The Problem of Output Measurement Feedback Control Under Set-valued - PowerPoint PPT Presentation

The Problem of Output Measurement Feedback Control Under Set-valued Uncertainty : from Theory to Computation A.B.KURZHANSKI (Moscow State Univ. and Univ. of California at Berkeley) Presentation at 44-th IEEE CDC and 28-th Chinese National Control Conference Shanghai, China, December 17, 2009 1

OUTLINE 1. Motivations 2. The Basic Problem. The Separation Property. 2. The GSE Problem of Guaranteed (Set-Membership) State Estimation 3. The GCS Problem OF Guaranteed Control Synthesis 4. Combination of GSE AND GCS: the Solution Strategy 5. Systems with Linear Structure: the system and its reconfiguration 6. Linear Systems : the Solution Scheme, reduction to finite-dimensions 7. Calculation: the Ellipsoidal and Polyhedral Techniques 8. Conclusion 2

MOTIVATIONS 3

flow, 0 ≤ v ≤ v t starting point end point measurements reachability set 4

Team Control Synthesis Complete measurements Container M W [ · ] W [ t 0 ] = W ( t 0 , t 1 , M ) Safety set � � x ( i ) ( t ) Safety zone B r 5

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The System Equations and the Uncertainties The uncertain system : dx dt = f 1 ( t , x , u )+ f 2 ( t , x , v ) , x ∈ R n , t ∈ [ t 0 , ϑ ] (1) with continuous right-hand sides satisfying conditions of uniqueness an extendibility of solutions. hard bounds on control u and unknown disturbance v ( t ) : u ∈ P ( t ) , v ( t ) ∈ Q ( t ) , (2) P ( t ) , Q ( t ) — compact sets in R p , R q , Hausdorff-continuous. 7

Measurement equation: y ( t ) = h ( t , x )+ ξ ( t ) , y ∈ R m , (3) measurements — y ( t ) , t ∈ T – (continuous or discrete) disturbance in measurement ξ ( t ) — unknown but bounded: ξ ( t ) ∈ R ( t ) , t ∈ [ t 0 , ϑ ] , (4) R ( t ) — similar to P ( t ) , h ( t , x ) — continuous. Initial condition: x ( t 0 ) ∈ X 0 , (5) X 0 — compact. Starting Position: { t 0 , X 0 } 8

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BASIC PROBLEM STEER SYSTEM dx dt = f 1 ( t , x , u )+ f 2 ( t , x , v ) , x ∈ R n , t ∈ [ t 0 , ϑ ] , ( 1 ) y ( t ) = h ( t , x )+ ξ ( t ) , ; y ∈ R m , ( 3 ) from starting position { t 0 , X 0 } to terminal position { ϑ , M } , by feedback control strategy U ( t , · ) , on the basis of available information: - system model : equations (1), (3), - starting position { t 0 , X 0 } , - available measurement y ( t ) , - given constraints on control u and uncertain disturbance inputs v ( t ) , ξ ( t ) 10

What should the NEW STATE of the SYSTEM be ? *** Classical case under complete information: Position (state) – { t , x } – single valued Closed-loop control : { u ( t , x ) } Trajectories – single-valued : x [ t ] = x ( t , t 0 , x 0 ) . *** Output feedback control under incomplete information: with – set-valued bounds (no statistical data available): Position (state) – set-valued: X [ t ] 11

On-line set-valued position (NEW STATE) of the system may be taken as: * { t , y t ( · ) } — memorize measurements, (in stochastic control this is done through observers and filters (Kalman)) ** { t , X [ t ] } — find set-valued information set consistent with measurements and constraints on uncertain items: find set-valued information tubes *** { t , V ( t , · ) } — find information state – function V ( t , x ) such that X [ t ] = { x : V ( t , x ) ≤ α } is the level set of V ( t , x ) , (found through Hamilton-Jacobi-Bellman (HJB) PDE equations). 12

