mpc

Model predictive controller

Description

A model predictive controller uses linear plant, disturbance, and noise models to estimate the controller state and predict future plant outputs. Using the predicted plant outputs, the controller solves a quadratic programming optimization problem to determine control moves.

For more information on the structure of model predictive controllers, see MPC Prediction Models.

Creation

Syntax

mpcobj = mpc(plant)

mpcobj = mpc(plant,ts)

mpcobj = mpc(plant,ts,P,M,W,MV,OV,DV)

mpcobj = mpc(model)

mpcobj = mpc(model,ts)

mpcobj = mpc(model,ts,P,M,W,MV,OV,DV)

Description

mpcobj = mpc(plant) creates a model predictive controller object based on the discrete-time prediction model plant. 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 dot notation.

If plant.Ts = -1, you must set the Ts property of the controller to a positive value before designing and simulating your controller.

mpcobj = mpc(plant,ts) creates a model predictive controller based on the specified plant model and sets the Ts property of the controller. If plant is:

• A continuous-time model, then the controller discretizes the model for prediction using sample time ts

• A discrete-time model with a specified sample time, the controller resamples the plant for prediction using sample time ts

• A discrete-time model with an unspecified sample time (plant.Ts = –1), it inherits the sample time ts when used for predictions.

mpcobj = mpc(plant,ts,P,M,W,MV,OV,DV) specifies the following controller properties. If any of these values are omitted or empty, the default values apply.

• P sets the PredictionHorizon property.

• M sets the ControlHorizon property.

• W sets the Weights property.

• MV sets the ManipulatedVariables property.

• OV sets the OutputVariables property.

• DV sets the DisturbanceVariables property.

mpcobj = mpc(model) creates a model predictive controller object based on the specified prediction model set, which includes the plant, input disturbance, and measurement noise models along with the nominal conditions at which the models were obtained. When you do not specify a sample time, the plant model, model.Plant, must be a discrete-time model. This syntax sets the Model property of the controller.

mpcobj = mpc(model,ts) creates a model predictive controller based on the specified plant model and sets the Ts property of the controller to ts. If model.Plant is a discrete-time LTI model with an unspecified sample time (model.Plant.Ts = –1), it inherits the sample time ts when used for predictions.

mpcobj = mpc(model,ts,P,M,W,MV,OV,DV) specifies additional controller properties. If any of these values are omitted or empty, the default values apply.

Input Arguments

plant — Plant prediction model

LTI model | identified linear model

Plant prediction model, specified as either an LTI model or a linear System Identification Toolbox™ model. The specified plant corresponds to the Model.Plant property of the controller.

If you do not specify a sample time when creating your controller, plant must be a discrete-time model. For prediction, plant is discretized or resampled if needed using mpcobj.Ts as sample time. Delays, if present, are incorporated in the resulting discrete time model.

For more information on MPC prediction models, see MPC Prediction Models.

Note

Direct feedthrough from manipulated variables to any output in plant is not supported.

Example: tf(10,[1 10])

model — Prediction model

structure

Prediction model, specified as a structure with the same format as the Model property of the controller. If you do not specify a sample time when creating your controller, model.Plant must be a discrete-time model.

For more information on MPC prediction models, see MPC Prediction Models.

Properties

Ts — Controller sample time

positive scalar

Controller sample time, specified as a positive finite scalar. The controller uses a discrete-time model with sample time Ts for prediction. The time unit is inherited from plant.TimeUnit.

Example: mpcobj.Ts = 0.1

PredictionHorizon — Prediction horizon

10 + delay-dependent steps (default) | positive integer

Prediction horizon steps, specified as a positive integer. The product of PredictionHorizon and Ts is the prediction time; that is, how far the controller looks into the future.

If you specify a prediction horizon that is not larger than the number of steps necessary to cover the longest delay among all the input-output channels, Nd = max(mpcobj.Model.Plant.IODelay(:))/mpcobj.Ts, then the software automatically increases the prediction horizon by floor(Nd).

Example: mpcobj.PredictionHorizon = 15

ControlHorizon — Control horizon

2 (default) | positive integer | vector of positive integers

Control horizon, specified as one of the following:

• Positive integer, m, between 1 and p, inclusive, where p is equal to PredictionHorizon. In this case, the controller computes m free control moves occurring at times k through k+m-1, and holds the controller output constant for the remaining prediction horizon steps from k+m through k+p-1. Here, k is the current control interval.

