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Nonlinear Multirate Model-Algorithmic Control. 1. Theory Michael P. Niemiec† and Costas Kravaris*,‡ Department of Chemical Engineering, The University of Michigan, Ann Arbor, Michigan 48109
This paper develops a systematic methodology for the synthesis of nonlinear multirate modelalgorithmic controllers for multivariable nonlinear systems. The control structure incorporates all available process measurements and forces each output to follow a prespecified reference trajectory regardless of the sampling period. The proposed multirate controller is a time-varying periodic system, which has properties that enable a rigorous stability analysis. Sufficient conditions for asymptotic stability of the closed-loop system are established. 1. Introduction A major class of industrial control problems involves measurements that are available at different rates. One common situation arises when one or more process outputs are measured with large sampling periods and possibly delays. These low sampling rates are primarily imposed by hardware limitations and/or measurement cost, where compositions and product properties are measured with various forms of analytical instruments. For example, gas chromatographs used for composition analysis have a cycle time of approximately 5-10 min, while gel permeation chromatographs used for polymer properties have a cycle time of approximately 15-30 min. Because of the long cycle times of the analytical instruments, fast-sampled secondary measurements such as temperature and pressure are often used in industry to control these important but slowly sampled process variables. Depending on the accuracy of the process model and the type of control algorithm used, the use of secondary measurements may lead to a poor control performance of the primary output. On the other hand, the use of a slowly acting controller, operating on the basis of the lowest sampling rate of all the outputs, will lead to an unnecessarily poor control performance of the faster-sampled outputs in terms of handling fast-varying disturbances. Research in multirate control systems began in the late 1950s but has received considerable attention in the past decade. The main advantage of a multirate controller is its ability to perform in common industrial situations where technological limitations require that measurements and control action calculations be performed at different rates and times. The majority of multirate control algorithms can be subdivided into two main categories.1 The first is the multirate-input controller, which changes the manipulated input several times during a period but receives the output measurement only once during that period.2-5 Pole assignment with a multirate “gain” controller was investigated in ref 2, while several multirate model predictive controllers (MPCs) were developed in the literature (see refs * To whom all correspondence should be addressed. Tel: +30-610996339. E-mail:
[email protected]. † Present address: Honeywell, 16404 N. Black Canyon Highway, Phoenix, AZ 85053. E-mail: michael.niemiec@ honeywell.com. ‡ Present address: Department of Chemical Engineering, University of Patras, GR-26500 Patras, Greece.
3-5). The MPC approaches include a linear state-space model-predictive technique that uses a suboptimal cascade filter for dual-rate systems in ref 3, a linear multirate controller based upon the model-algorithmic control (MAC) structure in ref 4, and a nonlinear predictive control strategy in ref 5. The second category, which is simpler to design, is the multirate-output controller. It receives the output measurement several times during a period but changes the manipulated input only once during that period.1,6,7 The pole assignment problem for linear multirate systems using a statespace description was considered in refs 1, 6, and 7. From an industrial point of view, the most relevant approach is the one of the first category (multirate-input controller), where the controller has enough information for the desired regulation of the slowly sampled output but changes the manipulated inputs at a rate sufficient for handling fast-varying disturbances. For this reason, the algorithm to be developed in the present paper falls under this category. The present work develops a multirate nonlinear model-algorithmic controller for the first time. The proposed multirate controller receives output measurements at rates that are generally slower than the manipulated input actuation rate and forces the controlled outputs to follow prespecified reference trajectories. While constructed in a similar spirit to the linear multirate MAC that utilizes linear step response models,4 the proposed nonlinear multirate MAC is based upon a nonlinear state-space model and is connected both conceptually and methodologically to the singlerate nonlinear MAC,8 which becomes a special case when all the outputs are sampled at the same rate. Furthermore, the nonlinear multirate MAC is amenable to rigorous closed-loop stability analysis. This series of papers has two parts, with part 1 developing the theory and part 2 experimentally applying the multirate controller to the free-radical polymerization of methyl methacrylate in a CSTR. Part 1 starts with some necessary mathematical notions and a review of nonlinear single-rate MAC. It is followed by a detailed derivation of the nonlinear multirate model-algorithmic controller. Sufficient conditions for closed-loop stability under the proposed controller are then derived. 2. Preliminaries Consider a nonlinear process described by a discretetime state-space model of the form
10.1021/ie010559n CCC: $22.00 © 2002 American Chemical Society Published on Web 07/16/2002
Ind. Eng. Chem. Res., Vol. 41, No. 16, 2002 4055
x(k+1) ) Φ[x(k),u(k)] y(k) ) h[x(k)]
yi(k+1) ) hi1[x(k)] (1)
