**OUTLINE**

8.1 Review of Integration and Differentiation

8.2 Numerical Integration

8.3 Numerical Differentiation

8.4 Analytical Solutions to Differential Equations

8.5 Numerical Methods for Differential Equations

8.6 Extension to Higher-Order Equations

8.7 ODE Solvers in the Control System Toolbox

8.8 Advanced Solver Syntax

8.9 Summary

Problems

This chapter covers numerical methods for computing integrals and derivatives and for solving ordinary differential equations. Some integrals cannot be evaluated analytically, and we need to compute them numerically with an approximate method. In addition, it is often necessary to use data to estimate rates of change, and this process requires a numerical estimate of the derivative. Finally, many differential equations cannot be solved analytically, and so we must solve them using appropriate numerical techniques.

When you have finished this you should be able to:

¤ Use MATLAB to numerically evaluate integrals.

¤ Use numerical methods with MATLAB to estimate derivatives.

¤ Use the analytical expressions for simple integrals and derivatives to check the accuracy of numerical methods.

¤ Use MATLAB’s numerical differential equation solvers to obtain solutions.

¤ Use the analytical solutions of simple differential equations to check the accuracy of numerical methods.