Real Uncertain Parameters Research Articles

Combined design of robust controller and disturbance observer in a fixed-order [formula omitted] control framework

The disturbance observer (DOB) and H∞ based robust control methods are widely used in feedback control applications due to their disturbance rejection capability and robustness improvement. However, the simultaneous synthesizing of DOB and H∞ controller considering the real uncertain parameters and dynamic uncertainties of the models is an open problem. This paper presents a new fixed-order μ-synthesis algorithm based optimization methodology for the combined design of DOB and robust controller in the presence of mixed uncertainty. The fixed-order robust H∞ control framework is adopted to optimize structured plant inverse, so-called Q filter and feedback controller. In order to guarantee the robust stability of the main feedback loop and inner DOB loop, the overall closed-loop system is considered with uncertainties during the design procedure. The proposed approach renders the need for an explicit plant inverse unnecessary. Thanks to this advantage and generality of μ-synthesis techniques, the proposed method can be directly applied to both the non-minimum phase and multiple-input and multiple-output (MIMO) uncertain systems without any approximation. The theoretical design approach is experimentally verified on an electromechanical control actuation system of an air vehicle and numerically on a non-minimum phase system.

ROBUST MULTIPLE MODEL ADAPTIVE CONTROL WITH FUZZY POSTERIOR PROBABILITY COMBINATION

This paper proposes a fuzzy posterior probability (FPP) combina-tion in a robust multiple model adaptive control (MMAC) frame-work, for linear systems subject to uncertain real parameters, eitherconstant or slowly time varying.

Robust model predictive control of uncertain fractional systems: a thermal application

In this study, a robust model predictive control of fractional-order systems with real uncertain parameters is presented. The Grunwald–Letnikov's method is used as an internal model to predict the plant future dynamic behaviour. This method consists in replacing the non-integer derivation operator of the adopted system representation by a discrete approximation. Therefore a robust fractional model predictive control is developed based on an uncertain fractional model. Moreover, the output deviation approach is adopted to design the j-step ahead output predictor, and the corresponding control law is obtained by the resolution of a min–max optimisation problem that takes into account the uncertainties of the fractional-order model parameters and under input constraints. The efficiency and the performances of the proposed predictive controller are illustrated with practical results of a thermal system.

LFT Modeling and Robust Stability Analysis of Missiles with Uncertain Parameters

The structured singular value (μ) analysis based method has many advantages for the robust stability analysis of missiles with uncertain parameters. Nevertheless, the present linear fractional transformation (LFT) modeling process, which is the basis of μ analysis, is complex, and not suitable for automatic implementation; on the other hand, μ analysis requires a large amount of computation, which is a burden for large-scale application. A constructive procedure, which is computationally more efficient, and which may lead to a lower order realization than existing algorithms, is proposed for LFT modeling. To reduce the calculation burden, an analysis method is developed, based on skew μ. On this basis, calculation of the supremum of μ over a fixed frequency range converts into a single skew μ value calculation. Two algorithms are given, to calculate the upper and lower bounds of skew μ, respectively. The validity of the proposed method is verified through robust stability analysis of a missile with real uncertain parameters.

Robust multiple model adaptive control: Modified using ν‐gap metric

AbstractA new robust adaptive control method is proposed, which removes the deficiencies of the classic robust multiple model adaptive control (RMMAC) using benefits of the ν‐gap metric. First, the classic RMMAC design procedure cannot be used for systematic design for unstable plants because it uses the Baram Proximity Measure, which cannot be calculated for open‐loop unstable plants. Next, the %FNARC method which is used as a systematic approach for subdividing the uncertainty set makes the RMMAC structure being always companion with the µ‐synthesis design method. Then in case of two or more uncertain parameters, the model set definition in the classic RMMAC is based on cumbersome ad hoc methods. Several methods based on ν‐gap metric for working out the mentioned problems are presented in this paper. To demonstrate the benefits of the proposed RMMAC method, two benchmark problems subject to unmodeled dynamics, stochastic disturbance input and sensor noise are considered as case studies. The first case‐study is a non‐minimum‐phase (NMP) system, which has an uncertain NMP zero; the second case‐study is a mass‐spring‐dashpot system that has three uncertain real parameters. In the first case‐study, five robust controller design methods (H2, H∞, QFT, H∞ loop‐shaping and µ‐synthesis) are implemented and it is shown via extensive simulations that RMMAC/ν/QFT method improves disturbance‐rejection, when compared with the classic RMMAC. In the second case‐study, two robust controller design methods (QFT and mixed µ‐synthesis) are applied and it is shown that the RMMAC/ν/QFT method improves disturbance‐rejection, when compared with RMMAC/ν/mixed−µ. Copyright © 2010 John Wiley & Sons, Ltd.

