methods in science / engineering / industry due to • methodological advances in statistics, optimization, & machine learning (ML) • unprecedented availability of brute force: deluge of data & computational power • ...and frenzy surrounding big data & ML 4/57
methods in science / engineering / industry due to • methodological advances in statistics, optimization, & machine learning (ML) • unprecedented availability of brute force: deluge of data & computational power • ...and frenzy surrounding big data & ML Make up your own opinion, but ML works too well to be ignored 4/57
methods in science / engineering / industry due to • methodological advances in statistics, optimization, & machine learning (ML) • unprecedented availability of brute force: deluge of data & computational power • ...and frenzy surrounding big data & ML Make up your own opinion, but ML works too well to be ignored – also in control ?!? 4/57
methods in science / engineering / industry due to • methodological advances in statistics, optimization, & machine learning (ML) • unprecedented availability of brute force: deluge of data & computational power • ...and frenzy surrounding big data & ML Make up your own opinion, but ML works too well to be ignored – also in control ?!? “ One of the major developments in control over the past decade – & one of the most important moving forward – is the interaction of ML & control systems. ” [CSS roadmap] 4/57
methods in science / engineering / industry due to • methodological advances in statistics, optimization, & machine learning (ML) • unprecedented availability of brute force: deluge of data & computational power • ...and frenzy surrounding big data & ML Make up your own opinion, but ML works too well to be ignored – also in control ?!? “ One of the major developments in control over the past decade – & one of the most important moving forward – is the interaction of ML & control systems. ” [CSS roadmap] 4/57
methods in science / engineering / industry due to • methodological advances in statistics, optimization, & machine learning (ML) • unprecedented availability of brute force: deluge of data & computational power • ...and frenzy surrounding big data & ML Make up your own opinion, but ML works too well to be ignored – also in control ?!? “ One of the major developments in control over the past decade – & one of the most important moving forward – is the interaction of ML & control systems. ” [CSS roadmap] 4/57
data SysID ! model + uncertainty ! control • growing trend: direct data-driven control by-passing models ...(again) hyped, why ? data-driven control u2 u1 y1 y2 5/57
data SysID ! model + uncertainty ! control • growing trend: direct data-driven control by-passing models ...(again) hyped, why ? data-driven control u2 u1 y1 y2 Central promise: It is often easier to learn a control policy from data rather than a model. 5/57
data SysID ! model + uncertainty ! control • growing trend: direct data-driven control by-passing models ...(again) hyped, why ? data-driven control u2 u1 y1 y2 Central promise: It is often easier to learn a control policy from data rather than a model. Example 1973: autotuned PID 5/57
data SysID ! model + uncertainty ! control • growing trend: direct data-driven control by-passing models ...(again) hyped, why ? The direct approach is viable alternative • for some applications : model-based approach is too complex to be useful ! too complex models, environments, sensing modalities, specifications (e.g., wind farm) data-driven control u2 u1 y1 y2 Central promise: It is often easier to learn a control policy from data rather than a model. Example 1973: autotuned PID 5/57
data SysID ! model + uncertainty ! control • growing trend: direct data-driven control by-passing models ...(again) hyped, why ? The direct approach is viable alternative • for some applications : model-based approach is too complex to be useful ! too complex models, environments, sensing modalities, specifications (e.g., wind farm) • due to (well-known) shortcomings of ID ! too cumbersome, models not identified for control, incompatible uncertainty estimates, ... data-driven control u2 u1 y1 y2 Central promise: It is often easier to learn a control policy from data rather than a model. Example 1973: autotuned PID 5/57
data SysID ! model + uncertainty ! control • growing trend: direct data-driven control by-passing models ...(again) hyped, why ? The direct approach is viable alternative • for some applications : model-based approach is too complex to be useful ! too complex models, environments, sensing modalities, specifications (e.g., wind farm) • due to (well-known) shortcomings of ID ! too cumbersome, models not identified for control, incompatible uncertainty estimates, ... • when brute force data/compute available data-driven control u2 u1 y1 y2 Central promise: It is often easier to learn a control policy from data rather than a model. Example 1973: autotuned PID 5/57
