An Example of a Globally Stabilizing Adaptive Controller with a ...
An Example of a Globally Stabilizing Adaptive Controller with a Generically Destabilizing Parameter Estimate S. TOWNLEY y School of Mathematical Sciences, University of Exeter, North Park Road, Exeter EX4 4QE, United Kingdom Email: [email protected] January 1998, Revised August 1998 and January 1999 Abstract: In this note we consider the question of whether an adaptive controller can
converge to a non-adaptive stabilizing controller. Speci cally, we show for a class of back-stepping controllers with adaptive tuning functions that the set of initial conditions in state and estimation parameter for which the estimation parameter converges to a parameter which produces a destabilizing controller can have non-empty interior and, consequently, non-zero Lebesgue measure. This surprising result is proved by way of a simple example with a quadratic nonlinearity. Keywords: adaptive control, parameter estimate, centre manifold. AMS subject classi cations: 93C10, 93C20, 93C25, 93D09, 93D10, 93D21.
1. Introduction and Problem Statement Consider a nonlinear system
x_ = F (x; u) y = G(x; u); (1.1) with state x 2 R n , input u 2 R m and output y 2 R m . Assume that F (0; 0) = 0. An adaptive controller, without persistency of excitation, which stabilizes (1.1) about the equilibrium x = 0, typically has the form: u = H (x; ^) (state feedback) or (1.2a) u = H (y; ^) (state output feedback); (1.2b) This work was supported by the Human Capital and Mobility programme (Project number CHRX-CT93-0402) and NATO (Grant CRG 950179). y Also with the Centre for Systems and Control Engineering, University of Exeter, UK.
where H (0; ^) = 0 for all ^ and where the parameter ^ is adapted (or estimated) according to (1.3) ^_ = W (x; ^) (or ^_ = W (y; ^)): The nonlinear system (1.1) is assumed to belong to a given class, which is de ned either via a parametrized family of systems with a xed, free and unknown parameter 2 R q , or by assuming that certain structural assumptions hold, for example that (1.1) is minimum phase. Design of the adaptive controller (1.2) depends on the class of systems to which (1.1) belongs. We focus on two examples:
Example 1.1 Adaptive-back stepping control for nonlinear systems in strict-feedback form Consider a nonlinear system (1.1) in strict feedback form: 9
x_ 1
= x2 + '1 (x1 )T > > > > > : > > > = : : > : > > > > x_ n?1 = xn + 'n?1(x1 ; :::; xn?1)T > > > ; T x_ n = u + 'n(x) ;
(1.4)
Here 2 R q is a vector of unknown constant parameters and for each i = 1; :::; n, 'i : R i 7! R q is a known smooth function satisfying 'i (0) = 0. An adaptive back-stepping controller for (1.4), see Krstic et. al. [5] and Krstic [6], is de ned recursively by:
zi = xi ? i?1 Pi?1 @ ?1 @ ?1 @ ?1 i = ?zi?1 ? cizi ? wiT + Pik?1 =1 @x xk+1 + @ ^ ? i + k=2 @ ^ ?wi zk i = i?1 +Pwizi @ ?1 wi = 'i ? ik?1 =1 @x 'k ; i
i
k
k
i
k
where 0 = 0, 0 = 0 and ? is any symmetric and positive de nite q by q matrix. With z = (z1; : : : ; zn)T , the control law is then given by u = n(x; ^) (1.5) with adaptation (estimation) ^_ = ? n(x; ^) = ?(w1; : : : ; wn)z:
(1.6)
3 Example 1.2 High-gain adaptive control for linear minimum-phase, relative degree one systems Suppose that (1.1) describes a linear minimum phase m-imput, m-output system
x_ = Ax + Bu y = Cx: 2
(1.7)
with (CB ) C + . A simple proportional adaptive control law with adaptive gain, see Willems and Byrnes [14], is given by (1.8) u = ?^y; ^_ = kyk2:
^ makes no attempt to estimate any of the parameters in (1.7), but merely attempts to seek out a stabilizing value of ^. The Willems-Byrnes controller is a simple example of so-called `Universal Adaptive Controllers' (see Martensson [7] for piecewise-constant dense searching universal adaptive controllers and Ilchmann [4] for a detailed survey of this area of adaptive control). 