Certainty Equivalence Adaptive Control Research Articles

Rigid-Body Noncertainty Equivalence Adaptive Control on the Tangent Bundle of SE(3)

This paper addresses an adaptive rigid-body pose-tracking control problem with unknown system parameters. The proposed control is based on a noncertainty equivalence (NCE) adaptive principle applied to the configuration manifold of rigid-body motion, i.e., special Euclidean group SE(3), and its tangent bundle TSE(3). The controller drives the system to follow an arbitrary reference trajectory with bounded time derivatives. The system states are composed of the elements of SE(3), consisting of rotations and translations, and the six-dimensional vector of angular and translational velocities in Euclidean space. Almost-global-asymptotic stability of the system is demonstrated using a Morse–Lyapunov stability analysis. Then, the performance of the proposed controller is verified and is compared to the results obtained by a certainty-equivalence (CE) adaptive control applied to exponential coordinates associated with the linear space of the Lie algebra . It is shown that the TSE(3)-based NCE adaptive control design can eliminate the performance degradation due to the cancellation of uncertain parameter effects that would otherwise appear when the CE adaptive controller is used. As a result, the proposed NCE-based adaptive pose-tracking controller recovers the closed-loop pose-tracking controller performance with the system parameters fully known to the controller.

Adaptive Certainty-Equivalence Control With Regulation-Triggered Finite-Time Least-Squares Identification

For general nonlinear control systems, we present a novel approach to adaptive control, which employs a certainty-equivalence (indirect) control law and an identifier with event-triggered updates of the plant parameter estimates, where the triggers are based on the size of the plant's state, and the updates are conducted using a nonrecursive least-squares estimation over certain finite time intervals, with updates employing delayed measurements of the state. With a suitable nonrestrictive parameter-observability assumption, our adaptive controller guarantees global stability, regulation of the plant state, and our identifier achieves parameter convergence, in finite time, even in the absence of persistent excitation, for all initial conditions other than those where the initial plant state is zero. The robustness of our event-triggered adaptive control scheme to vanishing and nonvanishing disturbances is verified in simulations with the assistance of a dead-zone-like modification of the update law. The major distinctions of our approach from supervisory adaptive schemes are that our approach is indirect and our triggering is related to the control objective (the regulation error). The major distinction from the classical indirect Lyapunov adaptive schemes based on tuning related to the regulation error is that our approach does not involve a complex redesign of the controller to compensate for the detrimental effects of rapid tuning on the transients by incorporating the update law into the control law. Instead, our approach allows for the first time to use a simple certainty-equivalence adaptive controller for general nonlinear systems.

Response improvement of Adaptive Trajectory Control of Rigid Link Robot Arm with Friction Compensation by Nonlinear PI Control Method

This paper proposes a new adaptive trajectory controller in order to realize a good transient performance of trajectory control system for a 2 degree of freedom planer rigid robot arm. This is an improved version of the controller proposed in literature(2) and consists of a new friction compensator and a dynamic certainty equivalent adaptive controller. The former does not use any friction models. It is designed as a nonlinear PI controller which directly compensate friction torque. The latter is designed so as to be robust to load change. Then, adjustable parameters are tuned by a dead zone adaptation law which is robust to the existence of bounded friction compensation error. The effectiveness of the proposed controller is verified by a simple analysis and numerical simulation results.

Certainty Equivalence Adaptive Control of Plants With Unmatched Uncertainty Using State Feedback

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> A systematic design procedure using state-feedback Certainty Equivalence Adaptive Control (CEAC) technique is developed for linear plants and a class of nonlinear plants with unmatched uncertainty. It is shown that a reduced order observer and adaptive laws with normalization in conjunction with the CEAC law result in a stable overall system in the case of linear plants of any relative degree, and a class of nonlinear plants of relative degree two. In the case of higher relative degrees a CEAC approach based on multiple observers is proposed. The proposed schemes guarantee overall system stability and asymptotic tracking. </para>

Approximate Fixed Point Iteration with an Application to Infinite Horizon Markov Decision Processes

