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  • 3.1 State Space ModelsIn this section we study state space models of continuous-time lin-ear systems. The corresponding results for discrete-time systems,obtained via duality with the continuous-time models, are given inSection 3.3.

    The state space model of a continuous-time dynamic systemcan be derived either from the system model given in the timedomain by a differential equation or from its transfer functionrepresentation. Both cases will be considered in this section. Fourstate space formsthe phase variable form (controller form), theobserver form, the modal form, and the Jordan formwhich areoften used in modern control theory and practice, are presented.

    3.1.1 The State Space Model and Differential EquationsConsider a general th-order model of a dynamic system repre-sented by an th-order differential equation

    (3.1)At this point we assume that all initial conditions for the abovedifferential equation, i.e.

    ,

    are equal to zero. We will show later how to take into account theeffect of initial conditions.

    In order to derive a systematic procedure that transforms adifferential equation of order to a state space form representinga system of first-order differential equations, we first start witha simplified version of (3.5), namely we study the case when no

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  • 96 STATE SPACE APPROACH

    derivatives with respect to the input are present

    (3.2)Introduce the following (easy to remember) change of variables

    .

    .

    .

    (3.3)

    which after taking derivatives leads to

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    .

    (3.4)

  • STATE SPACE APPROACH 97

    The state space form of (3.8) is given by

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    (3.5)with the corresponding output equation obtained from (3.7) as

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    (3.6)

    The state space form (3.9) and (3.10) is known in the literature asthe phase variable canonical form.

    In order to extend this technique to the general case definedby (3.5), which includes derivatives with respect to the input, weform an auxiliary differential equation of (3.5) having the form of(3.6) as

    (3.7)

  • 98 STATE SPACE APPROACH

    for which the change of variables (3.7) is applicable

    .

    .

    .

    (3.8)

    and then apply the superposition principle to (3.5) and (3.11). Sinceis the response of (3.11), then by the superposition property

    the response of (3.5) is given by

    (3.9)Equations (3.12) produce the state space equations in the formalready given by (3.9). The output equation can be obtained byeliminating

    from (3.13), by using (3.11), that is

    This leads to the output equation

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    (3.10)It is interesting to point out that for , which is almost alwaysthe case, the output equation also has an easy-to-remember form

  • STATE SPACE APPROACH 99

    given by

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    .

    .

    (3.11)

    Thus, in summary, for a given dynamic system modeled by dif-ferential equation (3.5), one is able to write immediately its statespace form, given by (3.9) and (3.15), just by identifying coeffi-cients and , and using them to form thecorresponding entries in matrices and .

    Example 3.1: Consider a dynamical system represented by thefollowing differential equation

    !#"%$ !#&%$ !(')$ !

    $ !

    $ !#*%$ !

    $

    where!

    $

    stands for the th derivative, i.e.!

    $

    .

    According to (3.9) and (3.14), the state space model of the abovesystem is described by the following matrices

  • 100 STATE SPACE APPROACH

    3.1.2 State Space Variables from Transfer FunctionsIn this section, we present two methods, known as direct andparallel programming techniques, which can be used for obtainingstate space models from system transfer functions. For simplicity,like in the previous subsection, we consider only single-inputsingle-output systems.

    The resulting state space models may or may not contain all themodes of the original transfer function, where by transfer functionmodes we mean poles of the original transfer function (beforezero-pole cancellation, if any, takes place). If some zeros andpoles in the transfer function are cancelled, then the resulting statespace model will be of reduced order and the corresponding modeswill not appear in the state space model. This problem of systemreducibility will be addressed in detail in Chapter 5 after we haveintroduced the system controllability and observability concepts.

    In the following, we first use direct programming techniques toderive the state space forms known as the controller canonical formand the observer canonical form; then, by the method of parallelprograming, the state space forms known as modal canonical formand Jordan canonical form are obtained.

