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A High-Gain Nonlinear Observer with Limited Gain Power
Daniele Astolfi, Lorenzo Marconi
TL;DR
High-gain observers for nonlinear systems offer tunable convergence but become difficult to implement when their gain grows to powers up to n. This paper introduces a 2n−2-dimensional observer whose gain grows only to power 2, while retaining the high-gain design paradigm and supporting disturbance bounds under stated assumptions. The paper reports benefits in implementation and high-frequency noise rejection, with confirmation in a nonlinear Van der Pol example.
Problem
Standard high-gain observers for n-dimensional nonlinear systems require gain powers up to n, creating implementation difficulties when n or ℓ is large.
Method
The paper designs a redundant high-gain observer of dimension 2n−2, with coefficients selected through a Hurwitz polynomial and bounded nonlinear saturation.
Results
The proposed observer preserves tunable state-estimate convergence while reducing gain growth to power 2 and shows high-frequency noise-rejection benefits in linear and nonlinear Van der Pol examples.
Takeaways & Limitations
The design exchanges higher observer dimension for lower gain power, offering a practical implementation benefit while retaining key high-gain features.
Abstract
from arXiv · showhide
In this note we deal with a new observer for nonlinear systems of dimension n in canonical observability form. We follow the standard high-gain paradigm, but instead of having an observer of dimension n with a gain that grows up to power n, we design an observer of dimension 2n-2 with a gain that grows up only to power 2.
I. INTRODUCTION
The paper studies high-gain observation for nonlinear systems in canonical observability form and proposes a redundant observer that retains high-gain benefits while limiting gain growth to power 2. The trade-off is an observer dimension of 2n−2 instead of n, motivated by implementation difficulties from powers up to n.
- Problem and setting: Uniform observability transforms the original nonlinear system into an n-dimensional canonical observability form with bounded disturbance and measurement noise.The transformed system is defined on X := φ(Z), with a prime-form triplet (A_n, B_n, C_n).
- Standard high-gain observer: The standard observer uses a high-gain parameter ℓ, a Hurwitz error-dynamics matrix, and a bounded saturation agreeing with ϕ on X.Under zero disturbance and noise, uniform Lipschitzness of ϕ on X supports exponential convergence for trajectories remaining in X.
- Standard high-gain observer: The exponential decay rate can be assigned through ℓ, while the gain exhibits polynomial peaking of order n−1.The convergence statement applies for arbitrary observer initial conditions as long as x(t) remains in X.
- Standard high-gain observer: With bounded disturbances or sensor noise, the standard observer provides asymptotic error bounds, but increasing ℓ worsens sensitivity to noise and the first n−2 disturbance components.A large ℓ can reduce the asymptotic gain on the n-th disturbance component while creating this sensitivity trade-off.
- Motivation: High-gain observers are important in output-feedback stabilization and semiglobal nonlinear separation principles because they offer tunable convergence and disturbance-estimation properties.These applications typically use an arbitrarily large compact invariant set X.
- Proposed approach: The proposed observer addresses implementation difficulty by limiting high-gain growth to power 2, at the cost of increasing observer dimension from n to 2n−2.The standard observer’s powers up to n become difficult to implement when n or ℓ is large.
II. MAIN RESULT
The proposed observer uses n−1 two-dimensional stages, yielding dimension 2n−2 while scaling each stage through D2(ℓ)=diag(ℓ, ℓ2). Its coefficients can assign the dynamics’ eigenvalues arbitrarily, and Proposition 1 establishes high-gain asymptotic properties for two extracted state estimates under bounded disturbances.
- Observer structure: The matrix M can realize the characteristic polynomial of any arbitrary Hurwitz polynomial through suitable choices of (ki1, ki2).This provides arbitrary eigenvalue assignment for the observer’s linear error dynamics.
- Observer structure: The observer has dimension 2n−2 and consists of n−1 coupled two-dimensional stages with gain scaling D2(ℓ)=diag(ℓ, ℓ2).Each stage uses a two-component state ξi and gain vector Ki.
- Asymptotic estimates: The redundant state ξ yields two extracted estimates, ˆx′ and ˆx′′, obtained through matrices L1 and L2.The first estimate extracts n components, while the second uses the observer’s redundancy to obtain an additional state estimate.
- Asymptotic estimates: For ℓ≥ℓ⋆ and bounded d(x,t), Proposition 1 guarantees the stated estimation bound for every initial observer state ξ(0)∈R2n−2 while x(t) remains in X.The coefficients are selected so that M is Hurwitz, and the nonlinear function is replaced by a bounded extension agreeing with ϕ on X.
