An Internal-Model Controller for a Class of Single-Input Single-Output Nonlinear Systems: Stability and Robustness

Dynamics and Control, 8, 123–144 (1998)c© 1998 Kluwer Academic Publishers. Manufactured in The Netherlands.

An Internal-Model Controller for a Class ofSingle-Input Single-Output Nonlinear Systems:Stability and Robustness*

JOAQUıN ALVAREZ [email protected]

SALVADOR ZAZUETA

Scientific Research and Advanced Studies Center of Ensenada (CICESE)Km. 107 Carretera Tijuana-Ensenada, 22860 Ensenada, BC, Mexico

Received October 31, 1995; Revised August 14, 1997

Editor: M.J. Corless

Abstract. A controller design procedure for a class of nonlinear systems is presented. The structure of thecontrol system corresponds to the so-called internal-model controller that, for linear systems, has exhibited goodperformance and stability robustness with respect to disturbances and to uncertainty in the plant parameters.The systems involved are single-input single-output and fully linearizable by coordinates transformation andstate feedback. It is shown that the plant output converges to a constant reference, even under the presence ofconstant disturbances and parameter uncertainties, provided the closed-loop system has an asymptotically stableequilibrium point placed anywhere. This scheme does not need an explicit design of a nonlinear observer; instead,it uses the state of a plant model. A conservative stability robustness margin is estimated by applying standardresults of Lyapunov theory.

Keywords: nonlinear systems, feedback linearization, internal-model control, Lyapunov theory, stability robust-ness

1. Introduction

The dynamics of an important class of systems can be described by the ordinary differentialequation

dnyP

dtn+ F

(yP ,

dyP

dt, . . . ,

dn−1yP

dtn−1

)= G

(yP ,

dyP

dt, . . . ,

dn−1yP

dtn−1

)u, (1)

whereF andG are smooth functions, andu is a control input. An adequate selection ofstate variables converts this system to the form

xPi = xPi+1, (i = 1, . . . , n− 1) ,

xPn = fP(xP)

+ gP(xP)u, (2)

yP = xP1 ,

wherexP =(xP1 , . . . , x

Pn

)is the state,xPi

def= di−1yP /dti−1 for i = 1, . . . , n; fP andgP

are functions given byfP (·) = −F(xP)

andgP (·) = G(xP). Suppose thatfP (0) = 0

* This work was partially supported by the CONACYT (M´exico).

124 ALVAREZ AND ZAZUETA

, gP(xP)6= 0 for xP ∈ XP , an open set in<n containing the origin, and thatyP is to be

regulated around a constanty∗. The control law

u(xP, y∗

)= −

fP(xP)

+ a ·(xP − y∗

)gP (xP)

, (3)

wherey∗ = (y∗, 0, . . . , 0) ∈ XP ,a = (a1, . . . , an) is a real vector such thatλn+anλn−1+· · ·+a1 is a strictly Hurwitz polynomial, yields a closed-loop, asymptotically stable, linearsystem that has the desired steady state outputy∗.

This linearization approach has been investigated since several years ago, giving importantresults in nonlinear control theory [12]. The application of this technique is simple andstraightforward; however, it needs an exact model and the availability of the full state.Using an approximate model do not produce a linear closed-loop system, and a stabilityanalysis of the controlled system becomes indispensable. This analysis is usually based onthe Lyapunov theory [14], which in turn leads to the redesign of the control law to improvethe stability robustness of the control system. Several results in this direction have beengiven in [3], [13], [15], [19].

In a practical implementation the full state is usually not available, so an observer isrequired. Unfortunately, the design of a nonlinear observer is not a trivial problem, and itneeds an accurate plant model to have a reasonable error [7]. Suppose that an approximatemodel of the plant is known, denoted byfM andgM . The use of an observed statexO inthe control law (3) yields the closed-loop system

xPi = xPi+1, (i = 1, . . . , n− 1) ,

xPn = fP(xP)−gP(xP)

gM (xO)[fM

(xO)

+ a ·(xO − y∗

)], (4)

xO = fO(xO, u

(xO, y∗

), yP

),

yP = xP1 ,

wherefO (·) represents the observer. Excepting some trivial cases, system (4) does not havethe desired output. It is also possible that system (4) would be unstable. IfyP converges toa value other thany∗, a second loop built with integral action may eliminate the steady stateerror, although this increases the order of the overall system. Depending on the observerstructure, the stability analysis may become quite involved.

