MA 108 Math The Language of Engineers. Stan Żak. School of Electrical and Computer Engineering

Transcription

1 MA 108Math The Language of Engineers p. 1/2 MA 108 Math The Language of Engineers Stan Żak School of Electrical and Computer Engineering October 14, 2010

2 MA 108Math The Language of Engineers p. 2/2 Outline What is math?

3 MA 108Math The Language of Engineers p. 2/2 Outline What is math? Some pure math problems

4 MA 108Math The Language of Engineers p. 2/2 Outline What is math? Some pure math problems Engineering challenge problems

5 MA 108Math The Language of Engineers p. 2/2 Outline What is math? Some pure math problems Engineering challenge problems Optimization and control apps

6 MA 108Math The Language of Engineers p. 2/2 Outline What is math? Some pure math problems Engineering challenge problems Optimization and control apps Some cool math applications

7 MA 108Math The Language of Engineers p. 2/2 Outline What is math? Some pure math problems Engineering challenge problems Optimization and control apps Some cool math applications Conclusions

8 MA 108Math The Language of Engineers p. 3/2 What Is Math? A part of the human search for understanding

9 MA 108Math The Language of Engineers p. 3/2 What Is Math? A part of the human search for understanding Mathematical discoveries have come both from the attempt to describe the natural world and from the desire to arrive at a form of inescapable truth from careful reasoning Kenyon College Math Department Web Page

10 MA 108Math The Language of Engineers p. 4/2 What Is Math Good For? In the last few decades mathematics has been successfully applied to many aspects of the human world: voting trends in politics

11 MA 108Math The Language of Engineers p. 4/2 What Is Math Good For? In the last few decades mathematics has been successfully applied to many aspects of the human world: voting trends in politics dating of ancient artifacts

12 MA 108Math The Language of Engineers p. 4/2 What Is Math Good For? In the last few decades mathematics has been successfully applied to many aspects of the human world: voting trends in politics dating of ancient artifacts analysis of automobile traffic patterns

13 MA 108Math The Language of Engineers p. 4/2 What Is Math Good For? In the last few decades mathematics has been successfully applied to many aspects of the human world: voting trends in politics dating of ancient artifacts analysis of automobile traffic patterns biology

14 MA 108Math The Language of Engineers p. 4/2 What Is Math Good For? In the last few decades mathematics has been successfully applied to many aspects of the human world: voting trends in politics dating of ancient artifacts analysis of automobile traffic patterns biology engineering

15 MA 108Math The Language of Engineers p. 5/2 Pure Math vs. Applied Math The Millennium Prize Problems

16 MA 108Math The Language of Engineers p. 5/2 Pure Math vs. Applied Math The Millennium Prize Problems

17 MA 108Math The Language of Engineers p. 6/2 Engineering Grand Challenges 14 grand challenges for engineering in the 21-st century identified by the National Academy of Engineering

18 MA 108Math The Language of Engineers p. 6/2 Engineering Grand Challenges 14 grand challenges for engineering in the 21-st century identified by the National Academy of Engineering

19 MA 108Math The Language of Engineers p. 7/2 Optimization An act, process, or methodology of making something (as a design, system, or decision) as fully perfect, functional, or effective as possible. The mathematical procedures (as finding the maximum of a function) involved in this.

20 MA 108Math The Language of Engineers p. 7/2 Optimization An act, process, or methodology of making something (as a design, system, or decision) as fully perfect, functional, or effective as possible. The mathematical procedures (as finding the maximum of a function) involved in this. Optimization making the best decision.

21 MA 108Math The Language of Engineers p. 7/2 Optimization An act, process, or methodology of making something (as a design, system, or decision) as fully perfect, functional, or effective as possible. The mathematical procedures (as finding the maximum of a function) involved in this. Optimization making the best decision. Engineering design, management, etc.

22 MA 108Math The Language of Engineers p. 7/2 Optimization An act, process, or methodology of making something (as a design, system, or decision) as fully perfect, functional, or effective as possible. The mathematical procedures (as finding the maximum of a function) involved in this. Optimization making the best decision. Engineering design, management, etc. What does best mean?

