Relaxing the stability limit of adaptive control systems in the presence of unmodelled dynamics

Dogan, K. Merve; Gruenwald, Benjamin C.; Yucelen, Tansel; Muse, Jonathan A.

doi:10.1080/00207179.2017.1330557

Cited by 30 publications

(23 citation statements)

References 17 publications

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“…This is without loss of generality since one can readily replace λ in with a positive‐definite diagonal matrix (see, for example, the works of Gruenwald et al) by following the system‐theoretical analysis steps given later in this paper. In addition, one can also consider an unknown control effectiveness matrix

normalΛ \in {double-struckR}_{+}^{m \times m} \cap {double-struckD}^{m \times m}

in by replacing the term “ v ( t )” with “Λ v ( t ).” This is also without loss of generality by following the system‐theoretical analysis steps of this paper along with the results, for example, in the work of Dogan et al…”

Section: Problem Formulationmentioning

confidence: 93%

“…In addition,

A \in {double-struckR}^{n \times n}

and

B \in {double-struckR}^{n \times m}

are matrices associated with the modeled dynamics with the pair ( A , B ) being controllable,

W \in {double-struckR}^{n \times m}

is an unknown weight matrix, and

F \in {double-struckR}^{p \times p}

G \in {double-struckR}^{p \times n}

, and

H \in {double-struckR}^{m \times p}

are matrices associated with the unmodeled dynamics. Since the state and output vectors of the unmodeled dynamics are unmeasurable, we consider (as in, for example, the works of Matsutani et al and Dogan et al) that F is Hurwitz for the solvability of the problem, and therefore, there is a unique

S \in {double-struckR}_{+}^{p \times p}

satisfying the Lyapunov equation 0= F T S + SF + I p . Furthermore,

v false(t false) \in {double-struckR}^{m}

in is the output of the actuator dynamics given by

\overset{v}{true} false(t false) = - λ false(v false(t false) - u false(t false) false), v false(0 false) = v_{0},

with

u false(t false) \in {double-struckR}^{m}

being the control input and

λ \in {double-struckR}_{+}

being the actuator bandwidth of all control channels.…”

Section: Problem Formulationmentioning

confidence: 99%

“…In this section, we focus on relaxing the second sufficient stability condition of the expanded MRAC architecture given by Assumption . Building on the results of Section 2.3, the main contribution here is to utilize a robustifying term for this purpose with this term inspired by the results in section 3 in the work of Dogan et al Note that our control procedure here is different than the one the work of Dogan et al in the sense that this paper considers actuator dynamics whereas this was not considered in the aforementioned work . To begin with, consider the feedback control algorithm given by

u false(t false) = - K_{1} x false(t false) + K_{2} c false(t false) - Ŵ^{normalT} false(t false) x false(t false) - μ B_{0}^{normalT} B_{0} P_{normalm}^{normalT} e false(t false) .

In ,

Ŵ false(t false) \in {double-struckR}^{n \times m}

is the estimate of W satisfying a weight update law given by .…”

Section: Problem Formulationmentioning

confidence: 99%

“…In this section, we provide the stability analyses of the adaptive control architectures given in Section 2. In order to have an additional level of flexibility in our following system‐theoretical analyses, we first introduce the state transformation z ( t )= βq ( t ) with

β \in {double-struckR}_{+}

. In this case, and can be equivalently written as

\overset{x}{true} false(t false) = A x false(t false) + B false[v false(t false) + W^{normalT} x false(t false) false] + β^{- 1} B H z false(t false), x false(0 false) = x_{0},

\overset{z}{true} false(t false) = F z false(t false) + β G x false(t false), z false(0 false) = β q_{0} .

…”

Section: Stability Analysesmentioning

confidence: 99%

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Adaptive architectures for control of uncertain dynamical systems with actuator and unmodeled dynamics

Dogan

Gruenwald

Yucelen

et al. 2019

Intl J Robust & Nonlinear

Self Cite

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Summary In adaptive control of uncertain dynamical systems, it is well known that the presence of actuator and/or unmodeled dynamics in feedback loops can yield to unstable closed‐loop system trajectories. Motivated by this standpoint, this paper focuses on the analysis and synthesis of multiple adaptive architectures for control of uncertain dynamical systems with both actuator and unmodeled dynamics. Specifically, we first analyze model reference adaptive control architectures with standard, hedging‐based, and expanded reference models for this class of uncertain dynamical systems and develop sufficient stability conditions. We then synthesize a robustifying term for the latter architecture and analytically show that this term can allow for a relaxed sufficient stability condition. The proposed theoretical treatments involve Lyapunov stability theory, linear matrix inequalities, and matrix mathematics. Finally, we compare the resulting sufficient stability conditions of the considered adaptive control architectures on a benchmark mechanical system subject to actuator and unmodeled dynamics.

