1 Relative degree and local normal forms
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- Briana Edwards
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1 THE ZERO DYNAMICS OF A NONLINEAR SYSTEM 1 Relative degree and local normal orms The purpose o this Section is to show how single-input single-output nonlinear systems can be locally given, by means o a suitable change o coordinates in the state space, a normal orm o special interest, on which several important properties can be elucidated. The point o departure o the whole analysis is the notion o relative degree o the system, which is ormally described in the ollowing way. The single-input single-output nonlinear system ẋ = (x) + g(x)u (1) y = h(x) is said to have relative degree r at a point x i 1 (i) L g L k h(x) = 0 or all x in a neighborhood o x and all k < r 1 (ii) L g L r 1 h(x ) 0. Note that there may be points where a relative degree cannot be deined. This occurs, in act, when the irst unction o the sequence L g h(x), L g L h(x),..., L g L k h(x),... which is not identically zero (in a neighborhood o x ) has a zero exactly at the point x = x. However, the set o points where a relative degree can be deined is clearly an open and dense subset o the set U where the system (1) is deined. Remark. In order to compare the notion thus introduced with a amiliar concept, let us calculate the relative degree o a linear system ẋ = Ax + Bu y = Cx. In this case, since (x) = Ax, g(x) = B, h(x) = Cx, it easily seen that and thereore L k h(x) = CAk x L g L k h(x) = CAk B. Thus, the integer r is characterized by the conditions CA k B = 0 or all k < r 1 CA r 1 B 0. 1 Let λ be real-valued unction and an n-vector-valued vector, both deined on a subset U o R n. The unction L λ is the real-valued unction deined as L λ(x) = n i=1 This unction is sometimes called derivative o λ along. λ i(x) := λ x i x (x). 1
2 It is well-known that the integer satisying these conditions is exactly equal to the dierence between the degree o the denominator polynomial and the degree o the numerator polynomial o the transer unction o the system. T (s) = C(sI A) 1 B We illustrate now a simple interpretation o the notion o relative degree, which is not restricted to the assumption o linearity considered in the previous Remark. Assume the system at some time t is in the state x(t ) = x and suppose we wish to calculate the value o the output y(t) and o its derivatives with respect to time y (k) (t), or k = 1, 2,..., at t = t. We obtain y(t ) = h(x(t )) = h(x ) y (1) (t) = h dx x dt = h ((x(t)) + g(x(t))u(t)) x = L h(x(t)) + L g h(x(t))u(t). I the relative degree r is larger than 1, or all t such that x(t) is near x, i.e. or all t near t, we have L g h(x(t)) = 0 and thereore This yields y (2) (t) = L h dx x dt y (1) (t) = L h(x(t)). = L h ((x(t)) + g(x(t))u(t)) x = L 2 h(x(t)) + L gl h(x(t))u(t). Again, i the relative degree is larger than 2, or all t near t we have L g L h(x(t)) = 0 and Continuing in this way, we get y (2) (t) = L 2 h(x(t)). y (k) (t) = L k h(x(t)) or all k < r and all t near t y (r) (t ) = L r h(x ) + L g L r 1 h(x )u(t ). Thus, the relative degree r is exactly equal to the number o times one has to dierentiate the output y(t) at time t = t in order to have the value u(t ) o the input explicitly appearing. Note also that i L g L k h(x) = 0 or all x in a neighborhood o x and all k 0 (in which case no relative degree can be deined at any point around x ) then the output o the system is not aected by the input, or all t near t. As a matter o act, i this is the case, the previous calculations show that the Taylor series expansion o y(t) at the point t = t has the orm y(t) = L k h(x ) (t t ) k k! k=0 2
