Guaranteed Control of Switched Control Systems Using Model Order - PowerPoint PPT Presentation

Sampled Switched Systems Example: Two-room apartment ˙ � T 1 � � � T 1 � � � � − α 21 − α e 1 − α f u α e 1 T e + α f T f u α 21 = + . − α 12 − α e 2 α 12 α e 2 T e T 2 T 2 Modes: u = 0 , 1 ; sampling period τ A pattern π is a finite sequence of modes (e.g. (1 · 0 · 0 · 0)) A state dependent control consists to select at each τ a mode (or a pattern) according to the current value of the state. NB: Each mode has its basic proper equilibrium point; by appropriate switching, one can drive the system to a specific stability zone A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 9 / 42

Sampled Switched Systems Safety and Stability Properties for the two-room apartment A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 10 / 42

Sampled Switched Systems Safety and Stability Properties for the two-room apartment Example of safety property to be checked: satisfactory temperature ∀ t ≥ 0 : T min ≤ T i ( t ) ≤ T max A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 10 / 42

Sampled Switched Systems Safety and Stability Properties for the two-room apartment Example of safety property to be checked: satisfactory temperature ∀ t ≥ 0 : T min ≤ T i ( t ) ≤ T max Example of stability property to be checked: temperature regulation | T i ( t ) − T reference | ≤ ε as t → ∞ A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 10 / 42

Sampled Switched Systems Affine Sampled Switched Systems (cont’d) We introduce the transition relation → u τ to denote the point reached at time t under mode u from initial condition x , A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 11 / 42

Sampled Switched Systems Affine Sampled Switched Systems (cont’d) We introduce the transition relation → u τ to denote the point reached at time t under mode u from initial condition x , defined by: x ′ = C u x + d u τ x ′ x → u iff A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 11 / 42

Sampled Switched Systems Affine Sampled Switched Systems (cont’d) We introduce the transition relation → u τ to denote the point reached at time t under mode u from initial condition x , defined by: x ′ = C u x + d u τ x ′ x → u iff with C u = e A u τ A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 11 / 42

Sampled Switched Systems Affine Sampled Switched Systems (cont’d) We introduce the transition relation → u τ to denote the point reached at time t under mode u from initial condition x , defined by: x ′ = C u x + d u τ x ′ x → u iff with C u = e A u τ � τ o e A u ( τ − t ) b u dt = ( e A u τ − I ) A − 1 and d u = u b u A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 11 / 42

Sampled Switched Systems Affine Sampled Switched Systems (cont’d) We introduce the transition relation → u τ to denote the point reached at time t under mode u from initial condition x , defined by: x ′ = C u x + d u τ x ′ x → u iff with C u = e A u τ � τ o e A u ( τ − t ) b u dt = ( e A u τ − I ) A − 1 and d u = u b u A sampled switched system can thus be viewed as a piecewise affine discrete-time system. A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 11 / 42

Sampled Switched Systems Post Set Operators Post u ( X ) = { x ′ | x → u τ x ′ for some x ∈ X } A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 12 / 42

Sampled Switched Systems Post Set Operators Post u ( X ) = { x ′ | x → u τ x ′ for some x ∈ X } Post π ( X ) = { x ′ | x → u 1 x ′ for some x ∈ X } τ · · · → u m τ if π is a pattern of the form ( u 1 · · · u m ) A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 12 / 42

Sampled Switched Systems Post Set Operators Post u ( X ) = { x ′ | x → u τ x ′ for some x ∈ X } Post π ( X ) = { x ′ | x → u 1 x ′ for some x ∈ X } τ · · · → u m τ if π is a pattern of the form ( u 1 · · · u m ) The unfolding of Post π ( X ) is the union of X , Post π ( X ) and the intermediate sets: X ∪ Post u 1 ( X ) ∪ Post u 1 · u 2 ( X ) ∪ · · · ∪ Post u 1 ··· u m − 1 ( X ) ∪ Post π ( X ) A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 12 / 42

State Space Decomposition Outline 1 Sampled Switched Systems 2 State Space Decomposition 3 Decomposition for Sampled Switched Systems with Output 4 Model Order Reduction and error bounding 5 Reduced Order Control A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 13 / 42

State Space Decomposition Bisection Method A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 14 / 42

State Space Decomposition Bisection Method Given a zone R (selected around a reference point Ω) Look for a pattern which maps R into R A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 14 / 42

