1 / 53

INTELLIGENT CONTROL SYSTEMS Day 1/3

The Warsaw School of Computer Science . INTELLIGENT CONTROL SYSTEMS Day 1/3. Prof. Marian S. Stachowicz Laboratory for Intelligent Systems ECE Department, University of Minnesota, USA Warsaw, Poland, May 16 – 22 , 2014. Linear and nonlinear systems.

lottie
Download Presentation

INTELLIGENT CONTROL SYSTEMS Day 1/3

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The Warsaw School of Computer Science INTELLIGENT CONTROL SYSTEMSDay 1/3 Prof. Marian S. Stachowicz Laboratory for Intelligent Systems ECE Department, University of Minnesota, USA Warsaw, Poland, May 16 –22, 2014

  2. Linear and nonlinear systems

  3. a. Linear systemb. Nonlinear system Intelligent Control Systems

  4. Some physical nonlinearities Intelligent Control Systems

  5. Linearization about a point A Intelligent Control Systems

  6. Linearization about a point A Intelligent Control Systems

  7. Taylor series Intelligent Control Systems

  8. Linearization of 5 cosxabout x= /2 Intelligent Control Systems

  9. Linearizing a differential equation Intelligent Control Systems

  10. Linearizing a differential equation Intelligent Control Systems

  11. Four steps… 1.The first step is to recognize the nonlinear component and write the nonlinear differential equation. 2.In the second step we linearized it for small-signal inputs about the steady-state solution when the small-signal input is equal to zero. This is equilibrium, for example when a pendulum is at rest. 3.Next we linearized the nonlinear differential equation, 4. We take the Laplace transform of the linearized differential equation, assuming zero initial conditions. Intelligent Control Systems

  12. Transfer function - nonlinear electricalnetwork Intelligent Control Systems

  13. 2. Equilibrium solution Intelligent Control Systems

  14. Linearity and superposition Definition: If all initial conditions in the system are zero, that is, if the system is completely at rest, then the system is a linear system if it has the following property: a. If an input u1(t) produces an output y1 (t), and b. an input u2(t) produces an output y2 (t), c. then input c1u1 (t) + c2u2 (t) produces an output c1y1 (t) + c2y2 (t) for all pairs of inputs u1 (t) and u2 (t) and all pairs of constants c1 and c2. Intelligent Control Systems

  15. Principle of Superposition The response y(t) of a linear system due to several inputs u1(t), u2(t), . . . , un(t) acting simultaneously is equal to the sum of the responses of each input acting alone, when all initial conditions in the system are zero. If yi(t) is the response due to the input ui(t), then Intelligent Control Systems

  16. Linearity and superposition Use the Principle of Superposition to determine the output y. Intelligent Control Systems

  17. Foru2 = u3 = 0, y1 = 5(d/dt)(sin t) = 5 cost. For u1 = u3 = 0, y2 = 5( d/dt)(cos 2t) = - 10 sin 2t. For u1 = u2 = 0, y3 = - 5t2. Therefore y = y1 + y2 + y3 = 5(cos t – 2 sin 2t – t2) Intelligent Control Systems

  18. Superposition of multiple inputs Step 1: Set all inputs except one equal to zero. Step 2: Transform the block diagram to canonical form, Step 3: Calculate the response due to the chosen input acting alone. Step 4: Repeat Steps 1 to 3 for each of the remaining inputs. Step 5: Algebraically add all of the responses (outputs) determined in Steps 1 to 4. This sum is the total output of the system with all inputs acting simultaneously. Intelligent Control Systems

  19. MULTIPLE INPUTS AND OUTPUTS Determine the output C due to U1, U2, and R. Intelligent Control Systems

  20. Let U1 = U2 = 0. After combining the cascaded blocks, we obtain the output C due to R acting alone. CR = [G1 G2 /(1 - G1G2 H1H2)]R Intelligent Control Systems

  21. Let R = U1 = 0. C1 is the response due to U1 acting alone. Intelligent Control Systems

  22. C1 = [G2/(1 – G1G2H1H2)]U1 Intelligent Control Systems

  23. Let R = U1 = 0. C2 is the response due to U2 acting alone. C2 = [G1G2H1/(1 – G1G2H1H2)]U2 Intelligent Control Systems

