A Graphical Understanding of Lyapunov-Based Nonlinear ... - CiteSeerX
Proceedings of the 41st IEEE Conference on Decision and Control Las Vegas, Nevada USA, December 2002
WeP10-1
A Graphical Understanding of Lyapunov-based Nonlinear Control J. Willard Curtis Randal W. Beard Department of Electrical and Computer Engineering Brigham Young University Provo, UT, 84602 fwilly, [email protected]
Abstract The class of continuous nonlinear controllers that are based on control Lyapunov functions (clfs) can be understood graphically as continuous selections from a set-valued map that takes states and maps them to control-value sets. The notion of inverse-optimality and robustness to input disturbances can also be graphically understood by this set-valued map. The graphical approach introduced in this paper makes clear the relationship between Sontag's formula, Freeman and Kokotovic' min-norm formula, and other `universal formulas', as well as shedding light on the meaning of Lyapunov redesign. Most importantly, this paper examines a null-space associated with these set-valued maps which o ers the potential for significant improvement in the large-signal performance of Lyapunov-based controllers.
1 Introduction The extension of Lyapunov's second method to dynamical systems with inputs was rst introduced by Artstein and Sontag in [1], [2] with the introduction of control Lyapunov functions (clfs). A constructive proof that the existence of a clf implies asymptotic stabilizability was provided in [3] based on Sontag's `universal formula' for stabilization. Another clf-based approach, the point wise minnorm formula (as presented in [4]) selects the minimum-norm control from a set in the control space at each state such that the Lyapunov derivative is less than a prescribed scalar function of the states. A similar approach is introduced in [5, 6] where clf-based satis cing is introduced as a complete parameterization of stabilizing clf-based control laws. Satis cing was also shown to completely parameterize the class
0-7803-7516-5/02/$17.00 ©2002 IEEE
of clf-based controls with Kalman-type gain margins, and its development led to the graphical approach presented in this paper. The large-signal performance of clf-based controls is sometimes a problem. To address this issue, [7] reintroduced the concept of inverse-optimality, where a control law is shown to be optimal with respect to some meaningful cost functional which is speci ed after the fact. Optimal control laws bene t from known stability margins [8, 9], and the fact that they minimize a meaningful cost functional ensures that control e ort is not wasted. Sontag's formula was shown to be optimal with respect to a meaningful cost function in [10], and a similar game theoretic result for the min-norm control of general uncertain systems can be found in [4]. In this paper, a graphical interpretation of clf-based nonlinear controllers is presented, where clf-based formulas are analyzed point-wise by an inspection of associated stabilizing control-value sets. This graphical approach to understanding Lyapunov stability aids in visualizing the meaning of stability margins and the property of being an optimal Lyapunov-based control. Most importantly though, this graphical approach reveals a large stabilizing region of the control space that has been, till now, unexploited in the design of clf-based control laws.
2 The Lyapunov Stabilizing ControlValue Set
2.1 Preliminaries
Consider the following ane nonlinear system with multiple inputs and a disturbance, i.e., x_ = f (x) + g(x)u; (1)
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where x 2 Rn , u 2 Rm and f (0) = 0. We will assume throughout the paper that f and g are locally Lipschitz functions, and the goal will be to regulate the state x to the origin. A C 1 function V (x) : Rn ! R is said to be a control Lyapunov function (clf) for the system if V is positive de nite, radially unbounded, and if inf V T (f + gu) < 0; 8x 6= 0: u x It has been shown in [1, 2] that system (1) is asymptotically controllable to the origin, if and only if there exists a clf for the system. In general, nding a clf is an open problem, however it is known that clfs exists for a large class of practically important systems [11] including feedback linearizable systems and systems susceptible to integrator backstepping. The clf V is said to satisfy the small control property [12] if there exists a control law c (x) continuous in Rn such that VxT f + VxT g c < 0 8x 6= 0. System (1) will be asymptotically controllable to the origin with continuous control if V satis es the small control property. In the remainder of this paper we will assume that V is known and that it satis es this property.
The result follows now directly from standard Lyapunov theory, (see for example [13, p.100 { 102]).
