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Feasibility of Simultaneous Input-Output Constraints for Tracking in a Class of LTI Systems: Part I

Junhyeok Yoon, Heather Hussain, Anuradha M. Annaswamy

arXiv:2610.02151v1eess.SYcs.SY

TL;DR

This paper studies simultaneous input and output constraint satisfaction during tracking for LTI systems with multiple inputs, state feedback, and integral action. It uses a CBF-based governor to modify the command and derives feasibility conditions and a parameter-design procedure. The conditions can require relaxing input constraints, and the analysis identifies cases where feasibility cannot be guaranteed on the full operating set.

  • Problem

    The paper addresses how to track a desired command while satisfying input and output constraints and keeping closed-loop solutions bounded for the stated class of LTI systems.

    Source: §I · p. 1, §II · p. 2

  • Method

    A CBF-based governor modifies the command, with barrier constraints and design parameters chosen to establish simultaneous input-output feasibility and forward invariance.

    Source: §I · p. 1, §II · p. 2, B. A Sufficient Condition for… · p. 4

  • Results

    Necessary and sufficient SIOCF conditions are established on Ω under the finiteness condition and on SE independently of it, while the design procedure can make the conditions hold by relaxing input constraints.

    Source: C. Construction of a Compact Set S · p. 5, §VI · p. 8

  • Takeaways & Limitations

    Some MIMO plants with state feedback and integral action may not permit input-output constraint satisfaction unless the design space is expanded to relax input constraints.

    Source: C. Construction of a Compact Set S · p. 5, §VI · p. 8

    For observable plants of order np > 2m, the finiteness condition is violated, so compactness of Ω cannot be guaranteed; a compact forward-invariant subset can be constructed when d > 0.

    Source: B. A Sufficient Condition for… · p. 4

Abstract

from arXiv · show

This paper addresses the problem of simultaneous satisfaction of input and output constraints for LTI systems with multiple inputs with state feedback and integral action using a Control Barrier Function based governor. Necessary and sufficient conditions for the CBF-based governor to have a feasible solution and for the closed-loop solutions to be bounded and forward-invariant are derived. A systematic design procedure for choosing the free parameters of the CBF-governor is also provided. These free parameters are associated with high-order CBFs, bounds on feasible command signals, and control input magnitude. A companion paper provides several numerical examples to illustrate the results of this paper, especially the feasibility (or infeasibility) of the CBF governor when the conditions are satisfied (or not satisfied).

I. INTRODUCTION

Prior approaches to simultaneous input-output constraints leave gaps in feasibility guarantees, computational burden, or system scope. This paper establishes necessary and sufficient conditions for a CBF-based governor and gives a procedure for designing its free parameters.

I. INTRODUCTION

  • Earlier methods provide only sufficient conditions, can be computationally demanding, or leave feasibility unestablished for the considered system classes.The related work also notes that one approach recovers the maximal safe set but suffers from the curse of dimensionality, while another does not establish forward invariance.Source: §I · p. 1
  • The paper derives necessary and sufficient conditions for simultaneous input-output constraint satisfaction during tracking with a CBF-based governor for LTI systems with multiple inputs, state feedback, and integral action.Under these conditions, closed-loop solutions remain bounded in a forward-invariant set while the governor modifies the command.Source: §I · p. 1
  • A systematic parameter-selection procedure can ensure the conditions hold, while making the relaxation of input constraints minimal.The free parameters are associated with the governor design, and the paper describes an efficient design procedure.Source: §I · p. 1
  • A companion paper provides examples and counterexamples illustrating when the conditions hold or fail and how governor design can address counterexamples.The examples are presented as support for the theoretical conditions.Source: §I · p. 1

II. PROBLEM SETTING

The problem is to track a desired output while keeping system states bounded and satisfying input and relevant-output constraints. A PI controller and CBF-based governor are proposed for a class of LTI plants with integral action.

II. PROBLEM SETTING

  • The plant has state xp, control input u, tracked output y, and integrator state eI; the governor chooses modified command yg to meet the input and output constraints.A PI controller supports boundedness, and a CBF-based governor addresses the tracking and constraint objectives.Source: §II · p. 2
  • The objective is bounded states, close tracking of ycmd, and satisfaction at all times of constraints on the control input u and relevant output z.The constraints are componentwise box bounds on u and z.Source: §II · p. 2
  • Problem 1 asks for a modified command that keeps every solution bounded and satisfies the constraints for every bounded piecewise-continuous ycmd.The requirement applies to the closed-loop system from its initial time onward.Source: §II · p. 2
  • The analysis assumes nonsingular constrained-output high-frequency gain, stable invariant zeros, and a Hurwitz closed-loop matrix; the first two assumptions do not restrict the tracked output.The paper states the Hurwitz condition can be achieved through gain choice whenever the plant is stabilizable.Source: §II · p. 2

III. I-O CBF GOVERNOR

The I-O CBF Governor selects a modified command subject to barrier inequalities for state and input constraints. When its quadratic program is feasible, the constraints hold and the selected command is unique and piecewise affine.

