structural controlbursi/download/phd_lectures_02july/... · 2012. 7. 10. · modern active control...

Earthquake Engineering 16.1

STRUCTURAL CONTROL

Many structures, such as tall buildings andflexible spacecrafts, are designed usingvibration suppression as a part of the totaldesign. Structural control provides animportant new tool for earthquake engineering.

First, a clarification of the difference betweenactive and passive control is in order.Basically, an active control system uses someexternal adjustable or active device (actuator)to provide control.

Passive control, on the other hand, dependsonly on a fixed (passive) change in thephysical parameters of the structure. Activecontrol depends on current measurements(feedback or feed-forward/closed-loop), unlikepassive control.

Control for civil engineering has a number ofdistinctive features, due to implementation.

In particular, when addressing civilengineering structures, there isconsiderable uncertainty, includingnonlinearity, associated with both physicalproperties and disturbances such asearthquakes and wind, the scale of theforces involved can be quite large, there areonly a limited number of sensors andactuators, the dynamics of the actuators canbe quite complex, the actuators are typicallyvery large, and the systems must be fail-safe (Soong 1990).

Modern active control of civil structuresofficially started in the late Eighties andconsolidated during the Nineties.

Active control systems can be grouped intothree categories:

- Purely active (force actuators);- Hybrid, combining active and passivecontrol technologies;- Semi-active (adjustable physicalparameters, no energy added).


PED: Passive Energy Dissipation


Full-scale example of hybrid control

The hybrid mass damper (HMD) is the mostcommon control device employed in full-scalecivil engineering applications. An HMD is acombination of a passive tuned mass damper(TMD) and an active control actuator. Theability of this device to reduce structuralresponses relies mainly on the natural motionof the TMD. The forces from the controlactuator are employed to increase theefficiency of the HMD and to increase itsrobustness to changes in the dynamiccharacteristics of the structure. The energyand forces required to operate a typical HMDare far less than those associated with a fullyactive mass damper system of comparableperformance.

An example of such an application is theHMD system installed in the SendagayaINTES building in Tokyo in 1991. the HMDwas installed atop the 11th floor and consistsof two masses to control transverse andtorsional motions, while hydraulic actuatorsprovide the active control capabilities.

The top view of the control system is alsoshown where ice thermal storage tanks areused as mass blocks so that no extra massmust be introduced. The masses are supportedby multi-stage rubber bearings intended forreducing the control energy consumed in theHMD and for insuring smooth mass movements


Full-scale example of active control

Design constraints, such as severe spacelimitations, can preclude the use of an HMDsystem. Such is the case in the active massdamper or active mass driver (AMD) systemdesigned and installed in the Kyobashi SeiwaBuilding in Tokyo and the NanjingCommunication Tower in Nanjing, China.The Kyobashi Seiwa Building, the first full-scale implementation of active controltechnology, is an 11-storey building with atotal floor area of 423 m2.

The control system consists of two AMDswhere the primary AMD is used for transversemotion and has a weight of 4 tons, while thesecondary AMD has a weight of 1 ton and isemployed to reduce torsional motion. The roleof the active system is to reduce buildingvibration under strong winds and moderateearthquake excitations and consequently toincrease comfort of occupants in the building.

In the case of the Nanjing Communicationtower, numerous physical constraints had tobe accounted for in the system design of themass damper. The physical size of thedamper was constrained to a ring-shapedfloor area with inner and outer radii of 3 mand 6.1 m, respectively. In addition, thedamper was by necessity elevated off thefloor on steel supports with Teflon bearings toallow free access to the floor area.


The final ring design allowed the damper tomove ± 750 mm from its rest position.Simulations indicate that this stroke issufficient to control the tower; however, agreater stroke would allow substantially moreimprovement in the response. The strength ofthe observation deck limited the weight of thedamper to 60 tons. Lack of sufficient lateralspace made the use of mechanical springsimpractical for restoring forces. Thus theactive control actuators provide restoringforce as well as the damping control forces.

The final design of the active mass damperuses three servo-controlled hydraulicactuators, each with a total stroke of ±1.50 mand a peak control force of 50 kN. Theseactuators are arranged 120o apart around thecircumference of the ring. The actuatorscontrol three degrees of freedom: twoorthogonal lateral directions of motion andtorsional rotation, which is held to zero.

