Parameter-Space Stability & D-Decomposition

Root locus varies one scalar gain and follows pole trajectories in the root plane. D-decomposition reverses the view: choose the acceptable root domain $D$, map its boundary into a space of controller or plant parameters, and label the resulting parameter regions by root count. For one or two gains this can reveal the complete stabilizing set, its disconnected components, and the exact mechanism at each boundary.

Root-count invariance #

Let

$$a(s,k)=a_n(k)s^n+\cdots+a_1(k)s+a_0(k)$$

be a real characteristic polynomial depending continuously on parameter vector $k$. Polynomial roots vary continuously as a multiset while degree remains fixed. Therefore the number of roots inside an open domain $D$ cannot change as $k$ moves continuously unless:

1. a root crosses the boundary $\partial D$; or
2. the leading coefficient vanishes and a root passes through infinity.

The parameter values causing those events form the D-boundary. Removing that boundary partitions parameter space into connected components. The root count inside $D$ is constant within each component, so one correctly evaluated interior point labels the entire component.

This is a topological argument, not a plotting heuristic. A numerical sweep is trustworthy only if it captures every boundary component, singular case and connected region required by the argument.

Continuous-time boundaries #

For Hurwitz stability, $\partial D$ is the imaginary axis. The ordinary crossing boundary satisfies

$$a(j\omega,k)=0,\qquad 0\le\omega<\infty.$$

Because coefficients are real, negative frequencies give the conjugate equation and need not be traced separately. Split into real equations:

$$\mathop{\mathrm{Re}}a(j\omega,k)=0,$$

$$\mathop{\mathrm{Im}}a(j\omega,k)=0.$$

Three mechanisms deserve separate handling:

The $\omega=0$ case can make one of the real/imaginary equations identically zero, changing rank. Treat it as its own boundary rather than trusting a generic solver at the endpoint. Likewise, the leading-coefficient boundary is invisible to any finite frequency sweep.

Multiple roots, tangencies and simultaneous crossings form singular points. Root count may change by more than one pair or not change across a tangent branch; evaluate neighboring regions directly.

Two affine parameters #

Suppose

$$a(s,k_1,k_2)=a_0(s)+k_1a_1(s)+k_2a_2(s).$$

At each ordinary frequency, solve

$$\begin{bmatrix} \mathop{\mathrm{Re}}a_1(j\omega) & \mathop{\mathrm{Re}}a_2(j\omega)\\ \mathop{\mathrm{Im}}a_1(j\omega) & \mathop{\mathrm{Im}}a_2(j\omega) \end{bmatrix} \begin{bmatrix}k_1\\k_2\end{bmatrix} =- \begin{bmatrix} \mathop{\mathrm{Re}}a_0(j\omega)\\ \mathop{\mathrm{Im}}a_0(j\omega) \end{bmatrix}.$$

When the $2\times2$ determinant is nonzero, this gives a parametric curve $(k_1(\omega),k_2(\omega))$. When it vanishes, the boundary may go to infinity, disappear, become a line, or contain a singular frequency. Do not silently drop those frequencies because a linear solve failed.

For nonlinear parameter dependence, the same boundary equations apply but may produce implicit algebraic curves, several branches or isolated components. Continuation and interval/root-count methods are safer than a coarse rectangular grid.

Worked cubic with every boundary type #

Consider

$$a(s;k_1,k_2)=(1+k_2)s^3+2s^2+k_1s+k_2.$$

Zero and infinity #

At $s=0$,

$$a(0;k_1,k_2)=k_2,$$

so the zero-root boundary is the entire line $k_2=0$. The leading coefficient is $1+k_2$, so degree loss occurs on $k_2=-1$.

Imaginary pair #

Substitute $s=j\omega$:

$$a(j\omega)=\left(k_2-2\omega^2\right) +j\left(k_1\omega-(1+k_2)\omega^3\right).$$

For $\omega>0$,

$$k_2=2\omega^2,$$

$$k_1=(1+k_2)\omega^2.$$

Eliminating $\omega$ gives

$$k_1=\frac{k_2(1+k_2)}{2},\qquad k_2>0.$$

This curve contains a conjugate pair at

$$\omega=\sqrt{\frac{k_2}{2}}.$$

Label the regions with Routh #

For a cubic $a_3s^3+a_2s^2+a_1s+a_0$, strict Hurwitz stability requires positive coefficients and $a_2a_1>a_3a_0$. Here that becomes

$$k_2>0,qquad k_1>\frac{k_2(1+k_2)}{2}.$$

The other coefficient conditions then follow: $1+k_2>0$ and $k_1>0$. The shaded component in the diagram is therefore the complete stable set.

