HICSS'01: Adaptive Control for Nonlinear Stochastic

Proceedings of the 34th Hawaii International Conference on System Sciences - 2001

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Adaptive Control for Nonlinear Stochastic Hybrid Systems with Input Saturation Shu-Mei Guo Department of Computer Science and Information Engineering, National Cheng-Kung University Tainan 701, Taiwan, R.O.C. Phone: 886-6-2757575ext62525 Fax: 886-6-2747076 E-mail: [email protected]

Leang-San Shieh Department of Electrical and Computer Engineering, University of Houston, Houston TX 77204-4793, U.S.A. Phone: (713) 743-4439; Fax: (713) 743-4444 E-mail: [email protected]

Ching-Fang Lin American GNC Corporation, 9131 Mason Avenue, Chastsworth, CA 91311, U.S.A. Phone: (818) 407-0092; Fax: (818) 407-0093 E-mail: [email protected]

Jagdish Chandra School of Engineering and Applied Sciences, George Washington University Washington DC 20052, U.S.A. Phone: (202) 994-0179; Fax: (202) 994-4606 E-mail: [email protected] y = h( X ) + v ,

Abstract

(2)

subject to This paper presents a new state-space self-tuning control scheme for adaptive digital control of continuous multivariable nonlinear stochastic hybrid systems with input saturation. Here, the continuous nonlinear stochastic system is assumed to have unknown system parameters, system and measurement noises, and inaccessible system states. The proposed method enables the development of a digitally implementable advanced control algorithm for chaotic stochastic hybrid systems.

1. Introduction Controls of complex interactive networks and systems present challenging problems in hybrid dynamic systems. The hybrid nature of these problems occurs at two different levels. First, at the process modeling level, the system could couple inherently continuous dynamics with network configurations that are event/logic driven. Examples of such systems include electrical power systems, transportation systems, and communication systems. At the second level, the hybrid nature may show up in the design and implementation of controls for such system. In this paper, we consider both these aspects, and provide an example of adaptive control for nonlinear stochastic hybrid systems with input saturation. We consider the following general model of hybrid systems

X&= f ( X ) + g ( X )u + w ,

(1)

min (max) FX ∈Ω = M ( X ) , (3) (4) H(X ) = 0 , (5) G( X ) ≤ 0 , X min ≤ X ≤ X max , (6) where w and v are system and output measurement noises, respectively, X is a vector with n unknown decision T variables, indicated as [ x1 , x 2 , Λ , x n ] , which may be continuous, discrete or hybrid variables; Ω is the solution space; F is a real-valued function in a real domain, M ( X ) is the objective function, which is a map from solution space to real field; X min and X max are the bounds of X ; H ( X ) and

G ( X ) are vector functions, represented as, H ( X ) = [h1 ( X ), h2 ( X ),Λ , hm1 ( X )]T , G ( X ) = [ g1 ( X ), g 2 ( X ),Λ , g m 2 ( X )]T ,

hi ( X ) and g i ( X ) for all i are system functions, m1 and m 2 are respectively the numbers of the equality and inequality constraints. A special case of this model was treated in [1], that describes interactions between continuous (smooth) dynamics and discrete events as it may occur in the stability analysis of electrical power and other transportation systems. The second aspect of hybrid nature of the problem under

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consideration occurs in the design and implementation of control laws. Specially, we establish computationally efficient digital implementations of analog control strategies with logic or event based constraints. Indeed, any system where saturation limits are routinely encountered can be thought as a hybrid system. These limits introduce discrete events that often have a significant influence on the overall behavior of the system under consideration [1].

2. Problem description In order to show the feasibility of this approach, we consider the class of continuous nonlinear stochastic hybrid systems x&(t ) = f ( x(t )) + g ( x(t ))u (t ) + w′(t ) , (7)

y (t ) = h( x(t )) + v′(t ) ,

(8)

subject to

u i ,min ≤ u i (t ) ≤ u i , max for i = 1,2, Λ , m,

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f : ℜ n → ℜ n , g : ℜ n → ℜ n×m , h : ℜn → ℜ p , u (t ) = [u1 (t ), u 2 (t ),Λ , u m (t )]T ∈ ℜ m is the control input,