3 i.e. that y(t) is a unction depending only on the initial state and not on the input. These calculations suggest that the unctions h(x), L h(x),..., L r 1 h(x) must have a special importance. As a matter o act, it is possible to show that they can be used in order to deine, at least partially, a local coordinates transormation around x (recall that x is a point where L g L r 1 h(x ) 0). This act is based on the ollowing property. Lemma 1 The row vectors 2 are linearly independent. dh(x ), dl h(x ),..., dl r 1 h(x ) Lemma 1 shows that necessarily r n and that the r unctions h(x), L h(x),..., L r 1 h(x) qualiy as a partial set o new coordinate unctions around the point x. As we shall see in a moment, the choice o these new coordinates entails a particularly simple structure or the equations describing the system. However, beore doing this, it is convenient to summarize the results discussed so ar in a ormal statement, that also illustrates a way in which the set o new coordinates can be completed in case the relative degree r is strictly less than n. Proposition 1 Suppose the system has relative degree r at x. Then r n. I r is strictly less than n, it is always possible to ind n r more unctions ψ 1 (x),..., ψ r r (x) such that the mapping Φ(x) = ψ 1 (x)... ψ n r (x) h(x) L h(x)... L r 1 has a jacobian matrix which is nonsingular at x and thereore qualiies as a local coordinates transormation in a neighborhood o x. The value at x o these additional unctions can be ixed arbitrarily. Moreover, it is always possible to choose ψ 1 (x),..., ψ n r (x) in such a way that L g ψ i (x) = 0 or all 1 i n r and all x around x. h(x) The description o the system in the new coordinates is ound very easily. Set ψ 1 (x) h(x) z = ψ 2 (x), ξ = L h(x) ψ n r (x) h(x) L r 1 2 Let λ be a real-valued unction deined on a subset U o R n. Its dierential, denote dλ(x) is the row vector ( λ dλ(x) = x 1 λ x 2 λ ) := λ x n x. 3
4 and x = (z, ξ). Looking at the calculations already carried out at the beginning, we obtain or ξ 1,..., ξ r dξ 1 dt dξ r 1 dt = h dx x dt = L h(x(t)) = ξ 2 (t) = (Lr 2 h) dx x dt = Lr 1 h(x(t)) = ξ r (t). For ξ r we obtain dξ r dt = Lr h(x(t)) + L gl r 1 h(x(t))u(t). On the right-hand side o this equation we must now replace x(t) with its expression as a unction o x(t), which will be written as x(t) = Φ 1 (z(t), ξ(t)). Thus, setting the equation in question can be rewritten as dξ r dt q(z, ξ) = L r h(φ 1 (z, ξ)) b(z, ξ) = L g L r 1 h(φ 1 (z, ξ)) = q(z(t), ξ(t)) + b(z(t), ξ(t))u(t). Note that at the point (z, ξ ) = Φ(x ), b(z, ξ ) 0 by deinition. Thus, the coeicient a(z, ξ) is nonzero or all (z, ξ) in a neighborhood o (z, ξ ). As ar as the other new coordinates are concerned, we cannot expect any special structure or the corresponding equations, i nothing else has been speciied. However, i ψ 1 (x),..., ψ n r (x) have been chosen in such a way that L g ψ i (x) = 0, then Setting dz i dt = ψ i x ((x(t)) + g(x(t))u(t)) = L ψ i (x(t)) + L g ψ i (x(t))u(t) = L ψ i (x(t)). L ψ 1 (Φ 1 (z, ξ)) 0 (z, ξ) = L ψ n r (Φ 1 (z, ξ)) the latter can be rewritten as dz dt = 0(z(t), ξ(t)). Thus, in summary, the state-space description o the system in the new coordinates will be as ollows ż = 0 (z, ξ) ξ 1 = ξ 2 ξ 2 = ξ 3 (2) ξ r 1 = ξ r ξ r = q(z, ξ) + b(z, ξ)u. 4
