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A novel adaptive switching function on fault tolerable sliding mode control for uncertain stochastic systems Seyed Ali Zahiripour n, Ali Akbar Jalali Department of Electrical Engineering, Iran University of Science and Technology, Tehran, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 18 March 2014 Received in revised form 15 May 2014 Accepted 22 May 2014 This paper was recommended for publication by Dr. A.B. Rad
A novel switching function based on an optimization strategy for the sliding mode control (SMC) method has been provided for uncertain stochastic systems subject to actuator degradation such that the closedloop system is globally asymptotically stable with probability one. In the previous researches the focus on sliding surface has been on proportional or proportionalintegral function of states. In this research, from a degree of freedom that depends on designer choice is used to meet certain objectives. In the design of the switching function, there is a parameter which the designer can regulate for specified objectives. A sliding-mode controller is synthesized to ensure the reachability of the specified switching surface, despite actuator degradation and uncertainties. Finally, the simulation results demonstrate the effectiveness of the proposed method. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Uncertain stochastic system Adaptive sliding-surface Optimization strategy Actuator degradation
1. Introduction This paper will investigate the design of reliable SMC for uncertain stochastic systems with possible occurrence of actuator faults. Sliding mode control is an effective robust scheme for incomplete modeled or uncertain systems, whose key feature relies on its complete insensitiveness to both parameter variations and external disturbances. Hence, in the past decades, a lot of important developments on SMC have been obtained [1–13]. As is well-known, the actuator degradation in actual physical systems is usually inevitable, and often yields performance degradation or even instability. Therefore, how to maintain an acceptable stability/performance for the closed-loop systems against actuator or sensor failures has been a long-standing and active research topic. In the literature, the design of reliable control systems can be broadly classified as the active approach and the passive approach. In the first type, faults are detected and identified by a fault detection and diagnosis (FDD) mechanism, and controllers are re-configured, such as those in [14–19]. In contrast, the passive methodology designs a reliable controller with a fixed structure by means of actuator redundancy. By taking possible actuator faults into account during the design of the controller, passive reliable systems can attain the stability and performance requirement even in the presence of
n
Corresponding author. Tel.: þ 98 77240487. E-mail addresses: [email protected], [email protected] (S.A. Zahiripour).
actuator faults. Moreover, various passive reliable techniques have also been developed, for example, linear-quadratic state-feedback control [20], pre-compensator [21], H1 disturbance attenuation [22], control allocation [23], and adaptive control [24–26]. It is known that a stochastic system has extensive applications in practice, and the SMC of stochastic systems receives much attention [27–29]. However, little work has been carried out on the reliable problem of SMC for uncertain stochastic systems subject to actuator faults. A common feature in the last corresponding works is related to choosing sliding surface which is mostly chosen as a proportional or proportional-integral function of states [30–32]. In the mentioned sliding surfaces we cannot discuss about the effect of changing a parameter of switching function on performance of the system, strongly. Furthermore these sliding manifolds have been proposed without expressing a reason and defining a specific goal. In this work, from a degree of freedom that depends on designer choice is used to meet certain objectives which were neglected in other recent researches. In other words, sliding surface is extracted based on a systematic strategy. In this method the problem of choosing sliding surface is converted to a virtual optimal control problem such that the constructed sliding surface minimizes a quadratic performance index. In this methodology there is a possibility of regulating the distance of state trajectories from sliding surface and the amount of energy consumption that has wealth importance in control strategies. Eventually, an adaptive sliding mode controller is synthesized, which can ensure the reachability of the aforementioned sliding surface. Additionally, sufficient conditions are derived via the stochastic Lyapunov method and linear matrix inequalities such
http://dx.doi.org/10.1016/j.isatra.2014.05.029 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Zahiripour SA, Jalali AA. A novel adaptive switching function on fault tolerable sliding mode control for uncertain stochastic systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.029i
S.A. Zahiripour, A.A. Jalali / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
that the closed-loop system is ensured to be asymptotically stable in probability one, not only when control component is in good working condition but also in the presence of actuator faults. Notations j:j, : denote, respectively absolute value and the Euclidean norm of a vector or its induced matrix norm. For a real matrix, M40 means that M is symmetric and positive definite, and I is used to represent an identity matrix of appropriate dimensions. λi (.) shows the i-th Eigen value of a matrix. (Ω, F, P) is a probability space with Ω the sample space, and F the s-algebra of subsets of the sample space, and P is the probability measure. Matrices, if not explicitly stated, are assumed to have compatible dimensions.
2. System description and preliminaries In this work, consider the stochastic nonlinear single-input systems which are established in the probability space (Ω, F, P) and described by the Itô stochastic differential equation as follows [32]: dx ¼ ½ðA þ ΔAðtÞÞx þ BðuF ðtÞ þ f ðtÞÞdt þ Dgðx; tÞdeðtÞ
ð1Þ
where x A ℝ , uF A ℝ and eðtÞ are, respectively, the states of system, the faulty control input, and l-dimensional Brownian motion. Here, A A ℝnnn ; B A ℝnn1 ; D A ℝnnm are known real constant matrices of appropriate dimensions. Without loss of generality, it is assumed that the pair ðA; BÞ is controllable. Moreover, matrix ΔA is for parameter uncertainty, f ðtÞ is an unknown time varying function and jf ðtÞj o r where r 40 is a known scalar and the unknown function gðx; tÞ A ℝml denotes nonlinear uncertainty, and it is assumed that the admissible uncertainties satisfy
3. Sliding surface construction In this work, sliding manifold is chosen as s¼
1 ρðtÞ R 1 BT PxðtÞ Ln ¼0 2 ρðtÞ þ R 1 BT PxðtÞ
ð5Þ
where ρðtÞ is a positive scalar to be regulated later so that ρðtÞ 4 jR 1 BT Px j which leads to sðtÞ that is real and feasible, R 40 is a scalar that is determined for some objectives and P 4 0 to be designed later is a symmetric matrix that meets ðA þ BZÞT P þPðA þ BZÞ PBR 1 BT P þ Q ¼ 0 where Q 40 is a symmetric matrix and Z is a row vector that is chosen so that ðA þBZÞ is Hurwitz and minðjλi ðA þ BZÞjÞ c i R 1 BT PBð1 μmax Þ. This sliding function is extracted from an optimization procedure as explained in the following: If uðtÞ ¼ ZxðtÞ þ ρðtÞtanhðsÞ is applied to the nominal system dxðtÞ ¼ ðAxðtÞ þ BuðtÞÞdt we have
n
ΔA ¼ EFðtÞH trace½g T ðx; tÞgðx; tÞ r NxðtÞ2
ð2Þ
where E, H and N are known as real constant matrices, and FðtÞ is an unknown matrix function satisfying F T ðtÞFðtÞ r I. In system (1) it is assumed that partial actuator degradation may occur, and this can be modeled as follows: uF ðtÞ ¼ ð1 μÞuðtÞ
ð3Þ
where μ satisfies 0 r μmin rμ r μmax o 1. The scalar μ usually termed as the effectiveness loss value of the actuator, which denotes the decrease in the effectiveness of the specified actuator. Remark 1. It can be seen that if μ ¼ 1 the control gain Bð1 μÞ will not satisfy the condition of full column rank that is a general assumption for the design of SMC. Hence, it is reasonable for SMC design to only consider the partial actuator degradation. Definition 1. The equilibrium solution, xt ¼ 0 of the stochastic differential Eq. (1) with uðtÞ ¼ 0 is said to be globally asymptotically stable (with probability one) if for any τ Z 0 and ε 40, limP supjxt τ;x j 4 ε ¼ 0; P lim jxt τ;x j ¼ 0 ¼ 1 x-0
τot
t-1
dxðtÞ ¼ ½Ax þBðZx þ ρðtÞtanhðsÞÞdt ¼ ½ðA þ BZÞx þ BρðtÞtanhðsÞdt If we define A þBZ ¼ A~
ð7Þ
ρðtÞtanhðsÞ ¼ vðtÞ Then dxðtÞ ¼ ðAxðtÞ þ BuðtÞÞdt can be rewritten as ~ dxðtÞ ¼ ðAxðtÞ þ BvðtÞÞdt
ð8Þ
For dynamical system (8), by choosing performance index Z 1 fxT ðtÞQ xðtÞ þ Rv2 ðtÞg dt J¼
ð9Þ
0
where Q 4 0 is a symmetric matrix and R 4 0 is a scalar. The virtual optimal controller is extracted as vðtÞ ¼ R 1 BT PxðtÞ
ð10Þ
T where P satisfies A~ P þ P A~ þPBR 1 BT P Q ¼ 0 Comparing (7) and (10)
ρðtÞtanhðsðtÞÞ ¼ R 1 BT PxðtÞ
ð11Þ
which yields sðtÞ ¼
1 ρðtÞ R 1 BT PxðtÞ Ln 2 ρðtÞ þ R 1 BT PxðtÞ
ð12Þ
This switching function has multiple benefits. By increasing R one may reduce control effort or distance of state trajectories from sliding surface. It follows from (1) and (12) that s¼
where xt τ;x denotes the solution at time t of a stochastic differential equation starting from the state x at time τ for τ o t.
