity, so the standard approach is to linearise the nonlinear model around an equilibrium point.

2.2.1 Linearisation via Taylor Expansion

2.3 Linear Time-Invariant (LTI) Systems

A linear time-invariant system in state-space form is:

\[ \dot{\mathbf{x}}(t) = A\mathbf{x}(t) + Bu(t) \]\[ y(t) = C\mathbf{x}(t) + Du(t) \]

where \( A \in \mathbb{R}^{n\times n} \), \( B \in \mathbb{R}^{n\times 1} \), \( C \in \mathbb{R}^{1\times n} \), \( D \in \mathbb{R} \). The solution to this ODE is given by the variation-of-parameters formula:

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

where \( e^{At} = \sum_{k=0}^\infty \frac{(At)^k}{k!} \) is the matrix exponential.

2.4 Laplace Transform Review

The Laplace transform converts differential equations into algebraic equations, which are far easier to manipulate. For a signal \( f(t) \) defined for \( t \geq 0 \):

\[ F(s) = \mathcal{L}\{f(t)\} = \int_0^\infty f(t)\,e^{-st}\,dt \]

Key transform pairs used throughout this course:

Time-domain signal	Laplace transform
\( \delta(t) \) (impulse)	\( 1 \)
\( 1(t) \) (unit step)	\( 1/s \)
\( e^{at}1(t) \)	\( 1/(s-a) \)
\( \sin(\omega t)1(t) \)	\( \omega/(s^2+\omega^2) \)
\( \cos(\omega t)1(t) \)	\( s/(s^2+\omega^2) \)
\( t^n 1(t) \)	\( n!/s^{n+1} \)

Key properties: linearity, differentiation (\( \mathcal{L}\{\dot{f}\} = sF(s) - f(0^-) \)), integration (\( \mathcal{L}\{\int_0^t f\} = F(s)/s \)), and convolution (\( \mathcal{L}\{f * g\} = F(s)G(s) \)).

2.5 Transfer Functions

The transfer function entirely characterises the input-output behaviour of an LTI system. Different state-space realisations can produce the same transfer function.

2.6 Block-Diagram Algebra

Block diagrams are graphical representations of LTI systems where blocks represent transfer functions and arrows represent signal flow. Three fundamental interconnections are:

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

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

Negative feedback loop: With plant \( P(s) \) in the forward path and controller \( C(s) \) in the feedback path, the closed-loop transfer function from reference \( R \) to output \( Y \) is:

\[ T(s) = \frac{C(s)P(s)}{1 + C(s)P(s)} \]

This is the most important formula in classical control. The product \( L(s) = C(s)P(s) \) is called the loop transfer function or loop gain.

2.7 Signal Flow Graphs and Mason’s Gain Rule

A signal flow graph (SFG) is an alternative to block diagrams. Nodes represent signals and directed branches represent gains. Mason’s gain formula provides a systematic method to compute the transfer function of an SFG directly.

Mason’s rule is particularly useful for multi-loop systems where repeated block-diagram reduction would be tedious.

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Chapter 3: Linear System Theory and Frequency Response

3.1 Response of LTI Systems

Given an LTI system with transfer function \( G(s) \) and input \( u(t) \), the output in the Laplace domain is \( Y(s) = G(s)U(s) \). In the time domain, this corresponds to the convolution:

\[ y(t) = (g * u)(t) = \int_0^t g(t - \tau)\,u(\tau)\,d\tau \]

where \( g(t) = \mathcal{L}^{-1}\{G(s)\} \) is the impulse response of the system.

The impulse response \( g(t) \) uniquely characterises the LTI system’s behaviour: knowledge of \( g(t) \) for all \( t \geq 0 \) is equivalent to knowledge of \( G(s) \).

