Systems theory; control

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• ConceptControllability

  A control system is controllable if every state can be steered to every other state by a suitable choice of the control. It is the first question of control theory: before asking how to steer well, one asks whether steering is possible at all. For linear systems $x' = A x + B u$ it is decided by the Kalman rank condition, and for systems without drift by the Chow–Rashevskii theorem.

• ConceptObservability

  A control system with an output is observable if the initial state can be recovered from the known input and the measured output over time. Observability matters because in practice only some quantities can be measured, while a feedback law usually needs the full state. For linear systems it is dual to controllability: the pair $(A, C)$ is observable exactly when $(A^T, C^T)$ is controllable.

• TheoremSeparation principle

  Consider the linear-quadratic-Gaussian control problem on $[0, T]$ with matrices $A, B, C, Q, R, F$, noise covariances $\Xi$ and $\Theta$, and initial mean $m_0$ and covariance $\Sigma_0$. Let $P$ solve the Riccati differential equation $-P' = A^{T} P + P A - P B R^{-1} B^{T} P + Q$ with $P(T) = F$. Let $\Sigma$ solve the filter Riccati equation $\Sigma' = A \Sigma + \Sigma A^{T} + \Xi - \Sigma C^{T} \Theta^{-1} C \Sigma$ with $\Sigma(0) = \Sigma_0$. Let $\hat{x}$ be the Kalman–Bucy filter, the solution of $$d\hat{x} = (A \hat{x} + B u) \, dt + \Sigma C^{T} \Theta^{-1} (dy - C \hat{x} \, dt), \qquad \hat{x}(0) = m_0.$$ Then the control $u^*(t) = -R^{-1} B^{T} P(t) \hat{x}(t)$ is admissible and optimal among all admissible controls. Here $\hat{x}(t)$ is the conditional expectation of $x(t)$ given the observations $y(s)$, $s \le t$.

• ConceptZ-transform

  The Z-transform turns a sequence $(x_n)$ of complex numbers into the function $X(z) = \sum_n x_n z^{-n}$ of a complex variable $z$, defined where the series converges. It is the discrete analogue of the Laplace transform. It turns shifts of sequences into multiplication by powers of $z$ and convolution into products. So it is the main tool for linear difference equations and for discrete-time signals and control systems.

• TheoremFinal value theorem

  Let $f \colon [0, \infty) \to \C$ be integrable on each interval $[0, T]$. Assume that the limit $f(\infty) = \lim_{t \to \infty} f(t)$ exists and is finite. Then the Laplace integral $F(s)$ of $f$ converges for every real $s > 0$, and $$\lim_{s \to 0^+} s F(s) = f(\infty),$$ where $s$ tends to $0$ through positive real values.

• ConceptHybrid system

  A hybrid system is a dynamical system whose state has a continuous part and a discrete part, called the mode. The continuous part flows by ordinary differential equations, and the mode changes by jumps. In each mode the continuous state follows its own vector field. When it reaches a switching condition, the mode changes and the continuous state may be reset. Hybrid systems model mechanical systems with impacts, digital controllers acting on physical plants, and switching circuits.