    The Direct Programming Technique and Controller Canon-ical Form

    This technique is convenient in the case when the plant transferfunction is given in a nonfactorized polynomial form

    +

    +

    +,-

    +,-

    - .

    +

    +,-

    +,-

    - .

    (3.12)

    For this system an auxiliary variable is introduced such that

  • STATE SPACE APPROACH 101

    the transfer function is split as

    /

    /01

    /01

    1 2

    (3.13a)

    /

    /

    /01

    /01

    1 2 (3.13b)

    The block diagram for this decomposition is given in Figure 3.1.

    U(s) V(s)V(s)/U(s) Y(s)/V(s)

    Y(s)

    Figure 3.1: Block diagram representation for (3.17)Equation (3.17a) has the same structure as (3.6), after the

    Laplace transformation is applied, which directly produces the statespace system equation identical to (3.9). It remains to find matricesfor the output equation (3.2). Equation (3.17b) can be rewritten as

    3

    3

    345

    345

    5 6 (3.14)indicating that is just a superposition of and its derivatives.Note that (3.17) may be considered as a differential equation in theoperator form for zero initial conditions, where . In thatcase, , , and are simply replaced with , ,and , respectively.

    The common procedure for obtaining state space models fromtransfer functions is performed with help of the so-called transferfunction simulation diagrams. In the case of continuous-time

  • 102 STATE SPACE APPROACH

    systems, the simulation diagrams are elementary analog computersthat solve differential equations describing systems dynamics. Theyare composed of integrators, adders, subtracters, and multipliers,which are physically realized by using operational amplifiers. Inaddition, function generators are used to generate input signals. Thenumber of integrators in a simulation diagram is equal to the orderof the differential equation under consideration. It is relatively easyto draw (design) a simulation diagram. There are many ways todraw a simulation diagram for a given dynamic system, and thereare also many ways to obtain the state space form from the givensimulation diagram.

    The simulation diagram for the system (3.17) can be ob-tained by direct programming technique as follows. Takeintegrators in cascade and denote their inputs, respectively, by7#8:9 7#8;=9 7?

  • STATE SPACE APPROACH 103

    xG 1xG

    1xG

    2xG

    2xG

    nxG n

    v (n)1/s1/s

    uH yIbJ 0

    bn

    bJ 2

    bJ 1

    -a0

    -a1

    -an-1

    vK (n-1) vK (1) v1/s

    Figure 3.2: Simulation diagram for the directprogramming technique (controller canonical form)

    A systematic procedure to obtain the state space form froma simulation diagram is to choose the outputs of integrators asstate variables. Using this convention, the state space model forthe simulation diagram presented in Figure 3.2 is obtained in astraightforward way by reading and recording information fromthe simulation diagram, which leads to

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    L M N OPM

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    (3.15)

  • 104 STATE SPACE APPROACH

    and

    Q Q R S S R RTS RTS R

    R

    (3.16)This form of the system model is called the controller canon-

    ical form. It is identical to the one obtained in the previous sec-tionequations (3.9) and (3.14). Controller canonical form playsan important role in control theory since it represents the so-calledcontrollable system. System controllability is one of the main con-cepts of modern control theory. It will be studied in detail inChapter 5.

    It is important to point out that there are many state space formsfor a given dynamical system, and that all of them are related bylinear transformations. More about this fact, together with thedevelopment of other important state space canonical forms, canbe found in Kailath (1980; see also similarity transformation inSection 3.4).

    Note that the MATLAB function tf2ss produces the statespace form for a given transfer function, in fact, it produces thecontroller canonical form.

    Example 3.2: The transfer function of the flexible beam fromSection 2.6 is given by

    U V W

    X Y

    U V W

    Using the direct programming technique with formulas (3.19) and

  • STATE SPACE APPROACH 105

    (3.20), the state space controller canonical form is given by

    and

    Direct Programming Technique and Observer CanonicalForm

    In addition to controller canonical form, observer canonicalform is related to another important concep

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