State Space Approach

8/11/2019 State Space Approach

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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



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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:



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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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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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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

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State Space Approach