Documentation

mpc

Create MPC controller

Syntax

  • MPCobj = mpc(Plant)
  • MPCobj = mpc(Plant,Ts)
  • MPCobj = mpc(Plant,Ts,p,m,W,MV,OV,DV)
  • MPCobj = mpc(Models,Ts,p,m,W,MV,OV,DV)

Description

MPCobj = mpc(Plant) creates a Model Predictive Controller object based on a discrete-time prediction model. The prediction model Plant can be either an LTI model with a specified sample time, or a System Identification model (see Identify Plant from Data). The controller, MPCobj, inherits its control interval from Plant.Ts, and its time unit from Plant.TimeUnit. All other controller properties are default values. After you create the MPC controller, you can set its properties using MPCobj.PropertyName = PropertyValue.

MPCobj = mpc(Plant,Ts) specifies a control interval of Ts. If Plant is a discrete-time LTI model with an unspecified sample time (Plant.Ts = –1), it inherits sample time Ts when used for predictions.

MPCobj = mpc(Plant,Ts,p,m,W,MV,OV,DV) specifies additional controller properties such as the prediction horizon (p), control horizon (m), and input, input increment, and output weights (W). You can also set the properties of manipulated variables (MV), output variables (OV), and input disturbance variables (DV). If any of these values are omitted or empty, the default values apply.

MPCobj = mpc(Models,Ts,p,m,W,MV,OV,DV) creates a Model Predictive Controller object based on a prediction model set, Models. This set includes plant, input disturbance, and measurement noise models along with the nominal conditions at which the models were obtained.

Input Arguments

Plant

Plant model to be used in predictions, specified as an LTI model (tf, ss, or zpk) or a System Identification Toolbox™ model. If the Ts input argument is unspecified, Plant must be a discrete-time LTI object with a specified sample time, or a System Identification Toolbox model.

Unless you specify otherwise, controller design assumes that all plant inputs are manipulated variables and all plant outputs are measured. Use the setmpcsignals command or the LTI InputGroup and OutputGroup properties to designate other signal types.

Ts

Controller sample time (control interval), specified as a positive scalar value.

p

Prediction horizon, specified as a positive integer. The control interval, Ts, determines the duration of each step. The default value is 10 + maximum intervals of delay (if any).

m

Control horizon, specified as a scalar integer, 1 ≤ mp, or as a vector of blocking factors such that sum(m)p (see Optimization Variables). The default value is 2.

W

Controller tuning weights, specified as a structure. For details about how to specify this structure, see Weights.

MV

Bounds and other properties of manipulated variables, specified as a 1-by-nu structure array, where nu is the number of manipulated variables defined in the plant model. For details about how to specify this structure, see ManipulatedVariables.

OV

Bounds and other properties of the output variables, specified as a 1-by-ny structure array, where ny is the number of output variables defined in the plant model. For details about how to specify this structure, see OutputVariables.

DV

Scale factors and other properties of the disturbance inputs, specified as a 1-by-nd structure array, where nd is the number of disturbance inputs (measured + unmeasured) defined in the plant model. For details about how to specify this structure, see DisturbanceVariables.

Models

Plant, input disturbance, and measurement noise models, along with the nominal conditions at which these models were obtained, specified as a structure. For details about how to specify this structure, see Model.

Construction and Initialization

To minimize computational overhead, Model Predictive Controller creation occurs in two phases. The first happens at construction when you invoke the mpc command, or when you change a controller property. Construction involves simple validity and consistency checks, such as signal dimensions and non-negativity of weights.

The second phase is initialization, which occurs when you use the object for the first time in a simulation or analytical procedure. Initialization computes all constant properties required for efficient numerical performance, such as matrices defining the optimal control problem and state estimator gains. Additional, diagnostic checks occur during initialization, such as verification that the controller states are observable.

By default, both phases display informative messages in the command window. You can turn these messages on or off using the mpcverbosity command.

Properties

All of the parameters defining the traditional (implicit) MPC control law are stored in an MPC object, whose properties are listed in the following table.

MPC Controller Object

Property

Description

ManipulatedVariables (or MV or Manipulated or Input)

Scale factors, input bounds, input-rate bounds, corresponding ECR values, target values, signal names, and units.

