From Robust Admissibility to Local Fragility: Sensitivity Mapping of Closed-Loop Root Contours

1. Introduction

1.1. Background and Research Gap

Parametric uncertainty is unavoidable in modeling for the practical design of control systems and may result from inaccurate identification, changes in operating conditions, manufacturing tolerances, and component aging. Therefore, the controller must preserve stability and acceptable dynamic performance of the controlled variable not only for the nominal model but also throughout the entire region of a priori uncertainty, [1,2,3,4,5].

Dynamic requirements or constraints are often specified through admissible regions for the closed-loop poles and stability regions in the parameter space of the controller parameters, [6,7,8]. When the plant model contains uncertain parameters, its nominal poles may be replaced by sets or root contours, which can be investigated using robust root-locus, parametric, and interval methods, [2,9,10,11].

These approaches show where the poles may be located, but they do not reveal how rapidly and in which direction the poles move under parameter variations.

The geometric interpretation of classical analysis tools provides a basis for a more detailed examination of this motion. Kurfess and Nagurka demonstrate geometric links among classical graphical tools and show that root-locus and frequency-domain representations contain complementary information about system dynamics, [12]. Cavicchi develops the phase-root locus as a graphical tool dual to the classical root locus and relates it to relative stability in the frequency domain, [13]. These studies justify the use of the modulus and argument of the complex variable as separate parametric characteristics of the closed-loop poles.

However, modulus and phase characteristics are considered either as separate classical representations or as elements of frequency-domain analysis rather than as coordinated parametric characteristics of root contours generated by bounded multiparametric uncertainty, [12,13].

The geometric representation of root sensitivity relates the magnitude and direction of root motion to the geometric properties of the characteristic equation, [14]. In early studies on pole placement, the sensitivity of the eigenvalues to variations in plant parameters was used as a criterion for obtaining a more robust closed-loop spectrum, [15]. Robust pole-assignment methods subsequently considered eigenvector conditioning and the sensitivity of the assigned poles as essential characteristics of the closed-loop control system, [16,17]. Analytical procedures have also been developed for determining the sensitivity of a prescribed pole spectrum to plant and controller parameters, [18], as well as optimization approaches for reducing sensitivity and improving the robustness of pole placement, [19].

In these studies, sensitivity is used primarily to minimize the variation of prescribed poles when the system parameters change.

Recent studies continue this line of research by using eigenvalue sensitivity in robust pole placement, controller tuning, and the analysis of critical operating conditions, [20,21,22]. Similar approaches are also applied to robotic and other complex mechanical systems to evaluate the influence of parameters on dynamics, stability, and transient performance, [23,24,25].

These studies confirm the practical value of sensitivity for both analysis and design. The focus is usually placed on controller tuning, identification of the most influential parameters, analysis of critical operating conditions, or comparison of transient-response indicators. They do not examine how pole sensitivity varies along the entire uncertainty-induced root contour.

Robust assessment is also applied to intelligent and nonlinear control structures, including model-based fuzzy logic control systems and mechanical systems with fuzzy uncertainties [26]. These studies confirm the need to complement formal stability with a robustness assessment, but they do not examine the local motion of the closed-loop poles or its mapping onto the root contours.

The author’s previous studies developed a geometric assessment of systems with parametric uncertainty using root contours. Indicators of robust performance and a combined assessment of stability and dynamic behavior were considered in [27,28], followed by a procedure for verifying and comparing robustly admissible tunings in [29]. The geometric relationship between frequency-domain robustness constraints and closed-loop pole locations was additionally investigated in [30]. These studies provide a global assessment of admissibility and pole dispersion, but they do not reveal the locally sensitive regions of the root contours.

Sensitive regions are known to occur near multiple roots, [14]. In such configurations, a small parameter variation may cause a large local displacement, although all pole locations remain within the admissible region. Therefore, a tuning may be globally robustly admissible while containing locally fragile regions. The concept of fragility is also used to distinguish stability and performance from sensitive behavior under small parametric variations, [31], but in the studies considered it is not associated with a local modulus–phase decomposition of pole motion.

The reviewed literature does not reveal an approach that relates local pole sensitivity to the geometry of uncertain root contours and to the time-domain behavior of the system. This is important because two controller tunings may be robustly admissible, while small parameter variations for one of them may cause considerably more abrupt pole motion and greater variation in the transient response. Without such an assessment, the global admissibility considered in [29] is not sufficient for a justified selection among the tunings.

The main objective of this study is to complement the global assessment of robust admissibility in [29] with an analysis of local pole sensitivity. Thus, the selection among robustly admissible tunings accounts not only for the pole locations but also for the rate and direction of their motion under parameter variations and its effect on the time-domain behavior. This provides a direct practical benefit for designers when several admissible controller tunings are available.

The approach combines modulus and phase characteristics of the tracked pole branches, a radial–tangential decomposition of pole sensitivity, a multiparametric assessment of the worst-case direction of variation, and validation using sensitivity measures and envelopes of the transient responses.

1.2. Contributions of the Paper

1. Use of modulus and phase characteristics of the tracked pole branches as a complement to the classical representation of uncertain root contours.

2. Decomposition of the total pole sensitivity into modulus and phase components, allowing radially and tangentially dominated regions of motion to be distinguished.

3. Mapping of the modulus, phase, and total sensitivities onto the corresponding locations of the real and complex pole branches.

4. Formulation of a multiparametric local indicator based on the largest singular value of the pole-sensitivity matrix, accounting for the worst-case combined direction of variation of the uncertain parameters.

5. Joint comparison of robustly admissible tunings using the global root-contour indicators, local pole sensitivity, and transient-response sensitivity.

6. Time-domain validation using families and envelopes of transient responses to evaluate the correspondence between local pole sensitivity and the spread of the output variable.

1.3. Structure of the Paper

Section 2 introduces the parametric representation of the poles, the modulus and phase characteristics, the analytical pole sensitivity, its radial–tangential decomposition, the behavior in the vicinity of multiple roots, and the numerical determination of the derivatives. Section 3 develops a sensitivity-analysis approach that includes tracking of the closed-loop pole branches, modulus and phase characteristics of the poles, radial–tangential decomposition of pole sensitivity, a Jacobian-based multiparametric indicator, and time-domain sensitivity indicators. Section 4 presents a numerical study in which the local sensitivity indicators are mapped onto the root contours, while the different admissible tunings are compared using the sensitivity measures and envelopes of the time-domain responses. Section 5 discusses the main results, limitations, and practical applications of the proposed approach. Section 6 summarizes the main conclusions

2. Formulation of Modulus, Phase, and Pole Sensitivities

2.1. Parametric Representation of Closed-Loop Poles

Let the characteristic equation of the closed-loop system be written as

P(s, p) = 0,

(1)

where s is the complex variable and p is a real parameter.

The parameter p may be a controller coefficient, a plant parameter, or another quantity whose variation affects the pole locations.

For each admissible value of p, the roots of the characteristic equation s_i(p) are represented as

s_i(p) = σ_i(p) + jω_i(p), i = 1,2,…,q,

(2)

where σ_i and ω_i are the real and imaginary parts of the i-th pole, respectively, and q is the order of the characteristic equation.

As the parameter p varies over the interval p∈[p⁻, p⁺], each root s_i(p) traces a trajectory in the complex plane:

Γ_i = {s_i(p)∈ℂ∣p∈[p⁻, p⁺]}.

(3)

Trajectory (3) corresponds to the classical tracking of pole motion and shows the location of the pole s_i for each value of the parameter p, but it does not provide a quantitative measure of the rate at which its position changes.

In addition to Cartesian coordinates, the pole s_i(p) can also be expressed in polar form:

$$s_i\left(p\right)=M_i\left(p\right)e^{j{\mathsf{\Phi }}_i\left(p\right)},$$

(4)

$$M_i\left(p\right)=\mid s_i\left(p\right)\mid =\sqrt{{\sigma }_i^2\left(p\right)+{\omega }_i^2\left(p\right)},$$

(5)

where M_i(p) is the modulus of the pole, and

Φ_i(p) = args_i(p) = atan2(ω_i(p), σ_i(p)),

(6)

is the argument of the pole, measured from the positive real axis in the complex plane.

For a more complete description of the motion of the pole s_i, the trajectory Γ_i is examined through its modulus and argument. Relationships (5) and (6) define the modulus and phase characteristics, respectively, and show how the radial and angular positions of the pole s_i change as the parameter p varies.

2.2. Modulus Characteristics M_i(p) of Closed-Loop Poles

The modulus characteristic of the i-th pole is represented by relationship (5). It is obtained by calculating the modulus M_i of the root s_i for each value of the parameter p along the same root trajectory, but it represents only the radial component of the pole motion. Therefore, it does not replace the trajectory in the complex plane but complements it with direct information about the variation of the pole modulus.

For the complex-conjugate pair s_1,2 = σ ± jω, the moduli M_i of the two poles are equal, i.e., $M_1=M_2=\sqrt{{\sigma }^2+{\omega }^2}$. For an equivalent second-order oscillatory mode with the transfer function $W\left(s\right)={\displaystyle \frac{{\omega }_n^2}{s^2+2\xi {\omega }_{n}s+{\omega }_n^2}}$, the poles are $s_{\mathrm{1,2}}=-\xi {\omega }_n\pm j{\omega }_n\sqrt{1-{\xi }^2}$, and their modulus is $∣s_{\mathrm{1,2}}∣=\sqrt{{\xi }^2{\omega }_n^2+{\omega }_n^2\left(1-{\xi }^2\right)}={\omega }_n$. Therefore,

M = ω_n.

(7)

Consequently, for an equivalent second-order oscillatory mode, the relationship M_i(p) also shows how the natural frequency ω_n changes as the parameter p varies.

For a real pole, s_i = σ_i and ω_i = 0; therefore, its modulus is M_i = |σ_i|. For a stable real pole, motion to the left leads to an increase in the modulus and faster decay of the corresponding aperiodic component. As the pole approaches the imaginary axis, the transient response in the time domain becomes slower.

2.3. Phase Characteristic Φ_i(p) of Closed-Loop Poles

The phase characteristic of the i-th pole is represented by relationship (6). It shows how the argument Φ_i of the pole s_i changes as the parameter p varies. Therefore, it does not replace the trajectory in the complex plane but complements it with direct information about the variation of the pole argument.

For the complex-conjugate pair s_1,2 = σ ± jω, the arguments are symmetric, Φ₂ = −Φ₁, for a suitably chosen range of the argument.

For a complex pole in the left half-plane and assuming an equivalent second-order mode, the damping ratio is ${\xi }_i=-{\displaystyle \frac{{\sigma }_i}{\sqrt{{\sigma }_i^2+{\omega }_i^2}}}=-{\displaystyle \frac{{\sigma }_i}{M_i}}$. From the geometric relationships in the complex plane, σ_i = M_icosΦ_i, it follows that

ξ_i = −cosΦ_i.

(8)

Consequently, for an equivalent second-order oscillatory mode, the relationship Φ_i(p) is directly related to the variation of the damping ratio.

As the pole approaches the imaginary axis, its argument approaches π/2 or −π/2, while the damping ratio ξ decreases. This leads to a more oscillatory transient response. As the pole moves away from the imaginary axis, the absolute value of the argument approaches π, while the damping increases.

2.4. Analytical Determination of Pole Sensitivity ∂s_i/∂p

The modulus M_i(p) and phase Φ_i(p) characteristics consider separately the radial and angular variations of the pole. To evaluate the local displacement of the pole in the complex plane, the total pole sensitivity is first determined and then decomposed into modulus and phase components.

For each root s_i(p), equation (1) is satisfied. As the parameter p varies, the position of the root s_i(p) also changes, while the characteristic equation remains satisfied. To trace this motion, equation (1) is differentiated with respect to the parameter p, yielding ${\displaystyle \frac{\partial P}{\partial s}}{\displaystyle \frac{\mathrm{d}{s}_i}{\mathrm{d}p}}+{\displaystyle \frac{\partial P}{\partial p}}=0$. In this way, the analytical pole sensitivity is defined as the derivative of the position of the i-th pole s_i with respect to the varying parameter p:

$${\displaystyle \frac{\mathrm{d}{s}_i}{\mathrm{d}p}}=-{{\displaystyle \frac{\partial P/\partial p}{\partial P/\partial s}}∣}_{s=s_i\left(p\right)}.$$

(9)

Relationship (9) is valid for a simple root s_i, for which ∂P/∂s ≠ 0; the behavior in the vicinity of multiple roots is considered separately in Section 2.8.

Since s_i(p) = σ_i(p) + jω_i(p), differentiation with respect to p gives ${\displaystyle \frac{\mathrm{d}{s}_i}{\mathrm{d}p}}={\displaystyle \frac{\mathrm{d}{\sigma }_i}{\mathrm{d}p}}+j{\displaystyle \frac{\mathrm{d}{\omega }_i}{\mathrm{d}p}}$. The real part dσ_i/dp represents the variation of the pole along the real axis, while the imaginary part dω_i/dp represents its variation along the imaginary axis.

The analytical sensitivity (9) indicates both the magnitude and the direction of the local pole displacement.

The absolute pole sensitivity is determined by the modulus of (9):

$$S_i^{\left(p\right)}=∣{\displaystyle \frac{\mathrm{d}{s}_i}{\mathrm{d}p}}∣=\sqrt{{\left({\displaystyle \frac{\mathrm{d}{\sigma }_i}{\mathrm{d}p}}\right)}^2+{\left({\displaystyle \frac{\mathrm{d}{\omega }_i}{\mathrm{d}p}}\right)}^2}.$$

(10)

A small value of $S_i^{\left(p\right)}$ means that the position of the pole s_i changes smoothly. A large value indicates that a small variation in p may cause a significant displacement of the pole s_i.

When comparing parameters with different scales, the relative sensitivity may be used:

$$S_{i,\mathrm{rel}}^{\left(p\right)}=∣{\displaystyle \frac{p}{s_i}}{\displaystyle \frac{\mathrm{d}{s}_i}{\mathrm{d}p}}∣.$$

(11)

This indicator is dimensionless and is suitable for comparisons between different parameters or different systems.

2.5. Modulus Sensitivity $S_{M_i}^{\left(p\right)}$

Differentiating (5) with respect to the parameter p gives

$${\displaystyle \frac{\mathrm{d}{M}_i}{\mathrm{d}p}}={\displaystyle \frac{{\sigma }_i\frac{\mathrm{d}{\sigma }_i}{\mathrm{d}p}+{\omega }_i\frac{\mathrm{d}{\omega }_i}{\mathrm{d}p}}{M_i}}.$$

(12)

where $M_i=\mid s_i\mid =\sqrt{{\sigma }_i^2+{\omega }_i^2}$ is the modulus of the pole under consideration.

