State Space Models The state space model represents a physical system as n first order differential equations. This form is better suited for computer simulation than an nth order input- output differential equation.

RLC serial network – second order system Select two state variables, State Space Model

RLC serial network – second order system Derivation of first state Equation,

State Space Model RLC serial network – second order system Derivation of second state Equation,

State equation in Matrix form, State Space Model

RLC serial network – second order system Output Equation,

State Space Model RLC serial network – second order system Output Equation in Matrix form,

State Space Models

Create state-space model - MATLAB >> a = [ ; ]; >> b = [1 -1;0 2]; >> c = [ ]; >> sys = ss(a,b,c,0); sys = ss(a,b,c,d) creates a state-space model object representing the continuous-time state-space model

Impulse response plot of dynamic system >> impulse(sys) impulse calculates the unit impulse response of a dynamic system model. impulse(sys) plots the impulse response of the dynamic system model sys. For state-space models, impulse assumes initial state values are zero. Plot the impulse response of the second-order state- space model

Left plot → impulse response of the first input channel Right plot → impulse response of the second input channel. Store the impulse response data in MATLAB arrays by >> [y,t] = impulse(sys) Impulse response plot of dynamic system

Step response plot of dynamic system step calculates the step response of a dynamic system. For the state space case, zero initial state is assumed. step(sys) plots the step response of an arbitrary dynamic system model sys. >> step(sys)

Step response plot of dynamic system Left plot → impulse response of the first input channel Right plot → impulse response of the second input channel.

Stability and Eigenvalues What do eigenvalues of system matrix tell us about stability? Eigenvalues tell us the exponential part of the solution of the differential equation system Three possible values for an eigenvalue 1) Positive value: system will increase exponentially 2) Negative value: system will decay exponentially 3) Imaginary value: system will oscillate If real component of at least one eigenvalue is positive, then the system is unstable

Stability and Eigenvalues EigenvalueEffect on system when disturbed Positive realDriven away from steady-state value Negative realDriven back to stead-state value 0Remains at position to which it was disturbed Complex, positive real Oscillates around steady-state value with increasing amplitude Complex, negative real Oscilate around steady-state value with decreasing amplitude ImaginaryOscillates around ss value with constant amplitude

Eigenvalue - MATLAB Stability? E = eig(X) is a vector containing the eigenvalues of a square matrix X [V,D] = eig(X) produces a diagonal matrix D of eigenvalues and matrix V whose columns are corresponding eigenvectors

Problem An RLC network is shown in figure. Define the state variable as :- X 1 = i 1, X 2 = i 2, X 3 = V c Let voltage across capacitor, V c is the output from the network. Input of the system is V a and V b Determine the state space representation of the RLC network in matrix form?

18 solution State variables and their derivatives

19 solution The derivatives equations for energy storage elements

20 solution For loop (1) ; For loop (2) ;

21 solution For current i C ; Substituting equation (4), (5) and (6) into equation (1), (2) and (3) yields

22 solution Substituting equation (4), (5) and (6) into equation (1), (2) and (3) yields

23 solution Substituting equation (4), (5) and (6) into equation (1), (2) and (3) yields

24 solution Rewrite equation (7), (8) and (9) in state space representation matrix form