Bilinear system control with exponential stability

Proceedings of the American Control Conference Philadelphia, Pennsylvania * June 1998

Bilinear System Control with Exponential Stability

Min-Shin Chin

Department of Mechanical Engineering

National Taiwan University

Taipei, Taiwan, Republic of China

TEL:02-23630231 ext. 2414

EMAIL:[email protected]

FAX102-23631755

Abstract- For a bilinear system that is open-loop neu- trally stable, a quadratic state feedback control has been proposed to ensure global asymptotical stability of the closed-loop system. In this paper, a new nonlinear con- trol is proposed so that the closed-loop system is not only globally stable but also exponentzully stable. The new control results in a much faster state convergence rate; furthermore, it can be applied to a construzned bilinear system which is subject to whatever tight satu- ration limits on the control input.

1

Introduction

This paper considers the control of a bilinear system i ( t ) = A z ( t )

+

~ ( t ) N i ~ ( t ) , ~ ( 0 ) = 2 0 , (1) where z ( t ) E

R"

is the system state vector, u ( t ) is a scalar control input, and A E Rnx" and N E R n x n are constant square matrices. It is assumed that there exists

is a positive definite matrix Q such that

In other words, the open-loop system is neutrally stable

[l]. Furthermore, the pair ( A , N) satisfies the following controllability assumption

[a]:

there exists an integer N ( > n - 1) such that

s p u n { u d " ( A , N ) z o , k = 0 , 1 , 2

,...,

N } = R" ₍₃₎ for any nonzero x o in R", where adk(A, N ) ' s are defined recursively by

U d 0 ( i 2 , N ) = N ,

u d k f l ( A , N ) = A . u d k ( A ,

N)

- adk(A, N ) . A ,

where

IC

= 0, 1, 2, ...

Conventionally, quadratic feedback control [3-51 has been proposed for the stabilization of the system (1):

u ( t )

=

- z T ( t ) Q N x ( t ) , (4) which ensures global asymptotic stability of the closed- loop system. However, it has been shown [6] that the closed-loop system is n o t exponentially stable, and the state converges as

(5) The objective of this work is to introduce a new non- linear control control which stabilizes the closed-loop system globally and, most importantly,exponentzally.

The exponential stability results in a much faster time response of the system state than (5); furthermore, it enhances the robustness of the controlled system [7].

2

Nonlinear Control

The proposed nonlinear control is as follows:

0 1 X ( t ) = 0,

where p is a positive control gain, and Q is as in ( 2 ) . Notice that the control (6) is uniformly bounded for whatever values of the state x ( t ) :

l4t)I

I

P n Q,

w

>

0 , (7) where q and n are respectively the matrix norms of Q and

N.

If the bilinear system (1) is subject to the control constraint

l 4 t ) l

5

%r”,

the control gain p in (6) will have to be chosen within the range:

3

Stability Analysis

The following lemma can be derived using a contradic- tion argument based on the controllability assumption (3).

Lemma 1 : If the system (1) satisfies the controllability assumption ( 3 ) , and there exists a constant vector zo

such that for all

t

E [ k T , kT

+

T )

Q N e A ( t - k T ) ~ o

0 ,

(9)

..,~~(t-k~)

for any T

>

0, then $ 0 must be the null vector. Given any time interval length T

>

0, define a scalar function B ( . ) : S -+

R

for the controlled system (1) and (6), where S is the unit sphere in R”, and z ( t )

#

0,

Note that given Q and

N ,

the function B ( . ) is deter- mined by the closed-loop trajectory

x(t)/llz(t)ll, t

E

[ k T , kT

+

T ) , which is in tern uniquely determined by its initial condition z(kT)/llz(kT)II. Therefore, B ( . )

in (10) is defined as a function of the initial condition

t(kT)/llz(kT)II. It can be shown that this function has t,he following property.

Lemma 2 : There exists some positive constant

P

such that

where the inf(zmum) is taken over all r(kT)/llz(kT)11 E

S (that is, over all z(!cT)

#

0).

One can now prove the global exponential stability of the controlled bilinear system.

Theorem : Consider the bilinear system (1) and the nonlinear control (6) subject to the constraint (8). Given any initial condition, the controlled state z ( t ) con- verges to zero exponentzally.

Proof: Define a Lyapunov function candidate

where Q is as in ( 2 ) . Notice that

x T ( t ) Q 4 t )

L

~ 1 1 ~ 1 1 2 ,

(12) where

x

is the maximum eigenvalue of the positive def- inite matrix Q. The time derivative of V ( t ) along (1) and (6) is given by

V ( t )

=

z T ( t ) ( A T Q

+

Q A ) z ( t )

+

2zT(t)QNz(t)u(t)

Since V ( t ) is non-increasing, one has

V ( k T

+

T )

5

V ( t ) , V t E [ k T , k T

+

T ) . ₍₁₄₎

Integrating the equation (13) from kT to ( k

+

l)T yields

V ( k T

+

T )

-

V ( k T )

=

5

-2-V(kT

P P

+

T ) ,

x

where the first inequality results from and the second from (11) in Lemma 2.

the above equation gives

1

12) and (14), Rearranging

proving that the Lyapunov function V ( k T ) decreases exponentially to zero as k approaches infinity, and so does z ( k T ) .

