Neural Comput & Applic (2013) 22:237–247 DOI 10.1007/s00521-011-0710-7
ISNN 2011
Robust receding horizon control of uncertain fuzzy systems Chonghui Song • Jinchun Ye
Received: 15 February 2011 / Accepted: 5 July 2011 / Published online: 28 July 2011 Ó Springer-Verlag London Limited 2011
Abstract The optimal control of uncertain fuzzy systems with constraints is still an open problem. One candidate to deal with this problem is robust receding horizon control (RRHC) schemes, which can be formulated as a differential game. Our focus concerns numerically solving Hamilton– Jacobi–Issac (HJI) equations derived from RRHC schemes for uncertain fuzzy systems. The developed finite difference approximation scheme with sigmoidal transformation is a stable and convergent algorithm for HJI equations. Accelerated procedures with boundary value iteration are developed to increase the calculation accuracy with less time consumption. Then, the state-feedback RRHC controller is designed for some class of uncertain fuzzy systems with constraints. The value function calculated by numerical methods acts as the design parameter. The closed-loop system is proven to be asymptotically stable. An engineering implementation of the controller is discussed. Keywords Differential game Dynamic programming Uncertain fuzzy systems Numerical methods Robust receding horizon control
1 Introduction Although fuzzy modeling and fuzzy control have been developed for many years [1, 2], the field of the optimal
C. Song (&) School of Information Science and Engineering, Northeastern University, Shenyang 110004, Liaoning, China e-mail:
[email protected] J. Ye CTC Holdings, Chicago, IL 60604, USA
control of uncertain fuzzy systems with constraints is still open. From the viewpoint of modeling, fuzzy models can be roughly classified into three types [2–4]. Since wellformed fuzzy systems can approximate continuous nonlinear systems to a specified degree [3] or precisely model continuous nonlinear systems in a specified zone [2], some paper [5, 6] has tried to use linear system methods to discuss the optimal control of fuzzy systems analytically. But even for linear time-invariant systems with constraints, solutions to the optimal control problem cannot be expressed analytically except for some special cases [7]. Song et al. [8] have discussed the optimal control problem of fuzzy systems with constraints derived from receding horizon control (RHC) schemes and proposed an infinitetime horizon state-feedback (SF) RHC scheme which can be implemented in real-time. The goal of this work is to propose a robust RHC (RRHC) scheme to control uncertain fuzzy systems with constraints and develop a numerical method to solve Hamilton–Jacobi–Isaac (HJI) equations derived from RRHC schemes with infinite-time horizon. In the modeling of fuzzy systems, modeling errors or parameter variations are treated as system uncertainties and disturbances. Uncertainties and disturbances are regarded to worse the closed-loop performance in the optimal control problem, but the control input is regarded to improve the closed-loop performance. It leads to a differential game. In an optimal control problem, there is only oneplayer control and a single criterion to be optimized. Differential game theory generalizes this to two-player controls. Many problems can be formulated as two-player differential games such as pursuit evasion games, queueing systems in heavy traffic, RRHC schemes [12–14]. Value functions of the optimal control of such systems lead to first-order equations including first-order Hamilton– Jacobi–Bellman (HJB) equations arising in optimal control
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and first-order HJI equations arising in differential games as well as second-order equations such as the ones arising in stochastic optimal control or stochastic differential games. In general, RHC schemes can deal with the optimal control problem of certain nonlinear system with constraints and are formulated as solving on-line a finite-time horizon open-loop optimal control problem subject to system dynamics and constraints and repeating this procedure at the new sample time. RHC schemes lead to firstorder HJB equations. Many RHC schemes have been developed [17–19, 22, 24, 26] and have found some successful applications especially in process industries [17, 22]. However, as shown in literatures [15, 18–26], in general, RHC schemes with finite-time horizon do not guarantee closed-loop stability. The closed-loop stability can only be achieved by a suitable tuning of design parameters such as predictive horizon, control horizon, weighting matrices, additional terminal constraints and penalties. Therefore, Bitmead [15] suggested an infinitetime horizon method for linear systems. Mayne and Michalska [24] presented a RHC scheme with an additional artificial terminal state equality constraint. In their successive paper [25], they suggested a dual mode controller and extended the terminal state equality constraint to a terminal state inequality constraint. Yang and Polak [23] presented a RHC scheme that the control horizon is also a minimizer. Moreover, inequality contraction constraints are added. Chen and Allgo¨wer [18] introduced a quasiinfinite-time horizon RHC scheme with the additional terminal penalty and the terminal inequality constraint. ElFarra [26] proposed a predictive control framework for switched nonlinear systems that transit between their constituent modes at prescribed switching times. For more RHC approaches and stability results, readers can refer to [17, 27] etc. Although conventional RHC schemes are quite powerful tools for the controller design of nonlinear systems, there are still some limitations: (1) Solutions to the finite-time horizon optimal control problem do not guarantee the closed-loop stability, so some additional artificial terminal constraints are added to guarantee the closed-loop stability. Those added artificial terminal constraints will definitely impact the feasibility of the optimal control problem. The closed-loop stability also depends on the feasibility at the initial state; (2) Conventional RHC schemes need to solve on-line the optimal control problem repeatedly. Optimization procedures are too time-consuming to meet the requirement for the real-time control; (3) Since the controller u ðtÞ obtained in conventional RHC schemes is expressed as a function of t but not an explicit state-feedback of x, it is very hard to adjust the controller. RRHC extends the design procedure of RHC to control problems where uncertainties occur. It uses the differential
