Uncertainty Principle in Control Theory, Part I: Analysis of Performance Limitations

Performance Limitations Ji King Abstract—This paper investigates performance limitations and tradeoffs in the control design for linear time-invariant systems. It is shown that control specifications in time domain and in frequency domain are always mutually exclusive determined by uncertainty relations. The uncertainty principle from quantum mechanics and harmonic analysis therefore embeds itself inherently in control theory. The relations among transient specifications, system bandwidth and control energy are obtained within the framework of uncertainty principle. If the control system is provided with a large bandwidth or great control energy, then it can ensure transient specifications as good as it can be. Such a control system could be approximated by prolate spheroidal wave functions. The obtained results are also applicable to filter design due to the duality of filtering and control. Index Terms — Time-frequency analysis, performance limitations, uncertainty principle, prolate spheroidal wave function, transient analysis, linear time-invariant system.