The constraint on can be visualized geometrically as follows. There is a region of the control space at every xed x that contains all control values that render V_ = VxT f + VxT gu negative. This region (denoted SV ) is an open half space, and it is bounded by a hyperplane which is perpendicular to the vector ;gT Vx . 2−D Control Space 5
4 Case where VTf = 2. x
3
2
1
0 −gTVx −1
−2
−4
−5 −5
2.2 The Stabilizing Control Value Set Note that given a clf V (x), any continuous control law can be written as u = ; (x)gT Vx + (x) (where (x) is continuous and (x) is continuous and orthogonal to gT Vx ). This decomposition into an element parallel to the vector ;gT Vx and an element perpendicular to it, will be instrumental in illuminating the structure of a point-wise stabilizing set. The following fundamental result shows that asymptotic stabilization is equivalent to a point-wise inner product constraint on the control. Theorem 2.1 An arbitrary continuous control law, u = ; (x)gT Vx + (x) (where (x) is orthogonal to gT Vx ) with u(0) = 0, is asymptotically stabilizing with respect to the known clf V (x) if and only if T xf : gT Vx 6= 0 =) (x) > V TVgg TV x
x
(2)
Proof: Suciency and necessity of the condition on
can be seen as follows:
T xf > V TVgg TV
x x T T () ; Vx gg Vx < ;VxT f () VxT gu < ;VxT f () VxT f + VxT gu < 0:
This shaded region contains all control values that render Lf+guV negative.
−3
−4
−3
−2
−1
0
1
2
3
4
5
Figure 1: Stabilizing Set in the Control Space This stabilizing region of Rm constitutes a statedependent set: De nition 2.2 The Stabilizing Set, denoted SV is
SV (x) = u 2 Rm : VxT gu < ;VxT f : Since VxT g and VxT f are both continuous in x, we know that the boundary of SV (x) also moves in Rm continuously with the state. An alternate way of viewing these results is to recognize that SV (x) is a (lower semi-continuous) set valued map which is parameterized by a clf. Thus, SV (x) maps a state x onto a subset of Rm whose points render V_ negative. From this perspective any clf-based stabilizing control must be, by Theorem 2.1, a continuous selection from the set-valued map, SV (x).
3 Sontag's and Min-norm Formula A fundamental contribution of this paper is an explanation of the relationships between Sontag's formula [3], Freeman and Kokotovic's min-norm controls [4], and all other continuous, clf-based stabilizing
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controls by a point wise control value analysis. It will be shown that at any xed x the point-wise min-norm control parameterizes a line in SV 2 Rm , and that a point on this line corresponds to Sontag's formula. As described in [3], Sontag's formula is a continuous control law given by
2−D Control Space 5
4
3
2
1
8 0 5 T For example u = [−8, 2] does not have a fifty percent gain reduction margin, reducing u by one half pushes it out of the stabilizing control value set.
4
3
u
2
.5u 1
0 T
−g Vx −1
Only controls beyond this dotted line will have a fifty percent gain reduction margin.
−2
−3
SR
4 Gain Margins and Inverse Optimality
De nition 4.1 An asymptotically stabilizing control
−4
Figure 4: In nite Gain Margins in the Control Space
is subject to an unknown disturbance d 2 Rr . Analogues of Sontag's formula and the min-norm approach can be applied to this problem, both of which result in control laws that are pointwise scalings of the ;gT Vx vector. The signi cance of this is that at any xed state, the projection of an arbitrary control vector onto ;gT Vx exactly predicts the rate of decrease the Lyapunov function will experience under that control authority. The next section will show how this simple idea can be used to ensure desirable gain margins and optimality with respect to a meaningful cost index.
Robustness to disturbances at the input can be measured in many ways, and Lyapunov redesign as discussed in the Section 3.1 addresses this issue, but a straightforward approach is to quantify the amount of ampli cation (and diminution) a control signal can experience before instability results. Toward that end, we de ne stability margins as follows.
This shaded region corresponds to stabilizing control values. However, only control values that lie on this side of the dashed line will have infinite gain margin.
