Model Predictive Control for T-S Fuzzy Markovian Jump Systems Using Dynamic Prediction Optimization

Let us consider a discrete-time FMJSs using IF-THEN rules:
Plant Rule $\hbar$: IF $f^{(1)}(x_{s})$ is $\mathcal{F}_{\hbar}^{(1)}$, $f^{(2)}(x_{s})$ is $\mathcal{F}_{\hbar}^{(2)}$, $\ldots$, and $f^{(r)}(x_{s})$ is $\mathcal{F}_{\hbar}^{(r)}$,

where $\hbar\in\mathcal{I}_{T}=\{1,2,\ldots,t\}$, $t$ symbolizes the number of IF-THEN rules;$f(x_{s})=[f^{(1)}(x_{s}),f^{(2)}(x_{s}),$ $\cdots,f^{(r)}(x_{s})]^{T}$ represents the premise variable of the system and $\{\mathcal{F}_{\hbar}^{(\alpha)},\alpha=1,2,\ldots,r\}$ represents a fuzzy set; $x_{s}\in{\mathbb{R}}^{n_{x}}$ and $u_{s}\in{\mathbb{R}}^{n_{u}}$ represent the system state and the system control input, respectively. $A_{\zeta_{s},\hbar}$ and $B_{\zeta_{s},\hbar}$ are the known constant matrices with appropriate dimensions.

The stochastic process $\zeta_{s}$ denotes a homogeneous Markov chain in a finite state space $\mathcal{I}_{N}$ with the transition probability

where $0\leq p_{\imath\jmath}\leq 1$. $\mathbb{P}=[p_{\imath\jmath}]$ is called the transition probability matrix. The initial state $x_{0}$ and mode $\zeta_{0}$ are indicated. For each specific $\zeta_{s}=\imath\in\mathcal{I}_{N}$, system matrices are represented by $A_{\imath,\hbar}$, $B_{\imath,\hbar}$.

Aided by the T-S fuzzy approach, we establish the overall FMJSs with $\zeta_{s}=\imath$ as follows:

where

with $\mathcal{F}_{\hbar}^{(\alpha)}(f^{(\alpha)}(x_{s}))$ referring to the grade of membership of $f^{(\alpha)}(x_{s})$ in $\mathcal{F}_{\hbar}^{(\alpha)}$. And $\theta_{\hbar}(f(x_{s}))$ implies the standard membership function of rule $\hbar$. For $\forall\hbar\in\mathcal{I}_{T}$, we have $\theta_{\hbar}(f(x_{s}))\geq 0$ and $\sum_{\hbar=1}^{t}\theta_{\hbar}(f(x_{s}))=1$. For notational brevity, $\theta_{\hbar}$ is denoted as $\theta_{\hbar}(f(x_{s}))$ for the subsequent analysis.

The FMJSs will be affected by the constraints imposed by engineering practices on inputs and states as follows:

where $\breve{u}$ and $\breve{x}$ are the known positive scalars, $\Phi$ is the matrix according to actual needs.

Considering the remarkable superiority demonstrated in terms of the applicability complexity, robustness and flexibility to the PMP method, the IPM method is used to design a fuzzy controller for discrete-time FMJSs (3). However, it is very likely to result in a huge computation burden due to the number of rules for the IPM fuzzy controller. To alleviate the computational load online and maintain effective system performance, we incorporate a perturbation generated by a dynamic controller into the mode-dependent state feedback fuzzy control law within the DPO-MPC framework. This integration results in the fuzzy predictive dynamic controller law outlined below:
Controller Rule $\upsilon$: IF $g^{(1)}(x_{s+\ell|s})$ is $\mathcal{H}_{\upsilon}^{(1)}$, $g^{(2)}(x_{s+\ell|s})$ is $\mathcal{H}_{\upsilon}^{(2)}$, $\ldots$, and $g^{(k)}(x_{s+\ell|s})$ is $\mathcal{H}_{\upsilon}^{(k)}$,

where $\upsilon\in\mathcal{I}_{V}=\{1,2,\ldots,v\}$ with $v$ symbolizing the number of inference rules; $\{\mathcal{H}_{\upsilon}^{(\kappa)},\kappa=1,2,\ldots,k\}$ represents a fuzzy set; $g(x_{s+\ell|s})=[g^{(1)}(x_{s+\ell|s}),g^{(2)}(x_{s+\ell|s}),$ $\cdots,g^{(k)}(x_{s+\ell|s})]^{T}$ represents the premise variable of the controller.

