## Effective Synthesis of Switching Controllers for Linear Systems (2000)

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Citations: | 76 - 8 self |

### BibTeX

@MISC{Asarin00effectivesynthesis,

author = {Eugene Asarin and Olivier Bournez and Thao Dang and Oded Maler and Amir Pnueli},

title = {Effective Synthesis of Switching Controllers for Linear Systems},

year = {2000}

}

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### Abstract

In this work we suggest a novel methodology for synthesizing switching controllers for continuous and hybrid systems whose dynamics are defined by linear differential equations. We formulate the synthesis problem as finding the conditions upon which a controller should switch the behavior of the system from one "mode" to another in order to avoid a set of bad states, and propose an abstract algorithm which solves the problem by an iterative computation of reachable states. We have implemented a concrete version of the algorithm, which uses a new approximation scheme for reachability analysis of linear systems.

### Citations

568 | The control of discrete event systems
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Citation Context ...is led eventually to the theory of discrete-event dynamical systems (DES), for which a controller synthesis methodology, essentially based on automata theory, has been developed by Ramadge and Wonham =-=[RW89]-=-. However these models, (with some exceptions, see [KASL00]) were kept apart from continuous models. It was only within computer science (in particular in the domains of verification and semantics), t... |

356 | HyTech: A Model Checker for Hybrid Systems
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Citation Context ...ing elementary linear algebra. This is the basis for reachability algorithms for timed automata and other hybrid systems with constant derivatives implemented in tools such as Kronos [Y97] and HyTech =-=[HHW97]-=-. In the less trivial case of linear systems, the definition reduces tos0 (F ) = fx : 9t e At x 2 Fg: In this case, quantifier-elimination techniques fail except for some special classes of matrices (... |

346 |
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Citation Context ...trol by switching (i.e. discrete change in the dynamics) has been used since the early days of control. This practice was sometimes accompanied by serious attempts of mathematical formalization, e.g. =-=[F88]-=-. However, the feeling is that in all these attempts, the discrete dynamics was a second-class citizen, 2 a kind of hacking to glue together several respectable continuous models. The reason for this ... |

332 | Mathematical Control Theory: Deterministic Finite Dimensional Systems (2nd edn
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Citation Context ...8]. In other words, control theory is concerned with finding systematic ways to influence the behavior of systems by observing their state and injecting appropriate control signals (see, for example, =-=[S90] for a mod-=-ern exposition). Classically, both the system to be controlled ("plant" in the control terminology) and the controller were modeled as continuous dynamical 1 systems, and the control signal ... |

210 |
Kronos: A Verification Tool for Real-time Systems
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Citation Context ...n be performed using elementary linear algebra. This is the basis for reachability algorithms for timed automata and other hybrid systems with constant derivatives implemented in tools such as Kronos =-=[Y97]-=- and HyTech [HHW97]. In the less trivial case of linear systems, the definition reduces tos0 (F ) = fx : 9t e At x 2 Fg: In this case, quantifier-elimination techniques fail except for some special cl... |

205 | An old-fashioned recipe for real time
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Citation Context ...ll not do it because there will be an explicit notion of the "good" and "bad" states in the formulations of the synthesis problem. The other anomaly is that of a Zeno behavior (a t=-=erm first coined in [AL92]-=-), namely a trajectory which switches infinitely often between discrete states during a bounded time interval. Similar phenomena have been studied extensively under the title sliding mode control [F88... |

204 | Basic problems in stability and design of switched systems - Liberzon, Morse - 1999 |

198 | Discrete abstractions of hybrid systems
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Citation Context ...edback map c : Q \Theta X ! 2 Q where H q = fx : q 2 c(q; x)g and G qq 0 = fx : q 0 2 c(q; x)g. The hybrid automata defined here are slightly more restricted than others appearing in this issue, e.g. =-=[AHLP00]-=-, [DN00] as there are no jumps or resets in the values of the continuous variables when transitions are taken. These more general models include also a reset map R : Q \Theta Q \Theta X ! 2 X , which ... |