where x ∈ R n denotes the vector of state variables, u ∈ R m denotes the manipulated input vector, and y ∈ R m denotes the controlled output vector. It is assumed that x ∈ X ⊂ R n, u ∈ U ⊂ R m, Φ(x,u) is an analytic vector function on X × U, and h(x) ) [h1(x), ..., hm(x)]T is an analytic vector function on X. 2.1. Open-Loop Stability. The local asymptotic stability of the open-loop process described by eq 1 can be checked via Lyapunov’s first method by calculating the eigenvalues of the Jacobian matrix evaluated at an equilibrium point (xs, us):
∂Φ JOL ) (x ,u ) ∂x s s
(2)
If all the eigenvalues of the Jacobian JOL are in the interior of the unit disk, the process is locally hyperbolically stable around the given equilibrium point. 2.2. Relative Order. The notion of relative order9 will play an instrumental role in the development of the control algorithm, and therefore it needs to be reviewed here. The ith output yi of a system of the form of eq 1 is said to have relative order 1 if
l yi(k+ri-1) ) hiri-1[x(k)] yi(k+ri) ) hiri-1[Φ[x(k),u(k)]]
which proves that ri is the smallest number of time steps after which the input vector u affects the ith output yi. The characteristic matrix of a system of the form of eq 1 with finite relative orders ri is defined as
[
∂ h r1-1[Φ(x,u)] ∂u 1 l C(x,u) ) ∂ rm-1 [Φ(x,u)] h ∂u m ∆
]
1
hi (x) ) hi[Φ(x,u)]
Throughout this work, it will be assumed that
det C(x,u) * 0 Because of the above assumption, the nonlinear algebraic equations
rm-1
hm
[Φ(x,u)] ) y˜ m
(6)
are locally solvable for the manipulated input vector u. The implicit function will be denoted by
u ) ΨO(x,y˜ )
If it so happens that
∂ 1 h [Φ(x,u)] ≡ / [0‚‚‚0] ∂u i then the output yi is said to have relative order 2. Otherwise, the function composition hi1[Φ(x,u)] is independent of u, and therefore it defines a function of x only:
hi2(x) ) hi1[Φ(x,u)]
}
x(k+1) ) Φ[x(k),u(k)] delay-free y˜i(k) ) hiri-1[Φ[x(k),u(k)]], i ) 1, ..., m pure delay (8)
This decomposition is a generalization of the factorization of linear discrete-time systems into an invertible part and a pure delay. The inverse of the delay-free subsystem can be constructed because of the solvability of eq 6:
[Φ(x,u)]
up until, for some integer ri:
∂ ri-1 [Φ(x,u)] ≡/ [0‚‚‚0] h ∂u i
(7)
and is assumed to be well-defined. 2.3. Minimum-Phase Behavior and Zero Dynamics. The notion of relative order motivates the decomposition of the process (1) into two subsystems in series.10 The subsystems are (i) a delay-free subsystem and (ii) a pure delay subsystem:
yi(k) ) y˜ i(k-ri), i ) 1, ..., m
This procedure continues, defining functions
hi (x) ) hi
(5)
l
Otherwise, the function composition hi[Φ(x,u)] is independent of u, and therefore it defines a function of x only:
l-1
(4)
hiri-1[Φ(x,u)] ) y˜ 1
∂ h [Φ(x,u)] ≡ / [0‚‚‚0] ∂u i
l
yi(k+2) ) hi2[x(k)]
(3)
in which case the output yi is said to have relative order ri. If no such integer exists, the output yi is said to have infinite relative order (ri ) ∞) and the manipulated input vector u does not affect the output yi. In a wellformulated control problem, every output yi must possess a finite relative order ri. Moreover, for a system of the form of eq 1 with finite relative order ri for the output yi, the following relations hold:9
x(k+1) ) Φ[x(k),ΨO[x(k),y˜ (k)]] u(k) ) ΨO[x(k),y˜ (k)]
(9)
If the system (1) is linear, the inverse system (9) has n - ∑ri poles at the transmission zeros of the delay-free subsystem and ∑ri poles at the origin. For the linear inverse system to be stable, the finite transmission zeros must be in the interior of the unit disk. Accordingly, the stability of the nonlinear inverse system can be defined.
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Definition 1. Given a discrete-time nonlinear process of the form of eq 1, its delay-free part is called locally minimum phase if the dynamics
x(k+1) ) Φ[x(k),ΨO[x(k),y˜ (k)]]
(10)
is locally asymptotically stable. Otherwise, the system will be called nonminimum phase. The local asymptotic stability of eq 10 can be checked by calculating the eigenvalues of the Jacobian evaluated at an equilibrium point via Lyapunov’s first method. Using eqs 6 and 7, along with the implicit function theorem, the Jacobian matrix of the inverse system evaluated at a reference equilibrium point (xs, us) is given by
JINV )
[
]
∂Φ(xs,us) ∂x
[
∂ h ri-1[Φ(xs,us)] ∂x 1 ∂Φ(xs,us) [C(xs,us)]-1 l ∂u ∂ rm-1 h [Φ(xs,us)] ∂x m
[
]
]
(11)
If all the eigenvalues of the Jacobian JINV are in the interior of the unit disk, the dynamics of eq 10 will be locally asymptotically stable around the reference equilibrium point. Definition 2. Given a discrete-time nonlinear process of the form of eq 1, its delay-free part is called locally hyperbolically minimum phase if all of the eigenvalues of the Jacobian matrix JINV are in the interior of the unit disk. It is important to note that if a system is hyperbolically minimum phase, then it will also be minimum phase. However, the converse of the statement may not hold. A system whose JINV has eigenvalues on the unit circle can be locally minimum phase. 2.4. Input/Output Linearizing State Feedback. When all the states of the process (1) are measurable at every time instant, a static state feedback law
u(k) ) Ψ[x(k),v(k)]
(12)
can be applied that induces linear closed-loop input/ output behavior. In particular, if eq 12 represents the function locally defined as the solution of the nonlinear algebraic equations
The local asymptotic stability of the closed-loop system under state feedback can be checked via Lyapunov’s first method by calculating the eigenvalues of the Jacobian:
JSF )
∂Φ ∂Φ ∂Ψ (x ,u ) + (x ,u ) (x ,u ) ∂x s s ∂u s s ∂x s s