control cost u, x subject to u, x satisfy state-space model where x estimated from u, y & model where model identified from ud, yd data 6/57 jopostoutpso
control cost u, x subject to u, x satisfy state-space model where x estimated from u, y & model where model identified from ud, yd data ! nested multi-level optimization problem ) outer optimization o middle opt. o inner opt. 6/57
control cost u, x subject to u, x satisfy state-space model where x estimated from u, y & model where model identified from ud, yd data ! nested multi-level optimization problem ) outer optimization o middle opt. o inner opt. 9 > > = > > ; separation & certainty equivalence (! LQG case) 6/57
control cost u, x subject to u, x satisfy state-space model where x estimated from u, y & model where model identified from ud, yd data ! nested multi-level optimization problem ) outer optimization o middle opt. o inner opt. 9 > > = > > ; separation & certainty equivalence (! LQG case) no separation (! ID-4-control) 6/57
control cost u, x subject to u, x satisfy state-space model where x estimated from u, y & model where model identified from ud, yd data ! nested multi-level optimization problem ) outer optimization o middle opt. o inner opt. 9 > > = > > ; separation & certainty equivalence (! LQG case) no separation (! ID-4-control) direct (black-box) data-driven control minimize control cost u, y subject to u, y consistent with ud, yd data 6/57
control cost u, x subject to u, x satisfy state-space model where x estimated from u, y & model where model identified from ud, yd data ! nested multi-level optimization problem ) outer optimization o middle opt. o inner opt. 9 > > = > > ; separation & certainty equivalence (! LQG case) no separation (! ID-4-control) direct (black-box) data-driven control minimize control cost u, y subject to u, y consistent with ud, yd data ! trade-o s modular vs. end-2-end suboptimal (?) vs. optimal convex vs. non-convex (?) 6/57
control cost u, x subject to u, x satisfy state-space model where x estimated from u, y & model where model identified from ud, yd data ! nested multi-level optimization problem ) outer optimization o middle opt. o inner opt. 9 > > = > > ; separation & certainty equivalence (! LQG case) no separation (! ID-4-control) direct (black-box) data-driven control minimize control cost u, y subject to u, y consistent with ud, yd data ! trade-o s modular vs. end-2-end suboptimal (?) vs. optimal convex vs. non-convex (?) Additionally: account for uncertainty (hard to propagate in indirect approach) 6/57
design & beyond • modular ! easy to debug & interpret • id = noise filtering • id = projection on model class • harder to propagate uncertainty through id • no robust separation principle ! suboptimal • ... ? x+ = f(x, u) y = h(x, u) y u • some models too complex to be useful • end-to-end ! suit- able for non-experts • design handles noise • harder to inject side info but no bias error • transparent: no unmodeled dynamics • possibly optimal but often less tractable • ... 7/57
design & beyond • modular ! easy to debug & interpret • id = noise filtering • id = projection on model class • harder to propagate uncertainty through id • no robust separation principle ! suboptimal • ... ? x+ = f(x, u) y = h(x, u) y u • some models too complex to be useful • end-to-end ! suit- able for non-experts • design handles noise • harder to inject side info but no bias error • transparent: no unmodeled dynamics • possibly optimal but often less tractable • ... lots of pros, cons, counterexamples, & no universal conclusions [discussion] 7/57
= · · · = 0 u0 = 1 x0 =0 y0 Now what if we had the impulse response recorded in our data-library? ⇥ g0 g1 g2 . . . ⇤ = ⇥ yd 0 yd 1 yd 2 . . . ⇤ 9/57 ↓ data : in pulse response g(t) = "all solutions can be parameterized by data"
: data-enabled predictive control 3. robustification via salient regularizations 4. cases studies from wind & power systems blooming literature (2-3 ArXiv / week) 10/57 5 #
: data-enabled predictive control 3. robustification via salient regularizations 4. cases studies from wind & power systems blooming literature (2-3 ArXiv / week) ! tutorial [link] to get started • [link] to graduate school material • [link] to survey • [link] to related bachelor lecture • [link] to related publications DATA-DRIVEN CONTROL BASED ON BEHAVIORAL APPROACH: FROM THEORY TO APPLICATIONS IN POWER SYSTEMS Ivan Markovsky, Linbin Huang, and Florian Dörfler I. Markovsky is with ICREA, Pg. Lluis Companys 23, Barcelona, and CIMNE, Gran Capitàn, Barcelona, Spain (e-mail: [email protected]), L. Huang and F. Dörfler are with the Automatic Control Laboratory, ETH Zürich, 8092 Zürich, Switzerland (e-mails: [email protected], dorfl[email protected]). modeling). Modeling using observed data, possibly incorporating some prior knowledge from the physical laws (that is, black-box 10/57
