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The Internal Model Principle for Systems with Unbounded Control and Observation

Lassi Paunonen, Seppo Pohjolainen

arXiv:1409.1773v1math.OC

TL;DR

Robust output regulation for distributed-parameter systems must handle plants with unbounded input and output operators, a setting arising in applications such as PDE boundary control and point measurements. The paper extends internal-model results to this setting, including p-copy and G-condition characterizations. It gives a heat-equation tracking example, while leaving closed-loop stabilization design open.

  • Problem

    The paper addresses extending robust output regulation and internal-model theory to plants with unbounded control and observation operators, which arise in settings such as PDE boundary control and pointwise measurements.

    Source: §1 · p. 2

  • Method

    The paper extends the p-copy internal model principle and G-condition characterization to this system class, with a new direct proof and internal-model characterization.

    Source: §1 · p. 3, §10 · p. 33

  • Results

    For strongly stabilizing controllers under the stated hypotheses, robust regulation is equivalent to incorporating a p-copy internal model when the output space is finite-dimensional; the paper also extends the G-condition results.

    Source: §6 · p. 17, §10 · p. 33

  • Takeaways & Limitations

    The heat-equation example illustrates robust tracking and rejection, with the periodic-signal controller solving the regulation problem provided its remaining parameters yield a strongly stable closed loop.

    Source: §1 · p. 3

    The paper does not provide techniques for stabilizing the closed loop, leaving the choice of the controller's remaining parameters open.

    Source: §9.2 · p. 31

Abstract

from arXiv · show

In this paper the theory of robust output regulation of distributed parameter systems with infinite-dimensional exosystems is extended for plants with unbounded control and observation. As the main result, we present the internal model principle for linear infinite-dimensional systems with unbounded input and output operators. We do this for two different definitions of an internal model found in the literature, namely, the p-copy internal model and the $\mathcal{G}$-conditions. We also introduce a new way of defining an internal model for infinite-dimensional systems. The theoretic results are illustrated with an example where we consider robust output tracking for a one-dimensional heat equation with boundary control and pointwise measurements.

1 Introduction

The paper extends robust output regulation to distributed-parameter plants with unbounded control and observation, addressing settings such as PDEs with boundary control or point measurements. It develops internal-model results for this setting, including a new proof and characterization, and illustrates the theory with heat-equation tracking.

1 Introduction

  • The reference and disturbance signals are generated by an exosystem, which can represent trigonometric or polynomial signals and, with an infinite-dimensional state space, nonsmooth periodic signals.Tracking and rejection of nonsmooth periodic and almost-periodic signals are motivated by applications including disk drives, robot arms, and power electronics.Source: §1 · pp. 1–2
  • The theory extends robust output regulation to plants with possibly unbounded input and output operators, under a well-defined closed-loop semigroup assumption.The paper notes that boundary control and pointwise measurements are practical settings where unbounded operators arise.Source: §1 · p. 2
  • The paper extends results from to unbounded input and output operators, infinite-dimensional exosystems, strongly stabilizable closed loops, and state disturbances.The earlier setting used finite-dimensional exosystems and exponentially stabilizable closed loops; its smaller uncertainty class could sometimes avoid a full internal model.Source: §1 · p. 3
  • The paper extends the p-copy internal model principle with a more direct proof that also yields a new characterization of controllers incorporating an internal model.The earlier proof contained parts that could not be extended to the considered system class.Source: §1 · p. 3
  • Robustness can also be characterized by the G-conditions, an alternative internal-model definition that applies when the plant output space is infinite-dimensional.The p-copy internal model and G-conditions have different strengths and weaknesses.Source: §1 · p. 3
  • For the heat-equation example, the paper considers constant-signal tracking and rejection with a 2-dimensional internal model, and nonsmooth periodic tracking using an infinite-dimensional diagonal exosystem.The latter controller solves the robust regulation problem provided its remaining parameters can be chosen to make the closed loop strongly stable.Source: §1 · p. 3

2 Mathematical Preliminaries

The preliminaries set the system and closed-loop assumptions for unbounded plant and controller operators, specify the allowed perturbations, and define operators used in the later internal-model results.

