Functions: Predictive Controllers

Abstract supertype of all predictive controllers.

(mpc::PredictiveController)(ry, d=[]; kwargs...) -> u

Functor allowing callable PredictiveController object as an alias for moveinput!.

Examples

julia> mpc = LinMPC(LinModel(tf(5, [2, 1]), 3), Nwt=[0], Hp=1000, Hc=1, direct=false);

julia> u = mpc([5]); round.(u, digits=3)
1-element Vector{Float64}:
 1.0

LinMPC(model::LinModel; <keyword arguments>)

Construct a linear predictive controller based on LinModel model.

The controller minimizes the following objective function at each discrete time $k$:

\[\begin{aligned} \min_{\mathbf{Z}, ϵ} \mathbf{(R̂_y - Ŷ)}' \mathbf{M}_{H_p} \mathbf{(R̂_y - Ŷ)} + \mathbf{(ΔU)}' \mathbf{N}_{H_c} \mathbf{(ΔU)} \\ + \mathbf{(R̂_u - U)}' \mathbf{L}_{H_p} \mathbf{(R̂_u - U)} + C ϵ^2 \end{aligned}\]

subject to setconstraint! bounds, and in which the weight matrices are repeated $H_p$ or $H_c$ times by default:

\[\begin{aligned} \mathbf{M}_{H_p} &= \text{diag}\mathbf{(M,M,...,M)} \\ \mathbf{N}_{H_c} &= \text{diag}\mathbf{(N,N,...,N)} \\ \mathbf{L}_{H_p} &= \text{diag}\mathbf{(L,L,...,L)} \end{aligned}\]

Time-varying and non-diagonal weights are also supported. Modify the last block in $\mathbf{M}_{H_p}$ to specify a terminal weight. The content of the decision vector $\mathbf{Z}$ depends on the chosen TranscriptionMethod (default to SingleShooting, hence $\mathbf{Z = ΔU}$). The $\mathbf{ΔU}$ includes the input increments $\mathbf{Δu}(k+j) = \mathbf{u}(k+j) - \mathbf{u}(k+j-1)$ from $j=0$ to $H_c-1$, the $\mathbf{Ŷ}$ vector, the output predictions $\mathbf{ŷ}(k+j)$ from $j=1$ to $H_p$, and the $\mathbf{U}$ vector, the manipulated inputs $\mathbf{u}(k+j)$ from $j=0$ to $H_p-1$. The slack variable $ϵ$ relaxes the constraints, as described in setconstraint! documentation. See Extended Help for a detailed nomenclature.

This method uses the default state estimator, a SteadyKalmanFilter with default arguments. This controller allocates memory at each time step for the optimization.

Arguments

• model::LinModel : model used for controller predictions and state estimations.
• Hp::Int=10+nk : prediction horizon $H_p$, nk is the number of delays in model.
• Hc::Union{Int, Vector{Int}}=2 : control horizon $H_c$, custom move blocking pattern is specified with a vector of integers (see move_blocking for details).
• Mwt=fill(1.0,model.ny) : main diagonal of $\mathbf{M}$ weight matrix (vector).
• Nwt=fill(0.1,model.nu) : main diagonal of $\mathbf{N}$ weight matrix (vector).
• Lwt=fill(0.0,model.nu) : main diagonal of $\mathbf{L}$ weight matrix (vector).
• M_Hp=Diagonal(repeat(Mwt,Hp)) : positive semidefinite symmetric matrix $\mathbf{M}_{H_p}$.
• N_Hc=Diagonal(repeat(Nwt,Hc)) : positive semidefinite symmetric matrix $\mathbf{N}_{H_c}$.
• L_Hp=Diagonal(repeat(Lwt,Hp)) : positive semidefinite symmetric matrix $\mathbf{L}_{H_p}$.
• Cwt=1e5 : slack variable weight $C$ (scalar), use Cwt=Inf for hard constraints only.
• transcription=SingleShooting() : SingleShooting or MultipleShooting.
• optim=JuMP.Model(OSQP.MathOptInterfaceOSQP.Optimizer) : quadratic optimizer used in the predictive controller, provided as a JuMP.Model object (default to OSQP optimizer).
• additional keyword arguments are passed to SteadyKalmanFilter constructor.