The derivative dM_i/dp in (12) determines the slope of the modulus characteristic M_i(p) and shows how the pole modulus M_i changes as the parameter p varies. The modulus sensitivity is defined by the absolute value of (12):

$$S_{M,i}^{\left(p\right)}=∣{\displaystyle \frac{\mathrm{d}{M}_i}{\mathrm{d}p}}∣.$$

(13)

When the complex-conjugate pair is considered as an equivalent second-order oscillatory mode, for which M_i = ω_n,i, the modulus sensitivity $S_{M,i}^{\left(p\right)}$ shows how strongly the natural frequency ω_n changes under a small variation of the parameter p.

Using the polar representation of the pole from (4) and the analytical pole sensitivity from (9), differentiation of (4) with respect to the parameter p gives $e^{-j{\mathsf{\Phi }}_i}{\displaystyle \frac{\mathrm{d}{s}_i}{\mathrm{d}p}}={\displaystyle \frac{\mathrm{d}{M}_i}{\mathrm{d}p}}+j{M}_i{\displaystyle \frac{\mathrm{d}{\mathsf{\Phi }}_i}{\mathrm{d}p}}$. The real part of this expression determines the derivative of the modulus; therefore, (12) can also be written as

$${\displaystyle \frac{\mathrm{d}{M}_i}{\mathrm{d}p}}=\mathrm{Re}\left\{e^{-j{\mathsf{\Phi }}_i}{\displaystyle \frac{\mathrm{d}{s}_i}{\mathrm{d}p}}\right\}.$$

(14)

Relationship (14) shows that dM_i/dp represents the radial component of the local pole displacement, while the modulus sensitivity $S_{M,i}^{\left(p\right)}$ determines its magnitude.

2.6. Phase Sensitivity $S_{\Phi ,i}^{\left(p\right)}$

Differentiating the phase characteristic (6) with respect to the parameter p gives

$${\displaystyle \frac{\mathrm{d}{\mathsf{\Phi }}_i}{\mathrm{d}p}}={\displaystyle \frac{{\sigma }_i\frac{\mathrm{d}{\omega }_i}{\mathrm{d}p}-{\omega }_i\frac{\mathrm{d}{\sigma }_i}{\mathrm{d}p}}{M_i^2}},$$

(15)

where $M_i=\left|s_i\right|=\sqrt{{\sigma }_i^2+{\omega }_i^2}$ is the modulus of the pole under consideration.

The derivative dΦ_i/dp determines the slope of the phase characteristic Φ_i(p) and shows how the pole argument changes as the parameter p varies. The same angular variation corresponds to different displacements in the complex plane depending on the distance of the pole s_i from the origin. The tangential displacement of the pole for a small variation of the argument dΦ_i is equal to M_idΦ_i. Therefore, the phase sensitivity is defined as

$$S_{\mathsf{\Phi },i}^{\left(p\right)}=M_i∣{\displaystyle \frac{\mathrm{d}{\mathsf{\Phi }}_i}{\mathrm{d}p}}∣.$$

(16)

Using the polar representation of the pole from (4) and the analytical pole sensitivity from (9), differentiation of (4) with respect to the parameter p gives $e^{-j{\mathsf{\Phi }}_i}{\displaystyle \frac{\mathrm{d}{s}_i}{\mathrm{d}p}}={\displaystyle \frac{\mathrm{d}{M}_i}{\mathrm{d}p}}+j{M}_i{\displaystyle \frac{\mathrm{d}{\mathsf{\Phi }}_i}{\mathrm{d}p}}$. The imaginary part of this expression determines the tangential component; therefore, (15) can also be written as

$$M_i{\displaystyle \frac{\mathrm{d}{\mathsf{\Phi }}_i}{\mathrm{d}p}}=\mathrm{Im}\left\{e^{-j{\mathsf{\Phi }}_i}{\displaystyle \frac{\mathrm{d}{s}_i}{\mathrm{d}p}}\right\}.$$

(17)

Relationship (17) shows that M_idΦ_i/dp represents the tangential component of the local pole displacement, while the phase sensitivity $S_{\mathsf{\Phi },i}^{\left(p\right)}$ determines its magnitude.

From the relationship between the pole argument (6) and the damping ratio (8), it follows that

$${\displaystyle \frac{\mathrm{d}{\xi }_i}{\mathrm{d}p}}=\sin {\mathsf{\Phi }}_i{\displaystyle \frac{\mathrm{d}{\mathsf{\Phi }}_i}{\mathrm{d}p}}.$$

(18)

A large value of ∣dΦ_i/dp∣ in (18) indicates that a small variation of the parameter p may lead to an abrupt change in the damping ratio ξ and to a substantial change in the oscillatory nature of the transient response.

2.7. Decomposition of the Total Pole Sensitivity $S_i^{\left(p\right)}$

Differentiating the polar representation (4) with respect to the parameter p gives ${\displaystyle \frac{\mathrm{d}{s}_i}{\mathrm{d}p}}=e^{j{\mathsf{\Phi }}_i}\left({\displaystyle \frac{\mathrm{d}{M}_i}{\mathrm{d}p}}+j{M}_i{\displaystyle \frac{\mathrm{d}{\mathsf{\Phi }}_i}{\mathrm{d}p}}\right)$.

Since $e^{j{\mathsf{\Phi }}_i}=\cos {\mathsf{\Phi }}_i+j\sin {\mathsf{\Phi }}_i$, its modulus is $∣e^{j{\mathsf{\Phi }}_i}∣=\sqrt{{\cos }^2{\mathsf{\Phi }}_i+{\sin }^2{\mathsf{\Phi }}_i}=1$. Consequently, the factor $e^{j{\mathsf{\Phi }}_i}$ determines only the direction without changing the magnitude of the pole sensitivity $S_i^{\left(p\right)}$. For the square of the modulus, it follows that ${∣{\displaystyle \frac{\mathrm{d}{s}_i}{\mathrm{d}p}}∣}^2={\left({\displaystyle \frac{\mathrm{d}{M}_i}{\mathrm{d}p}}\right)}^2+{\left(M_i{\displaystyle \frac{\mathrm{d}{\mathsf{\Phi }}_i}{\mathrm{d}p}}\right)}^2$. Using definitions (10), (13), and (16), it follows that

$${\left(S_i^{\left(p\right)}\right)}^2={\left(S_{M,i}^{\left(p\right)}\right)}^2+{\left(S_{\mathsf{\Phi },i}^{\left(p\right)}\right)}^2.$$

(19)

Relationship (19) shows that the total pole sensitivity $S_i^{\left(p\right)}$ consists of two mutually perpendicular components: a radial component $S_{M,i}^{\left(p\right)}$, related to the variation of the modulus, and a tangential component $S_{\mathsf{\Phi },i}^{\left(p\right)}$, related to the variation of the argument. If $S_{M,i}^{\left(p\right)}$ predominates, the pole moves mainly in the radial direction and, for a complex-conjugate pair, the natural frequency ω_n changes mainly. If $S_{\mathsf{\Phi },i}^{\left(p\right)}$ predominates, the pole moves mainly in the tangential direction and the damping ratio ξ changes mainly.

Thus, the decomposition shows not only the magnitude of the sensitivity but also the nature of the pole displacement.

2.8. Sensitivity in the Vicinity of Multiple Roots (∂P/∂s = 0)

A special case arises when the root s_i of (1) is multiple:

$${\displaystyle \frac{\partial P}{\partial s}}\left(s_i,p\right)=0.$$

(20)

It follows from (9) that, in the vicinity of such a point, the denominator becomes small and the pole sensitivity may increase significantly.

This explains why small variations of the parameter near the merging and splitting points of the root branches may lead to abrupt pole displacements.

At the multiple-root point itself, the condition ∂P/∂s ≠ 0, required for determining the sensitivity using (9), is not satisfied. Therefore, the derivative ds_i/dp may not exist or may become unbounded.

In a numerical study, the obtained value depends strongly on the selected step and the method used to track the root branches. When an exact multiple root is present, this value is interpreted as a discretization-dependent indicator of singularity rather than as a finite value of the pole sensitivity.

2.9. Numerical Determination of the Sensitivity Derivatives

When the analytical calculation of the derivatives is difficult, they can be determined numerically. For an interior point of the parameter grid, a central difference is used:

$${{\displaystyle \frac{\mathrm{d}{s}_i}{\mathrm{d}p}}∣}_{p=p_k}\approx {\displaystyle \frac{s_i\left(p_k+\mathsf{\Delta }p\right)-s_i\left(p_k-\mathsf{\Delta }p\right)}{2\mathsf{\Delta }p}}.$$

(21)

The derivatives of the modulus and phase are determined using the same central difference: ${{\displaystyle \frac{\mathrm{d}{M}_i}{\mathrm{d}p}}∣}_{p=p_k}\approx {\displaystyle \frac{M_i\left(p_k+\mathsf{\Delta }p\right)-M_i\left(p_k-\mathsf{\Delta }p\right)}{2\mathsf{\Delta }p}},$ ${{\displaystyle \frac{\mathrm{d}{\mathsf{\Phi }}_i}{\mathrm{d}p}}∣}_{p=p_k}\approx {\displaystyle \frac{{\mathsf{\Phi }}_i\left(p_k+\mathsf{\Delta }p\right)-{\mathsf{\Phi }}_i\left(p_k-\mathsf{\Delta }p\right)}{2\mathsf{\Delta }p}}.$

When calculating the phase derivative, the phase values must be represented as a continuous sequence, for example by adding or subtracting 2π when crossing between π and −π, to avoid artificial jumps of 2π.

Reliable tracking of the individual root branches is also essential. For a discrete variation of the parameter, the poles must be matched between two consecutive parameter values according to the continuity of the root branches, for example by minimizing the distance between the corresponding roots. Particular attention is required in the vicinity of close or multiple roots, where incorrect switching from one branch to another is possible.

3. Method for Sensitivity Mapping of Root Contours

3.1. Root Contours $𝒞$(θ) under Parametric Uncertainty ρ

Under parametric uncertainty of the plant, the parameter p is specified as one of the elements ρ_j of the vector ρ. Consequently, the derivative ds_i/dp is replaced by the partial derivative ∂s_i/∂ρ_j, while the modulus sensitivity $S_{M,i}^{\left(p\right)}$ and phase sensitivity $S_{\mathsf{\Phi },i}^{\left(p\right)}$ are determined separately with respect to each uncertain plant parameter ρ_j.

Let the closed-loop system be described by the characteristic equation

P(s, θ, ρ) = 0,

(22)

where $\theta ={\left[{\theta }_1,{\theta }_2,\dots ,{\theta }_r\right]}^T$ is the vector of controller parameters, while $\rho ={\left[{\rho }_1,{\rho }_2,\dots ,{\rho }_m\right]}^T$ is the vector of plant parameters.

The admissible sets are θ∈Ω_θ and ρ∈Ω_ρ.

For a fixed tuning θ and varying plant parameters ρ_j, the root contour is defined as

$𝒞$(θ) = {s∈ℂ∣∃ρ∈Ω_ρ:P(s, θ, ρ) = 0}.

(23)

If the characteristic equation is of order q, the individual root branch is defined as

$𝒞$_i(θ) = {s_i(ρ, θ)∈ℂ∣ρ∈Ω_ρ}, i = 1,…,q,

(24)

where the total contour (23) is the union of the individual branches (24).

The root contour $𝒞$(θ) shows where the poles s_i may be located but does not show how rapidly they move between neighboring positions as the plant parameters ρ_j vary. Therefore, for each point on the contour that is a solution of (22), the modulus M_i and argument Φ_i are calculated according to (5) and (6), together with the modulus sensitivity S_M,i,j and phase sensitivity S_Φ,i,j, defined by (13) and (16). Thus, the geometric position of the contour $𝒞$(θ) is complemented by a quantitative evaluation of the radial and tangential local motion of the pole s_i.

In [29], the robust admissibility of a fixed tuning is formulated by the condition $𝒞$(θ)⊂D, where D⊂ℂ is a prescribed region of dynamic admissibility. The position and dimensions of the root contours, characterized by Δσ and Δω, the maximum displacement d_max, and the geometric margin μ_D, are used to verify and compare the tunings [29]. Condition (25) guarantees that, for all ρ∈Ω_ρ, the closed-loop poles remain within the admissible region D, but it does not provide a quantitative evaluation of the total local variation of an individual pole under a small change in the plant parameters. Consequently, even when $𝒞$(θ)⊂D, a small variation of ρ_j at a particular parameter point may cause a significant local pole displacement.

3.2. Local Pole Sensitivity S_i,j with Respect to the Plant Parameters ρ_j

For a quantitative evaluation of the local variation of the i-th pole with respect to the parameter ρ_j, the pole sensitivity (26) is used. Setting p = ρ_j in the general relationship (9) gives ${\displaystyle \frac{\partial s_i}{\partial {\rho }_j}}=-{\displaystyle \frac{\partial P/\partial {\rho }_j}{\partial P/\partial s}}.$ According to definition (10), the pole sensitivity of the i-th pole with respect to the parameter ρ_j is defined in absolute form as

$$S_{i,j}=∣{\displaystyle \frac{\partial s_i}{\partial {\rho }_j}}∣.$$

(26)

A small value of S_i,j indicates a smooth variation of the pole with respect to ρ_j, while a large value indicates that a small change in the parameter ρ_j may lead to a significant displacement of s_i in the complex plane.

When comparing parameters with different dimensions or scales, the relative sensitivity from (11) is used, with p replaced by ρ_j.

For each pole s_i, the vector $S_i={\left[S_{i,1},S_{i,2},\dots ,S_{i,m}\right]}^T$ can be formed, combining the local pole sensitivities (26) with respect to all uncertain parameters. When the parameters have different dimensions or substantially different scales, the components of S_i are compared after parameter normalization or by using the relative sensitivity from (11).

3.3. Modulus S_M,i,j and Phase S_Φ,i,j Pole Sensitivities along the Root Contour

For each point of the root contour $𝒞$(θ), the modulus sensitivity S_M,i,j and phase sensitivity S_Φ,i,j are determined using relationships (13) and (16), with the parameter p replaced by the corresponding parameter ρ_j.

