Composite control with observer for switched stochastic systems subject to multiple disturbances and input saturation

This paper investigates the composite hierarchical anti-disturbance control problem for switched systems with saturating inputs, in which the multiple disturbances including normal-bounded disturbances, white noises and nonlinear uncertainties are considered. According to whether the states of the system be available or not, two types of disturbance observers and a state observer are designed, respectively. With the values of the observers, the observer-based composite controllers are constructed accordingly, under which an uniform form of the closed-loop system is presented. By utilising the minimum dwell-time technique and stochastic control theory, a new criterion of the mean-square exponential stability is presented for the closed-loop system with a weighted H ∞ performance level. The design conditions of the observers for the disturbances generated by exosystem and states of the system are proposed by propositions, respectively. Based on the gains of the observers, the solvable synthesis conditions of the composite controllers are presented for the cases of the available and unavailable states of the system. The simulations of GE-90 aero-engine model and a numerical example are employed to verify the applicability of the obtained results.


INTRODUCTION
Disturbances exist widely in all actual industrial processes and can degrade controlled performance or destroy even the stability of the systems. With the increasing demands of high control precision, it attracts considerable attention to antidisturbance control techniques [1]. There exist many advanced control schemes to achieve the disturbance attenuation and rejection, such as robust H ∞ control [2,3], sliding mode control [4], disturbance compensation control [6,7], and so on. The above-mentioned control strategies can be applied effectively to deal with one type of disturbances or a single equivalent disturbance. However, in practical engineering, the control system is usually subject to different types of multiple disturbances, such as bounded noises, stochastic noises, deterministic disturbances etc. Therefore, it could not achieve the required control accuracy for the system with multiple For fulfilling the need of high-precision control, a composite hierarchical anti-disturbance control (CHADC) scheme has been put forward, which mainly contains two parts, that is, classifying the disturbances based on the disturbance characteristics and constructing the targeted composite anti-disturbance controller [8]. When the exosystem generated disturbance and H 2 -norm bounded disturbance are considered simultaneously, a composite control strategy has been proposed for various control systems by integrating the disturbance-observer-based control (DOBC) strategy and H ∞ control strategy [9][10][11][12][13].
In [14,15], the above control scheme has been improved by introducing the technique of the fuzzy control or adaptive control for the system with the nonlinear uncertainty as well as the above two types of disturbances. Recently, the CHADC has been developed in the stochastic field when both the deterministic and stochastic disturbances are considered [16][17][18].
In actual engineering, the parameters of many practical control systems change in a specific pattern of abrupt and continual manner on an infinite time instant sequence. Switched system model composed with a set of subsystems is proposed to meet this type of systems, all subsystems of which are driven by a switching rule. In the past few decades, the research on switched control systems has gained extensive attentions and there exist a number of excellent results [19][20][21][22][23][24][25]. However, most of achievements of the research on anti-disturbance control for the switched systems are in the case of single disturbance [26][27][28][29]. Due to the complicated closed-loop dynamic under the CHADC scheme, it is difficult to design ideal composite anti-disturbance control strategy for the switched system with multiple disturbances, though a few results have been reported [30,31,33]. By combining the techniques of the l 2 -l ∞ control and DOBC, a composite anti-disturbance control scheme is given for the discretetime switched system with exosystem generated disturbance and norm bounded disturbance under an average dwell time switching rule in [30]. In [31], a similar composite DOB controller is designed for continuous-time switched system with the same two disturbances in [30], under which the disturbances can be rejected and attenuated with a mixed time-driven and state-dependent switching rule. Due to the growing demand for high-precision control, more disturbances for the switched system need to be considered in the process of anti-disturbance control research, such as white noises, nonlinear uncertainties. However, to the best of our knowledge, so far there has not a systematic anti-disturbance control method existed yet for the switched system with uncertainties, deterministic disturbances and stochastic noises, which is our motivation to undertake this work.
This paper studies a composite anti-disturbance control problem for the uncertain switched system with the normal bounded disturbances, the exosystem generated disturbances and the white noises. Meanwhile, the saturation on input channel is considered, which is an adverse effect factor of control performance or even stability loss. Firstly, for the case of available states of the system, a disturbance observer is designed to estimate of the disturbances generated by an exosystem, based on which a switching composite controller is constructed. When the system states are unavailable, a state observer and a disturbance observer are presented by using the system outputs, and a switching composite controller based on the estimated values of the states and exosystem generated disturbances is constructed. Under the above two control schemes, a uniform form of the closed-loop system is obtained. Under a minimum switching rule, a novel criterion is put forward to guarantee mean-square exponential stability of the closed-loop system with a weighted H ∞ performance level. The gains of three different observers can be solved based on the proposed propositions under arbitrary switching. Further, the composite controllers are designed based on the solution of linear matrix inequalities satisfying above criterion, respectively.