3.2 Stability

3.3 Frequency Response

A remarkable property of stable LTI systems is their response to sinusoidal inputs. If the input is \( u(t) = A\sin(\omega t) \) and the system is asymptotically stable, then the steady-state output is:

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

That is, the output is also sinusoidal at the same frequency \( \omega \), with amplitude scaled by \( |G(j\omega)| \) and phase shifted by \( \angle G(j\omega) \). The function \( G(j\omega) \) obtained by substituting \( s = j\omega \) into the transfer function is the frequency response.

3.4 Bode Plots

The Bode plot is a two-panel graphical representation of the frequency response:

• Magnitude plot: \( 20\log_{10}|G(j\omega)| \) in decibels (dB) vs. \( \log_{10}\omega \) (log-log scale)
• Phase plot: \( \angle G(j\omega) \) in degrees vs. \( \log_{10}\omega \) (log-linear scale)

Using logarithms turns multiplication into addition, making it easy to construct the Bode plot of a product of transfer functions by summing the individual Bode plots.

3.4.1 Bode Plot Rules for Standard Factors

Every rational transfer function can be factored into a product of four types of elementary terms. Their asymptotic Bode plots:

Constant gain \( K \): Magnitude is \( 20\log|K| \) dB (flat line); phase is \( 0° \) if \( K > 0 \) or \( -180° \) if \( K < 0 \).

Integrator \( 1/s \) (or differentiator \( s \)): Magnitude slope is \( -20 \) dB/decade (or \( +20 \) dB/decade); phase is \( -90° \) (or \( +90° \)), constant.

Real pole at \( s = -a \), i.e., \( 1/(1 + s/a) \): For \( \omega \ll a \): magnitude ≈ 0 dB, phase ≈ 0°. For \( \omega \gg a \): magnitude slope \( -20 \) dB/decade, phase ≈ \( -90° \). Transition at \( \omega = a \) (break frequency).

Complex conjugate poles at \( s = -\zeta\omega_n \pm j\omega_n\sqrt{1-\zeta^2} \): The standard second-order factor is \( \omega_n^2/(s^2 + 2\zeta\omega_n s + \omega_n^2) \). For \( \omega \ll \omega_n \): 0 dB, 0°. For \( \omega \gg \omega_n \): slope \( -40 \) dB/decade, phase \( -180° \). For small \( \zeta \), there is a resonance peak near \( \omega = \omega_n \) of height approximately \( 1/(2\zeta) \) (or \( -20\log(2\zeta) \) dB).

3.5 Nyquist Plots

The Nyquist plot (also called the polar plot) traces the curve \( G(j\omega) \) in the complex plane as \( \omega \) varies from \( -\infty \) to \( +\infty \). Since \( G(j(-\omega)) = \overline{G(j\omega)} \) for real-coefficient transfer functions, the plot for negative frequencies is the mirror image of the plot for positive frequencies about the real axis. In practice, we plot \( G(j\omega) \) for \( \omega \geq 0 \) and note the reflection.

Key features to identify on a Nyquist plot:

• The point \( (-1, 0) \) is special: if the loop gain \( L(j\omega) \) passes through \( -1 \), the denominator \( 1 + L(j\omega) = 0 \), indicating marginal stability of the closed-loop system.
• The distance from the curve to \( (-1, 0) \) provides a geometric measure of stability margins.

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Chapter 4: First- and Second-Order System Dynamics

4.1 First-Order Systems

A first-order LTI system has a single state and transfer function:

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

where \( K \) is the DC (static) gain and \( \tau > 0 \) is the time constant. The step response (response to input \( u(t) = 1(t) \)) is:

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

4.2 Second-Order Systems

The standard second-order transfer function is:

\[ G(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2} \]

where \( \omega_n > 0 \) is the natural frequency and \( \zeta \geq 0 \) is the damping ratio. The poles are:

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

The behaviour depends critically on \( \zeta \):

• \( \zeta > 1 \) (overdamped): Two distinct real negative poles; exponential decay, no oscillation.
• \( \zeta = 1 \) (critically damped): Repeated real pole at \( s = -\omega_n \); fastest non-oscillatory response.
• \( 0 < \zeta < 1 \) (underdamped): Complex conjugate poles \( s_{1,2} = -\sigma \pm j\omega_d \) where \( \sigma = \zeta\omega_n \) and \( \omega_d = \omega_n\sqrt{1-\zeta^2} \) is the damped natural frequency; oscillatory response.
• \( \zeta = 0 \) (undamped): Poles on the imaginary axis; sustained oscillation; marginally stable.

