# Zum Entwurf nichtlinearer zeitdiskreter Beobachter mittels Normalformen

## Abstract

Modern control design methods are based on the knowledge of all state variables of the considered system. Since a measurement of all states is in most cases not possible or too expensive, the use of observers is of great importance. Up to now, nonlinear observers have mainly been studied for continuous-time systems, however, discrete-time representations are of increasing interest. For a relatively small class of systems an observer design with linearizable error dynamics based on canonical forms is possible. This work gives an extension of the so called "Two-Step-Transformation" to nonlinear observer canonical form. This extension allows to enlarge the class of transformable systems considerably. Considering past measurements of the systems in- and output variables leads to the so called extended nonlinear observer canonical form which also allows to design an observer with linearizable error dynamics. The transformation into extended observer form includes several degrees of freedom which help to select the structure and the characteristics of the resulting observers. The extended observer form exists for every strongly locally observable system with one output. The transformation of a system with several outputs is subject to further conditions. Compared to the transformation into classical observer form, these conditions are noticeably less restrictive. The observers via extended observer form are compared to another design procedure, which can be found in the literature. For the latter, an alternative structure and an extension to systems with several outputs is presented in this work. The comparison of all considered observers includes the transient behaviour, robustness to noisy measurements, parameter sensitivity and the feasibility of the design procedure. One of the main tasks to use observers is the state feedback of dynamical systems. Since the separation principle which holds for all linear, timeinvariant systems does not hold in the nonlinear case, this work also focuses on the problem of nonlinear discrete-time observers for nonlinear state feedback. An experimental investigation of the closed loop dynamics was carried out for the stabilization of an inverted pendulum. The results show the general applicability to technical systems of all considered observers and furthermore significant differences between some observers in the closed loop were emphasized.

... In this paper we propose to use a general applicable non-linear observer theory in discrete time given in Lilge (1998Lilge ( , 1999 which uses, comparable to the continuous time observer theory used in Findeisen et al. (2003), a coordinate transformation of the original coordinates of the non-linear system equations. The state variable transformation results in another dynamic non-linear model, which has the same input/output behaviour. ...

... The state variable transformation results in another dynamic non-linear model, which has the same input/output behaviour. The model resulting from this state transformation is the so-called extended non-linear observer canonical form, which was studied in Lilge (1998Lilge ( , 1999. Designing an observer in this extended non-linear observer canonical form results in linear error dynamics, which means that the observer design can be performed using standard linear observer design techniques. ...

... The non-linear observer strategy proposed in Lilge (1998Lilge ( , 1999 is based on non-linear state equations that have a model structure which is called the extended non-linear canonical form. ...

Model predictive control in combination with discrete time non-linear observer theory is studied in this paper. Model predictive control, generally based on state space models, needs the complete state for feedback. In this paper the complete state is assumed not to be known and only outputs and inputs of the system are measured. To obtain knowledge of the full state an observer is used to obtain an estimate of the state. An extended non-linear observer is used for this purpose and potentially allows for successful output-based model predictive controllers.

... The nonlinear observer strategy proposed in [8] and [9] is based on nonlinear state equations that have a model structure which is called the Extended Nonlinear Canonical Form ...

... In [8] and [9] it has been proven that if the system equation given in (1) is strongly locally observable, then there exists a locally invertible coordinate transformation map S between the z-coordinates in which the proposed observer is defined and the x-coordinates on which the model of the Model Predictive Controller in section 2 is based (1). System (1) is strongly locally observable if the following condition is fulfilled ...

... The motivation for the coordinate transformation Z, can be explained by considering the structure of the Extended Nonlinear Observer Canonical Form (7) and taking into account that (19) For the derivation of (19) the reader is referred to [8] and [9]. ...

Model Predictive Control in combination with discrete time nonlinear observer theory is studied in this paper. Model Predictive Control, generally based on state space models, needs the complete state for feedback. In this paper the complete state is assumed not to be known and only outputs and inputs of the system are measured. To obtain knowledge of the full state an observer is used to obtain an estimate of the state. An extended nonlinear observer is used for this purpose and potentially allows for successful output based model predictive controllers.

... see [6], [7] and the references therein, to study the stability of the resulting closed-loop system. The extended observer design methodology from [8] is considered. The extended observer design has the advantage that it works (locally) under a very mild condition which is strong local observability of the system dynamics. ...

... First, some basic definitions and notations are given in Section II, together with basic NMPC notions. The observer theory of [8] is summarized in Section III. In Section IV we briefly explain the proposed NMPC scheme from which the problem set-up follows. ...

