Proceedings of the 25th Chinese Control Conference 7–10 August, 2006, Harbin, Heilongjiang

Further Results On Global Controllability of Affine Nonlinear Systems* Yimin SUN1 , Lei GUO2 , Qiang LU1 , Shengwei MEI1 1. Department of Electrical Engineering, Tsinghua University, Beijing, 100084 2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100080 E-mail: [email protected]

Abstract: Most of the existing results on controllability of nonlinear systems are concerned with local controllability. In this paper, we will present some further results on global controllability of two classes of affine nonlinear systems, by using the techniques developed recently in [1][2]. The first class is planar affine nonlinear systems with one singular point, the second class is high-dimensional affine nonlinear systems with special structure. Key Words: Nonlinear systems, global controllability, vector field.

1

INTRODUCTION

The controllability of nonlinear systems has been studied extensively over the past three decades, and considerable progress has been made in either analysis or synthesis by introducing some useful methods, including the well-known differential geometric method. However, most of the existing results in the literature on controllability of general nonlinear systems are concerned with local ones only, see, for example, [3]∼[6]. For the study of global controllability, although much effort has been made in the literature (see [7]∼ [13]), most of the related results are rather complicated and the complete characterization of global controllability is still lacking, due to the difficulty of this problem. Recently, a major advance has been made for the following planar affine nonlinear system, x˙ = f (x) + g(x)u,

(1)

by using a new method based on planar topology [1][2]. Earlier to [1][2], an important literature is the book [14], where under some hypotheses on f and/or g, some necessary and sufficient conditions on global controllability of planar affine nonlinear system were obtained. Unfortunately, the conditions on f and g used in [14] appear to be unnecessary and stringent since, for example, the main results in [14] (see, p. 44 and p. 109) cannot include the standard controllability criterion for even linear systems. The resent contribution [1] and [2] do not have these drawbacks, where the necessary and sufficient conditions on global controllability of the system (1) were obtained under some natural hypotheses on f and g. Moreover, the ideas *Please address all correspondence to Professor Lei GUO. This work was supported by the National Natural Science Foundation of China under Grant No. 50525721, 60221301 and 60334040.

in [1][2] provide a new approach to the study of more general planar systems and of some high dimensional systems. In this paper, we will investigate further the global controllability of affine nonlinear systems by using the methods in [1][2]. First, we will give a generalization of a result in [14], where the condition that the vector field g or −g is globally asymptotically stable seems to be unnecessary. Then, by two ways, we will generalize the main results in [1] to the high dimensional systems with special structure. One is the system with n − 1 inputs, the other is the singleinput system with triangular structure. Finally, we will give some physical examples in electric power systems to show the application of our theorems.

2

Main Results

First, we give the definition of global controllability of nonlinear systems. Consider the following affine nonlinear systems x˙ = F (x) + G(x)u, (2) where state vector x ∈ Rn , F (x) ∈ Rn×1 , G(x) ∈ Rn×m and F (x), G(x) are the C1 (Rn ) matrix functions, u ∈ Rm is the input vector. Definition 2.1 The control system (2) is said to be globally controllable, if for any two points x0 and x1 ∈ Rn , there exists a right continuous input vector function u(t) such that the trajectory of the system (2) under u(t) satisfies x(0) = x0 and x(T ) = x1 for some time T > 0. 2.1 Planar Affine Nonlinear with Singularity We consider the following planar affine nonlinear systems: x˙ 1 = f1 (x1 , x2 ) + g1 (x1 , x2 )u x˙ 2 = f2 (x1 , x2 ) + g2 (x1 , x2 )u,

(3)

where fi (x1 , x2 ), gi (x1 , x2 ), i = 1, 2 are smooth functions of the state x = (x1 , x2 )T in R2 , and u is the

control function taking values on R. We denote f (x) = T T (f1 (x), f2 (x)) , g(x) = (g1 (x), g2 (x)) and assume that g or −g is locally asymptotically stable at the origin and that g(0) = 0, g(x) 6= 0, ∀ x ∈ R2 \0. Without loss of generality, we suppose that g(x) is locally asymptotically stable (see [15]) and D is the domain of attraction of g(x). Definition 2.2 A control curve of the system (3) is defined as a solution (x1 (t), x2 (t)) of the following differential equation on the plane: x˙ 1 = g1 (x1 , x2 ) x˙ 2 = g2 (x1 , x2 ), where gi (x), i = 1, 2 are the same as those in (3). By Poincare-Bendixson Theorem and Jordan curve Theorem, every trajectory of the vector field g(x) tends to a singular point, or extends to infinite, or spirals around a limit cycle (see [16]). We are now in position to give the following definition. Definition 2.3 A solution trajectory of the vector field g(x) on the plane is called as a regular control curve of the system (3) if it is a nonzero closed curve or its two ends extend to infinite1 . Theorem 2.1 For the control systems (3), let f (0) 6= 0 and D is the domain of attraction of g(x). Then the system (3) is globally controllable, if and only if there are no points x+ , x− ∈ D\0 such that det[f (ϕ(x+ , t)), g(ϕ(x+ , t))] > 0