OutputVariables (or OV or Controlled or Output)

Scale factors, input bounds, input-rate bounds, corresponding ECR values, target values, signal names, and units.

DisturbanceVariables (or DV or Disturbance)

Disturbance scale factors, names, and units

Weights

Weights used in computing the performance (cost) function

Model

Plant, input disturbance, and output noise models, and nominal conditions.

Ts

Controller sample time

Optimizer

Parameters controlling the QP solver

PredictionHorizon

Prediction horizon

ControlHorizon

Number of free control moves or vector of blocking moves

History

Creation time

Notes

Text or comments about the MPC controller object

UserData

Any additional data

ManipulatedVariables

ManipulatedVariables (or MV or Manipulated or Input) is an nu-dimensional array of structures (nu = number of manipulated variables), one per manipulated variable. Each structure has the fields described in the following table, where p denotes the prediction horizon. Unless indicated otherwise, numerical values are in engineering units.

Manipulated Variable Structure

Field Name

Content

Default

ScaleFactor

Nonnegative scale factor for this MV

1

Min

1 to p length vector of lower bounds on this MV

-Inf

Max

1 to p length vector of upper bounds on this MV

Inf

MinECR

1 to p length vector of nonnegative parameters specifying the Min bound softness (0 = hard).

0 (dimensionless)

MaxECR

1 to p length vector of nonnegative parameters specifying the Max bound softness (0 = hard).

0 (dimensionless)

Target

1 to p length vector of target values for this MV

'nominal'

RateMin

1 to p length vector of lower bounds on the interval-to-interval change for this MV

-Inf

RateMax

1 to p length vector of upper bounds on the interval-to-interval change for this MV

Inf

RateMinECR

1 to p length vector of nonnegative parameters specifying the RateMin bound softness (0 = hard).

0 (dimensionless)

RateMaxECR

1 to p length vector of nonnegative parameters specifying the RateMax bound softness (0 = hard).

0 (dimensionless)

Name

Read-only MV signal name (character string)

InputName of LTI plant model

Units

Read-only MV signal units (character string)

InputUnit of LTI plant model

    Note   Rates refer to the difference Δu(k)=u(k)-u(k-1). Constraints and weights based on derivatives du/dt of continuous-time input signals must be properly reformulated for the discrete-time difference Δu(k), using the approximation du/dt ≅ Δu(k)/Ts.

OutputVariables

OutputVariables (or OV or Controlled or Output) is an ny-dimensional array of structures (ny = number of outputs), one per output signal. Each structure has the fields described in the following table. p denotes the prediction horizon. Unless specified otherwise, values are in engineering units.

Output Variable Structure

Field Name

Content

Default

ScaleFactor

Nonnegative scale factor for this OV

1

Min

1 to p length vector of lower bounds on this OV

-Inf

Max

1 to p length vector of upper bounds on this OV

Inf

MinECR

1 to p length vector of nonnegative parameters specifying the Min bound softness (0 = hard).

1 (dimensionless)

MaxECR

1 to p length vector of nonnegative parameters specifying the Max bound softness (0 = hard).

1 (dimensionless)

Name

Read-only OV signal name (character string)

OutputName of LTI plant model

Units

Read-only OV signal units (character string)

OutputUnit of LTI plant model

In order to reject constant disturbances due, for instance, to gain nonlinearities, the default measured output disturbance model used in Model Predictive Control Toolbox™ software is integrated white noise (see Output Disturbance Model).

DisturbanceVariables

DisturbanceVariables (or DV or Disturbance) is an (nv+nd)-dimensional array of structures (nv = number of measured input disturbances, nd = number of unmeasured input disturbances). Each structure has the fields described in the following table.

Disturbance Variable Structure

Field Name

Content

Default

ScaleFactor

Nonnegative scale factor for this DV

1

Name

Read-only DV signal name (character string)

InputName of LTI plant model

Units

Read-only DV signal units (character string)

InputUnit of LTI plant model

The order of the disturbance signals within the array DV is the following: the first nv entries relate to measured input disturbances, the last nd entries relate to unmeasured input disturbances.

