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4.1   Optimality of the compensator design

Webster's Collegiate Dictionary defines the word "optimal" as "most desirable," and practical engineering views optimality as a provision for best customer satisfaction.

In application to practical control systems, the controller performance means some combination of accuracy, speed of action, repeatability, reliability, and disturbance rejection. The trade-offs between these requirements are commonly quite clear and/or can be resolved with Bode integrals. The feedback controller design is incomplete without estimation of the theoretically best available performance.

The second kind of design trade-off is that of the controller performance versus complexity. As a rule, compensators and prefilters are many times less expensive than the plants. Hence, in order to improve system performance, it is worthwhile to make them very close to optimal, even at the price of making them complex. The order of the compensator to be reasonably close to the optimal is, typically, 8 to 15.

   

Example 1. Increasing the compensator order from 4 to 12 and including in the compensator several nonlinear links will only add 20 to 30 lines of code to the controller software, or several extra resistors, capacitors and operational amplifiers, if the compensator is analog and may substantially improve the system's performance. For example, if the settling time of some expensive manufacturing machinery with a short repetition cycle of operation can be reduced by 20% while retaining the same accuracy, the resulting time per operation might be reduced by, say, 5%, and the number of pieces of the equipment at the factory can be correspondingly reduced by 5%, with additional savings on maintenance. Or, a fighter's maneuverability can be noticeably improved. Or, the yield of a chemical process can be raised by 2% - etc. This is why the compensators should be designed to provide close to optimum performance, and not only the performance specified by a customer representative who doesn't know in advance what kind of performance might be available.

Even when the accuracy of the system with a simple controller suffices, it still pays to improve the controller, since with larger margins in accuracy, the system will remain operational when some of the system parameters degrade to the point that without the better controller, the system fails.

Linear compensators are fully defined by their frequency responses. Therefore, the problem of optimal linear compensator design is the problem of optimal loop response shaping. The theory of optimal feedback loop shaping should be able to provide the answers to the questions: (a) what performance is feasible, and (b) what loop response achieves this performance limit.

Commonly, control systems are initially designed as linear with some idealized plant model. Still, the design must result in a sound solution when the idealized plant is replaced by the physical plant and actuator models. Physical system models must reflect the uncertainty in the system parameters, the asymptotic behavior of the transfer functions at higher frequencies, the sensor noise, and the nonlinearities.

   

Example 2. In the paper "When is a linear control system optimal?" (see [128] in the bibliography to Ref. [9]) which was considered by many a cornerstone of the so called modern control, the following definition is given: "a feedback system is optimal if and only if the absolute value of the return difference is at least one at all frequencies". Meeting this condition assures dynamic optimality which is rigorously defined in the paper. The problem with this theory is, however, that it cannot be applied to physical systems. In physical systems the loop gain drops faster at higher frequencies than that of an integrator, and according to Bode integral (3.7), the absolute value of the return difference cannot be "at least one at all frequencies."

The process of control law design that deserves to be called optimal must provide timely information to the system engineers about the achievable control system performance and the related possibility of relieving certain requirements to the system hardware, which may permit replacing some initially chosen components and subsystems by simpler and cheaper ones. That is, the optimality requirements must relate to the entire engineering system and not only to the controller in the narrow sense.

   

Example 3. In Chapter 1, we analyzed some design of antenna elevation angle control - without proving that this design is the best possible. (It is not. The real-life controller is multiloop, and includes signal feedforward and high-order nonlinear compensators.) Reasonable questions for the customer to ask are: - Is this design the best? If not, by how much can it be improved?

Normally, we should not even ask: - and, at what cost? - since, as we already mentioned, the cost of the compensator is small compared with the cost of the antenna dish (although better or additional sensors, bearings, and servomotors can add to the system cost).

Also, the loop responses in this example do not include the plant structural modes. Given the modes, the loop gain at higher frequency should be rolled off fast to avoid instability.

The system engineer asked the control designer to find the shape of the control loop response which is optimal for some specific task, and to estimate the available performance without completing the lengthy compensator design. Can this be done?

     
   

Example 4. A flexible mode of a structure may be anywhere between 30 and 40 Hz. By this resonance, the environmental vibration noise is accentuated and the accuracy of the device under control decreases. The feedback system therefore must be able to reject the noise over the entire bandwidth 30 to 40 Hz. How to implement the maximum feedback over this frequency range, and what is the value of this feedback? Can it be quickly calculated without designing the controller?

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