A Computationally Efficient Robust Tube-Based MPC for Tracking of Linear Systems

Abstract

This paper addresses a computationally efficient robust tube-based model predictive control (RTBMPC) strategy of linear systems in the presence of bounded disturbance. In the RTBMPC strategy, a nominal system is introduced by ignoring the disturbances of uncertain system, and then the uncertain system will be controlled in a robust manner through its nominal system as well as an additional feedback term which rejects a bounded additive disturbance. In this paper, the tracking problem is converted into the regulation problem by introducing an extra system called regulation nominal system that its constraints are translated from tracking into regulation. It leads to a reduction in complexity of the objective function and simplification of driving the stability theory. On the other hand, RTBMPC strategy solves optimization problem for nominal system which ignores the disturbances. Since in the absence of disturbances, the state measured at the following sample will be the same as the one predicted by model, a variable prediction horizon is suggested to reduce the computational burden. In addition, new constraints are introduced to prove the recursive feasibility, local and asymptotic stability. The constrained sampled double integrator is presented to illustrate the effectiveness of the proposed RTBMPC.

Appendices

Appendix 1: Proof of Theorem 1

Proof

It is considered that RTBMPC problem with prediction horizon $N$ is feasible for an initial condition $\bar{z}_{k} \in \bar{\mathbb{Z}}$ at sample $k$. Therefore, solving RTBMPC problem yields the optimal input sequence $\bar{u}_{k}^{0:N - 1} = \left( {\begin{array}{*{20}c} {\bar{u}_{0\left| k \right.} } & \ldots & {\bar{u}_{N - 1\left| k \right.} } \\ \end{array} } \right)$ and the optimal state sequence $\bar{z}_{k}^{1:N} = \left( {\begin{array}{*{20}c} {\bar{z}_{1\left| k \right.} } & \ldots & {\bar{z}_{N\left| k \right.} } \\ \end{array} } \right)$.

According to the proposed algorithm, in each sample $horizon \, \to (horizon{-}1)$. Then, it is constructed the optimal input sequence $\bar{u}_{k + 1}^{1:N - 1} = \left( {\begin{array}{*{20}c} {\bar{u}_{{1\left| {k + 1} \right.}} } & \ldots & {\bar{u}_{N - 1\left| k \right. + 1} } \\ \end{array} } \right)$ at $k + 1{\text{th}}$ instant and injecting input sequence to system yields the state sequence $\bar{z}_{k + 1}^{1:N - 1} = \left( {\begin{array}{*{20}c} {\bar{z}_{{1\left| {k + 1} \right.}} } & \ldots & {\bar{z}_{{N - 1\left| {k + 1} \right.}} } \\ \end{array} } \right) .$ Since RTBMPC problem ignores the effect of disturbance, it can be concluded that $\bar{z}_{j + 1\left| k \right.} = \bar{z}_{{j\left| {k + 1} \right.}} ,\,j \in [1,N - 1]$. According to the relationship between $\bar{z}$ and $\bar{x}$ in Eq. (12), it can be resulted $\bar{x}_{j + 1\left| k \right.} = \bar{x}_{{j\left| {k + 1} \right.}} ,\,j \in [1,N - 1]$.

To prove the recursive feasibility of RTBMPC, first it is shown that $\bar{x}_{{N - 1\left| {k + 1} \right.}} \in X_{f}$ and then $x_{{k + 1}} \in \bar{x}_{{k + 1}} \oplus Z .$Since $\bar{z}_{N\left| k \right.} \in Z_{f}$ and $\bar{z}_{j + 1\left| k \right.} = \bar{z}_{{j\left| {k + 1} \right.}} ,\,j \in [1,N - 1]$, it is obvious $\bar{z}_{{N - 1\left| {k + 1} \right.}} \in Z_{f}$. Based on Eq. (12), $\bar{x}_{{N - 1\left| {k + 1} \right.}} = \bar{z}_{{N - 1\left| {k + 1} \right.}} \oplus x_{ref}$ and $\bar{z}_{{N - 1\left| {k + 1} \right.}} \in Z_{f}$ and it can be concluded $\bar{x}_{{N - 1\left| {k + 1} \right.}} \in Z_{f} \oplus x_{ref} \triangleq X_{f} \,.$ Also, based on the definition of disturbance invariant set $Z$ presented in Proposition 1 and $x_{k} \in \bar{x}_{k} \oplus Z$, it is clear that $x_{k + 1} \in \bar{x}_{k + 1} \oplus Z \triangleq (\bar{z}_{k + 1} \oplus x_{ref} ) \oplus Z$.

