Adaptive Robust Control: A Piecewise Lyapunov Function Approach

Transcription

1 Aaptive Robust Control: A Piecewise Lyapunov Function Approach Jianming Lian, Jianghai Hu an Stanislaw H. Żak Abstract The problem of output tracking control for a class of multi-input multi-output uncertain systems is consiere. A novel aaptive robust controller is propose, which incorporates a variable-structure raial basis function RBF network to approximate unknown system ynamics. The RBF network can etermine its structure on-line ynamically, where raial basis functions are ae or remove to ensure the esire tracking accuracy an to prevent the network reunancy simultaneously. The structure variation of the RBF network is taken into account in the stability analysis of the close-loop system. This is accomplishe by using the piecewise continuous quaratic Lyapunov function typically for the analysis of switche an hybri systems. I. INTRODUCTION In any controller esign, one of the essential elements is a mathematical moel of the plant to be controlle. However, the available mathematical moel often contains uncertainties resulting from unknown system ynamics or isturbances. Recently, several aaptive controller esign methoologies for uncertain systems have been introuce such as aaptive feeback linearization [], aaptive backstepping [2], nonlinear amping an swapping [3] an switching aaptive control [4]. Especially, a number of aaptive control strategies have been evelope for a class of feeback linearizable nonlinear systems incluing both single-input single-output SISO systems [5] [8] an multi-input multioutput MIMO systems [9] [2]. To improve the robustness of aaptive controllers, robustifying components have been use in [5], [7], [8], [] [2]. To eal with ynamical uncertainties, aaptive robust control strategies often involve certain types of function approximators to approximate unknown system ynamics. In particular, one-layer neural networks have been employe in [5], [], where raial basis function RBF networks were use to approximate unknown system ynamics. However, fixe-structure RBF networks require the off-line etermination of the appropriate network structure. In [6], [9], multilayer neural network MLNN base aaptive robust control strategies were propose. Although it is not require to efine the basis function sets for MLNNs, it is still necessary to pre-etermine the number of hien neurons. Moreover, compare with MLNNs, RBF networks are characterize by simpler structure, faster - computation time an superior aaptive performance. Variable structure RBF network base aaptive robust controllers have been propose for SISO This work was supporte by the Office of Naval Research Grant N an the National Science Founation Grant CNS Jianming Lian, Jianghai Hu an Stanislaw H. Żak are with the School of Electrical an Computer Engineering, Purue University, West Lafayette, IN , USA. {jlian,jianghai,zak}@purue.eu. feeback linearizable uncertain systems in [8], [3], [4], where the variable-structure RBF networks that can both grow an shrink were employe. However, they are subject to the problem of infinitely fast switching between ifferent structures. Moreover, the effect of the structure variation was not consiere in the stability analysis. In this paper, we consier, as in [9] [2], the output tracking control problem for a class of MIMO feeback linearizable uncertain systems. We propose a novel aaptive robust control strategy. The evelope controller incorporates a variable-structure RBF network, which is an improve version of the network consiere in [4], [5], to approximate unknown system ynamics. The employe variable-structure RBF network avois selecting basis functions off-line by etermining the network structure on-line ynamically. It can a or remove RBFs epening on the magnitue of the output tracking error to ensure the tracking accuracy an to prevent the network reunancy simultaneously. We impose a welling time requirement on the structure variation to avoi the problem of infinitely fast switching between ifferent structures as in [6]. The raisecosine RBF RCRBF presente in [7] is utilize because the RCRBF has compact support that results in significant reuction of computations require for the network s training an output evaluation [5]. By viewing the close-loop system as a switche system, we can apply the piecewise continuous quaratic Lyapunov function that has been use in the stability analysis of switche an hybri systems [8], [9]. This enables us to analyze the structure variation in the stability analysis without aitional restrictive assumptions. The journal version of this paper with all the etails an aitional results can be foun in [2]. II. SYSTEM DESCRIPTION AND PROBLEM STATEMENT We consier a class of uncertain systems consisting of p couple subsystems moele by the following equations, p y ni i = f i x + g ij xu j + i, i =,...,p, j= where y i, u i an i are the output, input an isturbance of the i-th subsystem, respectively, an f i x an g ij x are unknown functions with [ ] x = y y n y p y p np being the state vector of the overall system. The isturbance i can have the form of i t, i x or i t, x. Let A i, b i be the canonical controllable pair that represents a chain of n i integrators, an let c i = [ ] ni, y = [y y p ],

