Quantify the Unstable The Main Point Justifications I Li Qiu - PowerPoint PPT Presentation

Quantify the Unstable The Main Point Justifications – I Li Qiu Justifications – II Justifications – III Conclusions The Hong Kong University of Science and Technology with special thanks to: Collaborators: Jie Chen, Guoxiang Gu, Weizhou Su, Lihua Xie, Ling Shi; Students: Hui Sun, Wei Chen, Shuang Wan, Nan Xiao, Claire Rong Aug 2011 1

Outline The Main Point The Main Point Justifications – I Justifications – II Justifications – III Justifications – I Conclusions Justifications – II Justifications – III Conclusions 2

The Main Point Justifications – I Justifications – II Justifications – III The Main Point Conclusions 3

How Bad is an Unstable System? The Main Point ◮ Which one of the two systems Justifications – I Justifications – II x ( k + 1) = 2 x ( k ) and x ( k + 1) = 3 x ( k ) , Justifications – III Conclusions is more unstable? 4

How Bad is an Unstable System? The Main Point ◮ Which one of the two systems Justifications – I Justifications – II x ( k + 1) = 2 x ( k ) and x ( k + 1) = 3 x ( k ) , Justifications – III Conclusions is more unstable? ◮ Which one of the two systems   3 2 1 10 10 � � 2 x ( k +1) = x ( k ) and x ( k +1) = 0 3 2  x ( k ) ,  0 4 0 0 0 . 5 is more unstable? 4

Mahler Measure ◮ Consider a polynomial n a ( z ) = a 0 z n + a 1 z n − 1 + · · · + a n − 1 z + a n = a 0 � The Main Point ( z − r i ) . Justifications – I i =1 Justifications – II ◮ Kurt Mahler in 1960 defined the so-called Mahler measure Justifications – III Conclusions n � � M ( a ) = | a 0 | max { 1 , | r i |} = | a 0 | | r i | . i =1 | r i | > 1 5

Mahler Measure ◮ Consider a polynomial n a ( z ) = a 0 z n + a 1 z n − 1 + · · · + a n − 1 z + a n = a 0 � The Main Point ( z − r i ) . Justifications – I i =1 Justifications – II ◮ Kurt Mahler in 1960 defined the so-called Mahler measure Justifications – III Conclusions n � � M ( a ) = | a 0 | max { 1 , | r i |} = | a 0 | | r i | . i =1 | r i | > 1 ◮ He also observed that by using Jensen’s formula � 1 � 2 π � log | a ( e j ω ) | d ω M ( a ) = exp . 2 π 0 (Geometric mean) ◮ The Mahler measure of a square matrix A M ( A ) = M [det( zI − A )] . 5

Connection to Szeg¨ o’s Problem ◮ For a measurable f : T → C and p ∈ [0 , ∞ ], define The Main Point � 1 � 2 π � 1 / p Justifications – I | f ( e j ω ) | p d ω � f � p = Justifications – II 2 π 0 Justifications – III | f ( e j ω ) | � f � ∞ = lim p ր∞ � f � p = ess sup Conclusions ω ∈ [0 , 2 π ] � 1 � 2 π � log | f ( e j ω ) | d ω � f � 0 = lim p ց 0 � f � p = exp . 2 π 0 ◮ M ( a ) = � a � 0 . 6

Connection to Szeg¨ o’s Problem ◮ For a measurable f : T → C and p ∈ [0 , ∞ ], define The Main Point � 1 � 2 π � 1 / p Justifications – I | f ( e j ω ) | p d ω � f � p = Justifications – II 2 π 0 Justifications – III | f ( e j ω ) | � f � ∞ = lim p ր∞ � f � p = ess sup Conclusions ω ∈ [0 , 2 π ] � 1 � 2 π � log | f ( e j ω ) | d ω � f � 0 = lim p ց 0 � f � p = exp . 2 π 0 ◮ M ( a ) = � a � 0 . ◮ Theorem: (Durand, 1981) Let a be a given polynomial. Then inf q � aq � p = M ( a ) where the infimum is taken over all monic polynomials q . ◮ Is this useful? 6

Topological Entropy The Main Point Justifications – I Justifications – II ◮ Robert Bowen in 1971 defined a quantity h ( A ) to measure the Justifications – III Conclusions complexity and information content of system x ( k ) ∈ R n . x ( k + 1) = Ax ( k ) , 7

Topological Entropy The Main Point Justifications – I Justifications – II ◮ Robert Bowen in 1971 defined a quantity h ( A ) to measure the Justifications – III Conclusions complexity and information content of system x ( k ) ∈ R n . x ( k + 1) = Ax ( k ) , ◮ He called h ( A ) the topological entropy of A and proved that h ( A ) = log M ( A ). 7

Instability Measure The Main Point Justifications – I ◮ Main Point: Both M ( A ) and h ( A ) can serve as measures of Justifications – II Justifications – III instability. Conclusions 8

