denotes the class of nâtimes continuously differentiable func- tion; Df(x) = âf(x) ..... Jacobean of G, which is typically a large matrix needs to be inverted at least ...
Proceedings of the 40th IEEE Conference on Decision and Control Orlando, Florida USA, December 2001
FrA09-6
Trajectory Learning and Output Feedback Control of Nonlinear Discrete—Time Systems Nader Sadegh The George W. Woodruff School of Mechanical Engineering Georgia Institute of Technology, Atlanta Georgia 30332—0405
Abstract Output tracking control of a general class of nonlinear discrete—time systems is considered. A block input—output realization of the plant is first introduced that transforms the system into one with equal number of inputs and states. This realization is subsequently used to formulate a novel output feedback controller for regulation or tracking. The proposed controller only requires measurement of the system’s outputs, and is applicable to both minimum and nonminimum phase plants of arbitrary relative degree. For applications involving a finite—duration or periodic desired outputs, a new recursive algorithm based on the Newton—Rapson method is proposed to learn the feedforward input. A simulation example of a nonlinear non—minimum phase system is presented to further illustrate and evaluate the performance of the proposed control scheme.
1
Introduction
D f x, y
Many practical control applications particularly those in the motion control area demand precise tracking of a specified finite—duration or repetitive desired trajectory without requiring a precise knowledge of the system’s model. The learning control techniques evolved during the last decade offers an effective solution for many application ranging from robotics [3, 23] to disk drive actuation [26] and magnetic levitation [2]. The existing learning controllers can be divided into two broad categories: Iterative (ILC) [3, 6] and repetitive (RLC) [23, 7] learning controllers. Both controllers, adjusts the control input iteratively from cycle to cycle based on the information gathered from previous cycles in order to improve the future performance. Although similar in nature, the main difference between the two is that the ILC assumes resetting of the initial state of the system at the start of each training cycle while the RLC imposes no such restriction. Moreover, the RLC can be employed as an active component of the overall control system to account for changes that might take place during the operation of the system. The majority of the existing works on tracking control of nonlinear systems, particularly in the continuous—time domain, are based on feedback linearization; see for example [13, 12, 16, 2]. Although very powerful when applicable, they generally suffer from several drawbacks including stringent integrability conditions, in the case of input—state linearization, or the minimum phase requirement in the case of output 0-7803-7061-9/01/$10.00 © 2001 IEEE
feedback linearization. Both approaches also require measurements of the state variables. In this paper we formulate an integrated 2—tier tracking control system comprised of an output feedback controller and a higher level RLC for a fairly general class of nonlinear discrete—time plants. The only requirement we demand is that the plant be controllable and observable in the sense to be specified. The key approach exploited in formulating the output feedback part of our integrated controller is that of the ‘lifting’ or ‘block’ modeling technique [1]. This approach, which is mainly applicable to discrete—time or sampled data systems, has proven highly effective for many problems arising from control, estimation, and even realization of linear and nonlinear [10, 11, 20, 15, 22] dynamical systems. The role of the RLC will be to provide the desired state trajectory needed by the feedback controller. The effectiveness of the proposed control system is demonstrated by simulation. The following notations are freely used in the paper: C n denotes the class of n—times continuously differentiable func(x,y ) . The notation tion; Df (x) = ∂f∂x(x) ; x ( ) = ∂f∂x ( ) or ( ) means ( ) for a fixed or , respectively. y x Also, y ◦ ( ) = ( ( ) ) for a fixed . For matrices and (or vectors) 1 2 , define [ 1 2 ] := 1 2 1 ( 1 2 ) := . The notation denotes the vector 2 and matrix 2—norms, and ( )l denotes the —th row of matrix . h
x
h
h
y
h x, y
f x
y
M ,M
y
M
M ,M
M
M ,M
x
h f x ,y
M
.
