Showing papers in "International Journal of Robust and Nonlinear Control in 2003"

Sliding mode observers for robust detection and reconstruction of actuator and sensor faults

Abstract: This paper describes a method for designing sliding mode observers for detection and reconstruction of actuator and sensor faults, that is robust against system uncertainty. The method uses ℋ∞ concepts to design the sliding motion so that an upper bound on the effect of the uncertainty on the reconstruction of the faults will be minimized. The design method is first applied to the case of actuator faults, and then by some appropriate filtering, the method is extended to the case of sensor faults. A VTOL aircraft example taken from the fault detection literature is used to demonstrate the method and its effectiveness. Copyright © 2003 John Wiley & Sons, Ltd.

Robust model predictive control for nonlinear discrete-time systems

Abstract: This paper describes a model predictive control (MPC) algorithm for the solution of a state-feedback robust control problem for discrete-time nonlinear systems. The control law is obtained through the solution of a finite-horizon dynamic game and guarantees robust stability in the face of a class of bounded disturbances and/or parameter uncertainties. A simulation example is reported to show the applicability of the method. Copyright © 2003 John Wiley & Sons, Ltd.

A nonlinear set-membership filter for on-line applications

Abstract: A guaranteed estimator for a general class of nonlinear systems and on-line usage is developed and analysed. This filter bounds the linearization error, then applies a linear set-membership filter such that stability guarantees hold for nonlinear systems. A tight bound on the linearization error is found using interval analysis. This filter recursively estimates an ellipsoidal set in which the true state lies. General assumptions include the use of bounded noises and twice continuously differentiable dynamics. When the system is uniformly observable, it is proven that the nonlinear set-membership filter is stable. In addition, if no noise is present and the initial error is small, the error between the centre of the estimated set and the true value converges to zero. The result is an estimator which is computationally attractive and can be implemented robustly in real-time. The proposed method is applied to a two-state example to demonstrate the theoretical results. Copyright © 2003 John Wiley & Sons, Ltd.

Robust stability, stabilization and H-infinity control of time-delay systems with Markovian jump parameters

Abstract: In this paper, the problems of stochastic stability and stabilization for a class of uncertain time-delay systems with Markovian jump parameters are investigated. The jumping parameters are modelled as a continuous-time, discrete-state Markov process. The parametric uncertainties are assumed to be real, timevarying and norm-bounded that appear in the state, input and delayed-state matrices. The time-delay factor is constant and unknown with a known bound. Complete results for both delay-independent and delay-dependent stochastic stability criteria for the nominal and uncertain time-delay jumping systems are developed. The control objective is to design a state feedback controller such that stochastic stability and a prescribed H1-performance are guaranteed. We establish that the control problem for the time-delay Markovian jump systems with and without uncertain parameters can be essentially solved in terms of the solutions of a finite set of coupled algebraic Riccati inequalities or linear matrix inequalities. Extension of the developed results to the case of uncertain jumping rates is also provided.

Regional stabilization and H∞ control of time-delay systems with saturating actuators

Abstract: A linear parameter varying approach is introduced for the design of a constant state-feedback controller that locally stabilizes linear systems with state time-varying delays and saturating actuators and achieves a prescribed performance level for all disturbances with uniformly bounded magnitudes. A polytopic representation is used to describe the saturation behaviour. Delay-dependent sufficient conditions in terms of linear matrix inequalities (LMIs) are obtained for the existence of such a controller. An estimate is made of the domain of attraction for the disturbance-free system. The conditions for the stabilizability and H∞ performance of the system apply the Lyapunov–Krasovskii functional and the recent descriptor approach to the control of time-delay systems, whereas the conditions for finding an ellipsoid that bounds the set of the states (in the Euclidean space) that are reachable from the origin in finite time are obtained via the Razumikhin approach. The resulting conditions are expressed in terms of linear matrix inequalities, with some tuning parameters, and they apply a different Lyapunov function to each of the vertex points that stem from the polytopic description of the saturation in the actuators. Copyright © 2003 John Wiley & Sons, Ltd.