III. I-O CBF GOVERNOR

  • Two high-order CBFs address the state, while a first-order CBF addresses the input, reflecting the integrators in the plant and controller.The barriers are defined using positive gains α1, α2, and αu.Source: §III · p. 3
  • The governor imposes barrier inequalities and an additional constraint on yg to provide a feasible solution.The inequalities are expressed through the pointwise feasible set F(x̄).Source: §III · p. 3
  • If the governor’s quadratic program is feasible at every time, all input-output constraints are satisfied and yg is unique, continuous, piecewise affine, and Lipschitz in x̄ and ycmd.The feasible set is a compact convex polytope.Source: §III · p. 3

A. Restatement of Problem 1

The paper formalizes simultaneous input-output constraint feasibility on a closed subset of the operating set. The stated problem is to find such a subset for which the governor maintains feasibility, invariance, and constraint satisfaction.

A. Restatement of Problem 1

  • The operating set Ω is introduced for establishing forward invariance: an initial state in Ω remains in Ω for all future time.The design parameter α1 belongs to the positive reals.Source: A. Restatement of Problem 1 · p. 3
  • SIOCF on a nonempty closed set S requires a unique governor command, invariance of S, satisfaction of input and output constraints, and a bound on yg.These properties are required for every bounded piecewise-continuous command and every initial state in S.Source: A. Restatement of Problem 1 · p. 3
  • Problem 2 asks for a nonempty S contained in Ω on which the governor satisfies SIOCF for a given α1.Source: A. Restatement of Problem 1 · p. 3

A. Simplification of the Feasibility Condition

The section simplifies the feasibility requirement by representing a polyhedral cone through its extreme rays. The resulting condition is presented as a simpler necessary-and-sufficient test for feasibility.

A. Simplification of the Feasibility Condition

B. A Sufficient Condition for Feasibility

This section relates feasibility of the governor to a finiteness condition, ker Q = O, and gives a simpler test for that condition. It also identifies when compactness of Ω cannot be guaranteed and describes a forward-invariant subset for d > 0.

B. A Sufficient Condition for Feasibility

C. Construction of a Compact Set S

The section constructs a compact forward-invariant subset of the operating set and characterizes when the governor satisfies simultaneous input-output constraint feasibility there. Feasibility on the full operating set additionally requires the finiteness condition.

C. Construction of a Compact Set S

  • An RPI set E for the internal state η yields a compact set S_E that remains forward-invariant under the governor.The construction uses coordinates ξ = [zᵀ, żᵀ, ηᵀ]ᵀ and keeps η in E.Source: C. Construction of a Compact Set S · pp. 4–5
  • If d = 0, for example when n_p = m, Ω is compact; otherwise, the construction provides a compact subset of Ω.The paper notes that when d > 0, a compact subset S ⊆ Ω can be constructed and made forward-invariant.Source: C. Construction of a Compact Set S · p. 4
  • On Ω, SIOCF holds if and only if the SIOCF condition holds when the finiteness condition is satisfied.The finiteness condition requires ker Q to equal the unobservable subspace of (Ā, C).Source: C. Construction of a Compact Set S · p. 5
  • On the compact set S_E, the governor satisfies SIOCF if and only if the SIOCF condition holds there, regardless of the finiteness condition.The set is built from an RPI set E for the internal dynamics; the theorem also characterizes feasibility on Ω under the finiteness condition.Source: C. Construction of a Compact Set S · pp. 4–5
  • For some plants, input-output constraint conflict can make ν_k negative and cause finite-time output-constraint violation, motivating expansion of the design space beyond α1.The cited example describes a conflict that can occur under any control policy and any α1.Source: C. Construction of a Compact Set S · p. 5

V. SIOCF DESIGN PROCEDURE

The design procedure establishes feasibility by selecting parameters for the I-O CBF Governor and, when needed, relaxing the input constraint. It then seeks choices that produce a large operating set while keeping the relaxation near unity.

V. SIOCF DESIGN PROCEDURE

  • The efficient design chooses the first parameter class to make S_E as large as possible while keeping ϱ near unity.The first class is [α1, α2, αu]ᵀ; the second is [αg, ϱ]ᵀ.Source: §V · p. 6
  • For any positive α1, α2, and αu and an RPI set E, suitable αg and ϱ make the SIOCF condition hold on S_E(ϱ).The theorem supplies thresholds α⋆_g and ϱ⋆ for these choices.Source: A. Finding an α2 · pp. 6–7
  • The design parameters act on extreme rays: ϱ raises ν_k for input-weighting rays, while αg raises it for rays weighting the command bound.The remaining rays correspond to output pairs and have positive constant ν_k values.Source: A. Finding an α2 · p. 7
  • Input relaxation is needed because some extreme rays couple input and output constraints, leaving their ν_k values unaffected by α2, αu, and αg.The paper states that choosing ϱ ≥ ϱ⋆ renders these ν_k nonnegative.Source: A. Finding an α2 · p. 7

B. Efficient design of an α2

The procedure first tests whether α1 can support the SIOCF condition, then selects thresholds for the remaining CBF parameters or relaxes the input constraint. Its parameter choices are governed by feasibility tests on the operating set.

