Control With and Without Separation Elias Riedel Gårding Control over Gaussian Channels With and Setting Without Source–Channel Separation Background Problem setup Solutions Separation JSCC Elias Riedel Gårding Implementation Results Royal Institute of Technology (KTH), Stockholm, Sweden Further directions Supervisors: Victoria Kostina and Anatoly Khina Closing Electrical Engineering, Caltech August 24, 2017
Networked control Background—control theory Control With and Without Separation Control theory in a nutshell Elias Riedel Gårding Stabilize an unstable system (“plant”) using measurements in a Setting feedback loop. Background Problem setup Solutions Separation Traditional control JSCC Implementation Results Further directions Closing • Optimal strategies known
Networked control Background—control theory Control With and Without Separation Control theory in a nutshell Elias Riedel Gårding Stabilize an unstable system (“plant”) using measurements in a Setting feedback loop. Background Problem setup Solutions Separation Traditional control Networked control JSCC Implementation v t Results x t Sensor Plant w t y t Further directions c t u t Closing z t Controller Channel ˆ x t | t • Optimal strategies known • Channel limits communication • Sensor and controller co-located • Requires information theory
Networked control Applications Control With and Without Separation Elias Riedel Gårding Setting Background Problem setup Real-time wireless systems Solutions http://www.bbc.com/future/story/ 20140516-i-operate-on-people-400km-away Separation • Autonomous vehicles JSCC Implementation • Remote surgery Results • . . . Further directions Closing http://latam.pcmag.com/drones/1774/review/ yuneec-typhoon-h-pro
Networked control Problem setup v t Control With and Without Plant Separation x t w t x t +1 = αx t + w t + u t • Unstable if | α | > 1 Elias Riedel y t Gårding u t • Goal: Minimize Channel 1 � T x 2 Setting � � t = 1 E t T b i a i Controller/ Observer/ Background Problem setup Receiver Transmitter Solutions n i Separation JSCC Implementation Results Further directions Closing
Networked control Problem setup v t Control With and Without Plant Separation x t w t x t +1 = αx t + w t + u t • Unstable if | α | > 1 Elias Riedel y t Gårding u t • Goal: Minimize Channel 1 � T x 2 Setting � � t = 1 E t T b i a i Controller/ Observer/ Background Problem setup Receiver Transmitter Solutions n i Separation JSCC Channel model Implementation AWGN ( A dditive W hite G aussian N oise) Results Power constraint: E [ a 2 i ] ≤ 1 Further directions Closing
Networked control Problem setup v t Control With and Without Plant Separation x t w t x t +1 = αx t + w t + u t • Unstable if | α | > 1 Elias Riedel y t Gårding u t • Goal: Minimize Channel 1 � T x 2 Setting � � t = 1 E t T b i a i Controller/ Observer/ Background Problem setup Receiver Transmitter Solutions n i Separation JSCC Channel model Implementation AWGN ( A dditive W hite G aussian N oise) Results Power constraint: E [ a 2 i ] ≤ 1 Further directions Closing Signaling rate versus sampling rate We may use the channel K times per time step. We focus on K = 1 and K = 2.
Networked control Problem setup v t Control With and Without Plant Separation x t w t x t +1 = αx t + w t + u t • Unstable if | α | > 1 Elias Riedel y t Gårding u t • Goal: Minimize Channel 1 � T x 2 Setting � � t = 1 E t T b i a i Controller/ Observer/ Background Problem setup Receiver Transmitter Solutions n i Separation JSCC Channel model Implementation AWGN ( A dditive W hite G aussian N oise) Results Power constraint: E [ a 2 i ] ≤ 1 Further directions Closing Signaling rate versus sampling rate We may use the channel K times per time step. We focus on K = 1 and K = 2. How to encode/decode?