Instability Measure The Main Point Justifications – I ◮ Main Point: Both M ( A ) and h ( A ) can serve as measures of Justifications – II Justifications – III instability. Conclusions ◮ Back to the question in the beginning: For systems   3 2 1 10 10 � � 2 x ( k +1) = x ( k ) and x ( k +1) = 0 3 2  x ( k ) ,  0 4 0 0 0 . 5 the latter is more unstable than the former. 8

The Main Point Justifications – I Justifications – II Justifications – I Justifications – III Conclusions Difficulty in Stabilization 9

Sensitivity and Complementary Sensitivity d ❄ ✲ ❥ ✲ [ A | B ] v u The Main Point F Justifications – I Justifications – II ✻ x Justifications – III Conclusions ◮ � A B � is a system x ( k + 1) = Ax ( k ) + Bu ( k ) . � A ◮ Assume v ( k ) ∈ R m and B � is stabilizable. ◮ The sensitivity function S ( z ) = I + F ( zI − A − BF ) − 1 B : the transfer function from d to u . ◮ The complementary sensitivity function T ( z ) = F ( zI − A − BF ) − 1 B : the transfer function from d to v . 10

Sensitivity and Complementary Sensitivity d ❄ ✲ ❥ ✲ [ A | B ] v u The Main Point F Justifications – I Justifications – II ✻ x Justifications – III Conclusions ◮ � A B � is a system x ( k + 1) = Ax ( k ) + Bu ( k ) . � A ◮ Assume v ( k ) ∈ R m and B � is stabilizable. ◮ The sensitivity function S ( z ) = I + F ( zI − A − BF ) − 1 B : the transfer function from d to u . ◮ The complementary sensitivity function T ( z ) = F ( zI − A − BF ) − 1 B : the transfer function from d to v . ◮ The smallest achievable norms of S ( z ) and T ( z ) capture various � A B � difficulties in stabilizing system . 10

The Single-Input Case The Main Point ◮ Theorem: Let m = 1. For each p ∈ [0 , ∞ ], Justifications – I Justifications – II F : A + BF is stable � S ( z ) � p = M ( A ) . inf Justifications – III Conclusions Furthermore, in the regular case, the optimal F is the same for all p and the optimal S ( z ) is allpass. 11

The Single-Input Case The Main Point ◮ Theorem: Let m = 1. For each p ∈ [0 , ∞ ], Justifications – I Justifications – II F : A + BF is stable � S ( z ) � p = M ( A ) . inf Justifications – III Conclusions Furthermore, in the regular case, the optimal F is the same for all p and the optimal S ( z ) is allpass. ◮ Theorem: Let m = 1. M ( A ) 2 − 1 � F : A + BF is stable � T ( z ) � 2 = inf F : A + BF is stable � T ( z ) � ∞ = M ( A ) . inf 11

The Main Point Justifications – I ◮ Special cases of these single-input results appeared in Justifications – II • early performance limitation literature (e.g. Sung and Hara, Justifications – III 1988), and Conclusions • recent networked control systems literature (e.g. Elia and Mitter, 2001; Elia, 2005; Fu and Xie, 2005; Braslavsky, Middleton and Freudenberg, 2007). 12

The Main Point Justifications – I ◮ Special cases of these single-input results appeared in Justifications – II • early performance limitation literature (e.g. Sung and Hara, Justifications – III 1988), and Conclusions • recent networked control systems literature (e.g. Elia and Mitter, 2001; Elia, 2005; Fu and Xie, 2005; Braslavsky, Middleton and Freudenberg, 2007). ◮ Naive extension to the multiple-input case does not work. 12

The Main Point Justifications – I Justifications – II Justifications – III Justifications – II Conclusions Multivariable Networked Stabilization 13

Multivariable Networked Stabilization ✛ ✘ The Main Point Justifications – I v u ✲ ✲ [ A | B ] F Channels Justifications – II ✚ ✙ Justifications – III ✻ x Conclusions ◮ What is networked stabilization? • involving non-ideal channels. • involving channel/controller co-design. ◮ The smallest total “capacity” of the input channels needed so that the networked stabilization is possible also gives a difficulty in stabilization. 14

Channel Models The Main Point Justifications – I ✎ ☞ Justifications – II v i u i ✲ ✲ Justifications – III ✍ ✌ channel Conclusions ◮ What is a communication channel? What is its capacity? • Existing information-theoretical model is not very useful. • Models capturing individual channel features are available. • A timed information theoretical model is needed. 15

Model I: SER Model ✲ ∆ i e i ❄ ❥ ✲ v i u i ✲ The Main Point Justifications – I ◮ ∆ i is a nonlinear, time-varying, dynamic uncertain system. Justifications – II ◮ � ∆ i � ∞ ≤ δ i . Justifications – III ◮ δ − 1 Conclusions can be considered as the transmission accuracy or i signal-to-error ratio (SER). ◮ Channel capacity C i = log δ − 1 = − log δ i . i 16