The HyEQ Toolbox includes a series of blocks that model elements of a cyber-physical system (CPS). Those models are special instances of the hybrid systems blocks described in the Introduction to the Simulink-based Simulator , particularly the blocks that use embedded MATLAB functions. A cyber-physical system is given by the interconnection between a physical process, the plant , a computer algorithm used for control, the controller ; and the subsystems needed to interconnect the plant and the controller, i.e., the interfaces, converters, and signal conditioners . Most of the elements described in this section are presented in an extended form in [1].

In these notes, the temporal evolution of the variables of a cyber-physical system are captured using dynamical models. In this document, we advocate that hybrid dynamical system models can be employed to capture the behavior of cyber-physical systems. More precisely, the evolution of the continuous variables is captured by differential equations while the evolution of the discrete variables is captured by difference equations . These equations are typically nonlinear due to the complexity of the dynamics of those variables. Furthermore, conditions determining the change of the continuous and discrete variables according to the said equations/inclusions can be conveniently captured by functions of the variables, inputs, and outputs. The following Simulink blocks for cyber-physical components are provided with the HyEQ simulator:

Contents

Models of physical components

The physical components of a cyber-physical system include the analog elements, physical systems, and the environment. Consider a model of a physical system given by a differential equation \(\dot{x} = f(x,u)\) and a output map \(y = h(x,u)\) . Among the many possible models available, we capture the dynamics of the physical components by using the hybrid systems framework by

\[\begin{array}{ll} \dot{x} = f(x,u), & C := \mathbb{R}^n,\\ x^+ = g(x) =\emptyset, & D := \emptyset,\\ y = h(x,u) \end{array} \]

This model can be implemented with the Hybrid System with Inputs simulink block found at this link .

Models of cyber components

The cyber components of a cyber-physical system include those in charge of performing computations, implementing algorithms, and transmitting digital data over networks. The tasks performed by the code (at the software level) and the logic-based mechanisms (at the circuit level) involve variables that only change at discrete events, not necessarily periodically.

We denote the state variable of the cyber components by \(x_c \in \Upsilon\) , where \(\Upsilon \subset \mathbb{R}^{n_C}\) is the state space. The dynamics of \(\eta\) are defined by a difference equation with right-hand side defined by the map \(G_c\) . We let \(v \in {\cal V} \subset \mathbb{R}^{m_C}\) denote the input signals affecting the cyber components and \(\zeta\in\mathbb{R}^{r_C}\) to be the output defined by the output function \(\kappa\) , which is a function of the state \(x_c\) and of the input \(v\) . With these definitions, the general mathematical description of the cyber component is

\[x_c^+ = G_c(x_c,v), \quad \zeta = \kappa(x_c,v) \]

In certain cases, it would be needed to impose restrictions on the state and inputs to the cyber component. Such conditions can be modeled imposing that \(x_c\) and \(v\) belong to a subset of their state space, namely,

\[(x_c,v) \in D_c \subset \Upsilon \times {\cal V} \]

Next, we provide specific constructions of models of cyber components.

1. Pure Finite State Machines. A finite state machine (FSM) or deterministic finite automaton (DFA) is a system with inputs, states, and outputs taking values from discrete sets that are updated at discrete transitions (or jumps) triggered by its inputs. Then, given a FSM and an initial state \(q_0 \in Q\) , a transition to a state \(q_1 = \delta(q_0,v)\) is performed when an input \(v \in \Sigma\) is applied to it. After the transition, the output of the FSM is updated to \(\kappa(q_1)\) . This mechanism can be captured by the difference equation

\[ q^+ = \delta(q,v) \qquad \zeta = \kappa(q) \qquad (q,v) \in Q \times \Sigma \]

This model captures the dynamics the model of the cyber components given above with

\[ x_c = q, \qquad \Upsilon = Q, \qquad {\cal V} = \Sigma, \qquad G_c = \delta, \qquad D_c = \Upsilon\times{\cal V} \]

Note that there is no notion of ordinary time \(t\) associated with the FSM model above. An example of this model is presented in Finite State Machine . In addition, the FSM block shown at the top of this page can be used to model these type of systems.

