Some remarks on static-feedback linearization for time-varying systems

This work summarizes some results about static state feedback linearization for time-varying systems. Three different necessary and sufficient conditions are stated in this paper. The first condition is the one by [Sluis, W. M. (1993). A necessary condition for dynamic feedback linearization. Systems & Control Letters, 21, 277–283]. The second and the third are the generalizations of known results due respectively to [Aranda-Bricaire, E., Moog, C. H., Pomet, J. B. (1995). A linear algebraic framework for dynamic feedback linearization. IEEE Transactions on Automatic Control, 40, 127–132] and to [Jakubczyk, B., Respondek, W. (1980). On linearization of control systems. Bulletin del’Academie Polonaise des Sciences. Serie des Sciences Mathematiques, 28, 517–522]. The proofs of the second and third conditions are established by showing the equivalence between these three conditions. The results are re-stated in the infinite dimensional geometric approach of [Fliess, M., Lévine J., Martin, P., Rouchon, P. (1999). A Lie–Bäcklund approach to equivalence and flatness of nonlinear systems. IEEE Transactions on Automatic Control, 44(5), 922–937].     

Robustness bound for receding horizon finite memory control: Lyapunov–Krasovskii approach

This article presents a robustness bound (RB) of receding horizon finite memory control (RHFMC) (Kwon, W.H., and Han, S. (2004), ‘Receding Horizon Finite Memory Controls of Output Feedback Controls for State Space Systems’, IEEE Transactions on Automatic Control, 49, 1905–1915) for continuous-time state-space systems with norm-bounded uncertainties. The proposed RB is easily obtained by solving a convex problem in terms of a linear matrix inequality. We show through a numerical example that the RHFMC can guarantee the robust stabilisation for a larger class of uncertain systems than linear quadratic Gaussian controls when some poles of closed-loop systems are close to an imaginary axis in the complex plane.     

Robust supervisory control of timed discrete event systems under partial observation based on eligible time bounds: The existence conditions

This paper addresses a supervisory control problem for uncertain timed discrete event systems (DESs) under partial observation. An uncertain timed DES to be controlled is represented by a set of possible timed models based on the framework of Brandin and Wonham [(1994). Supervisory control of timed discrete event systems. IEEE Transactions on Automatic Control, 39(2), 329-342]. To avoid the state space explosion problem caused by tick events in the timed models, a notion of eligible time bounds is proposed for a single timed model obtained from the set of all possible timed models. Based on this notion, we present the necessary and sufficient conditions for the existence of a robust supervisor achieving a given language specification for the single timed model. Moreover, we show that the robust supervisor can also achieve the specification for any timed model in the set.     

A constructive solution for stabilization via immersion and invariance: The cart and pendulum system

Immersion and Invariance (I&I) is the method to design asymptotically stabilizing control laws for nonlinear systems that was proposed in [Astolfi, A., & Ortega, R. (2003). Immersion and invariance: A new tool for stabilization and adaptive control of nonlinear systems. IEEE Transactions on Automatic Control, 48, 590-606]. The key steps of I&I are (i) the definition of a target dynamics, whose order is strictly smaller than the order of the system to be controlled; (ii) the construction of an invariant manifold such that the restriction of the system dynamics to this manifold coincides with the target dynamics; (iii) the design of a control law that renders the manifold attractive and ensures that all signals are bounded. The second step requires the solution of a partial differential equation (PDE) that may be difficult to obtain. In this short note we use the classical cart and pendulum system to show that by interlacing the first and second steps, and invoking physical considerations, it is possible to obviate the solution of the PDE. To underscore the generality of the proposed variation of I&I, we show that it is also applicable to a class of n-dimensional systems that contain, as a particular case, the cart and pendulum system.     

Nonlinear stabilizability via encoded feedback: The case of integral ISS systems

In this note we observe how, using the same arguments of Liberzon and Hespanha [(2005). Stabilization of nonlinear systems with limited information feedback, IEEE Transactions on Automatic Control, 50, 910-915] , integral input-to-state stabilizability with respect to measurement errors is enough to prove stabilizability of nonlinear systems with limited information. The utility of the remark lies on the fact that, under a requirement on the data rate analogous to the one in the paper by Liberzon and Hespanha, the stabilizability result is proven for a larger class of systems.     

