On the stabilizability condition in indirect adaptive control

Many recent studies on overall stability of adaptive control systems require, among other conditions, that the identified model be stabilizable, at least asymptotically. In fact, the usual identification methods do not guarantee this property, in the passive approach (without exciting extra signals), or if the model order is subject to over-estimation. In order to solve this problem, it is shown in this paper that certain classical identification algorithms can be easily modified in order to ensure the above stabilizability condition, without affecting the other properties.     

Applying automated control synthesis methods to condition systems requiring state observers

Automated control synthesis methods have been presented in Holloway, Guan, Sundaravadivelu, and Ashley [(2000). Automated synthesis and composition of taskblocks for control of manufacturing systems. IEEE Transactions on Systems, Man, and Cybernetics Part B, 30(5)] for a class of fully observed plants modeled by condition systems, a class of discrete event systems. For plants which are not fully observed, an automated method to synthesize a state observer was presented in Gong and Holloway [(2000). State observer synthesis for a class of condition systems. In IEE international workshop on discrete event systems (WODES2000), Ghent, August 2000]. In this paper, it is shown how to apply automated control synthesis methods to plants which feed a state observer. In particular, the original plant model is transformed into an intermediate form on which automated control synthesis methods are applied. The control thus generated is then proven to be effective at achieving high-level control objectives when applied to the original plant composed with a state observer. The method is illustrated with a portion of a manufacturing automated assembly station.     

Algorithms for the synthesis of observers for nonlinear dynamic systems

For nonlinear dynamic systems, an approach based on cycle-by-cycle linearization is proposed to construct algorithms for state observers based on linear estimation theory. This approach makes it possible to analytically synthesize a full-rank and a reduced observer of the angular velocity of a spacecraft rotation relative to the body reference frame given the measurements of three angles determining the position of this reference frame in the inertial space. Simulation results are presented.     

On state observers for nonlinear systems

For linear systems, it is known that there exists a state observer if and only if the system is detectable, or in other words, asymptotically stabilizable by output injection. In this paper, it is shown that asymptotic stabilizability by output injections is neither necessary nor sufficient for the construction of a nonlinear observer which may not have an exponential error decay rate. The concept of uniform observers is introduced and necessary conditions and sufficient conditions are given for the construction of uniform observers.     

On the robust design of sliding observers for linear systems

The restrictive character of well-known structural constraints, related to the matching conditions, for the sliding mode feedforward state reconstruction problem in linear, time-invariant, perturbed systems is critically reexamined from a new perspective. It is shown that, in generalized state space coordinates, a matched canonical state space realization exists which always allows discontinuous asymptotic stabilization of the observation error dynamics. The well-known structural conditions thus become largely irrelevant and robust asymptotic state estimation is shown to be feasible, for any perturbed observable system, by means of sliding mode observers.     

Necessary and sufficient condition for stabilizability of discrete-time linear switched systems: A set-theory approach

In this paper, the stabilizability of discrete-time linear switched systems is considered. Several sufficient conditions for stabilizability are proposed in the literature, but no necessary and sufficient. The main contributions are the necessary and sufficient conditions for stabilizability based on the set-theory and the characterization of a universal class of Lyapunov functions. An algorithm for computing the Lyapunov functions and a procedure to design the stabilizing switching control law are provided, based on such conditions. Moreover, a sufficient condition for non-stabilizability for switched system is presented. Several academic examples are given to illustrate the efficiency of the proposed results. In particular, a Lyapunov function is obtained for a system for which the Lyapunov–Metzler condition for stabilizability does not hold.     

On semiglobal stabilizability of antistable systems by saturatedlinear feedback

It was recently established that a second-order antistable linear system can be semiglobally stabilized on its null controllable region by saturated linear feedback and a higher order linear system with two or more antistable poles can be semiglobally stabilized on its null controllable region by more general bounded feedback laws. We will show in this note that a system with three real-valued antistable poles cannot be semiglobally stabilized on its null controllable region by the simple saturated linear feedback     

Conditions for stabilizability of linear switched systems

This paper investigates some conditions that can provide stabilizability for linear switched systems with polytopic uncertainties via their closed loop linear quadratic state feedback regulator. The closed loop switched systems can stabilize unstable open loop systems or stable open loop systems but in which there is no solution for a common Lyapunov matrix. For continuous time switched linear systems, we show that if there exists solution in an associated Riccati equation for the closed loop systems sharing one common Lyapunov matrix, the switched linear systems are stable. For the discrete time switched systems, we derive a Linear Matrix Inequality (LMI) to calculate a common Lyapunov matrix and solution for the stable closed loop feedback systems. These closed loop linear quadratic state feedback regulators guarantee the global asymptotical stability for any switched linear systems with any switching signal sequence.     

On the robustness and performance of disturbance observers for second-order systems

The disturbance observer (DOB) has been widely utilized for high-precision and high-speed motion control applications. In this note, we suggest the robustness measure of the DOB as a criterion to design the robust DOB systems. Also, we suggest its design guidelines especially for second-order systems. Experimental results for an optical disk drive system show the validity of design guidelines.     