∀ t ∈ T+ ,

det[f (ϕ(x− , t)), g(ϕ(x− , t))] 6 0

∀ t ∈ T− ,

and the function g1 (x)f2 (x)−g2 (x)f1 (x) changes its sign over every regular control curve, where ϕ(x∗ , t) denotes the control curve passing through the point x∗ , and T∗ is the existence interval of the control curve ϕ(x∗ , t) and the star ∗ denotes either + or −. In particular, if the domain of attraction D is bounded, then the system (3) is globally controllable, if and only if the function g1 (x)f2 (x) − g2 (x)f1 (x) changes its sign over every regular control curve. Remark 2.1 Theorem 2.1 is a generalization of the Theorem 4.3 in [14], where the vector field g or −g is assumed to satisfy a stronger condition, i.e., the global asymptotical stability condition. Now, we consider a kind of the degenerate case of the control system (3), where the domain of attraction D of g(x) degenerates into a point 0. If we view the point 0 as a degenerate control curve of the systems (3), we have the following corollary. 1 The two ends of a curve Γ(t), t ∈ R extending to infinite means that: kΓ(t)k → +∞, when t → +∞ and −∞.

Corollary 2.1 Suppose f (0) 6= 0 and every control curve of the systems (3) is a closed curve. Then the control system (3) is globally controllable, if and only if the function g1 (x)f2 (x) − g2 (x)f1 (x) changes its sign over every nonzero control curve. For some systems, it is not difficult to verify if every control curve of the systems (3) is a closed curve. For example, if there is a Lyapunov function V (x), such that V (x) > 0 for any x ∈ R2 \ 0, V (0) = 0, V (x) → +∞ as kxk → ∂V ∂V g1 + ∂x g2 ≡ 0, then every +∞, and V (x) satisfies ∂x 1 2 trajectory of the vector field g(x) is a closed curve, namely every control curve of the systems (3) is a closed curve. 2.2 Some High Dimensional Systems First, we generalize the main theorem in [1] to n dimensional systems with n − 1 inputs. Here we should note that a similar but more general model had been ever studied in [13], however no necessary and sufficient condition is given there. Now, we consider the following system x˙ = f (x) +

n−1 X

bi ui ,

(4)

i=1

where f (x) ∈ C1 (Rn ), x ∈ Rn is the state vector, bi (i = 1, 2, . . . , n − 1) are constant vectors which are linearly independent, u = (u1 , u2 , . . . , un−1 )T is the control vector. Because bi (i = 1, 2, . . . , n − 1) are independent, there is a nonzero vector c such that hc, bi i = 0,

i = 1, 2, . . . , n − 1,

where h·, ·i denotes the inner product of two vectors. Now, we give the definition of control hyperplane as follows. Definition 2.4 The control hyperplane of the system (4) is a hyperplane which passes through any point x0 and takes the vector c as its normal vector, namely the hyperplane hx − x0 , ci = 0,

x ∈ Rn .

Theorem 2.2 The necessary and sufficient condition of global controllability of the system (4) is that the function det(f (x), b1 , b2 , . . . , bn−1 ) changes its sign over every control hyperplane. Now we give the generalization of main theorem in [1] to the systems with single input and triangular structure. Consider the following affine nonlinear control systems with triangular structure: x˙ 1 = f1 (x1 , x2 ) + g1 (x1 , x2 )x3 x˙ 2 = f2 (x1 , x2 ) + g2 (x1 , x2 )x3 x˙ 3 = f3 (x1 , x2 , x3 ) + g3 (x1 , x2 , x3 )x4 .. . x˙ n = fn (x1 , x2 , . . . , xn ) + gn (x1 , x2 , . . . , xn )u,

(5)

where fi , gi ∈ Cn−2 , i = 1, 2, . . . , n, namely they are the (n − 2) times smooth function, u is a right-continuous control function taking values on R. Let g(x1 , x2 ) = (g1 (x1 , x2 ), g2 (x1 , x2 ))T 6= 0 for any (x1 , x2 )T ∈ R2 and gi (x1 , x2 , . . . , xi ) 6= 0 for any (x1 , x2 , . . . , xi )T ∈ Ri , i = 3, . . . , n.

change its sign for any λ > 0, where every λ corresponds a control curve.

The key idea here is to take the advantage of the triangular structure and to apply the results on planar affine nonlinear systems to the following subsystem of (5)

∆ = cρ2 + 4(d − b)d(cλ)2 .

x˙ 1 = f1 (x1 , x2 ) + g1 (x1 , x2 )v x˙ 2 = f2 (x1 , x2 ) + g2 (x1 , x2 )v.