Dynamic anti-windup scheme for feedback linearizable nonlinear control systems with saturating inputs

This paper proposes a dynamic compensation scheme for input-constrained feedback linearizable nonlinear systems to cope with the windup phenomenon. Given a dynamic feedback linearizing controller designed without considering its input constraint, an additional dynamic compensator is proposed to account for the constraint. This dynamic anti-windup is based on the minimization of a reasonable performance index. The proposed strategy is a nonlinear extended version of [Park, J.-K., & Choi, C.-H. (1995). Dynamic compensation method for multivariable control systems with saturating actuators. IEEE Transactions on Automatic Control, 40(9), 1635–1640] with simplified derivation of an optimization solution under relaxed assumptions. The parameter matrices and structure of the solution are explicitly decided by mathematical optimization for infinite horizon without tuning of design parameters unlike previous schemes. During input saturation, the role of the anti-windup scheme with the proposed dynamic feedback compensator is to maintain the controller states to be exactly the same as those without input saturation. The local asymptotic stability and the total stability of the resulting systems are proved. The usefulness of the proposed design method is illustrated by comparative simulations for a constrained control system.     

Simple saturated designs for ANCBC systems and extension to feedforward nonlinear systems

This article re-examines the robust stabilisation of the asymptotically null-controllable with bounded controls (ANCBC) systems, and extends the established algorithm to a wide class of feedforward nonlinear systems whose nominal dynamics contains both multiple integrators and multiple oscillators. Based on the notion of ‘converging-input bounded-state’ (CIBS) rather than ‘small-input small-state’ (SISS), the computation burden in Sussmann et al. (Sussmann, H.J., Sontag, E.D., and Yang, Y. (1994), ‘A General Result on the Stabilization of Linear Systems using Bounded Controls’, IEEE Transactions on Automatic Control, 39, 2411–2425) is reduced and a class of simple saturated control laws is presented for the CIBS stabilisation of ANCBC systems. Then, by combining the technique of dealing with higher-order terms, the algorithm for ANCBC systems is extended to feedforward nonlinear systems.     

Passivity-based designs for synchronized path-following

We consider a formation control system where individual systems are controlled by a path-following design and the path variables are to be synchronized. We first show a passivity property for the path-following system, and next, combine this with a passivity-based synchronization algorithm developed in Arcak [2007. Passivity as a design tool for group coordination. IEEE Transactions on Automatic Control, in press]. The passivity approach expands the classes of synchronization schemes available to the designer. This generality offers the possibility to optimize controllers to, e.g., improve robustness and performance. Two designs are developed in the proposed passivity framework: the first employs the path error information in the synchronization loop, while the second only uses synchronization errors. A sampled-data design, where the path variables are updated in discrete-time and the path-following controllers are updated in continuous time, is also developed.     

A generalized multi-period mean-variance portfolio optimization with Markov switching parameters

In this paper, we deal with a generalized multi-period mean-variance portfolio selection problem with market parameters subject to Markov random regime switchings. Problems of this kind have been recently considered in the literature for control over bankruptcy, for cases in which there are no jumps in market parameters (see [Zhu, S. S., Li, D., & Wang, S. Y. (2004). Risk control over bankruptcy in dynamic portfolio selection: A generalized mean variance formulation. IEEE Transactions on Automatic Control, 49, 447-457]). We present necessary and sufficient conditions for obtaining an optimal control policy for this Markovian generalized multi-period mean-variance problem, based on a set of interconnected Riccati difference equations, and on a set of other recursive equations. Some closed formulas are also derived for two special cases, extending some previous results in the literature. We apply the results to a numerical example with real data for risk control over bankruptcy in a dynamic portfolio selection problem with Markov jumps selection problem.     