Fixed‐order H∞ control design via a partially augmented Lagrangian method

Abstract: In this paper we develop an augmented Lagrangian method to determine local optimal solutions of the reduced- and fixed-order H∞ synthesis problems. We cast these synthesis problems as optimization programs with a linear cost subject to linear matrix inequality (LMI) constraints along with nonlinear equality constraints representing a matrix inversion condition. The special feature of our algorithm is that only equality constraints are included in the augmented Lagrangian, while LMI constraints are kept explicitly in order to exploit currently available semi definite programming (SDP) codes. The step computation in the tangent problem is based on a Gauss–Newton model, and a specific line search and a first-order Lagrange multiplier update rule are used to enhance efficiency. A number of computational results are reported and underline the strong practical performance of the algorithm. Copyright © 2003 John Wiley & Sons, Ltd.

Input-output linearization with delay cancellation for nonlinear delay systems: the problem of the internal stability

Abstract: SUMMARY This paper investigates the issue of the internal stability of nonlinear delay systems controlled with a feedback law that performs exact input–output, linearization and delay cancellation. In previous works the authors showed that, unlike with the case of systems without state delays, when the relative degree is equal to the number of state variables and the output is forced to be identically zero, delay systems still possess a non-trivial internal state dynamics. Not only, in the same conditions delay systems are also characterized by a non-trivial input dynamics. Obviously, both internal state and input dynamics should give bounded trajectories, otherwise the exact input–output linearization and delay cancellation technique cannot be applied. This paper studies the relationships between the internal state and input dynamics of a controlled nonlinear delay system. An interesting result is that a suitable stability assumption on the internal state dynamics ensures that, when the output is asymptotically driven to zero, both the state and control variables asymptotically decay to zero. Copyright # 2003 John Wiley & Sons, Ltd.

Model predictive control of constrained piecewise affine discrete‐time systems

Abstract: The paper deals with model predictive control of a class of non-smooth nonlinear systems, namely piecewise affine systems. A novel procedure for solving the associated optimal control problem is presented and its convergence established. Stability properties of the closed-loop system using model predictive control with the new algorithm are derived. Examples illustrate the proposed algorithm for determining optimal control of constrained piecewise affine systems and the performance of the closed-loop system. Copyright © 2003 John Wiley & Sons, Ltd.

Efficient scheduled stabilizing model predictive control for constrained nonlinear systems

Abstract: We present a computationally efficient scheduled model predictive control (MPC) algorithm for constrained nonlinear systems with large operating regions. We design a set of local predictive controllers with estimates of their regions of stability covering the desired operating region, and implement them as a single scheduled MPC which on-line switches between the set of local controllers and achieves nonlinear transitions with guaranteed stability. This algorithm is computationally efficient and provides a general framework for the scheduled MPC design. The algorithm is illustrated with two examples. Copyright © 2003 John Wiley & Sons, Ltd.

An LMI approach to guaranteed cost control for uncertain linear neutral delay systems

Abstract: This paper considers the problem of guaranteed cost control for uncertain neutral delay systems with a quadratic cost function. The system under consideration is subject to norm-bounded time-varying parametric uncertainty appearing in all the matrices of the state-space model. The problem we address is the design of a state feedback controller such that the closed-loop system is not only stable but also guarantees an adequate level of performance for all admissible uncertainties. A sufficient condition for the existence of guaranteed cost controllers is given in terms of a linear matrix inequality (LMI). When this condition is feasible, the desired state feedback controller gain matrices can be obtained via convex optimization. An illustrative example is provided to demonstrate the effectiveness of the proposed approach. Copyright © 2002 John Wiley & Sons, Ltd.

