Stochastic Model Predictive Control for Linear Systems with Unbounded Additive Uncertainties

Consider a discrete-time LTI system subject to unbound additive uncertainties

where $x_{k}\in\mathbb{R}^{n}$, $u_{k}\in\mathbb{R}^{m}$ and $w_{k}\in\mathbb{R}^{n}$ are the system state, control input and uncertainty at time $k$, respectively. It is assumed that all the system states are measured perfectly, and the unbounded uncertainty $w_{k}\sim\mathcal{Q}^{w}$ is independently and identically distributed (i.i.d.). Moreover, the system in (1) is subject to the following constraints on states and inputs

where $\mathbb{X}\subset\mathbb{R}^{n}$, and $\mathbb{U}\subset\mathbb{R}^{m}$ are compact and each set contains the origin in its interior.

Taking advantage of the stochastic nature of the system dynamics, the chance constraints for all the predicted states and inputs in future horizon, which allow constraint violation with a probability no greater than $\varepsilon\in(0,1)$, is adopted to reduce the conservatism. The objective of this paper is to design SMPC algorithms to stabilize the system in (1) under the following chance constraints:

In this subsection, we recall some definitions which will facilitate algorithms design.

In order to derive a computationally tractable MPC algorithm, we make use of the techniques from DTMPC [11] to decouple the nominal trajectory planning and the uncertainty handling. Then, the system dynamics in (1) can be separated into a nominal dynamics and an error dynamics as

where the nominal state is denoted by $s_{k}\in\mathbb{R}^{n}$ and the nominal input by $v_{k}\in\mathbb{R}^{m}$. Without loss of generality, the feedback gain $K$ is chosen as the solution of the linear quadratic regulator (LQR) problem for the dynamics $s_{k+1}=As_{k}+Bv_{k}$ to ensure that $A_{cl}=A+BK$ is stable. The state feedback control law is used as in DTMPC

The state error $e_{k}$ between the actual state $x_{k}$ and the nominal state $s_{k}$, and the input error $e_{k}^{u}=Ke_{k}$ between the control input $u_{k}$ and the nominal input $v_{k}$ are defined as

Suppose that $\mathcal{E}^{w}_{1-\varepsilon}$ is a confidence region of probability level $1-\varepsilon$ for the uncertainty $w$, where $\varepsilon\in(0,1)$. Then

follows from Definition 3. The formulation methods for confidence region of uncertainty, such as the Chebyshev inequality method [15], the Scenario generation method [20], the Box-shaped method and the Ellipsoidal method [32], can be incorporated with the proposed tube-based SMPC scheme.

Suppose that the confidence region $\mathcal{E}^{e_{k}}_{1-\varepsilon}$ is a $k$-step PRS of probability level $1-\varepsilon$ for the state error dynamics in (4b), where $\varepsilon\in(0,1)$. Then,

follows from Definition 4 directly. The $k$-step PRS of probability level $1-\varepsilon$ for the input error can be calculated as $K\mathcal{E}^{e_{k}}_{1-\varepsilon}$ due to $e_{k}^{u}=Ke_{k}$. Then

is derived.

Any method to compute nested PRS, namely $\mathcal{E}^{e_{k}}_{1-\varepsilon}\subseteq\mathcal{E}^{e_{k+1}}_{1-\varepsilon}$, $k\in\mathbb{N}_{0}$, can be combined with the proposed tube-based SMPC scheme. For instance, the Ellipsoidal method and Half-space method [27] can be adopted to produce the PRS for uncertainties with mean and variance information. For uncertainty of any type, provided that samples can be obtained, the Scenario approach [35] can be used to formulate the PRS. And the above formulated $k$-step PRS has the property that $\mathcal{E}^{e_{k+1}}_{1-\varepsilon}\subseteq A_{cl}\mathcal{E}^{e_{k}}_{1-\varepsilon}\oplus\mathcal{E}^{w}_{1-\varepsilon}$, $k\in\mathbb{N}_{0}$ [32].

Especially, consider that the uncertainty $w$ follows the normal distribution $w\sim\mathcal{N}(0,W)$. Let $1-\varepsilon$ be the confidence coefficient of the uncertainty, where $\varepsilon\in(0,1)$. Then the quantile value corresponding to $1-\varepsilon$ is $\alpha=\Phi^{-1}(1-\varepsilon)$, where $\Phi^{-1}$ represents the quantile function of the standard normal distribution [16]. Then, the uncertainty confidence region of probability level $1-\varepsilon$ can be obtained as

Since the error dynamics in (4b) is linear, the distribution of $e_{k}$ is still normal. Then, the variance propagation of uncertainty can be evolved as

where $\Sigma_{0}=W$. In addition, $\Sigma_{k+1}\geq\Sigma_{k}$ follows from the expanded representation of above equation. The $k$-step PRS of probability level $1-\varepsilon$ can be defined as

which is the error confidence region of probability level $1-\varepsilon$, and $\mathcal{E}^{e_{0}}_{1-\varepsilon}=\mathcal{E}^{w}_{1-\varepsilon}$. Allowing for $\Sigma_{k+1}\geq\Sigma_{k}$, the corresponding probabilistic reachable sets are nested, namely, $\mathcal{E}^{e_{k}}_{1-\varepsilon}\subseteq\mathcal{E}^{e_{k+1}}_{1-\varepsilon}$. Another property of the $k$-step PRS see the following lemma.

