Certainty Equivalence Principle Research Articles

On Line Estimation and Multivariable Predictive Control of an Activated Sludge Process

An approach for estimating and controlling biological variables in a continuous flow biological wastewater treatment process is presented and analyzed. The goal in the control is to force the residual substrate and the dissolved oxygen concentrations to track a given reference model despite the disturbances and system parameter uncertainties. The methodology studied is based on direct exploitation of the non linear model representing the wastewater treatment process and is coupled with a joint observer estimator for on line tracking of unavailable states and time varying parameters. The estimated variables are used in the explicit design of the control algorithm according to certainty equivalence principle. A simulation study subject to measurement noise and abrupt jumps in the kinetic parameters shows the feasibility and robustness Of the Control Strategy.

On nonlinear H/sub ∞/ control under sampled measurements

This paper deals with the nonlinear H/sub /spl infin// control problem with sampled measurement feedback. This problem has already been studied in Suzuki et al. (1995), where, using a certainty equivalence principle, a control solution involving a state estimator with a linear injection gain is proposed. Using the same general framework, we propose a more refined estimator with a nonlinear injection gain. This gain is shown to be connected to a periodic solution of a Hamilton-Jacobi inequality with jumps.

Continuous-time indirect adaptive control of the electrohydraulic servo systems

A continuous-time robust indirect adaptive control (IAC) algorithm with a self-excitation capability is proposed for position control of an electrohydraulic servo system subject to parametric uncertainties and load disturbances. In this algorithm, a gradient least squares dead zone estimation is used to identify the plant parameters and then a linear pole-placement controller is designed using the estimate. By a coprimeness verification procedure, the proposed algorithm facilitates the establishment of the adaptive pole-placement control of the closed-loop system by using an additional nonlinear feedback signal implemented for supplying the system with sufficiently rich signals. An analysis shows this algorithm can guarantee parameter estimation convergence and system stability based on the certainty equivalence principle. The performance of the proposed algorithm is evaluated through both the simulation results and the experimental studies. Simulations and experiments are conducted to see how well the proposed algorithm compares with two existing control schemes in controlling the same process. The results show that the IAC scheme confirms the analysis and has considerable robustness subject to parametric uncertainties and load disturbances and has better performance than the other two controllers.

Robust implicit self-tuning regulator: Convergence and stability

A family of implicit self-tuning regulators (STR) is presented, based on Lyapunov analysis techniques for the control of a class of multi-input and multi-output (MIMO) dynamical systems. Linearity in the parameters is assumed to hold, but the ‘estimation error’ is considered to be nonzero; this allows control of a larger class of systems and also has the effect of producing a robust controller. Moreover, the certainty equivalence (CE) principle is not used in the STR design, overcoming a major problem in adaptive control. The structure of the STR is naturally derived using a tracking error/passivity approach. The role of persistency of excitation (PE) is explored, and is used to show the boundedness of parameter estimates when gradient-based parameter tuning of STR is performed in nonideal conditions. New on-line tuning algorithms for implicit STR are derived that are similar to the ε-modification approach for the case of continuous-time systems and that include a modification to the adaptation gain and a correction term to the standard gradient-based tuning. These improved parameter tuning algorithms guarantee tracking as well as bounded parameter estimates in nonideal situations, so that PE is not needed. The notions of a passive STR, a dissipative STR and a robust STR are introduced. Finally, this paper provides a comprehensive theory in the development of identification, prediction and adaptive control schemes for discrete-time systems based on Lyapunov analysis.

Min‐max certainty equivalence principle and differential games

This paper presents a version of the certainty equivalence principle, usable for nonlinear, variable end-time, partial observation zero-sum differential games, which states that under the unicity of the solution to the auxiliary problem, optimal controllers can be derived from the solution of the related perfect observation game. An example is provided where in one region, the new extended result holds, giving an optimal control, and in another region, the unicity condition if is not met, leading indeed to a non-certainty equivalent optimal controller.

Design of adaptive LQ regulators for MIMO systems based on multirate sampling of the plant output

In this paper, the certainty equivalence principle is used to combine a discrete-time adaptive law with a control structure derived from the linear-quadratic regulation problem. The proposed control structure is mainly based on multirate sampling of the output of the continuous-time plant under control and allows us to regulate the sampled closed-loop system, subject to a quadratic performance criterion, without making assumptions on the plant other than controllability and observability and the knowledge of two sets of structural indices. Using the proposed algorithm, the adaptive regulation problem is reduced to the determination of a fictitious static state feedback controller. Known techniques usually resort to the computation of fullorder adaptive state observers, thus introducing high-order exogenous dynamics in the control loop. Moreover, persistence of excitation and parameter convergence of the plant are provided without making any additional assumption on the system, as compared to known adaptive linear-quadratic regulation schemes.

