Functions: Predictive Controllers
All the predictive controllers in this module rely on a state estimator to compute the predictions. The default LinMPC
estimator is a SteadyKalmanFilter
, and NonLinMPC
with nonlinear models, an UnscentedKalmanFilter
. For simpler and more classical designs, an InternalModel
structure is also available, that assumes by default that the current model mismatch estimation is constant in the future (same approach as dynamic matrix control, DMC).
The nomenclature uses capital letters for time series (and matrices) and hats for the predictions (and estimations, for state estimators).
To be precise, at the $k$th control period, the vectors that encompass the future measured disturbances $\mathbf{d̂}$, model outputs $\mathbf{ŷ}$ and setpoints $\mathbf{r̂_y}$ over the prediction horizon $H_p$ are defined as:
\[ \mathbf{D̂} = \begin{bmatrix} \mathbf{d̂}(k+1) \\ \mathbf{d̂}(k+2) \\ \vdots \\ \mathbf{d̂}(k+H_p) \end{bmatrix} \: , \quad \mathbf{Ŷ} = \begin{bmatrix} \mathbf{ŷ}(k+1) \\ \mathbf{ŷ}(k+2) \\ \vdots \\ \mathbf{ŷ}(k+H_p) \end{bmatrix} \quad \text{and} \quad \mathbf{R̂_y} = \begin{bmatrix} \mathbf{r̂_y}(k+1) \\ \mathbf{r̂_y}(k+2) \\ \vdots \\ \mathbf{r̂_y}(k+H_p) \end{bmatrix}\]
The vectors for the manipulated input $\mathbf{u}$ are shifted by one time step:
\[ \mathbf{U} = \begin{bmatrix} \mathbf{u}(k+0) \\ \mathbf{u}(k+1) \\ \vdots \\ \mathbf{u}(k+H_p-1) \end{bmatrix} \quad \text{and} \quad \mathbf{R̂_u} = \begin{bmatrix} \mathbf{r̂_u}(k+0) \\ \mathbf{r̂_u}(k+1) \\ \vdots \\ \mathbf{r̂_u}(k+H_p-1) \end{bmatrix}\]
Defining the manipulated input increment as $\mathbf{Δu}(k+j) = \mathbf{u}(k+j) - \mathbf{u}(k+j-1)$, the control horizon $H_c$ enforces that $\mathbf{Δu}(k+j) = \mathbf{0}$ for $j ≥ H_c$. For this reason, the vector that collects them is truncated up to $k+H_c-1$:
\[ \mathbf{ΔU} = \begin{bmatrix} \mathbf{Δu}(k+0) \\ \mathbf{Δu}(k+1) \\ \vdots \\ \mathbf{Δu}(k+H_c-1) \end{bmatrix}\]
PredictiveController
ModelPredictiveControl.PredictiveController
— TypeAbstract 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
ModelPredictiveControl.LinMPC
— TypeLinMPC(model::LinModel; <keyword arguments>)
Construct a linear predictive controller based on LinModel
model
.
The controller minimizes the following objective function at each discrete time $k$:
\[\begin{aligned} \min_{\mathbf{ΔU}, ϵ} \mathbf{(R̂_y - Ŷ)}' \mathbf{M}_{H_p} \mathbf{(R̂_y - Ŷ)} + \mathbf{(ΔU)}' \mathbf{N}_{H_c} \mathbf{(ΔU)} \\ + \mathbf{(R̂_u - U)}' \mathbf{L}_{H_p} \mathbf{(R̂_u - U)} + C ϵ^2 \end{aligned}\]
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 $\mathbf{ΔU}$ 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.