1 Ln 2
Rt R 1 BT PfðA þ ΔAðτÞÞx þ BðuF þ f ðτÞÞgdτ 0 R 1 BT PDgðx; τÞdeðτÞ Rt 1 T B Pxðt 0 Þþ 0 R B P ðA þ ΔAðτÞÞx þ BðuF þ f ðτÞÞ dτ þ 0 R 1 BT PDgðx; τÞdeðτÞ
ρðtÞ R 1 BT Pxðt 0 Þ ρðtÞ þ R
1 T
Rt
R t0
ð13Þ
Lemma 1. The trivial solution of the stochastic differential equation dxðtÞ ¼ aðt; xÞdt þbðt; xÞdwðtÞ
with aðt; xÞ and bðt; xÞ sufficiently differentiable maps, is globally asymptotically stable (with probability one) if there exists a positive definite radially unbounded function V(t, x), and satisfies ∂V ∂V 1 ∂2 V þ aðt; xÞ þ trace bðt; xÞT 2 bðt; xÞ o 0 for x a 0 ð4Þ ℒV ¼ ∂t ∂x 2 ∂x
ð6Þ
where xðt 0 Þ is the initial state at t 0 Z 0 and the last term in numerator and denominator of logarithmic function is an Itô stochastic integral. Under the condition that BT PD ¼ 0, the sliding function (13) reduces to Rt ρðtÞ R 1 BT Pxðt 0 Þ 0 R 1 BT PfðA þ ΔAðτÞÞx þ BðuF þ f ðτÞÞgdτ 1 s ¼ Ln Rt 2 ρðtÞ þ R 1 BT Pxðt 0 Þ þ R 1 BT PfðA þ ΔAðτÞÞx þ BðuF þ f ðτÞÞgdτ 0
ð14Þ
Please cite this article as: Zahiripour SA, Jalali AA. A novel adaptive switching function on fault tolerable sliding mode control for uncertain stochastic systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.029i
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which shows sðx; tÞ is not explicitly involved in Brownian variable eðtÞ. That is, it is rational to take the time derivative of sðx; tÞ under the condition that BT PD ¼ 0 (will be ensured in Theorem 1).
4. Control scheme design
þ 2xT PBð1 μÞρðtÞsignðsÞ 2xT PBf ðx; tÞ þ traceðg T DT PDgÞ þ 2γ 1 μ~ μ_~ If the expression ð1 μÞð1 μ^ Þ may have
u ¼ ð1 μÞ ~ 1 Zx þ ρðtÞsignðsÞ ρ_ ðtÞ ¼ φψ
ð16Þ
where φ ¼ KB, ψ ¼ ϖ þr f ρðtÞ jKxðtÞjgð1 μmax Þ and ϖ 4 0 is an arbitrary scalar with K ¼ R 1 BT P and ϖ þr þ Kxð0Þj ð17Þ ρð0Þ 4 ð1 μmax Þ 0
if μ^ ¼ μmin and θ o0
θ
otherwise
¼ 1 þ μð1 ~ μ^ Þ
ð25Þ 1
is utilized, one
þ 2xðtÞT PBð1 μÞρðtÞsignðsðtÞÞ þ2xðtÞT PBf ðx; tÞ þ traceðgðx; tÞT DT PDgðx; tÞÞ þ 2xðtÞT PBμ~ ð1 μ^ Þ 1 ZxðtÞ þ 2γ 1 μ~ μ~_
ð15Þ
and the updating law for ρ and μ^ is given as
μ_^ ¼
1
ℒV 1 ðtÞ ¼ 2xðtÞT PAxðtÞ þ 2xðtÞT PΔAxðtÞ þ 2xðtÞT PBZxðtÞ
In this case, switching control law is synthesized as follows:
3
or μ^ ¼ μmax and θ Z0
ð18Þ
where: θ ¼ γxT PBð1 μ^ Þ 1 Zx, γ 4 0 is the arbitrary updating gain, μ^ is the estimated loss value of effectiveness for the actuator and P 4 0 will be designed in Theorem 1. Remark 2. It is noted that updating law (16) with ρð0Þ 4 ϖ þ r=ð1 μmax Þ þ jKxð0Þj gives o 1 n ρðtÞ ¼ ðKBÞ 1 ρ_ ðtÞ þ ϖ þ r þ KxðtÞj 1 μmax
By considering the definition of the sliding variable s in Section 3 2xðtÞT PBð1 μÞρðtÞsignðsðtÞÞ þ 2xðtÞT PBf ðx; tÞ ¼ 2ρðtÞ2 Rð1 μÞtanhðsðtÞÞsignðsðtÞÞ 2ρðtÞR tanhðsðtÞÞf ðx; tÞ r 2ρðtÞ2 Rð1 μmax Þj tanhðsðtÞÞj þ 2ρðtÞRj tanhðsðtÞÞj r
ð27Þ
For ρðtÞ given in (16) and Remark 2 one may have ρðtÞ 4
r 1 μmax
ð28Þ
therefore 2ρðtÞ2 Rð1 μmax Þj tanhðsðtÞÞj þ 2ρðtÞRj tanhðsðtÞÞj r r0
ð29Þ
Moreover, if μ^ ¼ μmin and θ o 0 or μ^ ¼ μmax and θ Z 0 from (18), μ_^ ¼ 0 and μθ ~ Z 0 which implies that the following expression (30) is satisfied. On the other hand, if, μ_^ ¼ θ the expression (30) is also satisfied. Thus, it can be concluded that the following expression is ensured by (18) 2xT PBμð1 ~ μ^ Þ 1 Zx þ 2γ 1 μ~ μ_~ r 0
Above equation and the condition minðjλi ðA þ BZÞjÞ c i KBð1 μmax Þ , leads to ρðtÞ that reduced slower than xðtÞ which means ρðtÞ 4 ϖ þ r=ð1 μmax Þ þ jKxðtÞj that yields ρ_ ðtÞ r 0 and ρðtÞ 4jKxðtÞj (as will be shown in simulation results, Fig. 4). Substituting (15) into (1) one obtains:
2xT PΔAx r δ 1 xT PEET Px þδxT H T Hx
dx ¼ ½ðA þ ΔAÞxðtÞ þ Bfð1 μÞ½ð1 μ^ Þ 1 ZxðtÞ þ ρðtÞsignðsðtÞÞ þf ðx; tÞgdt
ℒV 1 ðtÞ r xT PðA þ BZÞx þ xT ðA þ BZÞT Px þ δ 1 xT PEET Px
þ Dgðx; tÞdeðtÞ
ð26Þ
ð30Þ
Note that from (2), for δ 4 0
Thus, we have by (21) þ δxT H T Hx þ xT εN T Nx ¼ xT Σ x
ð19Þ where
Theorem 1. Consider the uncertain stochastic system (1). The SMC law is synthesized as (15) with adaptive laws (16) and (18). If there exist a matrix, P 4 0 and scalars ε 4 0 and δ 40 satisfying the following linear matrix inequalities (LMIs) with equality constraint Σ1 PE Ω¼ T o0 ð20Þ E P δI DT PD εI o 0
ð21Þ
BT PD ¼ 0
ð22Þ
with Σ 1 ¼ PðA þ BZÞ þ ðA þ BZÞT P þ δH T H þ εN T N
ð23Þ
Then overall closed-loop stochastic system is globally asymptotically stable (with probability one) despite the partial actuator degradation.
