State-space digital PI controller design for linear stochastic multivariable systems with input delay.

Zhou, Han-Qin ; Shieh, Leang-San ; Liu, Ce Richard 等

In this paper, a centralized digital PI control scheme is proposed for linear stochastic multivariable systems with input delay. The discrete linear quadratic regulator (LQR) approach with pole placement is used to achieve satisfactory set-point tracking with guaranteed closed-loop stability. In addition, the innovation form of Kalman gain is employed for state estimation with no prior knowledge of noise properties. Compared with existing designs, the proposed scheme provides an optimal closed-loop design via the digitally implementable PI controller for linear stochastic multivariable systems with input delay. Its effectiveness will be demonstrated by the simulation study on examples from both industrial process control and aircraft control.

Dans cet article, on propose un schema de controle PI numerique centralise pour des systemes multivariables stochastiques lineaires avec un retard d'entree. On utilise une approche a regulateur quadratique lineaire discret (LQR) avec placement de poles pour obtenir un suivi des points de consigne satisfaisant avec une stabilite en boucle fermee garantie. En outre, la forme innovante du gain de Kalman est employee pour l'estimation des etats sans connaissance prealable des proprietes de bruit. Compare aux concepts existants, le schema propose offre une conception en boucle fermee optimale grace au controleur PI qui peut etre implante numeriquement pour des systemes stochastiques multivariables avec un retard d'entree. Son efficacite sera demontree par l'etude de simulation sur des exemples venant du controle de procede industriel et du controle des aeronefs.

Keywords: input delay, multivariable systems, optimal control, PI controller, stochastic process

It is well-known that Proportional-Integral-Derivative (PID) controllers (Astrom and Hagglund, 1988) have dominated the practical control applications for over 50 years. And many design methods for its application on multi-input-multi-output (MIMO) processes have been reported in the literature, such as characteristic locus, inverse Nyquist array, internal model control, and optimization methods (Zgorzelski et al., 1990; Loh et al., 1993; Palmor et al., 1993; Zhuang and Atherton, 1994; Wang et al., 1997; Bao et al., 1999; Cha et al., 2002; Wang et al., 2002). However, most of these existing approaches would have to make either or both of the following assumptions: 1) The plant can be decoupled into the form of single-input-single-output (SISO) systems; (2) The plant can be described by first-order or second-order systems with time-delay. Although many industrial processes meet the above conditions sufficiently well, there do exist some plants that cannot be decoupled successfully or approximated satisfactorily by low-order models. Therefore, the design of effective PID controllers for high-order MIMO systems has been highly desired and remained an active research area (Zheng et al., 2002; Zhang et al., 2004).

To improve product quality and energy conservation, the reduction of process variance is always of primary concern in process control, which can be considered as the most important specification to assess the control performance of manufacturing plants. PID, as the most popular controller of simple fixed structure in the field, is not synthesized from process or disturbance model, and therefore subject to the question: How close do PID controllers achieve ideal performance in term of minimum variance under stochastic disturbance? To achieve such a goal, Miller et al. (1995) and Kowk et al. (2000) derived the stochastic discrete predictive PID control law by approximating the generalized predictive control (GPC) with steady-state weighting. Based on generalized minimum variance control (GMVC), some self-tuning PID schemes (Miura et al., 1998; Yamamoto et al., 1999; Sato et al., 2002) have been proposed for discrete-time systems in face of stochastic disturbances. The underlying property of the above approaches is to approximate advanced control strategies with a PID controller. However, there is no theoretical guarantee for the satisfaction of such approximations. Moreover, the extension of these methods from SISO to MIMO case is not yet readily available.