Many approximate iterative algorithms can be represented by the form $V_n=TV_{n-1}+U_n$, $n\geq1$, where $V_{n-1},U_n$, $n\geq1$, are elements of a seminormed linear space $(\mathcal{V},\|\cdot\|)$ and $T$ is a contractive operator. The objective is usually to calculate a fixed point $V^*$ of $T$. Here, the quantities $U_n$ may be interpreted as the errors obtained when the operator $T$ can be evaluated only approximately. As is well known, in the absence of such an approximation error, the algorithm converges to a fixed point with a geometric rate under general conditions. In this article the convergence properties of such algorithms in the presence of an approximation error $U_n$ are studied. It is shown that the error $\|V_n-V^*\|$ is dominated by $\|U_n\|$, so that convergence of $\|U_n\|$ to zero implies the convergence of $\|V_n-V^*\|$ to zero, at a rate determined by $\|U_n\|$. The results are naturally extended to a relative error of the form $\|U_n\|/\|V_{n-1}\|$, as well as to $J$-stage contractions. The utility of this general theory is then demonstrated by an extended application to the problem of model-based approximate and adaptive control of Markov decision processes. The theory is shown to permit a sharpening of known convergence rates under more general conditions. Additionally, bounds on regret for adaptive controls with forced exploration are calculated in terms of a stagewise exploration rate. This permits the determination of an optimal choice of exploration rate within the class of certainty-equivalence adaptive control policies.

Convergence analysis of cautious control

In this paper, we present a theoretical analysis on stability and convergence of the cautious control, which has advantages over the traditional certainty equivalence adaptive control, since it takes the parameter estimation error into account in the design, and is also one-step-ahead optimal in the mean square sense under Gaussian assumptions.

Vision-based adaptive tracking control of uncertain robot manipulators

This paper studies the problem of position-tracking adaptive control for planar robotic manipulators through visual servoing under a fixed-camera configuration. The uncertain parameters enter through two separate channels: 1) through the robot dynamics in terms of constant linearly appearing inertia values and 2) nonlinearly appearing camera calibration parameters such as the camera orientation angle and scale factor uncertainties. The associated adaptive control problem is challenged by the presence of the nonlinearly appearing uncertain camera-calibration parameters. We provide a novel solution to this problem by constructing two layers of control structures. To be specific, the inner layer is a conventional certainty-equivalence adaptive controller that stabilizes the faster robot dynamics, whereas the outer layer is a novel controller structure that is designed to handle the nonlinear parameterizations within the slower dynamics of the vision system. We prove global stability and boundedness for all of the closed-loop signals of the cascade interconnection under the relatively standard assumption of Jacobian nonsingularity. If the control objective is set-point regulation, we guarantee global asymptotic convergence for the tracking errors. On the other hand, if the desired trajectory is time-varying, we show that the closed-loop tracking errors under the action of the proposed controller converge to a residual set whose size is determined by the speed of the desired trajectory. Furthermore, we show that the size of this residual error set can be made arbitrarily small by appropriate choice of certain parameters within the controller structure. In contrast with other solutions for this problem that are documented in existing literature, we completely avoid overparameterizations and make no restrictions on the possible range of the camera orientation angle or any additional persistence of excitation requirements. The only piece of prior knowledge required for our solution is the availability of a lower bound on the camera scale factor. The effectiveness of the proposed scheme is illustrated through numerical simulations.

A novel parameter projection mechanism for smooth and stable adaptive control

A novel parameter projection technique for handling uncertain parameters with a priori available upper and lower bounds is the subject of this paper. It is a wellknown fact that the conventional parameter projection method employed in certainty equivalence adaptive control is able to guarantee global asymptotic stability only after compromising control smoothness. In contrast, the technique presented in this paper preserves both stability and smoothness guarantees for the closed-loop adaptive system. The novelty of the proposed adaptive control formulation comes through our deliberate introduction of a nonlinear re-parameterization of uncertainty, thereby amounting to a significant shift from the classical certainty equivalence methodology. The proposed controller is non-overparameterized and has the potential to significantly improve transient performance because of full and proper utilization of all the available prior information on the unknown parameters. The technical descriptions are further elaborated via the Duffing nonlinear oscillator example to highlight the advantages of this new approach.

Adaptive Control—A Departure from the Certainty-Equivalence Paradigm

A fundamental study of indirect adaptive control addressed towards a class of uncertain mechanical and aerospace systems is the subject of this paper. We utilize and build upon certain recent techniques within adaptive control that permit nonlinear parameterization of uncertainty, thereby amounting to a significant shift from the classical certainty equivalence methodology. Instead of the more popular direct adaptive control formulations, our focus is restricted to the indirect methods because imposing prior information on properties of physically meaningful plant parameters is convenient within this framework. For two different types of uncertainty parameterizations, namely scalar parameters and proper orthogonal matrix parameters, we present smooth, non-overparameterized (lower order), and stable adaptive control algorithms that provide improved transient performance while at the same time utilize all the available prior information on the unknown parameters. In the particular case when there exists a single scalar uncertain parameter, the most novel attribute of the proposed controller is that adaptation gets automatically frozen once the parameter estimate reaches the corresponding true value, a highly desirable feature that is clearly missing within all existing certainty equivalence adaptive control schemes. Through both analysis and simulations, we have been able to confirm that this “nice” feature amounts to having a proportional term within the parameter update law; and is in fact enabled by a crucial role played by the introduction of certain design functions that are embedded within the new controller. All technical descriptions will be further elaborated by illustrative physical examples as well as numerical simulations of the resulting closed-loop dynamics to demonstrate the potential advantages of this new approach.

