Exact PID and PI Gain Regions for Uncertain Non-Affine MIMO Systems

Cheng Zhao

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Exact PID and PI Gain Regions for Uncertain Non-Affine MIMO Systems

Cheng Zhao

arXiv:2609.11130v1Sep 10, 2026eess.SYcs.SYmath.OC

TL;DR

The paper addresses the unresolved problem of identifying all PID and PI gains that uniformly exponentially regulate nonlinear uncertainty classes, rather than only gains certified by a selected construction. It develops exact regions for second-order PID and first-order PI systems; the regions strictly enlarge earlier sufficient regions, while a compact linear example shows common-quadratic exactness is not general.

Problem
The paper seeks all PID or PI gains that guarantee uniform exponential regulation over an entire uncertainty class, beyond sufficient conditions tied to a particular certificate.
Source: §2.1 · p. 5, §2.2 · p. 7
Method
For PID, it matches an inner common-quadratic region to an outer linear-subclass test; for PI, it combines a lossless boundary-linear family with a common quadratic certificate.
Source: §1 · p. 3, §6 · p. 32
Results
The exact PID and PI regions strictly enlarge the corresponding sufficient regions in, and the PID result captures high-gain scalings excluded by the previous condition.
Source: §1 · p. 3, §6 · p. 32
Takeaways & Limitations
The results provide complete admissible gain domains for regulation and performance-oriented tuning within the prescribed uncertainty bounds.
Source: §2.1 · p. 5, §2.6 · p. 11
Common-quadratic losslessness is not general: a compact one-parameter linear family has a system-level PID region strictly larger than its common-quadratic counterpart.
Source: §4 · p. 14, §6 · p. 32

Abstract

from arXiv · show

Characterizing all PID gains that guarantee uniform exponential regulation of uncertain nonlinear systems remains a fundamental problem. Most existing results provide only sufficient conditions derived from a particular Lyapunov construction. This paper gives exact gain regions, defined directly by uniform regulation over the entire uncertainty class, for two classes of nonaffine MIMO systems with $m\ge2$ controlled coordinates and input channels. For systems of second order under PID control, we independently characterize a common quadratic inner region and an outer region based on a linear subclass. The two regions coincide and therefore characterize the PID gain region exactly at the system level. For systems of first order under PI control, a lossless reduction to a linear family on the uncertainty boundary, combined with a common quadratic certificate, yields the exact PI region. Both regions strictly enlarge existing sufficient regions, and the PID result also provides a sequential tuning rule. A compact linear family with one uncertain parameter shows that common quadratic certification is not generally lossless. These results establish endpoint coincidence and boundary family equivalence as two routes to exact gain regions and motivate the broader question of which nonlinear uncertainty classes admit such characterizations.

1 Introduction

Existing nonlinear PID/PI results mostly provide sufficient conditions tied to selected certificates, leaving the all-gains regulation region unresolved. The paper closes that gap through two exactness routes: coincident inner and outer regions for PID, and a lossless boundary-linear test for PI.

1 Introduction

A complete stabilizing gain region identifies robustness limits and provides a stability-preserving domain for performance-oriented tuning.Existing complete PID gain-region theories primarily address linear plants; characteristic-equation methods do not directly extend to general nonlinear uncertainty classes.Source: §1 · pp. 1–2
The unresolved problem is to characterize every PID or PI gain that guarantees uniform exponential regulation over an entire nonlinear uncertainty class without prespecifying a stability certificate.Certificate-based conditions can be useful but may certify only gains captured by the chosen construction.Source: §1 · p. 2
The paper uses a common-quadratic certificate for an inner region and a selected linear test family for necessary outer conditions, then establishes exactness by closing the gap.The PID route is endpoint coincidence; the PI route is boundary-linear equivalence.Source: §1 · p. 2

Exact PID region through endpoint coincidence

For second-order PID systems with m ≥2, independently derived common-quadratic inner and linear-subclass outer regions coincide, giving the exact system-level gain region.For m = 1, both endpoints remain exact but are generally distinct inner and outer bounds.Source: §1 · pp. 2–3

