Model Predictive Control Toolbox™

State Estimation

As the states x(k), x_d(k) are not directly measurable, predictions are obtained from a state estimator. In order to provide more flexibility, the estimator is based on the model depicted in Figure 2-2.

Figure 2-2: Model Used for State Estimation

Measurement Noise Model

We assume that the measured output vector y_m(k) is corrupted by a measurement noise m(k). The measurement noise m(t) is the output of the linear time-invariant system

The system described by these equations is driven by the random Gaussian noise n_m(k), having zero mean and unit covariance matrix.

Note    The objective of the model predictive controller is to bring yu(k) and [ym(k)-m(k)] as close as possible to the reference vector r(k). For this reason, the measurement noise model producing m(k) is not needed in the prediction model used for optimization described in Prediction Model.

Output Disturbance Model

In order to guarantee asymptotic rejection of output disturbances, the overall model is augmented by an output disturbance model. By default, in order to reject constant disturbances due for instance to gain nonlinearities, the output disturbance model is a collection of integrators driven by white noise on measured outputs. Output integrators are added according to the following rule:

1. Measured outputs are ordered by decreasing output weight (in case of time-varying weights, the sum of the absolute values over time is considered for each output channel, and in case of equal output weight the order within the output vector is followed).
2. By following such order, an output integrator is added per measured outputs, unless there is a violation of observability or the user forces it (through the OutputVariables.Integrators property described in OutputVariables).

An arbitrary output disturbance model can be specified through the function setoutdist. See also setoutdist for ways to remove the default output integrators.

State Observer

The state observer is designed to provide estimates of x(k), x_d(k), x_m(k), where x(k) is the state of the plant model, x_d(k) is the overall state of the input and output disturbance model, x_m(k) is the state of the measurement noise model. The estimates are computed from the measured output ym(k) by the linear state observer

where m denotes the rows of C,D corresponding to measured outputs.

To prevent numerical difficulties in the absence of unmeasured disturbances, the gain M is designed using Kalman filtering techniques (see kalman in the Control System Toolbox™ documentation) on the extended model

(2-6)

where n_u(k) and n_v(k) are additional unmeasured white noise disturbances having unit covariance matrix and zero mean, that are added on the vector of manipulated variables and the vector of measured disturbances, respectively, to ease the solvability of the Kalman filter design.

Note    The overall state-space realization of the combination of plant and disturbance models must be observable for the state estimation design to succeed. Model Predictive Control Toolbox™ software first checks for observability of the plant, provided that this is given in state-space form. After all models have been converted to discrete-time, delay-free, state-space form and combined together, observability of the overall extended model is checked (see setestim and Construction and Initialization). Note also that observability is only checked numerically. Hence, for large models of badly conditioned system matrices, unobservability may be reported by the toolbox even if the system is observable.

See also getestim and setestim for details on the methods that you can use to access and modify properties of the state estimator.

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