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game theory to deal with inputs and uncertainties. It is known as Min–Max RHC because it attempts to achieve robustness in the context of conventional RHC by taking all possible realizations of the uncertainty [12, 28–30] into consideration. In Min–Max RHC controllers, the value of the control signal to be applied is found by minimizing the worst case of a cost function (usually quadratic) which is in turn computed by maximizing over the possible expected values of disturbances and uncertainties. Min–Max RHC schemes lead to the first-order HJI equations. Such first-order HJI equations are usually nonlinear and difficult to solve in closed form. Thus, numerical methods become viable alternative. The initial development of the Markov chain approximation and the direct discretization numerical method with convergence proofs for stochastic systems refer to Kushner and Dupuis [35] and references therein. The numerical method for stochastic differential games refers to Kushner [36] and Song [37]. Viscosity solution methods provide another way to prove the convergence, see Evans and Souganidis [38] for differential games, Fleming and Soner [39] for stochastic controls, and Souganidis [40] for stochastic differential games. Ye [10, 11] used the Markov chain approximation with logarithmic transformation which can deal with both negative and nonnegative CRRA parameters in numerically solving the HJB equation in the area of insurance and finance. Song et al. [8] and Song [9] used a finite difference approximation with sigmoidal transformation (FDAST) to solve HJB equations arising in RHC schemes. In this paper, we extend the FDAST algorithm the in previous works [8, 9] to HJI equations derived from RRHC schemes of uncertain fuzzy systems with constraints. The FDAST algorithm is developed to get the value function of HJI equations. This numerical method is stable and convergent. Boundary value iteration (BVI) procedures are proposed to accelerate the numerical computation. Then, a state-feedback (SF) RRHC scheme is presented in a new way for some kind of uncertain fuzzy systems with constraints in that the finite-time horizon is extended to the infinite-time horizon, and the value function V(x) is used to design a state-feedback controller instead of using the optimal control u ðtÞ in conventional RRHC schemes as the current control action. The obtained controller can be implemented in real-time. Therefore, it can avoid on-line repeated optimization and the dependence on the feasibility of the initial state encountered in tractional RRHC schemes. This paper is organized as follows. Section 2 begins with a description of HJI equations derived from RRHC schemes of uncertain fuzzy systems with constraints. Section 3 presents an effective upwind FDAST algorithm to solve HJI equations. Accelerated procedures with BVI are proposed to improve the calculation accuracy with less time consumption. Section 4 designs the SF RRHC
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controller for uncertain fuzzy systems with constraints by using the value function as a design parameter. Section 5 provides a simulation example to demonstrate the proposed method and gives some comparisons with other controller design methods. Section 6 concludes the paper with further remarks.
VðxÞ ¼ min max JðxðtÞ; vðtÞÞ:
2 Problem formulation and preliminary results
The detail derivation of (7) is similar to derivation procedures of HJB equations in the previous works [8, 9], so it is omitted. Such HJI (7) covers RRHC schemes of uncertain fuzzy systems with constraints.
We consider following uncertain fuzzy systems _ Ri : IF g1 is Fgi 1 and . . . and gl is Fgi l ; THEN xðtÞ ¼ f i ðxðtÞ; uðtÞ; hðtÞÞ;
ð1Þ
with constraints uðtÞ 2 U c ; hðtÞ 2 U h ;
The above optimal control problem (6) satisfies following HJI equations ( h i 0 ¼ minu2U c maxh2U h oVðxÞ ox f ðx; vÞ þ Lðx; vÞ ; ð7Þ 0 ¼ Vð0Þ:
Remark 1 For finite-time horizon RRHC schemes without artificial terminal constraints or penalties, the cost function is
ð2Þ
where xðtÞ 2 Rn is the state, uðtÞ 2 Rm is the input, hðtÞ 2 Rq is unmodeled dynamics or uncertainties, U c is a nonempty compact convex subset including the original point 0; U h is a nonempty compact convex subset including the original point 0; gðtÞ ¼ ½g1 ðtÞ; . . .; gl ðtÞT 2 Rl is the premise variable and is some function of x(t) and u(t) (see [2]), Ri ði ¼ 1; . . .; MÞ denotes the ith rule of the fuzzy model, M is the number of fuzzy rules, Fgi 1 ; . . .; Fgi l are input fuzzy terms in the ith rule. To express concisely, let [ v ¼ ðu; hÞ 2 Rr ; r ¼ q þ m; U v ¼ U c U h ; ð3Þ and
ð6Þ
uðtÞ2U c hðtÞ2U h
JðxðtÞ; vðtÞÞ ¼
ZtþT LðxðsÞ; vðsÞÞds: t
The HJI function is oVðx; TÞ oVðx; TÞ þ min max f ðx; vÞ þ Lðx; vÞ : 0¼ u2U c h2U h oT ox Compared with finite-time horizon, infinite-time horizon enlarges the feasible zone (see Song et al. [8] and Song [9]). The feasibility of the optimal control problem with infinite-time horizon implies the stability of the closed-loop system without artificial terminal constraints or penalties. Another virtue of (7) is that it simplifies the numerical calculation of the value function.
