Specifications, closedloop

PERFORMANCE OF FEEDBACK CONTROLLERS 7.2.1 Specifications for Closedloop Response... 

There are a number of criteria by which the desired performance of a closedloop system can be spedlied in the time domain. For example, we could specify that the closedloop system be critically damped so that there is no overshoot or oscillation. We must then select the type of controller and set its tuning constants so that it will give, when coupled with the process, the desired closedloop response. Naturally the control specification must be physically attainable. We cannot make a Boeing 747 jumbo jet airplane behave like an F-IS fighter. We cannot... 

The dynamic performance of a system can be deduced by merely observing the location of the roots of the system characteristic equation in the s plane. The time-domain specifications of time constants and damping coefficients for a closedloop system can be used directly in the Laplace domain. 

Equation (11.65) is a general solution for any process and for any desired closedloop servo transfer function. Plugging in the values for and for the specific example gives... 

The most useful frequency-domain specification is the maximum closedloop fog modulus. The phase margin and gain margin spedfications can sometimes give poor results when the shape of the frequency-response curve is unusual. 

A commonly used maximum closedloop log modulus specification is 4 2 dB. The controller parameters are adjusted to give a maximum peak in the closedloop servo log modulus curve of -1-2 dB. This corresponds to a magnitude ratio of 1.3 and is approximately equivalent to an underdamped system with a damping coefficient of 0.4,... 

The lines of constant closedloop log modulus L, are part of the Nichols chart. If we are designing a closedloop system for an L specification, we merely have to adjust the controller type and settings so that the openloop B curve is tangent to the desired line on the Nichols chart. For example, the G B curve in Fig. 13,11b with X, = 20 is just tangent to the +2 dB line of the Nichols chart. The value of frequency at the point of tangency, 1.1 radians per minute, is the closedloop resonant frequency aif. The peak in the log modulus plot is clearly seen in the closedloop curves given in Fig. 13.12. 

A specific example will illustrate how the output of the closedloop system can be obtained in the z domain. 

These parameters are varied to achieve some desired performance criteria. In the z-plane root locus plots, the specifications of closedloop time constant and damping coefficient are usually used. The roots of the closedloop characteristic equation 1 -I- are modified by changing. ... 

Designing the secondary (slave) loop. We pick a closedloop damping coefficient specification for the secondary loop of 0.707 and calculate the required value of K. The closedloop characteristic equation for the slave loop is... 

There are two basic types of specifications commonly used in the frequency domain. The first type, phase margin and gain margin, specifies how near the openloop GM iu)Gc ia)) polar plot is to the critical (- 1,0) point. The second type, maximum closedloop log modulus, specifies the height of the resonant peak on the log modulus Bode plot of the closedloop servo transfer function. So keep the apples and the oranges straight. We make openloop transfer function plots and look at the (- 1, 0) point. We make closedloop servo transfer function plots and look at the peak in the log modulus curve (indicating an underdamped system). But in both cases we are concerned with closedloop stability. 

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