FundamentalsonSOC Estimation:ObservabilityandObserverDesign
Fundamentals on SOC Estimation: Observability and Observer Design
State ObservabilityAx +Bun = the dimension of xCx + DuSuppose that the initial state x(O) is unknowny(t) - C ['e 4(-t) Bu(t)dt - Du(t) = Ce 4^ x(0)If we measure the input u(t) and output y(t), can weuniquely determine x(O) (then x(t) can be alsoderived)?
State Observability y x = Ax + Bu = Cx + Du Suppose that the initial state x(0) is unknown. 0 t At e Bu( A(t− ) y(t) −C )d − Du(t) = Ce x(0) If we measure the input u(t) and output y(t), can we uniquely determine x(0) (then x(t) can be also derived)? n = the dimension of x
CCAObservability Matrix: Wo =CAn-1The systemis observableif and onlyifthe observability matrixis full rank
C CAn−1 CA Observability Matrix: WO = The system is observable if and only if the observability matrix is full rank
Example:BatteryModelsVpRRpNocvThis is a linear circuit, but Vocv Is a nonlinearfunction of the SocVoev = f(s)
Rp R v i vocv + vp - + Cp - This is a linear circuit, but Vocv Is a nonlinear function of theSOC Vocv = f (s) Example: Battery Models
TheStateSpaceModelState equation is lineartRv(t) = Vocv+ Ri(t)+ v,(t) = f(s(t))+V,(t)+ Ri(t)Output equation is nonlinearThis model is mainlyused forthe SOC estimation:From the measured terminal voltage v(t) and the charge ordischarge current i(t), estimate the internal states, especially s(twhich is the state of charge (SOC)This is a nonlinear state observer or state estimationproblem
The State Space Model State equation is linear 1 1 Output equation is nonlinear Q s(t) = 1 i(t) vp (t) =− v p (t) + i(t) RpCp Cp v(t) = vocv + Ri(t)+ vp (t) = f (s(t)) + vp (t) + Ri(t) This model is mainly used for the SOC estimation: From the measured terminal voltage v(t) and the charge or discharge current i(t), estimate the internal states, especially s(t) which is the state of charge (SOC). This is a nonlinear state observer or state estimation problem