23 MA 108Math The Language of Engineers p. 8/2 Objective function Measure goodness by a function f. (Cost function or objective function)

24 MA 108Math The Language of Engineers p. 8/2 Objective function Measure goodness by a function f. (Cost function or objective function) Want to minimize f. (Smaller = better)

25 MA 108Math The Language of Engineers p. 8/2 Objective function Measure goodness by a function f. (Cost function or objective function) Want to minimize f. (Smaller = better) Ω = set of all possible choices. (Feasible set)

26 MA 108Math The Language of Engineers p. 9/2 Notation R n denotes a set of real n-tuples

27 MA 108Math The Language of Engineers p. 9/2 Notation R n denotes a set of real n-tuples x R n means x = x 1 x 2. x n, x i R

28 MA 108Math The Language of Engineers p. 9/2 Notation R n denotes a set of real n-tuples x R n means x = x 1 x 2. x n, x i R f is a real-valued function on n variables, f : R n R

29 MA 108Math The Language of Engineers p. 10/2 Example f = f(x 1,x 2 ) = x 1 x 2 + 7, an example of a real-valued function of two variables, f : R 2 R

30 MA 108Math The Language of Engineers p. 10/2 Example f = f(x 1,x 2 ) = x 1 x 2 + 7, an example of a real-valued function of two variables, f : R 2 R x = [ x 1 x 2 ] a = f(x 1,x 2 ) R

31 MA 108Math The Language of Engineers p. 11/2 Optimization problem min subject to f(x) x Ω

32 MA 108Math The Language of Engineers p. 11/2 Optimization problem min subject to What about maximization? (Bigger = better) f(x) x Ω

33 MA 108Math The Language of Engineers p. 11/2 Optimization problem min subject to What about maximization? (Bigger = better) f(x) x Ω How to solve optimization problem?

34 MA 108Math The Language of Engineers p. 11/2 Optimization problem min subject to What about maximization? (Bigger = better) f(x) x Ω How to solve optimization problem? Analytically

35 MA 108Math The Language of Engineers p. 11/2 Optimization problem min subject to What about maximization? (Bigger = better) f(x) x Ω How to solve optimization problem? Analytically Numerically

36 MA 108Math The Language of Engineers p. 12/2 Example: Linear regression Given points on the plane: (t 0,y 0 ),...,(t n,y n )

37 MA 108Math The Language of Engineers p. 12/2 Example: Linear regression Given points on the plane: (t 0,y 0 ),...,(t n,y n ) Want to find the line of best fit through these points.

38 MA 108Math The Language of Engineers p. 12/2 Example: Linear regression Given points on the plane: (t 0,y 0 ),...,(t n,y n ) Want to find the line of best fit through these points. Best = minimize the average squared error.

39 MA 108Math The Language of Engineers p. 13/2 Minimizing the average squared error y mt 2 +c-y mt 1 +c-y mt o +c-y o t

40 MA 108Math The Language of Engineers p. 14/2 Line of best fit Equation of line: y = mt + c;

41 MA 108Math The Language of Engineers p. 14/2 Line of best fit Equation of line: y = mt + c; Optimization problem: Find m and c to min 1 n (mt i + c y i ) 2 n i=0

42 MA 108Math The Language of Engineers p. 14/2 Line of best fit Equation of line: y = mt + c; Optimization problem: Find m and c to min 1 n (mt i + c y i ) 2 n i=0 Solution: In this case we can find the solution analytically (using least-squares theory).

43 MA 108Math The Language of Engineers p. 14/2 Line of best fit Equation of line: y = mt + c; Optimization problem: Find m and c to min 1 n (mt i + c y i ) 2 n i=0 Solution: In this case we can find the solution analytically (using least-squares theory). Related application: system identification.

44 MA 108Math The Language of Engineers p. 15/2 Battery charger circuit R 1 R 3 R 5 I 1 I 3 I 5 30 Volts + I 2 R 2 I 4 R 4 Battery 10 Volts Battery 6 Volts Battery 20 Volts

45 MA 108Math The Language of Engineers p. 16/2 Charger circuit specifications Current I 1 I 2 I 3 I 4 I 5 Upper Limit (Amps) Lower Limit (Amps)

46 MA 108Math The Language of Engineers p. 17/2 Design objective Find I 1,...,I 5 to maximize power transferred to batteries, that is, max 10I 2 + 6I I 5 subject to I 1 = I 2 + I 3 I 3 = I 4 + I 5 I 1 4 I 2 3 I 3 3 I 4 2 I 5 2, I 1,I 2,I 3,I 4,I 5 0.