_ ¼ f i ðxðtÞ; vðtÞÞ xðtÞ is the ith subsystem. Define state space S ¼ fxðtÞ 2 Rn g: Assume that f i ð; Þ: S U v ! Rn is continuous. The origin (0,0) is assumed to be the balance point of the global model of the system (1), that is, if the system (1) is assembled into the global expression _ ¼ f ðxðtÞ; vðtÞÞ xðtÞ
ð4Þ
by using singleton fuzzifier, product inference and centeraverage defuzzifier, then f(0,0) = 0. In the following, when the context is clear, the time label t will be omitted. 2.1 HJI equations in RRHC of uncertain fuzzy systems with constraints Associated with (1) is the cost functional Z1 JðxðtÞ; vðtÞÞ ¼ LðxðsÞ; vðsÞÞds
ð5Þ
t
where Lð; Þ [ 0 if and only if L(0, 0) = 0, and the value functional is
2.2 Some notations Throughout the rest of this paper, we use following notations. For a scalar a 2 R; the sign function signðÞ is þ1 if a 0; signðaÞ , ð8Þ 1 if a\0; and the saturation function satðÞ is a if jaj /; satða; /Þ , signðaÞ/ if jaj [ /;
ð9Þ
where / 2 R and / [ 0. For a vector a ¼ ½a1 ; . . .; ar T 2 Rr and / ¼ ½/1 ; . . .; /r T [ 0 2 Rr ; the vector sign function SignðÞ is defined as SignðaÞ ¼ ½signða1 Þ; . . .; signðar ÞT ; the vector saturation function SatðÞ is defined as Satða; /Þ ¼ ½satða1 ; /1 Þ; . . .; satðar ; /r ÞT and Diag(a) is defined as 2 3 a1 6 7 .. ð10Þ DiagðaÞ , 4 5: . ar
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For the function f ðaÞ : Rr ! R and the function vector gðaÞ ¼ ½g1 ðaÞ; . . .; gp ðaÞ where gi ðaÞ ¼ ½g1i ðaÞ; . . .; gri ðaÞT 1 i p : Rr ! Rr ; Lg f is defined as (Isidori [16]) Lg f ,
of ðaÞ gðaÞ: oa
ð11Þ
3 Numerical solution to HJI equations 3.1 Limitations of direct discretization At this point, if we directly discretize (7), the numerical framework is as follows. Let ei be the unit basis of Rn for i ¼ 1; 2; . . .; n: For a given positive discretized parameter h; define discrete spaces in state by ( ) n X h R ¼ x2S:x¼ ji hei ; ji 2 Z : ð12Þ i¼1
Let V h ðxÞ denote the numerical solution to the value function. One candidate to obtain V h ðxÞ is the finite difference numerical scheme, that is, V V(x), oVðxÞ Vðx þ ei hÞ VðxÞ ; ¼ oxi h
if
oVðxÞ VðxÞ Vðx ei hÞ ; ¼ oxi h
if
fi ðx; vÞ 0; fi ðx; vÞ\0:
ð14Þ
fi ðx; vÞ ¼ max½fi ðx; vÞ; 0: ð15Þ
Substitute (13), (14) and (15) into (7). Then for a fixed control v = [cu, ch]T, the recursive (16) for the value function can be obtained after some basic manipulations (Kushner and Dupuis [35]) V h ðxjcÞ ¼
n X
wi ðx ei hjcÞV h ðx ei hjcÞ þ Lðx; cÞDt;
i¼1
ð16Þ where f ðx; cÞ h ; Dt ¼ Pn : hjcÞ ¼ Pn i wi ðx ei jf ðx; cÞj jf j¼1 j j¼1 j ðx; cÞj
i¼1
ð19Þ Here, we directly discretize (7) by approximating the partial derivative in (13) and (14). Here, (16) is the finite difference equation that must be solved iteratively. Its convergence condition (see Kushner and Dupuis [35]) is that 0 wi ðÞ 1 P and i wi ðÞ ¼ 1: But one important observation must be mentioned. For an interested area S~ (i.e., the area that needs to be computed in the optimal control problem), since fi(x, v) could be nonnegative or negative, the boundary condition in S~ involved in the computation should be determined. This means that not only V(0) but also Vð1Þ and Vð1Þ are involved in the computation. Boundary conditions in the interested area S~ cannot be determined reasonably under the direct discretization method. 3.2 Finite difference approximation with sigmoidal transformation (FDAST algorithm)
ð13Þ
Define fiþ ðx; vÞ ¼ max½fi ðx; vÞ; 0;
h ðxÞ ¼ min max VNþ1 cu 2U c ch 2U h " # n X h wi ðx ei hjcÞVN ðx ei hÞ þ Lðx; cÞDt :
ð17Þ
To solve these difficulties mentioned above, we will perform sigmoidal transformation on (7). For y ¼ ½y1 ; . . .; yn T 2 Rn and z ¼ ½z1 ; . . .; zn T 2 Rn ; let a a yi ,Wðxi Þ ¼ ; 1 þ eðbxi Þ 2 a 1 1 2 yi zi ,W ðyi Þ ¼ ln a ; ð20Þ b 2 þ yi ye,DiagðyÞ; y¼WðxÞ ,VðxÞ; Lðz;vÞ,Lðx;vÞ; f ðz;vÞ,f ðx;vÞ; VðyÞj where WðxÞ ¼ ½Wðx1 Þ; . . .; Wðxn ÞT ; a [ 0; and b [ 0. Here, we choose same a and same b in all dimensions to simplify the expressions, but they could be different in each dimension. b a a i From (7) and noting that dy dxi ¼ a ð2 þ yi Þð2 yi Þ; we have the equation for VðyÞ as follows ( h i b a eÞða2 yeÞf ðz;vÞ þ Lðz;vÞ ¼ 0; minu2U c maxh2U h oVðyÞ oy a ð2 þ y ¼ 0: Vð0Þ ð21Þ
The function of the approximation in policy space is vNþ1 ¼ arg min max cu 2U c ch 2U h " # n X wi ðx ei hjcÞV h ðx ei h; vN Þ þ Lðx; cÞDt : i¼1
ð18Þ
The boundary condition of x ! 1 moves to yi ¼ a2 via the coordinate transformation (20). The domain of the above HJI equation becomes ½ a2 ; þ a2 ½ a2 ; þ a2: Since the boundary condition of x ! 1 is involved in the numerical computation and Vð 1Þ ¼ þ1; it is nec y ¼ a ¼ C essary to specify the boundary condition VðyÞj i 2
The function of the approximation in value space is
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where C is a reasonable very large positive number.
