- Email: [email protected]
Feedback Control of Hyperbolic PDE Systems
Copyright @ IF AC Advanced Control of Chemical Processes, Pisa, Italy, 2000
FEEDBACK CONTROL OF HYPERBOLIC PDE SYSTEMS Huilan Shang * J. Fraser Forbes *,1 Martin Guay **
* Department of Chemical and Materials Engineering, University **
of Alberta, Edmonton, Alberta, Canada T6G 2G6 Department of Chemical Engineering, Queen's University, Kingston, Ontario, Canada K7L 3N6
Abstract: Many industrial processes are distributed parameter systems (DPS) that can be described by hyperbolic partial differential equations (e.g., some fixedbed reactors, sheet-forming and fibre-forming processes) . Conventionally, the control schemes for these systems have been designed by approximating the original PDEs as a finite number of ordinary differential equations (ODE) . To achieve better performance, some researchers have focused on control methods that deal explicitly with the original PDE model of the process. Unfortunately, most of this research considers only controller design in state feedback form or is restricted to specific model forms . This paper presents an approach for output feedback control of distributed parameter systems described by a hyperbolic partial differential equation. A state-feedback control law is developed employing a combination of the method of characteristics, concepts of sliding surface and geometric control schemes. An output feedback controller is obtained by combining the state feedback control with a state observer. The proposed control method is demonstrated with a heat exchanger. Copyright © 2000 IFAC
Keywords: Feedback Control, Distributed Parameter Systems, Sliding Surfaces
represented by parabolic PDEs, a finite number of modes can capture the dominant dynamics of the system (Curtain and Zwart, 1995). For hyperbolic PDEs, however, all of the eigenmodes of the spatial differential operator contain nearly the same amount of energy. Consequently, an infinite number of modes are required to accurately describe their dynamic behaviour (Christofides and Daoutidis, 1998). This characteristic prohibits the application of modal decomposition techniques to derive reduced-order ODE models that approximately describe the dynamics of the hyperbolic PDE system. Thus the control problem has to be addressed based on the infinite-dimensional model itself.
1. INTRODUCTION There are many processes in which states vary in both time and space (e.g., fixed-bed reactors, fibre spinlines, and sheet coating processes) . These are commonly termed Distributed Parameter Systems (DPS) . Mathematical models for distributed parameter processes typically consist of partial differential equations (PDE). All first-order PDEs are hyperbolic. Second-order PDEs can be classified into three categories: hyperbolic, parabolic and elliptic, according to the properties of the spatial differential operator. Higher-order PDEs can be decomposed into anyone or a combination of these three basic types. Controller design has focused on methods specific to the nature of the distributed parameter process model. For processes 1
A number of control approaches have been considered in the literature. A combination of the method of characteristics and sliding mode tech-
Corresponding Author: [email protected]
533
desired behaviour, termed the sliding surface, and is locally represented by (Sira-Ramirez, 1989):
niques was proposed for processes modeled by a first-order quasi-linear hyperbolic PDE using a switching feedback control law (Sira-Ramirez, 1989). This method was further developed to synthesize a continuous state feedback controller for a nonlinear hyperbolic PDE model (Hanczyc and Palazoglu, 1995). These methods focused on design of state feedback controller. A nonlinear distributed output feedback controller was synthesized for a specific quasilinear hyperbolic PDE system using geometric control methods (Christofides and Daoutidis, 1996) .
S={(V, x , t)ER n +Z
3.1 Quasilinear Systems Consider a dynamic system described by a firstorder quasilinear PDE:
ov ot
e
The vector = [1, al , · · · ,an ,b]T = [1,a,b]T defines a time-varying control-parametrized vector field. This vector field is called the characteristic vector field. By construction, the graph of any solution to (5) must be tangent to everywhere. Surfaces which are tangent at each point to a vector field are called integral surfaces of the vector field (McOwen, 1996).
(1)
e
By the method of characteristics, the quasilinear PDE system in Equation (5) can be transformed into a nonlinear ODE control system:
t=1
x = a(v,x,t ,u) v = b(v ,x ,t ,u)
Under fixed initial and boundary conditions, Equation (1) can be visualized as a surface in Rn+Z with (v, x , t) being coordinates and u being a parameter. The control objective is to have the output function track a specific trajectory by designing a controller that computes the proper value for manipulated variable u . Without loss of generality, assume the desired output trajectory is:
(6)
Y = h(v,x,t)
To simplify the notation, introduce the vector z = [t, x , vf. For the models investigated to date, the vector field can be split into f + g . u and the system of Equation (6) can be represented in an affine form:
e
z=f+g·u y=h(v ,x ,t)
(2)
(7)
The Lie derivative of the output function along the characteristic vector field g does not vanish for most of distributed parameter systems. Then the output function y = h( v, x , t) has relative degree of 1 globally. Its time derivative along the characteristic vector field is:
Then, this condition locally defines an isolated smooth manifold solution such that:
h(if>(x,t),x ,t) =0
(5)
where al, az , ... , an and b are continuous functions . Further, assume the condition Yr = 0 defines a smooth sliding surface, which allows the use of the Implicit Function Theorem.
where: t is time coordinate, x is the vector of local spatial coordinates Xi , defining points on an open set in Rn, V is the distributed state variable which changes in both time and space, u is the manipulated variable and y is scalarvalued output. For each smooth solution v of the system in Equation (1), and h are locally smooth functions of their arguments.
