**Feedback Control Systems (FCS)** Lecture-26-27-28-29 State Space Canonical forms Dr. Imtiaz Hussain email: imtiaz.hussain@faculty.muet.edu.pk URL :http://imtiazhussainkalwar.weebly.com/

**Lecture Outline** • Canonical forms of State Space Models • Phase Variable Canonical Form • Controllable Canonical form • Observable Canonical form • Similarity Transformations • Transformation of coordinates • Transformation to CCF • Transformation OCF

**Canonical Forms** • Canonical forms are the standard forms of state space models. • Each of these canonical form has specific advantages which makes it convenient for use in particular design technique. • There are four canonical forms of state space models • Phase variable canonical form • Controllable Canonical form • Observable Canonical form • Diagonal Canonical form • Jordan Canonical Form • It is interesting to note that the dynamics properties of system remain unchanged whichever the type of representation is used. Companion forms Modal forms

**Phase Variable Canonical form** • The method of phase variables possess mathematical advantage over other representations. • This type of representation can be obtained directly from differential equations. • Decomposition of transfer function also yields Phase variable form.

**Phase Variable Canonical form** • Consider an nthorder linear plant model described by the differential equation • Where y(t) is the plant output and u(t) is the plant input. • A state model for this system is not unique but depends on the choice of a set of state variables. • A useful set of state variables, referred to as phase variables, is defined as:

**Phase Variable Canonical form** • Taking derivatives of the first n-1state variables, we have

**Phase Variable Canonical form** • Output equation is simply

**Phase Variable Canonical form** … ∫ ∫ ∫ ∫ ＋ ＋

**Phase Variable Canonical form**

**Phase Variable Canonical form (Example-1)** • Obtain the state equation in phase variable form for the following differential equation, where u(t) is input and y(t) is output. • The differential equation is third order, thus there are three state variables: • And their derivatives are (i.e state equations)

**Phase Variable Canonical form (Example-1)** • In vector matrix form Home Work: Draw Sate diagram

**Phase Variable Canonical form (Example-2)** • Consider the transfer function of a third-order system where the numerator degree is lower than that of the denominator. • Transfer function can be decomposed into cascade form • Denoting the output of the first block as W(s), we have the following input/output relationships:

**Phase Variable Canonical form (Example-2)** • Re-arranging above equation yields • Taking inverse Laplace transform of above equations. • Choosing the state variables in phase variable form + +

**Phase Variable Canonical form (Example-1)** • State Equations are given as • And the output equation is

**Phase Variable Canonical form (Example-1)** • State Equations are given as • And the output equation is

**Phase Variable Canonical form (Example-1)** • State Equations are given as • And the output equation is • In vector matrix form

**Companion Forms** • Consider a system defined by • where u is the input and y is the output. • This equation can also be written as • We will present state-space representations of the system defined by above equations in controllable canonical form and observable canonical form.

**Controllable Canonical Form** • The following state-space representation is called a controllable canonical form:

**Controllable Canonical Form**

**…** … ∫ ∫ ∫ ∫ ＋ ＋ ＋ Controllable Canonical Form

**Controllable Canonical Form (Example)** • Let us Rewrite the given transfer function in following form

**Controllable Canonical Form (Example)**

**Controllable Canonical Form (Example)** • By direct decomposition of transfer function • Equating Y(s) with numerator on the right hand side and U(s) with denominator on right hand side.

**Controllable Canonical Form (Example)** • Rearranging equation-2 yields • Draw a simulation diagram using equations (1) and (3) -2 -3 3 U(s) Y(s) 1/s 1/s P(s) 1

**Controllable Canonical Form (Example)** • State equations and output equation are obtained from simulation diagram. -2 -3 3 U(s) Y(s) 1/s 1/s P(s) 1

**Controllable Canonical Form (Example)** • In vector Matrix form

**Observable Canonical Form** • The following state-space representation is called an observable canonical form:

**Observable Canonical Form**

**Observable Canonical Form (Example)** • Let us Rewrite the given transfer function in following form

**Observable Canonical Form (Example)**

**Similarity Transformations** • It is desirable to have a means of transforming one state-space representation into another. • This is achieved using so-called similarity transformations. • Consider state space model • Along with this, consider another state space model of the same plant • Here the state vector , say, represents the physical state relative to some other reference, or even a mathematical coordinate vector.

**Similarity Transformations** • When one set of coordinates are transformed into another set of coordinates of the same dimension using an algebraic coordinate transformation, such transformation is known as similarity transformation. • In mathematical form the change of variables is written as, • Where T is a nonsingular nxn transformation matrix. • The transformed state is written as

**Similarity Transformations** • The transformed state is written as • Taking time derivative of above equation

**Similarity Transformations** • Consider transformed output equation • Substituting in above equation • Since output of the system remain unchanged [i.e. ] therefore above equation is compared with that yields

**Similarity Transformations** • Following relations are used to preform transformation of coordinates algebraically

**Similarity Transformations** • Invariance of Eigen Values

**Transformation to CCF** • Transformation to CCf is done by means of transformation matrix P. • Where CM is controllability Matrix and is given as and W is coefficient matrix Where the ai’s are coefficients of the characteristic polynomial s+

**Transformation to CCF** • Once the transformation matrix Pis computed following relations are used to calculate transformed matrices.

**Transformation to CCF (Example)** • Consider the state space system given below. • Transform the given system in CCF.

**Transformation to CCF (Example)** • The characteristic equation of the system is

**Transformation to CCF (Example)** • Now the controllability matrix CM is calculated as • Transformation matrix P is now obtained as

**Transformation to CCF (Example)** • Using the following relationships given state space representation is transformed into CCf as

**Transformation to OCF** • Transformation to CCf is done by means of transformation matrix Q. • Where OM is observability Matrix and is given as and W is coefficient matrix Where the ai’s are coefficients of the characteristic polynomial s+

**Transformation to OCF** • Once the transformation matrix Qis computed following relations are used to calculate transformed matrices.

**Transformation to OCF (Example)** • Consider the state space system given below. • Transform the given system in OCF.

**Transformation to OCF (Example)** • The characteristic equation of the system is

**Transformation to OCF (Example)** • Now the observability matrix OM is calculated as • Transformation matrix Q is now obtained as

**Transformation to CCF (Example)** • Using the following relationships given state space representation is transformed into CCf as

**Home Work** • Obtain state space representation of following transfer function in Phase variable canonical form, OCF and CCF by • Direct Decomposition of Transfer Function • Similarity Transformation • Direct Approach

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