Fundamental Limitations in Networked Control Systems and - PowerPoint PPT Presentation

Fundamental Limitations in Networked Control Systems and Quantization Hid Hideaki Ishii ki I hii Tokyo Institute of Technology Tokyo Institute of Technology ishii@dis.titech.ac.jp HYCON2 PhD School on Control of HYCON2 PhD S h l C t l f Networked and Large-Scale Systems June 22, 2011

Networked control: The other perspective ? Communi- Control cation � Information technology & communication � Traditionally, for connecting humans & computers � Very mature area 2

Networked control: The other perspective Communi- Control cation � Networked control is new in communication also!! � For real-time control: Machine to machine � High reliability and robustness for wireless networks � Hi h li bilit d b t f i l t k � E g : Factory automation Medical devices � E.g.: Factory automation, Medical devices � Standardization for different layers Standardization for different layers 3

Example: Automotive LAN � In-vehicle electronic devices connected via networks � Protocols for real-time control: CAN, FlexRay,…‧ 4

Channel capacity considerations � NCS: Constraints on channel capacity � Even if the total capacity is large, each component may use only a (small) portion � New challenges: � Modeling of the capacity constraints in NCS � Modeling of the capacity constraints in NCS � Allocating bandwidth to each transmission g What is the necessary capacity for control? 5

Motivating example: Inverted pendulum g p p � Given a sampling period 10 ms p g p � Angle data is quantized: E.g. 8 bits � 800 bps � For stabilization, how much precision in bits is needed? 6

In NCS, two fields meet Communication / Information theory � Transmission of data generated by random processes � Not concerned with the content of the info and delay � Not concerned with the content of the info and delay Control theory Control theory � Information is specific, used for feedback control � Traditionally, assumed infinite bandwidth � Stochastic vs deterministic � Stochastic vs deterministic Sahai & Mitter (2006), Nair, Fagnani, Zampieri, & Evans (2007) 7

Interplay between control & communication Two directions: 1. Control over networks with capacity constraints 2. Systems analysis based on information theoretic tools � We will observe some fundamental limitations � We will observe some fundamental limitations arising in feedback control with networks � Analogous to Shannon’‚s source coding theory 8

Outline 0. Introduction 1. Control under capacity constraints a The minimum data rate for stabilization a. The minimum data rate for stabilization b. The coarsest quantization for stabilization 2. Information theoretic approach to Bode’‚s integral formula integral formula 3. Conclusion 9

1. Control under capacity constraints 10

Quantization and coding g � In control systems, signals take real values y , g � To be transmitted over finite capacity channels, they must be transformed to discrete values Quantizer Quantizer Channel Decoder & Encoder � Assume no error/delay in the channel Focus on quantization 11

Quantized control: Problem setup Plant Plant Controller Controller Channel Quantizer � Plant � Discrete-time, LTI � Unstable, but stabilizable (or controllable) ( ) 12 12

Uniform quantizer q � Quantization error: 13

General quantizer q if � Partition of � Output values � Index set � Index set : finite/countable : finite/countable Are there quantizer structures suitable for control? Are there quantizer structures suitable for control? 14

Quantized control Plant Plant Controller Controller Channel Quantizer � Control objective: Stabilization � Fundamental question: How much information is needed from the quantized signal to achieve this objective? 15 15

Control with quantized signals q g � Traditionally, quantized error is modeled as additive y, q white noise � This model becomes inaccurate when quantization resolution is coarse l ti i noise Bertram (1958), Curry (1970) 16

Delchamps’‚ observation 1 p � State info via uniform quantizer: q Result: Result: Assume is stable. Apply the control � There is a region around the origin into which each trajectory goes. (= Practical stability) � Inside there, behavior is chaotic. Delchamps (1990) 17

Delchamps’‚ observation 2 p Result: If the plant is not so unstable as then there is control such that each trajectory goes to 0. h h h j 0 Quantized info may be sufficient for precise control! 18

Outline 0. Introduction 1. Control under capacity constraints a The minimum data rate for stabilization a. The minimum data rate for stabilization b. The coarsest quantization for stabilization 2. Information theoretic approach to Bode’‚s integral formula integral formula 3. Conclusion 19