State Space Decomposition Bisection Method Given a zone R (selected around a reference point Ω) Look for a pattern which maps R into R If such a pattern exists, then uniform control over the whole R A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 14 / 42

State Space Decomposition Bisection Method Given a zone R (selected around a reference point Ω) Look for a pattern which maps R into R If such a pattern exists, then uniform control over the whole R Otherwise, A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 14 / 42

State Space Decomposition Bisection Method Given a zone R (selected around a reference point Ω) Look for a pattern which maps R into R If such a pattern exists, then uniform control over the whole R Otherwise, bisect of R into subparts, and search for patterns mapping these subparts into R A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 14 / 42

State Space Decomposition Bisection Method Given a zone R (selected around a reference point Ω) Look for a pattern which maps R into R If such a pattern exists, then uniform control over the whole R Otherwise, bisect of R into subparts, and search for patterns mapping these subparts into R In case of failure, iterate the bisection A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 14 / 42

State Space Decomposition Bisection Method Given a zone R (selected around a reference point Ω) Look for a pattern which maps R into R If such a pattern exists, then uniform control over the whole R Otherwise, bisect of R into subparts, and search for patterns mapping these subparts into R In case of failure, iterate the bisection Extension for safety: the unfolding must stay in the safety set S . A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 14 / 42

State Space Decomposition definition A decomposition ∆ of R is a set of couples { ( V i , π i ) } i ∈ I such that: A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 15 / 42

State Space Decomposition definition A decomposition ∆ of R is a set of couples { ( V i , π i ) } i ∈ I such that: � i ∈ I V i = R A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 15 / 42

State Space Decomposition definition A decomposition ∆ of R is a set of couples { ( V i , π i ) } i ∈ I such that: � i ∈ I V i = R ∀ i ∈ I Post π i ( V i ) ⊆ R A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 15 / 42

State Space Decomposition definition A decomposition ∆ of R is a set of couples { ( V i , π i ) } i ∈ I such that: � i ∈ I V i = R ∀ i ∈ I Post π i ( V i ) ⊆ R (Extension for safety: and ∀ i ∈ I Unf π i ( V i ) ⊆ S ). A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 15 / 42

State Space Decomposition definition A decomposition ∆ of R is a set of couples { ( V i , π i ) } i ∈ I such that: � i ∈ I V i = R ∀ i ∈ I Post π i ( V i ) ⊆ R (Extension for safety: and ∀ i ∈ I Unf π i ( V i ) ⊆ S ). A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 15 / 42

State Space Decomposition definition A decomposition ∆ of R is a set of couples { ( V i , π i ) } i ∈ I such that: � i ∈ I V i = R ∀ i ∈ I Post π i ( V i ) ⊆ R (Extension for safety: and ∀ i ∈ I Unf π i ( V i ) ⊆ S ). definition and property Let Post ∆ ( X ) = def � i ∈ I Post π i ( X ∩ V i ). A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 15 / 42

State Space Decomposition definition A decomposition ∆ of R is a set of couples { ( V i , π i ) } i ∈ I such that: � i ∈ I V i = R ∀ i ∈ I Post π i ( V i ) ⊆ R (Extension for safety: and ∀ i ∈ I Unf π i ( V i ) ⊆ S ). definition and property Let Post ∆ ( X ) = def � i ∈ I Post π i ( X ∩ V i ). We have: Post ∆ ( R ) ⊆ R A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 15 / 42

State Space Decomposition definition A decomposition ∆ of R is a set of couples { ( V i , π i ) } i ∈ I such that: � i ∈ I V i = R ∀ i ∈ I Post π i ( V i ) ⊆ R (Extension for safety: and ∀ i ∈ I Unf π i ( V i ) ⊆ S ). definition and property Let Post ∆ ( X ) = def � i ∈ I Post π i ( X ∩ V i ). We have: Post ∆ ( R ) ⊆ R (and Unf ∆ ( R ) ⊆ S ) A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 15 / 42 .

