[wpimath] Add RKF45 integration (#3047)

This is more stable than Runge-Kutta for systems with large elements in their A or B matrices.

Co-authored-by: Tyler Veness <calcmogul@gmail.com>

wpimath/src/test/native/cpp/system/NumericalIntegrationTest.cpp

@@ -0,0 +1,83 @@
+// Copyright (c) FIRST and other WPILib contributors.
+// Open Source Software; you can modify and/or share it under the terms of
+// the WPILib BSD license file in the root directory of this project.
+
+#include <gtest/gtest.h>
+
+#include <cmath>
+
+#include "frc/system/NumericalIntegration.h"
+
+// Tests that integrating dx/dt = e^x works.
+TEST(NumericalIntegrationTest, Exponential) {
+  Eigen::Matrix<double, 1, 1> y0;
+  y0(0) = 0.0;
+
+  Eigen::Matrix<double, 1, 1> y1 = frc::RK4(
+      [](Eigen::Matrix<double, 1, 1> x) {
+        Eigen::Matrix<double, 1, 1> y;
+        y(0) = std::exp(x(0));
+        return y;
+      },
+      y0, 0.1_s);
+  EXPECT_NEAR(y1(0), std::exp(0.1) - std::exp(0), 1e-3);
+}
+
+// Tests that integrating dx/dt = e^x works when we provide a U.
+TEST(NumericalIntegrationTest, ExponentialWithU) {
+  Eigen::Matrix<double, 1, 1> y0;
+  y0(0) = 0.0;
+
+  Eigen::Matrix<double, 1, 1> y1 = frc::RK4(
+      [](Eigen::Matrix<double, 1, 1> x, Eigen::Matrix<double, 1, 1> u) {
+        Eigen::Matrix<double, 1, 1> y;
+        y(0) = std::exp(u(0) * x(0));
+        return y;
+      },
+      y0, (Eigen::Matrix<double, 1, 1>() << 1.0).finished(), 0.1_s);
+  EXPECT_NEAR(y1(0), std::exp(0.1) - std::exp(0), 1e-3);
+}
+
+// Tests that integrating dx/dt = e^x works when we provide a U.
+TEST(NumericalIntegrationTest, ExponentialWithUAdaptive) {
+  Eigen::Matrix<double, 1, 1> y0;
+  y0(0) = 0.0;
+
+  Eigen::Matrix<double, 1, 1> y1 = frc::RKF45(
+      [](Eigen::Matrix<double, 1, 1> x, Eigen::Matrix<double, 1, 1> u) {
+        Eigen::Matrix<double, 1, 1> y;
+        y(0) = std::exp(x(0));
+        return y;
+      },
+      y0, (Eigen::Matrix<double, 1, 1>() << 0.0).finished(), 0.1_s);
+  EXPECT_NEAR(y1(0), std::exp(0.1) - std::exp(0), 1e-3);
+}
+
+namespace {
+Eigen::Matrix<double, 1, 1> RungeKuttaTimeVaryingSolution(double t) {
+  return (Eigen::Matrix<double, 1, 1>()
+          << 12.0 * std::exp(t) / (std::pow(std::exp(t) + 1.0, 2.0)))
+      .finished();
+}
+}  // namespace
+
+// Tests RungeKutta with a time varying solution.
+// Now, lets test RK4 with a time varying solution.  From
+// http://www2.hawaii.edu/~jmcfatri/math407/RungeKuttaTest.html:
+//   x' = x (2 / (e^t + 1) - 1)
+//
+// The true (analytical) solution is:
+//
+// x(t) = 12 * e^t / ((e^t + 1)^2)
+TEST(NumericalIntegrationTest, RungeKuttaTimeVarying) {
+  Eigen::Matrix<double, 1, 1> y0 = RungeKuttaTimeVaryingSolution(5.0);
+
+  Eigen::Matrix<double, 1, 1> y1 = frc::RungeKuttaTimeVarying(
+      [](units::second_t t, Eigen::Matrix<double, 1, 1> x) {
+        return (Eigen::Matrix<double, 1, 1>()
+                << x(0) * (2.0 / (std::exp(t.to<double>()) + 1.0) - 1.0))
+            .finished();
+      },
+      y0, 5_s, 1_s);
+  EXPECT_NEAR(y1(0), RungeKuttaTimeVaryingSolution(6.0)(0), 1e-3);
+}