5 In addition to these equations one has to speciy how the output o the system is related to the new state variables. But, being y = h(x), it is immediately seen that y = ξ 1. (3) The equations thus deined are said to be in normal orm. They are useul in understanding how certain control problems can be solved. 2 The Zero Dynamics In this section we introduce and discuss an important concept, that in many instances plays a role exactly similar to that o the zeros o the transer unction in a linear system. We have already seen that the relative degree r o a linear system can be interpreted as the dierence between the number o poles and the number o zeros in the transer unction. In particular, any linear system in which r is strictly less than n has zeros in its transer unction. On the contrary, i r = n the transer unction has no zeros; thus, a nonlinear system having relative degree r = n in some sense analogue to a linear systems without zeros. We shall see in this section that this kind o analogy can be pushed much urther. Consider a nonlinear system with r strictly less than n and look at its normal orm. Recall that, i x is such that (x ) = 0 and h(x ) = 0, then necessarily the set ξ o the last r new coordinates is 0 at x. Note also that it is always possible to choose arbitrarily the value at x o the irst n r new coordinates, thus in particular being 0 at x. Thereore, without loss o generality, one can assume that ξ = 0 and z = 0 at x. Thus, i x was an equilibrium or the system in the original coordinates, its corresponding point (z, ξ) = (0, 0) is an equilibrium or the system in the new coordinates and rom this we deduce that q(0, 0) = 0 0 (0, 0) = 0. Suppose now we want to analyze the ollowing problem, called the Problem o Zeroing the Output. Find, i any, pairs consisting o an initial state x and o an input unction u ( ), deined or all t in a neighborhood o t = 0, such that the corresponding output y(t) o the system is identically zero or all t in a neighborhood o t = 0. O course, we are interested in inding all such pairs (x, u ) and not simply in the trivial pair x = 0, u = 0 (corresponding to the situation in which the system is initially at rest and no input is applied). We perorm this analysis on the normal orm o the system. Recalling that in the normal orm y(t) = ξ 1 (t), we see that the constraint y(t) = 0 or all t implies that is ξ(t) = 0 or all t. ξ 1 (t) = ξ 2 (t) =... = ξ r (t) = 0, 5
6 Thus, we see that when the output o the system is identically zero its state is constrained to evolve in such a way that also ξ(t) is identically zero. In addition, the input u(t) must necessarily be the unique solution o the equation 0 = q(z(t), 0) + b(z(t), 0)u(t) (recall that b(z(t), 0) 0 i z(t) is close to 0). As ar as the variable z(t) is concerned, it is clear that, being ξ(t) identically zero, its behavior is governed by the dierential equation ż(t) = 0 (z(t), 0). (4) From this analysis we deduce the ollowing acts. I the output y(t) has to be zero, then necessarily the initial state o the system must be set to a value such that ξ(0) = 0, whereas z(0) = z can be chosen arbitrarily. According to the value o z, the input must be set as q(z(t), 0) u(t) = b(z(t), 0) where z(t) denotes the solution o the dierential equation ż(t) = 0 (z(t), 0) with initial condition z(0) = z. Note also that or each set o initial data ξ = 0 and z = z the input thus deined is the unique input capable to keep