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An Integral Sliding-Mode Parallel Control Approach for General Nonlinear Systems via Piecewise Affine Linear Models

The fundamental problem of stabilizing a general non-affine continuous-time nonlinear system is investigated via piecewise affine linear models (PALMs) in this paper. A novel integral sliding-mode parallel control (ISMPC) approach is developed, where an uncertain piecewise affine system (PWA) is constructed to model a non-affine continuous-time nonlinear system equivalently on a compact region containing the origin. A piecewise integral sliding-mode parallel controller is designed to globally stabilize the uncertain PWA and, consequently, to semi-globally stabilize the original nonlinear system. The proposed scheme enjoys two favorable features: i) some restrictions on the system input channel are eliminated, thus the developed method is more relaxed compared with the published approaches; and ii) it is convenient to be used to deal with both matched and unmatched uncertainties of the system. Moreover, we provide discussions about the universality analysis of the developed control strategy for two kinds of typical nonlinear systems. Simulation results from two numerical examples further demonstrate the performance of the developed control approach.


I. INTRODUCTION
P IECEWISE affine linear models (PALMs) possess the convenience of control design and simplicity of structure, which have resulted in its extensive employment in analysis and control of diverse industrial systems with nonlinearity [1]- [4]. By dividing the premise state space into a series of adjacent subspaces, PALMs express a nonlinear system in each subspace as an affine linear dynamic system with normbounded uncertainties that can be constructed arbitrarily small through an appropriate design. With the aid of the powerful linear system theory and a quadratic Lyapunov function, this comparatively simple framework promotes systematic analysis and controller synthesis for nonlinear systems significantly and research along this direction has been fruitful in the past decades [5]- [11]. As another research frontier in the robust control theory, sliding-mode control (SMC) [12] has attracted growing research interests from the control community and has been used in many industrial applications [13]- [15]. The SMC scheme enjoys various desirable advantages, including unique robustness against disturbances and uncertainties, excellent transient performance and quick response. The core idea of the SMC approach is to design the closed-loop control system such that the system trajectories are first driven onto a welldesigned linear sliding surface covering the equilibrium, and subsequently, are maintained on it with desired convergent characteristic towards the equilibrium [16]- [18]. The special dynamics the closed-loop control system behaves while its trajectories are moving on the sliding surface is called the sliding motion. An alternative approach is the integral slidingmode control (ISMC) scheme where an integral form sliding surface is used instead of a linear one. Different from the common SMC scheme, in the ISMC approach, the system trajectories maintain on the sliding surface during the whole time interval [19]. Consequently, the reaching phase can be removed from the dynamics w.r.t the controlled system, which demonstrates stronger robustness of the ISMC approach than the SMC approach [20]- [26].
By now, there have been few published results on the ISMC design for PWAs. Considering the equivalence between a T-S fuzzy system and an uncertain PWA [27], a convenient extension might be made such that the fuzzy ISMC strategy in [28] can be applied to PWAs. However, the fuzzy ISMC strategy in [28] suffers from a restrictive assumption that each submodel of the fuzzy system must hold an identical constant input matrix, thus it faces significant conservativeness when dealing with general nonlinear systems. Various attempts have been made to weaken or remove this restrictive assumption. The approach in [29] replaced this assumption by a less conservative assumption in the SMC design for fuzzy systems. A piecewise ISMC approach allowing different local input matrices can be found in the coauthor's previous work [30] where the state space was divided into a series of subspaces and each subspace has its unique design of integral sliding surface. However, the high complexity of proposed strategy in [30] obstructs its wide implementation in practice.
In this paper, motivated by the previous result in [31], an appropriate ISMC design for general nonlinear systems based on PALMs is developed. To be specific, an uncertain PWA is constructed on a compact region to express a given non-affine continuous-time nonlinear system first and then an integral sliding-mode parallel control (ISMPC) method is proposed for global asymptotic stabilization of the uncertain PWA and, consequently, for semi-global asymptotic stabilization of the original general nonlinear systems. In particular, the designed integral type sliding surface function depends on the system state and control signal. In each partition of the whole state space, the resulting sliding-mode controller used has a dynamical parallel compensator form and the corresponding control gain matrices are obtained by calculating a series of LMIs with the aid of a common quadratic Lyapunov function. This ISMPC strategy holds two favorable features: (i) Different input matrices w.r.t the PALMs are allowed, thus the ISMPC scheme is applicable for general nonlinear systems; and (ii) The uncertainties arising during the approximation procedure, either "matched" or "unmatched" [32], are eliminated in the control channel w.r.t the resulting controlled system, which introduces stronger robustness.
The universality discussion w.r.t the developed ISMPC scheme in this paper is another key contribution. The key concern is that, for any given stabilizable general nonlinear system with a smooth system function, can one always construct a piecewise integral sliding surface and a corresponding piecewise integral sliding-mode parallel controller such that the resulting closed-loop control system behaves a stable sliding motion since initially? This universality characteristic of the proposed ISMPC scheme is analyzed for two kinds of typical continuous-time nonlinear systems, i.e., globally asymptotically/exponentially stabilizable (GAS/GES) nonlinear systems, respectively. It is believed that these discussions may provide confidence in applying the developed approach to wider industrial practice.
The remaining of this paper has the following structure. Section II formulates the PLAM and the research problems. In Section III, an ISMPC strategy is developed to globally robustly stabilize an uncertain PWA and, consequently, to semi-globally stabilize the corresponding original nonlinear system on the predefined state space. Then, the universality discussion of the developed ISMPC strategy for GAS/GES non-affine nonlinear systems are presented respectively. Section IV presents the simulation studies. Section V concludes this paper.
Notations: The notation ⋆ in a matrix represents the entries that can be implied through matrix symmetry. Let A be a vector or a matrix, then A T indicates its transpose, A stands for its induced norm and A > 0 suggests A is positive definite. Let α : ℜ + → ℜ + be a continuous function, then α belongs to the class K, if it is strictly increasing, continuous and α (0) = 0; α belongs to the class K ∞ , if α (t) → ∞ as t → ∞ and it belongs to the class K.

