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Simulink Stability Analysis

Computing Steady-State Solutions. Matlab function trim ? finds steady state solutions for a Simulink system >> [x,u,y,dx]=trim(sys,x0,x0)Attempts to find values for x, u and y that set the state derivatives, dx, of the S-function sys to zero using a constrained optimization technique.Sets the ini

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Simulink Stability Analysis

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    1. Simulink Stability Analysis Computing steady-state solutions Constructing linearized models Biochemical reactor example

    2. Computing Steady-State Solutions Matlab function trim – finds steady state solutions for a Simulink system >> [x,u,y,dx]=trim(sys,x0,x0) Attempts to find values for x, u and y that set the state derivatives, dx, of the S-function sys to zero using a constrained optimization technique. Sets the initial starting guesses for x and u to x0 and u0, respectively.

    3. Computing Linearized Models Matlab function linearize – obtains a linear model from a Simulink model >> linsys = linearize(sys,sys_io) Takes a Simulink model, sys, and an I/O object, sys_io, as inputs and returns a linear state-space model, linsys. The linearization I/O object is created with the function linio. >> sys_io=linio(blockname,portnum,type) Creates a linearization I/O object for the signal that originates from the outport with port number, portnum, of the type given by type of the block, blockname, in a Simulink model. Available linearization I/O types include: 'in', linearization input point 'out', linearization output point

    4. Biochemical Reactor Example Continuous bioreactor model Parameter values KS = 1.2 g/L, mmax = 0.48 h-1, YX/S = 0.4 g/g D = 0.15 h-1, Si = 20 g/L Steady-state solutions Eigenvalues

    5. Simulink Model

    6. S-Function

    7. S-Function cont.

    8. S-Function cont.

    9. In-Class Exercise Use the Matlab function trim to find the two steady-state solutions Use the Matlab function linearize to find a linear model at the non-trivial steady state Use the Matlab function eig to check the stability of the non-trivial steady state

    10. Steady-State Solutions >> sys = 'bioreactor_stability'; >> load_system(sys); >> open_system(sys); >> [x1,u1,y1,dx1]=trim(sys,[1; 1],[]); >> x1 x1 = 7.7818 0.5455 >> [x2,u2,y2,dx2]=trim(sys,[0; 0],[]); >> x2 x2 = 0.0000 20.0000

    11. Linear Model Analysis >> sys_io(1)=linio('bioreactor_stability/Dilution',1,'in'); >> sys_io(2)=linio('bioreactor_stability/Bioreactor',1,'out'); >> linsys = linearize(sys,sys_io) a = Bioreactor(1 Bioreactor(2 Bioreactor(1 -8.596e-005 1.472 Bioreactor(2 -0.3748 -3.829 b = Dilution (pt Bioreactor(1 -7.78 Bioreactor(2 19.45 c = Bioreactor(1 Bioreactor(2 bioreactor_s 1 0 d = Dilution (pt bioreactor_s 0 >> lambda=eig(linsys.a) lambda = -0.1500 -3.6793

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