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Control for Networked Control Systems with Remote and Local Controllers over Unreliable Communication Channel

Xiao Liang, Juanjuan Xu

arXiv:1803.01336v1math.OC

TL;DR

This paper addresses optimal control and stabilization for networked control systems with remote and local controllers over unreliable communication channels, where packet dropout and asymmetric information complicate controller design. It uses Pontryagin’s maximum principle and Riccati equations to derive finite-horizon optimality and mean-square stabilization conditions, with additive-noise boundedness treated separately.

  • Problem

    For a similar networked-control setting, prior work gave only a sufficient optimal-control condition and omitted stabilization, while the remote and local controllers have asymmetric information.

    Source: §1 · p. 2

  • Method

    The paper applies Pontryagin’s maximum principle to forward-backward stochastic difference equations and develops a non-homogeneous relationship between state estimation and the costate.

    Source: §1 · p. 2

  • Results

    The paper gives a necessary and sufficient finite-horizon optimal-control condition and, without additive noise, a necessary and sufficient mean-square stabilization condition; with additive noise, it gives a sufficient mean-square boundedness condition.

    Source: §1 · p. 2, §5 · p. 7

  • Takeaways & Limitations

    The authors apply the results to a simple UAV system and report that the example illustrates the algorithm’s effectiveness.

    Source: §5 · p. 7

    With additive noise, the paper derives only a sufficient condition for mean-square boundedness, and its stabilization analysis removes the additive noise.

    Source: §3.1 · p. 4

Abstract

from arXiv · show

This paper is concerned with the problems of optimal control and stabilization for networked control systems (NCSs), where the remote controller and the local controller operate the linear plant simultaneously. The main contributions are two-fold. Firstly, a necessary and sufficient condition for the finite horizon optimal control problem is given in terms of the two Riccati equations. Secondly, it is shown that the system without the additive noise is stabilizable in the mean square sense if and only if the two algebraic Riccati equations admit the unique solutions, and a sufficient condition is given for the boundedness in the mean square sense of the system with the additive noise. Numerical examples about the unmanned aerial vehicle model are shown to illustrate the effectiveness of the proposed algorithm.

1 Introduction

The paper studies optimal control and stabilization for networked control systems with remote and local controllers communicating over an unreliable channel. It addresses the challenges of packet dropout and asymmetric controller information with Riccati-based conditions for finite-horizon control and mean-square stabilization.

1 Introduction

  • The system models a local controller sending the plant state to a remote controller over a channel where packets may be dropped, with an acknowledgement returned when the remote controller receives the signal.The local controller can observe the remote controller’s received signal, but the remote control action is not available to the local controller at time k.Source: §1 · pp. 1–2
  • Unlike the prior similar study, this work includes stabilization and gives necessary and sufficient rather than only sufficient conditions for optimal control.The authors identify asymmetric information and the inability of either controller to access the other’s current action as challenges.Source: §1 · p. 2
  • The paper gives a necessary and sufficient finite-horizon optimal-control condition using two Riccati equations, and a necessary and sufficient mean-square stabilization condition when additive noise is absent.With additive noise, it gives a sufficient condition for mean-square boundedness.Source: §1 · p. 2
  • The paper applies Pontryagin’s maximum principle to develop an approach based on forward-backward stochastic difference equations for the two-controller system.The introduction also situates the problem among Nash and Stackelberg strategies, noting that their information assumptions do not fit this setting.Source: §1 · p. 2

2 Finite Horizon Case

The finite-horizon problem models packet dropouts and distinct information available to the remote and local controllers, then derives optimal strategies using two Riccati equations. The finite-horizon solution is unique exactly when the stated positivity conditions hold at every time step.

