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The Effect of Delayed Side Information on Fundamental Limitations of Disturbance Attenuation Yingbo Zhao*, Vijay Gupta**, and Jorge Cort´ es* *: Department of Mechanical and Aerospace Engineering University of California, San Diego, **: Department of Electrical Engineering University of Notre Dame The 54th IEEE Conference on Decision and Control, Osaka, Japan Dec 17, 2015 Yingbo Zhao, Vijay Gupta, and Jorge Cort´ es The Effect of Delayed Side Information on Fundamental Limitations of Disturbance Attenuation 1

Disturbance attenuation in discrete-time feedback systems Σ O e y d Controller Plant - Measure of disturbance attenuation performance at frequency ω : � S d , e ( ω ) = Φ e ( ω ) / Φ d ( ω ) Φ x ( ω ) denotes the power spectral density of a wide sense stationary stochastic process x Yingbo Zhao, Vijay Gupta, and Jorge Cort´ es The Effect of Delayed Side Information on Fundamental Limitations of Disturbance Attenuation 2

Disturbance attenuation in discrete-time feedback systems Σ O e y d Controller Plant - Measure of disturbance attenuation performance at frequency ω : � S d , e ( ω ) = Φ e ( ω ) / Φ d ( ω ) Φ x ( ω ) denotes the power spectral density of a wide sense stationary stochastic process x If the controller is linear time-invariant, S d , e is the transfer function between d and e Yingbo Zhao, Vijay Gupta, and Jorge Cort´ es The Effect of Delayed Side Information on Fundamental Limitations of Disturbance Attenuation 2

Disturbance attenuation in discrete-time feedback systems Σ O e y d Controller Plant - Measure of disturbance attenuation performance at frequency ω : � S d , e ( ω ) = Φ e ( ω ) / Φ d ( ω ) Φ x ( ω ) denotes the power spectral density of a wide sense stationary stochastic process x If the controller is linear time-invariant, S d , e is the transfer function between d and e Small S d , e ( ω ) implies good disturbance attenuation performance Yingbo Zhao, Vijay Gupta, and Jorge Cort´ es The Effect of Delayed Side Information on Fundamental Limitations of Disturbance Attenuation 2

Disturbance attenuation in discrete-time feedback systems Σ O e y d Controller Plant - Measure of disturbance attenuation performance at frequency ω : � S d , e ( ω ) = Φ e ( ω ) / Φ d ( ω ) Φ x ( ω ) denotes the power spectral density of a wide sense stationary stochastic process x If the controller is linear time-invariant, S d , e is the transfer function between d and e Small S d , e ( ω ) implies good disturbance attenuation performance However, it is in general not possible to make S d , e ( ω ) small at all frequencies Yingbo Zhao, Vijay Gupta, and Jorge Cort´ es The Effect of Delayed Side Information on Fundamental Limitations of Disturbance Attenuation 2

Classical Bode integral formula (DT, SISO, LTI) Serious Design s.g 10 d k ( ) y k ( ) Plant Log Magnitude e k ( ) 1.0 1 π = ∑ ∫ ω ω λ log S ( ) d log d e , i 2 π − π i : λ ( ) 1 A > i u k ( ) 0.1 Controller 0.0 0.5 1.0 1.5 2.0 Frequency Figure: 1989 Bode lecture: respect the unstable, Gunter Stein Open-loop dynamics → achievable closed-loop performance. Controller can only shape the sensitivity integral. Important for controller design reference. Yingbo Zhao, Vijay Gupta, and Jorge Cort´ es The Effect of Delayed Side Information on Fundamental Limitations of Disturbance Attenuation 3

(Limited) literature review on Bode integral formula Bode (1945): Continuous, SISO, LTI, stable plant Freudenberg and Looze (1985): Unstable plant Freudenberg and Looze (1988), Chen and Nett (1995), Chen (2000), Ishii, Okano, and Hara (2011): MIMO system Iglesias (2001,2002), Sandberg and Bernhardsson (2005): Time-varying system Zhang and Iglesias (2003), Martins and Dahleh (2008), Yu and Mehta (2010): Nonlinear control Martins, Dahleh, and Doyle (2007): Bode integral formula with disturbance preview Zhao and Gupta (2014): DT linear periodic systems Yingbo Zhao, Vijay Gupta, and Jorge Cort´ es The Effect of Delayed Side Information on Fundamental Limitations of Disturbance Attenuation 4

Preview side information improves disturbance rejection z τ E Channel D ˆ d e y u d Controller Plant - Figure: Preview side information at the controller improves closed-loop disturbance rejection ( Martins, Dahleh, and Doyle (2007) ). � π 1 � log S d , e ( ω ) d ω ≥ log | λ i ( A ) | − C 2 π − π i : | λ i ( A ) | > 1 Yingbo Zhao, Vijay Gupta, and Jorge Cort´ es The Effect of Delayed Side Information on Fundamental Limitations of Disturbance Attenuation 5