In this paper we propose another technique that does not rely on adding explicit integratorsto eliminate the steady state error. This procedure yields a control system with some degreeof robustness with respect to parameter uncertainty and to other kinds of disturbances. Thecontroller has the form calledinternal-model-control(IMC) structure; a scheme extensivelyanalyzed in [17] for linear systems. This scheme uses a process model (“internal model”)connected in parallel to the plant, feeding back the difference between the outputs of the plantand the model (figure 1). Hence, an explicit design of a nonlinear observer is not needed.In the linear setting, this scheme has displayed some interesting properties related to thestability robustness with respect to model uncertainties. It also ensures the convergence ofthe plant output to a steady state constant reference [17].

Several procedures for extending this scheme to nonlinear systems have been proposed[1], [2], [4], [5], [9], [11]. Some of these procedures require stability of the plant, a discrete

AN INTERNAL-MODEL CONTROLLER 125

model, or an estimator [11]. In [8], an IMC is designed for a continuous-stirred-tankchemical reactor, using a nominal linear plant and theH2 andH∞-optimal theory to designthe control law. A robustness analysis of the scheme is performed, using some results ofthe structured singular value theory.

In this paper we propose a controller that combines the simplicity of the linearizationcontrol technique discussed before and the robustness the IMC structure has already shownin the linear framework. The class of nonlinear systems involved are those like (1) or (2),i.e., single-input, single-output, and fully linearizable by coordinates transformation andstate feedback. However, the control law does not linearize the plant, but an extendedsystem built with the plant, its approximate model, and a filter whose role will be explainedlater (figure 1). We propose a control law, and establish the conditions in order to the controlstructure have the following properties.

• Output regulation.For a given (nominal) plant, the output converges asymptotically toa tracking signal that approaches exponentially to a constant value.

• Robustness.If the property of output regulation holds for a nominal plant, this propertyis also satisfied, with the same control law, for any other plant “close enough” to thenominal one.

We show that the first property is fulfilled for plants having parameter uncertainty, pro-vided an asymptotically stable equilibrium point of the closed-loop system exists. There-fore, after proposing an adequate control law, we analyze the existence of equilibrium pointsof the closed-loop system and discuss the equilibrium stability. Finally, we apply the Lya-punov theory for analyzing the robustness of the control scheme to plant-model mismatches,estimating a conservative stability margin. We end the paper with the application of thiscontrol scheme to three models of physical systems: a pendulum, a chemical reactor, anda binary distillation column.

2. An internal-model controller for nonlinear systems

Let us consider a system whose dynamics are represented by

z = f (z) + g (z)u, (5)

y = h (z) ,

wherez (t) ∈ Z is the state,Z is a domain in<n that contains the origin;u (t) ∈ < is thecontrol input;f andg are smooth vector fields defined onZ, andh : Z → < is a smoothfunction. We suppose that the origin is an equilibrium point of the unforced system, i.e.,f (0) = 0, such thath (0) = 0. Let us define the Lie derivative of the scalar functionh

with respect to the vector fieldf asLfhdef= (∂h/∂z) f , andLjfh

def=(∂(Lj−1

f h)/∂z

)f .

We consider the following assumption.

A1. For allz ∈ Z, system (5) has a relative degree equal ton, i.e.,

LgLjfh (z) = 0 (j = 0, 1, . . . , n− 2) , LgL

n−1f h (z) 6= 0. (6)


It is known that, when condition (6) is satisfied, the map

xP = T (z) =(h, Lfh, . . . , L

n−1f h

)(z)

is a coordinates transformation [12]. Therefore, any system satisfying (6) can be describedusing the statexP. In these coordinates, (5) is given by the normal form

xPi = xPi+1, (i = 1, . . . , n− 1) ,

xPn = fP(xP)

+ gP(xP)u, (7)

yP = xP1 ,

where

xP ∈ XP = T (Z) , fP(xP)

= Lnf h(T−1

(xP)), gP

(xP)

= LgLn−1f h

(T−1

(xP)).

