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 fora) time-varying systems: ai , cj , explicit functions of time.b) multi-input-multi-output (MIMO) systems: l input-outputdifferential equations, l = # of outputs.c) nonlinear systems: differential equations includenonlinear terms.

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State VariablesTo 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: incompleteknowledge of the set prevents complete solutionbut additional initial conditions are not needed toobtain the solution.• Initial conditions provide a summary of theHistory of the system up to the initial time.

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DefinitionsSystem State: minimal set of numbers {xi(t),i = 1,2,...,n}, needed together with the inputu(t), t ∈ [t0,tf) to uniquely determine thebehavior of the system in the interval [t0,tf].n = order of the system.State Variables: As t increases, the state ofthe system evolves and each of thenumbers 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 pointmass 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 EquationsState 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 EquationsSolve the 1st order differential equations then substitute in y = x1

2 differential equations + algebraic expression areequivalent to the 2nd order differential equation.Feedback Control Law 2nd order underdamped system u /m = −3x −9x1. 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 outputin terms of the state variables.• Multi-output systems: a scalar output equation is needed to define each output.• Substitute from solution of state equationto 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 writestate (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 anduniqueness 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 forcem = mass of 1 Kgb(y) = nonlinear damper constantk(y) = nonlinear spring force.Find the equilibrium position correspondingto a force f0 in terms of the spring force,then linearize the equation of motion aboutthis equilibrium.

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SolutionEquilibrium 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 whosevalues depend on the equilibrium.• Originally linear terms do not change withlinearization.