State Variables

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Outline

• State variables.

• State-space representation.

• Linear state-space equations.

• Nonlinear state-space equations.

• Linearization of state-space equations.

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Input-output Description

The description is valid for

a) time-varying systems: ai , cj , explicit functions of time.

b) multi-input-multi-output (MIMO) systems: l input-output

differential equations, l = # of outputs.

c) nonlinear systems: differential equations include

nonlinear terms.

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State Variables

To solve the differential equation we need

(1) The system input u(t) for the period of interest.

(2) A set of constant initial conditions.

• Minimal set of initial conditions: incomplete

knowledge of the set prevents complete solution

but additional initial conditions are not needed to

obtain the solution.

• Initial conditions provide a summary of the

History of the system up to the initial time.

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Definitions

System State: minimal set of numbers {xi(t),

i = 1,2,...,n}, needed together with the input

u(t), t ∈ [t0,tf) to uniquely determine the

behavior of the system in the interval [t0,tf].

n = order of the system.

State Variables: As t increases, the state of

the system evolves and each of the

numbers xi(t) becomes a time variable.

State Vector: vector of state variables

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Notation

• Column vector bolded• Row vector bolded and transposed xT.

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Definitions

State Space: n-dimensional vector space where {xi(t), i = 1,2,...,n} represent the coordinate axes State plane: state space for a 2nd order system

Phase plane: special case where the state variables are proportional to the derivatives of the output.

Phase variables: state variables in phase plane. State trajectories: Curves in state space

State portrait: plot of state trajectories in the plane

(phase portrait for the phase plane).

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Example 7.1

• State for equation of motion of a point

mass m driven by a force f

• y = displacement of the point mass.

2 ⇒ system is second order

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Example 7.1 State Equations

State variables

State vector 2

Phase Variables: 2nd = derivative of the first.

Two first order differential equations

1. First equation: from definitions of state variables.

2. Second equation: from equation of motion.

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Solution of State Equations

Solve the 1st order differential equations then substitute in

y = x1

2 differential equations + algebraic expression are

equivalent to the 2nd order differential equation.

Feedback Control Law 2nd order underdamped system

u /m = −3x −9x

1. Solution depends only on initial conditions.

2. Obtain phase portrait using MATLAB command lsim,

3. Time is an implicit parameter.

4. Arrows indicate the direction of increasing time.

5. Choice of state variables is not unique.

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Phase Portrait

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State Equations

• Set of first order equations governing the state variables obtained from the input-output differential equation and the definitions of the state variables.

• In general, n state equations for a nth order system.

• The form of the state equations depends on the nature of the system (equations are time-varying for time-varying systems, nonlinear for nonlinear systems, etc.)

• State equations for linear time-invariant systems can also be obtained from their transfer functions.

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Output Equation

• Algebraic equation expressing the output

in terms of the state variables.

• Multi-output systems: a scalar output equation is needed to define each output.

• Substitute from solution of state equation

to obtain output.

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State-space Representation

• Representation for the system described by a differential equation in terms of state and output equations.

• Linear Systems: More convenient to write

state (output) equations as a single matrix equation

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Example 7.2

• The state-space equations for the system of Ex. 7.1

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General Form for Linear Systems

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State Space in MATLAB

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Linear Vs. Nonlinear State-Space

Example 7.3: The following are examples of state-space equations for linear systems a) 3rd order 2-input-2-output (MIMO) LTI

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Example 7.3 (b)2nd order 2-output-1-input (SIMO) linear time-varying

1. Zero direct D, constant B and C.2. Time-varying system: A has entries that are functions of t.

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Example 7.4: Nonlinear System

Obtain a state-space representation for the s-D.O.F. robotic manipulator from the equations of motion with output q.

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Solutionorder 2 s (need 2 s initial conditions to solve completely. State Variables

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Example 7.5

Write the state-space equations for the 2- D.O.F. anthropomorphic manipulator.

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Equations of Motion

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Solution

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Nonlinear State-space Equations

f(.) (n×1) and g(.) (l ×1) = vectors of functions satisfying mathematical conditions to guarantee the existence and

uniqueness of solution.

affine linear in the control: often encountered in practice

(includes equations of robotic manipulators)

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Linearization of State Equations

• Approximate nonlinear state equations by linear state equations for small ranges of the control and state variables.

• The linear equations are based on the first order approximation.

x0 constant, Δx = x - x0 = perturbation x0 .Approximation Error of order Δ2xAcceptable for small perturbations.

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Function of n Variables

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Nonlinear State-space Equations

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Perturbations Abt’ Equilibrium (x0, u0)

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Output Equation

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Linearized State-SpaceEquations

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Jacobians (drop "Δ"s)

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Example 7.6

Motion of nonlinear spring-mass-damper.

y = displacement f = applied force

m = mass of 1 Kg

b(y) = nonlinear damper constant

k(y) = nonlinear spring force.

Find the equilibrium position corresponding

to a force f0 in terms of the spring force,

then linearize the equation of motion about

this equilibrium.

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Solution

Equilibrium of the system with a force f0 (set all the time derivatives equal to zero and solve for y) Equilibrium is at zero velocity and the position y0.

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Linearize about the equilibrium

• Entries of state matrix: constants whose

values depend on the equilibrium.

• Originally linear terms do not change with

linearization.