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Nonlinear Stabilization under Sampled and Delayed Measurements, and with Inputs Subject to Delay and Zero-Order Hold
Iasson Karafyllis, Miroslav Krstic
TL;DR
The paper addresses stabilization with simultaneous sampling and input/output delays, where uncompensated delays restrict sampling and stability guarantees. It combines nominal delay-free feedback with predictor-based compensation to obtain global asymptotic stabilization under arbitrarily long delays and sampling periods.
Problem
Sampling with input and output delays leaves several global stabilization problems open, while uncompensated delays generally require short sampling periods and delays.
Method
The paper introduces two frameworks combining nominal feedback laws designed without delays with predictor-based compensation for delays and sampling effects.
Results
The frameworks achieve global asymptotic stabilization for forward complete systems with arbitrarily long input and output delays, and for sampled-data systems with arbitrarily long input and measurement delays.
Takeaways & Limitations
Global asymptotic stabilization can be obtained despite arbitrarily long delays and sampling periods within the stated system and feedback assumptions.
Takeaways & Limitations
The results require the existence of stabilizing feedback and involve technical issues concerning solution existence.
Abstract
from arXiv · showhide
Sampling arises simultaneously with input and output delays in networked control systems. When the delay is left uncompensated, the sampling period is generally required to be sufficiently small, the delay sufficiently short, and, for nonlinear systems, only semiglobal practical stability is generally achieved. For example, global stabilization of strict-feedforward systems under sampled measurements, sampled-data stabilization of the nonholonomic unicycle with arbitrarily sparse sampling, and sampled-data stabilization of LTI systems over networks with long delays, are open problems. In this paper we present two general results that address these example problems as special cases. First, we present global asymptotic stabilizers for forward complete systems under arbitrarily long input and output delays, with arbitrarily long sampling periods, and with continuous application of the control input. Second, we consider systems with sampled measurements and with control applied through a zero-order hold, under the assumption that the system is stabilizable under sampled-data feedback for some sampling period, and then construct sampled-data feedback laws that achieve global asymptotic stabilization under arbitrarily long input and measurement delays. All the results employ "nominal" feedback laws designed for the continuous-time systems in the absence of delays, combined with "predictor-based" compensation of delays and the effect of sampling.
1. Introduction
The paper addresses stabilization with simultaneous sampling, input delays, and measurement delays, where prior results impose delay-dependent restrictions and often provide only semiglobal practical stability. It introduces predictor-based frameworks that achieve global asymptotic stabilization under broad delay, sampling, and input-application conditions.
- Motivation: Sampling and input/output delays create open stabilization problems, including strict-feedforward systems, the nonholonomic unicycle, and LTI systems over networks.Prior nonlinear results generally guarantee only semiglobal practical stability, while existing networked-control results rely on delay-dependent conditions.
- First framework: The paper presents global asymptotic stabilizers for forward complete systems with arbitrarily long input and output delays, arbitrarily long sampling periods, and continuous control application.The first framework assumes the delay-free system is globally stabilizable and forward complete, with continuously adjustable input.
- Second framework: The paper also constructs sampled-data feedback laws for zero-order-hold control under arbitrarily long input and measurement delays.This framework assumes stabilizability under sampled-data feedback for some sampling period and uses sampled delayed outputs.
- Compensation method: Both frameworks combine nominal delay-free feedback laws with predictor-based compensation for delays and sampling effects.The approach is intended to reuse controllers designed for the corresponding delay-free systems.
- Stability guarantees: No restrictions are imposed on the input and measurement delays or the sampling period in the first framework.The result shows that continuous measurements are unnecessary for global asymptotic stabilization of stabilizable forward complete systems with arbitrary input and output delays.
- Applications: The results cover continuous input adjustment and zero-order-hold inputs, including applications to the nonholonomic integrator.For the nonholonomic integrator, the dynamic sampled-data controller has no restrictions on delay values or sampling-period size.
2. Dynamic Sampled-Data Feedback for Continuously Adjusted Input
The paper constructs a dynamic sampled-data feedback for forward-complete systems with continuously adjusted inputs, combining sampled outputs with predictor-based delay compensation. Under the stated hypotheses, the resulting closed loop is uniformly globally asymptotically stable, while the design avoids technical existence problems affecting a direct static predictor implementation.
- Assumptions: The design assumes forward completeness and a time-varying stabilizing feedback k(t,x) for the delay-free system.The feedback satisfies a class-K bound and yields a class-KL state estimate for the nominal system.
- Implementation implications: The dynamic controller avoids the existence difficulties that can arise when implementing the corresponding static predictor feedback from arbitrary initial histories.For sufficiently delayed times, its computed input coincides with the predictor-feedback value, while the dynamic construction avoids solving the problematic integral equation.
- Controller construction: The dynamic sampled-data controller interconnects a sampled-data subsystem with a functional difference-equation subsystem and uses output samples at t_i=t_0+τ+iT.Its input is the delayed output y(t)=x(t−τ), available only at discrete sampling instants.
- Main result: Theorem 2.1 establishes uniform global asymptotic stability of the closed loop for arbitrary positive sampling periods and nonnegative delays satisfying r+τ>0.The theorem provides a class-KL estimate for the closed-loop state, controller state, and input-history terms.
