Controlled Invariance of Discrete Time Systems Ren Vidal
- Slides: 15
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 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 • 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 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 : * . – 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 10/31/2020 6
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: – ! 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 – , 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 , , 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 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 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 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 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 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
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