The following are sufficient conditions for all the eigenvalues of eq 16 to be in the interior of the unit disk:10 (i) The closed-loop system is input/output stable. This is guaranteed if 0 e Ri < 1 for i ) 1, ..., m. (ii) The delay-free part of the process is locally hyperbolically minimum phase. This is guaranteed if all the eigenvalues of JINV are in the interior of the unit disk. The concepts presented in the preceding pages will be utilized in the following sections for the derivation and analysis of nonlinear model-algorithmic controllers, for both single-rate and multirate processes. 3. Nonlinear Single-Rate MAC: A Review The nonlinear multirate MPC to be developed in this work will be based upon the conceptual framework of nonlinear single-rate MAC. Therefore, it is useful to review the derivation of the nonlinear single-rate MAC.8 Given a process of the form of eq 1 with finite relative orders r1, ..., rm, future changes of the ith output can be predicted according to
yiM(k+1) - yiM(k) ) hi1[xM(k)] - hi[xM(k)] yiM(k+2) - yiM(k) ) hi2[xM(k)] - hi[xM(k)] l yiM(k+ri-1) - yiM(k) ) hiri-1[xM(k)] - hi[xM(k)] yiM(k+ri) - yiM(k) ) hiri-1[Φ[xM(k),u(k)]] - hi[xM(k)] (17) where the subscript M denotes variables from the online simulation of the process model. The predicted changes of the output can be added to the actual output measurement yi to obtain “closed-loop” predictions of the future output value:
yˆ i(k+1) ) yi(k) + hi1[xM(k)] - hi[xM(k)]
h1r1-1[Φ[x(k),u(k)]] ) (1 - R1)v1(k) + R1h1r1-1[x(k)]
yˆ i(k+2) ) yi(k) + hi2[xM(k)] - hi[xM(k)]
l
l
hm
rm-1
yˆ i(k+ri-1) ) yi(k) + hiri-1[xM(k)] - hi[xM(k)]
[Φ[x(k),u(k)]] ) (1 - Rm)vm(k) + Rmhmrm-1[x(k)] (13)
then the closed-loop system
x(k+1) ) Φ[x(k),Ψ[x(k),v(k)]] y(k) ) h[x(k)]
(16)
(14)
will have the following input/output behavior:
y1(k+r1) ) (1 - R1)v1(k) + R1y1(k+r1-1)
yˆ i(k+ri) ) yi(k) + hiri-1[Φ[xM(k),u(k)]] - hi[xM(k)] (18) By using the difference of the predicted output functions, any constant errors associated with the process model will be eliminated. The response of each process output is requested to follow a preassigned reference trajectory:
yˆ i(k+ri) ) (1 - Ri)yisp + Riyˆ i(k+ri-1)
(19)
l ym(k+rm) ) (1 - Rm)vm(k) + Rmym(k+rm-1)
(15)
where the subscript sp denotes the output set point and Ri is a tunable scalar parameter such that 0 e Ri < 1.
Ind. Eng. Chem. Res., Vol. 41, No. 16, 2002 4057
xM(k+1) ) Φ[xM(k),Ψ[xM(k),ysp - y(k) + h[xM(k)]]] x(k+1) ) Φ[x(k),Ψ[xM(k),ysp - y(k) + h[xM(k)]]] y(k) ) h[x(k)]
(23)
The structure of the closed-loop system is given in Figure 2. 3.1. Closed-Loop Stability. Utilizing Lyapunov’s first method, conditions for the local asymptotic stability of the closed-loop system can be established. The Jacobian of eq 23 is found to be Figure 1. Effect of the reference trajectory.
JSRM )
[
]
∂Φ ∂Φ ∂Ψ ∂Ψ ∂h + + ∂x ∂u ∂x ∂v ∂x
-
∂Φ ∂Ψ ∂Ψ ∂h + ∂u ∂x ∂v ∂x
∂Φ ∂Φ ∂Ψ ∂h ∂x ∂u ∂v ∂x (24)
(
)
(
)
∂Φ ∂Ψ ∂h ∂u ∂v ∂x
where all of the partial derivatives ∂Φ/∂x, ∂Φ/∂u, ∂Ψ/ ∂x, ∂Ψ/∂v, and ∂h/∂x are evaluated at the reference steady state. Applying the similarity transformation Figure 2. Single-rate closed-loop system structure.
T)
It is noted that this reference trajectory is exactly eq 15 with vi ) yisp. The effect of the tunable parameter on the output response is seen schematically in Figure 1. A parameter value of Ri ) 0 gives rise to a dead-beat controller which requests that the output yi reach the set point in ri sampling periods. This kind of response has poor robustness properties and is often infeasible because of constraints on the process inputs. Combining eqs 18 and 19, so that the “closed-loop” output prediction matches the reference trajectory, gives a set of nonlinear algebraic equations which depend on the manipulated input vector:
h1[xM(k)]] + R1h1
[xM(k)]
l hmrm-1[Φ[xM(k),u(k)]] ) (1 - Rm)[ymsp - ym(k) + hm[xM(k)]] + Rmhmrm-1[xM(k)] (20) If the process has finite relative orders and a nonsingular characteristic matrix, the above nonlinear algebraic equations are locally solvable via the implicit function theorem. The corresponding implicit algebraic function defined as the solution to eq 20 is given by
u(k) ) Ψ[xM(k),ysp - y(k) + h[xM(k)]]
(21)
where Ψ[*,*] is the same function as that defined by eqs 12 and 13. With the model states xM obtained by simulating xM(k+1) ) Φ[xM(k),u(k)] online, the resulting single-rate model-algorithmic controller can be expressed as
xM(k+1) ) Φ[xM(k),Ψ[xM(k),ysp - y(k) + h[xM(k)]]] u(k) ) Ψ[xM(k),ysp - y(k) + h[xM(k)]]
In -In 0 In
]
(22)
Under this single-rate MAC, the closed-loop system is as follows:
(25)
where In is the n × n identity matrix, the Jacobian becomes
JSRM ) T
-1
[
∂Φ ∂x
0
∂Φ ∂Ψ ∂Ψ ∂h + ∂u ∂x ∂v ∂x
∂Φ ∂Φ ∂Ψ + ∂x ∂u ∂x
(
)
]
T (26)
This form of the Jacobian shows that
{
}
∂Φ ∪ ∂x ∂Φ ∂Φ ∂Ψ eigenvalues of + ∂u ∂u ∂x
{eigenvalues of JSRM} ) eigenvalues of
h1r1-1[Φ[xM(k),u(k)]] ) (1 - R1)[y1sp - y1(k) + r1-1
[
{
}
) {eigenvalues of JOL} ∪ {eigenvalues of JSF} This immediately leads to the following sufficient conditions for local asymptotic stability of the closed-loop system:8,10 (i) The reference trajectories are stable. This is guaranteed if 0 e Ri < 1 for i ) 1, ..., m. (ii) The delay-free part of the process is locally hyperbolically minimum phase. This is guaranteed if all the eigenvalues of JINV are in the interior of the unit disk. (iii) The process is locally hyperbolically stable. This is guaranteed if all the eigenvalues of JOL are in the interior of the unit disk. 3.2. Input Constraints. With the previously described MAC controller, the outputs will follow the desired reference trajectories of eq 19 as long as the control actions are physically realizable. However, this will no longer hold if the inputs reach an active constraint. The input constraints of a process are usually due to the physical limitations of the process such as the pump size, flow valve limitations, and heating/cooling constraints. Because the controller implicitly possesses integral action, the presence of input
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Figure 3. Closed-loop system structure in the presence of input constraints.
