A New Hybrid Automaton Framework with Partial Differential Equation Dynamics

We first describe the behavior of the temperature of the rod. When the gas burner is OFF, the temperature of the rod, denoted by the variable $u$, decreases according to the following PDE: $u(x,t)_{t}-\alpha u(x,t)_{xx}=0$ where $\alpha$ is the thermal diffusivity. The thermal diffusivity measures the rate of heat transfer of a material from the hot side to the cold side. The boundary conditions are $u(0,t)=0$ and $u(L,t)=0$, which denote that at the location $x=0$ and $x=L$ the temperature is fixed to be $0$, and heat is absorbed. $X$ denotes the space coordinate and $t$ denotes time. This law, however, is only true when the temperature of the corresponding position on the rod is greater than 0.7. When the heater is OFF, and the temperature of a position on the rod is below 0.4, the heater at that position will be turned on. The heater function is defined as $f(x)=-\frac{x}{L}+1$. The maximum of the heater function appears at $x=0$ and the minimum is $0$ at $x=L$. During the heater ON mode, the temperature is governed by the equation $u(x,t)_{t}-\alpha u(x,t)_{xx}=f(x)$ where $f(x)$ is the function defined above. As one can see from the description of the evolution of the temperature, the system is not purely continuous. The system can switch from one mode to another by turning the heater on and off at a specific location on the rod.

The traffic model describes how traffic conditions may behave on a given highway. In free flow mode, which means the density is less than a specified critical density $\rho_{c}$, the density function is given by the following equation $u(x,t)_{t}+v_{1}u(x,t)_{x}=0$, where $v_{1}$ is always a positive number and denotes that the traffic flow propagates forward. However, if the traffic density achieves $\rho_{c}$, the system transitions into a high-density congestion mode. In this mode, the governing equation is described by $u(x,t)_{t}+v_{2}u(x,t)_{x}=0$, where $v_{2}$ is always a negative number and denotes that traffic congestion always moves backwards and propagates upstream. As an example, one of the causes of congestion may be a result of merging slow and fast-moving traffic, resulting in a high-density traffic flow. In this model, we can frequently switch between a free-flowing mode and a congestion mode, and as a result, a more powerful hybrid framework is needed to formalize these kinds of system behavior.

Additionally, the two models we have outlined present a mixture of space and time variables and are therefore quite complicated. In the following sections, we present a framework that is able to capture such a mixture of discrete and continuous behaviors, where PDEs describe the temporal and spatial dynamics.

Due to the properties of PDEs, the switching structure is more involved than in the case of ODEs [26]. One highway congestion alleviation model in [23] considers $N$ connected segments, where each segment is governed by an independent LWR PDE [24] with one side boundary condition. Each segment affects the boundary condition of the preceding segment and serves as an input function to the PDE. Additionally, the system is allowed to switch to another mode that incorporates a speed limit. One control approach for transport systems [40] proposes rules for switching based on criteria of minimizing a cost function. We outline two specific structures that are of interest to our framework:

• 1.

  Switching PDEs in time on the full spatial domain. This situation is portrayed in Figure 3 (left) and is the PDE counterpart of hybrid systems as defined in [41], where a game theory approach is used to determine when mode switching occurs. Boundary condition switching has been investigated in highway traffic applications in [42]. PDE switching in time appears in [40] and in [23] and the latter one contains switching between modes in each highway segment.

• 2.

  Switching PDEs on subsets of the time-space domain. This is illustrated in Figure 3 (right). Examples include the LWR PDE, in [23] where each highway segment is coupled with a boundary condition at the starting point of itself. This also appears in [24], where the LWR with triangular flux function can be decomposed into two modes (two one-dimensional wave equations) resulting in a partition of the $(x,t)$ space in regions with forward travelling waves, and regions with backward travelling waves.

In order to describe the partial domain switching case in Figure 3 (right), it is useful to partition the whole domain into several parts and assign each one a corresponding mode. One possible solution is to use the union of disjoint domains to represent the region governed by specific modes. For example, the heat equation $u_{t}-u_{xx}=0$ with $\alpha=1$ (mode 1) and without heat source can cover domain $[0,0.4)$. The equation $u_{t}-u_{xx}=f(x)$ with external source (mode 2) occupies domain $[0.4,1]$.