−4
−5 −9
−8
−7
−6
−5
−4
−3
−2
−1
0
1
Figure 5: Gain Reduction Margins in the Control Space
Theorem 4.2 The state-dependent control-value set which contains all controls that have stability margins of (; ; 1) is SR (x), where SR is de ned as follows: 1 2
4 u 2 Rm : ;uT gT V > max 0; 2V T f SR (x) = x x
law, u = q(x), has stability margins (m1 ; m2 ) where
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;
the null space of ;gT Vx , consider the following controllable linear system: x_ = Ax + Bu; (6) where 2 3 2 3 1 0 0 1 0 0 4 4 A = 4 0 ;1 15 and B = 4 1 1 05 : ;1 0 1 ;1 0 1 It is straightforward to verify that a valid clf for this system is V (x) = xT x. De ne a performance index: R 4 diag(2 1 1), J (x0 ; u) = 01 xT Qx + uT u, where Q = and denote the control that minimizes J as u = ;B T Px where P is the positive-de nite symmetric solution of the associated algebraic Riccati equation. Our rst clf-based control, u1(x) = ;(x)gT Vx + (x) (where (x) lies perpendicular to ;gT Vx ) will exploit the null space of gT Vx = B T x by using the optimal control to nd an appropriate (x) function:
Proof: First, any stabilizing control value which satis es ;uT gT Vx > 0 (or equivalently > 0) will have an in nite gain margin in the sense that amplifying the control value by any factor 1 < m2 < 1 will only push u farther into the stabilizing control-value set: V_ (m2 u) < V_ (u) 8 m2 > 1. This is shown in Figure 4. In order to have a gain reduction margin of 12 u, it is only necessary that the reduced control value still lie in the stabilizing region: VxT f + 12 VxT gu < 0 () VxT gu < ;2VxT f () ; VxT ggT Vx < ;2VxT f 2VxT f
() > V T ggT V x x () ;ugT Vx > 2VxT f:
This idea is shown in Figure 5.
4 u (x) + ;VxT gu gT V : (x) = x V T ggT V x
4.1 Optimality Optimal control laws can be related to stability margins. For example, in [8, 9] it is shown that optimal control laws have stability margins of (; 12 ; 1). In fact one of the primary motivations for considering inverse optimal control laws [4, 10] is that they have these desirable stability margins. It is shown in [10, p.108] that all clf-based control laws of the form u = ;k(x; Vx ) = ; 12 R;1 (x)gT Vx with gain margin of (; 12 ; 1) are optimal with respect to the meaningful cost function: Z 1 4 ;V T f + 1 gT V k: J (x) = l(x) + kT R(x)k; l(x) = x 2 x 0 This result was used in [5] to show that any continuous selection from the set-valued map SR (x) is inverse optimal (optimal with respect to a cost index that is constructed a posteriori instead of chosen a priori by the designer). The signi cance of this result is that optimality can be understood graphically as well: optimal controls lie in the control value set SR (x) at all nonzero states.
5 Example of Exploiting the Nullspace of VxT g As an illustration of the potential performance enhancement o ered by adding a control component in
x
Note that this operation merely selects the component of u (x) which is perpendicular to ;gT Vx and assigns that value to (x). We will ensure that u1 is stabilizing T f +xT x V x by choosing (x) = VxT ggT Vx , which clearly satis es Theorem 2.1. We will compare the performance of u1 (x) with the performance of another control, u2 (x) based on the same clf. u2 (x), however, will be constrained to lie in the direction of ;gT Vx . In order to be fair, we will set the magnitude of u2(x) to be equal to the magnitude of u1(x) so that u1 and u2 exert the same control authority, i. e. u2 = ; kkVuxT1gkk gT Vx . The cost associated with u1 (the control law which exploited the null-space) was J (x0 ; u1) = 296:3, whereas the cost associated with u2 was J (x0 ; u2 ) = 404:4. The initial state was chosen arbitrarily to be x0 = [8 ; 6 ; 2], however this simulation was repeated numerous times with a wide variation in initial states and u1 (x) was always found to o er superior performance. The signi cance of this simple example is that the performance of a control law is rarely measured in terms of its Lyapunov function. This simulation shows that high-performance, clf-based control laws should exploit the null space of gT Vx in some manner.
