Constrained Uncertain System Stabilization with Enlargement of Invariant Sets

An enhanced method able to perform accurate stability of constrained uncertain systems is presented. /e main objective of this method is to compute a sequence of feedback control laws which stabilizes the closed-loop system. /e proposed approach is based on robust model predictive control (RMPC) and enhanced maximized sets algorithm (EMSA), which are applied to improve the performance of the closed-loop system and achieve less conservative results. In fact, the proposed approach is split into two parts. /e first is a method of enhanced maximized ellipsoidal invariant sets (EMES) based on a semidefinite programming problem. /e second is an enhanced maximized polyhedral set (EMPS) which consists of appending new vertices to their convex hull to minimize the distance between each new vertex and the polyhedral set vertices to ensure state constraints. Simulation results on two examples, an uncertain nonisothermal CSTR and an angular positioning system, demonstrate the effectiveness of the proposed methodology when compared to other works related to a similar subject. According to the performance evaluation, we recorded higher feedback gain provided by smallest maximized invariant sets compared to recently studied methods, which shows the best region of stability. /erefore, the proposed algorithm can achieve less conservative results.


Introduction
Model predictive control (MPC) is a main concern for control design applied in different systems such as linear or nonlinear [1][2][3], continuous or discrete [4], and monovariable or multivariable [5]. Actually, MPC is a common technique for the dynamical systems' stabilization. is method is applicable already in numerous domains in industry [6] as regulation and control. Generally, real processes are nonlinear, complex, and uncertain [7][8][9][10].
erefore, a robust model predictive control (RMPC) has been introduced to guarantee robustness as well as constraint satisfaction against uncertainty. Moreover, model predictive control is an interesting approach to represent systems using fuzzy logic for designing controllers. Several works have focused on the use of fuzzy-model-based sliding mode control of nonlinear systems in combination with MPC algorithms [11][12][13][14]. In fact, the fuzzy logic technique is quite attractive in terms of time, simplicity of implementation, relatively low cost, and ability to rapidly model complex systems.
For constrained control problems processing, robust MPC is an effectual stabilization algorithm. is technique employs a specific model procedure based on input and output constraints, for each sampling time, in order to optimize system behavior through the prediction horizon. e controller implements merely the initial calculated input and reproduces these computations at the next sampling time, despite the fact that more than one input shift is calculated [15]. e major aim is to determine the state feedback control law to facilitate the minimization of the worst-case performance cost.
At each time phase, the convex problem is considered as an optimization problem including linear matrix inequalities (LMI). e main common current algorithm for RMPC is demonstrated to guarantee robust stability. But, due to the fact that the optimization problem is truly settled at each sampling time, it needs high computational time in online implementation. On the other hand, such problems rise appreciably with the size of the polytopic uncertainty set [16].
Many efforts have been made to design the state feedback control law which minimizes the worst-case performance cost. However, in future RMPC research [17], some constructive simulation experiences still remain. Several techniques have been performed in this field where practical treatments of RMPC are still a challenging task for model predictive control. Wan and Kothare [15] proposed an algorithm based on an offline robust constrained MPC by the use of ellipsoidal invariant sets subject to linear matrix inequality (LMI). is algorithm provides a detailed explicit control laws sequence corresponding to stable invariant ellipsoidal sequence asymptotically constructed offline in the state space. In the work of Bumroongsri and Kheawhom [18], the algorithm of Wan and Kothare [15] is developed in order to ensure the performance of the closedloop system focused on polyhedral invariant sets. An offline approach for the stabilization of constrained uncertain system is presented in this study. Various approaches have been proposed to investigate, estimate, or enlarge the maximum region of the state space where the system can operate without violating state and stabilization constraints. In fact, the obtainable difficulty is associated with the determination of controlled invariant sets [19,20]. e computation of the maximal controlled invariant set process introduced in [21] and the corresponding state feedback control laws for linear systems subject to polyhedral input and state constraints have been studied in [22,23]. Kouvaritakis et al. [24] developed an advanced method to enlarge the terminal invariant set using a linear programming approach. In the study by Henrion et al. [25], convex optimization problems are formulated for the region enlargement and hence tuning parameters for the positively invariant set improvement.
Many researchers [26][27][28] were interested in an automatic enlargement of invariant sets. In the work by Li and Lin [26], the characterization of the maximal contractively invariant ellipsoid associated with a given positive definite matrix is proposed for discrete-time linear systems. is description can be used to establish an algebraic computational approach and thus determine such maximal contractively invariant ellipsoids based on inputs from saturated linear feedback. In this field, the authors first divide the state space into several regions according to the saturation status of each input. Second, the possible maximal contractively invariant ellipsoids are computed in each region. Note that if none of the inputs saturate on their intersections, no region has been calculated. e minimal one among these possible maximal contractively invariant ellipsoids is the maximal contractively invariant ellipsoids of the system.
In this work, a new approach for maximizing ellipsoidal and polyhedral invariant sets associated with the determination of the corresponding state feedback control laws is developed. e contributions of this paper are twofold: firstly, to highlight the robust control of states, an RMPC algorithm [15,18] was applied. is approach is based on a computation method of maximal controlled invariant sets [21]. Secondly, the combination of MPC method and maximized invariant sets procedure is proposed in order to precisely advance the performance of the employed system. e considered techniques are realized to enlarge ellipsoidal and polyhedral invariant sets. For invariant ellipsoidal sets maximization, a semidefinite problem is used based on quadratic Lyapunov function. Besides, the proposed method for polyhedral sets enlargement consists of the iterative expansion of an initial invariant set precomputed by the LMI method, adding new vertices to its convex hull.
is is achieved by minimizing the distance between each new vertex from the vertices of the polyhedral set. Finally, an online implementation strategy has been applied.
So, in summary, using this proposed approach, we recorded these two contributions: Maximization of the invariant ellipsoidal and polyhedral sets in order to increase the region of stability Providing less conservative results and efficient system performance in terms of computational time is paper is organized as follows. Section 2 describes the proposed methodology based on robust model predictive control. In Section 3, simulation results and discussions of the whole proposed approach are reported using two examples: an uncertain nonisothermal CSTR and an angular positioning system. e conclusion is provided in Section 4. All preliminaries and notations used in this paper are revealed in Table 1.
Schur's Lemma 1 (see [16]). Let R, S, T be given matrices with appropriate sizes and assume that Q > 0; then the LMI R S S T Q > 0 (respectively, ≥ 0) is feasible if and only if the nonlinear constraint R − SQ −1 ST > 0 (respectively, ≥ 0) is feasible.

