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Reduction-related functions

Functions

MinimaxAdaptiveControl.bared — Function
bared(Ahat::AbstractMatrix{T}, Bhat::AbstractMatrix{T}, Ghat::AbstractMatrix{T}, H::AbstractMatrix{T}, model::GenericModel{T}) where T <: Real
bared(Ahat::AbstractMatrix{T}, Bhat::AbstractMatrix{T}, Ghat::AbstractMatrix{T}, H::AbstractMatrix{T}; method::Symbol = :Laub, iters::Int = 1000) where T <: Real

Solves the backward algebraic Riccati equation using the specified method.

Arguments

  • Ahat::AbstractMatrix{T}: State transition matrix.
  • Bhat::AbstractMatrix{T}: Control input matrix.
  • Ghat::AbstractMatrix{T}: Disturbance input matrix.
  • H::AbstractMatrix{T}: Cost matrix, symmetric.
  • model::GenericModel{T}: JuMP optimization model.
  • method::Symbol: Method to be used for solving the Riccati equation (default is :Laub). The available options are :Laub and :Iterate, see Details on the backward algebraic riccati equation for more information.
  • iters::Int: Number of iterations for the iterative method (default is 1000).

Returns

  • A tuple containing:
    • P::Matrix{T}: Backward riccati solution matrix, positive definite.
    • K::Matrix{T}: Gain matrix.
    • termination_status::Symbol: Status of the optimization.
source
MinimaxAdaptiveControl.fared — Function
fared(sys::OFLinMod{T}, γ::Real, model::GenericModel{T}; method::Symbol = :LMI2) where T <: Real
fared(sys::OFLinMod{T}, γ::Real; method::Symbol = :Iterate, iters::Int = 1000) where T <: Real
fared(A::AbstractMatrix{T}, B::AbstractMatrix{T}, C::AbstractMatrix{T}, D::AbstractMatrix{T}, G::AbstractMatrix{T}, Q::AbstractMatrix{T}, R::AbstractMatrix{T}, γ::Real, model::GenericModel{T}; method = :LMI2) where T <: Real
fared(A::AbstractMatrix{T}, B::AbstractMatrix{T}, C::AbstractMatrix{T}, D::AbstractMatrix{T}, G::AbstractMatrix{T}, Q::AbstractMatrix{T}, R::AbstractMatrix{T}, γ::Real; method = :Iterate, iters = 1000) where T <: Real

Solves the forward algebraic Riccati equation for an output-feedback linear model (or the given system matrices) using the specified method.

Arguments

  • sys::OFLinMod{T}: Output-feedback linear model.
  • A::AbstractMatrix{T}: State transition matrix.
  • B::AbstractMatrix{T}: Control input matrix.
  • C::AbstractMatrix{T}: Output matrix.
  • D::AbstractMatrix{T}: Measurement noise matrix.
  • G::AbstractMatrix{T}: Disturbance input matrix.
  • Q::AbstractMatrix{T}: State cost matrix.
  • R::AbstractMatrix{T}: Control input cost matrix.
  • γ::Real: Induced $\ell_2$-gain
  • model::GenericModel{T}: Generic optimization model.
  • method::Symbol: Method to be used for solving the Riccati equation (default is :LMI2 or :Iterate depending on the function signature). The options are :LMI1, :LMI2, :Iterate, and :Laub. See Details on the forward algebraic riccati equation for more information.
  • iters::Int: Number of iterations for the iterative method (default is 1000).

Returns

  • A tuple containing:
    • S::Matrix{T}: Forward riccati solution matrix, positive definite.
    • Ahat::Matrix{T}: Observer state transition matrix.
    • Ghat::Matrix{T}: Observer output injection matrix.
    • H::Matrix{T}: Cost matrix.
    • termination_status::Symbol: Status of the optimization.
source
MinimaxAdaptiveControl.reduceSys — Function
reduceSys(sys::Vector{SSLinMod{T}}, γ::T, models::AbstractVector{GenericModel{T}}) where T<: Real

The function computes the aggregated model matrices (Ahat, Bhat, Ghat) and the gain (Ks) and auxiliary (Hs) matrices for each of the input models.

Reduces state-feedback adaptive control with uncertain parameters

\[ \begin{aligned} x_{t+1} & = Ax_t + Bu_t + w_t \\ (A, B) & \in \{(A_1, B_1), \ldots, (A_M, B_M)\} \end{aligned}\]

with the soft-constrained objective function

\[ \min_u\max_{w, N, A, B}\sum_{t=0}^N\left( |x_t|^2_Q + |u_t|^2_R - \gamma^2 |w_t|^2 \right),\]

to a certain system

\[ \begin{aligned} z_{t + 1} & = \hat Az_t + \hat Bu_t + \hat G d_t \end{aligned}\]

with uncertain objective

\[\min_u \max_{H, d, N} \sum_{t = 0}^N \sigma_H(z_t, u_t, d_t)\]

Arguments

  • sys::Vector{SSLinMod{T}}: Vector of state-space linear models to be reduced.
  • γ::T: Induced $\ell_2$-gain.
  • models::AbstractVector{GenericModel{T}}: Vector of JuMP models, one per state-space model.

