Control theory

What is it?

Formulas

$$\dot x = Ax + Bu,\ \ u = -Kx \implies \dot x = (A - BK)x$$

state feedback: choose $K$ to place the eigenvalues of $A - BK$

$$\min_u \int_0^\infty\big(x^{\mathsf T}Qx + u^{\mathsf T}Ru\big)\,\dd t$$

linear-quadratic regulator (LQR)

The mathematics behind it

• Ordinary differential equations★★★★★fundamental

  Plants and controllers are described by ODEs; control designs their closed-loop behaviour.

• Dynamical systems★★★★★fundamental

  Control is the art of shaping the dynamics of a system by feedback.

• Equilibria and stability★★★★★fundamental

  Stability analysis (eigenvalues, Lyapunov functions) is the core of control design.

• Laplace transform★★★★★fundamental

  Classical control (Bode, Nyquist, root locus) works with transfer functions in the Laplace domain.

• Rational functions★★★★★frequent

  Linear time-invariant systems have rational transfer functions $G(s)$.

• Complex numbers★★★★★frequent

  A linear system is stable when the poles of its transfer function lie in the left half of the complex plane.

• Matrices and linear maps★★★★★frequent

  State-space models $\dot x = Ax + Bu$ are linear maps.

• Differential and linear approximation★★★★★frequent

  Nonlinear plants are linearized around an operating point and controlled with linear theory.

• Partial fractions★★★★★frequent

  The time response of a system is found by splitting $G(s)$ into partial fractions and inverting each term.

• Total differential and linearization★★★★★frequent

  Linearization around an equilibrium gives the matrices $A$, $B$ of a state-space model.

• First-order linear equations★★★★★frequent

  First-order lags with time constant $\tau = 1/a$ are the building blocks of system models.

• Z-transform★★★★★frequent

  Digital controllers running on microcontrollers are designed in the $z$-domain.

• Bifurcations★★★★★advanced

  Hopf bifurcations explain how increasing a gain turns a stable loop into an oscillating one.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

What depends on it

• Trajectory optimization and MPC