Showing papers in "International Journal of Robust and Nonlinear Control in 1998"
TL;DR: In this paper, the capabilities of the state-dependent Riccati equation (SDRE) technique are illustrated in producing two control designs for the benchmark problem, and the first design shows that in the absence of disturbances and uncertainties, the SDRE nonlinear feedback solution compares very favorably to the optimal open-loop solution of the posed nonlinear regulator problem.
Abstract: A nonlinear control problem has been posed by Bupp et al. to provide a benchmark for evaluating various nonlinear control design techniques. In this paper, the capabilities of the state-dependent Riccati equation (SDRE) technique are illustrated in producing two control designs for the benchmark problem. The SDRE technique represents a systematic way of designing nonlinear regulators. The design procedure consists of first using direct parameterization to bring the nonlinear system to a linear structure having state-dependent coefficients (SDC). A state-dependent Riccati equation is then solved at each point x along the trajectory to obtain a nonlinear feedback controller of the form u=−R-1(x)BT(x)P(x)x, where P(x) is the solution of the SDRE. Analysis of the first design shows that in the absence of disturbances and uncertainties, the SDRE nonlinear feedback solution compares very favorably to the optimal open-loop solution of the posed nonlinear regulator problem, the latter being obtained via numerical optimization. It is also shown via simulation that the closed-loop system has stability robustness against parametric variations and attenuates sinusoidal disturbances. In the second design it is demonstrated how a hard bound can be imposed on the control magnitude to avoid actuator saturation. © 1998 John Wiley & Sons, Ltd.
349 citations
TL;DR: In this article, a unified approach to linear controller synthesis that employs various LMI conditions to represent control specifications is proposed, where a general synthesis problem described by any LMI of the class is reduced to solving a certain LMI.
Abstract: This paper proposes a unified approach to linear controller synthesis that employs various LMI conditions to represent control specifications. We define a comprehensive class of LMIs and consider a general synthesis problem described by any LMI of the class. We show a procedure that reduces the synthesis problem, which is a BMI problem, to solving a certain LMI. The derived LMI condition is equivalent to the original BMI condition and also gives a convex parametrization of all the controllers that solve the synthesis problem. The class contains many of widely-known LMIs (for H∞ norm, H2 norm, etc.), and hence the solution of this paper unifies design methods that have been proposed depending on each LMI. Further, the class also provides LMIs for multi-objective performance measures, which enable a new formulation of controller design through convex optimization. © 1998 John Wiley & Sons, Ltd.
343 citations
TL;DR: In this article, the authors consider the problem of real-time trajectory generation and tracking for nonlinear control systems and employ a two-degree-of-freedom approach that separates the non-linear tracking problem into real time trajectory generation followed by local gain-scheduled stabilization.
Abstract: This paper considers the problem of real-time trajectory generation and tracking for nonlinear control systems. We employ a two-degree-of-freedom approach that separates the nonlinear tracking problem into real-time trajectory generation followed by local (gain-scheduled) stabilization. The central problem which we consider is how to generate, possibly with some delay, a feasible state space and input trajectory in real time from an output trajectory that is given online. We propose two algorithms that solve the real-time trajectory generation problem for differentially flat systems with (possibly non-minimum phase) zero dynamics. One is based on receding horizon point to point steering, the other allows additional minimization of a cost function. Both algorithms explicitly address the tradeoff between stability and performance and we prove convergence of the algorithms for a reasonable class of output trajectories. To illustrate the application of these techniques to physical systems, we present experimental results using a vectored thrust flight control experiment built at Caltech. A brief introduction to differentially flat systems and its relationship with feedback linearization is also included. © 1998 John Wiley & Sons, Ltd.
270 citations
TL;DR: In this article, the rotational/translational proof-mass actuator (RTAC) has been studied as a nonlinear control design problem involving the nonlinear interaction of a translational oscillator and an eccentric rotational proof mass.
Abstract: This paper describes a nonlinear control design problem involving the nonlinear interaction of a translational oscillator and an eccentric rotational proof mass. This problem provides a benchmark for examining nonlinear control design techniques within the framework of a nonlinear fourth-order dynamical system. The problem is posed in the spirit of the linear benchmark problem described in Reference 1. This system was originally studied as a simplified model of a dual-spin spacecraft to investigate the resonance capture phenomenon.2 More recently, it has been studied to investigate the utility of a rotational proof-mass actuator for stabilizing translational motion.3~5 Viewed in this way, the rotational/translational proof-mass actuator (RTAC) has the feature that the nonlinearities associated with the actuator stroke limitation are implicit in the system dynamics. In contrast, the stroke limitation constraint must be considered separately in linear translational proof-mass actuators.6 A similar system has been studied as a rotating unbalanced mass (RUM) actuator in References 7 and 8.
