chapter 3 system id

8/2/2019 Chapter 3 System ID

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Chapter 3 System ID

1. Scalar case, one unknown parameter, intuitive method 12. Scalar case, one unknown parameter, intuitive method 23. Scalar case, one unknown parameter, un-normalized4. Scalar case, one unknown parameter, normalized5. First order linear system-scalar case, un-normalized6. First order linear system-vector case, un-normalized7. First order linear system-vector case, normalized8. SPR Lyapunov approach, normalized9. Gradient method, normalized

10. Least squares, normalized

Chapter 3 System ID for adaptive control

1. Scalar case, one unknown parameter, intuitive method 12. Scalar case, one unknown parameter, intuitive method 23. Scalar case, one unknown parameter, un-normalized method4. Scalar case, one unknown parameter, normalized method5. First order linear system-scalar case, un-normalized6. First order linear system-vector case, un-normalized7. First order linear system-vector case, normalized8. SPR Lyapunov approach, normalized9. Gradient method, normalized

10. Least squares, normalized


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1. Scalar case, one unknown parameter, intuitive method 1System : y (t) = .u (t)Where : y (t) = output

u (t) = input

= unknown constantAssume that y (t) and u (t) are measurable t to and it is desired to obtain an estimate of.Solution : Get the 1/0 pair {u (t) , y (t)} over an interval [, + T] so that u (t) 0 forsome t [, + T] and choose as , [, + T]Remark :

u (t) maybe very close to zero = very big! UnboundedRemark :It would be worse if the system is subjected to noise

The system would result in a large spread in the value of at different t [, + T]Example : y (t) = . u (t) + d

Review

y (t) =

. u(t)

if u0 the problem is = of u0 ; may have measuring error/noise by u

2. IDScalar case, one unknown parameter, 2nd intuitive methodLets define an estimator with system : y = . u(t) t [, + T]The solution maybe considered as one that minimizes the performance index (p1) :

Where [ ] Its minimization is evaluated as :

[

]


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for u (t) = 0 , t [, + T]Remark : as in the previous method, can only be calculated once if no noise presentRemark : If noise shows up, it might not gine true ofRemark : Increasing of I will improve To anticipate it, we meed RLS : recursive lee square!

Because its impossible to increment the function to another time interval

3. IDScalar case, one unknown parameter, unvormalized (the system)System y(t) = Where u . is an unknown constant and u,y are measurableWe would like to determine from the measured datalets consider the estimated output : Define

output error

= = - ( ) u= -

Define J( performance J( is pointuise of J=

=

J( is convex over and hence the minimizationof J is global convex optimization problemECX is said to be convex if [0,1] such that

= * +=


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Stability of the estimator above can be proved as :

Let V (

V(x) is p.d if 0 V (x) = 0 x = 0 0 V (x) > 0 x

parameter error is bounded ( Check (proef)

=

<

Check =

If then 0 as t = y = u

J =

Because of : V =

If

=

as t =

When will as t = ?

as t = (Barbalats lemma)


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A necessary and sufficient condition is that u satisfies the PE condition

and for some PE = Persistent ExatationU should be rich enough to have parameter convergence

4. IDScalar case, one unknown parameter, normalizedSystem y(t) = Where is an unknown constant and u,y are measurable u

Since now u could be unbounded, we may not set

Because it would give an unbounded Let us divide the system equation by the normalizing signal m > 0To have Where and , and m2 = 1 + The signal ns is chose so that e.g. pick ns = u

m2

= 1 +

u

2

Consider the estimate Let normalized output error

= =

J( =

=

=


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= = m > 0

Using the gradient method

= * += * +

= Let V =

= = -

<

= if

then as t = as t =

For parameter convergence, me need PEPersistent excitation

ID of 1st

order linear systems (unnormalized)

Plant : Where Xp R is the output of the plant and u R is the bounded inputAp and Kp are unknown constants, but me know that ap < 0

The problem is to determine ap and kp from u and xp


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Here, we are going to solve this problem with 2 estimators :

1. Parallel estimatoroutput error methodonly for scalar system (not for vector)2. Seriesparallel estimatorequation error methodfor scalar and vector systemsMethod 1 : - output error method

Select the parallel estimator u - error dynamicsWhere e =

, (The output error dynamics is driven by the output error)And parameter error Y (t) = u not discussed yet

5.

ID of 1

st

order linear system (unnormalized)Plant = Where Xp IR , u R are measurableAp and Kp are unknown constants, ap < 0

1. Parallel estimatoroutput error method (only for scalar)2. Seriesparallel estimatorequation error method (also for method)

Method : - output error method

Select the parallel estimator

u - error dynamics


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Define a Lyapunov function candidate

Take the time derivate of V along the trajectory of the errors dynamics, we have : Pick

Bounded

U Xp

Stable

LTI System

BI

Bounded

Input

BIBO

Stable

BO

Bounded

Output


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We need PE for parameter Convergence

Summary :

For parameter Convergence, we need PE

For parameter Convergence, we need PE

Method 2 : - Equation error method

Select the series-paralel estimator : [ ] [ ]


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=[ ] Pick

t < 0 . We need PE for convergence of the parameters.

