Download State Variables Outline Linear Systems

Outline
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State Variables
M. Sami Fadali
Professor of Electrical Engineering
UNR
State variables.
State-space representation.
Linear state-space equations.
Nonlinear state-space equations.
Linearization of state-space equations.
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Linear Systems
Input-output Description
• Single-input-single-output (SISO)
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The description is valid for
a) time-varying systems: parameters are 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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• Time-dependent coefficients for timevarying system.
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State Variables
Definitions
To solve the differential equation we need
for the period of interest.
(1) The system input
(2) A set of constant initial conditions.
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• 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.
System State: minimal set of numbers
, needed together with
௜ ଴
the input
଴ ௙ to uniquely
determine the behavior of the system in
the interval ଴ ௙ . = order of the system.
State Variables: As increases, the state of
the system evolves and each of the
becomes a time variable.
numbers ௜
State Vector: vector of state variables
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Notation
Definitions
• Column vector bolded
• Row vector bolded and transposed
-dimensional vector space where
represent the coordinate axes
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State plane: state space for a 2nd order system
Phase plane: special case where the state
variables are proportional to the derivatives of
.
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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).
State Space:
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Example 7.1
Example 7.1
• State for equation of motion of a point mass
driven by a force
= displacement of the point mass.
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Write state-space equations for the springmass-damper system driven by a force
= mass
= spring constant
= damper constant
= displacement of the point mass.
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2  system is second order
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Solution (cont.)
Solution
• Obtain a complete solution given the
force, if 2 initial conditions are known
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଴ : state of the system at time
଴.
• 2 I.C.s  system is second order.
• State (phase)Variables:
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• State Vector:
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• Two first order differential equations
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1.First equation: from definitions of state
variables.
2.Second equation: from equation of motion
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State-space Representation
State Equations
• 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.
• 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
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Solution of State Equations
For the spring-mass-damper system
• Solve the 1st order differential equations
then substitute in
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• 2 differential equations + algebraic
expression are equivalent to the 2nd order
differential equation.
• Feedback Control Law 2nd order
underdamped system
• Algebraic equation expressing the output
in terms of the state variables and the
input.
• 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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More on the Solution
Phase Portrait
1.Solution depends only on initial conditions.
2.Obtain phase portrait using MATLAB
command lsim or initial
3.Time is an implicit parameter.
4.Arrows indicate the direction of increasing
time.
5.Choice of state variables is not unique.
x2
3
2
1
0
-1
-2
-3
-2
x1
-1.5
-1
-0.5
0
0.5
1
1.5
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Matrix Form
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General Form for Linear Systems
Example 7.2: Matrix Form
Scalar Eqns.
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real vectors.
Matrices:
state or system matrix
input or control matrix
output matrix
direct transmission or
feedforward matrix
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State Space in MATLAB
Linear Vs. Nonlinear State-Space
» a = [0, 1; 5, 4];b=[0;1];c=[1,1];d=0;
» p = ss(a,b,c,d) % State-space quadruple
Example 7.3: The following are examples of
state-space equations for linear systems
a) 3rd order 2-input-2-output (MIMO) LTI
a=
x1
x2
x1
0
-5
x1
x2
u1
0
1
y1
x1
1
x2
1
-4
b=
. 0.3 15
.   x1  01
.
0
 x1   11
 u1 


 x    01


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.
.  
2.2 x 2  0 11
2
  
 u 2 
  
.   x 3  10
. 10
. 
 x 3  0.4 2.4 11
c=
x2
1
d=
u1
y1
0
Continuous-time system.
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Example 7.3 (b)
2nd order 2-output-1-input (SIMO) linear
time-varying
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1. Zero D, constant
+
and .
2. Time-varying system:
are functions of .
has entries that
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 x1 
 y1  1 2 0   1 2  u1 
 y   0 0 1  x 2   0 1 u 


 2 
 2 
 x 3 
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