Formulation of the stability margins clearance criterion as a convex optimization problem

This paper presents the formulation of a flight clearance criterion as a convex optimization problem. The criterion is the stability margins criterion which is specified as an allowable phase and gain margin of a certain loop transfer function. The satisfaction of the criterion amounts to the Nichols plot of the loop transfer function being outside a specified trapezoidal region. It was shown previously that the exclusion condition from an ellipsoidal region is implied by using the generalized stability margin bPC and its calculation was performed frequency-wise by solving a sequence of convex optimization problems. In this paper we formulate the calculation of a lower bound on bPC as a convex optimization problem using Integral Quadratic Constraints (IQCs) and avoid the gridding procedure in the frequency domain. Furthermore, we formulate the problem of obtaining a lower bound on the perturbed stability margin, which is defined as the worst-case bPC over variations in real uncertain parameters.

Robust [formula omitted] output-feedback control of systems with time-delay

The problem of designing robust dynamic output-feedback controllers for linear, continuous, time-invariant systems with uncertain time-delay in the measured output and/or the control input and with polytopic type parameter uncertainties is considered. Given a transfer function matrix of a system with uncertain real parameters that reside in some known ranges, an appropriate, not necessarily minimal, state–space model of the system is described which permits reconstruction of its states. The resulting retarded model incorporates the uncertain parameters of the transfer function matrix in the state–space matrices and the uncertain time-delay that occurs in the control channel. To this model, the recent theory of robust H ∞ state-feedback control for retarded systems is applied. The theory is used to solve a benchmark problem of distillation column robust control design.

Robust [formula omitted] output-feedback control of linear discrete-time systems

The problem of designing H ∞ dynamic output-feedback controllers for linear discrete-time systems with polytopic type parameter uncertainties is considered. Given a transfer function matrix of a system with uncertain real parameters that reside in some known ranges, an appropriate, not necessarily minimal, state-space model of the system is described which permits reconstruction of all its states via the delayed inputs and outputs of the plant. The resulting model incorporates the uncertain parameters of the transfer function matrix in the state-space matrices. A recently developed linear parameter-dependent LMI approach to state-feedback H ∞ control of uncertain polytopic systems is then used to design a robust output-feedback controllers that are of order comparable to the one of the plant. These controllers ensure the stability and guarantee a prescribed performance level within the uncertainty polytope.

A direction method for stability robustness analysis with respect to real uncertain parameters

A direction method is developed in this study to characterize stability robustness of a nominal stable control system subject to perturbations of real uncertain parameters. In this direction method, characteristic loci of directional stability robustness matrices are used to infer directional stability margins. When only the number of the real uncertain parameters is given, these directional stability margins can be connected to form a stability region to characterize the stability robustness. When a parameter uncertainty template is given, these directional stability margins can also be used to obtain a stability margin with no conservation. The advantage of this method is that it can be utilized to obtain a stability margin for varieties of real parameter uncertainty templates which is generally not possible in other approaches.

Biquadratic stability of uncertain linear systems

Deals with the problem of stability analysis for linear systems with uncertain real, possibly time-varying, parameters. A robust stability approach based on a Lyapunov function which depends quadratically on the uncertain parameters as well as in the system state is proposed. This robust stability approach, referred to as biquadratic stability, is suited to deal with uncertain real parameters with magnitude and rate of change which are confined to a given convex region. A linear matrix inequality based sufficient condition for biquadratic stability is developed. The proposed robust stability analysis method includes quadratic stability and affine quadratic stability as particular cases.

Control of LPV systems with partly measured parameters

The control of linear parameter-varying (LPV) systems with time-varying real uncertain parameters, where only some of the parameters are measured and available for feedback, is considered. The control objectives are internal stability and disturbance attenuation in the sense of a bounded induced L/sub 2/-norm from the disturbance to the controlled output. Using parametric Lyapunov functions, the solvability conditions for dynamic output feedback controllers that depend on the measured parameters are expressed in terms of a set of linear matrix inequalities (LMIs) and an additional coupling constraint which destroys the convexity of the overall problem. By transforming the coupling constraint into a rank condition, the problem is recast into a rank-minimization problem with LMI constraints. An example is included that demonstrates the application of the results.

\boldmath Average $\cal H_2$ Performance and Maximal Parameter Perturbation Radius for Uncertain Systems

In this paper, methods are presented for calculating the maximal parameter perturbation bounds under ${\cal H}_2$ performance constraints for a family of uncertain systems and for calculating the average ${\cal H}_2$ performance under such parameter variations. The uncertain systems are described by state space models with nonlinear (polynomial) dependencies on real uncertain parameters. All results obtained are based on necessary and sufficient conditions. As a special virtue of the approach, the proposed algorithms for stability analysis and for performance analysis turn out to have exactly the same algebraic structure. An example illustrates the results and the algorithms.