47 MA 108Math The Language of Engineers p. 18/2 Solving example problem This is a linear programming problem.

48 MA 108Math The Language of Engineers p. 18/2 Solving example problem This is a linear programming problem. Solution: Can use the simplex algorithm see MA 511.

49 MA 108Math The Language of Engineers p. 19/2 Model-based Predictive Control Model-based Predictive Control MPC

50 MA 108Math The Language of Engineers p. 19/2 Model-based Predictive Control Model-based Predictive Control MPC MPC methodology is also referred to as the moving horizon control or the receding horizon control.

51 MA 108Math The Language of Engineers p. 19/2 Model-based Predictive Control Model-based Predictive Control MPC MPC methodology is also referred to as the moving horizon control or the receding horizon control. The idea behind this approach can be explained using an example of driving a car

52 MA 108Math The Language of Engineers p. 20/2 MPC analogy with driving a car The driver looks at the road ahead of him and taking into account the present state and the previous action predicts his action up to some distance ahead, which we refer to as the prediction horizon.

53 MA 108Math The Language of Engineers p. 20/2 MPC analogy with driving a car The driver looks at the road ahead of him and taking into account the present state and the previous action predicts his action up to some distance ahead, which we refer to as the prediction horizon. Based on the prediction, the driver adjusts the driving direction.

54 MA 108Math The Language of Engineers p. 21/2 MPC illustration Y v x v Lane direction X v y Prediction horizon The driver predicts future travel direction based on the current state of the car and the current position of the steering wheel

55 MPC strategy past future/prediction set-point y(t i ) y(t i + 2 t i ) u(t i + 2 t i ) u(t i ) t i t + 1 t i i t i + N p prediction horizon N p MPC construction MA 108Math The Language of Engineers p. 22/2

56 MA 108Math The Language of Engineers p. 23/2 Basic Structure of MPC cost function r(. ) predicted error + _ Optimizer predicted input u(t i ) y(t i ) Plant x predicted output Plant Model x(t i ) Model Predictive Controller

57 MA 108Math The Language of Engineers p. 24/2 Hypothalamic-pituitary-adrenal axis The hypothalamic-pituitary-adrenal HPA

58 MA 108Math The Language of Engineers p. 24/2 Hypothalamic-pituitary-adrenal axis The hypothalamic-pituitary-adrenal HPA The HPA axis is a set of interactions between the hypothalamus (a part of the brain), the pituitary gland (also part of the brain) and the adrenal or suprarenal glands (at the top of each kidney.)

59 Basic Structure of MPC MA 108Math The Language of Engineers p. 25/2

60 MA 108Math The Language of Engineers p. 26/2 HPA axis The HPA axis helps regulate our temperature, digestion, immune system, mood, sexuality and energy usage.

61 MA 108Math The Language of Engineers p. 26/2 HPA axis The HPA axis helps regulate our temperature, digestion, immune system, mood, sexuality and energy usage. It is also a major part of the system that controls our reaction to stress, trauma and injury.

62 MA 108Math The Language of Engineers p. 27/2 HPA axis therapeutic correction A problem related to human health how optimization can be used to find a therapeutic strategy

63 MA 108Math The Language of Engineers p. 27/2 HPA axis therapeutic correction A problem related to human health how optimization can be used to find a therapeutic strategy

64 MA 108Math The Language of Engineers p. 28/2 Math and Soccer?

65 MA 108Math The Language of Engineers p. 28/2 Math and Soccer? pdf

66 MA 108Math The Language of Engineers p. 29/2 Conclusions Math is powerful

67 MA 108Math The Language of Engineers p. 29/2 Conclusions Math is powerful Can do cool things using math

68 MA 108Math The Language of Engineers p. 29/2 Conclusions Math is powerful Can do cool things using math Need to consider the moral consequences of what we do!