4.3 Step Response Characteristics

For an underdamped second-order system (\( 0 < \zeta < 1 \)), the unit step response is:

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

where \( \phi = \arccos\zeta \). The key performance specifications are:

4.4 Poles in the Complex Plane

The location of the poles in the \( s \)-plane directly encodes the step response characteristics:

• Real part \( -\sigma = -\zeta\omega_n \): determines decay rate. The further left, the faster the decay. The settling time \( T_s \approx 4/\sigma \) depends only on the real part.
• Imaginary part \( \omega_d \): determines oscillation frequency. Peak time \( T_p = \pi/\omega_d \) depends only on the imaginary part.
• Distance from origin \( \omega_n = |s_{1,2}| \): natural frequency.
• Angle from negative real axis \( \theta = \arccos\zeta \): the overshoot depends only on \( \zeta \), which corresponds to this angle. Lines of constant \( \zeta \) are radial lines; lines of constant \( \sigma \) are vertical lines; lines of constant \( \omega_d \) are horizontal lines.

4.5 Effects of Adding Poles and Zeros

4.5.1 Dominant Poles

If one pair of complex poles is much closer to the origin than the others, the distant poles decay rapidly and the long-term response is dominated by the near poles. In this case, the system can be approximated by a second-order model. This is the dominant pole approximation: valid when the ratio of real parts is at least 5:1.

4.5.2 Adding a Zero

Adding a zero \( s = -z \) to the numerator of a second-order transfer function changes the step response even if the poles are unchanged. A zero in the LHP (minimum-phase zero, \( z > 0 \)) increases overshoot and decreases rise time. A zero in the RHP (non-minimum phase zero, \( z < 0 \)) causes an initial undershoot — the response first moves in the wrong direction before recovering. This is a fundamental limitation that cannot be overcome with feedback alone.

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Chapter 5: Feedback Stability and Steady-State Performance

5.1 Closed-Loop Transfer Function

Consider the standard unity-feedback loop with plant \( P(s) \) and controller \( C(s) \). The loop transfer function is \( L(s) = C(s)P(s) \). The closed-loop transfer function from reference \( R(s) \) to output \( Y(s) \) is:

\[ T(s) = \frac{L(s)}{1 + L(s)} = \frac{C(s)P(s)}{1 + C(s)P(s)} \]

The transfer function from disturbance \( D(s) \) (entering at the plant input) to output is:

\[ G_{d}(s) = \frac{P(s)}{1 + C(s)P(s)} = \frac{P(s)}{1 + L(s)} \]

The transfer function from reference to tracking error \( E(s) = R(s) - Y(s) \) is:

\[ S(s) = \frac{1}{1 + L(s)} \]

This function \( S(s) \) is called the sensitivity function and is fundamental to understanding performance and robustness.

5.2 Characteristic Equation

5.3 Routh-Hurwitz Stability Criterion

Directly computing the roots of the characteristic polynomial is cumbersome for high-order systems. The Routh-Hurwitz criterion allows us to determine stability algebraically from the polynomial coefficients alone, without finding the roots.