Description of Robust Model Predictive Control.
In this work, robust model predictive control (RMPC) analysis is the employed procedure to emphasize stability and effectively improve the performance of the uncertain discretetime linear systems. RMPC method is a typical scheme for minimizing the worst-case performance cost in order to determine the state feedback control law. is technique consists of two tasks: (i) offline part is introduced to search the feedback gain K i based on the resolution of Bumroongsri and Kheawhom problem [18]; (ii) online part, at each sampling time, determines the smallest invariant set containing the measured state and implements the corresponding state feedback control law to the process. In general, RMPC preprocessing strategy is suitable for stabilization process, decreasing computational time. Here, the regulated output is demonstrated to considerably evolve the system state faster to the origin. e step-by-step method of RMPC is described as follows.
e linear discrete-time system described by Wan and Kothare [15] is considered with the following polytopic uncertainty: 2 Complexity where x(k) ∈ R n x , u(k) ∈ R n u , and y(k) ∈ R n y are state, control, and output variables of the system, respectively. en, where conv is the convex hull, Ω is a polytope, and Step 2. Research to the feedback control law is as follows: Equation (3) stabilizes system (1) with the following cost: subject to where Θ > 0 and R > 0 are symmetric weighting matrices.
Step 4. For each K i , the corresponding polyhedral invariant set Figure 1, the proposed methodology is composed of three steps:

e Proposed Methodology. As illustrated in
Step 1. Enhanced maximized sets algorithm: by the combination of an RMPC technique proposed by Bumroongrsi and Kheawhom [18] and the enhanced maximized invariant sets approach, a successful progress of the closed-loop system performance was obtained. Two methods are developed to maximize the ellipsoidal and polyhedral invariant sets constructed by the RMPC algorithm. e ellipsoidal invariant sets approach referred to in Section 2.2.1 is a semidefinite programming method. Based on the work of Athanasopoulos and Bitsoris [21], a second linear programming approach is used to enlarge polyhedral sets.
It consists of adding new vertices to their convex hull by minimizing the distance between each new vertex and the polyhedral set vertices for securing the state constraints. e polyhedral invariant sets process is presented in Section 2.2.2.
Step 2. Online implementation of the feedback control law: at each sampling time, determine the smallest invariant set containing the measured state and implement the corresponding state feedback control law to the process.
Step 3. Evaluation criterion: the computational time (CT) required for the proposed approach has been reduced.