According to the decomposition (19), at each point of the contour, $S_{i,j}^2=S_{M,i,j}^2+S_{\mathsf{\Phi },i,j}^2.$

Thus, not only the total sensitivity but also the nature of the local motion can be determined.

When S_M,i,j > S_Φ,i,j, the radial variation of the pole s_i predominates. For complex roots, this means a stronger variation of the natural frequency ω_n.

When S_Φ,i,j > S_M,i,j, the tangential motion of s_i predominates, and its variation mainly affects the pole argument and the damping ratio ξ.

The modulus M_i and argument Φ_i, defined according to (5) and (6), describe the radial and angular positions of the pole s_i, respectively. The modulus sensitivity S_M,i,j and phase sensitivity S_Φ,i,j, defined by (13) and (16), characterize the radial and tangential components, respectively, of the local pole motion with respect to the uncertain parameter ρ_j.

To compare different fixed tunings θ, the maximum modulus and phase sensitivities over all pole branches $𝒞$_i(θ), (24), and the entire parametric uncertainty domain Ω_ρ are determined as S_M,max(θ) and S_Φ,max(θ), respectively. When the uncertain parameters have different dimensions or substantially different ranges, these indicators are determined after parameter normalization so that the sensitivities with respect to the different ρ_j are comparable. S_M,max(θ) determines the highest radial sensitivity, while S_Φ,max(θ) determines the highest tangential sensitivity attained by any pole with respect to any uncertain parameter within the domain Ω_ρ. The maxima are not necessarily attained for the same pole, parameter point, or uncertain parameter.

3.4. Multi-Parameter Local Sensitivity S_i,max and Overall Indicator S_max(θ)

For uncertain parameters with different dimensions or substantially different ranges, normalized coordinates ${\widehat{\rho }}_j=\left({\rho }_j-{\rho }_j^{-}\right)/\left({\rho }_j^{+}-{\rho }_j^{-}\right)$ can be used, where $\left[{\rho }_j^{-},{\rho }_j^{+}\right]$ is the variation interval of the j-th uncertain parameter. When normalized coordinates are used, the derivatives are determined with respect to ${\widehat{\rho }}_j$, so that the comparison of the influence of the individual parameters does not depend on their dimensions and numerical ranges.

The pole sensitivity S_i,j, defined in (26), characterizes the influence of each uncertain parameter ρ_j separately. When the plant parameters vary simultaneously, the local sensitivity matrix is used to evaluate their combined influence on the position of the pole s_i = σ_i + jω_i:

$$J_i=\left[\begin{array}{ccc}{\displaystyle \frac{\partial {\sigma }_i}{\partial {\widehat{\rho }}_1}}& \dots & {\displaystyle \frac{\partial {\sigma }_i}{\partial {\widehat{\rho }}_m}}\\ {\displaystyle \frac{\partial {\omega }_i}{\partial {\widehat{\rho }}_1}}& \dots & {\displaystyle \frac{\partial {\omega }_i}{\partial {\widehat{\rho }}_m}}\end{array}\right].$$

(27)

For a small parameter variation $\mathsf{\Delta }\widehat{\rho }={\left[\mathsf{\Delta },{\widehat{\rho }}_1,\dots \mathsf{\Delta }{\widehat{\rho }}_m\right]}^T$, the local displacement of the i-th pole is represented in a linear approximation as

$$\left[\begin{array}{c}\mathsf{\Delta }{\sigma }_i\\ \mathsf{\Delta }{\omega }_i\end{array}\right]\approx J_i\mathsf{\Delta }\widehat{\rho }.$$

(28)

To evaluate the strongest local pole variation under simultaneous parameter variations, the following indicator is introduced:

S_i,max = σ_max(J_i).

(29)

The indicator S_i,max characterizes the maximum local amplification of parameter variations into displacement of the i-th pole over all directions in the parameter space.

Since S_i,max is determined for a particular pole and a particular parameter point, the indicator S_max(θ) is used for an overall evaluation of the fixed tuning θ. It accounts for the highest value of S_i,max among all poles and all points in the domain Ω_ρ:

$$S_{\max }\left(\theta \right)={\max }_{\rho \in {\mathsf{\Omega }}_{\rho },\hspace{0.25em} i=1,\dots ,q}{S}_{i,\max }\left(\rho ,\theta \right).$$

(30)

The indicator S_max(θ) determines the highest local pole sensitivity attained by any pole over the entire parametric uncertainty domain Ω_ρ. Its high value indicates the presence of a locally sensitive pole configuration but does not necessarily imply an unsatisfactory transient response. Therefore, S_max(θ) is used as an indicator of local pole fragility rather than as a standalone criterion for the quality of the time-domain response.

3.5. Sensitivity of the Transient Response to the Plant Parameters ρ_j

The indicator S_i,max characterizes the strongest local variation in the position of the i-th pole under a small variation of the parameter vector ρ, but it does not show how this variation is manifested in the time-domain response of the system. To evaluate the direct influence of parametric uncertainty on the transient response, the sensitivity of the output variable with respect to the plant parameters ρ_j is introduced.

Let y(t, ρ, θ) be the output variable of the closed-loop system for a fixed tuning θ and parameter vector ρ, with its time dependence describing the corresponding transient response. For each uncertain parameter ρ_j, the local sensitivity of the output variable is defined as

$$S_{y,j}\left(t,\rho ,\theta \right)=∣{\displaystyle \frac{\partial y\left(t,\rho ,\theta \right)}{\partial {\widehat{\rho }}_j}}∣.$$

(31)

The indicator S_y,j(t, ρ, θ) characterizes the local sensitivity of the transient response to the individual uncertain parameter ρ_j at each instant of time. For m uncertain parameters, the total local sensitivity of the transient response is defined as

$$S_y\left(\rho ,\theta \right)={\max }_{t\in \left[0,T\right]}{\left[{{\displaystyle \sum }}_{j=1}^m{\left({\displaystyle \frac{\partial y\left(t,\rho ,\theta \right)}{\partial {\widehat{\rho }}_j}}\right)}^2\right]}^{1/2}.$$

(32)

where T is the time interval under consideration.

The indicator S_y(ρ, θ) simultaneously combines the influence of all uncertain parameters through the Euclidean norm of the vector of partial derivatives and determines its maximum value over the time interval under consideration.

When the plant parameters ρ have different dimensions or ranges, they must first be normalized. In this way, the contribution of the individual parameters in (32) does not depend primarily on their numerical scale. Let ρ^(k)∈Ω_ρ, k = 1,…,N_ρ, denote the k-th point of the discretized parametric uncertainty domain.

For an overall evaluation of the tuning θ over the entire domain Ω_ρ, the root-mean-square value is used:

$$S_{y,\mathrm{RMS}}\left(\theta \right)={\left[{\displaystyle \frac{1}{N_{\rho }}}{\sum }_{k=1}^{N_{\rho }}{S}_y^2\left({\rho }^{\left(k\right)},\theta \right)\right]}^{1/2}.$$

(33)

where N_ρ is the number of points in the discretized parameter domain Ω_ρ.

In addition, the maximum sensitivity from (32) is determined as

$$S_{y,\max }\left(\theta \right)={\max }_{\rho \in {\mathsf{\Omega }}_{\rho }}{S}_y\left(\rho ,\theta \right).$$

(34)

The indicator S_y,RMS characterizes the average level of the local transient-response sensitivity within the uncertainty domain, while S_y,max determines the worst local value. The obtained indicators are not a direct measure of the full width of the family of transient responses, which is additionally evaluated through their envelopes in the time domain.

The transient-response sensitivity defined by (31)–(34) does not replace the local pole sensitivity S_i,j defined in (26). The overall indicator S_max(θ), defined in (30), is used to detect locally sensitive pole configurations and regions close to multiple roots, while S_y,RMS(θ) and S_y,max(θ), defined in (33) and (34), respectively, evaluate the direct manifestation of small parameter variations in the time-domain response of the system.

The derivatives used in (31) and (32) are determined numerically using central differences because the output variable y(t, ρ, θ) is calculated for discrete values of the uncertain parameters. For an interior point of the parameter domain, the derivative with respect to the normalized parameter ${\widehat{\rho }}_j$ is determined as

$${\displaystyle \frac{\partial y\left(t,\widehat{\rho },\theta \right)}{\partial {\widehat{\rho }}_j}}\approx {\displaystyle \frac{y\left(t,\widehat{\rho }+\mathsf{\Delta }{\widehat{\rho }}_j{e}_j,\theta \right)-y\left(t,\widehat{\rho }-\mathsf{\Delta }{\widehat{\rho }}_j{e}_j,\theta \right)}{2\mathsf{\Delta }{\widehat{\rho }}_j}}.$$

(35)

Here, $\mathsf{\Delta }{\widehat{\rho }}_j$ is the numerical differentiation step for the j-th normalized uncertain parameter, and e_j is the j-th unit vector. When ${\widehat{\rho }}_j$ is varied, all other components of the parameter vector $\widehat{\rho }$ remain fixed. At the boundary points of the domain Ω_ρ, where a central difference cannot be applied without leaving the admissible parameter interval, the corresponding one-sided differences are used.

The calculations are performed for all robustly admissible tunings θ using the same parameter grid in Ω_ρ and the same time grid over the interval [0, T]. In this way, the obtained values of S_y,RMS(θ) and S_y,max(θ), determined by (33) and (34), can be directly compared among the individual tunings. The obtained values of S_y,RMS(θ) and S_y,max(θ), determined by (33) and (34), are used to compare the robustly admissible tunings, while the correspondence between the transient-response sensitivity evaluated by these indicators and the actual dispersion of the time-domain characteristics is additionally verified using the families and envelopes of the transient responses in Section 4.7.

3.6. Mapping of Local Pole Sensitivity S_i,j, Modulus Sensitivity S_M,i,j, Phase Sensitivity S_Φ,i,j, and Multi-Parameter Sensitivity S_i,max onto the Root Contours $𝒞$_i(θ)

For a fixed tuning θ, each point of a root branch of the contour $𝒞$(θ) is associated with the corresponding values of S_i,j, S_M,i,j, S_Φ,i,j, or S_i,max. In this way, the root contour is complemented by quantitative information about the local motion of the pole s_i.

For an individual uncertain parameter ρ_j, the following parametric mapping can be introduced:

ρ → (s_i(ρ, θ), S_i,j(ρ, θ)).

(36)

Similarly, parametric mappings can be introduced for the modulus and phase sensitivities:

ρ → (s_i(ρ, θ), S_M,i,j(ρ, θ)),
      ρ → (s_i(ρ, θ), S_Φ,i,j(ρ, θ)).

(37)

Under multi-parameter uncertainty, each parameter point ρ is associated with the pole position s_i(ρ, θ) and the corresponding value of the overall local indicator S_i,max(ρ, θ) from (29):

ρ → (s_i(ρ, θ), S_i,max(ρ, θ)).

(38)

Thus, each point of the root branch (24) of the contour $𝒞$(θ) is associated with a numerical value that characterizes the local influence of parametric uncertainty on the position of the corresponding pole.

For a visual representation of the sensitivity distribution over the root contour, the normalized indicator can be used:

$${\bar{S}}^{*}={\displaystyle \frac{S^{*}-S_{\min }^{*}}{S_{\max }^{*}-S_{\min }^{*}}}.$$

(39)

where S^* denotes the mapped indicator S_i,j, S_M,i,j, S_Φ,i,j, or S_i,max, while $S_{\min }^{*}$ and $S_{\max }^{*}$ are its minimum and maximum values, respectively, over the set under consideration. When several tunings are compared, common values of $S_{\min }^{*}$ and $S_{\max }^{*}$ are used for all tunings so that the same color corresponds to the same sensitivity level.

For each parameter point ρ, the mapping establishes a relationship between the corresponding position s_i(ρ, θ) of the i-th pole, its modulus M_i and argument Φ_i, and the local pole sensitivities S_M,i,j, S_Φ,i,j, S_i,j, and S_i,max, i.e.,

ρ→s_i(ρ, θ)→{M_i, Φ_i}→{S_M,i,j, S_Φ,i,j, S_i,j, S_i,max}.

(40)

In this way, each pole position on the root branch (24) of the contour (23) is associated with the value of the corresponding sensitivity indicator, which makes it possible to directly localize regions with increased radial, tangential, or total pole sensitivity.

The transient-response sensitivity indicator S_y(ρ, θ), defined in (32), refers to the entire time-domain response at a given parameter point ρ and is therefore not associated with an individual pole branch (24). It is considered over the parameter domain Ω_ρ, while S_y,RMS(θ) and S_y,max(θ), defined in (33) and (34), are used for an overall comparison of the tunings. Thus, the pole-sensitivity maps show where locally sensitive regions arise along the root contours $𝒞$(θ), while the transient-response sensitivity indicators evaluate the effect of parameter variations on the output variable in the time domain.

3.7. Robust Admissibility $𝒞$(θ)⊂D and Classification of Tunings According to S_max(θ), S_y,RMS(θ), and S_y,max(θ)

The set of robustly admissible tunings defined by condition (25) is

$${\mathsf{\Theta }}_D^r=\left\{\theta \in {\mathsf{\Omega }}_{\theta }\mid \mathcal{C}\left(\theta \right)\subset D\right\}.$$

(41)

In [29], this set is determined by the global position of the root contours $𝒞$(θ) with respect to the prescribed region D. The present method complements this evaluation with the maximum local pole sensitivity S_max(θ), defined in (30).

Let S_lim be an admissible threshold for the local sensitivity. The set of robustly admissible tunings without pronounced local pole fragility is then defined as

$${\mathsf{\Theta }}_{D,S}^r=\left\{\theta \in {\mathsf{\Theta }}_D^r\mid S_{\max }\left(\theta \right)\le S_{\lim }\right\}.$$

(42)

If a tuning belongs to ${\mathsf{\Theta }}_D^r$, but S_max(θ) > S_lim, it is classified as robustly admissible but locally pole-sensitive. A high value of S_max indicates the presence of a region in which small variations of the plant parameters may cause significant local pole displacement or rearrangement. It does not necessarily imply a large dispersion of the transient responses over the entire uncertainty domain.

