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Integrating Prescribed Performance and RISE Control for Asymptotic Tracking of Uncertain High-Order Nonlinear Systems
Athanasios K. Gkesoulis, Christos K. Verginis, George C. Karras, Charalampos P. Bechlioulis
TL;DR
Prescribed-performance and funnel methods typically confine tracking error to a residual set, while RISE approaches lack explicit transient guarantees. This paper combines a barrier-type transformation with a RISE mechanism for uncertain high-order systems. It establishes bounded closed-loop signals, funnel invariance, and asymptotic tracking under appropriate gain selection.
Problem
Prescribed-performance methods typically ensure funnel confinement rather than zero-error convergence, whereas RISE approaches provide asymptotic tracking without explicit transient guarantees.
Source: §I · p. 1Method
The controller combines a barrier-type prescribed-performance transformation of a filtered tracking error with a RISE integral mechanism.
Source: §I · p. 2, A. Control design · p. 3Results
All closed-loop signals remain bounded, the prescribed funnel is invariant for funnel-compatible initial conditions, and tracking error converges asymptotically to zero under appropriate gain selection.
Source: §I · p. 2Takeaways & Limitations
The design provides a continuous, low-complexity synthesis of prescribed transient performance and asymptotic tracking without global approximation structures.
Source: §I · pp. 1–2
Abstract
from arXiv · showhide
This paper addresses asymptotic tracking with prescribed transient and steady-state performance for a class of uncertain high-order nonlinear systems with uncertain dynamics. The proposed controller combines a barrier-type transformation of a filtered tracking error with a robust integral of the sign of the error (RISE) mechanism, yielding a continuous and low-complexity design. The proposed method does not employ neural, fuzzy, or other global approximation structures. It is shown that all closed-loop signals remain bounded, the prescribed performance funnel is preserved for all initial conditions in the funnel, and that the tracking error converges asymptotically to zero under appropriate gain selection. The analysis provides a continuous-feedback synthesis that integrates prescribed performance control (PPC) with the RISE methodology. Simulation results on a ship-steering example illustrate the effectiveness of the proposed approach.
I. INTRODUCTION
The paper targets asymptotic tracking with prescribed transient and steady-state performance for uncertain high-order nonlinear systems. It combines prescribed-performance barriers with a RISE mechanism and establishes boundedness, funnel invariance, and asymptotic tracking under appropriate gain selection.
I. INTRODUCTION
- PPC and funnel control typically confine tracking error to a funnel or residual set, while RISE-based approaches provide asymptotic tracking without explicit transient guarantees.The paper identifies low-complexity continuous control combining both properties as an open problem.Source: §I · p. 1
- Earlier RISE-based prescribed-performance designs use approximation mechanisms and gain-dependent initial-condition sets, while other approaches impose restrictions on dynamics or reference trajectories.The proposed design avoids approximation structures and covers funnel-compatible initial conditions under its stated gain conditions.Source: §I · p. 2
- The controller integrates a barrier-type prescribed-performance transformation with RISE to achieve continuous asymptotic tracking with transient and steady-state specifications.It considers uncertain high-order nonlinear systems without requiring knowledge of their dynamics or a vanishing funnel.Source: §I · p. 2
- The design establishes bounded closed-loop signals, prescribed-funnel invariance for funnel-compatible initial conditions, and asymptotic tracking under appropriate gain selection.These guarantees extend to all initial conditions satisfying the funnel constraints.Source: §I · p. 2
- The controller is low-complexity because it requires neither observers nor global approximators such as neural networks or fuzzy systems.Source: §I · p. 2
II. PRELIMINARIES
The preliminaries establish notation and solution concepts for the analysis. They specify the Filippov solution framework and the upper right Dini derivative.
II. PRELIMINARIES
- The Euclidean norm, L2 and L∞ spaces, and the meaning of “a.e.” are specified as analysis notation.Source: §II · p. 2
- A Filippov solution is absolutely continuous and its derivative belongs almost everywhere to the regularized dynamics.For a C1 function, the ordinary chain rule holds almost everywhere along these dynamics.Source: §II · p. 2
- The upper right Dini derivative is defined as the limsup of forward difference quotients as the increment approaches zero.Source: §II · p. 2
III. PROBLEM FORMULATION
The section models an uncertain high-order system as a chain of integrators and specifies continuous-input asymptotic tracking within a designer-chosen performance envelope.