Guaranteed State Estimation under Set-membership noise measurements 1 2 3 X ( τ ) information set t = t 0 t = τ Open-loop reachability tube 13

Problem I of Measurement Output Feedback Control : Specify feedback strategy (closed-loop controls) U ( t , X [ t ]) or U ( t , V ( t , · )) which steers overall system FROM any starting position { τ , X [ τ ] } , τ ∈ [ t 0 , ϑ ] TO given neighborhood M µ of target set M at time ϑ : { τ , X [ τ ] } → { ϑ , X [ τ ] } , X [ ϑ ] ⊆ M µ despite unknown disturbances and incomplete measurements. ATTENTION for MATHEMATICIANS: U = { U ( t , X [ t ]) } must ensure the existence and extendability of solutions to differential inclusion x ∈ f 1 ( t , x , U ( t , X [ t ]))+ f 2 ( t , x , v ) , ˙ within interval t ∈ [ t 0 , ϑ ] , whatever be v ( t ) . 14

(Measurement) Output Feedback Control Closed-loop (feedback) control strategies: U ( t , X ) , U ( t , V ( t , · )) , with state { t , X } , or { t , V ( t , · ) } , and trajectories –set-valued: X [ t ] = X ( t , t 0 , X 0 ) or single valued x [ t ] , with set-valued error-bound R [ t ] , with state { t , x [ t ] , Ω [ t ] } (external estimate E [ t ] ⊇ R [ t ] ). trajectories x [ t ] = x ( t , t 0 , x 0 ) , error bounds Ω [ t ] = Ω ( t , t 0 , X 0 − x 0 ) . 15

REMARK: Problem I may be separated into: Problem GSE of guaranteed state estimation(finite-dimensional) and Problem GCS of guaranteed control synthesis (infinite-dimensional) OUR AIM : (a) Find possibility of solutions while avoiding infinite-dimensional schemes. (b) Design feasible computational methods. 16

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SOLUTION METHODS (a) GENERAL METHOD: the HAMILTON-JACOBI-BELLMAN (HJB) EQUATIONS (b) USING INVARIANT SETS and AIMING METHODS SET-VALUED CALCULUS+ NONLINEAR ANALYSIS FOR LINEAR SYSTEMS: CONVEX ANALYSIS (c) THE H -INFINITY APPROACH (d) APPROXIMATE METHODS: THE COMPARISON PRINCIPLE, DISCRETIZATION METHODS (e) COMPUTATION METHODS FOR LINEAR SYSTEMS: ELLIPSOIDAL CALCULUS, POLYHEDRAL CALCULUS or BOTH (f) INTERTWINING THE ABOVE METHODS 19

Problem GSE of Guaranteed State Estimation The One-Stage Problem NOTE THAT THERE IS WORST CASE NOISE and BEST CASE NOISE 20

y = x + ξ , ξ ∈ R R noise bound worst case routine best case c ∗ 2 c ∗ 2 c ∗ 1 c ∗ 1 21

Examples: nonlinear maps   x ( k + 1 ) = f ( x ( k )) Gx ( k + 1 )+ ξ y ( k + 1 ) =    x 2  ax 1 2 Take x ∈ R 2 , f ( x ) =  , ax 2 x 2 1 X ( k ) = { x ∈ R 2 : | x i | ≤ 1; i = 1 , 2 } , y ( k + 1 ) = x 2 ( k )+ ξ , | ξ | ≤ µ , X Y ( k + 1 ) = { x : x ∈ [ y ( k + 1 )+ µ , y ( k + 1 ) − µ ] } , X ( k + 1 ) = f ( X ( k )) ∩ X Y ( k + 1 ) X Y X ( k ) f ( X ( k )) 22