where

x(k ) ∈ ℜ n is the state vector, y (t ) ∈ ℜ p is the measurable output vector, and w′(t ) and v′(t ) are uncorrelated white noise processes. It is assumed that f ( x (t )) , g ( x (t )) , h( x(t )) , and w′(t ) and v′(t ) with appropriate dimensions are unknown, and x (t ) is inaccessible. It's desired to design a digital tracker

ud (kT ) = ς (ud (•), yd (•), Γ(•), ui , min , ui , max ) (10) for kT ≤ t ≤ (k + 1)T , k = 0,1,2,Λ , so that it efficiently guides the controlled output y d (kT ) to track the desired reference orbit Γ(kT ) at each sampling time instant kT , where T is the sampling and holding period, and ς (∗) denotes some function of (∗) to be determined for the desired goal. In our recent work [2] (see also Section 7, Appendix), a state-space self-tuner has been newly developed to achieve satisfactory closed-loop performance of a class of multivariable nonlinear stochastic and chaotic hybrid systems by assuming no input-saturation constraint, described by a new state-space innovation form, and the results are extendable to the general multivariable case as well. It has been shown that the conventional state-space self-tuner used for linear stochastic systems is a special case of the one newly developed in this paper. The concept of an adjustable ARMA-based noise model is not new; however, this paper acts as a forerunner to embed state-space self-tuners for effective control of the multivariable nonlinear stochastic and chaotic hybrid systems. The advantages of utilizing state-space self-tuners based on innovation models are: (i) the equivalent internal states can be

sequentially estimated; (ii) the adaptive estimator gain (the adjustable ARMA -based noise model) or the Kalman gain ( the adjustable MA-based noise model) can subsequently be obtained; (iii) the self-tuners are simple, reliable and robust; and (iv) the changeable stable/unstable and minimum/non-minimum-phase systems during the process can be adaptively controlled. To take advantage of digital redesign methodology and take care of the non-negligible computation time delays in practice, an effective prediction-based digital tracker, especially suitable for chaotic orbit tracking, has been newly developed in this paper. This is based on the prediction-concept of digitally redesigned control and observer gains. The newly proposed control scheme in [2] has been successfully applied to Chen's chaotic attractor, which has recently been shown [5] to be topologically more complicated than the Lorenz attractor, and therefore is more challenging for control and for demonstrating its various advantages and properties, such as (i) adjustable ARMA-based noise model and (ii) digital redesign, and effects of (i) initialization of system parameters and (ii) consideration of computation time delay. Because of the nature of these flexible conditions [2], it's straightforward to show

u d (kT ) = u c (t v ) ≈

1 kT +T u c (t )dt , T ∫ kT

(11)

using the general Chebyshev quadrature formula [3], provided that the high-gain analog controller u c (t ) is used and some other appropriate conditions are assumed to be true, such as the selection of sampling period T satisfies the requirement of sampling theorem [4]. Here, u c (t ) is a function of the closed-loop system state

xc (t ) . It should be noted that (11)

implies the inter-sampling behavior of state matching is also considered here [3]. Furthermore, the absolute value of u d (kT ) is smaller than the absolute peak value of uc (t )

u d (kT ) is the geometric mean of the integral of u c (t ) for kT ≤ t < (k + 1)T , provided the desired

since

xd (kT ) = xc (kT ) holds for every k , where k = 0,1,2,Λ . However, in practice, to have the absolute value of ud (kT ) smaller than the absolute peak value of uc (t ) for every kT , k = 0,1,2,Λ , is too over-optimistic to reach, since there always exist some errors between xd (kT ) and xc (kT ) due to some unavoidable influences, such as (i) a objective

trade-off consideration between computation complexity/load and accuracy, (ii) the selection of the sampling period T does not satisfy the requirement of the sampling theorem and (iii) implementing the controller under the consideration of computation time delay and various limitations/constraints in practice. However, from the input saturation point of view, this

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is good enough as long as the maximum absolute value of ud (kT ) , k = 0,1,2,Λ is smaller than the maximum absolute peak value of

uc (t ) during the whole process and

satisfies the input saturation constraint. The structure of the state-space self-tuning control of nonlinear stochastic hybrid systems with input saturation is shown in Figs. 1-2.

