1/31/2007 Massachusetts Institute of Technology Context Hybrid - PDF document

1/31/2007 Massachusetts Institute of Technology Context – Hybrid Systems • Hybrid discrete-continuous models are convenient for many systems Active Estimation for Switching – Failure-prone components (Funiak03, Dearden02) – Piloted aircraft (Tomlin06) Linear Dynamic Systems – Insects (Oh05) • Example: Switching Linear Dynamic Systems Lars Blackmore, Senthooran 0.001 0.999 1 Rajamanoharan and Brian Williams x d,t = x d,t = nominal failed January 31, 2007 0 = + + υ = + + υ A B A B x x u + x x u c , t 1 nominal c , t nominal t nominal + c , t 1 failure c , t failure t failure = + + ω = + + ω C D C D y x u y x u t nominal c , t nominal t nominal t failure c , t failure t failure 2 Context – Hybrid State Estimation K-Best Hybrid State Estimation • Hybrid state estimation aims to determine: • Full hybrid estimation considers all mode sequences ( , | , ) p x x y u − c , t d , t 1 : T 0 : T 1 0.8 ok ok – Applications include fault detection, intent recognition… ok failed 0.05 ok 0.05 failed failed 0.1 • Exact hybrid estimation is intractable (Lerner01) • K-best enumeration retains the k mode sequences • Prior work has developed approximate approaches, with highest posterior probability for example: – Merging (Lerner00) • Problem: losing true mode sequence – Pruning (Hofbaur02) – Fault detection particularly problematic – Sampling (Doucet00) 3 4 Active Hybrid Estimation Problem Statement • System inputs greatly affect performance of hybrid estimator • Prior work has used control inputs for optimal • Design a finite sequence of control inputs u =[ u 0 …u h ] discrimination between linear dynamic models to minimize p(loss), the probability of losing the true discrete mode sequence – Subject to constraints on inputs and expected state • We present a novel method that uses control inputs to aid hybrid state estimation • For Switching Linear Dynamic Systems – Key idea: Minimize probability of losing true mode • Zero mean, Gaussian white process and observation noise sequence subject to explicit input and state • Assume pruning occurs at end of horizon constraints L16 5 6 1

Slide 6 L16 mention can't calculate in closed form now Lars, 12/12/2006

1/31/2007 Summary of Technical Approach Summary of Technical Approach 1. Express future mode sequences as multiple 1. Express future mode sequences as multiple known time-varying models known time-varying models 2. Bound p(loss) using Multiple-Model bound from 2. Bound p(loss) using Multiple-Model bound from (Blackmore06) (Blackmore06) 3. Make bound tractable using efficient pruning 3. Make bound tractable using efficient pruning approach approach 4. Minimize bound using constrained optimization 4. Minimize bound using constrained optimization 7 8 Mode Sequences as Models Bounding p(loss) • Each mode sequence corresponds to a known • Loss probability upper-bounded by probability that Linear Time Varying (LTV) model true mode sequence is not most likely sequence = – Finite number of ‘hypotheses’ ( ) ( true sequence not in top ) p loss p k ≤ – Hybrid estimation picks k most likely hypotheses ( true sequence not hypothesis ) p most likely • Extend bound in (Blackmore06) to SLDS to give: ok H 0 ok ∑∑ ≤ − ok k ( i , j ) H 1 ( | ) ( ) ( ) failed P loss P H P H e u i j > Σ + Σ i j i [ ] 1 1 ok H 2 − = − Σ + Σ 1 − + i j k ( i , j ) ( )' ( ) ln µ µ µ µ j i i j j i 4 2 Σ Σ 2 failed failed H 3 i j � Analytic upper bound on p(loss) • Idea: Use results from (Blackmore06) for multiple- • Problem: exponential number of hypotheses model discrimination to upper-bound p(loss) • Solution: find looser bound that considers subset 9 10 Summary of Technical Approach Considering a Subset of Sequences • Full bound with all terms: ∑∑ ≤ 1. Express future mode sequences as multiple ( ) ( ) P loss F u − ij 0 : T 1 known time-varying models � > i j i • Individual terms can be replaced by looser bound: = ≥ 1 / 2 1 / 2 2. Bound p(loss) using Multiple-Model bound from ( ) ( ) ( ) G P H P H F u − ij i j ij 0 : T 1 (Blackmore06) � • These terms do not depend on u 0:T-1 – Need not be considered in optimization 3. Make bound tractable using efficient pruning • Challenge: approach – Replace terms so resulting bound is as tight as possible ∑ ∑ ∑ ∑ ≤ + ( ) ( ) P loss F G u − ij 0 : T 1 ij 4. Minimize bound using constrained optimization ∈ > ∈ ∉ > ∉ i S j i , j S i S j i , j S 11 12 2