• Vector of positive integers [m₁, m₂, …], specifying the lengths of blocking intervals. By default the controller computes M blocks of free moves, where M is the number of blocking intervals. The first free move applies to times k through k+m₁-1, the second free move applies from time k+m₁ through k+m₁+m₂-1, and so on. Using block moves can improve the robustness of your controller. The sum of the values in ControlHorizon must match the prediction horizon p. If you specify a vector whose sum is:

  • Less than the prediction horizon, then the controller adds a blocking interval. The length of this interval is such that the sum of the interval lengths is p. For example, if p=10 and you specify a control horizon of ControlHorizon=[1 2 3], then the controller uses four intervals with lengths [1 2 3 4].

  • Greater than the prediction horizon, then the intervals are truncated until the sum of the interval lengths is equal to p. For example, if p=10 and you specify a control horizon of ControlHorizon= [1 2 3 6 7], then the controller uses four intervals with lengths [1 2 3 4].

  For more information on manipulated variable blocking, see Manipulated Variable Blocking.

Example: mpcobj.ControlHorizon = 3

Model — Prediction model and nominal conditions

structure

Prediction model and nominal conditions, specified as a structure with the following fields. For more information on the MPC prediction model, see MPC Prediction Models and Controller State Estimation.

Plant — Plant prediction model

LTI model | identified linear model

Plant prediction model, specified as either an LTI model or a linear System Identification Toolbox model.

Note

Direct feedthrough from manipulated variables to any output in plant is not supported.

Example: mpcobj.Model.Plant = ss(-1,1,1,0)

Disturbance — Model describing expected unmeasured disturbances

tf(1,[1 0]) (default) | LTI model

Model describing expected unmeasured disturbances, specified as an LTI model. This model is required only when the plant has unmeasured disturbances. You can set this disturbance model directly using dot notation or using the setindist function.

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.

Example: mpcobj.Model.Disturbance = tf(5,[1 5])

Noise — Model describing expected output measurement noise

tf(1) (default) | LTI model

Model describing expected output measurement noise, specified as an LTI model.

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.

Example: mpcobj.Model.Noise = zpk(0,-1,1)

Nominal — Nominal operating point at which plant model is linearized

structure

Nominal operating point at which plant model is linearized, specified as a structure with the following fields.

Field	Description	Default
X	Plant state at operating point, specified as a column vector with length equal to the number of states in Model.Plant.	zero vector
U	Plant input at operating point, including manipulated variables and measured and unmeasured disturbances, specified as a column vector with length equal to the number of inputs in Model.Plant.	zero vector
Y	Plant output at operating point, including measured and unmeasured outputs, specified as a column vector with length equal to the number of outputs in Model.Plant.	zero vector
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. Specify DX as a column vector with length equal to the number of states in Model.Plant.	zero vector

ManipulatedVariables — Manipulated variable information, bounds, and scale factors

structure array

Manipulated Variable (MV) information, bounds, and scale factors, specified as a structure array with N_mv elements, where N_mv is the number of manipulated variables. To access this property, you can use the alias MV instead of ManipulatedVariables.

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)/T_s.

Each structure element has the following fields.

Min — MV lower bound

-Inf (default) | scalar | vector

Lower bound for a given manipulated variable, specified as a scalar or vector. By default, this lower bound is -Inf.

To use the same bound across the prediction horizon, specify a scalar value.

To vary the bound over the prediction horizon from time k to time k+p-1, specify a vector of up to p values. Here, k is the current time and p is the prediction horizon. If you specify fewer than p values, the final bound is used for the remaining steps of the prediction horizon.

Example: mpcobj.ManipulatedVariables(1).Min = -5

Max — MV upper bound

Inf (default) | scalar | vector

Upper bound for a given manipulated variable, specified as a scalar or vector. By default, this upper bound is Inf.

To use the same bound across the prediction horizon, specify a scalar value.

To vary the bound over the prediction horizon from time k to time k+p-1, specify a vector of up to p values. Here, k is the current time and p is the prediction horizon. If you specify fewer than p values, the final bound is used for the remaining steps of the prediction horizon.

Example: mpcobj.ManipulatedVariables(1).Max = 5

MinECR — MV lower bound softness

0 (default) | nonnegative scalar | vector

Softness of the lower bound for a given manipulated variable. A larger equal concern for relaxation (ECR) value indicates a softer constraint, specified as a nonnegative scalar or vector. By default, MV lower bounds are hard constraints.