& provide pointers to more sophisticated research material ! study cutting-edge papers yourself • it’s a school: so we will spend time on the board ! take notes • We teach this material also in the ETH Z¨ urich bachelor & have plenty of background material + implementation experience ! please reach out to me or Saverio if you need anything • we will take a break after 90 minutes ! co ee , 11/57
put apply such a shaky data-driven method • on the world’s most complex engineered system (the electric grid), • using the world’s biggest actuators (Gigawatt-sized HVDC links), • and subject to real-time, safety, stability, constraints ...right? 13/57
is a 3-tuple (Z 0, W, B) where (i) Z 0 is the discrete-time axis, (ii) W is the signal space, & (iii) B ✓ WZ 0 is the behavior. 9 > > = > > ; B is the set of all trajectories 17/57 beYet of all time serieas
is a 3-tuple (Z 0, W, B) where (i) Z 0 is the discrete-time axis, (ii) W is the signal space, & (iii) B ✓ WZ 0 is the behavior. 9 > > = > > ; B is the set of all trajectories Definition: The dynamical system (Z 0, W, B) is (i) linear if W is a vector space & B is a subspace of WZ 0 (ii) & time-invariant if wt 2 B =) wt+1 2 B . 17/57
is a 3-tuple (Z 0, W, B) where (i) Z 0 is the discrete-time axis, (ii) W is the signal space, & (iii) B ✓ WZ 0 is the behavior. 9 > > = > > ; B is the set of all trajectories Definition: The dynamical system (Z 0, W, B) is (i) linear if W is a vector space & B is a subspace of WZ 0 (ii) & time-invariant if wt 2 B =) wt+1 2 B . y u 17/57
is a 3-tuple (Z 0, W, B) where (i) Z 0 is the discrete-time axis, (ii) W is the signal space, & (iii) B ✓ WZ 0 is the behavior. 9 > > = > > ; B is the set of all trajectories Definition: The dynamical system (Z 0, W, B) is (i) linear if W is a vector space & B is a subspace of WZ 0 (ii) & time-invariant if wt 2 B =) wt+1 2 B . LTI system = shift-invariant subspace of trajectory space ! abstract perspective suited for data-driven control y u 17/57
x+= A s 3 X(1) = Ax + Bu(0) x(2) = A3x + ABu(0) + 1(1(1) i y(t) = (A + x + At -T Bu(n) + Du(t) I I Ie ( x + as a I " : CAT- 3 B y(T) use m un Y Mendetit It : impulse matrix U
u7 y(t) t y4 y2 y1 y3 y5 y6 y7 Given: data ⇣ ud i yd i ⌘ 2 Rm+p & LTI complexity parameters ⇢ lag ` order n set of all T-length trajectories = n (u, y) 2 R(m+p)T : 9x 2 RnT s.t. x+ = Ax + Bu , y = Cx + Du o | {z } parametric state-space model 22/57
u7 y(t) t y4 y2 y1 y3 y5 y6 y7 Given: data ⇣ ud i yd i ⌘ 2 Rm+p & LTI complexity parameters ⇢ lag ` order n set of all T-length trajectories = n (u, y) 2 R(m+p)T : 9x 2 RnT s.t. x+ = Ax + Bu , y = Cx + Du o | {z } | {z } parametric state-space model raw data (every column is an experiment) colspan 2 6 6 6 6 6 6 6 6 4 ud 1,1 yd 1,1 ! ud 1,2 yd 1,2 ! ud 1,3 yd 1,3 ! ... ud 2,1 yd 2,1 ! ud 2,2 yd 2,2 ! ud 2,3 yd 2,3 ! ... . . . . . . . . . . . . ud T,1 yd T,1 ! ud T,2 yd T,2 ! ud T,3 yd T,3 ! ... 3 7 7 7 7 7 7 7 7 5 if and only if the trajectory matrix has rank m · T + n for all T > ` 22/57 /
R(m+p)T : 9x 2 RnT s.t. x+ = Ax + Bu , y = Cx + Du o | {z } | {z } parametric state-space model non-parametric model from raw data colspan 2 6 6 6 6 6 6 6 6 4 ud 1,1 yd 1,1 ! ud 1,2 yd 1,2 ! ud 1,3 yd 1,3 ! ... ud 2,1 yd 2,1 ! ud 2,2 yd 2,2 ! ud 2,3 yd 2,3 ! ... . . . . . . . . . . . . ud T,1 yd T,1 ! ud T,2 yd T,2 ! ud T,3 yd T,3 ! ... 3 7 7 7 7 7 7 7 7 5 all trajectories constructible from finitely many previous trajectories • standing on the shoulders of giants: classic Willems’ result was only “if” & required further assumptions: Hankel, persistency of excitation, controllability 23/57
R(m+p)T : 9x 2 RnT s.t. x+ = Ax + Bu , y = Cx + Du o | {z } | {z } parametric state-space model non-parametric model from raw data colspan 2 6 6 6 6 6 6 6 6 4 ud 1,1 yd 1,1 ! ud 1,2 yd 1,2 ! ud 1,3 yd 1,3 ! ... ud 2,1 yd 2,1 ! ud 2,2 yd 2,2 ! ud 2,3 yd 2,3 ! ... . . . . . . . . . . . . ud T,1 yd T,1 ! ud T,2 yd T,2 ! ud T,3 yd T,3 ! ... 3 7 7 7 7 7 7 7 7 5 all trajectories constructible from finitely many previous trajectories • standing on the shoulders of giants: classic Willems’ result was only “if” & required further assumptions: Hankel, persistency of excitation, controllability • terminology fundamental is justified : motion primitives, subspace SysID, dictionary learning, (E)DMD, ... all implicitly rely on this equivalence 23/57