Exact PI region through boundary-linear equivalence

For first-order PI systems with m ≥2, regulating a countable boundary-linear family is equivalent to regulating the full nonlinear uncertainty class.Combining this lossless boundary test with a common quadratic Lyapunov function yields the exact PI region.Source: §1 · p. 3

Strict improvements and structural limits

Both exact PID and PI regions strictly enlarge the corresponding sufficient regions in, while PID also captures previously excluded high-gain scalings and gives a sequential tuning rule.A compact one-parameter linear family shows that common-quadratic losslessness is specific to the considered uncertainty classes.Source: §1 · p. 3

2 PID control of second-order non-affine MIMO systems

This section characterizes robust PID gain regions for second-order non-affine MIMO systems, with coincident inner and outer regions determining the exact region when m ≥ 2. It also discusses the scalar case, where endpoint regions bracket but do not determine the system-level region, and presents an extension beyond the existing sufficient region and a sequential tuning rule.

2.1 Problem formulation and the system-level PID gain region

The system-level region comprises gains ensuring forward-complete, uniformly exponentially regulating closed loops for every admissible nonlinearity, reference, and initial condition.The uncertainty class includes coupled second-order systems with state-dependent nonlinearities and static input nonlinearities.Source: §2.1 · p. 4

2.2 Common-quadratic inner region

The common-quadratic region is an inner bound; for m ≥2 it has an explicit gain-only characterization using two auxiliary scalars, while the scalar case has a lossless four-vertex LMI test.Common-quadratic admissibility guarantees uniform exponential regulation, but does not alone establish that this region exhausts the system-level region.Source: §2.2 · pp. 6–7

2.3 Linear-subclass outer region

The linear-subclass region is an outer bound because robust regulation must stabilize every admissible linear member; for m = 1, its exact test is k̄1 > 0 and k̄1k̄2 > k̄0.Here k̄0 = bki, k̄1 = bkp − L1, and k̄2 = bkd − L2.Source: §2.3 · pp. 7–8

2.4 Endpoint coincidence and the exact PID gain region

For every m ≥2, the common-quadratic region, linear-subclass region, and system-level robust PID region coincide, making common-quadratic certification and the linear-subclass test lossless.This gives a complete gain-only characterization in terms of the PID gains and uncertainty bounds.Source: §2.4 · pp. 8–9

2.5 The endpoint gap in the scalar case

For m = 1, the endpoint regions remain exactly characterized but do not coincide, unlike the multivariable case; common-quadratic admissibility necessarily requires kd(bkd − L2) > kp.The section identifies this condition as one violated by a gain in the linear-subclass region.Source: §2.5 · p. 9
In the scalar case, the exact endpoints bracket but do not determine the system-level PID region.The paper establishes a proper inclusion between the relevant regions, rather than endpoint coincidence.Source: §2.5 · p. 9

2.6 Strict improvement and sequential PID tuning

For L1=L2=5, b=1, and ki=0.5, the exact PID region extends well beyond Zhao and Guo's sufficient region, admitting broader proportional and derivative gains.Figure 1 shows this comparison on the (kp, kd) cross-section with ki fixed at 0.5.Source: Figure 1 · p. 11, §2.6 · p. 11
The threshold c<b is sharp because it coincides with the Routh–Hurwitz condition for the admissible nominal system with D1=D2=0 and Θ=bIm.Source: §2.6 · p. 11
Choosing μ=ρ=1/2 trades some flexibility in the two-dimensional search for a transparent sequential PID tuning rule.The corollary gives this rule for every m≥2.Source: §2.6 · pp. 11–12

3 PI control of first-order non-affine MIMO systems

For first-order non-affine MIMO systems with m ≥2, the paper characterizes the exact PI gain region. A countable boundary-linear test family and a common quadratic certificate provide complementary lossless routes to that result.