=0
~ ov
+ ti' OXi ai (v ,x , t ,u) = b(v ,x ,t,u)
y = h(v ,x , t)
Any first-order PDE-based process model can be expressed as:
Yr
(4)
3. STATE FEEDBACK CONTROL
2. CONTROL OBJECTIVE
=0
v=if>(x,t)}
In the following section, a controller for the system described by a hyperbolic PDE is formulated such that the surface determined by Equation (1) converges to the sliding surface described by Equation (4) asymptotically.
In this paper, an output feedback control is derived for the processes represented by a first-order hyperbolic PDE. The resulting control law guarantees output tracking without offset for processes modeled by linear, quasilinear and nonlinear hyperbolic PDEs. The proposed approach yields a comparatively simple controller design technique and produces control laws that are easy to implement.
ov OV (v ,x,t , ox ' ot ,u) Y = h(v ,x , t)
:
(3)
The graph of v is assumed to be a smooth timevarying surface on which the system has the
iJ = Lrh + Lgh . u
534
(8)
The following theorem states the formulation of distributed state feedback control law which enforces output tracking.
nonlinear PDE control system can be described in the form of a nonlinear affine control system:
Theorem 1. Assume that the system modeled by the quasi-linear first-order PDE in Equation (5) has a minimum phase characteristic as given in Equation (6). The distributed state feedback control law, which guarantees asymptotical output tracking, is: 1
~r~[~
== Y -
(16) (9)
The formulation of a distributed state controller is stated in the following theorem.
Yr'
Theorem 2. Assume that the nonlinear first-order PDE in Equation(lO) has a minimum phase characteristic as given in Equation (12) . The state feedback control law, which guarantees asymptotical output tracking, is:
3.2 Nonlinear Systems For a first-order nonlinear PDE control system: ov ov ot +
U
(10)
Equation (10) can be interpreted as a hypersurface E2n+2 in the manifold M 2n +3 = Jl (vn+l, JR) of I-jets equipped with the standard contact structure. Let (x, t) be local coordinates on V n + 1 .. ov ov and v be the coordinate m JR, p = ox ' q = ot' The corresponding local coordinates are denoted by (x, t, v, p, q) in the space of I-jets. Then the differential equation can be written in the form: q +
«(t,x,v , p,u)
L'2 h
(17)
Considering that the number of measurements is finite, the quasilinear PDE system can be expressed as: ov
ov
Ft + oxa(v,x,t,u) = b(v,x,t,u)
(12)
y=h(v , x , t) w=Qv v(xo) = Vo
(18)
where: w is the set of measurements containing m elements, Q is an operator L 2 (O,I) ---> JRm, v(xo) = Vo is the boundary condition.
(13)
By splitting the term b(v ,x , t ,u) , the PDE in Equation (18) can be expressed as:
Substituting Equation (11) into Equation (13) yields:
z=
edr)-L h
___- - - -,\-
4.1 Quasilinear Systems
Denote vector z = [t, x, v, p), and vector field (' = [I,
z = ('(t,x , v,p ,q,u)
----~
(11)
-q
t
0
In the previous section, state feedback control that can enforce output tracking was formulated. However, the state of distributed parameter system is of infinite dimension while the measurements are always finite-dimensional. Then, a state observer is necessary in order to implement the designed state feedback control law.
1
x=
=
1
r[
-k(e+-~
4. OUTPUT FEEDBACK CONTROL
By the method of characteristics, the characteristic equations of system (10) in the manifold M2n+3 can be written as (Arnold, 1988):
i= q=
(15)
Since L(2h does not usually vanish, the output function y = h( v, x, t) has relative degree of 1 globally and its time derivative along the characteristic vector field is:
10t
-k(e + -- edr) - Lfh u= ______ _________ Lgh where e
+ (2(t,x,v,p) ' u
Z = (l(t,x,v,p) y=h(v,x,t)
ov ov ot =-oxa(v , x,t ,u)+b1(v,x,t,u)
(14)
(19)
+b2 (x, t , u) Choose Z = L2(O , I) as the state space. If an operator A on Z can be defined such that:
For the models investigated to date, the vector field ( can be split into (1 + (2 ' u and the original
535
dt.p At.p = --a(t.p, x , t, u) dx t.p E L2(0, 1)/ D(A)=
+ b1(t.p, x, t , u)
where: w is the set of measurements of m elements, Q is an operator L2(0, 1) ---+ ]Rm, v(Xo) = Vo is the boundary condition.
