INPUT-OUTPUT APPROACH NUMERICAL EXAMPLES

Transcription

1 INPUT-OUTPUT APPROACH NUMERICAL EXAMPLES EXERCISE s consider the linear dnamical sstem of order 2 with transfer fnction with Determine the gain 2 (H) of the inpt-otpt operator H associated with this sstem.

2 WEAKLY BOUNDED OPERATOR H casal Definition (weakl bonded operator): A casal operator is weakl bonded (or with finite gain) if Definition (gain of a weakl bonded operator): be a casal weakl bonded operator. The gain of H is given b EXAMPLE: LINEAR ASYMPTOTICALLY STABLE DYNAMICAL SYSTEM In conclsion: H is bonded and weakl bonded with gain norm of

3 s consider the linear dnamical sstem of order 2 with transfer fnction with Determine the gain 2 (H) of the inpt-otpt operator H associated with this sstem. The sstems is asmptoticall stable. We then need to determine the norm of. s consider the linear dnamical sstem of order 2 with transfer fnction with Determine the gain 2 (H) of the inpt-otpt operator H associated with this sstem. The sstems is asmptoticall stable. We then need to determine the norm of. is a transfer fnction with gain F(0)=1 and either real or complex conigate poles. In the latter case,we have a resonance pick if

4 s consider the linear dnamical sstem of order 2 with transfer fnction with Determine the gain 2 (H) of the inpt-otpt operator H associated with this sstem. The sstems is asmptoticall stable. We then need to determine the norm of. is a transfer fnction with gain F(0)=1 and either real or complex conigate poles. In the latter case,we have a resonance pick if Hence, we can conclde that: s consider the linear dnamical sstem of order 2 with transfer fnction with Determine the gain 2 (H) of the inpt-otpt operator H associated with this sstem. The sstems is asmptoticall stable. We then need to determine the norm of. is a transfer fnction with gain F(0)=1 and either real or complex conigate poles. In the latter case,we have a resonance pick if Hence, we can conclde that:

5 EXERCISE (n=2) B sing the small gain theorem, determine the vales of k>0 sch that the operator H with inpt and otpt is: 1) L -stable 2) L 2 -stable with finite gain. Provide also an estimate for the gain. STABILITY OF INTERCONNECTED SYSTEMS: FEEDBACK H 1 z 1 H z 2 2 Small gain theorem H be a well-posed casal operator obtained b connecting in feedback two casal and weakl bonded operators H 1 and. If then, H is weakl bonded. Frthermore, Remark: it holds irrespectivel of the signs at the smmation nodes.

6 STABILITY OF INTERCONNECTED SYSTEMS: FEEDBACK H 1 z 1 H z 2 2 Small gain theorem H be a well-posed casal operator obtained b connecting in feedback two casal and weakl bonded operators H 1 and. If then, H is weakl bonded. Frthermore, Remark: it holds irrespectivel of the signs at the smmation nodes. H 1 We next show that the two operators are weakl bonded and estimate their gain.

7 v w w k v j(v) v v w It is an nbiased operator becase:

8 v w It is an nbiased operator becase: Zero bias gain: (bonded operator weakl bonded operator) v w It is an nbiased operator becase: Zero bias gain: if then

9 v w Gain of the operator: Is the gain eqal to the zero bias gain? Not in general if, then Hence, (order n=2) asmptoticall stable linear sstem

10 ASYMPTOTICALLY STABLE LINEAR SYSTEM H is bonded since (zero bias gain) H is weakl bonded Given that the operator is affine, then: (n=2) asmptoticall stable linear sstem

11 (n=2) asmptoticall stable linear sstem since H 1 We next show that the two operators are weakl bonded and estimate their gain.

12 STABILITY OF INTERCONNECTED SYSTEMS: FEEDBACK H 1 z 1 H z 2 2 Small gain theorem H be a well-posed casal operator obtained b connecting in feedback two casal and weakl bonded operators H 1 and. If then, H is weakl bonded. Frthermore, Remark: it holds irrespectivel of the signs at the smmation nodes. H 1 s estimate for which vale of k the condition reqired b the small gain theorem is satisfied.