... In this paper we use the extended observer theory proposed in [8]. For notational brevity we consider the theory for the single input single output case, although the theory applies in the multiple input output case as well. ...

In this paper we present an asymptotically stabilizing output feedback control scheme for a class of nonlinear discrete-time systems. The presented scheme consists of an extended observer interconnected with an NMPC controller which represents a possible discontinuous state feedback control law. Local asymptotic stability of the resulting closed-loop system is proven

... In [38], new kinds of observer and its models described deeply. In [39], a new kinds of non-linear robotic system used which this article is in German language and in [40], sliding mode controller used for quadrotor's navigation and path planning. Another article proposed fractional-order of sliding mode controller for a quadrotors for obtaining some advantages in terms of less overshoot, weakening error and lower chattering effect [41]. ...

... Where 1 and 2 are the start-up (take-off) and system downtimes (flight time), respectively, is the row efficiency power, and is the theoretical maximum reliability and low cost of TR VTOL UAV as 3DoF robotic system with optimized load frequency control. To design and model TTR VTOL UAV non-linear adaptive fuzzy sliding mode-MPC observer based controller, a model structure which proposed in [2,3,[37][38][39][40][41][42] considered in this approach. This controller should examine reliability by using Lyapunov equation which divided into inner-loop and outer-loop controller which are similar. ...

Nowadays, Unmanned Aerial Vehicles (UAVs)-Robots use in many area. Flight control after takeoff is one of the main issue of UAVs-Robots in recent years. So proposing an optimal controller is so essential. Sliding Model controller is one of the controller and due to some internal and external disturbance such as noises and wind speed, considering uncertainty mode is important in adaptive mode. Also using an observer can reject these disturbances. In the other side, to satisfying some parameters, Model Predictive Control (MPC) considered. Combining observer with MPC can reject maximum disturbances. To tuning adaptive fuzzy sliding mode controller in observed-based MPC, a deep learning method applied in controller which is Deep Spiking Neural Network (DSNN). So, AFSM-DSNN (adaptive fuzzy sliding mode-deep spiking neural network) controller proposed for an UAV-Robot in flight mode with the maximum rejection of disturbances in robust forms. The stability and reliability of UAV-Robot modeled with Lyapunov equations which our model is three degree of freedom (3DoF). At the end, we use some performance measurement for evaluation criteria to guarantee the obtained results. We determine overshoot, undershoot, setting time, Sum Squared Error (SSE), Integral of Absolute Error (IAE), Integral of Square Error (ISE), and Time-weighted Absolute Error (ITAE) in additional load and drop load parts. That experimental results represent that we optimized about 0.9 % and 4% for overshoot, 1.2% and 0.3% for overshoot, 2.9% and 3.6% for setting time, 0.2% and 2.2% for SSE, 0.23% and 0.22% for IAE, 1.6% and 1.6% for ISE, and also 4.5% and 4.3% for ITAE In comparison to adaptive fuzzy sliding mode and MPC controller. Also two parameters of velocity tracking and control input in 150 seconds tested in simulation which proposed controller optimized 2% and 0.5%, respectively.

Synchronization of complex/chaotic systems is reviewed from a dynamical control perspective. It is shown that the notion of
an observer is essential in the problem of how to achieve synchronization between two systems on the basis of partial state
measurements of one of the systems. An overview of recent results on the design of synchronizing systems using observers is
given. Examples are given to demonstrate the main results.

Der Beitrag widmet sich dem Entwurf zeitdiskreter Beobachter für nichtlineare Systeme mit Hilfe der Transformation auf nichtlineare Beobachternormalform. Ausgehend von den Arbeiten von Brodmann 1994 sowie Lin und Byrnes 1995 wird durch die Verwendung von Altwerten des Systemausgangs die Systemklasse, für die ein Beobachterentwurf gelingt, beachtlich erweitert. Das angegebene Verfahren erlaubt zudem den Entwurf strukturell verschiedener Beobachter mit unterschiedlichen Eigenschaften. Als ein Beispiel für die praktische Anwendung der in diesem Beitrag vorgestellten Beobachterentwürfe dient ein Pendel auf einem Wagen. Die Betrachtungen beschränken sich auf Systeme mit einem Ausgang und mehreren Eingängen. Eine Anwendung des Verfahrens auf Systeme mit mehreren Ausgangsgrößen ist jedoch möglich.