Because $\mathcal{E}^{e_{k+1}}_{1-\varepsilon}\subseteq A_{cl}\mathcal{E}^{e_{k}}_{1-\varepsilon}\oplus\mathcal{E}^{w}_{1-\varepsilon}$, $k\in\mathbb{N}_{0}$, then $\mathcal{E}^{e_{k}}_{1-\varepsilon}\subseteq\sum\limits_{i=0}^{k}\oplus A_{cl}^{i}\mathcal{E}^{w}_{1-\varepsilon}$ follows from the recursively expanding. We define the relaxed PRS as

Then, $\mathcal{E}^{e_{k}}_{1-\varepsilon}\subseteq\mathcal{D}_{k}\subseteq\mathcal{D}_{k+1}$ follows directly. And

can be gained from (8), and

can be inferred from (9).

To guarantee the chance constraints, we propose a strategy to formulate the chance constraint sets as the time-varying deterministic constraints by using the relaxed PRS in (13). Construct the time-varying tightened state constraint set at step $k$ as

Then, $\tilde{\mathcal{C}}_{k+1}\subseteq\tilde{\mathcal{C}}_{k}$ holds owing to $\mathcal{D}_{k}\subseteq\mathcal{D}_{k+1}$. It can be seen that, if the nominal state $s_{k}\in\tilde{\mathcal{C}}_{k}$, then the state chance constraint in (3a) is satisfied, i.e., $\textbf{Pr}(x_{k}=s_{k}+e_{k}\in\mathbb{X})\geq 1-\varepsilon$ can be guaranteed by (14).

The satisfaction of input chance constraint in (3b), i.e., $\textbf{Pr}(u_{k}=v_{k}+e_{k}^{u}\in\mathbb{U})\geq 1-\varepsilon$ can be guaranteed by constructing the tightened input constraint set at step $k$ as

Similarly, if the nominal input $v_{k}\in\tilde{\mathcal{V}}_{k}$, then the input chance constraint in (3b) is satisfied according to (15).

Furthermore, let the minimal robust positively invariant (mRPI) set

for the error dynamics in (4b) with $w\in\mathcal{E}^{w}_{1-\varepsilon}$ be computed as the outer approximation using the strategy in [36]. Then, $\mathcal{D}_{k}\subseteq Z$ and $K\mathcal{D}_{k}\subseteq KZ$ can be derived from (13).

For the cost function, we adopt the stage cost $\ell(s_{k},v_{k})=\|s_{k}\|_{Q}^{2}+\|v_{k}\|_{R}^{2}$, and the terminal cost $V_{f}=\|s\|_{P}^{2}$, where $Q\succ 0$, $R\succ 0$, and $P$ is the solution of the algebraic Lyapunov equation

To ensure the feasibility, we construct the terminal constraint set as the maximal output admissible set [37]

In this case, $\tilde{\mathcal{X}}_{f}$ satisfies the axioms [38]:

• A1:

  $A_{cl}\tilde{\mathcal{X}}_{f}\subset\tilde{\mathcal{X}}_{f},\tilde{\mathcal{X}}_{f}\subset\mathbb{X}\ominus Z,K\tilde{\mathcal{X}}_{f}\subset\mathbb{U}\ominus KZ$.

• A2:

  $V_{f}(A_{cl}x)+\ell(x,Kx)\leq V_{f}(x),\forall x\in\tilde{\mathcal{X}}_{f}$.

On the basis of the time-varying constraint tightening method and the DTMPC framework in [11], the probabilistic time-varying tube-based stochastic finite horizon optimal control problem $\mathbb{P}^{ttsmpc}_{N}(x_{t})$ to be solved at each time instant $t$ is formulated as follows:

The optimal solution of $\mathbb{P}^{ttsmpc}_{N}(x_{t})$ consists of nominal state $s_{0|t}^{*}$ and input sequence $\textbf{{v}}^{*}(x_{t})=[v_{0|t}^{*},v_{1|t}^{*},\cdots,v_{N-1|t}^{*}]$, and the associated optimal state sequence for the nominal system is $\textbf{{s}}^{*}(x_{t})=[s_{0|t}^{*},s_{1|t}^{*},\cdots,s_{N|t}^{*}]$.

At each time instant, $\mathbb{P}^{ttsmpc}_{N}(x_{t})$ is solved to generate $\textbf{{v}}^{*}(x_{t})$ and $s_{0|t}^{*}$, and the optimal control law is designed as:

The entire process of the probabilistic time-varying tube-based SMPC is repeated for all $t\geqslant 0$ to yield a receding horizon control strategy.

