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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 {t=kT∣k ∈ I }, 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( t o) depends only on input at u(t o). A system is causal (nonanticipatory) iff output y(t o ) depends only on input u(t) for t o ≥ t. State The state of a system at t o , x( t o ), is the information required along with the input u(t) for t ≥ t o that uniquely determines the output y(t ) for t ≥ t o . Example: Find a state descriptions for the following system at t o =0 iL(t) u(t) Result: + 22 ΩΩ vc(t) _ 0.1H 0.5F 200 u t = ÿ t 101 ẏ t 120 y t x= [ ][ ] 10iL 0 y 0 = ẏ 0 100v c 0−10i L 0 + y(t) - 10Ω System Classes: Linear A system is linear iff for every t o and input-output pair (i =1,2) x i t o yi t , t ≥t o u i t , t ≥t o } then additivity holds: x 1 t o x 2 t o y1 t y 2 t ,t ≥t o u1 t u2 t , t≥t o and homogeneity holds: } } x i t o yi t , t ≥t o u i t , t ≥t o where α∈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: } x t o y zi t ,t≥t o ut≡0 , t ≥t o If state is zero, the response that results is due to the system input, known as the zero-state response: xt o =0 y zs t ,t≥t o ut , t ≥t o } In general for a linear system superposition holds between the contributions of the state and input to the response. Therefore, } x t o y zi t y zs ,t≥t o ut , t ≥t o Response Classes Zero-input response - system output due only to system state (or initial conditions). N 0=∑n=0 n n d y zi dt n Zero-state response - system output due only to the input of the system. N u=∑n=0 n n d y zs dt n , x=0 In general: Total response = Zero-input response + Zerostate response y= y zi y zs 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. • y=ku • u= ÿ ẏ y • y=ku10 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: ∞ y t =∫−∞ g t , u d where g t , is the system's time-varying impulse response at time τ. If system is zero-state (relaxed) at t o then integral can ∞ be written: y t=∫t g t , u d o If system also is causal, impulse response must be zero for τ>t: t y t=∫t g t ,u d o MIMO Input-Output Description For a p input and q output linear, causal, relaxed at t o , lumped or distributed system, the input-output relationship for a zerostate response can be expressed in terms of the convolution integral and the system's impulse response matrix: t y t =∫t G t ,ud o where G t , is the system's time-varying impulse response matrix describing the contribution of inputs at all p terminals to the q outputs: [ g 11 t , g 12 t , g t , g 22 t , Gt ,= 21 ⋮ ⋮ g q1 t , g q2 t , ... g 1p t , ... g 2p t , ... ⋮ ... g qp t , ] State-Space Description For a lumped system represented by an order N differential equation governing the state can be written as (p inputs): ẋ t=A t x tB t ut 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: y t =Ct x tD t ut 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 t o and input-output pair } x t o y t ,t ≥t o ut , t≥t o and any time shift T, the following also holds } x t oT y t T , t≥t oT ut−T , t≥t oT Time Invariance Linear systems that are time invariant are referred to as linear time invariant (LTI) systems. Their representations simplify to: t t y t =∫t G t ,ud y t=∫t G t−u d o o ẋ t=A t x tB t ut ẋ t=A x tB ut y t =Ct x tD t ut y t =C x tD ut 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. y s LT { g t }= = g s u s 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 statespace representation to obtain: y s =c s I−A x t o d c s I−A b u s −1 −1 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?