constraints may cause the controller to exhibit windup. The controller windup can be prevented by driving the open-loop state simulator with the actual control action that drives the physical process. This will give the most accurate model states and, accordingly, the most effective control of the process within the input constraints. The actual control action is obtained by clipping the unconstrained input values calculated from the controller:
{
uiL, if uiC < uiL ui ) uiC, if uiL e uiC e uiH uiH, if uiC > uiH
}
i ) 1, ..., m
(27)
4. Nonlinear Multirate MAC Consider a nonlinear process described by a discretetime state-space model of the form of eq 1:
x(k+1) ) Φ[x(k),u(k)] (28)
where x ∈ R n denotes the vector of state variables, u ∈ R m denotes the manipulated input vector, and y ∈ R m denotes the controlled output vector. For each time k∆t, where k is an integer and ∆t is the time step of the model, the manipulated inputs are actuated. In addition, for each time that is a multiple of Ni∆t, where Ni is an integer constant, the output measurement yi is received. Ni is defined as the ratio of the ith output sampling period to the input actuation period:
Ni )
nonlinear single-rate MAC, future changes of the output can be predicted by simulating the process model online:
yiM(k+1) - yiM yiM(k+2) - yiM
([ ] ) ([ ] )
[ ([ ] )] [ ([ ] )]
k k N ) hi1[xM(k)] - hi xM N Ni i Ni i k k N ) hi2[xM(k)] - hi xM N Ni i Ni i l
where ui is the physically realizable input to the process and state simulator, uiC is the calculated input from the MAC algorithm, uiL is the lower constraint of the input, and uiH is the upper constraint of the input. The structure of the closed-loop system with input constraints is shown in Figure 3. Clipping of the calculated control action, in the sense of eq 27 and Figure 3, does not change the local asymptotic stability analysis of the closed-loop system. However, as long as the input constraints are active, the outputs will not follow the reference trajectories (19), resulting in a loss of performance and possibly a loss of decoupling.
yi(k) ) hi[x(k)], i ) 1, ..., m
Figure 4. Multirate process.
sampling period of the output i period of input actuation
In other words, Ni is the number of input moves actuated in the time of one output measurement. A schematic of the multirate process is given in Figure 4. The following sections develop general multirate modelalgorithmic controllers for nonlinear processes. 4.1. Derivation of the Nonlinear Multirate ModelAlgorithmic Controller. In the same vein as the
yiM(k+ri-1) - yiM
([ ] )
k N ) hiri-1[xM(k)] Ni i
[ ([ ] )]
hi x M yiM(k+ri) - yiM
([ ] )
k N Ni i
k N ) hiri-1[Φ[xM(k),u(k)]] Ni i k h i xM N (29) Ni i
[ ([ ] )]
where [k/Ni] denotes the integer part of the real number k/Ni, xM is the vector of model states, and yiM is the ith output calculated on the basis of the model states. The predicted changes of the output can be added to the latest available output measurement to obtain the following “closed-loop” predictions of the output yi:
yˆ i(k+1) ) yi yˆ i(k+2) ) yi
([ ] ) ([ ] )
[ ([ ] )] [ ([ ] )]
k k N + hi1[xM(k)] - hi xM N Ni i Ni i k k N + hi2[xM(k)] - hi xM N Ni i Ni i l
yˆ i(k+ri-1) ) yi
([ ] )
k N + hiri-1[xM(k)] Ni i
[ ([ ] )]
h i xM yˆ i(k+ri) ) yi
([ ] )
k N Ni i
k N + hiri-1[Φ[xM(k),u(k)]] Ni i k N h i xM Ni i
[ ([ ] )]
(30)
Referring back to Figure 4, the output measurement yi is taken only at the time instants [k/Ni]Ni, but the manipulated inputs are actuated at every k. Equation 30 utilizes the best available information, which is the most recent measurement at [k/Ni]Ni, to predict the
Ind. Eng. Chem. Res., Vol. 41, No. 16, 2002 4059