The indicators S_y,RMS(θ) and S_y,max(θ), defined in (33) and (34), are used to evaluate the direct manifestation of parameter variations in the time-domain response. Let $S_{y,\lim }^{\mathrm{RMS}}$ and $S_{y,\lim }^{\max }$ be admissible thresholds for the transient-response sensitivity. These thresholds are also specified for the particular problem depending on the admissible sensitivity of the time-domain response. The value of S_lim is specified for the particular problem according to the adopted requirements for admissible local sensitivity and does not represent a universal limiting threshold. The following set can then be introduced:

$${\mathsf{\Theta }}_{D,S,y}^r=\left\{\theta \in {\mathsf{\Theta }}_{D,S}^r\mid S_{y,\mathrm{RMS}}\left(\theta \right)\le S_{y,\lim }^{\mathrm{RMS}},S_{y,\max }\left(\theta \right)\le S_{y,\lim }^{\max }\right\}.$$

(43)

Set (43) contains the tunings that are simultaneously robustly admissible, have no pronounced local pole fragility, and have limited transient-response sensitivity. A tuning θ is considered robustly stable if, for all ρ∈Ω_ρ, all closed-loop poles remain in the open left half-plane.

On this basis, the tunings can be classified sequentially according to robust stability, robust admissibility, local pole sensitivity, and transient-response sensitivity:

• Unstable tuning - there exists ρ∈Ω_ρ for which at least one closed-loop pole is unstable.
• Robustly stable but inadmissible tuning - the control system remains stable for all ρ∈Ω_ρ, but the robust-admissibility condition $𝒞$(θ)⊂D, (25), is not satisfied.
• Robustly admissible but locally pole-sensitive tuning - $𝒞$(θ)⊂D is satisfied, but S_max(θ) > S_lim, i.e., $\theta \in {\mathsf{\Theta }}_D^r$, but $\theta \notin {\mathsf{\Theta }}_{D,S}^r$, (42).
• Robustly admissible tuning with low pole sensitivity but increased transient-response sensitivity - S_max(θ) ≤ S_lim, but at least one of the indicators S_y,RMS(θ) or S_y,max(θ), defined in (33) and (34), exceeds the corresponding admissible threshold, i.e., $\theta \in {\mathsf{\Theta }}_{D,S}^r$, but $\theta \notin {\mathsf{\Theta }}_{D,S,y}^r$.
• Robustly admissible tuning with low pole sensitivity and low transient-response sensitivity - the robust-admissibility condition, S_max(θ) ≤ S_lim, and the constraints on S_y,RMS(θ) and S_y,max(θ) are simultaneously satisfied, i.e., $\theta \in {\mathsf{\Theta }}_{D,S,y}^r$, (43).

3.8. Algorithmic Sequence of the Proposed Method

The proposed method for pole-sensitivity mapping and comparison of robustly admissible tunings is summarized in Algorithm 1.

Algorithm 1. Pole-Sensitivity Mapping and Evaluation of Robustly Admissible Tunings

Input: characteristic equation P(s, θ, ρ) = 0, (22); parametric uncertainty domain Ωρ; set of tunings Ωθ; dynamic admissibility region D; parameter and time grids; admissible thresholds Slim, $S_{y,\lim }^{\mathrm{RMS}}$, and $S_{y,\lim }^{\max }$; minimum admissible geometric margin μD,lim.   Output: set of robustly admissible tunings and the indicators SM,max(θ), SΦ,max(θ), Smax(θ), Sy,RMS(θ), and Sy,max(θ). 1.  When the uncertain parameters ρj have different dimensions or substantially different ranges, they are normalized according to Section 3.4. For each tuning θ∈Ωθ, the poles si(ρ, θ) are calculated for all points ρ∈Ωρ. 2.  The poles at neighboring parameter points are tracked, and the root branches $𝒞$i(θ) and the total root contour $𝒞$(θ) are formed. 3.  The robust stability of the tuning is verified. If there exists ρ∈Ωρ for which at least one pole is unstable, the tuning is classified as unstable, and the procedure proceeds to the next tuning θ∈Ωθ. 4.  For a robustly stable tuning, the robust-admissibility condition $𝒞$(θ)⊂D, (25), is verified. If the condition is not satisfied, the tuning is classified as robustly stable but inadmissible, and the procedure proceeds to the next tuning θ∈Ωθ. 5.  For each robustly admissible tuning, the modulus Mi and argument Φi of the poles are determined using (5) and (6). 6.  For each pole si and uncertain parameter ρj, the modulus sensitivity SM,i,j, phase sensitivity SΦ,i,j, and local pole sensitivity Si,j are calculated using (13), (16), and (26). 7.  For simultaneous variations of the uncertain parameters, the matrix Ji, (27), is formed, and the multi-parameter local sensitivity Si,max, (29), is determined. 8.  The values of SM,i,j, SΦ,i,j, Si,j, and Si,max are mapped onto the corresponding root branches $𝒞$i(θ). 9.  For each tuning, SM,max(θ), SΦ,max(θ), and the overall indicator Smax(θ), (30), are determined. 10. For each parameter point, the transient response y(t, ρ, θ) is calculated. The derivatives with respect to ρj are determined using (35), after which Sy(ρ, θ), Sy,RMS(θ), and Sy,max(θ) are calculated using (32)–(34). 11. The tuning is classified according to its robust admissibility, Smax(θ), and the indicators Sy,RMS(θ) and Sy,max(θ), in accordance with (42) and (43). 12. The procedure is repeated for all θ∈Ωθ. 13. The robustly admissible tunings are compared using the global geometric indicators Δσ, Δω, dmax, and μD [29], the maximum pole sensitivities, and the transient-response sensitivity indicators. 14. $\mathrm{From}\mathrm{the}\mathrm{set}{\mathsf{\Theta }}_{D,S,y}^r$, defined in (43), the tunings providing a sufficient geometric margin μD(θ) ≥ μD,lim are selected. If more than one tuning satisfies these conditions, they are jointly compared according to SM,max(θ), SΦ,max(θ), Smax(θ), Sy,RMS(θ), Sy,max(θ), and the global geometric indicators from [29]. The final choice θ* is verified using the families and envelopes of the transient responses.

3.9. Presentation and Visualization of the Results

The results obtained using the proposed method can be presented through the root contours $𝒞$(θ), defined in (23), and their individual pole branches $𝒞$_i(θ), defined in (24), with respect to the dynamic admissibility region D; the modulus characteristics M_i(ρ_j), defined based on (5); the phase characteristics Φ_i(ρ_j), defined based on (6); maps of the modulus sensitivity S_M,i,j, phase sensitivity S_Φ,i,j, local pole sensitivity S_i,j, and multi-parameter local sensitivity S_i,max, introduced by (13), (16), (26), and (29), respectively; comparative diagrams of the global geometric, pole-sensitivity, and time-domain indicators; and families and envelopes of the transient responses.

The root contour $𝒞$(θ), (23), represents the set of possible pole locations as the parameters ρ∈Ω_ρ vary, while each root branch $𝒞$_i(θ), (24), traces the motion of a particular pole s_i(ρ, θ). The dynamic admissibility region D is plotted in the same complex plane, allowing direct visual verification of the robust-admissibility condition $𝒞$(θ)⊂D, (25), and evaluation of the position of the individual pole branches relative to the boundary of D.

The modulus and phase characteristics are presented as the relationships M_i(ρ_j) and Φ_i(ρ_j) with respect to the varying uncertain parameter ρ_j, where M_i and Φ_i are defined by (5) and (6), respectively. Each characteristic refers to a particular tracked root branch $𝒞$_i(θ), preserving the relationship between the pole motion in the complex plane and the variation of its modulus and argument. Thus, the plots of M_i(ρ_j) and Φ_i(ρ_j) show the radial and angular variations of the pole, respectively, along the parameter trajectory under consideration.

The pole-sensitivity maps are obtained by associating each point s_i(ρ, θ) of the root branch $𝒞$_i(θ) with the corresponding value of S_M,i,j, S_Φ,i,j, S_i,j, or S_i,max, according to the mapping relationships (36)–(38). The value of the selected indicator is color-coded along the root branch, making it possible to directly localize regions with increased radial, tangential, or total local pole sensitivity. If necessary, the normalized indicator from (39) can be used for visualization. When several tunings are compared, the same color scale is used so that the same color corresponds to the same sensitivity level. The sequential relationship between the parameter point, the pole position, its modulus and argument, and the corresponding sensitivities is summarized in (40).

The transient-response sensitivity S_y(ρ, θ), defined in (32), characterizes the entire time-domain response at a given parameter point and is therefore not mapped onto an individual pole branch. For an overall comparison of the tunings, S_y,RMS(θ) and S_y,max(θ), defined in (33) and (34), respectively, are used. The practical manifestation of parametric uncertainty in the time-domain response is visualized through the families and envelopes of the transient responses.

When several tunings are compared, the robust-admissibility condition (25) first identifies the robustly admissible solutions. Then, S_max(θ), defined in (30), together with S_M,max(θ) and S_Φ,max(θ), is used to evaluate the local pole fragility and the nature of the pole motion, while S_y,RMS(θ) and S_y,max(θ), (33) and (34), evaluate the manifestation of parameter variations in the time-domain response.

The final comparison is performed jointly using the global geometric indicators Δσ, Δω, d_max, and μ_D, defined in [29], the maximum pole sensitivities S_M,max(θ), S_Φ,max(θ), and S_max(θ), and the transient-response sensitivity indicators S_y,RMS(θ) and S_y,max(θ).

4. Numerical Study

4.1. System with a PI Controller under Parametric Uncertainty and Initial Assessment of Robust Admissibility, [29]

The plant is described by the transfer function $G\left(s,\rho \right)={\displaystyle \frac{1}{\left(s+{\rho }_1\right)\left(s+{\rho }_2\right)}}$, where $\rho ={\left[{\rho }_1{\rho }_2\right]}^T$ is the vector of uncertain plant parameters. The parameters vary within the intervals ρ₁∈[1.5, 2.5] and ρ₂∈[0.5, 1.5]; therefore, the parametric uncertainty domain is Ω_ρ = [1.5, 2.5] × [0.5, 1.5]. The nominal parameter vector is selected at the midpoint of the considered intervals as ρ⁰ = [2 1]^T.

A PI controller $R\left(s,\theta \right)=K_R+{\displaystyle \frac{K_I}{s}}$ is used to control the plant, where $\theta ={\left[K_R K_I\right]}^T$ is the vector of controller tunings. Under unity negative feedback, the characteristic equation of the closed-loop system is P(s, θ, ρ) = s³ + (ρ₁ + ρ₂)s² + (ρ₁ρ₂ + K_R)s + K_I = 0. For each fixed tuning θ and each admissible combination ρ∈Ω_ρ, it has three roots s_i = s_i(ρ, θ), i = 1,2,3.

According to definitions (23) and (24), their positions as ρ varies form three root branches $𝒞$_i(θ), while the total root contour is $\mathcal{C}\left(\theta \right)={\displaystyle {\bigcup }_{i=1}^3}{\mathcal{C}}_i\left(\theta \right).$

For the initial evaluation of the tunings, the following admissibility region is used:

D¹ = {s∣s = σ + jω∈ℂ, σ ≤ −0.1, ∣ω∣≤1.5}.

The constraint σ ≤ −0.1 requires all considered poles to be located at a prescribed distance to the left of the imaginary axis. The constraint ∣ω∣≤1.5 limits their position along the imaginary axis and, consequently, the excessive increase in the oscillatory component.

The robust admissibility of a fixed tuning θ is verified using condition (25). The condition must be satisfied for all pole branches (23), including the regions lying on the real axis. Therefore, a tuning is classified as robustly admissible only when the three pole branches obtained for ρ∈Ω_ρ remain within the region D¹.

In the present numerical study, condition (25) is verified on a two-dimensional 301 × 301 parameter grid, including the interior and boundary points of Ω_ρ. For each parameter combination, the three pole branches (24) are tracked, and a discrete approximation of the root contour (23) is formed. Therefore, robust admissibility is numerically confirmed on the grid used.

The considered set of candidate tunings is

$${\mathsf{\Omega }}_{\theta }^N=\left\{\left(K_R,K_I\right)\mid K_R\in \left\{0.5,1.0,2.0\right\},K_I\in \left\{0.5,1.0,2.0\right\}\right\}.$$

(44)

All nine tunings are robustly stable, but only seven satisfy the robust-admissibility condition $𝒞$(θ)⊂D¹:

$${\mathsf{\Theta }}_{D^1}^r=\left\{\left(0.5,0.5\right),\left(0.5,1.0\right),\left(1.0,0.5\right),\left(1.0,1.0\right),\left(1.0,2.0\right),\left(2.0,1.0\right),\left(2.0,2.0\right)\right\}.$$

(45)

The tunings (0.5, 2.0) and (2.0, 0.5) are robustly stable but are not robustly admissible with respect to D¹.

The global indicators defined in [29] do not provide an unambiguous ranking of the seven admissible tunings. For example, the tuning (K_R, K_I) = (0.5, 0.5) has the smallest vertical dispersion, Δω = 1.0289, but its geometric margin is relatively small, μ_D = 0.0304. The tuning (K_R, K_I) = (2.0, 2.0) has the smallest horizontal dispersion, Δσ = 1.3830, and the largest geometric margin, μ_D = 0.2435, but not the smallest maximum displacement relative to the nominal poles.

These differences show that the size of the root contour $𝒞$(θ), the maximum displacement, and the distance to the boundary of D¹ describe different properties. However, they do not show whether the contour is formed through smooth pole motion or contains regions in which a small variation of ρ₁ or ρ₂ leads to an abrupt displacement. Therefore, the condition $𝒞$(θ)⊂D¹ is retained as the first step in the selection.

4.2. Modulus M_i(ρ_j) and Phase Φ_i(ρ_j) Characteristics of the Poles under Parametric Uncertainty

For each tuning $\theta \in {\mathsf{\Theta }}_{D^1}^r$, where the set ${\mathsf{\Theta }}_{D^1}^r$ is defined in (45), the three roots s_i(ρ, θ), i = 1,2,3, of the closed-loop characteristic equation, representing a particular case of (22), are calculated. According to definitions (23) and (24), as ρ∈Ω_ρ varies, the root locations form three tracked pole branches $𝒞$_i(θ), whose union forms the root contour $𝒞$(θ). For each point s_i = σ_i + jω_i, defined according to (2), on the corresponding pole branch, the modulus M_i and argument Φ_i are determined using (5) and (6). The modulus and phase characteristics are not constructed independently of the root contours but are obtained by transforming each point of the tracked pole branch $𝒞$_i(θ), (24), from the rectangular coordinates (σ_i, ω_i) to the polar coordinates (M_i, Φ_i) according to representation (4).