Returns

  • Ahat::Matrix{T}: Aggregated state transition matrix.
  • Bhat::Matrix{T}: Aggregated control input matrix.
  • Ghat::Matrix{T}: Aggregated disturbance input matrix.
  • Ks::Vector{Matrix{T}}: Vector of $\gamma$-suboptimal $\mathcal H_\infty$ gain matrices.
  • Hs::Vector{Matrix{T}}: Vector of cost weights.
reduceSys(sys::Vector{OFLinMod{T}}, γ::T, models::AbstractVector{GenericModel{T}}, method::Symbol = :LMI2) where T <: Real

Reduces a set of output-feedback linear models into a single aggregated model using a specified method. The function computes the aggregated model matrices (Ahat, Bhat, Ghat) and the gain (Ks) and auxiliary (Hs) matrices for each of the input models.

Takes uncertain systems of the form

\[\begin{aligned} x_{t + 1} & = A x_t + B u_t + Gw_t,\quad t \geq 0 \\ y_t & = C x_t + D v_t \\ (A, B, C, D, G) & \in \mathcal M \end{aligned}\]

with soft-constrained objective function

\[ \min_u\max_{w, v, x_0 N, M \in \mathcal M}\sum_{t=0}^N\left( |x_t|^2_Q + |u_t|^2_R - \gamma^2 |(w_t, v_t)|^2 \right) - |x_0 - \hat x_0|^2_{S_{M, 0}},\]

to a certain system

\[ \begin{aligned} z_{t + 1} & = \hat Az_t + \hat Bu_t + \hat G d_t \end{aligned}\]

with uncertain objective

\[\min_u \max_{H, d, N} \sum_{t = 0}^N \sigma_H(z_t, u_t, d_t)\]

Arguments

  • sys::Vector{OFLinMod{T}}: Vector of output-feedback linear models to be reduced.
  • γ::T: Induced $\ell_2$-gain.
  • models::AbstractVector{GenericModel{T}}: Vector of JuMP models, one for each ouput-feedback model
  • method::Symbol: Method to be used for the reduction (default is :LMI2). See fared

Returns

  • A::Matrix{T}: Aggregated observer state transition matrix.
  • B::Matrix{T}: Aggregated observer control input matrix.
  • G::Matrix{T}: Aggregated observer measurement input matrix.
  • Ks::Vector{Matrix{T}}: Vector of $\gamma$-suboptimal $\mathcal H_\infty$ observer gain matrices.
  • Hs::Vector{Matrix{T}}: Vector of cost weights.
source

Details on the forward algebraic riccati equation

Given matrices $A \in \mathbb{R}^{n_x \times n_x}$, $C \in \mathbb{R}^{n_x \times n_y}$, $D \in \mathbb{R}^{n_y \times n_v}$, $G \in \mathbb{R}^{n_x \times n_w}$, $Q \in \mathbb{R}^{n_x \times n_x}$, $R \in \mathbb{R}^{n_u \times n_u}$, and $\gamma > 0$ the discrete-time forward algebraic Riccati equation (FARED) is defined as.

\[\begin{aligned} S & = (AX^{-1}A^\top + \gamma^{-2}GG^\top)^{-1} \\ X & = S + \gamma^2 C^\top (DD^\top)^{-1}C - Q \\ L & = \gamma^2 AX^{-1}C(DD^\top)^{-1}. \end{aligned}\]

The FARED may have multiple solutions. The function fared computes the solution associated with the maximal $S$ (over the positive semidefinite cone). The function also returns a matrix H which is given by

\[\begin{bmatrix} SX^{-1}S -S & 0 & \gamma^2 SX^{-1}C(DD^\top)^{-1} \\ 0 & R & 0 \\ \gamma^2 C^\top X^{-1}S(DD^\top)^{-1} & 0 & -\left((DD^\top)^{-1} + C(S - Q)^{-1}C^\top\right)^{-1} \end{bmatrix},\]

The observer matrices are constructed by $\hat A = A X^{-1} S$ and $\hat G = \gamma^2 A X^{-1} C (DD^\top)^{-1}$.