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Now for a given positive discretized parameter h, define discrete spaces by ( ) n X Rh ¼ y 2 S : y ¼ ji hei ; ji 2 Z ð22Þ
functions tanhðÞ can also work as the transformation function. This FDAST algorithm is summarized as follows. Step 1:
i¼1
where S , fy ¼ WðxÞg: Discretizing (21), we have 8 h P i ðy ei hjcÞVh ðy ei hjcÞ þ Lðz; vÞDt; < V ðyjcÞ ¼ i w h V ð0Þ ¼ 0; : Vh ðyÞj where C 0; yi ¼ a ¼ C; 2
Step 2: Step 3:
ð23Þ where i ðy ei hjcÞ ¼ w
Step 4: 2 ða4 y2i Þfi ðz; cÞ ; Pn a2 2 j¼1 ð 4 yj Þjfj ðz; cÞj
ah : Dt ¼ Pn a2 b j¼1 ð 4 y2j Þjfj ðz; cÞj
ð24Þ
Step 5:
Remark 2 The significant fact of our FDAST algorithm is that (23) specifies reasonable boundary conditions, in correspondence of x ! 1; in a finite domain ½ a2 ; a2: Here P i ðÞ ¼ 1; hence the solution to (21) i ðÞ 1 and i w 0w should be approximated by the solution to (23) as h ! 0 and C ! þ1:
Step 6:
Lemma 1 VðyÞ is the value function defined by (21) and Vh ðyÞ is the approximated value function in (23). Then Vh ðyÞ converges to VðyÞ as h ! 0 and C ! þ1: Fur thermore, VðxÞjx¼W1 ðyÞ ¼ VðyÞ: P i ðÞ 1 and i w i ðÞ ¼ Proof From (24), we have 0 w 1: When C ! þ1; Eq. (23) degenerates into h P i ðy ei hjcÞVh ðy ei hjcÞ þ Lðz; vÞDt; V ðyjcÞ ¼ i w h V ð0Þ ¼ 0: ð25Þ h
would follow the lines of As h ! 0; the result V ðyÞ ! VðyÞ the corresponding results of Kushner and Dupuis [35] or Song [37] for the deterministic case. Since WðÞ is a mono tone smooth invertible function, VðxÞjx¼W1 ðyÞ ¼ VðyÞ: Remark 3 By suitably choosing a, b and h, the border of the interested area S~ could lie in the area ½ a2 þ h2 ; a2 h2 ½ a2 þ h2 ; a2 h2: Thus, the boundary condition of the interested area S~ could be approximated when C ! þ1: In the practical calculation, C could be specified as a very large positive constant number to approximate the boundary condition. Remark 4 Equation (20) is some kind of logistic functions called Sigmoid function. Other kinds of monotone smooth invertible bounded functions such as Hyperbolic tangent
Referring to (18) and (19), construct the function of approximation in policy space and the function of the approximation in policy space for (23). Specify h, a, and b and give an initial guess of Vh ðyÞ as Vh ðy; v0 Þ: Starting from Vh ðy; vN Þ; solve the simplex optimization problem in the discrete space Rh to get vN?1. Fix vN?1 as the minimum, then iterate the function in value space obtained in Step 1 until h it converges to a specified degree to get VNþ1 ðyÞ h in the discrete space R : h ðyÞ as Vh ðy; vNþ1 Þ: Repeat from Step 3 Take VNþ1 until terminating criterions are satisfied. by the reverse transformation. Get V(x) from VðyÞ
3.3 FDAST with boundary value iteration (BVI) Suppose that the time consumption for a specified scalar approximation parameter h is tc(h), the time consumption for h2 is tc ðh2Þ 2n tc ðhÞ: That is, we can increase the solution’s accuracy to (21) as much as possible by choosing h ! 0 via h ¼ h2 ; h4 ; . . .; but the time consumption increases with 2n. By the above analysis, a major problem in getting a high accurate solution to (21) via any iterative procedure concerns how to increase the solution accuracy with the same parameter h. To some extent, the chosen parameter h indicates the affordable time consumption in the numerical calculation. Getting a high accurate solution with the same parameter h means getting a more accurate solution with almost the same affordable time consumption. Following procedures start at a basic idea that the more accurately the boundary values could be estimated, the more accurately the solution to the optimization could be obtained. The result obtained in Sect. 3.2 is used as a basis to get a more accurate boundary value estimation. Now suppose that h is very small positive number, we have þ ei hÞ VðyÞ Vðy ei hÞ Vðy VðyÞ : h h Then þ ei hÞ VðyÞ þ ðVðyÞ Vðy ei hÞÞ: Vðy