To obtain single-valued relationships with respect to one parameter, the characteristics are considered along the four boundary trajectories of the domain Ω_ρ: ρ₁∈[1.5, 2.5], ρ₂ = 0.5; ρ₁∈[1.5, 2.5], ρ₂ = 1.5; ρ₂∈[0.5, 1.5], ρ₁ = 1.5; and ρ₂∈[0.5, 1.5], ρ₁ = 2.5.

To avoid overloading the presentation, detailed modulus and phase characteristics are shown for four tunings with different global and local properties: θ_A = (0.5, 0.5), θ_B = (1.0, 1.0), θ_C = (2.0, 2.0), and θ_D = (2.0, 1.0). Tuning θ_A has the smallest vertical dispersion Δω among the robustly admissible tunings, θ_B has intermediate values of the global indicators Δσ, Δω, d_max, and μ_D, while θ_C has the largest geometric margin μ_D to the boundary of D¹ [29]. Tuning θ_D is included because, in the subsequent comparison, it shows a favorable combination of local pole sensitivity, evaluated by S_max(θ), (30), and transient-response sensitivity, characterized by S_y,RMS(θ) and S_y,max(θ), (33) and (34). Thus, the selected tunings allow cases with different global geometric indicators and different types of local pole motion to be compared.

4.2.1. Modulus Characteristics

For each fixed tuning θ, the three roots s_i(ρ, θ), i = 1,2,3, of the characteristic equation are first calculated and tracked along the selected boundary trajectory of the parametric domain Ω_ρ. According to (2), each root is represented as s_i = σ_i + jω_i, while its motion as the parameters vary forms the corresponding root branch $𝒞$_i(θ), defined in (24). The union of the three root branches forms the root contour $𝒞$(θ), defined in (23). For example, when ρ₁ varies and ρ₂ = 0.5 is fixed, the locations s_i(ρ₁, 0.5, θ) = σ_i(ρ₁) + jω_i(ρ₁), i = 1,2,3, are obtained for each value of ρ₁∈[1.5, 2.5]. The modulus characteristic of each tracked pole branch $𝒞$_i(θ) is obtained by sequentially calculating the modulus of the corresponding root, M_i(ρ₁)=∣s_i(ρ₁, 0.5, θ)∣, according to (5).

The modulus characteristics for ρ₂ = 1.5, as well as those obtained by varying ρ₂ for the fixed values ρ₁ = 1.5 and ρ₁ = 2.5, are determined in the same way. The varying uncertain parameter ρ_j is plotted along the horizontal axis, while the modulus M_i of the corresponding tracked pole is plotted along the vertical axis. Each point of the modulus characteristic M_i(ρ_j), defined by (5), corresponds to a particular location s_i(ρ, θ) on the same root branch $𝒞$_i(θ), (24), in the complex plane, as shown in Figure 1 and Figure 2.

For a complex-conjugate pair s_1,2 = σ ± jω, the moduli of the two poles coincide, M₁ = M₂, as follows directly from definition (5). Therefore, the corresponding modulus characteristics overlap and may appear as a single curve without filtering. When the complex-conjugate pair separates into two real poles, the modulus characteristics separate, and the two root branches $𝒞$_i(θ), (24), become distinguishable.

For complex poles, the modulus M_i, defined by (5), corresponds to the natural frequency ω_n of the equivalent oscillatory mode according to (7). For a stable real pole s_i = σ_i < 0, it follows from (5) that M_i = |σ_i|; consequently, an increase in M_i corresponds to a displacement to the left in the complex plane and faster decay of the aperiodic component, while a decrease in M_i indicates an approach to the imaginary axis and a slower transient response.

Figure 1 presents the variation of the moduli as ρ₁ varies. The left panels correspond to ρ₂ = 0.5, while the right panels correspond to ρ₂ = 1.5. Each row shows the results for one of the four selected tunings. When two poles form a complex-conjugate pair, their modulus characteristics coincide. Therefore, two distinct curves are observed in some regions, although three pole branches are tracked. When the poles transition to three real poles, the corresponding characteristics separate.

Figure 1 shows that the value of ρ₂ affects both the initial values and the slopes of the modulus characteristics. Consequently, the influence of ρ₁ on the poles also depends on the fixed value of the second parameter.

For θ_A, θ_B, and θ_C, differences are observed in the slopes and relative positions of the individual pole branches. For θ_D, the characteristics vary smoothly for both boundary values of ρ₂, without a clearly pronounced merging and subsequent separation of the branches. At ρ₂ = 0.5, the variations are relatively small, while at ρ₂ = 1.5, the modulus of the real pole branch increases, and the modulus of the complex-conjugate pair decreases smoothly as ρ₁ increases.

Figure 1 shows the range and nature of the modulus variations but does not provide a quantitative evaluation of the rate of these variations.

Figure 2 presents the modulus characteristics as ρ₂ varies. The left panels correspond to ρ₁ = 1.5, while the right panels correspond to ρ₁ = 2.5.

Compared with the results in Figure 1, more clearly pronounced regions of convergence and separation of the pole branches are observed here. In these regions, the nature of the roots may change from a complex-conjugate pair to two real poles or vice versa.

For θ_A at ρ₁ = 1.5, a critical region is observed around ρ₂≈1, where two modulus characteristics converge and then separate. At ρ₁ = 2.5, a similar transition occurs at a lower value of ρ₂, after which the individual branches move apart.

For θ_B at ρ₁ = 2.5, a clearly pronounced convergence and subsequent separation of two pole branches are also observed. For θ_C, the variation is smoother, although at ρ₁ = 2.5 the modulus characteristics intersect in the middle part of the interval.

For θ_D, the modulus characteristics vary smoothly at both boundary values of ρ₁. No abrupt changes in slope or clearly pronounced regions of branch merging and separation are observed. This indicates a more uniform variation of the poles with respect to ρ₂.

The observed differences are quantitatively evaluated in Section 4.3 using the modulus sensitivity.

4.2.2. Phase Characteristics

For each fixed tuning θ and each selected boundary trajectory of the parametric domain Ω_ρ, the three roots s_i(ρ, θ), i = 1,2,3, of the characteristic equation are first calculated and tracked. According to (2), each root is represented as s_i = σ_i + jω_i, while its motion as the parameters vary forms the corresponding root branch $𝒞$_i(θ), defined in (24). For example, when ρ₁ varies and ρ₂ = 0.5 is fixed, the locations s_i(ρ₁, 0.5, θ) = σ_i(ρ₁) + jω_i(ρ₁), i = 1,2,3, are obtained for each value of ρ₁∈[1.5, 2.5]. The phase characteristic of each tracked pole branch $𝒞$_i(θ) is obtained by sequentially calculating the argument of the corresponding root, Φ_i(ρ₁) = args_i(ρ₁, 0.5, θ), according to (6).

The phase characteristics for ρ₂ = 1.5, as well as those obtained by varying ρ₂ for the fixed values ρ₁ = 1.5 and ρ₁ = 2.5, are determined in the same way. The varying uncertain parameter ρ_j is plotted along the horizontal axis, while the argument Φ_i of the corresponding tracked pole is plotted along the vertical axis. Each point of the phase characteristic Φ_i(ρ_j), defined by (6), corresponds to a particular location s_i(ρ, θ) on the same root branch $𝒞$_i(θ), (24), in the complex plane, as shown in Figure 3 and Figure 4.

For a stable negative real pole s_i = σ_i < 0, the argument is Φ_i = π, or equivalently −π, depending on the selected interval for the argument, as follows from (6). For the complex-conjugate pair s_1,2 = σ ± jω, the arguments are symmetric, Φ₂ = −Φ₁, when the principal interval (−, ππ] is used. Therefore, for a complex-conjugate pair, only the phase characteristic of the pole branch with a positive imaginary part ω_i > 0 may be presented in the figures, while the characteristic of the conjugate branch is determined symmetrically.

When the complex-conjugate pair transitions into two real poles, the phase characteristic of the branch with a positive imaginary part is interrupted because no pole with ω_i > 0 exists in the corresponding parameter interval. This interruption is not a numerical error but reflects the change in the nature of the roots from complex to real.

For complex poles, the phase characteristic is directly related to the damping ratio through ξ_i = −cosΦ_i, (8). As the complex pole approaches the imaginary axis, Φ_i approaches π/2 and ξ_i decreases, whereas when it moves toward the negative real axis, Φ_i approaches π and ξ_i increases.

Figure 3 presents the phase characteristics as ρ₁ varies. The left panels correspond to ρ₂ = 0.5, while the right panels correspond to ρ₂ = 1.5. Each row shows the results for one of the four selected tunings.

An increase in the phase from 90^∘ to 180^∘ corresponds to the complex pole approaching the negative real axis and to an increase in damping. A comparison of the panels shows that the value of ρ₂ affects both the phase position and the slope of the characteristics.

For θ_A at ρ₂ = 1.5, the phase increases most strongly and approaches 180^∘ at the end of the interval. This indicates that the complex-conjugate pair approaches the negative real axis. For θ_B and θ_C, the phase characteristics vary more smoothly.

For θ_D, the phase also increases smoothly at both fixed values of ρ₂. At ρ₂ = 0.5, the variation is more pronounced, while at ρ₂ = 1.5, the characteristic has a smaller slope. No interruption of the complex pole branch is observed over the considered interval.

The slope of the phase characteristics is quantitatively evaluated in Section 4.3 using the phase sensitivity.

Figure 4 presents the phase characteristics as ρ₂ varies. The left panels correspond to ρ₁ = 1.5, while the right panels correspond to ρ₁ = 2.5.

In some cases, the phase rapidly approaches 180^∘, after which the characteristic is interrupted. The interruption indicates an interval in which no pole with a positive imaginary part exists because the complex-conjugate pair has transitioned into two real poles. Therefore, the absence of a curve is not a numerical error but results from the change in the nature of the roots.

A comparison of Figure 2 and Figure 4 shows that the convergence and separation of the modulus characteristics correspond to the variation or interruption of the phase characteristics. An increase in the phase and total pole sensitivities is expected around these critical values.

For θ_D, the phase characteristics vary smoothly at both fixed values of ρ₁, without interruption of the complex pole branch. At ρ₁ = 1.5, the phase increases as ρ₂ increases, whereas at ρ₁ = 2.5, the variation is weak.

In summary, the variations of the modulus and phase characteristics can be directly related to the nature of the transient response. For a complex pole s_i, an increase in the modulus M_i, defined by (5), corresponds to an increase in the natural frequency ω_n, according to (7), and, for approximately preserved damping, leads to a faster transient response. A decrease in M_i is associated with a lower natural frequency and a slower response. For a stable real pole, an increase in M_i indicates a displacement to the left in the complex plane and faster decay of the corresponding aperiodic component, while a decrease brings the pole closer to the imaginary axis and slows the transient response.

For a complex pole in the left half-plane, an increase in the argument Φ_i toward π leads to an increase in the damping ratio ξ_i, according to (8), which is associated with reduced oscillation and overshoot of the transient response. Conversely, a decrease in Φ_i toward π/2 leads to a decrease in ξ_i, weaker damping, and a more pronounced oscillatory nature of the time-domain response.

4.3. Modulus S_M,i,j, Phase S_Φ,i,j, and Total S_i,j Pole Sensitivities

For a quantitative assessment of the local slopes of the modulus and phase characteristics (5) and (6), the total pole sensitivity S_i,j, the modulus sensitivity S_M,i,j, and the phase sensitivity S_Φ,i,j are calculated according to (26), (13), and (16), respectively, while the relationship between them is given by the decomposition (19).

For the characteristic equation P(s, θ, ρ) = s³ + (ρ₁ + ρ₂)s² + (ρ₁ρ₂ + K_R)s + K_I, the required partial derivatives are

$${\displaystyle \frac{\partial P}{\partial s}}=3s^2+2\left({\rho }_1+{\rho }_2\right)s+{\rho }_1{\rho }_2+K_R.$$

(46)

$${\displaystyle \frac{\partial P}{\partial {\rho }_1}}=s\left(s+{\rho }_2\right),{\displaystyle \frac{\partial P}{\partial {\rho }_2}}=s\left(s+{\rho }_1\right).$$

(47)

By applying the general relationship (9) to each tracked pole s_i, the following expressions are obtained:

$$\begin{gathered} {\displaystyle \frac{\partial s_i}{\partial {\rho }_1}}=-{\displaystyle \frac{s_i\left(s_i+{\rho }_2\right)}{3s_i^2+2\left({\rho }_1+{\rho }_2\right)s_i+{\rho }_1{\rho }_2+K_R}}, \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\displaystyle \frac{\partial s_i}{\partial {\rho }_2}}=-{\displaystyle \frac{s_i\left(s_i+{\rho }_1\right)}{3s_i^2+2\left({\rho }_1+{\rho }_2\right)s_i+{\rho }_1{\rho }_2+K_R}}. \end{gathered}$$

(48)

For each parametric point, the procedure requires first calculating and tracking the roots s_i. Then, ∂s_i/∂ρ_j is obtained from (48), and finally S_i,j, S_M,i,j, and S_Φ,i,j are determined. High values may arise when the denominator in (48) approaches zero. This corresponds to proximity to a multiple root or to a merging and separation point of the pole branches, as discussed in Section 2.8.

4.3.1. Sensitivity with Respect to ρ₁

The sensitivity with respect to ρ₁ is calculated along the same two boundary trajectories used in Figure 1 and Figure 3, namely ρ₁∈[1.5, 2.5], ρ₂ = 0.5, and ρ₁∈[1.5, 2.5], ρ₂ = 1.5. For each value of ρ₁ and each tracked pole branch, the sensitivities S_i,1(ρ₁), S_M,i,1(ρ₁), and S_Φ,i,1(ρ₁) are determined according to (13), (16), and (26).

Figure 5 presents the three sensitivities with respect to ρ₁. Each row corresponds to one tuning, while the two columns correspond to the fixed values ρ₂ = 0.5 and ρ₂ = 1.5.

For θ_A at ρ₂ = 0.5, the sensitivities S_M,i,1, S_Φ,i,1, and S_i,1 vary smoothly, and the maximum value of the total pole sensitivity S_i,1 is 0.2806 at ρ₁ = 1.500. At ρ₂ = 1.5, the total pole sensitivity S_i,1 increases sharply and reaches 8.7487 at ρ₁ = 2.490. The maximum is predominantly phase-dominated, i.e., S_Φ,i,1 > S_M,i,1, and corresponds to a sharp variation in the pole argument Φ_i.

For θ_B, the maximum value of the total pole sensitivity S_i,1 is 0.3836 at ρ₂ = 0.5 and 0.7548 at ρ₂ = 1.5. No pronounced local peak is observed.