Nonlinear Examples   x ( k + 1 ) = f ( x ( k )) | ξ | ≤ ε a 2 x 2 1 + b 2 x 2 y ( k + 1 ) = 2 + ξ  X ( k ) = { x ∈ R 2 : | x i | ≤ 1; i = 1 , 2 } X Y ( k + 1 ) = { x : x ∈ [ y ( k + 1 ) − ε , y ( k + 1 )+ ε ] } X ( k + 1 ) = f ( X ( k )) ∩ X Y ( k + 1 ) X ( k + 1 ) disconnected 23

convex hull 24

Unkown but bounded noise (i)Measurements – at given time (continuous or discrete). Noise – unknown, with given bounds. Has a worst case when W [ t ] is largest possible and a best case when W [ t ] may even reduce to a point (ii) Measurements arrive at random instants of time, due to distribution of Poisson. Noise - with given bounds and given probabilistic density. With stochastic noise the worst and best cases arrive with probability zero. The statistical estimates of x are consistent. 25

The Dynamics of the Information Set t ∗ and t ∗ are the instants of discrete observations 26

Problem GSE of Guaranteed (“Minmax”) State Estimation Problem GSE may be formulated in two versions - E 1 and E 2 Problem E 1 : Given are equations dx dt = f 1 ( t , x , u )+ f 2 ( t , x , v ) , y ( t ) = h ( t , x )+ ξ ( t ) ( i ) position { t 0 , X 0 } , used control u [ s ] , s ∈ [ t 0 , τ ) , measurement y = y ∗ ( t ) , t ∈ [ t 0 , τ ] , and constraints u ∈ P , v ∈ Q , ξ ∈ R ( ii ) with P , Q , R given. Specify information set X [ τ ] , of solutions x ( τ ) to system (i), consistent with system equations, measurement y ∗ ( t ) , t ∈ [ t 0 , τ ] and constraints (ii). The information set X [ τ ] is the guaranteed estimate of x ( τ ) . 27

v d ( x ( t 0 ) , X 0 ) > 0 min V ( τ , x ) > 0 x ( τ ) �∈ X [ τ ] X 0 x ( τ ) ∈ X [ τ ] V ( τ , x ) = 0 v d ( x ( t 0 ) , X 0 ) = 0 min v d ( x ( t 0 ) , X 0 ) V ( τ , x ) = min 28

It is necessary not only to calculate set X [ τ ] , but to arrange on-line calculations , following the evolution of X [ t ] in time.!!! This leads to the problem of DYNAMIC OPTIMIZATION: Problem E 2 Given starting position { t 0 , X 0 } , and realization y ∗ ( s ) , s ∈ [ t 0 , τ ] , Find value function: v { d ( x ( t 0 ) , X 0 ) | v ( t ) ∈ Q ( t ) , t ∈ [ t 0 , τ ] } V ( τ , x ) = min due to equation (1), under additional conditions x ( τ ) = x ; y ∗ ( s ) − h ( s , x ( s )) ∈ R ( s ) , s ∈ [ t 0 , τ ] . The last condition is actually an on-line state constraint 29

The following relation is true X [ t ] = { x : V ( t , x ) ≤ 0 } !!! The value function V ( t , x ) may be found by solving an HJB equation! Introduce notation V ( τ , x ) = V ( τ , x | V ( t 0 , · )) , Then the principle of optimality for problem GSE reads: V ( τ , x | V ( t 0 , · )) = V ( τ , x | V ( t , ·| V ( t 0 , · ))) , t 0 ≤ t ≤ τ . ( ! ) This allows to derive an HJB (Dynamic Programming) equation, to calculate V ( t , x ) . 30

The HJB equation: �� ∂ V � ∂ V ∂ x , f 1 ( t , x , u ∗ ( t ))+ f 2 ( t , x , v ) ∂ t + max − v � � � − d 2 ( y ∗ ( t ) − h ( t , x ) , R ( t )) � v ( t ) ∈ Q ( t ) = 0 , � under boundary condition V ( t 0 , x ) = d 2 ( x , X 0 ) . Discretized scheme: X [ t + σ ] ∼ X [ t + σ − 0 ] ∩ Y ( t + σ ) 31

The Dynamics of the Information Set t ∗ and t ∗ are the instants of discrete observations 32