6 Discussion and Conclusion Lyapunov functions provide a powerful tool for analyzing the stability properties of nonlinear systems,
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and many formulas and techniques based on control Lyapunov functions have been proposed in the literature. All of these control strategies can be easily visualized in the control space at a xed state. The reason such a point-wise graphic has global meaning (despite the inherently local information provided by a graph of Rm at a xed state) is that the clf V (x) ensures global properties, while only requiring the local constraint that V_ < 0. In addition to being an aide in understanding Lyapunov-based controllers, such a state-dependent control value analysis opens new avenues of research into the construction of high-performance clf-based controls. For example, an asymptotically stable, inverse optimal model predictive control (MPC) strategy can be de ned as kMPC (x) = arg u2min J (x; u) S (x) MPC R
where JMPC (x; u) is a cost criteria based on model predictive strategies. This simple receding horizon strategy would combine the best of Lyapunov analysis and computational performance by searching the null-space of gT Vx for a performance enhancing augmentation to a standard clf-based strategy. Such a MPC would be guaranteed stable, optimal, robust to input disturbances, and no terminal constraint would be imposed on the optimization routine. Another interesting thing to note, is that the ef cacy of such an optimization would scale with the size of the control space. Stability, in the sense of Lyapunov, imposes a single dimensional constraint on the m-dimensional control vector, leaving m ; 1 dimensions free. Clearly for large values of m, choosing wisely the component of the control that lies in this `free' space becomes more critical; a single input system (m ; 1) has no such `free space', but for m = 4 the null space of gT Vx o ers three dimensions of freedom that could have a great impact on the large-signal performance of the closed-loop system. In the past, clf-based controls have neglected this null space of gT Vx . Since any component of a control that lies orthogonal to gT Vx will have no e ect on the rate V_ , this is understandable. However, if large-signal performance is sought in a control law, a component in this null space might signi cantly improve any performance measure that isn't speci cally clf-based.
Acknowledgments This work was supported by the National Science Foundation: award number 9732917.
References [1] Z. Artstein, \Stabilization with relaxed controls," Nonlinear Analysis, Theory, Methods, and Applications, vol. 7, no. 11, pp. 1163{1173, 1983. [2] E. D. Sontag, \A Lyapunov-like characterization of asymptotic controllability," SIAM Journal on Control and Optimization, vol. 21, pp. 462{471, May 1983. [3] E. D. Sontag, \A `universal' construction of Artstein's theorem on nonlinear stabilization," System and Control Letters, vol. 13, pp. 117{123, 1989. [4] R. A. Freeman and P. V. Kokotovic, Robust Nonlinear Control Design: State-Space and Lyapunov Techniques. Systems and Control: Foundations and Applications, Birkhauser, 1996. [5] J. W. Curtis and R. W. Beard, \Satis cing: A new approach to constructive nonlinear control," IEEE Transactions on Automatic Control, submitted 2001. [6] R. W. Beard, B. Young, and W. Stirling, \Nonlinear regulation using the satis cing paradigm," in 2001 American Control Conference, (Arlington, VA), June 25{27 2001. [7] R. A. Freeman and P. V. Kokotovic, \Optimal nonlinear controllers for feedback linearizable systems," in Proceedings of the American Control Conference, (Seattle, Washington), pp. 2722{2726, June 1995. [8] S. T. Glad, \On the gain margin of nonlinear and optimal regulators," IEEE Transactions on Automatic Control, vol. 29, pp. 615{620, July 1984. [9] J. N. Tsitsiklis and M. Athans, \Guaranteed robustness properties of multivariable nonlinear stochastic optimal regulators," IEEE Transactions on Automatic Control, vol. 29, pp. 690{696, August 1984. [10] R. Sepulchre, M. Jankovic, and P. Kokotovic, Constructive Nonlinear Control. Communication and Control Engineering Series, Springer-Verlag, 1997. [11] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design. Wiley, 1995. [12] M. Krstic and H. Deng, Stabilization of Nonlinear Uncertain Systems. Communications and Control Engineering, Springer Verlag, 1998. [13] H. K. Khalil, Nonlinear Systems. Upper Saddle River, NJ: Prentice Hall, second ed., 1996. [14] M. Corless, \Control of uncertain nonlinear systems," Journal of Dynamical Systems Measurement and Control, vol. 115, pp. 362{372, 1993. [15] M. Corless and G. Leitmann, \Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems," TAC, vol. 26, pp. 1139{ 1144, October 1981. [16] V. Jurdjevic and J. Quinn, \Controllability and stability," Journal of Di erential Equations, vol. 28, pp. 381{389, December 1978.
