- Date:15 Sep 2020
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PART 1 BASIC OBSERVABILITY CONTROLLABILITY RESULTS. Observability Theorem in Discrete Time, The linear discrete time system with the corresponding measurements. is observable if and only if the observability matrix. has rank equal to,Observability Theorem in Continuous Time. The linear continuous time system with the corresponding measurements. is observable if and only if the observability matrix. has full rank equal to,Controllability Theorem in Discrete Time. The linear discrete time system,Controllability Theorem in Continuous Time. The linear continuous time system, is controllable if and only if the controllability matrix defined by.

has full rank equal to,Similarity Transformation,For a given system. we can introduce a new state vector by a linear coordinate transformation as. where is some nonsingular matrix A new state space model is obtained as. Eigenvalue Invariance Under a Similarity Transformation. A new state space model obtained by the similarity transformation does not. change internal structure of the model that is the eigenvalues of the system remain. the same This can be shown as follows, Note that in this proof the following properties of the matrix determinant have been. 2 3 4 2 3 4, Controllability Invariance Under a Similarity Transformation. The pair is controllable if and only if the pair, Observability Invariance Under a Similarity Transformation. The pair is observable if and only if the pair is observable. The proof of this theorem is as follows,The nonsingularity of implies.

PART 2 INTRODUCTION TO LINEAR OBSERVERS, Sometimes all state space variables are not available for measurements or it is not. practical to measure all of them or it is too expensive to measure all state space. variables In order to be able to apply the state feedback control to a system all. of its state space variables must be available at all times Also in some control. system applications one is interested in having information about system state space. variables at any time instant Thus one is faced with the problem of estimating. system state space variables This can be done by constructing another dynamical. system called the observer or estimator connected to the system under consideration. whose role is to produce good estimates of the state space variables of the original. The theory of observers started with the work of Luenberger 1964 1966. 1971 so that observers are very often called Luenberger observers According. to Luenberger any system driven by the output of the given system can serve as an. observer for that system, Two main techniques are available for observer design. The first one is used for the full order observer design and produces an observer. that has the same dimension as the original system. The second technique exploits the knowledge of some state space variables. available through the output algebraic equation system measurements so that a. reduced order observer is constructed only for estimating state space variables. that are not directly obtainable from the system measurements. Full Order Observer Design, Consider a linear time invariant continuous system. where with constant matrices having, appropriate dimensions Since the system output variables are available. at all times we may construct another artificial dynamic system of order built. for example of capacitors and resistors having the same matrices. and compare the outputs and, These two outputs will be different since in the first case the system initial.

condition is unknown and in the second case it has been chosen arbitrarily. The difference between these two outputs will generate an error signal. which can be used as the feedback signal to the artificial system such that the. estimation observation error is reduced as much as possible. hopefully to zero at least at steady state This can be physically realized by. proposing the system observer structure as given in the next figure. System observer structure, In this structure represents the observer gain and has to be chosen such that. the observation error is minimized The observer alone is given by. Note that the observer has the same structure as the system plus the driving. feedback term that contain information about the observation error. The role of the feedback term is to reduce the observation error. to zero at steady state, The observer is usually implemented on line as a dynamic system driven by the. same input as the original system and the measurements coming from the original. Introduction to Linear and Nonlinear Observers Zoran Gajic Rutgers University Part 1 Review Basic Observability Controllability Results Part 2 Introduction to Full and Reduced OrderLinear Observers Part 3 Introduction to Full and Reduced OrderNonlinear Observers 1 PART 1 BASIC OBSERVABILITY CONTROLLABILITY RESULTS Observability Theorem in Discrete Time The linear discrete

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