(20)
t.p~~ are absolutely
continuous, -
By splitting the term
d~a(t.p, x,t, u)
+b1(t.p,x, t , u) E L 2(0 , 1) , t.p(Xo) = vo Equation (19) can be described through an abstract formulation on a Hilbert space Z :
dv dt = Av + b2 (x , t , u)
Choose Z = L 2 (0 , 1) as the state space. If an operator A on Z can be defined such that
(21)
Theorem 3. Consider a state observer for a firstorder quasilinear PDE system in Equation (18) of the form : d"1 dt
dt.p At.p =
= A"1 + b2(x , t , u) + C(w-Q"1)
(22 )
D(A)
dt.p L 2(0 , 1) /
is absolute1cpcontinuous,
where: "1 is an estimate of the state v . If an operator C : lR m -+ Z can be designed such that the composite operator (A - CQ) is an infinitesimal generator of a contraction semigroup, the state observer in Equation (22) guarantees that the estimated state converges to the true state asymptotically.
E
L2(0,1)
then the state observer for nonlinear PDE in Equation (25) can be constructed in the same way as quasi linear PDE.
Theorem 4. Consider a state observer for a nonlinear PDE system (24) of the form :
If the operator A defined by Equation (20) is an infinitesimal generator of a contraction semigroup, the simplest observer can be obtained by choosing C = O. The addit ion of the term C(w - Q"1) in the state observer accelerates the convergence of the estimation to the true state value.
~~
= A"1 +
+ C(w-Q"1)
(27)
If an operator C : lR m -+ Z can be designed such that the composite operator (A - CQ) is an infinitesimal generator of a contraction semigroup, the state observer (27) guarantees that the estimated state converges to the true state asymptotically.
By combining the developed state feedback controller in Equation (9) and state observer in Equation (22) , an output feedback controller for quasilinear system in Equation (18) is obtained:
0"1 0"1 at = - oxa("1 ,x,t,u) +b("1, x , t, u) + C(w - Q"1) k · e("1, x ,t) u( x) t - - -=--:--:-"'--~ , Lgh("1,x ,t)
=
E
(26)
By combining the developed state feedback controller (17) and state observer (27) , an output feedback controller for nonlinear system in Equation (24) is obtained:
(23)
-TJk 10 e("1, x , T)dT + Lfh("1 , x , t) t
0"1 0"1 at = -
Lgh( "1, x , t) where "1 is the estimated state.
k
L h("1, x ,t)
'2
where: "1 is the estimation of the state v .
The nonlinear PDE system can be expressed as:
ov
at +
y = h(v , x , t) w=Qv v(xo) = Vo
(28)
2
-TJ 10 e("1,X, T)dT + L([ h("1 ,X, t)
4.2 Nonlinear Systems
ov
t
In the controller (23) and (28), the first two terms are proportional and integral control, and the third term is a kind of feedforward control. So the developed controller has a simple form and is easy to implement.
=0 (24)
536
5. HEAT EXCHANGER Consider a forced-flow steam-jacketed tubular heat exchanger. The fluid within the tube is heated using steam without condensation. The dynamic model of the process in deviation form can be expressed as:
8T 8T -+u-+H(T-T)=O 8t 8x J
3.5
IV-\;'
2.5
... 2
(29)
V'
1.5
where T(x, t) denotes the temperature of the heat exchanger within the tube, x E [0,1]' T j denotes the jacket temperature, u is the fluid velocity through the tube and is the manipulated variable, and H is a positive constant. For simplicity, the steam temperature is considered to be spatially uniform. Fluid temperature T at the tube exit is the variable to be controlled. Then, the model given in Equation (29) is a quasilinear PDE.
o
Fig. 1. Setpoint response of temperature at x=0.2, 0.5, 0.8 and 1 with fluid velocity as the manipulated input t
= K[(T(I, t) -
where K = k/
(30)
where Tsp (x) denotes the set point temperature profile. The Lie derivative of output function y = T - Tsp(x) along the characteristic vector field ~ is:
8Tsp TJ ) - 8xu
k 8T. {x) p
[(T(x, t) - Tsp(x))
1
+ T[
(32)
8x
t
j(T(X, T) - Tsp(x))dT o
Tsp)
+ ~ j(T(I, T) (33) T[
~ (T(x, t) -1j) ]
where: 8Tir}x) is the spatial derivative of the setpoint temperature profile, which is specified. So this control law can be viewed as a simple PI control plus a kind of feedforward control. This is a spatially distributed controller; however, the manipulated variable u is spatially uniform throughout the heat exchanger when the steam is assumed to be well-mixed. The control objective is to have the exit temperature of the exchanger to reach its setpoint. So a modification can be made by replacing the spatially distributed variables by their values at x = 1. The resulting spatially uniform controller can be written as:
8
)]
J;Plx=1 .
