SECTION 4: STEADY STATE ERROR MAE 4421 Control of Aerospace & Mechanical Systems
2 Introduction
Steady State Error Introduction 3 Consider a simple unity feedback system The error is the difference between the reference and the output The input to the controller, Consider a case where: Reference input is a step Plant has no poles at the origin finite DC gain Controller is a simple gain block In steady state, the forward path reduces to a constant gain:
Steady State Error Introduction 4 In steady state, we d like: Output to be equal to the input: Zero steady state error: 0 Is that the case here? 1 1 No, if 0, then 0 Non zero steady state error to a step input for finite steady state forward path gain Finite DC gain implies no poles at the origin in or
Steady State Error Introduction 5 Now, allow a single pole at the origin An integrator in the forward path Now the error is For a step input 1 1 Applying the final value theorem gives the steady state error lim lim 0 Zero steady state error to a step input when there is an integrator in the forward path
Steady State Error Introduction 6 Next, consider a ramp input to the same system 1 and Now the error is 1 The steady state error is 1 lim lim 1 Non zero, but finite, steady state error to a ramp input when there is an integrator in the forward path
Steady State Error Introduction 7 Two key observations from the preceding example involving unity feedback systems: Steady state error is related to the number of integrators in the open loop transfer function Steady state error is related to the type of input We ll now explore both of these observations more thoroughly First, we ll introduce the concept of system type
8 System Type and Steady State Error
System Type 9 System Type The degree of the input polynomial for which the steadystate error is a finite, non zero constant Type 0: finite, non zero error to a step input Type 1: finite, non zero error to a ramp input Type 2: finite, non zero error to a parabolic input For the remainder of this sub section, and the one that follows, we ll consider only the special case of unity feedback systems
System Type Unity Feedback Systems 10 For unity feedback systems, system type is determined by the number of integrators in the forward path Type 0: no integrators in the open loop TF, e.g.: 6 4 48 Type 1: one integrator in the open loop TF, e.g.: 15 312 Type 2: two integrators in the open loop TF, e.g.: 5 3 7
Types of Inputs 11 When characterizing a control system s error performance we focus on three main inputs: Step Ramp Parabola We will derive expressions for the steady state error due to each Step: 1 For a positioning system, this represents a constant position
Types of Inputs 12 Ramp: 1 For a positioning system, this represents a constant velocity Parabola: 1 For a positioning system, this represents a constant acceleration
Steady State Error Unity Feedback 13 For unity feedback systems steady state error can be expressed in terms of the open loop transfer function, 1 Steady state error is found by applying the final value theorem lim lim 1 We ll now consider this expression for each of the three inputs of interest
Steady State Error Step Input 14 For a step input 1 1 Steady state error to a step input is lim lim lim 1 1 1 1
Steady State Error Step Input 15 1 1lim In order to have 0, as we d like, we must have lim That is, the DC gain of the open loop system must be infinite If has the following form then lim and we ll have non zero steady state error
Steady State Error Step Input 16 However, consider of the following form where That is, includes integrators It is a type system lim and 0 A type 1 or greater system will exhibit zero steadystate error to a step input
Steady State Error Ramp Input 17 For a ramp input 1 1 Steady state error to a ramp input is lim lim lim 1 1 1 1 lim
Steady State Error Ramp Input 18 1 lim In order to have 0, the following must be true lim If there are no integrators in the forward path, then and lim lim 0 A type 0 system has infinite steady state error to a ramp input
Steady State Error Ramp Input 19 If there is a single integrator in the forward path, i.e. a type 1 system then and lim A type 1 system has non zero, but finite, steadystate error to a ramp input
Steady State Error Ramp Input 20 If there are two or more integrators in the forward path, i.e. a type 2 or greater system then and, 2 lim lim 0 A type 2 or greater system has zero steady state error to a ramp input
Steady State Error Parabolic Input 21 For a Parabolic input 2 1 1 Steady state error to a parabolic input is lim lim lim 1 1 1 1 lim
Steady State Error Parabolic Input 22 1 lim In order to have 0, the following must be true lim If there are no integrators in the forward path, then and lim lim 0 A type 0 system has infinite steady state error to a parabolic input