A new autocovariance least-squares method for estimating noise covariances

Industrial implementation of model-based control methods, such as model predictive control, is often complicated by the lack of knowledge about the disturbances entering the system. In this paper, we present a new method (constrained ALS) to estimate the variances of the disturbances entering the process using routine operating data. A variety of methods have been proposed to solve this problem. Of note, we compare ALS to the classic approach presented in Mehra [(1970). On the identification of variances and adaptive Kalman filtering. IEEE Transactions on Automatic Control, 15(12), 175-184]. This classic method, and those based on it, use a three-step procedure to compute the covariances. The method presented in this paper is a one-step procedure, which yields covariance estimates with lower variance on all examples tested. The formulation used in this paper provides necessary and sufficient conditions for uniqueness of the estimated covariances, previously not available in the literature. We show that the estimated covariances are unbiased and converge to the true values with increasing sample size. The proposed method also guarantees positive semidefinite covariance estimates by adding constraints to the ALS problem. The resulting convex program can be solved efficiently.     

Asymptotic stabilisation of the ball and beam system: design of energy-based control law and experimental results

We present a new nonlinear control law to stabilise the ball and beam system at a desired operating point. The control law is based on the interconnection and damping assignment–passivity-based control (IDA-PBC) methodology developed in Ortega, Spong, Gomez-Estern, and Blankenstien (Ortega, R., Spong, M., Gomez-Estern, F., and Blankenstien, G. (2002), ‘Stabilization of Underactuated Mechanical Systems via Interconnection and Damping Assignment’, IEEE Transactions on Automatic Control, 47, 1218–1233) that guarantees stability in the sense of Lyapunov. We present a novel proof of the asymptotic stability of the desired operating point. The validity of the proposed control law is demonstrated through the experimental results.     

Digital redesign via the generalised bilinear transformation

A new controller discretisation approach, the generalised bilinear transformation (GBT), is proposed in Zhang, G., Chen, T., and Chen X. (2007a). Given an analog controller K, GBT generates a class of digital controllers K _gbt parameterised by a real number α ∈ (?∞, ∞). A geometric interpretation of GBT is first presented. Second, when the original analog feedback system is stable, two methods are proposed to find the value of the parameter α which provides an upper bound of sampling periods guaranteeing closed-loop stability of the resulting sampled-data system. Finally, several examples, namely, an IIR digital filter, an example studied in Rattan, K.S. (1984), ‘Digitization of Existing Continuous Control Systems,’ IEEE Transactions on Automatic Control, 29, 282–306, and Keller, J.P., and Anderson, B.D.O. (1992), ‘A New Approach to the Discretisation of Continuous-time Controllers,’ IEEE Transaction on Automatic Control, 37, 214–223, and an H _∞ control problem investigated in Chen, T., and Francis, B. (1995), Optimal Sampled-Data Control Systems, London: Springer, are used to demonstrate the strength of our discretisation approach. These examples show that GBT is able to retain the simplicity of the emulation methods such as the Tustin method, and simultaneously sustain closed-loop performance even at slow sampling.     

Adaptive state-feedback stabilization of high-order stochastic systems with nonlinear parameterization

This paper investigates the adaptive state-feedback stabilization of high-order stochastic systems with nonlinear parameterization. By using the parameter separation lemma in [Lin, W., & Qian, C. (2002a). Adaptive control of nonlinearly parameterized systems: A nonsmooth feedback framework. IEEE Transactions on Automatic Control, 47, 757-774.] and some flexible algebraic techniques, and choosing an appropriate Lyapunov function, a smooth adaptive state-feedback controller is designed, which guarantees that the closed-loop system has an almost surely unique solution for any initial state, the equilibrium of interest is globally stable in probability, and the state can be regulated to the origin almost surely.     

Adaptive boundary control for unstable parabolic PDEs—Part III: Output feedback examples with swapping identifiers

We develop output-feedback adaptive controllers for two benchmark parabolic PDEs motivated by a model of thermal instability in solid propellant rockets. Both benchmark plants are unstable, have infinite relative degree, and are controlled from the boundary. One plant has an unknown parameter in the PDE and the other in the boundary condition. In both cases the unknown parameter multiplies the measured output of the system, which is obtained with a boundary sensor located on the “opposite side” of the domain from the actuator. In comparison with the Lyapunov output-feedback adaptive controllers in Krstic and Smyshlyaev [(2005). Adaptive boundary control for unstable parabolic PDEs—Part I: Lyapunov design. IEEE Transactions on Automatic Control, submitted for publication], the controllers presented here employ much simpler update laws and do not require a priori knowledge about the unknown parameters. We show how our two benchmarks examples can be combined and illustrate the adaptive stabilization design by simulation.     

