依據改進的RICCATI 方程式之線性時變系統控制

依據改進的 Riccati 方程式之線性時變系統控制

A new Riccati equation design for LTV system control

計畫編號：NSC-88-2213-E-002-088 執行期間：87 年 8 月 1 日至 88 年 7 月 31 日 主持人：陳明新教授 台灣大學機械工程系 一、中文摘要 本計畫提出一種新的，對於線性時 變系統的控制方法。本方法是針對轉換 過的狀態變數作控制器設計。新的狀態 變數具有與轉換前的狀態變數大小成 反比的性質，因此新的控制方法是希望 能找到一種狀態回授增益，使得轉換過 的系統呈全域性指數形式發散。本文所 提出的控制方法與傳統控制時變系統 的方法比較，有兩個優點：一是計算狀 態回授增益時僅需要過去及現在的系 統資訊，不需預測系統未來的變化；另 一是不需對系統參數作微分，因此新的 控制方法也可以運用在具有片段連續 性時變參數的系統上。 關鍵詞：線性時變系統，控制設計， Riccati 方程式，LQ 控制。 Abstr act

This project proposes a new control design for linear time-varying systems. The design is based on an inversion state transfromation, which converts the prob-lem of stabilizing the system state into that of de-stabilizing the transformed state. The new design utilizes a forward Riccati equation while conventional con-trol ustilizes backward Riccati equation. The forward Riccati equation design has the advantage that only present and past information of the system matrices is required in calculating the control. The other advantage is that no differentiation of the time-varying system matrices is required so that the new control can be applied to systems whose time-varying parameters are piecewise continuous

only.

Keywor ds：linear time-varying systems,

control design, Riccati equation, LQ control.

二、Introduction

There are essentially two approaches to the state feedback control design of Linear Time-Varying (LTV) systems. In the first approach, one assumes knowl-edge of future information on the system matrices, and uses such information to synthesize the state feedback gain. A well-known example is the LQ control (Kalman, 1964; Kwakernaak and Sivan, 1972), in which the control Riccati equa-tion is solved backwards starting from some future time to the present time. However, their assumption is rarely met since prediction of how system matrices vary in the future can be very difficult in most practical applications. The second approach to the state feedback control design of LTV systems is the pole-placement-like control (Wolovich, 1968; Valasek and Olgac, 1993). In this ap-proach, one must differentiate thesystem matrices with respect to time consecu-tively up to the order of the system di-mension. In practice, information of the system matrices is often obtained through sensor measurement. Such measurement inevitably introduces noise that can easily destroy the fidelity of the

required information in a time differen-tiation process. Hence, it is generally not advisable to calculate the state feedback gain by differentiating a noise-corrupted system matrix.

control design that avoids the shortcom-ings of the above approaches. In other words, the control design should require neither time differentiation of the

system matrices nor prediction of future information of the system matrices. The calculation of the control law should be based soly on past and present infor-mation of the system matrices.

In this project, it will first be shown that by the introduction of an inversion state transformation, the problem of stabilizing the system state is converted into that of destabilizing the transformed system. Second, it is shown that the differential Riccati equation for destabilizing the transformed state becomes a forward Riccati equation with the boundary con-dition set at the initial time of operation instead of at some future time, similar to the differential Riccati equation used in the Kalman filter design. The require-ment of future information on the system matrices can thus be avoided.

三、Preliminar ies

Consider a multivariable linear time-varying system: x t C y x x t u t B x t A x ) ( ) 0 ( ), ( ) ( ) ( 0 = = + = (1) where x(t) is the system state, u(t) is

the control input, y(t) is the system out-put, the system matrix A(t) , the input matrix B(t), and the output matrix C(t) are time-varying matrices with piecewise continuous and bounded elements.

It is assumed that the system (1) is uniformly controllable as defined below.

Definition 1 : The pair (A(t),B(t)) is uniformly controllable if there exist a constant ∆ and another constant α de-pending on ∆ such that the controllability grammianI(t−∆,t) satisfies , 0 ) , ( ) ( ) ( ) , ( ) , ( Ψ Ψ ≥ > = −

∫

where Ψ(t,τ) is the state transition ma-trix of the free system in Eq.(1).

The goal is to find a state feedback gain )

, (t x

K such that the control input ) ( ) , ( ) (t K t x x t u =

stabilizes the closed-loop system expo-nentially.

四、Control Design

The proposed control design is based on the construction of an inversion state transformation . 0 ) ( , ) ( ) ( ) ( = ₂ ∀x t ≠ t x t x t z (2)

It follows from this definition that ) ( 1 ) ( t x t z = (3) By differentiating Eq.(2) along the tra-jectory (1), one obtains

), ( ) ( ) ( ) ( ) ( ) ( ) (t T x At z t T x B t vt z& = + (4)

where v(t) is the transformed input de-fined by , ) ( ) ( ) ( ₂ t x t u t v = (5) and T(x) is a Householder transforma-tion (Chen, 1984) . ) ( ) ( ) ( ) ( 2 ) ( t x t x t x t x I x T _T T − =

It is important to note that T(x) is sym-metric and remains uniformly bounded for any x(t)≠0 since

. 1 ) (x =

T (6) Equation (6) can be verified by checking that . ) ( 2 I x T =

Another consequence of the above equa-tion is that T(x) is invertible for any nonzero x(t): ). ( ) ( 1 x T x T− =

One can also verify thatT−1(x)z(t) ) (t z − = ;hence, z(t)=−T−1(x)z(t). Sub-stituting this relationship into the right-hand side of Eq.(4) yields

). ( ) ( ) ( ), ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( ) ( 1 t B x T t B x T t A x T t A t v t B t z t A t z = − = + = − & (7)

Here, both A(t) and B(t) are functions of x(t); however, the argument x is

omitted for the sake of brevity.