Instead of approximation, Huang and Huang (2004) proposed a state-space approach of multi-loop discrete PID design, where the covariance constraints on process variables are formulated into linear matrix inequalities (LMI), such that the controller parameters can be computed directly. However, in this scheme, numerical computation is heavily involved in solving the LMI and the resultant PID settings may depend on the initial searching point. Moreover, the controller setting optimized for covariance constraints does not necessarily give a satisfactory deterministic performance, where response speed, settling time, overshoot and damping ratio are concerned. Motivated by the pros and cons of this method, we propose a digital PI design by formulating the multivariable stochastic systems and correlated disturbances into a state-space innovation form and an associated ARMAX model (Shieh et al., 1983). Thus, the Kalman gain matrix for optimal state estimation under stochastic disturbance can be obtained without acquiring knowledge on noise properties or computing the discrete Riccati equation. Then the PI tuning is transformed into a discrete quadratic minimization problem with all closed-loop poles placed inside a circle centred at ([beta], 0) and of the radius [alpha], where stability is ensured for [alpha] + |[beta]| < 1 (Lee and Lee, 1986). Therefore, by user specified values of [alpha] and [beta], the closed-loop system can be adjusted for desired transient responses. In such a scheme, we try to reduce the trade-off existing on previous designs, which prevents the stochastic and deterministic performances from being satisfactorily achieved simultaneously (Qin, 1998).

As PI structure is much simpler to be designed and tuned, it is preferably adopted in industrial control. Besides, compared with most frequency-domain design methods for analogue PID controller, our proposed centralized digital PI design does not rely on low order models, and thus facilitates practical application on real systems of high dimension. The controller setting is determined by tuning the weighting matrices in the LQR performance indices as well as the parameters [alpha] and [beta] with guaranteed closed-loop stability. And there are no specific requirements on system stability, low-degree/low-dimension model, minimum-phase property, length of dead-time, information of noise properties and plant decoupling. Nevertheless, the resultant digital controller is much easier to be implemented.

The rest of this paper is organized as follows: The problem of linear stochastic multivariable control system is stated in the Problem Statement section; the digital MIMO PI LQR tuning is proposed in the Controller Design section; then simulation examples are given in the following section; conclusions are drawn in the final section.

PROBLEM STATEMENT

Consider a multivariable system under stochastic disturbances, which is described by the following continuous-time state-space innovation form:

[??] (t) = Ax(t) + Bu(t - [tau]) + V[eta](t)

y(t) = Cx(t) t [eta](t)

where [tau] is time delay, vectors x(t) [member of] [R.sup.n], u(t) [member of][R.sup.m], [eta](t)[member of][R.sup.p], y(t) [member of] [R.sup.p] are state, input, white noise, output, respectively, and A, B, C, V are constant matrices with appropriate dimensions. Here we assume the white noise process is immeasurable but with the statistical properties

E{[eta](t)} = O;E{[[eta].sub.i][[eta].sup.T.sub.j] = Q[delta](i - j), Q[greater than or equal to]

Denote input delay as

[tau] = (d - 1)T + [tau], (3)

where T is the sampling time, d is a positive integer, 0 < [tau'] [less than or equal to]T and [gamma]= [tau']/T. The continuous-time systems in Equation (1) can be formulated into a discrete-time model

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where

G = [e.sup.AT], (5)

[H.sub.0] = [[G.sup.(1-[gamma]]) - I] [A.sup.-1]B, (6)

[[H.sub.1] = [G - G.sup.(1-[gamma])] [A.sup.-1]B, (7)

[GAMMA] = [G - 1] [A.sup.-1]V. (8)

Note that the matrix function ([e.sup.XT]-I)[X.sup.-1] = (G-I)[X.sup.-1] shall be represented as [[summation].sup.[infinity].sub.i=1] T/i! [(XT).sup.i-1] when X is singular.

