trim

Compute steady-state value of MPC controller plant model state for given inputs and outputs

Description

Use the Model Predictive Control Toolbox™ trim function to calculate steady state values of LTI discrete-time plants controlled by an MPC controller (see mpc for background).

example

x = trim(MPCobj,y,u) returns a steady-state value for the plant state or the best approximation in a least squares sense such that:

$\begin{array}{l}x-{x}_{off}=A\left(x-{x}_{off}\right)+B\left(u-{u}_{off}\right)\\ y-{y}_{off}=C\left(x-{x}_{off}\right)+D\left(u-{u}_{off}\right)\end{array}$

Here, A,B, C, and D are the state space realization matrices of the discrete-time plant model used within MPCobj, xoff, uoff, and yoff are the nominal values of the extended state x, input u, and output y respectively.

Examples

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Create a plant, a corresponding MPC object, and calculate the steady state value of the plant model state.

mpcverbosity off;                           % turn off mpc messaging
plant=c2d(ss(zpk([],[-1 -10],20)),1);       % create plant (note the steady state gain)
mpcobj=mpc(plant,1);                        % create mpc object

x=trim(mpcobj,2,1)                          % caclulate trim point
MPCSTATE object with fields
Plant: [0.4000 0.4000]
Disturbance: 0
Noise: [1×0 double]
LastMove: 1
Covariance: [3×3 double]

% check whether the calculated value is actually an equilibrium point
mpcobj.Model.Plant.A*x.Plant+mpcobj.Model.Plant.B*1-x.Plant
ans =
1.0e-15 *
0.1110
0.0555
mpcobj.Model.Plant.C*x.Plant+mpcobj.Model.Plant.D*1-2
ans =
-2.2204e-16

The resulting state value is an equilibrium point because for the given output and input values, the state at the next time step is equal to the current state (except some numerical errors).

Input Arguments

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Model predictive controller, specified as an MPC controller object. To create an MPC controller, use mpc.

This is the plant output (including both measured and unmeasured signals) for which you want to find a stationary value of the extended plant state. If the plant has a finite steady state gain matrix G0 and y is equal to G0*u then the plant has a stationary state with output y and input u.

Example: [1 1]'

This is the plant input (including manipulated variables, measured disturbances, and unmeasured disturbances) for which you want to find a stationary value of the extended plant state. If unmeasured input disturbance variables exist, their value must be 0.

Example: [0 1]'

Output Arguments

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This is the best approximation, in a least squares sense, of the steady-state value for the plant state corresponding to the given input and output values.