EECE 574 - Adaptive Control - Laguerre-based Adaptive Control - Part I

EECE 574 - Adaptive ControlLaguerre-based Adaptive Control - Part I

Guy Dumont

Department of Electrical and Computer EngineeringUniversity of British Columbia

January 2013

Guy Dumont (UBC EECE) EECE 574 - Adaptive Control January 2013 1 / 39

History

History

Edmond Laguerre, Frenchmathematician,(Bar-le-Duc, 1834-1886)

He studied approximationmethods and is bestremembered for the specialfunctions: the Laguerrepolynomials


History

Mathematical Premise

Consider the set of square-(Lebesgue) integrable functions y(t):∫∞

0‖y(t)‖2dt < ∞

This set of signals is denoted as the Lebesgue space L2[0,∞).

The impulse response of an asymptotically stable system lies in L2.

Such a system is said to be L2-stable. Obviously, neither a pure oscillatoror an integrator is in L2.


History


Now, define the inner product as

< x,y >=∫

∞

0x(t)y(t)dt

We can define an orthonormal sequence {ψi} if it satisfies:

< ψi,ψj >= δi,j

where δi,j is the Dirac delta-function.


History


The set {ψi} is said to be complete if every y ∈ L2[0,∞) can be expandedinto the series:

y =∞

∑i=0

< y,ψi > ψi

Then, for any real ε > 0, there exists a positive integer N such that

‖y−N

∑i=0

< y,ψi > ψi‖< ε


History


An inner-product space with the property of completeness is called a Hilbertspace. Then we can state the classical Riesz-Fisher theorem:

Theorem (Riesz-Fisher Theorem)

Let {ψi} be a complete orthonormal set as specified above, and let {ci} be asequence of real numbers such that ∑

∞i=0 |ci|2 converges. Then, there exists a

unique y ∈ L2[0,∞) such that ci =< y,ψi >. Consequently,

y =∞

∑i=0

ciψi

in the sense of convergence in L2[0,∞).


History

History

Network synthesis using Laguerre functions (Wiener and Lee,1930’s-1960’s)

Data reduction in system identification (King and Paraskevopoulos,1970’s)

Description of nonlinear systems (Schetzen, 1980)

PID tuning (Zervos, Bélanger, Dumont) and adaptive control (Zervos andDumont, 1980’s)

Indutrial applications (Dumont et al., 1980’s-1990’s)


History

History

Theoretical investigations and extensions (Mäkilä, Wahlberg, Van denHof, Ninness, 1990’s)

Nonlinear adaptive control (Fu and Dumont, 1990’s)

Commercial adaptive controllers (Universal Dynamics, 1990’s)


Laguerre Functions Continuous Laguerre Functions

Laguerre Functions

In the time domain, the Laguerre functions are defined as:

li(t) =√

2pexp(pt)(i−1)!

di−1

dti−1 [ti−1 exp(−2pt)]

Because Laguerre functions form a complete set in L2[a,b], if given areal and continuous function y(t) in L2[a,b], and a positive real ε > 0then, there exists a positive integer N such that,

E =∫ b

a|y(t)−S(t)|2dt < ε

where,

S(t) =N

∑i=1

cili(t) = rT l

Where, cT = [ c1 c2 . . . cN ], and lT = [ l1 l2 . . . lN ].The constants ci’s are called the Laguerre spectrum gains.



Continuous Laguerre Functions

In the Laplace transform domain, the Laguerre filters are defined as:

Li(s) =√

2p(s−p)i−1

(s+p)i , i = 1, ..,N

where i is the order of the function (i = 1, ..N), and p > 0 is thetime-scale



Continuous Laguerre Functions

-√

2ps+p

U(s)- s−p

s+pL1(s)

- q q qL2(s)- s−p

s+pLN(s)p

?

p?

p?

c1 c2 cN

? ? ?

Summing Circuit�Y(s)

Figure: Representation of plant dynamics using a truncated continuous Laguerreladder network



Example of Laguerre Modelling

Figure: Laguerre Modelling


Laguerre Functions Discrete Laguerre Functions

Discrete Laguerre Models

An open loop stable linear system can be represented with arbitrary accuracy by:

y(k) =n

∑i=1

ciΦi(q−1)u(k)

where Φi(q−1), for i = 1, · · · ,n, are the discrete Laguerre functions:

Φi(q−1) =√

1−p2 q−1(q−1−p)i−1

(1−pq−1)i , i = 1, . . . ,n

ci =∞

∑k=0

h(k)Φi(q−1)δ (k)

where δ (k) is the unit impulse and h(k) is the system impulse response


Laguerre Functions Discrete Laguerre Functions

Modelling with Laguerre Functions

An open-loop stable system can be described by the stable, controllableand observable state-space model:

l(t+1) = Al(t)+bu(t)

y(t) = cT l(t)

withlT(t) = [ l1(t) l2(t) . . . lN(t) ]T

andcT = [c1 c2 . . .cN ]