Equation (3) suggests that stabiliza-tion of the original system state x(t) is achieved if one can find a transformed input v(t) in Eq.(7) to destabilize the transformed system (7). Such a destabi-lizing control input is given by

), ( ) ( ) ( ) ( ) (t R21 t B t P t z t v = − T (8) where P(t) satisfies the forward differ-ential Riccati Equation

), ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 2 1 t P t B t R t B t P t R t A t P t P t A t P T T − − + − − = , 0 ) 0 ( =P₀ > P (9) in which A(t), B(t) are defined in Eqs.(7), R₁(t)>0, R₂(t)>0, and

) (

1 t

R , R₂(t), R₁−1(t), R₂−1(t) are all uniformly bounded. It follows from Eqs.(8), (2), and (5) that the actual con-trol input u(t) is given by

). ( ) ( ) ( ) ( ) ( 1 2 t B t P t x t R t u = − T (10)

Observe that P(t), and hence the state feedback gain in the above control depend only on past information of the system matrices A(τ) and B(τ), and the past state x(τ), where τ∈

[ ]

in fact a nonlinear dynamic state feed-back control that does not require future information of system matrices.

Under the assumption of existence of solutions, the computation of P(t) is directly based on an on-line integration of

the forward differential Riccati equation (9) starting at t =0. This is quite differ-ent from the computation of the back-ward differential Riccati equation ap-pearing in the time-varying LQ control, where the integration has to be done off-line, starting from some future time.

The following lemmas state some properties of the transformed system ma-trices A(t) and B(t).

Lemma 1: If the pair (A(t),B(t)) in Eq.(1) is uniformly controllable, the pair

)) ( ), (

(At B t in Eq.(7) is also uniformly controllable.

The second lemma, whose proof is omitted, is an immediate consequence of Lemma 1 and Duality (Callier and Desoer, 1992).

Lemma 2: Under the Hypothesis of

Lemma 1, the pair (A(t),B(t)) is uni-formly observable in the sense of Defini-tion 1.

Observe that the differential Riccati Equation (9) can be re-written as

), ( )) ( )( ( )) ( )( ( ) ( )) ( )( ( ) ( )) ( ( ) ( 1 2 1 t P t B t R t B t P t R t A t P t P t A t P T T T T T − − + − + − =

which has exactly the same form as the observer Riccati Equation below:

), ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 2 1 t Q t C t V t C t Q t V t A t Q t Q t A t Q T T − − + + = & 0 ) 0 ( =Q₀ > Q

. 1 0 1 3 2 1 ) ( , sin 5 sin 0 1 sin 5 . 1 1 cos sin 5 . 1 1 1 cos sin 5 . 1 1 cos 5 . 1 1 ) ( 2 2           =           + − − + − − − − + − = t B t t t t t t t t t A

Therefore, based on Lemma 2, one can follow the procedure in Wonham, 1968 and Anderson and Moore, 1968, to show that P(t) in Eq.(9) is uniformly bounded above and below.

Lemma 3: Under the Hypothesis of

Lemma 1, there exist positive constants

1

γ and γ such that the solution 2 P(t)

of the Riccati Eq.(9) satisfies 0. t ) ( 2 1I ≤ P t ≤γ I ∀ > γ

One can now state the main theorem of this paper, which shows that the pro-posed control (10) is globally stabilizing for the system (1).

Theorem: Consider the system (1) and

the state feedback control (10) and (9). If the system (1) is uniformly controllable, the closed-loop system is globally expo-nentially stable.

五、Simulation Example

Example: Consider a multivariable

time-varying system (1) with

The initial condition of the system is

[

]

) 0

( =

T

x . From simulation studies, it is found that the open-loop system is unstable. The proposed control (10) is then applied to the system with the fol-lowing design parameters P(0)= I,

I t

R₁( )= and R₂(t)= I in Eq. (9). Figure 1 shows the time history of the System State, which converges to zero as guaranteed by Theorem 2, and Figure 2 shows the time history of the control in-puts.

六、References

[1] Anderson, B. D. O., and Moore, J. B., 1968, "Extensions of quadratic minimi-zation theory, II. Infinite time results," Int. J. Control, vol. 7, pp. 473-480.

[2] Callier, F., and Desoer, C. A., 1992, Linear System Theory, Springer-Verlag, Hong Kong.

[3] Chen, C. T., 1984, Linear System Theory and Design, Holt, Rinehart and Winston, New York.

[4] Kwakernaak, H., and Sivan, R., 1972, Linear Optimal Control Systems,Wiley, New York.

[5] Valasek, M., and Olgac, N., 1993, "Generalization of Ackermann's formula for linear MIMO time invariant and time varying systems," Proceedings of Con-ference on Decision and Control, pp. 827-831.

[6] Wolovich, W. A., 1968, "On the sta-bilization of controllable systems," IEEE Trans. Automat. Contr., vol. 13, pp. 569-572.

[7] Wonham, W. M., 1968, "On a matrix Riccati equation of stochastic control," SIAM J. Control, vol. 6, pp. 681-697.