Enhanced Maximized Ellipsoidal Invariant Sets (EMES).
Subsequent to the RMPC problem resolution and the feedback gains determination, an invariant ellipsoidal sets sequence is built. Let the following inequalities be which is equivalent to where P � Q −1 .
To maximize the ellipsoidal region by guaranteeing a wider stability domain, a semidefinite programming problem will be used. Let us consider the quadratic Lyapunov function V(x(k)) � x(k) T Px(k). en, we have Condition (14) is true if and only if Using Schur's lemma, the following condition with Q � P −1 and K � YQ −1 is obtained: A natural objective enables increasing the ellipsoid volume which is proportional to det(Q). Hence, if the maximal invariant ellipsoid volume corresponds to state feedback law, solving the following semidefinite programming is required: subject to

Enhanced Maximized Polyhedral Invariant Sets (EMPS). Given the state feedback gains
constructed. e enhanced maximized polyhedral invariant sets (EMPS) algorithm is given as follows.
Step 1. Let the polyhedral invariant sets S i be the convex hull of its vertices: Step 2. Consider new sets S i � 1, 2, . . . , N, q � 1, 2, . . . , n, and choose a point v ch ∉ S i .

Complexity
Step 3. Solve the following EMPS problem: subject to where p j+1 ∈ [0, 1]. Once the problem is solved, an optimal vertex v sup is obtained, and thus, the following maximized polyhedral set is constructed: Relations (21)-(23) imply the positive invariance and attractivity of S i ′ , while (24) and (25) guarantee constraint satisfaction.
Step 4. At each sampling time, determine the smallest maximized polyhedral invariant set containing the measured state and implement the corresponding state feedback control law u(k/k) � K i x(k/k) to the process. [15] is considered where the exothermic reaction A ⟶ B takes place. e reaction is irreversible and the rate of reaction is primary order with respect to component A. A cooling coil is employed to eliminate heat which is released in the exothermic reaction. e uncertain parameters are the reaction rate constant k 0 and the heat of reaction H rxn . e linearized model focused on the component balance and the energy balance is given by the following state equations:

Example 1. An uncertain nonisothermal CSTR
where C A T is the state vector x(t) and C A,F F C is the input control vector u(t). Matrices are defined by where C A is the concentration of A in the reactor, C A,F presents the feed concentration of A, T denotes the reactor temperature, and F C is the coolant flow. e operating parameters are as follows: F � 1 m 3 /min, V � 1 m 3 k 0 � 10 9 − 10 10 min − 1 E/R � 8330.1 K −ΔH rxn � 10 7 − 10 8 cal/kmol ρ � 10 6 g/m 3 U A � 5.3410 6 cal/(K min) , and F C � F C − F C,eq , where the subscript eq is used to denote the corresponding variable at equilibrium condition. By discretization, using a sampling time (ST � 0.15 min), the discrete-time model with , as state and control vectors, respectively, is given as follows: x(k + 1) � C A (k + 1) where 1 ≤ α(k) � k 0 /10 9 ≤ 10 and 1 ≤ β(k) � −ΔH rxn / 10 7 ≤ 10. e two parameters α(k) and β(k) are independent of each other. en, we consider the following polytopic uncertain model with four vertices: Complexity By manipulating C A,F and F C , the control of concentration C A and the reactor temperature T return to the origin. ese variables are constrained having |C A,F | ≤ 0.5 kmol/m 3 and |F C | ≤ 1.5 m 3 /min. e cost function is given by (4) ese sequences are used to compute six offline feedback gains K i , i � 1, 2, . . . , 6. is allows building an ellipsoidal and polyhedral invariant sets sequences.
Focused on the EMSA method, the maximized ellipsoidal and polyhedral invariant sets are larger compared to invariant sets [15,18]. e difference between these sets is, respectively, shown in Figures 2 and 3.
For both techniques, the invariant sets, ellipsoidal ( Figure 2) and polyhedral ones (Figure 3), are constructed based on the choice of the same states sequence x i , i � 1, . . . , 6. e maximized polyhedral invariant sets enable us to obtain an appreciably larger domain of stability compared to the polyhedral invariant ones in [18], for each chosen state x i .
is is due to the additional vertex of the obtained sets that have been added by the EMPS approach. Figure 3 reveals the comparison between the stabilizable sets of two feedback gains in terms of A and B points. As shown in Figure 3, it is clear that the maximized polyhedral invariant sets stabilize the states at point A by the use of feedback gain K 1 since the states are contained in the maximized sets S 1 . Contrariwise, the polyhedral sets [18] are not able to stabilize the states at point A because they are not contained in the initial invariant set. As illustrated in Figures 2 and 3, beginning at the point B, the polyhedral set can stabilize the states to the origin taking on the lowest feedback gain K 1 . In brief, the proposed approach EMPS algorithm can regulate the states at point B to the origin using a higher feedback gain K 6 in the fact that the points are contained in S 6 . In this case, EMPS method achieves the higher feedback gain, when compared to previous studies. Consequently, the proposed maximized approach attains less conservative results. To significantly clarify our results, Figures 4 and 5 demonstrate the regulated outputs. Here, we report that the considered EMSA method provides less conservative results and efficient system performance, when the state evolves faster to the origin. Compared to the previous work [15,18], the proposed strategy seems to be helpful for uncertain system control. As demonstrated in Table 2, we can deduce from the stabilization validation results that the EMSA technique is more efficient compared to the other model predictive control methods [15,18] in terms of stabilizable region and computational time (CT). EMSA strategy provides rigorous results in terms of CT (4.951 s) and larger stabilization region in different points. Although the construction of maximized polyhedral invariants sets requests more computational time than the standard ellipsoidal and polyhedral invariants sets, it is still more precise in enlargement of stability domain. Table 3 summarizes the cumulative cost obtained in Example 1.