2.1 Problem Formulation

  • The model uses an i.i.d. Bernoulli variable for state-packet delivery, and the remote controller observes received signals while the local controller also has access to the states.The local and remote controls are constrained to be measurable with respect to their respective information sets.Source: §2.1 · p. 3

2.2 Optimal Controllers Design

  • Pontryagin’s maximum principle is applied to the costate equations, and two Riccati equations are introduced as the route to obtaining the optimal controllers.The paper identifies one of the Riccati equations as a standard Riccati difference equation.Source: §2.2 · p. 3
  • Problem 1 has a unique solution if and only if Υk > 0 and Λk > 0 for k = N, . . . , 0.The theorem gives the condition in terms of the two Riccati-equation solutions.Source: §2.2 · p. 4
  • The optimal controls are specified using Kk, Λk, Mk and state estimates, with an associated optimal cost.The cited passage identifies ˆxk|k and ˜xk as the state estimate and estimation error.Source: §2.2 · p. 4
  • The authors contrast their finite-horizon conditions with, whose cost weighting matrices they describe as positive definite rather than positive semi-definite.They also distinguish their use of Pontryagin’s maximum principle from that study’s dynamic programming approach.Source: §2.2 · p. 4

3 Infinite Horizon Case

The section formulates infinite-horizon optimal control under mean-square stability, then gives Riccati-based conditions for noise-free stabilizability and noisy-system boundedness.

3.1 Problem Formulation

  • Because additive noise permits only a sufficient boundedness condition, the section removes that noise to derive a necessary-and-sufficient stabilization criterion.The noisy-system stabilization problem is discussed separately.Source: §3.1 · p. 4
  • The infinite-horizon problem seeks measurable remote and local controls that stabilize the closed loop in mean square and minimize its cost.The stated assumptions require positive controller weights and observability of (A, Q^1/2).Source: §3.1 · p. 4

3.2 Solution to Problem 2

  • Without additive noise, the system is mean-square stabilizable if and only if the unique Riccati solutions satisfy the stated positivity and initial-condition requirements.The theorem also states that the resulting optimal controllers stabilize the system and minimize the infinite-horizon cost.Source: §3.2 · p. 5
  • The estimator-error covariance converges to a unique positive-definite matrix if and only if the stated spectral condition holds, assuming mean-square stabilizability.The condition uses the largest-absolute-value eigenvalue of A − BLΛ^-1M.Source: §3.2 · p. 5
  • With additive noise, the stated spectral-radius condition together with unique Riccati solutions is sufficient for mean-square boundedness.Under these conditions, the optimal controllers also minimize the infinite-horizon cost.Source: §3.2 · p. 5

4 Numerical Examples

The examples apply the networked-control model to a UAV communicating with a ground-control center, then examine finite- and infinite-horizon behavior under packet dropout. In the finite-horizon case, energy cost rises greatly as dropout probability increases, while velocity shows no huge difference across the tested probabilities.

4 Numerical Examples

  • The UAV model targets reaching a destination with minimum energy cost, using location and velocity to describe straight-line flight and an unreliable UAV-to-ground channel.The UAV observes its location; the ground-control center returns commands and acknowledgements over a perfect channel.Source: §4 · pp. 5–6

4.2 Infinite horizon case

  • The UAV stabilization example omits additive noise ωk; Fig. 3 shows UAV velocity for p = 0, 0.5, and 1, while Fig. 4 shows optimal energy cost for different p.For Fig. 5, p = 0.5 and the other variables match the finite-horizon case.Source: §4.2 · p. 6, Figure 3 · p. 7, Figure 4 · p. 7

4.1 Finite horizon case

  • The optimal energy cost increases greatly with packet-dropout probability; UAV velocity υk shows no huge difference for p = 0, p = 0.5, and p = 1.Source: §4.1 · p. 6

5 Conclusion

The paper studies optimal control and stabilization for a linear plant operated by remote and local controllers. It derives Riccati-equation conditions for finite-horizon optimal control and mean-square stabilization without additive noise, and a sufficient condition for mean-square boundedness with additive noise.

5 Conclusion

  • The paper uses Pontryagin’s maximum principle to develop a non-homogeneous relationship between state estimation and the costate.Source: §5 · p. 7
  • For finite-horizon optimal control, the paper gives a necessary and sufficient condition in terms of Riccati-equation solutions.Source: §5 · p. 7
  • Without additive noise, the paper gives a necessary and sufficient mean-square stabilization condition; with additive noise, it gives a sufficient mean-square boundedness condition.Source: §5 · p. 7

A Proof of Lemma 1

The proof rewrites the costate equations and establishes the stated form by induction. Its induction uses a base case, a step at k = N, and a general step for k = n.