R(m+p)T : 9x 2 RnT s.t. x+ = Ax + Bu , y = Cx + Du o | {z } | {z } parametric state-space model non-parametric model from raw data colspan 2 6 6 6 6 6 6 6 6 4 ud 1,1 yd 1,1 ! ud 1,2 yd 1,2 ! ud 1,3 yd 1,3 ! ... ud 2,1 yd 2,1 ! ud 2,2 yd 2,2 ! ud 2,3 yd 2,3 ! ... . . . . . . . . . . . . ud T,1 yd T,1 ! ud T,2 yd T,2 ! ud T,3 yd T,3 ! ... 3 7 7 7 7 7 7 7 7 5 all trajectories constructible from finitely many previous trajectories • standing on the shoulders of giants: classic Willems’ result was only “if” & required further assumptions: Hankel, persistency of excitation, controllability • terminology fundamental is justified : motion primitives, subspace SysID, dictionary learning, (E)DMD, ... all implicitly rely on this equivalence • many recent extensions to other system classes (bi-linear, descriptor, LPV, delay, Volterra series, Wiener-Hammerstein, ...), other matrix data structures (mosaic Hankel, Page, ...), & other proof methods 23/57
u3 u5 u6 u7 y(t) t y4 y2 y1 y3 y5 y6 y7 Definition: The data signal ud 2 RmTd of length Td is persistently exciting of order T if the Hankel matrix 2 4 u1 ··· uTd T +1 . . . ... . . . uT ··· uTd 3 5 is of full rank. 24/57 un m
u3 u5 u6 u7 y(t) t y4 y2 y1 y3 y5 y6 y7 Definition: The data signal ud 2 RmTd of length Td is persistently exciting of order T if the Hankel matrix 2 4 u1 ··· uTd T +1 . . . ... . . . uT ··· uTd 3 5 is of full rank. 24/57 mT u" has to be sufficiently rich & long : To-T + = mT
u3 u5 u6 u7 y(t) t y4 y2 y1 y3 y5 y6 y7 Definition: The data signal ud 2 RmTd of length Td is persistently exciting of order T if the Hankel matrix 2 4 u1 ··· uTd T +1 . . . ... . . . uT ··· uTd 3 5 is of full rank. Input design [Willems et al, ’05]: Controllable LTI system & persistently exciting input ud of order T + n =) rank ⇣ H ⇣ ud yd ⌘⌘ = mT + n. 24/57
requires mana short independent experiments a page matrix H(w) = [ W We requires one long experiment · Hankel matrix H(v) = fr Wh . requires one short experiment but this matrix structured o any combination n... " block Hankel
X k=1 kyk rk k2 Q + kuk k2 R subject to xk+1 = Axk + Buk yk = Cxk + Duk 8k 2 {1, . . . , Tfuture } uk 2 U yk 2 Y 8k 2 {1, . . . , Tfuture } quadratic cost with R 0, Q ⌫ 0 & ref. r model for prediction with k 2 [1, Tfuture] hard operational or safety constraints 28/57
X k=1 kyk rk k2 Q + kuk k2 R subject to xk+1 = Axk + Buk yk = Cxk + Duk 8k 2 {1, . . . , Tfuture } xk+1 = Axk + Buk yk = Cxk + Duk 8k 2 { Tini 1, . . . , 0} uk 2 U yk 2 Y 8k 2 {1, . . . , Tfuture } quadratic cost with R 0, Q ⌫ 0 & ref. r model for prediction with k 2 [1, Tfuture] model for estimation with k 2 [ Tini 1, 0] & Tini lag (many flavors) hard operational or safety constraints 28/57
X k=1 kyk rk k2 Q + kuk k2 R subject to xk+1 = Axk + Buk yk = Cxk + Duk 8k 2 {1, . . . , Tfuture } xk+1 = Axk + Buk yk = Cxk + Duk 8k 2 { Tini 1, . . . , 0} uk 2 U yk 2 Y 8k 2 {1, . . . , Tfuture } quadratic cost with R 0, Q ⌫ 0 & ref. r model for prediction with k 2 [1, Tfuture] model for estimation with k 2 [ Tini 1, 0] & Tini lag (many flavors) hard operational or safety constraints “[MPC] has perhaps too little system theory and too much brute force – Willems ’07 28/57
X k=1 kyk rk k2 Q + kuk k2 R subject to xk+1 = Axk + Buk yk = Cxk + Duk 8k 2 {1, . . . , Tfuture } xk+1 = Axk + Buk yk = Cxk + Duk 8k 2 { Tini 1, . . . , 0} uk 2 U yk 2 Y 8k 2 {1, . . . , Tfuture } quadratic cost with R 0, Q ⌫ 0 & ref. r model for prediction with k 2 [1, Tfuture] model for estimation with k 2 [ Tini 1, 0] & Tini lag (many flavors) hard operational or safety constraints “[MPC] has perhaps too little system theory and too much brute force [...], but MPC is an area where all aspects of the field [...] are in synergy.” – Willems ’07 28/57
X k=1 kyk rk k2 Q + kuk k2 R subject to xk+1 = Axk + Buk yk = Cxk + Duk 8k 2 {1, . . . , Tfuture } xk+1 = Axk + Buk yk = Cxk + Duk 8k 2 { Tini 1, . . . , 0} uk 2 U yk 2 Y 8k 2 {1, . . . , Tfuture } quadratic cost with R 0, Q ⌫ 0 & ref. r model for prediction with k 2 [1, Tfuture] model for estimation with k 2 [ Tini 1, 0] & Tini lag (many flavors) hard operational or safety constraints “[MPC] has perhaps too little system theory and too much brute force [...], but MPC is an area where all aspects of the field [...] are in synergy.” – Willems ’07 Elegance aside, for an LTI plant, deterministic, & with known model, MPC is the gold standard of control. 28/57
k=1 kyk rk k2 Q + kuk k2 R subject to H ⇣ ud yd ⌘ · g = 2 6 6 4 uini yini u y 3 7 7 5 uk 2 U yk 2 Y 8k 2 {1, . . . , Tfuture } quadratic cost with R 0, Q ⌫ 0 & ref. r non-parametric model for prediction and estimation hard operational or safety constraints • real-time measurements (uini, yini) for estimation • trajectory matrix H ⇣ ud yd ⌘ from past experimental data updated online collected o ine (could be adapted online) 29/57 & online ?
k=1 kyk rk k2 Q + kuk k2 R subject to H ⇣ ud yd ⌘ · g = 2 6 6 4 uini yini u y 3 7 7 5 uk 2 U yk 2 Y 8k 2 {1, . . . , Tfuture } quadratic cost with R 0, Q ⌫ 0 & ref. r non-parametric model for prediction and estimation hard operational or safety constraints • real-time measurements (uini, yini) for estimation • trajectory matrix H ⇣ ud yd ⌘ from past experimental data updated online collected o ine (could be adapted online) ! equivalent to MPC in deterministic LTI case ... but needs to be robustified in case of noise / nonlinearity ! 29/57
In the Shallows of the DeePC Jeremy Coulson John Lygeros Florian D¨ orfler Abstract— We consider the problem of optimal trajectory tracking for unknown systems. A novel data-enabled predictive control (DeePC) algorithm is presented that computes optimal and safe control policies using real-time feedback driving the unknown system along a desired trajectory while satisfying system constraints. Using a finite number of data samples from the unknown system, our proposed algorithm uses a behavioural systems theory approach to learn a non-parametric system model used to predict future trajectories. The DeePC algorithm is shown to be equivalent to the classical and widely adopted Model Predictive Control (MPC) algorithm in the case of deterministic linear time-invariant systems. In the case of nonlinear stochastic systems, we propose regularizations to the DeePC algorithm. Simulations are provided to illustrate performance and compare the algorithm with other methods. approaches usually require a large number of data sam- ples to perform well, and are often sensitive to hyper- parameters leading to non-reproducible and highly variable outcomes [4]. Other approaches propose performing sequential system identification (ID) and control. System ID can be used to produce an approximate model and provide finite sample guarantees quantifying model uncertainty, allowing for robust control design [5]. In this spirit, an end-to-end ID and control pipeline is given in [6] and arrives at a data-driven control solution with guarantees on the sample efficiency, stability, performance, and robustness. The system identification step in these approaches disregards one of the main advantages of a data-driven approach: independence from an underlying DeePC = deep sea (making fun of ML & data science) … though the idea is really shallow 30/57
In the Shallows of the DeePC Jeremy Coulson John Lygeros Florian D¨ orfler Abstract— We consider the problem of optimal trajectory tracking for unknown systems. A novel data-enabled predictive control (DeePC) algorithm is presented that computes optimal and safe control policies using real-time feedback driving the unknown system along a desired trajectory while satisfying system constraints. Using a finite number of data samples from the unknown system, our proposed algorithm uses a behavioural systems theory approach to learn a non-parametric system model used to predict future trajectories. The DeePC algorithm is shown to be equivalent to the classical and widely adopted Model Predictive Control (MPC) algorithm in the case of deterministic linear time-invariant systems. In the case of nonlinear stochastic systems, we propose regularizations to the DeePC algorithm. Simulations are provided to illustrate performance and compare the algorithm with other methods. approaches usually require a large number of data sam- ples to perform well, and are often sensitive to hyper- parameters leading to non-reproducible and highly variable outcomes [4]. Other approaches propose performing sequential system identification (ID) and control. System ID can be used to produce an approximate model and provide finite sample guarantees quantifying model uncertainty, allowing for robust control design [5]. In this spirit, an end-to-end ID and control pipeline is given in [6] and arrives at a data-driven control solution with guarantees on the sample efficiency, stability, performance, and robustness. The system identification step in these approaches disregards one of the main advantages of a data-driven approach: independence from an underlying DeePC = deep sea (making fun of ML & data science) … though the idea is really shallow surprise: scheme has zero robustness ! need deeper thinking 30/57
X k=1 kyk rk k2 Q + kuk k2 R + y k kp + gh(g) subject to H ⇣ ud yd ⌘ · g = 2 6 6 4 uini yini u y 3 7 7 5 + 2 6 6 4 0 0 0 3 7 7 5 uk 2 U yk 2 Y 8k 2 {1, . . . , Tfuture } measurement noise ! infeasible yini estimate ! estimation slack ! moving-horizon least-square filter noisy or nonlinear (o ine) data matrix ! any (u y) feasible ! add regularizer h(g) 31/57 S in practice : neuer low runk
u, y subject to u y = H ⇣ ˆ u ˆ y ⌘ g where ⇣ ˆ u ˆ y ⌘ 2 argmin ⇣ ˆ u ˆ y ⌘ ⇣ ud yd ⌘ subject to rank H ˆ u ˆ y = mL + n 9 = ; optimal control 9 = ; low-rank approximation 32/57
