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Chapter 4

Feedback Control Systems. Chapter 4. A First Analysis of Feedback. Chapter 4. A First Analysis of Feedback. Control Specifications. While maintaining the essential property of stability , the control specifications include both static and dynamic requirements such as the following:.

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Chapter 4

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  1. Feedback Control Systems Chapter 4 A First Analysis of Feedback

  2. Chapter 4 A First Analysis of Feedback Control Specifications While maintaining the essential property of stability, the control specifications include both static and dynamic requirements such as the following: Steady-state response, static properties • The permissible steady-state error while performing regulation in the presence of constant (=“bias”) disturbance signal and sensor noise • The permissible steady-state error while tracking a polynomial reference signal such as a step or a ramp • The sensitivity of the system transfer function to changes in model/plant parameters • The permissible transient error in response to a step in either the reference or the disturbance input Transient response, dynamic properties

  3. Chapter 4 A First Analysis of Feedback The Basic Equations of Control • Open-loop control system can be given as: • The output is given by: W : Disturbance • The error, the difference between reference input and system output, is given by:

  4. Chapter 4 A First Analysis of Feedback The Basic Equations of Control • Feedback control system can be given as: V : Sensor noise W and Vare taken to be at the inputs of the process and the sensor ≡

  5. Chapter 4 A First Analysis of Feedback The Basic Equations of Control • Assuming the sensor to be fast and accurate, Hy can be taken to be a constant. The input shaping Hrcan be moved inside the loop to become:

  6. Chapter 4 A First Analysis of Feedback The Basic Equations of Control • If Hy is constant, it is standard practice to select equal input shaping factor so that Hr=Hyand a unity feedback gain is obtained:

  7. Chapter 4 A First Analysis of Feedback The Basic Equations of Control • In case W=0 and V=0, • In case R=0 and V=0, • In case R=0 and W=0,

  8. Chapter 4 A First Analysis of Feedback The Basic Equations of Control • The equation of output is: • The equation of error is:

  9. Chapter 4 A First Analysis of Feedback The Basic Equations of Control Or: where: S : Sensitivity function T : Complementary sensitivity function

  10. For closed-loop control: Chapter 4 A First Analysis of Feedback Tracking: Following the Reference Input • Tracking problem is to cause the output to follow the reference input as closely as possible. • For the special case, where only R is the input to the system, the ability of the system to track a reference can be compared. • For open-loop control: • R is said to be perfectly tracked if E is zero • In open-loop control, Dol (taking Hr = 1) cannot be freely chosen so that the error Eol is zero • In closed-loop control, D = HrDcl (taking Hr = 1) can be chosen to be large so that Ecl approaches zero

  11. Chapter 4 A First Analysis of Feedback Regulation: Disturbance Rejection • Regulation problem is to keep the error small when the reference is at most a constant set point and disturbances are present. • Suppose that disturbance W interacts with applied input R. • Now, compare open-loop control with feedback control on how well each system maintains a constant steady-state reference output in the face of external disturbance. Disturbance directly affect the output Disturbance can be substantially reduced by assigning Dcl properly

  12. Chapter 4 A First Analysis of Feedback Regulation: Sensor Noise Attenuation • Only applies for closed-loop control, where sensor noise V is defined as • To reduce steady-state error, Dcl is selected to be large • If Dcl is large, the transfer function of noise V tends to unity and the sensor noise is not reduced at all • Resolution: Frequency separation between reference R and disturbance W (very low frequency content)and sensor noise V (mostly high frequencies) • Each of R and V is a function of frequency, but with different frequency content

  13. The parameter changes may be caused by external factors from the surroundings such as temperature, pressure, etc. Internal factors may also contribute to parameter changes, in the form of imperfections in the system components such as static friction, amplifier drift, aging, deterioration, etc. Chapter 4 A First Analysis of Feedback Sensitivity

  14. Suppose that parameter changes happen in an operation, and make the gain of the plant Gdiffers from its original value toG+δG. As the result, the overall transfer function of the system Twill also change to to T+δT, and in time cause the change of system gain. By definition, the sensitivity of system gain T with respect to the plant gain G is given by: Chapter 4 A First Analysis of Feedback Sensitivity

  15. For the ideal case, only R is the input to the system, the original system gain T can be calculated. For open-loop control Chapter 4 A First Analysis of Feedback Sensitivity . • The output Yol is directly influenced by the plant gain G • Any change of the plant gain G will give impact to system gain T in the same fractional change (same magnitude)

  16. For closed-loop control Chapter 4 A First Analysis of Feedback Sensitivity . • The output Ycl is not directly influenced by the plant gain G • The fractional change of system gain can be reduced by properly assigning Dcl to make the sensitivity Scllow enough

  17. Chapter 4 A First Analysis of Feedback Advantages of Feedback Control • The permissible steady-state error in the presence of a constant disturbance signal (“bias”) can be reduced. • The permissible steady-state error while tracking a polynomial reference signal such as a step or a ramp can be reduced. • The sensitivity of the system transfer function to changes in model/plant parameters can be reduced. • The transient response can be speeded up to maintain permissible transient error in response to a step in either the reference or the disturbance input. • Unstable plants can be stabilized.

  18. Chapter 4 A First Analysis of Feedback Disadvantages of Feedback Control • Feedback control requires a sensor that can be very expensive and may introduce additional noise. • Feedback control systems are often more difficult to design and to operate than open-loop systems. • Feedback changes the dynamic response and often makes a system both faster and less stable.