To use the same ECR value across the prediction horizon, specify a scalar value.

To vary the ECR value over the prediction horizon from time k to time k+p-1, specify a vector of up to p values. Here, k is the current time and p is the prediction horizon. If you specify fewer than p values, the final ECR value is used for the remaining steps of the prediction horizon.

The i-th element of MinECR for the j-th manipulated variable corresponds to $$V_{j,min}^u\left(i\right)$$ in the constraints equations described in Constraints.

Example: mpcobj.ManipulatedVariables(1).MinECR = 0.01

MaxECR — MV upper bound

0 (default) | nonnegative scalar | vector

Softness of the upper bound for a given manipulated variable. A larger equal concern for relaxation (ECR) value indicates a softer constraint, specified as a nonnegative scalar or vector. By default, MV upper bounds are hard constraints.

To use the same ECR value across the prediction horizon, specify a scalar value.

To vary the ECR value over the prediction horizon from time k to time k+p-1, specify a vector of up to p values. Here, k is the current time and p is the prediction horizon. If you specify fewer than p values, the final ECR value is used for the remaining steps of the prediction horizon.

The i-th element of MaxECR for the j-th manipulated variable corresponds to $V_{j,max}^u(i)$ in the constraints equations described in Constraints.

Example: mpcobj.ManipulatedVariables(1).MaxECR = 0.01

RateMin — MV rate of change lower bound

-Inf (default) | nonpositive scalar | vector

Lower bound for the rate of change of a given manipulated variable, specified as a nonpositive scalar or vector. The MV rate of change is defined as MV(k) - MV(k-1), where k is the current time. By default, this lower bound is -Inf.

To use the same bound across the prediction horizon, specify a scalar value.

To vary the bound over the prediction horizon from time k to time k+p-1, specify a vector of up to p values. Here, k is the current time and p is the prediction horizon. If you specify fewer than p values, the final bound is used for the remaining steps of the prediction horizon.

Example: mpcobj.ManipulatedVariables(1).RateMin = -2

RateMax — MV rate of change upper bound

Inf (default) | nonnegative scalar | vector

Upper bound for the rate of change of a given manipulated variable, specified as a nonnegative scalar or vector. The MV rate of change is defined as MV(k) - MV(k-1), where k is the current time. By default, this lower bound is Inf.

To use the same bound across the prediction horizon, specify a scalar value.

To vary the bound over the prediction horizon from time k to time k+p-1, specify a vector of up to p values. Here, k is the current time and p is the prediction horizon. If you specify fewer than p values, the final bound is used for the remaining steps of the prediction horizon.

Example: mpcobj.ManipulatedVariables(1).RateMax = 2

RateMinECR — MV rate of change lower bound softness

0 (default) | nonnegative finite scalar | vector

Softness of the lower bound for the rate of change of a given manipulated variable. A larger equal concern for relaxation (ECR) value indicates a softer constraint, specified as a nonnegative finite scalar or vector. By default, MV rate of change lower bounds are hard constraints.

To use the same ECR value across the prediction horizon, specify a scalar value.

To vary the ECR values over the prediction horizon from time k to time k+p-1, specify a vector of up to p values. Here, k is the current time and p is the prediction horizon. If you specify fewer than p values, the final ECR values are used for the remaining steps of the prediction horizon.

The i-th element of RateMinECR for the j-th manipulated variable corresponds to $$V_{j,min}^{\Delta u}\left(i\right)$$ in the constraints equations described in Constraints.

Example: mpcobj.ManipulatedVariables(2).RateMinECR = 0.01

RateMaxECR — MV rate of change upper bound softness

0 (default) | nonnegative finite scalar | vector

Softness of the upper bound for the rate of change of a given manipulated variable. A larger equal concern for relaxation (ECR) value indicates a softer constraint, specified as a nonnegative finite scalar or vector. By default, MV rate of change upper bounds are hard constraints.

To use the same ECR value across the prediction horizon, specify a scalar value.

To vary the ECR values over the prediction horizon from time k to time k+p-1, specify a vector of up to p values. Here, k is the current time and p is the prediction horizon. If you specify fewer than p values, the final ECR values are used for the remaining steps of the prediction horizon.

The i-th element of RateMaxECR for the j-th manipulated variable corresponds to $V_{j,max}^{\Delta u}(i)$ in the constraints equations described in Constraints.