M
l
M
M
2
Problem Statement
Consider an -th order SISO discrete—time system given by the state-space representation ( k k) (1) k+1 = ( k) k = n
f x ,u
x
h x
y
where k ∈ IRn is the state vector, k ∈ IR is the input, and ( ) and ( ) are at least twice continuously differentiable ( 2 ) functions. We let the set of admissible desired states, denoted by X , be a bounded and convex open subset of IRn (e.g., X ⊂ { ∈IRn : } for some finite 0). The set of desired outputs is defined below. Definition 2.1 A bounded sequence dk ∈ IR is said to be a x
f ., .
u
h .
C
x
x
< r
r >
y
sequence of desired outputs if there exist bounded sequences dk ∈ X and dk ∈ IR, referred to as the desired state and input sequences, respectively, such that dk+1 = ( dk dk ) and dk = ( dk ) for all . x
4032
u
x
y
h x
k
f x
,u
The control objective is to formulate a controller that uses only output feedback to force the output, k , to follow a sequence of desired periodic outputs dk and keep k inside X starting from an arbitrary initial condition 0 ∈ X . y
y
x
x
open neighborhood of the equilibrium state (see for example [16, 18]). Assumption 3.1. is necessary in order for the state to be uniquely determined from the input and output blocks. In [20] the conditions imposed by the last two assumptions are referred to as the local uniform —observability and — observability rank conditions. The following proposition, which is a direct consequence of the implicit function theorem [19], shows that fulfillment of assumption 3.1 implies existence of a differentiable control and observation function for the system (also see [16, 18, 20] for similar results). Proposition 3.1 Consider a system whose block input— output relationship (2) satisfies assumption 3.1. There exist differentiable functions Ψ : X × X → U and Ω : Y × U → X with the following properties: ) Both the admissible input and output sets, U and Y , respectively, are bounded and open. ) For all x z ∈ X and u ∈ U we have u = Ψ(x z) ⇐⇒ z = (x u) ) For all x ∈ X , y ∈ Y , and u ∈ U , we have x = Ω(y u) ⇐⇒ y = (x u). ii
n
3
Block Realization
The block input—output realization is a convenient state realization of a dynamical system (1) as it ‘lifts’ the dimensions of the input and output vectors to that of the state vector resulting in a ‘square’ system. In this paper, we consider a realization that relates a block of outputs to a block of inputs defined by uk = ( nk nk+n−1 ) and yk = ( nk nk+n−1 ). Let the state transition map starting from an initial state via the input sequence 1 2 i be denoted by i ( u) := ui ◦ ui−1 ◦ · · · ◦ u2 ◦ u1 ( ). n
n
y
u
,... ,u
,... ,y
x
f
u ,u ,... ,u
x,
f
f
f
f
x
Applying the state equation (1), sequentially to evaluate nk+1 nk+n , yields the block input-output realization: xk+1 = (xk uk ) (2) yk = (xk uk ) where k , the state vector of the block realization, co(x ) = n (x u), and (x u) = incides with nk , x
,... ,x
F
,
H
,
x
x
F
,u
f
,
H
,
(x) ◦ u1 (x) ◦ un−1 (x) . Furthermore, the partial derivatives of and , which play a key role in our subsequent development, are denoted by J(x u) := x (x u), Q(x u) := u (x u), O(x u) := x (x u), and L(x u) := u (x u). These quantities may be computed by applying the chain rule to functions and or using a more efficient differential algebraic approach as in [22, 14]. The matrices Q and O, referred to as the controllability and observability matrices, respectively, are the generalizations of the well known controllability and observability matrices for linear systems. For example, for the linear system k+1 = A k +b k , k = c k , it can be easily seen that Q = An−1 b · · · Ab b and O = c cA · · · cAn−1 . To ensure controllability and observability of the system (1), we make the following assumptions: h
,h
f
,... ,h
f
F
H
,
D F
,
,
D F
D H
,
,
,
D H
F
x
x
u
y
,
