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 Similarity 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

DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS MATLAB Approach to the Discretization of Continuous Time State Equations Note: Default format is format short For more accuracy use format long Example: G and H differs for a different sampling period