Based on this idea, the main challenge is how to formalize the particular structure of the domain occupying behavior. One approach is to associate the union operator with the modes of the system. Thus, the system mode $q$ can be described as a composition of multiple modes $\cup q_{i}$, where each mode specifies the domain it occupies. Like in $q_{1}$, $u_{t}-u_{xx}=0$ when $x\in\mathcal{D}_{1}$ and in $q_{2}$, $u_{t}-u_{xx}=f(x)$ when $x\in\mathcal{D}_{2}$, where $\mathcal{D}_{i}$ is the region occupied by $q_{i}$. Thus, the space domain $X$ is composed of $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$.

This embodies our approach to represent the multiple mode occupation phenomenon, and we refer to the multiple mode occupation as the changing behavior of $\cup\mathcal{D}_{i}$. The remaining challenge is to how to describe the moving behavior of the modes. This can be realized by changing $\mathcal{D}_{i}$, together with its boundary condition. As the system evolves, points satisfying a given condition will be moved from one mode to another and cause $\mathcal{D}_{i}$ to grow or shrink. We call this point’s moving behavior a trajectory. The evolution of $\mathcal{D}_{i}$ is not easily depicted. It can be a function of time or a function of several other complicated variables. Additionally, it might be impossible to predict switching behavior due to the complexity exhibited by the PDE’s solution.

Based on the ideas discussed above, the concept of spatial control mode is given below.

Now, according to our definition, $\tilde{q}$ is a subset of $Q$ and not just an element in $Q$. We want to emphasize that the system could occupy several control modes simultaneously, rather than stay at a single mode as defined in classical hybrid automata. In order to accommodate $\tilde{q}$ and $Dom(q_{i})$ and for the purposes of formalizing domain division, we rely on partition defined in Definition 2. One can see that $\cup\{\mathcal{D}_{i}\}$ is a partition of domain $X$ since each $\mathcal{D}_{i}$ is nonempty, $\bigcup\{\mathcal{D}_{i}\}=X$ and $\mathcal{D}_{i}\cap\mathcal{D}_{j}=\emptyset$ for $i\neq j$. For convenience, we denote $p=\cup\{\mathcal{D}_{i}\}$ a specific partition of $X$ consisting of $\mathcal{D}_{i}$s and denote $P=\cup\{p\}$ a set containing all possible partitions of domain $X$.

It is worth noting that the new PDE dynamic hybrid automaton needs a function to describe its state value. The additional spatial variable requires us to specify the values at different space locations while, in classical hybrid automata, the system only has finite values since the system only considers finite continuous state variables. The values of these state variables and discrete modes can be determined as long as the time is fixed, and the system is deterministic. In our model, for any time $t$, we regard that the PDE solution $u(\cdot,t)$ lies in the state space $U=\mathbb{R}^{X}$. In other words, for any given $x\in X$, $u(x,\cdot)$ represents a real-valued trajectory.

After we obtain $\tilde{q}$, $p$ and $u(\cdot,t)$, the state can be defined now.

Bearing the above in mind, the next step is to formalize the flow function.

Similar to classical hybrid automata, PDE equations simulate the dynamics of the system. The flow consists of a piecewise PDE equation and the corresponding domain partition. A simple flow function for the heater example can be formally represented as

The ON mode, triggered by conditions defined in the set $E$, for example (Figure 4), $u(x,t)\geq r_{1}$, then $q=q_{1}$ so that $f(x,t;q_{1})=f(x)$ while the OFF mode happens when $u(x,t)\leq r_{2}$, we set $q=q_{2}$, then $f(x,t;q_{2})=0$.

With the above discussion in mind, we can now provide a formal definition of a partial differential hybrid automaton.

In our framework, each mode contains only one PDE instead of a system of equations as in classical hybrid automata. Boundary conditions must not be violated and can be treated as a kind of invariant. They are imposed on the boundary of $X$ and act like constraints within the mode. Additionally, they are forced to be satisfied even during mode discrete transitions.