H
x
,
,
,
,
Assumption 3.1 ) The system is strongly controllable (with respect to X )[10, 16]: ∀x z ∈ X , there exists a unique input vector u ∈ IRn such that (x u) = . We shall denote the corresponding set of admissible inputs and outputs by U = {u ∈IRn : z = (x u) for some x z ∈ X} and Y = {y ∈IRn : y = (x u) for some x ∈ X u ∈ U} i
,
F
,
,
H
,
z
,
,
,
,
which are both nonempty by the assumption. ) The system is strongly observable (with respect to X ) [11, 20]: For each u ∈ U the output function u is one to one: (x1 u) = (x2 u) =⇒ x1 = x2 ∀x1 x2 ∈ X , and u ∈U . ) The controllability and observability matrices, Q(x u) and O(x u), respectively, have full rank for all u ∈ U and x ∈ X . Remark 3.1 The assumptions made here about the system are rather standard controllability and observability assumptions for nonlinear systems also adopted by others [11, 20, 16, 18] in the literature. They are automatically satisfied if the linearized system is controllable and observable about an equilibrium point with X confined to an ii
H
H
iii
,
H
,
,
,
,
i
ii
,
F
,
,
iii
,
H
,
Remark 3.2 Evaluating the control and observation func-
tions, Ψ and Ω introduced in proposition 3.1, amounts to solving a system of nonlinear equations. For example, to determine Ψ(x z), we must find u that satisfies (x u) = z. A widely used method for this purpose is the iterative Newton— Raphson method. For other observer designs and numerical techniques for estimating the observed states see references [11, 25, 20]. An alternative computational tool for evaluating the control and observation functions is a perceptron neural network [21, 16, 18, 17]. Using analytical and/or experimental data, a network can be trained to learn the values of these functions at various operating points. ,
F
,
,
,
F
n
4
Control Law Formulation
In this section we formulate a nonlinear feedback control law that forces the state of the system x to follow a sequence of desired states, x ∈ X , using output feedback only. Given that x ∈ X and x +1 ∈ X , by proposition 3.1, setting the control input to u = Ψ(x +1 x ) (3) k
dk
k
d,k
k
d,k
,
k
gives x +1 = (x u ) = x +1 . Of course, as can be seen this control law requires knowledge of the state x . To relax this requirement, we formulate a state observer that determines x from y and u −1. Given that x ∈ X and u ∈ U , then by proposition 3.1, y = (x u ) =⇒ x = Ω(y u ). The state x +1 can now be determined using the block state equation (2): x +1 = Θ(y u ) := (Ω(y u ) u ) (4) F
k
k,
k
d,k
k
k
k
k
k
H
k
k,
k
k
k,
k
k
k
k,
k
F
k
k,
k ,
k
or equivalently x = Θ(y −1 u −1 ), which is equivalent to a dead-beat observer [20] as it provides the current state based on the previous inputs and outputs. The following theorem [24] shows that, if initial state and desired states are in X , the combined controller/observer given by (3) and (4) achieves the stated control objective.
4033
k
k
,
k
r
u
Ψ
y
System
z -1
in section 6, may lead to an unacceptably large final tracking error. In this paper we opt for a modified Newton—Raphson algorithm to learn the desired state trajectory. To this end, let X := (x 0, ··· , x −1), Y := (y 0 , ··· , y −1 ), and G(X ) := (G(x 0, x 1), ··· , G(x −1, x 0)) Assuming DG(X ) is invertible for any X ∈ X , the modified Newton—Raphson algorithm may be used to update X at the end of each cycle: −1 X +1 = X − γ DG(X ) E (6) d
d,
d,N
d
x
Θ
d
d,
d,
d,
d,N
d,N
d
d,
N
d
d
Figure 1: The Output Feedback Control Block Diagram
Theorem 4.1 Consider the block input—output system (2)
satisfying assumption 3.1. There exists an output feedback law of the form (5) u = (y −1 u −1 x +1 ) ≥ 1 k
K
,
k
,
k
,
d,k
k
where (y u z) := Ψ(z Θ(y u)) that guarantees xk = xdk ∀ ≥ 2 provided x0 and xdk ∈ X Remark 4.1 The output feedback control law (5) guaranK
,
,
,
,
k
.