3 One key issue in adaptive control is the question of whether the adaptation (estimation) parameter ^ is guaranteed to converge. In general this would depend on the system (1.1) and on the form of the controller (1.2)-(1.3). However, in Examples 1.1 and 1.2, the adaptation parameter converges for all initial conditions (x(0); ^(0)). For the remainder of this note we assume that this is true in our general case, that is: Assumption 1: For all x(0) 2 R n and ^(0) 2 Rq , the closed-loop system (1.1), (1.2) and (1.3) satis es q ^ ^ lim x(t) = 0 and tlim t!1 !1 (t) = 1 2 R ; where ^1 depends on x(0) and ^(0). We will call a system/adaptive controller (1.1) - (1.3) which satis es Assumption 1 a Stable Adaptive System. Given a stable adaptive system, any initial condition (x(0); ^(0)) determines a xed controller u = H (x; ^1) (or u = H (y; ^1)): (1.9) This non-adaptive controller is referred to as the limit controller, and, since F (0; 0) = 0 and H (0; ^1) = 0, it follows that x = 0 is an equilibrium of the corresponding limit system (1.1) and (1.9). A number of questions arise regarding this limit controller. Is the limit controller (1.9) guaranteed to stabilize (1.1) about x = 0, independently of (x(0); ^(0))? If not, then how big, in measure or topological terms, is the set U := f(x(0); ^(0)) 2 R n+q j x = 0 is not a stable equilibrium of (1:1) and (1:9)g; that is the totality of initial conditions for which the resulting limit controller (1.9) does not stabilize the system (1.1). In particular, is U guaranteed to have zero Lebesgue measure or its complement have dense interior. The latter would mean that the limit controller is generically stabilizing with respect to initial conditions. In the de nition of U , stability could mean asymptotic, exponential stability or some weaker notion of stability. In the context of the Willems-Byrnes adaptive controller these questions have been discussed comprehensively in Mestel and Townley [8], Hicks and Townley [3], Trianta llidis [11], Trianta llidis et. al. [12], [13] and for Martensson's dense searching piecewiseconstant universal adaptive controller in Townley [9], [10]. For example, in the case of 3
the Willems-Byrnes controller, if the root locus for (1.7) has a simple imaginary axis crossing at 0 2 C from left to right, then U 6= f0g and there is a one dimensional manifold of initial conditions in (x; ^) space for which the limit controller is stabilizing but not asymptotically stabilizing. The following proposition summarises the main ndings concerning U for the class of Universal Adaptive Controllers.
Proposition 1.3 The interior of the complement of U is dense, so that U has zero Les-
besgue measure, in
(i) all cases of piecewise-constant universal adaptive control of linear systems. (See [9], [10]); (ii) the Willems-Byrnes high-gain adaptive control of a generic subset of single-input, single-output systems. (See [3], [8], [11], [12] and [13].)
The techniques developed in analysing the two cases (i) and (ii) in Proposition 1.3 are quite di erent. In the case of piecewise-constant universal adaptive control, the approach relies on reformulating the closed-loop, stable adaptive system as a discrete-time nonlinear system. The approach used in the analysis of the stable adaptive system arising from the Willems-Byrnes controller relies on invariant and centre manifold theories. In this approach an essential role was played by a classi cation all possible normal forms for (1.7)-(1.8) about the globally attractive equilibrium manifold f(0; ^) j ^ 2 R g R n+1 . A centre manifold approach was also used by Krstic [6], to analyse the stabilizing properties of the limit controller (1.9) for the adaptive back-stepping controller (1.5). In particular, normal forms identical to those for (1.7)-(1.8) were obtained for (1.4), (1.5) and (1.6) about those equilibria (0; ^e) 2 R n+q for which the corresponding linearization of (1.1) and (1.9) has no imaginary axis eigenvalues. Following on from the ideas developed around Proposition 1.3 and addressed in [6], it is natural to consider the following question: Given a stable adaptive system is the interior of the complement of U , generically in some sense, always dense so that U has zero Lebesgue measure. In other words, do adaptation/estimation algorithms converge almost surely to stabilizing estimates? This issue is addressed in the next section by way of an example. Some of the claims made in [6] are clari ed by this example, see Remark 2.2 (iv).