Robust stabilization for continuous-time systems with slowly time-varying uncertain real parameters

The authors construct a class of parameter-dependent Lyapunov functions to guarantee robust stability in the presence of time-varying rate restricted plant uncertainty. Extensions to a class of time-varying nonlinear uncertainty that generalize the multivariable Popov criterion are also considered. These results are then used for controller synthesis to address the problem of robust stabilization in the presence of slowly time-varying real parameters.

Design of PID Controller Satisfying Robust Performances

In this paper, a method to compute an almost correct region of PID parameters which guarantees robust stability and several robust performances for systems with real uncertain parameters is proposed. The proposed method can be compute the region in a short computing time by adopting the following idea and technique : The first one is the computational strategy to avoid the unnecessary small splitting of the parameter space of the PID controller. The second one is adopting the idea of computational geometry approach to reduce the time for executing the algorithm. The third one is to use the Non-convex Polygon Interval Arithmetic (NPIA) to compute the “good” estimate of value sets of transfer functions.

Robust, reduced-order modeling for state-space systems via parameter-dependent bounding functions

One of the most important problems in dynamic systems theory is to approximate a higher-order system model with a low-order, relatively simpler model. However, the nominal high-order model is never an exact representation of the true physical system. In this paper the problem of approximating an uncertain high-order system with constant real parameter uncertainty by a robust reduced-order model is considered. A parameter-dependent quadratic bounding function is developed that bounds the effect of uncertain real parameters on the model-reduction error. An auxiliary minimization problem is formulated that minimizes an upper bound for the model-reduction error. The principal result is a necessary condition for solving the auxiliary minimization problem which effectively provides sufficient conditions for characterizing robust reduced-order models.

Affine parameter-dependent Lyapunov functions and real parametric uncertainty

This paper presents new tests to analyze the robust stability and/or performance of linear systems with uncertain real parameters. These tests are extensions of the notions of quadratic stability and performance where the fixed quadratic Lyapunov function is replaced by a Lyapunov function with affine dependence on the uncertain parameters. Admittedly with some conservatism, the construction of such parameter-dependent Lyapunov functions can be reduced to a linear matrix inequality (LMI) problem and hence is numerically tractable. These LMI-based tests are applicable to constant or time-varying uncertain parameters and are less conservative than quadratic stability in the case of slow parametric variations. They also avoid the frequency sweep needed in real-/spl mu/ analysis, and numerical experiments indicate that they often compare favorably with /spl mu/ analysis for time-invariant parameter uncertainty.

A convex parameterization of robustly stabilizing controllers

This paper treats synthesis of robust controllers for linear time-invariant systems. Uncertain real parameters are assumed to appear linearly in the closed loop characteristic polynomial. The main contribution is to give a convex parameterization of all controllers that simultaneously stabilize the system for all possible parameter combinations. With the new parameterization, certain robust performance problems can be stated in terms of quasi-convex optimization.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Some counterexamples in robust stability theory

This note deals with the problem of determining if a linear system whose characteristics polynomial depends multilinearly on n independent uncertain real parameters Δ i , i = 1,…, n, is robustly stable. It is shown by example that a polynomial in n variables may have a unique real root, and that this observation disposes of several natural conjectures in robust stability theory. In particular, we show that, in a certain sense, there are no ‘edge’ or ‘ m-dimensional face’ Kharitonov-like theorems for the general multilinear case. The result holds even when restricted to that subset of multilinear functions which can be written in the form f( Δ 1,…, Δ n ) = det( I + diag( Δ 1,…, Δ n ) M) for some complex matrix M.

Design of Tracking Systems with LQ Optimality and Quadratic Stability

This paper concerns a tracking problem for linear systems with real uncertain parameters. The objective here is to design a state feedback controller such that 1) the closed-loop system is optimal in the LQ sense for the nominal parameter values, 2) quadratically stable and 3) its output tracks the step reference input asymptotically under the parameter uncertainty. One such controller is obtained here by applying our "Inverse LQ design method" of extended type with quadratic stability requirement incorporated. The attractive feature of the proposed design method lies on one hand in the control structure of the designed tracking system with a low-pass filter placed before the control input, and on another hand in the design parameters with clear physical meaning in relation to this structure.

Does it suffice to check a subset of multilinear parameters in robustness analysis?

For polynomials with general multilinear dependence on uncertain real parameters it does not suffice to check a subset for Q for stability.