Since the frictional force between the Teflonbearings and mass can have a critical influenceon the response of the system, a detailedanalysis was performed to verify the systemperformance in the presence of this friction.


Full-scale example of semi-active control

Control strategies based on semi-activedevices combine the best features of bothpassive and active control systems. The closeattention received in this area in recent yearscan be attributed to the fact that semi-activecontrol devices offer the adaptability of activecontrol devices without requiring theassociated large power sources. In fact, manycan operate on battery power, which is criticalduring seismic events when the main powersource to the structure may fail. In additionsemi-active control devices do not have thepotential to destabilize (in the boundedinput/bounded output sense) the structuralsystem.

One means of achieving a semi-activedamping device is to use a controllable,electromechanical, variable-orifice valve toalter the resistance to flow of a conventionalhydraulic fluid damper. A schematic of such adevice is given in next figure.

Sack and Patten(1993) conductedexperiments in whicha hydraulic actuatorwith a controllableorifice wasimplemented in asingle-lane modelbridge to dissipate theenergy induced byvehicle traffic, followedby a full-scaleexperiment conductedon a bridge todemonstrate thistechnology. Thisexperiment constitutesthe first full-scaleimplementation ofstructural control inthe US.


Conceived as a variable-stiffness device,Kobori et al. (1993) and Kamagata andKobori (1994) implemented a full-scalevariable-orifice damper in a semi-activevariable-stiffness system (SAVS) toinvestigate semi-active control at the KoboriResearch Complex. The overall system isshown in figure where SAVS devices wereinstalled on both sides of the structure in thetransversal direction. The results of theseanalytical and experimental studies indicatethat this device is effective in reducingstructural responses.

A semi-active damper system was theninstalled in the Kajima Shizuoka Building inShizuoka, Japan. As seen in the secondfigure, semi-active hydraulic dampers areinstalled inside the walls on both sides of thebuilding to enable it to be used as a disasterrelief base in post-earthquake situations(Kobori, 1998; Kurata et al., 1999). Eachdamper contains a flow control valve, a checkvalve and an accumulator, and can develop amaximum damping force of 1000 kN.

It is seen that, under typical design earthquakemotion, both story shear forces and story driftsare greatly reduced with control activated. Inthe case of the shear forces, they are confinedwithin their elastic-limit values while, withoutcontrol, they would enter the plastic range.


Another class of semi-active devices usescontrollable fluids, schematically shown innext figure. In comparison with semi-activedamper systems described above, anadvantage of controllable fluid devices is thatthey contain no moving parts other than thepiston, which makes them simple andpotentially very reliable.

Two fluids that are viable contenders fordevelopment of controllable dampers are:

(a) electrorheological (ER) fluids(b) magnetorheological (MR) fluids

The essential characteristic of these fluids istheir ability to change reversibly from a free-flowing, linear viscous fluid to a semi-solidwith a controllable yield strength inmilliseconds when exposed to an electric (forER fluids) or a magnetic (for MR fluids) field.

Some obstacles remain in the development ofcommercially feasible damping devices usingER fluids. For example, the best ER fluidscurrently available have a yield stress of only3.0–3.5 kPa and cannot tolerate commonimpurities (e.g. water) that might beintroduced during manufacturing or use. Inaddition, safety, availability, and cost of thehigh-voltage required to control the ER fluidsneed to be addressed.

Recently developed MR fluids appear to bean attractive alternative to ER fluids for use incontrollable fluid dampers. MR fluids typicallyconsist of micronsized, magneticallypolarizable particles dispersed in a carriermedium such as mineral or silicone oil.


Achievable yield stress of an MR fluid is anorder of magnitude greater than its ERcounterpart and MR fluids can operate attemperatures from −40 to 150�C with onlymodest variations in the yield stress.Moreover, MR fluids are not sensitive toimpurities such as those commonlyencountered during manufacturing andusage, and little particle/carrier fluidseparation takes place in MR fluids undercommon flow conditions. The more,controllable devices using MR fluids have thepotential of being much smaller than ERdevices with similar capabilities.

