Instituto Superior Tecnico Loop Shaping (SISO case) 0db António Pascoal 2017 Instituto Superior Tecnico antonio@isr.ist.utl.pt

Feedback Control structure 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

Design the controller K(s) such that Key objectives Design the controller K(s) such that 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. iv) The actuation signal u is not driven beyond limits imposed by saturation values and bandwith of the plant´s actuator. v) The system meets stability and performance requirements in the face of plant parameter uncertainty and unmodeled dynamics (robust stability and robust performance).

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

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

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

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

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: constraints on the Loop Gain GK If Disturbace attenuation

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

Reference following _

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

If Reference following spectral contents of d concentrated in the frequency band 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

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

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

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

Noise reduction _

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

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

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

Noise reduction: constraints on the Loop Gain GK 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)

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

Suppose Actuator limits 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.

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

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

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

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

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

It is necessary to introduce some phase lead Loop Shaping – Design examples It is necessary to introduce some phase lead Minimum phase margin (specs): Additional phase required : real phase margin = 0 graus security factor (start by trying security factor = 0). Additional phase required: 450 Pure “PHASE LEAD” network z odb Phase of Phase lead

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

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

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

Loop Shaping – Design examples Specifications i) Static position error = 0. 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

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

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

Checking the constraints on Loop Gain Loop Shaping – Design examples Checking the constraints on Loop Gain +80db +40db 0db 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! Phase of The constraints on the loop gain are met, but … !

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)! New use +80db +40db 0db Phase of

Loop Shaping – Design examples +80db +40db 0db NOTICE: the LAG network must introduce a loss of -40dB at 1 rads-1. But .. the zero is introduced at -10-1rads-1, not -1rads-1! WHY?So that the extra phase introduced by the lag network will not “interfere too much” around 1 rads-1. Phase of

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

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

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

Loop Shaping – Design examples Specifications i) Static position error = 0. 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 v) Phase Margin v) Robustness of stability with respect to a total delay in the control channel of up to 0.5 sec

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

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

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

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

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

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

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

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

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

Intrinsic Limitations on Achievable Performance Controller Plant _ 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:

Intrinsic Limitations on Achievable Performance Controller Plant _ 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)

Intrinsic Limitations on Achievable Performance Controller Plant _ 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”. +ydb 0db -xdb High performance

Intrinsic Limitations on Achievable Performance Analytic Limitation “Barrier” approximation +20db z 10z 0db -40db High performance Objective: design a stabilizing controller K(s) such that 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: Reduce the performance bandwith and / or relax the level of performance plant zero 0db original spec

Intrinsic Limitations on Achievable Performance ii) Allow for increased gain over the complementary range of frequencies plant zero 0db 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) Open loop unstable system Must maintain a given closed loop bandwith (dangerous!) Before designing a controller, take a step back .. examine the system physics. Freudenberg and Looze, “Right half plane poles and zeros and design tradeoffs in feedback systems,” IEEE Trans. Automatic Control, Vol. 39(6), pp. 55-565, 1985.

Loop Shaping (SISO case) 0db António Pascoal 2017 antonio@isr.ist.utl.pt