  24. By superposition, the total output is Intelligent Control Systems

  25. Nonlinear electric circuit Intelligent Control Systems

  26. Subsystems of the antenna azimuth position control system Intelligent Control Systems

  27. Nonlinearities Intelligent Control Systems

  28. d"-'y +a,- dY + aoy = u - +an-,- + dt din-' dtd"Y (3.5) • nd'yd'u a'(?)- = b1(t)- iIodt' i-o dt'(3.4) • d2Y dy - +2- +y=odt2 dt Intelligent Control Systems

  29. Intelligent Control Systems

  30. Intelligent Control Systems

  31. Intelligent Control Systems

  32. System equations • A property common to all basic laws of physics is that certain fundamental quantities can be defined by numerical values. • The physical laws define relationships between these fundamental quantities and are usually represented by equations. Intelligent Control Systems

  33. The scalar versions • The scalar version of Newton's second law states that, if a force of magnitude f is applied to a mass of M units, the acceleration a of the mass is related to f by the equation f = Ma. Intelligent Control Systems

  34. Ohm's law states that, if a voltage of magnitude U is applied across a resistor of R units, the current i through the resistor is related to U by the equation U = Ri. Intelligent Control Systems

  35. Differential equations • A differential equation is any algebraic or transcendental equality which involves either differentials or derivatives. Intelligent Control Systems

  36. Newton's second law can be written alternatively as a relationship between force f, mass M, and the rate of change of the velocity v of the mass with respect to time t f = M (dv/dt) Intelligent Control Systems

  37. Ohm's law can be written alternatively as a relationship between voltage U, resistance R, and the time rate of passage of charge through the resistor, U = R (dq/dt) Intelligent Control Systems

  38. Partial differential equation • The diffusion equation in one dimension describes the relationship between the time rate of change of a quantity T in an object (e.g., heat concentration in an iron bar) and the positional rate of change of T: where k is a proportionality constant, x is a position variable, and t is time. Intelligent Control Systems

  39. Ordinary differential equation a0,a1,…, anare constants, y(t) andu(t) are dependent variables, t is the independent variable. Intelligent Control Systems

  40. Time invariance • Any differential equation of the form: a0,a1,…, an , b0,b1,…, bn , are constants, is time-invariant. The equation depends implicitly on t, via the dependent variables u and y and their derivatives. Intelligent Control Systems

  41. LINEAR AND NONLINEAR DIFFERENTIAL EQUATIONS • A linear term is one which is first degree in the dependent variables and their derivatives. • A linear equation is an equation consisting of a sum of linear terms. All others are nonlinear equations Intelligent Control Systems

  42. If any term of a differential equation contains higher powers - (dy/dt)3, products - u(dy/dt, or transcendental functions of the dependent variables - sin u, it is nonlinear. (5/cos t)( d2y/dt2) is a term of first degree in the dependent variable y, 2uy3(dy/dt) is a term of fifth degree in the dependent variables u and y. Intelligent Control Systems

  43. The ordinary differential equations (dy/dt)2 +y = 0 d2y/dt2 + cosy = 0 are nonlinear ( dy/dt)2is second degree cosyis not first degree, which is true of all transcendental functions. Intelligent Control Systems

  44. CLASSIFICATIONS OF DIFFERENTIAL EQUATIONS Intelligent Control Systems

  45. Classify the following differential equations accordingto whether they are ordinary or partial.Indicate the dependent and independent variables. Intelligent Control Systems

  46. Classify the following differential equations accordingto whether they are ordinary or partial. Indicate the dependent and independent variables. Intelligent Control Systems

  47. Classify the following linear differential equations according to whether they are time-variable or time-invariant. Indicate any time-variable terms. Intelligent Control Systems

  48. Classify the following linear differential equations according to whether they are time-variable or time-invariant. Indicate any time-variable terms. Intelligent Control Systems

  49. Classify the following differential equations according to whether they are linear or nonlinear. Indicate the dependent and independent variables and any nonlinear terms. Intelligent Control Systems

  50. Classify the following differential equations according to whether they are linear or nonlinear. Indicate the dependent and independent variables and any nonlinear terms. Intelligent Control Systems

More Related