- Predictor compensation: Predictor-based compensation maps delayed output and input histories to the predicted state needed by the nominal feedback.The predictor can be explicitly computed for linear and certain bilinear systems, and constructed inductively for a class of nonlinear systems.
- Applications: The result includes strict-feedforward systems with arbitrarily sparse state sampling and continuous control application as a special case.The paper identifies this as the first solution of that global asymptotic stabilization problem.
3. Sampled-Data Feedback for Input Applied with Zero Order Hold
The section develops sampled-data feedback with zero-order-held inputs for systems already stabilizable under sampled-data feedback. Predictor-based delay compensation yields global asymptotic or exponential stability despite input and measurement delays, with additional dead-beat conclusions in specified cases.
- Assumptions: The design assumes the undelayed system is globally stabilizable using zero-order-hold sampled-data feedback for some positive sampling period.This requirement is stated as hypothesis (H3), and the paper lists linear stabilizable and several nonlinear system classes satisfying it.
- Design: The feedback law composes a nominal feedback stabilizer with a predictor-based delay compensator evaluated at sampling times.The predictor mapping uses delayed state and input histories, while the resulting control is held constant between successive sampling times.
- Stability result: Under the stated delay and sampling alignment conditions, the resulting sampled-data closed loop is uniformly globally asymptotically stable.The theorem requires a positive sampling period, nonnegative input delay, positive total delay, and an integer relation between delay and sampling period.
- Stability result: For stabilizable LTI systems, the same construction gives global exponential stability, with an exponential bound on the state and input histories.The result applies when the relevant eigenvalues lie strictly inside the unit circle and includes the bound represented by equation (3.17).
- LTI example: For each admissible feedback gain and sampling period, a critical delay separates global exponential stability from exponentially growing solutions in the illustrated scalar system.The paper states stability when the delay is below the critical value and exponential growth when it exceeds that value.
- Further consequences: The predictor-based feedback extends the range of measurement delays permitting stabilization for fixed gain and sampling period.The paper also reports dead-beat properties for certain linear and linearizable controllable systems, including finite-time state vanishing under the stated conditions.
4. Stabilization of a Nonholonomic Mobile Robot Over a Long-Distance
The section applies predictor-based feedback to a nonholonomic mobile robot with sampled, delayed measurements and delayed inputs. It establishes uniform global stabilization without restrictions on delay magnitudes and also treats zero-order-hold control.
- Model and delays: The robot model uses measurements sampled every T, with measurement delay r and communication delay τ affecting the applied control.The delayed dynamics explicitly use x(t−τ), y(t−τ), and θ(t−τ), while measurements are available only at discrete instants.
- Predictor-based design: Predictor mappings, coordinate transformations, and nominal stabilizers for the nonholonomic integrator produce feedback laws for the delayed robot.The construction combines the robot transformations with an existing stabilizing feedback and the predictor-based theorem.
- Continuous-input case: Uniform global stabilization is achieved for the delayed robot using sampled-data dynamic feedback for every r ≥ 0, τ ≥ 0, and T > 0.The result imposes no restrictions on the magnitudes of the delays.
- Zero-order-hold case: The sampled implementation of the discontinuous feedback has a dead-beat property of order T^3 for every positive sampling period.The dead-beat property is used to establish the required sampled-data stabilizability condition.
- Zero-order-hold case: With zero-order-hold inputs and delayed sampled measurements, Proposition 4.2 establishes uniform global asymptotic stability when τ is an integer multiple of T.The applied velocities remain constant on each sampling interval.
5. Concluding Remarks
The paper studies nonlinear stabilization with input and measurement delays and sampled measurements in both continuously adjusted and zero-order-hold settings. Under forward-completeness and stabilizability assumptions, predictor-based feedback yields global asymptotic stability without restrictions on delay magnitude.
- Scope: Two cases are considered: continuously adjustable inputs and inputs applied through a zero-order hold, with measurements available only at sampling times.The zero-order-hold case requires a sampled-data stabilizability assumption for some sampling period.
- Main conclusion: Predictor-based delay compensation guarantees global asymptotic stability with no restrictions on the magnitude of the delays.This conclusion is stated for the closed-loop system under the paper’s forward-completeness and additional stabilizability assumptions.
- Main conclusion: When control is applied continuously and only measurements are sampled, the sampling time can be arbitrarily long.Applications include linear networked systems, strict-feedforward systems, and a nonholonomic mobile robot.
- Future work: Future work concerns robustness to actuator and measurement errors and extensions where delayed sampled outputs do not coincide with the state vector.These are identified as extensions rather than established results of the paper.
Appendix
The appendix supplies proof details for the main stabilization theorems. Its arguments establish solution existence, continuity, bounds, and the inequalities needed to complete the claims.
- Existence and bounds: The proofs establish existence of the relevant solutions on the required time intervals using the stated hypotheses and continuation bounds.The argument invokes the Fact together with hypotheses such as (H1) and inequalities (A1)–(A3).
- Claim 1: The appendix constructs bounds for the closed-loop variables and uses them to obtain a function G in K∞ satisfying the target inequality.The construction combines inequalities (A1), (A2), and (A3).
- Claim 3: For the proof of Claim 3, the cases τ < 0 and τ = 0 are handled separately before deriving inequality (3.9).The τ = 0 case uses inequalities (A4)–(A6), while the τ < 0 case uses (A3) and (A4).