Overcoming the detectability obstacle in certainty equivalence adaptive control

In this paper, we are interested in global adaptive stabilization of nonlinear systems where we know a linearly parametrized stabilizer and a Lyapunov function for the ideal closed-loop system such that the standard estimator is implementable, but this Lyapunov function is not strict, i.e., its derivative is only negative semi-definite. Our motivation to consider this situation stems from the fact that, for many physical systems, the natural Lyapunov function candidate is the total energy, which is always not strict. To complete the stability proof in this case it is necessary to add a—rather restrictive—detectability assumption. Our main contribution is to show that it is possible to overcome this obstacle by adding to the parameter estimator an identification error coming from an indirect identifier. We thus replace the detectability assumption by a new condition that essentially requires that the negative-definite terms appearing in the Lyapunov function derivative dominate, the uncertain terms that cannot be matched by the controller. No matching, or persistency of excitation assumptions are needed.

Adaptive Control for Semilinear Stochastic Systems

An adaptive, ergodic cost stochastic control problem for a partially known, semilinear, stochastic system in an infinite dimensional space is formulated and solved. The solutions of the Hamilton--Jacobi--Bellman equations for the discounted cost and the ergodic cost stochastic control problems require some special interpretations because they do not typically exist in the usual sense. The solutions of the parameter dependent ergodic Hamilton--Jacobi--Bellman equations are obtained from some corresponding discounted cost control problems as the discount rate tends to zero. The solutions of the ergodic Hamilton--Jacobi--Bellman equations are shown to depend continuously on the parameter. A certainty equivalence adaptive control is given that is based on the optimal controls from the solutions of the ergodic Hamilton--Jacobi--Bellman equations and a strongly consistent family of estimates of the unknown parameter. This adaptive control is shown to achieve the optimal ergodic cost for the known system.

On the control of dynamic systems with unknown operating point

For dynamic systems where the position of the operating point is unkown a certainty equivalence adaptive control scheme is proposed which extends and clarifies the wash-out filter practice. Stability properties of the adaptive controller are derived. For slow adaptation, a singular perturbation analysis is carried out to derive additional stability properties and design criteria. Conditions are given under which the proposed scheme achieves exponential stability of the unkown operating point.

Zero-State Performance Improvement and Analysis in a Dynamic Certainty Equivalent Adaptive Controller Using a Dynamic High Gain Feedback Compensator

A new structure of Morse's dynamic certainty equivalent adaptive controllers with a dynamic high gain feedback compensator in the continuous-time, single-input single-output linear time-invariant plant is proposed in this paper. The boundedness of all the signals in the closed-loop system and the convergence of the output error are proved. Furthermore, the mean square tracking error and L∞ bounds on the tracking error are obtained. According to the performance bounds, it is shown that an arbitrary improved zero-state transient performance can be obtained. Finally a numerical example is illustrated in order to show the effectiveness of the proposed method.

Robust Adaptive Control for Linear Systems with Unknown Parameters

In this paper, a robust adaptive control problem for a class of uncertain linear systems on a finite-time interval is formulated as a minimax game problem. For this class of systems, with uncertain control coefficients augmenting the state space, the system under consideration has a bilinear structure. Although the minimax solution allows for a finite-dimensional estimator, the resulting compensator structure remains much more complex than the linear-quadratic problem. For this class of partially observed systems, the minimax problem is reduced to an equivalent full information control problem where the state-space dynamics are composed of an estimator equation and its associated Riccati equation. The equivalence between the minimax adaptive controller and a saddle-point certainty equivalence adaptive controller is shown via Hamilton-Jacobi-Bellman theory where the optimal return function is differentiable. Points of discontinuity in the partial derivatives of the optimal return function are shown to form a manifold of Darboux points, from which multiple global optimal trajectories emanate. Therefore, the Dynamic Programming approach can be extended to be valid over the entire phase space, and the uniqueness of the value of the optimal return function is guaranteed. We finally show that with additional assumptions the finite-time problem can be extended to infinite time. © 1997 Elsevier Science Ltd.

