An Explicit Solution of Joint Estimation-Linear Model Predictive Control

Abstract

Model predictive control (MPC) represents an optimal strategy where constraints on inputs, outputs and system states can be implemented as part of the control law that takes the form of a mathematical program. MPC of linear systems with a quadratic objective function results in a quadratic program (QP) that needs to be solved online at each sampling instant. Multiparametric programming methods that attempt an explicit solution to QPs have been successfully used in context of MPC and is termed as explicit MPC (eMPC). eMPC for linear systems results in a piecewise affine-in-state feedback control law and is determined offline. During online implementation, the control law is selected from among the different pieces based on the real-time value of the states. An ability to verify the controller output over all possible state realizations of the feasible state-space is important in any critical application including health and aerospace and is a unique feature of eMPC. Since state feedback MPC requires full state information, it is always used in conjunction with a state estimator such as a Kalman Filter. Further, fault tolerant control methodologies depend on state and parameter estimation to detect and diagnose faults followed by compensation. Conventionally, the state filtering step is performed prior to MPC and thus the interaction between the estimator and controller remains hidden and can be analyzed only via numerical simulations. In this work, we propose using multiparametric programming to find the explicit solution of the joint estimator-MPC problem. In particular, the eMPC control law now depends linearly on the joint information of states and measurements. This allows explicitly obtaining the sensitivity of the MPC control law to the measurement as well as the estimator parameters. The proposed explicit solution of the joint estimator-MPC problem is demonstrated on a SISO 2-tank system. The effect of the estimator gain on the size of the feasible region is delineated.

Appendix

Prediction Equations: While Eq.(7) propagates in future, all future measurements within prediction horizon N are considered to be constant and equal to $y_{k}$. The prediction equations for the estimator dynamics are as follows.

$$\begin{aligned} \chi ={\tilde{A}}{\hat{x}}_{k\vert k-1}+{\tilde{B}}U+{\tilde{\Gamma }}y_{k} \end{aligned}$$

(1)

where ${\tilde{B}}=\begin{bmatrix} {\mathbf {0}}_{n\times m} &{} \cdots &{} \cdots &{} {\mathbf {0}}_{n\times m}\\ B &{} \ddots &{}&{} \vdots \\ AB &{} B &{} \ddots &{} \vdots \\ \vdots &{} &{} \ddots &{} {\mathbf {0}}_{n\times m}\\ A^{N-1}B &{} \cdots &{} AB &{} B \end{bmatrix}$, ${\hat{A}}=\begin{bmatrix} I_{n\times n}\\ A\\ A^{2}\\ A^{3}\\ \vdots \\ A^{N} \end{bmatrix}$, ${\tilde{A}}={\hat{A}}(I-K_{\infty }C)$ and ${\tilde{\Gamma }}={\hat{A}}K_{\infty }$

The prediction equations in Eq.(1) are linear in $K_{\infty }$.

Constraint Equations: The constraint matrices for Eq.(11) are as follows.

$G=\begin{bmatrix} {\tilde{B}}\\ -{\tilde{B}}\\ I_{mN\times mN}\\ -I_{mN\times mN}\\ \end{bmatrix}$, $S^{x}=S(I-K_{\infty }C)$,

$S^{y}=SK_{\infty }$, $S=\begin{bmatrix} -{\hat{A}}\\ {\hat{A}}\\ {\mathbf {0}}_{mN\times n}\\ {\mathbf {0}}_{mN\times n}\\ \end{bmatrix}$ and $W=\begin{bmatrix} \overline{X}\\ -{\underline{X}}\\ \overline{U}\\ -{\underline{U}}\\ \end{bmatrix}$

Objective Function: The matrices corresponding to ${\hat{J}}(U,{\hat{x}}_{k\vert k-1},y_{k})$ are as follows.

