Real Parametric Uncertainty Research Articles

Robust LQG control of uncertain linear systems via simultaneousH2 andH∞ approach

This communication is concerned with an optimal linear quadratic control design with guaranteed stability for systems with real time-varying parameter uncertainties and unstructured unmodelled dynamics. It is shown that this problem can be converted into a mixed H2 and H∞ control problem. A simultaneous H2 and H∞ approach is then applied to solve the latter problem. A numerical example is presented to demonstrate the significance of the proposed technique. © 1997 John Wiley & Sons, Ltd.

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.

Robust nonlinear-nonquadratic feedback control via parameterdependent Lyapunov functions

In this paper we develop a unified framework to address the problem of optimal nonlinear-nonquadratic robust control for systems with nonlinear time-invariant real parameter uncertainty. Specifically, we transform a given robust nonlinear control problem into an optimal control problem by modifying the performance functional to account for the system uncertainty. Robust stability of the closed loop nonlinear system is guaranteed by means of a parameter-dependent Lyapunov function composed of a fixed (parameter-independent) and variable (parameterdependent) part. The fixed part of the Lyapunov function can clearly be seen to be the solution to the steady-state Hamilton-Jacobi-Bellman equation for the nominal system. The overall framework generalizes the classical Hamilton-JacobiBellman conditions to address the design of robust optimal controllers for uncertain nonlinear systems via parameter-dependent Lyapunov functions and provides the foundation for extending robust linear-quadratic controller synthesis to robust nonlinear-nonquadratic problems.

Robust filtering for uncertain systems with sampled measurements

Abstract The problem of H∞ filtering for a class of nonlinear continuous-time systems subject to real time-varying parameter uncertainty with sampled-data measurements is considered. Linear and nonlinear filters are designed, respectively, that would guarantee a prescribed H∞ performance in the continuous-time context, irrespective of the parameter uncertainty and unknown initial states. Both the cases of finite and infinite horizon filtering are investigated in terms of two Riccati equations

Controller design with real parametric uncertainty

A number of techniques have been developed in recent years for the analysis and design of controllers which are robust with respect to structured complex uncertainty. In particular the complex μ synthesis procedure has been successfully applied to a number of engineering problems. However, the presence of real parametric uncertainty in the problem substantially complicates matters, so that standard complex μ synthesis techniques are no longer adequate. This paper develops a procedure to tackle the mixed (real and complex) μ synthesis problem. This procedure involves a (D, G)-K iteration between computing the mixed μ upper bound and solving an H ∞ optimal control problem. The resulting scheme has guaranteed convergence, though not necessarily to the global optimum of the (non-convex) problem. The (D, G)-K iteration has been implemented in software, and several controller designs are carried out and compared with the corresponding complex μ synthesis designs, to illustrate the advantages of the new technique.

Reduction of robust stabilization problems to standard H∞ problems for classes of systems with structured uncertainty

We consider the problem of robustly stabilizing a family of linear time-invariant systems with a linear time-invariant controller. For classes of robust stabilization problems with unmodelled dynamics, recent breakthroughs in H ∞ theory lead to solution via two Riccati equations. For systems with structured real uncertainty, however, a similar comprehensive theory does not exist. The objective of this paper is to delineate classes of linear time-invariant systems with structured uncertainty (either real or complex) for which a similar solution to the robust stabilization problem is possible. The theory has the following salient features. First, for classes of uncertainty structures, no conservatism is introduced. For structured real parametric uncertainty, we make a connection with H ∞ theory that does not amount to any crude overbounding via complex discs; the stabilizability conditions we obtain are not only sufficient but also necessary. The second salient feature in the theory is a special weighting function which we construct. For example, for the case when the structured uncertainty enters only in the denominator, the robust stabilization problem is replaced by a standard H ∞ sensitivity minimization problem with our special weighting function. A similar H ∞ complementary sensitivity problem is associated with numerator uncertainty.

Optimally scaled H-infinity full information control synthesis with real uncertainty

An algorithm to synthesize optimal controllers for the scaled Tioo full information problem with real and complex uncertainty is presented. The control problem is reduced to a linear matrix inequality, which can be solved via a finite dimensional convex optimization. This technique is compared with the optimal scaled 1-C^ full information with only complex uncertainty and D-K iteration control design to synthesize controllers for a missile autopilot. Directly including real parametric uncertainty into the control design results in improved robust performance of the missile autopilot. The controller synthesized via D-K iteration achieves results similar to the optimal designs.