Min-max certainty equivalence principle and differential games

This paper presents a version of the certainty equivalence principle, usable for nonlinear, variable end-time, partial observation zero-sum differential games, which states that under the unicity of the solution to the auxiliary problem, optimal controllers can be derived from the solution of the related perfect observation game. An example is provided where in one region, the new extended result holds, giving an optimal control and in another region, the unicity condition is not met, leading indeed to a non-certainty equivalent optimal controller.

Closed-loop identification for the best asymptotic performance of adaptive robust control

Discrete linear systems subjected to both additive and parametric deterministic disturbances are considered. The certainty equivalence principle is used for the design of adaptive control. Two different estimation algorithms are proposed to meet different control objectives in the regulation problem with the worst-case limsup performance index.

Singular nonlinearH∞ optimal control problem

The theory of nonlinear H∞ of optimal control for affine nonlinear systems is extended to the more general context of singular H∞ optimal control of nonlinear systems using ideas from the linear H∞ theory. Our approach yields under certain assumptions a necessary and sufficient condition for solvability of the state feedback singular H∞ control problem. The resulting state feedback is then used to construct a dynamic compensator solving the nonlinear output feedback H∞ control problem by applying the certainty equivalence principle.

Partially Observed Differential Games, Infinite-Dimensional Hamilton–Jacobi–Isaacs Equations, and Nonlinear $H_\infty $ Control

This paper presents new results for partially observed nonlinear differential games. Using the concept of information state, we solve this problem in terms of an infinite-dimensional partial differential equation, which turns out to be the Hamilton--Jacobi--Isaacs (HJI) equation for partially observed differential games. We give definitions of smooth and viscosity solutions and prove that the value function is a viscosity solution of the HJI equation. We prove a verification theorem, which implies that the optimal controls are separated in that they depend on the observations through the information state. This constitutes a separation principle for partially observed differential games. We also present some new results concerning the certainty equivalence principle under certain standard assumptions. Our results are applied to a nonlinear output feedback $H_\infty$ robust control problem.

Modular approach to adaptive nonlinear stabilization

A modular approach to adaptive control of nonlinear systems introduced previously is presented in a general framework. This approach abandons the certainty equivalence principle ubiquitous in the estimation-based approach. The adaptive stabilization problem is studied in the setting of control Lyapunov functions, and is reduced to the problem of input-to-state stabilization with respect to the parameter estimation error considered as the input.

On the application of minimum principle for solving partially observable risk-sensitive control problems

This paper is concerned with the application of a minimum principle derived for general nonlinear partially observable exponential-of-integral control problems, to solve linear-exponential-quadratic-Gaussian problems. This minimum principle is the stochastic analog of Pontryagin's minimum principle for deterministic systems. It consists of an information state equation, an adjoint process governed by a stochastic partial differential equation with terminal condition, and a Hamiltonian functional. Two methods are employed to obtain the optimal control law. The first method appeals to the well-known approach of completing the squares, by first determining the optimal control law that minimizes the Hamiltonian functional. The second method provides significant insight into relations with the HamiltoniJacobi approach associated with completely observable exponential-of-integral control problems. These methods of solution are particularly attractive because they do not assume a certainty equivalence principle, hence they can be used to solve nonlinear problems as well.

ARMA lattice identification: a new hereditary algorithm

This paper derives an identification solution of the ARMA-type optimal linear predictor as a time varying-lattice of arbitrarily fixed dimension for a process whose output signal only is known. The projection technique introduced here leads to a hereditary algorithm that is the adaptive extension to raw data of the authors' previous results on lattice realization from given autocorrelation functions. It produces a minimum-phase linear model of the signal whose nth order whiteness of the associated innovation has the following restricted meaning: orthogonality to an n-dimensional subspace memory of the past in a suitable Hilbert sequence space. The L/sup 2/ metric of that sequence space leads to a least-squares identification algorithm that possesses a certainty equivalence principle with respect to the corresponding realization algorithm (i.e., sample correlation products replace true correlation terms). Due to the detailed state-space time-varying computations, this is possible here while avoiding the well-known side errors from missing correlation products that usually occur in a blunt replacement of the output autocorrelation by averaged sample products. Application examples show the superiority of the hereditary algorithm over classical recursive and nonrecursive algorithms in terms of accuracy, adaptivity, and order reduction capabilities.