162 citations
TL;DR: In this article, the realness of the parameters is exploited using a "skew-symmetric scaling" technique, which can be applied to other problems such as anti-windup control, nonlinear control and model reduction.
Abstract: SUMMARY The gain scheduling problem considered in this paper concerns a linear system whose state-space equations depend rationally on real, time-varying parameters, which are measured in real time. A stabilizing, parameter-dependent controller is sought, such that a given L 2 -gain bound for the closed-loop system is ensured. Suƒcient linear matrix inequality (LMI) conditions are known, that guarantee the existence of such ‘gain-scheduled’ controllers. This paper improves these results in two directions. First, we show how to exploit the realness of the parameters using a ‘skew-symmetric scaling’ technique. Moreover, we show how to apply this technique in a time-varying and/or nonlinear setting. We first devise a general result pertaining to control synthesis of interconnection of dissipative operators, and apply it to the gain-scheduling problem. Owing to its generality, this result can be applied to other problems such as anti-windup control, nonlinear control and model reduction. ( 1998 John Wiley & Sons, Ltd.
156 citations
TL;DR: The problems of robust quadratic mean-square stability and stabilization for a class of uncertain discrete-time linear systems with both Markovian jumping parameters and Frobenius norm-bounded parametric uncertainities are studied and guaranteed cost control for the underlying systems is investigated.
Abstract: In this paper, we first study the problems of robust quadratic mean-square stability and stabilization for a class of uncertain discrete-time linear systems with both Markovian jumping parameters and Frobenius norm-bounded parametric uncertainities. Necessary and sufficient conditions for the above problems are proposed, which are in terms of positive-definite solutions of a set of coupled algebraic Riccati inequalities. Then, the problem of robust quadratic guaranteed cost control for the underlying systems is investigated. A guaranteed cost control is designed to ensure the cost function is within a certain bound, irrespective of all admissible uncertainities. © 1998 John Wiley & Sons, Ltd.
128 citations
TL;DR: The problem of constructing a controller which quadratically stabilizes an uncertain system and minimizes a guaranteed cost bound on a quadratic cost function is considered and the solution is obtained via a parameter-dependent linear matrix inequality problem.
Abstract: This paper considers the problem of constructing a controller which quadratically stabilizes an uncertain system and minimizes a guaranteed cost bound on a quadratic cost function. The solution is obtained via a parameter-dependent linear matrix inequality problem. © 1998 John Wiley & Sons, Ltd.
109 citations
TL;DR: In this paper, the application of various geometric methods of nonlinear control is shown by way of examples including DC motors (series, shunt and separately excited), induction motors, synchronous motors and DC-DC converters.
Abstract: SUMMARY The di⁄erential-geometric techniques of nonlinear control developed over the last 20 years or so include static and dynamic feedback linearization, input—output linearization, nonlinear state observers and disturbance decoupling. The theory has now reached a level of maturity where control practicioners are making e⁄ective use of the techniques for electric motors. Indeed, DC and AC motors have well-defined nonlinear mathematical models which often satisfy the structural conditions required of the di⁄erential-geometric theory. In this paper, the application of various di⁄erential-geometric methods of nonlinear control is shown by way of examples including DC motors (series, shunt and separately excited), induction motors, synchronous motors and DC—DC converters. A number of contributions are surveyed which show the benefits of the methods for the design of global control laws by systematic means. ( 1998 John Wiley & Sons, Ltd.
95 citations
TL;DR: In this paper, the authors present a new design procedure and stability analysis for robotic variable structure controllers with PID-like sliding surfaces, which can operate with an unknown dynamic model; the only information required is a bound on one parameter.