5.4 Steady-State Error Analysis

5.4.1 Tracking Error

For a unity-feedback system with loop gain \( L(s) \), the steady-state error to a reference input \( R(s) \) is found by applying the Final Value Theorem to the error signal \( E(s) = S(s)R(s) = R(s)/(1+L(s)) \):

\[ e_{\text{ss}} = \lim_{t\to\infty} e(t) = \lim_{s\to 0} s \cdot \frac{R(s)}{1 + L(s)} \]

The results are summarised in the following table:

System Type	Step error	Ramp error	Parabolic error
0	\( 1/(1+K_p) \)	\( \infty \)	\( \infty \)
1	0	\( 1/K_v \)	\( \infty \)
2	0	0	\( 1/K_a \)

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Chapter 6: Root-Locus Analysis

6.1 The Root-Locus Concept

The root-locus method, introduced by W.R. Evans in 1948, provides a graphical technique for visualising how the closed-loop poles move in the \( s \)-plane as a scalar gain \( K \) is varied from 0 to \( \infty \). This is enormously useful for understanding how gain affects stability and transient response.

The characteristic equation for a unity-feedback system with loop gain \( KG(s) \) is:

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

In polar form, this requires simultaneously:

\[ |G(s)| = \frac{1}{K} \quad \text{(magnitude condition)} \]\[ \angle G(s) = (2k+1)\cdot 180°,\quad k \in \mathbb{Z} \quad \text{(angle condition)} \]

The root locus is the set of all \( s \) satisfying the angle condition for some \( K \geq 0 \).

6.2 Rules for Constructing the Root Locus

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Chapter 7: PID and Classical Controller Design

7.1 PID Control

The Proportional-Integral-Derivative (PID) controller is by far the most widely used controller in industrial practice. Surveys consistently find that over 90% of industrial control loops use some form of PID. Its popularity stems from its simplicity, tuning intuitiveness, and effectiveness across a wide range of plants.

The PID control law in continuous time is:

\[ u(t) = K_P\,e(t) + K_I\int_0^t e(\tau)\,d\tau + K_D\,\dot{e}(t) \]

In transfer function form:

\[ C(s) = K_P + \frac{K_I}{s} + K_D s = K_P\left(1 + \frac{1}{T_I s} + T_D s\right) \]

where \( T_I = K_P/K_I \) is the integral time and \( T_D = K_D/K_P \) is the derivative time.

7.2 Effects of P, I, and D Terms

7.2.1 Proportional Action

The proportional term \( K_P e(t) \) produces a control signal proportional to the current error. Increasing \( K_P \) generally:

• Reduces steady-state error (though cannot eliminate it for a step reference in a type-0 plant)
• Speeds up the response (larger gains push poles further left or faster)
• Can destabilise the system if made too large

7.2.2 Integral Action

The integral term \( K_I \int e\,dt \) accumulates past errors and continues applying control effort as long as any nonzero error persists. This guarantees zero steady-state error to step references for any stable plant (raises system type by 1). However, integral action introduces phase lag and can destabilise or slow the response. Integral windup occurs when the actuator saturates: the integrator continues accumulating a large value that is difficult to unwind, causing overshoot when the setpoint is reached. Anti-windup mechanisms (conditional integration, back-calculation) address this.

7.2.3 Derivative Action

The derivative term \( K_D \dot{e}(t) \) anticipates future error by reacting to the rate of change. It provides phase lead, which can significantly improve stability margin and reduce overshoot. The major drawback is amplification of high-frequency noise: the derivative of a sensor noise signal is large even if the noise amplitude is small. In practice, the derivative is always filtered:

\[ C_D(s) = \frac{K_D s}{1 + s/N} \]

where \( N \) is the derivative filter coefficient (typically \( 5 \leq N \leq 20 \)).

7.3 Ziegler-Nichols Tuning

The Ziegler-Nichols methods provide systematic rules for setting PID parameters from simple open-loop or closed-loop experiments.