V 1 ¼ x Px þγ
1 2
μ~
ð24Þ
where μ~ ¼ μ^ μ is the estimate error of μ. Then, by Itô's formula, the infinitesimal generator ℒV 1 associated with system (19) is obtained as ℒV 1 ðtÞ ¼ 2xT PAx þ2xT PΔAx þ 2xT PBð1 μÞð1 μ^ Þ 1 Zx
hence if Σ o0, we have ℒV 1 ðtÞ o 0
xðtÞ a 0
ð31Þ
By Lemma 1, it can be seen from (31) that the closed-loop system (19) is globally asymptotically stable with probability one, despite faulty actuator. On the other hand, Σ o 0 is equivalent to condition (20). In the sequel, a simple algorithm is given to solve LMIs with equality constraint as (20)–(22). Consider the following matrix inequality for β 4 0 ðBT PDÞT ðBT PDÞ r βI By Schur's complement, (32) is equivalent to " # βI DT PB r0 I BT PD
ð32Þ
ð33Þ
Now, define the following minimization problem Min β subject to ð20Þ; ð21Þ and ð33Þ
Proof. Choose the Lyapunov function as T
Σ ¼ PðA þBZÞ þ ðA þ BZÞT P þ δ 1 PEET P þ δH T H þ εN T N
ð34Þ
which is a minimization problem involving linear objective and LMI constraints and can be solved by using LMI toolbox in Matlab. It can be seen that if the global infimum of problem (34) equals zero, the corresponding solutions P, ε and δ will satisfy (20)–(22). In the aforementioned theorem, it is proven that sliding mode controller (15) can ensure the closed-loop system (1) to be asymptotically stable in probability one. In the sequel, it is further proven that controller (15) can guarantee that the state trajectories
Please cite this article as: Zahiripour SA, Jalali AA. A novel adaptive switching function on fault tolerable sliding mode control for uncertain stochastic systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.029i
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are driven onto the specified sliding surface sðtÞ ¼ 0 in (5), that is, to ensure the reachability of the specified sliding surface sðtÞ ¼ 0. Theorem 2. Consider the uncertain stochastic system (1). The switching function is chosen as s ¼ ð1=2ÞLnðρðtÞ R 1 BT PxðtÞ= ρðtÞ þR 1 BT PxðtÞÞ where matrix P 4 0 is the solution of matrix inequalities and constrain equality (20)–(22). The reachability of sliding surface is guaranteed under the sliding-mode controller (15). Proof. Choose the Lyapunov function as 1 V 2 ¼ ðKBÞ 1 sðtÞ2 2
ð35Þ
with K ¼ R 1 BT P. By means of the following expression s_ ðtÞ ¼ aðtÞð_ρðtÞKxðtÞ ρðtÞK x_ ðtÞÞ
ð36Þ
by (16) and Remark 2 a 1 ðtÞ ¼ ρðtÞ2 ðKxðtÞÞ2 4 0 ℒV 2 ¼ asðtÞðKBÞ 1 ½_ρðtÞKxðtÞ ρðtÞKððA þ ΔAðtÞÞxðtÞ þ Bfð1 μÞð1 μ^ Þ 1 ZxðtÞ þ ð1 μÞρðtÞsignðsðtÞÞ þ f ðx; tÞgÞ
ð37Þ
¼ asðtÞ½ðKBÞ 1 ρ_ ðtÞKxðtÞ ρðtÞðKBÞ 1 KAxðtÞ ρðtÞðKBÞ 1 KΔAðtÞxðtÞ ρðtÞð1 μÞð1 μ^ Þ 1 ZxðtÞ ρðtÞ2 ð1 μÞsignðsðtÞÞ ρðtÞf ðx; tÞ
ð38Þ
ra½ ðKBÞ 1 jsðtÞjj_ρðtÞjjKxðtÞj þ ρðtÞðKBÞ 1 jsðtÞj AK xðtÞ þρðKBÞ 1 ð1 μmin Þ xðtÞ ð1 μmax Þρ2 ðtÞjsðtÞj jsðtÞj K EH xðtÞ þ ρðtÞsðtÞZ ð1 μmax Þ þ ρðtÞjsðtÞjr
ð39Þ
2
0:2
3
6 7 E ¼ 4 0:2 5 0:3
FðtÞ ¼ sin ðtÞ
H ¼ 0:1
0:2
N ¼ ½0:3 0:3 0:3
It is assumed that actuator faults may happen, and the faulty model is given as (3) with μmin ¼ 0:2, μmax ¼ 0:8 and r ¼ 1 Z jf ðtÞj, ϖ ¼ 0:1. By solving (34) we have 2 3 11:5545 0:4699 1:7913 6 0:4699 2:5581 2:9843 7 P¼ 4 5 ; δ ¼ 3:2664 ; 1:7913 ε ¼ 2:7193
2:9843
4:6432
and β 5 10 4 which is sufficiently small for the proposed method in practical application. The design parameter R is selected 0.5 at first and then 2.5 for discussing about its effect. We select vector Z so that ðA þ BZÞ is Hurwitz and minðjλi ðA þ BZÞjÞ c i KBð1 μmax Þ as follows:
Z ¼ 7:1096 0:8613 0:1947 It is expected from the definition of vðtÞ in (7) and performance index (9) that the increasing R leads to decreased in ρðtÞj tanhðsÞj. This effect may decrease ρðtÞ which helps to control signal to be less aggressive or may decrease sðtÞ which helps state trajectories to remain near the sliding surface which possibly results in higher quality and speed of system responses. In the following, we evaluate the above analytical results by simulation in two cases (a) and (b). a. For R ¼ 0:5 The desired sliding function (12) and the SMC law (15) can be obtained as follows: s¼
1 ρðtÞ ½ 1:5325 1:3472 1:1132xðtÞ Ln 2 ρðtÞ þ ½ 1:5325 1:3472 1:1132xðtÞ
Define the following domain
with
Φ ¼ fxðtÞ j
ρ_ ¼ 0:9857ð1:1 0:2fρ jKxjÞ
ηjjxðtÞjj o ϖg
0:2
and
where η ¼ ðKBÞ 1 jjAjj jjKjj þ ðKBÞ 1 jjK jj jjEjj jj Hjj þ jjZjj
ð1 μmin Þ ð1 μmax Þ
ð40Þ
uðtÞ ¼ ð1 μ^ Þ 1 ½ 7:1096
0:1947 x þ ρðtÞsignðsÞ
with
and ϖ is an arbitrary constant. In this domain
μ_^ ¼
ℒV 2 r aρðtÞjsðtÞj½ðKBÞ 1 j_ρðtÞj þ ϖ ρðtÞð1 μmax Þ þ r
0:8613
ð41Þ
8 0 > > > >
> > > : 0:5566
or μ^ ¼ 0:8 and θ Z 0
0:8613 0:1947x
otherwise
From the updating law (16) for ρ and Remark (2) ℒV 2 r aρjsðtÞj jKxðtÞjð1 μmax Þ r 0
ð42Þ
By utilizing the result in Theorem 1, it can be obtained that the state trajectories will enter the domain Φ in finite time. That is, the relation ηx o ϖ, can be ensured in finite time. Hence, it follows from (42) that the reachability of the sliding surface can be ensured in finite time. This concludes the proof.