ID VECTOR CASE UN-FORMALIZED

Plant = Where Assume that

is Hurwitz and u is bonded.

We would like to find and , based on the information of and u.Method : Output error methodDefine the parallel estimator :

The error dynamics can be as : Pick the Lyapinov function candidate : = =

?

Although is known to be stable, its elements are not known and hence a matrix p > 0 cant be found

s.t. with Q > 0.This approach doesnt work for veetos case but its fine for scalar case.

Method 2 : Equation error method

Define the series-parallel estimator Where and and is known Let V =

=


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= - Q

is stable s, t,

Method 2 :

Pick

because q is a constant

We need PE for parameter converage!

What about u 7. ID-1st order systems, normalized u

Plant : unknown constansLet + Define e = x - Where s is the laplace variable

Ns is to be designed s.t.

and

with

and ns =

Update laus


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(s-am)e = (S-am) . (x - )e. - ) am (x - )e.

-

= (axbu)(am

+ (

-am) x +

= am (x - ) + (a- x + (b-)u (x - ) + (a-x + (b-)uam (x - ) - e. = (am - e + (a-x + (b-)u = (am - e + +

Let V=

-

[ ] (am - ) xe ue (am - E,

||

For some n > 0

=

E = (am - e + + We can not conclude asympt stability of e

e.ns :v

Q.E.D


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e. m

= <


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= =

In general, we represent a linear parametric model as :

Y = G(s) . Review e = x- updated laws : Positive Linear System SPR : Strictly positive RealPositive linear systems are linear systems with positive real T.F.s

System :

(n-th order SISO linear system)Def : h(s) is positive real (PR) if Re [h(s)] 0 Re [s] 0h(s) is strictly positive real (SPR) if h (s-) is PR for some > 0

e.q. h(s) = > 0, S=

h( ) = =

If Re [h(s)] = Then h(s) is PR

For SPR :

Pick h(s - ) =

is P.R


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is SPR

Thm : h(s) is SPR iff

h(s) is strictly stable, and

Re [h(j)] > 0 Some useful necessary conditions to check SPR :1. h(s) strictly stable

e.g h(s) = is not SPR

because the evalue is not in the left hand plane

2. h(s) has relative degree 0 or 1e.g h(s) =

is not SPR

3. h(s) is strictly minimum phasee.g h(s) =

is not SPR4. the ngguist plot of h(j) lies entirely in RHP (RightHand - Plane)5. phaseshift in response to sinusoidal input is always less than 900the kalman-yakuborich lemma (positive real lemma)

lemma :

system :

y = (A,B) controllable

The T.F. h(s) = is SPRIff p > 0 and Q > 0 such that (s.t)

PB = C

Controllable stabilizable uncontrollable

Uncontrollable not stabilizable

PCstabilizable

Costabilizable is continuousstabilizable

stabilizable

stabilizable f is smooth


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p.c. stabilizable piece wise continuousmodified k-y-lemma

kaliman-Yakubovich-Meyer Lemma

given x

0 A asymptotically stable, symmetric p.d. matrix L

if the T.F. is SPR, then vector q and p = PT > 0 s.t

PB = C + 8. IDGeneral SISO plantSPR Lyapunov Approach, normalized

Plant : y = G(s) . , y

Choose L(s) s.t. L-1

(s) is a proper stable T.F. and G(s) . L(s)

is a proper SPR T.F

y = G (s) . L(s) . = G (s) . L(s) .

e.q : y =

= * += G(s).

Let L = s+2 L-1

=

Y = G L L-1

= * +

=

An estimator can be constructed as Case 1 : G(s) is minimum phase

Pick L (s) = G-1

(s) s.t . GL = 1


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Define

etc

Case 2 :

G(s) is not required to be minimum phase

Define the normalized error

GL e


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[ ] [ ]

Realization to state-space function


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1DGradient Method - normalized

System:

In parametic model

Two approaches are introduced below based on different cost-function

Case 1 :Instantaneous cost function

Convexity of guarantees the existense of a unique globalMinimum devine by :


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(1)Non-recursive algorithm 1 : * +

(2)Non recursive alogarithm 2:

Adapptive control of linear-time-varying system (phD Thesis)

Joannou and Tsaklis


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(3)Recursive Algorithm

March 30, 2006


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* +

( )

}**it means : to make


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Case 2 : Integral Cost Function

[ ] [

]

[ ]

{ } {

}

Homogeneous


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Note that R(t) and Q(t) are exactly solutions to the following two

Non-homogenous D.E. (differential equation) respectively

{ }

chapter 3 system id

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