3. Illustrative examples We now illustrate how the proposed state-space self-tuning control concepts and methodologies can be applied to facilitate the analysis and design of the stochastic chaotic Lorenz attractor [5]

 x&1 (t ) = − β x1 (t ) + x 2 (t ) x3 (t ) + w1′ (t ),  (12)  x&2 (t ) = −σ x 2 (t ) + σ x3 (t ) + w2′ (t ),  x&(t ) = − x (t ) x (t ) + ρ x (t ) − x (t ) + w′ (t ), 1 2 2 3 3  3

w′(t ) = [ w1 (t ), w2 (t ), has zero mean and covariance cov(w′)

where the white noise process

w3 (t )]T

= [0.02, 0.02, 0.02]T . The attractor is shown in Fig. 3 (a) where β = 8 3 , σ = 10, and ρ = 28 for the noise-free case. In the simulation, the sampling period T f = 0.01 sec., final simulation time

t f = 20 sec., and initial conditions

x1 (0) = 12 , x 2 (0) = 18 , x3 (0) = −18 . It has been noticed that whenever x1 (t ) , x 2 (t ) , and x3 (t ) are well solved, a plot command “plot3( x1 (t ) , x 2 (t ) , x3 (t ) )” in used are

MatLab yields a good view of the attractor, as seen in Fig. 3 (a). Let us represent the deterministic (i.e. noise-free) system of (12) by a simple state-space equation,

xo (t ) = f ( xo (t )),

xo (t ) ∈ ℜ 3 .

(13)

The controlled system is designed to be

 x&(t ) = f ( x(t )) + Bu (t ) + w′(t ), (14)   y (t ) = Cx (t ) + v′(t ), 3×m p×n where B ∈ ℜ and C ∈ ℜ , and the white noise T process v′(t ) = [v1′ (t ), v2′ (t ), v3′ (t )] has zero mean and cov(v ′) = [0.04, 0.04, 0.04]T . The goal here m is to find an analog optimal tracker, u (t ) ∈ ℜ , that would guide the system output y (t ) to match a pre-specified p reference signal, Γ(t ) ∈ ℜ , assuming system (12) is unknown and x (t ) is not accessible. The trajectory x o (t ) covariance

shown in Figs. 3 (a) and (b) can be viewed as some kind of deterministic chaotic motion trajectory before control is applied, based on which the design procedure is illustrated for a particular case B = I 3 and C = I 3 with m = n = p = 3 below: First, we specify the target reference

Γ(t ) . In our study,

we let Γ(t ) be a periodic component of the original deterministic chaotic orbit, as shown in Figs. 4 (a) and (b), and the objective is to guide the stochastic chaotic trajectory to settle on this deterministic orbit. We set a fast sampling time, T f = 0.01 sec., for the continuous-time simulation at

t = k f T f , k f = 0,1,2, Λ . We then collected xo (k f T f ) for

k f = 843, Λ ,1001 to construct the reference

Γ(k f T f ) as the target. Thus, for the system output to travel through

this

closed

orbit

£F( k f T f ) ,

it

takes

(1001 − 843 + 1) T f =1.590 sec. We then repeat the desired Γ(k f T f ) periodic curve cycle by cycle to fit the final simulation time

t f , for instance, t f = 10 sec. in our

illustrative example. The three-dimensional phase portrait and the time-domain response of reference Γ( k f T f ) are given in Figs. 4 (a) and (b), respectively. Appropriately using the above technique to any x o ( k f T f ) of the designed reference orbit

Γ(k f T f ) , we can control the speed of the system output trajectory to go through this reference orbit. Next, specify various initial values for the state-space self-tuning control scheme [2], as follows:

0 0  − 1.35  − 1.35 − 0.20 Gˆ o1 (0) = −( L1 + R1 ) =  0  0 − 0.20 − 1.35  0 0  0.455  Gˆ o 2 (0) = L1 R1 =  0 0.465 0.135,  0 0.135 0.465

,

0 0 0  Hˆ o1 (0) = −( L2 + R2 ) = 0 0.2 0  , 0 0 0.2 0 0 − 0.0676   Hˆ o 2 (0) = L2 R2 =  0 − 0.0900 − 0.6000 ,  0 − 0.6000 − 0.9000 where