b. For R ¼ 2:5 s¼
1 ρðtÞ ½ 0:3065 0:2694 0:226xðtÞ Ln 2 ρðtÞ þ ½ 0:3065 0:2694 0:226xðtÞ
with ρ_ ¼ 0:1971ð1:1 0:2fρ jKxjÞ and
5. Analysis of designing parameter (R) effect and numerical simulation Consider the uncertain stochastic system (1) with 2 3 2 3 0:2 0:2 0 0:2 6 7 6 7 0:3 5 B ¼ 4 1 5 f ðtÞ ¼ sin ðx1 ðtÞÞ A ¼ 4 0:2 0:6 2
0:2 0:04
0:4 3
6 7 D ¼ 4 0:03 5 0:04
0:2
g ¼ 0:1x2
0:6
uðtÞ ¼ ð1 μ^ Þ 1 ½ 7:1096
0:8613
0:1947 x þ ρðtÞsignðsÞ
with
μ_^ ¼
8 0 > > > >
> > > : 0:5566
0:8613
0:1947 x
otherwise
For practical purposes, it is assumed that the measurement of all states is polluted to a Gaussian white noise with zero mean value
Please cite this article as: Zahiripour SA, Jalali AA. A novel adaptive switching function on fault tolerable sliding mode control for uncertain stochastic systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.029i
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0 case(b) case(a)
0.3
-0.05
derivative of ro
0.25
u(t)
0.2 0.15 0.1 0.05
-0.1 -0.15 -0.2 -0.25 case(b) case(a)
-0.3
0 -0.05
5
0
5
10
15
20
-0.35
25
0
5
10
t (s)
0.1
25
20
25
0.1
case(b) case(a)
0.08
0.08
0.06
0.06
0.04
0.04
x1 (t)
s(t)
20
Fig. 4. Derivative of switching gain o 0.
Fig. 1. Control signals.
0.02
0.02
0
0
-0.02
-0.02
-0.04
15
t (s)
0
5
10
15
20
-0.04
25
0
5
10
15
t (s)
t (s)
Fig. 5. trajectory of 1(t).
Fig. 2. Curve of s(t).
4
0.3
case(b) case(a)
case(b) case(a)
0.25 0.2
ro (t)
x2 (t)
3.5
3
0.15 0.1 0.05 0
2.5
-0.05
0
5
10
15
20
25
0
10
15
20
25
t (s)
t (s)
Fig. 6. Trajectory of 2(t).
Fig. 3. Adaptive trajectory of switching gain.
and the variance s2 ¼ 10 6 . Furthermore we have used tanhðsÞ as a continuous function similar to signðsÞ, instead of signðsÞ for reducing chattering. The simulation results are shown in Figs. 1–8. Fig. 1 shows that in the case (b) controller reacts faster and it is less aggressive. Fig. 2 indicates that in the case (b) the state trajectories are closer to sliding surface which has some benefits as discussed previously. ρ_ ðtÞ Should have a negative value as mentioned in Remark 2. Figs. 3 and 4 display switching gain and derivative of it, respectively. Figs. 5–7 illustrate state responses of closed-loop system in two cases. As expected, increasing R made the responses
5
faster and with better quality. Finally, Fig. 8 shows that how adaptive trajectory of effectiveness loss value ðμ^ Þ changes.
6. Conclusion A robust SMC for single-input uncertain stochastic systems via a novel adaptive sliding surface has been studied in the presence of faulty actuator. The switching function has been extracted from a systematic strategy which minimizes a specific performance index and has multiple benefits. In the proposed sliding manifold there is a parameter which can be regulated for achieving the
Please cite this article as: Zahiripour SA, Jalali AA. A novel adaptive switching function on fault tolerable sliding mode control for uncertain stochastic systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.029i
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6 0.15
case(b) case(a)
0.1
x3 (t)
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-0.05
-0.1
0
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t (s) Fig. 7. Trjectory of 3(t). 0.65 case(b) case(a)
mu-hat (t)
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t (s) Fig. 8. Adaptive trajectory of effectiveness loss value.
special objectives of the designer that may lead to reduction of control effort or improving the quality of system responses. It is seen that the proposed SMC law ensures the reachability of sliding surface and global asymptotic stability (with probability one) of the closed-loop system. References [1] Huang L, Mao X. SMC design for robust H1 control of uncertain stochastic delay systems. Automatica 2010;46:405–12. [2] Lin Z, Xia Y, Shi P, Wu H. Robust sliding mode control for uncertain linear discrete systems independent of time-delay. Int J Innov Comput, Inf Control 2011;7:869–81. [3] Wu L, Ho DWC. Sliding mode control of singular stochastic hybrid systems. Automatica 2010;46:779–83. [4] Wu S, Sun Z, Deng H. Robust sliding mode controller design for globally fast attitude tracking of target spacecraft. Int J Innov Comput, Inf Control 2011;7:2087–98. [5] Xia Y, Yang H, Fu M, Shi P. Sliding mode control for linear systems with timevarying input and state delays. Circuits Syst Signal Process 2011;30:629–41.