188 | On the synthesis of discrete controllers for timed systems
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Citation Context ...les are clocks following the same dynamics,sx = 1, in all discrete states. Wong-Toi and Hoffmann [WH92] were the first to consider the application of supervisory control methodology to this class. In =-=[MPS95]-=- (see also [AMP95-b, AMPS98]) we have defined an algorithm similar to Algorithm 1 and have shown that the exact computation of P can be performed over the set of zones (a restricted class of polyhedra... |

178 |
A theory of timed automata, Theoretical Computer Science 126
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Citation Context ...plicit notion of an "oracle" that solves the hard computational problems? The first and simplest class of hybrid systems for which controller synthesis has been applied is the class of timed=-= automata [AD94]-=- where all continuous variables are clocks following the same dynamics,sx = 1, in all discrete states. Wong-Toi and Hoffmann [WH92] were the first to consider the application of supervisory control me... |

147 | From timed to hybrid systems - Maler, Manna, et al. - 1992 |

127 |
Nonlinear regulation: the piecewise linear approach
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Citation Context ...fectiveness and termination problems we propose an approximate variant of Algorithm 1, specialized for piecewise-linear systems i.e.sx = A q x for every q 2 Q (similar systems have been introduced in =-=[S81] for discr-=-ete time), and which uses a restricted class of subsets of X, orthogonal "griddy" polyhedra (unions of unit boxes defined by a fixed grid [BMP99]). The approximation techniques, described in... |

119 | Controller Synthesis for Timed Automata
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Citation Context ...linear system is approximated by a piecewise-linear one, a passage from one region of the state-space to another. There is a standard adaptation of thesoperator for such situations (see, for example, =-=[AMPS98]-=-). For continuous disturbances, we have extended our system to treat dynamics of the formsx = Ax+Bu where u ranges over a convex set. Using a modification of the procedure proposed in [V98], based on ... |

117 | Controllers for reachability specifications for hybrid systems - Lygeros, Tomlin, et al. - 1999 |

112 | Approximate reachability analysis of piecewise linear dynamical systems
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(Show Context)
Citation Context ...s called "flow pipes" or "tubes of trajectories") of continuous systems. In order to make this algorithm concrete we use a novel technique of approximating such sets for linear dif=-=ferential equations [ABDM00]-=- to give a switching controller with guaranteed correctness. This algorithm has been implemented. The bottom line of this work is that if you have a plant modeled as a piecewise-linear dynamical syste... |

110 | A.: Reachability analysis of dynamical systems having piecewise-constant derivatives. Theoretical Computer Science 138
- Asarin, Maler, et al.
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(Show Context)
Citation Context ...oint will not be reached in a finite number of iterations. This phenomenon is illustrated in Figure 7 using a PCD system (a special class of deterministic HA where all vector fields are constant, see =-=[AMP95-a]-=-). Given this state-ofaffairs we resort to a classical solution of continuous mathematicians and use numerical methods to compute approximately predecessors (and successors) in the continuous state-sp... |

109 | Symbolic controller synthesis for discrete and timed systems
- Asarin, Maler, et al.
- 1995
(Show Context)
Citation Context ...rollers that can avoid bad states only by generating such behaviors which "stop" Time. Our approach to this problem is to define HA which are non-Zeno by construction (as we did for timed au=-=tomata in [AMP95-b]) and-=- hence any synthesized controller cannot generate Zeno behaviors. Definition 6 (Non-Zeno Hybrid Automata) A Zeno cycle in a HA is a sequence q 1 ; : : : q s of states such that cl(G 12 ) " cl(G 2... |

93 | A.: Logics for hybrid systems
- Davoren, Nerode
- 2000
(Show Context)
Citation Context ... c : Q \Theta X ! 2 Q where H q = fx : q 2 c(q; x)g and G qq 0 = fx : q 0 2 c(q; x)g. The hybrid automata defined here are slightly more restricted than others appearing in this issue, e.g. [AHLP00], =-=[DN00]-=- as there are no jumps or resets in the values of the continuous variables when transitions are taken. These more general models include also a reset map R : Q \Theta Q \Theta X ! 2 X , which decompos... |

