In Celebration of Bill Heltons 56 th Birthday NSF Workshop in Honor - PowerPoint PPT Presentation

In Celebration of Bill Helton’s 56 th Birthday NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 1 / 17

Performance of Networked Feedback Systems: Best Tracking and Optimal Power Allocation Jie Chen Department of Electronic Engineering City University of Hong Kong Hong Kong, China (On leave from University of California, Riverside) In Honor of Professor Bill Helton San Diego, CA October 2-4, 2010 NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 2 / 17

Background Networked Control Systems y r Controller Plant Channel New issues and problems: packet loss, time delay, limited capacity, etc Channel models: quantized channel, AWGN channel, etc Degrading effects 1. Feedback stabilization 2. Control performance NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 3 / 17

Background AWN Feedback Channel K ( s ) P ( s ) + n ( t ) The AWN channel has a prescribed power constraint, which has a ready connection to channel capacity. For minimum phase systems, the channel signal-to-noise ratio (SNR) must satisfy the bound [Braslavsky 2004] � SNR > 2 p i for feedback stabilization. NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 4 / 17

Tracking Performance Over an Additive White Noise Channel Tracking Over an AWN Channel Objective: Find how the tracking performance may be constrained by the channel Configuration Parallel channel v 1 u 1 r ( t ) y ( t ) + [ K 1 K 2 ] P n 1 + v m u m + n ( t ) n m Assumptions 1. The plant is MIMO, right-invertible and minimum phase. 2. The noise and the reference input are uncorrelated. 3. Parallel AWN channel: n ( t ) is white and E [ n ( t ) n ( t ) T ] = diag (Φ 1 , . . . , Φ m ) . NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 5 / 17

Tracking Performance Over an Additive White Noise Channel Problem Statement Tracking performance: E [ � e � 2 ] = E [ � r − y � 2 ] . The reference signal: r ( t ) = ( r 1 ( t ) , . . . , r m ( t )) T is a WSS real vector valued random process. - Power of r i ( t ) : σ 2 ri = E [ r i ( t ) 2 ] . - Power spectrum of r ( t ) : positive real rational matrix G r ( j ω ) = ψ r ( j ω ) ψ T r ( − j ω ) . The channel has the input power constraint E [ � y � 2 ] ≤ Γ . Optimal tracking problem K stabilizing P E [ � e � 2 ] , subject to E [ � y � 2 ] ≤ Γ inf NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 6 / 17

Tracking Performance Over an Additive White Noise Channel Approach K stabilizing P E [ � e � 2 ] , subject to E [ � y � 2 ] ≤ Γ inf 1. Define H ( ǫ ) � ( 1 − ǫ ) E [ � e � 2 ] + ǫ E [ � y � 2 ] , 0 ≤ ǫ ≤ 1 2. Minimize H ( ǫ ) over all stabilizing controllers K stabilizing P H ( ǫ ) � H ∗ ( ǫ ) . inf 3. The optimal tracking error H ∗ e satisfies 1 H ∗ 1 − ǫ ( H ∗ ( ǫ ) − ǫ Γ) , e ≥ ∀ 0 ≤ ǫ ≤ 1 4. Because of the convexity of the objective and constraint functionals � � 1 H ∗ 1 − ǫ ( H ∗ ( ǫ ) − ǫ Γ) e = sup . 0 ≤ ǫ ≤ 1 NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 7 / 17

Tracking Performance Over an Additive White Noise Channel Approach K stabilizing P E [ � e � 2 ] , subject to E [ � y � 2 ] ≤ Γ inf 1. Define H ( ǫ ) � ( 1 − ǫ ) E [ � e � 2 ] + ǫ E [ � y � 2 ] , 0 ≤ ǫ ≤ 1 2. Minimize H ( ǫ ) over all stabilizing controllers K stabilizing P H ( ǫ ) � H ∗ ( ǫ ) . inf 3. The optimal tracking error H ∗ e satisfies 1 H ∗ 1 − ǫ ( H ∗ ( ǫ ) − ǫ Γ) , e ≥ ∀ 0 ≤ ǫ ≤ 1 4. Because of the convexity of the objective and constraint functionals � � 1 H ∗ 1 − ǫ ( H ∗ ( ǫ ) − ǫ Γ) e = sup . 0 ≤ ǫ ≤ 1 NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 7 / 17

Tracking Performance Over an Additive White Noise Channel Approach K stabilizing P E [ � e � 2 ] , subject to E [ � y � 2 ] ≤ Γ inf 1. Define H ( ǫ ) � ( 1 − ǫ ) E [ � e � 2 ] + ǫ E [ � y � 2 ] , 0 ≤ ǫ ≤ 1 2. Minimize H ( ǫ ) over all stabilizing controllers K stabilizing P H ( ǫ ) � H ∗ ( ǫ ) . inf 3. The optimal tracking error H ∗ e satisfies 1 H ∗ 1 − ǫ ( H ∗ ( ǫ ) − ǫ Γ) , e ≥ ∀ 0 ≤ ǫ ≤ 1 4. Because of the convexity of the objective and constraint functionals � � 1 H ∗ 1 − ǫ ( H ∗ ( ǫ ) − ǫ Γ) e = sup . 0 ≤ ǫ ≤ 1 NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 7 / 17

