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Lall Stanford 2001111201 Engr210a Lecture 13 Internal stability and coprime factorization Internal stability Stabilizing controllers Achievable closedloop maps Interpolation Parametrization of stabilizing controllers Division and coprimeness Euclids ID: 23331

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## Presentations text content in Internal stability and coprime factorization S

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13 - 1 Internal stability and coprime factorization S. Lall, Stanford. 2001.11.12.01 Engr210a Lecture 13: Internal stability and coprime factorization Internal stability Stabilizing controllers Achievable closed-loop maps Interpolation Parametrization of stabilizing controllers Division and coprimeness Euclid’s algorithm The Bezout equation Coprime factorization in

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13 - 2 Internal stability and coprime factorization S. Lall, Stanford. 2001.11.12.01 Alternative characterization of internal stability 22 This interconnection is equivalent to 22 Let 22 KP 22 22 22 22 22 Suppose the realizations for 22 and are stabilizable and detectable. Then the interconnection is internally stable

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13 - 3 Internal stability and coprime factorization S. Lall, Stanford. 2001.11.12.01 Stabilizing controllers Controller is called stabilizing if the interconnection of 22 and is internally stable. Characterizations Assume the realizations for 22 and are stabilizable and detectable. Then is stabilizing if and only if 22 KP 22 22 22 22 22 is stable Special case: if is stable, then is stabilizing 22 22 is stable Proof: Note that 22 22 22 +( 22 22 22 22 +( 22 22 Another special case: if is stable, then is stabilizing 22 is stable

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13 - 4 Internal stability and coprime factorization S. Lall, Stanford. 2001.11.12.01 Stable interconnections Recall the set of realizable maps is rlzbl RP P,K for some RP 11 12 RP 21 RP Deﬁne the set stable RP P,K for some RP the interconnection is internally stable the set of closed-loop maps achievable by stabilizing controllers. Theorem Suppose 22 is proper. Then stable is aﬃne.

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13 - 5 Internal stability and coprime factorization S. Lall, Stanford. 2001.11.12.01 Theorem Suppose 22 is proper. Then stable is aﬃne. Proof ∈H stable ifandonlyif 11 12 RP 21 ,and 22 KP 22 22 22 22 22 is stable Substituting 22 gives RP 22 22 22 22 Then =( RP 22 is stabilizing if and only if is stable. The map from to is aﬃne, and therefore the preimage of under this map is an aﬃne set in Hence the set of such that =( RP 22 is stabilizing is an aﬃne set. The map from to is aﬃne, and the image of an aﬃne set under an aﬃne map is aﬃne.

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13 - 6 Internal stability and coprime factorization S. Lall, Stanford. 2001.11.12.01 Interpolation conditions We have =( RP 22 is stabilizing RP 22 22 22 22 is stable For scalar plant and controller 22 and 22 ,let RP 22 .Then =( TP 22 is stabilizing TTG 22 is stable Let ,...,z be the unstable zeros and ,...,p be the unstable poles of 22 . Assume they are distinct. Then =( TP 22 is stabilizing )= for =1 ,...,m )=0 for =1 ,...,k relative degree of relative degree of 22 Then the closed loop map is P,K )= 11 12 TP 21 22 Note that the maximum modulus principle then implies that 11 and 22 if 22 has RHP zeroes; hence weights are essential.

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13 - 7 Internal stability and coprime factorization S. Lall, Stanford. 2001.11.12.01 Optimization and interpolation The general problem is minimize subject to P,K for some RP The closed-loop is stable Equivalent formulation for scalar 22 Let ,...,z be the unstable zeros and ,...,p be the unstable poles of 22 . Assume they are distinct. minimize 11 12 TP 22 21 subject to )= for =1 ,...,m )=0 for =1 ,...,k relative degree of relative degree of 22 This is an example of a Nevanlinna-Pick interpolation problem. In general, these problems are hard to solve (but it can be done).

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13 - 8 Internal stability and coprime factorization S. Lall, Stanford. 2001.11.12.01 Stabilizing controllers for stable plants Suppose is stable. Then is stabilizing =( RP 22 for some stable Then stable 11 12 RP 21 Proof is stable if and only if is stable, since RP 22 22 22 22 Notes If is stable, then the above gives a simple parametrization of all stabilizing con- trollers. What about when is unstable? We need the notion of coprime factorization

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13 - 9 Internal stability and coprime factorization S. Lall, Stanford. 2001.11.12.01 Optimization for stable The general problem is minimize subject to P,K for some RP The closed-loop is stable Equivalent formulation for stable minimize 11 12 RP 21 subject to Once the optimal is found, then the optimal is given by =( RP 22

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13 - 10 Internal stability and coprime factorization S. Lall, Stanford. 2001.11.12.01 Coprimeness Suppose n, d are integers. Then divides if there exists such that dq The integer is called the greatest common divisor (gcd) of n,m if divides and divides Every integer that divides both and also divides and are called coprime if their gcd is Examples 10 and 21 are coprime. 12 and 21 are not coprime. Their gcd is

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13 - 11 Internal stability and coprime factorization S. Lall, Stanford. 2001.11.12.01 Division Given n,m ,and . Then there exists a unique and with r such that nq is the quotient, is the remainder. Euclid’s algorithm Euclid’s algorithm gives a way to ﬁnd the gcd of n,m =1; Repeat Find and so that qb +1 until =0 The gcd is then 57 12 12

