Relaxing the high-frequency gain sign assumption in direct model reference adaptive control

doi:10.1016/j.ejcon.2018.06.002

European Journal of Control

Volume 43, September 2018, Pages 12-19

https://doi.org/10.1016/j.ejcon.2018.06.002 Get rights and content

Abstract

A new, high performance, solution to the classical problem of direct model reference adaptive control for linear time-invariant systems with unknown sign of the high frequency gain is reported in the paper. The proposed algorithm directly estimates this parameter with the only required prior knowledge of a lower bound on its absolute value. To avoid the possible appearance of singularities in the controller calculation a switched projection mechanism is introduced to change, if needed, the sign of the estimate. The recently introduced dynamic regressor extension and mixing estimator is used to ensure monotonicity of the estimation error of the high frequency gain, guaranteeing that the switching appears (at most) once and avoiding the possible appearance of chattering—that may happen in classical gradient-based algorithms. Comparative simulations with the Nussbaum gain-based and gradient estimators illustrate the dramatic performance improvement of the proposed controller.

Introduction

Model reference adaptive control (MRAC) is unquestionably the most widely studied problem in the adaptive literature that has a very long history going back to the 1950’s and extending to the present time. The first attempts to solve the MRAC problem followed the classical path of designing an observer, that had to be made adaptive because of the unknown plant parameters, and then feeding back the observed state (see [6]). Very little success was, however, obtained pursuing these lines—essentially because of the difficulty of simultaneously estimating state and parameters. A major breakthrough, essentially due to [2], [8], was the introduction of the so–called direct control parameterization (see Lemma 1 below), which revealed that the estimation of the plant state could be obviated and only a “good” estimation of the controller parameters was needed to achieve the asymptotic reference model output tracking objective. The intrinsic simplicity of this parametrization motivated the overwhelming majority of the researchers to pursue this line of reasoning and concentrated their efforts into the development of suitable parameter estimators. The interested reader is referred to [12] for a vivid description of the history of MRAC as well as to the existing textbooks [5], [14], [19] for further information on it.

As it is well-known, the direct control parameterization, referred to as output-error parameterization¹ in [19], leads to a bilinear regression form, where the parameter that corresponds to the high frequency gain—denoted k_p in the sequel—appears multiplying the controller parameters. This difficulty can be overcome assuming the knowledge of the sign(k_p), under which a globally convergent output-error MRAC may be designed introducing an overparameterisation of the regressor and a normalisation in, now classical, augmented error-based estimators [5], [14], [19]. It was shown that these algorithms enjoy the fundamental “self-tuning property”, that is, that global tracking is ensured for all reference signals—without imposing the stronger parameter convergence requirement. The use of normalisation and overparameterisation, however, comes with a very high tag for the overall performance of the scheme. Indeed, as thoroughly discussed in [10], [16], [19], overparameterisation hampers parameter convergence while normalisation “slows down” the adaptation and severely penalizes the parameter convergence rate. As shown in [16], this below par performance can be partially overcome using (the unnormalised) Morse’s high order tuners [11], but the additional information of an upper bound on k_p is required and the scheme is significantly more involved.

A major theoretical breakthrough for this problem is due to Nussbaum [15] who, motivated by a conjecture in [9], showed that the sign of k_p is not necessary for stabilization in MRAC. Nussbaum’s solution relies on the introduction of a function that changes periodically the sign of the estimator vector field in a “gain scheduling-like” fashion. It is clear that this kind of algorithms is only of theoretical interest since their transient performance is intrinsically bad and practically inadmissible—as it has been repeatedly reported in the literature.

Schemes that require the division by the estimate of k_p in the controller calculation, e.g., the one proposed in Section 4.5.2 of [5] and the other one presented in the paper [4], most incorporate a switched projection to avoid singularities. There are two drawbacks to this approach, on one hand, to the best of the authors’ knowledge, no proof of global tracking for this scheme has been reported in the literature without an unverifiable assumption of persistency of excitation (PE) of the regressor. On the other hand, there is no guarantee that the switching happens only a finite number of times nor the possible appearance of chattering phenomena.

In this paper, a new solution to the problem with improved transient performance is reported, which includes the following modifications:

(M1)
Abandon the bilinear model mentioned above, and adopt instead the overparameterized linear regression.
(M2)
Introduce a new factorization of the parameter estimates to update directly the controller parameters.
(M3)
Instead of classical gradient estimators we use the recently introduced dynamic regressor extension and mixing (DREM) estimator from [1].

The use of a linear parameterization is essential to apply the DREM estimator. Unfortunately, the estimation law still involves the division by an estimate of k_p. Therefore, similarly to the classical schemes, a switched projection of this estimate is added to keep it away from an a priori known band around the zero value. To avoid the undesirable chattering phenomena indicated above we exploit a key feature of DREM: that it ensures monotonicity of the estimation error of the parameter k_p, ensuring that the switching appears (at most) once. The monotonicity property holds for all reference signal. However, global tracking can only be ensured for reference signals that satisfy an excitation requirement, which holds true if the aforementioned PE assumption on the regressor of classical schemes is satisfied.

