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67*78 69*:8 ;8938?=

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Dynamic System Models

Mathematical Representations of

System Models

First-Order Equations State Representations Non-Linear Models Linearization

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First-Order Equations - I

Many, but not all, control design techniquesrequire that we have a mathematical model ofthe process.

Oftentimes a model is obtained by writing a setof (constitutive) equations that describe thephysical process.

For mechanical systems we frequently useNewtons Law:

net force = rate of change of momentum

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First-Order Equations - II

Assuming a constant mass, this law canbe written as the differential equation:

Differentiation with respect to time is oftenwritten with an overdot for each time

derivative:

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First-Order Equations - III

Models of dynamic processes are writtenas differential equations with time t asthe independent variable.

For mechanical systems, we typically have

x(t)=Position or Displacement

x(t)= Velocityx(t)=Acceleration

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First-Order Equations - IV

Example: Find the differential equation ofmotion for a mass-spring oscillatorexcited by an external force.

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First-Order Equations - V

Step 1: Draw the free-body diagram

Step 2: Apply Newtons law

The result is a second order diffeq in time:

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First-Order Equations - VI

Is there any way that we can write thissecond-order differential equation as aset of first-order differential equations?

What if we make the following definitions:

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First-Order Equations - VII

Substituting these into the original second-order differential equation:

mx2(t)+ kx1(t)= u(t)

Solving for the first derivative term:

The definition ofx1(t) andx2(t) provides thesecond differential equation:

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First-Order Equations - VIII

Now we can combine these two resultsinto one matrix expression:

This matrix expression is an alternative

representation of the original second-order differential equation.

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First-Order Equations - IX

Why is this important?

Any linear ordinary differential equation orset of differential equations can bewritten as a set of first-order coupleddifferential equations.

The original equations and the set of first-order equations represent the samedynamic system.

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In-Class Assignment - I

The dynamic equations for a DC motor are

Question #1: How many first-orderequations will be required to representthis dynamic system?

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In-Class Assignment - II

Question #2: Write the equations in first-order form using the following variables:

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Output Representation - I

What are outputs? The Outputs are quantities that we

measure either for feedback or for thepurpose of quantifying performance.

Outputs can always be written as linearcombinations of the Input and States (which

havent been formally defined yet)

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Output Representation - II

Lets go back to the spring-mass oscillatorexample:

What if we were interested in the spring

force and the acceleration as outputs?

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State Representations - I

We can now write out the full state-spacerepresentation for this example:

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State Representations - II

As indicated on the previous slide, thestandard notational convention for

describing a State-Space representation is

given by:

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State Representations - III

Conventional Nomenclature:

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State Representations - IV

What is the State of a dynamic system?The State is a set of variables which,

along with the current time, summarizes

the current configuration of a system.

States can be positions, velocities,

accelerations, forces, momentum,

torques, pressure, voltage, current,charge, etc.

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State Representations - V

Important State Representation facts:

The choice of state variables is not uniqueA model using a specific choice of state

variables is called a Realization

A system has infinitely many realizationsThe Dimension of a realization isN (the

number of states)

The Order of a system is the minimalnumber of state variables required todescribe it

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State Representations - VI

Example: Find an alternate realization forthe spring-mass oscillator example.

What if we choose momentum and spring

force as states? (Remember that we

chose position and velocity before)

x1(t)= mx(t)=

Momentumx2(t)= kx(t)= Spring force

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State Representations - VII

Substitute into the original diffeq:

Solve for the first derivative:

Relate the two new states to find the

second differential equation:

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State Representations - VIII

In state-space notation we have thefollowing new realization:

Compare this with the original realization:

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State Representations - IX

We can construct any new realization bytransforming the states using a non-

singular (i.e. full rank) transformation T:

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State Representations - X

Pre-multiply the state equations by T-1

Even though the state representation is notunique, the input-output relationships areinvariant.

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General State Representations

A set of first-order equations can bewritten in the following general form:

Where thefn andgp can be any function.

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General State Representations

There are three model classification types: LinearTime Invariant (LTI) systems

fn andgp are linear combinations of states

and inputs with no time-varying coefficients:x =Ax+Bu

y =Cx+Du

Linear Time Varying (LTV) systems

fn andgp are linear combinations of statesand inputs with time-varying coefficients:

x =A(t)x+B(t)u

y =C(t)x+D(t)u

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General State RepresentationsNon-Linear (NL) systems

fn and/orgp have terms with non-linearfunctions of states and/or inputs (i.e. sin,cos)

fn and/or gp have terms with states and/orinputs appearing as powers of somethingother than 1 or 0

fn and/orgp have terms with cross productsof states and/or inputs

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Linearization - I

Given a non-linear dynamic system, theequations can be Linearized using a Taylor-

series expansion, but only about some

specified operating point (x-( u-)

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Linearization - II

The linearized matrices are the Jacobianmatrices of f and g

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Linearization - III

Example: What if the original spring-mass

oscillator now has a non-linear spring?

The first-order diffeqs are:

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Linearization - IV

The Jacobian matrices for the stateequations evaluated at are:

Note: This result happens to be the sameas the original linear result because of a

careful choice of nonlinear spring!

.,