ð26Þ
Suppose that the point yb is at the boundary of the computed zone, the prediction of the boundary value is
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^ b Þ ¼ Vðy b ei hÞ þ Vðy b ei hÞ Vðy b 2ei hÞ: Vðy
b Þ ¼ Vðy ^ b Þ; then start the numerical algorithm Let Vðy defined by 8 P i ðy ei hjcÞVh ðy ei hjcÞ þ Lðz; vÞDt; < Vh ðyjcÞ ¼ i w h V ð0Þ ¼ 0; : h ^ b Þ: V ðyb Þ ¼ Vðy ð28Þ It worths reminding that if (16) is adopted in boundary value iteration procedures, after reverse coordinate transformation, the grid partition h is not uniform and should be jxii ¼ xiji þ1 xjii ¼ W1 ðyiji þ1 Þ W1 ðyjii Þ; h
ð29Þ
and the corresponding discrete spaces in state is redefined as ( ) n X h ji R ¼ x2S:x¼ ji hxi ei ; ji 2 Z : ð30Þ i¼1
Step 1:
Do the optimization via (23) to get the value function VðyÞ: ^ b Þ via (27). Do the Step 2: Predict the boundary value Vðy optimization defined by (28) to get the value function VðyÞ again. Step 3: Repeat Step 2 until the terminating criterion is satisfied. by the reverse transformation. Step 4: Get V(x) from VðyÞ
4 State-feedback (SF) RRHC of uncertain fuzzy systems We consider following uncertain fuzzy systems
ð34Þ
VðxÞ ¼ min max Jðx; u; hÞ u2U c h2U h
subject to _ ¼ fF ðxðtÞ; uðtÞ; hðtÞÞ; xðtÞ
xð0Þ ¼ x0 :
ð35Þ
Here Jðx; u; hÞ ¼
Z1
xðsÞT QxðsÞ þ uðsÞT RuðsÞds
ð36Þ
t
where Q 2 Rnn and R 2 Rm symmetric matrices, and
are positive-definite,
M X ðHi ðxÞAi x þ Bi uÞ þ hðx; uÞ; ð37Þ i¼1 Qn i l ðxj Þ ; and li(xj) is the fuzzy where Hi ðxÞ ¼ PM j¼1 Qn
fF ðx; u; hÞ ¼
j¼1
li ðxj Þ
i membership of Fx_j .
Further, let f ðxðtÞÞ ¼
N X
Hi ðxðtÞÞAi xðtÞ; gðxðtÞÞ ¼
i¼1
N X
Hi ðxðtÞÞBi :
i¼1
ð38Þ Then _ ¼ f ðxðtÞÞ þ gðxðtÞÞuðtÞ þ hðxðtÞ; uðtÞÞ : xðtÞ
ð39Þ
The key idea to design a stable controller for the system (31) in this paper is as following. The value function is calculated in the state domain. Then, it is used as controller design parameters to redesign the stable controller for this system. But in conventional RRHC schemes, the optimal control u*(t) calculated in the time domain is directly applied to the system. It grabs the key features of RRHC schemes but is given in a state-feedback form. Assumption 1 problem (34).
V(x) is calculated for the optimal control
ð31Þ
Remark 5 This assumption is reasonable, since the numerical method provided in Sect. 3 could compute V(x).
ð32Þ
Theorem 1 Suppose Assumption 1 is satisfied. Consider SF RRHC of the system (31) 1 1 T u ¼ Sat KR ðLg VÞ ð40Þ 2
with constraints uðtÞ 2 U c ; hðtÞ 2 U h
as
i¼1
For this case, the boundary value prediction also needs a slight modification. FDAST with BVI is summarized as follows.
_ Ri : IF x1 is Fxi 1 . . .xn is Fxi n ; THEN xðtÞ ¼ Ai xðtÞ þ Bi uðtÞ þ hðxðtÞ; uðtÞÞ
The SF RRHC scheme of the system (31) is formulated
ð27Þ
where definitions of x(t), u(t) and h(t) refer to (1), and Ai and Bi are matrices with proper dimensions. Detailed expressions of U c and U h are as follows
ð33Þ
where SatðÞ ¼ Satð; /u Þ and K ¼ Diagð½k1 ; . . .; km T Þ Im is the coefficient diagonal matrix. Then the closed-loop system is stable.
where /u 2 Rm ; and k(x, u) and Kðx; uÞ are known functions.