State Space Decomposition Control and Trajectories Induced by ∆ The decomposition ∆ = { ( V i , π i ) } i ∈ I induces a natural control: A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 16 / 42

State Space Decomposition Control and Trajectories Induced by ∆ The decomposition ∆ = { ( V i , π i ) } i ∈ I induces a natural control: 1 x ( t ) ∈ R , therefore ∃ i ∈ I such that x ( t ) ∈ V i A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 16 / 42

State Space Decomposition Control and Trajectories Induced by ∆ The decomposition ∆ = { ( V i , π i ) } i ∈ I induces a natural control: 1 x ( t ) ∈ R , therefore ∃ i ∈ I such that x ( t ) ∈ V i 2 Apply pattern π i to x ( t ) A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 16 / 42

State Space Decomposition Control and Trajectories Induced by ∆ The decomposition ∆ = { ( V i , π i ) } i ∈ I induces a natural control: 1 x ( t ) ∈ R , therefore ∃ i ∈ I such that x ( t ) ∈ V i 2 Apply pattern π i to x ( t ) 3 At the end of π i , x ( t ′ ) ∈ R , iterate by going back to step (1) A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 16 / 42

State Space Decomposition Control and Trajectories Induced by ∆ The decomposition ∆ = { ( V i , π i ) } i ∈ I induces a natural control: 1 x ( t ) ∈ R , therefore ∃ i ∈ I such that x ( t ) ∈ V i 2 Apply pattern π i to x ( t ) 3 At the end of π i , x ( t ′ ) ∈ R , iterate by going back to step (1) Property Under the ∆-control, any trajectory x 0 → π i 1 x 1 → π i 2 x 2 → π i 3 · · · always stays in R A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 16 / 42

State Space Decomposition Control and Trajectories Induced by ∆ The decomposition ∆ = { ( V i , π i ) } i ∈ I induces a natural control: 1 x ( t ) ∈ R , therefore ∃ i ∈ I such that x ( t ) ∈ V i 2 Apply pattern π i to x ( t ) 3 At the end of π i , x ( t ′ ) ∈ R , iterate by going back to step (1) Property Under the ∆-control, any trajectory x 0 → π i 1 x 1 → π i 2 x 2 → π i 3 · · · always stays in R The unfolding of the trajectory always stays in S A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 16 / 42

State Space Decomposition Decomposition for the two-room apartment For: α 12 = 5 × 10 − 2 , α 21 = 5 × 10 − 2 , α e 1 = 5 × 10 − 3 , α e 2 = 3 . 3 × 10 − 3 , α f = 8 . 3 × 10 − 3 , T e = 10 , T f = 50 and τ = 5. A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 17 / 42

State Space Decomposition Decomposition for the two-room apartment For: α 12 = 5 × 10 − 2 , α 21 = 5 × 10 − 2 , α e 1 = 5 × 10 − 3 , α e 2 = 3 . 3 × 10 − 3 , α f = 8 . 3 × 10 − 3 , T e = 10 , T f = 50 and τ = 5. Ω = (21 , 21), R = [20 . 25 , 21 . 75] × [20 . 25 , 21 . 75], S = [20 , 22] × [20 , 22] A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 17 / 42

State Space Decomposition Decomposition for the two-room apartment For: α 12 = 5 × 10 − 2 , α 21 = 5 × 10 − 2 , α e 1 = 5 × 10 − 3 , α e 2 = 3 . 3 × 10 − 3 , α f = 8 . 3 × 10 − 3 , T e = 10 , T f = 50 and τ = 5. Ω = (21 , 21), R = [20 . 25 , 21 . 75] × [20 . 25 , 21 . 75], S = [20 , 22] × [20 , 22] A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 17 / 42

State Space Decomposition Decomposition for the two-room apartment For: α 12 = 5 × 10 − 2 , α 21 = 5 × 10 − 2 , α e 1 = 5 × 10 − 3 , α e 2 = 3 . 3 × 10 − 3 , α f = 8 . 3 × 10 − 3 , T e = 10 , T f = 50 and τ = 5. Ω = (21 , 21), R = [20 . 25 , 21 . 75] × [20 . 25 , 21 . 75], S = [20 , 22] × [20 , 22] A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 17 / 42

State Space Decomposition Decomposition for the two-room apartment For: α 12 = 5 × 10 − 2 , α 21 = 5 × 10 − 2 , α e 1 = 5 × 10 − 3 , α e 2 = 3 . 3 × 10 − 3 , α f = 8 . 3 × 10 − 3 , T e = 10 , T f = 50 and τ = 5. Ω = (21 , 21), R = [20 . 25 , 21 . 75] × [20 . 25 , 21 . 75], S = [20 , 22] × [20 , 22] A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 17 / 42

