Introduction

Introduction

What Is CDPSim

CDPSim is a toolkit for modeling and creating real-time simulations of physical systems described by differential equations. This add-on makes it possible to integrate a simulation model with CDP control systems, user interfaces and virtual 3D models – all running in real-time. One important application is to replace real hardware in Hardware In the Loop testing (HIL testing).

CDPSim is built on CDP technology, and the methodology is basically the same as for building CDP components. As for CDP components, you are not restricted when is comes to creating you own models. A high degree of accuracy is obtained by running the simulations at very small time steps, much smaller than the periodic processes of standard CDP components.

Integration with CDP

CDPSim uses state variables that are specializations of standard CDP signals, allowing monitoring by standard CDP tools, and allowing routing to other CDP components on the network. The simulation model behaves like any other CDP component, and can run either as a stand-alone application, as part of a control application or a mix of both.

This means that CDPSim models can easily replace real hardware in Hardware In the Loop tests (HIL tests), simply by routing the signals from the simulated signals instead of from the i/o signals.

How CDPSim Works

A simulation model is created by inheriting the class CDPSim::DynamicSimComponent, and then implementing the virtual method EvaluteDiffEquations(). This function defines the behavior of the simulated (sub)system in form of differential equations. To define the new values of the derivatives, the model can use the state variables of the model itself, the state variables of other models, standard CDP signals, CDP properties etc. The implementation is done in C++.

The simulation is performed by the SimulatorManager, calling this virtual method in the simulator model to update the derivatives of the state variables. CDPSim then integrates the derivatives to calculate new values for the state variables, using the selected integration method and time step.

Integration Algorithms

CDPSim comes with three different integration algorithms: Euler, Heun and Runge-Kutta. The Runge-Kutta algorithm supports adaptive time-step calculation. The user is able to add more algorithms by inheriting from IntegrationBase class.

The integration method can be selected individually for each simulation application. The choice of algorithm and simulation time step depends on the nature of the simulated system. Stiff systems require smaller time steps, and to prevent instability, these (sub)systems should be implemented as separate applications to enable running them at smaller time steps, as running the complete system at a sufficiently small time step could be too CPU intensive for the host computer.

Foundation

Describing a Physical System

A physical system can be complex. To be able to simulate it, it is necessary to formulate a simplified model of the system, so simple that it is understandable for your mind and solvable for the computer, and so complex that it really can be used to tell something about the actual physical system you are investigating. Example 3: Long Elastic Wire is an example of how to make a model.

The behavior of the model is described in terms of differential equations. Deriving these requires some insight to physics, and is not always straight-forward. Two useful methods are presented.

Newtons Second Law

This is a well-known and widely used law. It states that the sum of all forces acting on a body is equal to the mass of the body times its acceleration, i.e. `sum F = ma`. `F` is a force, `m` is the mass, and `a` is the acceleration, which is the double time derivate of the position, `a = ddot x`. Then it is just a matter of finding the forces.

Consider for example a mass hanging from a spring, as one of the examples is simulating. The forces acting on the mass are the gravity force `G` and the spring force `S`. Gravity force is given by `G = mg`, where `m` is body mass and `g` is the gravity acceleration. Hooke's law states that the spring force is given by `S = -kx`, where `k` is a spring constant describing the stiffness of the spring, and `x` is the displacement from equilibrium (where the spring is at rest). The sum of forces is `sum F = G + S = mg - kx`. Newtons second law now yields that `mg - kx = m ddot x` which is a differential equation. Expressed in terms of `ddot x`, this yields

`ddot x = g - (k)/(m) x`

Euler-Lagrange Equations

In more complex systems, in particular rotating systems, it can be hard to identify all forces. Another way of deriving differential equations is to consider the energy of the system, and then apply the Euler-Lagrange equations. There are two main types of mechanical energy: kinetic energy `T` (energy from movement, for example a driving car) and potential energy `V` (for example energy associated with position in the gravitational field, like a bike on the top of a hill). The Lagrangian of a system is defined as the difference between these two energy types, i.e. `L = T - V`. The Euler-Lagrange equations is now

`(d)/(dt) ((del L) / (del dot q_i)) - (del L) / (del q_i) = 0`

where `q_i` are generalized coordinates, for example `x` in a one-dimensional movement. The challenge is thus to find the energy of the system.

Consider again the mass hanging from a spring. The kinetic energy of the mass is `T = 1/2 mv^2` . The potential energy due to displacement from spring equilibrium and gravitational energy is `V = 1/2 kx^2 - mgx`. Then the Lagrangian is

`L = T - V = 1/2 m dot x^2 - 1/2 k x^2 + mgx`

Now the Euler-Lagrange equation yields

`m ddot x + kx - mg = 0 rArr ddot x = g - k/m x`

just as we found by Newtons second law.

One might point out that in this example, Newtons second law seems to be by far the easiest way to derive the equations. In other cases, however, like the rotary inverted pendulum, it would be close to impossible to find the equations from Newtons second law. The Euler-Lagrange equations is thus a tangled, but more powerful method for deriving differential equations.