Examples

julia> model = LinModel([tf(3, [30, 1]); tf(-2, [5, 1])], 4);

julia> mpc = LinMPC(model, Mwt=[0, 1], Nwt=[0.5], Hp=30, Hc=1)
LinMPC controller with a sample time Ts = 4.0 s:
├ estimator: SteadyKalmanFilter
├ model: LinModel
├ optimizer: OSQP
├ transcription: SingleShooting
└ dimensions:
  ├ 30 prediction steps Hp
  ├  1 control steps Hc
  ├  1 slack variable ϵ (control constraints)
  ├  1 manipulated inputs u (0 integrating states)
  ├  4 estimated states x̂
  ├  2 measured outputs ym (2 integrating states)
  ├  0 unmeasured outputs yu
  └  0 measured disturbances d

Extended Help

LinMPC(estim::StateEstimator; <keyword arguments>)

Use custom state estimator estim to construct LinMPC.

estim.model must be a LinModel. Else, a NonLinMPC is required. See ManualEstimator for linear controllers with nonlinear state estimation.

Examples

julia> estim = KalmanFilter(LinModel([tf(3, [30, 1]); tf(-2, [5, 1])], 4), i_ym=[2]);

julia> mpc = LinMPC(estim, Mwt=[0, 1], Nwt=[0.5], Hp=30, Hc=1)
LinMPC controller with a sample time Ts = 4.0 s:
├ estimator: KalmanFilter
├ model: LinModel
├ optimizer: OSQP
├ transcription: SingleShooting
└ dimensions:
  ├ 30 prediction steps Hp
  ├  1 control steps Hc
  ├  1 slack variable ϵ (control constraints)
  ├  1 manipulated inputs u (0 integrating states)
  ├  3 estimated states x̂
  ├  1 measured outputs ym (1 integrating states)
  ├  1 unmeasured outputs yu
  └  0 measured disturbances d

ExplicitMPC(model::LinModel; <keyword arguments>)

Construct an explicit linear predictive controller based on LinModel model.

The controller minimizes the following objective function at each discrete time $k$:

\[\begin{aligned} \min_{\mathbf{ΔU}} \mathbf{(R̂_y - Ŷ)}' \mathbf{M}_{H_p} \mathbf{(R̂_y - Ŷ)} + \mathbf{(ΔU)}' \mathbf{N}_{H_c} \mathbf{(ΔU)} \\ + \mathbf{(R̂_u - U)}' \mathbf{L}_{H_p} \mathbf{(R̂_u - U)} \end{aligned}\]

See LinMPC for the variable definitions. This controller does not support constraints but the computational costs are extremely low (array division), therefore suitable for applications that require small sample times. The keyword arguments are identical to LinMPC, except for Cwt, transcription and optim, which are not supported. It uses a SingleShooting transcription method and is allocation-free.

This method uses the default state estimator, a SteadyKalmanFilter with default arguments.

Examples

julia> model = LinModel([tf(3, [30, 1]); tf(-2, [5, 1])], 4);

julia> mpc = ExplicitMPC(model, Mwt=[0, 1], Nwt=[0.5], Hp=30, Hc=1)
ExplicitMPC controller with a sample time Ts = 4.0 s:
├ estimator: SteadyKalmanFilter
├ model: LinModel
└ dimensions:
  ├  1 manipulated inputs u (0 integrating states)
  ├  4 estimated states x̂
  ├  2 measured outputs ym (2 integrating states)
  ├  0 unmeasured outputs yu
  └  0 measured disturbances d

ExplicitMPC(estim::StateEstimator; <keyword arguments>)

Use custom state estimator estim to construct ExplicitMPC.

estim.model must be a LinModel. Else, a NonLinMPC is required.

NonLinMPC(model::SimModel; <keyword arguments>)

Construct a nonlinear predictive controller based on SimModel model.

