Categoría: Time Domain

The mechanical system shown in Figure P5.52(a) is used as part of the unity feedback system shown in Figure P5.52(b). Find the values of M and D to yield 20% overshoot and 2 seconds settling time.

1. System Dynamic

where:

2. Laplace Transform

3. Motor&Load Transfer Function (θm(s)/Ea(s))

4. Direct Transfer Function

For the system:

The open-loop transfer function Ga(s) is:

5. Closed-loop transfer function

The closed-loop transfer function Gc(s) is:

That is to say:

6. Calculation of M and D

According to:

Besides:

In this way:

Meanwhile:

7. Matlab verification

We use Matlab to corroborate replacing all the values calculated in the original transfer function:

Find the values of M and D to yield 20% overshoot and 2 seconds settling time.

>> stepinfo (sys)

RiseTime: 0.3554

SettlingTime: 1.8989

SettlingMin: 0.9331

SettlingMax: 1.1999

Overshoot: 19.9890

Undershoot: 0

Peak: 1.1999

PeakTime: 0.8059

Written by Larry Francis Obando – Technical Specialist – Educational Content Writer

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Errors in a control system can be attributed to many factors. Changes in the reference input will cause unavoidable errors during transient periods and may also cause steady-state errors. Imperfections in the system components, such as static friction, backlash, and amplifier drift, as well as aging or deterioration, will cause errors at steady state. In this section, however, we shall not discuss errors due to imperfections in the system components. Rather, we shall investigate a type of steady-state error that is caused by the incapability of a system to follow particular types of inputs.

Steady-state error is the difference between the input and the output for a prescribed test input as time tends to infinity. Test inputs used for steady-state error analysis and design are summarized in Table 7.1. In order to explain how these test signals are used, let us assume a position control system, where the output position follows the input commanded position.

Step inputs represent constant position and thus are useful in determining the ability of the control system to position itself with respect to a stationary target. An antenna position control is an example of a system that can be tested for accuracy using step inputs.

Ramp inputs represent constant-velocity inputs to a position control system by their linearly increasing amplitude. These waveforms can be used to test a system’s ability to follow a linearly increasing input or, equivalently, to track a constant velocity target. For example, a position control system that tracks a satellite that moves across the sky at a constant angular velocity.

Parabolas inputs, whose second derivatives are constant, represent constant acceleration inputs to position control systems and can be used to represent accelerating targets, such as a missile.

Any physical control system inherently suffers steady-state error in response to certain types of inputs. A system may have no steady-state error to a step input, but the same system may exhibit nonzero steady-state error to a ramp input. (The only way we may be able to eliminate this error is to modify the system structure.) Whether a given system will exhibit steady-state error for a given type of input depends on the type of open-loop transfer function of the system.

Definition of the error in steady state depending on the configuration of the system.

The steady-state errors of linear control systems depend on the type of the reference signal and the type of system. Before undertaking the error in steady state, it must be clarified what is the meaning of the system error.

The error can be seen as a signal that should quickly be reduced to zero, if this is possible. Consider the system of Figure 7-5:

Where r (t) is the input signal, u (t) is the acting signal, b (t) is the feedback signal and y (t) is the output signal. The error e (t) of the system can be defined as:

We must remember that r (t) and y (t) do not necessarily have the same dimensions. On the other hand, when the system has unit feedback, H (s) = 1, the input r (t) is the reference signal and the error is simply:

That is, the error is the acting signal, u (t). When H (s) is not equal to 1, u (t) may or may not be the error, depending on the form and purpose of H (s). Therefore, the reference signal must be defined when H (s) is not equal to 1.

The error in steady state is defined as:

To establish a systematic study of the error in steady state for linear systems, we will classify the control systems as follows:

1. Unit feedback systems,
2. Non-unit feedback systems.

Steady-State Error in Unity-feedback control systems

Consider the system shown in Figure 5-49:

The closed-loop transfer function for this can be obtained as:

The transfer function between the error signal e(t) and the input signal r(t) is:

Where the error e(t) is the difference between the input signal and the output signal. The final-value theorem provides a convenient way to find the steady-state performance of a stable system. Since:

The steady-state error is:

This last equation allows us to calculate the steady-state error Ess, given the input R(s) and the transfer function G(s). We then substitute several inputs for R(s) and then draw conclusions about the relationship that exists between the open-loop system G(s)  and the nature of the steady-state error Ess.

• Step Input: Using R(s) =1/S, we obtain:

Where:

Is the gain of the forward transfer function. In order to have zero steady-state error,

To satisfy the last condition, G(s) must have the following form:

And for the limit to be infinite the denominator must be equal to zero a S goes to zero. So n>=1, that is, at least one pole must be at the origin, equal to say that at least one pure integration must be present in the forward path. The steady-state respond for this case of zero steady-state  error is similar to that shown in Figure 7-2a, ouput 1.

If there are no integrations, the n=0, and it yields a finite error. This is the case shown in Figure 7-2a, output 2.

In summary, for a step input to a unity feedback system, the steady-state error will be zero if there is at least one pure integration in the forward path.  