Proof The first step is to show whether the value function V(x), which is obtained by solving the optimal control
U c , fuðtÞ: juðtÞj /u ; /u [ 0g; U h , fhðtÞ: kðx; uÞ hðtÞ Kðx; uÞg;
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problem (34), can work as a Lyapunov function. From (34), we get that VðxÞ [ 0; VðxÞ ! þ1 when x ! 1 and V(0) = 0. Therefore, V(x) can work as a Lyapunov function. It is obvious that the control law (40) satisfies the constraint |u| B /u. Since K C Im, let K ¼ Im þ dK ; dK 0:
ð41Þ
Consider now the closed-loop system of (31) and (40). Evaluating the time derivative of the Lyapunov function along the closed-loop trajectory, we obtain _ ¼ Lf V þ Lg Vu þ Lh V VðtÞ 1 1 T ¼ Lf V Lg VSat KR ðLg VÞ þ Lh V: 2
ð42Þ
The optimization problem (34) satisfies the following HJI equation 0 ¼ min max½xT Qx þ uT Ru þ Lf V þ Lg Vu þ Lh V: u2U c h2U h
ð43Þ
Suppose u* and h* is the optimal solution to (34). If use first-order condition for the optimization 0¼
oðxT Qx þ uT Ru þ Lf V þ Lg Vu þ Lh VÞ ; ou
ð44Þ
we get 1 u ¼ R1 ðLg VÞT : 2 For the case
j 12 KR1 ðLg VÞT j /u ;
we have
ð45Þ
ð46Þ
and index sets jc , i n j:
½f jc , f ½f j :
ð48Þ
For the case |[v]j| [ [/u]j and noticing that u* is the solution to the optimization problem (34), we have /u ju j 0;
ð49Þ
and substituting (41) and (49) into (42), we have ! 1 T 1 KR ðLg VÞ V_ ¼ Lf V þ Lh V Lg VSat 2 jc þ Lg Vð½u j DiagðSignð½Lg Vj ÞÞ½/u j ½u j Þ 1 ¼ Lf V þ Lh V þ Lg V½u jc ½dK Lg VR1 ðLg VÞT jc 2 þ Lg V½u j Lg VDiagðSignð½Lg Vj ÞÞ½/u j Lg V½u j 1 ¼ Lf V þ Lh V þ Lg Vu ½dK Lg VR1 ðLg VÞT jc 2 j½Lg Vj j½/u j Lg V½u j 1 ¼ Lf V þ Lh V þ Lg Vu ½dK Lg VR1 ðLg VÞT jc 2 j½Lg Vj jð½/u j j½u j jÞ 1 ¼ xT Qx uT Ru ½dK Lg VR1 ðLg VÞT jc 2 j½Lg Vj jð½/u j j½u j jÞ 1 xT Qx Lg VR1 ðLg VÞT 4 \0: ð50Þ
Remark 6 Since V(x) is calculated by the numerical method, computational accuracy will influence the closedloop performance. Thus, the saturation function SatðÞ and the proportional coefficient K are introduced to tolerate the computational errors. The obtained controller does not explicitly include the uncertainty h, since h has been incorporated into the value function V(x).
Before study another case, first define
j , fi; vi [ /ui g;
fi if fi [ /ui ; 0 if fi /ui
_ Summarizing all the cases, we have V\0:
_ ¼ Lf V 1 KLg VR1 ðLg VÞT þ Lh V VðtÞ 2 1 ¼ Lf V ðIn þ dK ÞLg VR1 ðLg VÞT þ Lh V 2 1 ¼ Lf V þ Lg Vu þ Lh V dK Lg VR1 ðLg VÞT 2 1 ¼ xT Qx uT Ru dK Lg VR1 ðLg VÞT 2 1 T 1 x Qx Lg VR ðLg VÞT 4 \0:
1 v ¼ KR1 ðLg VÞT : 2
½f j ,
ð47Þ
where i ¼ f1; . . .; ng: Let us define two parts of the function vector f by
Remark 7 If K [ I, K = I ? dK and u = u*(x) ? dK u*(x). From the viewpoint of the RRHC, the controller is a RRHC controller with some kind of compensators. Why V(x) is chosen as the controller design parameter in this paper has the following reasons. 1) In conventional RRHC schemes, the optimal control u*(t) at time t is expressed as the function of t explicitly. It is very hard to judge the closed-loop stability at different initial states except by try and trial. Since V(x) is a function of x, the controller designed by V(x) is expressed as a function of x explicitly and the unstabilized zone could be indicated explicitly. 2) Since the optimization process is quite time-consuming, conventional RRHC controllers implemented by repeated on-line open-loop optimization could not be applied in
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real-time control. In order to avoid the repeated on-line optimization in RRHC schemes, the optimization calculation in this paper is once for all computed off-line.