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A Graphical Understanding of Lyapunov-based Nonlinear Control J. Willard Curtis Randal W. Beard Department of Electrical and Computer Engineering Brigham Young University Provo, UT, 84602 fwilly, [email protected]
Abstract The class of continuous nonlinear controllers that are based on control Lyapunov functions (clfs) can be understood graphically as continuous selections from a set-valued map that takes states and maps them to control-value sets. The notion of inverse-optimality and robustness to input disturbances can also be graphically understood by this set-valued map. The graphical approach introduced in this paper makes clear the relationship between Sontag's formula, Freeman and Kokotovic' min-norm formula, and other `universal formulas', as well as shedding light on the meaning of Lyapunov redesign. Most importantly, this paper examines a null-space associated with these set-valued maps which o ers the potential for significant improvement in the large-signal performance of Lyapunov-based controllers.
1 Introduction The extension of Lyapunov's second method to dynamical systems with inputs was rst introduced by Artstein and Sontag in [1], [2] with the introduction of control Lyapunov functions (clfs). A constructive proof that the existence of a clf implies asymptotic stabilizability was provided in [3] based on Sontag's `universal formula' for stabilization. Another clf-based approach, the point wise minnorm formula (as presented in [4]) selects the minimum-norm control from a set in the control space at each state such that the Lyapunov derivative is less than a prescribed scalar function of the states. A similar approach is introduced in [5, 6] where clf-based satis cing is introduced as a complete parameterization of stabilizing clf-based control laws. Satis cing was also shown to completely parameterize the class
0-7803-7516-5/02/$17.00 ©2002 IEEE
of clf-based controls with Kalman-type gain margins, and its development led to the graphical approach presented in this paper. The large-signal performance of clf-based controls is sometimes a problem. To address this issue, [7] reintroduced the concept of inverse-optimality, where a control law is shown to be optimal with respect to some meaningful cost functional which is speci ed after the fact. Optimal control laws bene t from known stability margins [8, 9], and the fact that they minimize a meaningful cost functional ensures that control e ort is not wasted. Sontag's formula was shown to be optimal with respect to a meaningful cost function in [10], and a similar game theoretic result for the min-norm control of general uncertain systems can be found in [4]. In this paper, a graphical interpretation of clf-based nonlinear controllers is presented, where clf-based formulas are analyzed point-wise by an inspection of associated stabilizing control-value sets. This graphical approach to understanding Lyapunov stability aids in visualizing the meaning of stability margins and the property of being an optimal Lyapunov-based control. Most importantly though, this graphical approach reveals a large stabilizing region of the control space that has been, till now, unexploited in the design of clf-based control laws.
2 The Lyapunov Stabilizing ControlValue Set
2.1 Preliminaries
Consider the following ane nonlinear system with multiple inputs and a disturbance, i.e., x_ = f (x) + g(x)u; (1)
2278
where x 2 Rn , u 2 Rm and f (0) = 0. We will assume throughout the paper that f and g are locally Lipschitz functions, and the goal will be to regulate the state x to the origin. A C 1 function V (x) : Rn ! R is said to be a control Lyapunov function (clf) for the system if V is positive de nite, radially unbounded, and if inf V T (f + gu) < 0; 8x 6= 0: u x It has been shown in [1, 2] that system (1) is asymptotically controllable to the origin, if and only if there exists a clf for the system. In general, nding a clf is an open problem, however it is known that clfs exists for a large class of practically important systems [11] including feedback linearizable systems and systems susceptible to integrator backstepping. The clf V is said to satisfy the small control property [12] if there exists a control law c (x) continuous in Rn such that VxT f + VxT g c < 0 8x 6= 0. System (1) will be asymptotically controllable to the origin with continuous control if V satis es the small control property. In the remainder of this paper we will assume that V is known and that it satis es this property.
The result follows now directly from standard Lyapunov theory, (see for example [13, p.100 { 102]).