2. Analog-to-Digital Converters. Analog-to-digital converters (ADCs), also called sampling devices , are commonly used to provide measurements of the physical systems to the cyber components. Their main function is to sample their input, which is usually the output of the sensors measuring the output \(y\) , at a given periodic rate \(T^*_s\) and to make these samples available to the embedded computer. A basic model for a sampling device consists of a timer state and a sample state. When the timer reaches the value of the sampling time \(T^*_s\) , the timer is reset to zero and the sample state is updated with the inputs to the sampling device.

The model for the sampling device we propose has both continuous and discrete dynamics. If the timer state has not reached \(T^*_s\) , then the dynamics are such that the timer state increases continuously with a constant, unitary rate. When \(T^*_s\) is reached, the timer state is reset to zero and the sample state is mapped to the inputs of the sampling device. To implement this mechanism, we employ a timer state \(\tau_s \in \mathbb{R}_{\geq}\) and a sample state \(m_s \in \mathbb{R}^{r_P}\) . The input to the sampling device is denoted by \(v_s \in \mathbb{R}^{r_p}\) . The model of the sampling devices is

\[\begin{array}{ll} \left[\begin{array}{cc} \dot{\tau}_s, \\ \dot{m}_s \end{array}\right] = \left[\begin{array}{cc} 1 \\ 0 \end{array}\right] & \textrm{ when } \tau_s \in [0,T^*_s] \\ \left[\begin{array}{cc} \tau_s^+ \\ m_s^+ \end{array}\right] = \left[\begin{array}{cc} 0 \\ v_s \end{array}\right] & \textrm{ when } \tau_s \geq T^*_s \end{array} \]

In practice, there exists a time, usually called the ADC acquisition time , between the triggering of the ADC with the sampling device and the update of its output. Such a delay limits the number of samples per second that the ADC can provide. Additionally, an ADC can store and process finite-length digital words, which causes quantization. The model above omits effects such as acquisition delays and quantization effects, but those can be incorporated if needed. In particular, quantization effects can be added to the ADC model by replacing the update law for \(m_s\) to \(m_s^+ = \textrm{round}(v_s)\) , where \(\textrm{round}(v_s)\) is the closest number to \(v_s\) that the machine precision can represent.

3. Digital-to-Analog Converters. The digital signals in the cyber components need to be converted to analog signals for their use in the physical world. A Digital-to-analog converter (DAC) performs such a task by converting digital signals into analog equivalents. One of the most common models for a DAC is the zero-order hold (ZOH) model. The output of a ZOH is updated at discrete time instants, typically periodically, and held constant in between updates, until new information is available at the next sampling time. We will model DACs as ZOH devices with dynamics similar to the ADC model. Let \(\tau_h\in\mathbb{R}_{\geq}\) be the timer state, \(m_h \in \mathbb{R}^{r_C}\) be the sample state (note that the value of \(h\) indicates the number of DACs in the interface), and \(v_h \in \mathbb{R}^{r_C}\) be the inputs of the DAC. Its operation is as follows. When \(\tau_h \leq 0\) , the timer state is reset to \(\tau_r\) and the sample state is updated with \(v_h\) (usually the output of the embedded computer), where \(\tau_r\in[T^{\textrm{min}},T^{\textrm{max}}]\) is a random variable that models the time in-between communication instants and \(T^{\textrm{min}}\leq T^{\textrm{max}}\) . A model that captures this mechanism is given by

\[\begin{array}{ll} \left[\begin{array}{cc} \dot{\tau}_h \\ \dot{m}_h \end{array}\right] \left[\begin{array}{cc} -1 \\ 0 \end{array}\right] &\textrm{ when } \tau_h \in [T^{\textrm{min}},T^{\textrm{max}}] \\ \left[\begin{array}{cc} \tau_h^+ \\ m_h^+ \end{array}\right] = \left[\begin{array}{cc} \tau_r \\ v_h \end{array}\right] & \textrm{ when } \tau_h \leq T^{\textrm{min}}. \end{array} \]