2. An Example in which U has non-empty interior It is clear that the main diculty in describing the topology or measure of U is caused by those equilibria (0; ^e) of (1.1) and (1.9) about which the linearization has (possibly multiple) imaginary axis eigenvalues. Now it is well known from nonlinear dynamics (see Guckenheimer and Holmes [1]) that equilibria of second order systems can have two-dimensional centre-stable manifolds. In the context of our problem, this could give 4
int(U ) 6= ;. However, it is not obvious that such degenerate cases can arise from global adaptive stabilization of an interesting class of nonlinear systems. This brings us to the main result of this note which, by example, shows that U having non-empty interior, and consequently non-zero Lebesgue measure, can occur quite easily.
Example 2.1 Consider the nonlinear system y_ = u + (y ? by2); y(0) 2 R :
Here is an unknown scalar parameter to be estimated by the adaptive controller and b > 0 is a known constant to be speci ed later. This is the simplest case of a system in strict-feedback form (1.4). Here '(y) = y ? by2 so that '(0) = 0 and '() is C 1. Applying the adaptive back-stepping approach yields an adaptive controller of the form u = ?y ? ^(y ? by2 ) (2.1) ^_ = y2 ? by3; (2.2) and, with ~ = ? ^, a closed-loop, stable adaptive system y_ = ?y + ~(y ? by2 ) (2.3a) ~_ = ?(y2 ? by3): (2.3b) At the equilibrium y = 0, ~ = 1 the linearization of (2.3) is y_ = 0 ~_ = 0;
i.e. a marginally stable double zero pole. Moreover, if limt!1 ~(t) = 1, i.e. ^1 = + 1, then the corresponding limit system is y_ = ?by2 : For all y(0) < 0, this limit system has nite escape time ?1=by(0). Our aim is to determine the initial conditions (y(0); ^(0)) which give limt!1 ~(t) = 1. Such initial conditions clearly belong to U . For convenience, introduce a variable k = 1 ? ~. Then (2.3) becomes y_ = ?ky + b(k ? 1)y2 (2.4a) 2 3 k_ = (y ? by ): (2.4b) We need to determine (y(0); k(0)) pairs with limt!1(y(t); k(t)) = (0; 0). Let b 2. De ne four curves, C1 ; : : : ; C4 , in the (k; y) plane by C1 = f(k; y) j (k ? 1)2 + y2 = constantg p2 2 g C2 = f(k; y) j y 1b ; k = ? p b2(b + b ? 4) b ? 4) k; 0 y < 1 g ( b + C3 = f(k; y) j y = ? 2 b k C4 = f(k; y) j y = ? b(1 ? k) ; k < 0g :
Since n = 1, there is no back-stepping and the adaptive controller only involves parameter estimation
5
Note that in this context we are thinking of k as the `independent' variable and accordingly plot y against k. It is easy to show that the region enclosed by these curves is an invariant set for (2.4). For example, if (k; y) 2 C4, then k_ = y2 ? by3 > 0; and y_ = 0 ; so that along C4 the ow of (2.4) is into the region and, by construction of this adaptive control scheme, d (k ? 1)2 + y2 = ?2y2 < 0; for all (k; y) 2 R 2 : dt 0.5
0.45
0.4
0.35
0.3 y 0.25
0.2
0.15
0.1
0.05
0 −1
−0.9
−0.8
−0.7
−0.6
−0.5 k
−0.4
−0.3
−0.2
−0.1
0
Figure 1: Phase Plane Picture for (2.4) in the (k; y) plane Since every solution of (2.4) converges to a point on the k-axis, it follows that the centre-stable manifold of the equilibrium (0; 0) of (2.4), and therefore U , contains the region enclosed by C1 ; : : : ; C4. Clearly this region has non-empty interior. In fact, as b increases this region increases and converges to the upper-left quadrant in the (k; y) plane. Note furthermore, as is shown in Figure 1 for b = 2:1, that for any k(0) xed with k(0) negative and large enough U (k(0)) := fy(0)j(y(0); (0)) 2 Ug = (0; 1) ! It is worth noting that dk = y2 ? by3 = y ? by2 : dy b(k ? 1)y2 ? ky b(k ? 1)y ? k 6
This shows that the phase plane curves of (2.4) in the (y; k) plane agree with those of Y_ = ?K + b(K ? 1)Y (2.5a) 2 K_ = Y ? bY ; (2.5b) in the (Y; K ) plane. The planar system (2.5) can be analysed using standard linearization techniques to elaborate a detailed phase plane analysis of (2.4). Of course, in comparing solution curves of (2.4) and (2.5), if y(0) < 0, then we must reverse the direction of ow of a solution curve for (2.5) to obtain a solution curve for (2.4).