A new approach to generate weighted fuzzy rules using genetic algorithms for estimating null values

In this paper, we present a new method to generate weighted fuzzy rules using genetic algorithms for estimating null values in relational database systems, where there are negative functional dependency relationships between attributes. The proposed method can get higher average estimated accuracy rates than the method presented in [Chen, S. M., & Huang, C. M. (2003). Generating weighted fuzzy rules from relational database systems for estimating null values using genetic algorithms. IEEE Transactions on Fuzzy Systems, 11(4), 495–506].     

Optimal discrete systems with prescribed eigenvalues

In this article, the problem of the numerical computation of the stabilising solution of the game theoretic algebraic Riccati equation is investigated. The Riccati equation under consideration occurs in connection with the solution of the H _∞ control problem for a class of stochastic systems affected by state-dependent and control-dependent white noise and subjected to Markovian jumping. The stabilising solution of the considered game theoretic Riccati equation is obtained as a limit of a sequence of approximations constructed based on stabilising solutions of a sequence of algebraic Riccati equations of stochastic control with definite sign of the quadratic part. The proposed algorithm extends to this general framework the method proposed in Lanzon, Feng, Anderson, and Rotkowitz (Lanzon, A., Feng, Y., Anderson, B.D.O., and Rotkowitz, M. (2008), ‘Computing the Positive Stabilizing Solution to Algebraic Riccati Equations with an Indefinite Quadratic Term Viaa Recursive Method,’ IEEE Transactions on Automatic Control, 53, pp. 2280–2291). In the proof of the convergence of the proposed algorithm different concepts associated the generalised Lyapunov operators as stability, stabilisability and detectability are widely involved. The efficiency of the proposed algorithm is demonstrated by several numerical experiments.     

Stability and robust stability for systems with a time-varying delay

To concern the stability and robust stability criteria for systems with time-varying delays, this note uses not only the time-varying-delayed state x(t-h(t)) but also the delay-upper-bounded state to exploit all possible information for the relationship among a current state x(t), an exactly delayed state x(t-h(t)), a marginally delayed state , and the derivative of the state , when constructing Lyapunov-Krasovskii functionals and some appropriate integral inequalities, originally suggested by Park (1999. A delay-dependent stability criterion for systems with uncertain time-invariant delays. IEEE Transactions on Automatic Control, 44(4), 876-877). Two fundamental criteria are provided for the cases where no bound of delay derivative is assumed and where an upper bound of delay derivative is assumed. Examples show the resulting criteria outperform all existing ones in the literature.     

A note on static output‐feedback for affine nonlinear systems

In this paper we present sufficient conditions for the solvability of the static‐output feedback stabilization problem for affine nonlinear systems using a factorization approach. We extend the results of (IEEE Trans. Autom. Control, Vol. 47, No. 12, pp. 2038–2041 (2002)) to include disturbance‐attenuation and singular control. The sufficient conditions given are also less stringent than the ones given in (IEEE Trans. Autom. Control Vol. 47, No. 12, pp. 2038–2041 (2002)). The usefulness of the results are also illustrated with some examples. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Further results on global stabilization of uncertain nonlinear systems by output feedback

For a family of uncertain nonlinear systems dominated by a triangular system that satisfies linear growth condition with an output dependent growth rate, we prove that global robust stabilization can be achieved by smooth output feedback. This conclusion has incorporated and generalized the recent output feedback stabilization results, for instance, the work (IEEE Trans. Automat. Control 2002; 47 :2068–2073) where the same conclusion was already shown to be true for planar systems, and the work (Proceedings of the 42nd IEEE, CDC, 2003; 1544–1549) where the growth rate is required to be a polynomial function of the system output. There are two key ingredients in the present contribution. One of them is the introduction of a rescaling transformation with a dynamic factor that is tuned on‐line through a Riccati‐like differential equation, which turns out to be extremely effective in dealing with the system uncertainty. The other one is the development of a recursive observer design algorithm making it possible to assign the robust observer gains in a step‐by‐step fashion. Both a smooth state feedback controller and a robust observer are explicitly constructed for the rescaled system using only the knowledge of the bounding system. Copyright © 2005 John Wiley & Sons, Ltd.