Data-Enabled Predictive Control of Autonomous Energy Systems - PowerPoint PPT Presentation

Data-Enabled Predictive Control of Autonomous Energy Systems Florian D¨ orfler Automatic Control Laboratory, ETH Z¨ urich

Acknowledgements Jeremy Coulson Linbin Huang Paul Beuchat John Lygeros Ivan Markovsky Ezzat Elokda 1/37

Perspectives on model-based control From experiment design to closed-loop control � Single system level: H˚ akan Hjalmarsson ∗ • modeling & system ID Department of Signals, Sensors and Systems, Royal Institute of Technology, S-100 44 Stockholm, Sweden 1. Introduction be justified is for the commissioning of model predictive controllers. are very expensive Ever increasing productivity demands and environmental It has also been recognized that models for control pose special considerations. Again quoting (Ogunnaike, 1996): standards necessitate more and more advanced control meth- ods to be employed in industry. However, such methods usu- “ There is abundant evidence in industrial practice that • models not always ally require a model of the process and modeling and system when modeling for control is not based on criteria related identification are expensive. Quoting (Ogunnaike, 1996): to the actual end use , the results can sometimes be quite “ It is also widely recognized , however , that obtaining the disappointing. ” useful for control process model is the single most time consuming task in the Hence, efficient modeling and system identification tech- application of model-based control. ” niques suited for industrial use and tailored for control de- sign applications have become important enablers for indus- In Hussain (1999) it is reported that three quarters of the • need for end-to-end total costs associated with advanced control projects can trial advances. The Panel for Future Directions in Control, (Murray, ˚ be attributed to modeling. It is estimated that models exist Aström, Boyd, Brockett, & Stein, 2003), has iden- for far less than one percent of all processes in regulatory tified automatic synthesis of control algorithms , with inte- automation solutions control. According to Desborough and Miller (2001), one of grated validation and verification as one of the major future the few instances when the cost of dynamic modeling can challenges in control. Quoting (Murray et al., 2003): “ Researchers need to develop much more powerful design tools that automate the entire control design process from model development to hardware-in-the-loop simulation. ” Critical infrastructure level: (especially in energy)  • subsystem (device) models & controls are proprietary   nobody has  any dynamic • infrastructure (network) owned by many entities/countries   models ...  • operating points/modes are in flux & constantly changing 2/37

Control in a data-rich world • ever-growing trend in CS & applications: data-driven control by-passing models • canonical problem: black/gray-box u 2 y 2 system control based on I/O samples u 1 y 1 Q: Why give up physical modeling and reliable model-based algorithms ? data-driven control Data-driven control is viable alternative when • models are too complex to be useful Central promise: It (e.g., fluids, wind farms, & building automation) is often easier to learn • first-principle models are not conceivable control policies directly from data, rather than (e.g., human-in-the-loop, biology, & perception) learning a model. • modeling & system ID is too cumbersome Example: PID (e.g., robotics & electronics applications) 3/37

Snippets from the literature unknown system 1. reinforcement learning / stochastic adaptive control / dual control / approximate dynamic programming observation ø not suitable for physical, real-time, & safety-critical action reinforcement learning control ? estimate reward y u 2. gray-box safe learning & control (adaptive) robust/adaptive ø limited applicability: need a-priori safety control u 2 y 2 3. sequential system ID + UQ + control → recent finite-sample & end-to-end ID + u 1 y 1 UQ + control pipelines out-performing RL ø ID seeks best but not most useful model + ? → “easier to learn policies than models” 4/37

Colorful idea u 1 = 1 y 3 y 5 y 1 y 7 u 2 = u 3 = · · · = 0 x 0 =0 y 4 y 6 y 2 If you had the impulse response of a LTI system, then ... • can identify model (e.g., transfer function or Kalman-Ho realization) • ...but can also build predictive model directly from raw data :   u future ( t )   u future ( t − 1) � �   y future ( t ) = y 1 y 2 y 3 . . . ·   u future ( t − 2)   . . . • model predictive control from data: dynamic matrix control (DMC) • today: can we do so with arbitrary, finite, and corrupted I/O samples ? 5/37