Remark 2.2 (i) The same closed-loop behaviour will be observed if, in Example 2.1, the nonlinearity y ? by2 is replaced by any nonlinearity '(y) whose Taylor expansion at 0 agrees to quadratic order with y ? by2. (ii) The simple scalar example above exhibits the worst behaviour possible for the limit system. Such undesirable behaviour will obviously arise in higher order cases. (iii) If an adaptation gain is used in Example 2.1 so that ^_ = (y2 ? by3) ; then a phase plane analysis of the corresponding (2.5) can be used to show that for all b > 0 there exists > 0 large enough so that U f(0; ) j 2 R g and int(U ) = ;. Therefore if b is known, then the possibility of destabilizing limit gains can be avoided. However, b is not known in the controller design, so that in this context the usefulness of an adaptation gain is limited. Moreover, it is unclear whether this can help in higher dimensional cases. (iv) Finally, it has been claimed in [6] Theorem 5.1 that, for the adaptive back-stepping controller (which would include Example 2.1), the set U of all initial conditions leading to destabilizing limit controllers has zero Lebesgue measure. Example 2.1 shows that this claim is not true in general. However, Theorem 5.1 [6] is partially true in that the subset of U comprising those initial conditions leading to a limit system with an unstable linearization does have zero Lebesgue measurey. Since Theorem 5.1 [6] is needed in Corollary 6.1 and Theorem 7.1 in [6], similar comments would apply to these results as well.
3
Acknowledgements: The author would like to thank Gene Ryan for helpful comments
on an early version of this paper and an anonymous referee for pointing out the comment in Remark 2.2 (iii).
This partial result was obtained, albeit in a di erent context, by Byrnes et al. in [2] for a general class of adaptive controllers using the Baire Category Theorem and was observed in [8] for the Willems-Byrnes adaptive controller via an application of invariant manifold theory. y
7
References [1] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, (1983). [2] C. I. Byrnes, U. Helmke and A. S. Morse, Necessary conditions in adaptive control, in: C. I. Byrnes and A. Lindquist, eds., Modelling, Identi cation and Robust Control, North-Holland Press, pp. 3-14, (1986). [3] A. C. Hicks and S. Townley, On exact solutions of di erential equations arising in universal adaptive stabilization, Systems and Control Letters, 20, pp. 117-129, (1993). [4] A. Ilchmann, Non-identi er based adaptive control of dynamical systems: A survey, IMA J. Math. Control and Information, 8, pp. 321-366, (1991). [5] M. Krstic, I. Kanellakopoulos and P. V. Kokotovic, Adaptive nonlinear control without overparametrization, Systems and Control Letters, 19 pp. 177-185, (1992). [6] M. Krstic, Invariant manifolds and asymptotic properties of adaptive nonlinear stabilizers, IEEE Trans. Aut. Control, 41, pp. 817-829, (1996). [7] B. Martensson, Adaptive Stabilization, Ph.D Thesis, Lund Institute of Technology, Lund, Sweden, (1986). [8] B. D. Mestel and S. Townley, Topological classi cation of universal adaptive dynamics, in: Proc. 2nd European Control Conference, Groningen, pp. 1597-1602, (1993). [9] S. Townley, Generic properties of universal adaptive stabilization schemes, in: Proc. 31st IEEE CDC, Tuscon pp. 3630-3631, (1992). [10] S. Townley, Topological aspects of universal adaptive stabilization, SIAM J. Control & Optimization, 34, pp. 1044-1070, (1996). [11] S. Trianta llidis, Robust and Adaptive Control for Uncertain Dynamical Systems, PhD. Thesis, University of Exeter, (1994). [12] S. Trianta llidis, J.A. Leach, D. H. Owens and S. Townley, Limit systems, limit gains and non-generic behaviours arising in adaptive stabilizing control, Dynamics and Stability of Systems, 10, pp. 233-254, (1995). [13] S. Trianta llidis, J. A. Leach, D. H. Owens and S. Townley, Qualitative dynamics, root loci and asymptotic structure of universal adaptive stabilizers, IMA J. Contr. Inf Sci., 13, pp. 251-278, (1996). [14] J. C. Willems and C. I. Byrnes, Global adaptive stabilization in the absence of information on the sign of the high frequency gain, LNCIS, 62, pp. 49-67, Berlin, Springer-Verlag, (1984). 8
converge to a non-adaptive stabilizing controller. Speci cally, we show for a class of back-stepping controllers with adaptive tuning functions that the set of initial conditions in state and estimation parameter for which the estimation parameter converges to a parameter which produces a destabilizing controller can have non-empty interior and, consequently, non-zero Lebesgue measure. This surprising result is proved by way of a simple example with a quadratic nonlinearity. Keywords: adaptive control, parameter estimate, centre manifold. AMS subject classi cations: 93C10, 93C20, 93C25, 93D09, 93D10, 93D21.