State Space Decomposition Decomposition for the two-room apartment For: α 12 = 5 × 10 − 2 , α 21 = 5 × 10 − 2 , α e 1 = 5 × 10 − 3 , α e 2 = 3 . 3 × 10 − 3 , α f = 8 . 3 × 10 − 3 , T e = 10 , T f = 50 and τ = 5. Ω = (21 , 21), R = [20 . 25 , 21 . 75] × [20 . 25 , 21 . 75], S = [20 , 22] × [20 , 22] Figure : Decomposition (left) ; unfolding (middle) ; unfolded trajectory (right) in plane ( T 1 , T 2 ) A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 17 / 42

State Space Decomposition Decomposition for the two-room apartment For: α 12 = 5 × 10 − 2 , α 21 = 5 × 10 − 2 , α e 1 = 5 × 10 − 3 , α e 2 = 3 . 3 × 10 − 3 , α f = 8 . 3 × 10 − 3 , T e = 10 , T f = 50 and τ = 5. Ω = (21 , 21), R = [20 . 25 , 21 . 75] × [20 . 25 , 21 . 75], S = [20 , 22] × [20 , 22] Figure : Decomposition (left) ; unfolding (middle) ; unfolded trajectory (right) in plane ( T 1 , T 2 ) Decomposition found for k = 4 , d = 3. A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 17 / 42

Decomposition for S.S.S. with Output Outline 1 Sampled Switched Systems 2 State Space Decomposition 3 Decomposition for Sampled Switched Systems with Output 4 Model Order Reduction and error bounding 5 Reduced Order Control A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 18 / 42

Decomposition for S.S.S. with Output A Sampled Switched System with Output Described by the differential equation: � ˙ x ( t ) = Ax ( t ) + Bu ( t ) y ( t ) = Cx ( t ) Constraint: x of “high” dimension. A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 19 / 42

Decomposition for S.S.S. with Output A Sampled Switched System with Output Described by the differential equation: � ˙ x ( t ) = Ax ( t ) + Bu ( t ) y ( t ) = Cx ( t ) x ∈ R n : state variable y ∈ R m : output u ∈ R p : control input, takes a finite number of values (modes) A , B , C : matrices of appropriate dimensions Constraint: x of “high” dimension. A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 19 / 42

Decomposition for S.S.S. with Output A Sampled Switched System with Output Described by the differential equation: � ˙ x ( t ) = Ax ( t ) + Bu ( t ) y ( t ) = Cx ( t ) x ∈ R n : state variable y ∈ R m : output u ∈ R p : control input, takes a finite number of values (modes) A , B , C : matrices of appropriate dimensions Idea: impose the right u ( t ) such that x and y verify some properties (stability, reachability...) Constraint: x of “high” dimension. A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 19 / 42

Decomposition for S.S.S. with Output A Sampled Switched System with Output Described by the differential equation: � ˙ x ( t ) = Ax ( t ) + Bu ( t ) y ( t ) = Cx ( t ) x ∈ R n : state variable y ∈ R m : output u ∈ R p : control input, takes a finite number of values (modes) A , B , C : matrices of appropriate dimensions Idea: impose the right u ( t ) such that x and y verify some properties (stability, reachability...) Objectives: 1 x-stabilization : make all the state trajectories starting in a compact interest set R x ⊂ R n return to R x ; 2 y-convergence : send the output of all the trajectories starting in R x into an objective set R y ⊂ R m ; Constraint: x of “high” dimension. A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 19 / 42

Decomposition for S.S.S. with Output A Sampled Switched System with Output A distillation column A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 20 / 42

Decomposition for S.S.S. with Output Output Post Set Operators Post u,C ( X ) = { y = Cx ′ | x → u τ x ′ for some x ∈ X } A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 21 / 42

Decomposition for S.S.S. with Output Output Post Set Operators Post u,C ( X ) = { y = Cx ′ | x → u τ x ′ for some x ∈ X } Post Pat,C ( X ) = { y = Cx ′ | x → u 1 x ′ for some x ∈ X } τ · · · → u m τ if Pat is a pattern of the form ( u 1 · · · u m ) A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 21 / 42

Decomposition for S.S.S. with Output New Decomposition definition A decomposition ∆ of R x is a set of couples { ( V i , Pat i ) } i ∈ I such that: A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 22 / 42

