Invariant Sets and Control Synthesis for Switching ... - Semantic Scholar

Invariant Sets and Control Synthesis for Switching Systems with Safety Specifications Luca Berardi University of Rome “La Sapienza”

Elena De Santis University of L’Aquila

Maria Domenica Di Benedetto University of L’Aquila; University of California, Berkeley 1 UCB, Nov.3, 1999

Presentation’s Outline • Motivation: Engine Control • Formulation • Description of algorithm • Extensions and comparisons with existing procedures • Construction of Invariant Sets and Approximations

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1

Motivation: Gear Change Problem Gear 1

Gear 2 Idle

Gear 3

Gear 4

Gear 5

3

Gear Change Problem (cont.) Gear Change

The problem:

Control objective:

Change in powertrain parameters describing the continuous evolution of the system

Sudden change in powertrain parameters causes acceleration oscillations, source of discomfort for driver. Maintain oscillations within a given range 4

2

Idle Speed Control Problem Clutch insertion/release

Clutch is inserted

S1

S2 Clutch is released

Problem and control objective:

Change in powertrain parameters describing continuous evolution of system

While in idle mode, maintain the engine speed within a given range, rejecting torque disturbances and avoiding engine shut off.

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Goal Find Computationally Efficient Procedures to Solve the Synthesis Problem

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3

The Model Simplified class of hybrid systems: • no controllable and “invariance” transitions

Switching Systems

• guard conditions only on “timer” state variables • Discrete Structure •Switching Systems represented by an FSM at top level. •Transitions between FSM states determined by external uncontrollable events. • Continuous Structure •A dynamical system corresponds to each state of the FSM •Evolution in time of dynamical systems determined by differential equations or difference equations. 7

The Model (cont.) Example:

•FSM states and the associated dynamics = configurations. S2 S1

•Switching times unknown. •If ta and tb are two switching times :

S3

tb-ta ≥ ∆ i

8

4

Formulation Suppose switching times unknown. Find all initial states such that output of system can be maintained in a given set for some control input.

Safety condition: the output of the system must belong to a given set. X0 = safe set = set of all possible initial states such that the constraints on the output can be satisfied for some control input.

9

Our Approach New procedure for determination of set of “good” initial states (safe set) that exploits the FSM structure.

Complexity of problem reduced by decomposition into simpler sub-problems. 10

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Two-step methodology: • Decompose the “hybrid” problem into a set of simpler “continuous” sub-problems. • Solve each of these “continuous” sub-problems separately. Subproblem:

Find the maximal controlled invariant set for a continuous dynamical system given some constraints on the output.

Numerical Procedure + Matlab implementation for discrete time linear systems & linear constraints. 11

Our Approach • simple and general procedure: can be extended to general hybrid systems

Advantages:

• (relative) insensitivity to the number of continuous variables; • computational efficiency; •convergence properties.

• Berardi, De Santis, Di Benedetto “Control of Switching Constrained Systems”, AAAI Workshop, Stanford, March 1999. • Berardi, De Santis, Di Benedetto “Control of Switching Systems under State and Input Constraints”, European Control Conference 1999. •Berardi, De Santis, Di Benedetto “A Structural Approach to the Control of Switching Systems with an Application to Automotive Engines”, CDC99. •Berardi, De Santis, Di Benedetto, Research Rep. n.99-35, Univ. of L’Aquila, 1999. 12

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Some Useful Definitions Definition: Σ is controlled invariant if, starting from any point of Σ, a control law exists such that the state evolution remains in Σ and the output satisfies constraints. We denote by Ii(Λ) the maximal controlled invariant set contained in Λ and such that output constraints are satisfied (yi ∈ Ω).

Σ

Λ 13

Three Particular Cases • Serial • Star • Cyclic

Remark: For the sake of notational simplicity, we first consider the case where the reset function is the identity function. The results obtained for this simplified case can be immediately extended to the case where resets are present. 14

7

Serial Case S1

S2

...

