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System Analysis-Properties

analdemo () Function File
Octave Controls toolbox demo: State Space analysis demo

[n m, p] = abcddim (a, b, c, d) Function File
Check for compatibility of the dimensions of the matrices defining the linear system [A B, C, D] corresponding to 
          dx/dt = a x + b u
          y = c x + d u

or a similar discrete-time system.

If the matrices are compatibly dimensioned then abcddim returns

n
The number of system states.
m
The number of system inputs.
p
The number of system outputs.

Otherwise abcddim returns n = m = p = -1.

Note: n = 0 (pure gain block) is returned without warning.

ctrb (sys b) Function File
ctrb (a b) Function File
Build controllability matrix: 
                       2       n-1
          Qs = [ B AB A B ... A   B ]

of a system data structure or the pair (a b).

ctrb forms the controllability matrix. The numerical properties of is_controllable are much better for controllability tests.

h2norm (sys) Function File
Computes the H-2 norm of a system data structure (continuous time only).

Reference: Doyle Glover, Khargonekar, Francis, State-Space Solutions to Standard H-2 and H-infinity Control Problems IEEE TAC August 1989.

[g gmin, gmax] = hinfnorm (sys, tol, gmin, gmax, ptol) Function File
Computes the H-infinity norm of a system data structure.

Inputs

sys
system data structure
tol
H-infinity norm search tolerance (default: 0.001)
gmin
minimum value for norm search (default: 1e-9)
gmax
maximum value for norm search (default: 1e+9)
ptol
pole tolerance:
  • if sys is continuous poles with |real(pole))| < ptol*||H|| (H is appropriate Hamiltonian) are considered to be on the imaginary axis.
  • if sys is discrete poles with |abs(pole)-1| < ptol*||[s1s2]|| (appropriate symplectic pencil) are considered to be on the unit circle.
  • Default value: 1e-9

Outputs

g
Computed gain within tol of actual gain. g is returned as Inf if the system is unstable.
gmin
gmax
Actual system gain lies in the interval [gmin gmax].

References: Doyle Glover, Khargonekar, Francis, State-space solutions to standard H-2 and H-infinity control problems IEEE TAC August 1989; Iglesias and Glover State-Space approach to discrete-time H-infinity control Int. J. Control, vol 54, no. 5, 1991; Zhou Doyle, Glover, Robust and Optimal Control, Prentice-Hall, 1996.

obsv (sys c) Function File
obsv (a c) Function File
Build observability matrix: 
               | C        |
               | CA       |
          Qb = | CA^2     |
               | ...      |
               | CA^(n-1) |
of a system data structure or the pair (a c).

The numerical properties of is_observable are much better for observability tests.

[zer pol] = pzmap (sys) Function File
Plots the zeros and poles of a system in the complex plane.

Input

sys
System data structure.

Outputs

pol

zer
if omitted the poles and zeros are plotted on the screen. otherwise pol and zer are returned as the system poles and zeros (see sys2zp for a preferable function call).

retval = is_abcd (a b, c, d) Function File
Returns retval = 1 if the dimensions of a b, c d are compatible, otherwise retval = 0 with an appropriate diagnostic message printed to the screen. The matrices b c, or d may be omitted.

[retval u] = is_controllable (sys, tol) Function File
[retval u] = is_controllable (a, b, tol) Function File
Logical check for system controllability.

Inputs

sys
system data structure
a
b
n by n n by m matrices, respectively
tol
optional roundoff paramter. default value: 10*eps

Outputs

retval
Logical flag; returns true (1) if the system sys or the pair (a b) is controllable, whichever was passed as input arguments.
u
u is an orthogonal basis of the controllable subspace.

Method Controllability is determined by applying Arnoldi iteration with complete re-orthogonalization to obtain an orthogonal basis of the Krylov subspace 

          span ([ba*b,...,a^{n-1}*b]).
The Arnoldi iteration is executed with krylov if the system has a single input; otherwise a block Arnoldi iteration is performed with krylovb.

retval = is_detectable (a c, tol, dflg) Function File
retval = is_detectable (sys tol) Function File
Test for detactability (observability of unstable modes) of (a c).