5 Simulation
4.1 SF RRHC synthesis—an engineering implementation method
Example 1 We consider a nonlinear system in [18, 24, 9] described by
This SF RRHC strategy, as defined in (40), is not directly implementable since it involves the exact solution of the value function V(x). Generally speaking, the value function and the optimal control for nonlinear systems cannot be solved analytically except some special demos. If numerical methods are used to solve V(x), the solution is discrete. Suppose that V h ðxÞ for x 2 Rh ; which are defined by (29) and (30), is the numerical solution of the value function. The exact solution V(x) means that the grid partition hjxii ! 0 (i.e., infinitely small often). Since oVðxÞ ox is involved in the controller design, an implementable strategy could be to approximate oVðxÞ ox by some kind of continuous interpolation of
h
oV ðxÞ 1 ox jx¼W ðyÞ oV c ðxÞ ox
Define
where
h
oV ðxÞ oxi
V
h
j ðxþhxii ei ÞV h ðxÞ j hxii
: oV h ðxÞ ox : ðxj11 ; . . .; xjnn Þ
as the continuous interpolation of h
Mesh a data grid in R : Suppose that xj1 jn ¼ is a joint point of the data grid, and choose the value at the joint xj1 jn as
oV h ðxÞ xj1 jn : ox jx¼
The continuous interpolation
can be written in a concise form as the following fuzzy system Ri : IF x1 is Fxj11 and and xn is Fxjnn ; THEN ¼ ej1 jn ;
oV c ðxÞ ox
ð51Þ
h
where ej1 jn ¼ oVoxðxÞ jx¼xj jn ; Fxjii is the jith fuzzy set defined 1
on variable xi :lðxi ; xiji 1 ; xjii ; xiji þ1 ) is the triangle membership function of Fxjii and has the following expression 8 j 1 xi xii > xi 2 ½xjii 1 ; xjii Þ > ji > < xi xjii 1 j þ1 ð52Þ lðxi ; xiji 1 ; xjii ; xjii þ1 Þ ¼ xi xii xi 2 ½xjii ; xiji þ1 : ji ji þ1 > > x x > i i : 0 xi 62 ½xiji 1 ; xjii þ1 Since h jxii is very small, let oVðxÞ oV c ðxÞ : ox ox
ð53Þ
SF RRHC synthesis procedures are as follows: Step 1:
Solve the discrete value function V h ðxÞ:
Step 2:
Get oVðxÞ ox by the continuous interpolation (51), and then implement the controller according to (40).
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5.1 Simulation results
x_ 1 ¼ x2 þ uðl þ ð1 lÞx1 Þ x_ 2 ¼ x1 þ uðl 4ð1 lÞx2 Þ
ð54Þ
where |u(t)| B 4, and l is the uncertain parameter and l 2 ½0:5; 0:7: The system (54) can be modeled by the following T-S type fuzzy system with uncertainties in the zone X 2 ½1; 1 ½1; 1 _ Ri : IF x1 is Fxi 1 . . .xn is Fxi n ; THEN xðtÞ ¼ Ai xðtÞ þ Bi uðtÞ þ hðxðtÞ; uðtÞÞ;
ð55Þ
where
uðl 0:6 þ ð0:6 lÞx1 Þ hðx; u; lÞ ¼ : uðl 0:6 þ 4ðl 0:6Þx2 Þ To express precisely, the zone X 2 ½1; 1 ½1; 1 is divided into four cells C11, C12, C21 and C22, where C11 2 ½0; 1 ½0; 1; C12 2 ½0; 1 ½1; 0; C21 2 ½1; 0 ½0; 1; C22 2 ½1; 0 ½1; 0: The triangle membership functions on Fx1 and Fx2 are 1
l ðx1 ðtÞÞ ¼
x1
x1 2 ½0; 1
0 x1 62 ½0; 1; 8 x1 2 ½0; 1 1 > < x1 2 l ðx1 ðtÞÞ ¼ 1 þ x1 x1 2 ½1; 0Þ > : 0 x1 62 ½1; 1; x1 x1 2 ½1; 0 l3 ðx1 ðtÞÞ ¼ 0 x1 62 ½1; 0; x2 x2 2 ½0; 1 l1 ðx2 ðtÞÞ ¼ 0 x2 62 ½0; 1; 8 x2 2 ½0; 1 > < 1 x2 2 l ðx2 ðtÞÞ ¼ 1 þ x2 x2 2 ½1; 0Þ > : 0 x2 62 ½1; 1; x x 2 2 2 ½1; 0 l3 ðx2 ðtÞÞ ¼ 0 x2 62 ½1; 0:
Parameters in fuzzy rules are in Table 1 and in the below. 1 1 0 0 ; A2 ¼ ; A3 ¼ ; A4 ¼ ; A1 ¼ 1 0 1 0 1 1 ; A6 ¼ ; A5 ¼ 1 0
Neural Comput & Applic (2013) 22:237–247
245
0 0 1 1 ; A8 ¼ ; A9 ¼ ; A10 ¼ ; 1 0 1 0 0 0 A11 ¼ ; A12 ¼ ; 1 0 1 1 0 ; A14 ¼ ; A15 ¼ ; A13 ¼ 1 0 1 0 1 A16 ¼ ; B1 ¼ ; 0 1 0:6 1 0:6 ; B3 ¼ ; B4 ¼ ; B2 ¼ 1 0:6 0:6 1 0:6 ; B6 ¼ ; B5 ¼ 2:2 2:2 1 0:6 0 ; B8 ¼ ; B9 ¼ ; B7 ¼ 0:6 0:6 1 0:6 0 B10 ¼ ; B11 ¼ : 1 0:6 0:6 0 0:6 : B13 ¼ ; B14 ¼ ; B12 ¼ 0:6 2:2 2:2 0 0:6 ; B16 ¼ : B15 ¼ 0:6 0:6 Choose a = 2, b = 4.8, h = 0.05, P = I2 and R = 1 in optimization procedures. Figures 1 and 2 show the numerical solution to the value function V(x) in x 2 ½0:77; 0:77 ½0:77; 0:77 and its contour with the indicated unstabilized zone, respectively. Choose K = 1 and hjxii ¼ xiji þ1 xjii ¼ W1 ðyiji þ1 Þ A7 ¼
in the controller design. W1 ðyjii Þ to implement oVðxÞ ox Figures 3, 4, and 5 show the simulation results of the closedloop trajectory with initial states x(0) = [0.5, -0.2]T, x(0) = [0.5, 0.1]T, x(0) = [-0.5, 0.5]T and x(0) = [-0.5, -0.5]T for l = 0.5, l = 0.6 and l = 0.7, respectively.