2 u = [u u p ] an = [ p ]. We represent the system in a compact form as ẋ = Ax + B fx + Gxu + 2 y = Cx, where A = iag[a A p ], B = iag[b b p ], C = iag[c c p ], fx = [f x f p x] an Gx = [g ij x] p p. The above system is often referre to as square system, because there are same numbers of inputs an outputs. As pointe out in [], many physical systems, such as natural Lagrangian systems an circuit systems, can be moele in the form of. In this paper, we assume that fx an Gx are Lipschitz continuous vector- an matrixvalue functions of x, respectively, is Lipschitz continuous in x an piecewise continuous in t, an o. In aition, we consier the case where the input matrix Gx is efinite with eigenvalues boune away from zero for all x of interest. Without loss of generality, we assume that Gx is positive efinite with gi p Gx gi p, where g an g are positive constants. Our goal is to evelop a tracking control strategy such that the i-th system output y i, i =,...,p, tracks a reference signal y i. We assume that y i has boune erivatives up to the n i -th orer. Thus, we have y n = [y n y np p ] Ω y, where Ω y is a compact subset of R p. Then we efine the esire system state vector as x = [y y n y p y np p ]. We have x Ω x, where Ω x is a compact subset of R n. Let e i = y i y i enote the i-th output tracking error. We efine the output tracking error as e y = [e e p ] an then the state tracking error as e = x x. It follows from 2 that the state tracking error ynamics are given by ė = Ae + B fx + Gxu y n +. 3 In the next section, we first escribe the variable-structure RBF network that is employe to approximate the unknown function fx in the aaptive robust controller esign. III. VARIABLE-STRUCTURE RBF NETWORK The employe variable-structure RBF network is an improve version of the self-organizing RBF network use in [4], which, in turn, was aapte from [5]. The major improvement is in the RBF aing an removing operations. Moreover, we introuce a welling time T, which is a esign parameter, into the structure variation of the RBF network to prevent fast switching between ifferent structures. Our variable-structure RBF network has N ifferent amissible structures, where N is etermine by the esign parameters iscusse later. For each amissible structure illustrate in Fig., the RBF network consists of n input neurons, M v hien neurons, where v {,...,N} enotes the scalar inex, an p output neurons. The k-th output of the RBF network with the v-th amissible structure can be represente as M v ˆf k,v x = ω kj,v ξ j,v x, 4 j= Fig.. Self-organizing raial basis function network. where ω kj,v is the weight from the j-th hien neuron to the k-th output neuron an ξ j,v x is the raial basis function for the j-th hien neuron. Let W v = [ω,v ω p,v ] with ω i,v = [ω i,v ω imv,v] an ξ v x = [ξ,v x ξ Mv,vx]. Then we have ˆf v x = W v ξ v x. We employ the raise-cosine RBF propose in [7] instea of the commonly use Gaussian RBF. The one-imensional RCRBF is efine as { 2 + cos πx c δ if x c δ ξx = if x c > δ, where c is the center an δ is the raius. The n-imensional RCRBF can be represente as the prouct of n oneimensional RCRBFs. Although the Gaussian RBF is commonly use for the construction of the raial basis function network, we prefer the raise-cosine RBF to the Gaussian RBF because of the compact support associate with the raise-cosine RBF. As iscusse in [5], [7], the compact support of the RCRBF enables fast an efficient training an output evaluation of the RBF network. In the following subsections, we provie a etaile escription of the improve variable-structure RBF network. A. Center Gri Recall that the unknown function fx is approximate over a compact set Ω x R n. Without loss of generality, it is assume that Ω x can be represente as Ω x = {x R n : x l x x u } = {x R n : x li x i x ui, i n}, where the n-imensional vectors x l an x u enote lower an upper bouns of x, respectively. To locate the centers of RBFs insie the approximation region Ω x, we utilize an n- imensional center gri with layer hierarchy, where each gri noe correspons to the center of one RBF. The gri noes of the center gri are locate at S S n, where S i is the set of locations of the gri noes in the i-th coorinate an enotes the Cartesian prouct. The center gri is initialize insie the approximation region Ω x with S i = {x li, x ui }, i =,...,n. The 2 n gri noes of the initial gri are referre to as the bounary gri noes, an they cannot be remove.

