Feedback Control

Output Feedback

With plant $G(s)$ and controller $C(s)$ in a unity negative feedback loop, the closed-loop and sensitivity functions are

$$T(s) = \frac{G(s)C(s)}{1 + G(s)C(s)}, \qquad S(s) = \frac{1}{1 + G(s)C(s)}$$

Goal 1: Stabilization All closed-loop poles, the roots of $1 + GC = 0$, in the open left half-plane.

Goal 2: Robustness to parameters How much the closed loop changes when the plant changes,

$$\frac{\partial T}{\partial G} = \frac{C}{(1 + GC)^2} \approx \frac{\Delta \text{ closed loop}}{\Delta \text{ open loop}}, \qquad \frac{G}{T} \cdot \frac{\partial T}{\partial G} = S(s)$$

so the relative sensitivity of $T$ to $G$ is $S$.

Goal 3: Tracking With $e(t) = r(t) - y(t)$, tracking means $e(t) \to 0$. For step references this requires $T(0) = 1$, equivalently $S(0) = 0$.

Goal 4: Disturbance rejection With a disturbance $D$ entering at the plant input,

$$Y(s) = \frac{CG}{1 + CG} R(s) + \frac{G}{1 + CG} D(s)$$

The simplest controller is proportional, $C(s) = K_p$.

State Feedback

With $u(t) = -Kx(t)$,

$$\dot{x} = (A - BK)x, \qquad y = (C - DK)x$$

so the closed-loop poles are the eigenvalues of $A - BK$.

• Controllability If $\mathrm{rank}\begin{bmatrix} B & AB & A^2B & \cdots & A^{n-1}B \end{bmatrix} = n$, the eigenvalues of $A - BK$ can be placed anywhere.
• Stabilizability There is a stabilizing $K$ iff $\mathrm{rank}\,[\lambda I - A \mid B] = n$ for every eigenvalue $\lambda$ with $\mathrm{Re}(\lambda) \ge 0$, i.e. every uncontrollable mode is already stable.

Output Tracking

To make $y(t) \to y_d$, use $u(t) = -Kx(t) + N y_d$ with $K$ placing the eigenvalues. Then

$$\dot{x} = (A - BK)x + BN y_d, \qquad y = (C - DK)x + DN y_d$$

At steady state $x_\infty = -(A - BK)^{-1}BN y_d$, so requiring $y_\infty = y_d$ gives (single output)

$$N = \frac{1}{D - (C - DK)(A - BK)^{-1}B}$$

Or find $T(s) = Y(s)/Y_d(s)$ and choose $N$ such that $T(0) = 1$.