Abstract: SUMMARY Dynamic behaviour of a system in sliding mode is entirely defined by the sliding surface. Customarily, the surface is selected as a hyperplane in the system’s state-space resulting in a PD-type sliding surface. This is not the only possible structure, and other designs with more complex or time-varying surfaces may provide definite advantages. Slotine and Spong1 included an integral term in the sliding surface expression that resulted in a type of PID sliding surface. However, the advantages of such a design were not elaborated in following publications of these or other researchers. In this paper we present a new design procedure and stability analysis for robotic variable structure controllers with PID-like sliding surfaces. Two versions of the controller are presented: regular and adaptive. The former is very simple and can operate with an unknown dynamic model; the only information required is a bound on one parameter. The latter provides an on-line estimation for this bound. Both controllers are robust with respect to bounded external disturbances and some unmodelled dynamic e⁄ects. The simulation results have demonstrated stability, with minimum transient responses that may be significantly faster than responses of traditional PD-manifold controllers under the same conditions. ( 1998 John Wiley & Sons, Ltd.
78 citations
TL;DR: In this paper, a series expansion solution to the Hamilton-Jacobi-Isaacs Equation associated with the nonlinear disturbance attenuation problem was obtained for a nonlinear controller.
Abstract: In this paper, we use the theory of L2 disturbance attenuation for linear (H1) and nonlinear systems to obtain solutions to the Nonlinear Benchmark Problem (NLBP) proposed in the paper by Bupp et. al.1. By considering a series expansion solution to the Hamilton-Jacobi-Isaacs Equation associated with the nonlinear disturbance attenuation problem, we obtain a series expansion solution for a nonlinear controller. Numerical simulations compare the performance of the third order approximation of the nonlinear controller with its rst order approximation (which is the same as the linear H1 controller obtained from the linearized problem.)
75 citations
TL;DR: In this article, the stability radii of positive linear discrete-time systems under affine parameter perturbations are studied and the results are derived for arbitrary perturbation norms induced by monotonic vector norms.
Abstract: In this paper we study stability radii of positive linear discrete-time systems under affine parameter perturbations. It is shown that real and complex stability radii of positive systems coincide for arbitrary perturbation structures, in particular, for blockdiagonal disturbances as considered in μ-analysis. Estimates and computable formulae are derived for these stability radii. The results are derived for arbitrary perturbation norms induced by monotonic vector norms (e.g. p-norms, 1⩽p⩽∞). © 1998 John Wiley & Sons, Ltd.
TL;DR: The RTAC experimental testbed as discussed by the authors provides a means for implementing and evaluating nonlinear controllers and includes all physical parameters of the device that are relevant to the design and implementation of controllers.
Abstract: This paper fully describes the RTAC experimental testbed which provides a means for implementing and evaluating nonlinear controllers. The description of the testbed includes all physical parameters of the device that are relevant to the design and implementation of controllers. Next, four nonlinear controllers are considered. The first controller is a static, full-state-feedback, globally asymptotically stabilizing control law developed using partial feedback linearization and integrator backstepping. Next, three nonlinear controllers based upon passivity principles are presented, two of which are encompassed by the classical passivity framework, while the third is based upon the novel concept of virtual resetting absorbers. The passive controllers do not require either translational position or velocity measurements. All of the controllers are implemented on the RTAC testbed and their performance is examined. © 1998 John Wiley & Sons, Ltd.
TL;DR: In this paper, a multi-objective control methodology based on linear matrix inequalities and robust control is proposed to solve the RTAC nonlinear control design benchmark problem, which is based on the search for a common quadratic Lyapunov function.
Abstract: The RTAC nonlinear control design benchmark problem is addressed using a multi-objective control methodology based on linear matrix inequalities and robust control. The approach hinges on the search for a common quadratic Lyapunov function ensuring various specifications (stability, L 2 -gain, command input and output peak bounds) for the closed-loop system. The resulting output-feedback controller is measurement-scheduled; precisely, its state-space matrices depend on the measurement vector, in a nonlinear fashion. We evaluate the performance of our design with various simulations and predicted trade-off curves. ( 1998 John Wiley & Sons, Ltd.
TL;DR: In this article, a robust digital linear controller is designed using the pole placement method combined with sensitivity function shaping in the frequency domain, which is based on identified discrete-time models.