3 In each coorinate, aitional gri noes will be ae into an then can be remove from the set S i as the controlle system evolves in time. However, new gri noes can only be place at the potential locations. The potential gri noes are etermine coorinate-wise. In each coorinate, the potential gri noes of the first layer are the two fixe bounary gri noes. The secon layer has only one potential gri noe in the mile of the bounary gri noes. Then the potential gri noes of the subsequent layers are in the mile of the ajacent potential gri noes of all the previous layers. B. Aing RBFs As the controlle system evolves in time, the output tracking error e y is measure. If the Eucliean norm of e y excees a prespecifie threshol e max, the network attempts to a new RBFs, that is, a new gri noes. First, the nearest neighboring gri noe in the center gri, enote c nearest, to the current input x is locate among existing gri noes. Then the nearer neighboring gri noe in the center gri enote c nearer is locate, where c inearer is etermine such that x i is between c inearest an c inearer. Next, the aing operation is performe for each coorinate inepenently. A new gri noe is ae into S i if the following conitions are satisfie: e y > e max ; 2 the elapse time since the last operation, aing or removing, is greater than T ; 3 xi c inearest > cinearest c inearer /4; 4 x i c inearest > ithreshol, where ithreshol is a esign parameter that specifies the minimum gri istance in the i-th coorinate an thus etermines the number of amissible structures enote by N. C. Removing RBFs When the Eucliean norm of the output tracking error e y is less than or equal to the prespecifie threshol e max, the network attempts to remove some of the existing RBFs to prevent network reunancy. The RBF removing operation is also implemente for each coorinate inepenently. The gri noe locate at c inearest is remove from S i if e y e max ; 2 the elapse time since the last operation, aing or removing, is greater than T ; 3 c inearest / {x li, x ui }; 4 the the gri noe in the i-th coorinate with its coorinate equal to c inearest is in the higher than or in the same layer as the highest layer of the two neighboring gri noes in the same coorinate; 5 xi c inearest < τ cinearest c inearer, for τ,.5. D. Uniform Gri Transformation The etermination of the raius of the RBF is much easier in a uniform gri than in a nonuniform gri because the RBF is raially symmetric with respect to its center. To simplify the raius etermination, the one-to-one mapping zx = [z x z n x n ], propose in [7], is use to transform the center gri into a uniform gri. Suppose that the RBF network is now with the v-th amissible structure after the aing or removing operation an there are M i,v istinct elements in S i orere as c i < < c imi,v, where c ik is the k-th element with c i = x li an c imi,v = x ui. Then the mapping function z i x i : [x li, x ui ] [, M i,v ] takes the following form, z i x i = k + x i c ik c ik+ c ik, c ik x i < c ik+ 5 which maps c ik into the integer k. Thus, the transformation zx : Ω x R n maps the center gri into a gri with unit spacing between ajacent gri noes such that the raius of the RBF can be easily chosen. For the raise-cosine RBF, the raius in every coorinate is selecte to be equal to one unit, that is, the raius will touch but not exten beyon the neighboring gri noes in the uniform gri. This particular choice of the raius guarantees that for any given input x, the number of nonzero raise-cosine RBFs in the uniform gri is at most 2 n. IV. ADAPTIVE ROBUST CONTROLLER DEVELOPMENT The propose aaptive robust controller has the form u = u a,v + u s,v = G ˆf v x + y n Ke + u s,v, 6 where G = g I