This paper presents a geometric study of controllability for discrete-time nonlinear systems. Various accessibility properties are characterized in terms of Lie algebras of vector fields. Some of the results obtained are parallel to analogous ones in continuous-time, but in many respects the theory is substantially different and many new phenomena appear.

This paper focuses on the design of nonlinear observers for discrete-time systems by means of a so called extended nonlinear observer canonical form which is computed via a nonlinear observability canonical form. In contrast to other approaches by Brodmann 1994 and Lin and Byrnes 1995 using a two-step-transformation, past measurements of the system output are used. This allows to extend the class of systems for which an observer can be designed and leads to several observers with different characteristics. An application to a pendulum on a cart shows the efficiency of the design method.

This paper focuses on the design of non-linear observers for discrete-time systems by means of a transformation into non-linear observer canonical form based on recent results presented by Brodmann and by Lin and Byrnes. In contrast to their approaches, past measurements of the system output are used. This allows extension of the class of systems for which an observer can be designed and leads to several observers with different characteristics. Simulations show the efficiency of the design method and the differences between possible observers. In this paper, the observer design is limited to multi-input and single-output systems.

We describe an adaptive observer/identifier for single input single output observable nonlinear systems that can be transformed to a certain observable canonical form. We provide sufficient conditions for stability of this observer. These condi tions are in terms of the structure of the system and its canoni cal form, the boundedness of the parameter variations and the sufficient richness of some signals. We motivate the scope of our canonical form and the use of our observer/identifier by presenting applications to a number of nonlinear systems. In each case we present the specific stability conditions.

We consider the problems of stability and stabilization of implicit discretetime nonlinear scalar systems. Linear factorization is defined and results on local stability in terms of this concept are given. Special attention is given to polynomial systems. Necessary and sufficient conditions for stability of polynomial autonomous systems and for stabilizability of polynomial control systems are obtained.

A linear differential operator enables a compact description of the observability matrix for discrete-time analytic systems. For systems in observer canonical form local observability is ensured. Recursive equations that determine the transformation into this canonical form are derived. For systems linear with respect to the states this transformation always exists, if the system is locally observable. As a technical application a continuous-time bilinear model of a translatory hydraulic drive is considered, which has a state affine discrete-time equivalent. A canonical form observer design for this example is outlined.

Invariant distributions are defined for discrete-time nonlinear control systems, and necessary and sufficient conditions are given for their controlled invariance. This extends to discrete-time systems the basic tool which has been so important in solving the various synthesis problems for continuous-time systems. To indicate their utility in the discrete-time setting, they are used to locally solve the disturbance decoupling problem.

The factorization of rational matrix-valued functions of the complex variable s is frequently employed in designing robust controllers. Various factorization algorithms developed, e.g., by Youla, D.C., 1961,. Youla, D.C., et al. 1976; Francis, B.A., 1987,. Safonov, M.G., and Verma, M.S., 1985,. Vidyasagar, M., and Kimura, H., 1986 are presented in this chapter.

A decentral observer design for multiple Output nonlinear systems offers degrees offreedom that can be used to fulfill more easily the st riet requirements for the application of nonlinear design methods. To apply this technique, the system must be decomposed into observable subsystems which are connected by the unmeasured states in a cascade or a block-triangular form, respectively. If the system is not in a blocktriangular form, the form may be obtained by a nonlinear State transformation. This transformation, including the decentral observer design, is shown for the ball & beam example.

A compensator designed in the form of static state feedback ensures invariance of the output of a closed-loop system under disturbances for a subclass of discrete-time nonlinear systems.

Als anschauliches Beispiel für ein nichtlineares System mit mehreren Gleichgewichtslagen wurde am Institut für Regelungstechnik der Universität Hannover ein Laborversuch aufgebaut, bei dem ein Pendel aus beliebiger Anfangslage aufgeschwungen und in der labilen Gleichgewichtslage gehalten wird, Die Regelung erfolgt dabei mittels eines Pro-zeßrechners. Automatisierungstechnik at 38 (1990) 3.

The methods presented in this set of three books are based on the state concept. In this context, the actual implementation of a control law requires at each instant the knowledge of the state or of a part of it. We assume that the following variables are known:
The system inputs that are computed from the control algorithm,
The “outputs” measured by sensors.

In this paper, the stabilization problem for discrete-time systems by means of an output feedback law is considered. Sufficient
Lyapunov-like conditions as well as necessary conditions for nonlinear feedback stabilization are provided. Special emphasis
is given to the linear case.