II. SYSTEM DESCRIPTION AND PRELIMINARIES
This paper will principally concentrate on the following general non-affine continuous-time nonlinear system defined on a compact region X × U ⊂ ℜ n × ℜ m : where x := [x 1 , ..., x n ] T ∈ X, u := [u 1 , ..., u m ] T ∈ U . The following assumption is necessary: Assumption 2.1. 1). X × U contains the origin.
2). The system function f is continuously differentiable on X × U and has the origin as its only zero point.
In view of the universal approximation capability of the PALMs, the following PALM can be constructed to express the nonlinear system in (1) equivalently on the compact region X × U : where x (t) = x T (t) , u T (t) T , the approximation error bounds ε f 0 , ε f , and ε g can be made arbitrarily small [1], X i and U i are the subspaces of X and U respectively,X i = X i × U i are the adjacent partitions of X × U , the origin or the equilibrium is uniquely inX 0 , and l is the amount of the partitions besides X 0 . Note that the partitions induced by the uncertain PWA in (2) are polyhedral regions in the state space. Thus, the resulting partitions of the uncertain PWA in (2) are inherently polyhedra. An ellipsoid E i can work well to outer approximate each polyhedral region S i . Specifically, assume that there exist some matrices Q i and f i satisfying Suppose the polyhedral regions S i are slabs in the following form: where β i1 , β i2 ∈ ℜ. Then each region can be precisely illustrated as a degenerate ellipsoid in (5) with Based on (5), the statex (t) within an ellipsoid region S i satisfies By regarding the approximation errors arising during the modeling procedure as norm-bounded uncertainty terms, one can conclude that an ISMC scheme robustly stabilizing the uncertain PWA in (2) can stabilize the original nonlinear system in (1) simultaneously. By now, there have been few results on the ISMC design for PWAs. Considering the equivalence between a T-S fuzzy system and an uncertain PWA [27], one may extend the fuzzy ISMC approach in [28] to our case. However, the approach in [28] is useful only when the nonlinear system in (1) has a constant and linear input channel. This motivates us to develop a new ISMPC scheme that stabilizes the general non-affine nonlinear system in (1) based on its corresponding PALM in (2) and to remove those restrictions.