Both NonLinModel and LinModel are supported (see Extended Help). The controller minimizes the following objective function at each discrete time $k$:

\[\begin{aligned} \min_{\mathbf{Z}, ϵ}\ & \mathbf{(R̂_y - Ŷ)}' \mathbf{M}_{H_p} \mathbf{(R̂_y - Ŷ)} + \mathbf{(ΔU)}' \mathbf{N}_{H_c} \mathbf{(ΔU)} \\& + \mathbf{(R̂_u - U)}' \mathbf{L}_{H_p} \mathbf{(R̂_u - U)} + C ϵ^2 + E J_E(\mathbf{U_e}, \mathbf{Ŷ_e}, \mathbf{D̂_e}, \mathbf{p}) \end{aligned}\]

subject to setconstraint! bounds, and the custom inequality constraints:

\[\mathbf{g_c}(\mathbf{U_e}, \mathbf{Ŷ_e}, \mathbf{D̂_e}, \mathbf{p}, ϵ) ≤ \mathbf{0}\]

with the decision variables $\mathbf{Z}$ and slack $ϵ$. By default, a SingleShooting transcription method is used, hence $\mathbf{Z=ΔU}$. The economic function $J_E$ can penalizes solutions with high economic costs. Setting all the weights to 0 except $E$ creates a pure economic model predictive controller (EMPC). As a matter of fact, $J_E$ can be any nonlinear function as a custom objective, even if there is no economic interpretation to it. The arguments of $J_E$ and $\mathbf{g_c}$ include the manipulated inputs, predicted outputs and measured disturbances, extended from $k$ to $k+H_p$ (inclusively, see Extended Help for more details):

\[ \mathbf{U_e} = \begin{bmatrix} \mathbf{U} \\ \mathbf{u}(k+H_p-1) \end{bmatrix} , \quad \mathbf{Ŷ_e} = \begin{bmatrix} \mathbf{ŷ}(k) \\ \mathbf{Ŷ} \end{bmatrix} , \quad \mathbf{D̂_e} = \begin{bmatrix} \mathbf{d}(k) \\ \mathbf{D̂} \end{bmatrix}\]

The argument $\mathbf{p}$ is a custom parameter object of any type, but use a mutable one if you want to modify it later e.g.: a vector. See LinMPC Extended Help for the definition of the other variables.

This method uses the default state estimator:

• if model is a LinModel, a SteadyKalmanFilter with default arguments;
• else, an UnscentedKalmanFilter with default arguments.

This controller allocates memory at each time step for the optimization.

Arguments

• model::SimModel : model used for controller predictions and state estimations.
• Hp::Int=10+nk : prediction horizon $H_p$, nk is the number of delays if model is a LinModel (must be specified otherwise).
• Hc::Union{Int, Vector{Int}}=2 : control horizon $H_c$, custom move blocking pattern is specified with a vector of integers (see move_blocking for details).
• Mwt=fill(1.0,model.ny) : main diagonal of $\mathbf{M}$ weight matrix (vector).
• Nwt=fill(0.1,model.nu) : main diagonal of $\mathbf{N}$ weight matrix (vector).
• Lwt=fill(0.0,model.nu) : main diagonal of $\mathbf{L}$ weight matrix (vector).
• M_Hp=Diagonal(repeat(Mwt,Hp)) : positive semidefinite symmetric matrix $\mathbf{M}_{H_p}$.
• N_Hc=Diagonal(repeat(Nwt,Hc)) : positive semidefinite symmetric matrix $\mathbf{N}_{H_c}$.
• L_Hp=Diagonal(repeat(Lwt,Hp)) : positive semidefinite symmetric matrix $\mathbf{L}_{H_p}$.
• Cwt=1e5 : slack variable weight $C$ (scalar), use Cwt=Inf for hard constraints only.
• Ewt=0.0 : economic costs weight $E$ (scalar).
• JE=(_,_,_,_)->0.0 : economic or custom cost function $J_E(\mathbf{U_e}, \mathbf{Ŷ_e}, \mathbf{D̂_e}, \mathbf{p})$.
• gc=(_,_,_,_,_,_)->nothing or gc! : custom inequality constraint function $\mathbf{g_c}(\mathbf{U_e}, \mathbf{Ŷ_e}, \mathbf{D̂_e}, \mathbf{p}, ϵ)$, mutating or not (details in Extended Help).
• nc=0 : number of custom inequality constraints.
• p=model.p : $J_E$ and $\mathbf{g_c}$ functions parameter $\mathbf{p}$ (any type).
• transcription=SingleShooting() : a TranscriptionMethod for the optimization.
• optim=JuMP.Model(Ipopt.Optimizer) : nonlinear optimizer used in the predictive controller, provided as a JuMP.Model object (default to Ipopt optimizer).
• gradient=AutoForwardDiff() : an AbstractADType backend for the gradient of the objective function, see DifferentiationInterface doc.
• jacobian=default_jacobian(transcription) : an AbstractADType backend for the Jacobian of the nonlinear constraints, see gradient above for the options (default in Extended Help).
• oracle=JuMP.solver_name(optim)=="Ipopt": use the efficient VectorNonlinearOracle for the nonlinear constraints (not supported by most optimizers for now).
• additional keyword arguments are passed to UnscentedKalmanFilter constructor (or SteadyKalmanFilter, for LinModel).