Revisiting the examples outlined at the very beginning, we construct two models to illustrate the idea of PDHA in Figures 1 and 2. In each model, the system is composed of two modes and each mode is governed by a PDE, the given boundary conditions, and the initial condition. In the heater model, we have $f_{i}(\Delta u,\nabla u,x,t)=\Delta u+f(x,t;q_{i})$.

The continuous trajectory is realized by moving a point satisfying a guard condition $G$ to another mode. In the heater model, a point in the mode ON where the temperature exceeds the threshold will be transferred to the mode OFF, meaning the heater located at this point is turned off while the other areas remain unchanged. At the same time, as the temperature at some points drops below some given bound, the heater at that location is turned on and the point at mode OFF will be moved to mode ON.

The same rule applies to the traffic model as well. The points in the free flow mode satisfying the guard condition $G_{1}$ will be moved to a congestion mode, and points in the congestion mode satisfying $G_{2}$ will be moved to the free flow mode.

A discrete treatment of PDEs requires the use of new tools to realize the functions we need. One of the most widely used sets of methods is called finite difference methods (FDMs). Generally speaking, FDMs are a family of numerical approaches for solving differential equations by replacing them with an approximation using difference equations. Combining FDMs and PDHA produces the following definition:

We present several schemes below to demonstrate how this technique approximates spatial derivatives.

After applying (2) and dropping the redundant part $O(h^{2})$, the heat equation becomes

We replace $u(i\times h,t)$ by $u_{i}(t)$ since the derivative with respect to space has been removed. Thus, $\dot{u}_{i}(t)$ is used to represent the time derivative at the grid point $i\times h$, and $f(i\times h,t)$ represents the input function valued at the same location.

Once the scheme has been chosen, the relation between our models can be described by the following definition.

Using a discretization scheme, we obtain discrete models of the state, transition, flow and reset function of the original PHDA. These discrete models are defined below.

In contrast to the continuous domain hybrid automaton which uses $\tilde{q}\times p$ to describe the current mode that the system occupies, a discrete partition combines the two notations and simplifies the expression.

A discrete state gives the whole picture of the system by combining a discrete partition and state values.

Everything listed above leads us to discrete flow.

A key observation is $DPH\preceq^{\mathcal{R}}PH$ if the discrete space partial differential hybrid automaton $DPH$ is obtained from a partial differential hybrid automaton $PH$ through $\mathcal{R}$.

The DSPHDA provides us with a set of properties that can easily be formalized and stated. Additionally, similar concepts that appear in HA are outlined below.

Now that we have defined the syntax of DSPDHA, we define the semantics in terms of executions.

An execution $\chi=(\tau,p_{d},u_{d})$ is called finite if $\chi$ is a finite sequence ending with a closed interval, infinite if $\tau$ is either an infinite sequence, or if $\sum_{i=0}^{\infty}(\tau_{i}^{{}^{\prime}}-\tau_{i})=\infty$, and Zeno if it is infinite but $\sum_{i=0}^{\infty}(\tau_{i}^{{}^{\prime}}-\tau_{i})<\infty$.

We use $\mathcal{E}_{DPH}(p_{d,0},u_{d,0})$ to denote the set of all executions of DSPDHA with initial condition $(p_{d,0},u_{d,0})\in Init$, $\mathcal{E}_{DPH}^{*}(p_{d,0},u_{d,0})$ to denote the set of all finite executions, $\mathcal{E}_{DPH}^{\infty}(p_{d,0},u_{d,0})$ to denote the set of all infinite executions, and $\mathcal{E}_{DPH}$ to denote the union of $\mathcal{E}_{DPH}(p_{d,0},u_{d,0})$ over all $(p_{d,0},u_{d,0})\in Init$.

The set of states reachable by a DSPDHA, $Reach_{DPH}$, is defined as

where $\tau=\{[\tau_{i},\tau_{i}^{{}^{\prime}}]\}_{i=0}^{N}$.

More definitions about DSPDHA are similar to those in the classical HA theory, and we omit them here for the simplicity of the presentation.