A simulation was performed to evaluate the performance of the state feedback controller of Equation (33). The parameters used for simulation were: H = I, T j = 10, k = 5, T[ = 2. The initial temperature profile was specified as T(x, 0) = 1e- 5x and the boundary condition as T(O, t) = O. The specification of the exit temperature and its spatial derivative was taken as Tsp = 3 and 8T. p 8x -
(31)
Applying the control law given in Equation (9) yields an expression for u:
u(x, t) =
10
6
o H -Tsp)dT - "k(T(I , t) - Tj
The characteristic vector field for Equation (29) is ~ = [1, u, - H (T - T j )] and the control variable in which we are interested is the temperature profile. Then the sliding surface can be defined as:
= - H(T -
,.tl2
V 0.5
5.1 State Feedback Control
L
.tl,
1
U
T-Tsp(x) =0
lA
3.
For simulation purposes, the method of finite differences was used to derive a finite-dimensional approximation of the original PDE equation, with a choice of 10 discretization points. The results of simulation are shown in Fig.1. It was observed that the exit temperature, i.e., T at x = I, tracks the set point well and the temperature at other points reaches its steady state after some time. It can be concluded that the controller (33) has good setpoint tracking behaviour and the closedloop response is stable.
5.2 Output Feedback Control Assume temperature in the heat exchanger can be measured at 5 points x = 0.2,0.4,0.6,0.8, 1.0. A state observer for system (29) can be designed as:
8T at. -8t = - u8x- - H(T -
.
T) - C(J: - QT)(34) J
m
where Tm is the temperature measurements of R5,
T is the estimated temperature profile in the heat
537
2.6r----....----~----..----.---___,
o
+
0.80~---:0.... .0:-5--~0.-,-1---:-0~ . 15:----0.....2----,J0.25
0.5L----'-----'-----'-------1 o 0.5 1.5
Fig. 2. Comparison of true temperature evolution (+) and temperature evolution from the state observer at x=0.5
Fig. 3. Comparison of temperature evolution at x=1 by output feedback control and state feedback control identified to define a desired closed-loop system behaviour. Based on the characteristic ODE, nonlinear output tracking control schemes were used to drive the system toward the sliding surface. The proposed state feedback control and output feedback control were shown, via simulation, to provide good performance. In comparison to other available methods, the proposed controller has a simple form. It is easy to implement and does not require the calculation of spatial derivatives of high degree in distributed state variable. Although the results show promise, more work is required in the area of robustness and stability analysis for the proposed approach.
exchanger. The operator Q maps the estimated temperature profile into temperature at 5 points. An operator C : ~5 --+ Z can be designed as follows :
-"'IT c( Cy - L.. 5
5
i
j"oi
(x-Xj)
)Yi
Xi -
(35)
Xj
where: parameter c is adjusted to reach the desired convergence of estimation to true state. For simulation, assume actual initial temperature profile is: T(x,O) = 1 - exp( -5x) and estimated initial temperature profile is T(x,O) = 5x. In this example, the estimated temperature profile converges to the true temperature profile for any negative value of the parameter c (see Fig.2) . The more negative the value of c, the faster convergence of estimation to the true temperature; however, large absolute values of c can cause large oscillations in the temperature estimates, which is undesirable. This phenomena has been discussed in the literature (Isidori, 1995).
7. REFERENCES Arnold, V.1. (1988). Geometric Methods in the Theory of Ordinary Differential Equations. Springer-Verlag. New York. Christofides, P.D. and P. Daoutidis (1996). Feedback control of hyperbolic pde systems. AIChE ]ournal42(11), 3063-3086. Christofides, P.D. and P. Daoutidis (1998). Robust control of hyperbolic pde systems. Chemical Engineering Science 53(1),85-105. Curtain, R.F. and H.J. Zwart (1995) . An introduction to Infinite-dimensional Linear Systems Theory. Springer-Verlag. New York. Hanczyc, E.M. and A. Palazoglu (1995). Sliding mode control of nonlinear distributed parameter chemical processes. Ind. Eng. Chem. Res. 34, 557-566. Isidori, A. (1995). Nonlinear Control Systems. Springer-Verlag. McOwen, R. (1996) . Partial Differential Equations. Prentice-Hall Inc. Sira-Ramirez, H. (1989) . Distributed sliding mode control in systems described by quasilinear partial differential equations. Systems and Control Letters 13, 117-181.
Fig.3 compares the state feedback control and output feedback control performance using c = -10. From this figure it appears that the output feedback controller performs very well despite its use of spatial limited information.