System description
Consider the following switched stochastic system: (1) where x(t ) ∈ ℝ n are system states; u(t ) ∈ ℝ m are control inputs; y(t ) ∈ ℝ q are system outputs; f (x(t ), t ) ∈ ℝ n is nonlinear vector function which describes the system uncertainty and satisfies: ; the multiplicative noise (t ) is a white noise; the disturbances d 0 (t ) are matched with control inputs and generated by the exogenous system described as: where (t ) ∈ ℝ r are disturbance system states; d 2 (t ) is a additional noise in ℒ 2 [0, ∞]. The piecewise constant function (t ) is called as switching rule taking the value in finite set S = {1, 2, … , N }. For any switching time instant t k , (t ) = (t k ) = i k ∈ S when t ∈ [t k , t k+1 ). The notation S [T * ] stands for a set of switching signals in which the successive switching times satisfy t i+1 − t i ≥ T * , where T * is usually called as minimum dwell time. The saturation function sat : ℝ m → ℝ m is defined as where sat (u i ) = sgn(u i )min{ , |u i |} with a saturation level .
Here, we define a complete probability space as (Ω, , t , ) with a filtration t satisfying the usual conditions. From [34], the white noise (t ) can be described as , where (t ) is standard Wiener process in (Ω, , t , ). Therefore, for i ∈ S , system (1) and exogenous system (2) can be rewritten as For system (3), we make an assumption as: In what follows, the composite controllers will be constructed for the cases of available and unavailable states of system (1), respectively.

2.2
The case of available states of the system Firstly, all of the states of system (1) are assumed to be available. To estimate the disturbance d 0 (t ), a disturbance observer is designed as follows: Based on the disturbance estimationd 0 (t ), a composite controller is constructed for system (1) as where K i , i ∈ S are gains of the controller.

The case of unavailable states of the system
Next, under the assumption of the unavailable states of system (1), a state observer should be constructed as wherex(t ) ∈ ℝ n is the estimate of x(t ),ŷ(t ) ∈ ℝ p are the observer output, L 1i ∈ ℝ n×q are the gains of the observer and d 0 (t ) can be observed by the following observer Based on the observers (6) and (7), a composite controller is designed for system (1) as where K i , i ∈ S are gains of the controller.
Remark 1. In this paper, the disturbance observer (4) and composite controller (5) have been designed based on the information of states of the system (1). On the other hand, for the case of unavailable states of the system, we first proposed the composite control scheme (8) by designing a state observer (6) and disturbance observer (7) based on the outputs of the system. Therefore, a systematic anti-disturbance control problem would be studied for the switched system with multiple disturbances under the available or unavailable states of the system.
Remark 2. Comparing the existing results of anti-disturbance control of the switched system [30,31], more types of disturbances and input saturation are considered in this paper. For the cases of available and unavailable states of the system, the corresponding composite controllers has been designed based on the observer technique.

Closed-loop system description
Denoting a function as (•) = • − sat(•), the augmented closed-loop system can be presented as wherez(t ) is control output, and the matrices in the closedloop system (9) are described, respectively, under controller (5) as ; and under controller (8) as Remark 3. By slight abuse of notation, the notation (t ) is used to denote the augmented states vectors of the closed-loop system under the controllers (5) and (8).
In this paper, some definitions and lemmas are introduced for the later development.

Definition 2.
Given a scalar > 0, system (9) under external disturbance d (t ) is said to be mean-square exponentially stable with a weighted H ∞ performance level ( , ) if it is meansquare exponentially stable when d (t ) ≡ 0, and under zero initial condition and for all nonzero d (t ) ∈ ℒ 2 [0, ∞), the following holds: where z(t ) ∈ ℝ q is control output.
Lemma 1 [35]. For the above defined function Lemma 2 [36]. Given any constant > 0 and any matrices M, Γ, U of compatible dimensions, then for all x ∈ ℝ n , where Γ is an uncertain matrix satisfying Γ T Γ ≤ I .