tees exact tracking in finite time (i.e., dead-beat control) when the model of the system is perfectly known. In the case that there is a mismatch between the actual system and its model, the convergence of xk to xdk is at best asymptotic. Later in this paper, we will demonstrate the robustness of the proposed controller to modeling mismatch by means of a simulation case study.
i
Learning Tracking Control
,
G
,
H
,
G
D G
,
,
,
,
D G
,
,
,
period and xdk the corresponding periodic desired states. By proposition 5.1 there exists a differentiable function G : n N
X × X → IR
such that y = G(x , x +1), j = 0, . . . , N − 2 y −1 = G(x −1, x 0) Solving the preceding system of equations (nN equations and nN unknowns) for the desired states, x ’s, yields the information needed in the output feedback control law (5). There are several different ways of solving this system of equations numerically [4, 9, 5, 24], the two most commonly used of which are gradient descent and Newton’s method. The gradient descent investigated in [24] generally has a slower convergence rate and, as can be seen from the simulation results d,j
d,N
d,j
d,N
d,j
d,
d,j
i
i
i
d
i
d
k
G
,
G
G
Gj X
G
,
j,
,
j
,
X
,··· ,
Gj X
G
,
,
,
D G
N
j,
,
j
,
j,k
The output feedback control law (5) requires knowledge of the desired state trajectory, which in most cases is difficult to determine from the desired output trajectory. In this section, we will develop a learning algorithm for on—line estimation of the desired state sequence corresponding to a periodic or finite duration desired output. To begin our formulation, we need the following proposition [24]: Proposition 5.1 Applying the output feedback control law (5) to the input—output block system (2) results in the following state—output equation: yk = (xdk xdk+1 ) where (x z) = (x Ψ(x z)). Furthermore, the partial derivatives of , denoted by R := x and S := z , are given by R(x z) = O(x u) − LQ−1 J(x u) and S(x z) = LQ−1(x u)where u = Ψ(x z). Let ydk ∈ Y be a periodic sequence of desired outputs of G
i
where X is the estimate of X at the end of the i—th cycle, E = G(X ) − Y , and 0 < γ ≤ 1 is a positive scalar. This algorithm may be difficult to implement because the value G must be evaluated based on the plant outputs, some of which may not be available at the updating instant, and that the Jacobean of G, which is typically a large matrix needs to be inverted at least once per cycle. To avoid these problems, we propose a causal recursive version of (6), which distributes the required computations over the entire period by slightly modifying the desired state trajectory at each sampling instant . Before presenting our learning algorithms, we need to introduce a few additional notations regarding the state— output (x z) in proposition 5.1. First, we let ˆ (x z) denote an estimate of (x z) to be used for computing its Jacobians, and for = (x0 x −1 ), we define ˆ (x z) = ( ) := (x x +1) ˆ ( ) := ˆ (x x +1), R ˆ ˆ ˆ x (x z) and S(x z) = z (x z). To bookkeep the estimated desired states, let xˆ denote the estimate of x at time for 0 , and ˆ := (xˆ0 x ˆ −1 ). For indexes outside this range xˆ is interpreted as a circular buffer of length : xˆ = xˆ where = mod (i.e., = + ) for some . Finally, we denote the estiˆ := R ˆ (x ˆ := mate dependent Jacobians by R ˆ x ˆ +1 ), S ˆ (x S ˆ x ˆ +1 ), and Φ := ˆ ( ˆ ) = mod , where we note that Φ , which plays the role of our regressor, is a ˆ k , Sˆk , 0, . . . , 0 . highly sparse matrix of the form 0, . . . , 0, R Remark 5.1 Let us examine more closely the invertibility requirement for the Jacobian of G(Xd ) in the linear time— invariant case, where DG(X ) is a constant matrix and Y = DG(Xd )X . In this case DG(X ) is singular if and only if there exists a nonzero X for which Y = 0. This amounts to having a zero output sequence in the presence of a nonzero periodic state (and consequently a nonzero periodic input), which in turn indicates the presence of a zero at e2πkj/N for some k. Thus in the linear case, DG(X ) is invertible if and only if the uncontrolled system has no zeros at any e2πkj/N , k = 0, 1, . . . , N − 1. Obviously, this situation can always be avoided by readjusting the repetition period N . The following orthogonalized projection algorithm [8] which shall be referred to as the learning algorithm, provides a recursive variation of the modified Newton—Raphson algorithm in (6): ˆ k+1 = Xˆ k − γ Pk ΦTk Φk Pk ΦTk −1 ¯e X (7) −1 ˆ ˆ ˆ ¯e = e + γ − 1 Φ X − X + R (xˆ − xˆ −1 ) (8) −1 P +1 = P − P Φ Φ P Φ Φ P , P = I × (9) D G
5
i
4034
k
< j
d,j
< N
Xk
,k , · · · ,
N
,k
p,k
N
p
Ni
p,k
j
j
j,k
j,k ,
k
j,k ,
j
p
N
i
,k
DGj Xk , j
k
j
k
,k
k
N
k
k
k
k
k
k
k
k
T k
k
k
k
iN
T k
k
k
k
j,k
j,k
iN
N
N
for i = 0, 1, 2, . . . , j = 0, 1, . . . , N − 1, and k = iN + j where e =y −y . Remark 5.2 The recursive algorithm given by (7)—(9) is a causal one as at time k it updates Xˆ based on the past output vector y −1 = (y − , . . . , y −1 ). The linear dependence of Φ ’s (i.e., invertibility of the Jacobian) guarantees that the projection matrix P is well defined. As can be seen, P is reset to identity at the beginning of each repetition period. The following lemma shows that combining the output feedback control law (5) with the learning algorithm is equivalent to the Newton—Raphson updating at the beginning of each period with a slightly modified error vector: Lemma 5.1 Suppose that the estimated desired state vector, instead of the actual one, is used in the output feedback control law (5) at the time instant k: u = K (y −1 , u −1, xˆ +1 ). Letting X = Xˆ , then the learning algorithm given by (7)—(9) gives X +1 = X − γ Φ¯ −1 Eˆ where Φ¯ = (Φ , . . . , Φ + −1 ), Eˆ = (ˆe , . . . , ˆe + −1 ), ˆ (x ˆe = e + R ˆ − xˆ −1 ) + Sˆ (xˆ +1 − xˆ +1 ) k
k
d,k
k
k
nk
n
nk
k
k
k
k
i
iN
k
k
j
,k
iN i
iN
k
k
i
N
j,iN
k
i
i
iN
iN
j,k
i
N
j
k
i
,iN
j
,k
and e = G(xˆ −1, xˆ +1 ) − y . More generally, the intermediary estimates are given by Xˆ +1 = X − γ (Φ , . . . , Φ )+ (ˆe , . . . , ˆe ) (10) k
j,k
j
,k
i
k
d,k
iN
k
iN
k
k = iN, . . . , iN + N − 1 where A+ is pseudo—inverse of A, i.e., A+ = A (AA )−1 . T
T
For ease of notation we only prove the lemma for
Proof.