III. PROBLEM FORMULATION
- The plant is a kth-order integrator chain, with unknown drift f and uncertain input gain g acting on the final state equation.The drift represents modeling uncertainties, unknown parameters, and bounded exogenous effects.Source: §III · p. 2
- The unknown functions f and g are C2 with bounded partial derivatives through order two on each compact state set, and g(x,t) ≥ gmin > 0 globally.The positive lower bound on g is the stated sufficient controllability condition.Source: §III · p. 2
- The desired reference has bounded derivatives through order k+2, and the tracking errors are defined as e_i = x_i − y_d^(i−1).These errors form a chain: ė_i = e_(i+1) for i < k, while ė_k includes the plant dynamics and the kth reference derivative.Source: §III · p. 2
- The objective is continuous-input tracking with e1(t) converging to zero while |e1(t)| remains below the prescribed envelope φ(t) for all t ≥ 0.The envelope is positive, continuously differentiable, strictly decreasing, and has a positive limit; initially, φ(0) > |e1(0)|.Source: §III · pp. 2–3
A. Control design
The design combines a barrier transformation of filtered tracking errors with a RISE integral term to enforce a funnel bound and asymptotic tracking.
A. Control design
- Reciprocal barriers retain e1 within its funnel, while an integral of its sign compensates for upper bounds on the drift dynamics within the funnel.The proposed design uses continuous feedback.Source: A. Control design · p. 3
- Auxiliary filtered errors η_i are constructed by repeatedly applying d/dt + λ to e1, with λ > 0.The filtered errors provide the coordinates used in the control design.Source: A. Control design · p. 3
- The design bounds η_k within a prescribed funnel ρ(t), then applies a barrier-type map that grows without bound as its argument approaches ±1.A bounded transformed error implies |η_k(t)| < ρ(t).Source: A. Control design · p. 3
- Bounds on η_k yield exponentially decaying-plus-constant bounds on lower-order errors, which can be used to select ρ(t) for prescribed e1 performance.For ρ(t) = (ρ0 − ρ∞)e^(−δt) + ρ∞ with λ > δ, each lower-order error approaches a stated residual set at least as fast as e^(−δt).Source: A. Control design · p. 3
- The RISE integral accumulates compensation for f and the kth reference derivative, allowing η_k to converge to zero without requiring ρ(t) to converge to zero.This holds for every funnel-compatible initial condition.Source: A. Control design · p. 3
B. Stability proof
The proof establishes bounded closed-loop behavior, preservation of the filtered-error funnel, and asymptotic convergence under suitable gain selection. It uses Filippov solution existence, boundedness, and a Lyapunov argument to obtain convergence.
B. Stability proof
- All closed-loop signals remain bounded and |ηk(t)| < ρ(t) for all t ≥ 0; with sufficiently large k1 and k2, every tracking error converges to zero.The theorem also gives exponentially decaying bounds plus a residual bound for errors before the asymptotic convergence claim.Source: B. Stability proof · p. 3
- Filippov existence theory supplies a solution from initial conditions inside the funnel, and the boundedness argument rules out finite escape.The proof considers a maximal interval first, then shows the solution extends for all time.Source: B. Stability proof · pp. 3–4
- A Lyapunov derivative bound yields χ ∈ L2 ∩ L∞; bounded derivatives make χ uniformly continuous, so Barbalat’s lemma gives χ(t) → 0.The resulting convergence propagates through the stable cascade to ηk and all tracking errors.Source: B. Stability proof · p. 5
- The controller’s plant input is continuous because the signum term acts through the integral state, although the augmented controller dynamics are discontinuous at ε = 0.The augmented dynamics are interpreted in the Filippov sense.Source: B. Stability proof · pp. 4–5
V. SIMULATION RESULTS
The Norrbin ship-steering simulation illustrates asymptotic tracking and confinement of the filtered error within its prescribed bounds. The reported input oscillation is not classical chattering because the plant input is continuous.
V. SIMULATION RESULTS
- The test uses the Norrbin ship-steering model with a sinusoidal reference of amplitude 20° and period 50 s.The initial heading and heading rate are zero; the design uses λ = 0.6, ρ0 = 1, ρ∞ = 0.1, ℓ = 0.5, k1 = k2 = 20, and α = 5.Source: §V · p. 5
- The simulation shows e1 and e2 tending to zero while η2 stays within ±ρ(t).The reported transient oscillation in u is not classical chattering because the plant input is continuous.Source: §V · p. 5
- The transient input oscillation is attributed to nonsmooth integral action, the gains used to enforce performance and convergence, and possible numerical-integration amplification.The authors distinguish this oscillation from classical chattering because the plant input is continuous.Source: §V · p. 5
VI. CONCLUSION AND FUTURE WORK
The paper concludes that its continuous, low-complexity controller combines prescribed performance with asymptotic tracking for uncertain high-order nonlinear systems. The work demonstrates the design in ship steering and identifies several directions for future study.
VI. CONCLUSION AND FUTURE WORK
- The paper establishes boundedness, funnel invariance, and zero tracking error for its controller combining prescribed performance and asymptotic tracking.The stated scope is uncertain high-order nonlinear systems.Source: §VI · p. 5
- Ship steering provides the paper’s demonstration of the proposed approach.The conclusion describes the controller as continuous and low-complexity.Source: §VI · pp. 5–6
- Future work will address multi-input multi-output systems, input constraints, and adaptive gain selection.Source: §VI · p. 6