future values of the slowly sampled output yi. Using this structure, a controller can be derived that not only includes the necessary information for good control of outputs sampled at the lowest rates but also exhibits good disturbance rejection for outputs sampled at faster rates. It should be noted that for measurements available at a sampling period equal to the input actuation period (Ni ) 1), [k/Ni]Ni will reduce to k in the notation. For each of the outputs, a reference trajectory can be defined according to
yˆ i(k+ri) ) (1 - Ri)yisp + Ri yˆ i(k+ri-1)
(31)
where sp denotes the output set point and Ri is a tunable scalar parameter such that 0 e Ri < 1. Matching the predictions with the reference trajectories gives a set of nonlinear algebraic equations that must be solved numerically online for the input vector:
[
([ ] )
h1r1-1[Φ[xM(k),u(k)]] ) (1 - R1) y1sp - y1
[ ([ ] )]] k N N1 1
h1 xM
k N + N1 1
+ R1h1r1-1[xM(k)]
[
([ ] ) [ ([ ] )]]
+
Rmhmrm-1[xM(k)] (32)
The corresponding implicit algebraic function defined as the solution to eq 32 is given by
u(k) ) Ψ[xM(k),w(k)]
wi(k) ) yisp - yi
([ ] ) [ ([ ] )]
(34)
With the model states xM obtained by simulating xM(k+1) ) Φ[xM(k),u(k)] online, the resulting nonlinear multirate model-algorithmic controller can be expressed as
(35)
where w(k) ) [w1(k), ..., as defined by eq 34. Note that the calculation of wi(k) involves a hold operation on both yi(k) and hi[xM(k)]: wm(k)]T,
wi(k) ) H(yisp - yi(k) + hi[xM(k)])
([ ] ) [ ([ ] )]
) yisp - yi
k k N + hi xM N Ni i Ni i
{
1, if N divides k δi(k) ) 0, if Ni does not divide k i
}
for i ) 1, ..., m
Equivalently, the time-varying system can be written in state-space form as follows:
Υi(k+1) ) (1 - δi(k))Υi(k) + δi(k) (yisp - yi(k) + hi[xM(k)]) wi(k) ) (1 - δi(k))Υi(k) + δi(k) (yisp - yi(k) + hi[xM(k)]) (38) When the above equations for i ) 1, ..., m are combined with the state feedback u(k) ) Ψ[xM(k),w(k)] and the model dynamics xM(k+1) ) Φ[xM(k),u(k)], the multirate model-algorithmic controller (35) can be described as the time-varying dynamic system:
xM(k+1) ) Φ[xM(k),Ψ[xM(k),(Im-∆(k))Υ(k) + ∆(k)(ysp-y(k)+h[xM(k)])]] u(k) ) Ψ[xM(k),(Im-∆(k))Υ(k) + ∆(k)(ysp-y(k) + h[xM(k)])] (39) where
[
δ1(k)
∆(k) )
0
‚
0 ‚
‚ δm(k)
]
(40)
Observe that the function ∆(k) is a periodic function, with period N equal to the least common multiple of N1, ..., Nm. Therefore, the multirate controller of eq 39 is a time-varying periodic system with the same period:
N ) lcm(N1,...,Nm) Under the multirate MAC, the closed-loop system is given by
xM(k+1) ) Φ[xM(k),Ψ[xM(k),w(k)]] u(k) ) Ψ[xM(k),w(k)]
(37)
where vi(k) ) yisp - yi(k) + hi[xM(k)] and
(33)
where Ψ[*,*] is the same function as that in the singlerate MAC controller and
k k N + h i xM N Ni i Ni i
wi(k) ) (1 - δi(k))wi(k-1) + δi(k) vi(k)
Υ(k+1) ) (Im - ∆(k))Υ(k) + ∆(k)(ysp - y(k) + h[xM(k)])
l hmrm-1[Φ[xM(k),u(k)]] ) k k N + hm xM N (1 - Rm) ymsp - ym Nm m Nm m
of a time-varying dynamic system
(36)
where H is a hold operator. The structure of the resulting closed-loop system is given in Figure 5. 4.1.1. Nonlinear Multirate Model-Algorithmic Controller as a Time-Varying Periodic System. To be able to write the nonlinear multirate model-algorithmic controller as a dynamic system, observe that the action of the hold operator can be represented in terms
Υ(k+1) ) (Im - ∆(k))Υ(k) + ∆(k)(ysp - h[x(k)] + h[xM(k)]) xM(k+1) ) Φ[xM(k),Ψ[xM(k),(Im-∆(k))Υ(k) + ∆(k)(ysp-h[x(k)]+h[xM(k)])]] x(k+1) ) Φ[x(k),Ψ[xM(k),(Im-∆(k))Υ(k) + ∆(k)(ysp-h[x(k)]+h[xM(k)])]] y(k) ) h[x(k)]
(41)
4.2. Closed-Loop Stability Analysis. Expressing the multirate controller as a nonlinear time-varying system will facilitate the analysis of the closed-loop system and lead to conditions that guarantee local asymptotic stability. Because much of the time-varying difference equation mathematics is not well-known, two
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Ind. Eng. Chem. Res., Vol. 41, No. 16, 2002
[
]
JMRM(k) ) ∂h ∂x ∂Φ ∂Ψ ∂h ∂Φ ∂Φ ∂Ψ ∂Ψ (I - ∆(k)) + + ∆(k) ∂u ∂v m ∂x ∂u ∂x ∂v ∂x ∂Φ ∂Ψ ∂h ∂Φ ∂Ψ ∂Ψ (I - ∆(k)) + ∆(k) ∂u ∂v m ∂u ∂x ∂v ∂x
Im - ∆(k)
Figure 5. Multirate closed-loop system structure.