6. CONCLUSION In this work, an output feedback control method for systems described by a single first-order hyperbolic PDE was developed. The central idea of this control approach is the combination of the method of characteristics with concepts from the sliding mode control and geometric control. The method of characteristics can be employed on general first-order PDE system to derive a nonlinear ODE control system. The sliding surface was
538
FEEDBACK CONTROL OF HYPERBOLIC PDE SYSTEMS Huilan Shang * J. Fraser Forbes *,1 Martin Guay **
* Department of Chemical and Materials Engineering, University **
of Alberta, Edmonton, Alberta, Canada T6G 2G6 Department of Chemical Engineering, Queen's University, Kingston, Ontario, Canada K7L 3N6
Abstract: Many industrial processes are distributed parameter systems (DPS) that can be described by hyperbolic partial differential equations (e.g., some fixedbed reactors, sheet-forming and fibre-forming processes) . Conventionally, the control schemes for these systems have been designed by approximating the original PDEs as a finite number of ordinary differential equations (ODE) . To achieve better performance, some researchers have focused on control methods that deal explicitly with the original PDE model of the process. Unfortunately, most of this research considers only controller design in state feedback form or is restricted to specific model forms . This paper presents an approach for output feedback control of distributed parameter systems described by a hyperbolic partial differential equation. A state-feedback control law is developed employing a combination of the method of characteristics, concepts of sliding surface and geometric control schemes. An output feedback controller is obtained by combining the state feedback control with a state observer. The proposed control method is demonstrated with a heat exchanger. Copyright © 2000 IFAC
Keywords: Feedback Control, Distributed Parameter Systems, Sliding Surfaces
represented by parabolic PDEs, a finite number of modes can capture the dominant dynamics of the system (Curtain and Zwart, 1995). For hyperbolic PDEs, however, all of the eigenmodes of the spatial differential operator contain nearly the same amount of energy. Consequently, an infinite number of modes are required to accurately describe their dynamic behaviour (Christofides and Daoutidis, 1998). This characteristic prohibits the application of modal decomposition techniques to derive reduced-order ODE models that approximately describe the dynamics of the hyperbolic PDE system. Thus the control problem has to be addressed based on the infinite-dimensional model itself.
1. INTRODUCTION There are many processes in which states vary in both time and space (e.g., fixed-bed reactors, fibre spinlines, and sheet coating processes) . These are commonly termed Distributed Parameter Systems (DPS) . Mathematical models for distributed parameter processes typically consist of partial differential equations (PDE). All first-order PDEs are hyperbolic. Second-order PDEs can be classified into three categories: hyperbolic, parabolic and elliptic, according to the properties of the spatial differential operator. Higher-order PDEs can be decomposed into anyone or a combination of these three basic types. Controller design has focused on methods specific to the nature of the distributed parameter process model. For processes 1
A number of control approaches have been considered in the literature. A combination of the method of characteristics and sliding mode tech-
Corresponding Author: [email protected]
533
desired behaviour, termed the sliding surface, and is locally represented by (Sira-Ramirez, 1989):
niques was proposed for processes modeled by a first-order quasi-linear hyperbolic PDE using a switching feedback control law (Sira-Ramirez, 1989). This method was further developed to synthesize a continuous state feedback controller for a nonlinear hyperbolic PDE model (Hanczyc and Palazoglu, 1995). These methods focused on design of state feedback controller. A nonlinear distributed output feedback controller was synthesized for a specific quasilinear hyperbolic PDE system using geometric control methods (Christofides and Daoutidis, 1996) .
S={(V, x , t)ER n +Z
3.1 Quasilinear Systems Consider a dynamic system described by a firstorder quasilinear PDE:
ov ot
e
The vector = [1, al , · · · ,an ,b]T = [1,a,b]T defines a time-varying control-parametrized vector field. This vector field is called the characteristic vector field. By construction, the graph of any solution to (5) must be tangent to everywhere. Surfaces which are tangent at each point to a vector field are called integral surfaces of the vector field (McOwen, 1996).
(1)
e
By the method of characteristics, the quasilinear PDE system in Equation (5) can be transformed into a nonlinear ODE control system:
t=1
x = a(v,x,t ,u) v = b(v ,x ,t ,u)
Under fixed initial and boundary conditions, Equation (1) can be visualized as a surface in Rn+Z with (v, x , t) being coordinates and u being a parameter. The control objective is to have the output function track a specific trajectory by designing a controller that computes the proper value for manipulated variable u . Without loss of generality, assume the desired output trajectory is:
(6)
Y = h(v,x,t)
To simplify the notation, introduce the vector z = [t, x , vf. For the models investigated to date, the vector field can be split into f + g . u and the system of Equation (6) can be represented in an affine form:
e
z=f+g·u y=h(v ,x ,t)
(2)
(7)
The Lie derivative of the output function along the characteristic vector field g does not vanish for most of distributed parameter systems. Then the output function y = h( v, x , t) has relative degree of 1 globally. Its time derivative along the characteristic vector field is:
Then, this condition locally defines an isolated smooth manifold solution such that:
h(if>(x,t),x ,t) =0
(5)
where al, az , ... , an and b are continuous functions . Further, assume the condition Yr = 0 defines a smooth sliding surface, which allows the use of the Implicit Function Theorem.
where: t is time coordinate, x is the vector of local spatial coordinates Xi , defining points on an open set in Rn, V is the distributed state variable which changes in both time and space, u is the manipulated variable and y is scalarvalued output. For each smooth solution v of the system in Equation (1), and h are locally smooth functions of their arguments.