u, y subject to u y = H ⇣ ˆ u ˆ y ⌘ g where ⇣ ˆ u ˆ y ⌘ 2 argmin ⇣ ˆ u ˆ y ⌘ ⇣ ud yd ⌘ subject to rank H ˆ u ˆ y = mL + n # sequence of convex relaxations # minimizeu,y,g control cost u, y subject to u y = H ⇣ ˆ u ˆ y ⌘ g where ⇣ ˆ u ˆ y ⌘ 2 argmin ⇣ ˆ u ˆ y ⌘ ⇣ ud yd ⌘ subject to rank H ˆ u ˆ y = mL + n , kgk0 mL + n 9 = ; optimal control 9 = ; low-rank approximation 32/57 low-rank - ML + n "horizon -sparse
u, y subject to u y = H ⇣ ˆ u ˆ y ⌘ g where ⇣ ˆ u ˆ y ⌘ 2 argmin ⇣ ˆ u ˆ y ⌘ ⇣ ud yd ⌘ subject to rank H ˆ u ˆ y = mL + n # sequence of convex relaxations # minimizeu,y,g control cost u, y subject to u y = H ⇣ ˆ u ˆ y ⌘ g where ⇣ ˆ u ˆ y ⌘ 2 argmin ⇣ ˆ u ˆ y ⌘ ⇣ ud yd ⌘ subject to rank H ˆ u ˆ y = mL + n , kgk0 mL + n 9 = ; optimal control 9 = ; low-rank approximation 32/57
u, y subject to u y = H ⇣ ˆ u ˆ y ⌘ g where ⇣ ˆ u ˆ y ⌘ 2 argmin ⇣ ˆ u ˆ y ⌘ ⇣ ud yd ⌘ subject to rank H ˆ u ˆ y = mL + n # sequence of convex relaxations # minimizeu,y,g control cost u, y subject to u y = H ⇣ ˆ u ˆ y ⌘ g where ⇣ ˆ u ˆ y ⌘ 2 argmin ⇣ ˆ u ˆ y ⌘ ⇣ ud yd ⌘ subject to rank H ˆ u ˆ y = mL + n , kgk0 mL + n = ⇣ ud yd ⌘ 9 = ; optimal control 9 = ; low-rank approximation 32/57
u, y subject to u y = H ⇣ ˆ u ˆ y ⌘ g where ⇣ ˆ u ˆ y ⌘ 2 argmin ⇣ ˆ u ˆ y ⌘ ⇣ ud yd ⌘ subject to rank H ˆ u ˆ y = mL + n # sequence of convex relaxations # minimizeu,y,g control cost u, y subject to u y = H ⇣ ud yd ⌘ g , kgk0 mL + n 9 = ; optimal control 9 = ; low-rank approximation 32/57
u, y subject to u y = H ⇣ ˆ u ˆ y ⌘ g where ⇣ ˆ u ˆ y ⌘ 2 argmin ⇣ ˆ u ˆ y ⌘ ⇣ ud yd ⌘ subject to rank H ˆ u ˆ y = mL + n # sequence of convex relaxations # minimizeu,y,g control cost u, y subject to u y = H ⇣ ud yd ⌘ g , kgk1 mL + n 9 = ; optimal control 9 = ; low-rank approximation 32/57
u, y subject to u y = H ⇣ ˆ u ˆ y ⌘ g where ⇣ ˆ u ˆ y ⌘ 2 argmin ⇣ ˆ u ˆ y ⌘ ⇣ ud yd ⌘ subject to rank H ˆ u ˆ y = mL + n # sequence of convex relaxations # minimizeu,y,g control cost u, y + g · kgk1 subject to u y = H ⇣ ud yd ⌘ g 9 = ; optimal control 9 = ; low-rank approximation 32/57
u, y subject to u y = H ⇣ ˆ u ˆ y ⌘ g where ⇣ ˆ u ˆ y ⌘ 2 argmin ⇣ ˆ u ˆ y ⌘ ⇣ ud yd ⌘ subject to rank H ˆ u ˆ y = mL + n # sequence of convex relaxations # minimizeu,y,g control cost u, y + g · kgk1 subject to u y = H ⇣ ud yd ⌘ g `1 -regularization = relaxation of low-rank approximation & smoothened order selection 9 = ; optimal control 9 = ; low-rank approximation 32/57
u, y subject to u y = H ⇣ ˆ u ˆ y ⌘ g where ⇣ ˆ u ˆ y ⌘ 2 argmin ⇣ ˆ u ˆ y ⌘ ⇣ ud yd ⌘ subject to rank H ˆ u ˆ y = mL + n # sequence of convex relaxations # minimizeu,y,g control cost u, y + g · kgk1 subject to u y = H ⇣ ud yd ⌘ g `1 -regularization = relaxation of low-rank approximation & smoothened order selection 9 = ; optimal control 9 = ; low-rank approximation !"#$%&'"##()*#$+ realized closed-loop cost g 32/57 "
[ii] [i] = Hig = ( where K is learnt via u y = Yeg wheresolves K = urymin 114-i . [ - k = Ye . [Y - 32.% Cl - > to recover the model-based solution we need to penalize grom
ID: H ⇣ ud yd ⌘ ⇠ 2 6 6 4 Up Yp Uf Yf 3 7 7 5 (m + p)Tini (m + p)Tfuture ! indirect SysID + control problem minimizeu,y control cost(u, y) subject to y = K? 2 4 uini yini u 3 5 where K? = argmin K YF K 2 4 Up Yp Uf 3 5 34/57
ID: H ⇣ ud yd ⌘ ⇠ 2 6 6 4 Up Yp Uf Yf 3 7 7 5 (m + p)Tini (m + p)Tfuture ID of optimal multi-step predictor as in SPC: K? = YF Up Yp Uf † 8 < : ! indirect SysID + control problem minimizeu,y control cost(u, y) subject to y = K? 2 4 uini yini u 3 5 where K? = argmin K YF K 2 4 Up Yp Uf 3 5 34/57 subspace predictor y = y[ : ] (ii)
ID: H ⇣ ud yd ⌘ ⇠ 2 6 6 4 Up Yp Uf Yf 3 7 7 5 (m + p)Tini (m + p)Tfuture ID of optimal multi-step predictor as in SPC: K? = YF Up Yp Uf † 8 < : ! indirect SysID + control problem minimizeu,y control cost(u, y) subject to y = K? 2 4 uini yini u 3 5 where K? = argmin K YF K 2 4 Up Yp Uf 3 5 The above is equivalent to regularized DeePC minimizeg,u,y control cost(u, y) + g Proj ⇣ ud yd ⌘ g p subject to H ⇣ ud yd ⌘ · g = 2 6 6 4 uini yini u y 3 7 7 5 34/57
ID: H ⇣ ud yd ⌘ ⇠ 2 6 6 4 Up Yp Uf Yf 3 7 7 5 (m + p)Tini (m + p)Tfuture ID of optimal multi-step predictor as in SPC: K? = YF Up Yp Uf † 8 < : ! indirect SysID + control problem minimizeu,y control cost(u, y) subject to y = K? 2 4 uini yini u 3 5 where K? = argmin K YF K 2 4 Up Yp Uf 3 5 The above is equivalent to regularized DeePC where Proj ⇣ ud yd ⌘ projects orthogonal to ker Up Yp Uf minimizeg,u,y control cost(u, y) + g Proj ⇣ ud yd ⌘ g p subject to H ⇣ ud yd ⌘ · g = 2 6 6 4 uini yini u y 3 7 7 5 34/57
closed-loop data leveraging a transient predictor – single-step – causal – unbiased – with optimal regularizers • subspace tracking for LTV systems • extensions to nonlinear system class: conic/convex/a ne behaviors, LPV, RHKS formulations, or ... • robust problem formulations to robustify objectives & constraints • behavioral uncertainty quantification via subspace angles online system identificat Andr´ as Sasfi1, Alberto Padoan1, Ivan Markovsky2, Florian D¨ orfle 1 Automatic Control Laboratory, ETH Z¨ urich, Switzerland, 2 CIMN Motivation dsfsdgsdafgvsdfvsdfvsdzv Data-driven methods based on behavioral systems theory are powerful, yet difficult to extend to time-varying or nonlinear systems Subspaces as system models U1 U2 Behavioral systems theory: linear time-varying behaviors are time-varying subspaces Illustration: LTV state-space equations xt+1 = At xt + Bt ut , yt = Ct xt + Dt ut Th As W 1.W 2.P Co su 36/57