- Proof strategy: The proof rescales the transformed error so that its dynamics have leading term ℓMε, with disturbance and nonlinear terms multiplied by ℓ−(n−1).A quadratic Lyapunov function based on P satisfying PM+M^TP=−I yields exponential estimates that are transferred back to the state errors.
FREQUENCY NOISE IN THE LINEAR CASE
For linear systems, the proposed observer is analyzed against the standard high-gain observer under high-frequency measurement noise. Its asymptotic error ratio is strictly decreasing with noise frequency for state estimates i=2,...,n, while the first-state exception remains.
- High-gain observers trade faster state estimation against greater sensitivity to measurement noise.
- The comparison specializes the proposed and standard observers to linear systems with sinusoidal measurement noise.The noise is ν(t)=a_N sin(ω_N t+f_N).
- For state variables i=2,...,n, the new-to-standard asymptotic estimation-error ratio is a strictly decreasing polynomial function of noise frequency.
- At sufficiently high frequencies, the proposed observer has better asymptotic properties for those state estimates, except the first one.The formal result applies under Hurwitz error dynamics and the stated index conditions.
- The proof compares harmonic transfer functions from measurement noise to each state-estimation error in the two observer error systems.Both error systems are described as Hurwitz, and the relevant outputs have relative degrees used in the frequency comparison.
- The frequency-domain conclusion follows from bounds on the corresponding transfer functions for sufficiently large noise frequency.
IV. EXAMPLE: OBSERVER FOR THE UNCERTAIN VAN DER POL OSCILLATOR
The example applies the proposed observer to an uncertain Van der Pol oscillator immersed in a five-dimensional canonical observability form. Simulations examine noiseless estimation and compare high-frequency-noise sensitivity with a standard high-gain observer.
- Model and immersion: The uncertain Van der Pol oscillator has uncertain constant parameters and evolves on a compact invariant limit-cycle set.The parameters are μ=(α^2,β)^T∈U, and the state belongs to W⊂R^2.
- Model and immersion: The oscillator augmented with constant-parameter dynamics is immersed into a dimension-five prime-form system with measured output y=C_5x+ν(t).
- Observer implementation: The proposed observer is implemented with four two-dimensional subsystems, producing two state estimates through ξ and the matrices L_1 and L_2.
- Simulation results: With α=1, β=0.5, gain ℓ_1=100, and no sensor noise, Figures 1 and 2 show the first two state-estimation errors.
- Simulation results: Under ν(t)=10^-2 sin(10^3t), the table compares normalized asymptotic errors of the proposed and standard high-gain observers.The normalized error is the limiting supremum of estimation error divided by the noise amplitude a_N.
- Simulation results: The numerical table shows a remarkable improvement in sensitivity to high-frequency measurement noise for the new observer relative to the standard observer.The analytical result cited for this comparison is stated only for linear systems.
V. CONCLUSIONS
The proposed observer reduces the high-gain power from n to 2 at the cost of increasing state dimension from n to 2n−2, while retaining tunable convergence speed and improving high-frequency noise rejection in tested cases.
- The observer has dimension 2n−2 instead of n, while its high-gain power is 2 instead of n.
- Its state-estimate convergence speed remains tunable through the high-gain design.
- For linear systems, the proposed observer improves high-frequency measurement-noise rejection relative to the standard high-gain observer.
- The noise-rejection benefit is also confirmed numerically for the nonlinear Van der Pol example.
- A complete characterization of the new observer’s sensitivity to sensor noise remains under investigation.
- The proposed structure does not prevent the peaking caused by incorrect initial conditions and fast convergence, although existing mitigation techniques can be adopted.
- The work does not consider the general multi-output case; extension is immediate only for specified block-triangular systems and otherwise remains under investigation.
A. Procedure to assign the eigenvalues of M
The eigenvalue-assignment procedure recursively constructs characteristic polynomials for matrices M_i, selecting gain pairs K_i so that the final matrix M has a desired polynomial.
- The procedure defines matrices M_i recursively and tracks the characteristic polynomials of M_i and M_{i−1}.
- The gain pair K_i affects the matrices Q_i and E_i, while M_i depends on the preceding gains K_1 through K_i.
- The matrix relations use the invertibility of I_{2i−2}+k_i1F to obtain further equations from the first relation.
- The matrix F is a zero matrix with I_{2i−3} in its lower-left block, and v_1 has a 1 in its first position.
- Because σ_1 is an odd-order polynomial in k_i1, the procedure guarantees at least one real choice of k_i1 for the relevant coefficient assignment.
- The basic assignment algorithm recursively maps a desired polynomial for M_i to a gain pair K_i and a preceding polynomial for M_{i−1}.
- Starting from the desired polynomial of M_{n−1}, the algorithm proceeds down to i=2 and then selects K_1 to match the remaining quadratic polynomial.