The first figure comes from the design of afull-scale, 20-ton MR damper (Carlson andSpencer, 1996), see figure showing that thistechnology is scalable to devices appropriatefor civil engineering.

In 2001, two 30-ton MR fluid dampers wereinstalled between the 3rd and 5th floors of theTokyo Natural Museum of Engineering.

In 2002, MR dampers were used to retrofit acable-stayed bridge crossing the DongtingLake in China to mitigate rain–wind-inducedvibration of the cables (see figure below).


Modern active control methods operate intothe space state. In more detail, a state spacemodel for control should satisfy two basicrequirements:

- being a reduced order model;- being discrete-time, as output and input arein form of discrete-time samples.

Model reduction

A difficulty with many control methods is thatthey work with a small number of DOFs, whilststructural systems have a large number ofDOFs. One approach to this problem is toreduce the size of the original model torelevant DOFs.

Relevant DOFs are usually those that are notloaded and may be considered as slaveDOFs, according to the static reductionproposed by Guyan (1965) .

Models for active control

Hence, the second order equation of motionmay be rewritten in the reduced form, byintroducing reduced mass and stiffnessmatrices:

T T TP m P u P k P u P F F1 1 1 =

The first order state space equation may beconstructed from the reduced matrices andvectors.

By defining a slave DOFs vector u2:

K K u FK K u11 12 1 1

21 22 2 0

one gets a coordinate transformation from thefull coordinate system u and the reducedcoordinate system, u1:

1 11

22 21

Iu u P u

Κ K


Continuous time models for control

Active control is most often formulated in thestate space by:

where 2nx1 state vectors and 2nx2n matriceshave a similar definition as in Lesson 6. Theonly difference regards excitation consistingnow of two parts associated to dynamicloading {w} and control force {u}, respectively.

x A x B E +u w

Furthermore:

1 1

B Em B m E1 10 0

;

In which [B1] and [E1] matrices are nxq andnxr connectivity matrices, depending on the qand r locations of control and external forces,respectively.

t ty D x

where [D] is the px2n output matrix. In theabsence of time delay, the control forces arecalculated directly from the multiplication ofoutput measurements by constant feedbackgains, or

where [G] is the qxp time-invariant feedbackgain matrix.

Since the complete set of states is difficult tomonitor in practical applications, the measuredoutput vector {y(t)} from a limited number ofsensors, say p(p‹‹2n), is some combination ofthe state vector, i.e.

t t G yu


Controllability and observability

A formal definition of controllable system isoften given in the state space system:

Controllability and observability represent twomajor concepts of modern control systemtheory. These concepts were introduced byKalman in 1960.

In very simple systems, controllability is anintuitive concept. The figure represents anuncontrollable system versus a controllableone.

x A x B

y D x

t t

u

2 2n-1R B A B A B A B...

has rank 2n. In this case the pair of matrices[A], [B] is said to be controllable and [R] iscalled the controllability matrix for the matrixpair.

The standard check for controllability of asystem is a rank test of a certain matrix. Theprevious state space system may bedemonstrated to be controllable if and only ifthe 2n x 2nq matrix, defined by

A similar concept is the idea that every statevariable in the system has some effect on theoutput of the system. This is calledobservability. A system is observable if byexamining the output of the system about thestate variables can be determined, whichhappens when the 2n x 2np matrix

T2 2n-1O = D D A D A ... D A

has rank 2n.


Discrete time models for control

The equation above gives the relationship ofstates at two consecutive discrete times.

The discrete time version of the state spaceequation for control reads

having posed

AΔt

Ασ

Ασ

A' e

B' e B

E' e E

0

0

;

;

.

t

t

d

d

' ' + 'k k k k x A x B E1 u w

Implementation issues of control

When only the structural response variablesare measured, the control configuration isreferred to as feedback control since thestructural response is continually monitoredand this information is used to make continualcorrections to the applied control forces.

A feedforward control results when thecontrol forces are regulated only by themeasured excitation, which can be achieved,for earthquake inputs, by measuringaccelerations at the structural base.

In the case where the information on both theresponse quantities and excitation are utilizedfor control design, the term feedback-feedforward control is used.