i = 0. The proof for i > 0 is completely parallel to that of
= 0. First, from (9), it can be seen that the symmetric matrix P +1 is an orthogonal projection onto the null space of (Φ0 , . . . , Φ ) implying that P +1 Φ = 0, j = 1, . . . , k . ˆ +1 − Xˆ = X from left yields Φ Multiplying (7) by Φ −γ¯e and Φ Xˆ +1 − Xˆ = 0 for j = 0, . . . , k − 1. Thus (Φ0, . . . , Φ ) Xˆ +1 − Xˆ 0 = −γ (ˆe0 , . . . , ˆe ) k = 1, . . . , N − 1, where ˆ e = ¯e − γ −1 Φ Xˆ − Xˆ 0 . Using the expression for ¯e from 8 and Φ Xˆ − Xˆ0 yields ˆe = e +Rˆ (xˆ 0 − xˆ −1)+Sˆ (xˆ +1 0 − xˆ +1 ). The proof of the lemma is completed by noting that y = G(xˆ −1 , xˆ +1 ), from theorem 4.1 and propositions and 5.1. The following theorem guarantees the convergence of the proposed algorithm for sufficiently small learning gain and initial tracking error: Theorem 5.1 Consider the control law (5) together with the learning control law given by (7)—(8) applied after the first cycle to the block input—output system (2) satisfying assump tion 3.1. Let α := sup ∈X DGˆ (x, z) − DG(x, z) , and 2 1 2 e0 := . There exist α¯ > 0, e¯0 > 0, =1 (y − y ) and 0 < γ < 1 such that if α < α¯ , e0 < e¯0 the output tracking over a cycle approaches zero asymptotically, i.e., lim →∞ y y = 0. Proof. Let X ∈ X denote a solution to Y = G(X ), and as in lemma 5.1, X = Xˆ +1 . Since G is C 1 , by the mean-value theorem [19], it can see that G(X +1) = G(X )+ +1 = X − X , and ∆ is an nN × nN such ∆ δX where δX that ∆ = DG(ξ ) , where ξ is a point lying on the line i
k
k
T j
k
j
j
k
k
k
k
k
j
j
j
j
j
j,
k
k
k
j
k
j
j,j
j
j
,
j
j
j
j
,j
k
j,k
j
,k
x,z
nN k
| k −
k
/
k
k,d
k,d |
N
d
d
i
i
i
i
i
i
l
i
i l
l
d
iN
i
i l
i
segment joining X to X +1. By the convexity ofX , ξ ∈ X , ∀l = 1, . . . , nN . Let σ > 0 be such that ∆ < σ for all possible ξ ∈ X¯ , which always exists by the compactness of X¯ . Using E = G(X )−Y and δX = −γ Φ¯ −1 Eˆ given by lemma 5.1 in the preceding equation gives E +1 = E −γ Λ Eˆ −1 ¯ where Λ = ∆ Φ . Applying the mean—value theorem to ˆ +1 ) in the error estimate expression provided G(x ˆ −1 , x by lemma 5.1, after some algebra, we get Eˆ = E + E˜ , ˜ = (˜e , . . . , ˜e + −1) where E ˜ + δ −1 + S ˜ + δ , δ =x ˜e + = R ˆ +1 + − x ˆ +1 ˆ ˜ = R(η , η +1 ) − R R + ˜ + S = S(η , η +1 ) − Sˆ i
i
i l
i
i l
N
i
i
i
d
i
i
i
i
i
j,k
j
i
iN iN
iN
j
j
i
i
N
iN
j
i j
i j,l
l
j
i
,k
iN
iN
i
i
i
i
N
i j
i j,l
l
iN
i j
i j
j
,l
k
,l
k
i j
j
,iN
j
j
l
l