1/31/2007 Considering a Subset of Sequences Considering a Subset of Sequences • Worst-case difference between F ij ( u 0:T-1 ) and G ij is: • Idea: Use graph search to find highest priors 1 / 2 1 / 2 P ( H ) P ( H ) i j • Include in S the mode sequences that maximize: ∑ ∑ 1 / 2 1 / 2 P ( H ) P ( H ) i j ∈ > ∈ , i S j i j S • S must include hypotheses with highest prior p(H i ) • Challenge: efficient enumeration of mode x d,0 x d,1 x d,2 x d,3 … x d,h-1 x d,h h sequences with highest prior probability ∏ = max p ( x : x ) p ( x ) T d , 0 d , t d , 0 x x − d , i d , i 1 = i 1 13 14 Summary of Technical Approach Optimization: Constraints • As in many trajectory design problems, we may want to: 1. Express future mode sequences as multiple i = known time-varying models � – Ensure fulfillment of task defined in terms of E [ x ] x task expected state i ≤ E [ x ] x 2. Bound p(loss) using Multiple-Model bound from max – Bound expected state of the system (Blackmore06) � i ≤ u u max – Model actuator saturation k ∑ i ≤ 3. Make bound tractable using efficient pruning u fuel – Restrict total fuel usage approach � = i 1 • All of these are linear constraints 4. Minimize bound using constrained optimization 15 16 Optimization: Overall Formulation Summary of Technical Approach • Resulting nonlinear optimization Cost function that is nonlinear, nonconvex 1. 1. Express future mode sequences as multiple ∑∑ − known time-varying models � ≤ k ( i , j ) P ( loss | ) P ( H ) P ( H ) e u i j > i j i 2. Constraints that are linear in the control inputs 2. Bound p(loss) using Multiple-Model bound from – E.g. k ≤ ≤ ∀ (Blackmore06) � E [ x ] x u u i max i max • Can solve using Sequential Quadratic Programming 3. Make bound tractable using efficient pruning approach � – Local optimality • Now constrained active hybrid estimation possible: 4. Minimize bound using constrained optimization � – Use constraints for control, optimization for discrimination 17 18 3

1/31/2007 Simulation Results – Active Approach Results: Box-Constrained Maneuver L7 • Satellite dynamics linearized about nominal circular orbit (Hill’s equations) 60 40 • Motion in two dimensions considered (in-track and radial) Displacement(m) 20 0 • Sensors: −20 In−track Radial – Radial and in-track velocity −40 −60 0 100 200 300 400 500 600 • Actuators – Radial and in-track thrusters 10 Change in Velocity(mm/s) • Hybrid model has 4 discrete modes: 5 Mode 0: Nominal (no faults) 0 Mode 1: Radial velocity sensor failure (zero mean noise observed) In−track Mode 2: In-track velocity sensor failure (zero mean noise observed) −5 Radial Mode 3: Radial thruster failure (no response) −10 0 100 200 300 400 500 600 • Horizon of 10 time steps, dt = 60s Time(s) ≤ p ( loss ) 0 . 12 19 20 Results: Box-Constrained Maneuver Results: Displacement Maneuver Discrimination-optimal Maneuver Discrimination−Optimal Maneuver 200 In−track Radial 150 Displacement(m) 1.4 100 ≤ Bound on Probability of Pruning p ( loss ) 0 . 10 1.2 50 0 1 −50 0.8 −100 0 100 200 300 400 500 600 0.6 Fuel-optimal Maneuver 0.4 Fuel−Optimal Maneuver 200 In−track Radial 0.2 150 Displacement(m) 0 100 ≤ 0 20 40 60 80 100 120 140 160 180 200 ( ) 0 . 87 p loss Size of Box Constraint (m) 50 0 −50 −100 0 100 200 300 400 500 600 21 Time(s) 22 Conclusion Questions? • A novel approach for active hybrid estimation – Minimize upper bound on probability of losing true mode sequence, subject to constraints on inputs and state 23 24 4

Slide 20 L7 mention constraints explicitly Lars, 12/8/2005

1/31/2007 Summary of Approach 1. Hybrid Estimation calculates approximate belief state – Distribution over k mode sequence – Continuous distribution conditioned on mode sequence 2. Best first search enumerates s most likely future mode sequences 3. Form cost function with s most likely sequences 4. Optimize subject to constraints, using SQP 5. Execute control inputs, while estimating hybrid state 25 5