Example: mpcobj.ManipulatedVariables(2).RateMaxECR = 0.01

Target — MV targets

'nominal' (default) | scalar | vector

Targets for a given manipulated variable, specified as a scalar, vector, or as 'nominal' (default). When Target is 'nominal', then the manipulated variable targets correspond to mpcobj.Model.Nominal.U.

To use the same target across the prediction horizon, specify a scalar value.

To vary the target over the prediction horizon from time k to time k+p-1, specify a vector of up to p values. Here, k is the current time and p is the prediction horizon. If you specify fewer than p values, the final value is used for the remaining steps of the prediction horizon.

You might want to set target values for some manipulated variables, along with corresponding nonzero cost function weights, for economic or operational reasons, whenever you have more manipulated variables than plant outputs. For more information, see Design MPC Controller for Nonsquare Plants and Setting Targets for Manipulated Variables.

Example: mpcobj.ManipulatedVariables(2).Target = [0.3 0.2]

Name — MV name

string | character vector

Name of a given manipulated variable, specified as a string or character vector. This is a read-only property. To modify the names of manipulated variables, use mpcobj.Model.Plant.InputName.

Example: mpcobj.ManipulatedVariables(2).Name

Units — MV units

string | character vector

Units of a given manipulated variable, specified as a string or character vector. This is a read-only property. To modify the units of manipulated variables, use mpcobj.Model.Plant.InputUnit.

Example: mpcobj.ManipulatedVariables(2).Units

ScaleFactor — MV scale factor

1 (default) | positive finite scalar

Scale factor of a given manipulated variable, specified as a positive finite scalar. Specifying the proper scale factor can improve numerical conditioning for optimization. In general, use the amplitude of the operating range of the manipulated variable. For more information, see Specify Scale Factors.

Example: mpcobj.ManipulatedVariables(1).ScaleFactor = 10

Type — MV type

'continuous' (default) | 'integer' | 'binary' | vector

Type of a given manipulated variable, specified as:

• 'continuous' — This indicates that the manipulated variable is continuous.

• 'binary' — This restricts the manipulated variable to be either 0 or 1.

• 'integer' — This restricts the manipulated variable to be an integer.

• A vector containing all the possible values — This restricts the manipulated variable to the specified values, for example mpcobj.MV(1).Type = [-1,0,0.5,1,2];.

By default, the type is set to 'continuous'.

For more information, see Finite Control Set MPC.

Example: mpcobj.ManipulatedVariables(1).Type = 'binary'

OutputVariables — Output variable information, bounds, and scale factors

structure array

Output variable (OV) information, bounds, and scale factors, specified as a structure array with N_y elements, where N_y is the number of output variables. To access this property, you can use the alias OV instead of OutputVariables.

Each structure element has the following fields.

Min — OV lower bound

-Inf (default) | scalar | vector

Lower bound for a given output variable, specified as a scalar or vector. By default, this lower bound is -Inf.

To use the same bound across the prediction horizon, specify a scalar value.

To vary the bound over the prediction horizon from time k+1 to time k+p, specify a vector of up to p values. Here, k is the current time and p is the prediction horizon. If you specify fewer than p values, the final bound is used for the remaining steps of the prediction horizon.

Example: mpcobj.OutputVariables(1).Min = -10

Max — OV upper bound

Inf (default) | scalar | vector

Upper bound for a given output variable, specified as a scalar or vector. By default, this upper bound is Inf.

To use the same bound across the prediction horizon, specify a scalar value.

To vary the bound over the prediction horizon from time k+1 to time k+p, specify a vector of up to p values. Here, k is the current time and p is the prediction horizon. If you specify fewer than p values, the final bound is used for the remaining steps of the prediction horizon.

Example: mpcobj.OutputVariables(1).Max = 10

MinECR — OV lower bound softness

1 (default) | nonnegative finite scalar | vector

Softness of the lower bound for a given output variable. A larger equal concern for relaxation (ECR) value indicates a softer constraint, specified as a nonnegative finite scalar or vector. By default, OV upper bounds are soft constraints.

To avoid creating an infeasible optimization problem at run time, it is best practice to use soft OV bounds.

To use the same ECR value across the prediction horizon, specify a scalar value.

To vary the ECR value over the prediction horizon from time k+1 to time k+p, specify a vector of up to p values. Here, k is the current time and p is the prediction horizon. If you specify fewer than p values, the final ECR value is used for the remaining steps of the prediction horizon.

The i-th element of MinECR for the j-th output variable corresponds to $$V_{j,min}^y\left(i\right)$$ in the constraints equations described in Constraints.