Robust H∞ control for uncertain singular systems with state delay

Abstract: This paper addresses the problem of robust H∞ control for uncertain continuous singular systems with state delay. The singular system under consideration involves state time delay and time-invariant norm-bounded uncertainty. Based on the linear matrix inequality (LMI) approach, we design a memoryless state feedback controller law, which guarantees that, for all admissible uncertainties, the resulting closed-loop system is not only regular, impulse free and stable, but also meets an H∞-norm bound constraint on disturbance attenuation. A numerical example is provided to demonstrate the applicability of the proposed method. Copyright © 2003 John Wiley & Sons, Ltd.

Gain-scheduling MPC of nonlinear systems

Abstract: Predictive control of nonlinear systems is addressed by embedding the dynamics into an LPV system and by computing robust invariant sets. This mitigates the on-line computational burden by transferring most of the computations off-line. Benefits and conservatism of this approach are discussed in relation with the control of a critical mechanical system. Copyright © 2003 John Wiley & Sons, Ltd.

Output feedback stabilization of constrained systems with nonlinear predictive control

Abstract: We present an output feedback stabilization scheme for uniformly completely observable nonlinear MIMO systems combining nonlinear model predictive control (NMPC) and high-gain observers. The control signal is recalculated at discrete sampling instants by an NMPC controller using a system model for the predictions. The state information necessary for the prediction is provided by a continuous time high-gain observer. The resulting ‘optimal’ control signal is open-loop implemented until the next sampling instant. With the proposed scheme semi-global practical stability is achieved. That is, for initial conditions in any compact set contained in the region of attraction of the NMPC state feedback controller, the system states will enter any small set containing the origin, if the high-gain observers is sufficiently fast and the sampling time is small enough. In principle the proposed approach can be used for a variety of state feedback NMPC schemes. Copyright © 2003 John Wiley & Sons, Ltd.

An improved stability criterion for systems with distributed delays

Abstract: This article improves the formulation of discretized Lyapunov functional method for systems with distributed delays. The main idea is to apply Jensen's inequality and variable elimination in matrix inequalities. The resulting stability criterion is much simpler. Experience in numerical calculation indicates that it generally converges to the analytical result as grid size decreases, and the speed of convergence is much faster than the previous method. The new formulation is also applicable to a wider class of systems. Copyright © 2003 John Wiley & Sons, Ltd.

Predictive control of constrained nonlinear systems via LPV linear embeddings

Abstract: This paper analyzes the applicability of convex MPC schemes, synthesized for LPV polytopic systems, to nonlinear plants. The nonlinear systems under consideration are those whose trajectories can be embedded within those of a polytopic LPV discrete-time system. It is postulated that the latter belongs to a polytopic family of linear systems, each member of which is parameterized by the value that a parameter vector assumes in the unit simplex. Such a parameter can be measured on-line and exploited for feedback while a bound on its rate of change is known and exploited for predictions. Different customizations and improvements of a recently introduced MPC scheme for LPV systems are presented and contrasted in terms of their numerical burdens and control performance. The proposed predictive controllers are proved to quadratically stabilize LPV polytopic systems, as well as any other embedded non-linear system, in the presence of input and state constraints. Copyright © 2003 John Wiley & Sons, Ltd.

Nonlinear receding horizon control of an underactuated hovercraft

Abstract: This paper describes nonlinear receding horizon control of an underactuated hovercraft with discrete-valued inputs and its hardware implementation. The hovercraft has two fixed thrusters, and the thrusters generate thrusts of only three discrete values, that is, forward, backward and zero thrusts. Nonlinear receding horizon control is applied to position control of the hovercraft with the discrete-valued inputs approximated by constrained continuous-valued inputs. Although nonlinear receding horizon control requires real-time constrained optimization, a fast algorithm is successfully implemented for the hardware experiment. Copyright © 2003 John Wiley & Sons, Ltd.

