Minimal unknown-input functional observers for multi-input multi-output LTI systems

Designing minimum possible order (minimal) disturbance-decoupled proper functional observers for multi-input multi-output (MIMO) linear time-invariant (LTI) systems is studied. It is not necessary that a minimum-order unknown-input functional observer (UIFO) exists in our proposed design procedure. If the minimum-order observer cannot be attained, the observer's order is increased sequentially through a recursive algorithm, so that the minimal order UIFO can be obtained. To the best of our knowledge, this is the first time that this specific problem is addressed. It is assumed that the system is unknown-input functional detectable, which is the least requirement for the existence of a stable UIFO. This condition also is a certificate for the convergence of our observer's order-increase algorithm. Two methodologies are demonstrated to solve the observer design equations. The second presented scheme, is a new design method that based on our observations has a better numerical performance than the first conventional one. Numerical examples and simulation results in the MATLAB/Simulink environment describe the overall observer design procedure, and highlight the efficacy of our new methodology to solve the observer equations in comparison to the conventional one.     

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 exponential observers for nonlinear systems

The aim of the paper is to identify the class of nonlinear systems that have exponential observers - a concept introduced by previous authors. It is shown that a necessary condition for the existence of an exponential observer for a nonlinear system is that the corresponding linearized system is detectable, and for local exponential stabilization problems, the condition is also sufficient.This paper gives also a theorem on the separation property for the exponential design problem, and it enables us to tell exactly to what extent the classical local linearization approach is applicable.     

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     

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.     

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 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.     

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.     

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.     

线性系统区域稳定的可靠控制

针对线性系统,提出了抵御系统执行器故障和不确定性的状态反馈区域稳定可靠控制设计问题.通过对具有连续增益故障控制系统的分析,给出了区域稳定可靠控制器存在的充分条件和设计方法.所设计的区域稳定可靠控制器可以消除执行器故障对闭环系统极点配置的影响.使用线性矩阵不等式(LMI)表示的结果能够容易得到可靠控制增益的数值解.通过CH-47双旋翼直升机的实例仿真验证了该结果的可行性.通过比较区域稳定正常控制系统和可靠控制系统,进一步说明对系统进行可靠控制设计的必要性.