Example: mpcobj.OutputVariables(1).MinECR = 5

MaxECR — OV upper bound softness

1 (default) | nonnegative finite scalar | vector

Softness of the upper bound for a given output variable. A larger equal concern for relaxation (ECR) value indicates a softer constraint, specified as a nonnegative finite scalar or vector. By default, OV lower bounds are soft constraints.

To avoid creating an infeasible optimization problem at run time, it is best practice to use soft OV bounds.

To use the same ECR value across the prediction horizon, specify a scalar value.

To vary the ECR value over the prediction horizon from time k+1 to time k+p, specify a vector of up to p values. Here, k is the current time and p is the prediction horizon. If you specify fewer than p values, the final ECR value is used for the remaining steps of the prediction horizon.

The i-th element of MaxECR for the j-th output variable corresponds to $V_{j,max}^y(i)$ in the constraints equations described in Constraints.

Example: mpcobj.OutputVariables(1).MaxECR = 10

Name — OV name

string | character vector

Name of a given output variable, specified as a string or character vector. This is a read-only property. To modify the names of output variables, use mpcobj.Model.Plant.OutputName.

Example: mpcobj.OutputVariables(2).Name

Units — OV units

string | character vector

Units of a given output variable, specified as a string or character vector. This is a read-only property. To modify the names of output variables, use mpcobj.Model.Plant.OutputUnit.

Example: mpcobj.OutputVariables(2).Units

ScaleFactor — OV scale factor

1 (default) | positive finite scalar

Scale factor of a given output variable, specified as a positive finite scalar. Specifying the proper scale factor can improve numerical conditioning for optimization. In general, use the operating range of the output variable. For more information, see Specify Scale Factors.

Example: mpcobj.OutputVariables(1).ScaleFactor = 20

DisturbanceVariables — Input disturbance variable information and scale factors

structure array

Disturbance variable (DV) information and scale factors, specified as a structure array with N_d elements, where N_d is the total number of measured and unmeasured disturbance inputs. The order of the disturbance signals within DisturbanceVariables is the following: the first N_md entries relate to measured input disturbances, the last N_ud entries relate to unmeasured input disturbances.

To access this property, you can use the alias DV instead of DisturbanceVariables.

Each structure element has the following fields.

Name — DV name

string | character vector

DV name, specified as a string or character vector. This is a read-only property. To modify the names of disturbance variables, use mpcobj.Plant.Inputname.

Example: mpcobj.DisturbanceVariables(1).Name

Units — OV units

"" (default) | string | character vector

OV units, specified as a string or character vector. This is a read-only property. To modify the units of disturbance variables, use mpcobj.Model.Plant.InputUnit.

Example: mpcobj.DisturbanceVariables(1).Units

ScaleFactor — DV scale factor

1 (default) | positive finite scalar

DV scale factor, specified as a positive finite scalar. Specifying the proper scale factor can improve numerical conditioning for optimization. For more information, see Specify Scale Factors.

Example: mpcobj.DisturbanceVariables(1).ScaleFactor = 15

Weights — Standard cost function tuning weights

structure

Standard cost function tuning weights, specified as a structure. The controller applies these weights to the scaled variables. Therefore, the tuning weights are dimensionless values.

The format of OutputWeights must match the format of the Weights.OutputVariables property of the controller object. For example, you cannot specify constant weights across the prediction horizon in the controller object, and then specify time-varying weights using mpcmoveopt.

Weights has the following fields. The values of these fields depend on whether you use the standard or alternative cost function. For more information on these cost functions, see QP Optimization Problem for Linear MPC.

ManipulatedVariables — Manipulated variable tuning weights

row vector | array

Manipulated variable tuning weights, which penalize deviations from MV targets, specified as a row vector or array of nonnegative values. The default weight for all manipulated variables is 0.

To use the same weights across the prediction horizon, specify a row vector of length N_mv, where N_mv is the number of manipulated variables.

To vary the tuning weights over the prediction horizon from time k to time k+p-1, specify an array with N_mv columns and up to p rows. Here, k is the current time and p is the prediction horizon. Each row contains the manipulated variable tuning weights for one prediction horizon step. If you specify fewer than p rows, the weights in the final row are used for the remaining steps of the prediction horizon.

If you use the alternative cost function, specify Weights.ManipulatedVariables as a cell array that contains the N_mv-by-N_mv R_u matrix. For example, mpcobj.Weights.ManipulatedVariables = {Ru}. R_u must be a positive semidefinite matrix. Varying the R_u matrix across the prediction horizon Is not supported. For more information, see Alternative Cost Function.