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normalΛ \in {double-struckR}_{+}^{m \times m} \cap {double-struckD}^{m \times m}

Section: Problem Formulationmentioning

confidence: 93%

“…In addition,

A \in {double-struckR}^{n \times n}

and

B \in {double-struckR}^{n \times m}

are matrices associated with the modeled dynamics with the pair ( A , B ) being controllable,

W \in {double-struckR}^{n \times m}

is an unknown weight matrix, and

F \in {double-struckR}^{p \times p}

G \in {double-struckR}^{p \times n}

, and

H \in {double-struckR}^{m \times p}

S \in {double-struckR}_{+}^{p \times p}

satisfying the Lyapunov equation 0= F T S + SF + I p . Furthermore,

v false(t false) \in {double-struckR}^{m}

in is the output of the actuator dynamics given by

\overset{v}{true} false(t false) = - λ false(v false(t false) - u false(t false) false), v false(0 false) = v_{0},

with

u false(t false) \in {double-struckR}^{m}

being the control input and

λ \in {double-struckR}_{+}

being the actuator bandwidth of all control channels.…”

Section: Problem Formulationmentioning

confidence: 99%

u false(t false) = - K_{1} x false(t false) + K_{2} c false(t false) - Ŵ^{normalT} false(t false) x false(t false) - μ B_{0}^{normalT} B_{0} P_{normalm}^{normalT} e false(t false) .

In ,

Ŵ false(t false) \in {double-struckR}^{n \times m}

is the estimate of W satisfying a weight update law given by .…”

Section: Problem Formulationmentioning

confidence: 99%

β \in {double-struckR}_{+}

. In this case, and can be equivalently written as

\overset{x}{true} false(t false) = A x false(t false) + B false[v false(t false) + W^{normalT} x false(t false) false] + β^{- 1} B H z false(t false), x false(0 false) = x_{0},

\overset{z}{true} false(t false) = F z false(t false) + β G x false(t false), z false(0 false) = β q_{0} .

…”

Section: Stability Analysesmentioning

confidence: 99%

See 2 more Smart Citations

Adaptive architectures for control of uncertain dynamical systems with actuator and unmodeled dynamics

Dogan

Gruenwald

Yucelen

et al. 2019

Intl J Robust & Nonlinear

Self Cite

View full text Add to dashboard Cite

show abstract

“…By introducing a dynamical signal and using dynamic surface control approach, an adaptive neural network control problem was studied for strict‐feedback nonlinear systems with full‐state constraints and unmodeled dynamics in the work of Zhang et al For compensating actuator failures problem in nonlinear systems with time‐varying delay and unmodeled dynamics, Yin et al proposed an adaptive neural network fault‐tolerant control method. A fuzzy adaptive control issue was solved by employing small‐gain scheme for nonlinear stochastic switched systems with unmodeled dynamics and arbitrary switchings in the work of Li et al When an unmodeled dynamics satisfies a set of conditions, the proposed approach in the work of Dogan et al allows that the closed‐loop dynamical system remains stable in the presence of large system uncertainties.…”

Section: Introductionmentioning

confidence: 99%

Nussbaum gain adaptive backstepping control of nonlinear strict‐feedback systems with unmodeled dynamics and unknown dead zone

Liang

et al. 2018

Intl J Robust & Nonlinear

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Summary This paper investigates an adaptive state feedback control problem for a class of single‐input and single‐output nonlinear systems with strict‐feedback form. A novel adaptive control strategy is developed for nonlinear systems with unmodeled dynamics, unknown control directions, dynamic disturbances and unknown dead zone. To stabilize this category of systems, a dynamic signal is introduced to dominate the unmodeled dynamics. Nussbaum gain technique is used to overcome the unknown control directions problem. Nonlinear damping terms are framed to counteract the dynamic disturbances, and the assistant signal is constructed to compensate the influence of symmetric dead zone. The adaptive control approach ensures all signals in the closed‐loop system to be bounded. Finally, a numerical example is used to verify the effectiveness of the theoretical results.

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Model Reference Adaptive Control

Yucelen

2019

Wiley Encyclopedia of Electrical and Electronics Engineering

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Adaptive control methods present effective system‐theoretical tools in order to achieve closed‐loop system stability and performance in the presence of exogenous disturbances and system uncertainties, where they are generally classified as either direct or indirect. A well‐known class of direct adaptive control methods is model reference adaptive control architectures. In particular, these architectures employ two major components – a reference model and a parameter adjustment mechanism. A desired closed‐loop dynamical system response is captured by the reference model for which its response is compared with the response of the uncertain dynamical system. The system error signal resulting from this comparison drives the parameter adjustment mechanism. This mechanism then adjusts the controller parameters in a real‐time (i.e., online) manner for driving the trajectories of the uncertain dynamical system to the trajectories of the reference model. This article discusses these components in a basic state feedback setting in order to provide an introduction to the model reference adaptive control design procedure. The article also makes connections to several other, relatively advanced model reference adaptive control methods for interested readers.

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Relaxing the stability limit of adaptive control systems in the presence of unmodelled dynamics

Cited by 30 publications

References 17 publications

Adaptive architectures for control of uncertain dynamical systems with actuator and unmodeled dynamics

Adaptive architectures for control of uncertain dynamical systems with actuator and unmodeled dynamics

Nussbaum gain adaptive backstepping control of nonlinear strict‐feedback systems with unmodeled dynamics and unknown dead zone

Model Reference Adaptive Control

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