Henceforth, we consider the class of nonlinear plants described by (7).Now suppose that a model of (7) is available,

xMi = xMi+1, (i = 1, . . . , n− 1) ,

xMn = fM(xM)

+ gM(xM)u, (8)

yM = xM1 ,

wherexM =(xM1 , . . . , xMn

)∈ XM ⊂ <n, XP ⊂ XM , and fM , gM are smooth

functions. Also, we suppose that (8) has a strong relative degree equal ton, i.e.,

A2. gM(xM)6= 0 for all xM ∈ XM . Moreover,fM (0) = 0.

Let ePM be the plant-model output error

ePM = yP − yM , (9)

and consider another system whose role is to filter this error, having the linear form

xFi = xFi+1, (i = 1, . . . , n− 1) ,

xFn = −aF · xF + aF1 ePM , (10)

yF = xF1 ,

wherexF =(xF1 , . . . , x

Fn

)∈ <n andaF =

(aF1 , . . . , a

Fn

)∈ <n is such thatλn+aFnλ

n−1+· · ·+ aF1 is strictly Hurwitz.

A schematic diagram of this control structure is shown in figure 1. We suppose that thesignal available from the plant is only the outputyP , and assume the following controlobjective.

A3. The plant output is to track a smooth signaly∗ that converges exponentially to aconstanty∗.


Figure 1. Schematic diagram of the internal-model controller.

We propose a control law that linearizes a global input-output relationship. Let us defineauxiliary variables

yi+1 =diyM

dti+diyF

dti− diy∗

dti, (i = 0, . . . , n− 1) ,

and a global outputyg ≡ y1. Let the control law be

u = −fM

(xM)− aF · xF + aF1 e

PM − dny∗/dtn + a · ygM (xM)

, (11)

wherey = (y1, . . . , yn) anda = (a1, . . . , an) is a real vector such thatλn+ anλn−1+

· · ·+ a1 is strictly Hurwitz. This control law linearizes the dynamics of the global outputyg, leading to

y =

0 1 · · · 0...

......

0 0 · · · 1−a1 −a2 · · · −an

y. (12)

An important fact related to the exponential convergence of the global outputy1 to zero(system (12)) is established by the next lemma, which can be proved very easily.

Lemma 1 Suppose that assumptionsA2 andA3 are satisfied and the closed-loop systemgiven by (7), (8), (10), and (11) has an equilibrium pointx =

(xP, xM, xF

)∈ <3n. Then

the plant output (yP ) corresponding to this equilibrium isyP = y∗ = limt→∞ y∗ (t).


This lemma remarks the importance of the existence of asymptotically stable equilibriumpoints of the closed-loop system. When this type of equilibria exists and the initial stateis in the basin of attraction of any of these equilibria, the plant output converges to thereference. Note that these equilibria can be placed anywhere, and a perfect plant-modelmatching is not needed.

Let us interpret the meaning of the global outputyg. The main control objective is to makethe plant outputyP converge to the referencey∗, in spite of disturbances or plant-modelmismatches. This means that the plant output erroreP = yP − y∗, which can be expressedas

eP = ePM + xM1 − y∗, (13)

must converge to zero. Now suppose that the filter used in the IMC is similar to thatproposed in [17],

F (s) =1

(aF s+ 1)n=

(aF)−n∑n

j=0

(nn−j)

(aF )−j sn−j(14)

with aF > 0. A state representation of this filter is

xFi = xFi+1, (i = 1, . . . , n− 1) , (15)

xFn = −n∑j=1

(n

n− j

)(aF)−j

xFn−j+1 +(aF)−n

ePM .

