ss
Convert unconstrained MPC controller to state-space linear system form
Syntax
Description
Use the Model Predictive Control Toolbox™
ss
function to convert an unconstrained MPC controller with defined
sample time to state space form (see mpc
for background). The returned controller is equivalent to the original MPC
controller mpcobj
when no constraints are active. You can then use
Control System Toolbox™ software for sensitivity analysis and other diagnostic
calculations.
To create or convert a generic LTI dynamical system to state space form, see ss
and
Dynamic System Models.
specifies whether the returned controller has preview action, that is if it uses the whole
reference and measured disturbance sequences as input signals.kssFullPv
= ss(mpcobj
,signals
,refPreview
,mdPreview
)
Examples
Convert Unconstrained MPC Controller to State-Space Model
Create the plant model.
plant = rss(5,2,3); plant.D = 0; plant = setmpcsignals(plant,'mv',1,'md',2,'ud',3,'mo',1,'uo',2);
Configure the MPC controller with a sample time of 0.1 seconds and nonzero nominal values, weights, and input targets.
mpcobj = mpc(plant,0.1);
-->"PredictionHorizon" is empty. Assuming default 10. -->"ControlHorizon" is empty. Assuming default 2. -->"Weights.ManipulatedVariables" is empty. Assuming default 0.00000. -->"Weights.ManipulatedVariablesRate" is empty. Assuming default 0.10000. -->"Weights.OutputVariables" is empty. Assuming default 1.00000. for output(s) y1 and zero weight for output(s) y2
mpcobj.Model.Nominal.U = [0.7 0.8 0]; mpcobj.Model.Nominal.Y = [0.5 0.6]; mpcobj.Model.Nominal.DX = rand(5,1); mpcobj.Weights.MV = 2; mpcobj.Weights.OV = [3 4]; mpcobj.MV.Target = [0.1 0.2 0.3];
Specifying mpcobj.Model.Nominal.DX
as nonzero means that the nominal values are not at steady state mpcobj.MV.Target
specifies three preview steps.
Convert mpcobj
to a state-space model.
Kss = ss(mpcobj)
-->Converting model to discrete time. -->The "Model.Disturbance" property is empty: Assuming unmeasured input disturbance #3 is integrated white noise. Assuming no disturbance added to measured output #1. -->"Model.Noise" is empty. Assuming white noise on each measured output. Kss = A = xMPC1 xMPC2 xMPC3 xMPC4 xMPC5 xMPC6 prev.MV1 xMPC1 0.76 0.2216 -0.006469 0.3508 0.02977 -0.04783 3.019e-05 xMPC2 -0.231 1.135 -0.4594 0.5914 0.01656 -0.1007 7.773e-08 xMPC3 -0.03007 0.1185 0.5467 0.1386 0.01752 -0.11 0.0001122 xMPC4 -0.1927 -0.2434 -0.1331 0.2121 -0.02496 -0.1596 0.0001298 xMPC5 -0.1399 -0.2179 0.09667 -0.3748 0.5926 0.1385 -2.941e-05 xMPC6 -0.09826 0 0.1393 -0.1346 -0.09578 1 0 prev.MV1 -2.511 0.02726 -2.984 -1.2 -0.08729 -2.102 0.001735 B = MO1 xMPC1 0.05903 xMPC2 0.1294 xMPC3 0.03619 xMPC4 -0.0589 xMPC5 -0.08955 xMPC6 -0.09022 prev.MV1 0.09002 C = xMPC1 xMPC2 xMPC3 xMPC4 xMPC5 xMPC6 prev.MV1 MV1 -2.511 0.02726 -2.984 -1.2 -0.08729 -2.102 0.001735 D = MO1 MV1 0.09002 Sample time: 0.1 seconds Discrete-time state-space model.
The output, sys
, is a seventh-order SISO state-space model. The seven states include the five plant model states, one state from the default input disturbance model, and one state from the previous move, u(k-1)
.
Set mpcobj
to use custom state estimation.
setEstimator(mpcobj,'custom');
Convert mpcobj
to a static gain matrix for state-feedback.
K = ss(mpcobj)
-->Converting model to discrete time. -->The "Model.Disturbance" property is empty: Assuming unmeasured input disturbance #3 is integrated white noise. Assuming no disturbance added to measured output #1. -->"Model.Noise" is empty. Assuming white noise on each measured output. K = D = xMPC1 xMPC2 xMPC3 xMPC4 xMPC5 xMPC6 prev.MV1 MV1 -2.609 0.02726 -2.845 -1.334 -0.1829 -2.102 0.001735 Static gain.