[6] Zhang J, Shi P, Xia Y. Robust adaptive sliding mode control for fuzzy systems with mismatched uncertainties. IEEE Trans Fuzzy Syst 2010;18:700–11. [7] Zong Q, Zeng F, Liu W, Ji Y, Tao Y. Sliding mode observer-based fault detection of distributed networked control systems with time delay. Circuits Syst Signal Process 2012;31:203–22. [8] Li Yongming, Tong Shaocheng, Li Tieshan, Jing Xingjian. Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach. Fuzzy Sets and Systems 2014;235:1–24. [9] Tong Shaocheng, Li Han-Xiong. Fuzzy adaptive sliding-mode control for MIMO nonlinear systems. IEEE Trans Fuzzy Syst 2003;11:354–60. [10] Li Han-Xiong, Tong Shaocheng. A hybrid adaptive fuzzy control for a class of nonlinear MIMO systems. IEEE Trans Fuzzy Syst 2003;11:24–34. [11] Shaocheng Tong Yue Li, Li Yongming, Liu Yanjun. Observer-based adaptive fuzzy backstepping control for a class of stochastic nonlinear strict-feedback systems, IEEE Transactions on Systems. Man Cybern Part B 2011;41:1693–704. [12] Li-Ying Hao Guang-Hong Yang. Robust fault tolerant control based on sliding mode method for uncertain linear systems with quantization. ISA Trans 2013;52:600–10. [13] Singla Mithun, Shieh Leang-San, Song Gangbing, Xie Linbo, Zhang Yongpeng. sliding mode controller design using scalar sign function. ISA Trans 2014;53:267–79. [14] Madonski R, Herman P. On the usefulness of higher-order disturbance observers in real control scenarios based on perturbation estimation and mitigation. In: Proceedings of the international workshop on robot motion and control; 2013. p. 252–7. [15] Madonski R, Kordasz M, Sauer P. Application of a disturbance-rejection controller for robotic-enhanced limb rehabilitation trainings. ISA Trans 2013. [16] Han J. From PID to active disturbance rejection control. IEEE Trans Ind Electron 2009;56:900–6. [17] Jiang J. Design of reconfigurable control systems using eigenstructure assignment. Int J Control 1994;59:395–410. [18] El-Farra NH, Giridhar A. Fault detection and fault-tolerant control of particulate processes using population balance models. In: Proceedings of 2007 American control conference. IEEE conference publications; 2007. p. 4148–53. [19] Fekih A. Effective fault tolerant control design for nonlinear systems: application to a class of motor control system. IET Control Theory Appl 2008;2:762–72. [20] Veillette RJ. Reliable linear-quadratic state-feedback control. Automatica 1995;31:137–43. [21] Zhao Q, Jiang J. Reliable state feedback control system design against actuator failures. Automatica 1998;34:1267–72. [22] Yang G-H, Wang JL, Soh YC. Reliable H1 controller design for linear systems. Automatica 2001;37:717–25. [23] Harkegard O, Glad ST. Resolving actuator redundancy—optimal control vs. control allocation. Automatica 2005;41:137–44. [24] Tao G, Chen S, Joshi SM. An adaptive actuator failure compensation controller using output feedback. IEEE Trans Autom Control 2002;47:506–11. [25] Ye D, Yang G-H. Adaptive fault-tolerant tracking control against actuator faults with application to flight control. IEEE Trans Control Syst Technol 2006;14:1088–96. [26] Jiang B, Gao Z, Shi P, Xu Y. Adaptive fault-tolerant tracking control of near space vehicle using Takagi-Sugeno fuzzy models. IEEE Trans Fuzzy Syst 2010;18:1000–7. [27] Niu Y, Ho DWC, Lam J. Robust integral sliding mode control for uncertain stochastic systems with time-varying delay. Automatica 2005;41:873–80. [28] Ma S, Boukas E-K. A singular system approach to robust sliding mode control for uncertain Markov jump systems. Automatica 2009;45:2707–13. [29] Huang L, Mao X. SMC design for robust H1 control of uncertain stochastic delay systems. Automatica 2010;46:405–12. [30] Niu1 Yugang, Liu Yonghui, Jia Tinggang. Reliable control of stochastic systems via sliding mode technique. Optim Control Appl Methods 2012;34:712–27. [31] Chen Bei, Niu Yugang, Zou Yuanyuan, Jia Tinggang. Reliable sliding-mode control for Markovian jumping systems subject to partial actuator degradation. Circuits Syst Signal Process 2013;32:601–14. [32] Chen Bei, Niu Yugang, Zou Yuanyuan. Adaptive sliding mode control for stochastic Markovian jumping systems with actuator degradation. Automatica 2013;49:1748–54.
Please cite this article as: Zahiripour SA, Jalali AA. A novel adaptive switching function on fault tolerable sliding mode control for uncertain stochastic systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.029i
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A novel adaptive switching function on fault tolerable sliding mode control for uncertain stochastic systems Seyed Ali Zahiripour n, Ali Akbar Jalali Department of Electrical Engineering, Iran University of Science and Technology, Tehran, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 18 March 2014 Received in revised form 15 May 2014 Accepted 22 May 2014 This paper was recommended for publication by Dr. A.B. Rad
A novel switching function based on an optimization strategy for the sliding mode control (SMC) method has been provided for uncertain stochastic systems subject to actuator degradation such that the closedloop system is globally asymptotically stable with probability one. In the previous researches the focus on sliding surface has been on proportional or proportionalintegral function of states. In this research, from a degree of freedom that depends on designer choice is used to meet certain objectives. In the design of the switching function, there is a parameter which the designer can regulate for specified objectives. A sliding-mode controller is synthesized to ensure the reachability of the specified switching surface, despite actuator degradation and uncertainties. Finally, the simulation results demonstrate the effectiveness of the proposed method. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Uncertain stochastic system Adaptive sliding-surface Optimization strategy Actuator degradation
1. Introduction This paper will investigate the design of reliable SMC for uncertain stochastic systems with possible occurrence of actuator faults. Sliding mode control is an effective robust scheme for incomplete modeled or uncertain systems, whose key feature relies on its complete insensitiveness to both parameter variations and external disturbances. Hence, in the past decades, a lot of important developments on SMC have been obtained [1–13]. As is well-known, the actuator degradation in actual physical systems is usually inevitable, and often yields performance degradation or even instability. Therefore, how to maintain an acceptable stability/performance for the closed-loop systems against actuator or sensor failures has been a long-standing and active research topic. In the literature, the design of reliable control systems can be broadly classified as the active approach and the passive approach. In the first type, faults are detected and identified by a fault detection and diagnosis (FDD) mechanism, and controllers are re-configured, such as those in [14–19]. In contrast, the passive methodology designs a reliable controller with a fixed structure by means of actuator redundancy. By taking possible actuator faults into account during the design of the controller, passive reliable systems can attain the stability and performance requirement even in the presence of
n
Corresponding author. Tel.: þ 98 77240487. E-mail addresses: [email protected], [email protected] (S.A. Zahiripour).
actuator faults. Moreover, various passive reliable techniques have also been developed, for example, linear-quadratic state-feedback control [20], pre-compensator [21], H1 disturbance attenuation [22], control allocation [23], and adaptive control [24–26]. It is known that a stochastic system has extensive applications in practice, and the SMC of stochastic systems receives much attention [27–29]. However, little work has been carried out on the reliable problem of SMC for uncertain stochastic systems subject to actuator faults. A common feature in the last corresponding works is related to choosing sliding surface which is mostly chosen as a proportional or proportional-integral function of states [30–32]. In the mentioned sliding surfaces we cannot discuss about the effect of changing a parameter of switching function on performance of the system, strongly. Furthermore these sliding manifolds have been proposed without expressing a reason and defining a specific goal. In this work, from a degree of freedom that depends on designer choice is used to meet certain objectives which were neglected in other recent researches. In other words, sliding surface is extracted based on a systematic strategy. In this method the problem of choosing sliding surface is converted to a virtual optimal control problem such that the constructed sliding surface minimizes a quadratic performance index. In this methodology there is a possibility of regulating the distance of state trajectories from sliding surface and the amount of energy consumption that has wealth importance in control strategies. Eventually, an adaptive sliding mode controller is synthesized, which can ensure the reachability of the aforementioned sliding surface. Additionally, sufficient conditions are derived via the stochastic Lyapunov method and linear matrix inequalities such
http://dx.doi.org/10.1016/j.isatra.2014.05.029 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Zahiripour SA, Jalali AA. A novel adaptive switching function on fault tolerable sliding mode control for uncertain stochastic systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.029i
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2
that the closed-loop system is ensured to be asymptotically stable in probability one, not only when control component is in good working condition but also in the presence of actuator faults. Notations j:j, : denote, respectively absolute value and the Euclidean norm of a vector or its induced matrix norm. For a real matrix, M40 means that M is symmetric and positive definite, and I is used to represent an identity matrix of appropriate dimensions. λi (.) shows the i-th Eigen value of a matrix. (Ω, F, P) is a probability space with Ω the sample space, and F the s-algebra of subsets of the sample space, and P is the probability measure. Matrices, if not explicitly stated, are assumed to have compatible dimensions.
