Control Systems

Routh-Hurwitz stability criterion

The Routh-Hurwitz criterion tells whether every root of a characteristic polynomial lies in the open left half of the s-plane without computing the roots. For a polynomial a_n s^n + a_{n-1}s^{n-1} + \cdots + a_0 with a_n>0, the Routh array is built row by row, and each new entry is -\dfrac{1}{p}\det\begin{bmatrix} x_1 & x_{k+1} \\ y_1 & y_{k+1}\end{bmatrix}, where x and y are the two rows above and p=y_1 is the first entry of the row just above. The number of sign changes in the first column equals the number of right-half-plane roots, so the system is stable only when all first-column entries are positive and nonzero. A necessary condition is that every coefficient is present and has the same sign, so a missing term already signals instability or marginal stability. Special cases need care. If only a first-column entry becomes zero, the usual remedy is to replace it by a small positive \epsilon and take the limit. If an entire row vanishes, the auxiliary polynomial is formed from the row directly above, its roots are symmetric about the origin and lie on the imaginary axis, and the zero row is replaced by the coefficients of its derivative. For a cubic s^3+a_2s^2+a_1s+a_0 the criterion reduces to a simple test: all coefficients are positive and a_2a_1>a_0. Marginal stability occurs when a_2a_1=a_0, and the auxiliary equation a_2s^2+a_0=0 then gives poles at \pm j\sqrt{a_0/a_2}. For example, s^3+2s^2+4s+c is stable for 0<c<8 and has poles at \pm j2 when c=8. Routh-Hurwitz is also used to find a gain range, since the gain at which a first-column entry crosses zero is the limiting gain for stability. When a polynomial is even in s, its roots are symmetric about both axes. Substituting x=s^2 reduces the problem to a quadratic in x, and all roots of the original polynomial lie on the imaginary axis only when the x-roots are real and non-positive. This requires a nonnegative discriminant, a nonnegative product and a nonpositive sum of the x-roots. For instance, s^4+3s^2+c has all roots on the imaginary axis only if 9\ge 4c and c\ge 0.

Block diagram reduction

Block diagram reduction replaces a network of blocks, summing junctions and take-off points with a single equivalent transfer function. Three basic rules cover most cases. Blocks in series multiply, giving G_1G_2; blocks in parallel with the same sign add, giving G_1+G_2; and a negative feedback loop with forward gain G and feedback gain H reduces to G/(1+GH). When an input enters at more than one point, superposition applies: find the transfer function from each input to the output with the other inputs set to zero, then add the results. These transfer functions share the same denominator, the loop characteristic term, so a factor such as 1+G+GH found for one input must appear for the other input as well. A feedback signal that passes through a forward block before it is summed adds GH to the denominator, not H alone, so the loop gain must be traced carefully through each junction. For a disturbance entering at a point other than the reference, the error caused by that disturbance is found from the transfer function between the disturbance and the output with the reference set to zero. If the error is defined as reference minus output, its Laplace transform is -Y(s), and the steady-state error follows from the final value theorem, e(\infty)=\lim_{s\to 0}sE(s). Evaluating the disturbance gain at s=0 therefore gives the steady-state effect of a constant disturbance. Parallel branches with opposite signs can cancel exactly, so that an internal feedback path vanishes. In such diagrams the reduction must follow the correct order: combine parallel paths and feedback loops first, then form the overall ratio. A signal flow graph is an alternative description of the same linear equations, and Mason's gain formula gives its overall transfer function as T=\dfrac{\sum_k P_k\Delta_k}{\Delta}, where P_k are the forward path gains, \Delta=1-\sum L_i+\sum L_iL_j-\cdots is built from the loop gains, and \Delta_k is the same determinant with the loops touching path k removed. A graph represents a given block diagram only if Mason's formula gives the same overall transfer function, so the signs of the loop gains and the forward path gains must be checked.

Nyquist stability criterion

The Nyquist stability criterion counts closed-loop poles in the right half-plane from the open-loop frequency response. Let N be the number of clockwise encirclements of the critical point -1 by the plot of G(j\omega)H(j\omega) as \omega increases, and let P be the number of open-loop poles in the right half-plane. The number of closed-loop right-half-plane poles is then Z=N+P, and Z can never be negative, which constrains the value of N. Zeros of G(s)H(s) never enter this count, because only the open-loop poles and the encirclements matter. When the open-loop gain is a parameter K, the critical point in the plane of G(j\omega)H(j\omega) moves to -1/K, so stability for a chosen gain is found by counting encirclements of -1/K. Gain crossover is the frequency at which |G(j\omega)H(j\omega)|=1, that is, where the locus meets the unit circle. Phase crossover is the frequency at which the phase equals -180^\circ, that is, where the locus crosses the negative real axis. Both frequencies are read directly from the Nyquist plot and are used to define the gain and phase margins.

State-space controllability and observability

A linear system \dot{x}=Ax+Bu, y=Cx of order n is completely state controllable if the controllability matrix [B\;AB\;\cdots\;A^{n-1}B] has rank n. It is completely observable if the observability matrix [C;\,CA;\,\cdots;\,CA^{n-1}] has rank n. The two properties are independent, so a system can be controllable but not observable, observable but not controllable, both, or neither. For a diagonal A with distinct eigenvalues, a mode loses controllability when the corresponding row of B is zero, and it loses observability when the corresponding column of C is zero. With repeated eigenvalues, these row and column tests are not sufficient, and the rank tests must be applied to the full matrices, since rows of B can become proportional even when no row is zero. A transfer function obtained from a state model loses a pole-zero pair exactly when the realization is not controllable or not observable. A minimal realization, with the smallest number of states, is both controllable and observable.

Root locus construction from pole-zero pattern

A root locus is the path traced by the closed-loop poles as the gain K varies from zero to infinity in 1+KG(s)=0. Its branches start at the open-loop poles when K=0 and end at the open-loop zeros or at infinity when K\to\infty. The number of branches equals the number of open-loop poles n, and the number of branches that go to infinity is n-m, where m is the number of finite zeros. Real-axis segments of the locus lie to the left of an odd number of real poles and zeros, counted from the right. The asymptotes have angles \dfrac{(2k+1)180^\circ}{n-m} and meet the real axis at the centroid \sigma_a=\dfrac{\sum p_i-\sum z_j}{n-m}. When r poles coincide, r branches leave that point with equal angular spacing 360^\circ/r, so a cluster of five coincident poles gives departure angles 72 degrees apart. A given point s_0 lies on the locus only if the angle condition \sum\angle(s_0-z_j)-\sum\angle(s_0-p_i)=(2k+1)180^\circ holds. If it does, the gain at that point is K=1/|G(s_0)|, which must be real and positive. A point that fails the angle condition, or that gives a complex gain, is not a closed-loop pole for any gain.

Steady-state error for ramp input

For a unity feedback system, the steady-state error depends on the type of the open-loop transfer function G(s), which is the number of pure integrators at the origin. For a ramp input r(t)=At, the steady-state error is e_{ss}=A/K_v, where the velocity constant is K_v=\lim_{s\to 0}sG(s). A type 0 system has K_v=0, so its ramp error is infinite; a type 1 system has a finite nonzero K_v; and a type 2 system has zero ramp error. Only the factor s is removed when forming K_v, so the remaining poles and the gain enter only through their value at s=0. For example, if G(s)=\dfrac{5}{s(s+1)}, then K_v=5 and a ramp of slope 2 gives e_{ss}=0.4. The formula is valid only when the closed loop is stable, so stability must be checked before the error is computed.