As expected this is a row vector with seven elements.
Reset mpcobj
to use the default estimator.
setEstimator(mpcobj,'default');
You can use 'rv'
as a second argument to return a system with 3 additional inputs (reference of measured plant output, reference of unmeasured plant output and measured disturbance).
Kss = ss(mpcobj, 'rv')
-->Converting model to discrete time. -->The "Model.Disturbance" property is empty: Assuming unmeasured input disturbance #3 is integrated white noise. Assuming no disturbance added to measured output #1. -->"Model.Noise" is empty. Assuming white noise on each measured output. Kss = A = xMPC1 xMPC2 xMPC3 xMPC4 xMPC5 xMPC6 prev.MV1 xMPC1 0.76 0.2216 -0.006469 0.3508 0.02977 -0.04783 3.019e-05 xMPC2 -0.231 1.135 -0.4594 0.5914 0.01656 -0.1007 7.773e-08 xMPC3 -0.03007 0.1185 0.5467 0.1386 0.01752 -0.11 0.0001122 xMPC4 -0.1927 -0.2434 -0.1331 0.2121 -0.02496 -0.1596 0.0001298 xMPC5 -0.1399 -0.2179 0.09667 -0.3748 0.5926 0.1385 -2.941e-05 xMPC6 -0.09826 0 0.1393 -0.1346 -0.09578 1 0 prev.MV1 -2.511 0.02726 -2.984 -1.2 -0.08729 -2.102 0.001735 B = MO1 ref.MO1 ref.UO1 MD1 xMPC1 0.05903 -0.003162 -0.03311 -0.06524 xMPC2 0.1294 -8.142e-06 -8.526e-05 -0.0008242 xMPC3 0.03619 -0.01175 -0.123 0.08255 xMPC4 -0.0589 -0.01359 -0.1423 0.03386 xMPC5 -0.08955 0.003081 0.03226 -0.005793 xMPC6 -0.09022 0 0 0 prev.MV1 0.09002 -0.1817 -1.903 1.503 C = xMPC1 xMPC2 xMPC3 xMPC4 xMPC5 xMPC6 prev.MV1 MV1 -2.511 0.02726 -2.984 -1.2 -0.08729 -2.102 0.001735 D = MO1 ref.MO1 ref.UO1 MD1 MV1 0.09002 -0.1817 -1.903 1.503 Sample time: 0.1 seconds Discrete-time state-space model.
Use 't'
as a second argument to return a system with 3 additional inputs (input target channels corresponding to three preview steps).
Kss = ss(mpcobj, 't')
Kss = A = xMPC1 xMPC2 xMPC3 xMPC4 xMPC5 xMPC6 prev.MV1 xMPC1 0.76 0.2216 -0.006469 0.3508 0.02977 -0.04783 3.019e-05 xMPC2 -0.231 1.135 -0.4594 0.5914 0.01656 -0.1007 7.773e-08 xMPC3 -0.03007 0.1185 0.5467 0.1386 0.01752 -0.11 0.0001122 xMPC4 -0.1927 -0.2434 -0.1331 0.2121 -0.02496 -0.1596 0.0001298 xMPC5 -0.1399 -0.2179 0.09667 -0.3748 0.5926 0.1385 -2.941e-05 xMPC6 -0.09826 0 0.1393 -0.1346 -0.09578 1 0 prev.MV1 -2.511 0.02726 -2.984 -1.2 -0.08729 -2.102 0.001735 B = MO1 MV1.target(0 MV1.target(1 MV1.target(2 xMPC1 0.05903 0.01207 -0.001223 -0.009782 xMPC2 0.1294 3.109e-05 -3.149e-06 -2.519e-05 xMPC3 0.03619 0.04487 -0.004544 -0.03635 xMPC4 -0.0589 0.05191 -0.005257 -0.04206 xMPC5 -0.08955 -0.01176 0.001191 0.00953 xMPC6 -0.09022 0 0 0 prev.MV1 0.09002 0.6938 -0.07026 -0.5621 C = xMPC1 xMPC2 xMPC3 xMPC4 xMPC5 xMPC6 prev.MV1 MV1 -2.511 0.02726 -2.984 -1.2 -0.08729 -2.102 0.001735 D = MO1 MV1.target(0 MV1.target(1 MV1.target(2 MV1 0.09002 0.6938 -0.07026 -0.5621 Sample time: 0.1 seconds Discrete-time state-space model.
You can use the second output argument to return the vector of manipulated variables target values. This vector corresponds to C.MV.Target
- C.Model.Nominal.U(1)
.