Adaptive decoupling control of linear systems without a persistent excitation requirement

In the present paper, the certainty equivalence principle is used to combine a discrete-time adaptive law with a control structure derived from the decoupling control problem. The proposed control structure is mainly based on a special multirate sampling mechanism of the ouputs of the continuous-time plant under control. Such a control strategy allows us to regulate the outputs of the MIMO sampled closed-loop system independently and does not make assumptions on the plant other than controllability and observability and the knowledge of two sets of structural indices, namely the controllability and the observability indices. On the basis of the proposed indirect adaptive algorithm, the adaptive decoupling control problem is reduced to the determination of a fictitious static state feedback controller, due to the merits of the proposed multirate sampling mechanism. Known techniques usually resort to the direct computation of dynamic controllers. The controller determination reduces to the simple problem of solving a linear algebraic system of equations, whereas in known techniques matrix polynomial Diophantine equations usually need to be solved. Moreover, in the present technique, persistent excitation and, therefore, parameter convergence, of the continuous-time plant is provided without making any special assumption either on the richness of the reference signals or on the existence of specific convex sets in which the estimated parameters belong, or finally on the coprimeness of the polynomials describing the ARMA models, as compared to known adaptive decoupling control schemes.

CERTAINTY EQUIVALENCE AND LOGARITHMIC UTILITIES IN CONSUMPTION/INVESTMENT PROBLEMS

We investigate an optimal consumption/investment decision problem with partially observable drift. Logarithmic utilities are shown to be necessary and sufficient for the certainty equivalence principle to hold. For the sufficiency part of the proof, we allow a general stochastic structure about the unobservable drift. On the other hand, a simple Bayesian structure is assumed for the necessity part in order to utilize the Hamilton‐Jacobi‐Bellman equations.

Self-Optimality of Adaptive Control Systems Based on the Certainty Equivalence Principle

In this paper, we consider a general adaptive control scheme based on the recursive least squares identification technique. The control is performed by applying any control law in a certainty equivalent fashion. First, a stability result is proven, which does not require the systea to be minimum-phase. Then, it is shown that on the grounds of the stability property the control system attains the asymptotic optimality. These results constitute a fundamental step forward in the comprehension of general adaptive control schemes of possibly nonminimum phase plants.

Adaptive H∞ Control Using Coprime Factors and Set-Membership Identification: The Nonlinear Case

The aim of this paper is twofold. We are first interested in developing tools for control and robustness analysis in a nonlinear context : an incremental approach is especially proposed for the extension of the H∞ control concept to a nonlinear context. The second aim of this paper is to combine H∞ control, robustness analysis and set-membership estimation theories for the adaptive control of a nonlinear plant. As the robust estimator provides a measure of the uncertainties in the estimated model, one can bypass the certainty equivalence principle and adjust on line the achievable performance of the control law with the model uncertainties. The finite-time convergence of the estimator enables to ensure the stability and robustness of the adaptive scheme. The methodology is applied to the synthesis of an adaptive H∞ missile autopilot.

A New Type of HJBI Inequality Governing Situations where Certainty Equivalence Fails

This paper deals with the generalization ofHcontrol methods to non-linear systems. The classical certainty equivalence principle says that the measurement feedback can be obtained by applying the state feedback law to a certain ”worst case” estimate of the state. However, all the results to date relied heavily on the assumption that the ”worst case” state is unique. In this paper we study in detail the special case when there are precisely two ”worst case” estimates of the state. Our analysis gives not only sufficient but also necessary conditions for the special case of two ”worst case” estimates of the state. The key element of our construction is a new ”two-point” Hamilton- Jacobi-Isaac-Belman equation which deals with two states at the same time. The results announced here are part of a longer paper which includes more details on smoothness assumption and how to avoid them.

Sensitivity Minimisation for the Class of Self-Tuning Control Systems

A new method for designing self-tuning control systems is proposed. The ultimate goal of the presented approach is to construct a self-tuner capable of coping with situations when the process under control cannot be modelled accurately within the structure provided to the least-squares identification method. Since in such circumstances the certainty-equivalence principle must be applied with extreme caution, an extra care is taken to preserve quality of control and robustness of stability. These are achieved by using control synthesis algorithm that minimises certain, so called, sensitivity functions.

A discrete-time min-max certainty equivalence principle

We extend a result of (Bernhard, submitted) to a setup comparable to that provided in (Bernhard and Rapaport, submitted) for the continuous-time case, thus obtaining the most comprehensive certainty equivalence principle for the discrete-time problem to date.