2. System description and preliminaries In this work, consider the stochastic nonlinear single-input systems which are established in the probability space (Ω, F, P) and described by the Itô stochastic differential equation as follows [32]: dx ¼ ½ðA þ ΔAðtÞÞx þ BðuF ðtÞ þ f ðtÞÞdt þ Dgðx; tÞdeðtÞ
ð1Þ
where x A ℝ , uF A ℝ and eðtÞ are, respectively, the states of system, the faulty control input, and l-dimensional Brownian motion. Here, A A ℝnnn ; B A ℝnn1 ; D A ℝnnm are known real constant matrices of appropriate dimensions. Without loss of generality, it is assumed that the pair ðA; BÞ is controllable. Moreover, matrix ΔA is for parameter uncertainty, f ðtÞ is an unknown time varying function and jf ðtÞj o r where r 40 is a known scalar and the unknown function gðx; tÞ A ℝml denotes nonlinear uncertainty, and it is assumed that the admissible uncertainties satisfy
3. Sliding surface construction In this work, sliding manifold is chosen as s¼
1 ρðtÞ R 1 BT PxðtÞ Ln ¼0 2 ρðtÞ þ R 1 BT PxðtÞ
ð5Þ
where ρðtÞ is a positive scalar to be regulated later so that ρðtÞ 4 jR 1 BT Px j which leads to sðtÞ that is real and feasible, R 40 is a scalar that is determined for some objectives and P 4 0 to be designed later is a symmetric matrix that meets ðA þ BZÞT P þPðA þ BZÞ PBR 1 BT P þ Q ¼ 0 where Q 40 is a symmetric matrix and Z is a row vector that is chosen so that ðA þBZÞ is Hurwitz and minðjλi ðA þ BZÞjÞ c i R 1 BT PBð1 μmax Þ. This sliding function is extracted from an optimization procedure as explained in the following: If uðtÞ ¼ ZxðtÞ þ ρðtÞtanhðsÞ is applied to the nominal system dxðtÞ ¼ ðAxðtÞ þ BuðtÞÞdt we have
n
ΔA ¼ EFðtÞH trace½g T ðx; tÞgðx; tÞ r NxðtÞ2
ð2Þ
where E, H and N are known as real constant matrices, and FðtÞ is an unknown matrix function satisfying F T ðtÞFðtÞ r I. In system (1) it is assumed that partial actuator degradation may occur, and this can be modeled as follows: uF ðtÞ ¼ ð1 μÞuðtÞ
ð3Þ
where μ satisfies 0 r μmin rμ r μmax o 1. The scalar μ usually termed as the effectiveness loss value of the actuator, which denotes the decrease in the effectiveness of the specified actuator. Remark 1. It can be seen that if μ ¼ 1 the control gain Bð1 μÞ will not satisfy the condition of full column rank that is a general assumption for the design of SMC. Hence, it is reasonable for SMC design to only consider the partial actuator degradation. Definition 1. The equilibrium solution, xt ¼ 0 of the stochastic differential Eq. (1) with uðtÞ ¼ 0 is said to be globally asymptotically stable (with probability one) if for any τ Z 0 and ε 40, limP supjxt τ;x j 4 ε ¼ 0; P lim jxt τ;x j ¼ 0 ¼ 1 x-0
τot
t-1
dxðtÞ ¼ ½Ax þBðZx þ ρðtÞtanhðsÞÞdt ¼ ½ðA þ BZÞx þ BρðtÞtanhðsÞdt If we define A þBZ ¼ A~
ð7Þ
ρðtÞtanhðsÞ ¼ vðtÞ Then dxðtÞ ¼ ðAxðtÞ þ BuðtÞÞdt can be rewritten as ~ dxðtÞ ¼ ðAxðtÞ þ BvðtÞÞdt
ð8Þ
For dynamical system (8), by choosing performance index Z 1 fxT ðtÞQ xðtÞ þ Rv2 ðtÞg dt J¼
ð9Þ
0
where Q 4 0 is a symmetric matrix and R 4 0 is a scalar. The virtual optimal controller is extracted as vðtÞ ¼ R 1 BT PxðtÞ
ð10Þ
T where P satisfies A~ P þ P A~ þPBR 1 BT P Q ¼ 0 Comparing (7) and (10)
ρðtÞtanhðsðtÞÞ ¼ R 1 BT PxðtÞ
ð11Þ
which yields sðtÞ ¼
1 ρðtÞ R 1 BT PxðtÞ Ln 2 ρðtÞ þ R 1 BT PxðtÞ
ð12Þ
This switching function has multiple benefits. By increasing R one may reduce control effort or distance of state trajectories from sliding surface. It follows from (1) and (12) that s¼
where xt τ;x denotes the solution at time t of a stochastic differential equation starting from the state x at time τ for τ o t.
1 Ln 2
Rt R 1 BT PfðA þ ΔAðτÞÞx þ BðuF þ f ðτÞÞgdτ 0 R 1 BT PDgðx; τÞdeðτÞ Rt 1 T B Pxðt 0 Þþ 0 R B P ðA þ ΔAðτÞÞx þ BðuF þ f ðτÞÞ dτ þ 0 R 1 BT PDgðx; τÞdeðτÞ
ρðtÞ R 1 BT Pxðt 0 Þ ρðtÞ þ R
1 T
Rt
R t0
ð13Þ
Lemma 1. The trivial solution of the stochastic differential equation dxðtÞ ¼ aðt; xÞdt þbðt; xÞdwðtÞ
with aðt; xÞ and bðt; xÞ sufficiently differentiable maps, is globally asymptotically stable (with probability one) if there exists a positive definite radially unbounded function V(t, x), and satisfies ∂V ∂V 1 ∂2 V þ aðt; xÞ þ trace bðt; xÞT 2 bðt; xÞ o 0 for x a 0 ð4Þ ℒV ¼ ∂t ∂x 2 ∂x
ð6Þ
where xðt 0 Þ is the initial state at t 0 Z 0 and the last term in numerator and denominator of logarithmic function is an Itô stochastic integral. Under the condition that BT PD ¼ 0, the sliding function (13) reduces to Rt ρðtÞ R 1 BT Pxðt 0 Þ 0 R 1 BT PfðA þ ΔAðτÞÞx þ BðuF þ f ðτÞÞgdτ 1 s ¼ Ln Rt 2 ρðtÞ þ R 1 BT Pxðt 0 Þ þ R 1 BT PfðA þ ΔAðτÞÞx þ BðuF þ f ðτÞÞgdτ 0
ð14Þ
Please cite this article as: Zahiripour SA, Jalali AA. A novel adaptive switching function on fault tolerable sliding mode control for uncertain stochastic systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.029i
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which shows sðx; tÞ is not explicitly involved in Brownian variable eðtÞ. That is, it is rational to take the time derivative of sðx; tÞ under the condition that BT PD ¼ 0 (will be ensured in Theorem 1).