Global Stability/Instability of LS-Based Discrete-Time Adaptive Nonlinear Control

Global stability and instability of a class of LS based discrete-time adaptive nonlinear control systems are investigated in this paper. The systems to be controlled are assumed to be linear in unknown parameters but nonlinear in dynamics which are characterized by a nonlinear function f(x). In the scalar parameter case, it is shown that the certainty equivalence adaptive control is globally stable whenever f(x) has a growth rate |f(x)| = O(||x||b) with 6 < 8 and is not globally stable in general when 6 ≥ 8. Both the results found and the new analytical methods introduced in this paper may be regarded as a basic step for further study.

An energy amplification condition for decentralized adaptive stabilization

The author is interested here in the problem of global decentralized adaptive regulation (of the plant output to zero) of square multivariable linear time-invariant systems without any restrictions on relative degrees or matching assumptions. The first solution to this problem was recently reported by the author using Morse's new dynamic certainty equivalent adaptive controller to prove that global stabilization is possible if the unmodeled interconnections do not induce amplification of the energy of the signals in all channels. In this paper the author shows that to preserve global convergence, it is actually enough to have only one nonamplifying channel. Instrumental for the establishment of the authors' result is the fundamental S-procedure losslessness theorem of Megretsky and Treil, together with some basic loop transformations and D-scalings.

On the Adaptive Control of Jump Parameter Systems via Nonlinear Filtering

In this paper we first present an error analysis for the process of estimates generated by the Wonham filter when it is used for the estimation of the (finite set-valued) jump-Markov parameters of a random parameter linear stochastic system and further give bounds on certain functions of these estimates. We then consider a certainty equivalence adaptive linear-quadratic Gaussian feedback control law using the estimates generated by the nonlinear filter and demonstrate the global existence of solutions to the resulting closed-loop system. A stochastic Lyapunov analysis establishes that the certainty equivalence law stabilizes the Markov jump parameter linear system in the mean square average sense. The conditions for this result are that certain products of (i) the parameter process jump rate and (ii) the solution of the control Riccati equation and its second derivatives should be less than certain given bounds. An example is given where the controlled linear system has state dimension 2. Finally, the stabilizing properties of certainty equivalence laws which depend on (i) the maximum likelihood estimate of the parameter value and (ii) a modified version of this estimate are established under certain conditions.

Directly computable L2 and L∞ performance bounds for morse's dynamic certainty equivalence adaptive controller

AbstractIn this paper we consider a model reference adaptive control scheme where the classical error augmentation and standard tuning error normalization are avoided through the use of Morse's high‐order tuner. We consider the particular scheme of Morse where the concept of dynamic certainty equivalence is used to reduce the error equation to one that involves only first‐order dynamics. With such an error equation, it is first shown that one can directly obtain computable L∞ and L∞ bounds on the tracking error. This is an improvement over some earlier results where either only local L∞ bounds were obtained or the calculation of the global bounds required additional computation. Second, inserting an adaptive gain into Morse's high‐order tuner, we show that fast adaptation improves both the L2 and L∞ bounds on the tracking error, in the sense that the effect of the parametric uncertainty on these bounds is attenuated. Finally, using a simple example, we demonstrate how an earlier attempt to use the adaptive gain to simultaneously attenuate the effect of the parametric uncertainty as well as the initial conditions on the L2 bound for the tracking error has led to an incorrect result.

Using Persistent Excitation with Fixed Energy to Stabilize Adaptive Controllers and Obtain Hard Bounds for the Parameter Estimation Error

Two important instability problems in certainty equivalence adaptive control are solved by external excitation. The first instability is parameter drift along an unstable manifold when the excitation level is not high enough. The second instability is numerical and due to a division with zero in the adaptive law. Global methods based on excitation have been developed to solve this problem, but the energy of the excitation has been tuned on-line. The main contribution of the current paper is in showing that the estimator is stabilized when we apply excitation with fixed and finite energy. The level of excitation should be sufficiently high relative to the magnitudes of the external disturbances and the unmodeled dynamics. The approach can be generalized to more complex adaptive laws. This, together with the fact that we obtain hard bounds for the parameter estimation error, opens up for the possibility of designing robust controllers that are adaptive.

Transient Bounds of Dynamic Certainty Equivalent Adaptive Controllers

In this paper we show that the transient bounds of Morse’s dynamic certainty equivalent adaptive controller [1] established in [2] can be considerably strenghtened. Specifically, we derive a computable bound on the L∞ norm of the tracking error which, in contrast with the local bound obtained in [2], holds for all systems initial conditions. Also, we add an “adaptation gain” in the high order estimator to prove that, by increasing this gain, the L2 norm of the tracking error can be made arbitrarily small without increasing its L∞ norm. This is an improvement over the results obtained with backstep-ping designs [6] where these performance measures must be traded-off in the selection of the adaptation speed.