Using (15) into (13) we obtain

eP = xF1 + xM1 − y∗ +

(aF )n xFn +n−1∑j=1

(n

n− j

)(aF)n−j

xFn−j+1

. (16)

If aF is very small, (16) can be approximated byeP ≈ xF1 + xM1 − y∗ = y1. Now from(16) we can obtain derivatives of higher order. IfaF ≈ 0 thendieP /dti ≈ xFi+1 + xMi+1 −diy∗/dti = yi+1 for i = 0, 1, . . . , n − 1. Therefore, whenaF is small, the auxiliaryvariables defined before (y) can be seen as an estimation of the plant output error and its(n− 1)-time derivatives. This kind of high-gain estimator is often used elsewhere [13]. Inthe limit case (aF → 0) we see that, if the model is identical to the plant, and they startfrom the same initial condition, then the control law (11) reduces to (3).

3. Equilibrium points and stability

The existence and stability of closed-loop system’s equilibria guarantee the fulfillment ofthe regulation property. In this section these two points are analyzed. For this analysis

we define new coordinatesξ = (ξ1, ξ2, ξ3) def=(e,xF,y

), wheree = (e1, . . . , en) def=(

ePM , dePM/dt, . . . , dn−1ePM/dtn−1). The new coordinates are given by

ξ1 = e = xP − xM, ξ2 = xF, ξ3 = y = xM + xF − y∗, (17)


wherey∗ =(y∗, dy∗/dt, . . . , dn−1y∗/dtn−1

). The original coordinates are given by

xP = e− xF + y + y∗ = ξ1 − ξ2 + ξ3 + y∗,

xM = −xF + y + y∗ = −ξ2 + ξ3 + y∗, (18)

xF = ξ2.

In theξ-coordinates the overall system has the form

ei = ei+1, en = φe (ξ) , (19)

xFi = xFi+1, xFn = φxF (ξ) , (20)

yi = yi+1, yn = φy (ξ) , (21)

for i = 1, . . . , n− 1, with

φe (·) = fP(xP)− fM

(xM)

+[gP(xP)− gM

(xM)]u(xP1 ,x

M,xF),

φxF (·) = −aF · xF + aF1(xP1 − xM1

),

φy (·) = −a ·(xM + xF − y∗

),

u (·) = −fM

(xM)− aF · xF + aF1 e1 − dny∗/dtn + a · y

gM (xM), (22)

wherexP, xM, andxF are given by (18). We have the following result regarding theexistence of equilibrium points of system (19-22).

Theorem 1 Suppose that assumptionsA2 andA3 are satisfied. If the linear approxima-tion about the origin of the plant model (8), withu = 0, has no eigenvalues at zero, thenthere are open real intervalsU andV containing the origin, and a functionµ : U → Vsuch that, for each numbery∗ ∈ U , there is a unique valuee1 ∈ V given bye1 = µ (y∗)such that

ξ = (ν (y∗) , ν (y∗) ,0) , (23)

whereν (·) = (µ (y∗) , 0, . . . , 0) ∈ <n, 0 ∈ <n, is an equilibrium point of (19-22).

Proof: The closed-loop system is given by the equations (19-22), represented in compactform by

ξ = φ (ξ) , (24)

whereφ (ξ) =(e2, . . . , en, φe (ξ) , xF2 , . . . , x

Fn , φxF (ξ) , y2, . . . , yn, φy (ξ)

). Let ξ =(

e, xF, y)

be a point in<3n. ξ is an equilibrium point of (24) if, and only if,φ(ξ)

= 0.Therefore, using (10), if an equilibrium exists, it must satisfy

e = (e1, 0, . . . , 0) def= e1, xF = e1, y = 0, (25)


wheree1 is a real number given by the solution of the equation

φe

(ξ)

= fP(xP)− fM

(xM)

+[gP(xP)− gM

(xM)]u(xP1 , x

M, xF)

= 0, (26)

with xP, xP1 , xM, andxF obtained from (18) and (25) as

xP = y∗, xP1 = y∗, xM = y∗ − e1, xF = e1, (27)

wherey∗ = (y∗, 0, . . . , 0). The steady state control input is obtained, from (22) and (27),as

u (·) = − fM

gM

∣∣∣∣xM=y∗−e1

def= u (e1, y∗) . (28)

Taking into account these values in (26) we obtain

ψ (e1, y∗) def= φe

(ξ)

= fP (y∗) + gP (y∗) u (e1, y∗) = 0. (29)

Therefore, there exists an equilibrium point of the closed-loop system if, and only if, for agiveny∗ ∈ < a solutione1 ∈ < of (29) exists. Note thatψ (0, 0) = 0; furthermore, becausethe linear approximation of the unforced plant model has no eigenvalues at the origin, then

∂ψ

∂e1

∣∣∣∣(0,0)

=gP (y∗)gM (xM)

∂

∂xM1fM

(xM)∣∣∣∣

xM=0

6= 0.