important theorems will be reviewed. Lyapunov’s first method for nonlinear time-varying discrete-time systems can be outlined in the following theorem:11 Theorem 1. Consider a nonlinear nonautonomous system:
x(k+1) ) f[k,x(k)]
|
∂f[k,x] ∂x
x)0
and assume that
f[k,x] - A(k) x(k) ) o(||x||) as ||x|| f 0 If the origin is a uniformly asymptotically stable equilibrium point of the linear system
z(k+1) ) A(k) z(k)
)
(
)
∂h ∂x ∂h ∂Φ ∂Ψ ∆(k) ∂u ∂v ∂x ∂h ∂Φ ∂Φ ∂Ψ ∆(k) ∂x ∂u ∂v ∂x
-∆(k)
(45)
where all of the partial derivatives ∂Φ/∂x, ∂Φ/∂u, ∂Ψ/ ∂x, ∂Ψ/∂v, and ∂h/∂x are evaluated at the reference steady state. Notice that the linearization of the original nonlinear time-varying system is also a time-varying system, and therefore JMRM depends on k. Also, JMRM is a periodic function of k with the same period N as the original nonlinear system. When the following similarity transformation is applied
[ ]
0 In -In T ) Im 0 0 0 0 In
(46)
the Jacobian becomes
[
JMRM(k) ) ∂Φ ∂x ∂h -1 T ∆(k) ∂x ∂Φ ∂Ψ ∂Ψ ∂h + ∆(k) ∂u ∂x ∂v ∂x
(
)
0
0
Im - ∆(k)
0
]
T
∂Φ ∂Ψ ∂Φ ∂Φ ∂Ψ (I - ∆(k)) + ∂u ∂v m ∂x ∂u ∂x
(47)
Now the monodromy matrix N-1
(43)
then it is an exponentially stable equilibrium point of the nonlinear system (42). Moreover, if the time-varying linear system of eq 43 is periodic, stability can be assessed with the following theorem:11 Theorem 2. Consider a nonautonomous linear periodic system
x(k+1) ) A(k) x(k)
(
(42)
where the function f[k,x] is continuously differentiable with respect to x. Without loss of generality, suppose that the origin is an equilibrium point of the system: f[k,0] ) 0, for every k. Define
A(k) )
∆(k)
JMRM(k) ) JMRM(N-1) JMRM(N-2) ... JMRM(0) ∏ k)0
can be conveniently expressed as follows: N-1
([
∏J k)0
MRM(k)
)
∂Φ
(44)
with period N, i.e., A(k+N) ) A(k). Then the equilibrium point 0 is (i) uniformly stable if each eigenvalue of the monodromy matrix M ) A(N-1) A(N-2)‚‚‚A(0) has a modulus of less than or equal to 1 and (ii) uniformly asymptotically stable if each eigenvalue of the monodromy matrix M ) A(N-1) A(N-2)‚‚‚A(0) has a modulus of less than 1. It is important to note that, for the time-invariant linear system x(k+1) ) Ax(k), the eigenvalues of A determine the stability properties of the system. However, for the periodic system x(k+1) ) A(k) x(k), the eigenvalues of A(k) do not play any role in the determination of the stability properties of the system. Instead, the eigenvalues of monodromy matrix M determine those properties. Having reviewed the important general results for time-varying discrete-time systems, they can now be applied to the closed-loop system under multirate MAC given by eq 41. The Jacobian of the closed-loop system is
T-1
0
0
Im - ∆(k)
0
∂x
N-1
∏ ∆(k) ∂x
∂h
k)0
(
∂Φ ∂Ψ ∂u
+
∂x
∂Ψ ∂v
∆(k)
)
∂h ∂x
])
T
∂Φ ∂Ψ ∂Φ ∂Φ ∂Ψ (Im - ∆(k)) + ∂u ∂v ∂x ∂u ∂x
(48)
Expanding the product yields N-1
[
JMRM(k) ) ∏ k)0
( ) ∂Φ ∂x
N
0
0
N-1
T-1 * *
(Im - ∆(k)) ∏ k)0
*
(
∂Φ ∂x
]
T (49)
0 +
)
∂Φ ∂Ψ ∂u ∂x
N
where * denotes expressions which are not used in the analysis and
Ind. Eng. Chem. Res., Vol. 41, No. 16, 2002 4061 N-1
(Im - ∆(k)) ) ∏ k)0
[
(Im - ∆(N-1))(Im - ∆(N-2)) ... (Im - ∆(0)) ) N)1
(1 - δ1(k)) ∏ k)0
0 ‚
‚
‚ N)1
(1 - δm(k)) ∏ k)0
0
]
(ii) The delay-free part of the process is locally hyperbolically minimum phase. This is guaranteed if all the eigenvalues of JINV are in the interior of the unit disk. (iii) The process is locally hyperbolically stable. This is guaranteed if all the eigenvalues of JOL are in the interior of the unit disk. 4.3. Nonlinear Multirate MAC in the Presence of Measurement Delays. Significant time delays are often associated with slowly sampled outputs because of the analytical nature of the measurements. In this case, the process dynamics can be described as follows:
Recalling that ∂Φ/∂x ) JOL and ∂Φ/∂x + (∂Φ/∂u)(∂Ψ/ ∂x) ) JSF from eqs 2 and 16, respectively, the result for the monodromy matrix can be expressed as follows: N-1
[
JMRM(k) ) ∏ k)0 JOLN
0
0
N-1
(1 - δ1(k)) ∏ k)0
T-1 *
0 ‚
‚
N-1
(1 - δm(k)) ∏ k)0
0 *
*
]
T
0
‚
JSFN
x(k+1) ) Φ[x(k),u(k)] yi(k) ) hi[x(k-φi)], i ) 1, ..., m
The system (51) can be easily converted to the form of eq 28 by introducing additional state variables, and in this way, all previous results are directly applicable. However, because measurement delays for some outputs are often quite large relative to the input actuation period, it will be useful to express the control law in terms of the description (51) to avoid handling an excessively large state vector. Observe that the system (51) can be decomposed into the subsystems
x(k+1) ) Φ[x(k),u(k)] yji(k) ) hi[x(k)], i ) 1, ..., m
(50)