From the fuzzy predictive dynamic controller described by (6), the mode-dependent fuzzy feedback gain $K_{\imath,\upsilon}$ is designed on terminal constraint set. $\{c_{s+\ell|s,\upsilon}\}$ represents perturbation variables for fine-tuning control inputs, which are generated by the dynamic fuzzy controller. $\eta_{s|s}\triangleq\eta_{s}\in\mathbb{R}^{n_{x}}$ represents the dynamic controller state to be determined. The estimator gain $\mathcal{A}_{\imath,\upsilon}$ and dynamic feedback gain $\mathcal{C}_{\imath,\upsilon}$ are needed to be designed. Then, the integrated T-S fuzzy predictive controller with system mode $\zeta_{s}=i$ is given by

where

$\mathcal{H}_{\upsilon}(g(x_{s+\ell|s}))$ refers to the grade of membership of $g^{(\kappa)}(x_{s+\ell|s})$ in $\mathcal{H}_{\upsilon}^{(\kappa)}$. And $\vartheta_{\upsilon}(g(x_{s+\ell|s}))$ implies the standard membership function of rule $\upsilon$. For $\forall\upsilon\in\mathcal{I}_{V}$, we have $\vartheta_{\upsilon}(g(x_{s+\ell|s}))\geq 0$ and $\sum_{\upsilon=1}^{v}\vartheta_{\upsilon}(g(x_{s+\ell|s}))=1$. For notational brevity, $\vartheta_{\upsilon}$ is denoted as its abbreviation of $\vartheta_{\upsilon}(g(x_{s+\ell|s}))$.

Before delving into the main results, we provide some definitions to facilitate the derivation of subsequent results.

The primary goal of this paper is to design a mode-dependent fuzzy predictive dynamic control law for the FMJS (3) subject to hard constraints. Specifically, for any $x_{0}\in\Sigma_{\zeta_{0}}$ (referred to as the initial feasible region, which will be defined in subsequent discussions), our aim is to solve an optimization problem at each time instant $s$. This optimization problem seeks to determine the mode-dependent fuzzy feedback gain $K_{\imath,\upsilon}$, estimator gain $\mathcal{A}_{\imath,\upsilon}$, dynamic feedback gain $\mathcal{C}_{\imath,\upsilon}$ and the optimization controller state $\eta_{s}$, ensuring the mean-square stability of the FMJS (3). This optimization problem is formulated as follows:

where $J^{\infty}_{s}\triangleq\mathbb{E}\Big{\{}\sum_{\ell=0}^{\infty}\big{(}\|x_{s+\ell|s}\|^{2}_{S}+\|u_{s+\ell|s}\|^{2}_{R}\big{)}\Big{\}}$, $S>0$ and $R>0$represent known weighting matrices. The term $\mathbb{E}\{\Delta V_{s+\ell|s}\}$ is defined as the expected difference in $V_{s+\ell|s}$, where $V_{s+\ell|s}$ represents a Lyapunov-like function. We assume that $V_{s+\ell|s}$ satisfies condition (14), also known as the terminal cost function condition [31]. This condition aids in constructing the upper bound of the objective function and achieving mean-square stability for the closed-loop system.

In the subsequent steps, our objective is to devise the fuzzy control law (7)-(9) by addressing OP1. Since directly minimizing the cost function over an infinite horizon, especially considering mode jumps and fuzzy rules, can be challenging, we opt to minimize a certain upper bound instead. It’s worth noting that the fuzzy control law (7) consists of two components: one related to the mode-dependent fuzzy feedback gain $K_{\imath\vartheta}$ and the other dependent on $c_{s+\ell|s,\vartheta}$ determined by prediction dynamics. Therefore, our approach involves breaking down the optimization problem OP1 into several auxiliary optimization problems to achieve our objective.

To begin, we establish the definition of a set known as the terminal constraint set as follows:

where the scalar $\sigma>0$, the matrix $P_{\imath\theta}\triangleq\sum_{\hbar=1}^{t}\theta_{\hbar}P_{\imath,\hbar}$ with $P_{\imath,\hbar}>0,\forall~{}\imath\in\mathcal{I}_{N}$. In the terminal constraint set, the mode-dependent state feedback controller is constructed without using perturbation variables by:

Applying (16) to FMJS (3) gives rise to

With regard to the cost function $J^{\infty}_{s}$, a min-max problem is employed to design a set of mode-dependent fuzzy controllers, outlined as follows:

where $\hat{J}_{s}=\mathbb{E}\Big{\{}\sum_{\ell=0}^{\infty}\big{(}\|x_{s+\ell|s}\|^{2}_{S}+\|u^{\ast}_{s+\ell|s}\|^{2}_{R}\big{)}\Big{\}}$.