Generalized mixed-μ bounds for real and complex multiple-block uncertainty with internal matrix structure

New absolute stability results that unify and extend existing structured singular value bounds for mixed uncertainty are developed using frequency-domain arguments. These bounds generalize prior upper bounds for mixed-μ by permitting the treatment of non-diagonal real uncertain blocks, as well as accounting for internal matrix structure in the uncertainty. Several specializations to well known absolute stability criteria from the classical literature are given, which allow for a systematic comparison of these results in addressing the problem of robust stability for real parameter uncertainty.

Nonconservative and Noniterative Solution to the Robust Stabilization Problem for Classes of Systems with Real Parametric Uncertainties

This paper considers the problem of robustly stabilizing a family of linear time-invariant systems with a linear time-invariant controller. More specifically, systems with structured real parametric uncertainties are considered. By taking advantage of the special uncertainty structure due to certain important system's configurations and modeling, a nonconservative and noniterative solution to the posed robust stabilization problem in terms of a standard H∞ optimization problem is given.

H∞ CONTROL OF A CLASS OF UNCERTAIN NONLINEAR SYSTEMS: AN OBSERVER-BASED CONTROLLER DESIGN

This paper is concerned with the problem of H∞ control of a class of continuous-time nonlinear systems with or without real parameter uncertainty. We address the problem of designing observer-based nonlinear dynamic output feedback controllers which achieve a prescribed performance in an H∞ sense. A Riccati equation approach is proposed for solving the above control design problems.

Robust filtering for a class of discrete-time uncertain nonlinear systems: AnH∞ approach

This paper deals with the H∞ filtering problem for a class of discrete-time nonlinear systems with or without real time-varying parameter uncertainty and unknown initial state. For the case when there is no parametric uncertainty in the system, we are concerned with designing a nonlinear H∞ filter such that the induced l2 norm of the mapping from the noise signal to the estimation error is within a specified bound. It is shown that this problem can be solved via one Riccati equation. We also consider the design of nonlinear filters which guarantee a prescribed H∞ performance in the presence of parametric uncertainties. In this situation, a solution is obtained in terms of two Riccati equations.

Parametric robustness evaluation of a H-infinity missile autopilot

A multivariable missile autopilot is synthesized using a H^ loop shaping procedure by McFarlane and Glover. Because missile flight control systems must guarantee stability and performance in the presence of large aerody- namic uncertainties, advanced p, tools are next used to analyze robust stability and performance of the control law in the presence of real parametric uncertainties in the stability derivatives. The multivariable H^ autopilot is proved to exhibit good parametric robustness properties. The control law is finally validated on a nonlinear simulator. New \JL tools are nevertheless also available: IJL sensitivities7 can bring further information in addition to the basic application of jit analysis, whereas the v tool811 can directly solve skewed ^ problems that would otherwise involve a recursive application of \JL analysis (e.g., robust performance problems or direct computation of the maximal s.s.v. over the frequency range). The use of these new JJL tools consequently enables further in- vestigation of the robustness properties of a controller beyond the basic application of the real/mixed \ji tool. We first illustrate on the //oo missile autopilot that the v measure can advantageously replace the s.s.v. in robust performance analysis. We then illustrate that the s.s.v., the v measure, and the JJL sensitivities can be combined so as to maximize in a rather systematical way the guaranteed domain of stability of the closed loop in the space of uncertainties. The paper is organized as follows. Section II describes the design objectives and the missile model. Section III is a general presen- tation of the //oo design with coprime factors and loop shaping. Section IV briefly reviews the /z tools used in this paper, whereas Sec. V describes the design of the autopilot and a first analysis of the control law. Section VI is devoted to the parametric robustness anal- ysis and to the validation of the autopilot on a nonlinear simulator. Concluding remarks end the paper.

Polynomial methods for the structured singular value with real parameters

We consider the structured singular value problem with real parametric uncertainty only. Using techniques from algebraic geometry, we propose two algorithms that in principle can yield the precise value of the structured singular value at a fixed frequency. Their ability to do so depends upon their ability to find all common roots to a system of polynomial equations. The first algorithm is applicable to problems with two real parameters each of multiplicity two. The second algorithm is applicable to problems with n distinct real parameters. These algorithms have proved useful in applications to aerospace control law analysis.

Optimal Popov controller analysis and synthesis for systems with real parameter uncertainties

Robust performance analysis plays an important role in the design of controllers for uncertain multivariable systems. Recent research has investigated the use of absolute stability criteria to develop less conservative analysis tests for systems with linear and nonlinear real parameter uncertainties. This note extends previous work on optimal /spl Hscr/2 performance analysis using the Popov criterion by presenting a numerical homotopy algorithm that can be used to analyze systems with less restrictive assumptions on the structure of the uncertainty block. The technique is used to compare relative robustness capabilities of the various control algorithms that have been designed for the Middeck active control experiment (MACE). The analysis is combined with the previously presented Popov controller synthesis to yield compensators that guarantee robust performance for systems with real parameter uncertainty.