13 H 1 s estimate for which vale of k the condition reqired b the small gain theorem is satisfied. For k< ½, the operator H is weakl bonded and, hence, L -stable with finite gain H 1 s estimate the gain of the operator H via the small gain theorem.

14 EXERCISE (n=2) B sing the small gain theorem, determine the vales of k>0 sch that the operator H with inpt and otpt is: 1) L -stable 2) L 2 -stable with finite gain. Provide also an estimate for the gain. H 1 We next show that the two operators are weakl bonded and estimate their gain.

15 v w w k v j(v) v v w It is an nbiased operator becase: Zero bias gain: (bonded operator weakl bonded operator)

16 v w It is an nbiased operator becase: Zero bias gain: If then and hence (n=2) asmptoticall stable linear sstem

17 ASYMPTOTICALLY STABLE LINEAR SYSTEM H is bonded becase (zero bias gain) H is weakl bonded Given that the operator is affine, then: (n=2) asmptoticall stable linear sstem perchè

18 H 1 We next show that the two operators are weakl bonded and estimate their gain. H 1 s estimate for which vale of k the condition reqired b the small gain theorem is satisfied. For k< ½, the operator H is weakl bonded and hence L 2 -stable with finite gain

19 H 1 H s estimate the gain of the operator H via the small gain theorem. EXERCISE (n=2) B the circle criterion, determine the vales of k>0 sch that the operator H with inpt and otpt is L 2 -stable.

20 L 2 STABILITY IN SECTOR [k 1,k 2 ] S : w e G(s) Theorem (Circle criterion for L 2 stabilit of a Lr e sstem) Sstem S is L 2 -stable for an if the nmber of encirclements of G(s) Nqist plot arond O(k 1,k 2 ) is eqal to the nmber of poles of G(s) with positive real part. Im Im Im Re -1 k 2-1 k 1 Re -1 k 1-1 k 2 Re 0 k 1 < k 2 k 1 < 0 < k 2 k 1 < k 2 0 L 2 STABILITY IN SECTOR [k 1,k 2 ] S : w G(s) Theorem (Circle criterion for L 2 stabilit of a Lr e sstem) Sstem S is L 2 -stable for an if the nmber of encirclements of G(s) Nqist plot arond O(k 1,k 2 ) is eqal to the nmber of poles of G(s) with positive real part. In or setting: e S :

21 L 2 STABILITY IN SECTOR [k 1,k 2 ] Im O(0,k) 0 k 1 < k 2 Re S : Im O(0,k) Re

22 EXERCISE k j() (n=2) Vale of k>0 sch that the operatore H with inpt and otpt is L 2 -stable: small gain theorem k<1/2 circle criterion k<4 EXERCISE k j() (n=2) Vale of k>0 sch that the operatore H with inpt and otpt is L 2 -stable: small gain theorem k<1/2 circle criterion k<4 conservative reslt obtained b the small gain theorem. Bt the circle criterion derives from the small gain theorem

23 LUR E SYSTEM: SMALL GAIN THEOREM 1 1 z 1 H S: H 1 2 z 2 2 Sstem S (operator H): is L 2 -stable [for an fnction ] in sector [-k, k] if sstem with as transfer fnction is asmptoticall stable and In this case, same condition bt with in sector [0,k] EXERCISE The figre below reports Nqist plot of the transfer fnction G(s) of a linear sstem of order 3:

24 EXERCISE G(s) Determine k 1 and k 2 sch that the Lr e sstem is L 2 -stable in the sector [k 1,k 2 ] b sing the circle criterion G(s) Determine k 1 and k 2 sch that the Lr e sstem is L 2 -stable in the sector [k 1,k 2 ] b sing the circle criterion Sol: G(s) has a real positive pole Nqist plot of G(s) shold encircle once the circle O(k 1,k 2 ) anticlockwise.

25 If we draw the circle centered in -3 and with radis 0.7, then, the circle criterion is satisfies. We then get -1/k 1 = k 1 = /k 2 = k 2 = 0.43