• Ramp Input:  Using R(s) =1/Sˆ2, we obtain:

To have zero steady-state error to a ramp input we must have:

To satisfy this G(s) must take the form where n>=2. In other words, there must be at least two integrations in the forward path. An example of a steady-state error for a ramp input is shown in Figure 7.2b, output 1:

If only one integrator exist in the forward path then lim sG(s) is finite rather tan infinite and this lead to a constant error, as shown in Figure 7.2b, output 2. If there is only one integrator in the forward path then lim sG(s) =0, and the steady-state error will be infinite and lead to diverging ramp, as shown in Figure 7.2b, output 3.

• Parabolic Input: Using R(s) =1/Sˆ3, we obtain:

In order to have zero steady-state error for a parabolic input, we must have:

To satisfy this G(s) n must be n>=3. In other words, there must be at least three integrations in the forward path. If there only two integrators in the forward path then lim sˆ2G(s) is finite rather tan infinite and this lead to a constant error. If there one or zero integrators in the forward path then e() is infinite.

Classification of Control Systems (System Types) and Static Errors Constant.

System Type.  Control system may be classified according to their ability to follow step inputs, ramp inputs or parabolic inputs and so on. This is a reasonable classification scheme because most of the actual inputs can be considered a combination of such inputs. Consider the unity-feedback control system with the following open-loop transfer function G(s):

It involves the term SˆN in the denominator, representing a pole of multiplicity N at the origin. A system is called type 0, type 1, type 2,…if N=0, 1, 2…respectively. As the type increases, accuracy is improved. However, this agraves the stability problem. If  G(s) is written so that each term in the numerator and denominator, except the term SˆN, approaches unity as s approaches zero, then the open-loop gain K is directly related to the steady-state error.  

Static Error Constant. The Static Error Constants defined in the following are figures of merit of control systems. The higher the constants, the smaller the steady-state error.

• Static Position Error Constant Kp. The steady-state error of a system for a unit-step input is:

The Static Position Error Constant Kp is defined by:

Thus the steady-state error in terms of the Static Position Error Constant Kp is given by:

For a type 0 system:

For a type 1 or higher system:

• Static Velocity Error Constant Kv. The steady-state error of a system for a unit-ramp input is given by:

The Static Velocity Error Constant Kv is defined by:

Thus the steady-state error in terms of the Static Velocity Error Constant Kv is given by:

For a type 0 system:

For a type 1 system:

For For a type 2 system or higher:

• Static Acceleration Error Constant Ka. The steady-state error of a system for a unit-parabolic input is given by:

The Static Acceleration Error Constant Kv is defined by:

Thus the steady-state error in terms of the Static Acceleration Error Constant Ka is given by:

For a type 0 system:

For a type 1 system:

For a type 2 system:

For a type 3 system or higher:

Table 7.2 ties together the concepts of steady-state error, static error constants and system type. The table shows the static error constants and the steady-state error as a functions of the input waveform and the system type.

Steady-State Error for Non-unity Feedback Systems.

Control systems often do not have unity feedback because of the compensation used to improve performance or because of the physical model of the system. In these cases the most practical way to analyze the steady-state error is to take the system and form a unity feedback system by adding and subtracting unity feedback paths as shown in Figure 7.15:

Donde G(s)=G1(s)G2(s) y H(s)=H1(s)/G1(s). Notice that these steps require that input and output signals have the same units.

BEFORE: Control System Stability

NEXT: PID -Basic control system actions

Sources:

1. Control Systems Engineering, Nise pp 340, 353
2. Sistemas de Control Automatico Benjamin C Kuo pp 390, 395
3. Modern_Control_Engineering, Ogata 4t pp 301,305

Written by: Larry Francis Obando – Technical Specialist – Educational Content Writer.

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Related:

The Block Diagram – Control Engineering

Dinámica de un Sistema Masa-Resorte-Amortiguador

Block Diagram of Electromechanical Systems – DC Motor

Transient-response Specifications

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:

[2]

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:

[2]

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):

[3]

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:

[3]

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):

[1]

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:

1. Delay time, Td
2. Rise time, Tr
3. Peak time, Tp
4. Percent overshoot (%OS) or Maximum overshoot (Mp)
5. 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:

[3]

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.

Analytically:
Rise time (Tr):

where ωd is the damped natural frequency:

and ß is defined by the Figure 5-9:

[3]

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).

[1]

If the input is a unit step, where R(s)=1/s, 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:

[1]

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

Source:

1. Control Systems Engineering, Nise pp 177-181
2. Sistemas de Control Automatico Benjamin C Kuo p 385,
3. Modern_Control_Engineering, Ogata 4t pp 224, 232

Written by: Larry Francis Obando – Technical Specialist – Educational Content Writer.

Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, Valle de Sartenejas.

Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, Caracas.

Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.

Contact: Caracas, Quito, Guayaquil, Jaén- Telf. +34 633129287

WhatsApp: +34 633129287

email: dademuchconnection@gmail.com

Attention: If what you need is to solve urgently a problem of a “Mass-Spring-Damper System ” (find the output X (t), graphs in Matlab of the 2nd Order system and relevant parameters, etc.), or a “System of Electromechanical Control “… to deliver to your teacher in two or three days, or with greater urgency … or simply need an advisor to solve the problem and study for the next exam … send me the problem… , …I will give you the answer in digital and I give you a video-conference to explain the solution … it also includes simulation in Matlab. In the link above, you will find the description of the service and its cost.

Related:

The Block Diagram – Control Engineering

Dinámica de un Sistema Masa-Resorte-Amortiguador

Block Diagram of Electromechanical Systems – DC Motor

Control System Stability

Steady-state error in Control Systems