Example: mpcobj.Weights.ManipulatedVariables = [0.1 0.2]

ManipulatedVariablesRate — Manipulated variable rate tuning weights

row vector | array | cell array

Manipulated variable rate tuning weights, which penalize large changes in control moves, specified as a row vector or array of nonnegative values. The default weight for all manipulated variable rates is 0.1.

To use the same weights across the prediction horizon, specify a row vector of length N_mv, where N_mv is the number of manipulated variables.

To vary the tuning weights over the prediction horizon from time k to time k+p-1, specify an array with N_mv columns and up to p rows. Here, k is the current time and p is the prediction horizon. Each row contains the manipulated variable rate tuning weights for one prediction horizon step. If you specify fewer than p rows, the weights in the final row are used for the remaining steps of the prediction horizon.

Note

It is best practice to use nonzero manipulated variable rate weights. If all manipulated variable rate weights are strictly positive, the resulting QP problem is strictly convex. If some weights are zero, the QP Hessian could be positive semidefinite. To keep the QP problem strictly convex, when the condition number of the Hessian matrix K_ΔU is larger than 10¹², the quantity 10*sqrt(eps) is added to each diagonal term. See Cost Function.

If you use the alternative cost function, specify Weights.ManipulatedVariablesRate as a cell array that contains the N_mv-by-N_mv R_Δu matrix. For example, mpcobj.Weights.ManipulatedVariablesRate = {Rdu}. R_Δu must be a positive semidefinite matrix. Varying the R_Δu matrix across the prediction horizon Is not supported. For more information, see Alternative Cost Function.

Example: mpcobj.Weights.ManipulatedVariablesRate = [0.1 0.1]

OutputVariables — Output variable tuning weights

vector | array

Output variable tuning weights, which penalize deviation from output references, specified as a row vector or array of nonnegative values. The default weight for all output variables is 1.

To use the same weights across the prediction horizon, specify a row vector of length N_y, where N_y is the number of output variables.

To vary the tuning weights over the prediction horizon from time k+1 to time k+p, specify an array with N_y columns and up to p rows. Here, k is the current time and p is the prediction horizon. Each row contains the output variable tuning weights for one prediction horizon step. If you specify fewer than p rows, the weights in the final row are used for the remaining steps of the prediction horizon.

If you use the alternative cost function, specify Weights.OutputVariables as a cell array that contains the N_y-by-N_y Q matrix. For example, mpcobj.Weights.OutputVariables = {Q}. Q must be a positive semidefinite matrix. Varying the Q matrix across the prediction horizon Is not supported. For more information, see Alternative Cost Function.

Example: mpcobj.Weights.OutputVariables = [1 1]

ECR — Slack variable tuning weight

1e5 (default) | positive scalar

Slack variable tuning weight, specified as a positive scalar. Increase or decrease the equal concern for relaxation (ECR) weight to make all soft constraints harder or softer, respectively. This dimensionless weight corresponds to ρ_ε in Standard Cost Function. For more information, see , Constraints and Constraint Softening.

Example: mpcobj.Weights.ECR = 1e4

Optimizer — Optimization information

structure

Optimization information for the solver, specified as a structure with the following fields. The first three fields, OptimizationType, Solver, and SolverOptions, relate only to the built-in solvers. If you use a custom solver for simulation (by setting CustomSolver to true), these three fields are ignored for simulation. Likewise, if you use a custom solver for code generation (by setting CustomSolverCodeGen to true) these three fields are ignored for code generation.

For more information on the supported QP solvers, see QP Solvers for Linear MPC.

OptimizationType — Optimization type

Read-only: 'QP' (default) | 'MIQP'

This property is read-only.

Optimization type, returned as either 'QP' for continuous problems or 'MIQP' for finite-set problems. This property is selected automatically by the software according to the ManipulatedVariables.Type properties of the mpc object. For more information on finite set MPC, see Finite Control Set MPC.

Example: 'MIQP'

Solver — Solver algorithm

'active-set' (default) | 'interior-point' | 'admm' | 'branch-and-bound'

Solver algorithm, specified as one of the following:

• 'active-set' — Solve the QP problem using the KWIK active-set algorithm.

• 'interior-point' — Solve the QP problem using a primal-dual interior-point algorithm with the Mehrotra predictor-corrector.