Decomposition for S.S.S. with Output New Decomposition definition A decomposition ∆ of R x is a set of couples { ( V i , Pat i ) } i ∈ I such that: � i ∈ I V i = R x A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 22 / 42

Decomposition for S.S.S. with Output New Decomposition definition A decomposition ∆ of R x is a set of couples { ( V i , Pat i ) } i ∈ I such that: � i ∈ I V i = R x ∀ i ∈ I Post Pat i ( V i ) ⊆ R x ( x -stabilization) A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 22 / 42

Decomposition for S.S.S. with Output New Decomposition definition A decomposition ∆ of R x is a set of couples { ( V i , Pat i ) } i ∈ I such that: � i ∈ I V i = R x ∀ i ∈ I Post Pat i ( V i ) ⊆ R x ( x -stabilization) ∀ i ∈ I Post Pat i ,C ( V i ) ⊆ R y ( y -convergence) A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 22 / 42

Decomposition for S.S.S. with Output New Decomposition definition A decomposition ∆ of R x is a set of couples { ( V i , Pat i ) } i ∈ I such that: � i ∈ I V i = R x ∀ i ∈ I Post Pat i ( V i ) ⊆ R x ( x -stabilization) ∀ i ∈ I Post Pat i ,C ( V i ) ⊆ R y ( y -convergence) definition and property � Let Post ∆ ( X ) = def i ∈ I Post π i ( X ∩ V i ). A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 22 / 42

Decomposition for S.S.S. with Output New Decomposition definition A decomposition ∆ of R x is a set of couples { ( V i , Pat i ) } i ∈ I such that: � i ∈ I V i = R x ∀ i ∈ I Post Pat i ( V i ) ⊆ R x ( x -stabilization) ∀ i ∈ I Post Pat i ,C ( V i ) ⊆ R y ( y -convergence) definition and property � Let Post ∆ ( X ) = def i ∈ I Post π i ( X ∩ V i ). We have: Post ∆ ( R x ) ⊆ R x and Post ∆ ,C ( R x ) ⊆ R y . A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 22 / 42

Decomposition for S.S.S. with Output New Decomposition definition A decomposition ∆ of R x is a set of couples { ( V i , Pat i ) } i ∈ I such that: � i ∈ I V i = R x ∀ i ∈ I Post Pat i ( V i ) ⊆ R x ( x -stabilization) ∀ i ∈ I Post Pat i ,C ( V i ) ⊆ R y ( y -convergence) definition and property � Let Post ∆ ( X ) = def i ∈ I Post π i ( X ∩ V i ). We have: Post ∆ ( R x ) ⊆ R x and Post ∆ ,C ( R x ) ⊆ R y . Computational cost of decomposition: at most in O (2 nd N k ). A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 22 / 42

Model Order Reduction and error bounding Outline 1 Sampled Switched Systems 2 State Space Decomposition 3 Decomposition for Sampled Switched Systems with Output 4 Model Order Reduction and error bounding 5 Reduced Order Control A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 23 / 42

Model Order Reduction and error bounding Model Order Reduction by Projection Construction of a reduced order system ˆ Σ of order n r : � ˙ = ˆ x ( t ) + ˆ ˆ x ( t ) A ˆ Bu ( t ) , ˆ Σ : = ˆ y r ( t ) C ˆ x ( t ) . A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 24 / 42

Model Order Reduction and error bounding Model Order Reduction by Projection Construction of a reduced order system ˆ Σ of order n r : � ˙ = ˆ x ( t ) + ˆ ˆ x ( t ) A ˆ Bu ( t ) , ˆ Σ : = ˆ y r ( t ) C ˆ x ( t ) . Reduction by a projection (constructed by balanced truncation) π = π L π R , π L ∈ R n × n r , π R ∈ R n r × n : ˆ ˆ ˆ A = π R Aπ L , B = π R B, C = Cπ L . A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 24 / 42

Model Order Reduction and error bounding Model Order Reduction by Projection Construction of a reduced order system ˆ Σ of order n r : � ˙ = ˆ x ( t ) + ˆ ˆ x ( t ) A ˆ Bu ( t ) , ˆ Σ : = ˆ y r ( t ) C ˆ x ( t ) . Reduction by a projection (constructed by balanced truncation) π = π L π R , π L ∈ R n × n r , π R ∈ R n r × n : ˆ ˆ ˆ A = π R Aπ L , B = π R B, C = Cπ L . Goal: design a controle rule u ( · ) at the low-order level and apply it at the full-order level. A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 24 / 42