7.3.1 Open-Loop (Step Response) Method

Apply a step input to the open-loop plant and record the step response. Identify the S-curve shape and draw a tangent line at the inflection point. Read off:

• \( L \): apparent dead time (time before the step response begins to rise)
• \( \tau \): time constant (slope parameter)

Then:

Controller	\( K_P \)	\( T_I \)	\( T_D \)
P	\( \tau/(KL) \)	\( \infty \)	0
PI	\( 0.9\tau/(KL) \)	\( 3.3L \)	0
PID	\( 1.2\tau/(KL) \)	\( 2L \)	\( 0.5L \)

7.3.2 Closed-Loop (Ultimate Gain) Method

With the controller set to proportional only, increase \( K_P \) until the closed-loop system is at the verge of instability (sustained oscillations). The gain at this point is the ultimate gain \( K_u \) and the period of oscillation is the ultimate period \( T_u \). Then:

Controller	\( K_P \)	\( T_I \)	\( T_D \)
P	\( 0.5 K_u \)	\( \infty \)	0
PI	\( 0.45 K_u \)	\( T_u/1.2 \)	0
PID	\( 0.6 K_u \)	\( T_u/2 \)	\( T_u/8 \)

7.4 Pole-Placement Design

Pole placement (also called direct design) specifies the desired closed-loop pole locations and solves for the controller that achieves them. For a PID controller acting on a second-order plant \( P(s) = \omega_n^2/(s^2 + 2\zeta\omega_n s + \omega_n^2) \), the loop transfer function is:

\[ L(s) = C_{\text{PID}}(s) \cdot P(s) = \frac{K_P s^2 + K_I s + K_I K_D/K_P}{s} \cdot \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2} \]

The closed-loop characteristic polynomial (of degree 3 or 4 depending on the plant order) is matched to a desired polynomial with specified poles. Equating coefficients yields equations for \( K_P \), \( K_I \), \( K_D \).

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Chapter 8: Frequency-Domain Stability — Nyquist and Bode

8.1 The Nyquist Stability Criterion

The Nyquist stability criterion provides a graphical method for assessing closed-loop stability from the open-loop frequency response, without needing to find closed-loop poles explicitly. It is based on the argument principle from complex analysis.

8.1.1 Application to Feedback Systems

8.2 Stability Margins

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Chapter 9: Lead-Lag Compensation and Loop Shaping

9.1 The Loop-Shaping Philosophy

Loop shaping is a systematic design methodology that works directly on the open-loop frequency response \( L(j\omega) \) to achieve desired closed-loop properties. The key insight is that closed-loop performance is encoded in the open-loop:

• Tracking bandwidth is related to the gain crossover frequency \( \omega_{gc} \): a higher \( \omega_{gc} \) means the loop has high gain at higher frequencies, enabling better tracking of faster references and rejection of lower-frequency disturbances.
• Stability margins are determined by the phase and gain of \( L \) near \( \omega_{gc} \).
• High-frequency noise rejection requires \( |L(j\omega)| \) to be small for large \( \omega \) (low gain at high frequencies).
• Low-frequency disturbance rejection and steady-state accuracy require \( |L(j\omega)| \) to be large for small \( \omega \).

These requirements pull in opposing directions: we want the loop gain to be large at low frequencies and small at high frequencies, with a clean transition near the bandwidth. The ideal loop shape is approximately:

\[ |L(j\omega)| \approx \frac{\omega_{gc}}{\omega} \]

(a single integrator slope of \( -20 \) dB/decade) over a range around \( \omega_{gc} \).

9.2 Performance Specifications in the Frequency Domain

The bandwidth is closely related to (though not identical to) the gain crossover frequency \( \omega_{gc} \). For a well-designed loop with adequate phase margin, \( \omega_B \approx \omega_{gc} \).

9.3 Lead Compensator Design

A lead compensator adds positive phase (phase advance) near the gain crossover frequency, improving phase margin and allowing for higher gain crossover frequency (faster response).