Therefore, from the implicit function theorem we conclude that there are two real intervalscontaining the origin,U andV , and a functionµ : U → V such thatψ (µ (y∗) , y∗) = 0 forall y∗ ∈ U andψ (e1, y

∗) 6= 0 if (e1, y∗) ∈ V × U but e1 6= µ (y∗). This is equivalent to

have, for each valuey∗ ∈ U , a unique equilibrium point of the closed-loop system aboutthe origin, and (23) follows.

Corollary 1 (i) Suppose that the hypothesis given in theorem 1 are satisfied. IffP (x)= fM (x), gP (x) = gM (x) for all x ∈ XP , thenξ = 0 is an isolated equilibrium pointof the closed-loop system for any referencey∗ small enough. (ii) Ify∗ = 0 then the originis an isolated equilibrium point for any plant, model, and filter satisfying the requirementsof theorem 1.

When the closed-loop system has equilibrium points, the stability analysis can be carriedout using the indirect method of Lyapunov. Inξ-coordinates, the Jacobian matrix of theclosed-loop system (19-22) is given by

A =

Aee AexF Aey

AxFe AxFxF 00 0 Ayy

, (30)

where

Aee =

0 1 · · · 0...

......

0 0 · · · 1∂φe

∂e1

∂φe

∂e2· · · ∂φe

∂en

, AexF =

0 · · · 0...

...∂φe

∂xF1· · · ∂φe

∂xFn

,


Aey =

0 · · · 0...

...∂φe

∂y1· · · ∂φe

∂yn

, AxFe =

0 · · · 0...

...aF1 · · · 0

,

AxFxF =

0 1 · · · 0...

......

0 0 · · · 1−aF1 −aF2 · · · −aFn

, Ayy =

0 1 · · · 0...

......

0 0 · · · 1−a1 −a2 · · · −an

.

All the derivatives in these matrices are evaluated at the equilibrium point (23). The

characteristic polynomial of matrixA is pA (λ) = det (λI −Ayy) det(λI − A

), where

A =(

Aee AexF

AxFe AxFxF

). (31)

The matrixA hasn eigenvalues given by the roots ofλn+ anλn−1+ · · ·+ a1 that, by

design, have negative real part. The other2n eigenvalues are those of matrixA (31). Thecharacteristic polynomial of this matrix is

pA (λ) = λ2n + q2nλ2n−1 + · · ·+ q2λ+ q1, (32)

where

qj =j∑

k=1

bkdj−k+1 − cjaF1 , (j = 1, . . . , n) ; (33)

qn+i =n∑

k=i+1

bkdn+i−k+1 − bi − di, (i = 1, . . . , n− 1) ; (34)

q2n = −bn − dn; (35)

bi =∂φe

∂ei

∣∣∣∣ξ

, ci =∂φe

∂xFi

∣∣∣∣ξ

, di = −aFi , (i = 1, . . . , n) . (36)

In many cases it is possible to make polynomial (32) be strictly Hurwitz. We summarizethese results in the next theorem.

Theorem 2 Suppose the conditions established in theorem 1 are satisfied. Then theequilibrium point given by (23) is asymptotically stable if the polynomial defined by (32-36) is strictly Hurwitz.

