Control System Analysis

Nyquist stability criterion

The Nyquist stability criterion links the closed-loop stability of a feedback system to the way the Nyquist plot of the open-loop function GH(j\omega) encircles the critical point -1+j0. Let P be the number of poles of GH(s) in the right half of the s-plane, Z the number of zeros of 1+GH(s) in that half-plane, and N the net number of clockwise encirclements of -1+j0. The relation is N = Z - P. A closed loop is stable (Z=0) only when the plot wraps the critical point P times in the anticlockwise direction.

Two details decide most questions. First, the sign of N depends on the convention and on the direction in which the contour is traversed. A contour drawn counterclockwise reverses the roles, so the encirclement count must be read against that orientation. Second, the criterion can be applied to the function 1+GH about the origin, which is equivalent to applying it to GH about -1+j0. Counting encirclements of the origin by a rational function F(s) is simple: the net clockwise count equals Z-P, where Z and P are the numbers of zeros and poles of F(s) in the right half-plane.

Example. For F(s)=\dfrac{s-3}{s+1} we have Z=1 and P=0, so the plot of F encircles the origin once clockwise. Right-half-plane poles of the open loop must always be included in the count P.

Gain and phase margin

Gain margin and phase margin measure how far a loop is from instability at the frequencies that matter most. The gain crossover frequency \omega_{gc} is where |GH(j\omega)|=1, and the phase margin is PM = 180^\circ + \angle GH(j\omega_{gc}). The phase crossover frequency \omega_{pc} is where \angle GH(j\omega)=-180^\circ, and the gain margin is GM = 1/|GH(j\omega_{pc})|, or GM_{dB} = -20\log_{10}|GH(j\omega_{pc})| in decibels.

The point where the Nyquist plot meets the unit circle gives the phase margin, and the point where it meets the negative real axis gives the gain margin. Phase values must be read with their sign: a phase below -180^\circ at gain crossover gives a negative phase margin, which means instability.

A zero in the right half-plane contributes the same magnitude as a left-half-plane zero but a phase lag instead of a lead, so it reduces the phase margin. Example: for GH(s)=\dfrac{2}{(s+1)^3}, the phase is -3\arctan\omega, which equals -180^\circ at \omega=\sqrt{3} rad/s. There |GH|=2/8=1/4, so GM=4, or about 12 dB. When a gain K must give a specified margin, invert |GH(j\omega_{pc})|=1/GM for K.

Pole location and transient response

The location of the poles in the s-plane determines the shape of the transient response. A real pole s=-a gives an exponential term e^{-at}. A complex pair s=-\sigma\pm j\omega_d gives a damped oscillation with frequency \omega_d and decay rate \sigma. A decaying oscillation therefore requires a complex-conjugate pair with negative real part, and such a pair can exist only in a system of order two or higher.

For a standard second-order system with 0<\xi<1, the poles are s=\omega_n(-\xi\pm j\sqrt{1-\xi^2}), where \omega_n is the natural frequency and \xi the damping ratio. The distance from the origin is |s|=\omega_n, and the angle \theta measured from the negative real axis satisfies \cos\theta=\xi. A region bounded by a ray from the origin therefore fixes a minimum damping ratio, and a circular boundary fixes a minimum natural frequency.

Real poles give non-oscillatory terms, complex pairs give oscillation, and the real part sets the decay rate, so a pole farther left decays faster. Additional real poles may accompany an oscillating pair, so their absence cannot be inferred from the oscillation alone.

Steady-state error of unit step

For a unity-feedback system with forward path G(s), the steady-state error to a unit step depends on the dc gain of the loop. The final-value theorem y(\infty)=\lim_{s\to0}sY(s) gives the steady-state output whenever all poles of sY(s) lie in the open left half-plane. The error is then the final output value minus the reference, and the sign convention must be fixed before the limit is taken.

In a closed loop with constant feedback, the steady-state output equals the closed-loop dc gain, so the error to a unit step is 1 minus that dc gain.

Example. For G(s)=\dfrac{K}{s+2} in unity feedback, the closed-loop dc gain is \dfrac{K}{2+K}, so e_{ss}=\dfrac{2}{2+K}. For K=3 the error is 0.4.

Routh-Hurwitz stability criterion

The Routh-Hurwitz criterion decides stability from the coefficients of a polynomial without computing its roots. Arrange the coefficients in alternating rows, compute the first-column entries, and count sign changes. The number of sign changes equals the number of roots in the right half-plane, and a zero in the first column signals roots on the imaginary axis.

For a cubic a_3s^3+a_2s^2+a_1s+a_0 with a_3>0, the stability conditions are that all coefficients are positive and a_2a_1>a_3a_0. Failure of either condition means that at least one root lies in the right half-plane or on the imaginary axis. Since the second condition involves a product of coefficients, the test is often used to find a range of a gain.

The criterion also applies to numerator polynomials. The zeros of a transfer function are the roots of its numerator, so a Routh array of the numerator counts zeros in the right half-plane. When the first column has no sign change and no zero entries, all numerator zeros lie in the left half-plane.

Example. For s^3+3s^2+4s+2 all coefficients are positive and 3\cdot4>2, so all roots lie in the left half-plane.

Closed-loop stability with proportional gain

With a plant G(s) and a proportional controller C=K in unity feedback, the characteristic equation is 1+KG(s)=0. Writing this as a polynomial in s whose coefficients contain K lets the Routh-Hurwitz conditions give the range of K for which the loop is stable. The constant term often sets the sign requirement: if it is proportional to K, stability is possible only for one sign of K.

Example. For G(s)=\dfrac{1}{s(s+1)(s+2)} the characteristic polynomial is s^3+3s^2+2s+K. The conditions are that all coefficients are positive and 3\cdot2>K, giving 0<K<6.

Sinusoidal steady-state response

For a stable linear system with transfer function G(s) driven by x(t)=A\sin\omega t, the steady-state output is y_{ss}(t)=A|G(j\omega)|\sin\left(\omega t+\angle G(j\omega)\right). The amplitude ratio is |G(j\omega)| and the phase shift is \angle G(j\omega), both obtained by substituting s=j\omega. The transient terms decay with time, so only this sinusoid remains.

Example. For G(s)=\dfrac{10}{s+10} and input 2\sin 10t, |G(j10)|=\dfrac{10}{\sqrt{200}}\approx0.707. The steady-state amplitude is therefore about 1.41, and the phase shift is -45^\circ.

Closed-loop stability analysis

For a unity negative-feedback loop with loop gain L(s), the closed-loop transfer function is T(s)=\dfrac{L(s)}{1+L(s)}, and its poles are the roots of 1+L(s)=0. Stability is judged from these poles, while causality is judged from the region of convergence of the inverse transform. A causal system has its region of convergence to the right of its rightmost pole.

Causality and stability are independent properties. A proper rational transfer function with all poles in the right half-plane is causal and unstable. Example: T(s)=\dfrac{1}{s-1} with the region of convergence \text{Re}(s)>1 is causal, and it is unstable because the imaginary axis is not inside that region.

State-space realization order

The number of state variables in a minimal state-space realization equals the degree of the transfer function's denominator after all common factors between numerator and denominator are cancelled. Pole-zero cancellation reduces the order, so the reduced form must be used.

Example. G(s)=\dfrac{s+2}{(s+1)(s+4)} needs two states, whereas G(s)=\dfrac{s+1}{(s+1)(s+3)} reduces to \dfrac{1}{s+3} and needs one. The zeros do not set the number of states, so a numerator of the same degree as the denominator requires a direct feedthrough term but no extra states.