4. Control scheme design
þ 2xT PBð1 μÞρðtÞsignðsÞ 2xT PBf ðx; tÞ þ traceðg T DT PDgÞ þ 2γ 1 μ~ μ_~ If the expression ð1 μÞð1 μ^ Þ may have
u ¼ ð1 μÞ ~ 1 Zx þ ρðtÞsignðsÞ ρ_ ðtÞ ¼ φψ
ð16Þ
where φ ¼ KB, ψ ¼ ϖ þr f ρðtÞ jKxðtÞjgð1 μmax Þ and ϖ 4 0 is an arbitrary scalar with K ¼ R 1 BT P and ϖ þr þ Kxð0Þj ð17Þ ρð0Þ 4 ð1 μmax Þ 0
if μ^ ¼ μmin and θ o0
θ
otherwise
¼ 1 þ μð1 ~ μ^ Þ
ð25Þ 1
is utilized, one
þ 2xðtÞT PBð1 μÞρðtÞsignðsðtÞÞ þ2xðtÞT PBf ðx; tÞ þ traceðgðx; tÞT DT PDgðx; tÞÞ þ 2xðtÞT PBμ~ ð1 μ^ Þ 1 ZxðtÞ þ 2γ 1 μ~ μ~_
ð15Þ
and the updating law for ρ and μ^ is given as
μ_^ ¼
1
ℒV 1 ðtÞ ¼ 2xðtÞT PAxðtÞ þ 2xðtÞT PΔAxðtÞ þ 2xðtÞT PBZxðtÞ
In this case, switching control law is synthesized as follows:
3
or μ^ ¼ μmax and θ Z0
ð18Þ
where: θ ¼ γxT PBð1 μ^ Þ 1 Zx, γ 4 0 is the arbitrary updating gain, μ^ is the estimated loss value of effectiveness for the actuator and P 4 0 will be designed in Theorem 1. Remark 2. It is noted that updating law (16) with ρð0Þ 4 ϖ þ r=ð1 μmax Þ þ jKxð0Þj gives o 1 n ρðtÞ ¼ ðKBÞ 1 ρ_ ðtÞ þ ϖ þ r þ KxðtÞj 1 μmax
By considering the definition of the sliding variable s in Section 3 2xðtÞT PBð1 μÞρðtÞsignðsðtÞÞ þ 2xðtÞT PBf ðx; tÞ ¼ 2ρðtÞ2 Rð1 μÞtanhðsðtÞÞsignðsðtÞÞ 2ρðtÞR tanhðsðtÞÞf ðx; tÞ r 2ρðtÞ2 Rð1 μmax Þj tanhðsðtÞÞj þ 2ρðtÞRj tanhðsðtÞÞj r
ð27Þ
For ρðtÞ given in (16) and Remark 2 one may have ρðtÞ 4
r 1 μmax
ð28Þ
therefore 2ρðtÞ2 Rð1 μmax Þj tanhðsðtÞÞj þ 2ρðtÞRj tanhðsðtÞÞj r r0
ð29Þ
Moreover, if μ^ ¼ μmin and θ o 0 or μ^ ¼ μmax and θ Z 0 from (18), μ_^ ¼ 0 and μθ ~ Z 0 which implies that the following expression (30) is satisfied. On the other hand, if, μ_^ ¼ θ the expression (30) is also satisfied. Thus, it can be concluded that the following expression is ensured by (18) 2xT PBμð1 ~ μ^ Þ 1 Zx þ 2γ 1 μ~ μ_~ r 0
Above equation and the condition minðjλi ðA þ BZÞjÞ c i KBð1 μmax Þ , leads to ρðtÞ that reduced slower than xðtÞ which means ρðtÞ 4 ϖ þ r=ð1 μmax Þ þ jKxðtÞj that yields ρ_ ðtÞ r 0 and ρðtÞ 4jKxðtÞj (as will be shown in simulation results, Fig. 4). Substituting (15) into (1) one obtains:
2xT PΔAx r δ 1 xT PEET Px þδxT H T Hx
dx ¼ ½ðA þ ΔAÞxðtÞ þ Bfð1 μÞ½ð1 μ^ Þ 1 ZxðtÞ þ ρðtÞsignðsðtÞÞ þf ðx; tÞgdt
ℒV 1 ðtÞ r xT PðA þ BZÞx þ xT ðA þ BZÞT Px þ δ 1 xT PEET Px
þ Dgðx; tÞdeðtÞ
ð26Þ
ð30Þ
Note that from (2), for δ 4 0
Thus, we have by (21) þ δxT H T Hx þ xT εN T Nx ¼ xT Σ x
ð19Þ where
Theorem 1. Consider the uncertain stochastic system (1). The SMC law is synthesized as (15) with adaptive laws (16) and (18). If there exist a matrix, P 4 0 and scalars ε 4 0 and δ 40 satisfying the following linear matrix inequalities (LMIs) with equality constraint Σ1 PE Ω¼ T o0 ð20Þ E P δI DT PD εI o 0
ð21Þ
BT PD ¼ 0
ð22Þ
with Σ 1 ¼ PðA þ BZÞ þ ðA þ BZÞT P þ δH T H þ εN T N
ð23Þ
Then overall closed-loop stochastic system is globally asymptotically stable (with probability one) despite the partial actuator degradation.
V 1 ¼ x Px þγ
1 2
μ~
ð24Þ
where μ~ ¼ μ^ μ is the estimate error of μ. Then, by Itô's formula, the infinitesimal generator ℒV 1 associated with system (19) is obtained as ℒV 1 ðtÞ ¼ 2xT PAx þ2xT PΔAx þ 2xT PBð1 μÞð1 μ^ Þ 1 Zx
hence if Σ o0, we have ℒV 1 ðtÞ o 0
xðtÞ a 0
ð31Þ
By Lemma 1, it can be seen from (31) that the closed-loop system (19) is globally asymptotically stable with probability one, despite faulty actuator. On the other hand, Σ o 0 is equivalent to condition (20). In the sequel, a simple algorithm is given to solve LMIs with equality constraint as (20)–(22). Consider the following matrix inequality for β 4 0 ðBT PDÞT ðBT PDÞ r βI By Schur's complement, (32) is equivalent to " # βI DT PB r0 I BT PD
ð32Þ
ð33Þ
Now, define the following minimization problem Min β subject to ð20Þ; ð21Þ and ð33Þ
Proof. Choose the Lyapunov function as T
Σ ¼ PðA þBZÞ þ ðA þ BZÞT P þ δ 1 PEET P þ δH T H þ εN T N
ð34Þ
which is a minimization problem involving linear objective and LMI constraints and can be solved by using LMI toolbox in Matlab. It can be seen that if the global infimum of problem (34) equals zero, the corresponding solutions P, ε and δ will satisfy (20)–(22). In the aforementioned theorem, it is proven that sliding mode controller (15) can ensure the closed-loop system (1) to be asymptotically stable in probability one. In the sequel, it is further proven that controller (15) can guarantee that the state trajectories
Please cite this article as: Zahiripour SA, Jalali AA. A novel adaptive switching function on fault tolerable sliding mode control for uncertain stochastic systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.029i
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4
are driven onto the specified sliding surface sðtÞ ¼ 0 in (5), that is, to ensure the reachability of the specified sliding surface sðtÞ ¼ 0. Theorem 2. Consider the uncertain stochastic system (1). The switching function is chosen as s ¼ ð1=2ÞLnðρðtÞ R 1 BT PxðtÞ= ρðtÞ þR 1 BT PxðtÞÞ where matrix P 4 0 is the solution of matrix inequalities and constrain equality (20)–(22). The reachability of sliding surface is guaranteed under the sliding-mode controller (15). Proof. Choose the Lyapunov function as 1 V 2 ¼ ðKBÞ 1 sðtÞ2 2
ð35Þ
with K ¼ R 1 BT P. By means of the following expression s_ ðtÞ ¼ aðtÞð_ρðtÞKxðtÞ ρðtÞK x_ ðtÞÞ
ð36Þ
by (16) and Remark 2 a 1 ðtÞ ¼ ρðtÞ2 ðKxðtÞÞ2 4 0 ℒV 2 ¼ asðtÞðKBÞ 1 ½_ρðtÞKxðtÞ ρðtÞKððA þ ΔAðtÞÞxðtÞ þ Bfð1 μÞð1 μ^ Þ 1 ZxðtÞ þ ð1 μÞρðtÞsignðsðtÞÞ þ f ðx; tÞgÞ
ð37Þ
¼ asðtÞ½ðKBÞ 1 ρ_ ðtÞKxðtÞ ρðtÞðKBÞ 1 KAxðtÞ ρðtÞðKBÞ 1 KΔAðtÞxðtÞ ρðtÞð1 μÞð1 μ^ Þ 1 ZxðtÞ ρðtÞ2 ð1 μÞsignðsðtÞÞ ρðtÞf ðx; tÞ
ð38Þ
ra½ ðKBÞ 1 jsðtÞjj_ρðtÞjjKxðtÞj þ ρðtÞðKBÞ 1 jsðtÞj AK xðtÞ þρðKBÞ 1 ð1 μmin Þ xðtÞ ð1 μmax Þρ2 ðtÞjsðtÞj jsðtÞj K EH xðtÞ þ ρðtÞsðtÞZ ð1 μmax Þ þ ρðtÞjsðtÞjr
ð39Þ
2
0:2
3
6 7 E ¼ 4 0:2 5 0:3
