Dr. Boris J. Lurie


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High-order nonlinear controllers for SISO and MIMO systems provide substantially better performance than low-order linear controllers, even when the plant is low-order and linear.

Classical control in many textbooks is persistently misrepresented as only a collection of primitive design rules (root locus method, PID control) resulting in low performance servos.

A better definition of the classical control is: "best frequency-domain design methods developed before 1960, and their extentions". This definition includes high-order feedback loop design based on Bode integrals which was fully developed in the 1940s and 50s and widely employed in the telecommunication industry; frequency-domain nonlinear methods developed in the 50s and in later years; and the design methods based on the frequency-domain Popov criterion developed in the 60s-80s. A framework integrating these methods enables designing high-performance controllers.

The following lists some critical issues related to the classical control scope, applicability, methods, performance, and inferiority/superiority to the time-domain (modern, optimal) control. (See also Appendix 11 of Classical Feedback Control, where many more discussion issues are presented in question and answer format.)

  1. The major goal for the application feedback in control is, typically, disturbance rejection, and
  2. in well designed systems, typically, the feedback bandwidth is limited by the plant uncertainty.
  3. In this case the loop response must have Bode step for higher performance.

2. Well-designed high-order compensation provides better approximation to those loop shapes which are optimal (when judged using Bode integrals) and, consequently, substantially better control system performance.

3. High-order compensation is robust when the prescribed stability margins are defined by appropriate continuous boundaries and not by guard-point margins, and the compensator is cascaded appropriately from several links.

4. Numerical problems do not prohibit using high-order transfer functions in compensators. (In analog telecommunication systems, compensators with the order over 20 have been used routinely, with the approximation errors smaller than 0.002dB.)

5. Bessel-filter performance of the closed-loop transient response (nearly no overshoot and small rise time) can be achieved without sacrificing the disturbance rejection and sensitivity. This can be done by application of a prefilter, or a feedforward path, or a nonlinear dynamic compensator in the feedback loop.

6. Bode integrals allow precise and rapid estimation of the best available performance in advance, without designing the control system. This greatly simplifies conceptual design of engineering systems of which the control systems are subsystems.

7. Using Bode integrals, the problem of optimal loop shaping can be divided into two simple sequential steps: shaping the loop first in the range of positive feedback and then in the range of negative feedback. This drastically simplifies the design and is an important advantage over the H method.

8. In accordance with the Bode integrals, better disturbance rejection in most cases should be expected from Nyquist-stable systems.

9. A Nyquist-stable system is not (generally) conditionally stable. It can (and should) easily be made globally stable by application of nonlinear dynamic compensation.

10. Nonlinear dynamic compensation allows optimizing the system performance in both acquisition and tracking regimes, without a trade-off sacrifice.

11. The accuracy of the describing functions analysis and synthesis of a system to be well designed is more than adequate since, in accordance with the Bode integrals, its loop response must be filter-type.

12. The Popov criterion can be used for rigorous design of high-performance high-order nonlinear dynamic compensation with close to the best possible disturbance rejection and, at the same time, nearly best possible transient responses to large commands and disturbances.

13. In the MIMO system stability analysis, the largest uncertainty of the loop transfer function to be taken into account results from nonlinear properties of the actuators. (When an/the actuator saturates/not saturatres, its equivalent gain reduces by orders of magnitude.)

14. The often heard statement that classical control methods are not applicable to MIMO control systems design is incorrect. (All of the more than 300 feedback loops in a common TV set are designed with classical feedback methods.)

  1. There exist convenient rules for response shaping that guarantee minimum-phase character of parallel connection of minimum phase links while providing best selectivity.
  2. Without application of these rules, the failure of the design of a MIMO system including various sensors and actuators, is most probable.
  3. Ignorance of these rules is one of the reasons classical control is not considered by some to be applicable to MIMO system design.

16. The number of engineering analog feedback systems exceeds that of digital control systems by more than an order of magnitude. (The number of operational amplifiers manufactured yearly exceeds the number of digital processors much more than by an order of magnitude. Each op-amp has several internal and at least one external feedback loop.)

17. It is important to use nested feedback loops for shaping the output impedance of the actuator, in order to reduce disturbances and the plant uncertainty, and to substantially increase the available feedback about the plant.

  1. Systems with multiple coupled actuators using compound feedback in elementary loops are much more robust and tolerant of large plant parameter variations than the systems using only force or only position sensors.
  2. Biological motion control systems use compound feedback.

In the opinion of the authors of Classical Feedback Control, all these statements are correct. They are discussed in some depth in Classical Feedback Control and in Feedback Maximization.