89 | A game theoretic approach to controller design for hybrid systems - Tomlin, Lygeros, et al. - 2000 |

83 |
Differential Games: A mathematical theory with applications to warfare and pursuit, control and optimization
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- 1965
(Show Context)
Citation Context ...?" can be rephrased as asking whether a particular given strategy is winning. Similarly, many problems in control can be viewed as simple instances of finding winning strategies in differential g=-=ames [I65]-=-. For finite-state discrete systems, all variants of these problems are algorithmically solvable, and we survey various attempts to extend such results to deal with continuous and hybrid dynamics. The... |

83 | Temporal Verification of Reactive Systems
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(Show Context)
Citation Context ... system will never reach a set of bad states" or "every occurrence of event a will be followed by an occurrence of event b" etc. These properties can be expressed in formalisms such as =-=temporal logic [MP95]-=-. Performance criteria used in control have similar yet somewhat different flavor, partly due to historical reasons, partly due to the different nature of the time domains and state-spaces (in discret... |

82 | S.: A new class of decidable hybrid systems
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(Show Context)
Citation Context ...ues fail except for some special classes of matrices (recent results concerning the applicability of algebraic manipulation techniques for the reachability analysis of linear hybrid systems appear in =-=[PLY99]-=-, [SPS00] and [AHLP00]). The geometry ofs0 (F ) for constant and non-constant dynamics is illustrated in Figures 6-(a) and 6-(b). Even if we were equipped with an effective precise procedure to comput... |

72 | O.: Reachability analysis via face lifting - Dang, Maler - 1998 |

63 |
The synthesis of controllers for linear hybrid automata
- Wong-Toi
- 1997
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Citation Context ...ar algebra (polyhedra are closed under these operations). A controller synthesis procedure for fixed slope hybrid automata, which also treats the problem of Zeno controllers, was given by Wong-Toi in =-=[W97]-=- and implemented in HyTech. Earlier work concerning controller synthesis for such systems was reported in [TE94]. Moving to non-trivial continuous dynamics one faces the problems mentioned in this pap... |

59 | Verification of hybrid systems with linear differential inclusions using ellipsoidal approximations - Botchkarev, Tripakis - 2000 |

57 | Hybrid automata with finite bisimulations
- Henzinger
- 1995
(Show Context)
Citation Context ...lem a-la Ramadge-Wonham is solved. Our work in [MPS95] solves the same problem in a direct manner. In verification of systems with constant slopes, the indirect approach is represented by papers like =-=[H95]-=- and [AHLP00], which prove that for certain classes of systems such reductions are possible. The verification procedure in [ACH + 95] and in [AMP95-a] (the latter proves decidability for a class of sy... |

56 | Discrete-time control for rectangular hybrid automata
- HENZINGER, KOPKE
- 1999
(Show Context)
Citation Context ...itialized rectangular hybrid automata, systems where the dynamics in each state is of the form csx isd for every variable x i and for which controller synthesis was proved to be solvable for discrete =-=[HK99]-=- and continuous [HHM99] time by Henzinger et al. In the rest of the constant slope world, exact verification and synthesis problems are undecidable, algorithms such as Algorithm 1 are not guaranteed t... |

51 | Interconnected automata and linear systems: A theoretical framework in discrete-time
- Sontag
- 1996
(Show Context)
Citation Context ...aler@imag.fr 1 Throughout this paper the term "continuous dynamical system" is used in the narrow sense of a system whose behaviors are solutions to differential equations, e.g. [HS74]. 1 be=-= found in [S96]). Note th-=-at this is not an argument about the "real" nature of the world but rather about the utility of certain classes of mathematical models. People designing real control systems have always know... |

49 | Synthesis of supervisory controllers for hybrid systems based on approximating automata
- Cury, Krogh, et al.
- 1998
(Show Context)
Citation Context ... hybrid models to the supervisory control framework. Since the dynamics treated by these authors is non-trivial, they do not look for exact finite quotient but rather for approximating automata (e.g. =-=[CKN98]-=-). In the more recent works such and [CK99, MR99], backward methods similar to ours are used for computing and refining finite partitions of such systems. In [KASL00] the problem of interface design, ... |