Weights

Weights is the structure defining the QP weighting matrices. It contains four fields. The values of these fields depend on whether you are using the standard quadratic cost function (see Standard Cost Function) or the alternative cost function (see Alternative Cost Function).

Standard Cost Function

The following table lists the content of the four structure fields. In the table, p denotes the prediction horizon, nu the number of manipulated variables, and ny the number of output variables.

For the MV, MVRate and OV weights, if you specify fewer than p rows, the last row repeats automatically to form a matrix containing p rows.

Weights for the Standard Cost Function

Field Name (Abbreviations)

Content

Default (dimensionless)

ManipulatedVariables (or MV or Manipulated or Input)

(1 to p)-by-nu dimensional array of nonnegative MV weights

zeros(1,nu)

ManipulatedVariablesRate (or MVRate or ManipulatedRate or InputRate)

(1 to p)-by-nu dimensional array of MV-increment weights

0.1*ones(1,nu)

OutputVariables (or OV or Controlled or Output)

(1 to p)-by-ny dimensional array of OV weights

1 (The default for output weights is the following: if nuny, all outputs are weighted with unit weight; if nu<ny, nu outputs default to 1, with preference given to measured outputs, and the rest default to 0.)

ECR

Scalar weight on the slack variable ɛ used for constraint softening

1e5*(max weight)

    Note   If all MVRate weights are strictly positive, the resulting QP problem is strictly convex. If some MVRate weights are zero, the QP Hessian could be positive semidefinite. In order to keep the QP problem strictly convex, when the condition number of the Hessian matrix KΔU is larger than 1012, the quantity 10*sqrt(eps) is added to each diagonal term. See Cost Function.

Alternative Cost Function

You can specify off-diagonal Q and R weight matrices in the cost function. To do so, define the fields ManipulatedVariables, ManipulatedVariablesRate, and OutputVariables as cell arrays, each containing a single positive-semi-definite matrix of the appropriate size. Specifically, OutputVariables must be a cell array containing the ny-by-ny Q matrix, ManipulatedVariables must be a cell array containing the nu-by-nu Ru matrix, and ManipulatedVariablesRate must be a cell array containing the nu-by-nu RΔu matrix (see Alternative Cost Function and the mpcweightsdemo example). You can use diagonal weight matrices for one or more of these fields. If you omit a field, the MPC controller uses the defaults shown in the table above.

For example, you can specify off-diagonal weights, as follows

MPCobj.Weights.OutputVariables = {Q};
MPCobj.Weights.ManipulatedVariables = {Ru};
MPCobj.Weights.ManipulatedVariablesRate = {Rdu};

where Q = Q. Ru = Ru, and Rdu = RΔu are positive semidefinite matrices.

    Note   You cannot specify non-diagonal weights that vary at each prediction horizon step. The same Q, Ru, and Rdu weights apply at each step.

Model

The property Model specifies plant, input disturbance, and output noise models, and nominal conditions, according to the model setup described in Controller State Estimation. It is a 1-D structure containing the following fields.

Models Used by MPC

Field Name

Content

Default

Plant

LTI model or identified linear model of the plant

No default

Disturbance

LTI model describing expected unmeasured input disturbances

[ ] (By default, input disturbances are expected to be integrated white noise. To model the signal, an integrator with dimensionless unity gain is added for each unmeasured input disturbance, unless the addition causes the controller to lose state observability. In that case, the disturbance is expected to be white noise, and so, a dimensionless unity gain is added to that channel instead.)

Noise

LTI model describing expected noise for output measurements

[ ] (By default, measurement noise is expected to be white noise with unit variance. To model the signal, a dimensionless unity gain is added for each measured channel.)

Nominal

Structure containing the state, input, and output values where Model.Plant is linearized

The default values of the fields are shown in the following table:

Field

Description

Default

X

Plant state at operating point

0

U

Plant input at operating point, including manipulated variables, measured and unmeasured disturbances

0

Y

Plant output at operating point

0

DX

For continuous-time models, DX is the state derivative at operating point: DX=f(X,U). For discrete-time models, DX=x(k+1)-x(k)=f(X,U)-X.

0

    Note   Direct feedthrough from manipulated variables to any output in Model.Plant is not allowed. See MPC Modeling.