Considering the control law (16), condition (14) in OP1 can be reformulated as follows:

We will now endeavor to identify the solvability condition for the terminal constraint (19). To begin, we introduce the following lemma, which is essential for deriving our main results.

In this subsection, we will try to find an upper bound of $\hat{J}_{s}$ based on (19) to design the fuzzy controller in the terminal constraint set.

To ensure the objective remains finite, it is necessary that $x_{\infty|s}=0$, leading to $V_{\infty|s}(x_{\infty|s})=0$. Adding both sides of equation (19) from $\ell=0$ to $\infty$ and considering $\lim_{\ell\rightarrow\infty}\mathbb{E}\{V_{s+\ell|s}\}=0$, we obtain

indicating $\max_{[A_{\imath\theta},B_{\imath\theta}],~{}\imath\in\mathcal{I}_{N}}\hat{J}_{s}\leq x^{T}_{s}P_{\imath\theta}x_{s}$. This sets an upper bound for the objective function across the infinite horizon $\hat{J}_{s}$. Therefore, our aim regarding the terminal constraint set of MPC is to formulate a series of mode-specific constant control laws, as defined by (16), to minimize the upper bound $V_{s}$.

Assuming the state at time $s$ lies within the terminal constraint set as defined by (15), specifically,

it is evident from (19) that the predicted state at any future time remains within the set $\Sigma_{x,\imath}$ in terms of mean-square, i.e., $x_{s+\ell|s}\in\Sigma_{x,\imath}$. Condition (30) is also referred to as the positive control invariant set condition. Additionally, we derive

Clearly, $\sigma$ serves as an upper bound for $V_{s}$ and consequently for the objective function $\hat{J}_{s}$.

Next, we will address hard constraints on inputs and states.

Following the preceding discussions, given condition (30), we utilize the upper bound $\sigma$ of $\hat{J}_{s}$ to formulate an optimization problem for deriving the mode-dependent controller gain $K_{\imath\vartheta}$ regarding the terminal constraint set $\Sigma_{x,\imath}$, outlined as:

Considering equations (3) and (7)-(9), and defining $\xi_{s}=[x_{s}^{T}~{}\eta_{s}^{T}]^{T}\in\mathbb{R}^{2n_{x}}$, we describe the closed-loop system as follows:

where

Define a set for system (34) as follows:

where $\mathcal{P}_{\imath\vartheta}=\sum_{\upsilon=1}^{v}\vartheta_{\upsilon}\mathcal{P}_{\imath,\upsilon}$ and $\mathcal{P}_{\imath,\upsilon}$ denotes a collection of symmetric and positive-definite matrices. Subsequently, we demonstrate that $\Sigma_{\xi,\imath}$ serves as a positive control invariant set for (34) and is admissible under constraints (4) and (5). According to [18], the following conditions are formulated to provide theoretical support for a domain attracting constraints for $\xi_{s+\ell|s}$.

where

The matrices $\mathbf{U}$ and $\mathbf{X}$ are auxiliary and positive.

Up to this point, we formulate an optimization challenge based on (36)-(38) to expand the constraint attraction region. Nevertheless, it is important to note that (36) exhibits non-convexity due to the interaction between $\Xi_{\imath\theta\vartheta}$ and $\mathcal{P}_{\jmath\vartheta_{+}}$. Therefore, addressing this non-convex issue is essential to ensure solvability of the forthcoming optimization problem. For this purpose, we introduce the following variable transformation.

To demonstrate this, consider $E_{\imath,\upsilon},~{}F_{\imath,\upsilon}\in\mathbb{R}^{n_{x}\times n_{x}}$, and positive matrices $M_{\imath,\upsilon},~{}L_{\imath,\upsilon}\in\mathbb{R}^{n_{x}\times n_{x}}$ defined by

Thus, $\mathcal{P}_{\imath,\upsilon}\mathcal{P}_{\imath,\upsilon}^{-1}=\mathbb{I}_{2n_{x}}$ implies that

By virtue of (39)-(42), the following Lemma is presented to help derive the sufficient conditions to (36)-(38).