Because of the lower block-triangular structure, it follows that the set of eigenvalues of eq 50 consists of (a) the eigenvalues of JOLN, (b) the eigenvalues of JSFN, N-1 (1 - δi(k)), i ) 1, ..., m. Each and (c) the numbers ∏k)0 subset of eigenvalues can now be examined separately: (a) If the eigenvalues of JOL are denoted by λi, then the eigenvalues of JOLN are exactly λiN, i ) 1, ..., n. Therefore, if all λi are in the interior of the unit disk, λiN will also be in the interior of the unit disk. (b) If the eigenvalues of JSF are denoted by ζi, then the eigenvalues of JSFN are exactly ζiN, i ) 1, ..., n. Therefore, if all ζi are in the interior of the unit disk, ζiN will also be in the interior of the unit disk. (c) Because N ) lcm(N1, ..., Nm), at least one of the factors in N-1
(1 - δi(k)) ) ∏ k)0
and
yi(k) ) yji(k-φi), i ) 1, ..., m The first subsystem describes the process dynamics, including delays which are not associated with the measurements, whereas the second subsystem describes the measurement delays. Denote by ri the relative order of the ith output of the first subsystem; i.e., ri is the smallest number of time steps beyond the measurement delay after which the input vector u(k) affects the measurement yi. The following relations will now hold:
yi(k+φi+1) ) hi1[x(k)] yi(k+φi+2) ) hi2[x(k)] l
(1 - δi(N-1))(1 - δi(N-2)) ... (1 - δi(0)) will vanish (because δi(k) will equal 1 for at least one k N-1 (1 - δi(k)) within the range 0 to N - 1). Therefore, ∏k)0 ) 0 for all i ) 1, ..., m. Hence, there will be m N-1 JMRM(k) at the origin. eigenvalues of ∏k)0 The foregoing observations immediately lead to sufficient conditions for closed-loop stability. The result is summarized in the following theorem: Theorem 3. Consider the nonlinear process described by eq 28 under the nonlinear multirate model-algorithmic controller of eq 35 or eq 39. The closed-loop system will be locally asymptotically stable if the following conditions are met: (i) The reference trajectories are stable. This is guaranteed if 0 e Ri < 1 for i ) 1, ..., m.
(51)
yi(k+φi+ri-1) ) hiri-1[x(k)] yi(k+φi+ri) ) hiri-1[Φ[x(k),u(k)]]
(52)
Again, the functions
hi1(x) ) hi[Φ(x,u)], hi2(x) ) hi1[Φ(x,u)], ..., hiri-1(x) ) hiri-2[Φ(x,u)] are predetermined by the process history and are only functions of the state vector x. Moreover, predictions of the future output changes can be obtained by simulating the process model online:
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yiM(k+φi+1) - yiM
([ ] )
k N ) Ni i
[ ([ ]
hi1[xM(k)] - hi xM yiM(k+φi+2) - yiM
([ ] )
k N - φi Ni i
[ ([ ]
2
hi [xM(k)] - hi xM
)]
k N - φi Ni i
)]
l
([ ] )
k N ) Ni i
ri-1
hi
([ ] )
k N ) Ni i
[ ([ ]
k N - φi Ni i
[ ([ ]
[Φ[xM(k),u(k)]] - hi xM
k N - φi Ni i
)]
)]
(53)
([ ] )
k N + Ni i
([ ] )
k N + Ni i
[ ([ ]
)]
[ ([ ]
)]
[ ([ ]
)]
hi2[xM(k)] - hi xM
k N - φi Ni i
k N - φi Ni i
l yˆ i(k+φi+ri-1) ) yi
([ ] )
k N + Ni i
hiri-1[xM(k)] - hi xM yˆ i(k+φi+ri) ) yi
([ ] )
k N + Ni i
)]]
k N + N1 1
+ R1h1r1-1[xM(k)]
[
(1 - Rm) ymsp - ym
[ ([ ]
)]]
k N - φm Nm m
([ ] )
k N + Nm m
+ Rmhmrm-1[xM(k)] (56)
The implicit function defined as the solution to eq 56 is given by
u(k) ) Ψ[xM(k),w(k)]
k N - φi Ni i
[ ([ ]
hiri-1[Φ[xM(k),u(k)]] - hi xM
k N - φi Ni i
)]
(54)
Again, eq 54 utilizes the best available information of the system, which is the measurement received at [k/Ni]Ni, to predict the future of the slowly sampled and delayed output yi. For each of the outputs, a reference trajectory can be defined:
yˆ i(k+φi+ri) ) (1 - Ri)yisp + Riyˆ i(k+φi+ri-1)
(55)
where Ri is a tunable scalar parameter. When the predictions are matched with the reference trajectories, the input vector can be calculated by solving the following nonlinear algebraic equations online:
(57)
whereΨ[*,*] is the same function as that in the singlerate MAC controller and
([ ] ) [ ([ ]
k k N + hi xM N - φi Ni i Ni i
)]
(58)
Simulating xM(k+1) ) Φ[xM(k),u(k)] online to obtain the model states xM results in the nonlinear multirate model-algorithmic controller
xM(k+1) ) Φ[xM(k),Ψ[xM(k),w(k)]] u(k) ) Ψ[xM(k),w(k)]
hi1[xM(k)] - hi xM yˆ i(k+φi+2) ) yi
k N - φ1 N1 1
hmrm-1[Φ[xM(k),u(k)]] )
wi(k) ) yisp - yi
where the subscript M denotes, as before, variables from the online simulation of the model. The predicted changes of the output can be added to the latest available delayed output measurement yi to obtain the “closed-loop” predictions of the outputs:
yˆ i(k+φi+1) ) yi
[ ([ ]
h 1 xM
hm x M
hiri-1[xM(k)] - hi xM yiM(k+φi+ri) - yiM
([ ] )
l
k N ) Ni i
yiM(k+φi+ri-1) - yiM
[
h1r1-1[Φ[xM(k),u(k)]] ) (1 - R1) y1sp - y1
(59)
