Controlled Invariance of Discrete Time Systems Ren Vidal

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Controlled Invariance of Discrete Time Systems René Vidal, Shawn Schaffert, John Lygeros and Shankar

Controlled Invariance of Discrete Time Systems René Vidal, Shawn Schaffert, John Lygeros and Shankar Sastry EECS, University of California at Berkeley 10/31/2020 1

Outline • Controlled invariance problem (CIP) • CIP for definable systems • CIP for

Outline • Controlled invariance problem (CIP) • CIP for definable systems • CIP for linear discrete time systems (LDTS) • Decidable special cases • Experimental results • Conclusions and future work EECS, University of California at Berkeley 10/31/2020 2

Introduction • Controller synthesis for hybrid systems – Hierarchical control, distributed control, optimal control

Introduction • Controller synthesis for hybrid systems – Hierarchical control, distributed control, optimal control • Control problems with reachability specifications – Extend game theory methods to discrete, continuous and hybrid systems • Little attention devoted to discrete time systems EECS, University of California at Berkeley 10/31/2020 3

Discrete Time System (DTS) • DTS , , , Init, : – : Finite

Discrete Time System (DTS) • DTS , , , Init, : – : Finite collection of state variables, – , : Finite collection of input variables, – Init : Set of initial states, – : : Reset relation. • Execution of a DTS: – , , * , – [ ] Init and [ ] ( [ ], [ ]) . EECS, University of California at Berkeley 10/31/2020 4

Controllers & Safety Properties • A controller is a map : * . –

Controllers & Safety Properties • A controller is a map : * . – A controller is non-blocking if for all *. – A controller is memoryless if , ’ * ending at the same state ’. • A controller solves the problem , if and only if stays in forever. • Proposition 1: in solving , one can restrict attention to memoryless controllers. EECS, University of California at Berkeley 10/31/2020 5

Controlled Invariance Problem ( , , ) = EECS, University of California at Berkeley

Controlled Invariance Problem ( , , ) = EECS, University of California at Berkeley 10/31/2020 6

Controlled Invariant Set and Least Restrictive Controller • is a controlled invariant set of

Controlled Invariant Set and Least Restrictive Controller • is a controlled invariant set of if – There is a non-blocking controller that solves ’, , where ’ , , . • Many memoryless controllers may solve the problem. • A controller : that solves , is called maximal if ’ and ’ solving , . EECS, University of California at Berkeley 10/31/2020 7

Controlled Invariance Problem • Theorem 1: , can be solved if and only if:

Controlled Invariance Problem • Theorem 1: , can be solved if and only if: – ! maximal controlled invariant set , and – ! least restrictive controller that renders invariant. • Theorem 2: is the fixed point of the operator Pre { , , }. • There is no straightforward way of implementing 10/31/2020 this calculation in general. 8 EECS, University of California at Berkeley

CIP for Definable Systems • A CIP is definable in a theory if –

CIP for Definable Systems • A CIP is definable in a theory if – , and , Init, , , and are definable in the same theory. – Examples of theories are Lin( ) and OF( ). • Theorem 3: The class of CIP definable in OF is semi-decidable. • CIP can be solved by quantifier elimination (QE), but the complexity is doubly exponential. EECS, University of California at Berkeley 10/31/2020 9

CIP for Linear DTS (LCIP) • A LCIP is a CIP where , ,

CIP for Linear DTS (LCIP) • A LCIP is a CIP where , , Init and are convex polygons and , , { }. • QE can be performed more efficiently for LCIP – QE of is equivalent to a series of Linear Programs. – QE of is done by Fourier Elimination. – A redundancy algorithm is required. – Complexity is worst case exponential. • Theorem 4: The LCIP is semi-decidable. EECS, University of California at Berkeley 10/31/2020 10

Decidable special cases • The LCIP is decidable for { } if – rank

Decidable special cases • The LCIP is decidable for { } if – rank min{ , } or – one of the columns of is positive (negative). • The LCIP is decidable in steps if – The LDTS is SISD, – The LDTS is in canonical controllable form, – , [ , ] and is a rectangle. EECS, University of California at Berkeley 10/31/2020 11

Experimental Results Iteration 1 Iteration 2 100 50 50 0 0 -50 -100 -50

Experimental Results Iteration 1 Iteration 2 100 50 50 0 0 -50 -100 -50 0 50 100 -100 -50 EECS, University of California at Berkeley 0 50 10/31/2020 12

Conclusions • The CIP for DTS is well posed. • The general algorithm can

Conclusions • The CIP for DTS is well posed. • The general algorithm can be implemented for definable systems via Quantifier Elimination. • The CIP is semi-decidable in Lin( ) and OF( ). • For LDTS the algorithm can be implemented by Linear Programming and Fourier Elimination. • The problem is decidable in some cases. EECS, University of California at Berkeley 10/31/2020 13

Current & Future Work • CIP for LDTS with ellipsoidal constraints – CIP can

Current & Future Work • CIP for LDTS with ellipsoidal constraints – CIP can be posed as a robust semi definite programming problem Safe Set Ellipsoidal approximation – Approximations can be computed in polynomial time EECS, University of California at Berkeley 10/31/2020 14

Current & Future Work • CIP for Discrete Time Hybrid Systems – Special case

Current & Future Work • CIP for Discrete Time Hybrid Systems – Special case of DTS, hence results of existence and uniqueness are valid. – Computation is harder because includes disjunctions. – Still it is possible to have a worst case exponential algorithm. EECS, University of California at Berkeley 10/31/2020 15