ME 176Control Systems Engineering

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Mechanical Engineering

Steady-State Errors

Background: Design Process

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Background: Analysis & Design Objectives

"Analysis is the process by which a system's performance is determined."

"Design is the process by which a systems performance is created or changed."

Transient ResponseSteady State Response

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Mechanical Engineering

Steady State Error Stability

Background: Steady-State Error

Definition : is the difference between the input and the output for a prescribed test input as t approaches infinity.

Scope :

Linear - the relationship between the input and the output of the system satisfies the superposition property. If the input to the system is the sum of two component signals:

In general:

If, then,

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Mechanical Engineering

Background: Steady-State Error

Scope :

Time invariant systems - are systems that can be modeled with a transfer function that is not a function of time except expressed by the input and output.

"Meaning, that whether we apply an input to the system now or T seconds from now, the output will be identical, except for a time delay of the T seconds. If the output due to input x (t ) is y (t ), then the output due to input x (t − T ) is y (t − T ). More specifically, an input affected by a time delay should effect a corresponding time delay in the output, hence time-invariant."

STABLE

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Background: Steady-State Error

Test Inputs :

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Evaluating: Steady-State Error

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1. Step Input:Output 1 : No Steady-State ErrorOutput 2 : Constant Steady-State Error of e2

2. Ramp InputOutput 1 : No Steady-State ErrorOutput 2 : Constant Steady-State Error of e2

Output 3 : Infinite Steady-State Error

Representation: Steady-State Error

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R(s) and C(s) : Input and Output Respectively E(s) : Steady-State Error

a) General Representation:

T(s) : Closed loop transfer function

b) Unity Feedback SystemsG(s): Open loop transfer function

Sources: Steady-State Error

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Mechanical Engineering

Scope : Errors arising from configuration of the system itself and the type of applied input.

a) Pure Gain : there will always be a steady state error for a step input

b) Integrator : can have a zero steadystate error for a step input

Defining: Steady-State Error for Unity Feedback

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Example: Steady-State Error for Unity Feedback Steady-state error for a unit step input:

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Defining: Steady-State Error for Unity Feedback

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Example: Steady-State Error for Unity Feedback Find the steady-state errors for inputs of 5u(t), 5tu(t), and 5t^2u(t). The function u(t) is the step function.

Note Laplace transforms:

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Defining: Static Error Constants for Unity Feedback Position ConstantVelocity Constant

Acceleration Constant

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Example: Static Error Constants for Unity Feedback

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Example: Static Error Constants for Unity Feedback

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Mechanical Engineering

Example: Static Error Constants for Unity Feedback

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Mechanical Engineering

System Types forUnity Feedback: Given the system shown, the "system type" is defined as thevalue of "n" in the denominator;or, equivalently the number of pure integrations in the feedforward path.

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Specifications: Steady-State Error

"Static error constants can be used to specificy the steady-state error characteristics of a control system."

Knowing Kp = 1000 what can be learned of the system:

1. System is stable.2. System is Type 03. Input Test signal is step.4. Error per unit step:

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Example: Steady-State Error Specification Find K so that there is a 10% error in steady state.

Since system is Type 1, error stated must apply to ramp function.

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Analysis: Steady-State Error for Disturbances

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"Steady-state error produced by a step function can be reduced by increasing the gain of G1(s) or decreasing the gain of G2(s)."

Example: Steady-State Error for Disturbances Find the steady-state error component due to a step disturbance.

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Mechanical Engineering

Definition: Steady-State Error for Nonunity Feedback

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Move R(s) to right of summing junction.

Compute resulting G(s) and H(s).

Add and subtract unity feedback paths.

Combine negative feedback path to H(s).

Combine feedback system consisting of G(s) and [H(s)-1].

Example: Steady-State Error for Nonunity Feedback Find system type, appropriate error constant, steady-state error for unit step input.

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Definition: Steady-State Error for Nonunity Feedback w/ Disturbances General form: For step input and step distrubances:

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Definition: Steady-State Error for Nonunity Feedback w/ Disturbances For zero error:

1. System is stable2. G1(s) is type 1.3. G2(s) is type 0.4. H(s) is type 0 with a dc gain of unity.

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Definition: Steady-State Error for Nonunity Feedback w/ Disturbances Steady-state value of the actuating signal Ea1(s)::

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Example: Steady-State Error for Nonunity Feedback w/ Disturbances Find the steady-state actuating signal for unity step input. Repeat for unit ramp input: Step: Ramp:

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Definition: Sensitivity

"The degree to which changes in system parameters affect system transfer functions, and hence performance."

A system with zero sensitivity is ideal.Greater the sensitivity, the less desirable.

"The ratio of the fractional change in the function to the fractional change in parameter as the fractional change of parameters approaches zero"

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Example: Sensitivity Calculate sensitivity of the closed-loop transfer function to changes in parameter a:

Closed-loop transfer function:

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Example: Sensitivity Calculate sensitivity of the closed-loop transfer function to changes in parameter K and a, with ramp inputs:

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