SYSTEM PERFORMANCE ANALYSIS
In this part, we will develop a new criterion of the mean-square exponential stability with the weighted H ∞ performance for the closed-loop system in (9) under the minimum dwell-time switching rule. where Proof. Please see the Appendix A. □ Remark 4. In [38], the research has been made on antidisturbance control of the nonlinear systems with exosystem generated disturbance and input saturation. A locally sector bounded nonlinearity and a locally convex polynomial approximation are chosen to treat the saturation function, which require the states of the system stay in a bounded set. However, it might not fulfill this requirements for the switched stochastic system due to effect of the system switching and stochastic factors. In this paper, by using a global condition in Lemma 1 and matrix inequality technique, the input saturation is effectively handled.
Remark 5. Here, a minimum dwell-time approach is chosen to design the switching rule, which might be conservative than the average dwell-time approach from a theoretical point of view. That is because the switching rule satisfying the minimum dwelltime requirement might be easy to be applied in practice, when the asymptotic stability or stabilisation is considered. On the other hand, the result in Theorem 1 can be generalised to the case under the average dwell-time based on the result in [39].

COMPOSITE CONTROLLER DESIGN
Now, by using convex linearisation method, we will give some solvable design conditions of the controller (5) and (8) based on the result in Theorem 1.

4.1
The design method of controller (5) Proposition 1. Given scalars i > 0, the disturbance error systeṁ e (t ) = (W (t ) + LB (t ) V (t ) )e (t ) is exponentially stable under arbitrary switching signal, if there exist matrices G and P > 0 such that Moreover, the gain of disturbance observer (4) can be chosen by L = P −1 G.
Proof. Setting G = LP, (16) is equivalent to A common Lyapunov function is chosen as where P > 0. The time derivative along the state trajectory of the disturbance error system is computed aṡ For any i ∈ S , combing (17) with (19) From [37], it is obtained that (20) guarantees the disturbance error system to be exponentially stable under arbitrary switching signal. □ Theorem 2. Given scalars i > 0, > 0 and matrix L satisfying Proposition 1, if there exist scalars > 1, i > 0, matrices X 1i > 0, X 2i > 0, M i such that for i ∈ S, where , then the argument system in (9) is mean-square exponentially stable with a weighted H ∞ performance level for any Moreover, the gains of the composite controller can be chosen as Proof. Setting P i = X −1 i , applying Schur complement lemma and pre-and post-multiplying the inequality (21) by diag{P i , I }, one has that (21) is equivalent to (14). Noticing X i = diag{X 1i , X 2i }, it is easy to get that (22) can guarantee (15). Next, the proof can follow Theorem 1.
Moreover, the gains of state observer (7)can be chosen by L 1i = P −1 G 1i .

Proposition 3.
Given scalars i > 0, the disturbance error systeṁ e (t ) = (W (t ) + L 2 CB (t ) V (t ) )e (t ) is exponentially stable under arbitrary switching signal, if there exist matrices G and P > 0 such that Moreover, the gain of disturbance observer (6) can be chosen by L 2 = P −1 G 2 .
Theorem 3. Given scalars i > 0, > 0, matrices L 1i and L 2 satisfying Proposition 2 and Proposition 3, respectively, if there exist scalars > 1, i > 0, matrices X 1i > 0, X 2i > 0, M i such that for i ∈ S, , , then the argument system in (7) is mean-square exponentially stable with a weighted H ∞ performance level for any Moreover, the gains of the composite controller can be chosen as Proof. Given constant matrices U 1 and U 2 , by Assumption 1, we can have the following inequality Denoting P ri = X −1 ri , r = 1, 2, pre-and post-multiplying the inequality (40) by P i = diag{P 1i , P 1i , P 2i }, and using Schur complement lemma yield that It is obvious that (30) has the same form with (14). On the other hand, one has (22) can guarantee (15). The proof can follow Theorem 1. □ Remark 6. In this paper, we choose two steps strategy to obtain the solvable design conditions of the composite controllers, that is, solving the gains of the observer to guarantee the error system be exponentially stable and proposing the design conditions for the controller based on the observers satisfying Theorem 1, which is inspired by [17].

SIMULATED EXAMPLES
In this section, the simulations of two examples are shown to illustrate the effectiveness of the proposed composite control schemes for the cases of available and unavailable states of system (1), respectively.