  19. Chapter 4 A First Analysis of Feedback System Type • In the previous study we assumed both reference and disturbance to be constants, and also D(0) and G(0) to be finite constants. • In this section, we will consider the possibility that either or both of D(s) and G(s) have poles at s=0. • An example for D(s) is the well-known structure for the control equation of the form: • With we can deduct the corresponding transfer function Proportional Integral Derivative (PID) control

  20. Chapter 4 A First Analysis of Feedback System Type • To accommodate various inputs that may be fed to a control system, the polynomial inputs of different degrees will now be fed to the system and the resulting steady-state tracking errors will be evaluated. “ A stable system can be classified as a type ksystem, with k defined to be the degree of the input polynomial for which the steady-state system error is a nonzero finite constant. ” “ Stable systems are classified into “types” according to the degree of the polynomial that they can reasonably track. ” • For example, a system that can track a polynomial of degree 1 with a constant error is called Type 1.

  21. Chapter 4 A First Analysis of Feedback System Type: Unity Feedback Case • If we consider only the reference input R alone, set W=V=0, where

  22. Chapter 4 A First Analysis of Feedback System Type: Unity Feedback Case • To consider polynomial inputs, let step input(“position” input) ramp input(“velocity” input) parabola input(“acceleration” input)

  23. Chapter 4 A First Analysis of Feedback System Type: Unity Feedback Case • Applying the Final Value Theorem, • We consider first a system for which L has no pole at the origin. For this system, only step input, R(s)=1/s (or k=0),will guarantee that the system error is a nonzero finite constant. Type 0 “Position error constant” Nonzero & finite

  24. Chapter 4 A First Analysis of Feedback System Type: Unity Feedback Case • For general expression of the steady-state errors, we collect all terms except the pole(s) at the origin into a function L0(s) Finite at s = 0 and define 00 = 1 01 = 0 02 = 0, … • Substituting this expression to calculate the steady-state error, • If n > k, then ess = 0 • If n < k, then ess→ ∞ • If n = k, then essis nonzero finite • n = k =0, ess = 1/(1+Kp) • n = k ≠ 0, ess = 1/Kn

  25. Chapter 4 A First Analysis of Feedback System Type: Unity Feedback Case • A unity feedback system is defined to be of type k if for for • For type 0 system, the error constant Kp, position constant, is given by • For type 1 system, the error constant Kv, velocity constant, is given by • For type 2 system, the error constant Ka, acceleration constant, is given by

  26. Chapter 4 A First Analysis of Feedback System Type: Unity Feedback Case The higher the constant, the smaller the steady-state error

  27. Chapter 4 A First Analysis of Feedback System Type: Unity Feedback Case “ Classifying a system as type k indicates the ability of the system to achieve zero steady-state error to any polynomial input r(t) of degree lessthan k. ” “ The system is of type k if the steady-state error is zero to all polynomials r(t) of degree less than k, nonzero finite for all polynomial of degree k, and infinite for all polynomial of degree more than k. ”

  28. Chapter 4 A First Analysis of Feedback System Type: Unity Feedback Case “ An unity feedback system is of type k if the open-loop transfer function of the system has k poles at s=0. ” With pi ≠ 0, Type 0 Type 1 Type 2 Type n

  29. Chapter 4 A First Analysis of Feedback System Type: Unity Feedback Case • Example: • A temperature control system is found to have zero steady-state error to a constant tracking input and a steady-state error of 0.5°C to a ramp tracking input, rising at the rate of 40°C/s. • What is the system type? Finite error of ess=0.5 °C to a ramp tracking input  Type 1 • What is the error constant? For type 1, ess= 1/Kv for unit ramp input ess= 40/Kv for the input in this example Kv = 40/ess= 80 s–1

  30. Chapter 4 A First Analysis of Feedback System Type: Unity Feedback Case • Example: • Determine the system type and the relevant error constant for the speed-control with unity feedback and PI-control, if the plant transfer function is G=A/(τs+1) and the controller transfer function is Dc=kp+ki/s. • System type? L(s) = D(s)·G(s) = A(kps + ki)/{s(τs+1)}  One pole at origin  Type 1 • Error constant? For type 1, Kv = lims·L(s) = Aki s→0 • Try to check using Final Value Theorem, ess=1/Kv

  31. Chapter 4 A First Analysis of Feedback System Type: Unity Feedback Case Example: Find the steady-state error in terms of K and Kt when the system above are subjected to a unit step and a unit ramp function. ≡ Type 1

  32. Chapter 4 A First Analysis of Feedback System Type: Unity Feedback Case ≡ 1st Way

  33. Chapter 4 A First Analysis of Feedback System Type: Unity Feedback Case The steady-state error for unit step input R(s)=1/s: The steady-state error for unit ramp input R(s)=1/s2:

  34. Chapter 4 A First Analysis of Feedback System Type: Unity Feedback Case 2nd Way • 1 pole at the origin owned by L(s)=D(s)·G(s) • System type: Type 1 • Steady-state error for a unit step input: • Steady-state error for a unit ramp input:

  35. Chapter 4 A First Analysis of Feedback Homework 4 • No.1.a • For a plant having the transfer function 1/(s2+3s+9), it is proposed to use a controller in a unity feedback system and having the transfer function (c2s2+c1s+c0)/(s2+d1s). • Solve for the parameters of this controller {c0,c1,c2,d1} so that the closed-loop will have the characteristic equation (s+6)(s+3)(s2+3s+9). • No.1.b • Show that if the reference input to the system of the above problem is a step of amplitude A, then steady-state error will be zero. • Hint: Error is the difference between the desired output and the actual one. Steady-state error is an error that still persists in steady-state condition, at t →∞. • No.2, FPE (5th Ed.), 4.20. • Deadline: 02.10.2012, at 07:30.

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