Unstabilized Zone
1 10 0.5 5 0 0 −0.8
−0.5 −0.6
−0.4
−0.2
0
0.2
0.4
−1
0.6
Fig. 1 Optimization result of value function V(x) with x 2 ½0:77; 0:77 ½0:77; 0:77 for l 2 ½0:4; 0:6
0.6
Unstabilized Zone
0.4 0.2 0 −0.2 −0.4 −0.6
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Fig. 2 2D contour of V(x)
0.5 0.4
Table 1 Parameters in the ith fuzzy rule x1 is
x2 is
Ai
Bi
x1 is
x2 is
Ai
Bi
0.3 0.2
In the cell C11
In the cell C12 0.1
Fx11
Fx12
A1
B1
Fx11
Fx32
A5
B5
Fx21
Fx12
A2
B2
Fx21
Fx32
A6
B6
0
Fx11
Fx22
A3
B3
Fx11
Fx22
A7
B7
−0.1
Fx21
Fx22
B4
Fx21
Fx22
A8
B8
−0.2
A4
In the cell C21
In the cell C22
−0.3
Fx21
Fx12
A9
B9
Fx31
Fx32
A13
B13
Fx31
Fx12
A10
B10
Fx21
Fx32
A14
B14
Fx21
Fx22
A11
B11
Fx31
Fx22
A15
B15
Fx31
Fx22
A12
B12
Fx21
Fx22
A16
B16
−0.4 −0.5 −0.5
0
0.5
Fig. 3 Closed-loop trajectory with l = 0.5
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5.2 Comparison with other methods
0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.5
0
0.5
Fig. 4 Closed-loop trajectory with l = 0.6
6 Conclusion
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8 −0.5
Since the field of SF RRHC of uncertain nonlinear systems with constraints is still nearly open, there are rare results in literatures to be compared with. So we compare our SF RRHC scheme with RHC schemes of certain nonlinear systems in literatures. When compared with other existing RHC approaches, the proposed SF RRHC scheme has significant advantages in the computation and in the application. Since the simulation is also used in papers [18, 24], we can use simulation results as a reference. To show this, we compared the proposed SF RRHC (named Controller A) with other two RHC controllers (named Controller B and Controller C) in Table 2. Controller B is used in the paper [18] and Controller C is used in the paper [24].
0
0.5
Fig. 5 Closed-loop trajectory with l = 0.7
In this paper, HJI equations in RRHC schemes of uncertain fuzzy system with constraints are solved by the FDAST algorithm. The value function is used in SF RRHC design of some kind of uncertain fuzzy systems with constraints. This procedure absorbs the basic ideas of RRHC schemes, but the setup differs from conventional ones in that it extends the finite-time horizon to infinite-time horizon, the value function V(x) acts as the design parameter but not the current optimal control action u*(t) is directly used as the control action, and the controller is a state-feedback controller. The closed-loop stability does not depend on the feasibility at the initial state which is a prerequisite in conventional RRHC schemes. There are no additional artificial terminal constraints and penalties which definitely impact the feasibility of the optimal problem. On-line
Table 2 Comparison of computational burden and other features Items
Controller A
Controller B
Controller C
Could be used with input constraints
Yes
Yes
Yes
On-line computational burden
Real-time
Indicate the unstabilized zone
Yes
No (by try)
No (by try)
Stability depends on the feasibility of initial states
No
Yes
Yes
Added artificial constraints to guarantee the stability
No
Yes
Yes
Could deal with uncertainties
Yes
No
No
Optimization algorithm
FDAST programmed in MATLAB
Large-scale commercial software
Large-scale commercial software
Applicable in engineering
Yes
In simulation
In simulation
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C570 s
C1,472 s
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repeated optimization procedures, which are too timeconsuming, can be avoided. The state-feedback controller can be absolutely implemented in real-time. Unstabilized zones are indicated in the optimization result without try and trial. Current researches focus on a further reduction on the computational burden in optimization, on adaptive control schemes for uncertain nonlinear system, and on control structures of more generalized nonlinear systems. Acknowledgments The authors would like to acknowledge Mr. Zhuo Wang Ph.D. and Xiuchong Liu for their useful discussion. This work was supported by the National Natural Science Foundation of China (Grant Nos. 60974141, 60504006, 60621001, 60728307, 60774093), the Natural Science Foundation of Liaoning Province (Grant No. 20092007) and the Fundamental Research Funds for the Central Universities (Grant Nos. N100404015, N100404012).
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