Contents I. Data-Enabled Predictive Control (DeePC): Basic Idea J. Coulson, J. Lygeros, and F. D ¨ orfler. Data-Enabled Predictive Control: In the Shallows of the DeePC . arxiv.org/abs/1811.05890. II. From Heuristics & Numerical Promises to Theorems J. Coulson, J. Lygeros, and F. D¨ orfler. Regularized and Distributionally Robust Data-Enabled Predictive Control . arxiv.org/abs/1903.06804. III. Application: End-to-End Automation in Energy Systems L. Huang, J. Coulson, J. Lygeros, and F. D ¨ orfler. Data-Enabled Predictive Control for Grid-Connected Power Converters . arxiv.org/abs/1903.07339.

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Behavioral view on LTI systems Definition: A discrete-time dynamical system is a 3 -tuple ( Z ≥ 0 , W , B ) where (i) Z ≥ 0 is the discrete-time axis, (ii) W is a signal space, and (iii) B ⊆ W Z ≥ 0 is the behavior. Definition: The dynamical system ( Z ≥ 0 , W , B ) is (i) linear if W is a vector space & B is a subspace of W Z ≥ 0 y (ii) and time-invariant if B ⊆ σ B , where σw t = w t +1 . u B = set of trajectories & B T is restriction to t ∈ [0 , T ] 7/37

LTI systems and matrix time series foundation of state-space subspace system ID & signal recovery algorithms u ( t ) y ( t ) y 3 u 4 y 4 u 1 u 3 y 1 u 7 y 5 y 7 t u 5 u 6 t y 2 u 2 y 6 � � [ b 0 a 0 b 1 a 1 ... b n a n ] spans left nullspace u ( t ) , y ( t ) satisfy recursive of Hankel matrix (collected from data) difference equation � u T − L +1 �   b 0 u t + b 1 u t +1 + . . . + b n u t + n + ( u 1 y 1 ) ( u 2 y 2 ) ( u 3 y 3 ) · · · y T − L +1   a 0 y t + a 1 y t +1 + . . . + a n y t + n = 0 . . ⇒  ( u 2 y 2 ) ( u 3 y 3 ) ( u 4  y 4 ) · · · .     . H L ( u (ARMA / kernel representation) y ) = .   ( u 3 y 3 ) ( u 4 y 4 ) ( u 5 y 5 ) · · · .     . . ... ... ... ⇐  . .  . . ( u L ( u T y L ) · · · · · · · · · y T ) under assumptions 8/37

The Fundamental Lemma Definition : The signal u = col ( u 1 , . . . , u T ) ∈ R mT is persistently   u 1 ··· u T − L +1   is of full row rank, exciting of order L if H L ( u ) = . . ... . . . . u L ··· u T i.e., if the signal is sufficiently rich and long ( T − L + 1 ≥ mL ) . Fundamental Lemma [Willems et al, ’05] : Let T, t ∈ Z > 0 , Consider • a controllable LTI system ( Z ≥ 0 , R m + p , B ) , and • a T -sample long trajectory col ( u, y ) ∈ B T , where • u is persistently exciting of order t + n (prediction span + # states). colspan ( H t ( u Then y )) = B t . 9/37

Cartoon of Fundamental Lemma u ( t ) y ( t ) y 3 u 4 y 4 u 1 u 3 y 1 u 7 y 5 y 7 t u 5 u 6 t y 2 u 2 y 6 persistently exciting controllable LTI sufficiently many samples   ( u 1 ( u 2 ( u 3 y 1 ) y 2 ) y 3 ) . . .   ( u 2 ( u 3 ( u 4 y 2 ) y 3 ) y 4 ) x k +1 = Ax k + Bu k   . . . colspan   ( u 3 ( u 4 ( u 5 y 3 ) y 4 ) y 5 )   . . . y k = Cx k + Du k . ... ... ... . . � �� � � �� � parametric state-space model non-parametric model from raw data all trajectories constructible from finitely many previous trajectories 10/37