The constraint on can be visualized geometrically as follows. There is a region of the control space at every xed x that contains all control values that render V_ = VxT f + VxT gu negative. This region (denoted SV ) is an open half space, and it is bounded by a hyperplane which is perpendicular to the vector ;gT Vx . 2−D Control Space 5
4 Case where VTf = 2. x
3
2
1
0 −gTVx −1
−2
−4
−5 −5
2.2 The Stabilizing Control Value Set Note that given a clf V (x), any continuous control law can be written as u = ; (x)gT Vx + (x) (where (x) is continuous and (x) is continuous and orthogonal to gT Vx ). This decomposition into an element parallel to the vector ;gT Vx and an element perpendicular to it, will be instrumental in illuminating the structure of a point-wise stabilizing set. The following fundamental result shows that asymptotic stabilization is equivalent to a point-wise inner product constraint on the control. Theorem 2.1 An arbitrary continuous control law, u = ; (x)gT Vx + (x) (where (x) is orthogonal to gT Vx ) with u(0) = 0, is asymptotically stabilizing with respect to the known clf V (x) if and only if T xf : gT Vx 6= 0 =) (x) > V TVgg TV x
x
(2)
Proof: Suciency and necessity of the condition on
can be seen as follows:
T xf > V TVgg TV
x x T T () ; Vx gg Vx < ;VxT f () VxT gu < ;VxT f () VxT f + VxT gu < 0:
This shaded region contains all control values that render Lf+guV negative.
−3
−4
−3
−2
−1
0
1
2
3
4
5
Figure 1: Stabilizing Set in the Control Space This stabilizing region of Rm constitutes a statedependent set: De nition 2.2 The Stabilizing Set, denoted SV is
SV (x) = u 2 Rm : VxT gu < ;VxT f : Since VxT g and VxT f are both continuous in x, we know that the boundary of SV (x) also moves in Rm continuously with the state. An alternate way of viewing these results is to recognize that SV (x) is a (lower semi-continuous) set valued map which is parameterized by a clf. Thus, SV (x) maps a state x onto a subset of Rm whose points render V_ negative. From this perspective any clf-based stabilizing control must be, by Theorem 2.1, a continuous selection from the set-valued map, SV (x).
3 Sontag's and Min-norm Formula A fundamental contribution of this paper is an explanation of the relationships between Sontag's formula [3], Freeman and Kokotovic's min-norm controls [4], and all other continuous, clf-based stabilizing
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controls by a point wise control value analysis. It will be shown that at any xed x the point-wise min-norm control parameterizes a line in SV 2 Rm , and that a point on this line corresponds to Sontag's formula. As described in [3], Sontag's formula is a continuous control law given by
2−D Control Space 5
4
3
2
1
8 0 5 T For example u = [−8, 2] does not have a fifty percent gain reduction margin, reducing u by one half pushes it out of the stabilizing control value set.
4
3
u
2
.5u 1
0 T
−g Vx −1
Only controls beyond this dotted line will have a fifty percent gain reduction margin.
−2
−3
SR
4 Gain Margins and Inverse Optimality
De nition 4.1 An asymptotically stabilizing control
−4
Figure 4: In nite Gain Margins in the Control Space
is subject to an unknown disturbance d 2 Rr . Analogues of Sontag's formula and the min-norm approach can be applied to this problem, both of which result in control laws that are pointwise scalings of the ;gT Vx vector. The signi cance of this is that at any xed state, the projection of an arbitrary control vector onto ;gT Vx exactly predicts the rate of decrease the Lyapunov function will experience under that control authority. The next section will show how this simple idea can be used to ensure desirable gain margins and optimality with respect to a meaningful cost index.
Robustness to disturbances at the input can be measured in many ways, and Lyapunov redesign as discussed in the Section 3.1 addresses this issue, but a straightforward approach is to quantify the amount of ampli cation (and diminution) a control signal can experience before instability results. Toward that end, we de ne stability margins as follows.
This shaded region corresponds to stabilizing control values. However, only control values that lie on this side of the dashed line will have infinite gain margin.