FðtÞ ¼ sin ðtÞ
H ¼ 0:1
0:2
N ¼ ½0:3 0:3 0:3
It is assumed that actuator faults may happen, and the faulty model is given as (3) with μmin ¼ 0:2, μmax ¼ 0:8 and r ¼ 1 Z jf ðtÞj, ϖ ¼ 0:1. By solving (34) we have 2 3 11:5545 0:4699 1:7913 6 0:4699 2:5581 2:9843 7 P¼ 4 5 ; δ ¼ 3:2664 ; 1:7913 ε ¼ 2:7193
2:9843
4:6432
and β 5 10 4 which is sufficiently small for the proposed method in practical application. The design parameter R is selected 0.5 at first and then 2.5 for discussing about its effect. We select vector Z so that ðA þ BZÞ is Hurwitz and minðjλi ðA þ BZÞjÞ c i KBð1 μmax Þ as follows:
Z ¼ 7:1096 0:8613 0:1947 It is expected from the definition of vðtÞ in (7) and performance index (9) that the increasing R leads to decreased in ρðtÞj tanhðsÞj. This effect may decrease ρðtÞ which helps to control signal to be less aggressive or may decrease sðtÞ which helps state trajectories to remain near the sliding surface which possibly results in higher quality and speed of system responses. In the following, we evaluate the above analytical results by simulation in two cases (a) and (b). a. For R ¼ 0:5 The desired sliding function (12) and the SMC law (15) can be obtained as follows: s¼
1 ρðtÞ ½ 1:5325 1:3472 1:1132xðtÞ Ln 2 ρðtÞ þ ½ 1:5325 1:3472 1:1132xðtÞ
Define the following domain
with
Φ ¼ fxðtÞ j
ρ_ ¼ 0:9857ð1:1 0:2fρ jKxjÞ
ηjjxðtÞjj o ϖg
0:2
and
where η ¼ ðKBÞ 1 jjAjj jjKjj þ ðKBÞ 1 jjK jj jjEjj jj Hjj þ jjZjj
ð1 μmin Þ ð1 μmax Þ
ð40Þ
uðtÞ ¼ ð1 μ^ Þ 1 ½ 7:1096
0:1947 x þ ρðtÞsignðsÞ
with
and ϖ is an arbitrary constant. In this domain
μ_^ ¼
ℒV 2 r aρðtÞjsðtÞj½ðKBÞ 1 j_ρðtÞj þ ϖ ρðtÞð1 μmax Þ þ r
0:8613
ð41Þ
8 0 > > > >
> > > : 0:5566
or μ^ ¼ 0:8 and θ Z 0
0:8613 0:1947x
otherwise
From the updating law (16) for ρ and Remark (2) ℒV 2 r aρjsðtÞj jKxðtÞjð1 μmax Þ r 0
ð42Þ
By utilizing the result in Theorem 1, it can be obtained that the state trajectories will enter the domain Φ in finite time. That is, the relation ηx o ϖ, can be ensured in finite time. Hence, it follows from (42) that the reachability of the sliding surface can be ensured in finite time. This concludes the proof.
b. For R ¼ 2:5 s¼
1 ρðtÞ ½ 0:3065 0:2694 0:226xðtÞ Ln 2 ρðtÞ þ ½ 0:3065 0:2694 0:226xðtÞ
with ρ_ ¼ 0:1971ð1:1 0:2fρ jKxjÞ and
5. Analysis of designing parameter (R) effect and numerical simulation Consider the uncertain stochastic system (1) with 2 3 2 3 0:2 0:2 0 0:2 6 7 6 7 0:3 5 B ¼ 4 1 5 f ðtÞ ¼ sin ðx1 ðtÞÞ A ¼ 4 0:2 0:6 2
0:2 0:04
0:4 3
6 7 D ¼ 4 0:03 5 0:04
0:2
g ¼ 0:1x2
0:6
uðtÞ ¼ ð1 μ^ Þ 1 ½ 7:1096
0:8613
0:1947 x þ ρðtÞsignðsÞ
with
μ_^ ¼
8 0 > > > >
> > > : 0:5566
0:8613
0:1947 x
otherwise
For practical purposes, it is assumed that the measurement of all states is polluted to a Gaussian white noise with zero mean value
Please cite this article as: Zahiripour SA, Jalali AA. A novel adaptive switching function on fault tolerable sliding mode control for uncertain stochastic systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.029i
S.A. Zahiripour, A.A. Jalali / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 0.35
0 case(b) case(a)
0.3
-0.05
derivative of ro
0.25
u(t)
0.2 0.15 0.1 0.05
-0.1 -0.15 -0.2 -0.25 case(b) case(a)
-0.3
0 -0.05
5
0
5
10
15
20
-0.35
25
0
5
10
t (s)
0.1
25
20
25
0.1
case(b) case(a)
0.08
0.08
0.06
0.06
0.04
0.04
x1 (t)
s(t)
20
Fig. 4. Derivative of switching gain o 0.
Fig. 1. Control signals.
0.02
0.02
0
0
-0.02
-0.02
-0.04
15
t (s)
0
5
10
15
20
-0.04
25
0
5
10
15
t (s)
t (s)
Fig. 5. trajectory of 1(t).
Fig. 2. Curve of s(t).
4
0.3
case(b) case(a)
case(b) case(a)
0.25 0.2
ro (t)
x2 (t)
3.5
3
0.15 0.1 0.05 0
2.5
-0.05
0
5
10
15
20
25
0
10
15
20
25
t (s)
t (s)
Fig. 6. Trajectory of 2(t).
Fig. 3. Adaptive trajectory of switching gain.
and the variance s2 ¼ 10 6 . Furthermore we have used tanhðsÞ as a continuous function similar to signðsÞ, instead of signðsÞ for reducing chattering. The simulation results are shown in Figs. 1–8. Fig. 1 shows that in the case (b) controller reacts faster and it is less aggressive. Fig. 2 indicates that in the case (b) the state trajectories are closer to sliding surface which has some benefits as discussed previously. ρ_ ðtÞ Should have a negative value as mentioned in Remark 2. Figs. 3 and 4 display switching gain and derivative of it, respectively. Figs. 5–7 illustrate state responses of closed-loop system in two cases. As expected, increasing R made the responses
5
faster and with better quality. Finally, Fig. 8 shows that how adaptive trajectory of effectiveness loss value ðμ^ Þ changes.
6. Conclusion A robust SMC for single-input uncertain stochastic systems via a novel adaptive sliding surface has been studied in the presence of faulty actuator. The switching function has been extracted from a systematic strategy which minimizes a specific performance index and has multiple benefits. In the proposed sliding manifold there is a parameter which can be regulated for achieving the
Please cite this article as: Zahiripour SA, Jalali AA. A novel adaptive switching function on fault tolerable sliding mode control for uncertain stochastic systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.029i
S.A. Zahiripour, A.A. Jalali / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6 0.15
case(b) case(a)
0.1
x3 (t)
0.05
0
-0.05
-0.1
0
5
10
15
20
25
t (s) Fig. 7. Trjectory of 3(t). 0.65 case(b) case(a)
mu-hat (t)
0.6
0.55
0.5
0.45
0.4
0
5
10
15
20
25
t (s) Fig. 8. Adaptive trajectory of effectiveness loss value.
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Please cite this article as: Zahiripour SA, Jalali AA. A novel adaptive switching function on fault tolerable sliding mode control for uncertain stochastic systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.029i