The remainder of the paper is organized as follows. Section 2 formulates the MRAC problem addressed in the paper and briefly reviews the current literature available on this topic. An MRAC, with a gradient-based procedure to estimate the controller parameters using the new factorization mentioned above, is given in Section 3. Section 4 contains our main result, namely, the description of the DREM estimator and its stability properties when applied in a MRAC scheme. Comparative simulations with the classical Nussbaum gain-based and gradient estimators, which illustrate the significant performance improvement of the proposed controller, are presented in Section 5. The paper is wrapped-up with concluding remarks in Section 6.

Section snippets

Problem formulation

We are interested in the classical problem of relaxing the knowledge of the high frequency gain in MRAC of the scalar linear time-invariant (LTI) continuous-time plant $D (p) y = k_{p} N (p) u,$ where y, u are the plant output and input, respectively, D(p) and N(p) are monic and coprime polynomials of degree n and m, respectively, $p : = \frac{d}{d t}$ , $ρ : = n - m \geq 1$ and $k_{p} \in R$ is the high frequency gain. The parameters of D(p) and N(p) are unknown.

We make the following assumptions regarding the plant.

(A.1)
N(p) is a Hurwitz

Gradient-based MRAC

As indicated in the previous section in Section 4.5.2 of [5] it is proposed to use the linear regression model (7) to estimate directly the parameters col(μ, k_p) ≔ col(k_pθ, k_p) using a standard gradient estimator⁴ $\begin{matrix} \dot{\hat{μ}} & = & - Γ_{θ} ϕ_{f} (e + ϕ_{f}^{⊤} \hat{μ} - u_{f} {\hat{k}}_{p}), \\ {\dot{\hat{k}}}_{p} & = & γ_{p} u_{f} (e + ϕ_{f}^{⊤} \hat{μ} - u_{f} {\hat{k}}_{p}) \end{matrix}$ with normalized adaptation gains $Γ_{θ} = Γ_{θ 0} / (1 + ψ^{⊤} ψ), Γ_{θ 0} \in R^{2 n \times 2 n}$ positive-definite and $γ_{p} = γ_{p 0} / (1 + ψ^{⊤} ψ), γ_{p 0} \in R_{+},$ $ψ = col (- ϕ_{f}, u_{f}),$ and compute the controller parameters via

DREM-based MRAC

As discussed in [1] DREM has several advantages over gradient (or least-squares) estimators, including a provable transient performance improvement. The feature of DREM that we exploit in this paper is that it guarantees monotonicity of each element of the parameter error. This is a much stronger property than monotonicity of the Euclidean norm of the parameter error vector ensured by standard estimators [5], [19]. However, this monotonicity property does not preclude the possible appearance of

Simulations

In this section simulation results of three MRAC schemes for the second order plant $(p^{2} + d_{1} p + d_{0}) y = k_{p} u$ with unknown parameters $d_{1} = 2,$ $d_{0} = 1$ and $k_{p} = 3$ are demonstrated and compared. The first scheme is based on the use of a Nussbaum gain, cf. (see Chapter 9 of [14]). The second and the third schemes use the gradient and DREM identifiers described in Sections 3 and 4, respectively.

The objective of MRAC is to ensure boundedness of all signals and asymptotic convergence to zero of the tracking error $e = y - k$

Concluding remarks and future research

An alternative solution to the problem of MRAC of LTI, minimum phase systems with unknown sign of the high frequency gain has been presented. The proposed scheme replaces the latter knowledge by the availability of a lower bound on the absolute value of this gain, required to implement a switched projection of the estimate. The main novelty is the use of a DREM estimator that guarantees, via the monotonicity of the parameter estimation error, that the switching happens at most once. Moreover,

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^☆: This article is supported by Government of Russian Federation (grant 08-08), the Ministry of Education and Science of Russian Federation (project 14.Z50.31.0031) and the Russian Science Foundation (grant 17-19-01422).

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Relaxing the high-frequency gain sign assumption in direct model reference adaptive control☆

Abstract

Introduction

Section snippets

Problem formulation

Gradient-based MRAC

DREM-based MRAC

Simulations

Concluding remarks and future research

Automatica

Eur. J. Control

Syst. Control Lett.

Syst. Control Lett.

Automatica

Performance enhancement of parameter estimators via dynamic regressor extension and mixing

IEEE Trans. Autom. Control

On the need of projections in input-error model reference adaptive control

Int. J. Adapt. Control Signal Process.

Adaptive tracking of unknown multi-sinusoidal signal in linear systems with arbitrary input delays and unknown sign of high frequency gain

IFAC-PapersOnLine

Robust Adaptive Control

Design of a self optimizing control system

Trans. ASME