1. Introduction and Problem Statement Consider a nonlinear system
x_ = F (x; u) y = G(x; u); (1.1) with state x 2 R n , input u 2 R m and output y 2 R m . Assume that F (0; 0) = 0. An adaptive controller, without persistency of excitation, which stabilizes (1.1) about the equilibrium x = 0, typically has the form: u = H (x; ^) (state feedback) or (1.2a) u = H (y; ^) (state output feedback); (1.2b) This work was supported by the Human Capital and Mobility programme (Project number CHRX-CT93-0402) and NATO (Grant CRG 950179). y Also with the Centre for Systems and Control Engineering, University of Exeter, UK.
where H (0; ^) = 0 for all ^ and where the parameter ^ is adapted (or estimated) according to (1.3) ^_ = W (x; ^) (or ^_ = W (y; ^)): The nonlinear system (1.1) is assumed to belong to a given class, which is de ned either via a parametrized family of systems with a xed, free and unknown parameter 2 R q , or by assuming that certain structural assumptions hold, for example that (1.1) is minimum phase. Design of the adaptive controller (1.2) depends on the class of systems to which (1.1) belongs. We focus on two examples:
Example 1.1 Adaptive-back stepping control for nonlinear systems in strict-feedback form Consider a nonlinear system (1.1) in strict feedback form: 9
x_ 1
= x2 + '1 (x1 )T > > > > > : > > > = : : > : > > > > x_ n?1 = xn + 'n?1(x1 ; :::; xn?1)T > > > ; T x_ n = u + 'n(x) ;
(1.4)
Here 2 R q is a vector of unknown constant parameters and for each i = 1; :::; n, 'i : R i 7! R q is a known smooth function satisfying 'i (0) = 0. An adaptive back-stepping controller for (1.4), see Krstic et. al. [5] and Krstic [6], is de ned recursively by:
zi = xi ? i?1 Pi?1 @ ?1 @ ?1 @ ?1 i = ?zi?1 ? cizi ? wiT + Pik?1 =1 @x xk+1 + @ ^ ? i + k=2 @ ^ ?wi zk i = i?1 +Pwizi @ ?1 wi = 'i ? ik?1 =1 @x 'k ; i
i
k
k
i
k
where 0 = 0, 0 = 0 and ? is any symmetric and positive de nite q by q matrix. With z = (z1; : : : ; zn)T , the control law is then given by u = n(x; ^) (1.5) with adaptation (estimation) ^_ = ? n(x; ^) = ?(w1; : : : ; wn)z:
(1.6)
3 Example 1.2 High-gain adaptive control for linear minimum-phase, relative degree one systems Suppose that (1.1) describes a linear minimum phase m-imput, m-output system
x_ = Ax + Bu y = Cx: 2
(1.7)
with (CB ) C + . A simple proportional adaptive control law with adaptive gain, see Willems and Byrnes [14], is given by (1.8) u = ?^y; ^_ = kyk2:
^ makes no attempt to estimate any of the parameters in (1.7), but merely attempts to seek out a stabilizing value of ^. The Willems-Byrnes controller is a simple example of so-called `Universal Adaptive Controllers' (see Martensson [7] for piecewise-constant dense searching universal adaptive controllers and Ilchmann [4] for a detailed survey of this area of adaptive control). 3 One key issue in adaptive control is the question of whether the adaptation (estimation) parameter ^ is guaranteed to converge. In general this would depend on the system (1.1) and on the form of the controller (1.2)-(1.3). However, in Examples 1.1 and 1.2, the adaptation parameter converges for all initial conditions (x(0); ^(0)). For the remainder of this note we assume that this is true in our general case, that is: Assumption 1: For all x(0) 2 R n and ^(0) 2 Rq , the closed-loop system (1.1), (1.2) and (1.3) satis es q ^ ^ lim x(t) = 0 and tlim t!1 !1 (t) = 1 2 R ; where ^1 depends on x(0) and ^(0). We will call a system/adaptive controller (1.1) - (1.3) which satis es Assumption 1 a Stable Adaptive System. Given a stable adaptive system, any initial condition (x(0); ^(0)) determines a xed controller u = H (x; ^1) (or u = H (y; ^1)): (1.9) This non-adaptive controller is referred to as the limit controller, and, since F (0; 0) = 0 and H (0; ^1) = 0, it follows that x = 0 is an equilibrium of the corresponding limit system (1.1) and (1.9). A number of questions arise regarding this limit controller. Is the limit controller (1.9) guaranteed to stabilize (1.1) about x = 0, independently of (x(0); ^(0))? If not, then how big, in measure or