$H=2({\tilde{B}}^{T}{\tilde{Q}}{\tilde{B}}+{\tilde{R}})$, $F_{x}=2(I-K_{\infty }C)^{T}{\hat{A}}^{T}{\tilde{Q}}{\tilde{B}}$ and $F_{y}=2K^{T}_{\infty }{\hat{A}}^{T}{\tilde{Q}}{\tilde{B}}$

where ${\tilde{Q}}=diag(Q,Q,\cdots ,Q,P_{f})$ and ${\tilde{R}}=diag(R,R,\cdots ,R,R)$.

Constraint Matrices for Standard MpQP: The constraint matrices for the RHS of Eq.(14) are given below

$E^{x}={\tilde{E}}(I-K_{\infty }C)$, $E^{y}={\tilde{E}}K_{\infty }$ and

${\tilde{E}}=(S+2GH^{-1}{\tilde{B}}^{T}{\tilde{Q}}{\hat{A}})$

Details related to Theorem 1: The matrices corresponding to Eq.(23) are as follows.

${\mathcal {F}}_{{\mathcal {A}}_{j},z_{x}}=\tilde{{\mathcal {F}}}_{z,j}(I-K_{\infty }C)$, ${\mathcal {F}}_{{\mathcal {A}}_{j},z_{y}}=\tilde{{\mathcal {F}}}_{z,j}K_{\infty }$ and $g_{{\mathcal {A}}_{j},z}=H^{-1}G^{T}_{{\mathcal {A}}_{j}} (G_{{\mathcal {A}}_{j}}H^{-1}G^{T}_{{\mathcal {A}}_{j}})^{-1}W_{{\mathcal {A}}_{j}}$.

where $\tilde{{\mathcal {F}}}_{x,j}=H^{-1}G^{T}_{{\mathcal {A}}_{j}} (G_{{\mathcal {A}}_{j}}H^{-1}G^{T}_{{\mathcal {A}}_{j}})^{-1}{\tilde{E}}_{{\mathcal {A}}_{j}}$.

The matrices corresponding to Eq.(24) are as follows.

${\mathcal {F}}_{{\mathcal {A}}_{j},\lambda _{x}}=\tilde{{\mathcal {F}}}_{\lambda ,j}(I-K_{\infty }C)$, ${\mathcal {F}}_{{\mathcal {A}}_{j},\lambda _{y}}=\tilde{{\mathcal {F}}}_{\lambda ,j}K_{\infty }$ and $g_{{\mathcal {A}}_{j},\lambda }=-(G_{{\mathcal {A}}_{j}}H^{-1}G^{T}_{{\mathcal {A}}_{j}})^{-1}W_{{\mathcal {A}}_{j}}$

where $\tilde{{\mathcal {F}}}_{\lambda ,j}=(G_{{\mathcal {A}}_{j}}H^{-1}G^{T}_{{\mathcal {A}}_{j}})^{-1}{\tilde{E}}_{{\mathcal {A}}_{j}}$.

The matrices corresponding to Eq.(28) are as follows.

${\tilde{H}}_{{\mathcal {A}}_{j},x}={\tilde{H}}_{j}(I-K_{\infty }C)$, ${\tilde{H}}_{{\mathcal {A}}_{j},y}={\tilde{H}}_{j}K_{\infty }$ and ${\tilde{L}}_{j}=\begin{bmatrix} g_{{\mathcal {A}}_{j},\lambda }\\ W_{{\mathcal {I}}_{j}}-G_{{\mathcal {I}}_{j}}g_{{\mathcal {A}}_{j},z}\\ \end{bmatrix}$

where ${\tilde{H}}_{j}=\begin{bmatrix} -(G_{{\mathcal {A}}_{j}}H^{-1}G^{T}_{{\mathcal {A}}_{j}})^{-1}{\tilde{E}}_{{\mathcal {A}}_{j}}\\ G_{{\mathcal {I}}_{j}}H^{-1}G^{T}_{{\mathcal {A}}_{j}} (G_{{\mathcal {A}}_{j}}H^{-1}G^{T}_{{\mathcal {A}}_{j}})^{-1}{\tilde{E}}_{{\mathcal {A}}_{j}}-{\tilde{E}}_{{\mathcal {I}}_{j}}\\ \end{bmatrix}$