• 'admm' — Solve the QP problem using a first-order alternating direction method of multipliers (ADMM) solver.

• 'branch-and-bound' — Solve the MIQP problem using a KWIK algorithm with branch and bound. This method is automatically selected when the Type property of any manipulated variable is set to anything other than 'continuous'.

For custom applications that require solving QP problems, you can also access the active-set and interior-point algorithms using the mpcActiveSetSolver and mpcInteriorPointSolver functions, respectively.

Example: mpcobj.Optimizer.Solver = 'interior-point'

SolverOptions — Solver options

ActiveSet object (default) | InteriorPoint object | ADMM object | BranchBound object

Solver options, specified as an ActiveSet, InteriorPoint, ADMM, or BranchBound options object. The options object that applies corresponds to the solver selected in the Solver field of the Optimizer property. For more information, see QP Optimization Problem for Linear MPC and QP Solvers for Linear MPC.

MinOutputECR — Minimum value allowed for output constraint ECR values

0 (default) | nonnegative scalar

Minimum value allowed for output constraint equal concern for relaxation (ECR) values, specified as a nonnegative scalar. A value of 0 indicates that hard output constraints are allowed. If either of the OutputVariables.MinECR (ECR for the lower bounds) or OutputVariables.MaxECR (ECR for the upper bounds) properties of an MPC controller are less than MinOutputECR, a warning is displayed and the value is raised to MinOutputECR during computation.

Example: mpcobj.Optimizer.MinOutputECR = 1e-10

UseSuboptimalSolution — Option indicating whether a suboptimal solution is acceptable

false (default) | true

Option indicating whether a suboptimal solution is acceptable, specified as a logical value. When the QP solver reaches the maximum number of iterations without finding a solution (the exit Option is 0), the controller:

• Freezes the MV values if UseSuboptimalSolution is false

• Applies the suboptimal solution found by the solver after the final iteration if UseSuboptimalSolution is true

To specify the maximum number of iterations, depending on the value of Algorithm, use either ActiveSetOptions.MaxIterations or InteriorPointOptions.MaxIterations.

Example: mpcobj.Optimizer.UseSuboptimalSolution = true

CustomSolver — Option indicating whether to use a custom QP solver for simulation

false (default) | true

Option indicating whether to use a custom QP solver for simulation, specified as a logical value. If CustomSolver is true, the user must provide an mpcCustomSolver function on the MATLAB^® path.

This custom solver is not used for code generation. To generate code for a controller with a custom solver, use CustomSolverCodeGen.

If CustomSolver is true, the controller ignores the OptimizationType, Solver and SolverOptions properties for simulation.

You can also use the function setCustomSolver to automatically configure mpcobj to use the active-set algorithm of quadprog (Optimization Toolbox) as a custom QP solver for both simulation and code generation.

For more information on using a custom QP solver see, QP Solvers for Linear MPC.

Example: mpcobj.Optimizer.CustomSolver = true

CustomSolverCodeGen — Option indicating whether to use a custom QP solver for code generation

false (default) | true

Option indicating whether to use a custom QP solver for code generation, specified as a logical value. If CustomSolverCodeGen is true, you must provide an mpcCustomSolverCodeGen function on the MATLAB path.

This custom solver is not used for simulation. To simulate a controller with a custom solver, use CustomSolver.

You can also use the function setCustomSolver to automatically configure mpcobj to use the active-set algorithm of quadprog (Optimization Toolbox) as a custom QP solver for both simulation and code generation.

For more information on using a custom QP solver, see QP Solvers for Linear MPC.

Example: mpcobj.Optimizer.CustomSolverCodeGen = true

Notes — User notes

{} (default) | cell array of character vectors

User notes associated with the MPC controller, specified as a cell array of character vectors.

Example: mpcobj.Notes = {'Longitudinal Controller'; 'Version 2.1'}

UserData — User data

[] (default) | any MATLAB data

User data associated with the MPC controller, specified as any MATLAB data, such as a cell array or structure.

Example: mpcobj.UserData = {'Parameters',0.2,[3 4]'}

History — Controller creation date and time

Read-only: vector

This property is read-only.

Controller creation date and time, specified as a vector with the following elements:

• History(1) — Year

• History(2) — Month

• History(3) — Day

• History(4) — Hours

• History(5) — Minutes

• History(6) — Seconds

Use datestr(mpcobj.History) to display the controller creation date as a character vector.