y(t) identically zero or all times. The dynamics o (4) correspond to the dynamics describing the internal behavior o the system when input and initial conditions have been chosen in such a way as to constrain the output to remain identically zero. These dynamics, which are rather important in many o our developments, are called the zero dynamics o the system. Remark. In order to understand why we used the terminology zero dynamics in dealing with the dynamical system (4), it is convenient to examine how these dynamics are related to the zeros o the transer unction in a linear system. Let T (s) = K b 0 + b 1 s + + b n r 1 s n r 1 + s n r a 0 + a 1 s + + a n 1 s n 1 + s n denote the transer unction o a linear system (where r characterizes, as expected, the relative degree). Suppose the numerator and denominator polynomials are relatively prime and consider a minimal realization o T (s) ẋ = Ax + Bu y = Cx with A = a 0 a 1 a 2 a n 1 C = ( b 0 b 1 b n r ). 6 B = K
7 Let us now calculate its normal orm. For the last r new coordinates we know we have to take ξ 1 = Cx = b 0 x 1 + b 1 x b n r 1 x n r + x n r+1 ξ 2 = CAx = b 0 x 2 + b 1 x b n r 1 x n r+1 + x n r+2 ξ r = CA r 1 x = b 0 x r + b 1 x r b n r 1 x n 1 + x n. For the irst n r new coordinates we have some reedom o choice (provided that the conditions stated in Proposition 1 are satisied), but the simplest one is z 1 = x 1 z 2 = x 2 z n r = x n r. This is indeed an admissible choice because the corresponding coordinates transormation x = Φ(x) has a jacobian matrix with the ollowing structure Φ x = ( ) and thereore nonsingular. In the new coordinates we obtain equations in normal orm, which, because o the linearity o the system, have the ollowing structure ż = F z + Gξ ξ 1 = ξ 2 ξ 2 = ξ 3 ξ r 1 = ξ r ξ r = Hz + Kξ + bu where H and K are row vectors and F and G matrices, o suitable dimensions. The zero dynamics o this system, according to our previous deinition, are those o ż = F z. The particular choice o the irst n r new coordinates (i.e. o the elements o z) entails a particularly simple structure or the matrices F and G. As a matter o act, is easily checked 7
8 that dz 1 dt dz n r 1 dt dz n r dt = dx 1 dt = x 2(t) = z 2 (t) = dx n r 1 dt = x n r (t) = z n r (t) = dx n r = x n r+1 (t) = b 0 x 1 (t) b n r 1 x n r (t) + ξ 1 (t) dt = b 0 z 1 (t) b n r 1 z n r (t) + ξ 1 (t) rom which we deduce that F = , G = b 0 b 1 b 2 b n r From the particular orm o this matrix, it is clear that the eigenvalues o F coincide with the zeros o the numerator polynomial o T (s), i.e. with the zeros o the transer unction. Thus it is concluded that in a linear system the zero dynamics are linear dynamics with eigenvalues coinciding with the zeros o the transer unction o the system. 3 Global Normal Forms We address, in this section, the problem o deriving the global version o the coordinates transormation and normal orm introduced in section 1. Consider a single-input singleoutput system described by equations o the orm ẋ = (x) + g(x)u y = h(x) in which (x) and g(x) are smooth vector ields, and h(x) is a smooth unction, deined on R n. Assume, as usual, that (0) = 0 and h(0) = 0. This system is said to have uniorm relative degree r i it has relative degree r at each x R n. I system (5) has uniorm relative degree r, the r dierentials dh(x), dl h(x),..., dl r 1 h(x) are linearly independent at each x R n and thereore the set Z = {x R n : h(x) = L h(x) =... = L r 1 h(x) = 0} (which is nonempty in view o the hypothesis that (0) = 0 and h(0) = 0) is a smooth embedded submaniold o R n, o dimension n r. In particular, each connected component o Z is a maximal integral maniold o the (nonsingular and involutive) distribution = (span{dh, dl h,..., dl r 1 h}). The submaniold Z is the point o departure or the construction a globally deined version o the coordinates transormation considered in section 1. 8 (5)