simple script adapted from our ETH Z¨ urich bachelor course on Computational control : https://colab.research.google.com/ drive/1URdRqr-Up0A6uDMjlU6gwmsoAAPl1GId?usp=sharing 39/57
system to large/1-dimensional bi-/linear system ! Carleman, Volterra, Fliess, Koopman, Sturm-Liouville methods ! nonlinear dynamics can be approximated by LTI on finite horizon 40/57
system to large/1-dimensional bi-/linear system ! Carleman, Volterra, Fliess, Koopman, Sturm-Liouville methods ! nonlinear dynamics can be approximated by LTI on finite horizon regularization singles out relevant features / basis functions in data 40/57
system to large/1-dimensional bi-/linear system ! Carleman, Volterra, Fliess, Koopman, Sturm-Liouville methods ! nonlinear dynamics can be approximated by LTI on finite horizon regularization singles out relevant features / basis functions in data https://www.research-collection.ethz.ch/handle/20.500.11850/493419 40/57
b ⇠, x = minx2X E ⇠⇠b P ⇥ c (⇠, x) ⇤ where b ⇠ denotes measured data with empirical distribution b P = b ⇠ 42/57 ↑ ↑ ↑ sample distribution Sege Sug Seg
b ⇠, x = minx2X E ⇠⇠b P ⇥ c (⇠, x) ⇤ where b ⇠ denotes measured data with empirical distribution b P = b ⇠ ) poor out-of-sample performance of above sample-average solution x? for real problem: E⇠⇠P ⇥ c (⇠, x?) ⇤ where P is the unknown distribution of ⇠ 42/57
b ⇠, x = minx2X E ⇠⇠b P ⇥ c (⇠, x) ⇤ where b ⇠ denotes measured data with empirical distribution b P = b ⇠ ) poor out-of-sample performance of above sample-average solution x? for real problem: E⇠⇠P ⇥ c (⇠, x?) ⇤ where P is the unknown distribution of ⇠ • distributionally robust formulation accounting for all (possibly nonlinear) stochastic processes that could have generated the data 42/57
b ⇠, x = minx2X E ⇠⇠b P ⇥ c (⇠, x) ⇤ where b ⇠ denotes measured data with empirical distribution b P = b ⇠ ) poor out-of-sample performance of above sample-average solution x? for real problem: E⇠⇠P ⇥ c (⇠, x?) ⇤ where P is the unknown distribution of ⇠ • distributionally robust formulation accounting for all (possibly nonlinear) stochastic processes that could have generated the data inf x2X sup Q2B✏(b P) E⇠⇠Q ⇥ c (⇠, x) ⇤ 42/57 - maximize over all distributions Q which are -close to the samples ↑ design
b ⇠, x = minx2X E ⇠⇠b P ⇥ c (⇠, x) ⇤ where b ⇠ denotes measured data with empirical distribution b P = b ⇠ ) poor out-of-sample performance of above sample-average solution x? for real problem: E⇠⇠P ⇥ c (⇠, x?) ⇤ where P is the unknown distribution of ⇠ • distributionally robust formulation accounting for all (possibly nonlinear) stochastic processes that could have generated the data inf x2X sup Q2B✏(b P) E⇠⇠Q ⇥ c (⇠, x) ⇤ where B✏(b P) is an ✏-Wasserstein ball centered at empirical sample distribution b P : B✏(b P) = ⇢ P : inf ⇧ Z ⇠ b ⇠ p d⇧ ✏ ˆ ⇠ ⇠ ˆ P P ⇧ 42/57
control cost u, y subject to u, y satisfy parametric model where model 2 argmin id cost ud, yd subject to model 2 LTI(n, `) class ) ID ID projects data on the set of LTI models • with parameters (n, `) • removes noise & thus lowers variance error • su ers bias error if plant is not LTI(n, `) 45/57
control cost u, y subject to u, y satisfy parametric model where model 2 argmin id cost ud, yd subject to model 2 LTI(n, `) class ) ID ID projects data on the set of LTI models • with parameters (n, `) • removes noise & thus lowers variance error • su ers bias error if plant is not LTI(n, `) direct regularized data-driven control minimize control cost u, y + · regularizer subject to u, y consistent with ud, yd data 45/57 [i] E image Hagel
control cost u, y subject to u, y satisfy parametric model where model 2 argmin id cost ud, yd subject to model 2 LTI(n, `) class ) ID ID projects data on the set of LTI models • with parameters (n, `) • removes noise & thus lowers variance error • su ers bias error if plant is not LTI(n, `) direct regularized data-driven control minimize control cost u, y + · regularizer subject to u, y consistent with ud, yd data • regularization robustifies ! choosing makes it work • no projection on LTI(n, `) ! no de-noising & no bias 45/57
control cost u, y subject to u, y satisfy parametric model where model 2 argmin id cost ud, yd subject to model 2 LTI(n, `) class ) ID ID projects data on the set of LTI models • with parameters (n, `) • removes noise & thus lowers variance error • su ers bias error if plant is not LTI(n, `) direct regularized data-driven control minimize control cost u, y + · regularizer subject to u, y consistent with ud, yd data • regularization robustifies ! choosing makes it work • no projection on LTI(n, `) ! no de-noising & no bias hypothesis: ID wins in stochastic (variance) & DeePC in nonlinear (bias) case 45/57 Istochastic LTI of her on complexity
LQR control of 5th order LTI system • Gaussian noise with varying noise to signal ratio (100 rollouts each case) • `1 -regularized DeePC, SysID via N4SID, & judicious hyper-parameters 46/57 of known order
