SN 5 Robust Stability Ha Context

7/30/2019 SN 5 Robust Stability Ha Context

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Robust stability in the Context

Dr. S. Ushakumari

Associate Professor

Department of Electrical Engineering

College of Engineering Trivandrum

H

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Derive conditions under which a system

remains stable for all perturbations in an

uncertainty set

Robust stability with multiplicative uncertainty

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)(swm )(sm)(sHp

)(sH)(sK

Fig 1. Closed loop system with multiplicative uncertainty

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Contd..

Consider a feedback system as follows.

H(s) - plant

K(s) - controller

wm(s) - multiplicative uncertainty of magnitude

Open loop transfer function of the feedback

system is given by

)K(s)Gp(s)=Hp(s

(s)])s(w1=H(s)K(s)[ mm

(s)(s)G(s)wGp(s)=G(s) mm

1(jw)m 4


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Contd..

Assumptions

Stability of the nominal closed loop systemis guaranteed.

Open loop transfer function Gp(s)is stable

Test the robust stability of the system,Use the Nyquist stability criterion

Robust stability is equivalent to

the stability of the system for all Gp(s)

Gp(s)should not encircle the point (-1+j0)for allGp(s)

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Re

Im

0

G(jw)

(-1, j0)

1+G(jw)

wm(jw)G(jw)

Fig. 2 Graphical derivation of the robust stability condition through

Nyquist plot.

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Robust stability contd..

From the Nyquist plot ofGp(s)

Distance from the point (-1+j0)to the centre

of the disk, which represents Gp(s) is

Radius of the disk is

To avoid encirclement of(-1+j0), none of the

disks should cover the critical point.

G(s)1

(s)G(s)wm

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Robust stability contd..

It may be concluded that the encirclement is

avoided if and only if,

Or, equivalently, iff

,)s(G1(s)G(s)wm

,1)s(G1

(s)G(s)wm

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Definitions

S(s)= Sensitivity function

T(s) = Complementary function

Using the above definitions, it can be concluded

that the encirclement is avoided iff

1G(s)1S(s)

1

K(s)H(s)1)s(H)s(K)s(T

1s(T)s(S

,1)s(T)s(wm

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Contd..

From definition 1, it can be concluded that

robust stability under multiplicative

perturbation is assumed iff,

mw

1T

1)s(T)s(wm

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Contd..

The robust stability condition for the case ofmultiplicative uncertainty gives an upperbound on the complementary sensitivityfunction

Or

To guarantee robust stability in the case ofmultiplicative uncertainty, make T(s)small at

frequencies, where the uncertainty weightexceeds 1 in magnitude.

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Contd..

Condition given above is necessary and

sufficient, provided that at each frequency,

all perturbations satisfying

If this is not the case, the condition is only

sufficient.

1)jw(m

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Since Gp(s)is assumed to be stable and the

nominal closed loop system is stable by design,

then the nominal open loop system does notencircle the critical point -1+j0

Consequently,

Since the family of uncertain plant is normbounded, it then follows that if the source

Gp1(s)is family of uncertain plants, we have

encirclement of-1+j0at some frequency

Gp2(s) isanother set of uncertain family whichpasses through -1+j0 at some frequency.

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Contd..

To guarantee robust stability, the following

condition must hold.

Hence, robust stability is guaranteed iff

The last condition is most easily violated at

each frequency when has magnitude

1 and the phase is such that the term 1+G(s)and have opposite signs

w,Gp0)s(Gp1

w,1)s(,0)s()s(G)s(w)s(G1 mmm

)jw(

)s(G)s(m)s(wm

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Contd..

The robust stability is guaranteed iff

w,0)s(G)s(w)s(G1 m

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Example

Consider the uncertain feedback control system

shown in fig. 1. with Multiplicative uncertainty.Assume that the uncertain plant transfer

function is given by

, and))s()s(w1)(s(H)s(Hmm

1s

1)s(H

10s

2)s(wm

The controller K(s) is a constant, Given

controller is of the form K(s)=10. Determine,

whether the system is robustly stable

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Solution

The complementary sensitivity function T(s)

is given by

9s

10)s(T

)s(H)s(K1

)s(H)s(K)s(T

1s1

1s1

10110

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Solution contd..

1/wm(jw)

T(jw)

Fig 3. Robust stability with Multiplicative uncertainty

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Contd..

Fig 3. Gives the magnitude of T(s) as a

function of frequency versus magnitude of

Here, the magnitude of overboundsthe magnitude of T(s). Hence the robust

stability condition is satisfied.

Hence system is robust ly stable

2

)10s(

)s(w

1

m

)s(w

1

m

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Case 2. Robust stability with an

inverse multiplicative uncertainty

Fig. 4. Closed loop feedback system with inverse multiplicative uncertainty

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H(s) - Plant

K(s) - Controller

- Inverse multiplicative uncertainty

Suppose the open loop transfer functionGp(s)is stable and that the nominal closed

loop system is also stable

Robust stability is guaranteed if

encirclements of the point -1+j0 are

avoided.

(jw)wim

1imim )s()s(w1)s(H)s(Hp

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Robust stability is guaranteed, iff the

following four equivalent equalit ies holds.

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The last condition is most easily violated at

each frequency when has magnitude 1

and the phase is such that andhave opposite signs.

Thus, robust stability is guaranteed, iff,

Considering the sensitivity function S(s)of the

norm, the robust stability with inverse

multiplicative uncertainty is guaranteed, iff

and

)j(im

)s(G1 )s()s(w immi

,0)s(w)s(G1mi

1)s(S)s(wmi

)s(w

1)s(SRs

mi

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Robust stability tests

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Robust performance in the context

H

The general notion of robust performance

is, internal stability and performance of a

specific type should hold for all plants in

family P.

Before dealing with robust performance,it is necessary to study the nominal

performance and its relation to the

sensitivity function.

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Nominal Performance

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Closed loop feed back system with disturbance and noise


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Nominal Performance contd..

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Common specifications in terms of S(s) are

Maximum tracking error at

prespecified frequencies

Minimum steady state tracking error

A

Maximum peak magnitude M of S(s)

Minimum band width *B31


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Therefore, S(s) is used as a performance indicator.

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Re

Im

0

G(jw)

(-1, j0)

1+G(jw)

wp(jw)

Fig. 5. Nominal performance in the Nyquist plot35


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Robust performance

)s(R)s(K

)s(wm )s(m

)s(H

)s(D

)s(wp

*)s(Y

Fig. 6. Block diagram for robust performance in the Multiplicative uncertainty case

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Re

Im

0

G(jw)

(-1, j0)

1+G( jw)

wp( jw)

)j(G)j(wm

Robust performance in the Nyquist plot40


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Contd..

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Contd..

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Remarks on Nominal performance, robust stability

and robust performance

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SN 5 Robust Stability Ha Context