The present paper is devoted to an analysis of the observability of discrete nonlinear stationary dynamical systems. This analysis is based on the algebraic properties of the functions describing the system. These properties are similar to the properties of partitions used in the analysis and synthesis of finite automata, a fact that enables us to establish a number of correspondences between the results of the theories of dynamical systems and finite automata.

Symbolic programming languages enable the computer-aided analysis and synthesis of time-variant and nonlinear control systems. By use of the symbolic language MACSYMA, a program is developed for the observability analysis and the observer design of nonlinear systems, and the application is shown for a simple example.

An attempt is made to design a globally stabilizing robust observer for an arbitrarily nonlinear plant with several equilibria and noisy but bounded output. The construction uses the Liapunov technique and avoids forming the 'error equation'. The known Luenberger observer follows as a particular case. The results are interpreted in terms of the total energy flow of the system and illustrated by the example of a nonlinear oscillator.

We characterize the equivalence of single-input single-output discrete-time nonlinear systems to linear ones, via a state-coordinate change and with or without feedback. Four cases are distinguished by allowing or disallowing feedback as well as by including the output map or not; the interdependence of these problems is analyzed. An important feature that distinguishes these discrete-time problems from the corresponding problem in continuous-time is that the state-coordinate transformation is here directly computable as a higher composition of the system and output maps. Finally, certain connections are made with the continuous-time case.

A necessary and sufficient condition is obtained for a discrete-time nonlinear system to be state equivalent to the nonlinear observer form in the multi-output cases. The result looks similar to the continuous counterpart except for the fact that Ad-operation is utilized instead of ad-operation

We investigate the effect of sampling on linearization for continuous time systems. It is shown that the discretized system is linearizable by state coordinate change for an open set of sampling times if and only if the continuous time system is linearizable by state coordinate change. Also, it is shown that linearizability via digital feedback imposes highly nongeneric constraints on the structure of the plant, even if this is known to be linearizable with continuous-time feedback. For n = 2, we show, under the assumption of completeness of , that if the discretized system is lineariable by state coordinate change and feedback, then the continuous time affine complete analytic system is linearizable by state coordinate change only. Also, we suggest a method of proof when n ≥ 3.

We consider a nonlinear discrete-time system of the form Σ: x(t+1)=f(x(t), u(t)), y(t) =h(x(t)), where x ε{lunate} RN, u ε{lunate} Rm, y ε{lunate} Rq and f and h are analytic. Necessary and sufficient conditions for local input-output linearizability are given. We show that these conditions are also sufficient for a formal solution to the global input-output linearization problem. Finally, we show that zeros at infinity of ε{lunate} can be obtained by the structure algorithm for locally input-output linearizable systems.

The concept of immersion, just recently introduced by Fliess and Kupka, is related to the capability of reproducing the input-output maps of a given system by means of another one. In this paper we give necessary and sufficient conditions under which a polynomial analytic system, i.e. a system whose dynamics are polynomial with respect to the input and nonlinear analytic with respect to the state, is immersed into a polynomial affine system, i.e. a system whose dynamics are polynomial in the input and linear in the state. An application to the problem of the reproducibility of a linear analytic state feedback acting on a given system by means of a preprocessor suitably initialized is also performed.

The purpose of the paper is twofold. First, based on a set of invariants, a local canonical form for a locally weakly observable system without inputs is derived. Second, a class of nonlinear control systems is introduced for which this set of invariants is unaffected by an arbitrary input function.

A new technique for constructing decoupled reduced-order discrete-time slow and fast observers for singularly perturbed continuous-time linear systems with piecewise-constant inputs is presented. The resulting state estimates are shown to converge to a neighborhood of the true states, in the sense that the state estimation error vanishes as the singular perturbation parameter tends to sero. The reduced-order multirate implementation results in significant computational savings over competing full-order techniques. Moreover, in the event that the singular perturbation parameter is sufficiently small with respect to the input sampling period, a simplification can be made which further reduces the computational requirements.

The authors address the problem of observability of polynomial discrete-time systems. The ideal theoretic definition is translated to effective computations in terms of Grobner bases. Linear system observability is a special case, and for general polynomial systems n samples are needed to determine observability, where n is the state dimension. The formulation yields a decision criterion as well as an implicit form of an observer

Systems theory and some canonical representations are introduced for a class of nonlinear systems. Techniques are devised to identify, synthesize, and model such systems and their signals. Nonlinear systems theory is introduced at a fundamental level. To identify a system from input and output (or just output) data, parameters are estimated for a canonical state-space or difference-equation representation, depending on which representation is most convenient for further analysis

The state-of-the-art of nonlinear state estimators or “observers” is reviewed. The use of these observers in real time nonlinear compensators is evaluated in terms of their on-line computational requirements. Their robustness properties are evaluated in terms of the extent to which the design requires a “perfect” model.