Developing First Order Differential Equations

The numerical integration methods available in CDPSim only deal with first order differential equations. Many physical systems, like the mass hanging from a spring, are however described by second order differential equations. Fortunately, all second order differential equations can easily be rewritten as two first order equations, simply by introducing a new state variable equal to the time derivative of the existing state variable.

Consider the second order differential equation describing the mass hanging from a spring:

`ddot x = g - k/m x`

Introduce a new variable `v` equal to the time derivative of `x`, and substitute the time derivatives of `x` in the equation with this new variable:

`dot x = v`

`dot v = g - k/m x`

We now have first order differential equations, ready to be implemented in CDPSim.

Numerical Integration

There are many algorithms for making a numeric solution to ordinary differential equations on the form

`dot y(t) = f(t, y(t)),`

`y(t_0) = y_0`

The main idea is to assume that the solution is linear within small enough time steps, i.e.

`y_{n+1} = y_n + a dt`

where `dt` is the time step, and `a` is the slope. The challenge is to find a good prediction of the slope.

CDPSim comes with three different integration algorithms: the first order Euler-method, the second order Heun method and the fourth order Runge-Kutta method. The Runge-Kutta supports adaptive time steps. Higher order methods are heavier to run, but produces smaller errors, and can thus be run at larger time steps.

Euler Method

The Euler method simply states that the next `y`-value approximately equals the previous value plus the time derivative of `y` times a chosen time step `dt`:

`y_{n+1} = y_n + f(t_n,y_n) dt`

This method introduces an error per time step proportional to `dt^2`, which results in a total error in the order of magnitude `dt` when solving over a given range of `t`. The Euler method is thus said to be a first order method.

Heun Method

Heun's method is a so-called predictor-corrector method, using Euler's method as a predictor and a trapezoidal method as a corrector. The idea is to first calculate an intermediate value, and then weight this value equally with the previous value when calculating the value at next time step:

`tilde y_{n+1} = y_n + f(t_n,y_n) dt` (intermediate value)

`y_{n+1} = y_n + 1/2 ( f(t_n,y_n) + f(t_{n+1}, tilde y_{n+1}) ) dt`

The error is here proportional to `dt^3`.

Runge-Kutta Method

The Runge-Kutta method has a more comprehensive way of predicting the slope, given by the following equation:

`y_{n+1} = y_n + 1/6 ( k_1 + 2 k_2 + 2 k_3 + k_4 ) dt`

where

`k_1 = f (t_n, y_n)`

`k_2 = f (t_n + 1/2 dt, y_n + 1/2 k_1 dt)`

`k_3 = f (t_n + 1/2 dt, y_n + 1/2 k_2 dt)`

`k_4 = f (t_n + dt, y_n + k_3 dt)`

This method produces an error of order `dt^5`.

Real-time Performance

One of the main advantages with CDPSim, is that the simulations can run in real time. However, if the simulation gets too CPU intensive, real-time simulation can not be achieved. This occurs if the integration time step is too small, or the code is too heavy to run, either because of a large number of components, or long loops in the code due to large state variable arrays. There are two ways of dealing with this problem: either by adjusting integration time step, or by distributing the components to run on different computers.

Adjusting Integration Time Step in CDPSim

The integration time step can, just as the choice of integration algorithm, be adjusted in the SimulatorManager component by changing the TimeStep property.

Choosing a too small time step results in loss of real-time performance, while when using a too large time step, the solution of the differential equation becomes inaccurate, or the integration even blows up, and the program breaks down (for oscillating systems).

The limit where the integration becomes unstable is related to the highest underlying frequency in the system. Consider Example 3: Long Elastic Wire. Here, this highest frequency `f` is given by

`f = n sqrt{k / m}`

It is found that for the Runge-Kutta algorithm, the integration becomes unstable when

`dt approx sqrt{2}/f`

where `dt` is the integration time step.

Distribution of Components

A system consisting of many components can easily be distributed to run on different computers, simply by putting them to different CDP applications and routing the signals between them. Be aware that the routing frequency is limited to the process frequency of your SimulatorManager component. This is controlled by the fs property (which is different from simulation frequency controlled by TimeStep). The fs is usually around 100 Hz but may be increased to 10 kHz. On the other hand, if simulator components within the same application route SimSignals and StateVariables, then data is up to date regardless of process frequency.

A single component can also be split up for distributing purposes. For example, the mass-spring system (example 1) could be programmed such that one component handles the first half of the masses, and the second component handling the rest, routing the state variables through some additional signals. A major problem arises though, if the underlying frequency of the system is getting close to the process frequency, which might be the case for stiff systems. The process frequency can be increased if necessary, and it can also be an idea to find the slowest changing point in the system, and make the split here.

Further Reading

For further information, see the Getting Started, CDP Simulator Examples and Configuration Manual.