Preview side information improves disturbance rejection z τ E Channel D ˆ d e y u d Controller Plant - Figure: Preview side information at the controller improves closed-loop disturbance rejection ( Martins, Dahleh, and Doyle (2007) ). � π 1 � log S d , e ( ω ) d ω ≥ log | λ i ( A ) | − C 2 π − π i : | λ i ( A ) | > 1 What about delayed side information? Yingbo Zhao, Vijay Gupta, and Jorge Cort´ es The Effect of Delayed Side Information on Fundamental Limitations of Disturbance Attenuation 5

Can DSI improve disturbance rejection? z τ − Channel D E ˆ d e y u d Controller Plant - Figure: Feedback system configuration when the controller has delayed side information. � π 1 log S d , e ( ω ) d ω ≥ ? 2 π − π Intuitively, delayed side information about an i.i.d. disturbance process is not useful since it contains no information about the current or future disturbance. Yingbo Zhao, Vijay Gupta, and Jorge Cort´ es The Effect of Delayed Side Information on Fundamental Limitations of Disturbance Attenuation 6

Can DSI improve disturbance rejection? z τ − E Channel D ˆ d e y u d Controller Plant - Figure: Feedback system configuration when the controller has DSI. However, we will show that DSI improves disturbance rejection if the plant is unstable � π 1 � + � � log S d , e ( ω ) d ω ≥ log | λ i ( A ) | − C 2 π − π i : | λ i ( A ) | > 1 where ( x ) + � max( x , 0) and C represents the Shannon capacity of the side channel. Yingbo Zhao, Vijay Gupta, and Jorge Cort´ es The Effect of Delayed Side Information on Fundamental Limitations of Disturbance Attenuation 7

Problem setup z τ − E Channel D ˆ d e y u d Controller Plant - Plant: � x ( k + 1) � A � � x ( k ) � � B = y ( k ) H 0 u ( k ) where x ( k ) ∈ R n , u ( k ) , y ( k ) , e ( k ) ∈ R , ∀ k ∈ Z + . Yingbo Zhao, Vijay Gupta, and Jorge Cort´ es The Effect of Delayed Side Information on Fundamental Limitations of Disturbance Attenuation 8

Problem setup z τ − E Channel D ˆ d e y u d Controller Plant - Plant: � x ( k + 1) � A � � x ( k ) � � B = y ( k ) H 0 u ( k ) where x ( k ) ∈ R n , u ( k ) , y ( k ) , e ( k ) ∈ R , ∀ k ∈ Z + . Controller: u ( k ) = f k ( k , ˆ d k , e k ) where f k is a time-varying, possibly nonlinear, function. Yingbo Zhao, Vijay Gupta, and Jorge Cort´ es The Effect of Delayed Side Information on Fundamental Limitations of Disturbance Attenuation 8

Assumptions z τ − E Channel D ˆ d y e u d Controller Plant - The closed-loop system is mean-square stable. The disturbance process d is a zero-mean Gaussian process with i.i.d. r.v. d ( k ). The plant’s initial condition x (0) is a zero-mean r.v. with finite differential entropy, and independent of d . Yingbo Zhao, Vijay Gupta, and Jorge Cort´ es The Effect of Delayed Side Information on Fundamental Limitations of Disturbance Attenuation 9

DSI is useful for unstable plants Theorem (DSI can reduce the log integral of sensitivity) Denote the transfer function from the disturbance d to the error e by S d , e � π 1 � + . � � log S d , e ( ω ) d ω ≥ log | λ i ( A ) | − C 2 π − π i : | λ i ( A ) | > 1 Unlike PSI, the contribution of DSI to the disturbance attenuation performance is upper bounded by � i : | λ i ( A ) | > 1 log | λ i ( A ) | . Yingbo Zhao, Vijay Gupta, and Jorge Cort´ es The Effect of Delayed Side Information on Fundamental Limitations of Disturbance Attenuation 10

DSI is useful for unstable plants Theorem (DSI can reduce the log integral of sensitivity) Denote the transfer function from the disturbance d to the error e by S d , e � π 1 � + . � � log S d , e ( ω ) d ω ≥ log | λ i ( A ) | − C 2 π − π i : | λ i ( A ) | > 1 Unlike PSI, the contribution of DSI to the disturbance attenuation performance is upper bounded by � i : | λ i ( A ) | > 1 log | λ i ( A ) | . DSI can only help to stabilize the open-loop system but cannot reduce the controller’s uncertainty about the disturbance. Yingbo Zhao, Vijay Gupta, and Jorge Cort´ es The Effect of Delayed Side Information on Fundamental Limitations of Disturbance Attenuation 10

DSI can help to stabilize an unstable plant z τ − E Channel D ˆ d y e u d Controller Plant - � π 1 � + . � � log S d , e ( ω ) d ω ≥ log | λ i ( A ) | − C 2 π − π i : | λ i ( A ) | > 1 The power in e comes from 2 sources: disturbance d and stabilizing information about x (0). Yingbo Zhao, Vijay Gupta, and Jorge Cort´ es The Effect of Delayed Side Information on Fundamental Limitations of Disturbance Attenuation 11