III. INTEGRAL SLIDING-MODE PARALLEL CONTROL
A new ISMPC approach will be developed in this section to robustly control the uncertain PWA in (2) and, consequently, to control the nonlinear system in (1).

A. An Integral Sliding-Mode Parallel Controller
Considering the uncertain PWA in (2), or equivalently (1), we propose a novel piecewise integral sliding surface as where S x ∈ ℜ m×n , S u ∈ ℜ m×m represent the sliding surface matrices to be designed and S u is required to be nonsingular. Calculation of the matrices F i ∈ ℜ m×n , G i ∈ ℜ m×m and D i ∈ ℜ m×1 will be presented later with D 0 ≡ 0.
The following theorem provides an appropriate design of the sliding-mode controller to guarantee that the integral sliding surface in (9) can be maintained since initially.
Theorem 3.1: For the uncertain PWA in (2), or correspondingly, the original nonlinear system in (1), if the following piecewise integral sliding-mode parallel controller is used: with u (0) = 0, where F i and G i are defined in (9), γ > 0 is a scalar variable, and the approximation error bounds ε f 0 , ε f and ε g are defined in (3) and (4), then the piecewise integral sliding surface in (9) is reached and maintained since initially in potential. Remark 3.1: Notice that the equivalence between the uncertain PWA in (2) and the non-affine continuous-time nonlinear system in (1) is ensured only within X × U . Put another way, the closed-loop control system consisting of (1) and (10), which is named the practical closed-loop control system in this paper, behaves the sliding motion since the initial time only when its trajectories keep moving within X × U during the time interval of interest. This is, however, often not the case in practice even for a stable closed-loop control system. When the initial states are very close to the boundary of X × U , the system trajectories are highly possible to move out of the approximation region. In this case, the developed ISMPC approach no longer works. Therefore, the ideal sliding mode can be realized "in potential". Practically, designing a compact region where the PLAMs is constructed big enough can improve this situation.
Remark 3.2: The proposed sliding-mode controller in (10) has a dynamical parallel compensator form which distinguishes the approach from the existing methods in [33]- [35]. One can observe an important advantage of this ISMPC strategy is that it can be applied to the uncertain PWA in (2) without requiring that all subsystems have an identical input matrix.
Proof: The Lyapunov function candidate w.r.t the piecewise integral sliding surface in (9) is chosen as For the sake of simplicity, the case thatx (t) ⊆X i , i ∈ Φ is considered exclusively in this proof. The case thatx(t) ⊆X 0 can be analyzed similarly.
Combining (9) and (10) yieldṡ It then follows from (12) and (13) thaṫ Substituting (11) and (13) into (14) yieldṡ from which one concludes that the integral sliding surface variable s (t) can decrease to zero in finite time. Since initially s (0) = 0 and, consequently, the piecewise integral sliding surface in (9) is maintained subsequently. Denote Then, one has a more compact form of the integral sliding surface in (9) as In the literature, the PWA is usually named the "nominal closed-loop control system". One observes that the integral sliding surface variable s (t) in (16) is in fact the real time difference between the trajectories of the practical closed-loop control system defined in Remark 3.1 and those of (17), multiplying by a weight matrixS. This represents the core idea of the proposed ISMC approach, i.e., to achieve a sliding motion that is as close to (17) as possible. In other words, one can design (17) according to desired control criteria to force the practical closed-loop control system to behave satisfied control performance.
The following lemma provides a constructive procedure for designing (17): Given a series of matrices D i , i ∈ Φ of (17), the PWA in (17) is asymptotically stable, if the following LMIs are feasible w.r.t a matrix W ∈ ℜ (m+n)×(m+n) > 0, a series of scalars λ i > 0 and matrices Y j ∈ ℜ m×(m+n) , j ∈ ϕ and furthermore, the control gains can be calculated asK j = Y j W −1 : where Q i and f i are defined in (5). Proof. Lemma 3.1 follows directly from Lemma 4.1 in [6] and the proof is omitted here.
The detailed design procedure to calculate the control matricesK i and D i in (17) is summarized as follows: Algorithm 3.1 [6] (Sample Method): Step 1: Define a grid for the domain of the vector [D 1 D 2 · · · D l ] and sample its value at N points.
Step 2: Solve (18) for each point in the grid. If a feasible solution is found, stop; Step 3: Increase the sampling density of the grid and go back to Step 2.