Therefore, RTBMPC problem is recursively feasible. □

Appendix 2: Proof of Theorem 2

Proof

According to assumption of feasibility of problem and the existence of disturbance invariant set $Z$, it is clear $\bar{z}_{N\left| k \right.} \in Z_{f}$. Based on Eq. (12) $(\bar{x}_{k} = \bar{z}_{k} \oplus x_{ref} )$, it is resulted $\bar{x}_{N\left| k \right.} \in x_{\text{ref}} \oplus X_{f}$ and consequently $x_{N\left| k \right.} \in \bar{x}_{N\left| k \right.} \oplus Z \Rightarrow x_{N\left| k \right.} \in (x_{\text{ref}} \oplus Z_{f} ) \oplus Z \triangleq X_{f} \oplus Z$. In other words, the feasibility assumption guarantees convergence of the regulation nominal system into $Z_{f}$ and results $\bar{z}_{N\left| k \right.} \in Z_{f}$. Based on Eq. (12), $\bar{x}_{N\left| k \right.} \triangleq \bar{z}_{N\left| k \right.} \oplus x_{\text{ref}} \in Z_{f} \oplus x_{\text{ref}} \triangleq X_{f}.$ On the other hand, it is concluded from Proposition 1 that $x \in \bar{x} \oplus Z$. Then, $x_{N} \in \left\{ {x\,\left| {\,\exists \,\bar{x}\,\, \in \,\,X_{f} \,\,such\,\,that\,\,x \in \bar{x} \oplus Z} \right.} \right\} = X_{f} \oplus Z.$□

Appendix 3: Proof of Theorem 3

Proof

First, the convergence of the regulation nominal system to zero setpoint is proven and then, based on Eq. (12), the convergence of the nominal system to $x_{\text{ref}}$ is resulted. Finally, the convergence of uncertain system to $x_{\text{ref}} \oplus Z$ is concluded.

Based on proposed algorithm, the prediction horizon is changing over time and after $N - 1$ instants is fixed on one. Therefore, the proof is divided into two steps: variable prediction horizon and invariable prediction horizon. Now, its robust stability in horizon control is discussed.

Step 1. (variable prediction horizon)

Solving the RTBMPC problem yields the optimal input sequence $\bar{u}_{k}^{0:N - 1} = \left( {\begin{array}{*{20}c} {\bar{u}_{0\left| k \right.} } & \ldots & {\bar{u}_{N - 1\left| k \right.} } \\ \end{array} } \right)$ at $k{\text{th}}$ instant and $\bar{u}_{k + 1}^{0:N - 2} = \left( {\begin{array}{*{20}c} {\bar{u}_{{0\left| {k + 1} \right.}} } & \ldots & {\bar{u}_{{N - 2\left| {k + 1} \right.}} } \\ \end{array} } \right)$ at $k + 1{\text{th}}$ instant for the nominal system. Then,

$$\begin{aligned} \varvec{V}_{N - 1} (\bar{z}_{k + 1} ) - \varvec{V}_{N} (\bar{z}_{k} ) & = \varvec{L}(\bar{z}_{{0\left| {k + 1} \right.}} ,\bar{u}_{{0\left| {k + 1} \right.}} ) + \cdots + \varvec{L}(\bar{z}_{{N - 2\left| {k + 1} \right.}} ,\bar{u}_{{N - 2\left| {k + 1} \right.}} ) + \varvec{F}(\bar{z}_{{N - 1\left| {k + 1} \right.}} ) \hfill \\ & & & - \left( {\varvec{L}(\bar{z}_{0\left| k \right.} ,\bar{u}_{0\left| k \right.} ) + \cdots + \varvec{L}(\bar{z}_{N - 1\left| k \right.} ,\bar{u}_{N - 1\left| k \right.} ) + \varvec{F}(\bar{z}_{N\left| k \right.} )} \right) \hfill \\ \end{aligned}$$

(20)