Identification of Laguerre Models


The classical method is to compute the Laguerre spectrum of a signalusing the correlation method, i.e.

ci =∫ b

ah(t)li(t)dt

where h(t) is the plant impulse response and li(t) is the ith Laguerrefunction

In practice, it is more realistic to use a least-squares type algorithm




Consider the real plant described by

y(t) =N

∑i=1

ciLi(q)+∞

∑i=N+1

ciLi(q)+w(t)

where w(t) is a disturbance

This model has an output-error structure, is linear in the parameters, andgives a convex identification problem




With the following standard notations:

φT(k) = [ l1(kT) l2(kT) . . . lN(kT) ]

φTU(k) = [ lN+1(kT) lN+2(kT) . . . . . . ]

This is somewhat of an abuse of notation, as in theory the true plant isrepresented as an infinite series. However, for all practical purposes, the trueplant can be exactly described by a large but truncated series.

ΦT = [ φ(T) φ(2T) . . . φ(nT) ]

ΦTU = [ φ

U(T) φ

U(2T) . . . φ

U(nT) ]

YT = [ y(T) y(2T) . . . y(nT) ]

WT = [ w(T) w(2T) . . . w(nT) ]




Then, we can write:Y = Φ

Tθ 0 +Φ

TUθ U +W

The least-squares estimate θ of θ 0 is given by:

θ = [ΦΦT ]−1

ΦY

We can write the estimate as:

θ = θ 0 +[ΦΦT ]−1

ΦΦUθ U +[ΦΦT ]−1

ΦW

Taking the expectation yields:

E[θ ] = θ 0 +E{[ΦΦT ]−1

ΦΦUθ U}+E{[ΦΦT ]−1

ΦW}

It is easy to see that, for large n and zero-mean, white input u:

E[θ ] = θ 0




Even if w(t) it coloured and non-zero mean, simple least-squares provideconsistent estimates of the ci’s.

The estimate of the nominal plant, i.e. of ci, for i = 1, · · · ,N is unaffectedby the presence of the unmodelled dynamics represented by ci, fori = N +1, · · · ,∞.

This provides us with a robust estimator, one of the two key ingredientsin a robust adaptive controller.




The mapping (1+aeiω)/(eiω +a) improves the condition number of theleast-squares covariance matrix.

The implicit assumption that the system is low-pass in nature reduces theasymptotic covariance of the estimate at high frequenciesFor least-squares, the mean square error of the transfer function estimatecan be approximated by

π(eiω ) =12

(N(1−λ )

1−a2

|eiω −a|2Φv(eiω )

Φu(eiω )+

µ2

1−λr1(eiω )

)




The case a = 0 corresponds to a FIR model. The MSE is proportional tothe number of parameters. Compared with a FIR model, an orthonormalseries representation requires less parameters and thus gives a smallerMSE.The disturbance spectrum is scaled by

1−a2

|eiω −a|2

thus reducing the detrimental effect of disturbances at high frequencies.


Non Linear Systems

Extension to Nonlinear Systems

Special case of a Wiener model, where the linear dynamic partrepresented by a series of Laguerre filters is followed by a memorylessnonlinear mapping

Can be derived from the Volterra series input-output representation,where the Volterra kernels are expanded via truncated Laguerre functions

A finite-time observable nonlinear system can be approximated as atruncated Wiener-Volterra series:

y(t) = h0(t)+N

∑n=1

∫· · ·∫

hn(τ1, · · · ,τn)n

∏i=1

u(t− τi)dτi


Non Linear Systems


For instance, truncating the series after the second-order kernel:

y(t) = h0(t)+∫

∞

0h1(τ1)u(t− τ1)dτ1 +∫

∞

0

∫∞

0h2(τ1,τ2)u(t− τ1)u(t− τ2)dτ1dτ2

Assuming that the Volterra kernels are in L2[,∞), they can be expandedand approximated as:

h1(τ1) =N

∑k=1

ckφk(τ1)

h2(τ1,τ2) =N

∑n=1

N

∑m=1

cnmφn(τ1)φn(τ2)


Non Linear Systems


Using Laguerre functions, this second-order nonlinear system can beexpressed as the nonlinear state-space model:

l(t) = Al(t)+bu(t)

y(t) = c0 + cT l(t)+ lT(t)Dl(t)

where c = {ck} and D = {cnm}.Note that since the Volterra kernels are symmetric, cnm = cmn and thus Dis symmetric. A discrete model can be derived in a similar form. Notethat this model is linear in the parameters, and can thus be easilyidentified.