9.3.1 Lead Compensator Design Procedure

1. Determine the gain crossover frequency \( \omega_{gc,0} \) and phase margin \( PM_0 \) of the uncompensated loop \( L_0(s) = K P(s) \).
2. Determine the additional phase needed: \( \phi_{\text{need}} = PM_{\text{desired}} - PM_0 + \epsilon \) (add 5–10° to account for the phase lost from the shift in \( \omega_{gc} \)).
3. Compute \( \alpha \) from: \( \sin\phi_{\text{need}} = (1-\alpha)/(1+\alpha) \Rightarrow \alpha = (1 - \sin\phi_{\text{need}})/(1 + \sin\phi_{\text{need}}) \).
4. Place \( \omega_{\text{max}} \) at the new desired gain crossover frequency \( \omega_{gc}^* \). At \( \omega_{gc}^* \), the uncompensated loop gain must equal \( 1/\sqrt{\alpha} \) (to ensure \( |L(\omega_{gc}^*)| = 1 \) after compensation):

\[ |L_0(j\omega_{gc}^*)| = \frac{1}{\sqrt{\alpha}} \Rightarrow \omega_{gc}^* = |L_0^{-1}(1/\sqrt{\alpha})| \]

5. Set \( z = \omega_{gc}^*/\sqrt{\alpha^{-1}} \) and \( p = z/\alpha \).
6. Set the compensator gain \( K_c \) to place \( |C_{\text{lead}}(j\omega_{gc}^*)P(j\omega_{gc}^*)| = 1 \).

9.4 Lag Compensator Design

A lag compensator increases the DC gain without significantly affecting the phase near the crossover frequency, thereby reducing steady-state error.

9.4.1 Lag Compensator Design Procedure

The lag compensator is placed so that its zero and pole are both much lower than the gain crossover frequency (typically a decade or more lower), so they contribute negligible phase near \( \omega_{gc} \):

1. Design for the desired phase margin using only gain (proportional control): choose \( K \) so that \( PM = PM_{\text{desired}} \).
2. If the steady-state error specification is not met, determine the required additional low-frequency gain \( \beta = \) (required gain increase).
3. Place the lag zero at \( z = \omega_{gc}/10 \) and the lag pole at \( p = z/\beta \).

9.5 Lead-Lag Compensator

In many practical problems, neither lead nor lag alone is sufficient. A lead-lag compensator combines both:

\[ C_{\text{lead-lag}}(s) = K_c \frac{(s+z_1)(s+z_2)}{(s+p_1)(s+p_2)} \]

with \( p_1 < z_1 \) (lag section) and \( z_2 < p_2 \) (lead section). This provides increased low-frequency gain (lag) and increased phase margin (lead) simultaneously.

9.6 Trade-offs Between Bandwidth and Stability Margins

A fundamental limitation of feedback control is captured in the waterbed effect (Bode’s sensitivity integral): for a minimum-phase system with at least two more poles than zeros:

\[ \int_0^\infty \ln|S(j\omega)|\,d\omega = 0 \]

where \( S(j\omega) = 1/(1 + L(j\omega)) \) is the sensitivity. This integral constraint means that if the sensitivity is reduced (loop gain increased) over some frequency range, it must increase over another range. Pushing the bandwidth up to track faster references necessarily increases the sensitivity peak at higher frequencies, making the system less robust to high-frequency model uncertainty and noise.

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Chapter 10: Beyond Classical Control

10.1 Limitations of Classical Control

Classical feedback control — the material of Chapters 1–9 — is centred on single-input single-output (SISO) linear time-invariant systems with a single control loop. It provides elegant graphical tools (root locus, Bode, Nyquist) and effective controller structures (PID, lead-lag). However, it has inherent limitations:

• SISO only: Real systems often have multiple inputs and multiple outputs (MIMO). Classical methods do not extend gracefully to multi-loop systems due to cross-coupling.
• LTI assumption: Linearisation is valid only locally, near the operating point. Highly nonlinear systems or systems operating over large ranges require nonlinear control.
• Manual tuning: Classical design requires significant engineering judgment and iteration. High-dimensional systems make this infeasible.
• No formal performance optimisation: Classical design meets specifications (phase margin, bandwidth) but does not minimise any cost function.