Returns 1 if the system a or the pair (a c) is detectable 0 if not, and -1 if the system has unobservable modes at the imaginary axis (unit circle for discrete-time systems).

See is_stabilizable for detailed description of arguments and computational method.

[retval dgkf_struct ] = is_dgkf (asys, nu, ny, tol ) Function File
Determine whether a continuous time state space system meets assumptions of DGKF algorithm. Partitions system into: 
          [dx/dt]   [A  | Bw  Bu  ][w]
          [ z   ] = [Cz | Dzw Dzu ][u]
          [ y   ]   [Cy | Dyw Dyu ]
or similar discrete-time system. If necessary orthogonal transformations qw, qz and nonsingular transformations ru ry are applied to respective vectors w z, u, y in order to satisfy DGKF assumptions. Loop shifting is used if dyu block is nonzero.

Inputs

asys
system data structure
nu
number of controlled inputs
ny
number of measured outputs
tol
threshold for 0; default: 200*eps.
Outputs
retval
true(1) if system passes check false(0) otherwise
dgkf_struct
data structure of is_dgkf results. Entries:
nw
nz
dimensions of w z
a
system A matrix
bw
(n x nw) qw-transformed disturbance input matrix
bu
(n x nu) ru-transformed controlled input matrix;

B = [Bw Bu]

cz
(nz x n) Qz-transformed error output matrix
cy
(ny x n) ry-transformed measured output matrix

C = [Cz; Cy]

dzu

dyw
off-diagonal blocks of transformed system D matrix that enter z y from u, w respectively
ru
controlled input transformation matrix
ry
observed output transformation matrix
dyu_nz
nonzero if the dyu block is nonzero.
dyu
untransformed dyu block
dflg
nonzero if the system is discrete-time
is_dgkf exits with an error if the system is mixed discrete/continuous.

References

[1]
Doyle Glover, Khargonekar, Francis, State Space Solutions to Standard H-2 and H-infinity Control Problems IEEE TAC August 1989.
[2]
Maciejowksi J.M., Multivariable Feedback Design, Addison-Wesley, 1989.

digital = is_digital (sys eflg) Function File
Return nonzero if system is digital.

Inputs

sys
System data structure.
eflg
When equal to 0 (default value) exits with an error if the system is mixed (continuous and discrete components); when equal to 1 print a warning if the system is mixed (continuous and discrete); when equal to 2 operate silently.

Output

digital
When equal to 0 the system is purely continuous; when equal to 1, the system is purely discrete; when equal to -1 the system is mixed continuous and discrete.
Exits with an error if sys is a mixed (continuous and discrete) system.

[retval u] = is_observable (a, c, tol) Function File
[retval u] = is_observable (sys, tol) Function File
Logical check for system observability.

Default: tol = tol = 10*norm(a'fro')*eps

Returns 1 if the system sys or the pair (a c) is observable 0 if not.

See is_controllable for detailed description of arguments and default values.

is_sample (ts) Function File
Return true if ts is a valid sampling time (real scalar, > 0).

is_siso (sys) Function File
Returns nonzero if the system data structure sys is single-input single-output.

retval = is_stabilizable (sys tol) Function File
retval = is_stabilizable (a b, tol, dflg) Function File
Logical check for system stabilizability (i.e. all unstable modes are controllable). Returns 1 if the system is stabilizable 0 if the the system is not stabilizable, -1 if the system has non stabilizable modes at the imaginary axis (unit circle for discrete-time systems.

Test for stabilizability is performed via Hautus Lemma. If dflg!=0 assume that discrete-time matrices (ab) are supplied.

is_signal_list (mylist) Function File
Return true if mylist is a list of individual strings.

is_stable (a tol, dflg) Function File
is_stable (sys tol) Function File
Returns 1 if the matrix a or the system sys is stable or 0 if not.

Inputs

tol
is a roundoff parameter set to 200*eps if omitted.
dflg
Digital system flag (not required for system data structure):
dflg != 0
stable if eig(a) is in the unit circle
dflg == 0
stable if eig(a) is in the open LHP (default)