9 Proposition 2 Suppose (5) has uniorm relative degree r. Set α(x) = Lr h(x) L g L r 1 h(x) and consider the (globally deined) vector ields β(x) = 1 L g L r 1 h(x) Suppose the vector ields (x) = (x) + g(x)α(x), g(x) = g(x)β(x). τ i = ( 1) i 1 ad i 1 g(x), 1 i r (6) are complete. Then Z is connected. Moreover, the smooth mapping Φ : Z R r R n (z, (ξ 1,..., ξ r )) Φ τ 1 ξ r Φ τ 2 ξ r 1 Φ τ r ξ 1 (z), in which as usual Φ τ t (x) denotes the low o the vector ield τ, has a globally deined smooth inverse (z, (ξ 1,..., ξ r )) = Φ 1 (x) (8) in which z = Φ τr h(x) Φτ 2 ξ i = L i 1 h(x) 1 i r. L r 2 h(x) Φτ 1 (x) L r 1 h(x) The globally deined dieomorphism (8) changes system (5) into a system described by equations o the orm (7) ż = 0 (z, ξ 1,..., ξ r ) ξ 1 = ξ 2 ξ r 1 = ξ r ξ r = q(z, ξ 1,..., ξ r ) + b(z, ξ 1,..., ξ r )u (9) where y = ξ 1 q(z, ξ 1,..., ξ r ) = L r h Φ(z, (ξ 1,..., ξ r )) b(z, ξ 1,..., ξ r ) = L g L r 1 h Φ(z, (ξ 1,..., ξ r )). I, and only i, the vector ields (6) are such that 3 [τ i, τ j ] = 0 or all 1 i, j r, 3 Let and g be vector ields on R n. Their Lie bracket, denoted [, g], is the vector ield o R n deined as [, g](x) = g (x) x x g(x). 9
10 then the globally deined dieomorphism (8) changes system (5) into a system described by equations o the orm ż = 0 (z, ξ 1 ) ξ 1 = ξ 2 ξ r 1 = ξ r ξ r = q(z, ξ 1,..., ξ r ) + b(z, ξ 1,..., ξ r )u y = ξ 1. (10) Note also that, i r < n, the submaniold Z is the largest (with respect to inclusion) smooth submaniold o h 1 (0) with the property that, at each x Z, there is u (x) such that (x) = (x) + g(x)u (x) is tangent to Z. Actually, or each x Z there is only one u (x) rendering this condition satisied, namely, u (x) = Lr h(x) L g L r 1 h(x). The submaniold Z is an invariant maniold o the (autonomous) system ẋ = (x) (11) and the restriction o this system to Z, which can be identiied with (n r)-dimensional system ż = 0 (z, 0,..., 0) o Z, is the system which characterizes the zero dynamics o (5). 4 Robust stabilization via partial-state eedback An typical setting in which normal orms are useul is a systematic method or stabilization in the large o certain classes o nonlinear system, even in the presence o parameter uncertainties. We discuss irst the case o systems having relative degree 1, i.e. we consider systems modelled by equations o the orm ż = 0 (z, ξ) ξ = q(z, ξ) + b(z, ξ)u (12) y = ξ 1 in which z R n 1 and ξ R. All unctions are smooth unctions o their arguments. About this system we assume the ollowing. The coeicient b(z, ξ) satisies b(z, ξ) b 0 > 0 or some b 0. Moreover, there is a positive deinite and proper smooth real-valued unction V (z) that satisies V z 0(z, 0) α( z ) z R n 1 10
