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State-space approach
Contents:State space representationPole placement by state feedback
LQR (Linear Quadratic Regulator)Observer designKalman Filter LQGSeparation Principle
SpilloverFrequency Shaped LQGHAC-LAC strategy
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Transfer function approach:
State variable form:
State spaceEquation:
Feedthrough
Plant noise
Measurement noise
(Ch.7, p.138)
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The choice of state variable is not unique
Example: s.d.o.f. oscillator:
AccelerationOutput:
C
1.
D (feedthrough)
A B
2.
A is dimensionallyhomogene
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Inverted Pendulum
Equation of motion:
Linearization:
Change of variable:
with(natural frequencyOf the pendulum)
State variable form:
Output equation:
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System transfer function
s.d.o.f. oscillator:
Inverted Pendulum:
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For SISO systems, one can write:
poles
Zeros
Poles: such that, for some initial condition, the free response is
Free response:
are the eigenvalues of A, solution of
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An input applied from appropriate initial conditions
produces no output:
2.Zeros:
The state vector has the form:
If:
That is if:
Then:
Y = 0 if
(1)
(2)
(1) And (2)
dtm = 0
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Pole placement by state feedback
Statefeedback
If the system is controllable, the closed-loop polescan be placed arbitrarily in the complex plane.
The gain G can be chosen such that
Closed-loop characteristicequation
Selected arbitrarily
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Example: s.d.o.f. oscillator (1)
Relocating the polesDeeper in le left-half plane
Example: s d o f oscillator (2)
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State-space equation:
State feedback:
Closed-loop characteristic equation:
Desired behaviour:
Example: s.d.o.f. oscillator (2)
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Linear Quadratic Regulator (SISO)
u such that the performance index J is minimized
Controlled variable: Control force: u
Weighing coefficient
Solution: The closed-loop poles are the stable roots of:
where
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Characteristic equation:
-Identical to that of:
Symmetric with respect to the imaginary axis
As well as the real axisOnly the roots in the left half plane have to beconsidered
Symmetric root locus
WeighingOn the control
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Example: Inverted pendulum (1)
ControlledVariable:
Selected poles
Example: Inverted pendulum (2)
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1. Select the poles on the left side of theSymmetric root locus
2. Compute the gains so as to match the desired poles:
Example: Inverted pendulum (2)
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Observer design
Full stat observer (Luenberger observer):
Duplicates
the system(perfect modeling !!) Innovation
Error: Error equation:
If the system is observable, the poles of theError equation can be assigneg arbitrarily byAppropriate choice of kiIn practice, the poles of the observer shouldBe taken 2 to 6 times faster than the regulatorpoles
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In practice, there are modeling errors and measurement noise;These should be taken into account in selecting the observer gains
One way to assign the observer poles: KALMAN filter
(minimum variance observer)
The optimal poles location minimizing the variance of theMeasurement error are the stable roots of thesymmetric root locus:
ScalarWhite noiseprocesses
Where is the T.F. between w andy and
Plant noise intensity (w) a
Measurement noise intensity (v)
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Example: Inverted pendulum (1)
1. Assume that the noise enters the system at the input (E = B)
proportional to
The same root locus can beused for the regulator and
the observer design
E l I t d d l (2)
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Example: Inverted pendulum (2)
2. Assume
Observer poles
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Separation Principle
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Separation Principle
Compensator
Reconstructed statelosed-loop
equations:
2n state variablesWith
Block triangular the eigenvalues are decoupled
Transfer function of the compensator
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Transfer function of the compensator
The poles of the compensator H(s) are solutions of the characteristic equation:
They have not been specified anywhere in the designThey may be unstableH(s) is always of the same order as the system
Th t bl (1)
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u
X1 =yX3The two-mass problem (1)
u
State-space equation:
LQG design with symmetric root locus based on
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Two-mass problem (2): Symmetric root-locus
Open-looppoles
Two-mass problem (3)
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Design procedure:
Select the regulator poles on the locusCompute the corresponding gains GSelect the observer poles (2 to 6 times faster)Compute the corresponding gains KCompute the compensator H(s)
One finds: Notch filter !
Two-mass problem (3)
T bl (4) R l f h LQG ll
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Two-mass problem (4): Root locus of the LQG controller
Optimum design for g= 1
Compensator
Notchfilter
Two-mass problem (5): robustness analysis
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Two mass problem (5) robustness analysisEffect of doubling the natural frequency
The notch filterbecomes useless
This frequency
has been doubled
Unstable loop !
T bl (6) R b t l i
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Two-mass problem (6): Robustness analysisEffect of lowering the natural frequency by 20%
Pole/zero Flipping !
The notch is unchanged
Spillover (1)(Ch.9, p.206)
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Crossover
Phase
stabilized
Bandwidth
Gain stabilized
The residual modesNear crossover mayBe destabilized by
Spillover
Spillover (2): mechanism
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Actuators Sensors
Controlledmodes
Residual
modes
Flexible structure dynamics
Spillover (3): Equations
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Structure dynamics:
Controlled modes:
Residual modes:
Output:
Full state observer:
Full state feedback:
ControlSpillover
ObservationSpillover
Spillover (4): Eigen value problem
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p p
Observation spilloverControl spillover
If either Br=0 or Cr=0, the eigen values remain decoupled
If both Br and Cr exist, there is Spillover
Spillover (5): Closed-loop poles
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The residual modeshave a small stabilitymargin (damping !)and can be destabilizedby Spillover
Integral control with state feedback(Ch.9, p.211)
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Constant disturbance
Non-zero steady state error on y
Introduce the augmented state p such that :
State feedback:
Closed loop equation:
If G and Gp are chosen so as to stabilize the system,
without knowledge of the disturbance w
Frequency Shaped LQR (1)(Ch.9, p.212)
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LQR:
Parsevals theorem: Frequency independent
Frequency-shaped LQR:
To achieve P + I action
At low frequency
To increase the roll-off
At high frequency
Frequency shaped LQR (2): weight specification
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P + I Increased roll-off
Frequency shaped LQR (3): Augmented system
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Frequency independent cost functional
State space realization of the augmented system
Frequency shaped LQR (4): Augmented system
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The state feedback of the augmented systemis designed with the frequency independent
Cost functional:
Frequency shaped LQR (5): Architecture of the controller
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Augmentedstates
Only the states of theStructure must be reconstructed
HAC / LAC strategy (1)
Th l i f h i b dd d l
(Ch.13, p.295)
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The control system consists of tho imbedded loops:
1) The inner loop (LAC: Low Authority Control) consists of adecentralized active damping with collocated actuator/sensor pairs(no model necessary).
2) The outer loop (HAC: High Authority Control) consists of amodel-based non-collocated controller (based on a model of theactively damped structure).
HAC / LAC strategy (2)
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Advantages:
The active damping extends outside the bandwidth of the HAC(reduces the settling time of the modes beyond the bandwidth)
The active damping makes it easier to gain-stabilize the modesoutside the bandwidth of the HAC loop (improved gain margin)
The larger damping of the modes within the controller bandwidth makesthem more robust to parametric uncertainty (improved phase margin)
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HAC / LAC strategy (4): ExampleWide-band position control of a truss
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Bode plot of the controller H Open-loop FRF of the design model: GH
HAC / LAC strategy (5): ExampleWide-band position control of a truss
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Open-loop FRF of the full system: G*H Nyquist plot
Step response
t (sec)
High frequency dynamics