fared :LMI1

Solves (if possible) the SDP

\[\begin{aligned} \text{maximize} & \quad \text{trace}(S) \\ \text{subject to} & \quad \begin{bmatrix} S & SA & SG \\ A^\top S & X & 0_{n_x \times n_w} \\ G^\top S & 0_{n_w \times n_x} & \gamma^2 I_{n_w} \end{bmatrix} \succeq 0 \\ & \quad X = S + \gamma^2 C^\top (DD^\top)^{-1}C - Q, \end{aligned}\]

where $S$ is the decision variable. The function returns the optimal $S$ and the corresponding $X$, $L$, and $H$.

fared :LMI2

Directly constructs an SDP whose solution gives the optimal $(S, \hat A, \hat G, H)$. Let

\[Y = \begin{bmatrix} I_{n_x} & 0 \\ 0 & C^\top \\ 0 & 0 \\ 0 & D^\top \end{bmatrix}\]

The SDP is given by

\[\begin{aligned} \underset{S, \hat A, \hat B, H}{\text{minimize}} & \quad \text{trace}(H) \\ \text{subject to} & \quad \begin{bmatrix} S & -S & 0 & 0 & -\hat A^\top \\ -S & S - Q & 0 & 0 & A^\top S - C^\top \hat G \\ 0 & 0 & \gamma^2 I_{n_w} & 0 & G^\top S \\ 0 & 0 & 0 & \gamma^2 I_{n_v} & -\hat G^\top \\ -\hat A & A^\top S & G & -G & -\hat G D S \end{bmatrix} - \begin{bmatrix} Y H Y^\top & 0 \\ 0 & 0 \end{bmatrix} \succeq 0\\ & S \succeq 0, \\ & H - H^\top = 0. \end{aligned}\]

fared :Iterate

This method solves the FARED by iterating starting with $S_0 = \gamma^2 I_{n_x}$. The iteration is given by

\[\begin{aligned} S_{t + 1}& = (AX_t^{-1}A^\top + \gamma^{-2}GG^\top)^{-1} \\ X_t & = S_t + \gamma^2 C^\top (DD^\top)^{-1}C - Q \end{aligned}\]

fared :Laub

Uses MatrixEquations.jl function ared to solve the FARED. ared implements W. F. Arnold and A. J. Laub, "Generalized eigenproblem algorithms and software for algebraic Riccati equations," in Proceedings of the IEEE, vol. 72, no. 12, pp. 1746-1754, Dec. 1984, doi: 10.1109/PROC.1984.13083.

Details on the backward algebraic riccati equation

Fix $\gamma > 0$ and consider the transformation

\[ \begin{aligned} G(\gamma) & = \gamma\hat G\sqrt{-H^{-dd}}, \qquad R = H^{uu} - H^{ud}H^{-dd}H^{du}, \\ Q & = H^{dd} - H^{zd}H^{-dd}H^{dz}\\ & \qquad -(H^{zu} - H^{zd}H^{-dd} H^{du})R^{-1}(*), \\ A & = \hat A - BR^{-1}(H^{zu} - H^{zd}H^{-dd}H^{du}) \\ & \quad - GH^{-dd}\left(H^{dz} - H^{du}R^{-1}\left(H^{uz} - H^{ud}H^{-dd}H^{dz}\right)\right)\\ B & = \hat B - G(H^{dd})^{-1}H^{du}. \end{aligned}\]

As we are interested in the upper value, $d_t$ is allowed to depend causally on $z$ and $u$, but $u$ is required to depend strictly causally on $d$. The following parameterization of $d_t$ and $u_t$ respect the causalilty structure of the problem,

\[\begin{aligned} d_t & = -H^{-dd}(H^{dz}z_t + H^{du}u_t) + \gamma\sqrt{-H^{-dd}}\delta_t, \quad t \geq 0 \\ u_t & = -R^{-1}(H^{uz} - H^{ud}H^{-dd}H^{dz})z_t + v_t. \quad t \geq 0, \end{aligned}\]

and leads to the follow equivalent dynamic game. Compute

\[ \inf_{\nu}\sup_{\delta, N} \sum_{t=0}^{N-1} \left( |z_t|^2_{Q} + |v_t|^2_{R} - \gamma^2|\delta_t|^2\right),\]

subject to the dynamics

\[\begin{aligned} z_{t + 1} & = Az_t + Bv_t + G(\gamma)\delta_t, \quad t \geq 0, \\ v_t & = \nu_t(z_0, \ldots, z_t), \quad t \geq 0. \end{aligned}\]

This reformulation puts the problem on standard $\gamma$-suboptimal $\Hinf$ form, and the value function, if bounded, is well known to be a quadratic function of the initial state, and the optimal controller is of the form $v_t = -K_\dagger z_t$.

The value function, $V_\star(z_0)$, has the form $V_\star(z_0) = |z_0|^2_{P_\star}$, where $P_\star$ is the minimal fixed point of the generalized algebraic riccati equation. $P_\star$ and $K_\dagger$ and can be computed, for example, through value iteration, Arnold and Laub's Schur methods and convex optimization.