Following Part I, this article continues to describe the calculation of the reach sets and tubes for linear control systems with time-varying coefficients and ellipsoidal hard bounds on the controls and initial states. It deals with parametrized families of internal ellipsoidal approximations constructed such that they touch the reach sets at every point of their boundary at any instant of time. The reach tubes are thus touched internally by ellipsoidal tubes along some curves. The ellipsoidal tubes are chosen here in such a way that the touching curves do not intersect and that the boundary of the reach tube would be entirely covered by such curves. This allows exact parametric representation of reach tubes through unions of tight internal ellipsoidal tubes as compared with earlier methods based on constructing one or several isolated approximating tubes. The method of external and internal ellipsoidal approximations is then propagated to systems with box-valued hard bounds on the controls and initial states. It appears that the proposed technique may well work for nonellipsoidal, box-valued constraints. This broadens the range of applications of the approach and opens new routes to the arrangement of efficient numerical algorithms.

We present a method to design controllers for safety specifications in hybrid systems. The hybrid system combines discrete event dynamics with nonlinear continuous dynamics: the discrete event dynamics model linguistic and qualitative information and naturally accommodate mode switching logic, and the continuous dynamics model the physical processes themselves, such as the continuous response of an aircraft to the forces of aileron and throttle. Input variables model both continuous and discrete control and disturbance parameters. We translate safety specifications into restrictions on the system’s reachable sets of states. Then, using analysis based on optimal control and game theory for automata and continuous dynamical systems, we derive Hamilton–Jacobi equations whose solutions describe the boundaries of reachable sets. These equations are the heart of our general controller synthesis technique for hybrid systems, in which we calculate feedback control laws for

Reach set computation is a basic component of many verification and control synthesis procedures. Effective computation schemes are available for discrete systems described by finite state machines and continuous-variable systems described by linear differential inequalities. This paper suggests an approach based on the Pontryagin maximum principle of optimal control theory. The approach is elaborated for linear systems, and it may prove useful for more general continuous-variable systems. 1 Introduction Consider a state machine S = (X; U; T ) where X is the finite set of states, U is the finite set of control values, and T : X \Theta U ! X is the transition function: for x 2 X and u 2 U , x 0 = T (x; u) is the next state. A trajectory of S is a pair of sequences x 0 ; :::; x t and u 0 ; :::; u t\Gamma1 such that x i+1 = T (x i ; u i ). When it is unimportant to recall the control sequence, the sequence x 0 ; :::; x t by itself is called a trajectory. For a set of initial states ...

Reach set computations are of fundamental importance in control theory. We consider the reach set problem for open-loop systems described by parametric inhomogeneous linear dierential systems and use real quanti er elimination methods to get exact and approximate solutions. For simple elementary functions we give an exact calculation of the cases where exact semialgebraic transcendental implicitization is possible. For the negative cases we provide approximate alternating using discrete point checking or safe estimations of reach sets and control parameter sets. The method employs a reduction of forward and backward reach set and control parameter set problem to the transcendental implicitization problem for the components of special solutions of simpler non-parametric systems. Numerous examples are computed using the redlog and qepcad packages.

This paper presents a new approach to finite-horizon guaranteed state prediction for discrete-time systems affected by bounded noise and unknown-but-bounded parameter uncertainty. Our framework handles possibly nonlinear dependence of the state-space matrices on the uncertain parameters. The main result is that a minimal confidence ellipsoid for the state, consistent with the measured output and the uncertainty description, may be recursively computed in polynomial time, using interior-point methods for convex optimization. With n states, l uncertain parameters appearing linearly in the state-space matrices, with rank-one matrix coefficients, the worst-case complexity grows as O(l(n + l) 3:5 ). With unstructured uncertainty in all system matrices, the worst-case complexity reduces to O(n 3:5 ).

Abstract. The tolerable windows (TW) approach is presented as a novel scheme for integrated assessment of climate change. The TW approach is based on the specification of a set of guardrails for climate evolution which refer to various climate-related attributes. These constraints, which definewhatwecalltolerable windows, can be purely systemic in nature – like critical thresholds for the North Atlantic Deep Water formation – or of a normative type – like minimum standards for per-capita food production worldwide. Starting from this catalogue of knock-out criteria and using appropriate modeling techniques, those policy strategies which are compatible with all the constraints specified are sought to be identified. In addition to the discussion of the basic elements and the general theory of the TW approach, a modeling exercise is carried out, based on simple models and assumptions adopted from the German Advisory Council on Global Change (WBGU). The analysis shows that if the global mean temperature is restricted to 2 ◦C beyond the preindustrial level, the cumulative emissions of CO2 are asymptotically limited to about 1550 Gt C. Yet the temporal distribution of these emissions is also determined by the climate and socio-economic constraints: using, for example, a maximal tolerable rate of temperature change of 0.2 ◦C/dec and a smoothly varying emissions profile, we obtain the maximal cumulative emissions, amounting to 370 Gt C in 2050 and 585 Gt C in 2100. 1.

In this paper, we discuss semidefinite relaxation techniques for computing minimal size ellipsoids that bound the solution set of a system of uncertain linear equations. The proposed technique is based on the combination of a quadratic embedding of the uncertainty, and the S-procedure. This formulation leads to convex optimization problems that can be essentially solved in O(n 3)—n being the size of unknown vector — by means of suitable interior point barrier methods, as well as to closed form results in some particular cases. We further show that the uncertain linear equations paradigm can be directly applied to various state-bounding problems for dynamical systems subject to set-valued noise and model uncertainty.

Abstract. We discuss a procedure for synthesizing controllers for safety specifications for hybrid systems. The procedure depends on the construction of the set of states of a continuous dynamical system that can be driven to a subset of the state space, avoiding another subset of the state space (the Reach-Avoid set). We present a new characterization of the Reach-Avoid set in terms of the solution of a pair of coupled Hamilton-Jacobi partial differential equations. We also discuss a computational algorithm for solving such partial differential equations and demonstrate its effectiveness on numerical examples. 1

Abstract. This work is concerned with the problem of computing the set of reachable states for linear time-invariant systems with bounded inputs. Our main contribution is a novel algorithm which improves significantly the computational complexity of reachability analysis. Algorithms to compute over and under-approximations of the reachable sets are proposed as well. These algorithms are not subject to the wrapping effect and therefore our approximations are tight. We show that these approximations are useful in the context of hybrid systems verification and control synthesis. The performance of a prototype implementation of the algorithm confirms its qualities and gives hope for scaling up verification technology for continuous and hybrid systems. 1

For a class of dynamical systems, with uncertain nonlinear terms considered as "unknown inputs", we give sufficient conditions for observability. We show also that there does not exist any exact observer independent of the unknown inputs. Under the additional assumption that the uncertainty is bounded, we build practical observers whose error converges exponentially towards an arbitrarily small neighborhood of the origin. Under the hypothesis that bounds are available for the uncertain terms, we build parallelotopic observers providing time-varying bounds for the state variables, even when the system is not observable for unknown inputs These results are illustrated on a biological model of a structured population.