digital control systems state space analysis(2). state space representations of discrete-time sys...
TRANSCRIPT
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Digital Control Systems
State Space Analysis(2)
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STATE SPACE REPRESENTATIONS OF DISCRETE-TIME SYS
Nonuniqueness of State Space Representations
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STATE SPACE REPRESENTATIONS OF DISCRETE-TIME SYS
Nonuniqueness of State Space Representations
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SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
Solution of LTI Discrete-Tim State Equations
x(k) or any positive integer k may be obtined directly by recursion, as follows:
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SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
State Transition Matrix
It is possible to write the solution of the homogeneous state equation
as
state transition matrix(fundamental matrix) :
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SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
State Transition Matrix
![Page 7: Digital Control Systems State Space Analysis(2). STATE SPACE REPRESENTATIONS OF DISCRETE-TIME SYS Nonuniqueness of State Space Representations](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649d755503460f94a56354/html5/thumbnails/7.jpg)
SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
z Transform Approach to the Solution of Discrete-Time State Equations
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SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
z Transform Approach to the Solution of Discrete-Time State Equations
Example:
a)
b)
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SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
z Transform Approach to the Solution of Discrete-Time State Equations
Example:
a)
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SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
z Transform Approach to the Solution of Discrete-Time State Equations
Example:
a)
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SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
z Transform Approach to the Solution of Discrete-Time State Equations
Example:
a)
![Page 12: Digital Control Systems State Space Analysis(2). STATE SPACE REPRESENTATIONS OF DISCRETE-TIME SYS Nonuniqueness of State Space Representations](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649d755503460f94a56354/html5/thumbnails/12.jpg)
SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
z Transform Approach to the Solution of Discrete-Time State Equations
Example:
a)
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SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
Solution of LTV Discrete-Time State Equations
solution of x(k) may be found easily by recusion
State transition matrix
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SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
Solution of LTV Discrete-Time State Equations
![Page 15: Digital Control Systems State Space Analysis(2). STATE SPACE REPRESENTATIONS OF DISCRETE-TIME SYS Nonuniqueness of State Space Representations](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649d755503460f94a56354/html5/thumbnails/15.jpg)
SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
Solution of LTV Discrete-Time State Equations
![Page 16: Digital Control Systems State Space Analysis(2). STATE SPACE REPRESENTATIONS OF DISCRETE-TIME SYS Nonuniqueness of State Space Representations](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649d755503460f94a56354/html5/thumbnails/16.jpg)
SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
Solution of LTV Discrete-Time State Equations
Properties of
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PULSE TRANSFER FUNCTION MATRIX
Pulse Transfer function matrix:
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PULSE TRANSFER FUNCTION MATRIXSimilarity Transformation:
The pulse transfer function matrix is invariant under simiarity transformation.
The pulse transfer function does not depend on the particular state vector.
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DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS
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DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS
Solution of Continuous Time State Equations
Properties of matrix exponential
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DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS
Solution of Continuous Time State Equations
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DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS
Discrete-time representation of
Discretization of Continuous Time State Equations
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DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS
Discretization of Continuous Time State Equations
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DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS
Multiplying (2) by eAT and subtracting it from (1) gives:
Discretization of Continuous Time State Equations
Remember:
(1)
(2)
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DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS
Discretization of Continuous Time State Equations
G(T),H(T) depend on the sampling period C and D are constant matrices and do not depend on the sampling period T.
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DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS
Discretization of Continuous Time State Equations
Example:
This result agrees with the z transform of G(s), where it is preceded by a sampler and zero order hold
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DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS
Discretization of Continuous Time State Equations
Example:
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DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS
Discretization of Continuous Time State Equations
Example:
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DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS
Discretization of Continuous Time State Equations
Example:
When T=1
ZOH G(s)T
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DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS
MATLAB Approach to the Discretization of Continuous Time State Equations
Note:Default format is format shortFor more accuracy use format long
Example:
G and H differs for a different sampling period