Dear Liming,

As a doctoral student, you know the basics of calculus differentiation, integration, multiple integrals, elementary linear algebra, and ordinary differential equations. You may know something about partial differential equations and complex variables.

This is a good start to improve your skills in some advanced topics in math.

My advice is to start reading more about matrices and how to locate their eigenvalues( Roots of a characteristic polynomial) using the Gershgorin's circles, the quadratic forms, semi-positive and semi-negative matrices and their relations with the eigenvalues. Also, calculus of several variables, Jacobian matrices, Hessian matrices and the system of linear differential equations, equilibrium points, and stability theory. All mentioned topics are readable.

The following textbook is excellent to start.

[PDF]Differential Equations, Dynamical Systems, and Linear Algebra

https://www.math.upatras.gr/~bountis/files/def-eq.pdf

Hirsch, Morris W., 1933-. Differential equations, dynamical systems, and an introduction to chaos/Morris W. Hirsch, Stephen Smale, Robert L. Devaney. p. cm.

Later you will see the difference when you read any topic related to control theory.

Best regards

To develop the interest in control theory, you have to find your passions that you always dreamed of doing or knowing. It starts out with your curiosity, the desire to learn or know about something. You don't need to know math to become curious.

For example, you wonder how Segway could balance itself. Then, you find out that the self-balancing problem is similar to the Inverted Pendulum, and this leads you to explore further on how to derive the mathematical model and to develop a simple control system to balance the inverted pendulum. The CTMS is a very good website for beginners in Modern Control Theory.

http://ctms.engin.umich.edu/CTMS/index.php?example=InvertedPendulum§ion=SystemModeling

Technically, you should master the fundamentals of Differential Equations as they are the prerequisites for Modern Control Theory. You probably find the formulas for Laplace transforms, Closed-loop transfer function, and R.O.S.E. (Rise time, Overshoot, Settling time, steady state Error) difficult to understand because you don't know where they come from.

You can learn a lot just from watching YouTube tutorials on the subject. Also, learning controls using an Arduino board to control a motor via simple PID loops is an excellent, hands-on approach.

Dear Liming,

As a doctoral student, you know the basics of calculus differentiation, integration, multiple integrals, elementary linear algebra, and ordinary differential equations. You may know something about partial differential equations and complex variables.

This is a good start to improve your skills in some advanced topics in math.

My advice is to start reading more about matrices and how to locate their eigenvalues( Roots of a characteristic polynomial) using the Gershgorin's circles, the quadratic forms, semi-positive and semi-negative matrices and their relations with the eigenvalues. Also, calculus of several variables, Jacobian matrices, Hessian matrices and the system of linear differential equations, equilibrium points, and stability theory. All mentioned topics are readable.

The following textbook is excellent to start.

[PDF]Differential Equations, Dynamical Systems, and Linear Algebra

https://www.math.upatras.gr/~bountis/files/def-eq.pdf

Hirsch, Morris W., 1933-. Differential equations, dynamical systems, and an introduction to chaos/Morris W. Hirsch, Stephen Smale, Robert L. Devaney. p. cm.

Later you will see the difference when you read any topic related to control theory.

Best regards