NEXT: Control System Stability
The response in time of a control system is usually divided into two parts: the transient response and the steady-state response. Let y (t) be the response of a system in continuous time, then:
where yt (t) is the transient response, while yss (t) is the steady state response.
The transient response of a control system is important since both its amplitude and its duration must be kept within tolerable or prescribed limits. It is defined as the part of the response in time that tends to zero when the time becomes very large. Thus,
All real stable control systems present a transient phenomenon before reaching the steady state response. For analysis and design purposes it is necessary to assume some basic types of test inputs to evaluate the performance of a system. The proper selection of these test signals allows the prediction of system performance with other more complex inputs. The following signals are used: Step function, which represents an instantaneous change in the reference input; Ramp function, which represents a linear change over time; Parabolic function, which represents a faster order than the ramp. These signals have the common characteristic that they are simple to write in mathematical form, it is rarely necessary or feasible to use faster functions. In Figure 7-1 you can see these functions:
For a linear control system, the analysis and characterization of the transient response is performed frequently using the unit step function Us (t), shown in Figure 7-1a with R = 1. A typical response of a control system to a unit step input is shown in Figure 7-11:
The transient response of a practical control system often exhibits damped oscillations before reaching the steady state. That’s happens because systems have energy storage and cannot responds immediately. The transient-response to a unit step input depends on the initial conditions. That’s why it is a common practice to use the standard initial conditions that the system is at rest initially with the output an all time derivatives thereof zero.
Second-order systems and Transient-response specifications.
Figure 5-5a shows a Servo System as an example of a second-order system. It consists of a proportional controller and load elements (inertia and viscous friction elements):
The closed-loop Transfer Function of the system shown in Figure 5-5c is:
In the transient-response analysis it is convenient to write:
Where σ is called the attenuation; ωn is the undamped natural frequency; and ζ the damping ratio of the system. ζ is the ratio of the actual damping B to the critical damping Bc equal to two times the square-root of JK:
In terms of ωn y σ, the system shown in Figure 5-5c can be expressed as Figure 5-6:
Now, the Transfer Function C(s)/R(s) can be written as:
This form is called The Standard Form. The dynamic behavior of a second-order system can be now described in terms of the two parameters ωn and σ. In short, the cases of second-order response as a function of σ are summarized in Figure 4.11 (for a better review see FIRST and SECOND ORDER SYSTEMS):
In specifying the transient-response characteristics of a control system to a unit-step input, it is common to specify the following parameters associated with the underdamped response:
- Delay time, Td
- Rise time, Tr
- Peak time, Tp
- Percent overshoot (%OS) or Maximum overshoot (Mp)
- Settling time, Ts
These specifications are defined as follows:
Delay time (Td): it is the time required for the response to reach half the final value the very first time.
Rise time (Tr): it is the time required for the response to rise from 10% to 90%. In other words, to go from 0.1 of the final value to 0.9 of the final value.
Peak time (Tp): it is the time required for the response to reach the first peak of the overshoot.
Maximum overshoot (Mp): it is the maximum peak value of the response curve measured from unity. It is also the amount that the waveform overshoots the final value, expressed as a percentage of the steady-state value.
Settling time (Ts): it is the time required for the transient damping oscillations to reach and stay within ±2% or ±5% of the final or steady-state value.
These specifications are graphically shown in Figure 5-8:
It is important to remark that these specifications don’t necessarily apply to any given case. For example, the terms peak time and maximum overshoot do not apply to overdamped systems.
Except for certain applications where oscillations can’t be tolerated, it is desirable that the transient-response be sufficiently fast and sufficiently damped. Thus, for a desirable transient response of a second-order system, the damping ratio must be between 0.4 and 0.8. Small values of σ (σ<0.4) yields excessive overshoot in the transient response, and systems with a large value of σ (σ>0.8) responds sluggishly. We will also see that the maximum overshoot and the rise time conflict with each other. In other words, they cannot be made smaller simultaneously.
Rise time (Tr):
where ωd is the damped natural frequency:
and ß is defined by the Figure 5-9:
Peak time (Tp):
Settling time (Ts):
Transient-response of Higher-order systems.
It could be seen that the transient response of a system higher than a second-order is the sum of the responses of first-order and second order systems.
Transient-response of a First-order system.
We briefly discuss the transient response of a first-order system. A first-order system without zeros can be described by the transfer function shown in Figure 4.4(a).
If the input is a unit step, where R(s)=1, the Laplace transform of the step response is C(s), where:
Taking the inverse transform:
Figure 4-5 shows a typical response of this system to a unit step input:
We call 1/a the time constant of the response. The parameter a is the only one needed to describe the transient response for a first-order system. Thus, the time constant can be considered a transient response specification for a first order system, since it is related to the speed at which the system responds to a step input. Since the pole of the transfer function is at a, we can say the pole is located at the reciprocal of the time constant, and the farther the pole from the imaginary axis, the faster the transient response.
The other specifications for a first-order system are:
Rise time (Tr):
Settling time (Ts):
NEXT: Control System Stability
- Control Systems Engineering, Nise pp 177-181
- Sistemas de Control Automatico Benjamin C Kuo p 385,
- Modern_Control_Engineering, Ogata 4t pp 224, 232
Written by: Larry Francis Obando – Technical Specialist – Educational Content Writer.
Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, Caracas.
Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, Valle de Sartenejas.
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