When d = 1, [tau] = [tau] = [gamma]T, the augmented discrete-time system of Equation (1) which accommodates the input delay can be represented as

[X.sub.a] (kT + T) = [G.sub.p][X.sub.a] (kT) + [H.sub.p]u (kT) + [[GAMMA].sub.p][eta](kT), y (kT) = [C.sub.p][X .sub.a](kT) + [eta] (kT), (9)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For more general case, i.e., d > 1, the dimension of augmented discrete-time system of Equation (1) will be increased as follows

[X.sub.a] (kT + T) = [G.sub.p][X.sub.a] (kT) + [H.sub.p]u (kT) + [[GAMMA].sub.p][eta](kT), y (kT) = [C.sub.p][X .sub.a](kT) + [eta] (kT), (9)

where

Apparently, if the continuous-time system in Equation (1) is delay free, i.e., [tau] = 0, then Equation (4) can be reduced to

x (kT + T) = Gx(kT) + Hu(kT) + [GAMMA][eta](kT), y (kT) = Cx(kT) + [eta](kT), (11)

where H = [H.sub.0] + [H.sub.1] = (G - I)[A.sup.-1]B.

If the discrete-time state equations in Equation (11) is block observable, i.e.,

rank [Q(G,C)] = n, (12)

where

Q(G,C) = [[C[G.sup.r-1]).sup.T], (C[G.sup.r-2]).sup.T], ..., [(CG).sup.T], [C.sup.T]].sup.T],

r = n/m is an integer, and n and m are dimensions of x(kT) and u(kT), respectively.

Then the class of MIMO system in Equation (11) can be transformed into the following observable block companion form (Shieh et al., 1983) as

[X.sub.o](kT + T) = [G.sub.0][x.sub.0](kT) + [H.sub.o]u(kT) + [K.sub.0][eta](kT), y(kT) = [C.sub.o][X.sub.o](kT) + [eta](kT),

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that [K.sub.o] is the Kalman gain matrix. The corresponding transfer function matrix can thus be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [D.sub.i] = [G.sub.oi] + [K.sub.oi], for i = 1, 2, ..., r.

The block elements ([K.sub.oi] for i = 1, 2, ..., r) of the matrix [K.sub.o] and the state [X.sub.o](kT) in Equation (13) are the Kalman gain matrices and the optimally estimated state. Equation (14) is the multivariable ARMAX model. When the delay-free continuous-time system in Equation (1) ([tau] = 0) is unknown, the corresponding discrete-time model in Equation (14) with the unknown parameters ([G.sub.oi],[H.sub.oi],[D.sub.i], for i = 1, 2, ..., r) can be identified by the extended least square estimation algorithm with no prior knowledge of noise properties in Equation (2), and thus the Kalman gain matrix [K.sub.o] can be constructed from [K.sub.oi] = [D.sub.i] - [G.sub.oi] for i = 1, 2, ..., r. The Kalman gain matrices for a general MIMO system, i.e., r [not equal to] n/m, a non-integer, can be found in Shieh et al. (1989).

With the above discrete-time model, the controller design can be then derived in the next section.

CONTROLLER DESIGN

Consider a multivariable continuous-time plant Equation (1) cascaded with a digital PI controller as depicted in Figure 1, where the transfer function of the filtered multivariable PI controller is given as

K(z) = [K.sub.I]z/z - 1 + [K.sub.p] = [K.sub.2]/z - 1 + [K.sub.1] (15)

[FIGURE 1 OMITTED]

where [K.sub.I] [member of] [R.sup.mxp] and [K.sub.P] [member of] [R.up.mxp] are integral and proportional gain matrices respectively, and [K.sub.2] = [K.sub.I] and [K.sub.1] = [K.sub.I] + [K.sub.P]. Suppose that the plant is square and its static gain matrix is nonsingular. Note that static decoupling for the plant is usually helpful for control performance enhancement and adopted by Goodwin et al. (2001) and Chen and Seborg (2002). Thus, [K.sub.1] can be chosen for static decoupling as [K.sub.1] = [G.sup.-1.sub.p] (0), where [G.sub.p](s) is the Laplace domain transfer function matrix of MIMO plant. [K.sub.2] is the gain matrix to be determined in PI tuning.