On designing observers for time-delay systems with non-linear disturbances

In this paper, the observer design problem is studied for a class of time-delay non-linear systems. The system under consideration is subject to delayed state and non-linear disturbances. The time-delay is allowed to be time-varying, and the non-linearities are assumed to satisfy global Lipschitz conditions. The problem addressed is the design of state observers such that, for the admissible time-delay as well as non-linear disturbances, the dynamics of the observation error is globally exponentially stable. An effective algebraic matrix inequality approach is developed to solve the non-linear observer design problem. Specifically, some conditions for the existence of the desired observers are derived, and an explicit expression of desired observers is given in terms of some free parameters. A simulation example is included to illustrate the practical applicability of the proposed theory.     

On emulated nonlinear reduced-order observers for networked control systems

We consider a general class of nonlinear reduced-order observers and show that the global asymptotic convergence of the observation error in the absence of network-induced constraints is maintained for the emulated observer semiglobally and practically (with respect to the maximum allowable transmission interval) when system measurements are sent through a communication channel. Networks governed by a Lyapunov uniformly globally asymptotically stable protocol are investigated. Our results can be used to synthesize various observers for networked control systems for a range of network configurations, as we illustrate it by considering classes of immersion and invariance observers which include the circle-criterion observers.     

Open-loop stabilizability of infinite-dimensional systems

In this paper we study open-loop stabilizability, a general notion of stabilizability for linear differential equations =Ax+Bu in an infinite-dimensional state space. This notion is sufficiently general to be implied by exact controllability, by optimizability, and by various general definitions of closedloop stabilizability. Here,A is the generator of a strongly continuous semigroup, and we make very few a priori restrictions on the class of controlsu. Our results hinge upon the control operatorB being smoothly left-invertible, which is a very mild restriction when the input space is finite-dimensional. Since open-loop stabilizability is a weak concept, lack of open-loop stability is quites strong. A focus of this paper is to give necessary conditions for open-loop stabilizability, thus identifying classes of systems which are not open-loops stabilizable. First we give useful frequency domain conditions that are equivalent to our definitions of open-loop stabilizability, and lead to a version of the Hautus test for open-loop stabilizability. When the input space is finite-dimensional, we give necessary conditions for open-loop stabilizability which involve spectral properties ofA. We show that these results are not true if the conditions onB are weakened. We obtain analogous results for discrete-time systems. We show that, for a class of systems without spectrum determined growth, optimizability is impossible. Finally, we show that a system is open-loop stabilizable with a class of controlu if and only if the system with the sameA but a more boundedB is open-loop stabilizable with a larger class of controls. This work was partially supported by NSF Grant DMS-9623392.     

Exponential observers for nonlinear systems

Exponentially converging observers are exhibited for a large class of multi-input multi-output (MIMO) nonlinear systems. Potential applications of these observers to mechanical systems and distillation columns are presented     

Finite-dimensional observers for delay systems

A finite-dimensional observer theory is formulated for a class of point delayed systems. The observer is characterized by the fact that it directly generates a finite-dimensional feedback controller for this class of systems. The theory is based on a preliminary approximation theory which allows the delay system to be exponentially approximated by a system of ordinary differential equations.     

On semi-global stabilizability of MIMO nonlinear systems by output feedback

For multi-input multi-output (MIMO) nonlinear systems, we prove that global stabilizability via smooth state feedback and uniform observability imply semi-global stabilizability by dynamic output feedback. This result incorporates and generalizes Teel-Praly's theorem established previously for single-input single-output (SISO) nonlinear systems. The generalization is made possible by employing, in contrast to the complicated proof in Teel and Praly [(1994). Global stabilizability and observability imply semi-global stabilizability by output feedback. Systems and Control Letters, 22, 313-325], a simple and intuitive argument that involves no intricate Lyapunov functions.     

High-gain observers for non-linear systems

State estimation for non-linear dynamic systems is discussed. A high-gain injection from the output variables is used to attenuate to any desired degree, the effect of the non-linear terms on the estimation errors, which can be made arbitrarily small, from a particular observable form. Some Liapunov arguments are used to study the stability properties of the resulting error dynamics. It is shown that any completely observable multi-input multi-output non-linear system, not necessarily linear in the input variables, can be transformed, by means of a change of coordinates depending on the input variables, in such an observable form. In particular, the solvability of partial differential equations is not needed for the design of this transformation.     

On LMI conditions to design observers for Lipschitz nonlinear systems

This paper addresses the problem of observer design for Lipschitz nonlinear systems via LMI. The goal of this note is to clarify some recent results and to propose a new design methodology. This is based on the reformulation of the Lipschitz property using some mathematical tools. This reformulation is a relevant and useful Lipschitz condition, which leads to less restrictive LMI conditions. To show the superiority of this latter, two numerical examples are presented.     

Modal synthesis of multivariable systems with observers of specified order

The problem of pole assignment in multivariable linear systems with observers of a given order v is considered. By way of a simple proof, a very general result is established which constitutes an extension of all theorems known in the literature on pole assignment via observers. On its basis, an efficient design algorithm is worked out.