p with g >, ˆfv x = W v ξ v x, K = iag[k... k p ] is selecte such that A mi = A i b i k i is Hurwitz, an u s,v is the robustifying component to be esigne later. Note that the controller architecture varies as the structure of the RBF network changes. A particular controller architecture is referre to as a moe. Because there are N amissible network structures, the propose controller has N ifferent moes. Let Ω e be a compact set incluing all possible initial state tracking errors an let c e = max e Ω e 2 e P m e, where P m = iag[p m P mp ] is the solution to the continuous Lyapunov matrix equation A mp m + P m A m = 2Q m for Q m = iag[q m Q mp ] with Q mi = Q mi >. Choose c e > c e an let Ω e = {e : 2 e P m e c e }. Then we efine Ω x = {x : x = e + x, e Ω e, x Ω x }, over which the unknown function fx is approximate. To procee, recall that W v = [ω,v ω p,v ]. For the practical implementation, we constrain the weight vectors ω i,v to resie in the compact sets Ω i,v = {ω i,v : ω i ω ij,v ω i, j M v }, where ω i an ω i, i =,...,p, are esign parameters. Let W v = [ω,v ω p,v ] enote the optimal constant weight matrix corresponing to each amissible network structure, which are use only in the analytical analysis an efine as W fx v = argmin max W v ξ v x. x Ω x ω i,v Ω i,v

4 Let Φ v = W v W v = [φ,v φ p,v ] with φ i,v = ω i,v ω i,v, an let c = p i= c i, where c i = max max v ω i,v,ω 2η φ i,v φ i,v, 7 i,v Ωi,v η > is a esign parameter, an max v enotes the maximization taken over all the amissible structures of the RBF networks. It is obvious that c i ecreases as η increases. Let σ = B P m e. We employ the following projection base weight matrix aaptation law Ẇ v = Proj W v, ηξ v xσ, 8 where ProjW v,θ v enotes Projω ij,v, θ ij,v for i =,...,p an j =,...,M v an Projω ij,v, θ ij,v is the iscontinuous projection operators use in [4]. For the above projection operator, we have η trace Φ v Ẇ v ηξ v xσ. 9 Furthermore, the aaptation law 8 guarantees that if ω i,v t Ω i,v, then ω i,v t Ω i,v for t t. Substituting the propose aaptive robust controller given by 6 into the state tracking error ynamics 3 gives ė = Ae + B fx + Gxu a,v + u s,v y n + = Ae + B ˆf v x + G u a,v y n + BGxu s,v + B fx ˆf v x + Gx G u a,v + = A m e BΦ v ξ v x + BGxu s,v + B fx W v ξ v x + B Gx G u a,v + B. As can be seen from, the robustifying component u s,v has to counteract the effects of the approximation error as well as the boune isturbance to force the state tracking error e converge or, at least, be boune. Let f = max v u s,v = max fx W v ξ v x x Ω x an g = max x Ω x Gx G. We propose a robustifying component u s,v of the form { k s,v σ g σ if σ ν where ks,v g σ ν if σ < ν, k s,v = f + g u a,v + o an ν > is a esign parameter. Remark: Let the increasing sequence {t i } i= be a partition of the interval [t, such that v = v i over [t i, t i+ ]. During the i-th time interval [t i, t i+ ], the controller u given by 6 an has a fixe architecture. Thus, as iscusse in [2], there exists a unique solution x vi t to 2 starting at x vi t i over [t i, t i+ ]. On the other han, we have impose the welling time requirement on each moe so that t i+ t i T. Therefore, we can piece together the solutions x vi t over [t i, t i+ ] to establish the existence of a unique solution xt to 2 starting at xt over [t,, where x v t = xt an x vi+ t i+ = x vi t i+. Now we consier the following piecewise continuous quaratic Lyapunov function caniate whenever the propose aaptive robust controller 6 is in the v-th moe, V v = 2 e P m e + 2η trace Φ v Φ v. 2 This Lyapunov function has jump iscontinuities when the propose controller switches between ifferent moes. Evaluating the time erivative of V v on the