Precise bounds for the petersen time-varying real parameter uncertainty system stabilized via linear static controllers

Using a class of linear static controllers, we stabilize the Petersen open-loop two-dimensional linear system (Ref. 1), which consists of one time-varying uncertainty in the state matrixA and one timevarying uncertainty in the input matrixB. We show that the worst-case uncertainty strategy for the closed-loop system is a piecewise constant strategy of the angular state θ with three switches on the half-turn, −π/2≤θ≤π/2; it is unique with respect to a set of measure zero. Formulas are derived for the worst-case half-turn radius gainr HT as a function of the parameters of the class of stabilizing linear static controllers. Using the class of scalar-quadratic Lyapunov functions, we show that a necessary and sufficient condition for the closed-loop system to be robustly stable against all time-varying admissible uncertainties is thatr HT be less than unity. The bound on the time-varying real parameter uncertainties for the closed-loop system to be robustly stable is derived for the class of linear static feedback controllers. We obtain stabilizing linear static controllers such that the bound is as close to infinity as desired. The derived results are compared with numerical results obtained using commerical robust-control software.

A Unified State-Space Solution to Unstructured Robustness and Disturbance Attenuation

In this paper, a unified framework for the robustness against various kinds of unmodelled uncertainty is presented. It is based on the H∞ optimization for a particular nominal feedback matrix. Within such a framework, disturbance attenuation is also considered. Furthermore, the more difficult problem, i.e., when there exists real parameter uncertainty for the plant, is studied.

A popov criterion for uncertain linear multivariable systems

The Popov absolute stability criterion is traditionally proved using a Lyapunov function and the positive real lemma. In this paper a simplified proof of the multivariable Popov criterion is given for the case of one-sided, sector-bounded real parameter uncertainty. A loop-shifting transformation is then used to extend the Popov criterion to two-sided, sector-bounded uncertain matrices. Specialization of this result to norm-bounded uncertain matrices leads to an upper bound for the structured singular value for block-structured, real parameter uncertainty.

Robust game theoretic synthesis in the presence of uncertain initial states

In this paper a performance robust control synthesis problem is considered; the performance measure v represents the sensitivity of the output to the nonzero initial state. The problem is to find a controller, out of a prescribed set, that minimizes v such that internal stability and a prescribed level of disturbance attenuation for all real parameter uncertainties in a given set are achieved. The primary motivation for considering this problem is that v can be given a direct physical interpretation in terms of the transient response of a closed-loop system. Linear time-invariant systems with output feedback are considered. A game theoretic approach is employed to solve the synthesis problem and necessary conditions for optimality are given.

The octomorphic criterion for real parameter uncertainty: Real-μ bounds without circles and D, N-scales

In this paper we introduce new bounds for robust stability analysis with real parameter uncertainty. The approach is based on an absolute stability criterion that excludes the Nyquist plot from a paraboloidal region containing the point −1 + J0. Transformation of this criterion to the case of norm-bounded uncertainty leads to a stability criterion in terms of the octomorphic, or figure-eight shaped, region. The requirement that the Nyquist plot lie inside the octomorphic region thus yields a bound on the allowable real parameter uncertainty. This stability criterion is distinct from recent bounds for real-μ which involve frequency-dependent scales having a frequency-dependent, off-axis circle interpretation. Since the octomorphic region includes both upper and lower halves, it is able to encompass the entire Nyquist plot without using frequency-dependent scales.

Optimum structural design and robust active control using singular value constraints

This paper introduces a methodology for simultaneously designing a minimum weight structure and robust active controls to reduce vibrations in an aircraft structure due to external disturbances. The design problem is posed as a mathematical optimization problem with the principal objective function being the weight of the structure. The robust control design is achieved by specifying appropriate constraints on singular values of the closed-loop transfer matrices. The control approach selected for this purpose is based on designing a dynamic compensator that simultaneously minimizes the upper bound of a quadratic performance index H2 and the H∞ norm of a disturbance transfer function of a multi-input/multi-output system. The controller can tolerate both real parameter uncertainty in the structural frequencies and damping, and unmodelled dynamics. The design variables are the crosspsectional areas of the structure and the parameters used in the design of a control system. The method was applied to three structures idealized with membrane elements, shear panels and bar elements with embedded actuators and sensors simulating an active flexible aircraft wing.