=0
~ ov
+ ti' OXi ai (v ,x , t ,u) = b(v ,x ,t,u)
y = h(v ,x , t)
Any first-order PDE-based process model can be expressed as:
Yr
(4)
3. STATE FEEDBACK CONTROL
2. CONTROL OBJECTIVE
=0
v=if>(x,t)}
In the following section, a controller for the system described by a hyperbolic PDE is formulated such that the surface determined by Equation (1) converges to the sliding surface described by Equation (4) asymptotically.
In this paper, an output feedback control is derived for the processes represented by a first-order hyperbolic PDE. The resulting control law guarantees output tracking without offset for processes modeled by linear, quasilinear and nonlinear hyperbolic PDEs. The proposed approach yields a comparatively simple controller design technique and produces control laws that are easy to implement.
ov OV (v ,x,t , ox ' ot ,u) Y = h(v ,x , t)
:
(3)
The graph of v is assumed to be a smooth timevarying surface on which the system has the
iJ = Lrh + Lgh . u
534
(8)
The following theorem states the formulation of distributed state feedback control law which enforces output tracking.
nonlinear PDE control system can be described in the form of a nonlinear affine control system:
Theorem 1. Assume that the system modeled by the quasi-linear first-order PDE in Equation (5) has a minimum phase characteristic as given in Equation (6). The distributed state feedback control law, which guarantees asymptotical output tracking, is: 1
~r~[~
== Y -
(16) (9)
The formulation of a distributed state controller is stated in the following theorem.
Yr'
Theorem 2. Assume that the nonlinear first-order PDE in Equation(lO) has a minimum phase characteristic as given in Equation (12) . The state feedback control law, which guarantees asymptotical output tracking, is:
3.2 Nonlinear Systems For a first-order nonlinear PDE control system: ov ov ot +
U
(10)
Equation (10) can be interpreted as a hypersurface E2n+2 in the manifold M 2n +3 = Jl (vn+l, JR) of I-jets equipped with the standard contact structure. Let (x, t) be local coordinates on V n + 1 .. ov ov and v be the coordinate m JR, p = ox ' q = ot' The corresponding local coordinates are denoted by (x, t, v, p, q) in the space of I-jets. Then the differential equation can be written in the form: q +
«(t,x,v , p,u)
L'2 h
(17)
Considering that the number of measurements is finite, the quasilinear PDE system can be expressed as: ov
ov
Ft + oxa(v,x,t,u) = b(v,x,t,u)
(12)
y=h(v , x , t) w=Qv v(xo) = Vo
(18)
where: w is the set of measurements containing m elements, Q is an operator L 2 (O,I) ---> JRm, v(xo) = Vo is the boundary condition.
(13)
By splitting the term b(v ,x , t ,u) , the PDE in Equation (18) can be expressed as:
Substituting Equation (11) into Equation (13) yields:
z=
edr)-L h
___- - - -,\-
4.1 Quasilinear Systems
Denote vector z = [t, x, v, p), and vector field (' = [I,
z = ('(t,x , v,p ,q,u)
----~
(11)
-q
t
0
In the previous section, state feedback control that can enforce output tracking was formulated. However, the state of distributed parameter system is of infinite dimension while the measurements are always finite-dimensional. Then, a state observer is necessary in order to implement the designed state feedback control law.
1
x=
=
1
r[
-k(e+-~
4. OUTPUT FEEDBACK CONTROL
By the method of characteristics, the characteristic equations of system (10) in the manifold M2n+3 can be written as (Arnold, 1988):
i= q=
(15)
Since L(2h does not usually vanish, the output function y = h( v, x, t) has relative degree of 1 globally and its time derivative along the characteristic vector field is:
10t
-k(e + -- edr) - Lfh u= ______ _________ Lgh where e
+ (2(t,x,v,p) ' u
Z = (l(t,x,v,p) y=h(v,x,t)
ov ov ot =-oxa(v , x,t ,u)+b1(v,x,t,u)
(14)
(19)
+b2 (x, t , u) Choose Z = L2(O , I) as the state space. If an operator A on Z can be defined such that:
For the models investigated to date, the vector field ( can be split into (1 + (2 ' u and the original
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dt.p At.p = --a(t.p, x , t, u) dx t.p E L2(0, 1)/ D(A)=
+ b1(t.p, x, t , u)
where: w is the set of measurements of m elements, Q is an operator L2(0, 1) ---+ ]Rm, v(Xo) = Vo is the boundary condition.
