Control Systems

Closed-loop transfer function reduction

A closed-loop system is reduced to a single transfer function by naming the intermediate signal at each summing junction, writing one equation per block, and eliminating the intermediate signals. For a negative feedback loop with forward path G(s) and feedback path H(s), the transfer function from the reference R(s) to the output C(s) is \dfrac{C(s)}{R(s)}=\dfrac{G(s)}{1+G(s)H(s)}. The transfer function from R(s) to the error E(s) is different, and it equals \dfrac{E(s)}{R(s)}=\dfrac{1}{1+G(s)H(s)}. Therefore, the signal named in a question must be identified before the reduction is written. The sign at each junction determines the denominator. Negative feedback gives 1+GH, while positive feedback gives 1-GH, and a positive loop gain can place a closed-loop pole in the right half plane. When a reference path carries a net minus sign, the numerator picks up that sign, and the loop gain appears in the denominator with its own sign. For example, if the feedback loop gain is 2G with positive feedback, the denominator is 1-2G. For a unity-feedback second-order loop written as \dfrac{\omega_n^2}{s^2+2\zeta\omega_n s+\omega_n^2}, the damped natural frequency is \omega_d=\omega_n\sqrt{1-\zeta^2}, and this is the frequency of the oscillation in the step response.

Bode magnitude plot analysis

The Bode magnitude plot is drawn in decibels, 20\log_{10}|G(j\omega)|, and its asymptotes are straight lines with slopes that change at the corner frequencies. A simple pole or zero changes the slope by \mp 20 dB/decade at its corner frequency, and a pure integrator 1/s gives a slope of -20 dB/decade over the whole range. A zero in the numerator raises the slope by +20 dB/decade, and a pole lowers it by -20 dB/decade. The low-frequency asymptote of a type-1 system with a single integrator gives the gain constant. If the magnitude falls at -20 dB/decade and crosses 0 dB at \omega_c, then |G(j\omega)|=K/\omega near that region, so K=\omega_c. The velocity constant is K_v=\lim_{s\to 0}sG(s)=K, and for a unity-feedback system with a unit ramp input the steady-state error is e_{ss}=1/K_v. A second-order section with a damping ratio \zeta<0.707 shows a resonant peak at \omega_0 whose height is approximately 1/(2\zeta), which is the quality factor Q. For an RLC circuit, Q=\dfrac{1}{R}\sqrt{\dfrac{L}{C}}, so a measured peak gain fixes the resistance. A magnitude plot that is constant at all frequencies identifies an all-pass system, whose poles and zeros are mirror images about the imaginary axis.

Nyquist plot and contour mapping

The Nyquist criterion relates the number of encirclements of the critical point -1+j0 by the Nyquist plot of G(s)H(s) to the number of poles and zeros in the right half plane. It is written as Z=N+P, where Z is the number of closed-loop poles in the right half plane, P is the number of open-loop poles in the right half plane, and N is the number of clockwise encirclements of -1. An anticlockwise encirclement therefore counts as N=-1. A contour that passes through a pole on the imaginary axis, such as a pole at the origin, must be indented around that pole. The indentation is usually taken through the right half plane, and the small arc it creates is mapped into a large arc on the Nyquist plot, which can add encirclements of -1. For a transfer function G(s)H(s) whose numerator and denominator have the same degree, the large arc of the contour maps to a single point, namely the limit of G(s)H(s) as s\to\infty. This limit is zero only when the transfer function is strictly proper.

Routh-Hurwitz stability criterion

The Routh-Hurwitz criterion determines the number of roots of a polynomial in the right half plane without solving for the roots. The characteristic equation of a closed-loop system is 1+G(s)H(s)=0, and it is written as a polynomial in s. A necessary condition for stability is that all coefficients of the polynomial be present and have the same sign. For a second-order polynomial a_2s^2+a_1s+a_0 with a_2>0, this condition is also sufficient, so the roots lie in the left half plane if and only if a_1>0 and a_0>0. The constant term often depends on a gain K, and the stability range of K is found by requiring that constant term to be positive. If the constant term becomes negative for a range of K, one real root moves into the right half plane, and the system is unstable in that range. For higher-order polynomials, the Routh array is built from the coefficients, and each sign change in the first column counts one root in the right half plane. A zero in the first column or a complete row of zeros needs special treatment before the array can be read.