SN

Proposition SERIAL: The safe set X0 is equal to Σ1 :

Σ i = Ω ix∆i ( I i (Σ i +1 ) )

( )

i = 1,K , N − 1

ΣN = IN Rn

where : Ω ix∆ i (Λ ) = set of x : ∃u () ⋅ such that the output of the i - th configuration, initialized at x, belongs to Ω, ∀t 0 ≤ t ≤ t 0 + ∆ i and x(t0 + ∆ i )∈ Λ

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Star Case S2 S1

... SN

Proposition STAR: The safe set X0 is given by:

( (

( )) )

X 0 = Ω1∆x1 I1 I j = 2LN I j R n

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8

Cyclic Case S1

S2

...

SN

Proposition CYCLIC I: The safe set X0 is equal to Σ1 where { Σi }, i=1…N are the least fixed points of:

( ( )) ( I (Σ ) )

Σ ik +1 = Ω ix∆ i I i Σ ik+1 Σ kN+1 = Ω ∆NxN

N

i = 1, K , N − 1

k 1

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Cyclic Case Proposition CYCLIC II: If ∆ i = 0 (for all i=1…N) the safe set X0 is equal to the maximal controlled invariant set with respect to all configurations S1,…,SN contained in the set Ω1x ∩ Ω2x ∩ ... ∩ ΩNx, ,, where: Ωix= {x: yi∈ Ω, for some u }

Remark: If a switching system is described by a strongly connected graph and if ∆ i = 0 (for all i=1…N), the safe set X0 is equal to the one computed applying Proposition Cyclic II to the maximal cycle contained in the graph.

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General Case with ∆i=0 • F : connected FSM (from all state bi-partitions, ∃ a transition from one set of the bipartition to the other). • Strongly connected components of F: maximal sets of mutually reachable states. • Each strongly connected component Fj of F can be replaced by a unique FSM state fj. The invariant set corresponding to fj is obtained by applying proposition CYCLIC II to the maximal cycle contained in Fj

We obtain a DAG (Directed Acyclic Graph) 19

General Case with ∆i=0 The algorithm: Start from terminal nodes of DAG and proceed backwards: • solve all SERIAL cases stopping when a node containing two or more outgoing arcs is found; • solve all STAR case; • repeat the two previous steps until the root node is found.

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An Example

1. Serial

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An Example

2. Star

3. Star (stop) 22

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General Case with ∆i> 0 The algorithm: Start from terminal nodes of DAG and proceed backwards: • solve all SERIAL cases stopping when a strongly connected component is found; • solve all strongly connected components found so far; • repeat the two previous steps until the root node is found. 23

General Case with ∆i> 0 ( ) ( ) ( )

Σ 0a = I a R n  0 n Σ b = I b R Σ 0 = I R n c  c k =0

Init :

b

e

a

c

Repeat d

( ( )) Σ = Ω (I (Σ ∩ Σ )) Σ = Ω (I (Σ ∩ Σ ∩ Σ )) Until < a fixed point (Σ , Σ , Σ ) Σ ka +1 = Ω ∆a a I a Σ bk k +1 b

∆b b

b

k c

k e

k +1 c

∆c c

c

k a

k b

* a

is found >

k d

* b

* c

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Comparison (Tomlin, et al. (1998))

For example, consider serial case: 1

...

2

N

Solution with our procedure:

( )

ΣN = IN Rn

Hp : ∆ i = 0,

For i ← N − 1 to 1

i = 1,K , N

Computation of N inv.sets

Σ i = I i (Σ i +1 )

End For 25

Solution with procedure in Tomlin et al. (1998):

(

( ) )

W i +1 = W i − Reach Pre d W i , ∅ Let :

w ki = W i

q = Qk

For i ← 1 to N For k ← 1 to N − i + 1

(

i −1 k

w = Ik w i k

End For

i −1 N

∩K∩ w

)

Computation of N(N-1)/2 inv.sets

End For 26

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Extensions Presence of (continuous) disturbances.

General Hybrid Systems

Robust Invariant Sets

Introduction of controllable transitions, “invariance” and guard conditions.