Abstract: SUMMARY It is shown in this paper that based on identified discrete-time models a robust digital linear controller can be designed using the pole placement method combined with sensitivity function shaping in the frequency domain. Two di⁄erent design techniques are presented. The first one is based on the shaping of the sensitivity functions using the fixed parts in the controller and the auxiliary poles of the closed loop while keeping the dominant poles in the desired places. The main idea of the second one is to determine a weighting filter for the output sensitivity function in an H = optimization approach which assures partial pole placement and desired performances. In this technique the weighting filter is interpreted as the inverse of the desired output sensitivity function and is computed using a constrained optimization program. The application of this technique on a flexible transmission system is presented. ( 1998 John Wiley & Sons, Ltd. The controller design methodology considered in this paper is based on pole placement combined with the shaping of the sensitivity function. The computation of the controller in the pole placement technique requires the specification of the desired closed-loop poles (the nominal stability problem) and of some fixed parts of the controller for the rejection of disturbances at various frequencies (the nominal performance problem). However, this is not enough to guarantee the robustness of the design with respect to the plant model uncertainties (the robust stability and robust performance problem). A robust controller design requires also the shaping of the sensitivity functions. The sensitivity functions, particularly the output sensitivity function, are key indicators for the nominal and robust performance as well as for the robust stability of the closed-loop system. The inverse of the maximum value of the output sensitivity function, i.e., the inverse of its H = norm, gives the minimum distance between the Nyquist plot of the open-loop system and the critical point [!1, j0]. This quantity, called the modulus margin, is a much more significant robustness indicator than the phase and gain margins. On the other hand, conditions for assuring a certain delay margin which is also a very important robustness indicator, particularly in the high frequency region, can also be expressed in terms of the shape of the output sensitivity function. It seems therefore reasonable to combine the pole placement with the shaping of the output
TL;DR: In this paper, the authors consider the benchmark nonlinear control problem and use backstepping to design several active controllers for tracking and disturbance attenuation; these two problems are much more challenging than stabilization.
Abstract: SUMMARY In this paper we consider the benchmark nonlinear control problem and use backstepping to design several active controllers for tracking and disturbance attenuation; these two problems are much more challenging than stabilization. We show that the significant flexibility of backstepping can be exploited to reduce the required control e⁄ort and to eliminate the winding problem. ( 1998 John Wiley & Sons, Ltd.
TL;DR: In this article, the authors present the solution to the mixed µ synthesis problem, and how to design gain scheduling controllers with linear fractional transformations (LFTs), where the system is assumed to have a parametrization.
Abstract: This paper presents the solution to the mixed µ synthesis problem, and how to design gain scheduling controllers with linear fractional transformations (LFTs). The system is assumed to have a param ...
TL;DR: In this article, the authors proposed a robust stability condition for an input-output delay model, which guarantees that the output vector remains in a given domain deflned by a set of symmetrical linear constraints.
Abstract: A delay system is represented by a linear difierence equation. The system parameters and the delays are assumed to be unperfectly known. The output vector is perturbed by a bounded external disturbance vector. The addressed problem is to characterize conditions which guarantee that the output vector remains in a given domain deflned by a set of symmetrical linear constraints. This problem is solved by imposing positive invariance conditions. These conditions also imply delay independent asymptotic stability of the associated deterministic system. The notion of distance to instability is then analyzed through the concept of stability radius. The possible use of these new robust stability conditions for controlling an input-output delay model is then presented. An application is flnally proposed ; it concerns an inventory control problem for a simple production loop subject to constraints on inventory levels.
TL;DR: The problem of robust, finite-time, H∞-tracking for linear, discrete, time-varying systems is considered from the game theory point of view, and it is shown that its saddle-point equilibrium, if it exists, guarantees a prescribed H ∞-norm performance of the tracker, for all possible parameters.
Abstract: The problem of robust, finite-time, H∞-tracking for linear, discrete, time-varying systems is considered from the game theory point of view. No a priori knowledge of the dynamic model of the reference signal to be tracked is assumed, and the parameters of the system are not completely known. Two tracking problems are investigated, depending on whether the reference signal is perfectly known in advance, or previewed in a fixed-interval of time ahead. An augmented state-space description that converts the parameter uncertainty to bounded energy signal is used. A tracking game is then defined and solved. It is shown that its saddle-point equilibrium, if it exists, guarantees a prescribed H∞-norm performance of the tracker, for all possible parameters. © 1998 John Wiley & Sons, Ltd.
TL;DR: In this paper, the problem of computing the maximal Lyapunov exponent (generalized spectral radius) of a discrete inclusion is formulated as an average yield optimal control problem, and it is shown that the maximal value of this problem can be approximated by the maximum value of discounted optimal control problems, where for irreducible inclusions the convergence is linear in the discount rate.