Since $\bar{z}_{j + 1\left| k \right.} = \bar{z}_{{j\left| {k + 1} \right.}} ,\,j \in [1,N - 1]$, it is concluded:

$$\begin{aligned} \varvec{V}_{N - 1} (\bar{z}_{k + 1} ) - \varvec{V}_{N} (\bar{z}_{k} ) & = \varvec{L}(\bar{z}_{1\left| k \right.} ,\bar{u}_{1\left| k \right.} ) + \cdots + \varvec{L}(\bar{z}_{N - 1\left| k \right.} ,\bar{u}_{N - 1\left| k \right.} ) + \varvec{F}(\bar{z}_{N\left| k \right.} ) \hfill \\ & & & - \left( {\varvec{L}(\bar{z}_{0\left| k \right.} ,\bar{u}_{0\left| k \right.} ) + \cdots + \varvec{L}(\bar{z}_{N - 1\left| k \right.} ,\bar{u}_{N - 1\left| k \right.} ) + \varvec{F}(\bar{z}_{N\left| k \right.} )} \right) \hfill \\ & & & = - \varvec{L}(\bar{z}_{0\left| k \right.} ,\bar{u}_{0\left| k \right.} ) \hfill \\ & & & = - 0.5(\bar{z}_{0\left| k \right.}^{T} Q\bar{z}_{0\left| k \right.} ) - 0.5(\bar{u}_{0\left| k \right.}^{T} R\bar{u}_{0\left| k \right.} ) \le 0 \hfill \\ \end{aligned}$$

(21)

Also, for the next instant $k + 1$, the Lyapunov function is resulted as:

$$\varvec{V}_{N - 2} (\bar{z}_{k + 2} ) - \varvec{V}_{N - 1} (\bar{z}_{k + 1} ) = - \varvec{L}(\bar{z}_{{1\left| {k + 1} \right.}} ,\bar{u}_{{1\left| {k + 1} \right.}} ) = - 0.5(\bar{z}_{1\left| k \right. + 1}^{T} Q\bar{z}_{{1\left| {k + 1} \right.}} ) - 0.5(\bar{u}_{1\left| k \right. + 1}^{T} R\bar{u}_{1\left| k \right. + 1} ) \le 0$$

(22)

So, the $\varvec{V}(\bar{z})$ is nonincreasing sequence and bounded. Then, the stability of the regulation nominal system is proven.

Step 2. (invariable prediction horizon)

In theorem 2, it was proven that $\bar{z}_{{0\left| {k + N} \right.}} \in Z_{f}$. Based on the constraint introduced on terminal constraint set $Z_{f}$ in Remark 4 (Eq. 17), it can be concluded $\bar{z}_{{0\left| {k + N + i} \right.}} \in Z_{f} ,\,\forall i \ge 0$. Then, by constraint in Eq. (16), it is resulted:

$$\varvec{V}_{{\mathbf{1}}} (\bar{z}_{k + N + 2} ) \le \varvec{V}_{{\mathbf{1}}} (\bar{z}_{k + N + 1} ) \le \varvec{V}_{{\mathbf{1}}} (\bar{z}_{k + N} )$$

(23)

Solving the RTBMPC problem for the control horizon more than $k + N$ instant yields the optimal input sequence $\bar{u}_{k}^{0} = \bar{u}_{0\left| k \right. + N}$ at $k + N$ instant and $\bar{u}_{k + 1}^{0} = \bar{u}_{{0\left| {k + N + 1} \right.}}$ at $k + N + 1$ instant for the regulation nominal system. From Eq. (16), it is concluded:

$$\varvec{V}_{1} (\bar{z}_{k + N + 1} ) - \varvec{V}_{1} (\bar{z}_{k + N} ) \le - \alpha_{1} \left\| {\bar{z}_{k + N} } \right\|^{2}$$

(24)

On other hand, the terminal cost is upper-bounded in Remark 6(A3). Hence, $\mathop {\lim }\limits_{N \to \infty } \,\,\,\bar{z}_{k + N} = 0$.

So, the $V(\bar{z})$ is nonincreasing sequence and bounded and also, $\mathop {\lim }\limits_{N \to \infty } \,\,\,\bar{z}_{k + N} = 0$.

Nevertheless, it can be concluded from Eqs. (22) and (24) that the origin is robustly asymptotically stable for the regulation nominal system. Based on the relationship between $\bar{z}$ and $\bar{x}$$(\bar{x}_{k} = \bar{z}_{k} \oplus x_{\text{ref}} )$ in Eq. (12), $x_{\text{ref}}$ is robustly asymptotically stable for the nominal system. Since $x_{k} \in \bar{x}_{k} \oplus Z$, the uncertain system asymptotically converges to $x_{\text{ref}} \oplus Z$.□