Steady-state error

The total response of the system is always the sum of the transient and steady-state eomponents. Figure 3.1 shows the transient and steady-state periods of time response. Differenees between the input funetion X[ t) (in this ease a ramp funetion) and system response Xo t) are ealled transient errors during the transient period, and steady-state errors during the steady-state period. One of the major objeetives of eontrol system design is to minimize these errors. 

The first term in equation (3.39) represents the input quantity, the seeond is the steady-state error and the third is the transient eomponent. When time t is expressed as a ratio of time eonstant T, then Table 3.3 and Figure 3.15 ean be eonstrueted. In Figure 3.15 the distanee along the time axis between the input and output, in the steady-state, is the time eonstant. 

In practice, there will always be transient errors, but the transient period should be kept as small as possible. It is usually possible to design the controller so that steady-state errors are minimized, or ideally, eliminated. 

Henee, for the system to have zero steady-state error, the terms in equation (4.64) should be... 

This ean only happen if the open-loop gain eonstant K K is infinite. In praetiee this is not possible and therefore the proportional eontrol system proposed in Figure 4.23 will always produee steady-state errors. These ean be minimized by keeping the open-loop gain eonstant K K as high as possible. 

For a first-order plant, proportional eontrol will always produee steady-state errors. This is diseussed in more detail in Chapter 6 under system type elassifieation where equations (6.63)-(6.65) define a set of error eoeffieients. Inereasing the open-loop... 

Including a term that is a function of the integral of the error can, with the type of plant shown in Figure 4.23, eliminate steady-state errors. 

Thus, when r t) and f2(f) are unehanging, or have step ehanges, there are no steady-state errors as ean be seen in Figure 4.25. The seeond-order dynamies of the elosed-loop system depend upon the values of T[, T, K and K. Again, a high value of K will provide a fast transient response sinee it inereases the undamped natural frequeney, but with higher order plant transfer funetions ean give rise to instability. 

The root locus method provides a very powerful tool for control system design. The objective is to shape the loci so that closed-loop poles can be placed in the. v-plane at positions that produce a transient response that meets a given performance specification. It should be noted that a root locus diagram does not provide information relating to steady-state response, so that steady-state errors may go undetected, unless checked by other means, i.e. time response. 

Without stabilization, the step response of the roll dynamies produees a 45% overshoot and a settling time of 10 seeonds. The stabilization eontrol system is required to provide a step response with an overshoot of less than 25%, a settling time of less than 2 seeonds, and zero steady-state error. 

Frequeney domain analysis is eoneerned with the ealeulation or measurement of the steady-state system output when responding to a eonstant amplitude, variable frequeney sinusoidal input. Steady-state errors, in terms of amplitude and phase relate direetly to the dynamie eharaeteristies, i.e. the transfer funetion, of the system. 

From the final value theorem given in equation (3.10) it is possible to define a set of steady-state error coefficients. 

Table 6.4 Relationship between input funetion, system type and steady-state error...

Table 6.4 shows the relationship between input funetion, system type and steady-state error. From Table 6.4 it might appear that it would be desirable to make most systems type two. It should be noted from Figure 6.22(e) that type two systems are unstable for all values of K, and will require some form of eompensation (see Example 6.6). 

Set Kto a. suitable value so that any steady-state error eriteria are met. 

The compensated and uncompensated open-loop frequency response is shown in Figure 6.41. From this Figure the compensated gain margin is 12.5 dB, and the phase margin is 48°. In equation (6.117), K does not need to be adjusted, and can be set to unity. When responding to a step input, the steady-state error is now 4.6%. 

Figure 9.13 indicates the burner temperature time response The temperature falls from its initial value, since the gas valve is closed, and then climbs with a response indicated by the eigenvalues in equation (9.93) to a steady-state value of 400 °C, or a steady-state error of 50 °C. 

As the band dryer is a type zero system, and there are no integrators in the controller, steady-state errors must be expected. ITowever, the selection of the elements in the Q matrix, equation (9.90), focuses the control effort on control-... 

