Matlab Controller Design

1
Matlab Controller Design

• Control system toolbox
• State feedback and observer design
• Biochemical reactor example

2
Control System Toolbox

• Provides industry-standard algorithms and tools
  for systematically analyzing, designing and
  tuning linear control systems.
• System can be specified as a transfer function,
  state-space and pole-zero-gain model.
• Provides tools for model representation
  conversion and low-order approximation of
  high-order systems
• Allows series, parallel, feedback and general
  block-diagram connection of linear models.
• Interactive tools and command-line functions,
  such as the step response plot, allow
  visualization of system behavior in the time
  domain.
• Provides tools for automatic PID controller
  tuning, root locus analysis, and other
  interactive and automated techniques.
• Controller design can be validated by verifying
  rise time, overshoot, settling time and other
  requirements.

3
Selected Functions

• Model creation conversion
• tf create transfer function (TF) model
• zpk create zero/pole/gain (ZPK) model
• ss create state-space (SS) model
• System gain dynamics
• dcgain steady-state gain
• pole system poles
• zero system zeros
• pzmap pole-zero map

• Time domain analysis
• step step response
• stepinfo step response characteristics (rise
  time, ...)
• impulse impulse response
• lsim response to user-defined input signal
• lsiminfo linear response characteristics
• gensig generate input signal for lsim

4
Selected Functions cont.

• Compensator design
• place pole placement
• estim form estimator given estimator gain
• reg form regulator given state-feedback and
  estimator gains
• State-space models
• ctrb controllability matrix
• obsv observability matrix

• Time delays
• pade pade approximation of time delay
• Model dimensions characteristics
• class model type ('tf', 'zpk', 'ss', or 'frd')
• size model size and order
• isproper true for proper models
• isstable true for models with stable dynamics

5
Linear System Simulation

• y,t step(sys)
• Plots the step response of the model sys (created
  with either tf, zpk, or ss). The time range and
  number of points are chosen automatically.
• gtgt num-2 1
• gtgt den1 6 5
• gtgt gtf(num,den)
• Transfer function
• -2 s 1
• -------------
• s2 6 s 5
• gtgt step(g)

6
State Feedback and Observer Design

• CO ctrb(A,B) returns the controllability matrix
  B AB A2B ...
• OB obsv(A,C) returns the observability matrix
  C CA CA2 ...
• K place(A,B,P) computes a state-feedback matrix
  K such that the eigenvalues of A-BK are those
  specified in vector P. No eigenvalue should have
  a multiplicity greater than the number of inputs.
• L place(A,C,P) computes an observer matrix L
  such that the eigenvalues of A-LC are those
  specified in vector P. No eigenvalue should have
  a multiplicity greater than the number of inputs.

7
Biochemical Reactor Example

• Continuous bioreactor model
• 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
• gtgt sys 'bioreactor_stability'
• gtgt load_system(sys)
• gtgt open_system(sys)
• gtgt x1,u1,y1,dx1trim(sys,1 1,)
• gtgt x1
• x1
• 7.7818
• 0.5455

8
Linear Model Generation

• gtgt sys_io(1)linio('bioreactor_stability/Dilution'
  ,1,'in')
• gtgt sys_io(2)linio('bioreactor_stability/Bioreacto
  r',1,'out')
• gtgt 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

• gtgt lambdaeig(linsys.a)
• lambda
• -0.1500
• -3.6793

9
In-Class Exercise

• Construct the linear state-space model
• Check controllability and observability of the
  state-space model
• Compute the state feedback gains that place the
  eigenvalues of A-BK at l -0.05, -0.1
• Compute the observer gains that place the
  eigenvalues of A-LC at l -0.01, -0.02
• Simulate the performance of the output feedback
  controller

10
Controller and Observer Gains

• gtgt a-8.596e-005 1.472 -0.3748 -3.829
• gtgt b-7.78 19.45
• gtgt c1 0
• gtgt d0
• gtgt rank(ctrb(a,b))
• ans 2
• gtgt rank(obsv(a,c))
• ans 2

• gtgt p-0.25 -0.3
• gtgt kplace(a,b,p)
• k
• 1.8971 0.5903
• gtgt p-2.5 -3.0
• gtgt lplace(a',c',p)'
• l
• 1.6709
• 0.3737

11
Simulink Model