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0 0 0  0.7 0 0.65    L1 =  0 0.7 0.1 , R1 =  0 0.65 0.10 ,  0 0.1 0.7   0 0.10 0.65 0 0 0  0.26 0 − 0.26    L2 =  0 0.2 0.1 , R2 =  0 − 0.4 − 0.1 ,  0  0 0.1 0.2 − 0.1 − 0.4 Dˆ = 0 , i = 1,2 ; oi

3

−3

S (0) = 10 I18 ,

λ0 = 0.95

and

λ (0) = 0.91 ;

x(0) = [12, 18, − 18] , xˆ (0 | −1) = [0 0 0 0 0 0]T ; Q = 104 I 6 and R = I 3 , Qob = 104 I 6 and Rob = I 3 . T

[2] S. M. Guo, L. S. Shieh, C. F. Lin, and J. Chandra, “State-space self-tuning control for nonlinear stochastic and chaotic hybrid systems,” International Journal of Bifurcation and Chaos (to appear in vol. 11, no. 4, April 2001). [3] L. S. Shieh, W. Wang, and J. S. H. Tsai, “Optimal digital redesign of hybrid uncertain systems using genetic algorithms,” IEE Proc. D, Control Theory & Applications vol. 146, no. 2, 1999, pp. 119-130. [4] Lewis, F. L., Applied Optimal Control and Estimation: Digital and Implementation, prentice-Hall, Englewood Cliffs, New Jersey, 1992. [5] Chen, G., Controlling Chaos and Bifurcations in Engineering Systems CRC Press, Boca Raton, 1999.

Following the design procedure shown in our work [2], in which no input saturation is considered, the deterministic reference orbit, stochastic chaotic trajectories of Lorenz's system, and time series of control input are shown in Fig. 4 for no input saturation and in Fig. 5 for the input saturation

[6] Tsay, Y. T., L. S. Shieh, and S. Barnett, Structural Analysis and Design of Multivariable Control Systems: an Algebraic Approach, Heidelberg: Springer-Verlag, Berlin, 1988.

− 300 ≤ ud 1 (t ) ≤ 300, − 250 ≤ ud 2 (t ) ≤ 250,−100 ≤ ud 3 (t ) ≤ 100 , for comparison. The simulation results have

7. Appendix

shown that the proposed state-space self-tuner is effective in guiding the chaotic trajectories of the stochastic chaotic Lorenz's system with input saturation to the intended orbit.

4. Conclusion We have investigated in this paper a challenging control problem for a hybrid dynamical system. The hybrid nature of the problem is encountered at both the system model level where the continuous dynamics is coupled with network configuration that could be event/logic driven, and in the design and implementation of control procedures. In addition, we assume the continuous dynamics to be controlled to be nonlinear stochastic system with unknown system parameters, unknown system and measurement noises, and inaccessible system states. The adaptive control procedures and the implementation schemes used in an illustrative example show the effectiveness of the proposed design methodology.

5. Acknowledgement This work was supported by the US Army Research Office under the Grant DAAG55-98-1-0198 and by EPRI and ARO under WO8333-04.

6. References [1] I. A. Hiskens and M. A. Pai, “Trajectory sensitivity analysis of hybrid systems,” IEEE Transactions on Circuits and Systems I, vol. 47, no. 2, February 2000, pp. 204-220.

STATE-SPACE SELF-TUNER FOR NONLINEAR HYBRID SYSTEMS WITHOUT THE INPUT-SATURATION CONSTRAINT [2]

Given the class of continuous-time nonlinear stochastic systems (A1a) x&(t ) = f ( x(t )) + g ( x(t ))u (t ) + w′(t ) ,

(A1b) y (t ) = h( x(t )) + v′(t ) , n p n n n n× m where f : ℜ → ℜ , g : ℜ → ℜ , h:ℜ → ℜ , u (k ) ∈ ℜ m is the control input, x(k ) ∈ ℜ n is the state p vector, y (t ) ∈ ℜ is the measurable output vector, and w′(t ) and v′(t ) are uncorrelated white noise processes. It is assumed that f ( x (t )) , g ( x (t )) , h( x (t )) , and w′(t ) and v′(t ) with appropriate dimensions are unknown, and x(t ) is

inaccessible. For the sake of simplicity, in this paper we only consider a special class of multivariable nonlinear stochastic systems, where n = qp = qm , i.e., p = m , and q is an integer. These results, however, are extendable to the general multivariable casel. The design procedure is summarized as follows: Step A1: Choose an appropriate ARMAX model as the local discrete-time linear model for the given continuous-time nonlinear stochastic system (A1). The associated state-space innovation form, for instance p = m = n = n′ , is given by

xˆ o (k + 1 | k ) = Gˆ o (k ) xˆ o (k | k − 1) + Hˆ o (k )u (k ) + Ld ( k ) e o ( k ) , (A2a)