Abstract: The problem of calculating the maximal Lyapunov exponent (generalized spectral radius) of a discrete inclusion is formulated as an average yield optimal control problem. It is shown that the maximal value of this problem can be approximated by the maximal value of discounted optimal control problems, where for irreducible inclusions the convergence is linear in the discount rate. This result is used to obtain convergence rates of an algorithm for the calculation of time-varying stability radii. © 1998 John Wiley & Sons, Ltd.
TL;DR: In this paper, the design and implementation of robust multivariable controllers for a compact disc mechanism is considered, with the objective of achieving good track-following and focusing performance in the presence of track disturbances, external mechanical shocks and structured norm-bounded plant uncertainty.
Abstract: SUMMARY This paper considers the design and implementation of robust multivariable controllers for a compact disc mechanism.The design objective is to achieve good track-following and focusing performance in the presence of track disturbances, external mechanical shocks and structured norm-bounded plant uncertainty. This robust performance problem has been solved in the k-framework using the DK-iteration scheme. Limits of implementation necessitate the use of sophisticated model reduction techniques and the use of two parallel digital signal processors. ( 1998 John Wiley & Sons, Ltd.
TL;DR: In this article, the authors considered the control of a bank-to-turn missile with very lightly damped bending modes and synthesized the autopilot using the H∞ loop-shaping design procedure of McFarlane and Glover.
Abstract: This paper considers the control of a bank to turn missile (developed by Aerospatiale) with very lightly damped bending modes. The autopilot is synthesized using the H∞ loop-shaping design procedure of McFarlane and Glover. Robustness against large modelling uncertainties (including parameters and bending modes) is then investigated using the ν-tool, which circumvents the drawback of frequency gridding occurring in μ-analysis. The autopilot of the missile is finally validated on a nonlinear simulator: the performance and the robustness of the design are clearly underlined. © 1998 John Wiley & Sons, Ltd.
TL;DR: In this article, a hybrid feedback control law is proposed for the RTAC system, expressed in terms of a continuous feedback part and a part that includes switched parameters determined according to a logic-based switching rule.
Abstract: SUMMARY A hybrid feedback control law is proposed for the RTAC system. This hybrid feedback control law is expressed in terms of a continuous feedback part and a part that includes switched parameters determined according to a logic-based switching rule. By appropriate selection of the switching rule, previous theoretical results guarantee that the origin is globally asymptotically stable. Some comments are made about the closed-loop properties, and experiments confirm that good responses are obtained for the case studied. ( 1998 John Wiley & Sons, Ltd.
TL;DR: In this paper, the inverted pendulum is used as an example of nonlinear systems which are not exactly linearizable, and a controller design method for the system based on approximate linerization is given.
Abstract: In this paper, taking the inverted pendulum as an example of nonlinear systems which are not exactly linearizable, we give a controller design method for the system based on approximate linerization. In the method, we try to suppress the effect of the higher order residual terms in choosing the new co-ordinate for the approximate linearization. Furthermore, we show its effectiveness by experiments. © 1998 John Wiley & Sons, Ltd.
TL;DR: In this paper, an explicit expression for computing the robustness margin for a set of systems whose real and complex coeƒcients are related by an ellipsoidal constraint was derived, which is an application of a result by Chen, Fan, and Nett for rank-one generalized structured singular value problems.
Abstract: SUMMARY This note derives an explicit expression for computing the robustness margin for aƒne systems whose real and complex coeƒcients are related by an ellipsoidal constraint. The expression, which is an application of a result by Chen, Fan, and Nett for rank-one generalized structured singular-value problems, extends and unifies previous results on robustness margin computation for systems with ellipsoidal uncertainty. ( 1998 John Wiley & Sons, Ltd.
TL;DR: In this article, a robust stability criterion for linear systems is proposed by combining the passivity and small gain theorems in different frequency bands, and a controller synthesis method based on the new criterion is also developed.
Abstract: In this paper, a new robust stability criterion for linear systems is proposed by combining the passivity and small gain theorems in different frequency bands. A controller synthesis method based on the new criterion is also developed. The controller can achieve both good performance and robustness in the same frequency band, if the uncertainty in that frequency band is passive or near passive. For processes with near passive lumped uncertainties with large gain in the frequency region in which good performance is required, the proposed controller can have better performance than that of H∞ control for the same robustness specification.