# Some Performance Is Just Not PossibleSome Performance Is Just Not Possible

DN Staff

December 11, 2009

What is feasible and what is infeasible in a given set of circumstances? Can we say something about achievable performance of all possible designs? This is a fundamental engineering question. Central to the design of real systems are inescapable performance limitations that evolve from the laws of nature. The ability to identify performance trade-offs and limitations (at both the component and integrated-system level) and their sources, and quantify their impact, is essential in mechatronics design and control. This information helps one compare potential designs and then optimize a system design to either remove some of the limitations or stretch the bounds imposed by these limitations to a higher level. The importance of fundamental performance limitations in the design, analysis and control of mechatronic systems cannot be overemphasized. Unfortunately, this subject is rarely addressed in any academic courses. Professor Kamal Youcef-Toumi of MIT and Professor Karl Astrom of Lund University have been strong proponents, and the 1997 book "Fundamental Limitations in Filtering and Control" by M. Seron, J. Braslavsky and G. Goodwin is a key reference.

A fundamental example of a performance limitation is given by the well-known Sampling Theorem of Shannon and Nyquist, which states the highest frequency that can be unambiguously represented by discrete samples is half the sampling rate. The X-29 flight control design, cited by Gunter Stein in his Bode Lecture, is an excellent example of the importance of knowing these fundamental limitations. At one flight condition, it was desired to have a 45-degree phase margin. Unfortunately, the system possessed an open-loop unstable pole and a nonminimum-phase zero, and a 45-degree phase margin was infeasible! Much effort was wasted trying to meet this design goal when clearly (at least to the well-grounded controls engineer) it was not attainable!

These fundamental limitations typically take three forms. Some are in the form of algebraic equations, e.g., T(s) + S(s) = 1 holds at all frequencies, where T(s) and S(s) are the complementary and sensitivity transfer functions, respectively. One cannot achieve good disturbance rejection (s) and attenuation of measurement noise (t) both at the same frequency. Other constraints take the form of a frequency-domain integral on a closed-loop transfer function such as S(s). The Bode integral constraint is an example. Reducing the sensitivity at some prescribed frequency range results in large values of sensitivity at other frequency ranges. This trade-off is worse in the presence of open-loop unstable poles and nonminimum-phase zeros. The third type takes the form of time-domain integral constraints on a system signal such as the feedback error.

So what is the big picture here? The core of any mechatronic system design team will consist of mechanical, electronics and computer systems engineers. Control is pervasive and all need to thoroughly understand its benefits and application. Developing simple, integrated design models to evaluate concepts will necessarily involve physical system dynamics, and most likely compliance, as well as choices of sensor and actuator locations. This will be represented mathematically by poles and zeros of transfer functions. These locations, coupled with fundamental laws, will inform what is possible and what is not in performance.

Ignorance is no excuse! My eyes have been opened wide to this most important aspect of mechatronic system design. The references are there and we are always on a learning journey.