46 | Beyond HYTECH: Hybrid systems analysis using interval numerical methods
- Henzinger, Horowitz, et al.
- 2000
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Citation Context ...x 1 = \Gammax 1 + u u 2 [\Gamma0:5; 0:5] andsx 1 = \Gammax 1 + u u 2 [3:5; 4:5]: Note that these are equivalent to differential inclusions due to uncontrolled (but bounded) disturbances. Similarly to =-=[HHMW99] we augmen-=-t the system with an additional "clock" variable x 2 , such that in every statesx 2 = 1, every transition is guarded by the condition x 2s0:5 and x 2 is reset to zero after a transition is t... |

42 | I.: Reachability analysis using polygonal projections - Greenstreet, Mitchell - 1999 |

40 | Set Computation Using Optimal Control
- Reach
- 1998
(Show Context)
Citation Context ...xample, [AMPS98]). For continuous disturbances, we have extended our system to treat dynamics of the formsx = Ax+Bu where u ranges over a convex set. Using a modification of the procedure proposed in =-=[V98]-=-, based on the maximum principle, we can compute an appropriate variant ofsand solve the synthesis problem in the presence of such disturbances as Example 1 will show. Our techniques can be adapted, v... |

37 | Supervisory control of hybrid systems - Koutsoukos, Antsaklis, et al. - 2000 |

33 | Orthogonal polyhedra: Representation and computation
- Bournez, Maler, et al.
- 1999
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Citation Context ...q 2 Q (similar systems have been introduced in [S81] for discrete time), and which uses a restricted class of subsets of X, orthogonal "griddy" polyhedra (unions of unit boxes defined by a f=-=ixed grid [BMP99]-=-). The approximation techniques, described in the next section (and in more detail in [ABDM00]), when plugged into Algorithm 1, yields a sequence ~ P k of polyhedra which converges after finitely many... |

33 |
The Control of Dense Real-Time Discrete Event Systems
- Wong-Toi, Hoffmann
- 1991
(Show Context)
Citation Context ...controller synthesis has been applied is the class of timed automata [AD94] where all continuous variables are clocks following the same dynamics,sx = 1, in all discrete states. Wong-Toi and Hoffmann =-=[WH92]-=- were the first to consider the application of supervisory control methodology to this class. In [MPS95] (see also [AMP95-b, AMPS98]) we have defined an algorithm similar to Algorithm 1 and have shown... |

29 | Optimization-based verification and stability characterization of piecewise affine and hybrid systems - Bemporad, Torrisi, et al. |

28 |
Analysis of digital circuits through symbolic reduction
- Kurshan, McMillan
- 1991
(Show Context)
Citation Context ...ter proves decidability for a class of systems not having a finite quotient) and the synthesis algorithm of [W97] are examples of the direct approach. More complex continuous dynamics were treated by =-=[KM91]-=- and [G96] who tried to compute approximate discrete abstractions for electrical circuits modeled at the transistor level. We believe that in practice, except timed automata, very few interesting hybr... |

20 |
Supervisory Hybrid Systems
- Lemmon, He, et al.
- 1999
(Show Context)
Citation Context ...ate abstraction as a pre-processing step. Other works based on a mixture of direct and indirect approaches have been proposed by various authors from the control and DES communities (see the tutorial =-=[LHM99]-=- and the survey in [KASL00]). Some of these works try to relate continuous and hybrid models to the supervisory control framework. Since the dynamics treated by these authors is non-trivial, they do n... |

19 | Verifying safety properties of differential equations
- Greenstreet
- 1996
(Show Context)
Citation Context ...decidability for a class of systems not having a finite quotient) and the synthesis algorithm of [W97] are examples of the direct approach. More complex continuous dynamics were treated by [KM91] and =-=[G96]-=- who tried to compute approximate discrete abstractions for electrical circuits modeled at the transistor level. We believe that in practice, except timed automata, very few interesting hybrid systems... |

