EE611 Deterministic Systems System Descriptions, State, Convolution Kevin D. Donohue Electrical and Computer Engineering University of Kentucky

Input-Output Systems SISO- systems with a single input and single output. MIMO- systems with multiple inputs and multiple outputs. Continuous-time systems – inputs and output are defined over all time t (SISO, input u(t), output y(t), MIMO, input u(t), output y(t)) Discrete-time systems – inputs and output are defined at discrete points in time, with sampling interval T (u(kT) = u[k], y(kT) = y[k] ).

System Classes A system is memoryless (instantaneous) iff (if and only if) output y( ) depends only on input at u( ). A system is causal (nonanticipatory) iff output y( ) depends only on input u(t) for ≥ t.

State The state of a system at, x( ), is the information required along with the input u(t) for t ≥ that uniquely determines the output y( ) for t ≥. Example: Find a state descriptions for the following system at =0 Result:

System Classes: Linear A system is linear iff for every  and input-output pair (i =1,2) then additivity holds: and homogeneity holds: where  R

Zero-State, Zero-Input Response If input is zero, the response that results is due to the system state, known as the zero-input response: If state is zero, the response that results is due to the system input, known as the zero-state response: In general for a linear system superposition holds between the contributions of the state and input to the response. Therefore,

Response Classes Zero-input response - system output due only to system state (or initial conditions). Zero-state response - system output due only to the input of the system. In general: Total response = Zero-input response + Zero- state response

Examples of Linearity Determination Determine whether or not each system described below is linear. Assume inputs and outputs are functions of time denoted by u and y, and constants are denoted k. 

Input-Output Description Convolution For a linear lumped or distributed system, the input-output relationship for a zero-state response can be expressed in terms of the convolution integral and the system's impulse response: where is the system's time-varying impulse response at time . If system is zero-state (relaxed) at then integral can be written: If system also is causal, impulse response must be zero for  t:

MIMO Input-Output Description For a p input and q output linear, causal, relaxed at, lumped or distributed system, the input-output relationship for a zero- state response can be expressed in terms of the convolution integral and the system's impulse response matrix: where is the system's time-varying impulse response matrix describing the contribution of inputs at all p terminals to the q outputs:

State-Space Description For a lumped system represented by an order N differential equation governing the state can be written as (p inputs): where A is an NxN matrix, x is a Nx1 vector, B is a Nxp matrix and u is a 1xp vector. The output (q outputs) is a linear combination of the states and inputs and can be written as: where C is an qxN matrix, x is an Nx1 vector, D is a qxp matrix and u is a 1xp vector.

Time Invariance If system is not changing over time, it is referred to as time invariant and results in significant simplifications. More formally stated: A system is time invariant iff for every and input-output pair and any time shift T, the following also holds

Time Invariance Linear systems that are time invariant are referred to as linear time invariant (LTI) systems. Their representations simplify to:

Transfer Functions The transfer function TF of an LTI system can be derived from the Laplace Transform of its input-output description. Show for a relaxed system, the Laplace Transform of the impulse response is its transfer function.

Transfer Functions and State Space For a SISO system, derive the relationship between TF and the zero-state and zero-input responses by taking the LT of a state- space representation to obtain: Find the formula to convert a state-space representation to a TF for the zero-state case. ➢ Is it possible for a TF to represent the case when the state is not zero (not a relaxed system)? ➢ What is the significance of the d parameter?