Model Order Reduction and error bounding Model Order Reduction by Projection Construction of a reduced order system ˆ Σ of order n r : � ˙ = ˆ x ( t ) + ˆ x ( t ) ˆ A ˆ Bu ( t ) , ˆ Σ : = ˆ y r ( t ) C ˆ x ( t ) . Reduction by a projection (constructed by balanced truncation) π = π L π R , π L ∈ R n × n r , π R ∈ R n r × n : ˆ ˆ ˆ A = π R Aπ L , B = π R B, C = Cπ L . Goal: design a controle rule u ( · ) at the low-order level and apply it at the full-order level. Requirements: projection of the interest set ˆ R x = π R R x A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 24 / 42

Model Order Reduction and error bounding Model Order Reduction by Projection Construction of a reduced order system ˆ Σ of order n r : � ˙ = ˆ x ( t ) + ˆ ˆ x ( t ) A ˆ Bu ( t ) , ˆ Σ : = ˆ y r ( t ) C ˆ x ( t ) . Reduction by a projection (constructed by balanced truncation) π = π L π R , π L ∈ R n × n r , π R ∈ R n r × n : ˆ ˆ ˆ A = π R Aπ L , B = π R B, C = Cπ L . Goal: design a controle rule u ( · ) at the low-order level and apply it at the full-order level. Requirements: projection of the interest set ˆ R x = π R R x error bounding of the state and output trajectory A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 24 / 42

Model Order Reduction and error bounding Output trajectory error [4] Defined by (for a pattern Pat ): e y ( | Pat | τ ) = � CPost Pat ( x ) − ˆ CPost Pat ( π R x ) � A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 25 / 42

Model Order Reduction and error bounding Output trajectory error [4] Defined by (for a pattern Pat ): e y ( | Pat | τ ) = � CPost Pat ( x ) − ˆ CPost Pat ( π R x ) � For a pattern of length j , bounded by ε j y : e y ( jτ ) ≤ ε j ∀ t = jτ > 0 , y A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 25 / 42

Model Order Reduction and error bounding Output trajectory error [4] Defined by (for a pattern Pat ): e y ( | Pat | τ ) = � CPost Pat ( x ) − ˆ CPost Pat ( π R x ) � For a pattern of length j , bounded by ε j y : e y ( jτ ) ≤ ε j ∀ t = jτ > 0 , y where: � � B � jτ � e tA � ε j y = � u ( · ) � [0 ,jτ ] − ˆ � � � � dt + C C e t ˆ ∞ ˆ A B 0 � � � e jτA � x 0 − ˆ � � sup � � . C C e jτ ˆ A π R x 0 x 0 ∈ R x A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 25 / 42

Model Order Reduction and error bounding State trajectory error Defined by (for a pattern Pat ): e x ( | Pat | τ ) = � π R Post Pat ( x ) − Post Pat ( π R x ) � A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 26 / 42

Model Order Reduction and error bounding State trajectory error Defined by (for a pattern Pat ): e x ( | Pat | τ ) = � π R Post Pat ( x ) − Post Pat ( π R x ) � For a pattern of length j , bounded by ε j x : e x ( jτ ) ≤ ε j ∀ t = jτ > 0 , x A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 26 / 42

Model Order Reduction and error bounding State trajectory error Defined by (for a pattern Pat ): e x ( | Pat | τ ) = � π R Post Pat ( x ) − Post Pat ( π R x ) � For a pattern of length j , bounded by ε j x : e x ( jτ ) ≤ ε j ∀ t = jτ > 0 , x where: � � B � jτ � e tA � ε j x = � u ( · ) � [0 ,jτ ] � � � − I n r � dt + π R e t ˆ ∞ ˆ A B 0 � � � e jτA � x 0 � � sup � π R − I n r � . e jτ ˆ A π R x 0 x 0 ∈ R x A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 26 / 42

Reduced Order Control Outline 1 Sampled Switched Systems 2 State Space Decomposition 3 Decomposition for Sampled Switched Systems with Output 4 Model Order Reduction and error bounding 5 Reduced Order Control Guaranteed offline control Guaranteed online control A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 27 / 42

