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Anderson


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 -- Example Script: Anderson

     Frequency-weighted coprime factorization controller reduction [1].


     *References*
     [1] Anderson, B.D.O.: ‘Controller Reduction: Concepts and
     Approaches’, IEEE Transactions of Automatic Control, Vol.  34, No.
     8, August 1989.


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Frequency-weighted coprime factorization controller reduction [1].



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BMWengine


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 -- Function File: SYS = BMWengine ()
 -- Function File: SYS = BMWengine ("SCALED")
 -- Function File: SYS = BMWengine ("UNSCALED")
     Model of the BMW 4-cylinder engine at ETH Zurich's control
     laboratory.

          OPERATING POINT
          Drosselklappenstellung     alpha_DK = 10.3 Grad
          Saugrohrdruck              p_s = 0.48 bar
          Motordrehzahl              n = 860 U/min
          Lambda-Messwert            lambda = 1.000
          Relativer Wandfilminhalt   nu = 1

          INPUTS
          U_1 Sollsignal Drosselklappenstellung   [Grad]
          U_2 Relative Einspritzmenge             [-]
          U_3 Zuendzeitpunkt                      [Grad KW]
          M_L Lastdrehmoment                      [Nm]

          STATES
          X_1 Drosselklappenstellung     [Grad]
          X_2 Saugrohrdruck              [bar]
          X_3 Motordrehzahl              [U/min]
          X_4 Messwert Lamba-Sonde       [-]
          X_5 Relativer Wandfilminhalt   [-]

          OUTPUTS
          Y_1 Motordrehzahl              [U/min]
          Y_2 Messwert Lambda-Sonde      [-]

          SCALING
          U_1N, X_1N   1 Grad
          U_2N, X_4N, X_5N, Y_2N   0.05
          U_3N   1.6 Grad KW
          X_2N   0.05 bar
          X_3N, Y_1N   200 U/min


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Model of the BMW 4-cylinder engine at ETH Zurich's control laboratory.



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Boeing707


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 -- Function File: SYS = Boeing707 ()
     Creates a linearized state-space model of a Boeing 707-321 aircraft
     at V=80 m/s (M = 0.26, GA0 = -3 deg, ALPHA0 = 4 deg, KAPPA = 50
     deg).

     System inputs:
       1. Thrust
       2. Elevator angle
     System outputs:
       1. Airspeed
       2. Pitch angle

     *Reference*: R. Brockhaus: ‘Flugregelung’ (Flight Control),
     Springer, 1994.


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Creates a linearized state-space model of a Boeing 707-321 aircraft at
V=80 m...



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MDSSystem


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 -- Function File: MDSSystem ()

     Run the example for robust control of a mass-damper-spring system.

   Run ‘edit MDSSystem’ to open the example file.


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Run the example for robust control of a mass-damper-spring system.



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Madievski


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 -- Example Script: Madievski
     Demonstration of frequency-weighted controller reduction.

     The system considered in this example has been studied by Madievski
     and Anderson [1] and comprises four spinning disks.  The disks are
     connected by a flexible rod, a motor applies torque to the third
     disk, and the angular displacement of the first disk is the
     variable of interest.  The state-space model of eighth order is
     non-minimumphase and unstable.  The continuous-time LQG controller
     used in [1] is open-loop stable and of eighth order like the plant.
     This eighth-order controller shall be reduced by frequency-weighted
     singular perturbation approximation (SPA). The major aim of this
     reduction is the preservation of the closed-loop transfer function.
     This means that the error in approximation of the controller K by
     the reduced-order controller KR is minimized by
          min ||W (K-Kr) V||
           Kr               inf
     where weights W and V are dictated by the requirement to preserve
     (as far as possible) the closed-loop transfer function.  In
     minimizing the error, they cause the approximation process for K to
     be more accurate at certain frequencies.  Suggested by [1] is the
     use of the following stability and performance enforcing weights:
                       -1                      -1
          W = (I - G K)   G,      V = (I - G K)
     This example script reduces the eighth-order controller to orders
     four and two by the function call ‘Kr = spaconred (G, K, nr,
     'feedback', '-')’ where argument NR denotes the desired order (4 or
     2).  The key-value pair ‘'feedback', '-'’ allows the reduction of
     negative feedback controllers while the default setting expects
     positive feedback controllers.  The frequency responses of the
     original and reduced-order controllers are depicted in figure 1,
     the step responses of the closed loop in figure 2.  There is no
     visible difference between the step responses of the closed-loop
     systems with original (blue) and fourth order (green) controllers.
     The second order controller (red) causes ripples in the step
     response, but otherwise the behavior of the system is unaltered.
     This leads to the conclusion that function ‘spaconred’ is well
     suited to reduce the order of controllers considerably, while
     stability and performance are retained.