11 in which α( z ) is a class K unction. This is equivalent to assume that the equilibrium z = 0 o ż = 0 (z, 0) is globally asymptotically stable. Consider now or (12) a control law u = ky in which k > 0, which yields the closed loop system ż = 0 (z, ξ) ξ = q(z, ξ) b(z, ξ)kξ. (13) For convenience, we set and rewrite the latter as in which x = ( ) z ξ ẋ = F k (x) (14) ( ) F k (x) = 0 (z, ξ). q(z, ξ) b(z, ξ)kξ Consider, or this system, the candidate Lyapunov unction W (x) = V (z) ξ2 which is positive deinite and proper. For any real number a 0, let Ω a = {x R n : W (x) a} denote the sublevel set consisting o all points o R n at which the value o W (x) is less than or equal to a and let B a = {x R n : x a} denotes the closed ball consisting o all points o R n whose norm does not exceed a. Since W (x) is proper, the set Ω a is a compact set or any a. Pick any arbitrarily large number R, any arbitrarily small number r. Since W (x) is positive deinite and proper, there exist numbers numbers 0 < d < c such that Ω d B r B R Ω c. Consider also the compact annular region S c d = {x Rn : d V (x) c}. Our goal is to show that i the gain coeicient k is large enough the unction Ẇ (x) := W x F k(x) 11
12 is negative at each point o Sd c. Observe, in this respect, that Ẇ (x) = V x 0(z, ξ) + ξq(z, ξ) b(z, ξ)kξ 2. To this end, we proceed as ollows. Consider the compact set Z = S c d {x R n : ξ = 0}. At each point o Z Ẇ (x) = V x 0(z, 0) α( z ) Since z 0 at each point o Z, there is a number a > 0 such that Ẇ (x) a x Z. Hence, by continuity, there is an open set Z ε containing Z such Ẇ (x) a/2 x Z ε. (15) Consider now the set S = {x S c d : x Z ε}. which is a compact set, let M = max{ V x S x 0(z, ξ) + ξq(z, ξ)} m = min{b(z, ξ)ξ 2 } x S and observe that m > 0 because b(z, ξ) b 0 > 0 and ξ cannot vanish at any point o S. Thus, since k > 0, we obtain Ẇ (x) M km x S. Let k 1 be such that M k 1 m = a/2. Then, i k k 1, This, together with (15) shows that Ẇ (x) a/2 x S. (16) k k 1 Ẇ (x) a/2 x S c d, as requested. This being the case, we see that or any trajectory with initial condition in Sd c, so long as z(t) Sd c we have W (x(t)) W (x(0)) (a/2)t. As a consequence, the trajectory in inite time enters the set Ω d, and remains there (because Ẇ (x) is negative on the boundary o this set). We summarize the result as ollows. 12
13 Proposition 3 Consider system (12). Suppose that b(z, ξ) b 0 > 0 or some b 0 and suppose that the equilibrium z = 0 o ż = 0 (z, 0) is globally asymptotically stable. Then, or every choice o two positive numbers R, r, with R r, there is a number k 1 such that, or all k k 1 there is a inite time T such that all trajectories o (13) with initial condition x(0) R remain bounded and satisy x(t) r or all t T. This property is commonly known as property o semi-global practical stabilizability, o the equilibrium (z, ξ) = (0, 0) o (12). To obtain asymptotic stability, an extra assumption is required. The simplest case in which this can be achieved is when the equilibrium z = 0 o ż = 0 (z, 0) is not just asymptotically, but also locally exponentially, stable, i.e. when the eigenvalues o the matrix F 0 := 0 (0, 0) (17) z have negative real part. Linear arguments can be invoked to prove the existence o a number k 2 such that, i k k 2, the equilibrium x = 0 o (12) is locally asymptotically stable. It is also possible to show that there is a number r such that, or all k k 2, the closed ball B r is always contained in the domain o attraction o x = 0. To check that this is the case, let P be the positive deinite solution o P F 0 + F T 0 P = 2I, which exists because all eigenvalues o F 0 have negative real part and consider, or the system the candidate quadratic Lyapunov unction yielding U(x) = z T P z ξ2 U(x) = 2z T P 0 (z, ξ) + ξq(z, ξ) b(z, ξ)kξ 2. Expand 0 (z, ξ) as in which Then, there is a number δ such that, and this yields 0 (z, ξ) = F 0 z + g(z) + [ 0 (z, ξ) 0 (z, 0)] g(z) lim = 0. z 0 z z δ g(z) 1 2 P z 2z T P [F 0 z + g(z)] = 2 z 2 + 2z T P g(z) z 2 z B δ. Since the unction [ 0 (z, ξ) 0 (z, 0)] is a continuously dierentiable unction that vanish at ξ = 0, there is a number M 1 such that 2z T P [ 0 (z, ξ) 0 (z, 0)] M 1 z ξ or all (z, ξ) such that z B δ and ξ δ. 13