For θ_C, the maximum values of S_i,1 are 0.4641 at ρ₂ = 0.5 and 0.7162 at ρ₂ = 1.5, respectively. At ρ₂ = 0.5, the dominant component between S_M,i,1 and S_Φ,i,1 changes along the trajectory, whereas at ρ₂ = 1.5, the phase sensitivity S_Φ,i,1 remains higher without a sharp increase.

For θ_D, the maximum value of the total pole sensitivity S_i,1 is 1.067 at ρ₂ = 0.5 and ρ₁ = 2.500, whereas at ρ₂ = 1.5, it is 0.571, again at ρ₁ = 2.500. In both cases, S_M,i,1, S_Φ,i,1, and S_i,1 vary smoothly without local peaks. At ρ₂ = 0.5, the total pole sensitivity S_i,1 increases as ρ₁ increases, whereas at ρ₂ = 1.5, it remains lower and relatively uniform.

Therefore, a significant local maximum of S_i,1 with respect to ρ₁ is observed only for θ_A at ρ₂ = 1.5. The tunings θ_B, θ_C, and θ_D exhibit smoother local behavior, with no sharp variations observed for θ_D along the two considered trajectories.

4.3.2. Sensitivity with Respect to ρ₂

The sensitivity with respect to ρ₂ is calculated along the boundary trajectories ρ₂∈[0.5, 1.5], ρ₁ = 1.5, and ρ₂∈[0.5, 1.5], ρ₁ = 2.5. For each value of ρ₂, the sensitivities S_i,2(ρ₂), S_M,i,2(ρ₂), and S_Φ,i,2(ρ₂) are determined according to (13), (16), and (26).

Figure 6 presents the sensitivities with respect to ρ₂. The left panels correspond to ρ₁ = 1.5, and the right panels correspond to ρ₁ = 2.5. If the individual panels use different vertical scales, the quantitative comparison should be made using the indicated maximum values.

For θ_A at ρ₁ = 1.5, the total pole sensitivity S_i,2 reaches 7.5466 at ρ₂ = 0.963. At ρ₁ = 2.5, the maximum value of S_i,2 is 7.2940 at ρ₂ = 0.747. In both cases, the phase sensitivity S_Φ,i,2 exceeds the modulus sensitivity S_M,i,2 and determines the total sensitivity, with the maximum occurring near a transition between complex and real poles.

For θ_B at ρ₁ = 1.5, the maximum total pole sensitivity S_i,2 is 1.2116 at ρ₂ = 1.120. At ρ₁ = 2.5, the highest value among the considered cases is obtained, S_i,2 = 9.0884 at ρ₂ = 0.937. The maximum is phase-dominated, S_Φ,i,2 > S_M,i,2, and is located immediately before the interruption of the complex pole branch.

For θ_C at ρ₁ = 1.5, the total pole sensitivity S_i,2 reaches 0.7185 at ρ₂ = 1.407. At ρ₁ = 2.5, the maximum value is S_i,2 = 2.4038 at ρ₂ = 0.953 and is determined mainly by the modulus sensitivity S_M,i,2, i.e., S_M,i,2 > S_Φ,i,2.

For θ_D at ρ₁ = 1.5, the total pole sensitivity S_i,2 reaches 0.681 at ρ₂ = 0.500, whereas at ρ₁ = 2.5, the maximum value is S_i,2 = 1.504, also at ρ₂ = 0.500. In both cases, S_M,i,2, S_Φ,i,2, and S_i,2 decrease smoothly as ρ₂ increases, and no local peaks are observed. At ρ₁ = 1.5, the modulus and phase sensitivities S_M,i,2 and S_Φ,i,2 are close in value, whereas at ρ₁ = 2.5, the modulus sensitivity S_M,i,2 is dominant.

The results show that the high local maxima of the total pole sensitivity S_i,2 for θ_A and θ_B are predominantly phase-dominated, S_Φ,i,2 > S_M,i,2, and occur near a change in the nature of the poles. For θ_C, the sensitivity is lower and more uniform, while the local maximum at ρ₁ = 2.5 has a radial nature, S_M,i,2 > S_Φ,i,2. The tuning θ_D contains no local peaks and exhibits a smooth variation in S_M,i,2, S_Φ,i,2, and S_i,2 along the two considered trajectories.

4.3.3. Real Pole Sensitivity

For a negative real pole, the phase is constant. Therefore, the phase sensitivity is zero, and the total sensitivity coincides with the modulus sensitivity. To quantify the influence of ρ₁ and ρ₂ on the real pole, S_i,1 and S_i,2 are calculated according to (26) along the same boundary trajectories. This sensitivity is presented separately because the phase sensitivity of the real pole is zero.

Figure 7 presents the sensitivity of the tracked real pole branch. The left panels show its variation with respect to ρ₁ at ρ₂ = 0.5 and ρ₂ = 1.5, while the right panels show its variation with respect to ρ₂ at ρ₁ = 1.5 and ρ₁ = 2.5.

As ρ₁ varies, the sensitivity of the tracked real pole S_i,1, determined according to (26), changes smoothly for all four tunings. For θ_A, θ_B, and θ_C, no sharp local maxima are observed, and the values of S_i,1 remain approximately no greater than one. For θ_D at ρ₂ = 0.5, S_i,1 = 0, because along this trajectory the tracked real pole remains at s_i = −0.5 and does not change its position as ρ₁ varies. At ρ₂ = 1.5, the sensitivity S_i,1 decreases smoothly from 0.126. Therefore, variation in ρ₁ leads to a smooth displacement of the real pole along all considered trajectories.

More significant differences are observed as ρ₂ varies, i.e., for the sensitivity S_i,2.

For θ_A at ρ₁ = 1.5, S_i,2 increases sharply around ρ₂≈1, where the denominator ∂P/∂s in the analytical relationship (9) is equal to zero. The resulting very high numerical value indicates a multiple root and practically singular behavior rather than an ordinary local maximum. At ρ₁ = 2.5, the maximum value is S_i,2 = 4.000 at ρ₂ = 1.500.

For θ_B at ρ₁ = 1.5, the sensitivity S_i,2 reaches 1.409 at ρ₂ = 1.131. At ρ₁ = 2.5, a clearly pronounced maximum of S_i,2 = 29.082 is observed at ρ₂ = 1.187. This shows a significant dependence of the local sensitivity with respect to ρ₂ on the fixed value of ρ₁.

For θ_C at ρ₁ = 1.5, the sensitivity S_i,2 varies smoothly and reaches 0.413 at ρ₂ = 1.426. At ρ₁ = 2.5, a local maximum of S_i,2 = 3.806 is observed around ρ₂ = 0.953.

For θ_D, the sensitivity S_i,2 decreases monotonically as ρ₂ increases. The maximum values occur at the beginning of the interval: S_i,2 = 0.333 at ρ₁ = 1.5 and S_i,2 = 1.000 at ρ₁ = 2.5, in both cases at ρ₂ = 0.500. No local peaks or regions of sharp variation are observed.

A comparison of the eight panels shows that the tracked real pole is more sensitive to ρ₂ than to ρ₁. The highest values of S_i,2 for θ_A and θ_B occur near a change in the nature of the roots, whereas θ_D exhibits the smoothest and most uniform behavior of the real pole branch.

From the time-domain perspective, a high local pole sensitivity S_i,j, determined according to (26), means that a small variation in the uncertain parameter ρ_j may cause a significant displacement of the real pole s_i. For a stable real pole s_i = σ_i < 0, the modulus is M_i=∣σ_i∣, according to (5). Therefore, a more substantial variation in s_i appears as a variation in the decay rate of the corresponding aperiodic component of the transient response. When the real pole moves to the left, i.e., as ∣σ_i∣ and M_i increase, the decay becomes faster. When the pole approaches the imaginary axis, ∣σ_i∣ and M_i decrease, and the corresponding component of the transient response becomes slower. Conversely, the smooth variation in S_i,j for θ_D indicates lower local sensitivity of the real pole position and implies a smaller variation in the decay rate under small parametric variations.

This interpretation is based on the local motion of an individual pole and does not constitute an independent assessment of the entire response time. The direct influence of the parametric variations on the output variable is evaluated using the transient-response sensitivity S_y(ρ, θ), defined in (32), and the aggregate indicators S_y,RMS(θ) and S_y,max(θ), defined in (33) and (34), respectively. The correspondence between the local pole sensitivity and the actual spread of the time responses is additionally verified using the families and envelopes of the transient responses in Section 4.7.

The maximum values obtained for the complex pole branch along the considered boundary trajectories are summarized in Table 1.

The numerical comparison of SM,i,j, S_Φ,i,j, and S_i,j is used not only to detect local maxima but also to determine whether the expected variation in the transient response primarily affects the speed of response, the damping and oscillatory behavior, or both simultaneously. The extent to which these local pole variations appear in the output variable is assessed further using S_y(ρ, θ), S_y,RMS(θ), and S_y,max(θ), defined in (32)–(34), as well as the families and envelopes of the transient responses.

The obtained values of S_M,i,j, S_Φ,i,j, and S_i,j are mapped onto the corresponding positions s_i(ρ, θ) on the root contours in Section 4.4 according to the mapping relationships (36)–(40).

4.4. Sensitivity Mapping onto the Root Contours

For the geometric localization of the identified sensitivity maxima, the obtained values are associated with the corresponding pole positions s_i(ρ, θ) in the complex plane.

For each fixed tuning θ, parametric point ρ∈Ω_ρ, and tracked pole branch C_i(θ), defined in (24), the values of the total pole sensitivity S_i,j, modulus sensitivity S_M,i,j, and phase sensitivity S_Φ,i,j, determined according to (26), (13), and (16), respectively, are assigned to the position s_i(ρ, θ), where j = 1,2. The mapping is performed according to relationships (36)–(38), so that the same geometric trajectory can be represented according to S_i,j, S_M,i,j, or S_Φ,i,j.

The sensitivity with respect to ρ₁, i.e., S_i,1, is mapped along the boundary trajectories ρ₁∈[1.5, 2.5] at ρ₂ = 0.5 and ρ₂ = 1.5. At each parametric point, the numerically determined value S_i,1 is mapped onto the corresponding pole position s_i(ρ₁, ρ₂, θ).

Similarly, the sensitivity with respect to ρ₂, i.e., S_i,2, is mapped along the trajectories ρ₂∈[0.5, 1.5] at ρ₁ = 1.5 and ρ₁ = 2.5.

Separate maps of S_i,1 and S_i,2 are obtained. They show the influence of the corresponding uncertain parameter on the different pole branches C_i(θ) and allow direct localization of regions with increased local pole sensitivity.

For visualization, a common color scale is used for all compared tunings, and, when necessary, the values are represented using the normalized indicator from (39). Thus, the same color S_Φ,i,jorresponds to the same sensitivity level and enables a quantitative comparison between the different tunings.

4.4.1. Sensitivity Maps with Respect to ρ₁

The mapping is performed for the four representative tunings θ_A = (0.5, 0.5), θ_B = (1.0, 1.0), θ_C = (2.0, 2.0), and θ_D = (2.0, 1.0). While Figure 5 presents S_M,i,1, S_Φ,i,1, and S_i,1 for the complex pole branch with a positive imaginary part, Figure 8 presents the total pole sensitivity S_i,1 mapped onto all pole branches C_i(θ), defined in (24).

Figure 8 presents the geometric location of the sensitive regions as ρ₁ varies. The color along the root contours encodes the local value of the total pole sensitivity S_i,1. To provide a clearer visualization of the wide range of values, the transformation C = log₁₀(1 + S_i,1) is used. The dark-blue and blue regions correspond to low sensitivity, the light-blue and green regions to intermediate values, and the yellow, orange, and red regions to increased local sensitivity. A common color scale is used for all panels so that the same color corresponds to the same level of S_i,1. The points marked with a star indicate the maxima of S_i,1 along the corresponding boundary trajectories.

For θ_A at ρ₂ = 0.5, the complex pole branches exhibit low and smoothly varying sensitivity. The maximum value over all pole branches is max_iS_i,1 = 1.036 at ρ₁ = 2.500 and occurs on the real pole branch. At ρ₂ = 1.5, a local increase is observed at the end of the complex pole branches, where max_iS_i,1 = 8.749 at ρ₁ = 2.490. This value coincides with the maximum of the complex pole in Figure 5.

For θ_B at ρ₂ = 0.5, the maximum sensitivity is max_iS_i,1 = 1.071 at ρ₁ = 2.500 and belongs to the real pole branch. At ρ₂ = 1.5, the maximum is max_iS_i,1 = 0.755 at ρ₁ = 1.500.

For θ_C at ρ₂ = 0.5, the highest value is max_iS_i,1 = 1.062 at ρ₁ = 2.500 and occurs on the real pole branch. At ρ₂ = 1.5, the maximum is max_iS_i,1 = 0.716 at ρ₁ = 1.500 and is located on the complex pole branch.

For θ_D at ρ₂ = 0.5, the maximum sensitivity is max_iS_i,1 = 1.067 at ρ₁ = 2.500 and occurs on the complex pole branch. At ρ₂ = 1.5, the maximum is max_iS_i,1 = 0.571 at ρ₁ = 2.500. No sharp local peak is observed along either trajectory.

The mapping shows that the maximum sensitivity with respect to ρ₁ is not always determined by the complex poles but may also occur on the real pole branch. The most pronounced local maximum is obtained for θ_A at ρ₂ = 1.5, where the map shows a clearly localized transition toward warmer colors. For θ_D, the color distribution remains more uniform and does not indicate a sharp local increase in S_i,1.

4.4.2. Sensitivity Maps with Respect to ρ₂

The mapping is performed for the same four representative tunings θ_A = (0.5, 0.5), θ_B = (1.0, 1.0), θ_C = (2.0, 2.0), and θ_D = (2.0, 1.0). While Figure 6 presents the modulus sensitivity S_M,i,2, phase sensitivity S_Φ,i,2, and total pole sensitivity S_i,2 for the complex pole branch with a positive imaginary part, Figure 9 presents the total pole sensitivity S_i,2 mapped onto all pole branches C_i(θ), defined in (24), as ρ₂ varies. The left panels correspond to the boundary trajectory ρ₂∈[0.5, 1.5] at ρ₁ = 1.5, and the right panels correspond to the same trajectory at ρ₁ = 2.5.

Figure 9 presents the geometric location of the sensitive regions as ρ₂ varies. The color along the root contours encodes the local value of the total pole sensitivity S_i,2.

