Introduction to Computer Control Systems
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1 Introduction to Computer Control Systems Lecture 1: Introduction Dave Zachariah Div. Systems and Control, Dept. Information Technology, Uppsala University October 28, 2014 (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
2 Course information Main teacher: Dave Zachariah Credits: 5 hp Course code: 1RT485 Period: 2 (Weeks ) Webpage: (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
3 Course information Teachers: Dave Zachariah (Lectures) Jesus Zambrano (Tutorials) Ruben Cubo (Labs) Structure: 10 lectures (20h) 10 problem solving sessions (20h) 2 computer labs (4h) 3 process labs (12h) 5hp 400/3h 133h in total. Remaining (400/3 56)h of self/group-study, readings and homework assignments. (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
4 Course information Textbooks: Torkel Glad and Lennart Ljung: Control Theory - Multivariable and Nonlinear Methods, Taylor and Francis, Karl Johan, Åström Richard M. Murray, Feedback Systems, Princeton, Available as pdf free of charge! Examination: Written examination on Wednesday, January 7th Passed laboratory is also required. (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
5 Course information What does the course offer you?: Preparing you for analysis, design and implementation of advanced control systems How will we achieve that?: Participatory lectures, problem solving sessions, process and computer labs (LEGO NXT, Matlab) (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
6 Application areas Figure: Biomedicine and biomolecular interactions (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
7 Application areas Figure: Automotive control and emission reduction (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
8 Application areas Figure: Aircraft control and stabilization (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
9 Application areas Figure: Robotics and autonomous systems (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
10 Application areas Figure: Industrial processes and power systems (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
11 Problem formulation y + u (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
12 Problem formulation y + u Q: What is the input u to the electrical motor such that the output y is stable around the desired level or reference r = 0? (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
13 Problem formulation y + u Q: What is the input u to the electrical motor such that the output y is stable around the desired level or reference r = 0? Computer should figure that out! This is the practical goal of control theory (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
14 The feedback approach input System output (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
15 The feedback approach disturbance input System output (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
16 The feedback approach disturbance reference Controller input System output (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
17 The feedback approach disturbance reference + Controller input System output (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
18 The feedback approach r + Controller G u y Feedback approach: Use the difference (reference output) to adjust the controller The controller needs a representation of the system G Approximate model is sufficient! (+) The stability of the feedback system becomes critical! ( ) (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
19 Mathematical model of system: Concepts u(t) G y(t) Figure: Graphical representation of a system G with input u(t) and output y(t). Models are not right or wrong but more or less accurate useful representations of underlying mechanisms. (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
20 Mathematical model of system: Concepts u(t) G y(t) Figure: Graphical representation of a system G with input u(t) and output y(t). System G is causal: if y(t) at t depends on current and previous values of u(τ). That is, τ t. Otherwise, noncausal. static: if y(t) at t depends only on current value u(t). Otherwise, dynamical, i.e. has memory. (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
21 Mathematical model of system: Concepts u(t) G y(t) Figure: Graphical representation of a system G with input u(t) and output y(t). System G is discrete-time: y(t) and u(t) are only defined at time points t = t 0 +, t 0 + 2, t 0 + 3,.... Otherwise, continuous-time. time invariant: properties not depend on absolute time t, they are the same each time the system starts up. Otherwise, time-varying. (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
22 Mathematical model of system: Concepts u(t) G y(t) Figure: Graphical representation of a system G with input u(t) and output y(t). System G is linear: Combined input a 1 u 1 (t) + a 2 u 2 (t) yields combined output a 1 y 1 (t) + a 2 y 2 (t) as if generated separately and added ( superposition ). Otherwise, non-linear. (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
23 Mathematical model of system: Concepts u(t) G y(t) Figure: Graphical representation of a system G with input u(t) and output y(t). For prediction and control 1 Linear, 2 Time-Invariant, 3 (Dynamical) models are often useful and accurate enough. LTI system models is how we will approximate many systems in this course. (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
24 Example On the board: apply concepts to a central heater (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
25 LTI system models: Mathematical prerequisites u(t) G y(t) Figure: LTI system model G with input u(t) and output y(t). #1: Complex-valued numbers: E.g. ρe jφ = ρcosφ + jρ sin φ Magnitude and phase of a complex number? How to view a number on complex plane? (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
26 LTI system models: Mathematical prerequisites u(t) G y(t) Figure: LTI system model G with input u(t) and output y(t). #2: Ordinary differential equations: E.g. d dt y(t) + a 1y(t) = b 0 d dt u(t) + b 1u(t) Solve y(t) when u(t) is a given function. (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
27 LTI system models: Mathematical prerequisites u(t) G y(t) Figure: LTI system model G with input u(t) and output y(t). #3: Approximation by first- and second-order Taylor series: E.g. Given some (differentiable) function f(t), we have can describe it around a fixed point t = t 0 by a sum: f(t) = n=0 d n dt n f(t) (t t 0 ) n t=t 0 n! = f(t 0 ) + d dt f(t 0) (t t 0) 1 1! + d2 dt 2 f(t 0) (t t 0) 2 + 2! (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
28 LTI system models: Mathematical prerequisites u(t) G y(t) Figure: LTI system model G with input u(t) and output y(t). #4: Basic vector/matrix algebra and eigenvalues: E.g. y = Ax Y = AX Perform small matrix/vector multiplication by hand. Av = λv What are the eigenvalues of a square matrix A? (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
29 LTI system models: Mathematical prerequisites u(t) G y(t) Figure: LTI system model G with input u(t) and output y(t). Homework: Refresh your memory by studying the math tutorial (UU/Info Technology/SysCon) Intro. Computer Control Sys. October 28, / 8
Intro. Computer Control Systems: F9
Intro. Computer Control Systems: F9 State-feedback control and observers Dave Zachariah Dept. Information Technology, Div. Systems and Control 1 / 21 dave.zachariah@it.uu.se F8: Quiz! 2 / 21 dave.zachariah@it.uu.se
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