| Using Simulink | |
Choosing a Variable-Step Solver
When the Type control of the Solver configuration pane is set to variable-step, the configuration pane's Solver control allows you to choose one of the set of variable-step solvers that Simulink provides. As with fixed-step solvers in Simulink, the set of variable-step solvers comprises a discrete solver and a subset of continuous solvers. Both types compute the time of the next time step by adding a step size to the time of the current time that varies depending on the rate of change of the model's states. The continuous solvers, in addition, use numerical integration to compute the values of the model's continuous states at the next time step. Both types of solvers rely on blocks that define the model's discrete states to compute the values of the discrete states that each defines.
The choice between the two types of solvers depends on whether the blocks in your model defines states and, if so, the kind of states that they define. If your model defines no states or defines only discrete states, you should select the discrete solver. In fact, if a model has no states or only discrete states, Simulink will use the discrete solver to simulate the model even if the model specifies a continuous solver.
About Variable-Step Continuous Solvers
Simulink variable-step solvers vary the step size during the simulation, reducing the step size to increase accuracy when a model's states are changing rapidly and increasing the step size to avoid taking unnecessary steps when the model's states are changing slowly. Computing the step size adds to the computational overhead at each step but can reduce the total number of steps, and hence simulation time, required to maintain a specified level of accuracy for models with rapidly changing or piecewise continuous states.
Simulink provides the following variable-step continuous solvers:
ode45 is based on an explicit Runge-Kutta (4,5) formula, the Dormand-Prince pair. It is a one-step solver; that is, in computing y(tn), it needs only the solution at the immediately preceding time point, y(tn-1). In general, ode45 is the best solver to apply as a first try for most problems. For this reason, ode45 is the default solver used by Simulink for models with continuous states.
ode23 is also based on an explicit Runge-Kutta (2,3) pair of Bogacki and Shampine. It can be more efficient than ode45 at crude tolerances and in the presence of mild stiffness. ode23 is a one-step solver.
ode113 is a variable-order Adams-Bashforth-Moulton PECE solver. It can be more efficient than ode45 at stringent tolerances. ode113 is a multistep solver; that is, it normally needs the solutions at several preceding time points to compute the current solution.
ode15s is a variable-order solver based on the numerical differentiation formulas (NDFs). These are related to but are more efficient than the backward differentiation formulas, BDFs (also known as Gear's method). Like ode113, ode15s is a multistep method solver. If you suspect that a problem is stiff, or if ode45 failed or was very inefficient, try ode15s.
ode23s is based on a modified Rosenbrock formula of order 2. Because it is a one-step solver, it can be more efficient than ode15s at crude tolerances. It can solve some kinds of stiff problems for which ode15s is not effective.
ode23t is an implementation of the trapezoidal rule using a "free" interpolant. Use this solver if the problem is only moderately stiff and you need a solution without numerical damping.
ode23tb is an implementation of TR-BDF2, an implicit Runge-Kutta formula with a first stage that is a trapezoidal rule step and a second stage that is a backward differentiation formula of order two. By construction, the same iteration matrix is used in evaluating both stages. Like ode23s, this solver can be more efficient than ode15s at crude tolerances.
Specifying Variable-Step Solver Error Tolerances
The solvers use standard local error control techniques to monitor the error at each time step. During each time step, the solvers compute the state values at the end of the step and also determine the local error, the estimated error of these state values. They then compare the local error to the acceptable error, which is a function of the relative tolerance (rtol) and absolute tolerance (atol). If the error is greater than the acceptable error for any state, the solver reduces the step size and tries again:
The error for the ith state, ei, is required to satisfy
The following figure shows a plot of a state and the regions in which the acceptable error is determined by the relative tolerance and the absolute tolerance.
If you specify auto (the default), Simulink sets the absolute tolerance for each state initially to 1e-6. As the simulation progresses, Simulink resets the absolute tolerance for each state to the maximum value that the state has assumed thus far times the relative tolerance for that state. Thus, if a state goes from 0 to 1 and reltol is 1e-3, then by the end of the simulation the abstol is set to 1e-3 also. If a state goes from 0 to 1000, then the abstol is set to 1.
If the computed setting is not suitable, you can determine an appropriate setting yourself. You might have to run a simulation more than once to determine an appropriate value for the absolute tolerance.
The Integrator, Transfer Fcn, State-Space, and Zero-Pole blocks allow you to specify absolute tolerance values for solving the model states that they compute or that determine their output. The absolute tolerance values that you specify for these blocks override the global settings in the Configuration Parameters dialog box. You might want to override the global setting in this way, if the global setting does not provide sufficient error control for all of your model's states, for example, because they vary widely in magnitude.
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