Other practice-based important issues:

-custom-designed overall systemconfiguration, software/hardware integration,digital/analog controller integration-custom-designed system status monitoring-automatic operations of force generators-multi-protection fail-safe measures-control performance verification

Basic components of an active control systemare combination of analog and digitalelectronics, as well as electro-servo-hydraulics:

-Sensors-Digital controller-Control algorithms-Control force generation devices

Typical issues of active control :

-limited number of sensors andcontrollers/actuators-modeling errors-spillover effect (possible excitation of theunmodeled dynamics)-discrete-time implementation-time delay-control-structure interaction

Time delay, in particular, has always been amajor issue. Delay is not due to computation(only a second’s fraction) but to the dynamicsof actuators. Hence the actuator dynamicshould be included in the model of thestructure.


Time models for control: observers

Recall the discrete equation of a controllablesystem in the state space:

In the general case, however, states are notmeasured directly, so the model should beable to distinguish between estimated (withhat) and measured quantities.

A model accounting for this distinction is thestate space observer model:

where:

' 'k k k

k k

x A x B

y D x

1 u

ˆ ˆ' '

ˆ

ˆ ˆ

k k k

k k

k k

x A x B

r y y

y D x

1 u

Substitution of the second equation into thefirst gives:

ˆ ˆ ' '

ˆ

ˆ ' '

ˆ

k k k k

k

k k

k

k k

x A x B r y

r D x

A r D x B

r y

A x B v

1 u

u

' ;

' ;

k

k

A A r D

B B r

vy

u

Thus one has:

ˆ ˆk k k 1x A x B v


The equation governing the estimation errorreads:

If the observer state matrix is asymptoticallystable, then for large k the estimated statetends to the true state.

In principle, one could choose the observergain matrix [r] to make the state estimatorerror to diminish as quick as possible (e.g.deadbeat observer gain).

In the presence of unmeasured excitationand noise the quickest observer is theKalman filter.

ˆ ' '

ˆ ' '

'

k k k k k

k k k

k

e x x A x B

A r D x B r y

A r D e

1 1 1 u

u

Kalman filter

To make life a little easier, let’s simplify notation.

Estimate the state x of a linear stochasticdifference equation:

k k k k' ' x A x B F w1 1

k k k y Dx v

where process noise w is white zero-meanGaussian, with covariance matrix Q.

Define a measurement vector state y:

where process noise v is white zero-meanGaussian, with covariance matrix R.

The Kalman filter is a recursion that provides the“best” estimate of the state vector x.


Errors at time k read:

with error covariance matrices:

Time update (predictor):

- Update expected value of x:e k x k x̂ k

e k x k x̂ k

Pk E[ek

ekT ]

Pk E[ek ekT ]

where apex “-” means after prediction andbefore observation.

Kalman filter is employed to update theestimate, as well as the covariance matrix,based on new information coming frommeasurements.

k k kˆ ˆ' '

x A x B F1

- Update error covariance matrix P:T

k k' '

P A P A Q1

Measurement update (corrector):

- Update expected value of x:

k k k k kˆ ˆ ˆ x x K y Dxwhere innovation is: k kˆ

y Dx

- Update error covariance matrix P:

k k k P (I K D)P


The optimal Kalman gain can be easilydemonstrated to be: b) identify a state-space model as an

intermediate step for subsequent modelbased controller development. In deatail,create a model of sufficient order (preservingstability and performance of a full-ordermodel) to serve as a “starting point” fordesign of an observer-based state feedbackcontroller.

c) create a model for simulation andcontroller verification, with accuracy (minimalerror to measured I/O data).

T Tk k k K P D DP D R 1

a) Utilise methods and results fromRealization Theory to extract the state-spacemodel and observer gains from noisecorrupted dynamic input/output data sets.