Robust adaptive fuzzy controller for non‐affine nonlinear systems with dynamic rule activation

Abstract: This paper describes the design of a robust adaptive fuzzy controller for an uncertain single-input single-output nonlinear dynamical systems. While most recent results on fuzzy controllers considers affine systems with fixed rule-base fuzzy systems, we propose a control scheme for non-affine nonlinear systems and a dynamic fuzzy rule activation scheme in which an appropriate number of the fuzzy rules are chosen on-line. By using the proposed scheme, we can reduce the computation time, storage space, and dynamic order of the adaptive fuzzy system without significant performance degradation. The Lyapunov synthesis approach is used to guarantee a uniform ultimate boundedness property for the tracking error, as well as for all other signals in the closed loop. No a priori knowledge of an upper bounds on the uncertainties is required. The theoretical results are illustrated through a simulation example. Copyright © 2002 John Wiley & Sons, Ltd.

ℋ︁∞ filtering for discrete‐time linear systems with Markovian jumping parameters†

Abstract: This paper investigates the problem of ℋ∞ filtering for discrete-time linear systems with Markovian jumping parameters. It is assumed that the jumping parameter is available. This paper develops necessary and sufficient conditions for designing a discrete-time Markovian jump linear filter which ensures a prescribed bound on the l2-induced gain from the noise signals to the estimation error. The proposed filter design is given in terms of linear matrix inequalities. Copyright © 2003 John Wiley & Sons, Ltd.

Adaptive identification of linear time‐delay systems

Abstract: Synthesis of an adaptive parameter identifier is developed for linear dynamic systems with finitely many lumped delays in the state vector and control input. These systems are governed by linear functional differential equations with uncertain time-invariant parameters and delays. The state of the system is assumed to be available for measurements. Constructive necessary and sufficient conditions for the system parameters and delays to be identifiable are provided. Once the parameter identifiability is guaranteed the simultaneous on-line identification of the system parameters and delays is achieved by the adaptive identifier proposed. Theoretical results are supported by numerical simulation. Copyright © 2003 John Wiley & Sons, Ltd.

The Liapunov's second method for continuous time difference equations

Abstract: Among many other cases such as economic and lossless propagation models, continuous time difference equations are encountered as the internal dynamics in a class of non-linear time delay systems, when controlled by a suitable state feedback which drives the output exponentially to zero. The Liapunov's second method for these infinite dimensional systems has not been extensively investigated in the literature. This paper has the aim of filling this gap. Liapunov's second method theorems for checking the stability and the asymptotic stability of this class of infinite dimensional systems are built up, in both a finite and an infinite dimensional setting. In the finite dimensional setting, the Liapunov function is defined on finite dimensional sets. The conditions for stability are given as inequalities on continuous time. No derivatives are involved, as in the dynamics of the studied systems. In the infinite dimensional setting, the continuous time difference equation is transformed into a discrete time system evolving on an infinite dimensional space, and then the classical Liapunov theorem for the system in the new form is written. In this paper the very general case is considered, that is non-linear continuous time difference equations with multiple non commensurate delays are considered, and moreover the functions involved in the dynamics are allowed to be discontinuous, as well as the initial state. In order to study the stability of the internal dynamics in non-linear time delay feedback systems, an exogenous disturbance is added, which goes to zero exponentially as the time goes to infinity. An example is considered, from non-linear time delay feedback theory. While the results available in the literature are inconclusive as far as the stability of that example is concerned, such stability is proved to hold by the theorems developed in this paper, and is validated by simulation results. Copyright © 2003 John Wiley & Sons, Ltd.

Stabilization of a solid propellant rocket instability by state feedback

Abstract: SUMMARY In this paper a globally stabilizing feedback boundary control law for an arbitrarily fine discretization of a one-dimensional nonlinear PDE model of unstable burning in solid propellant rockets is presented. The PDE has a destabilizing boundary condition imposed on one part of the boundary. We discretize the original nonlinear PDE model in space using finite difference approximation and get a high order system of coupled nonlinear ODEs. Then, using backstepping design for parabolic PDEs, properly modified to accommodate the imposed destabilizing nonlinear boundary condition at the burning end, we transform the original system into a target system that is asymptotically stable in l 2 -norm with the same type of boundary condition at the burning end, and homogeneous Dirichlet boundary condition at the control end. Copyright # 2003 John Wiley & Sons, Ltd.