A Proof of Lemma 1

  • The proof rewrites the costate equations using conditional expectations and related equations before deriving the claimed form.Source: A Proof of Lemma 1 · p. 9
  • The claimed form is established first at k = N + 1 and k = N, then extended by induction to k = n.Source: A Proof of Lemma 1 · pp. 9–10
  • In the induction step, substituting (A.11) and (A.12) into (A.10) yields a result; plugging (A.13) into (A.6), using (A.3), yields a further result.Source: A Proof of Lemma 1 · p. 10

The optimal controller

The proof substitutes earlier relations into the costate equation to establish a state-estimate relation, completing the induction step.

The optimal controller

B Proof of Theorem 1

The proof establishes that the finite-horizon problem has a unique solution when the Riccati-derived quantities are positive, and that the resulting controllers attain the stated optimal cost.

B Proof of Theorem 1

  • Necessity is proved by induction, showing Υk and Λk are positive at each step when the problem has a unique solution.The proof concludes with Λn > 0.Source: B Proof of Theorem 1 · pp. 11–12
  • For sufficiency, assuming Υk and Λk are positive across the horizon allows the cost to be rewritten and the unique solution to be established.The proof identifies the resulting value with (22).Source: B Proof of Theorem 1 · pp. 11–13
  • Positivity of Υk and Λk for every k from 0 to N yields unique optimal controllers and the corresponding optimal cost.The controllers are given by (20) and (21), and the cost equals (22).Source: B Proof of Theorem 1 · p. 13

C Proof of Theorem 2

The proof derives the Riccati solutions from mean-square stabilizability and establishes that the resulting feedback controllers stabilize the system and minimize its cost.

C Proof of Theorem 2

  • Mean-square stabilizability implies convergence of the finite-horizon quantities used to construct the Riccati solutions.The proof shows convergence of Ψk(N), Zk(N), and Xk(N).Source: C Proof of Theorem 2 · pp. 13–14
  • The proof establishes uniqueness of Z and X by comparing the cost for arbitrary initial states with the solutions of the Riccati equations.Uniqueness of Z, together with (12) and (C.9), yields uniqueness of X.Source: C Proof of Theorem 2 · pp. 14–15
  • The feedback controllers (32) and (33) stabilize system (23) in the mean-square sense and minimize cost function (24).Their optimal cost is given by (34).Source: C Proof of Theorem 2 · p. 16

D Proof of Lemma 2

The proof uses the Riccati equations and a spectral condition to establish boundedness of the estimator error covariance.

D Proof of Lemma 2

  • Under the assumed mean-square stabilizability, the algebraic Riccati equations hold.The proof invokes this to proceed with the covariance argument.Source: D Proof of Lemma 2 · p. 16
  • The estimator error covariance is shown to have a unique limiting solution P, completing the sufficiency proof.The passage identifies P as the unique solution of the following equation.Source: D Proof of Lemma 2 · p. 16
  • p|λmax(A−BLΛ−1M)| < 1 makes the associated covariance transient term vanish as k tends to infinity.The proof then obtains the limiting covariance relation.Source: D Proof of Lemma 2 · p. 16

E Proof of Corollary 1

With the stated Riccati solutions and spectral condition, the proof shows bounded mean-square state behavior under additive noise and verifies optimality of the controllers.

E Proof of Corollary 1

  • Boundedness of the state second moment follows when the system formed with the estimation error and additive noise is mean-square stable.The proof uses bounded estimation-error covariance and additive-noise covariance Qω.Source: E Proof of Corollary 1 · p. 17
  • The controllers (36) and (37) make system (1) bounded in the mean-square sense.The conclusion is that limk→∞ E[x′kxk] is bounded.Source: E Proof of Corollary 1 · p. 17
  • The controllers (36) and (37) also minimize cost function (35), with the corresponding optimal cost as stated there.The proof relies on positive definiteness of Υ and Λ.Source: E Proof of Corollary 1 · p. 18
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