Control Systems/System Representations

System Representations This is a table of times when it is appropriate to use each different type of system representation: Properties State-SpaceEquations TransferFunction TransferMatrixLinear, Distributed no no noLinear, Lumped yes no noLinear, Time-Invariant, Distributed no yes noLinear, Time-Invariant, Lumped yes yes yes General Description These are the general external system descriptions. y is the system output, h is the system response characteristic, and x is the system input. In the time-variant cases, the general description is also known as the convolution description. General DescriptionTime-Invariant, Non-causalTime-Invariant, CausalTime-Variant, Non-CausalTime-Variant, Causal State-Space Equations These are the state-space representations for a system. y is the system output, x is the internal system state, and u is the system input. The matrices A, B, C, and D are coefficient matrices. Analog State Equations State-Space EquationsTime-InvariantTime-Variant These are the digital versions of the equations listed above. All the variables have the same meanings, except that the systems are digital. Digital State Equations State-Space EquationsTime-InvariantTime-Variant Transfer Functions These are the transfer function descriptions, obtained by using the Laplace Transform or the Z-Transform on the general system descriptions listed above. Y is the system output, H is the system transfer function, and X is the system input. Analog Transfer Function Transfer Function Digital Transfer Function Transfer Function Transfer Matrix This is the transfer matrix system description. This representation can be obtained by taking the Laplace or Z transforms of the state-space equations. In the SISO case, these equations reduce to the transfer function representations listed above. In the MIMO case, Y is the vector of system outputs, X is the vector of system inputs, and H is the transfer matrix that relates each input X to each output Y. Analog Transfer Matrix Transfer Matrix Digital Transfer Matrix Transfer Matrix