−4
−5 −9
−8
−7
−6
−5
−4
−3
−2
−1
0
1
Figure 5: Gain Reduction Margins in the Control Space
Theorem 4.2 The state-dependent control-value set which contains all controls that have stability margins of (; ; 1) is SR (x), where SR is de ned as follows: 1 2
4 u 2 Rm : ;uT gT V > max 0; 2V T f SR (x) = x x
law, u = q(x), has stability margins (m1 ; m2 ) where
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;
the null space of ;gT Vx , consider the following controllable linear system: x_ = Ax + Bu; (6) where 2 3 2 3 1 0 0 1 0 0 4 4 A = 4 0 ;1 15 and B = 4 1 1 05 : ;1 0 1 ;1 0 1 It is straightforward to verify that a valid clf for this system is V (x) = xT x. De ne a performance index: R 4 diag(2 1 1), J (x0 ; u) = 01 xT Qx + uT u, where Q = and denote the control that minimizes J as u = ;B T Px where P is the positive-de nite symmetric solution of the associated algebraic Riccati equation. Our rst clf-based control, u1(x) = ;(x)gT Vx + (x) (where (x) lies perpendicular to ;gT Vx ) will exploit the null space of gT Vx = B T x by using the optimal control to nd an appropriate (x) function:
Proof: First, any stabilizing control value which satis es ;uT gT Vx > 0 (or equivalently > 0) will have an in nite gain margin in the sense that amplifying the control value by any factor 1 < m2 < 1 will only push u farther into the stabilizing control-value set: V_ (m2 u) < V_ (u) 8 m2 > 1. This is shown in Figure 4. In order to have a gain reduction margin of 12 u, it is only necessary that the reduced control value still lie in the stabilizing region: VxT f + 12 VxT gu < 0 () VxT gu < ;2VxT f () ; VxT ggT Vx < ;2VxT f 2VxT f
() > V T ggT V x x () ;ugT Vx > 2VxT f:
This idea is shown in Figure 5.
4 u (x) + ;VxT gu gT V : (x) = x V T ggT V x
4.1 Optimality Optimal control laws can be related to stability margins. For example, in [8, 9] it is shown that optimal control laws have stability margins of (; 12 ; 1). In fact one of the primary motivations for considering inverse optimal control laws [4, 10] is that they have these desirable stability margins. It is shown in [10, p.108] that all clf-based control laws of the form u = ;k(x; Vx ) = ; 12 R;1 (x)gT Vx with gain margin of (; 12 ; 1) are optimal with respect to the meaningful cost function: Z 1 4 ;V T f + 1 gT V k: J (x) = l(x) + kT R(x)k; l(x) = x 2 x 0 This result was used in [5] to show that any continuous selection from the set-valued map SR (x) is inverse optimal (optimal with respect to a cost index that is constructed a posteriori instead of chosen a priori by the designer). The signi cance of this result is that optimality can be understood graphically as well: optimal controls lie in the control value set SR (x) at all nonzero states.
5 Example of Exploiting the Nullspace of VxT g As an illustration of the potential performance enhancement o ered by adding a control component in
x
Note that this operation merely selects the component of u (x) which is perpendicular to ;gT Vx and assigns that value to (x). We will ensure that u1 is stabilizing T f +xT x V x by choosing (x) = VxT ggT Vx , which clearly satis es Theorem 2.1. We will compare the performance of u1 (x) with the performance of another control, u2 (x) based on the same clf. u2 (x), however, will be constrained to lie in the direction of ;gT Vx . In order to be fair, we will set the magnitude of u2(x) to be equal to the magnitude of u1(x) so that u1 and u2 exert the same control authority, i. e. u2 = ; kkVuxT1gkk gT Vx . The cost associated with u1 (the control law which exploited the null-space) was J (x0 ; u1) = 296:3, whereas the cost associated with u2 was J (x0 ; u2 ) = 404:4. The initial state was chosen arbitrarily to be x0 = [8 ; 6 ; 2], however this simulation was repeated numerous times with a wide variation in initial states and u1 (x) was always found to o er superior performance. The signi cance of this simple example is that the performance of a control law is rarely measured in terms of its Lyapunov function. This simulation shows that high-performance, clf-based control laws should exploit the null space of gT Vx in some manner.