B. The Sliding Motion
How to obtain the sliding surface matrixS through stability analysis of the sliding motion will be further shown in this subsection. For the sliding motion w.r.t (9), two conditions must be fulfilled simultaneously: Since S u is nonsingular, (20) yields which is usually named the equivalent sliding-mode control.
The corresponding sliding motion is then referred to the closed-loop control system consisting of (2) and (21): or equivalently One has the following theorem: Theorem 3.2: The sliding motion in (23) is asymptotically stable, if the LMIs in (24) and the one in (25) are feasible w.r.t a series of positive scalars η 0 , η i1 , η i2 , η i3 , i ∈ Φ and a matrix P ∈ ℜ (m+n)×(m+n) > 0. Additionally, the integral sliding surface matrix isS = R T 2 P :  where Proof: The following quadratic Lyapunov function is used for stability analysis: Its derivative along the trajectories of the sliding motion in (23) is given as follows: Let M be a positive definite matrix satisfying M = √ P . Then, it follows from the factS = R T 2 P that and Then, combining (29) and (30), we can obtaiṅ It follows from (31) Besides, the inequality in (32) is equivalent to where Furthermore, the norm-bound of approximation errors defined in (3) and (4) satisfy and For the partitionsX i , i ∈ Φ, (8) is rewritten as By applying the S-procedure in [36] and combining the LMIs in (33), (35), (36) and (37), one can conclude that (33) is fulfilled if there exists a series of positive scalars Based on the Schur's complement in [36], the LMIs in (38) are proved to be equivalent to those in (24). One can obtain that the derivative of V (x (t)) is strictly negative when the LMIs in (24) and (25) are both satisfied. Therefore, the sliding motion in (23) is asymptotically stable when the LMIs in (24) and (25) are both satisfied.
Corollary 3.1: If the conditions in Theorem 3.2 are fulfilled simultaneously, the practical closed-loop control system consisting of (1) and (10) behaves a semi-globally asymptotically stable sliding motion since initially.
Proof: Corollary 3.1 can be concluded based on the fact that only within X × U , the uncertain PWA in (2) is equivalent to the original nonlinear system in (1). The proof will be thus omitted here.
A systematic algorithm for implementing the ISMPC approach concerning the non-affine continuous-time nonlinear systems in (1) can be summarized as follows: Algorithm 3.2: For any non-affine continuous-time nonlinear system in (1), a piecewise integral sliding-mode parallel controller in (10) can be constructed such that a semi-globally asymptotically stable sliding motion can be achieved since initially, by conducting the following design procedure: Step 1: To obtain the PALM through linearizing the given nonlinear system at multi-operating points on X × U .
Step 2: To obtain the control matricesK i = [F i , G i ] and D i based on Algorithm 3.1.
Step 3: To obtain the sliding surface matrixS based on Theorem 3.2.
Step 4: To choose a suitable parameter γ > 0 to ensure that the trajectories of the practical closed-loop control system are moving within X × U . In case Algorithm 3.1 can not return a feasible solution, raise the amount of partitions l and repeat the loop until l exceeds the threshold.