Finally, it remains to show that the contznuous state

z ( t ) remains bounded and also converges to zero expo- nentially. To this end, note from (1) and (7) that

IlWll

5

( a

+

Pn2q)114t)lll (16)

where a is the matrix norm of the open-loop system matrix A . Using (16) and Proposition 1.4.1 in [8], one can derive that the continuous state

z ( t )

is bounded by the discrete state t ( k T ) by

Ilt(t)ll

5

,(atpn2q)(t-eT)IIz(kT)II,

V t E

[kT,

kT

+

T ) ,

and this shows that the continuous state z ( t ) remains bounded and converges exponentially to zero as the dis-

Crete state z ( k T ) does. 0

Remark : Notice that the Theorem holds for whatever

value of the control saturation limit umaz

>

0 as long as

the control gain p satisfies (8). As a result, even if the control actuator can provide only a small amount of en- ergy (i.e.: a tight saturation limite U,,,), the proposed

control can still stabilize the system globally. Such a property is not shared by the conventional quadratic control (4) where the amount of energy required is pro- portional to the square of Ilz(t)ll. Hence, large control energy is required if ~ ( t ) is initially far from the origin.

4

Simulation Examples

Example 1: Consider the bilinear system (1) with

and the initial condition ~ ~ ( 0 ) = [5, -21. Figure 1 shows the state response of the system with the conventional quadratic control ( 4 ) with Q = 31, and Figure 2 the state response with the new nonlinear control (5) with

p = 3, Q = 1. It is obvious that the new nonlinear control results in a much faster time response since the state now decays exponentially.

Example 2: Consider same system as in the previous example but with a perturbation on the open-loop sys-

The perturbed open-loop system becomes slightly un- stable, but still controllable in the sense of (3). The initial condition is ~ ~ ( 0 )

=

[5, -21. Figure 3 shows the closed-loop system becomes unstable under the conven- tional quadratic control (4). However, the new nonlinear control ( p = 3, Q = I ) can still stabilize the perturbed unstable system as is shown in Figure 4.

The reason why the new control can stabilize a slightly perturbed system is as follows. Since the closed-loop system with the proposed control is exponentzal stable, one can show, following Theorem 121 in Chapter 7 of [7], that the exponential stability is retained given any small perturbation in the open-loop system matrix A in (1). Note t h a t such slightly perturbed system may not be stabilized by the conventional quadratic control (4) since it does not provide exponential stability for the nominal closed-loop system.

one as p approaches zero or infinity. Hence, according t o (15), a too small or too large control gain p can re- sult in slow state convergence. In other words, there exists an optimal value of the control gain p , which can best expedites the state convergence. However, a t this moment, there is no analytic method t o predict this op- timal value; it can be searched only throulgh computer simulations.

5

Conclusions

In this paper, a new nonlinear control different from the conventional quadratic feedback control is proposed to stabilize a homogeneous-in-the-state bilinear system. The new control results in exponential stability of the closed-loop system, and hence a much faster time re- sponse than with the quadratic control.

References

[l] M. Slemrod, ”Stabilization of Bilinear Control Sys- tems with Applications t o Nonconservative Problems in Elasticity,” SIAM J . Contr. and Optimization, ~01.16,

[ 2 ] M. Vidyasagar, Nonlanear Systems Analyszs, Prentice Hall, New Jersey, 1993.

[3] V. Jurdjevic and J . P. Quinn, ”Controllability and Stability,” Journal of Differential Equations, vo1.28, [4] E. P. Ryan and N. J . Buckingham, ” O n Asymptoti- cally Stabilizing Feedback Control of Bilinear Systems,” IEEE Trans. Auto. Contr., vol. AC-28, pp.863-864, 1983.

[5] S. N . Singh, ”Stabilizing Feedback Controls for Non- linear Hamiltonian Systems and Nonconservative Bilin- ear Systems in Elasticity,” J . of Dyn. Syst. Meas. and Contr., vol. 104, pp.27-32, 1982.

[6]

J .

P. Quinn ”Stabilization of Bilinear Systems by Quadratic Feedback Controls,” Journal of Mathematical Analysis and Applications, vol. 75, pp.66-80, 1980. [7] F. Callier and C. A. Desoer, Lznear System Theory, Springer-Verlag, Hong Kong, 1992.

[8]

S.

Sastry and M. Bodson, Adaptave Control, S t a b d aty, Convergence, and Robustness, Prentice-Hall, Lon- don, 1989.

pp.131-141, 1978.

pp.381-389, 1978.

Finally, note that the constant /3 (defined in (11)) also depends on p. Consequently, the number 1

+

2pp/x in (15) may not be a monotonically increasing function of p . A plot of 1

+

2 p p / x versus p is given in Figure 5 for the controlled system in Example 1. The simulation results indicate t h a t the number 1

+

2p,f3/x approaches

- I t

1

-2 I I I I I I I I I

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 Time (second)

.

Figure 1. State response with quadratic c3ntrol

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 -18.0 20.0 Time (second)

Figure 2. State response with new nonlinear control 125 100 75 50 2s 0 I I I I I I I -25

'

I I I I I I I I 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Time (second)

F i r e 3. Perturbed response with auadratic-control

5 4 3 2 I 0 - I -2 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 Time (second)

Ffgure 4. Perturbed response with new nonlinear control

3.5

-

3.0

-

2.5

-

2 0

-

15

-

1.0 L I I I I I 1 I I 0.0 25 5.0 7.5 10.0 125 15.0 175 20.0 P ~ i g v e 5. I

+

2p$ l i versus

p

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