Control Response Patterns

Control Response Patterns Eng. R. L. Nkumbwa Copperbelt University 2010

System Metrics and Time-Domain Analysis • System Metrics • Time-Domain Analysis • Time Response • Poles and Zeros • Transient Response Eng. R. L. Nkumbwa Coperbelt University, School of Technology

System Metrics • So, What are System Metrics? • When a system is being designed and analyzed, it doesn't make any sense to test the system with all manner of strange input functions or to measure all sorts of arbitrary performance metrics. • Instead, it is in everybody's best interest to test the system with a set of standard, simple, reference functions. Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Control Systems Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Eng. R. L. Nkumbwa Coperbelt University, School of Technology

System Metrics • Once the system is tested with the reference functions, there are a number of different metrics that we can use to determine the system performance. • So, what are the examples of such metrics? Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Time-Response Analysis • Since time is used as an independent variable in most control systems, it is usually of interest to evaluate the state and the output responses with respect to time or simply, the Time-Response. • In control system design analysis, a reference input signal is applied to a system and the performance of the system is evaluated by studying the system response in the time-domain. Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Time-Response • The time-response of a control system is usually divided into two parts namely; the Steady-State Response and the Transient Response. • In other words, the output response of a system is the sum of two responses: the forced response (steady-state response) and the natural response (zero-input response). Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Time-Response of an Elevator Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Transient Response • Defined as the part of the time response that goes to zero as time goes to infinity. Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Steady-State Response • Defined as the part of the total response that remains after the transient has died out. Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Poles and Zeros • The poles of a transfer function are: • (1) The values of the Laplace transform variable, s , that cause the transfer function to become infinite, or • (2) Any roots of the denominator of the transfer function that are common to roots of the numerator. • The zeros of a transfer function are: • (1) The values of the Laplace transform variable, s , that cause the transfer function to become zero, or • (2) Any roots of the numerator of the transfer function that are common to roots of the denominator. Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Response blueprint • A pole of the input function generates the form of the forced response. • A pole of the transfer function generates the form of the natural response. • A pole on the real axis generates an exponential response. • The zeros and poles generate the amplitudes for both the forced and natural responses. Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Natural response Forced response Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Standard Input Signals • All of the standard inputs are zero before time zero. All the standard inputs are causal. • So, what is causal? • Causal: A system whose output does not depend on future inputs. All physical systems must be causal. Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Standard Input Signals • There are a number of standard inputs that are considered simple enough and universal enough that they are considered when designing a control system. • These inputs are known as a unit step, a ramp, and a parabolic input functions. Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Unit Step Function • A unit step function is defined piecewise as such: • The unit step function is a highly important function, not only in control systems engineering, but also in signal processing, systems analysis, and all branches of engineering. Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Unit Step Function Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Ramp Input Function • A unit ramp is defined in terms of the unit step function, as such: r(t) = tu(t). • It is important to note that the ramp function is simply the integral of the unit step function: Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Ramp Input Function Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Parabolic Input Function • A unit parabolic input is similar to a ramp input: • Notice also that, the unit parabolic input is equal to the integral of the ramp function: Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Parabolic Input Function Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Steady State • To be more precise, we should have taken the limit as t approaches infinity. However, as a shorthand notation, we will typically say "t equals infinity", and assume the reader understands the shortcut that is being used. Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Steady State • When a unit-step function is input to a system, the steady state value of that system is the output value at time t = ∞. • Since it is impractical (if not completely impossible) to wait till infinity to observe the system, approximations and mathematical calculations are used to determine the steady-state value of the system. Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Steady State • Most system responses are asymptotic, that is, the response approaches a particular value. Systems that are asymptotic are typically obvious from viewing the graph of that response. Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Step Response • The step response of a system is most frequently used to analyze systems and there is a large amount of terminology involved with step responses. • When exposed to the step input, the system will initially have an undesirable output period known as the transient response. • The transient response occurs because a system is approaching its final output value. • The steady-state response of the system is the response after the transient response has ended. Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Step Response • It is common for a systems engineer to try and improve the step response of a system. • In general, it is desired for the transient response to be reduced. Eng. R. L. Nkumbwa Coperbelt University, School of Technology

First-Order Systems Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Initial Conditions are zero Eng. R. L. Nkumbwa Coperbelt University, School of Technology

First-Order Systems Response Eng. R. L. Nkumbwa Coperbelt University, School of Technology

System Response K (1 − e−t /τ ) System response. K = gain Response to initial condition Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Unit Step Response Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Unit Step Response • The time constant can be described as the time for to decay to 37% of its initial value. Alternately, the time is the time it takes for the step response to rise to 67% of its final value. • The reciprocal of the time constant has the units (1/seconds), or frequency. Thus, we call the parameter a the exponential frequency. Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Time Constant Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Second-Order Systems Response ζ = 0 Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Step Response Vs. Pole Location Eng. R. L. Nkumbwa Coperbelt University, School of Technology

System Response Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Target Value • The target output value is the value that our system attempts to obtain for a given input. • This is not the same as the steady-state value, which is the actual value that the target does obtain. • The target value is frequently referred to as the reference value, or the "reference function" of the system. • In essence, this is the value that we want the system to produce. Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Example of an Elevator • When we input a "5" into an elevator, we want the output (the final position of the elevator) to be the fifth floor. • Pressing the "5" button is the reference input, and is the expected value that we want to obtain. • If we press the "5" button, and the elevator goes to the third floor, then our elevator is poorly designed. Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Time-Domain Specifications • So, what are Time-Domain Specifications? Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Time-Domain Specifications Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Rise Time • Is the amount of time that it takes for the system response to reach the target value from an initial state of zero. • Rise time is defined as the time for the waveform to go from 0.1 to 0.9 of its final value. • Rise time is typically denoted tr, or trise. • This is because some systems never rise to 100% of the expected target value and therefore, they would have an infinite rise-time. Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Settling Time • After the initial rise time of the system, some systems will oscillate and vibrate for an amount of time before the system output settles on the final value. • The amount of time it takes to reach steady state after the initial rise time is known as the settling time • Which is defined as the time for the response to reach and stay within, 2% (or 5%) of its final value. • Damped oscillating systems may never settle completely. Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Settling time Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Peak Time • The time required to reach the first or maximum peak. Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Percent Overshoot • The amount that the waveform overshoots the steady-state or final value at the peak time, expressed as a percentage of the steady-state value. Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Pole-Zero Plots Eng. R. L. Nkumbwa Coperbelt University, School of Technology

Control Response Patterns