Direct roots cross-check three components:

At $k_2=1$, the oscillatory boundary is $k_1=1$ and the crossing frequency is $\omega=1/\sqrt2$. A direct substitution gives $a(\pm j/\sqrt2)=0$, while the third root remains negative.

Boundary orientation #

At a regular boundary point, differentiate

$$a(s(k),k)=0$$

with respect to a scalar transverse parameter $\rho$:

$$\frac{ds}{d\rho}=-\frac{\partial a/\partial\rho}{\partial a/\partial s}.$$

The sign of $\mathop{\mathrm{Re}}(ds/d\rho)$ tells which side moves the root into the right or left half-plane. For two parameters, a Jacobian or chosen normal direction supplies the transverse motion.

Orientation can propagate root counts efficiently across a regular curve, but it is not a substitute for test points near tangencies, self-intersections, singular frequencies, multiple roots or degree loss. A robust implementation computes roots at representative interior points and verifies continuity under small parameter perturbations.

Designing more than stability #

Replace the imaginary axis by another root boundary to impose a geometric performance condition:

• shifted line $s=-\sigma+j\omega$ for minimum exponential decay rate $\sigma$;
• rays of a damping sector for a minimum relative damping ratio;
• circles or other contours for pole magnitude/natural-frequency bounds;
• unit disk or contracted disk for discrete-time stability and settling-rate requirements.

The resulting D-decomposition labels root placement relative to that domain. It still does not constrain zeros, residues, control effort, sensitivity peaks, noise amplification or saturation. A pole-feasible parameter region must be intersected with those separate requirements.

For a PI or PID controller, substitute controller gains into the closed-loop characteristic polynomial and build the region in $(K_p,K_i)$ or a chosen two-dimensional slice of $(K_p,K_i,K_d)$. This finds all stabilizing gains in the slice. It does not prove robust stability for an uncertain plant unless the uncertainty is included and the boundary construction covers the entire family.

Discrete-time case #

For Schur stability, $D$ is the open unit disk and the ordinary boundary is

$$a(e^{j\omega},k)=0,qquad 0\le\omega\le\pi.$$

The endpoints $z=1$ and $z=-1$ are real-root boundaries and should be treated explicitly. Real coefficients again provide conjugate symmetry. Unlike the continuous-time case, loss of polynomial degree does not represent a root crossing the finite unit-circle boundary in the same way; use the discrete root-count argument rather than copying the continuous infinity rule.

A bilinear transform can map the unit disk to a half-plane and reuse Hurwitz machinery, but parameter dependence may become more complicated and the sample period remains part of the physical interpretation.

Numerical workflow #

1. Derive the uncancelled internal characteristic equation and declare the root domain $D$.
2. List adjustable parameters, allowed ranges and any coefficient singularities.
3. Derive ordinary $\partial D$ equations symbolically where possible.
4. Add zero/end-point, infinity/degree-loss and singular-frequency boundaries explicitly.
5. Trace every branch with continuation or validated root finding; refine near intersections and high curvature.
6. Build connected components of the complement inside the allowed parameter set.
7. Evaluate one or more interior points per component with direct roots or a separate stability criterion.
8. Cross-check boundary points by substituting the claimed crossing root.
9. Intersect the stable set with performance, effort, uncertainty and implementation constraints.
10. Validate selected controllers in the full nonlinear, delayed and sampled loop.

Failure modes and limits #

• Coarse frequency grid: misses narrow loops, cusps or disconnected branches.
• Cancelled characteristic polynomial: hides unstable internal modes before decomposition begins.
• Omitted degree-loss line: merges regions whose root count differs through infinity.
• One test point for a disconnected color patch: assumes connectivity that the plot did not establish.
• Axes clipped too tightly: boundary branches leave and re-enter the displayed window.
• More than two or three parameters: surfaces and component enumeration become difficult to visualize and computationally expensive.
• Uncertain coefficients: a nominal stable region is not the intersection of stable regions for all plants unless that intersection is actually computed or certified.

D-decomposition is strongest when a low-order controller has one or two meaningful gains and the characteristic equation is available. Its value is completeness within that declared slice: it shows why a region is stable, which root crosses at its edge, and how much geometric room a chosen controller has before losing the root-domain requirement.