New results for the analysis of linear systems with time‐invariant delays

Abstract: This paper presents a comparison system approach for the analysis of stability and ℋ∞ performance of linear time-invariant systems with unknown delays. The comparison system is developed by replacing the delay elements with certain parameter-dependent Pade approximations. It is shown using the special properties of the Pade approximation to e−s that the value sets of these approximations provide outer and inner coverings for that of each delay element and that the robust stability of the outer covering system is a sufficient condition for the stability of the original time delay system. The inner covering system, in turn, is used to provide an upper bound on the degree of conservatism of the delay margin established by the sufficient condition. This upper bound is dependent only upon the Pade approximation order and may be made arbitrarily small. In the single delay case, the delay margin can be calculated explicitly without incurring any additional conservatism. In the general case, this condition can be reduced with some (typically small) conservatism to finite-dimensional LMIs. Finally, this approach is also extended to the analysis of ℋ∞ performance for linear time-delay systems with an exogenous disturbance. Copyright © 2003 John Wiley & Sons, Ltd.

Tracking control of second-order chained form systems by cascaded backstepping

Abstract: A design methodology is presented for tracking control of second-order chained form systems. The methodology separates the tracking-error dynamics, which are in cascade form, into two parts: a linear subsystem and a linear time-varying subsystem. The linear time-varying subsystem, after the first subsystem has converged, can be treated as a chain of integrators for the purposes of a backstepping controller. The two controllers are designed separately and the proof of stability is given by using a result for cascade systems. The method consists of three steps. In the first step we apply a stabilizing linear state feedback to the linear subsystem. In the second step the second subsystem is exponentially stabilized by applying a backstepping procedure. In the final step it is shown that the closed-loop tracking dynamics of the second-order chained form system are globally exponentially stable under a persistence of excitation condition on the reference trajectory. The control design methodology is illustrated by application to a second-order non-holonomic system. This planar manipulator with two translational and one rotational joint (PPR) is a special case of a second-order non-holonomic system. The simulation results show the effectiveness of our approach. Copyright © 2002 John Wiley & Sons, Ltd.

Stabilization of electromechanical systems via interconnection and damping assignment

Abstract: In this article, we propose a general controller structure for asymptotic position regulation of electromechanical systems derived using the Interconnection and Damping Assignment Passivity-Based Control methodology recently proposed in the literature. The controller is applicable to arbitrary fully actuated electromechanical systems with linear magnetic materials consisting of inductances, permanent magnets, and one mechanical co-ordinate. We assume linear magnetic materials and fully actuated electrical dynamics; however, no restrictions are imposed on the particular form of the parameters that define the system dynamics, i.e. the inductance matrix, the magnetic coupling or the potential energy function. This allows us to treat—in a unified framework and without any additional simplifying assumptions—very diverse applications, including magnetic suspensions, and stepper and permanent magnet synchronous motors. Instrumental for our developments is the inclusion of ‘virtual’ couplings between the electrical and the mechanical subsystem, which is naturally suggested in this control methodology. Copyright © 2003 John Wiley & Sons, Ltd.

Discontinuous feedbacks, discontinuous optimal controls, and continuous-time model predictive control