topological terms, is the set U := f(x(0); ^(0)) 2 R n+q j x = 0 is not a stable equilibrium of (1:1) and (1:9)g; that is the totality of initial conditions for which the resulting limit controller (1.9) does not stabilize the system (1.1). In particular, is U guaranteed to have zero Lebesgue measure or its complement have dense interior. The latter would mean that the limit controller is generically stabilizing with respect to initial conditions. In the de nition of U , stability could mean asymptotic, exponential stability or some weaker notion of stability. In the context of the Willems-Byrnes adaptive controller these questions have been discussed comprehensively in Mestel and Townley [8], Hicks and Townley [3], Trianta llidis [11], Trianta llidis et. al. [12], [13] and for Martensson's dense searching piecewiseconstant universal adaptive controller in Townley [9], [10]. For example, in the case of 3
the Willems-Byrnes controller, if the root locus for (1.7) has a simple imaginary axis crossing at 0 2 C from left to right, then U 6= f0g and there is a one dimensional manifold of initial conditions in (x; ^) space for which the limit controller is stabilizing but not asymptotically stabilizing. The following proposition summarises the main ndings concerning U for the class of Universal Adaptive Controllers.
Proposition 1.3 The interior of the complement of U is dense, so that U has zero Les-
besgue measure, in
(i) all cases of piecewise-constant universal adaptive control of linear systems. (See [9], [10]); (ii) the Willems-Byrnes high-gain adaptive control of a generic subset of single-input, single-output systems. (See [3], [8], [11], [12] and [13].)
The techniques developed in analysing the two cases (i) and (ii) in Proposition 1.3 are quite di erent. In the case of piecewise-constant universal adaptive control, the approach relies on reformulating the closed-loop, stable adaptive system as a discrete-time nonlinear system. The approach used in the analysis of the stable adaptive system arising from the Willems-Byrnes controller relies on invariant and centre manifold theories. In this approach an essential role was played by a classi cation all possible normal forms for (1.7)-(1.8) about the globally attractive equilibrium manifold f(0; ^) j ^ 2 R g R n+1 . A centre manifold approach was also used by Krstic [6], to analyse the stabilizing properties of the limit controller (1.9) for the adaptive back-stepping controller (1.5). In particular, normal forms identical to those for (1.7)-(1.8) were obtained for (1.4), (1.5) and (1.6) about those equilibria (0; ^e) 2 R n+q for which the corresponding linearization of (1.1) and (1.9) has no imaginary axis eigenvalues. Following on from the ideas developed around Proposition 1.3 and addressed in [6], it is natural to consider the following question: Given a stable adaptive system is the interior of the complement of U , generically in some sense, always dense so that U has zero Lebesgue measure. In other words, do adaptation/estimation algorithms converge almost surely to stabilizing estimates? This issue is addressed in the next section by way of an example. Some of the claims made in [6] are clari ed by this example, see Remark 2.2 (iv).
2. An Example in which U has non-empty interior It is clear that the main diculty in describing the topology or measure of U is caused by those equilibria (0; ^e) of (1.1) and (1.9) about which the linearization has (possibly multiple) imaginary axis eigenvalues. Now it is well known from nonlinear dynamics (see Guckenheimer and Holmes [1]) that equilibria of second order systems can have two-dimensional centre-stable manifolds. In the context of our problem, this could give 4
int(U ) 6= ;. However, it is not obvious that such degenerate cases can arise from global adaptive stabilization of an interesting class of nonlinear systems. This brings us to the main result of this note which, by example, shows that U having non-empty interior, and consequently non-zero Lebesgue measure, can occur quite easily.