Natural candidates are:

State-space Identification for modelbased control

Observer/Kalman Filter Identification (OKID):an ERA Method

Eigensystem Realization Algorithm (ERA)

Observer/Controller Identification (OCID):closed loop ERA method

Other observer methods

Observer Kalman Filter Identification (OKID)

experimental excitation (typically white noise) and response data is collected from the structure

from these data sets, the Markov parameters, linearly related to the I/O data, are computed.

the system state-space and observer gain (non-linearly related to the Markov parameters) areextracted.

the output of the OKID algorithm is a state-space model for the structure and an observer gainthat is optimal with respect to the process and measurement noise of the I/O data

method does not require physicalmodeling of the system

no a priori knowledge of system isrequired

modal balanced model minimizeserror

OKID advantages


Observer Kalman Filter Identification (OKID)Earthquake Engineering 16.20

OKID flowchart (Juang et al 1992)

OCID is a system Identification Method (extension of OKID) that simultaneouslyidentifies: Open Loop State-Space System Model [A’,B’, D] Observer (Kalman) Gain [K] Existing Controller Gain [G]

from the closed-loop input/output data of the combined structure and existingcontroller

Observer Controller Identification (OCID)Earthquake Engineering 16.21

Observer Controller Identification (OCID)Earthquake Engineering 16.22

OCID flowchart (Juang et al 1992)


Extended Kalman filter

Suppose the state-evolution and measurementequations are non-linear:

k k k kf , x x F w1 1 k k kh y x v

where process noise w and v are white zero-meanGaussian, with covariance matrices Q and R,respectively.

f(.) and h(.) functions may be linearized by using theJacobian, e.g.:

then Kk=J, and similarly for f and A’, so the Kalmanfilter equations are almost the same as before.

Time update (predictor):


- Update error covariance matrix P:T

k k' '

P A P A Q1

Measurement update (corrector):


- Update error covariance matrix P:

k k k P (I K D)P

k k kˆ ˆf , x x F1

k k k k kˆ ˆ ˆh x x K y x

Kalman gain: K k PkHT (HPk

HT R)1

nn

nn

n

n x

x

)(xh)(

xh

)(xh)(

xh

y

y

1

1

1

1

11

xx

xx

xJy

The UKF represents a derivative-free alternative to the EKF, and provides superiorperformances at an equivalent level of complexity.

The basic idea is a non-linear transformation.

Sigma point

Unscented Kalman filterEarthquake Engineering 16.24

– The n-dimensional random variable with meancovariance is approximated by 2n+1 weighted point by

x xxxP

nWn

nWn

nW

niixxni

iixxi

2/1

2/1

/00

Pxχ

Pxχ

xχ

constant:weight /:point /sigma: Wχ

Unscented Kalman filter


• The Unscented Transform– Instantiate each point through the function to yield the set of transformed

sigma points

– The mean and covariance are given by the weighted average and theweighted outer product of the transformed points,

ii f χς

n

iiiW

2

0ςy

Tin

iiiyy W yςyςP

2

0



• Example

=2nd order accuracy



• Overall process– Time Update (“predict”)

n

ikiik W

2

0,ˆ χx

n

i

Tkkikkiik W

2

0,, ˆˆ xχxχP

111, ,ˆ kkki Pxχ

i ,k i ,k kh , χ χ u1 1 i ,k i ,kD yχ

1

nk i i ,k

iˆ W

y y2

0



• Overall process– Measurement Update (“correct”)

Tkkk kk

KPKPP zzk

k k k k kˆ ˆ ˆ x x K y y

k kn T

i i ,k k i ,k ki

ˆ ˆW

y yP y y y y2

0

k kn T

i i ,k k i ,k ki

ˆ ˆW

x yP

χ x y y2

0

k k k k

x y y yK P P1



• The Extended Kalman Filter– Linearized version of Kalman filter– Non-linear system (O)– Linearization error problem

• The Unscented Kalman Filter– Approximates the distribution– Accurate to at least the 2nd order.– No Jacobians or Hessian are calculated.– Efficient “sampling” approach.– The UKF consistently achieves a better level of accuracy than the EKF at

a comparable level of complexity.

Extended/Unscented Kalman filter


For on-line identification purposes, EKF and UKF retain the same implementation asfor correcting the state vector of the structure, but the state vector is considered anexplicit function of the parameters to be estimated.

Therefore, based on the state vector, a system should be solved in order to recoverrecursive solutions for the structural parameters, or better the structural parametersshould be included into an augmented state vector.

Extended/Unscented Kalman filter and nonlinear identification


structural controlbursi/download/phd_lectures_02july/... · 2012. 7. 10. · modern active control...

Documents