Control System Analysis, Time Domain

Steady-State error – Control Systems

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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