In this section, the feasibility of the designed pTTSMPC scheme is first presented, then the closed-loop chance constraint satisfaction is verified, and the stability results of the closed-loop system are also provided. Before that, a lemma for developing the feasibility result is presented.

Based on Lemma 2, the probabilistic resolvability result is reported as follows.

Suppose that the problem $\mathbb{P}_{N}^{ttsmpc}(x_{t})$ is feasible, then the satisfaction of closed-loop chance constraints follows directly, which is shown in the Theorem 2.

Finally, the stability result is presented in the following theorem.

In order to avoid possible infeasibility when violation occurs, the slack variable can be imposed on the state constraints [40, 16, 39]. However, here, we only impose the slack variable $\lambda$ on the flexible initial state constraint in (20e), such that

And a finite penalty term

is added to the cost function in (20a) to force $\lambda$ to tend towards 1, so that the distance between the actual state and the nominal state will be as close as possible while maintaining the feasibility. The weight $\gamma$ is chosen large enough so that $\psi(\lambda)$ makes the best of the penalty function [41]. As a result, the probabilistic resolvability of the optimal control problem $\mathbb{P}^{ttsmpc}_{N}(x_{t})$ is enhanced to strong feasibility by using the soft initial state constraint in (22) and the penalty term in (23). Moreover, this approach allows for solving a single optimisation problem instead of having to verify feasibility and decide which MPC problem to be solved accordingly, as is the case in [25, 15, 26].

The relationship between the RPI set and the PPI set is presented in the following proposition.

Let $Z$ be the mRPI set for the error dynamics in (4b) with $w\in\mathcal{E}^{w}_{1-\varepsilon}$ computed as in (18). If $\mathcal{E}^{w}_{1-\varepsilon}$ is a box-shaped or an ellipsoidal confidence region, then $Z$ is also the minimal probabilistic positively invariant (mPPI) set of probability level $1-\varepsilon$ for the error dynamics in (4b) with $w\sim\mathcal{Q}^{w}$ [32].

Especially, when the uncertainty $w$ follows the normal distribution $w\sim\mathcal{N}(0,W)$, the following property holds.

Since $Z$ is the mPPI set of probability level $1-\varepsilon$ for the error dynamics in (4b) with $w\sim\mathcal{Q}^{w}$, then

follows from Definition 2 directly. And

can be derived from $e_{k}^{u}=Ke_{k}$ subsequently.

Construct the constantly tightened state constraint set as

then, the chance constraint in (3a) is satisfied. The reason is that, if the nominal state $s_{k}\in\bar{\mathcal{C}}$, then, $\textbf{Pr}(x_{k}=s_{k}+e_{k}\in\mathbb{X})\geq 1-\varepsilon$ can be guaranteed by (24).

The satisfaction of $\textbf{Pr}(u_{k}=v_{k}+e_{k}^{u}\in\mathbb{U})\geq 1-\varepsilon$ can be guaranteed by constructing the constantly tightened input constraint set as

Thus, if the nominal input $v_{k}\in\bar{\mathcal{V}}$, then the input chance constraint in (3b) is satisfied according to (25).

Furthermore, to ensure the feasibility, we construct the terminal constraint set as

which satisfies the axioms A1 and A2 similar with (19).

On the basis of the constant constraint tightening method and the DTMPC framework in [11], the probabilistic constant tube-based stochastic finite horizon optimal control problem $\mathbb{P}^{ctsmpc}_{N}(x_{t})$ to be solved at each time instant $t$ is formulated as follows:

The optimal control law is designed as in (21). At each time instant, $\mathbb{P}^{ctsmpc}_{N}(x_{t})$ is solved to generate input sequence $\textbf{{v}}^{*}(x_{t})$ and nominal state $s_{0|t}^{*}$. The entire process of the probabilistic constant tube-based SMPC is repeated for all $t\geqslant 0$ to yield a receding horizon control strategy.

Using the similar argument as in pTTSMPC, we can prove that the optimal control problem $\mathbb{P}^{ctsmpc}_{N}(x_{t})$ in (29) is probabilistically resolvable with the probability no less than $1-\varepsilon$. Suppose that the problem $\mathbb{P}^{ctsmpc}_{N}(x_{t})$ is recursively feasible, then the satisfaction of closed-loop chance constraints can be guaranteed, and $Z$ is probabilistically stable for the closed-loop system. Moreover, if the uncertainty has zero mean, the closed-loop system is almost surely asymptotically stable in the mean.

Besides, in order to avoid possible infeasibility when violation occurs, we impose the slack variable $\lambda$ on the flexible initial state constraint in (29e), such that

And a finite penalty term as in (23) is added to the cost function in (29a) to force $\lambda$ to tend towards 1. As a result, the probabilistic resolvability of the optimal control problem $\mathbb{P}^{ctsmpc}_{N}(x_{t})$ is enhanced to strong feasibility by using the soft initial state constraint in (30).