Automatic Control III (Reglerteknik III) fall 20 4. Nonlinear systems, Part 3 (Chapter 4) Hans Norlander Systems and Control Department of Information Technology Uppsala University
OSCILLATIONS AND DESCRIBING FUNCTIONS The describing function method Autotuning of PID controllers Fall 20 Automatic Control III:4 part 3 Page
DESCRIBING FUNCTIONS An approximate method for examining existence of periodic solutions for systems of the form () ( ) ¹ Û Ý ¹ () ( ) a static nonlinearity linear system of low-pass character (dampens high frequencies more than low ones) Fall 20 Automatic Control III:4 part 3 Page 2
CONDITIONS FOR OSCILLATION () ( ) ¹ Û Ý ¹ Assume (Ø) = sin Ø. As () is static, Û(Ø) will be periodic. Use a Fourier series Û(Ø) = ( sin Ø) = 0 () + () sin(ø + ()) + 2 () sin(2ø + 2 ()) + 3 () sin(3ø + 3 ()) + Assume that ( ) = () µ 0 () = 0. Fall 20 Automatic Control III:4 part 3 Page 3
Amplitude =, phase shift = 80 Æ ¾ CONDITIONS FOR OSCILLATION Assume that the higher frequencies can be neglected. Thus Ý(Ø) ()() sin (Ø + () + arg(())) Conditions for an undamped oscillation: ()() = () + arg(()) = + 2 Fall 20 Automatic Control III:4 part 3 Page 4
DESCRIBING FUNCTION Definition of describing function () = () () The condition for an undamped oscillation ()() = ( ) Procedure:. Determine the describing function (). 2. Investigate whether or not equation ( ) has a solution. 3. Determine if the oscillation will be maintained. Fall 20 Automatic Control III:4 part 3 Page 5
COMPUTATION OF THE DESCRIBING FUNCTION Set «= Ø. The nonlinear element sin «) = () sin(«+ ()) + ( = () «cos + () «sin + The describing function () = ()() = () + () Fourier coefficients () = 2 0 ( sin «) cos ««() = ( sin «) sin ««0 Hysteresis gives a complex-valued describing function. 2 Fall 20 Automatic Control III:4 part 3 Page 6
ANALYZING THE OSCILLATION If possible, solve the equation ()() = with respect to and. ( =the amplitude of an oscillation, before the nonlinearity; = the frequency of the oscillation). The equation can be treated graphically using a Nyquist curve and a plot of (): () = () intersection By reasoning as for the Nyquist theorem, one can examine if the oscillation is stable (the amplitude tends to remain constant) or not. Fall 20 Automatic Control III:4 part 3 Page 7
WILL THE OSCILLATION BE STABLE? Apply the Nyquist theorem ( () corresponds to in the linear case). For the case when ( ) has no poles in the right half plane: If () encloses the point () [ fixed], then will increase. If () does not enclose the point () [ fixed], then will decrease. Fall 20 Automatic Control III:4 part 3 Page 8
WILL THE OSCILLATION BE STABLE? Examples stable oscillation unstable oscillation G(iw) G(iw) Y f (C) C Y f (C) C Fall 20 Automatic Control III:4 part 3 Page 9
2 DESCRIBING FUNCTION Example Ideal relay () = 0 0 No phase shift, a gain depending on the amplitude. () = 0 cos ««+ ( ) cos ««= 0 () = () = 4 0 sin ««+ 2 ( ) sin ««= 4 Fall 20 Automatic Control III:4 part 3 Page 0
DESCRIBING FUNCTION Example, cont d ( ) = Ã + ) 2 () ideal relay ( Condition for oscillation () = () Ã + ) = 2 4 ( =µ = Ã 2 = 4 = 2Ã µ Fall 20 Automatic Control III:4 part 3 Page
DESCRIBING FUNCTION 2 0 Example, cont d C 2 3 4 5 6 5 0 5 0 5 The oscillation is expected to be stable Fall 20 Automatic Control III:4 part 3 Page 2
DESCRIBING FUNCTION, Example, cont d Σ ¹ ¹ ¹ Ù 3 ( +) 2 Ý Æ Simulation 2.5 0.5 y, u 0 0.5.5 2 0 5 0 5 20 25 30 t Fall 20 Automatic Control III:4 part 3 Page 3
DESCRIBING FUNCTION Relay with hysteresis a complex-valued describing function: ÁÑ Ê À () increases 4À À () = where sin = 4À () = = 4À Ô 2 2 µ 4À 4À Fall 20 Automatic Control III:4 part 3 Page 4
PID DESIGN: AUTO-TUNING Idea: Identify and modify one point on the Nyquist curve. Force the system to self-oscillation by relay feedback (ideal or with hysteresis). + ÑΣ ¹ PID Relay ( ) ¹ ¹ ¹ Ö Ù Ý ¹ Õ Õ Õ If Ý oscillates with frequency Ó and amplitude Ó, then ( Ó Ó ) should be a solution to µ ( Ó ) = ( Ó )( Ó ) = ( Ó ) = Ó 4À Ó where sin Ó = Ó Fall 20 Automatic Control III:4 part 3 Page 5
AUTO-TUNING, cont d The parameters Ó and Ó can easily be determined experimentally. Let Ì + + È Á ( ) = Ã Choose the parameters to get the ( point ) È Á ( Ó Ó ) on the Nyquist curve to some desired location in. Ì Á A commonly used choice is to let = Ó, and then choose È Á ( ) to have a desired phase margin Ñ µ ( Ó ) È Á ( Ó ) = Ñ is desired. Fall 20 Automatic Control III:4 part 3 Page 6
Ì Ó Ì Ó AUTO-TUNING, cont d Since ( Ó ) = Ó Ó we must have 4À È ( Ó Á ) = 4À ( Ñ Ó Ó ) = 4À Ó cos( Ñ Ó ) ( + tan( Ñ Ó )) and since Ì + Ó È Á ( Ó ) = Ã the parameters should be chosen so that = Ã 4À Ó Ñ cos( ) Ó = Ñ tan( Ó ) Ó Ì Fall 20 Automatic Control III:4 part 3 Page 7