Steady State Error Parabolic Input 23 If there is a single integrator in the forward path, i.e. a type 1 system then and lim 0 A type 1 system has infinite steady state error to a parabolic input
Steady State Error Parabolic Input 24 If there are two integrators in the forward path, i.e. a type 2 system then and lim lim A type 2 system has non zero, but finite, steady state error to a parabolic input
Steady State Error Parabolic Input 25 If there are three or more integrators in the forward path, i.e. a type 3 or greater system then and, 3 lim lim 0 A type 3 or greater system has zero steady state error to a parabolic input
26 Static Error Constants
Static Error Constants Unity Feedback 27 We ve seen that the steady state error to each of the inputs considered is Step: Ramp: Parabola: The limit term in each expression is the static error constant associated with that particular input: Position constant: Velocity constant: Acceleration constant: lim lim lim
Steady State Error vs. System Type 28 Steady state error vs. input and system type System Input Type Step Ramp Parabola 0 1 1 1 2 0 1 0 0 3 0 0 0 Note that the given steady state error is for inputs of unit magnitude Actual error is scaled by the magnitude of the reference input 1
29 Non Unity Feedback Systems
Non Unity Feedback Systems 30 So far, we ve focused on the special case of unityfeedback systems System type determined by # of integrators in the forward path i.e., # of open loop poles at the origin Steady state error determined using static error constants Static error constants determined from the open loop transfer function
Non Unity Feedback Systems 31 More general approach to determining steady state error is to use the closed loop transfer function Applicable to non unity feedback systems, e.g.: 1 The error is
Non Unity Feedback Systems 32 Apply the final value theorem to determine the steady state error: Here, system type is determined by using the more general definition: System type is the degree of the input polynomial for which the steady state error is a finite, non zero constant
Non Unity Feedback Systems 33 Alternatively, find steady state error by converting to a unity feedback configuration, e.g.: Add and subtract unity feedback paths:
Non Unity Feedback Systems 34 Combine the two upper parallel feedback paths: Collapsing the inner feedback form leaves a unityfeedback system Can now apply unityfeedback error analysis techniques
35 Steady State Error Examples
Steady State Error Example 1 36 What is the steady state error to a constant reference input, 3 1, for the following feedback positioning system? A type 0 system Non zero error to a constant reference Position constant: lim 10 Steady state error: 1 3 1 1 110 0.27
Steady State Error Example 1 37 0.27
Steady State Error Example 1 38 What is the same system s steady state error to a unit ramp input, 1? A type 0 system, so error to a ramp reference will be infinite Verify using closed loop transfer function 1 10 11 Steady state error is lim 1 lim 1 10 1 11 1 lim 1 11
Steady State Error Example 1 39
Steady State Error Example 2 40 Design the controller,, for error of 0.05 to a unit ramp input Plant is type 0 Forward path must be type 1 for finite error to a ramp input must be type 1, so one very simple option is: Forward path transfer function is 2 1 5
Steady State Error Example 2 41 The velocity constant is Steady state error is For error of 0.05: lim lim 1 5 2 0.05 5 2 50 2 1 5 2 5
Steady State Error Example 2 42 0.05
Steady State Error Example 3 43 Next, consider a non unity feedback system: Determine controller gain,, to provide a 2% steady state error to a constant reference input First, convert to a unityfeedback system Combine forward path blocks Simultaneously add and subtract unity feedback paths
Steady State Error Example 3 44 Combine the top two parallel feedback paths Simplifying the inner feedback form leaves a unity feedback system
Steady State Error Example 3 45 Steady state error for this type 0 system is 1 1 where For 2% steady state error The controller gain is lim 0.02 24.5 20 10 2 1 12
Steady State Error Example 3 46 Note that the controller gain has been set to satisfy a steady state error requirement only Closed loop poles are very lightlydamped Dynamic response is likely unacceptable 0.02
Steady State Error Example 4 47 Now, consider a unity feedback system with a disturbance input where and Determine the controller gain,, such that error due to a constant disturbance is 1% of For this value of, what is the steady state error to a constant reference input?
Steady State Error Example 4 48 The total error is given by 1 1 1 1 Substituting in controller and plant transfer functions gives 5 5 1 5
Steady State Error Example 4 49 Error due to a constant disturbance can be found by applying the final value theorem, lim 1 5, lim 1 1 5 1 5 We can calculate the required gain for 1% error, 0.01 1 5 95 At this gain value, the error due to a constant reference is, lim 1 5 5 5 100 5%