António Pascoal 2011 Instituto Superior Tecnico Loop Shaping (SISO case) 0db

_ Controller Plant r – reference signal ( to be tracked by the output y) d – external perturbation (referred to the output) n – sensor noise e – error y – output signal u – actuation signal Feedback Control structure

Key objectives i) K(s) stabilizes G(s) ii) The output y follows the reference signals r. iii) The system reduces the effect of external disturbance d and noise n on the output y. v) The system meets stability and performance requirements in the face of plant parameter uncertainty and unmodeled dynamics (robust stability and robust performance). Design the controller K(s) such that iv) The actuation signal u is not driven beyond limits imposed by saturation values and bandwith of the plant´s actuator.

Control objectives External disturbance attenuation (reducing the impact of d on y) _ Linear system superposition principle

D(s) Y(s) S(s) – Sensitivity Function Disturbance attenuation S(s) – possible Bode diagram 0db -x db below the ‘ barrier ’ of –x db for

Attenuation of at least –x db Attenuation of sinusoidal disturbances d – sinusoidal signals Performance specs on disturbance attenuation Upper limit on Performance bandwith Upper limit –x db and performance bandwith are problem dependent

What happens when d is not a sinusoid? d- modeled as a stationary stochastic process with spectral density Energy y - stationary stochastic process with spectral density Disturbance attenuation

If spectral contents of d concentrated in the frequency band Basic technique to reduce the energy of y: reduce Its is up to the system designer to select the level of attenuation Disturbance attenuation

Disturbance attenuation: constraints on the Loop Gain GK If Disturbace attenuation

0db Lower bound ( “ barrier ” ) on shaped by proper choice of controller K(s) Disturbance attenuation: constraints on the Loop Gain GK

Reference following _

Energia e - stationary stochastic process with spectral density r- modeled as a stationary stochastic process with spectral density Reference following

If spectral contents of d concentrated in the frequency band Reference following Technique to reduce the energy of the tracking error e Reduce Its is up to the system designer to select the level of error reduction

0db below the “ barrier ” of db for Geometric constraint db Reference following

If reference following: Reference following: constraints on the Loop Gain GK

0db Reference following: constraints on the Loop Gain GK Lower bound ( “ barrier ” ) on shaped by proper choice of controller K(s)

Noise reduction _

y - stationary stochastic process with spectral density n- modeled as a stationary stochastic process with spectral density Energy Noise reduction

Noise reduction (high frequency noise) If spectral contents of n concentrated in the frequency band Technique to reduce the energy of y caused by the noise n: Reduce Its is up to the system designer to select the level of error reduction

0db upper bound ( “ barrier ” ) on shaped by proper choice of controller K(s) Noise reduction (high frequency noise)

If noise reduction Noise reduction: constraints on the Loop Gain GK

0db Upper bound ( “ barrier ” ) on loop gain shaped by proper choice of K(s) Noise reduction: constraints on the Loop Gain GK

Actuator limits _ Suppose (plant gain rolls off at high frequencies)

Actuator limits Suppose Actuation signals too high unless the loop gain starts rolling off at frequencies below Golden rule: never try to make the closed loop bandwidth extend well above the region where there the plant gain starts to roll off below 0db.

0db Upper bound ( “ barrier ” ) on loop gain shaped by proper choice of K(s) Actuator limits Technique for limiting actuation signals Its is up to the system designer to select the parameters

Putting it all together Loops Gain restrictions 0db Low frequency barriers r, d High frequency barriers n, u Goal: Shape (by appropriate choice of K(s) the LOOP GAIN G(s)K(s)so that it will meet the barrier constraints while preserving closed loop stability.