Example 2.1 Consider the nonlinear system y_ = u + (y ? by2); y(0) 2 R :
Here is an unknown scalar parameter to be estimated by the adaptive controller and b > 0 is a known constant to be speci ed later. This is the simplest case of a system in strict-feedback form (1.4). Here '(y) = y ? by2 so that '(0) = 0 and '() is C 1. Applying the adaptive back-stepping approach yields an adaptive controller of the form u = ?y ? ^(y ? by2 ) (2.1) ^_ = y2 ? by3; (2.2) and, with ~ = ? ^, a closed-loop, stable adaptive system y_ = ?y + ~(y ? by2 ) (2.3a) ~_ = ?(y2 ? by3): (2.3b) At the equilibrium y = 0, ~ = 1 the linearization of (2.3) is y_ = 0 ~_ = 0;
i.e. a marginally stable double zero pole. Moreover, if limt!1 ~(t) = 1, i.e. ^1 = + 1, then the corresponding limit system is y_ = ?by2 : For all y(0) < 0, this limit system has nite escape time ?1=by(0). Our aim is to determine the initial conditions (y(0); ^(0)) which give limt!1 ~(t) = 1. Such initial conditions clearly belong to U . For convenience, introduce a variable k = 1 ? ~. Then (2.3) becomes y_ = ?ky + b(k ? 1)y2 (2.4a) 2 3 k_ = (y ? by ): (2.4b) We need to determine (y(0); k(0)) pairs with limt!1(y(t); k(t)) = (0; 0). Let b 2. De ne four curves, C1 ; : : : ; C4 , in the (k; y) plane by C1 = f(k; y) j (k ? 1)2 + y2 = constantg p2 2 g C2 = f(k; y) j y 1b ; k = ? p b2(b + b ? 4) b ? 4) k; 0 y < 1 g ( b + C3 = f(k; y) j y = ? 2 b k C4 = f(k; y) j y = ? b(1 ? k) ; k < 0g :
Since n = 1, there is no back-stepping and the adaptive controller only involves parameter estimation
5
Note that in this context we are thinking of k as the `independent' variable and accordingly plot y against k. It is easy to show that the region enclosed by these curves is an invariant set for (2.4). For example, if (k; y) 2 C4, then k_ = y2 ? by3 > 0; and y_ = 0 ; so that along C4 the ow of (2.4) is into the region and, by construction of this adaptive control scheme, d (k ? 1)2 + y2 = ?2y2 < 0; for all (k; y) 2 R 2 : dt 0.5
0.45
0.4
0.35
0.3 y 0.25
0.2
0.15
0.1
0.05
0 −1
−0.9
−0.8
−0.7
−0.6
−0.5 k
−0.4
−0.3
−0.2
−0.1
0
Figure 1: Phase Plane Picture for (2.4) in the (k; y) plane Since every solution of (2.4) converges to a point on the k-axis, it follows that the centre-stable manifold of the equilibrium (0; 0) of (2.4), and therefore U , contains the region enclosed by C1 ; : : : ; C4. Clearly this region has non-empty interior. In fact, as b increases this region increases and converges to the upper-left quadrant in the (k; y) plane. Note furthermore, as is shown in Figure 1 for b = 2:1, that for any k(0) xed with k(0) negative and large enough U (k(0)) := fy(0)j(y(0); (0)) 2 Ug = (0; 1) ! It is worth noting that dk = y2 ? by3 = y ? by2 : dy b(k ? 1)y2 ? ky b(k ? 1)y ? k 6
This shows that the phase plane curves of (2.4) in the (y; k) plane agree with those of Y_ = ?K + b(K ? 1)Y (2.5a) 2 K_ = Y ? bY ; (2.5b) in the (Y; K ) plane. The planar system (2.5) can be analysed using standard linearization techniques to elaborate a detailed phase plane analysis of (2.4). Of course, in comparing solution curves of (2.4) and (2.5), if y(0) < 0, then we must reverse the direction of ow of a solution curve for (2.5) to obtain a solution curve for (2.4).