Arguments
model::LinModel
: model used for controller predictions and state estimations.Hp=10+nk
: prediction horizon $H_p$,nk
is the number of delays inmodel
.Hc=2
: control horizon $H_c$.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=diagm(repeat(Mwt,Hp))
: positive semidefinite symmetric matrix $\mathbf{M}_{H_p}$.N_Hc=diagm(repeat(Nwt,Hc))
: positive semidefinite symmetric matrix $\mathbf{N}_{H_c}$.L_Hp=diagm(repeat(Lwt,Hp))
: positive semidefinite symmetric matrix $\mathbf{L}_{H_p}$.Cwt=1e5
: slack variable weight $C$ (scalar), useCwt=Inf
for hard constraints only.optim=JuMP.Model(OSQP.MathOptInterfaceOSQP.Optimizer)
: quadratic optimizer used in the predictive controller, provided as aJuMP.Model
(default toOSQP
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, OSQP optimizer, SteadyKalmanFilter estimator and:
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
Extended Help
Manipulated inputs setpoints $\mathbf{r_u}$ are not common but they can be interesting for over-actuated systems, when nu > ny
(e.g. prioritize solutions with lower economical costs). The default Lwt
value implies that this feature is disabled by default.
The objective function follows this nomenclature:
VARIABLE | DESCRIPTION | SIZE |
---|---|---|
$H_p$ | prediction horizon (integer) | () |
$H_c$ | control horizon (integer) | () |
$\mathbf{ΔU}$ | manipulated input increments over $H_c$ | (nu*Hc,) |
$\mathbf{Ŷ}$ | predicted outputs over $H_p$ | (ny*Hp,) |
$\mathbf{U}$ | manipulated inputs over $H_p$ | (nu*Hp,) |
$\mathbf{R̂_y}$ | predicted output setpoints over $H_p$ | (ny*Hp,) |
$\mathbf{R̂_u}$ | predicted manipulated input setpoints over $H_p$ | (nu*Hp,) |
$\mathbf{M}_{H_p}$ | output setpoint tracking weights over $H_p$ | (ny*Hp, ny*Hp) |
$\mathbf{N}_{H_c}$ | manipulated input increment weights over $H_c$ | (nu*Hc, nu*Hc) |
$\mathbf{L}_{H_p}$ | manipulated input setpoint tracking weights over $H_p$ | (nu*Hp, nu*Hp) |
$C$ | slack variable weight | () |
$ϵ$ | slack variable for constraint softening | () |
LinMPC(estim::StateEstimator; <keyword arguments>)
Use custom state estimator estim
to construct LinMPC
.
estim.model
must be a LinModel
. Else, a NonLinMPC
is required.
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, OSQP optimizer, KalmanFilter estimator and:
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
ModelPredictiveControl.ExplicitMPC
— TypeExplicitMPC(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
and optim
which are not supported.
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, SteadyKalmanFilter estimator and:
30 prediction steps Hp
1 control steps Hc
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
ModelPredictiveControl.NonLinMPC
— TypeNonLinMPC(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 $k$:
\[\begin{aligned} \min_{\mathbf{ΔU}, ϵ}\ & \mathbf{(R̂_y - Ŷ)}' \mathbf{M}_{H_p} \mathbf{(R̂_y - Ŷ)} + \mathbf{(ΔU)}' \mathbf{N}_{H_c} \mathbf{(ΔU)} \\& + \mathbf{(R̂_u - U)}' \mathbf{L}_{H_p} \mathbf{(R̂_u - U)} + C ϵ^2 + E J_E(\mathbf{U_e}, \mathbf{Ŷ_e}, \mathbf{D̂_e}, \mathbf{p}) \end{aligned}\]
subject to setconstraint!
bounds, and the custom inequality constraints:
\[\mathbf{g_c}\Big(\mathbf{U_e}, \mathbf{Ŷ_e}, \mathbf{D̂_e}, \mathbf{p}, ϵ(k)\Big) ≤ \mathbf{0}\]
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 to customize the 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):
\[ \mathbf{U_e} = \begin{bmatrix} \mathbf{U} \\ \mathbf{u}(k+H_p-1) \end{bmatrix} , \quad \mathbf{Ŷ_e} = \begin{bmatrix} \mathbf{ŷ}(k) \\ \mathbf{Ŷ} \end{bmatrix} , \quad \mathbf{D̂_e} = \begin{bmatrix} \mathbf{d}(k) \\ \mathbf{D̂} \end{bmatrix}\]
since $H_c ≤ H_p$ implies that $\mathbf{Δu}(k+H_p) = \mathbf{0}$ or $\mathbf{u}(k+H_p)= \mathbf{u}(k+H_p-1)$. The vector $\mathbf{D̂}$ comprises the measured disturbance predictions over $H_p$. 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.