Reduced Order Control Reduced Order Control Two systems: Full-order system: Σ, R x , R y � ˙ x ( t ) = Ax ( t ) + Bu ( t ) Σ : y ( t ) = Cx ( t ) Reduced-order system: ˆ Σ, ˆ R x , R y � ˙ = ˆ x ( t ) + ˆ ˆ x ( t ) A ˆ Bu ( t ) , ˆ Σ : = ˆ y r ( t ) C ˆ x ( t ) . A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 28 / 42

Reduced Order Control Reduced Order Control Two systems: Full-order system: Σ, R x , R y � ˙ x ( t ) = Ax ( t ) + Bu ( t ) Σ : y ( t ) = Cx ( t ) Reduced-order system: ˆ Σ, ˆ R x , R y � ˙ = ˆ x ( t ) + ˆ ˆ x ( t ) A ˆ Bu ( t ) , ˆ Σ : = ˆ y r ( t ) C ˆ x ( t ) . Control synthesis (decomposition) for the reduced-order system. A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 28 / 42

Reduced Order Control Reduced Order Control Two systems: Full-order system: Σ, R x , R y � ˙ x ( t ) = Ax ( t ) + Bu ( t ) Σ : y ( t ) = Cx ( t ) Reduced-order system: ˆ Σ, ˆ R x , R y � ˙ = ˆ x ( t ) + ˆ ˆ x ( t ) A ˆ Bu ( t ) , ˆ Σ : = ˆ y r ( t ) C ˆ x ( t ) . Control synthesis (decomposition) for the reduced-order system. ⇒ reduced-order control A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 28 / 42

Reduced Order Control Reduced Order Control Two systems: Full-order system: Σ, R x , R y � ˙ x ( t ) = Ax ( t ) + Bu ( t ) Σ : y ( t ) = Cx ( t ) Reduced-order system: ˆ Σ, ˆ R x , R y � ˙ = ˆ x ( t ) + ˆ x ( t ) ˆ A ˆ Bu ( t ) , ˆ Σ : = ˆ y r ( t ) C ˆ x ( t ) . Control synthesis (decomposition) for the reduced-order system. ⇒ reduced-order control ⇒ application of the reduced-order control to the full-order system A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 28 / 42

Reduced Order Control Reduced Order Control Two systems: Full-order system: Σ, R x , R y � ˙ x ( t ) = Ax ( t ) + Bu ( t ) Σ : y ( t ) = Cx ( t ) Reduced-order system: ˆ Σ, ˆ R x , R y � ˙ = ˆ x ( t ) + ˆ x ( t ) ˆ A ˆ Bu ( t ) , ˆ Σ : = ˆ y r ( t ) C ˆ x ( t ) . Control synthesis (decomposition) for the reduced-order system. ⇒ reduced-order control ⇒ application of the reduced-order control to the full-order system Questions: A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 28 / 42

Reduced Order Control Reduced Order Control Two systems: Full-order system: Σ, R x , R y � ˙ x ( t ) = Ax ( t ) + Bu ( t ) Σ : y ( t ) = Cx ( t ) Reduced-order system: ˆ Σ, ˆ R x , R y � ˙ = ˆ x ( t ) + ˆ ˆ x ( t ) A ˆ Bu ( t ) , ˆ Σ : = ˆ y r ( t ) C ˆ x ( t ) . Control synthesis (decomposition) for the reduced-order system. ⇒ reduced-order control ⇒ application of the reduced-order control to the full-order system Questions: How is it applied? A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 28 / 42

Reduced Order Control Reduced Order Control Two systems: Full-order system: Σ, R x , R y � ˙ x ( t ) = Ax ( t ) + Bu ( t ) Σ : y ( t ) = Cx ( t ) Reduced-order system: ˆ Σ, ˆ R x , R y � ˙ = ˆ x ( t ) + ˆ ˆ x ( t ) A ˆ Bu ( t ) , ˆ Σ : = ˆ y r ( t ) C ˆ x ( t ) . Control synthesis (decomposition) for the reduced-order system. ⇒ reduced-order control ⇒ application of the reduced-order control to the full-order system Questions: How is it applied? Is the reduced-order control effective at the full-order level? A. Le Co¨ ent, F. de Vuyst, L. Fribourg Guaranteed Switched Control April 11, 2015 28 / 42