[Kss, ut] = ss(mpcobj); ut
ut = 3×1
-0.6000
-0.5000
-0.4000
Input Arguments
mpcobj
— Model predictive controller
mpc
object
Model predictive controller, specified as an MPC controller
object. To create an MPC controller, use mpc
.
signals
— Signal selection
''
(default) | character array | string
Specify signals
as a character vector or string with any
combination that contains one or more of the following characters:
'r'
— Output references
'v'
— Measured disturbances
'o'
— Offset terms
't'
— Input targets
For example, to obtain a controller that maps [ym; r; v] to u, use:
kss = ss(mpcobj,'rv');
Example: 'r'
refPreview
— use whole reference sequence as input
'off'
(default) | 'on'
If this flag is 'on'
, then the input matrices of the returned
controller have a larger size to multiply the whole reference sequence.
Example: 'on'
mdPreview
— use whole measured disturbance sequence as input
'off'
(default) | 'on'
If this flag is 'on'
, then the input matrices of the returned
controller have a larger size to multiply the whole disturbance sequence.
Example: 'on'
Output Arguments
kss
— state space form of the unconstrained MPC controller
ss
object
The discrete-time state space form of the unconstrained MPC controller has the following structure:
x(k + 1) = Ax(k) + Bym(k)
u(k) = Cx(k) + Dym(k)
where A, B, C, and
D are the matrices forming a state space realization of the
controller kss
, ym is the
vector of measured outputs of the plant, and u is the vector of
manipulated variables. The sampling time of controller kss
is
mpcobj.Ts
.
Note
Vector x includes the states of the observer (plant + disturbance + noise model states) and the previous manipulated variable u(k-1).
Note
Only the following fields of mpcobj
are used when computing
the state-space model: Model
,
PredictionHorizon
, ControlHorizon
,
Ts
, Weights
.
Note
If mpcobj
is set to use custom state estimation, then
ss
returns a static gain feedback matrix from the states of
the augmented discrete-time plant, including previous values of the manipulated
variables, to the manipulated variables.
kssFull
— full state space form of the unconstrained MPC controller
ss
object
The full discrete-time state space form of the unconstrained MPC controller has the following structure:
x(k + 1) = Ax(k) + Bym(k) + Brr(k) + Bvv(k) + Bututarget(k) + Boff
u(k) = Cx(k) + Dym(k) + Drr(k) + Dvv(k) + Dututarget(k) + Doff
Here:
A, B, C, and D are the matrices forming a state space realization of the controller from measured plant output to manipulated variables
r is the vector of setpoints for both measured and unmeasured plant outputs
v is the vector of measured disturbances.
utarget is the vector of preferred values for manipulated variables.
In the general case of nonzero offsets,
ym, r,
v, and utarget must be
interpreted as the difference between the vector and the corresponding offset. Offsets
can be nonzero is mpcobj.Model.Nominal.Y
or
mpcobj.Model.Nominal.U
are nonzero.
Vectors Boff and
Doff are constant terms. They are nonzero
if and only if mpcobj.Model.Nominal.DX
is nonzero (continuous-time
prediction models), or
mpcobj.Model.Nominal.Dx
-mpcobj.Model.Nominal.X
is nonzero (discrete-time prediction models). In other words, when
Nominal.X
represents an equilibrium state,
Boff,
Doff are zero.
kssFullPv
— full state space form of the unconstrained MPC controller
ss
object
If the flag refPreview = 'on'
, then matrices
Br and
Dr multiply the whole reference
sequence:
x(k + 1) = Ax(k) + Bym(k) + Br[r(k);r(k + 1);...;r(k + p – 1)] +...
u(k) = Cx(k) + Dym(k) + Dr[r(k);r(k + 1);...;r(k + p– 1)] +...
Similarly, if the flag mdPreview='on'
, then matrices
Bv and
Dv multiply the whole measured disturbance
sequence:
x(k + 1) = Ax(k) +...+ Bv[v(k);v(k + 1);...;v(k + p)] +...
u(k) = Cx(k) +...+ Dv[v(k);v(k + 1);...;v(k + p)] +...
ut
— target values
column vector
ut
is returned as a vector of doubles, [utarget(k);
utarget(k+1); ... utarget(k+h)]
.
Here:
h — Maximum length of previewed inputs; that is,
h = max(length(mpcobj.ManipulatedVariables(:).Target))
utarget
— Difference between the input target and corresponding input offsets; that is,mpcobj.ManipulatedVariables(:).Targets - mpcobj.Model.Nominal.U
Version History
Introduced before R2006a
See Also
Functions
Objects
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