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13 - 12 Internal stability and coprime factorization S. Lall, Stanford. 2001.11.12.01 Polynomials Let be the set of polynomials in the variable Suppose n, d are polynomials. Then divides if there exists such that dq The polynomial is called a greatest common divisor (gcd) of n,m if divides and divides Every that divides both and also divides and are called coprime if their gcd is a scalar. Examples 1)( 2) and 3) are coprime. 1)( +2) and 1) are not coprime. A gcd is any scalar multiple of 1)

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13 - 13 Internal stability and coprime factorization S. Lall, Stanford. 2001.11.12.01 Polynomials Given two polynomials and , we can apply Euclid’s algorithm to ﬁnd their gcd. Euclid’s algorithm Euclid’s algorithm gives a way to ﬁnd the gcd of n,m =1; Repeat Find and so that qb +1 until =0 A gcd is then +5 +6 +3 +1 +3 +2 +1 +3 +2 +1 +4 +2 +4 +2 +1 +1 +4 +4

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13 - 14 Internal stability and coprime factorization S. Lall, Stanford. 2001.11.12.01 Euclid’s algorithm ( 300 B.C.) is the gcd of and Proof We have ,and =0 ,where for =2 ,...,k We know divides , and the above equation implies that if divides then divides . Hence by induction, divides and ;thatis, divides and Also, implies that xa ya for some x,y for =2 ,...,k .Thatis, is a linear combination of and where the coeﬃcients are integers. By induction again, we have xa ya xm yn for some x,y hence any divisor of and is also a divisor of . Hence is a gcd.

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13 - 15 Internal stability and coprime factorization S. Lall, Stanford. 2001.11.12.01 The Bezout equation The integers m, n are coprime if and only if there exists x,y such that xm yn =1 This equation is called the Bezout equation Proof The proof follows immediately from the above proof for Euclid’s algorithm. Notes Euclid’s algorithm works for The integers Polynomials Scalar, stable, proper rational functions in RH Matrix-valued stable, proper rational functions in RH The general algebraic structure for which this works is called a ring The if direction is easy; e.g. for polynomials, if and have a common zero, then their cannot exist a solution to the Bezout equation.

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13 - 16 Internal stability and coprime factorization S. Lall, Stanford. 2001.11.12.01 Scalar stable proper transfer functions Suppose m, n RH .Then divides if there exists RH such that dq Notes divides ifandonlyif RH Examples )= +1 +2) )= +1 )= +1) )= +4 )= )= +2) divides and , but not divides and , but not divides , but not or

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13 - 17 Internal stability and coprime factorization S. Lall, Stanford. 2001.11.12.01 Scalar stable proper transfer functions RH is called a greatest common divisor (gcd) of n, m RH if divides and divides Every RH that divides both and also divides and are called coprime if and are stable and proper for all gcds Notes and are coprime if and only if they have no common zeros in the right-half-plane, or at inﬁnity. Examples +1) and +1 are coprime. xm yn =1 is satisﬁed for (2 +4)( +1) +3 +3 +1 and 2)( +1) +3 +3 +1 +3) and +3) are not coprime.

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13 - 18 Internal stability and coprime factorization S. Lall, Stanford. 2001.11.12.01 Coprime factorization Rational numbers Given ,ﬁnd n,m such that and n,m are coprime Rational functions; factorization over Given RP ﬁnd n,m such that and n,m are coprime n,m always exist; just cancel any common zeros. Rational functions; factorization over RH Given RP ﬁnd n,m RH such that and n,m are coprime In contrast to above: n,m must be stable proper transfer functions.

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13 - 19 Internal stability and coprime factorization S. Lall, Stanford. 2001.11.12.01 Coprime factorization over RH Given RP ﬁnd n,m RH such that and n,m are coprime Notes n,m must be stable proper transfer functions. A coprime factorization always exists; make all stable poles of poles of , all stable zeros of poles of , and add zeros to and as necessary. Example Suppose is )= 1)( +2) 3)( +4) A coprime-factorization is )= +4 )= +2

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13 - 20 Internal stability and coprime factorization S. Lall, Stanford. 2001.11.12.01 Coprime transfer functions in RH Suppose M,N RH ,andlet RH be square. Then right-divides if there exists RH such that QD The square RH us called a right greatest common divisor of M,N if right-divides and right-divides Every RH that right-divides both and also right-divides and are called right-coprime if and are stable and proper for all gcds The Bezout equation M,N RH are right-coprime if and only if there exists X,Y RH such that XM YN

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Right-coprime factorization Given RP , a factorization such that NM N,M RH and are right-coprime is called a right-coprime factorization of Left-coprime factorization Given RH , a factorization such that N, RH and are left-coprime is called a left-coprime factorization of Notes Left and right coprime factoriza- tions always exist. Example Suppose is )= +1)( 1) A coprime-factorization is )= +1) )= +1 13 - 21 Internal stability and coprime factorization S. Lall, Stanford. 2001.11.12.01 Coprime factorization in RH

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13 - 22 Internal stability and coprime factorization S. Lall, Stanford. 2001.11.12.01 Stabilization via coprime factorization Scalar example Suppose 22 RH .Let 22 )= be a coprime factorization, and x, RH satisfy the Bezout equation ) ) )=1 Theorem )= is a stabilizing controller. Proof 22 22 22 which is stable.

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13 - 23 Internal stability and coprime factorization S. Lall, Stanford. 2001.11.12.01 Every stabilizing controller Suppose 22 RH .Let 22 )= ) be a coprime factorization, and x, RH satisfy the Bezout equation ) ) )=1 Theorem Every stabilizing controller has the form for some RH Proof The proof that is stabilizing is the same as before, since ( ) ( )=1 Then ( ) ( ) ( ) ( ) which is stable. We will prove that every has this form in the matrix case.