Abstract: It is known that there is a class of nonlinear systems that cannot be stabilized by a continuous time-invariant feedback. This class includes systems with interest in practice, such as nonholonomic systems, frequently appearing in robotics and other areas. Yet, most continuous-time model predictive control (MPC) frameworks had to assume continuity of the resulting feedback law, being unable to address an important class of nonlinear systems. It is also known that the open-loop optimal control problems that are solved in MPC algorithms may not have, in general, a continuous solution. Again, most continuous-time MPC frameworks had to artificially assume continuity of the optimal controls or, alternatively, impose some demanding assumptions on the data of the optimal control problem to achieve the desired continuity. In this work we analyse the reasons why traditional MPC approaches had to impose the continuity assumptions, the difficulties in relaxing these assumptions, and how the concept of ‘sampling feedbacks’ combines naturally with MPC to overcome these difficulties. A continuous-time MPC framework using a strictly positive inter-sampling time is argued to be appropriate to use with discontinuous optimal controls and discontinuous feedbacks. The essential features for the stability of such MPC framework are reviewed. Copyright © 2003 John Wiley & Sons, Ltd.

A theoretical framework for gain scheduling

Abstract: The weighted incremental norm approach was originally introduced as a natural framework for extending well-known H∞ linear control concepts into the nonlinear context. In this paper, we investigate the numerous links between this new approach and the classical gain-scheduling technique. Although based on heuristic rules, gain-scheduled control is probably the most widespread nonlinear technique. In this paper, we point out that the control objectives of the gain-scheduled controller design can be expressed as the weighted incremental norm minimization of a nonlinear operator. The result interest is twofold: it first provides a rigorous mathematical formulation of the gain-scheduling problem. Furthermore, existing gain-scheduling techniques can be interpreted as approximate solutions to the weighted incremental norm minimization of a nonlinear operator. Copyright © 2003 John Wiley & Sons, Ltd.

Characterizing the solution set of polynomial systems in terms of homogeneous forms: an LMI approach

Abstract: This paper considers the problem of determining the solution set of polynomial systems, a well-known problem in control system analysis and design. A novel approach is developed as a viable alternative to the commonly employed algebraic geometry and homotopy methods. The first result of the paper shows that the solution set of the polynomial system belongs to the kernel of a suitable symmetric matrix. Such a matrix is obtained via the solution of a linear matrix inequality (LMI) involving the maximization of the minimum eigenvalue of an affine family of symmetric matrices. The second result concerns the computation of the solution set from the kernel of the obtained matrix. For polynomial systems of degree m in n variables, a basic procedure is available if the kernel dimension does not exceed m+1, while an extended procedure can be applied if the kernel dimension is less than n(m−1)+2. Finally, some application examples are illustrated to show the features of the approach and to make a brief comparison with polynomial resultant techniques. Copyright © 2003 John Wiley & Sons, Ltd.

Simulation based strategy for nonlinear optimal control: application to a microbial cell reactor

Abstract: Optimal control of systems with complex nonlinear behaviour such as steady state multiplicity results in a nonlinear optimization problem that needs to be solved online at each sample time. We present an approach based on simulation, function approximation and evolutionary improvement aimed towards simplifying online optimization. Closed loop data from a suboptimal control law, such as MPC based on successive linearization, are used to obtain an approximation of the ‘cost-to-go’ function, which is subsequently improved through iterations of the Bellman equation. Using this offline-computed cost approximation, an infinite horizon problem is converted to an equivalent single stage problem—substantially reducing the computational burden. This approach is tested on continuous culture of microbes growing on a nutrient medium containing two substrates that exhibits steady state multiplicity. Extrapolation of the cost-to-go function approximator can lead to deterioration of online performance. Some remedies to prevent such problems caused by extrapolation are proposed. Copyright © 2003 John Wiley & Sons, Ltd.

LMI based stability analysis and robust controller design for discrete linear repetitive processes

Abstract: Discrete linear repetitive processes are a distinct class of 2D linear systems with applications in areas ranging from long-wall coal cutting through to iterative learning control schemes. The main feature which makes them distinct from other classes of 2D linear systems is that information propagation in one of the two independent directions only occurs over a finite duration. This, in turn, means that a distinct systems theory must be developed for them. In this paper, the major new development is that an LMI based re-formulation of the stability conditions can used to enable the design of a family of control laws which have a well defined physical basis. It is also noted that this setting can be used to investigate robustness aspects.