Polynomial Method And Its Applications To Complex Systems

Polynomial method is a straightforward control design method based on the closed-loop char-acteristic polynomial of a system. Its design parameters, characteristic ratios γ’s and timeconstant τ, are directly related to the transient response of a control system. This thesispresents the theoretical discussion of polynomial method and its application to complex sys-tems, such as non-all-pole system, high-order systems and single-input multi-output (SIMO)system.First, the assignment of time constant for a non-all-pole system is discussed. The timeconstant not only influences the speed of response but also damping (i.e. overshoot) of asystem. For the ease of explanation, a general two-mass system is introduced as a case study,which has one pair of jw-axis nonminimum-phase zeros. Under an ideal two-parameter controlconfguration, the exact lower bound of the generalized time constant is determined that resultsin monotonic step responses, while a moderate generalized time constant is shown to be desirablefor robustness purpose. A m-IPD control confguration is then adopted for the implementationof the idea two-parameter controller. It is found that specifc control confguration may alsoimpose limits on the assignment of the time constant and characteristic ratios.Second, a target characteristic polynomial matching (TCPM) method to design controllerfor high-order systems is developed. The problem of controller design is formulated as an opti-mization problem with a quadratic objective function and a stability constraint. The objectivefunction is the weighted sum of characteristic coefcient variation. The weighted influencesof the characteristic coefcient related to the step response are also discussed and quantifed.A mIPD controller for a3-mass system is successfully designed via the TCPM method as anillustrative example.Third, the application of the polynomial method is extended to a single-input multiple-output (SIMO) system. A state-feedback controller is designed via the polynomial method tobalance a double inverted pendulum system which is a typical SIMO system. The feedbackgain matrix is calculated via Ackermann’s formula. Compared with the LQR controller, thepolynomial-method-based controller has a simpler and more straightforward design procedure.All the proposed polynomial-method-based controllers are validated by simulation and ex-perimental results. These successful applications of the polynomial method have demonstratedthe efectiveness of the polynomial method for controller design for complex systems.