The PI controller in Equation (15) is represented in the state-space form as

z(kT + T) = [G.sub.c]z(kT) + [H.sub.c]e(kT), w(kT) = [K.sub.2]z(kT) + [K.sub.1]e(kT), (16)

where

e(kT) = -y(kT) + r (kT)

is the error signal between reference input and plant output, and [G.sub.c] = [I.sub.m] and [H.sub.c] [member of] [R.sup.mxp]. Note in such a control system (Figure 1), the PI design is based on the set-point response, that is, there is no disturbance, d(t) = 0. The plant with input delay is transformed to either Equation (9) or (10), depending on the size of delay. Then, the overall augmented cascaded discrete-time system of plant Equation (10) and PI controller Equation (16) is shown in Figure 2 and is described as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

[FIGURE 2 OMITTED]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

And the control signal is

u(kT) F[x.sub.a] = -(kT) - [K.sub.2]z(kT) - [K.sub.1]e(kT), (18)

where F is the state-feedback gain matrix of the plant and [K.sub.2] is the controller gain matrix to be designed. As Equation (18) involves e(kT), it looks not suitable for LQR design. Note e(kT) = r(kT) - y(kT) = r(kT) - [C.sub.p][x.sub.a](kT). It follows from Equation (18) that

U(kT) = -F[x.sub.a](kT) - [K.sub.2]z(kT) - [K.sub.1] [r(kT) - [C.sub.p][x.sub.a](kT) = -(F - (K.sub.1][C.sub.p])[x.sub.a](kT) - [K.sub.2]z(kT) - [K.sub.1]r(kT). (19)

Note in control signal Equation (19), [K.sub.1] is pre-assigned and r(kT) is the reference signal, both having no effect for LQR design. Therefore, neglecting the last term in Equation (19), we can use the LQR to find the optimal control law:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

and then determine F and [K.sub.2] by

F = [K.sub.1][C.sub.p] + [[bar.K].sub.1], [[bar.K].sub.2] [K.sub.2]. (21)

Hence, the desired control signal u(kT) can be practically implemented by Equation (18). And the overall designed close-loop system is shown in Figure 2, which is represented in the following discrete state-space equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

To ensure all the closed-loop poles at a specific region inside the unit circle, as shown in Figure 3, the control signal [u.sup.*](kT) will be revealed in LQR optimal design with regional pole assignment by the following lemma proposed by Lee and Lee (1986).

[FIGURE 3 OMITTED]

Lemma 1

Consider a linear time-invariant discrete-time controllable system

X(kT + T) = GX (kT) + HU(kT). (23)

The optimal control law

U(kT) = -KX(kT) (24)

is to be found such that all the closed-loop poles inside a circle centred at ([beta],0) with a radius [alpha] in Figure 3 and the following performance index

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

is minimized, where Q = [Q.sup.T] [greater than or equal to]0 and R = [R.sup.T] > 0. With specified values of [alpha] and [beta], it can be determined that

[G.sub.[alpha][beta]] = (1/[alpha])(G - [beta]I) [H.sub.[alpha]] = (1/[alpha])H.

Therefore, by solving for P = [P.sup.T] > 0 the discrete Riccati equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (26)

the optimal controller gain is given as

K = [(R + [H.sup.T.sub.[alpha]] P[H.sub.[alpha]]).sup.-1] [H.sup.T.sub.[alpha]]P[G.sub.[alpha][beta] (27)

For more details, readers may refer to Lee and Lee (1986).

Therefore, with user specified values of [alpha] and [beta], it follows from Equations (17), (26) and (27) that [bar.K] can be computed. Then the gains matrices [K.sub.2] and F can be recovered from Equation (21). The control system design procedure is thus accomplished.