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Controllable transitions Σ1

1

2

Σ2

• safe-set Σ2 for state 2 is: I2(Rn)

( )

(

)

n −1 • safe-set Σ1 for state 1 is: Σ1 = I1 R ∪ Z1 R12 (Σ 2 )

Extensions to more complex FSM topologies handled similarly. 28

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Construction of Invariant Sets To be able to effectively compute the safe set X0, we consider: • discrete-time linear systems in each node of the FSM (or configuration). • polyhedral constrained sets : F y ≤ w

Dynamical system:

x(t+1) = Ai x(t)+Bi u(t)+ Ri d(t)

(1)

y(t) = Ci x(t)+Di u(t) Fy≤ w

Output constraints:

(2) 29

Computation of the Maximal Controlled Invariant Set I 0 ← Λ Init :  k ← 0

( )

I k +1 ← Ω1ix I k

I

k +1

=I ?

NO

k

YES

End : I i (Λ ) ← I k

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Construction of Invariant Sets • Outer Approximations

• Inner Approximations

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Outer Approximations  xt +1 = Axt + But + Rδ t , δ t ∈ Ξ (where Ξ is a polytope)  1. Init : Wx ≤ M (state constraints)  Du ≤ d (input constraints)  2.

W (Ax + Bu ) ≤ M ′ = M − Rδ MAX where : δ MAX = arg max Rδ (LP) δ ∈Ξ

3.

WB   M − Ax   D u ≤  d     

4.

 M − Ax  Q ≥0  d 

(from extended Farkas Lemma) 32

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5.

Q = [Q1 Q2 ];

Q1M − Q1WAx + Q2 d ≥ 0 ;

Q1WAx ≤ Q1M + Q2 d  Q WA Q M + Q2 d  dove : W ′ =  1  M ′ =  1  M  W    

6.

W ′x ≤ M ′

7.

If

8.

(maximal controlled invariant set has been found) W ←W ′ ; M ← M ′

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Go To 2.

{x : Wx ≤ M }= {x : W ′x ≤ M ′}

STOP

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Control Laws For each discrete state (or configuration) q, the algorithm finds the set of all possible “good” (continuous) states, described in terms of linear inequalities: Gq x ≤ w q

Then all possible control laws are of the form:

xt +1 = Aq xt + Bq ut

Gq x t +1 ≤ wq

Gq (Aq xt + Bq ut )≤ wq ; Gq Bq ut ≤ wq − Gq Aq xt G q ut ≤ w q (x t ) 34

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Resets Σ1

1

2

Σ2

A) No resets (the reset function is the identity function):

( )

Σ2 = I 2 Rn

Σ1 = Ω1∆x1 (I1 (Σ 2 ))

Σ 2 = {x : Fx ≤ v}

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Resets B) Resets (the reset function is an affine function): reset function :

( )

x( 2) = f (x(1) ) → x( 2) = M x(1) + N

Σ2 = I 2 R n → Σ2 ={x( 2) : F x(2 ) ≤ v} Σ′2 = f −1 (Σ 2 ) = {x(1) : F f (x(1) ) ≤ v} Σ1 = Ω1∆x1 (I1 (Σ′2 )) Σ′2: F f (x(1) )≤v ;

FM x(1) ≤ v − FN ;

F (M x(1) + N )≤v F ′ x(1) ≤ v′ 36

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Inner Approximations Hypotheses: • Constraining Sets ΩX,ΩU: polyhedral C-sets • (A,B) : controllable • A : full rank Σ 0 = {0}

[

]

Σ t = A−1 Σ t −1 − A−1 B ΩU ∩ Ω X

U Σt = Σ MAX lim t →∞ 37

Idle Speed Control Constraints:

800 − 30 RPM ≤ n ≤ 800 + 30 RPM

Nonlinear Continuous-Time Systems:

[

]

30 1  n& = π J i k1 pη (AV )− (T friction + Tload ) eq   p& = n π 1 [pred (α , n )− p ] 30 τ 

Linearization Sampling

Linear Discretex& = Ai x + Bi u + Riδ Time Systems

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Idle Speed Control: Safe-Set 800

P re s s ure (+ 300 m Ba r)

600

400

200

0

-200

-400 -30

-20

-10

0 RP M (+ 800)

10

20

30

39

Conclusions •Developed a procedure for the computation of safe initial states (and corresponding control laws) for switching systems extensible to general hybrid systems. •Procedure exploits structure of FSM. •Complexity of former (hybrid) problem reduced by decomposition into simpler (continuous type) sub-problems. •Numerical methods proposed for the determination of controlled invariant sets and their approximations. 40

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