Specify input and output signal types via the InputGroup and OutputGroup properties of Model.Plant, or, more conveniently, use the setmpcsignals command. Valid signal types are listed in the following tables.

Input Groups in Plant Model

Name (Abbreviations)

Value

ManipulatedVariables (or MV or Manipulated or Input)

Indices of manipulated variables in Model.Plant

MeasuredDisturbances (or MD or Measured)

Indices of measured disturbances in Model.Plant

UnmeasuredDisturbances (or UD or Unmeasured)

Indices of unmeasured disturbances in Model.Plant

Output Groups in Plant Model

Name (Abbreviations)

Value

MeasuredOutputs (or MO or Measured)

Indices of measured outputs in Model.Plant

UnmeasuredOutputs (or UO or Unmeasured)

Indices of unmeasured outputs in Model.Plant

By default, all Model.Plant inputs are manipulated variables, and all outputs are measured.

The structure Nominal contains the values (in engineering units) for states, inputs, outputs, and state derivatives/differences at the operating point where Model.Plant applies. This point is typically a linearization point. The fields are reported in the following table (see also MPC Modeling).

Nominal Values at Operating Point

Field

Description

Default

X

Plant state at operating point

0

U

Plant input at operating point, including manipulated variables, measured and unmeasured disturbances

0

Y

Plant output at operating point

0

DX

For continuous-time models, DX is the state derivative at operating point: DX=f(X,U). For discrete-time models, DX=x(k+1)-x(k)=f(X,U)-X.

0

Ts

Sample time of the MPC controller. By default, if Model.Plant is a discrete-time model, Ts = Model.Plant.ts. For continuous-time plant models, specify a controller Ts. The Ts measurement unit is inherited from Model.Plant.TimeUnit.

Optimizer

Parameters for the QP optimization. Optimizer is a structure with the fields reported in the following table.

Optimizer Properties

Field

Description

Default

MaxIter

Maximum number of iterations allowed in the QP solver

'Default'

MinOutputECR

Minimum value allowed for OutputMinECR and OutputMaxECR

0

MaxIter is either a positive scalar integer or the string ‘Default'. If set to an integer value, QP solver stops when the iteration number reaches this value. If set to ‘Default', the MPC controller automatically determines the maximum iteration number based on the controller specifications.

MinOutputECR is a nonnegative scalar used to specify the minimum allowed ECR for output constraints. By default, MinOutputECR is 0, which means that output constraints are allowed to be hard. If a custom MinOutputECR is specified, when a value smaller than MinOutputECR is provided in the OutputVariables.MinECR or OutputVariables.MaxECR properties of MPC objects, a warning message is issued and the value is raised to MinOutputECR during computation.

PredictionHorizon

PredictionHorizon is the integer number of prediction horizon steps. The control interval, Ts, determines the duration of each step. The default value is 10 + maximum intervals of delay (if any).

ControlHorizon

ControlHorizon is either a number of free control moves, or a vector of blocking moves (see Optimization Variables). The default value is 2.

History

History stores the time the MPC controller was created (read only).

Notes

Notes stores text or comments as a cell array of strings.

UserData

Any additional data stored within the MPC controller object.

Examples

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Create MPC Controller with Specified Prediction and Control Horizons

Define an MPC controller based on the transfer function model s+1/(s2+2s), with time Ts = 0.1 s. Define bounds on the manipulated variable at –1 ≤ u ≤ 1, use a 20-interval prediction horizon, and a 3-interval control horizon.

Plant = tf([1 1],[1 2 0]);

The plant is SISO, so its input must be a manipulated variable and its output must be measured. In general, it is good practice to designate all plant signal types using the setmpcsignals command (or the LTI InputGroup and OutputGroup properties).

In addition, the plant is a continuous-time LTI model, so it is necessary to specify the time of the MPC controller.

Ts = 0.1;
MV = struct('Min',-1,'Max',1); 
p = 20;
m = 3; 

Here, MV contains only the upper and lower bounds on the manipulated variable. In general, you can specify additional MV properties in general. When you do not specify other properties, their default values apply. Similarly, the fifth input argument (W) is empty, so default tuning weights apply.

Create the controller using the values you have specified.

MPCobj = mpc(Plant,Ts,p,m,[],MV);

Related Examples

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