14 Likewise, since q(z, ξ) is a continuously dierentiable unction that vanish at (z, ξ) = (0, 0), there are numbers N 1 and N 2 such that ξq(z, ξ) N 1 z ξ + N 2 ξ 2 or all (z, ξ) such that z B δ and ξ δ. Finally, since b(z, ξ) is positive and nowhere zero, there is a number b 0 such that (recall that k > 0) kb(z, ξ) kb 0 or all (z, ξ) such that z B δ and ξ δ. Putting all these inequalities together, one inds that, or all or all (z, ξ) such that z B δ and ξ δ U(x) z 2 + (M 1 + N 1 ) z ξ (kb 0 N 2 ) ξ 2. It is easy to check that i k is such that (2kb 0 2N 2 1) (M 1 + N 1 ) 2, U(x) 1 2 x 2. This shows that there is a number k 2 such that, i k k 2, the unction U(x) is negative deinite or all x satisying z B δ and ξ δ. Pick now any (nontrivial) sublevel set Ω c o U(x) entirely contained in the set o all x satisying z B δ and ξ δ and let r be such that B r Ω c. Then, the argument above shows that, or all k k 2, the equilibrium x = 0 is asymptotically stable with a domain o attraction that contains B r, which is a set independent o the choice o k. Pick now r r, pick any R > 0 and use the result o the Proposition above. There is a number k 1 such that, i k k 1 all trajectories with initial condition in B R in inite time enter the region B r, and hence enter the region o attraction o x = 0. It can be concluded, setting k = max{k 1, k 2 }, that or all k k the equilibrium x = 0 o (13) is asymptotically stable, with a domain o attraction that contains B R. The case o higher relative degree ż = 0 (z, ξ 1 ) ξ 1 = ξ 2 ξ r 1 = ξ r ξ r = q(z, ξ 1,..., ξ r ) + b(z, ξ 1,..., ξ r )u can be reduced to the case already studied. To this end, it suices to consider the dummy output variable θ = ξ r + g r 1 a 0 ξ 1 + g r 2 a 1 ξ 2 + ga r 2 ξ r 1 with a 0,..., a r 2 such that the polynomial d(λ) = λ r 1 + a r 2 λ r a 1 λ + a 0 is Hurwitz. The system thus obtained has relative degree 1 and a normal orm z = 0 ( z, θ) θ = q( z, θ) + b( z, θ)u 14
15 in which z = col(z, ξ 1,..., ξ r 1 ) 0 (z, ξ 1 ) ξ 2 0 ( z, θ) = ξ r 2 [g r 1 a 0 ξ 1 + g r 2 a 1 ξ 2 + ga r 2 ξ r 1 ] + θ and q( z, θ), b( z, θ) suitable unctions. The zero dynamics o this system are characterized by ż = 0 (z, ξ 1 ) ξ 1 = ξ 2 ξ r 1 = [g r 1 a 0 ξ 1 + g r 2 a 1 ξ 2 + ga r 2 ξ r 1 ] Changing variables as yields a system o the orm ζ i = 1 g i 1 ξ i, i = 1, 2,..., i 1, ż = 0 (z, ζ 1 ) ζ = gaζ in which A = a 0 a 1 a 2 a r 2 is a Hurwitz matrix. Arguments similar to those used above can be invoked to show that i the equilibrium z = 0 o ż = 0 (z, 0) is globally asymptotically and locally exponentially stable the equilibrium z = 0 o z = 0 ( z, 0) (18) is locally exponentially stable, with a domain o attraction that can be made arbitrarily large by increasing the design parameter g. In act, letting Q to be the positive deinite solution o QA + A T Q = I and considering the candidate Lyapunov unction W ( z) = V (z) + ζ T Qζ one proves, in exactly the same way, that or every choice o two positive numbers R, r, with R r, there is a number g 1 such that, or all g g 1 there is a inite time T such that all 15
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