Example 1
Consider a GE-90 aero-engines model borrowed in [31] with the fan speed increment x 1 (t ) , the core speed increment x 2 (t ), the fuel flow increment u(t ) and the engine pressure ratio z 1 (t ), the states of which can be available and the parameters of which are given as follows Here, the disturbance d 0 (t ) is generated by system (2) with the following parameters In what follows, we set 1 = 2 = 0.5, = 2, = 1.1 and U = [0.1 0.2]. By Proposition 1, we can obtain the gain of the Next, the minimum dwell time and the feedback control gain K i can be solved by Theorem 2 with (33) as T * = 0.1906, This simulation is made under f (x(t ), t ) = 0.1 * x 2 e −0.1t , d 1 (t ) = 5e −t , d 2 (t ) = e −t and the switching signal in Figure 1 satisfying the minimum dwell time T ≥ T * . Figure 2 shows the states of system (1) under controller (5) with parameters (31), (33) and (34). In Figure 3, we plot not only the disturbance d 0 (t ) and its estimationd 0 (t ) but also the error between them, which makes clear that the observer (4) can estimate the disturbance d 0 instantly. The signals of the control output are shown in Figure 4. From Figures 2-4, it is verified that the GE-90 aero-engines system with (31) under the composite controller (5) based on the disturbance observer (4) and minimum switching rule is mean-square exponentially stable with a prescribed H ∞ performance level.

FIGURE 3
Curves of disturbance d 0 (t ), estimationd 0 (t ) and errors e d (t ) under observer (4) with parameters in (32) and (33) FIGURE 4 Curves of control output for augmented system (9) Remark 7. In [31], a composite control was investigated for the switched system with the norm-bounded disturbance and exosystem generated disturbance, in which the disturbance observer based controller is designed under the assumption of available states of the system. In this example, we consider not only the above two types of disturbances but also the uncertainties of the system, the stochastic disturbance and input saturation. The control scheme might be ineffective for the switched system subject to above mentioned multiple disturbances and input saturation.

Example 2
The system (1) with two subsystems is presented to illustrating the effectiveness of the proposed control design scheme for the case of the unavailable systems states, the parameters of which are shown as  (37) and (38) The parameters of the disturbance d 0 (t ) are given as follows First, the following gains of the observers (6) and (7) can be obtained by solving Propositions 2 and 3, respectively ] .
By setting the scalars as 1 = 2 = 0.1, = 3, and = 1.1, we can obtain the parameters of the controller The simulation results are shown in Figures 5-9. In Figure 5, a switching signal satisfying the minimum dwell time T * ≥ 0.1906 is given. From Figures 7-8, the states x(t ) and the disturbance d 0 (t ) can be estimated effectively by observers (6) and (7) with gains (37). From Figure 6, it can be verified the system (1) with the assumption of unavailable state information can be stochastically exponentially stabilised in mean-square under the composite control scheme (8) Figure 9 show that the disturbance attenuation and rejection has been achieved.

CONCLUSION
The composite anti-disturbance control problem of switched system with nonlinear uncertainties, multiple disturbances and input saturation has been studied by combining strategies of the observer based control, robust stochastic control and H ∞ control. For the cases of the available or unavailable states of the system, the proposed composite control schemes can guarantee the closed-loop system to be mean-square exponentially stable with a weighted H ∞ performance level under a minimum

FIGURE 9
Curves of control output for augmented system (9) switching rule, which are based on the disturbance and state observers.
, one has (t ) = i k . Choose a Lyapunov function for the mode i k as follows It is easy to get that where a i k = min (P i k ) and b i k = max (P i k ). For a fixed i k and by Itô formula, the stochastic differential can be computed along the solution of argument system (9) as Under Assumption 1 with a given matrix U , using lemmas (1) and (2), there exist positive scalar i k and real number ∈ (0, 1) such that When d (t ) ≡ 0, (14) implies It is easy to further get that Under the switching signal ∈ S [T * ] and the corresponding switching sequence, for t ∈ [t k , t k+1 ), integrating both sides of (A.6) from t k > 0 to t and taking expectation yield that Noting that P i = X −1 i , the inequality (15) is equivalent to P i < P j , ∀i, j ∈ S . At the switching time instant t k , one can obtain that From (A.7) and (A.8), the following inequality can hold with a = min{a i } and b = max{b i } for i ∈ S . By definition 1, (A.12) implies system (9) with d (t ) ≡ 0 is mean-square exponentially stable.
When system (9) is considered to be with disturbances d (t ), the following performance index with positive constant is introduced as t ∈ [t k , t k+1 ). < −(z T (t )z(t ) − 2 d T (t )d (t )). (A.14) From (A.3) and (A.14), it is easy to see that dV i k ( (t )) < 2 T (t )P iDi (t )d (t ) − i k V i k ( (t )) −z T (t )z(t ) + 2 d T (t )d (t ). (A.15) As the above process, for t ∈ [t k , t k+1 ), one has which implies the mean-square exponential stability of the augmented system in (9) with a weighted H ∞ performance level under the minimum switching rule in S [T * ] with T * ≥ max i∈S { ln i }. □