1a. The minimum data rate for stabilization 20

Control under capacity constraint p y Plant Plant Controller Controller Channel Quantizer � How small can the data rate be for stabilization? � Total # of discrete values = � D t � Data rate = [bits/sample] t [bit / l ] � First studied by Wong & Brockett (1999) � First studied by Wong & Brockett (1999) 21 21

Minimum data rate: Scalar case Plant � : Partitions [ 1 , 1 ] into N intervals � 0 1 1 Corresponding control inputs control inputs � To keep within , how large should be? x ( k ) [ 1 , 1 ] N 22

Minimum data rate: Scalar case Theorem Theorem � � � � � Data Data rate rate [bits/samp [bits/samp le] le] � Minimum data rate depends on the unstable pole � More unstable plants require more data rate Wong & Brockett (1999) 23

Minimum data rate: Scalar case ( � Let ) Proof Then, , its width must satisfy y On On the the other other hand, hand Th Thus, � 1 1 0 0 1 1 � � ( ) With a uniform quantizer on [ 1 , 1 ] , let Then 24

Minimum data rate: General case Necessary part Plant : Bounded subset of Partitions Partitions into into cells cells : : N N Wong & Brockett (1999) 25

Minimum data rate: General case Necessary part Plant : Bounded subset of Partitions Partitions into into cells cells : : N N Theorem � � � Data Rate [bits/sample] : Unstable eigenvalues of Wong & Brockett (1999) 26

Dynamic quantizer y q � Time-varying quantizers with memory � Example: Digital camera � When the state is outside the range, zoom out Zoom out Brockett & Liberzon (2000) 27

Dynamic quantizer y q � When the state is inside the range, hold and then apply control � To locate the state more precisely, zoom in � T l t th t t i l i Zoom in � Global asymptotic stabilization may be achieved � Additional 2 bits: Zoom in/out, Hold 28

Minimum data rate: General setup Plant Plant Controller Controller Channel Encoder � Encoder � Encoder # of code words � Average data rate � Controller � Controller 29 29

The minimum data rate Theorem Theorem as � Ave data rate [bits/samp le] � General results: Control is impossible if the bound does not hold! d t h ld! � Deterministic case: Tatikonda & Mitter (2004) � Stochastic case: Nair & Evans (2004) � P � Proof by construction: f b t ti � Quantizer transmission scheme controller � Quantizer, transmission scheme, controller 30

Structure of the controller Plant Plant One-step Controller Controller ahead Dynamic Estimator Estimator Q Channel Channel : Coarse estimate of state from quantized signal : Coarse estimate of state from quantized signal ˆ k x x ( ( k ) ) : Estimate of one-step ahead 31 31

At the encoder 32

At the encoder The overall quantizer x ˆ � ˆ x Q ( x , c , E , L , R ) x x 2 R 2 R 2 i � 4 states � Determine region of � Determine region of c quantization l � Shared by encoder � Shared by encoder 2 2 and decoder e 2 2 2 R 2 R R : Bits assigned to : Bits assigned to � � 1 1 1 i 1 the i th mode l e 1 1 1 � � � � u u R R log log i 2 i x 1 33

Comments and further studies � Average data rate [bits/sample] � Stochastic control case � General class of noise and disturbance � Stabilization in the mean-square sense � Stabilization in the mean-square sense � The same minimum rate: Does not depend on noise statistics Nair & Evans (2004), Tatikonda et al. (2004), Matveev & Savkin (2004), Yuksel & Basar (2006), Ishii, Ohyama, & Tsumura (2008) ( ) y ( ) � With multiple sensors & controllers Tatikonda (2003), Yuksel & Basar (2007) 34

Outline 0. Introduction 1. Control under capacity constraints a The minimum data rate for stabilization a. The minimum data rate for stabilization b. The coarsest quantization for stabilization 2. Information theoretic approach to Bode’‚s integral formula integral formula 3. Conclusion 35

1b. The coarsest quantization for stabilization 36