where w(k) ) [w1(k)‚‚‚wm(k)]T, as defined by eq 58. The above multirate model-algorithmic controller is equivalent to the one derived in the previous subsection, in the sense that it collapses to eq 35 when the previous description (51) is converted to eq 28 through additional state variables. Consequently, the closed-loop stability analysis of the previous subsection and its conclusion (theorem 3) remain valid. 4.4. Final Remarks. (i) The multirate model-algorithmic controller will be able to force the outputs to follow the desired reference trajectories (31) or (55) as long as the calculated control actions lie within the physical limits of the manipulated process inputs. Like in single-rate MAC, calculated control action must be “clipped” according to eq 27 before it drives the model simulator. Clipping does not affect the local closed-loop stability analysis detailed in the previous section as long as the nominal conditions correspond to the interior of feasible input ranges. However, it must be noted that while input constraints are active, the outputs will no longer follow the desired reference trajectories, resulting in a loss of performance and possibly a loss of decoupling. This point will be discussed further in part 2 of this series, in the context of the specific polymerization control application. (ii) What is tunable in the multirate model-algorithmic controller is the m reference trajectory parameters Ri which directly shape the speed of the response of the system outputs in closed loop. Decreasing the values of the parameters Ri makes the controller more aggressive, as it tries to enforce faster output response, and therefore less robust. Increasing the values of the parameters Ri makes the controller more robust, at the expense of losing performance. The issue of tuning of the controller parameters Ri will be discussed further
Ind. Eng. Chem. Res., Vol. 41, No. 16, 2002 4063
in part 2 of this series, in the context of the specific polymerization control application. (iii) Like in single-rate MAC, the proposed multirate model-algorithmic controller is a predictive controller, involving output predictions one step ahead, beyond the effective dead time of each output. It would be possible to extend the proposed approach, allowing the possibility of using larger prediction horizons and/or optimizing the controller moves. However, in such an extension, the closed-loop stability analysis could be much more difficult. 5. Conclusion This paper developed a nonlinear multirate modelalgorithmic controller for multivariable processes. The proposed control structure incorporates all available process measurements and forces each output to follow a prespecified reference trajectory regardless of sampling period. It was shown that the multirate controller is a time-varying periodic system, which has properties that enable rigorous stability analysis. The closed-loop system was then found to be asymptotically stable if the process is stable and minimum phase and, in addition, the reference trajectories are stable. Acknowledgment Financial support from the National Science Foundation through Grant CTS-9403432 is gratefully acknowledged. Notation A(k) ) matrix in a linearized time-varying system C(x,u) ) characteristic matrix H ) hold operator I ) identity matrix J ) Jacobian matrix of a nonlinear system m ) number of input and output variables M ) monodromy matrix n ) number of state variables N ) period of the multirate system Ni ) ratio of output sampling period to input actuation period ri ) relative order of the ith output with respect to the manipulated input vector
T ) transformation matrix u ) manipulated input vector v ) driving input for the state feedback component in single-rate MAC w ) driving input for the state feedback component in multirate MAC x ) process state vector y ) process output vector Greek Letters Ri ) tunable controller parameter δi ) time-varying parameter, 0 or 1 ∆t ) time step or input actuation period φi ) measurement time delay Υ ) state variable of the time-varying system
Literature Cited (1) Hagiwara, T.; Fujimura, T.; Araki, M. Generalized Multirate-Output Controllers. Int. J. Control 1990, 52, 597. (2) Araki, M.; Hagiwara, T. Pole Assignment by Multirate Sampled-Data Output Feedback. Int. J. Control 1986, 44, 1661. (3) Lee, J. H.; Gelormino, M.; Morari, M. Model Predictive Control of Multi-Rate Sampled-Data Systems: A State-Space Approach. Int. J. Control 1992, 55, 153. (4) Ohshima, M.; Hashimoto, I.; Ohno, H.; Takeda, M.; Yoneyama, T.; Gotoh, F. Multirate Multivariable Model Predictive Control and its Application to a Polymerization Reactor. Int. J. Control 1994, 59, 731. (5) Bequette, B. W. Nonlinear Predictive Control Using Multirate Sampling. Can. J. Chem. Eng. 1991, 69, 136. (6) Hagiwara, T.; Araki, M. Design of a Stable State Feedback Controller Based on the Multirate Sampling of the Plant Output. IEEE Trans. Autom. Control 1988, 33, 812. (7) Er, M. J.; Anderson, B. D. O. Practical Issues in Multirate Output Controllers. Int. J. Control 1991, 53, 1005. (8) Soroush, M.; Kravaris, C. MPC formulation of GLC. AIChE J. 1996, 42, 2377. (9) Nijmeijer, H.; van der Schaft, A. J. Nonlinear Dynamical Control; Springer-Verlag: New York, 1990. (10) Soroush, M.; Kravaris, C. Discrete-Time Nonlinear Feedback Control of Multivariable Processes. AIChE J. 1996, 42, 187. (11) Elaydi, S. N. An Introduction to Difference Equations; Springer-Verlag: New York, 1996.
Received for review June 28, 2001 Revised manuscript received January 7, 2002 Accepted April 5, 2002 IE010559N