solutions to an taking into account 9, we obtain V v = e P m ė + η trace Φ v Φ v = e Q m e + σ Gxu s,v + σ + σ fx W v ξ v x + Gx G u a,v + η trace Φ Φ v v σ Φ v ξ v x λ min Q m e 2 + k s,v σ + σ Gxu s,v. 3 If σ > ν, we have k s,v σ + σ Gxu s,v k s,v σ k s,v σ = ; 4 if σ ν, we have k s,v σ + σ Gxu s,v k s,v σ k s,v σ 2 ν k s,v 4 ν. 5 Recall that k s,v = f + g u a,v + o. Let k s = f + g max v max u a,v + o, where the inner maximization is taken over e Ω e, x Ω x, y n Ω y an ω i,v Ω i,v. Combining 4 an 5, we obtain k s,v σ + σ Gxu s,v k s,v 4 ν k s ν. 6 4 It follows from 7 that p 2η trace Φ p v Φ v = 2η φ i,vφ i,v c i = c. 7 i= i= Taking into account 6 an 7 in 3 gives V v λ min Q m e 2 + k s 4 ν 2µ m V v + 2µ m c + k s 4 ν µ m V v µ m V v 2 c, 8 where µ m = λ min Q m /λ max P m an c = c+k s ν/8µ m. Let t,v an t f,v enote the initial an the final time instant, respectively, of a continuous time perio uring which the controller is in the v-th moe. It follows from 8 that if V v t 2 c for t [t,v, t f,v ], we have V v t µ m V v t, which implies that for t [t,v, t f,v ]. V v t exp µ m t t,v V v t,v 9

5 Theorem : Let t, t 2 an t 3 be three consecutive switching time instants so that v = v for t [t, t 2 ] an v = v 2 for t [t 2, t 3 ]. Suppose that V v t 2 c for t [t, t 3 ]. If the welling time T of the variable-structure RBF network is selecte such that T µ m ln 3, 2 2 then V v2 t 2 < V v t an V v2 t 3 < V v t 2. Proof: When the controller switches from the v -th to the v 2 -th moe, there exists a jump between V v2 t 2 an V v t 2, which is enote as = V v2 t 2 V v t 2. 2 We know that e is continuous because x is continuous. Hence, it follows from 2 an 7 that = 2η trace Φ v 2 t 2 Φ v2 t 2 2η trace Φ v t 2 Φ v t 2 an c. Because V v2 t 2 c for t [t 2, t 3 ], it follows from 9 an 2 that V v2 t 3 exp µ m t 3 t 2 V v2 t 2 = exp µ m t 3 t 2 V v t 2 + exp µ m t 3 t 2 V v t 2 + c. Due to the welling time requirement on the variablestructure RBF network, we have t 3 t 2 T, which implies that exp µ m t 3 t 2 exp µ m T. Thus, we have V v2 t 3 exp µ m T V v t 2 + c. In orer to have V v2 t 3 < V v t 2, it is sufficient to have exp µ m T V v t 2 + c < V v t Solving 22 for T, we obtain T > µ m ln Vv t 2 + c V v t 2. It is easy to verify that the right-han sie of the above inequality is a ecreasing function of V v t 2 when V v t 2 2 c. Recall that c > c. It follows that ln Vv t 2 + c V v t 2 2 c + c ln < ln 2 c 3. 2 Therefore, if we choose T to satisfy the conition 2, then 22 is satisfie, which implies that V v2 t 3 < V v t 2. On the other han, because V v t 2 c for t [t, t 2 ], it follows from 9 that V v t 2 exp µ m t 2 t V v t, which implies that V v t expµ m t 2 t V v t 2 expµ m T V v t 2, because t 2 t T. Note that, for the conition given by 2, we have V v t 2 > exp µ m T V v t 2 + c. Then it follows that V v t expµ m T V v t 2 > V v t 2 + c V v t 2 + = V v2 t 2. Hence, if the welling time T satisfies the conition 2, then V v2 t 2 < V v t an V v2 t 3 < V v t 2. The proof of the theorem is complete. Theorem 2: Consier the system 2 riven by the propose aaptive robust controller 6 an with the aaptation laws 8. Suppose that T satisfies 2. If c e max {c e + c, 2 c + c}, then et Ω e an xt Ω x for t t. Moreover, there exists a finite time T t such that 2 e tp m et 2 c + c for t T. If, in aition, there exists a finite time T s t such that v = v s for t T s, then there exists a finite time T 2 T s such that 2 e tp m et 2 c for t T 2. Proof: If T satisfies 2, it follows from Theorem that V v t will visit the interval [, 2 c] infinitely often. That is, for any T t, there exists a time t T such that V v t 2 c. Let T t be the first time such that V v T 2 c. The