15 | Decidable Controller Synthesis for Classes of Linear Systems
- Shakernia, Pappas, et al.
- 2000
(Show Context)
Citation Context ...except for some special classes of matrices (recent results concerning the applicability of algebraic manipulation techniques for the reachability analysis of linear hybrid systems appear in [PLY99], =-=[SPS00]-=- and [AHLP00]). The geometry ofs0 (F ) for constant and non-constant dynamics is illustrated in Figures 6-(a) and 6-(b). Even if we were equipped with an effective precise procedure to compute (F ), A... |

13 | Reachability analysis of a class of switched continuous systems by integrating rectangular approximation and rectangular analysis - Preussig, Stursberg, et al. - 1999 |

12 |
Verification of polyhedral invariant hybrid automata using polygonal flow pipe approximations
- Chutinan, Krogh
- 1999
(Show Context)
Citation Context ... among a class of new techniques for computing reachable states of continuous systems, e.g. [G96, GM99, DM98, KV97, V98, CK99, PSK99, HHMW99, BT00, BTM00]. Among these, the work of Chutinan and Krogh =-=[CK99], also cen-=-tered on linear differential equations, is the closest to ours. Their goal is to find discrete abstractions, and they use the approximate "flow pipes" as means to achieve this goal. Their ap... |

8 | Vályi I. Ellipsoidal Calculus for Estimation and Control - Kurzhanski - 1997 |

6 | Controllability and control-law synthesis of linear hybrid systems
- Tittus, Egardt
- 1994
(Show Context)
Citation Context ...id automata, which also treats the problem of Zeno controllers, was given by Wong-Toi in [W97] and implemented in HyTech. Earlier work concerning controller synthesis for such systems was reported in =-=[TE94]-=-. Moving to non-trivial continuous dynamics one faces the problems mentioned in this paper, and the class of systems for which results on synthesis can be given an exact computational content is extre... |

6 | A unified approach for studying discrete and continuous dynamical systems - Maler - 1998 |

5 | Phase-space nonlinear control toolbox: The maglev experience
- Zhao, Loh, et al.
- 1997
(Show Context)
Citation Context ...tate automata or for all timed automata. On the other hand, in the study of continuous systems, it is sometimes hard enough to treat one instance of a problem. For example, the work of Zhao et al. in =-=[ZLM99]-=- combines knowledge of dynamics with computation-geometrical algorithms in order to synthesize controllers that navigate in the phase-space of one particular non-linear system. In our work we have tri... |

3 |
A Unified Approach for Studying Discrete
- Maler
- 1998
(Show Context)
Citation Context ... real theory of hybrid control which can pick useful ingredients from both theories of dynamical systems and pass on the rest with silence (a primal sketch of a basis of such a theory can be found in =-=[M98]-=-). In addition to the theory of automata, an important potential contribution of computer science to control is in the emphasis on computation in both design and implementation of control systems. Com... |

2 |
Global Simulation via Partial Differential Equations, Unpublished note, Verimag
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- 1993
(Show Context)
Citation Context ...ton-Jacobi-Bellman-Isaacs partial differential equation. In fact, all problems of tracking the evolution of a subset of X under differential flows can be rephrased as an initial-value problem for PDE =-=[C93]-=-, but no evidence has been given so far of the computational advantages of this point of view. In [SPS00] it has been shown that for the sub-class of linear systems where the matrices are either diago... |

2 |
Differential Automata and their Simulators, Nonlinear Analysis
- Tavernini
- 1987
(Show Context)
Citation Context ...and the archaic "uniqueness of solutions" is traded for higher expressive power. Non-determinism can express uncertainty 5 In fact, differential automata have been used to model hysteresis a=-=lready in [T87]. 6 By &qu-=-ot;set-valued" we mean non-determinism which specifies a set of possibilities but does not define probabilities on this set. 3 Plant + continuous controller Plant + continuous controller Plant + ... |