Example 2.
We consider the angular positioning system described by the following discrete-time equation [29]: where θ(k) is the angular position of the antenna, θ • (k) is the angular velocity, and u(k) is the input voltage of the motor. It is assumed that the uncertain parameter is arbitrarily time varying: 0.1 ≤ α(k) � k 0 /10 9 ≤ 10. Let , and u � u − u eq , where the subscript eq denotes the corresponding variable at equilibrium condition Figure 6.
e obtained system can be written as follows: System (34) has the following polytopic structure: e input constraint is e weighting matrices Θ and R are given by Let us choose the following seven states sequence: In this example, the sequence of seven states x i , i � 1, . . . , 7, is used to compute seven state feedback gains K i corresponding to seven ellipsoidal and polyhedral invariant sets. Using the EMSA algorithm, the maximized ellipsoidal and polyhedral invariant sets are drawn compared to invariant sets [15,18]. Figure 7 exemplifies the comparison between the maximized ellipsoidal and polyhedral invariant sets.

Complexity
Compared to the invariant set [18], the maximized invariant set has a significantly larger domain of stability, for each chosen state x i , i � 1, . . . , 7. Figure 7 reveals the comparison between the stabilizable sets of three feedback gains in terms of A, B, and C points. Simulation results illustrated in Figure 7 highlight the robustness of the proposed method using maximized polyhedral invariant sets which stabilize the states at point A employing the feedback gain K 1 (the states are contained in the maximized set S 1 ). On the other hand, the polyhedral [18] and the maximized ellipsoidal sets are not able to stabilize the states at point A (the states are not contained in the original polyhedral and maximized ellipsoidal sets). Concerning the point B, the polyhedral set [18] can stabilize the states to the origin corresponding to the lowest feedback gain K 1 . In addition, the proposed EMPS approach can regulate the states at point B to the origin utilizing a higher feedback gain K 5 (points contained in S 5 ). On the contrary, the maximized ellipsoidal set and the ellipsoidal set [15] cannot control the states at point B because they are not situated in these invariant sets. Also, starting by the point C, it is obvious that the maximized invariant set obtained from EMES approach can stabilize the states to the origin containing the lowest feedback gain K 1 . e proposed EMSA model can control the states at this point to the origin exploiting higher feedback gain K 7 (points contained in S 7 ). Note that previous studies [18] are capable of stabilizing these states at points C from the feedback gain K 6 . Figures 8 and 9 display the regulated outputs. In this case, it is evident that the projected EMSA method supplies less conservative results. If the state evolves faster to the origin, the     Table 4, we can assume that the proposed scheme is more successfully having a larger stabilizable region. Table 5 resumes the cumulative cost in the second example.   Figure 7: (a) Maximized ellipsoidal invariant sets compared to [15] and (b) maximized polyhedral invariant sets compared to [18].

Conclusion
In this paper, we described an enhanced method which can be used for constrained uncertain discrete-time linear systems stabilization. A useful RMPC technique was applied to emphasize the robust control and improve the state stabilization. e proposed procedure gives appropriate optimization and notable precision when compared to existing model predictive control results. en, we have suggested the combined RMPC method and maximized invariant sets process that can accurately progress the performance of the closed-loop system. e included methods are used to enlarge ellipsoidal and polyhedral invariant sets constructed by the RMPC algorithm. An online implementation for the obtained feedback control laws has been made. e proposed method has been compared with some existing algorithms in order to enlarge stability domain. Experiment results demonstrate that the proposed method can permanently control system states having a larger stabilizable region. erefore, the performance of the proposed strategy furnishes a rigid basis in support of solving the control problem. As future works, we propose to use deep learning to obtain flexible models for nonlinear model predictive control (MPC).

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Methods
Cumulative cost Cumulative equation Wan and Kothare [15] 0.19 ∞ i�0 x T i Θx i + u T i Ru i Bumroongsri and Kheawhom [18] 0.12 Proposed approach 0.9 10 Complexity