Loop Shaping – Design examples Exemple 1 G(s). Plant (system to) be controlled. Control specifications _ Controller Plant Design K(s) so as to stabilize G(s) and meet the following performance specifications:

Specifications i) Reduce by at least –80db the influence of d on y in the frequency band ii) Follow with error less than or equal to -40db the reference signals r in the frequency band iii) Attenuate by at least –20db the noise n in the frequency band iv) Static error in response to a unit parabola reference v) Phase Margin vi) Gain Margin Loop Shaping – Design examples

Geometrical constraints; conditions i), ii), iii) i) ii) iii Loop Shaping – Design examples

0db Loop Gain Constraints Low frequency barriers r, d High frequency barrier n Loop Shaping – Design examples

Condition iv) Let Loop Shaping – Design examples (possible to achieve, because G(s) has two poles at the origin) Static error in response to a unit parabola reference

A simple controller candidate: Checking the constraints on Loop Gain 0db Phase of The constraints are met but ….. ! Loop Shaping – Design examples

It is necessary to introduce some phase lead Minimum phase margin (specs): Additional phase required : security factor real phase margin = 0 graus (start by trying security factor = 0). Additional phase required: 45 0 Pure “PHASE LEAD” network z z odb Phase of Loop Shaping – Design examples Phase lead

0db Phase of Loop Gain constraints are met and ….. New Loop Shaping – Design examples Checking the constraints on Loop Gain Phase lead NOTICE: phase lead “opens-up” the loop gain! The new loop gain barely avoids violating the noise-barrier!

Final check on stability and Gain Margin Use Nyquist ’ s Theorem Nyquist contour x x Number of open loop poles inside the Nyquist contour P=0 x Number of encirclements around –1 N=0 Stable! Gain Margin equals infinity! Loop Shaping – Design examples Phase lead

Example 2 G(s). Plant (simple torpedo model). Control objectives _ Controller Plant Design K(s) so as to stabilize G(s) and meet the following performance specifications: Loop Shaping – Design examples

Specifications ii) Attenuate by at least –40db the signals d in the frequency band iii) Follow with error smaller than or equal to -100db the signals r in the frequency band iv) Attenuate by at least –40db the noise n in the frequency band v) Phase Margin vi) Gain Margin i) Static position error = 0. Loop Shaping – Design examples

Geometrical constraints; conditions i), ii), iii) ii) iii) iv) Loop Shaping – Design examples

Condition i) Static position error (1 pure integrator in the direct path) A simple controller candidate: Loop Gain Loop Shaping – Design examples

Checking the constraints on Loop Gain 0db Phase of The constraints on the loop gain are met, but … ! +40db +80db Loop Shaping – Design examples Notice! Now it is not possible to use a phase-lead network because the open-loop plot would “open-up” and violate the noise barrier!

0db +40db +80db New Phase of use Loop Shaping – Design examples The high frequency barrier does not allow for the use of a lead network – use a lag network (“gain-loss” network)! Force a new 0dB crossing point such that if the phase were not changed, the gain margin would meet the specifications (must loose -40dB at 1.0 rads-1)!

0db +40db +80db Phase of Loop Shaping – Design examples NOTICE: the LAG network must introduce a loss of -40dB at 1 rads -1. But.. the zero is introduced at rads -1, not -1rads -1 ! WHY?So that the extra phase introduced by the lag network will not “interfere too much” around 1 rads -1.

Final check on stability and Gain Margin Nyquist Theorem Nyquist Contour x Number of open loop poles inside the Nyquist contour P=0 x Number of encirclements around -1 N=0 Stable! Gain Margin equals infinity! x x -p -z Loop Shaping – Design examples

Example 3 (Lunar Excursion Module – LEM) Loop Shaping – Design examples

Example 3 (Lunar Excursion Module – LEM) G(s). Plant (vehicle controlled in attitude by gas jets and actuator; J=100 Nm/(rads -2 )). Control objectives (attitude control) _ Controller Plant Design K(s) so as to stabilize G(s) and meet the following performance specifications: Loop Shaping – Design examples Torque Input Voltage Attitude

Specifications ii) Follow with error smaller than or equal to -40db the signals r in the frequency band iii) Attenuate by at least –40db the noise n in the frequency band iv) Gain Margin i) Static position error = 0. Loop Shaping – Design examples v) Phase Margin v) Robustness of stability with respect to a total delay in the control channel of up to 0.5 sec

