by the following grammar:
ϕ := s(x1, x2, . . . , xn) ∼ 0 | ¬ϕ | ϕ ∧ ϕ | ϕUIϕ (2.9)
where s : Rn → R, ∼∈ {<, ≤}, and I is a closed non-singular interval of R≥0. There
are two elements that distinguish STL from other logics:
• the predicates s(x1, x2, . . . , xn) ∼ 0 are evaluated on real-values, that in our case
are the states of the dynamical system;
• the temporal operators ϕUIϕ are decorated with intervals that determine the tem-
poral windows on which the operators are defined.
From these basic operators, other classical temporal operators can be defined in the usual way, such as true >, false ⊥, eventually/future FIϕ ≡ >UIϕ, or always/globally
GIϕ ≡ ¬FI¬ϕ.
With these elements, we can formalize our requirement expressed in human language “Always between time 0 and 30, the number of infected individuals i is below 0.70” using the STL formula G[0,30](i < 0.70).
2.2.1
The Parameter Synthesis Problem
Now that we have defined Signal Temporal Logic formulas, we are ready to formalize the parameter synthesis problem for a generic dynamical system.
Definition 5 (Parameter Synthesis Problem). Let S = (X , P, f ) be a dynamical system, X0⊆ X be a set of initial conditions, P ⊆ P be a set of parameters, and ϕ be an STL
specification. Find the largest subset Pϕ∗⊆ P such that: ∀x0∈ X0, ∀p ∈ Pϕ∗, ξ
p
x0 satisfies ϕ (2.10)
where ξpx0 is a trajectory of S.
The notion of formula satisfaction and the parameter synthesis problem will be formalized in Section 5.1.
2.3
Two Important Questions
An intuitive way to generate a valid parameter set is to check the parameters one by one and populate the set Pϕ. In general, this naive algorithm is incomplete and incorrect.
Incomplete, because the parameter set might be infinite and uncountable, hence we will never be able to consider all the possible parameter values.
Incorrect, because when we have established the validity of a parameter value, we have done it considering a finite number of initial conditions and trajectories. If the set of initial conditions is infinite, there might be a point that we have missed such that the correspondent trajectory does not satisfy the specification.
These two observations suggest us that in order to solve the parameter synthesis problem, in the worst case we would need to compute all the trajectories starting from all the initial conditions with all the parameters. Moreover, even once we have all the
22 2. Dynamical Systems and Parameters
trajectories, we would need to produce a set that contains an infinite number of valid parameters. Then, the crucial questions are:
1. How to compute all the parametric trajectories generated from infinite sets of initial conditions and parameters?
2. How to compute and represent a valid refinement of the parameter set dealing with infinite sets?
The objective of this work is to give a possible solution to these questions. In Chapter 3 we will clarify the problem of computing all the trajectories generated from an infinite set of initial conditions, while in Chapter 4 we will define some techniques to over-approximate such computation. Later, in Chapter 5, we will define a method to synthesize valid parameter sets.
3
Parametric Reachability
In this chapter we define and discuss the parametric reachability problem for parametric dynamical systems, i.e., the problem of computing all the states visited by the trajec- tories of a dynamical system starting from a set of initial conditions and being biased by a set of parameters. This problem plays a central role in the parameter synthesis problem, since we will be able to determine valid parameter sets only once we are able to compute the evolution of the system under the influence of the treated parameter set.
The chapter begins with the definition of the reachability problem (Section 3.1), then it presents the technique of the numerical integration (Section 3.2) and two different approaches for the computation of reachable sets (Section 3.3). Finally, there will be some considerations on the decidability of the reachability problem (Section 3.4).
3.1
Parametric Reachability Problem
The problem of computing the states visited by the trajectories of a dynamical system starting from an initial set and having a particular parameter set is called the parametric reachability problem.
Let S = (X , P, f ) be a dynamical system. Given two states x, x0∈ X , we say that x0 is reachable from x in time 0 ≤ t < +∞ if there are a parameter p ∈ P and a trajectory ξxp of S starting in x such that x0 = ξpx(t). The set of all the states reached by the
system from x0∈ X with parameter p ∈ P is defined as:
Reachp(x0) = {x0 | x0= ξpx0(t), t ∈ T} (3.1)
where ξxp
0 is a trajectory of S and T is the set of non-negative reals R≥0 or the set of
naturals N, depending on whether S is a continuous-time or discrete-time dynamical system, respectively.
We can extend the notion of reachability to sets, that is, given a set of initial condi- tions X0⊆ X and a parameter set P ⊆ P, the reachable set is the set of all the states
24 3. Parametric Reachability
reachable by the system:
ReachP(X0) = [ x0∈X [ p∈P Reachp(x0). (3.2)
The definition of reachable set reflects the behavior of the dynamical system for an infinite amount of time. However, we might be interested in studying a model for a bounded time horizon. Thus, the set of states reachable in a bounded amount of time T ∈ T is defined as: ReachpT(x0) = {x0| x0 = ξxp0(t), 0 ≤ t ≤ T } (3.3) ReachPT(X0) = [ x0∈X0 [ p∈P ReachpT(x0). (3.4)
Reachable Set Computation
The computation of the reachable set of a dynamical system, in both its bounded or unbounded time versions, might be problematic. The first issue concerns the numerical computation of the states visited by a trajectory. With the exception of the cases where the trajectories can be characterized by explicit solutions (e.g., {x0eAt | t ∈
R≥0} for linear systems ˙x = Ax), the usual way to compute the reachable states is
to use numerical integration. The second issue interests the possible infinite number of trajectories we have to deal with, since we might consider infinite sets of initial conditions and parameters. There are several techniques that try to cope with these problems. They can be grouped in two classes:
• Trajectory-Based Reachability: a finite number of initial conditions and parame- ters, called nominal values, are chosen. Usually, the nominal values are the result of a discretization or some statistical assumptions on the state-parameter space. In general, the number of nominal values necessary to reach a certain level of coverage of the state-parameter space grows drastically in the dimension of the system.
• Set-Based Reachability: considering all the given initial conditions and parame- ters at once, an exhaustive set of trajectories, called flowpipe, is generated. This approach is strongly related to formal verification and set-based computation. In this case it is necessary to deal with image computation and manipulation of sets, problems that are mathematically and computationally nontrivial.
In this work we focus exclusively on set-based reachability and on the computation of valid flowpipes for dynamical systems. Before going into the details of our techniques, we become familiar with the notions of numerical integration, trajectory-based and set- based reachability techniques, providing an overview on the existing methods for the reachability problem.