Prediction Model

The linear model used in Model Predictive Control Toolbox™ software for prediction and optimization is depicted in Figure 2-1.

Figure 2-1: Model Used for Optimization

The model consists of:

• A model of the plant to be controlled, whose inputs are the manipulated variables, the measured disturbances, and the unmeasured disturbances
• A model generating the unmeasured disturbances

Note    When defining a model predictive controller, you must specify a plant model. You do not need to specify a model generating the disturbances, as the controller setup assumes by default that unmeasured disturbances are generated by integrators driven by white noise (see Output Disturbance Model and setindist).

The model of the plant is a linear time-invariant system described by the equations

• 

where x(k) is the n_x-dimensional state vector of the plant, u(k) is the n_u-dimensional vector of manipulated variables (MV), i.e., the command inputs, v(k) is the n_v-dimensional vector of measured disturbances (MD), d(k) is the n_d-dimensional vector of unmeasured disturbances (UD) entering the plant, y_m(k) is the vector of measured outputs (MO), and y_u(k) is the vector of unmeasured outputs (UO). The overall n_y-dimensional output vector y(k) collects y_m(k) and y_u(k).

Model Predictive Control Toolbox software accepts both plant models specified as LTI objects, and models obtained from input/output data using System Identification Toolbox (IDMODEL objects), see Using Identified Models.

In the above equations d(k) collects both state disturbances (Bd0) and output disturbances (Dd0).

Note    A valid plant model for Model Predictive Control Toolbox software cannot have direct feedthrough of manipulated variables u(k) on the output vector y(k).

The unmeasured disturbance d(k) is modeled as the output of the linear time invariant system:

(2-2)

The system described by the above equations is driven by the random Gaussian noise n_d(k), having zero mean and unit covariance matrix. For instance, a step-like unmeasured disturbance is modeled as the output of an integrator. Input disturbance models as in the equations above can be manipulated by using the methods getindist and setindist.

Note    If continuous-time models are supplied, they are internally sampled with the controller's sampling time.

Offsets

In many practical applications, the matrices A, B, C, D of the model representing the process to control are obtained by linearizing a nonlinear dynamical system, such as

• 

• ,

at some nominal value x=x₀, u=u₀, v=v₀, d=d₀. In these equations x´ denotes either the time derivative (continuous time model) or the successor x(k+1) (discrete time model). As an example, x₀, u₀, v₀, d₀ may be obtained by using TRIM on a Simulink® model describing the nonlinear dynamical equations, and A, B, C, D by using LINMOD. The linearized model has the form:

• 

• 

The matrices A, B, C, D of the model are readily obtained from the Jacobian matrices appearing in the equations above.

The linearized dynamics are affected by the constant terms F=f(x₀, u₀, v₀, d₀) and H=h(x₀, u₀, v₀, d₀). For this reason the model predictive control algorithm internally adds a measured disturbance v=1, so that F and H can be embedded into B_v and D_v, respectively, as additional columns.

Note    Nonzero offset values d0 for unmeasured disturbances, while relevant for obtaining the linearized model matrices, are not relevant for the model predictive control problem setup. In fact, only d-d0 can be estimated from output measurements.

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Model Predictive Control Problem Setup

Optimization Problem

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