Solution approaches JSCC versus separation Control With and Without Source–channel separation Separation Elias Riedel Split encoding and decoding into smaller subproblems. Gårding • Standard approach in information theory: Bits! Setting Background • Optimal for long messages (but control needs short ones) Problem setup Solutions • No reason it should work well for control Separation JSCC Implementation Joint source–channel coding (JSCC) Results Design the encoder and decoder holistically. Further directions • Much simpler and less computationally intensive Closing • Hypothesis: Gives much better control performance My task: Implement, simulate and compare
Solution approaches Separation: source coding (quantization) v t Control With and Without Plant x t Separation w t x t +1 = α x t + w t + u t y t = x t + v t Elias Riedel u t Gårding Channel Controller/ Observer/ Setting Receiver Transmitter R Background Problem setup Solutions • Separation = ⇒ noiseless digital channel Separation JSCC • Encode the measurement as a fixed number of bits Implementation • Quantization errors blow up Results Further • Fixed quantizer won’t work directions • Optimal strategy known Closing
Solution approaches Separation: source coding (quantization) v t Control With and Without Plant x t Separation w t x t +1 = α x t + w t + u t y t = x t + v t Elias Riedel u t Gårding Channel Controller/ Observer/ Setting Receiver Transmitter R Background Problem setup Solutions • Separation = ⇒ noiseless digital channel Separation JSCC • Encode the measurement as a fixed number of bits Implementation • Quantization errors blow up Results Further • Fixed quantizer won’t work directions • Optimal strategy known Closing
Solution approaches Separation: source coding (quantization) v t Control With and Without Plant x t Separation w t x t +1 = α x t + w t + u t y t = x t + v t Elias Riedel u t Gårding Channel Controller/ Observer/ Setting Receiver Transmitter R Background Problem setup Solutions • Separation = ⇒ noiseless digital channel Separation JSCC • Encode the measurement as a fixed number of bits Implementation • Quantization errors blow up Results Further • Fixed quantizer won’t work directions • Optimal strategy known Closing
Solution approaches Separation: source coding (quantization) v t Control With and Without Plant x t Separation w t x t +1 = α x t + w t + u t y t = x t + v t Elias Riedel u t Gårding Channel Controller/ Observer/ Setting Receiver Transmitter R Background Problem setup Solutions • Separation = ⇒ noiseless digital channel Separation JSCC • Encode the measurement as a fixed number of bits Implementation • Quantization errors blow up Results Further • Fixed quantizer won’t work directions • Optimal strategy known Closing
Solution approaches Separation: channel coding (tree codes) Recall x t + 1 = α x t + w t + u t Control With and Without Separation Anytime reliability Elias Riedel Gårding • Errors get magnified as α t ( | α | > 1) Setting • Error probabilites must shrink as α − t Background Problem setup Solutions • Each sent bit must depend on the entire input history Separation JSCC • Optimal decoding computationally infeasible Implementation • Sequential decoding Results Further directions Closing
Solution approaches JSCC and spirals Control With and Without Separation For K = 1: Optimal encoding map R → R known: Transmit as is! Elias Riedel Gårding Setting Background Problem setup Solutions Separation JSCC Implementation Results Further directions Closing
Solution approaches JSCC and spirals Control With and Without Separation For K = 1: Optimal encoding map R → R known: Transmit as is! Elias Riedel Gårding Try the same for K = 2? Setting Background Problem setup � a 1 = s Solutions • Repetition: Separation a 2 = s JSCC Implementation Results Further directions Closing
Solution approaches JSCC and spirals Control With and Without Separation For K = 1: Optimal encoding map R → R known: Transmit as is! Elias Riedel Gårding Try the same for K = 2? Setting Background Problem setup � a 1 = s Solutions • Repetition: Separation a 2 = s JSCC Implementation • Waste of space! Stretch? Results Further directions Closing
Solution approaches JSCC and spirals Control With and Without Separation For K = 1: Optimal encoding map R → R known: Transmit as is! Elias Riedel Gårding Try the same for K = 2? Setting Background Problem setup � a 1 = s Solutions • Repetition: Separation a 2 = s JSCC Implementation • Waste of space! Stretch? Results √ � a 1 = 2 s cos( ω | s | ) Further • Spiral: √ directions a 2 = 2 s sin( ω | s | ) Closing
Solution approaches JSCC and spirals Control With and Without Separation For K = 1: Optimal encoding map R → R known: Transmit as is! Elias Riedel Gårding Try the same for K = 2? Setting Background Problem setup � a 1 = s Solutions • Repetition: Separation a 2 = s JSCC Implementation • Waste of space! Stretch? Results √ � a 1 = 2 s cos( ω | s | ) Further • Spiral: √ directions a 2 = 2 s sin( ω | s | ) Closing • Much better! • Tightness–crossover tradeoff
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