Transfer function approach of system modeling provides 64257nal relation between output variable and input variable However a system may have other internal variables of import ance State variable representa tion takes into account of all such inter ID: 24615 Download Pdf

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Transfer function approach of system modeling provides 64257nal relation between output variable and input variable However a system may have other internal variables of import ance State variable representa tion takes into account of all such inter

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Digital Control Module 7 Lecture 1 Module 7: Discrete State Space Models Lecture Note 1 1 Introduction to State Variable Model In the preceding lectures, we have learned how to design a sam pled data control system or a digital system using the transfer function of the system to be controlled. Transfer function approach of system modeling provides ﬁnal relation between output variable and input variable. However, a system may have other internal variables of import ance. State variable representa- tion takes into account of all such internal variables. More over,

controller design using classical methods, e.g., root locus or frequency domain method are lim ited to only LTI systems, partic- ularly SISO (single input single output) systems since for M IMO (multi input multi output) systems controller design using classical approach become s more complex. These limitations of classical approach led to the development of state variable approach of system modeling and control which formed a basis of modern control theory. State variable models are basically time domain models wher e we are interested in the dynamics of some characterizing variables called

state variables wh ich along with the input represent the state of a system at a given time. State: The state of a dynamic system is the smallest set of var iables, , such that given ) and , t > t , t > t can be uniquely determined. Usuallyasystemgovernedbya th orderdiﬀerentialequationor th ordertransferfunction is expressed in terms of state variables: ,x ,...,x The generic structure of a state-space model of a th order continuous time dynamical system with input and output is given by: ) = )+ ) : State Equation (1) ) = )+ ) : Output Equation where, ) is the dimensional state vector, )

is the dimensional input vector, is the dimensional output vector and Example Consider a nth order diﬀerential equation dt dt ... I. Kar

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Digital Control Module 7 Lecture 1 Deﬁne following variables, dy dt . = dt dt ... The nth order diﬀerential equation may be written in the form of ﬁrst order diﬀerential equations as . = ... or in matrix form as, Bu where 0 1 0 ... 0 0 1 ... ... 0 0 0 ... The output can be one of states or a combination of many states . Since, = [1 0 0 0 ... 0] 1.1 Correlation between state variable and transfer functio ns

models The transfer function corresponding to state variable mode l (1), when and are scalars, is: ) = sI (2) sI where sI is the characteristic polynomial of the system. 1.2 Solution of Continuous Time State Equation The solution of state equation (1) is given as ) = )+ Bu d where At = Φ( ) is known as the state transition matrix and ) is the initial state of the system. I. Kar

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Digital Control Module 7 Lecture 1 2 State Variable Analysis of Digital Control Systems The discrete time systems, as discussed earlier, can be clas siﬁed in two types. 1. Systems that

result from sampling the continuous time sys tem output at discrete instants only, i.e., sampled data systems. 2. Systems which are inherently discrete where the system st ates are deﬁned only at discrete time instants and what happens in between is of no concern to u s. 2.1 State Equations of Sampled Data Systems Let us assume that the following continuous time system is su bject to sampling process with an interval of ) = )+ Bu ) : State Equation (3) ) = )+ Du ) : Output Equation We know that the solution to above state equation is: ) = Φ( )+ Φ( Bu d Since the inputs are

constants in between two sampling insta nts, one can write: ) = kT ) for , kT +1) which implies that the following expression is valid within the interval kT +1) if we consider kT ) = Φ( kT kT )+ kT Φ( Bu kT d Let us denote kT Φ( Bd by KT ). Then we can write: ) = Φ( kT kT )+ KT kT If = ( +1) (( +1) ) = Φ( kT )+ kT ) (4) where Φ( ) = AT and ) = +1) kT Φ(( +1) Bd . If kT , we can rewrite ) as ) = Φ( Bdt . Equation (4) has a similar form as that of equation (3) if we consider ) = and ) = . Similarly by setting kT , one can show that the output equation

also has a similar form as that of the continuous tim e one. I. Kar

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Digital Control Module 7 Lecture 1 When = 1, +1) = Φ(1) )+ (1) ) = )+ Du 2.2 State Equations of Inherently Discrete Systems When a discrete system is composed of all digital signals, th e state and output equations can be described by +1) = )+ Bu ) = )+ Du 2.3 Discrete Time Approximation of A Continuous Time State Spa ce Model Let us consider the dynamical system described by the state s pace model (3). By approximating the derivative at kT using forward diﬀerence, we can write: kT (( +1) kT )] ((

+1) kT )] = kT )+ Bu kT and , y kT ) = kT )+ Du kT Rearranging the above equations, (( +1) ) = ( TA kT )+ TBu kT If, = 1 +1) = ( )+ Bu and ) = )+ Du We can thus conclude from the discussions so far that the disc rete time state variable model of a system can be described by +1) = )+ Bu ) = )+ Du where are either the descriptions of an all digital system or obtai ned by sampling the continuous time process. I. Kar

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