B. Efficient design of an α2

  • If Steps 1 and 2 cannot satisfy the SIOCF condition, Step 3 increases ϱ to relax the input constraint and enlarge S_E(ϱ).The design holds ϱ = ϱ1 := 1 in the first two steps.Source: B. Efficient design of an α2 · pp. 7–8
  • When ᾱΘ(α1, ϱ1) > 0, thresholds for α2, αu, and αg can be chosen so the SIOCF condition holds.Step 2 sets αu = cα2 and selects αg at least as large as its parameter-dependent threshold.Source: B. Efficient design of an α2 · pp. 7–8
  • No α1 exists when ˆΘ < 0, and any such choice cannot exceed ¯α1; for the chosen α1, thresholds for (α2, αu, αg) can be determined so the SIOCF condition holds.Source: B. Efficient design of an α2 · p. 8
  • For a polyhedral approximation of E, checking the SIOCF condition has O(N) computational burden.The procedure concludes feasibility on S_E(ϱ) through Proposition 6(d) or Theorem 2, depending on which step is reached.Source: B. Efficient design of an α2 · p. 8

VI. CONCLUSIONS

The paper derives necessary and sufficient conditions for simultaneous input-output constraint satisfaction in a class of closed-loop LTI systems with integral action using a CBF-based command governor, and proposes systematic governor-parameter design for a close-to-minimal relaxation of input constraints. The supplied proofs establish compactness, convexity, and robust positive invariance of E, and show finite-time violation of the scalar system’s output box under the stated conditions.

VI. CONCLUSIONS

  • For some MIMO plants, input-output constraint satisfaction may be impossible; relaxing input constraints can make the SIOCF condition satisfiable, with a systematic design for close-to-minimal relaxation.The proposed CBF governor modifies the command and guarantees simultaneous satisfaction when the necessary and sufficient conditions hold.Source: §VI · p. 8

A. Proof of Proposition 1

B. Proof of the bound on N in Proposition 1

  • The proof establishes at least 9m extreme rays in Λ for every plant and gain.It derives this lower bound from 3m antipodal-pair rays and six additional families, each with m rays.Source: B. Proof of the bound on N in… · p. 9

C. Proof of Proposition 2

D. Proof of Proposition 3

E. Proof of Proposition 4

F. Proof of Lemma 1

  • In the coordinates (z, η̃), the invariant zeros are the eigenvalues of Aη, which is Hurwitz.Source: F. Proof of Lemma 1 · p. 11
  • The proof derives (18) by differentiating η = ˜η + Bdz and establishes Tξ nonsingular by showing that Tξv = 0 implies v = 0.The kernel argument first yields vp = 0 and then ve = 0.Source: F. Proof of Lemma 1 · p. 12

G. Proof that E of Section IV-C is RPI

  • E is compact and convex, contains 0, is robust positively invariant, and includes Emin.The proof establishes compactness and 0 ∈ E, verifies invariance under inputs in W, and concludes Emin ⊆ E.Source: G. Proof that E of Section IV-C is RPI · p. 12

H. Proof of Proposition 5

  • Under a > 0 and z(t0) − umax/a > 0, the lower-bound argument shows z(t) > zmax for t > t1, violating the output box in finite time.The lower bound equals zmax at t1 and exceeds it thereafter.Source: H. Proof of Proposition 5 · p. 12
  • For the scalar integrator with no internal dynamics, d = 0 and SE = Ω; at the constructed x*, ψ1+(x*) = (azmax − umax)/(a + α1) > 0.The construction uses Cz x* = z* and −Kfull x* = −umax.Source: H. Proof of Proposition 5 · p. 12

I. Proof of Proposition 6

The proof characterizes parameter thresholds for meeting a target lower bound on the CBF quantities and shows their best attainable bound is Θ̄.

I. Proof of Proposition 6

  • For every ε < Θ̄, the target bound holds exactly when α2 ≥ α̲2 and αg ≥ αg(α2).The required αg is nonincreasing in α2.Source: §I · p. 13
  • The strict condition ε < Θ̄ matters because at or above Θ̄, s_k may vanish on D_k and the required α2 may be infinite.Source: §I · p. 13
  • The supremum of the minimum CBF quantity over the design parameters equals Θ̄.The proof bounds the supremum above by Θ̄ and approaches that bound for every ε < Θ̄.Source: §I · p. 13
  • Increasing α1 cannot raise Θ̄(α1, ϱ) above the bound obtained by restricting the comparison to H_k.Source: §I · p. 13
  • Solutions of the construction equations exist generically when |J+_k| ≤ np − m; for each admissible k, the proof constructs q and A with q2 < 0.Source: §I · pp. 13–14
  • When Θ̄ > 0, ε = 0 meets the target bound, which is the SIOCF condition on SE(ϱ).Source: §I · p. 14

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