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eo (k ) = y (k ) − Cˆ o xˆ o (k | k − 1) ,

(A2b)

where

 − Gˆ (k ) I n  2 n×2 n Gˆ o (k ) =  o1  ∈ℜ ˆ − Go 2 (k ) 0n   Hˆ (k )  Hˆ o (k ) =  o1  ∈ ℜ2 n× n , ˆ  H o 2 ( k ) Cˆ o = [ I n , 0 n ] ∈ ℜ n×2 n , xˆ o (k | k − 1) ∈ ℜ 2 n .

,

Step A4: Generate the digital control input at each sampling period T: (i) Compute the digital control gains K d (k ) , Ed (k ) , and

Ld (k ) from the current analog control gains K c (k ) , Ec (k ) , and Lc (k ) , using the digital redesign formula (Eq. (45) in [2] and Eq. (49) in [2]) as

K d (k ) = [ I m + K c (k ) Hˆ o (k )] −1 K c (k )Gˆ o (k ), (A3a) E (k ) = [ I + K (k ) Hˆ (k )]−1 Eˆ (k ) , (A3b) d

Step A2: Parameter identification and state estimation at each sampling period T: (i). Set the initial conditions for parameter identification used for the state-space RLS algorithm (Eq. (23)-Eq. (24) in [2]) and state estimation (A2) as follows:

Gˆ o1 (0) = −( L1 + R1 ) , Hˆ (0) = −( L + R ) , o1

where

2

2

Gˆ o 2 (0) = L1R1 , Hˆ (0) = L R , o2

2

2

Li ∈ ℜ n×n and Ri ∈ ℜ n×n for i = 1,2 are some

appropriate left and right solvents [6], respectively. Also, set

m

c

o

c

Ld (k ) = Gˆ o (k ) Lc (k )[ I p + CLc (k )] −1 .

(A3c)

The digital control input is generated at each T for kT ≤ t ≤ (k + 1)T by

u d (k ) = − K d {Gˆ o xˆ d (k − 1 | k − 2) + Hˆ o u d (k − 1)

+ Ld [ y (k − 1) − Cxˆ d (k − 1 | k − 2)]} + E d £F(k + 1). (A4) (ii) Set k := k + 1 . Go to Step A2 and continue the adaptive control process. •

Dˆ oi (0) = 0 for i = 1,2 to have a dead-beat-like

S (0) ∈ ℜ 3( 2 n )×3( 2 n ) > 0 , 0 < λ0 < 1 , 0.9 < λ (0) < 1 , xˆ (0 | −1) . Then, select appropriate

property;

weighting matrices {Q, R} (Eqn. (32) in [2]) and {Qob , Rob } (Eq. (50) in [2]) to have the high-gain property analog control law (Eq. (30) in [2]) and analog estimator gain (Eq. (48) in [2]). (ii). At each sampling interval T, apply the RLS algorithm (Eq. (23)- Eq. (24) in [2]) to the given continuous-time nonlinear stochastic system (A1) with a piecewise-constant control input (see Step A4), to be determined, and obtain the updated parameters θˆ( k ) and the predicted state (A2), based on the updated digital estimator gain (see Step A3). Step A3: Compute the analog control gains at each sampling period T: When t = kT , compute the desired analog control gains, based on the updated identified parameter

θˆ(k ) in Step A2,

Gˆ o (k ) and Hˆ o (k ) , associated with θˆ(k ) in Step A2, into the corresponding Aˆ (k ) and Bˆ (k )

as follows: Convert

using formula (Eq. (47) in [2]); follow the analog linear quadratic tracker/observer (Sec. 4 in [2]) to compute the analog control gains {K c ( k ), Ec ( k )} (Eq. (31) in [2]) and

Lc (k ) (Eq. (48) in [2]).

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