10.2 State-Space Methods and Modern Control

Modern control theory, developed from the 1960s onward, works directly in the state-space representation and handles MIMO systems systematically.

10.2.1 State Feedback and Pole Placement

With full state access, the control law \( u = -K\mathbf{x} \) (state feedback) places the closed-loop poles at arbitrary locations (provided the system is controllable) by choosing the gain matrix \( K \in \mathbb{R}^{1\times n} \). This generalises the root-locus pole placement of Chapter 6 to MIMO systems.

10.2.2 Linear Quadratic Regulator (LQR)

Rather than prescribing pole locations, LQR minimises an infinite-horizon quadratic cost:

\[ J = \int_0^\infty \left[\mathbf{x}^T Q\,\mathbf{x} + u^T R\,u\right] dt \]

where \( Q \succeq 0 \) penalises state deviations and \( R > 0 \) penalises control effort. The optimal control law is linear: \( u^*(t) = -K_{\text{LQR}}\mathbf{x}(t) \), where the gain matrix \( K_{\text{LQR}} = R^{-1}B^T P \) and \( P \) is the solution to the algebraic Riccati equation:

\[ PA + A^T P - PBR^{-1}B^T P + Q = 0 \]

LQR provides a principled way to tune the trade-off between fast response (large \( Q \)) and small control effort (large \( R \)). Moreover, LQR-designed systems have guaranteed stability margins: gain margin \( \geq 6 \) dB and phase margin \( \geq 60° \) in the single-input case.

10.3 Model Predictive Control (MPC)

Model predictive control (MPC), also called receding-horizon control, is now the dominant advanced control strategy in chemical process control and is rapidly growing in automotive and aerospace applications.

At each time step \( t \), MPC:

1. Measures (or estimates) the current state \( \mathbf{x}(t) \).
2. Solves a finite-horizon optimal control problem over a prediction horizon \( T_p \):

\[ \min_{u(\cdot)} \int_t^{t+T_p} \left[\mathbf{x}^T Q\,\mathbf{x} + u^T R\,u\right] d\tau + \mathbf{x}(t+T_p)^T P_f \mathbf{x}(t+T_p) \]

subject to state dynamics, input constraints (\( u_{\min} \leq u \leq u_{\max} \)), and state constraints.

3. Applies only the first control input of the optimal trajectory.
4. Repeats at the next time step (receding horizon).

The key advantage of MPC is that it handles constraints explicitly and naturally — something classical control cannot do. Its main limitation is computational: the optimisation must be solved in real time at each sample period.

10.4 Nonlinear Control

10.4.1 Feedback Linearisation

For certain classes of nonlinear systems, it is possible to find a change of coordinates and a control law that exactly cancels the nonlinearity, making the closed-loop system behave like a linear system. This approach, called feedback linearisation (or exact linearisation), is conceptually distinct from the approximate linearisation of Chapter 2: it works exactly rather than approximately.

For the single-input nonlinear system \( \dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}) + g(\mathbf{x})u \), if a change of variables \( \mathbf{z} = \Phi(\mathbf{x}) \) exists such that the system in \( z \)-coordinates is linear, then a linear control law in \( z \)-coordinates gives exact tracking of arbitrary references. The difficulty is that the required change of variables does not always exist, and even when it does, cancellation of nonlinearities requires exact model knowledge.

10.4.2 Lyapunov-Based Control

Lyapunov stability theory provides a general framework for analysing and designing controllers for nonlinear systems. A function \( V(\mathbf{x}) \) is a Lyapunov function for the system \( \dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}) \) if:

1. \( V(\mathbf{0}) = 0 \) and \( V(\mathbf{x}) > 0 \) for \( \mathbf{x} \neq 0 \) (positive definite).
2. \( \dot{V}(\mathbf{x}) = \nabla V \cdot \mathbf{f}(\mathbf{x}) < 0 \) for \( \mathbf{x} \neq 0 \) (negative definite derivative along trajectories).