(20)
t.p~~ are absolutely
continuous, -
By splitting the term
d~a(t.p, x,t, u)
+b1(t.p,x, t , u) E L 2(0 , 1) , t.p(Xo) = vo Equation (19) can be described through an abstract formulation on a Hilbert space Z :
dv dt = Av + b2 (x , t , u)
Choose Z = L 2 (0 , 1) as the state space. If an operator A on Z can be defined such that
(21)
Theorem 3. Consider a state observer for a firstorder quasilinear PDE system in Equation (18) of the form : d"1 dt
dt.p At.p =
= A"1 + b2(x , t , u) + C(w-Q"1)
(22 )
D(A)
dt.p L 2(0 , 1) /
is absolute1cpcontinuous,
where: "1 is an estimate of the state v . If an operator C : lR m -+ Z can be designed such that the composite operator (A - CQ) is an infinitesimal generator of a contraction semigroup, the state observer in Equation (22) guarantees that the estimated state converges to the true state asymptotically.
E
L2(0,1)
then the state observer for nonlinear PDE in Equation (25) can be constructed in the same way as quasi linear PDE.
Theorem 4. Consider a state observer for a nonlinear PDE system (24) of the form :
If the operator A defined by Equation (20) is an infinitesimal generator of a contraction semigroup, the simplest observer can be obtained by choosing C = O. The addit ion of the term C(w - Q"1) in the state observer accelerates the convergence of the estimation to the true state value.
~~
= A"1 +
+ C(w-Q"1)
(27)
If an operator C : lR m -+ Z can be designed such that the composite operator (A - CQ) is an infinitesimal generator of a contraction semigroup, the state observer (27) guarantees that the estimated state converges to the true state asymptotically.
By combining the developed state feedback controller in Equation (9) and state observer in Equation (22) , an output feedback controller for quasilinear system in Equation (18) is obtained:
0"1 0"1 at = - oxa("1 ,x,t,u) +b("1, x , t, u) + C(w - Q"1) k · e("1, x ,t) u( x) t - - -=--:--:-"'--~ , Lgh("1,x ,t)
=
E
(26)
By combining the developed state feedback controller (17) and state observer (27) , an output feedback controller for nonlinear system in Equation (24) is obtained:
(23)
-TJk 10 e("1, x , T)dT + Lfh("1 , x , t) t
0"1 0"1 at = -
Lgh( "1, x , t) where "1 is the estimated state.
k
L h("1, x ,t)
'2
where: "1 is the estimation of the state v .
The nonlinear PDE system can be expressed as:
ov
at +
y = h(v , x , t) w=Qv v(xo) = Vo
(28)
2
-TJ 10 e("1,X, T)dT + L([ h("1 ,X, t)
4.2 Nonlinear Systems
ov
t
In the controller (23) and (28), the first two terms are proportional and integral control, and the third term is a kind of feedforward control. So the developed controller has a simple form and is easy to implement.
=0 (24)
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5. HEAT EXCHANGER Consider a forced-flow steam-jacketed tubular heat exchanger. The fluid within the tube is heated using steam without condensation. The dynamic model of the process in deviation form can be expressed as:
8T 8T -+u-+H(T-T)=O 8t 8x J
3.5
IV-\;'
2.5
... 2
(29)
V'
1.5
where T(x, t) denotes the temperature of the heat exchanger within the tube, x E [0,1]' T j denotes the jacket temperature, u is the fluid velocity through the tube and is the manipulated variable, and H is a positive constant. For simplicity, the steam temperature is considered to be spatially uniform. Fluid temperature T at the tube exit is the variable to be controlled. Then, the model given in Equation (29) is a quasilinear PDE.
o
Fig. 1. Setpoint response of temperature at x=0.2, 0.5, 0.8 and 1 with fluid velocity as the manipulated input t
= K[(T(I, t) -
where K = k/
(30)
where Tsp (x) denotes the set point temperature profile. The Lie derivative of output function y = T - Tsp(x) along the characteristic vector field ~ is:
8Tsp TJ ) - 8xu
k 8T. {x) p
[(T(x, t) - Tsp(x))
1
+ T[
(32)
8x
t
j(T(X, T) - Tsp(x))dT o
Tsp)
+ ~ j(T(I, T) (33) T[
~ (T(x, t) -1j) ]
where: 8Tir}x) is the spatial derivative of the setpoint temperature profile, which is specified. So this control law can be viewed as a simple PI control plus a kind of feedforward control. This is a spatially distributed controller; however, the manipulated variable u is spatially uniform throughout the heat exchanger when the steam is assumed to be well-mixed. The control objective is to have the exit temperature of the exchanger to reach its setpoint. So a modification can be made by replacing the spatially distributed variables by their values at x = 1. The resulting spatially uniform controller can be written as:
8
)]
J;Plx=1 .