Remark 2.2 (i) The same closed-loop behaviour will be observed if, in Example 2.1, the nonlinearity y ? by2 is replaced by any nonlinearity '(y) whose Taylor expansion at 0 agrees to quadratic order with y ? by2. (ii) The simple scalar example above exhibits the worst behaviour possible for the limit system. Such undesirable behaviour will obviously arise in higher order cases. (iii) If an adaptation gain is used in Example 2.1 so that ^_ = (y2 ? by3) ; then a phase plane analysis of the corresponding (2.5) can be used to show that for all b > 0 there exists > 0 large enough so that U f(0; ) j 2 R g and int(U ) = ;. Therefore if b is known, then the possibility of destabilizing limit gains can be avoided. However, b is not known in the controller design, so that in this context the usefulness of an adaptation gain is limited. Moreover, it is unclear whether this can help in higher dimensional cases. (iv) Finally, it has been claimed in [6] Theorem 5.1 that, for the adaptive back-stepping controller (which would include Example 2.1), the set U of all initial conditions leading to destabilizing limit controllers has zero Lebesgue measure. Example 2.1 shows that this claim is not true in general. However, Theorem 5.1 [6] is partially true in that the subset of U comprising those initial conditions leading to a limit system with an unstable linearization does have zero Lebesgue measurey. Since Theorem 5.1 [6] is needed in Corollary 6.1 and Theorem 7.1 in [6], similar comments would apply to these results as well.
3
Acknowledgements: The author would like to thank Gene Ryan for helpful comments
on an early version of this paper and an anonymous referee for pointing out the comment in Remark 2.2 (iii).
This partial result was obtained, albeit in a di erent context, by Byrnes et al. in [2] for a general class of adaptive controllers using the Baire Category Theorem and was observed in [8] for the Willems-Byrnes adaptive controller via an application of invariant manifold theory. y
7
References [1] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, (1983). [2] C. I. Byrnes, U. Helmke and A. S. Morse, Necessary conditions in adaptive control, in: C. I. Byrnes and A. Lindquist, eds., Modelling, Identi cation and Robust Control, North-Holland Press, pp. 3-14, (1986). [3] A. C. Hicks and S. Townley, On exact solutions of di erential equations arising in universal adaptive stabilization, Systems and Control Letters, 20, pp. 117-129, (1993). [4] A. Ilchmann, Non-identi er based adaptive control of dynamical systems: A survey, IMA J. Math. Control and Information, 8, pp. 321-366, (1991). [5] M. Krstic, I. Kanellakopoulos and P. V. Kokotovic, Adaptive nonlinear control without overparametrization, Systems and Control Letters, 19 pp. 177-185, (1992). [6] M. Krstic, Invariant manifolds and asymptotic properties of adaptive nonlinear stabilizers, IEEE Trans. Aut. Control, 41, pp. 817-829, (1996). [7] B. Martensson, Adaptive Stabilization, Ph.D Thesis, Lund Institute of Technology, Lund, Sweden, (1986). [8] B. D. Mestel and S. Townley, Topological classi cation of universal adaptive dynamics, in: Proc. 2nd European Control Conference, Groningen, pp. 1597-1602, (1993). [9] S. Townley, Generic properties of universal adaptive stabilization schemes, in: Proc. 31st IEEE CDC, Tuscon pp. 3630-3631, (1992). [10] S. Townley, Topological aspects of universal adaptive stabilization, SIAM J. Control & Optimization, 34, pp. 1044-1070, (1996). [11] S. Trianta llidis, Robust and Adaptive Control for Uncertain Dynamical Systems, PhD. Thesis, University of Exeter, (1994). [12] S. Trianta llidis, J.A. Leach, D. H. Owens and S. Townley, Limit systems, limit gains and non-generic behaviours arising in adaptive stabilizing control, Dynamics and Stability of Systems, 10, pp. 233-254, (1995). [13] S. Trianta llidis, J. A. Leach, D. H. Owens and S. Townley, Qualitative dynamics, root loci and asymptotic structure of universal adaptive stabilizers, IMA J. Contr. Inf Sci., 13, pp. 251-278, (1996). [14] J. C. Willems and C. I. Byrnes, Global adaptive stabilization in the absence of information on the sign of the high frequency gain, LNCIS, 62, pp. 49-67, Berlin, Springer-Verlag, (1984). 8