3.1 Problem formulation and the system-level PI gain region

The system-level PI region requires uniform exponential regulation over every system in the uncertainty class, while the common-quadratic region requires a shared quadratic certificate.Source: §3.1 · pp. 12–13

3.2 Boundary-linear equivalence and the exact PI gain region

For m ≥2, uniform exponential regulation of the countable boundary-linear family is equivalent to regulation of the full nonlinear class, and the common-quadratic certificate is lossless for the PI region.Source: §3.2 · pp. 13–14
For m ≥2, the exact PI region consists of gains satisfying kp > 0, ki > 0, and ki + Lkp < bk^2.Source: §3.2 · pp. 13–14
The exact condition removes the additional L^2/(4b) term from the sufficient condition in; for m = 1, the available conditions only bracket the uniform-exponential region.Source: §3.2 · p. 14

4 Scope and Limits of Common-Quadratic Losslessness

The paper shows that common-quadratic certification is lossless for the two nonlinear uncertainty classes considered, but not generally for robust stability. A compact one-parameter linear family has a strictly larger system-level PID region than its common-quadratic region.

Common-quadratic losslessness holds for the two nonlinear classes considered when m ≥2, but is not automatic for compact parametric families.Source: §4 · pp. 14, 16
At L = 5 and b = 1, the exact PI condition admits gains beyond the region from [28], with a vertical boundary difference of 6.25.Source: Figure 2 · p. 15
For every L > 0, the common-quadratic PID gain region is a strict subset of the system-level region for the compact one-parameter linear family.Source: §4 · p. 15
A gain triple uniformly stabilizes the family across a ∈ [0, L] yet admits no common quadratic Lyapunov matrix for the endpoint pair.Source: §4 · pp. 15–16

5 Proofs of the main results

This section proves the PID characterizations, establishing the linear-subclass stability test and showing that its boundary yields the common-quadratic region. It also proves the exact PI gain condition through sufficiency and necessity arguments.

5.1 Common-quadratic PID characterization

For m≥2, rank-two Gram realizability and minimax elimination yield the exact multivariable common-quadratic PID conditions.The proof also shows that the certificate can be restricted losslessly to P0 ⊗ I_m; feasibility forces positive gains and permits normalization P0e3 = κ.Source: §5.1 · pp. 17, 18, 22
For m=1, the four-vertex condition is equivalent to common-quadratic admissibility, and it implies gain positivity rather than assuming it.The proof extends the vertex condition to all admissible uncertainty values by convexity.Source: §5.1 · pp. 18–19

5.2 Linear-subclass PID characterization

For the linear subclass, the boundary choice d1 = L1, d2 = L2, θ = b yields necessary Routh–Hurwitz inequalities, while conditions (15) ensure stability throughout the admissible set.The proof checks the characteristic polynomial using the Routh–Hurwitz criterion.Source: §5.2 · p. 23
For the linear subclass, PID stability is equivalent to positive gains, bkpkd > ki, and the frequency inequality (16).Necessity follows from Routh–Hurwitz and exclusion of imaginary-axis roots; sufficiency follows by continuing from a Hurwitz nominal plant without an imaginary-axis crossing.Source: §5.2 · pp. 24–25

5.3 PID endpoint coincidence

The proof establishes F(t) > 0 for every t > 0 iff kd > D*, then uses the region inclusions to prove PID endpoint coincidence.At the boundary kd = D*, feasibility is attained; increasing kd makes the integral-gain inequality strictly feasible for the selected multipliers.Source: §5.3 · pp. 28–29

6 Conclusion

The paper establishes exact system-level gain regions for second-order PID and first-order PI control of two classes of non-affine MIMO systems. It also identifies a boundary to these results: common-quadratic losslessness does not hold in general.

6 Conclusion

For second-order PID control, the common-quadratic inner region and linear-subclass outer region coincide, exactly characterizing the system-level gain region.The PID region strictly enlarges the corresponding existing sufficient region and yields a sequential tuning rule.Source: §6 · p. 32
For first-order PI control, regulation of a countable boundary-linear family is equivalent to regulation of the full nonlinear class, with an exact condition admitting a common quadratic certificate.The PI condition relaxes the corresponding existing bound by L2/(4b).Source: §6 · p. 32
A compact linear counterexample shows that common-quadratic certification is not generally lossless, leaving structural conditions for exact inner–outer coincidence open.The paper identifies finding verifiable conditions that guarantee coincidence as a direction for future research.Source: §6 · p. 32