LQR control of 5th order LTI system • Gaussian noise with varying noise to signal ratio (100 rollouts each case) • `1 -regularized DeePC, SysID via N4SID, & judicious hyper-parameters deterministic noisy 46/57
indirect ID wins • LQR control of 5th order LTI system • Gaussian noise with varying noise to signal ratio (100 rollouts each case) • `1 -regularized DeePC, SysID via N4SID, & judicious hyper-parameters deterministic noisy 46/57
indirect ID wins • LQR control of 5th order LTI system • Gaussian noise with varying noise to signal ratio (100 rollouts each case) • `1 -regularized DeePC, SysID via N4SID, & judicious hyper-parameters deterministic noisy nonlinear case • Lotka-Volterra + control: x+ = f(x, u) • interpolated system x+ = ✏·flinearized(x, u)+(1 ✏)·f(x, u) • same ID & DeePC as on the left & 100 initial x0 rollouts for each ✏ 46/57
indirect ID wins • LQR control of 5th order LTI system • Gaussian noise with varying noise to signal ratio (100 rollouts each case) • `1 -regularized DeePC, SysID via N4SID, & judicious hyper-parameters deterministic noisy nonlinear case • Lotka-Volterra + control: x+ = f(x, u) • interpolated system x+ = ✏·flinearized(x, u)+(1 ✏)·f(x, u) • same ID & DeePC as on the left & 100 initial x0 rollouts for each ✏ nonlinear linear 46/57
indirect ID wins • LQR control of 5th order LTI system • Gaussian noise with varying noise to signal ratio (100 rollouts each case) • `1 -regularized DeePC, SysID via N4SID, & judicious hyper-parameters deterministic noisy nonlinear case ! direct DeePC wins • Lotka-Volterra + control: x+ = f(x, u) • interpolated system x+ = ✏·flinearized(x, u)+(1 ✏)·f(x, u) • same ID & DeePC as on the left & 100 initial x0 rollouts for each ✏ nonlinear linear 46/57
0.4 0.6 0.8 0 5 10 15 20 25 30 0.0 0.2 0.4 0.6 0 5 10 15 20 25 30 0.0 0.2 0.4 0.6 time (s) a control; —– with PEM-MPC (s = 60); —– with DeePC Closed‐loop cost Number of simulations DeePC PEM‐MPC ost comparison of DeePC and PEM-MPC under the practic = Prediction Error Method (PEM) System ID + MPC t < 10 s : open loop data collection with white noise excitat. t > 10 s : control 48/57
• for distributional robustness ⇡ radius of Wasserstein ball • wide range of sweet spots ! choose g = 20 estimation horizon Tini • for model complexity ⇡ lag • Tini 50 is su cient & low computational complexity ! choose Tini = 60 50/57 1 En
horizon Tfuture • nominal MPC is stable if horizon Tfuture long enough ! choose Tfuture = 120 & apply first 60 input steps data length T • long enough for low-rank condition but card(g) grows ! choose T = 1500 (data matrix ⇡ square) 51/57 as far as computation well conditioned
k Averaged Closed‐loop Cost S0=1 ⌅ Hankel matrix ⌅ Hankel matrix with SVD ( threshhold = 1) ⌅ Page matrix ⌅ Page matrix with SVD ( threshhold = 1) • comparison baseline: Hankel and Page matrices of same size • perfomance : Page consistency beats Hankel matrix predictors • o ine denoising via SVD threshholding works wonderfully for Page though obviously not for Hankel (entries are constrained) • e ects very pronounced for longer horizon (= open-loop time) • price-to-be-paid : Page matrix predictor requires more data 53/57
0.2 0.4 0.6 0.8 0 5 10 15 20 25 30 0.0 0.2 0.4 0.6 0 5 10 15 20 25 30 0.0 0.2 0.4 0.6 time (s) • colors correspond to di erent hyper- parameter settings (not discernible) • ambiguity set W is 1-ball (box) • for computational e ciency W is downsampled (piece-wise linear) • solver time ⇡ 2.6 s ) implementable 55/57
• robustness & side-info by regularization • method that works in theory & practice • focus is robust prediction not predictor ID ongoing work ! certificates for adaptive & nonlinear cases ! applications with a true “business case”, push TRL scale, & industry collaborations SG 1 SG 2 SG 3 1 2 3 4 5 6 7 9 8 IEEE nine-bus system wind farm 1 2 3 4 5 6 7 8 9 10 56/57 Lar (nuclear fusion
• robustness & side-info by regularization • method that works in theory & practice • focus is robust prediction not predictor ID ongoing work ! certificates for adaptive & nonlinear cases ! applications with a true “business case”, push TRL scale, & industry collaborations SG 1 SG 2 SG 3 1 2 3 4 5 6 7 9 8 IEEE nine-bus system wind farm 1 2 3 4 5 6 7 8 9 10 questions we should discuss • catch? violate no-free-lunch theorem ? ! more real-time computation 56/57 sparse MPC constraint / = dense Dee PC constraint
• robustness & side-info by regularization • method that works in theory & practice • focus is robust prediction not predictor ID ongoing work ! certificates for adaptive & nonlinear cases ! applications with a true “business case”, push TRL scale, & industry collaborations SG 1 SG 2 SG 3 1 2 3 4 5 6 7 9 8 IEEE nine-bus system wind farm 1 2 3 4 5 6 7 8 9 10 questions we should discuss • catch? violate no-free-lunch theorem ? ! more real-time computation • when does direct beat indirect ? ! Id4Control & bias/variance issues ? 56/57
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