6 Discussion and Conclusion Lyapunov functions provide a powerful tool for analyzing the stability properties of nonlinear systems,
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and many formulas and techniques based on control Lyapunov functions have been proposed in the literature. All of these control strategies can be easily visualized in the control space at a xed state. The reason such a point-wise graphic has global meaning (despite the inherently local information provided by a graph of Rm at a xed state) is that the clf V (x) ensures global properties, while only requiring the local constraint that V_ < 0. In addition to being an aide in understanding Lyapunov-based controllers, such a state-dependent control value analysis opens new avenues of research into the construction of high-performance clf-based controls. For example, an asymptotically stable, inverse optimal model predictive control (MPC) strategy can be de ned as kMPC (x) = arg u2min J (x; u) S (x) MPC R
where JMPC (x; u) is a cost criteria based on model predictive strategies. This simple receding horizon strategy would combine the best of Lyapunov analysis and computational performance by searching the null-space of gT Vx for a performance enhancing augmentation to a standard clf-based strategy. Such a MPC would be guaranteed stable, optimal, robust to input disturbances, and no terminal constraint would be imposed on the optimization routine. Another interesting thing to note, is that the ef cacy of such an optimization would scale with the size of the control space. Stability, in the sense of Lyapunov, imposes a single dimensional constraint on the m-dimensional control vector, leaving m ; 1 dimensions free. Clearly for large values of m, choosing wisely the component of the control that lies in this `free' space becomes more critical; a single input system (m ; 1) has no such `free space', but for m = 4 the null space of gT Vx o ers three dimensions of freedom that could have a great impact on the large-signal performance of the closed-loop system. In the past, clf-based controls have neglected this null space of gT Vx . Since any component of a control that lies orthogonal to gT Vx will have no e ect on the rate V_ , this is understandable. However, if large-signal performance is sought in a control law, a component in this null space might signi cantly improve any performance measure that isn't speci cally clf-based.
Acknowledgments This work was supported by the National Science Foundation: award number 9732917.
References [1] Z. Artstein, \Stabilization with relaxed controls," Nonlinear Analysis, Theory, Methods, and Applications, vol. 7, no. 11, pp. 1163{1173, 1983. [2] E. D. Sontag, \A Lyapunov-like characterization of asymptotic controllability," SIAM Journal on Control and Optimization, vol. 21, pp. 462{471, May 1983. [3] E. D. Sontag, \A `universal' construction of Artstein's theorem on nonlinear stabilization," System and Control Letters, vol. 13, pp. 117{123, 1989. [4] R. A. Freeman and P. V. Kokotovic, Robust Nonlinear Control Design: State-Space and Lyapunov Techniques. Systems and Control: Foundations and Applications, Birkhauser, 1996. [5] J. W. Curtis and R. W. Beard, \Satis cing: A new approach to constructive nonlinear control," IEEE Transactions on Automatic Control, submitted 2001. [6] R. W. Beard, B. Young, and W. Stirling, \Nonlinear regulation using the satis cing paradigm," in 2001 American Control Conference, (Arlington, VA), June 25{27 2001. [7] R. A. Freeman and P. V. Kokotovic, \Optimal nonlinear controllers for feedback linearizable systems," in Proceedings of the American Control Conference, (Seattle, Washington), pp. 2722{2726, June 1995. [8] S. T. Glad, \On the gain margin of nonlinear and optimal regulators," IEEE Transactions on Automatic Control, vol. 29, pp. 615{620, July 1984. [9] J. N. Tsitsiklis and M. Athans, \Guaranteed robustness properties of multivariable nonlinear stochastic optimal regulators," IEEE Transactions on Automatic Control, vol. 29, pp. 690{696, August 1984. [10] R. Sepulchre, M. Jankovic, and P. Kokotovic, Constructive Nonlinear Control. Communication and Control Engineering Series, Springer-Verlag, 1997. [11] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design. Wiley, 1995. [12] M. Krstic and H. Deng, Stabilization of Nonlinear Uncertain Systems. Communications and Control Engineering, Springer Verlag, 1998. [13] H. K. Khalil, Nonlinear Systems. Upper Saddle River, NJ: Prentice Hall, second ed., 1996. [14] M. Corless, \Control of uncertain nonlinear systems," Journal of Dynamical Systems Measurement and Control, vol. 115, pp. 362{372, 1993. [15] M. Corless and G. Leitmann, \Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems," TAC, vol. 26, pp. 1139{ 1144, October 1981. [16] V. Jurdjevic and J. Quinn, \Controllability and stability," Journal of Di erential Equations, vol. 28, pp. 381{389, December 1978.
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