2 Mathematical Preliminaries

  • The plant and controller may have unbounded input and output operators, while the closed-loop operator on the composite state space must generate a strongly continuous semigroup.The maximal domain is chosen so the closed-loop operator maps into the composite state space; the error output operator is unbounded and relatively bounded with respect to the closed-loop operator.Source: §2 · pp. 4, 6, 7

2.1 The Class of Perturbations

  • The perturbation class requires the perturbed plant and closed loop to satisfy semigroup and resolvent conditions, with strong stability and a solvable Sylvester equation.For exponentially stable nominal closed loops, sufficiently small bounded perturbations satisfy the stated conditions; finite-dimensional exosystems admit the Sylvester solution when the relevant spectra are disjoint.Source: §2.1 · p. 7

2.2 Special Operators

  • The special operators encode Jordan-block structure at each exosystem frequency and use the perturbed plant's resolvent and transfer function there.The kernel of J_G1(iω_k) characterizes Jordan chains of G1 associated with iω_k.Source: §2.2 · pp. 7–8

3 Control Objectives

The control objective is asymptotic tracking and disturbance rejection with a strongly stable closed loop. The robust version additionally requires regulation to persist under the specified plant and exosystem perturbations.

3 Control Objectives

  • The controller must make the closed-loop semigroup strongly stable and ensure the regulation error tends to zero for initial exosystem states in W_α+1.The required error decay is posed for signals associated with those exosystem initial states, whose smoothness relates to the periodic signals tracked or rejected.Source: §3 · p. 8
  • Under perturbations in the specified class, the robust objective requires the perturbed closed loop to be strongly stable and its regulation error to tend to zero for the same exosystem-state class.The perturbations must also meet the additional assumptions stated for that class.Source: §3 · p. 9

4 Characterizing the Solvability of the Output Regulation Problem

The section characterizes output regulation through the regulator equations and establishes properties of the Sylvester equation used in that characterization. Under the stated stability and solvability assumptions, regulation is equivalent to satisfying both regulator equations.

4 Characterizing the Solvability of the Output Regulation Problem

  • Under the theorem’s assumptions, the controller solves output regulation on Wα if and only if the regulator equations have a solution Σ.The equations require ΣS = AeΣ + Be and CeΣ + De = 0.Source: §4 · p. 9
  • If 1 ∈ ρ(Ae), the Sylvester solution Σ makes CeΣ a bounded operator from Wα+1 to Y.This regularity supports the error-continuity argument.Source: §4 · p. 9
  • For xe0 ∈ D(Ae) and v0 ∈ Wα+1, the regulation error is continuous and converges to (CeΣ + De)TS(t)v0.Thus, when CeΣ + De = 0, the error tends to zero.Source: §4 · pp. 10–11

4.1 Properties of the Sylvester Equation ΣS = AeΣ + Be

  • For a strongly stable closed loop, the Sylvester equation has at most one solution; under Theorem 8(b)’s conditions, it has a unique solution in L(Wα, Xe).When iωk ∈ ρ(Ae) for every k and a solution exists, it is given by (7).Source: §4.1 · pp. 11–12

5 Characterizing Robustness with Respect to Given Perturbations

This section gives a test for robustness against a specified set of perturbations. It characterizes robustness through frequency-indexed equations and the associated perturbed regulator equations.

5 Characterizing Robustness with Respect to Given Perturbations

  • Theorem 9 extends a robustness test from, which treated bounded input and output operators, a finite-dimensional exosystem, and an exponentially stable closed loop.Theorem 9 and its corollaries are also used in the proofs of the internal model results.Source: §5 · p. 13
  • A controller that solves output regulation is robust to the specified perturbations if and only if equations (8) hold.For every k ∈ Z, the solution of (8) is unique.Source: §5 · p. 13
  • Robustness to the perturbations is equivalent to the perturbed regulator output condition ˜Ce˜Σ + ˜De = 0 on Wα+1.This equivalence assumes the controller already solves output regulation.Source: §5 · p. 13

6 The p-Copy Internal Model Principle

For finite-dimensional output spaces, the p-copy internal model characterizes robust output regulation under the stated stability, resolvent, and Sylvester-equation assumptions. The section also gives an equivalent operator-based characterization for infinite-dimensional controllers.