ILLUSTRATIVE EXAMPLES

Example 1

Consider the example of a dry process rotary cement kiln with a capacity of 1000t of clinker per day. The outputs [y.sub.1], [y.sub.2] are temperatures of the pre-heater and the kiln drive power, respectively. The inputs are: [u.sub.1], kiln exhaust fan speed; [u.sub.2], flow of raw material into the kiln. The kiln uses coal as fuel, and the oxygen concentration in the exhaust gas is controlled by the fuel rate. Therefore, the control variable u1 correlates with the energy flow into the kiln. A detailed description can be found in Westerlund (1981).

The continuous-time model is

[??](t) = Ax(t) + Bu(t) + V[eta](t),

y(t) = Cx(t) + t [eta](t), (28)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In discrete-time description, the ARMAX model is identified in sampling time T = 5 as

x(kT + T) = Gx(kT) + Hu(kT) + [GAMMA][eta](kT),

y(kT) = Cx(kT) + [eta](kT), (29)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that the above equation is already in Kalman innovation form with [T.sub.o] = [I.sub.2]. Due to the utilization of zero-order hold, an input delay of [tau] = T/2 will be introduced (Chidambaran, 2002). Therefore, Equation (28) becomes

[??](t) = Ax(t) + Bu(t - [tau]) + V[eta](t), y(t) = Cx(t) + [eta](t), (30)

To accommodate the input delay in the following design procedure, the augmented discrete-time model Equation (9) will be employed, i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To formulate Equation (31)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

are computed according to Equations (6) and (7), respectively.

The observer for optimal state estimation under stochastic disturbance can be found in Kalman filter innovation form as

x(kT + T) = (G - [GAMMA]C) x (kT) + [H.sub.0]u (kT) + [H.sub.0]u + (H.sub.1]u(Kt - T) + [GAMMA]y(kT).

The PI controller in Equation (15) is

K(z) = [K.sub.I]z/z - 1 + [K.sub.p], (32)

whose state-space equation is

z (kT + T) = [G.sub.c]z(kT) + [H.sub.c]e(kT) w(kT) = [K.sub.2]z(kT) + [K.sub.1]e(kT), (33)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[K.sub.1] is chosen for static decoupling as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [G.sub.p](s) is the transfer function matrix of plant. Therefore, only [K.sub.2] is left to be determined.

It follows from Equations (17), (31) and (33) that the cascaded system becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

Note [bar.x](kT) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the system dimension is now increased to 6 x 6.

To employ Lemma 1 for discrete-time optimal control with closed-loop pole placement, there will be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, Lemma 1 is ready to be applied. Choose weighting matrices Q = 10[I.sub.6] and R = [I.sub.2] for the performance index of Equation (25), and [alpha] = 0.2, [beta] = 0.7 for closed-loop pole assignment.

By solving the discrete Riccati Equation (26), it follows from Equation (27) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Decomposing [bar.K] according to Equation (21), the corresponding feedforward and feedback gain matrices respectively are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, it follows from Equation (18) that the control law is

u(kT) = -F[x.sub.a](kT) - [K.sub.2]z(kT) [K.sub.1]e(kT), (36)

The implementation of this control law is shown in Figure 1.

The centralized discrete-time PI controller is therefore given as

K(z) = [K.sub.1] + [K.sub.2](zI - [G.sub.c].sup.-1] [H.sub.c]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The closed-loop characteristic equation of such a 6 x 6 digital system is

(z - 0.17) (z - 0.7002) (z - 0.6984) ([z.sup.2] - 1.604z + 0.6442) = 0,

which gives the poles {0.71, 0.7002, 0.6984, 0.6768, 0.8020 [+ or -] 0.0316i}. Apparently, all these closed-loop poles are stable and located in the desired region, i.e., a circle centred at 0.7 with a radius of 0.2 as shown in Figure 3.