4. Digital Networks. The information transfer between the physical and cyber components, or between subsystems within the cyber components, might occur over a digital communication network. The communication links bridging each of these components are not capable of continuously transmitting information, but rather, they can only transmit sampled (and quantized) information at discrete time instants. Combining the ideas in the models of the converters in the previous items, we propose a model of a digital network link that has a variable that triggers the transfer of information provided at its input, and that stores that information until new information arrives. We assume that the transmission of information occurs at instants \(\{t_i\}_{i=1}^{i^*}\) , \(i^* \in \mathbb{N}\cup \{\infty\}\) , satisfying \(T^{*\min}_{N} \leq t_{i+1}-t_i\,\leq T^{*\max}_{N}\) for all \(i\in \{1,2,\ldots, i^*-1\}\) where \(T^{*\min}_{N}\) and \(T^{*\max}_{N}\) are constants satisfying \(T^{*\min}_{N}, T^{*\max}_{N} \in [0,\infty]\) , \(T^{*\min}_{N} \leq T^{*\max}_{N}\) , and \(i^*\) is the number of transmission events, which might be finite or infinite. The constant \(T^{*\min}_{N}\) determines the minimum possible time in between transmissions while the constant \(T^{*\max}_{N}\) defines the maximum amount of time elapsed between transmissions. In this way, a communication channel that allows transmission events at a high rate would have \(T^{*\min}_{N}\) small (zero for infinitely fast transmissions), while one with slow data rate would have \(T^{*\min}_{N}\) large. The constant \(T^{*\max}_{N}\) determines how often transmissions may take place. Note that the constants \(T^{*\min}_{N}\) and \(T^{*\max}_{N}\) can be generalized to functions so as to change according to other states or inputs.

At every \(t_i\) , the information at the input \(v_N\) of the communication link is used to update the internal variable \(m_N\) , which is accessible at the output end of the network and remains constant between communication events. This internal variable acts as an information buffer, which can contain not only the latest piece of information transmitted but also previously transmitted information. A record of the number of communication events is logged in the internal variable \(j_N\) . At communication events, the value of \(j_N\) gets updated and remains constant in-between events. A mathematical model capturing the said mechanism is given by

\[\begin{array}{ll} \left[\begin{array}{c} \dot{\tau}_N \\ \dot{m}_N \\ \dot{j}_N \end{array}\right] = \left[\begin{array}{c} -1 \\ 0 \\ 0 \end{array}\right] & \textrm{ when } \tau_N \in [0,T^{*\max}_{N}] \\ \\ \left\{\begin{array}{c} \tau_N^+ \in [T^{*\min}_{N},T^{*\max}_{N}] \\ \left[\begin{array}{c} m_N^+ \\ j_N^+ \end{array}\right] = \left[\begin{array}{c} v_N \\ j_N+1 \end{array}\right] \end{array}\right. & \textrm{ when } \tau_N \leq 0 \end{array} \]

Note that the update law for \(\tau_N\) at jumps is given in terms of a difference inclusion, which implies that the new value of \(\tau_N\) is taken from the set \([T^{*\min}_{N},T^{*\max}_{N}]\) . The dimension of the states and the input would depend on the type of components that connect to and from it, and also the size of data transmitted and buffered. Similar to the models proposed for conversion, the model of the digital link does not include delays nor quantization, but such effects can be incorporated if needed.

References

[1] R. G. Sanfelice. CMPE142 Class Notes - Introduction to Cyber-Physical Systems. https://hybrid.soe.ucsc.edu/cmpe149-249-2016

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