Report

Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo Stability Margins Outline of Today’s Lecture Review Open Loop System Nyquist Plot Simple Nyquist Theorem Nyquist Gain Scaling Conditional Stability Full NyquistTheorem Is stability enough? Margins from Nyquist Plots Margins from Bode Plot Non Minimum Phase Systems Loop Nomenclature Reference Input R(s) Prefilter F(s) +- Error signal E(s) Disturbance/Noise Controller C(s) Open Loop Signal B(s) +- Plant G(s) Output y(s) Sensor H(s) The plant is that which is to be controlled with transfer function G(s) The prefilter and the controller define the control laws of the system. The open loop signal is the signal that results from the actions of the prefilter, the controller, the plant and the sensor and has the transfer function F(s)C(s)G(s)H(s) The closed loop signal is the output of the system and has the transfer function F ( s )C ( s )G ( s ) 1 C ( s )G ( s ) H ( s ) Note: Your book uses L(s) rather than B(s) To avoid confusion with the Laplace transform, I will use B(s) Open Loop System Input r(s) ++ Error signal E(s) Controller C(s) Open Loop Signal B(s) Sensor -1 Plant P(s) Output y(s) nc s n p s b( s) The open loop transfer function is B( s) C s P s r( s) dc s d p s If in the closed loop, the input r(s) were sinusoidal and if the signal were to continue in the same form and magnitude after the signal were disconnected, it would be necessary for B(i0 ) nc s n p s 1 dc s d p s Simple Nyquist Theorem Error signal E(s) Input r(s) ++ Controller C(s) Open Loop Signal B(s) Imaginary -1 is called the -1 critical point Plant P(s) Sensor -1 -B(i) Plane of the Open Loop Transfer Function B(i) Stable B(0) Real B(i) Unstable Simple Nyquist Theorem: For the loop transfer function, B(i), if B(i) has no poles in the right hand side, expect for simple poles on the imaginary axis, then the system is stable if there are no encirclements of the critical point -1. Output y(s) Nyquist Gain Scaling The form of the Nyquist plot is scaled by the system gain Conditional Stabilty While most system increase stability by decreasing gain, some can be stabilized by increasing gain Show with Sisotool K (0.25s 2 0.12 s 1) B( s ) s 1.69 s 2 1.09 s 1 Definition of Stable A system described the solution (the response) is stable if that system’s response stay arbitrarily near some value, a, for all of time greater than some value, tf. b a x(t; b) x(t; a) for all t 0 Full Nyquist Theorem Assume that the transfer function B(i) with P poles has been plotted as a Nyquist plot. Let N be the number of clockwise encirclements of -1 by B(i) minus the counterclockwise encirclements of -1 by B(i)Then the closed loop system has Z=N+P poles in the right half plane. Determination of Stability from Eigenvalues The eigenvectors of A are i i Continuous Time x Ax Unstable Stable Asymptotic Stability Discrete Time x(k 1) Ax(k ) If i 0 for any simple root If i 0 for any simple root Or i 0 for any repeated root Or i 0 for any repeated root If i 0 for any simple root If i 0 for any simple root Or i 0 for any repeated root Or i 0 for any repeated root i 0 for all roots i 0 for all roots Is Stability Enough? If not, why not? Margins Margins are the range from the current system design to the edge of instability. We will determine Gain Margin How much can gain be increased? Formally: the smallest multiple amount the gain can be increased before the closed loop response is unstable. If the gain margin is expressed in dB, then the multiple gain is G 100.05gm Phase Margin How much further can the phase be shifted? Formally: the smallest amount the phase can be increased before the closed loop response is unstable. Stability Margin How far is the system from the critical point? Gain and Phase Margin Definition Nyquist Plot 1 gm -1 m Example 45( s 600) ( s 6)( s 2 60 s 900) From the plot, the gain margin is 10.2 dB G( s) The gain multiple is G 100.05 gm G 100.05*10.2 3.2359 Using Matlab command nyquist(gs) Example Here the gain from the previous plot has been multiplied by 3.2359 The result is that stability is about to be lost Example Unstable Example: 270( s 600) G( s) ( s 6)( s 2 60s 900) Using Matlab command nyquist(gs) Magnitude, dB Gain and Phase Margin Definition Bode Plots 0 Positive Gain Margin Phase, deg -180 Phase Margin Phase Crossover Frequency Example G( s) 45( s 600) ( s 6)( s 2 60s 900) Using Matlab command bode(gs) Example G( s) 3.2359 * 45( s 600) ( s 6)( s 2 60s 900) Again, stability is about to be lost. Example G( s) 270( s 600) ( s 6)( s 2 60s 900) Using Matlab command bode(gs) Note The book does not plot the Magnitude of the Bode Plot in decibels. Therefore, you will get different results than the book where decibels are required. Matlab uses decibels where needed. Gain 10 gm 20 where gm is measured in dB Stability Margin It is possible for a system to have relatively large gain and phase margins, yet be relatively unstable. Stability margin, sm Non-Minimum Phase Systems Non minimum phase systems are those systems which have poles on the right hand side of the plane: they have positive real parts. This terminology comes from a phase shift with sinusoidal inputs s 1 s 1 G ( s ) and G 2 Consider the transfer functions 1 s s 2 s( s 2) The magnitude plots of a Bode diagram are exactly the same but the phase has a major difference: Another Non Minimum Phase System A Delay Delays are modeled by the function which multiplies the T.F. y (t ) x (t T ) G( s) eTs Summary Is stability enough? Margins from Nyquist Plots Margins from Bode Plot Non Minimum Phase Systems Next Class: PID Controls