To assess the control effect of the proposed digital PI scheme, the deterministic performance is first explored: a unit-step signal is given as reference input [r.sub.1] at t = 0, after the system settles down, [r.sub.2] changes from zero to unit-step at t = 750. Then a step size disturbance of magnitude -1 is injected on [y.sub.1] at t = 1500 and another same disturbance acts on [y.sub.2] at t = 2250. The system response is exhibited in Figure 4. To evaluate the stochastic performance, PI controller is used as a regulator when the system is in face of white noise. Without loss of generality, the reference input is set zero in this scenario. The output variables and input control signals are presented in Figure 5 and Figure 6, respectively.

[FIGURES 4-6 OMITTED]

For comparison, the result given by the decentralized digital PI controller reported by Huang and Huang (2004) is also included. The controller setting is obtained as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)

which is computed based on the variance constraints of

E{[y.sup.2.sub.1](t)} [less than or equal to] 0.939, E{[y.sub.2.sup.2](t)} [less than or equal to] 0.345;

E{[u.sup.2.sub.1](t)} [less than or equal to] 0.004, E{[u.sub.2.sup.2](t)} [less than or equal to] 1.5. (40)

Note that E{[u.sup.2.sub.1](t)} [less than or equal to] 0.004, E{[u.sup.2.sub.2](t)} [less than or equal to] 1.5 are the practical restrictions on input signals (Makila et al., 1984).

It can be observed that our proposed PI control gives much better tracking response for both step set-point change and disturbance rejection. This is largely due to the more control freedom by specifying the closed-loop poles' location, i.e., the choice of [alpha] and [beta]. With different design objective, Huang's LMI based computational algorithm concerns primarily about the variable constraints in a delay-free system. Therefore, the close-loop set-point response is not optimally designed.

However, as for regulation of stochastic disturbance, Huang's method gives the result of

E{[y.sup.2.sub.1](t)} = 0.0079, E{[y.sub.2.sup.2](t)} = 0.0279;

E{[u.sup.2.sub.1](t)} = 0.0002, E{[u.sub.2.sup.2](t)} = 0.0946 (41)

While our proposed scheme gives the result of

E{[y.sup.2.sub.1](t)} = 0.0051, E{[y.sub.2.sup.2](t)} = 0.0306;

E{[u.sup.2.sub.1](t)} = 0.0005, E{[u.sub.2.sup.2](t)} = 0.05232. (42)

In this comparison, which presents almost the same regulation effects on output variables, Huang's method gives less variance in control inputs. But as a computational algorithm, in Huang's method, the successful finding of a PID setting depends on the constraint criteria, i.e., as stated by the author, it is largely due to the performance limitation imposed by the decentralized controller structure. Moreover, the different initial search point in LMI optimization may affect controller settings. Our proposed scheme, although does not consider process variance specifications, is designed in the framework of deterministic performance and thus easy to be used. Since the computation of Kalman gain is an integrated part of the overall design procedure, our proposed method can give considerably satisfactory stochastic disturbance regulation. Changing the weighting matrices Q and R may further adjust the variances of input and output variables. Also the delay effect in digital implementation has been considered in our scheme. Nevertheless, the PI control is also less costly for implementation and easier to be tuned than PID.

Example 2

To further demonstrate the applicability of proposed scheme, the following abstracted longitudinal control design example (Doyle and Stein, 1981) of a CH-47 tandem rotor helicopter will be investigated. The design objective is to control two measured outputs--vertical velocity and pitch attitude--by manipulating collective and differential collective rotor thrust commands. The nominal model for the dynamics relating these variables at 40 knot airspeed is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (43)

which is a 4th-order unstable non-minimum phase system with 3 RHP poles located at 0.0652, 0.4913 [+ or -] 0.4151j, respectively. Suppose it is under stochastic disturbance, its state-space equation is represented as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (44)