6 The p-Copy Internal Model Principle

  • A p-copy internal model requires at least dim Y independent Jordan chains of length at least nk for each frequency iωk.The p-copy concept is meaningful only for finite-dimensional output spaces.Source: §6 · pp. 16–17
  • Under strong closed-loop stability, iωk ∈ ρ(Ae) for all k, and solvability of the Sylvester equation, robust output regulation holds if and only if the controller incorporates a p-copy internal model.The p-copy definition applies when dim Y is finite.Source: §6 · pp. 16–17
  • The proof supplies a new, more direct argument for the p-copy principle and yields an operator-based way to characterize controllers containing an internal model.The earlier proof in [16] contains parts that do not extend to systems with unbounded input and output operators.Source: §6 · p. 17
  • The restriction of the perturbed transfer operator ˜P(iωk)K to N(JG1(iωk)) being surjective for every k implies that the controller has a p-copy internal model.The paper presents invertibility of this restricted operator as an equivalent characterization under its stated conditions.Source: §6 · p. 17

7 The G-Conditions

The G-conditions provide a characterization of controllers solving robust output regulation, including when the output space is infinite-dimensional. With the stated range, stability, and Sylvester-equation assumptions, they are necessary and sufficient.

7 The G-Conditions

  • The G-conditions also characterize robust output regulation when the output space Y is infinite-dimensional.The paper notes that they are straightforward to verify for certain triangular controllers.Source: §7 · p. 20
  • With strong closed-loop stability, Z = R(iωk − G1) + R(G2) for every k, and a solution to the Sylvester equation, robust output regulation holds if and only if the controller satisfies the G-conditions.The conditions are the range intersection, kernel, and generalized-kernel inclusions in (17a)–(17c).Source: §7 · p. 21
  • If the controller satisfies the G-conditions, the closed loop is strongly stable, and the Sylvester equation has a solution, then the controller solves robust output regulation on Wα.The sufficiency result is stated for the perturbation class O.Source: §7 · p. 23

8 Regular Linear Systems

The section shows that regularity of both the plant and controller is sufficient for Assumption 3. It establishes this by forming a composite open-loop system and applying static output feedback.

8 Regular Linear Systems

  • Admissible plant and controller input/output operators, together with the stated transfer-function boundedness, define regular linear systems.The plant requires the resolvent-input range to lie in D(CΛ) and its transfer function to be uniformly bounded on a right half-plane.Source: §8 · pp. 24–25
  • If the plant and controller are regular linear systems and G2 is bounded, then Assumption 3 is satisfied.The proof establishes regularity of the composite system and its closed-loop system under static output feedback.Source: §8 · pp. 25–27

9 Robust Output Tracking for a Heat Equation

The example applies robust output regulation to a heat equation with Neumann boundary control and point measurements. It treats constant references and continuous periodic signals, with stability design remaining an open part of the periodic case.

9 Robust Output Tracking for a Heat Equation

  • The example uses a stable one-dimensional heat equation with Neumann boundary control and point measurements.The state space is L2(0, 1), with two control inputs and two measured outputs.Source: §9 · p. 27

9.1 Robust Tracking of Constant Reference Signals

  • A two-dimensional p-copy controller achieves robust asymptotic tracking of constant references under perturbations that preserve closed-loop stability and Sylvester-equation solvability.The robustness includes sufficiently small bounded plant-operator perturbations and arbitrary E and F perturbations.Source: §9.1 · pp. 30–31

9.2 Robust Tracking of Continuous Periodic Signals

  • The paper does not provide techniques for stabilizing the periodic-signal closed loop, leaving the choice of free controller parameters for that purpose open.Those parameters must yield strong stability and a solution to the Sylvester equation.Source: §9.2 · p. 31
  • For continuous periodic signals, the infinite-dimensional exosystem generates signals with Fourier coefficients weighted by |k|^-3/5, and the controller is constructed to satisfy the G-conditions.The controller solves robust output regulation if closed-loop stability and Sylvester-equation solvability are achieved.Source: §9.2 · p. 32

10 Conclusions

The paper extends robust output regulation and internal-model results to infinite-dimensional systems with unbounded input and output operators. Robust controller design and closed-loop stabilization remain important future research topics.

10 Conclusions

  • The paper extends the p-copy internal model principle and the G-conditions to infinite-dimensional systems with unbounded input and output operators and block diagonal infinite-dimensional exosystems.The proofs use Theorem 9, which also tests controller robustness against specific perturbations.Source: §10 · p. 33
  • Robust controller design and closed-loop stabilization are identified as key future research topics.Techniques used previously in are not applicable without modifications for plants with unbounded control and observation operators.Source: §10 · p. 33

A Properties of the Exosystem and the Proof of Lemma 11

The appendix proves equivalences among operator conditions in Lemma 11 using Sylvester equations. It also establishes that an operator satisfying property (20) must be zero.

A Properties of the Exosystem and the Proof of Lemma 11

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