Relaxing the high-frequency gain sign assumption in direct model reference adaptive control☆
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 kp in the sequel—appears multiplying the controller parameters. This difficulty can be overcome assuming the knowledge of the sign(kp), 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 kp 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 kp 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 kp 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 kp. 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 kp, 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 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, , and 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, kp) ≔ col(kpθ, kp) using a standard gradient estimator4 with normalized adaptation gains positive-definite and 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 with unknown parameters and 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
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,
References (20)
- et al.
On self–tuning regulators
Automatica
(1973) Overcoming the obstacle of high-relative degree
Eur. J. Control
(1996)Some remarks on a conjecture in parameter adaptive control
Syst. Control Lett.
(1983)- et al.
Immersion and invariance adaptive control of linear multivariable systems
Syst. Control Lett.
(2003) Multivariable adaptive control: a survey
Automatica
(2014)- et al.
Performance enhancement of parameter estimators via dynamic regressor extension and mixing
IEEE Trans. Autom. Control
(2017) - et al.
On the need of projections in input-error model reference adaptive control
Int. J. Adapt. Control Signal Process.
(2018) - et al.
Adaptive tracking of unknown multi-sinusoidal signal in linear systems with arbitrary input delays and unknown sign of high frequency gain
IFAC-PapersOnLine
(2017) - et al.
Robust Adaptive Control
(1996) Design of a self optimizing control system
Trans. ASME
(1958)
Cited by (33)
Anti-saturation fault-tolerant control for Markov jump nonlinear systems with unknown control coefficients and unmodeled dynamics
2023, Nonlinear Analysis: Hybrid SystemsSignal-parametric discrete-time adaptive controller for pneumatically actuated Stewart platform
2023, Control Engineering PracticeExponentially Stable Adaptive Control of MIMO Systems with Unknown Control Matrix
2023, IFAC-PapersOnLineSelf-tuning methodology for adaptive controllers based on genetic algorithms applied for grid-tied power converters
2023, Control Engineering PracticeAugmented error based adaptive control with improved parametric convergence
2022, IFAC-PapersOnLineImprovement of transient performance in MRAC by memory regressor extension
2021, European Journal of Control
- ☆
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).