Second-order damping ratio and natural frequency

A standard second-order system is written as \dfrac{\omega_n^2}{s^2+2\zeta\omega_n s+\omega_n^2}, where \omega_n is the undamped natural frequency and \zeta is the damping ratio. Comparing the coefficients of a closed-loop denominator with this form gives \omega_n from the constant term and \zeta from the coefficient of s. When the poles are drawn on the s-plane, the pole has real part -\zeta\omega_n and imaginary part \omega_n\sqrt{1-\zeta^2}. The angle \theta of the pole measured from the negative real axis satisfies \cos\theta=\zeta, and its distance from the origin is \omega_n. The 2% settling time of an underdamped second-order system is approximately T_s\approx\dfrac{4}{\zeta\omega_n}, so a larger real part of the pole gives a shorter settling time. Comparing two systems therefore requires comparing their values of \zeta\omega_n, not their natural frequencies alone.

Lead compensator frequency and gain

A lead compensator has the form K(s)=\dfrac{1+s/a}{1+s/(\beta a)} with \beta>1. Its zero lies at \omega=a and its pole lies at \omega=\beta a, so the phase is positive between the two corner frequencies. The maximum phase lead \phi_m occurs at the geometric mean of the corner frequencies, \omega_m=a\sqrt{\beta}, and it satisfies \sin\phi_m=\dfrac{\beta-1}{\beta+1}. At \omega_m the magnitude of the lead compensator is \sqrt{\beta}, so the gain amplification in decibels is 20\log_{10}\sqrt{\beta}. Specifying the frequency of maximum lead and the gain at that frequency gives two equations that determine a and \beta. In a design problem, the derivative-type compensator D(s)=1+K_D s gives a phase lead of \tan^{-1}(\omega K_D) at frequency \omega, and its magnitude is \sqrt{1+\omega^2K_D^2}. Choosing K_D so that the magnitude at the desired crossover frequency is the required value fixes the compensator gain.

First-order system time constant from step response

A stable first-order system with DC gain K has the transfer function \dfrac{K}{\tau s+1}, and its unit step response is y(t)=K\left(1-e^{-t/\tau}\right). The time constant \tau is the time at which the response reaches 63.2\% of its final value, and the response is within 2% of its final value after about 4\tau. The final value y(\infty)=K can be read from the data, and then \tau can be found from any point with 0<y(t)<y(\infty) using \tau=\dfrac{t}{\ln\left(\dfrac{y(\infty)}{y(\infty)-y(t)}\right)}. For the same system, the 10% to 90% rise time is T_r=\ln 9\,\tau\approx 2.2\tau, which is the time between the two crossings. The pole of the system is s=-1/\tau, so the time constant is found directly from the pole location. Two or three data points with small errors can give slightly different values of \tau, so the closest value is taken from the data.

State-space to transfer function conversion

A linear time-invariant state-space model \dot{\mathbf{x}}=A\mathbf{x}+B u, y=C\mathbf{x}+Du has the transfer function G(s)=C(sI-A)^{-1}B+D. The poles of the system are the roots of the characteristic polynomial \det(sI-A)=0, and they are the eigenvalues of A. The matrices B and C do not change the poles, although they can cancel a pole in the transfer function. To compute the transfer function, the adjugate form (sI-A)^{-1}=\dfrac{\operatorname{adj}(sI-A)}{\det(sI-A)} is used, and then the product C\,\operatorname{adj}(sI-A)\,B is simplified. For a 2\times 2 matrix A=\begin{bmatrix}a & b\\ -a & 0\end{bmatrix}, the characteristic polynomial is s^2-as+ab, and the numerator follows from the same product. When the desired poles are given, the characteristic polynomial is formed by multiplying the factors (s-p_1)(s-p_2), and its coefficients are matched with those of \det(sI-A) to find the unknown parameters of A.