Slide contents
Every slide of every chapter — use the search box (top right) to find a topic
1. Mathematical Preliminaries & Dynamic Response
1.1 Process Models and Deviation Variables
- A simple differential-equation model
- Linearization of nonlinear models
- Example 1.1 Linearize a mixed vessel
- Example 1.2 Stirred-tank heater
- Example 1.3 Non-isothermal CSTR
- Problem #1 (homework)
- Definition and the look-up table approach
- Transforms of common functions
- Inputs common to control problems
- Initial- and final-value theorems
1.3 Partial-Fraction Expansion
- Breaking \(F(s)\) into table entries
- Example 1.5 Case 1: distinct real roots
- Example 1.6 Case 2: complex roots
- Example 1.7 Case 3: repeated roots
- Partial fractions by computer
- Problem #2 (homework)
1.4 Transfer Functions, Poles and Zeros
- The transfer function
- Poles tell the shape of the response
- Back to the first-order tank: step and impulse
- First-order model
- Effect of \(\tau\) and \(K\)
- Ramp input and the integrating process
- Example 1.8 Two stirred tanks in series
- Problem #3 (homework)
- Second-order model
- Example 1.9 Oscillations of a damped manometer
- Step response of a second-order model
- Features of the underdamped step response
- Second-order responses in code
- Problem #4 (homework)
- Dead time (time delay, transport lag)
- Example 1.10 Padé approximation of \(e^{-3s}/(10s+1)\)
- Problem #5 (homework)
2. Analysis of Single-Loop Control Systems
2.1 Classical Feedback Control System
- A liquid-level loop
- Block-diagram algebra
- The standard loop: stirred-tank heater
- Servo and regulator problems
- Choosing the controlled and manipulated variables
- Problem #1 (homework)
- Hardware around the loop
- Sensor and transmitter: \(G_m\)
- Actuator: the control valve, \(G_a\)
- The heater loop assembled
- Proportional control
- Proportional-integral control
- The price of integral action
- Derivative action
- Derivative kick
- PID control and its forms
- The three modes side by side
- Problem #2 (homework)
2.3 Closed-Loop System Response
- Unity-feedback servo loop
- Example 2.1 P control, first-order process
- Example 2.2 P control, second-order overdamped process
- Example 2.3 PI control, first-order process
- Example 2.4 PD control, first-order process
- Example 2.5 P control, integrating process
- Examples 2.1–2.5 at a glance
- Example 2.6 Closed-loop simulation
- Interactive: try the modes yourself
3. Tuning of Single-Loop Control Systems
3.1 Tuning with Empirical Relations
- Empirical tuning in practice
- Process-reaction curve: the open-loop step test
- Fitting the FOPDT model
- Ziegler–Nichols and Cohen–Coon (open-loop test)
- Minimum error-integral criteria
- Minimum-ITAE tuning relations
- Ziegler–Nichols ultimate-cycle method (closed-loop test)
- Ziegler–Nichols ultimate-cycle settings
- Example: the Van de Vusse reactor
- Van de Vusse in code
- Example 3.6 Same plant, five tuning rules
- Example 3.6: responses
- Tuning rules in code
- Interactive: compare the rules yourself
3.2 Direct Synthesis and Internal Model Control
- Design the closed-loop response first
- The catch, and two more target responses
- Example 3.1 First-order process → PI
- Example 3.2 Second-order overdamped process → PID
- Example 3.3 FOPDT process → PI
- Example 3.4 Overdamped process, underdamped target
- Internal model control (IMC)
- IMC closed loop and the model controller
- Example 3.5 IMC for a FOPDT process
- Choosing \(\tau_c\)
- Example 3.7 IMC for the Van de Vusse reactor
- Summary: which method when?
- Problem #1 (homework)
- Resources
4. Stability of Closed-Loop Systems
- Closed-loop characteristic equation
- Pole location ↔︎ time response
- Example 4.1 Stabilizing an unstable process with P control
- (1) Hurwitz test: a necessary condition
- (2) Routh array: necessary and sufficient
- (3) Special cases
- Example 4.2 P control of a third-order process
- Example 4.2 at \(K_c=60\): a row of zeros
- Example 4.3 PI control of a critically damped process
- Routh array in code
4.3 Direct-Substitution Analysis
- Finding the stability boundary with \(s=j\omega\)
- Example 4.4 First-order process with dead time
- Dead time: exact vs. Padé approximation
- Dead time in code
- Which technique? (decision chart)
- From \(K_{cu}, P_u\) to controller tuning
- What is a root locus?
- Angle and magnitude conditions
- Sketching rules (1)–(5)
- Sketching rules (6)–(8)
- Example 4.2B Root locus of Ex 4.2
- Example 4.6 Representative root-locus shapes
- Example 4.6 (c), (d): the calculations
- Effect of adding a zero
- Effect of adding a pole: PI control (Example 4.3B)
- RHP zero: inverse response
5. Frequency-Response Analysis & Design
- A sine in, a sine out
- … and \(G(j\omega)\) gives AR and \(\phi\) directly
- Products of transfer functions
- Bode and Nyquist plots
- Example 5.2 First-order lag \(\;K_p/(\tau_ps+1)\)
- Example 5.3 Second-order \(\;K/(\tau^2s^2+2\zeta\tau s+1)\)
- Example 5.3: Bode and Nyquist for \(\zeta=1.2\ldots0.2\)
- Examples 5.4, 5.7, 5.8: lead, integrator, lag + integrator
- Examples 5.5, 5.6: dead time
- Building blocks at a glance
- Sketching a Bode plot by asymptotes
- Example 5.9 \((5s+1)/[(10s+1)(2s+1)]\) by asymptotes
- Bode and Nyquist in code
- Two frequency-domain tests
- Reading the Nyquist criterion
- Examples 5.10, 5.11 Inverse-response process, P control
- Example 5.11 by frequency response
- Ex 4.4 revisited: dead time handled exactly
- Example 5.14 Nyquist for an open-loop unstable process
- Gain and phase margins
- What the margins protect against
- Phase margin and damping
- Example 5.12 Ziegler–Nichols PI for the inverse-response process
- Margins in code
6. Multi-loop Systems
- The fired-heater problem
- Master and slave loops
- Block diagram
- Reduced block diagram
- Example 6.1 Choosing the slave gain \(K_{c2}\)
- Example 6.1 (d): choosing the master \(\tau_I\)
- Example 6.1 (e): without the cascade
- Example 6.1: responses (replaces the Simulink runs)
- Example 6.1 in code
- Act on the disturbance, not on the error
- The feedforward controller
- Example 6.7 Static or dynamic feedforward?
6.3 Feedforward–Feedback Control
- Dead time in the loop
- The Smith predictor
- Example 6.8 Smith predictor on the Ex 4.4 plant
- Smith predictor in code
6.6 Multiple-Input Multiple-Output Control
- A blending process
- MIMO closed loop
- Process gain matrix
- Relative gain array (RGA)
- Example 6.4 RGA of the blending process
- Reading the RGA: pairing rules
- Example 6.5 Pairing depends on the operating point
- RGA in code
6.7 Decoupling of Interacting Systems
- Feedforward-like decouplers
- Example 6.6 Distillation column
- Example 6.6: PI tuning (ITAE)
- Example 6.6: does decoupling help? (replaces the Simulink runs)
- Decoupling in code
- Interactive: cascade vs. single loop
- Summary: which structure for which problem?
- Problem #1 (homework)
- Resources
- A final word — Neal R. Amundson (University of Minnesota, circa 1970)