The dark-blue and blue regions correspond to low sensitivity, the light-blue and green regions to intermediate values, and the yellow, orange, and red regions to increased local sensitivity. A common color scale is used for all panels so that the same color corresponds to the same level of S_i,2. The points marked with a star indicate the maxima of S_i,2 along the corresponding boundary trajectories.

For θ_A at ρ₁ = 1.5, the maximum sensitivity over all pole branches is max_iS_i,2 = 14.1276 at ρ₂ = 0.967. At ρ₁ = 2.5, the maximum is max_iS_i,2 = 9.2765 at ρ₂ = 0.750. In both cases, the sensitive regions are located near a transition between real and complex poles.

For θ_B at ρ₁ = 1.5, the maximum sensitivity is max_iS_i,2 = 1.4084 at ρ₂ = 1.130. At ρ₁ = 2.5, it increases to max_iS_i,2 = 13.2000 at ρ₂ = 1.187, with the maximum occurring on the real pole branch.

For θ_C at ρ₁ = 1.5, the maximum sensitivity is max_iS_i,2 = 0.7185 at ρ₂ = 1.407. At ρ₁ = 2.5, the maximum is max_iS_i,2 = 3.8037 at ρ₂ = 0.953.

For θ_D at ρ₁ = 1.5, the maximum sensitivity is max_iS_i,2 = 0.6807 at ρ₂ = 0.500, whereas at ρ₁ = 2.5, it is max_iS_i,2 = 1.5038, also at ρ₂ = 0.500. Along both trajectories, the maximum occurs at the beginning of the interval, and no internal local peaks are observed.

The mapping shows that the maximum sensitivity with respect to ρ₂ is not always determined by the considered complex pole branch. For θ_B at ρ₁ = 2.5, as well as for θ_C at ρ₁ = 2.5, the maximum over all pole branches is higher than that of the considered complex pole branch. A comparison of the eight cases shows that the highest values of S_i,2 occur near merging and separation points of the pole branches, whereas the tuning θ_D exhibits the most uniform sensitivity distribution along the considered trajectories.

4.4.3. Overall Local Indicator for Simultaneous Parameter Variation

When the parameters ρ₁ and ρ₂ vary simultaneously, the local sensitivity matrix J_i, defined in (27), is formed for each pole s_i. For the considered example, ${\rho }_1^{+}-{\rho }_1^{-}=1$ and ${\rho }_2^{+}-{\rho }_2^{-}=1$. Therefore, the numerical values of the derivatives with respect to the normalized coordinates ${\widehat{\rho }}_1$ and ${\widehat{\rho }}_2$ coincide with those with respect to ρ₁ and ρ₂.

The multi-parameter local indicator S_i,max is determined by the largest singular value of J_i, according to (29), and characterizes the maximum local amplification of the joint parametric variations in the displacement of the i-th pole.

For each fixed tuning θ, the indicator S_i,max(ρ, θ) is calculated at every point of the two-dimensional parametric region Ω_ρ and is associated with the corresponding pole position s_i(ρ, θ) on the pole branch C_i(θ), defined in (24), according to the mapping relationship (38). Thus, Figure 10 presents the distribution of S_i,max over all pole branches and allows localization of the regions in which the simultaneous variation in ρ₁ and ρ₂ causes the greatest local change in the pole positions.

Unlike Figure 8 and Figure 9, where S_i,1 and S_i,2, determined according to (26), assess the influence of each uncertain parameter separately, S_i,max accounts for their joint local influence and the most unfavorable direction of the parametric variation. For an overall comparison of the different tunings over the entire region Ω_ρ, the indicator S_max(θ), defined in (30), is used.

The numerical results for the four tunings, obtained on a 301 × 301 parametric grid, are: S_max(θ_A) = 16.2017 at ρ_cr,A = (1.500, 0.967), S_max(θ_B) = 43.9509 at ρ_cr,B = (2.003, 1.000), S_max(θ_C) = 3.8541 at ρ_cr,C = (2.500, 0.950), and S_max(θ_D) = 1.5799 at ρ_cr,D = (2.500, 0.500). At each parametric point, the multi-parameter local indicator S_i,max is determined through the largest singular value of the matrix J_i, according to (27) and (29), while S_max(θ), defined by (30), represents its highest value over all poles and the entire considered parametric region Ω_ρ.

Figure 10 summarizes the joint effect of the two uncertain parameters ρ₁ and ρ₂, whose individual effects are presented in Figure 8 and Figure 9 through S_i,1 and S_i,2. Therefore, it does not represent a simple sum of the two sensitivities but follows the multi-parameter formulation in Section 3.4, in which the matrix J_i simultaneously accounts for variations with respect to ρ₁ and ρ₂, while S_i,max characterizes the most unfavorable local direction of their joint variation.

The comparison reveals substantial differences among the four robustly admissible tunings. The highest numerical value on the considered grid is obtained for θ_B, with S_max(θ_B) = 43.9509. For θ_B = (1, 1) and ρ = (2, 1), P(s) = (s + 1)³, i.e., an exact triple root exists. This is followed by θ_A, for which S_max(θ_A) = 16.2017. Significantly lower values are obtained for θ_C and θ_D, with the lowest value being S_max(θ_D) = 1.5799. This means that for θ_A, and especially for θ_B, small joint variations in ρ₁ and ρ₂ can locally cause considerably larger pole displacements, whereas for θ_D the pole positions are less sensitive to the joint parametric uncertainty.

From the viewpoint of the transient response, this suggests a greater possibility of changes in response speed, damping, and oscillatory behavior for tunings with high values of S_max(θ), whereas the low value for θ_D suggests less dispersion of the dynamic characteristics. Whether this local pole sensitivity is manifested in the time response is examined further through S_y(ρ; θ), S_y,RMS(θ), and S_y,max(θ), defined by (32)–(34), as well as through the families and envelopes of the transient responses.

4.4.4. Map of Modulus and Phase Sensitivities

The total pole sensitivity S_1,2, defined by (26), characterizes the magnitude of the local pole displacement when ρ₂ varies but does not indicate which of its components predominates. Therefore, for the tuning θ_B = (1.0, 1.0), for which a pronounced local maximum with respect to ρ₂ was identified, the modulus sensitivity S_M,1,2, defined by (13), and the phase sensitivity S_Φ,1,2, defined by (16), are mapped separately onto the same complex pole branch. This determines whether the sensitive segment is predominantly radial, i.e., associated mainly with a change in the modulus M_i, or phase-dominated, i.e., associated mainly with a change in the argument Φ_i. This separation provides a clearer interpretation of the local maximum of S_1,2 and facilitates relating it to the expected change in the transient response.

In the left panel, the color indicates the modulus sensitivity S_M,i,2, defined by (13), while in the right panel it indicates the phase sensitivity S_Φ,i,2, defined by (16). The two components are obtained from the same numerically evaluated derivative ∂s_i/∂ρ₂ and satisfy the decomposition in (19). The maximum error in the numerical verification of the decomposition is 1.421 × 10⁻¹⁴.

The maximum modulus sensitivity is $S_{M,i,2}^{\max }=1.051$ at ρ₂ = 1.193, where s_i = −1.667 + j0.084. The maximum phase sensitivity is $S_{\mathsf{\Phi },i,2}^{\max }=9.084$ at ρ₂ = 0.937, where s_i = −0.700 + j0.037. This value is practically identical to the maximum total pole sensitivity $S_{i,2}^{\max }=9.088$, obtained at the same parametric point. Therefore, the local maximum of S_i,2 is phase-dominated and coincides with the maximum identified on the total-sensitivity map in Figure 9. In contrast, the maximum of S_M,i,2 is obtained at a higher value of ρ₂ and on another segment of the same tracked pole branch.

Between the two complex segments, the pole passes through real positions. Therefore, the map is interrupted over the corresponding parametric interval. This separate mapping provides a more specific interpretation of the results in Figure 8 and Figure 9: Figure 8 shows the distribution of the total sensitivity over all pole branches, Figure 9 localizes the segments with the highest sensitivity with respect to ρ₂, and Figure 11 shows which component modulus or phase determines the local maximum on the selected complex branch.

From the viewpoint of the transient response, this means that for the tuning θ_B and near ρ₂≈0.937, the change in the time response is expected to be associated mainly with a change in the argument Φ_i and, accordingly, in the damping ratio ξ_i, related through (8), rather than with a radial displacement of the pole. Therefore, a more substantial change in oscillatory behavior and overshoot can be expected, while the effect on response speed is secondary. This interpretation is used as a local pole diagnostic and is subsequently verified through the transient-response sensitivity S_y(ρ; θ), S_y,RMS(θ), and S_y,max(θ), defined by (32)–(34), as well as through the families and envelopes of the transient responses.

4.5. Transient-Response Sensitivity

The indicator S_max(θ), defined by (30), characterizes the largest local pole sensitivity over the parametric uncertainty region Ω_ρ. It enables the detection of sensitive pole configurations and segments near multiple roots but does not directly assess the variation of the output y(t; ρ; θ) in the time domain.

Therefore, the robustly admissible tunings $\theta \in {\mathsf{\Theta }}_D^r$ are also compared using the transient-response sensitivity indicators S_y(ρ; θ), S_y,RMS(θ), and S_y,max(θ), introduced by (32), (33), and (34), respectively. For the considered system, the transient response is determined from the closed-loop output y(t; ρ; θ) under a unit-step input.

The partial derivatives of the output with respect to the uncertain parameters ρ₁ and ρ₂ are evaluated numerically on the same parametric grid in Ω_ρ used for the pole sensitivity analysis. The central difference in (35) is used for the interior points, whereas the corresponding one-sided differences are used for the boundary points. For all compared tunings, the same 51 × 51 parametric grid in Ω_ρ, with increments Δρ₁ = Δρ₂ = 0.02, and the same time grid t = 0:0.01:25 s, containing 2501 points, are used. Since ${\rho }_1^{+}-{\rho }_1^{-}=1$ and ${\rho }_2^{+}-{\rho }_2^{-}=1$, the numerical values of the derivatives with respect to the normalized coordinates ${\widehat{\rho }}_1$ and ${\widehat{\rho }}_2$ coincide with those with respect to ρ₁ and ρ₂.

The indicator S_y,RMS(θ), defined by (33), characterizes the average level of the local transient-response sensitivity over the uncertainty region Ω_ρ, whereas S_y,max(θ), defined by (34), gives the most unfavorable local value. These indicators do not replace pole sensitivity but complement it with a direct assessment of the influence of parameter variations on the time-domain characteristics of the system.

4.6. Comparison of the Robustly Admissible Tunings

The aim is to determine whether the tunings $\theta \in {\mathsf{\Theta }}_D^r$, whose root contours $𝒞$(θ) satisfy the robust admissibility condition $𝒞$(θ)⊂D in (25), are equally suitable with respect to the global geometry of the root contours, local pole sensitivity, and transient-response sensitivity. The global geometric indicators Δσ, Δω, d_max, and μ_D, introduced in the geometric analysis of the root contours in [29], are used for the comparison. They characterize, respectively, the pole dispersion along the real and imaginary axes, the maximum displacement from the nominal configuration, and the minimum geometric margin of the root contour $𝒞$(θ) from the boundary of the dynamic admissibility region D. These are supplemented by the maximum modulus sensitivity S_M,max(θ), the maximum phase sensitivity S_Φ,max(θ), the overall multi-parameter indicator S_max(θ), defined by (30), and the transient-response sensitivity indicators S_y,RMS(θ) and S_y,max(θ), defined by (33) and (34), respectively. For all tunings, the indicators are evaluated over the same parametric uncertainty region Ω_ρ using the same discretization, which allows their direct quantitative comparison.

These indicators provide complementary information: the global geometric indicators characterize the overall distribution of the root contours, the local sensitivity indicators identify fragile pole configurations, and the transient-response indicators show how parameter variations are manifested in the time domain.

The results for the seven robustly admissible tunings $\theta \in {\mathsf{\Theta }}_D^r$ are presented in Table 2. The values of S_M,max(θ), S_Φ,max(θ), and S_max(θ) are determined over all pole branches $𝒞$_i(θ) in (24) and over the entire region Ω_ρ, while S_y,RMS(θ) and S_y,max(θ) are used to verify whether the identified pole sensitivity is also manifested in the time domain.

The results show that the maximum local pole sensitivity and transient-response sensitivity do not yield the same ranking of the robustly admissible tunings. The indicator S_max(θ), defined by (30), identifies the tunings with high local pole sensitivity and the regions near nearly multiple roots, whereas S_y,RMS(θ) and S_y,max(θ), defined by (33) and (34), respectively, show the extent to which the parameter variations are directly manifested in the output variable y(t; ρ; θ).

The tuning (K_R, K_I) = (1.0, 2.0) has the lowest maximum local pole sensitivity, with S_max = 1.388. However, the values S_y,RMS = 0.594724 and S_y,max = 1.163005 are relatively high. Therefore, low value of S_max(θ) and smooth local pole motion are not sufficient conditions for low variability of the transient response over the entire region of parametric uncertainty Ω_ρ.

The most favorable combination of pole and time-domain sensitivity is obtained for the tuning (K_R, K_I) = (2.0, 1.0), for which S_max = 1.580, S_y,RMS = 0.428249, and S_y,max = 0.654165. This tuning combine low maximum local pole sensitivity with the lowest values of both transient-response sensitivity indicators and is therefore identified as the most balanced among the robustly admissible tunings considered.

The tuning (K_R, K_I) = (2.0, 2.0) also shows relatively low transient-response sensitivity, with S_y,RMS = 0.484100 and S_y,max = 0.722000, but its value of S_max = 3.854, with the modulus component S_M,max = 3.854 predominating over S_Φ,max = 1.723. The tunings (0.5, 0.5), (0.5, 1.0), (1.0, 0.5), and (1.0, 1.0) have considerably higher values of S_max(θ) and, accordingly, more pronounced locally sensitive pole configurations, although the indicators S_y,RMS(θ) and S_y,max(θ) remain moderate for some of them.

For an overall graphical comparison, the indicators are normalized to the interval [0, 1] according to (39). The normalization does not change the physical meaning of the individual indicators: for the geometric margin μ_D, defined in [29], a higher value is favorable, whereas for d_max, S_M,max, S_Φ,max, S_max(θ), S_y,RMS(θ), and S_y,max(θ), a lower value indicates less dispersion or lower sensitivity. Figure 12 compares the normalized values of the global geometric indicators, the maximum pole sensitivities, and the transient-response sensitivity indicators.