C. Universality Discussion
In Subsections III-A and III-B, we have developed an ISMPC scheme to stabilize a general nonlinear system as in (1) through PALMs. It is shown that the scheme works if a series of LMIs is fulfilled simultaneously. Therefore, the universality of such a scheme is questionable, which motivates the study in the remaining of this section. To be specific, we will answer the following question in this subsection: for any given stabilizable nonlinear system in (1), can one always construct a piecewise integral sliding-mode parallel controller as in (10) such that the resulting closed-loop control system behaves a stable sliding motion since initially?
In the remaining of this paper, we say that a general nonaffine continuous-time nonlinear system in (1) is GAS/GES, if there exists a control law in the form ofu (t) = g (x (t) , u (t)) such that is globally asymptotically/exponentially stable. For brevity, we rewrite (39) aṡ (40) where R 1 and R 2 are defined in Section III-A below Equation (15).
The subsequent two theorems (Theorems 3.3 and 3.4) summarize the main results of this subsection. Theorem 3.3: For a GES nonlinear system in (1), one can always construct a piecewise integral sliding-mode parallel controller as in (10) such that the resulting closed-loop control system behaves a semi-globally exponentially stable sliding motion since the initial time.
Proof. See Appendix-A. Before proceeding with the more general case, a preliminary result from the Lyapunov converse theorem in [37] is presented. Suppose the nonlinear system in (40) is globally asymptotically stable, then along the trajectories of (40), there exist two functions α 1 (·) and α 2 (·) belonging to the class K ∞ , a function α 3 (·) belonging to the class K, a scalar h > 0 and a Lyapunov function V (x) satisfying Theorem 3.4: For a GAS nonlinear system in (1), one can always construct a piecewise integral sliding-mode parallel controller as in (10) such that the resulting closed-loop control system behaves a semi-globally asymptotically stable sliding motion since initially, if for the function α 3 (·) defined in (42), the inequality holds for a K function α 4 (·) and a scalar ρ > 0.
Proof. See Appendix-B.

IV. SIMULATION STUDIES
Two numerical examples are given in this section to demonstrate the developed ISMPC approach.

A. Nonlinear Chua's Circuit
The famous Chua's circuit has the following dynamical equation [38]: where x 1 and x 2 are the voltages across the capacitors C 1 and C 2 respectively, x 3 is the current through the inductor L, u is the control current used to stabilize the nonlinear circuit and v d is the voltage loss R 0 · x 3 or the external disturbance. The nonlinearity of the resistor g (x 1 ) is described by Here we use the approximation model built in [39] for the control design. Based on Algorithm 3.2, one can obtain the controller matrices and the integral sliding surface matrix as In order to avoid singular problem and reduce chattering phenomenon, sgn (s (t)) in (10) is replaced by its approximation function: The currents trajectories In this simulation, we setx (0) = 4 1 0 0 . The voltages across the capacitors C 1 and C 2 are shown in Fig. 1 respectively and the currents in the circuit are presented in Fig. 2. One can observe that the control input is pretty smooth during the simulation time. It is also observed from Fig. 3 that the sliding mode is realized prior to t = 5s and maintained afterwards. While the convergence of x (t) and u (t) happens after t = 25s. That is, the resulting closed-loop Chua's circuit achieves the sliding mode first and then behaves asymptotic stability. Note that theoretically, the practical closed-loop control system should enter and keep in the sliding mode since the beginning of this simulation. The approximation signum function utilized in the constructed control law in (10) results in the deviation from the ideal sliding mode.