Replace any of the arguments of $J_E$ and $\mathbf{g_c}$ functions with _
if not needed (see e.g. the default value of JE
below).
See LinMPC
for the definition of the other variables. This method uses the default state estimator :
- if
model
is aLinModel
, aSteadyKalmanFilter
with default arguments; - else, an
UnscentedKalmanFilter
with default arguments.
See Extended Help if you get an error like: MethodError: no method matching Float64(::ForwardDiff.Dual)
.
Arguments
model::SimModel
: model used for controller predictions and state estimations.Hp=nothing
: prediction horizon $H_p$, must be specified forNonLinModel
.Hc=2
: control horizon $H_c$.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=diagm(repeat(Mwt,Hp))
: positive semidefinite symmetric matrix $\mathbf{M}_{H_p}$.N_Hc=diagm(repeat(Nwt,Hc))
: positive semidefinite symmetric matrix $\mathbf{N}_{H_c}$.L_Hp=diagm(repeat(Lwt,Hp))
: positive semidefinite symmetric matrix $\mathbf{L}_{H_p}$.Cwt=1e5
: slack variable weight $C$ (scalar), useCwt=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
orgc!
: 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).optim=JuMP.Model(Ipopt.Optimizer)
: nonlinear optimizer used in the predictive controller, provided as aJuMP.Model
(default toIpopt
optimizer).- additional keyword arguments are passed to
UnscentedKalmanFilter
constructor (orSteadyKalmanFilter
, forLinModel
).
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=1, Cwt=1e6)
NonLinMPC controller with a sample time Ts = 10.0 s, Ipopt optimizer, UnscentedKalmanFilter estimator and:
20 prediction steps Hp
1 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
Extended Help
NonLinMPC
controllers based on LinModel
compute the predictions with matrix algebra instead of a for
loop. This feature can accelerate the optimization, especially for the constraint handling, and is not available in any other package, to my knowledge.
The optimization relies on JuMP
automatic differentiation (AD) to compute the objective and constraint derivatives. Optimizers generally benefit from exact derivatives like AD. However, the NonLinModel
state-space functions must be compatible with this feature. See Automatic differentiation for common mistakes when writing these functions.
If LHS
represents the result of the left-hand side in the inequality $\mathbf{g_c}(\mathbf{U_e}, \mathbf{Ŷ_e}, \mathbf{D̂_e}, \mathbf{p}, ϵ) ≤ \mathbf{0}$, the function gc
can be implemented in two ways:
- Non-mutating function (out-of-place): define it as
gc(Ue, Ŷe, D̂e, p, ϵ) -> LHS
.
This syntax is simple and intuitive but it allocates more memory.
- Mutating function (in-place): define it as
gc!(LHS, Ue, Ŷe, D̂e, p, ϵ) -> nothing
.
This syntax reduces the allocations and potentially the computational burden as well.
The keyword argument nc
is the number of elements in the LHS
vector, and gc!
, an alias for the gc
argument (both accepts non-mutating and mutating functions). Note that if Cwt≠Inf
, the attribute nlp_scaling_max_gradient
of Ipopt
is set to 10/Cwt
(if not already set), to scale the small values of $ϵ$.
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, Hc=1, Cwt=1e6)
NonLinMPC controller with a sample time Ts = 10.0 s, Ipopt optimizer, UnscentedKalmanFilter estimator and:
20 prediction steps Hp
1 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
Move Manipulated Input u
ModelPredictiveControl.moveinput!
— Functionmoveinput!(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 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.
Calling a PredictiveController
object calls this method.
See also LinMPC
, ExplicitMPC
, NonLinMPC
.
Arguments
Keyword arguments with emphasis
are non-Unicode alternatives.
mpc::PredictiveController
: solve optimization problem ofmpc
.ry=mpc.estim.model.yop
: current output setpoints $\mathbf{r_y}(k)$.d=[]
: current measured disturbances $\mathbf{d}(k)$.D̂=repeat(d, mpc.Hp)
orDhat
: 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)
orRhaty
: 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
orRhatu
: predicted manipulated input setpoints, 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