and η is on the line segment joining ˆ + −1 to x ˆ . Tak x ˜ ≤ r˜ δ−1 + δ E ing the norm of E˜ it follows that ˜ , S ˜ and δ = δ1 , . . . , δ −1 where r˜ = 2 max R (note that δ1 = 0). Using that δ , j = 1, . . . , N , is a subvector of ˆ + − X = γ (Φ , . . . , Φ + −1)+ (ˆe , . . . , ˆe + −1) X as in lemma 5.1, it follows that there exists a constant ˆ . Similarly, since β1 > 0 such that δ ≤ γβ1 E δ−1 = δ −−11 − δ −1 , there exists a constant β2 > 0 such that δ ≤ γβ2 Eˆ −1 . Let β be an upper bound for max {β1 , β2 } for all possible X , X +1 ∈ X , which does exist by the compactness of X . Thus ˜ ≤ r˜ βγ Eˆ −1 + Eˆ E (11) Substituting for E +1 and E into E +1 = E − γ Λ Eˆ in terms of Eˆ +1 and Eˆ yields ˆ +1 = (I N − γ Λi )Eˆ i + E˜ i+1 − E˜ i E (12) Let > 0 be such that < min{IN N − Λi , r˜i β } for all possible X i , X i+1 ∈ X N . From the definition of ∆i and the hypothesis it follows that → 0 as α, γ → 0. Thus there exist α ¯, and 0 < γ¯ < 1 such that if α ≤ α¯, γ ≤ γ¯ then < 1/5, which will be needed in subsequent analysis. This choice will also make IN N − γ Λi ≤ 1 − γ (1 − ) < 1 and r˜i βγ < 1, which upon using in (12) yields i E ≤ (1 + γ) Eˆ i + γ Eˆ i 1 i i 1 ˆ ˆ i+1 Eˆ ≤ [1 − γ (1 − 3)] E + γ E (13) 1 − γ Let r > 0 be such that Br (Xd ) ⊂ X N ,where Br (X ) denotes 0 the open ball of radius r centered at X . Choosing 0 X ∈ Br (Xd ), setting γ < γ¯ prescribed earlier, and E = e¯0 < i 0 i 5r 7σ , we shall show that Eˆ ≤ E and X ∈ Br (Xd ), ∀i ≥ 0, using an induction argument. Suppose that Eˆ i ≤ 0 E and X i ∈ Br (Xd ) for all i ≤ i. We first show that X i+1 ∈ Br (Xd ): If that is not the case, then it is possible to choose γ < γ¯ such that X i+1 − Xd = r. On the other hand, by inequality (13) and the induction hypothesis, i+1 [1 − γ (1 − 3)] E 0 + γ E 0 0 Eˆ ≤ ≤ E 1 − γ
4035
i j,l
j,iN
i
i
k
i
iN
iN
j
iN
i
i N
i
i
i
i N
i j
i
j
j,iN
i
i
k
i
iN
j
i
iN
j
i
i N
i
i
i
i
i
i
i
i
N
i
i
i
i
i
i
i
i
i
N×
×
×
−
−
,iN
Desired Tip Output (rad)
mp u y
4 2 0 -2 -4 0
0.2
0.4
0.6
0.8
1 1.2 Time(sec)
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1 1.2 Time(sec)
1.4
1.6
1.8
2
200 Desired Input (N.m)
Figure 2: Single Flexible Arm since < 1/5. Using E i+1 = G(X i+1) − G(Xd ), e¯0 < 75σr , 1, and < 1/5 yield i+1 X − Xd ≤ σ (1 + γ) Eˆ i+1 + γ Eˆ i < r ≤ E 0 . This proves that Thus X i+1 ∈ Br (Xd )andEˆ i+1 i+1 ∈ B (X ) andE ˆ i ≤ E 0 for i = 0, 1, 2, .... As a X r d result, i+1 ˆ ≤ c1 Eˆ i + c2 Eˆ i−1 E where c1 = 1−γ1−(1γ−3) and c2 = 1−γγ , and by the choice i+1 i ˆ of γ and , c1 + c2 < 1. Defining zi = E , Eˆ , then it follows that zi+1 ≤ Czi , where C = c11 c02 . Since c1 + c2 < 1 the eigenvalues of C are less than 1 and |zi | ≤ |C i ||z0 | converges to zero exponentially. Thus E i also γ