The existence of such a \( V \) guarantees asymptotic stability of the origin. For control design, one can choose a desired \( V \) and design \( u \) to make \( \dot{V} < 0 \) — this is called control Lyapunov function (CLF) design.

10.5 Data-Driven and Learning-Based Control

Classical control design requires a mathematical model of the plant. In many modern applications — robotics, autonomous systems, biological systems — deriving accurate first-principles models is difficult or impossible. This has motivated data-driven approaches.

10.5.1 System Identification

System identification uses input-output data to fit a parametric model (e.g., a transfer function of known order, or a state-space model) to the observed behaviour. The identified model can then be used as the plant model for classical or modern control design. Methods include least-squares, prediction error methods (PEM), and subspace identification.

10.5.2 Reinforcement Learning for Control

Reinforcement learning (RL) treats control as a sequential decision problem: an agent interacts with an environment, receives scalar rewards, and learns a policy mapping observations to actions that maximises cumulative reward. In continuous control problems, RL algorithms such as Proximal Policy Optimisation (PPO), Soft Actor-Critic (SAC), and TD3 have demonstrated impressive results on robotic manipulation, locomotion, and game-playing.

The connection to control theory is deep: the Bellman optimality equation in RL corresponds to the Hamilton-Jacobi-Bellman (HJB) equation in optimal control theory, and the LQR solution is the exact solution to the RL problem when the system is linear and the reward is quadratic.

Challenges of RL for control include:

• Sample efficiency: Learning requires many interactions with the plant, which is expensive or unsafe for physical systems.
• Safety: RL policies may violate constraints or explore dangerous regions during training.
• Interpretability: Learned neural network policies are difficult to analyse or certify for safety-critical applications.
• Generalisation: Policies trained in simulation may not transfer to the real system (the “sim-to-real gap”).

Current research actively addresses these challenges through safe RL, model-based RL (which learns a model of the environment and plans within it — analogous to MPC), and constrained policy optimisation.

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Summary of Key Formulas

Closed-loop TF: \( T(s) = \frac{C(s)P(s)}{1 + C(s)P(s)} \)

Sensitivity: \( S(s) = \frac{1}{1 + L(s)} \)

Stability: All poles of \( T(s) \) in open LHP \( \Leftrightarrow \) all roots of \( 1 + L(s) = 0 \) in open LHP.

Routh criterion: First column of Routh array all positive \( \Leftrightarrow \) stable.

Steady-state errors (unity feedback):

\[ e_{\text{ss, step}} = \frac{1}{1 + K_p}, \quad e_{\text{ss, ramp}} = \frac{1}{K_v}, \quad e_{\text{ss, parabola}} = \frac{1}{K_a} \]

Second-order specs:

\[ \%OS = 100\,e^{-\pi\zeta/\sqrt{1-\zeta^2}}, \quad T_p = \frac{\pi}{\omega_n\sqrt{1-\zeta^2}}, \quad T_s \approx \frac{4}{\zeta\omega_n} \]

Nyquist criterion: \( Z_{\text{RHP}} = P_{\text{RHP}} - N \) (closed-loop stable iff \( Z_{\text{RHP}} = 0 \)).

Phase margin: \( PM = 180° + \angle L(j\omega_{gc}) \)

Gain margin: \( GM = 1/|L(j\omega_{pc})| \)

Lead compensator max phase: \( \phi_{\text{max}} = \arcsin\frac{1-\alpha}{1+\alpha} \) at \( \omega_{\text{max}} = \sqrt{zp} \)

LQR Riccati equation: \( PA + A^TP - PBR^{-1}B^TP + Q = 0 \)