trajectory of V v t starting at t = T will stay insie the interval [, 2 c] until it jumps outsie when the controller switches the moe. However, the jump of V v t between ifferent moes satisfies that c. Hence, we have V v t 2 c + c for t T. On the other han, we know that V v t c e + c. Therefore, if c e max {c e + c, 2 c + c}, then V v t c e for t t. Because 2 e tp m et V v t, we have 2 e tp m et c e, which implies that et Ω e an xt Ω x, for t t. Moreover, we have 2 e tp m et 2 c + c for t T. If, in aition, we have v = v s for t T s, then it follows from 8 that Vvs µ m V vs µ m V vs 2 c for t T s. Thus, there exists a finite time T 2 T s such that V vs 2 c, which implies that 2 e tp m et 2 c, for t T 2. The proof of the theorem is complete. V. EXAMPLE We illustrate the effectiveness of the propose variablestructure RCRBF network base aaptive robust controller with a planar articulate two-link manipulator, whose moel is given in [22, p. 394]. Recall that q an q 2 enote the angular positions of joint an 2, respectively, an τ an τ 2 enote the applie torques. We assume that there exist input isturbances η an η 2 associate with the applie torques τ an τ 2, respectively. The ynamics of this two-link rigi robot are escribe by [ ] [ ] H H 2 q H 22 q 2 H 2 + [ h q2 h q + q 2 h q ] [ ] [ ] q τ + η =, q 2 τ 2 + η 2 where H = a + 2a 3 cosq 2 + 2a 4 sinq 2, H 2 = H 2 = a 2 +a 3 cosq 2 +a 4 sinq 2, H 22 = a 2 an h = a 3 sinq 2 a 4 cosq 2 with a = I + m l 2 c + I e + m e l 2 ce + m el 2, a 2 = I e + m e l 2 ce, a 3 = m e l l ce cosδ e an a 4 = m e l l ce sinδ e. The same plant moel was also use in [9], [] to test the propose controllers there but without input isturbances, that is, η =. In our simulation, we use the same numerical values as in [9], [], [22, p. 396], that is, m =. m e = 2. I =.2 I e =.25 l c =.5 l ce =.6 l = δ e = π/6.

6 ..5.5 Output trakcing error: Joint Output tracking error: Joint time sec Fig. 2. Tracking errors e an e 2. Number of hien neurons time sec Fig. 3. Variation of the number of gri noes. We select the input isturbances η an η 2 to be the banlimite white noise signals. The manipulator is initially at rest, that is, q = q 2 = an q = q 2 =. We consier the same reference signals as in [], which are efine as q t = π 6 cos2πt an q 2t = π 4 cos2πt. Thus, we choose the gri bounaries for q, q, q 2 an q 2 to be [..], [ 4. 4.], [..] an [ ], respectively. For the controller, we choose k = [ 2], k 2 = [4 4], Q = Q 2 =.5I 2, f = g = o = 5, ν =.25, g =. an G = 2I 2. The rest of the esign parameters of the variable-structure RCRBF network are selecte as e max =.5, threshol = [ ], ω = ω 2 = 25, ω = ω 2 = 25, η = 5 an T =.5. In Fig. 2 an Fig. 3, we show the performance of the propose aaptive robust controller. VI. CONCLUSIONS A novel aaptive robust controller has been propose for the output tracking control of a class of MIMO uncertain systems, where a variable-structure RBF network is use to approximate unknown system ynamics. The RBF network can grow or shrink on-line ynamically accoring to the tracking performance. The raise-cosine RBF was employe in orer to guarantee the overall computational efficiency. To account for the effects of the structure variation of the RBF network in the stability analysis of the close-loop system, the piecewise continuous Lyapunov function for switche an hybri systems was use. Simulation results confirm the effectiveness of the propose aaptive robust controller. REFERENCES [] I. Kanellakopoulos, P. V. Kokotovic, an R. Marino, An extene irect scheme for robust aaptive nonlinear control, Automatica, vol. 27, no. 2, pp , 99. [2] A. Kojic an A. M. 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