Geometrical constraints; conditions ii), iii) ii) iii) Loop Shaping – Design examples

Condition i) Static position error (there are already two integrators in the direct path) A simple controller candidate: Loop Gain Loop Shaping – Design examples

Candidate Loop Gain Checking the constraints on Loop Gain (with ) 0db Fase de Loop Shaping – Design examples -40db

Checking the stability of the closed-loop system Use Nyquist ’ s Theorem Nyquist contour x x Number of open loop poles inside the Nyquist contour P=0 x Number of encirclements around –1 N=+2 Unstable! Loop Shaping – Design examples x

Possible strategy: introduce some phase lead Pure Phase Lead network z z 0 dB Phase of Loop Shaping – Design examples Phase lead What value of z should be adopted? Try z =1 rads -1 ; that is, frequency at which

New candidate Loop Gain Checking the constraints on the Loop Gain 0db Loop Shaping – Design examples -40db “old” loop gain “new” loop gain

Final check on stability and Gain Margin Use Nyquist ’ s Theorem Nyquist contour x x Number of open loop poles inside the Nyquist contour P=0 x Loop Shaping – Design examples Phase lead Number of encirclements around –1 N=0 Stable! Gain Margin equals infinity! Phase Margin

New candidate Loop Gain Checking the constraints on the Loop Gain 0db Loop Shaping – Design examples -40db “old” loop gain “new” loop gain

Robustness of stability with respect to a delay in the control channel Loop Shaping – Design examples Transfer function of a pure delay  exp (-s  ) Only change in the Bode diagram! Danger: if the gain margin of 45º is completely lost! Maximum  allowed is app. 0.75s >0.5 sec!

_ Controller Plant Intrinsic Limitations on Achievable Performance Simple algebraic limitation Find (if at all possible) a controller K(s) that will stabilize G(s) and such that (reference following spec) (noise attenuation spec) Notice:

_ Controller Plant Intrinsic Limitations on Achievable Performance If then There is no controller that will meet the specs! (cannot expect good performance over a frequency band where there is significant sensor noise: buy a better sensor, or relax the specs)

_ Controller Plant Intrinsic Limitations on Achievable Performance Analytic Limitation Find (if at all possible) a controller K(s) that will stabilize G(s) and such that the sensitivity function S(s) will “ acquire a desired target shape ”. 0db High performance -xdb +ydb

Intrinsic Limitations on Achievable Performance Analytic Limitation 0db High performance -40db +20db “Barrier” approximation Objective: design a stabilizing controller K(s) such that z 10z stable with a stable inverse is analytic in the right half complex plane (RHP)

Intrinsic Limitations on Achievable Performance If K(s) stabilizes G(s), then S(s) is analytic in the RHP (maximum modulus principle) Suppose the plant G(s) has an “unstable” zero (no unstable pole-zero cancellations) condition to be satisfied!

Intrinsic Limitations on Achievable Performance Case 1. Z=2rad/s (plant zero “inside” the high performance bandwidth region) Impossible to meet the specifications! Case 1. Z=0.05rad/s (plant zero “outside” the high performance bandwidth region) The specs are met.

Intrinsic Limitations on Achievable Performance Analytic Limitations (extension) Case 1. Z=2rad/s (plant zero “inside” the high performance bandwidth region) Impossible to meet the specifications! Possible strategies: i)Reduce the performance bandwith and / or relax the level of performance 0db plant zero original spec

Intrinsic Limitations on Achievable Performance ii) Allow for increased gain over the complementary range of frequencies 0db plant zero original spec Waterbed effect 0db plant zero original spec

Intrinsic Limitations on Achievable Performance Open loop (unstable) zeros and poles place fundamental restrictions on what can be done with feedback! (not “textbook” examples) Freudenberg and Looze, “Right half plane poles and zeros and design tradeoffs in feedback systems,” IEEE Trans. Automatic Control, Vol. 39(6), pp , Before designing a controller, take a step back.. examine the system physics. Open loop unstable system Must maintain a given closed loop bandwith (dangerous!)

António Pascoal 2011 Loop Shaping (SISO case) 0db