2. Classical Control System Design
2.5. Classical control system design
For SISO systems we have the following partial list of typical classical performance specifica- tions. Consider the feedback loop of Fig.2.20. These are the basic requirements for a well- designed control system:
1. The transient response is sufficiently fast.
2. The transient response shows satisfactory damping.
3. The transient response satisfies accuracy requirements, often expressed in terms of the error constants of §2.2(p.60).
4. The system is sufficiently insensitive to external disturbances and variations of internal parameters.
These basic requirements may be further specified in terms of both a number of frequency-domain specifications and certain time-domain specifications.
Figures2.12(p.73) and2.21illustrate several important frequency-domain quantities:
0 dB -3 dB |H | ωr M frequency B
Figure 2.21: Frequency-domain performance quantities
Gain margin. The gain margin — see §1.4(p.20) — measures relative stability. It is defined as the reciprocal of the magnitude of the loop frequency response L, evaluated at the frequencyωπ at which the phase angle is−180 degrees. The frequency ωπ is called the phase crossover frequency.
Phase margin. The phase margin — again see §1.4— also measures relative stability. It is defined as 180◦plus the phase angleφ1of the loop frequency response L at the frequency ω1where the gain is unity. The frequencyω1is called the gain crossover frequency. Bandwidth. The bandwidth B measures the speed of response in frequency-domain terms. It
is defined as the range of frequencies over which the closed-loop frequency response H has a magnitude that is at least within a factor 12√2 = 0.707 (3 dB) of its value at zero frequency.
Resonance peak. Relative stability may also be measured in terms of the peak value M of the magnitude of the closed-loop frequency response H (in dB), occurring at the resonance frequencyωr.
Figure2.22shows five important time-domain quantities that may be used for performance spec- ifications for the response of the control system output to a step in the reference input:
2.5. Classical control system design 100 % 50 % 0 % 10 % 90 % FVE PO Td Tr Ts time Step response
Figure 2.22: Time-domain quantities
Delay timeTd. delay time measures the total average delay between reference and output. It may for instance be defined as time where the response is at 50% of the step amplitude. Rise timeTr. The rise time expresses the “sharpness” of the leading edge of the response. Var-
ious definitions exist. One defines TRas the time needed to rise from 10% to 90% of the final value.
Percentage overshoot PO. This quantity expresses the maximum difference (in % of the steady-state value) between the transient and the steady-state response to a step input. Settling timeTs. The settling time is often defined as time required for the response to a step
input to reach and remain within a specified percentage (typically 2 or 5%) of its final value.
Final value of error FVE. The FVE is the steady-state position error.
This list is not exhaustive. It includes no specifications of the disturbance attenuating properties. These specifications can not be easily expressed in general terms. They should be considered individually for each application.
Horowitz(1963, pp. 190–194) lists a number of quasi-empirical relations between the time domain parameters Td, Tr, Ts and the overshoot on the one hand and the frequency domain parameters B, M and the phase at the frequency B on the other. The author advises to use them with caution.
Exercise 2.5.1. Cruise control system Evaluate the various time and frequency performance indicators for the integral cruise control system design of Example2.3.3(p.67).
2.5.2. Compensator design
In the classical control engineering era the design of feedback compensation to a great extent relied on trial-and-error procedures. Experience and engineering sense were as important as a thorough theoretical understanding of the tools that were employed.
In this section we consider the basic goals that may be pursued from a classical point of view. In the classical view the following series of steps leads to a successful control system design:
• Investigate the shape of the frequency response P(jω), ω ∈ R, to understand the properties of the system fully.
• Consider the desired steady-state error properties of the system (see §2.2, p.60). Choose a compensator structure — for instance by introducing integrating action or lag compen- sation — that provides the required steady-state error characteristics of the compensated system.
• Plot the Bode, Nyquist or Nichols diagram of the loop frequency response of the compen- sated system. Adjust the gain to obtain a desired degree of stability of the system. M- and N -circles are useful tools. The gain and phase margins are measures for the success of the design.
• If the specifications are not met then determine the adjustment of the loop gain frequency response function that is required. Use lag, lead, lag-lead or other compensation to realize the necessary modification of the loop frequency response function. The Bode gain-phase relation sets the limits.
The graphic tools essential to go through these steps that were developed in former time now are integrated in computer aided design environments.
The design sequence summarizes the main ideas of classical control theory developed in the period 1940–1960. It is presented in terms of shaping loop transfer functions for single-input, single-output systems.
In §2.6(p.82) we consider techniques for loop shaping using simple controller structures — lead, lag, and lead-lag compensators. In § 2.8 (p. 90) we discuss the Guillemin-Truxal design procedure. Section2.9(p.93) is devoted to Horowitz’s Quantitative Feedback Theory (Horowitz and Sidi,1972), which allows to impose and satisfy quantitative bounds on the robust- ness of the feedback system.