Examples

julia> model = NonLinModel((x,u,_,_)->0.5x+u, (x,_,_)->2x, 10.0, 1, 1, 1, solver=nothing);

julia> mpc = NonLinMPC(model, Hp=20, Hc=10, transcription=MultipleShooting())
NonLinMPC controller with a sample time Ts = 10.0 s:
├ estimator: UnscentedKalmanFilter
├ model: NonLinModel
├ optimizer: Ipopt 
├ transcription: MultipleShooting
├ gradient: AutoForwardDiff
├ jacobian: AutoSparse (AutoForwardDiff, TracerSparsityDetector, GreedyColoringAlgorithm)
└ dimensions:
  ├ 20 prediction steps Hp
  ├ 10 control steps Hc
  ├  1 slack variable ϵ (control constraints)
  ├  1 manipulated inputs u (0 integrating states)
  ├  2 estimated states x̂
  ├  1 measured outputs ym (1 integrating states)
  ├  0 unmeasured outputs yu
  └  0 measured disturbances d

Extended Help

AutoSparse(
    AutoForwardDiff(); 
    sparsity_detector  = TracerSparsityDetector(), 
    coloring_algorithm = GreedyColoringAlgorithm()
)

NonLinMPC(estim::StateEstimator; <keyword arguments>)

Use custom state estimator estim to construct NonLinMPC.

Examples

julia> model = NonLinModel((x,u,_,_)->0.5x+u, (x,_,_)->2x, 10.0, 1, 1, 1, solver=nothing);

julia> estim = UnscentedKalmanFilter(model, σQint_ym=[0.05]);

julia> mpc = NonLinMPC(estim, Hp=20, Cwt=1e6)
NonLinMPC controller with a sample time Ts = 10.0 s:
├ estimator: UnscentedKalmanFilter
├ model: NonLinModel
├ optimizer: Ipopt
├ transcription: SingleShooting
├ gradient: AutoForwardDiff
├ jacobian: AutoForwardDiff
└ dimensions:
  ├ 20 prediction steps Hp
  ├  2 control steps Hc
  ├  1 slack variable ϵ (control constraints)
  ├  1 manipulated inputs u (0 integrating states)
  ├  2 estimated states x̂
  ├  1 measured outputs ym (1 integrating states)
  ├  0 unmeasured outputs yu
  └  0 measured disturbances d

moveinput!(mpc::PredictiveController, ry=mpc.estim.model.yop, d=[]; <keyword args>) -> u

Compute the optimal manipulated input value u for the current control period.

Solve the optimization problem of mpc PredictiveController and return the results $\mathbf{u}(k)$. Following the receding horizon principle, the algorithm discards the optimal future manipulated inputs $\mathbf{u}(k+1), \mathbf{u}(k+2), ...$ Note that the method mutates mpc internal data (it stores u - mpc.estim.model.uop at mpc.lastu0 for instance) but it does not modifies mpc.estim states. Call preparestate!(mpc, ym, d) before moveinput!, and updatestate!(mpc, u, ym, d) after, to update mpc state estimates. Setpoint and measured disturbance previews can be implemented with the R̂y, R̂u and D̂ keyword arguments.

Calling a PredictiveController object calls this method.

See also LinMPC, ExplicitMPC, NonLinMPC.

Arguments

• mpc::PredictiveController : solve optimization problem of mpc.
• ry=mpc.estim.model.yop : current output setpoints $\mathbf{r_y}(k)$.
• d=[] : current measured disturbances $\mathbf{d}(k)$.
• D̂=repeat(d, mpc.Hp) or Dhat : predicted measured disturbances $\mathbf{D̂}$, constant in the future by default or $\mathbf{d̂}(k+j)=\mathbf{d}(k)$ for $j=1$ to $H_p$.
• R̂y=repeat(ry, mpc.Hp) or Rhaty : predicted output setpoints $\mathbf{R̂_y}$, constant in the future by default or $\mathbf{r̂_y}(k+j)=\mathbf{r_y}(k)$ for $j=1$ to $H_p$.
• R̂u=mpc.Uop or Rhatu : predicted manipulated input setpoints $\mathbf{R̂_u}$, constant in the future by default or $\mathbf{r̂_u}(k+j)=\mathbf{u_{op}}$ for $j=0$ to $H_p-1$.

