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 Modeling.

Creation

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 controller sample time to ts, which corresponds to 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 sample time ts when used for predictions

example

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 controller sample time to ts. If model.Plant is a discrete-time LTI model with an unspecified sample time (model.Plant.Ts = –1), it inherits 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

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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 more information on MPC prediction models, see MPC Modeling.

Note

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

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 Modeling.

Properties

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Controller sample time, specified as a positive finite scalar. The controller uses a discrete-time model with sample time Ts for prediction.

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.

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.

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

The Model structure has the following fields.

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.

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.

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.

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

FieldDescriptionDefault
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

Manipulated Variable (MV) information, bounds, and scale factors, specified as a structure array with Nmv elements, where Nmv 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)/Ts.

Each structure element has the following fields.

MV lower bound, specified as a scalar or vector. By default, this lower bound is unconstrained.

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.

MV upper bound, specified as a scalar or vector. By default, this upper bound is unconstrained.

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.

MV lower bound softness, where 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.

MV upper bound softness, where 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.

MV rate of change lower bound, 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 unconstrained.

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.

MV rate of change upper bound, 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 unconstrained.

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.

MV rate of change lower bound softness, where 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.

MV rate of change upper bound softness, where 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.

MV name, specified as a string or character vector.

MV units, specified as a string or character vector.

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

Output variable (OV) information, bounds, and scale factors, specified as a structure array with Ny elements, where Ny 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.

OV lower bound, specified as a scalar or vector. By default, this lower bound is unconstrained.

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.

OV upper bound, specified as a scalar or vector. By default, this upper bound is unconstrained.

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.

OV lower bound softness, where 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.

OV upper bound softness, where 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.

OV name, specified as a string or character vector.

OV units, specified as a string or character vector.

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

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

DV name, specified as a string or character vector.

OV units, specified as a string or character vector.

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.

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 Optimization Problem.

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 Nmv, where Nmv 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 Nmv 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 Nmv-by-Nmv Ru matrix. For example, mpcobj.Weights.ManipulatedVariables = {Ru}. Ru must be a positive semidefinite matrix. Varying the Ru matrix across the prediction horizon Is not supported. For more information, see Alternative Cost Function.

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 Nmv, where Nmv 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 Nmv 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.

To improve the numerical robustness of the optimization problem, the software adds the quantity 10*sqrt(eps) to each zero-valued weight.

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 KΔU 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 Nmv-by-Nmv RΔu matrix. For example, mpcobj.Weights.ManipulatedVariablesRate = {Rdu}. RΔu must be a positive semidefinite matrix. Varying the RΔu matrix across the prediction horizon Is not supported. For more information, see Alternative Cost Function.

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 Ny, where Ny 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 Ny 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 Ny-by-Ny 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.

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.

QP optimization parameters, specified as a structure with the following fields.

Maximum number of iterations allowed in the QP solver, specified as one of the following:

  • 'Default' — The MPC controller automatically computes the maximum number of QP solver iterations as:

    MaxIter=4(nc+nv)

    Here,

    • nc is the total number of constraints across the prediction horizon.

    • nv is the total number of optimization variables across the control horizon.

    The default MaxIter value has a lower bound of 120.

  • Positive integer — The QP solver stops after MaxIter iterations. If the solver fails to converge in the final iteration, the controller:

    • Freezes the controller movement if UseSuboptimalSolution is false.

    • Applies the suboptimal solution reached after the final iteration if UseSuboptimalSolution is true.

If CustomSolver or CustomSolverCodeGen is true, the controller does not require the custom solver to honor MaxIter.

Note

The default MaxIter value can be very large for some controller configurations, such as those with large prediction and control horizons. When simulating such controllers, if the QP solver cannot find a feasible solution, the simulation can appear to stop responding, since the solver continues searching for MaxIter iterations.

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 or OutputVariables.MaxECR properties of an MPC controller are less than MinOutputECR, a warning is displayed and the value is raised to MinOutputECR during computation.

Flag 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 flag 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, use Optimizer.MaxIter.

Flag indicating whether to warm start each QP solver iteration by passing in a list of active inequalities from the previous iteration, specified as a logical value. Inequalities are active when their equal portion is true.

If CustomSolver or CustomSolverCodeGen is true, the controller does not require the custom solver to honor UseWarmStart.

Flag 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 does not require the custom solver to honor MaxIter and UseWarmStart.

For more information on specifying custom solvers, see Custom QP Solver.

Flag indicating whether to use a custom QP solver for code generation, specified as a logical value. If CustomSolverCodeGen is true, the user 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.

If CustomSolverCodeGen is true, the controller does not require the custom solver to honor MaxIter and UseWarmStart.

For more information on specifying custom solvers, see Custom QP Solver.

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

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

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

Object Functions

reviewExamine MPC controller for design errors and stability problems at run time
mpcmoveCompute optimal control action
simSimulate closed-loop/open-loop response to arbitrary reference and disturbance signals for implicit or explicit MPC
mpcstateMPC controller state
getCodeGenerationDataCreate data structures for mpcmoveCodeGeneration
generateExplicitMPCConvert implicit MPC controller to explicit MPC controller

Examples

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Create a plant model with the transfer function (s+1)/(s2+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 -1u1.

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);
-->The "Weights.ManipulatedVariables" property of "mpc" object is empty. Assuming default 0.00000.
-->The "Weights.ManipulatedVariablesRate" property of "mpc" object is empty. Assuming default 0.10000.
-->The "Weights.OutputVariables" property of "mpc" object 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.

Compatibility Considerations

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Errors starting in R2018b

Introduced before R2006a