In this section, we explore the relations between classical hybrid automata, DSPDHA and PDHA. The central basis of introducing PDHA is extending HA with ODE dynamics to discretize PDE dynamics. To achieve this goal, some adjustments need to be made to accommodate the new framework within the classical hybrid automata scheme. It is our hope, that after these changes, the properties that hold for HA will hold for PDHA as well. This is the basis for the motivation to add an intermediate level (DSPDHA) lying between PDHA and HA, which captures the idea of PDHA and also stays close to classical HA. Using this framework, we make comparisons and illustrate useful results in the form of theorems.

In PDHA, we use space control modes and partitions $\tilde{q}\times p$ to replace a discrete mode $q$, a state function $\varphi(x)$ to replace state variables $X$. The weakness of this scheme is determining executions and transitions because $\tilde{q}\times p$ evolves continuously. DSPDHA overcomes these shortcomings by introducing a discrete state partition, $p_{d}$ which functions similarly to $q$ in HA. Along the same lines, a discrete flow function obtained from PDHA by FDM is essentially the same as a HA flow function. This allows us to assert some PDE system properties where we cannot otherwise reason with certainty due to a lack of information. We lay out several points below to illustrate how these connections are built.

1. (1)

   Any DSPDHA is a HA if $p_{d}$ has identical elements. Based on the knowledge that elements in $p_{d}$ represent the modes where each location resides, identical elements indicate all the locations are assigned to the same modes and the system stays at a single mode. Therefore, the given DSPDHA is a HA. Thus, the general dynamical properties of HA, for instance, the existence, uniqueness, continuous dependence, reachability, stability and invariant sets follows. (See [45]). On the other, the DSPDHA inherits the physical feature (like diffusion) from the PDE model, which we demonstrate in the heat model below.

2. (2)

   If the order of the time derivative of the PDE considered in a PDHA is $1$, the number of points $m$, in a discrete domain $X_{d}$ is equivalent to the size of flow functions in the corresponding DSPDHA. According to Definition 13, the discretization is made on each discrete point in $X_{d}$. As a result, we have $m$ equations eventually.

3. (3)

   If the order of the highest time derivative of a PDE is $n$, then the PDE can be reduced to a system of $n$ PDEs with a $1^{st}$ order time derivative. We begin the proof by a change of variables. Let us replace the $k^{st}$ order time derivative of $u$ by $v_{k}$ and add a new equation to the system once the replacement is done. In the end, we obtain a system of equations in the form below.

   $$\begin{cases}\dot{u}=v_{1},\\ \dot{v}_{1}=v_{2},\\ \dot{v}_{2}=v_{3},\\ ...,\\ \dot{v}_{n-2}=v_{n-1},\\ \dot{v}_{n-1}=f(u,u_{x},u_{xx},...,t).\end{cases}$$

   (9)

   where the original equation is $u^{(n)}=f(u,u_{x},u_{xx},...,t)$ and $u^{(n)}$ is its $n^{th}$ time derivative. For example, we can deal with the wave equation $u_{tt}-u_{xx}=0$.

4. (4)

   If the order of time derivative of the PDE considered in a PDHA is $n$ and the number of points in discrete domain $X_{d}$ is $m$, the size of flow functions in the corresponding DSPDHA is $n\cdot m$. Since $m$ determines the number of mesh points and each point is associated with an ODE ($1^{st}$ order PDE after spatial discretization), $m$ equals the dimension of the system of ODEs which is the flow function.

Besides investigating the relation and distinction between models above, an interesting observation which occurs in classical hybrid automata are discontinuity issues, especially under PDE dynamics. [40] discusses the propagation of discontinuities when the system switches mode. They are numerous reasons why discontinuities occur. For the transport equation with a nonlinear flux function, there does not always exist a smooth solution, and a weak solution has to be constructed to handle a shock or rarefaction wave [46]. Jump discontinuities for hyperbolic system are discussed in [47].

The situation discussed in [40] is likely to happen if the discrete transition takes place in our framework. Consider a segment of road governed by a transport equation and the model is $u_{t}+u_{x}=0$. Supposing the switching is triggered, and a new model $u_{t}+2u_{x}=0$ is introduced, the latter model which involves fast wave speed will push the front flow to move fast, and the traffic will start to accumulate at the boundary of the road which directly leads to a discontinuity.