Example: mpcobj.History = datevec(now)

Object Functions

buildMEX	Build MEX file that solves an MPC control problem
cloffset	Compute closed-loop DC gain from output disturbances to measured outputs assuming constraints are inactive at steady state
compare	Compare two MPC objects
generateExplicitMPC	Convert implicit MPC controller to explicit MPC controller
generatePlotParameters	Parameters for plotSection
generateExplicitOptions	Optimization options for explicit MPC generation
generateExplicitRange	Bounds on explicit MPC control law parameters
get	Get property values from MPC object
getCodeGenerationData	Create data structures for mpcmoveCodeGeneration
getEstimator	Obtain Kalman gains and model for estimator design
getconstraint	Obtain mixed input/output constraints from model predictive controller
getindist	Retrieve unmeasured input disturbance model
getname	Retrieve I/O signal names from MPC plant model
getoutdist	Retrieve unmeasured output disturbance model
mpcmove	Compute optimal control action and update controller states
mpcmoveAdaptive	Compute optimal control with prediction model updating
plot	Plot responses generated by MPC simulations
review	Examine MPC controller for design errors and stability problems at run time
sensitivity	Calculate the value of a performance metric and its sensitivity to the diagonal weights of an MPC controller
set	Set or modify MPC object properties
setCustomSolver	Configures an MPC object to use the QP solver from Optimization Toolbox as a custom solver
setEstimator	Modify a model predictive controller’s state estimator
setconstraint	Set mixed input/output constraints for model predictive controller
setindist	Modify unmeasured input disturbance model
setname	Set I/O signal names in MPC plant model
setoutdist	Modify unmeasured output disturbance model
setterminal	Terminal weights and constraints
sim	Simulate an MPC controller in closed loop with a linear plant
size	Size and order of MPC Controller
ss	Convert unconstrained MPC controller to state-space linear system form
tf	Convert unconstrained MPC controller to linear transfer function form
trim	Compute steady-state value of MPC controller plant model state for given inputs and outputs

Examples

Create MPC Controller with Specified Prediction and Control Horizons

Create a plant model with the transfer function $$(s+1)/(s^2+2s)$$.

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 either the setmpcsignals command, or the LTI InputGroup and OutputGroup properties.

Specify a sample time for the controller.

Ts = 0.1;

Define bounds on the manipulated variable, $$u$$, such that $$-1\le u\le 1$$.

MV = struct(Min=-1,Max=1);

MV contains only the upper and lower bounds on the manipulated variable. In general, you can specify additional MV properties. When you do not specify other properties, their default values apply.

Specify a 20-interval prediction horizon and a 3-interval control horizon.

p = 20;
m = 3;

Create an MPC controller using the specified values. The fifth input argument is empty, so default tuning weights apply.

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

-->"Weights.ManipulatedVariables" is empty. Assuming default 0.00000.
-->"Weights.ManipulatedVariablesRate" is empty. Assuming default 0.10000.
-->"Weights.OutputVariables" is empty. Assuming default 1.00000.

Algorithms

To minimize computational overhead, model predictive controller creation occurs in two phases. The first happens at creation when you use the mpc function, or when you change a controller property. Creation includes basic validity and consistency checks, such as signal dimensions and nonnegativity 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 function.

Alternative Functionality

You can also create model predictive controllers using the MPC Designer app.

Version History

Introduced before R2006a

R2018b: Support for implementing economic MPC using a linear MPC controller has been removed

Support for implementing economic MPC using a linear MPC controller has been removed. Implement economic MPC using a nonlinear MPC controller instead. For more information on nonlinear MPC controllers, see Nonlinear MPC.

Update Code

If you previously saved a linear MPC object configured with custom cost or constraint functions, the software generates a warning when the object is loaded and an error if it is simulated. To suppress the error and warning messages and continue using your linear MPC controller, mpcobj, without the custom costs and constraints, set the IsEconomicMPC option to false.

mpcobj.IsEconomicMPC = false;

To implement your economic MPC controller using a nonlinear MPC object:

1. Create an nlmpc object.

2. Convert your custom cost function to the format required for nonlinear MPC. For more information on nonlinear MPC cost functions, see Specify Cost Function for Nonlinear MPC.

3. Convert your custom constraint function to the format required for nonlinear MPC. For more information on nonlinear MPC constraints, see Specify Constraints for Nonlinear MPC.

4. Implement your linear prediction model using state and output functions. For more information on nonlinear MPC prediction models, see Specify Prediction Model for Nonlinear MPC.