A pole is a value of s that causes a system’s transfer function move toward infinity. \(H(s)=\infty\) Therefore, they are the roots of the denominator of a transfer function. Closed-loop poles provide a binary assessment of system stability. If all poles have negative real parts, then the system is stable. if the polls of the system are positive, then the eigenvalues of the system are also positive. If one of the states has a positive eigenvalue, then the solution of that state will be in the form of an exponential rise to a positive number. This means the solution will approach infinity; therefore, the system, in general, is unstable. The number of poles is the order of the system. In the discrete domain, the pole of a system is the value at which the exponential growth of an input signal matches the exponential decay of the system. For aircraft, the inertial properties of the aircraft affect the open-loop poles of the system.
Therefore the sum of the signals, or the z-transform, is infinity.
[[Eigenvalue]]
[[Effects on Root-Locus Plots of Poles and Zeros]]
[[Z-Transform Method]] – used to match exponentials to find the poles and zeros of a system
[[Lead and Lag Compensators]] – the location of the poles and zeros in relation to the origin determines if it is a lead or a lag compensator.
[[FBW Jaguar Program]] – the time-to-double of an unstable system can be used to find the pole location.
[[LC Circuit]] – is a two-pole system
F-22 Flight Control System – integrator poles was the dominant mode for pitch response
[[Nyquist Stability Criterion]] – number of encirclements of -1+0j is the number of open-loop poles
[[Root-Locus plot of an Integrator]] – pole at origin
[[SMC with PI-PD Sliding Surface]] – use when complex poles are poorly located
[[Bode’s Integral]] – relates pole placement to the magnitude of the log of the sensitivity function
Pilot-Induced Oscillations – loss of asymptotic stability occurs when the poles or pair of poles cross from the left hand plane to the axis.
[[AC45 3DOF Model]] – Shows the poles of the system
[[Multivariable Plant Inverse Equations]] – the multivariable zeros of a system are the poles of the inverse
[[VAAC FCL005 Weighting Function]] – the poles of the controller is a tradeoff between tracking inputs and not propagating sensor noise
[[Zeros of a System]] – Zeros of a controller can affect the poles of the closed-loo feedback systems
Sources
- StatespaceModelMATLAB
- briandouglasBodePlotsHand2012a(incomplete source review)
- philslabZTransformPracticalApplications2021 (incomplete source review)
- UnderstandingZTransformYouTube
- fieldApplicationFlightControl1993