Non Linear Systems


Figure: Structure of the nonlinear Laguerre model


Laguerre-based Control A Simple Predictive Control Law

Laguerre EHC

Given the discrete Laguerre model

l(k+1) = Al(k)+bu(k)

y(k) = Cl(k)

We can predict the plant output for the future d-steps based on using thecurrently measured output y(k), which is:

y(k+d) = y(k)+C[l(k+d)− l(k)]



Laguerre EHC

Recursively using the state equation in Equation (1.2) gives the futureLaguerre model states:

l(k+1) = Al(k)+Bu(k)

l(k+2) = A2l(k)+ABu(k)+Bu(k+1)

· · · = · · ·l(k+d) = Adl(k)+Ad−1Bu(k)+ · · ·+Bu(k+d−1)



Laguerre EHC

By assumingu(k) = u(k+1) = · · ·= u(k+d−1)

The future control inputs are held constant, the d-step ahead Laguerremodel state prediction is then:

l(k+d) = Adl(k)+(Ad−1 + · · ·+ I)Bu(k)



Laguerre EHC

Substitutingl(k+d) = Adl(k)+(Ad−1 + · · ·+ I)Bu(k)

intoy(k+d) = y(k)+C[l(k+d)− l(k)]

we finally obtain the d-steps ahead output prediction:

y(k+d) = y(k)+C(Ad− I)l(k)+βu(k)

where,β4= C(Ad−1 + · · ·+ I)B



Laguerre EHC

Setting the d-steps ahead output y(k+d) equal to the desired plant output yr,we obtain the control u(k) as:

u(k) =yr− y(k)−C(Ad− I)l(k)

β



Laguerre EHC

It is worth mentioning that unless d is larger than the plant delay, β willbe zero.

Also, if the plant is non-minimum phase, a non-zero β is not guaranteedeven when d is larger than the plant delay.

Thus d should be sufficiently large. One must choose c such that β is ofthe same sign as the process static gain and of sufficiently largeamplitude.



Laguerre EHC

Therefore a possible criterion to be satisfied when choosing the horizon dis:

β (k)sign(C(I−A)−1B)≥ ε|C(I−A)−1B|

with ε = 0.5.

Note that the matrix (I−A)−1B can be computed off-line as it dependsonly on the Laguerre filters.

In adaptive mode, additional computation has to be carried on-line sincethe identified model (i.e the Laguerre coefficients: C) is changing.



Laguerre EHC

Sometimes, the control signal is expressed in velocity form. The controllercan be rewritten as:

u(t) = (yr− y(t))/β −CS(Al(t)−Al(t−1)−Bu(t−1))/β (1)

Using the definition of β and rearranging, one gets

∆u(t) =yr− y(t)− f T∆l(t)

β

where S = (Ad−1 + · · ·+ I), f T = cTSA, ∆u(t) = u(t)−u(t−1), and∆l(t) = l(t)− l(t−1).


Laguerre-based Control Feedforward Control

Feedforward Control

Inclusion of a feedforward variable uf requires an additional Laguerrenetwork

lf (k+1) = Af lf (k)+Bf uf (k)

The plant output is then described by

y(k) = CT l(k)+CTf lf (k)

Following the same derivations as before but using the above modifiedmodel will give a control law that automatically incorporatesfeedforward compensation.


Laguerre-based Control Integrating Systems

Integrating Systems

Time delay integrating systems are common in the process industries.

Although Laguerre functions are particularly appealing to describe stabledynamic systems, they cannot be used to represent dynamics that containan unstable or an integrating mode.



Integrating Systems

In case of an integrator, we can generally assume knowledge of itspresence. We can then simply remove the effect of the integrator fromthe data by differentiating it.

The Laguerre network is then employed to model only the stable part ofthe plant as in

l(k) = Al(k)+Bu(k)

∆y(k) = Cl(k)

where ∆y(k) = y(k)− y(k−1)



Integrating Systems

Good adaptive control of those systems requires estimation of thedisturbance

Assuming both known and unknown disturbances, the model states areupdated based on:

l(k+1) = Al(k)+Bu(k)

lf (k+1) = Af lf (k)+Bf uf (k)

ld(k+1) = Adld(k)+Bdud(k)



Integrating Systems

The output estimations are defined as:

y(k) = y(k−1)+C(k)l(k)+ yf (k)+ yd(k)

yf (k) = yf (k−1)+Cf (k)lf (k)

yd(k) = yd(k−1)+Cd(k)ld(k)

where f denotes a feedforward variable and d a variable associated withthe unmeasured disturbance model.



Integrating Systems

The disturbance is then estimated using an extended Kalman filter with

ud(k) = (y(k)+ yff (k)+ yd(k))− y(k)

where y(k), yd(k) and yff (k) are the estimated plant, known and unknowndisturbance model outputs.

A GPC type control law can then be derived.


EECE 574 - Adaptive Control - Laguerre-based Adaptive Control - Part I

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