Documentation Center |
Convert unconstrained MPC controller to state-space linear system
sys=ss(MPCobj)
sys = ss(MPCobj,signals)
sys = ss(MPCobj,signals,ref_preview,md_preview)
[sys,ut]
= ss(MPCobj)
The ss command returns a linear controller in the state-space form. The controller is equivalent to the MPC controller MPCobj when the constraints are not active. The purpose is to use the linear equivalent control in Control System Toolbox™ software for sensitivity analysis and other linear analysis.
sys=ss(MPCobj) returns the linear discrete-time dynamic controller sys
x(k + 1) = Ax(k) + By_{m}(k)
u(k) = Cx(k) + Dy_{m}(k)
where y_{m} is the vector of measured outputs of the plant, and u is the vector of manipulated variables. The sampling time of controller sys 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). |
sys = ss(MPCobj,signals) returns the linearized MPC controller in its full form and allows you to specify the signals that you want to include as inputs for sys.
The full form of the MPC controller has the following structure:
x(k + 1) = Ax(k) + By_{m}(k) + B_{r}r(k) + B_{v}v(k) + B_{ut}u_{target}(k) + B_{off}
u(k) = Cx(k) + Dy_{m}(k) + D_{r}r(k) + D_{v}v(k) + D_{ut}u_{target}(k) + D_{off}
Here, r is the vector of setpoints for both measured and unmeasured plant outputs, v is the vector of measured disturbances, u_{target} is the vector of preferred values for manipulated variables.
Specify signals as a single or multicharacter string constructed using any of the following:
'r' — Output references
'v' — Measured disturbances
'o' — Offset terms
't' — Input targets
For example, to obtain a controller that maps [y_{m}; r; v] to u, use:
sys = ss(MPCobj,'rv');
In the general case of nonzero offsets, y_{m} (as well as r, v, and u_{target}) 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 B_{off}, D_{off} 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, B_{off}, D_{off} are zero.
Only the following fields of MPCobj are used when computing the state-space model: Model, PredictionHorizon, ControlHorizon, Ts, Weights.
sys = ss(MPCobj,signals,ref_preview,md_preview) specifies if the MPC controller has preview actions on the reference and measured disturbance signals. If the flag ref_preview='on', then matrices B_{r} and D_{r} multiply the whole reference sequence:
x(k + 1) = Ax(k) + By_{m}(k) + B_{r}[r(k);r(k + 1);...;r(k + p – 1)] +...
u(k) = Cx(k) + Dy_{m}(k) + D_{r}[r(k);r(k + 1);...;r(k + p– 1)] +...
Similarly if the flag md_preview='on', then matrices B_{v} and D_{v} multiply the whole measured disturbance sequence:
x(k + 1) = Ax(k) +...+ B_{v}[v(k);v(k + 1);...;v(k + p)] +...
u(k) = Cx(k) +...+ D_{v}[v(k);v(k + 1);...;v(k + p)] +...
[sys,ut] = ss(MPCobj) additionally returns the input target values for the full form of the controller.
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