B. Inverted Pendulum
Stabilization of the inverted pendulum is used to demonstrate the advantages and effectiveness of various control approaches. We choose an inverted pendulum system in [5] without disturbance. The dynamics of the inverted pendulum are described bẏ where x 1 is the angle of pendulum from the vertical, x 2 denotes the angular velocity, and u is the input force from the motor. The gravity constant g = 9.8m/s 2 , M = 4.0kg is the mass of the cart, the mass and length of the pendulum is m = 2.0kg and l = 0.5m respectively, a = 1/(M + m).
The compact region X × U is chosen to be − π 2 , π 2 × [−3, 3] × [−300, 300]. To obtain the corresponding PLAM of the inverted pendulum by linearization around the operating points (0; 0; 0) , ± π 3 ; 0; 0 and ± 13π 30 ; 0; 0 . One obtains Base on Algorithm 3.2, one can obtain the controller matrices and the sliding surface matrix as follows: It is noted that the constructed PALM has different input matrices in each partition. Therefore, the method in [28] can not be extended to our case trivially.
Similarly, the function sgn (s (t)) is approximated by the function s (t) s (t) + 0.020 .
In the simulation, we setx (0) as 82 • 0 0 . Fig. 4 and Fig. 5 show the inverted pendulum trajectories. One can observe that both x (t) and u (t) are moving within X × U while behaving the asymptotic stability. It is also observed from Fig. 6 that the sliding mode is achieved prior to t = 1s while the system trajectories converge to the origin after t = 3s. That is, the closedloop inverted pendulum enters the sliding mode before its trajectories converge to the equilibrium and behaves the ideal sliding motion afterwards. A new ISMPC scheme has been proposed to stabilize the general non-affine continuous-time nonlinear systems via PALMs. The proposed control strategy removes a restrictive assumption that is required in relevant research and shows significant convenience in dealing with general nonlinear systems with the construction of a piecewise integral sliding surface and a corresponding piecewise integral sliding-mode parallel control law. Moreover, results on the universality of the proposed ISMPC scheme have been provided, which further demonstrates its usefulness. Future research topics include conservatism reduction and practical applications of the developed method.

A. Proof of Theorem 3.3
For a GES nonlinear system in (1), one can always construct a globally exponentially stable system as in (39) or (40). In view of the universal approximation capability of the PALM [1], for any positive scalars ε f , ε g , ε h and ε i , one can build on X × U the following approximation system of (39): such that In particular, for the partitionX 0 , we have max {ε f , ε h } ≡ 0.
The system in (46) is rewritten aṡ Then, the piecewise integral sliding surface will be designed as while the piecewise integral sliding-mode parallel controller is constructed as It can be concluded from the proof procedure of Theorem 3.1 that (50) is reached and maintained since initially and the resulting sliding motion iṡ It follows from the Lyapunov converse theorem in [37] that, along the trajectories of (40), there exist a Lyapunov function V (x) and four positive scalars b 1 , b 2 , b 3 and b 4 satisfying On the other hand, the derivative of V (x) along the trajectories of (48) satisfieṡ Case i: For the trajectories that move within the partition X 0 , one has that max {ε g , ε i } ≡ 0. It then follows from (56) thatV by choosing the appropriate approximation error bounds ε f and ε h such that Case ii: For trajectories that move within the partitions X i , i ∈ Φ, one can conclude that the norm x(t) is lower bounded.
Given a scalar 0 < λ < 1/2, one can always choose the approximation error bounds ε g , ε i , ε f and ε h such that and By submitting (59) and (60) into (56), we havė (61) From (57) and (61), one has that along the trajectories of (46) moving within X × U , Therefore, the semi-global exponential stability of (48) is concluded based on the Lyapunov stability theory.
We are now ready to analyze the stability of (52) that can be treated as a perturbation system of (48). By following the similar procedure of (53)-(63) and based on the following inequalities: the semi-global exponential stability of (52) is guaranteed, by appropriately designing the approximation error bounds ε f , ε h , ε g and ε i such that where 0 < λ < 1 2 + R 2 S −1 u S x , b 3 and b 4 are defined in (54) and (55) respectively. The proof is then completed.

B. Proof of Theorem 3.4
By using the piecewise integral sliding-mode parallel controller in (51), we have shown in Appendix-A that the piecewise integral sliding surface in (50) can be reached and maintained since initially. So now our main goal is to show that, for a GAS nonlinear system in (1), its corresponding sliding motion in (52) will be controlled to behave the semiglobal asymptotic stability. This can be done with the aid of V (x (t)) defined in (41)-(43).
The derivative of V (x (t)) defined in (41)-(43) along the trajectories of (48) satisfieṡ V (x (t)) = ∂V (x (t)) ∂xF (x (t)) = ∂V (x (t)) ∂x F (x (t)) Case i: For the trajectories that move within the partition X 0 , one haṡ if (44) is satisfied and the approximation error bounds ε f and ε g satisfy Case ii: For trajectories that move within the partitions X i , i ∈ Φ, the norm x(t) is lower bounded.