A simulation was performed to evaluate the performance of the state feedback controller of Equation (33). The parameters used for simulation were: H = I, T j = 10, k = 5, T[ = 2. The initial temperature profile was specified as T(x, 0) = 1e- 5x and the boundary condition as T(O, t) = O. The specification of the exit temperature and its spatial derivative was taken as Tsp = 3 and 8T. p 8x -
(31)
Applying the control law given in Equation (9) yields an expression for u:
u(x, t) =
10
6
o H -Tsp)dT - "k(T(I , t) - Tj
The characteristic vector field for Equation (29) is ~ = [1, u, - H (T - T j )] and the control variable in which we are interested is the temperature profile. Then the sliding surface can be defined as:
= - H(T -
,.tl2
V 0.5
5.1 State Feedback Control
L
.tl,
1
U
T-Tsp(x) =0
lA
3.
For simulation purposes, the method of finite differences was used to derive a finite-dimensional approximation of the original PDE equation, with a choice of 10 discretization points. The results of simulation are shown in Fig.1. It was observed that the exit temperature, i.e., T at x = I, tracks the set point well and the temperature at other points reaches its steady state after some time. It can be concluded that the controller (33) has good setpoint tracking behaviour and the closedloop response is stable.
5.2 Output Feedback Control Assume temperature in the heat exchanger can be measured at 5 points x = 0.2,0.4,0.6,0.8, 1.0. A state observer for system (29) can be designed as:
8T at. -8t = - u8x- - H(T -
.
T) - C(J: - QT)(34) J
m
where Tm is the temperature measurements of R5,
T is the estimated temperature profile in the heat
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2.6r----....----~----..----.---___,
o
+
0.80~---:0.... .0:-5--~0.-,-1---:-0~ . 15:----0.....2----,J0.25
0.5L----'-----'-----'-------1 o 0.5 1.5
Fig. 2. Comparison of true temperature evolution (+) and temperature evolution from the state observer at x=0.5
Fig. 3. Comparison of temperature evolution at x=1 by output feedback control and state feedback control identified to define a desired closed-loop system behaviour. Based on the characteristic ODE, nonlinear output tracking control schemes were used to drive the system toward the sliding surface. The proposed state feedback control and output feedback control were shown, via simulation, to provide good performance. In comparison to other available methods, the proposed controller has a simple form. It is easy to implement and does not require the calculation of spatial derivatives of high degree in distributed state variable. Although the results show promise, more work is required in the area of robustness and stability analysis for the proposed approach.
exchanger. The operator Q maps the estimated temperature profile into temperature at 5 points. An operator C : ~5 --+ Z can be designed as follows :
-"'IT c( Cy - L.. 5
5
i
j"oi
(x-Xj)
)Yi
Xi -
(35)
Xj
where: parameter c is adjusted to reach the desired convergence of estimation to true state. For simulation, assume actual initial temperature profile is: T(x,O) = 1 - exp( -5x) and estimated initial temperature profile is T(x,O) = 5x. In this example, the estimated temperature profile converges to the true temperature profile for any negative value of the parameter c (see Fig.2) . The more negative the value of c, the faster convergence of estimation to the true temperature; however, large absolute values of c can cause large oscillations in the temperature estimates, which is undesirable. This phenomena has been discussed in the literature (Isidori, 1995).
7. REFERENCES Arnold, V.1. (1988). Geometric Methods in the Theory of Ordinary Differential Equations. Springer-Verlag. New York. Christofides, P.D. and P. Daoutidis (1996). Feedback control of hyperbolic pde systems. AIChE ]ournal42(11), 3063-3086. Christofides, P.D. and P. Daoutidis (1998). Robust control of hyperbolic pde systems. Chemical Engineering Science 53(1),85-105. Curtain, R.F. and H.J. Zwart (1995) . An introduction to Infinite-dimensional Linear Systems Theory. Springer-Verlag. New York. Hanczyc, E.M. and A. Palazoglu (1995). Sliding mode control of nonlinear distributed parameter chemical processes. Ind. Eng. Chem. Res. 34, 557-566. Isidori, A. (1995). Nonlinear Control Systems. Springer-Verlag. McOwen, R. (1996) . Partial Differential Equations. Prentice-Hall Inc. Sira-Ramirez, H. (1989) . Distributed sliding mode control in systems described by quasilinear partial differential equations. Systems and Control Letters 13, 117-181.
Fig.3 compares the state feedback control and output feedback control performance using c = -10. From this figure it appears that the output feedback controller performs very well despite its use of spatial limited information.
6. CONCLUSION In this work, an output feedback control method for systems described by a single first-order hyperbolic PDE was developed. The central idea of this control approach is the combination of the method of characteristics with concepts from the sliding mode control and geometric control. The method of characteristics can be employed on general first-order PDE system to derive a nonlinear ODE control system. The sliding surface was
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