Examples

julia> mpc = LinMPC(LinModel(tf(5, [2, 1]), 3), Nwt=[0], Hp=1000, Hc=1);

julia> preparestate!(mpc, [0]); ry = [5];

julia> u = moveinput!(mpc, ry); round.(u, digits=3)
1-element Vector{Float64}:
 1.0

Abstract supertype of all transcription methods of PredictiveController.

The module currently supports SingleShooting, MultipleShooting and TrapezoidalCollocation transcription methods.

SingleShooting()

Construct a direct single shooting TranscriptionMethod.

The decision variable in the optimization problem is (excluding the slack $ϵ$ and without any custom move blocking):

\[\mathbf{Z} = \mathbf{ΔU} = \begin{bmatrix} \mathbf{Δu}(k+0) \\ \mathbf{Δu}(k+1) \\ \vdots \\ \mathbf{Δu}(k+H_c-1) \end{bmatrix}\]

This method computes the predictions by calling the augmented discrete-time model recursively over the prediction horizon $H_p$ in the objective function, or by updating the linear coefficients of the quadratic optimization for LinModel. It is generally more efficient for small control horizon $H_c$, stable and mildly nonlinear plant model/constraints.

MultipleShooting(; f_threads=false, h_threads=false)

Construct a direct multiple shooting TranscriptionMethod.

The decision variable is (excluding $ϵ$):

\[\mathbf{Z} = \begin{bmatrix} \mathbf{ΔU} \\ \mathbf{X̂_0} \end{bmatrix}\]

thus it also includes the predicted states, expressed as deviation vectors from the operating point $\mathbf{x̂_{op}}$ (see augment_model):

\[\mathbf{X̂_0} = \mathbf{X̂ - X̂_{op}} = \begin{bmatrix} \mathbf{x̂}_i(k+1) - \mathbf{x̂_{op}} \\ \mathbf{x̂}_i(k+2) - \mathbf{x̂_{op}} \\ \vdots \\ \mathbf{x̂}_i(k+H_p) - \mathbf{x̂_{op}} \end{bmatrix}\]

where $\mathbf{x̂}_i(k+j)$ is the state prediction for time $k+j$, estimated by the observer at time $i=k$ or $i=k-1$ depending on its direct flag. Note that $\mathbf{X̂_0 = X̂}$ if the operating point is zero, which is typically the case in practice for NonLinModel.

This transcription computes the predictions by calling the augmented discrete-time model in the equality constraint function recursively over $H_p$, or by updating the linear equality constraint vector for LinModel. It is generally more efficient for large control horizon $H_c$, unstable or highly nonlinear models/constraints. Multithreading with f_threads or h_threads keyword arguments can be advantageous if $\mathbf{f}$ or $\mathbf{h}$ in the NonLinModel is expensive to evaluate, respectively.

Sparse optimizers like OSQP or Ipopt and sparse Jacobian computations are recommended for this transcription method.

TrapezoidalCollocation(h::Int=0; f_threads=false, h_threads=false)

Construct an implicit trapezoidal TranscriptionMethod with hth order hold.

This is the simplest collocation method. It supports continuous-time NonLinModels only. The decision variables are the same as for MultipleShooting, hence similar computational costs. See the same docstring for descriptions of f_threads and h_threads keywords. The h argument is 0 or 1, for piecewise constant or linear manipulated inputs $\mathbf{u}$ (h=1 is slightly less expensive). Note that the various DiffSolver here assume zero-order hold, so h=1 will induce a plant-model mismatch if the plant is simulated with these solvers.

This transcription computes the predictions by calling the continuous-time model in the equality constraint function and by using the implicit trapezoidal rule. It can handle moderately stiff systems and is A-stable. Note that the built-in StateEstimator will still use the solver provided at the construction of the NonLinModel to estimate the plant states, not the trapezoidal rule (see supersample option of RungeKutta for stiff systems). See Extended Help for more details.

Sparse optimizers like Ipopt and sparse Jacobian computations are recommended for this transcription method.

Extended Help