Digital Control Systems

State Space Analysis(2)

STATE SPACE REPRESENTATIONS OF DISCRETE-TIME SYS

Nonuniqueness of State Space Representations

STATE SPACE REPRESENTATIONS OF DISCRETE-TIME SYS

Nonuniqueness of State Space Representations

≡

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:

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) :

SOLVING DISCRETE TIE STATE-SPACE EQUATIONS

State Transition Matrix

SOLVING DISCRETE TIE STATE-SPACE EQUATIONS

z Transform Approach to the Solution of Discrete-Time State Equations

SOLVING DISCRETE TIE STATE-SPACE EQUATIONS

z Transform Approach to the Solution of Discrete-Time State Equations

Example:

a)

b)

SOLVING DISCRETE TIE STATE-SPACE EQUATIONS

z Transform Approach to the Solution of Discrete-Time State Equations

Example:

a)

SOLVING DISCRETE TIE STATE-SPACE EQUATIONS

z Transform Approach to the Solution of Discrete-Time State Equations

Example:

a)

SOLVING DISCRETE TIE STATE-SPACE EQUATIONS

z Transform Approach to the Solution of Discrete-Time State Equations

Example:

a)

SOLVING DISCRETE TIE STATE-SPACE EQUATIONS

z Transform Approach to the Solution of Discrete-Time State Equations

Example:

a)

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

SOLVING DISCRETE TIE STATE-SPACE EQUATIONS

Solution of LTV Discrete-Time State Equations

SOLVING DISCRETE TIE STATE-SPACE EQUATIONS

Solution of LTV Discrete-Time State Equations

SOLVING DISCRETE TIE STATE-SPACE EQUATIONS

Solution of LTV Discrete-Time State Equations

Properties of

PULSE TRANSFER FUNCTION MATRIX

Pulse Transfer function matrix:

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.

DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS

DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS

Solution of Continuous Time State Equations

Properties of matrix exponential

DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS

Solution of Continuous Time State Equations

DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS

Discrete-time representation of

Discretization of Continuous Time State Equations

DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS

Discretization of Continuous Time State Equations

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)

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.

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

DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS

Discretization of Continuous Time State Equations

Example:

DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS

Discretization of Continuous Time State Equations

Example:

DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS

Discretization of Continuous Time State Equations

Example:

When T=1

ZOH G(s)T

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