(6)

Then we have the following theorem. Theorem 2.3 The control system (5) is globally controllable if and only if its subsystem (6) is globally controllable, namely the function g1 (x1 , x2 )f2 (x1 , x2 ) − g2 (x1 , x2 )f1 (x1 , x2 ) changes its sign over every control curve of (6) in the plane (x1 , x2 ). Remark 2.2 The global controllability of the system (6) was established in [1]. Similarly, if the subsystem (6) of the system (5) satisfies the conditions of Corollary 2.1, then the Theorem 2.3 is also valid.

3

If d − b = 0, obviously, as long as λ is large enough, (9) will be negative for any s ∈ [−1, 1]. If d − b < 0, we have

Obviously, as long as λ is large enough, ∆ will be negative. Therefore (9) will be negative for any s ∈ [−1, 1]. √ If d−b > 0, the equation (d−b)cλs2 + cρs−cdλ = 0 has 2 a positive and a negative roots. √ √ Because (d − b)cλ(−1) + cρ(−1)−cdλ = −bcλ− cρ < 0, the negative root must be less than −1. Similarly, as long as λ is √ √ large enough, we have (d − b)cλ + cρ − cdλ = −bcλ + cρ < 0, therefore the positive root must be larger than 1. Hence, (9) will be negative for any s ∈ [−1, 1] as long as λ is large enough. In summary, the system (8) is not globally controllable. Finally, by Theorem 2.3 and Remark 2.2, the system (7) is not globally controllable. Example 3.2 A field-controlled DC motor with negligible shaft damping term −dx3 may be represented by a thirdorder model of the form

Some Examples

x˙ 1 = −ax1 + u

First, we will show the application of our theorems by the following three practical models of electrical power systems. Example 3.1 A field-controlled DC motor can be described by x˙ 1 = −ax1 + u x˙ 2 = −bx2 + ρ − cx1 x3

(7)

x˙ 3 = θx1 x2 − dx3 ,

where x1 , x2 , x3 and u represent the stator current, the rotor current, the angular velocity of the motor shaft and the stator voltage respectively, a, b, c, d, θ and ρ are positive constants (see [15] pp. 51-52). According to Theorem 2.3 and Remark 2.2, we need only to investigate the global controllability of the following system: x˙ 2 = −bx2 + ρ − cx3 v x˙ 3 = −dx3 + θx2 v.

(8)

It is easy to know that the system (8) satisfies the condition of Corollary 2.1 and the trajectoT ries√ of the √ vector √ field√ (−cx3 , θx2 ) are ellipses (λ c cos( cθt), λ θ sin( cθt)), t ∈ R, for any λ > 0. By Corollary 2.1, the control system (8) is globally controllable if and only if the following function √ (d − b)cλs2 + cρs − cdλ, s ∈ [−1, 1] (9)

x˙ 2 = −bx2 + ρ − cx1 x3 x˙ 3 = θx1 x2 ,

(10)

where a, b, c, θ and ρ are positive constants (see [15] p. 530). Similar to the above example, the control system (10) is globally controllable if and only if the following function √ √ −bcλs2 + cρs = s(−bcλs + cρ), s ∈ [−1, 1] (11) change its sign for any λ > 0, where every λ corresponds a control curve. It is easy to know that (11) change its sign for any λ > 0. Hence, the control system (10) is globally controllable. Example 3.3 Consider the following model of a permanent magnet synchronous motor: did = −Rs id + np ωLq iq + ud dt diq Lq = −Rs iq − np ωLd id − np ωΦ + uq dt dω 3 J = np [(Ld − Lq )id iq + Φiq ] − τL , dt 2

Ld

(12)

where id and iq are d − q axis currents, ω is the motor speed, ud and uq are d − q axis voltages, Ld , Lq and J are nonzero constants, Rs , np , Φ and τL are constants, id , iq and ω are state variables (see [17]).

The control hyperplane of the system (12) is ω = c, and it is easy to know that the function 3 2 np [(Ld

− Lq )id iq + Φiq ] − τL J changes its sign over every control hyperplane if and only if np [(Ld −Lq )2 +Φ2 ] 6= 0. Hence, by Theorem 2.2 we know that the necessary and sufficient condition for the global controllability of the system (12) is np [(Ld − Lq )2 + Φ2 ] 6= 0. det(f (x), b1 , b2 ) =

4

Concluding Remarks

In this paper, by using the method in [1] and [2], we presented a necessary and sufficient condition for global controllability of planar affine nonlinear systems with a singular point and of some high dimensional systems with special structure. These conditions are imposed on the system nonlinear structure only. In addition, we also give some examples to show the application of these conditions in some physical systems. For future investigation, it is desirable to extend the main results of this paper to more general high dimensional nonlinear control systems.

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