As the controller bandwidth is under constraint of [w.sub.b] [less than or equal to]10, we chose sampling period T = [pi]/30[W.sub.b] [??] 0.01. The corresponding discrete-time model is obtained as follows:

x(kT + T) = Gx(kT) + Hu(kT) + [GAMMA][eta](kT), y(kT) = Cx(kT) + [eta](kT), (45)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Consider the input-delay of T/2 imposed by zero-order hold, H is decomposed to [H.sub.0] and [H.sub.1], which are computed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The 2 x 2 PI controller described by Equation (33) is decided with the pre-assignment of [G.sub.c] = [I.sub.2], [H.sub.c] = [I.sub.2] and [K.sub.1] [G.sup.-1.sub.P](0) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for static decoupling. Note [G.sub.p](s) is unstable with no open-loop steady state. However, for our LQR closed-loop design, it will be stabilized by feedback and thus reach steady-state.

Therefore, the rest of the design procedures are similar to apply Lemma 1. We chose Q = 10[I.sub.6], R = [I.sub.2] for the performance index and [alpha] = 1, [beta] = 0 for the general case. The resultant parameters are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (46)

Hence, the following PI controller setting is obtained

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (46)

And the 8 x 8 closed-loop designed system has the poles of {0.9998, 0.4697 [+ or -] 0.0613j, 0.4990 [+ or -] 0.0100j, 0.4799, 0.0001, 0}.

To implement to proposed control system configuration, the observer

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (47)

needs to be employed to recover the states of plant. It follows from Equations (13) and (45) that [G.sub.o] = [T.sup.-1.sub.o] G[T.sub.o], [H.sub.0o] = [T.sup.-1.sub.o] [H.sub.0], [H.sub.1o] = [T.sup.-1.sub.o] [H.sub.1], [K.sub.o] = [T.sup.-1.sub.o] [GAMMA][C.sub.o] = [[I.sub.2], [0.sub.2]] can be computed with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The deterministic performance is exhibited in Figure 7: a unit-step signal is given as reference input [r.sub.1] at t = 0, after the system settles down, [r.sub.2] changes from zero to unit-step at t = 1. Then a step size disturbance of magnitude -1 is injected on output [y.sub.1] at t = 2 and another same disturbance acts on [y.sub.2] at t = 3. To evaluate the stochastic performance, PI controller is used as a regulator when the system is in face of a random process of white noise and the reference input is set to zero. The output variables are presented in Figure 8.

[FIGURES 7-8 OMITTED]

CONCLUSION

In this paper, a digital PI control scheme is proposed for linear multivariable stochastic system with input delay. A LQR based design will be employed to determine the centralized PI controller setting. Four design parameters will provide considerable control freedoms on deterministic performance, i.e., [alpha] and [beta] will specify the region for closed-loop poles, while the traditional Q and R will change the weighting in performance index as well as effect the variance of system variables. Furthermore, the innovation form of Kalman gain matrix for optimal state estimation under disturbance will be integrated in the design procedure. The resulting digital PI controller can achieve both satisfactory set-point response and disturbance regulation. Nevertheless, the proposed optimal design will guarantee the closed-loop stability with no specific requirements on system order/dimension, minimum phase, knowledge of noise properties, length of dead-time and decoupling property.

ACKNOWLEDGEMENT

This work was supported in part by the U.S. Army Research Office under Grant DAAD 19-02-1-0321, NASA Johnson Space Center under Grant No. NNJ04HF32G and TXDOT Contract 466PV1A003.

The authors would also like to thank the anonymous reviewers for their valuable comments and suggestions, as well as the editorial staff for their help.

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Manuscript received July 29, 2005; revised manuscript received November 2, 2005; accepted for publication November 2, 2005.

Han-Qin Zhou (1), Leang-San Shieh (1) *, Ce Richard Liu (1) and Qing-Guo Wang (2)

(1.) Department of Electrical and Computer Engineering, University of Houston, Houston, TX, U.S. 77204-4005

(2.) Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260

* Author to whom correspondence may be addressed.

E-mail address: [email protected]