With the trolley displaeement and veloeity shown in Figures 10.15(e) and (d), the state feedbaek, although oseillatory, give the best results sinee there is no steady-state error. The positional error for both rulebases inereases with time, and there is a eonstant veloeity steady-state error for the 11 set rulebase, and inereasing error for the 22 set rulebase. Figure 10.15(e) shows the eontrol foree for eaeh of the three strategies. 

We expect a system with only a proportional controller to have a steady state error (or an offset). A formal analysis will be introduced in the next section. This is one simplistic way to see why. Let s say we change the system to a new set point. The proportional controller output, p = ps + Kce, is required to shift away from the previous bias ps and move the system to a new steady state. For p to be different from ps, the error must have a finite non-zero value.3... 

On the plus side, the integration of the error allows us to detect and eliminate very small errors. To make a simple explanation of why integral control can eliminate offsets, refer back to our intuitive explanation of offset with only a proportional controller. If we desire e = 0 at steady state, and to shift controller output p away from the previous bias ps, we must have a nonzero term. Here, it is provided by the integral in Eq. (5-5). That is, as time progresses, the integral term takes on a final nonzero value, thus permitting the steady state error to stay at zero. 

We now take a formal look at the steady state error (offset). Let s consider a more general step change in set point, R = M/s. The eventual change in the controlled variable, via the final value theorem, is... 

Let s take another look at the algebra of evaluating the steady state error. The error that we have derived in the example is really the difference between the change in controlled variable and the change in set point in the block diagram (Fig. 5.6). Thus we can write ... 

Now if we have a unit step change R = 1/s, the steady state error via the final value theorem is (recall that e = e )... 

We name Kerr the position error constant.1 For the error to approach zero, Kcrr must approach infinity. In Example 5.1, the error constant and steady state error are... 

Lastly, we should see immediately that the system steady state gain in this case is the same as that in Example 5.1, meaning that this second order system will have the same steady state error. 

The time integral is from t = 0 to t = °°, and we can only minimize it if it is bounded. In other words, we cannot minimize the integral if there is a steady state error. Only PI and PID controllers are applicable to this design method.1... 

If you come across a proportional controller here, it is only possible if the derivation has ignored the steady state error, or shifted the reference such that the so-called offset is zero. 

Not only is this equation identical to the form in Eq. (9-16), but we also can interpret the analysis as equivalent to a problem where we want to find K such that the steady state error e(t) approaches zero as quickly as possible. 

Example 4.7C Add integral action to the system in Example 4.7B so we can eliminate the steady state error. 

Do the time response simulation in Example 7.5B. We found that the state space system has a steady state error. Implement integral control and find the new state feedback gain vector. Perform a time response simulation to confirm the result. 

The modest 10% offset that we have in the slave loop is acceptable under most circumstances. As long as we have integral action in the outer loop, the primary controller can make necessary adjustments in its output and ensure that there is no steady state error in the controlled variable (e.g., the furnace temperature). 

Derivative cannot be used alone as a control mode. This is because a steady-state input produces a zero output in a differentiator. If the differentiator were used as a controller, the input signal it would receive is the error signal. As just described, a steady-state error signal corresponds to any number of necessary output signals for the positioning of the final control element. Therefore, derivative action is combined with proportional action in a manner such that the proportional section output serves as the derivative section input. 

The performance of the single-ended analog proportional temperature controller in the stabilization mode is shown in Fig. 6.5. The ambient temperature was ramped from -40 to 120 °C in steps of 5 °C. A control voltage of 1.53 V was applied, which produced a microhotplate temperature of 331 °C. The steady-state error of the proportional temperature controller over the operating temperature range is less than 1% of the preset microhotplate temperature. 

The steady-state error of the system is the deviation at infinite time after the disturbance. For a unit change in set-point the error is... 

The same result obtains with three-mode control. On the other hand the combination of proportional and derivative control gives the same steady state error as proportional alone, since the derivative contribution disappears at low frequency. 

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