Figure 12 shows that the performance of the tunings cannot be assessed unambiguously using only one indicator. The tuning (K_R, K_I) = (1.0, 2.0) has the lowest value of S_max(θ), but not the lowest values of S_y,RMS(θ) and S_y,max(θ). The tuning (K_R, K_I) = (2.0, 1.0) shows the most balanced behavior because it combines a low value of S_max(θ) with the lowest values of S_y,RMS(θ) and S_y,max(θ). The tuning (K_R, K_I) = (2.0, 2.0) has a more favorable geometric margin but higher modulus and total pole sensitivity than (2.0, 1.0).

Therefore, the global geometric indicators μ_D and d_max, defined in [29], assess the location and dispersion of the root contours $𝒞$(θ); S_M,max, S_Φ,max, and S_max(θ), defined by (30), characterize the local sensitivity and the nature of the pole motion; and S_y,RMS(θ) and S_y,max(θ), defined by (33) and (34), assess the direct effect of parametric uncertainty on the time-domain response. Their combined use makes it possible to distinguish between tunings that are equally robustly admissible according to the condition $𝒞$(θ)⊂D, (25), but differ substantially in their local pole and time-domain sensitivities.

Based on this combined comparison, the tuning (K_R, K_I) = (2.0, 1.0) is selected for subsequent verification in the time domain. The tuning (K_R, K_I) = (1.0, 2.0), which has the lowest value of S_max(θ) but higher values of S_y,RMS(θ) and S_y,max(θ), is used for comparison. Thus, Section 4.7 uses the families and envelopes of the transient responses to verify whether minimizing only the maximum local pole sensitivity is sufficient to limit the dispersion of the time-domain response.

4.7. Verification in the Time Domain

The time-domain verification is performed for two robustly admissible tunings $\theta \in {\mathsf{\Theta }}_D^r$ satisfying the condition $𝒞$(θ)⊂D, (25), but characterized by different combinations of pole sensitivity and transient-response sensitivity.

The tuning θ_D = (K_R, K_I) = (2.0, 1.0) was selected in Section 4.6 as the most balanced because it combines a low value of the maximum local pole sensitivity S_max(θ), defined by (30), with the lowest values of S_y,RMS(θ) and S_y,max(θ), defined by (33) and (34), respectively. For comparison, the tuning θ_E = (K_R, K_I) = (1.0, 2.0) was selected because it has the lowest value of S_max(θ) among the tunings considered but higher values of S_y,RMS(θ) and S_y,max(θ).

The comparison between θ_D and θ_E makes it possible to verify whether minimizing S_max(θ) alone is sufficient to limit the dispersion of the transient responses or whether the local pole sensitivity and transient-response sensitivity must be considered jointly.

For each tuning, a family of transient responses y(t, ρ, θ) to a unit-step input is calculated using the same parametric grid ρ∈Ω_ρ and the same time grid t∈[0, T] used to determine S_y(ρ, θ), S_y,RMS(θ), and S_y,max(θ) according to (32)–(34). The lower and upper envelopes are determined from the resulting family and define the complete range of variation of the output y(t, ρ, θ) at each instant. The nominal transient response and the response at the parameter point ρ at which the local indicator S_y(ρ, θ), defined by (32), reaches S_y,max(θ), defined by (34), are also highlighted, as shown in Figure 13 and Figure 14.

Figure 13 and Figure 14 show different degrees of dispersion of the transient responses y(t, ρ, θ) over the region Ω_ρ.

For the tuning θ_D = (K_R, K_I) = (2.0, 1.0), the family of transient responses is more compact, and the distance between the upper envelope y_U(t, θ_D) and the lower envelope y_L(t, θ_D) is smaller. This is consistent with the lowest values among the tunings considered, S_y,RMS(θ_D) = 0.428249 and S_y,max(θ_D) = 0.654165, defined by (33) and (34), respectively. For the tuning θ_E = (K_R, K_I) = (1.0, 2.0), the maximum local pole sensitivity S_max(θ_E) = 1.388, defined by (30), is lower than S_max(θ_D) = 1.580, but the dispersion of the time-domain responses is greater. Therefore, minimizing S_max(θ) alone does not guarantee minimum dispersion of the transient response over the entire region Ω_ρ.

For a quantitative assessment of this dispersion, the instantaneous envelope width is introduced as w_env(t, θ) = y_U(t, θ) − y_L(t, θ), where y_U(t, θ) and y_L(t, θ) are the upper and lower envelopes of the family of transient responses, respectively. The maximum envelope width is defined as J_env(θ) = max_{t∈[0, T]}w_env(t, θ), while its average level over the time grid t_n, n = 1,…,N_t, is evaluated by $J_{\mathrm{env},\mathrm{RMS}}\left(\theta \right)=\sqrt{{\displaystyle \frac{1}{N_t}}{\sum }_{n=1}^{N_t}{w}_{\mathrm{env}}^2\left(t_n,\theta \right)}$. The resulting values of J_env(θ), the corresponding time t at which the maximum width is reached, and J_env,RMS(θ) are indicated directly in Figure 13 and Figure 14 for the tunings θ_D and θ_E.

For θ_D, J_env = 0.565166 at t = 2.340 s and J_env,RMS = 0.192698, whereas for θ_E, the corresponding values are J_env = 0.779993 at t = 2.730 s and J_env,RMS = 0.257335. The lower values for θ_D confirm the smaller dispersion of the time-domain response observed visually in Figure 13 and Figure 14.

Direct comparison in Figure 15 shows that, over practically the entire significant portion of the transient response, the envelope for θ_D is narrower than that for θ_E. The maximum width J_env(θ_D) = 0.565166 is lower than J_env(θ_E) = 0.779993, and the same relationship is observed for J_env,RMS. This result is consistent with the lower values of S_y,RMS(θ_D) and S_y,max(θ_D) and shows that the tuning θ_D provides lower variability of the output y(t, ρ, θ) under the admissible variations of ρ₁ and ρ₂.

The difference between the results for S_max(θ) and the envelopes is not a contradiction. The indicator S_max(θ), defined in (30), characterizes the most adverse local pole displacement, whereas S_y,RMS(θ), S_y,max(θ), and J_env(θ) assess the manifestation of parametric uncertainty in the entire time response. The latter is determined not only by the local sensitivity of the individual poles, but also by their absolute locations, the relative contributions of the individual modes, and the other characteristics of the closed-loop system. Therefore, tuning with a minimum value of S_max(θ) does not necessarily have the narrowest family of transient responses.

4.7.1. Main result of the time-domain verification.

The time-domain verification does not introduce a new criterion for robust admissibility, which remains defined by the condition $𝒞$(θ)⊂D in (25), but verifies the engineering significance of the comparison performed using the local pole sensitivity and the transient-response sensitivity indicators. The tuning θ_E = (1.0, 2.0) has the lowest value of S_max(θ), but does not provide the smallest spread of transient responses. The tuning θ_D = (2.0, 1.0) combines low maximum local pole sensitivity, the lowest values of S_y,RMS(θ) and S_y,max(θ), and the narrowest envelope of the time responses. Therefore, θ_D is confirmed as the most balanced among the investigated robustly admissible PI tunings, and the result shows consistency in both the complex plane and the time domain.

5. Discussion

The results show that robust admissibility, formulated by the condition $𝒞$(θ)⊂D, is necessary but not sufficient for a reasoned selection among multiple admissible controller tunings. The seven PI tunings considered are robustly admissible, but they differ in the nature of the pole motion, local sensitivity, and dispersion of the transient responses. This confirms the need to complement the global verification of the root-contour locations with a local analysis.

The first result is the augmentation of the root contours with modulus and phase characteristics of the tracked pole branches. A root contour shows the possible pole locations under parametric uncertainty but does not separate their motion into radial and tangential components. The modulus M_i and argument Φ_i provide additional information. For a complex pole, the variation of M_i is mainly related to a change in the natural frequency, whereas the variation of Φ_i is related to a change in the damping ratio. For a stable real pole, an increase in the modulus corresponds to a displacement further to the left in the complex plane and faster decay of the corresponding no oscillatory component.

The modulus and phase characteristics are particularly useful in regions where a complex-conjugate pair becomes two real poles or vice versa. In these regions, the modulus characteristics converge or separate, while the phase characteristic of the branch with a positive imaginary part reaches 180^∘ and terminates. This termination is not a numerical error but reflects the change in the nature of the roots. The modulus–phase representation therefore supports the engineering interpretation of the root-contour geometry.

The second main result is the quantitative decomposition of the local pole sensitivity S_i,j into a modulus component S_M,i,j and a phase component S_Φ,i,j. According to relation (19), these components are mutually perpendicular and describe the radial and tangential pole motion, respectively. This decomposition is important because the same value of the total sensitivity may have different engineering meanings. When the modulus sensitivity dominates, the change mainly affects the natural frequency or the decay rate. When the phase sensitivity dominates, the damping ratio and the oscillatory nature of the transient response are mainly affected.

The numerical results confirm that large local maxima of the total pole sensitivity are often phase-driven and occur near transitions between complex and real poles. For other tunings, the modulus component dominates, and the poles vary more smoothly in the radial direction. Therefore, the total sensitivity S_i,j determines the magnitude of the local pole displacement, but the joint consideration of S_M,i,j and S_Φ,i,j is necessary to determine the nature and expected dynamic effect of this displacement.

The third practical result is the mapping of the local sensitivity onto the corresponding pole locations. The maps in Figure 8 and Figure 9 localize the regions where a small variation in a particular uncertain parameter may cause a significant pole displacement. In this way, the sensitivity is not considered only as a function of the parameter but is directly related to the root-contour geometry. This is particularly important for higher-order systems because the maximum sensitivity may occur on a real pole branch rather than on the visually more pronounced complex contour.

The mapping shows that locally sensitive regions are usually located near points of convergence, separation, or rearrangement of the pole branches. In these regions, the denominator in the analytical relation (9) becomes small, and the sensitivity increases. A root contour may remain entirely within the admissible region D while still containing a region where a small parameter variation produces a sharp pole motion. Thus, tuning may be globally robustly admissible but locally fragile.

The mapping is supplemented by multi-parameter assessment based on the matrix J_i, the indicator S_i,max, and the overall indicator S_max(θ). The largest singular value accounts for the most adverse combined direction of variation of the uncertain parameters, which cannot be identified by considering them individually. Figure 10 shows that the joint influence of the parameters is not simply the sum of the individual sensitivities. Low sensitivity along the individual one-parameter trajectories does not guarantee low sensitivity when the parameters vary simultaneously.

The comparison in Figure 12 shows that pole sensitivity and transient-response sensitivity are not interchangeable. The indicator S_max(θ) characterizes the most adverse local pole displacement, whereas S_y(ρ, θ), S_y,RMS(θ), and S_y,max(θ) assess the direct manifestation of the parameter variations in the output. The transient response depends not only on the local sensitivity of the poles but also on their absolute locations and the relative contributions of the individual modes to the time response.

This explains why the tuning (K_R, K_I) = (1.0, 2.0), which has the lowest value of S_max(θ), does not provide the smallest dispersion of the transient responses. The tuning (K_R, K_I) = (2.0, 1.0) provides a more balanced combination of low local pole sensitivity, the lowest transient-response sensitivity, and a narrower envelope of the transient responses. The time-domain verification in Figure 13, Figure 14 and Figure 15 confirms that minimizing only the maximum local pole sensitivity does not guarantee minimum variation of the transient response over the entire uncertainty domain.

From a practical point of view, the proposed method is applied after the verification of robust stability and robust admissibility. It does not replace these checks but supports a reasoned selection among multiple tunings that satisfy $𝒞$(θ)⊂D while differing in their local pole sensitivity and time-domain behavior. The procedure distinguishes tunings with locally fragile regions, tunings with radially or tangentially dominated pole motion, tunings with low pole sensitivity but high time-response sensitivity, and tunings with balanced behavior according to both types of indicators.

In engineering practice, this makes it possible first to identify all robustly admissible tunings and then to compare them according to the nature of the pole motion, the most adverse combined influence of the uncertain parameters, and the dispersion of the transient responses. In addition to supporting the selection of a controller tuning, the sensitivity maps may identify plant parameters that require more accurate identification, tighter manufacturing tolerances, or stricter control of the operating conditions.

The main limitation associated with an increasing number of uncertain parameters is the growth in computational and visual complexity. For m uncertain parameters, the principle of the method remains unchanged, and the matrix J_i is extended by additional columns. For a hyperrectangular uncertainty domain, however, the number of one-parameter boundary trajectories increases as m 2^m−1. Their complete simultaneous representation may therefore become difficult to interpret. For a larger number of parameters, the visualization should be limited to the critical regions and the most influential parameters, while the final comparison should be performed numerically over a discretized uncertainty domain Ω_ρ. Adaptive grids and local refinement around the detected maxima may be used to reduce the computational effort.

The accuracy of the local indicators also depends on reliable tracking of the pole branches and on the selected parameter-grid step. Near repeated or nearly repeated roots, the numerical values may depend on the discretization. In such cases, a large value should be interpreted as an indicator of local fragility or proximity to a singular configuration rather than as an absolute limiting value. Moreover, S_i,max is a local indicator based on a linear approximation for small parameter variations. Verification through the families and envelopes of the transient responses therefore remains necessary.

The pragmatic value of the procedure lies in its clear sequence of application. First, global robust admissibility is verified. The sensitive regions are then localized, and the nature of the pole motion is determined. The most adverse combined influence of the uncertain parameters is assessed, and the selected tunings are finally verified in the time domain. The method transforms the availability of multiple robustly admissible tunings into a reasoned engineering selection procedure.

6. Conclusions

The proposed approach extends the analysis of robustly admissible tunings by jointly considering the modulus and phase characteristics of the poles, local pole sensitivity, and transient-response sensitivity. Thus, the assessment is not limited to the location of the root contours within the admissible region but also accounts for the nature, direction, and magnitude of the motion of the individual pole branches.

The results show that robust admissibility alone is insufficient for a reasoned selection among multiple tunings. Low local pole sensitivity does not necessarily guarantee small dispersion of the transient response. Therefore, pole and time-domain sensitivities should be assessed jointly. Mapping the modulus, phase, and total sensitivities allows critical regions to be localized and determines whether the sensitive pole motion is predominantly radial or tangential.

The practical value of the method lies in distinguishing robustly admissible but locally sensitive tunings and selecting a solution with more balanced behavior in both the complex plane and the time domain. The method provides an additional engineering stage for selecting among multiple robustly admissible controller tunings.