   *Reference*
[1] Madievski, A.G. and Anderson, B.D.O. ‘Sampled-Data Controller
Reduction Procedure’, IEEE Transactions of Automatic Control, Vol.  40,
No.  11, November 1995


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Demonstration of frequency-weighted controller reduction.



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VLFamp


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 -- Function File: VLFamp
 -- Function File: RESULT = VLFamp (VERBOSE)
     Calculations on a two stage preamp for a multi-turn, air-core
     solenoid loop antenna for the reception of signals below 30kHz.

     The Octave Control Package functions are used extensively to
     approximate the behavior of operational amplifiers and passive
     electrical circuit elements.

     This example presents several 'screen' pages of documentation of
     the calculations and some reasoning about why.  Plots of the
     results are presented in most cases.

     The process is to display a 'screen' page of text followed by the
     calculation and a 'Press return to continue' message.  To proceed
     in the example, press return.  ^C to exit.

     At one point in the calculations, the process may seem to hang,
     but, this is because of extensive calculations.

     The returned transfer function is more than 100 characters long so
     will wrap in screens that are narrow and appear jumbled.


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Calculations on a two stage preamp for a multi-turn, air-core solenoid
loop a...



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WestlandLynx


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 -- Function File: SYS = WestlandLynx ()
     Model of the Westland Lynx Helicopter about hover.
          INPUTS
          main rotor collective
          longitudinal cyclic
          lateral cyclic
          tail rotor collective

          STATES
          pitch attitude           theta       [rad]
          roll attitude            phi         [rad]
          roll rate (body-axis)    p           [rad/s]
          pitch rate (body-axis)   q           [rad/s]
          yaw rate                 xi          [rad/s]
          forward velocity         v_x         [ft/s]
          lateral velocity         v_y         [ft/s]
          vertical velocity        v_z         [ft/s]

          OUTPUTS
          heave velocity           H_dot       [ft/s]
          pitch attitude           theta       [rad]
          roll attitude            phi         [rad]
          heading rate             psi_dot     [rad/s]
          roll rate                p           [rad/s]
          pitch rate               q           [rad/s]

     *References*
     [1] Skogestad, S. and Postlethwaite I. (2005)
‘Multivariable
     Feedback Control: Analysis and Design:
Second Edition’.  Wiley.
     <http://www.nt.ntnu.no/users/skoge/book/2nd_edition/matlab_m/matfiles.html>


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Model of the Westland Lynx Helicopter about hover.



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acker


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 -- Function File: K = acker (A, B, P)
     Calculates the state feedback matrix of a completely controllable
     SISO system using Ackermann's formula

     Given the state-space system
                .
                x = Ax + Bu

     and the desired eigenvalues of the closed loop in the vector P, the
     state feedback vector k is calculated in the form

                K = (k1 k2 ... kn)

     such that the closed loop system matrix

                A - BK

     has the eigenvalues given in P.

     See also: place.


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Calculates the state feedback matrix of a completely controllable SISO
system...



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append


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 -- Function File: SYS = append (SYS1, SYS2, ..., SYSN)
     Group LTI models by appending their inputs and outputs.


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Group LTI models by appending their inputs and outputs.



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arx


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 -- Function File: [SYS, X0] = arx (DAT, N, ...)
 -- Function File: [SYS, X0] = arx (DAT, N, OPT, ...)
 -- Function File: [SYS, X0] = arx (DAT, OPT, ...)
 -- Function File: [SYS, X0] = arx (DAT, 'NA', NA, 'NB', NB)
     Estimate ARX model using QR factorization.

          A(q) y(t) = B(q) u(t) + e(t)

     *Inputs*
     DAT
          iddata identification dataset containing the measurements,
          i.e.  time-domain signals.
     N
          The desired order of the resulting model SYS.
     ...
          Optional pairs of keys and values.  ‘'key1', value1, 'key2',
          value2’.
     OPT
          Optional struct with keys as field names.  Struct OPT can be
          created directly or by function ‘options’.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          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