Tracking Performance Over an Additive White Noise Channel Approach K stabilizing P E [ � e � 2 ] , subject to E [ � y � 2 ] ≤ Γ inf 1. Define H ( ǫ ) � ( 1 − ǫ ) E [ � e � 2 ] + ǫ E [ � y � 2 ] , 0 ≤ ǫ ≤ 1 2. Minimize H ( ǫ ) over all stabilizing controllers K stabilizing P H ( ǫ ) � H ∗ ( ǫ ) . inf 3. The optimal tracking error H ∗ e satisfies 1 H ∗ 1 − ǫ ( H ∗ ( ǫ ) − ǫ Γ) , e ≥ ∀ 0 ≤ ǫ ≤ 1 4. Because of the convexity of the objective and constraint functionals � � 1 H ∗ 1 − ǫ ( H ∗ ( ǫ ) − ǫ Γ) e = sup . 0 ≤ ǫ ≤ 1 NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 7 / 17

Tracking Performance Over an Additive White Noise Channel Preliminaries Coprime factorization P = NM − 1 = ˜ M − 1 ˜ N All stabilizing two-parameter controllers: = (˜ X − R ˜ Y − R ˜ ˜ N ) − 1 × � � � � K = { K : K = K 1 K 2 Q M , Q , R ∈ R H ∞ } The power can be expressed in terms of the H 2 norm as E [ � e � 2 ] = � ( I − NQ ) ψ r � 2 2 + � T Θ � 2 2 E [ � y � 2 ] = � NQ ψ r � 2 2 + � T Θ � 2 2 T � I − N (˜ X ˜ N r − R ) ˜ M Θ � diag ( � � Φ 1 , . . . , Φ m ) NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 8 / 17

Tracking Performance Over an Additive White Noise Channel Allpass Factorization Factorize ˜ M Θ � ˜ M Θ : M ( m ) M Θ ( s ) = ˜ ˜ Θ ( s )˜ B Θ ( s ) , M ( m ) where ˜ is the outer factor of ˜ M Θ and Θ B ( N p ) B ( N p − 1 ) B ( 1 ) B Θ = ˜ ˜ ˜ · · · ˜ Θ , Θ Θ with Θ ( s ) = I − 2Re { p i } B ( i ) ˜ ζ i ζ H i . s + ¯ p i The pole direction vector ζ i : � T � ζ ( 1 ) , . . . , ζ ( m ) ζ H ζ i = , i ζ i = 1 i i NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 9 / 17

Tracking Performance Over an Additive White Noise Channel The Optimal Tracking Performance Theorem 1 Assume that P ( s ) has simple unstable poles p i ∈ C + , i = 1 , . . . , N p . Define N p m | ζ ( j ) � � i | 2 Φ j . η � 2 p i i = 1 j = 1 Then the system is stabilizable if and only if Γ > η and the best tracking performance under the channel input power constraint E [ � y � 2 ] ≤ Γ is  r − √ Γ − η � 2 �� if η < Γ < η + σ 2 σ 2 + η  H ∗ r e = if Γ ≥ η + σ 2 η  r r = E [ r ( t ) T r ( t ) ] : reference input power. σ 2 NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 10 / 17

Tracking Performance Over an Additive White Noise Channel The channel input power Total channel input power � if η < Γ < η + σ 2 Γ , r , E [ � y � 2 ] = η + σ 2 if Γ ≥ η + σ 2 r , r . Power allocation of the parallel channel The power distributed to the k th sub-channel is  N p  p i | ζ ( k ) Γ − η � | 2 Φ k ,  r σ 2 if η < Γ < η + σ 2 r k + 2 r ,   σ 2 i   i = 1 N p  p i | ζ ( k ) � σ 2 | 2 Φ k , if Γ ≥ η + σ 2  r k + 2 r .  i    i = 1 The channel is not exploited to the maximum extent allowable. NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 11 / 17

Tracking Performance Over an Additive White Noise Channel Power Allocation vs Water Filling Water Filling for Transmission (Shannon) Maximize the capacity ( ν − N i ) + = P P i = ( ν − N i ) + , X Power Allocation for Tracking:“Fire Quenching” Minimize the tracking error N p 8 p i | ζ ( k ) Γ − η X | 2 Φ k , > r σ 2 if η < Γ < η + σ 2 r k + 2 r , > > σ 2 i > < i = 1 N p > X p i | ζ ( k ) σ 2 | 2 Φ k , if Γ ≥ η + σ 2 > r k + 2 r . > > i : i = 1 NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 12 / 17

Joint Channel/Controller Design Joint Channel/Controller Design: Scaling r ( t ) y ( t ) [ K 1 K 2 ] P y λ ( t ) Λ − 1 + Λ n ( t ) Λ = λ I : uniform scaling for all channels. The optimal design problem: λ> 0 , K ∈K E [ � e � 2 ] , subject to E [ � y λ � 2 ] ≤ Γ inf K : the set of all stabilizing controllers. NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 13 / 17