Necessary and sufficient conditions for local solvability of the title problem around a given trajectory are obtained. The proposed conditions are less restrictive than those obtained by Lee and Marcus for the problem of immersion a nonlinear system into a linear system via static state feedback. Instrumental in the problem solution is the inversion (structure) algorithm for a discrete-time nonlinear system. The solvability conditions are expressed in terms of the inversion algorithm. Moreover, the construction of both the dynamic state feedback and the immersion map relies on this algorithm. Finally, it is shown that, for locally right-invertible systems, the considered problem, unlike the problem of immersion via static state feedback, is always solvable.

This paper studies the problem of linearizing the input-output map of an analytic discrete-time nonlinear system locally around a given trajectory. Necessary and sufficient conditions are given for the existence of a regular dynamic state feedback control law under which the input-dependent part of the response of a nonlinear system becomes linear in the input and independent of the initial state. The proposed conditions are less restrictive than those obtained by Lee and Marcus for linearizing the input-output map via a static-state feedback. Instrumental in the problem solution is the inversion (structure) algorithm for a discrete-time nonlinear system. Firstly, the solvability conditions are expressed in terms of the inversion algorithm. Secondly, the proof of the existence and construction of the dynamic state feedback compensator relies on this algorithm.

This paper studies the non-linear observer design problem by observer canonical forms. We present necessary and sufficient conditions in terms of the ranks of matrices for the existence of a non-linear state coordinate change under which a time-variable system is transformed into an observer canonical form. These conditions give the multiple versions of Li and Tao (1986) for the single-output case. Then a comparison is made with the Lie algebraic conditions obtained by Krener and Respondek (1985), and modified by the authors in a recent paper (Xia and Gao 1987).

This document considers a single-input nonlinear discrete-time system of a certain form. Many authors have studied (local or global) linearization (Cheng et. al. 1985, Hunt and Su 1981, Jakubczyk and Respondek 1980, Krener 1973, Su 1982) and approximate linearization (Krener 1984) by state feedback and coordinate change for nonlinear continuous-time systems. This paper discusses necessary conditions and sufficient conditions for local linearization and approximate linearization by state feedback and coordinate change for nonlinear discrete-time systems. Other related work on nonlinear discrete-time systems can be found in (Grizzle 1985a, 1985b, Grizzle and Nijmeijer 1985, Monaco and Normand-Cyrot 1983a, 1983b). Keywords: Matrices(Mathematris).

An observer of canonical (phase-variable) form for non-linear time-variable systems is introduced. The development of this non-linear time-variable form requires regularity of the non-linear time-variable- observability matrix of the system. From the relationships derived during the development, it follows that a non-linear time-variable observer can be dimensioned by an eigenvalue assignment with respect to the canonical state coordinates if a linearization of system non-linearities about the reconstructed state trajectories is permissible. This is an assumption similar to that of the extended Kalman filter based on a linearization about the current estimate.

A controllability canonical form for non-linear time-variable systems with a control variable appearing linearly is introduced. The global controllability of this nonlinear phase-variable form is given independently of the non-linearities. A transformation to the controllability canonical form requires the regularity of a matrix which can be considered the controllability matrix of a non-linear time-variable system. A problem which still remains unsolved is the actual computation of the non-linear controllability canonical form for a given system.

An observer design method for a certain class of non-linear single output systems is introduced. The characteristic feature of this method consists of the fact that it does not require any linearization in the way that the given non-linear system is approximated by a linear one. The present paper deals with the derivation of a transformation of the considered non-linear system into a generalized observer canonical form (GOCF) which enables a systematic observer design similar to the linear one based on the well-known linear observer canonical form. To assign conditions for its existence, the transformation into the GOCF is carried out in two steps via a generalized observability canonical form (GOBCF). In contrast to previous forms used for linear systems, the two non-linear canonical forms presented here also depend on the time derivatives of the input variables. This means that the resulting observer has to be supplied, not only with the input and output variables of the given system, but also with derivatives of the input variables. However, a final example shows that it is possible to eliminate those derivatives in special cases.