Example: Consider a first-order system. Then (32) has the formλ2 + q2λ+ q1, with

q1 = −aF1∂φe

∂e1− aF1

∂φe

∂xF1, q2 = −∂φe

∂e1+ aF1 . (37)


Therefore, this polynomial will be Hurwitz ifq1 andq2 are both positive, so the equilibriumpoint (e, e, 0) is asymptotically stable if

∂fP

∂xP1+∂gP

∂xP1u− gP

gMaF1 < 0,

gP

gM

(∂fM

∂xM1+∂gM

∂xM1u

)< 0, (38)

where all the expressions are evaluated at(xP1 , x

M1 , xF1

)= (y∗, y∗ − e, e). Note that it

is possible, with an appropriate selection of the filter, to tolerate a certain plant-modelmismatch.

Example: Now consider a second-order plant. In this case, (32) is given byλ4 +q4λ

3 + q3λ2 + q2λ+ q1, with q1 = −aF1 ∂φe/∂e1 − aF1 ∂φe/∂x

F1 , q2 = −aF2 ∂φe/∂e1 −

aF1 ∂φe/∂e2−aF1 ∂φe/∂xF2 , q3 = −∂φe/∂e1 +aF1 −aF2 ∂φe/∂e2, andq4 = −∂φe/∂e2 +

aF2 . All the derivatives are evaluated at the equilibrium pointξ (23). By applying theRout-Hurwitz criterion it can be shown that this equilibrium is asymptotically stable ifq1,q4, c ≡ q3q4 − q2, andq2c− q1q

24 are positive.

Remark WhenfP (·) = fM (·), gP (·) = gM (·), the matrixAexF is zero. This meansthat equation (32) has the formpA (·) = det (λI −Aee) det (λI −AxFxF), whereAee isevaluated atxP = y∗ and u = −fP (y∗) /gP (y∗). Therefore, when the control inputis constant, ifAee is strictly Hurwitz then the closed-loop system will have the point(y∗, y∗,0) as an asymptotically stable equilibrium. In particular, ify∗ = 0 and the origin isa stable equilibrium point of the unforced plant, then the closed-loop system will also havethe origin as a stable equilibrium, for any stable filterF . This is exactly the same propertyfound in the design of internal-model controllers for linear systems [17].

The results obtained in this section give the conditions to have a local asymptoticallystable equilibrium point. In some cases these results give complete information about thestability robustness of the system. However, in general they do not give an estimation of therobustness degree. In the next section we present a Lyapunov stability approach to estimatethis parameter [14]. Although this procedure yields a very conservative estimation, it maybe helpful to get a first idea about this important property.

4. Robustness

Suppose that the proposed control law, designed with a given nominal model, yields aclosed-loop system with an exponentially stable equilibrium point at the origin of a suitablecoordinates system. The robust nature of hyperbolic equilibria makes the closed-loopsystem exhibit stability near or at the origin when small perturbations exist [18]. In thissection we use the Lyapunov theory to estimate perturbation bounds that preserve stability.For that, we suppose that the closed-loop system (19-22) has an asymptotically stableequilibrium pointξ given by (23).

Let us define the coordinates∆ξ =(∆e,∆xF,∆y

)=(e− e,xF − e,y

), wheree =

(µ (y∗) , 0, . . . , 0) andµ is established by theorem 1, so the equilibrium point of the nominalsystem, in the new coordinates, is the origin of<3n. Now suppose that there exists someuncertainty in the parameter values of the plant, andfP

(xP; θ

), gP

(xP; θ

)are given as

functions of a parameterθ ∈K ⊂<k, whereK is a compact set containing the origin. The


functionsfP(xP; 0

)andgP

(xP; 0

)correspond to the nominal system. Suppose that, for

each parameter valueθ ∈ K, there exists a pointx ∈ <n such thatfP (x; θ) = 0 , andgP(xP; θ

)6= 0 for all xP ∈ XP . In the∆ξ-coordinates, the uncertain closed-loop system

has the form

∆ξ = A∆ξ + g (∆ξ; θ) , (39)

whereA is given by (30), and the perturbation term is

g (∆ξ; θ) = (0, . . . , ge(∆ξ; θ), 0, . . . , 0), ge (∆ξ; θ) = φe

(ξ + ∆ξ; θ

)− k ∆ξ,

where thege-term occupies then-th place andk = (∂φe/∂ξ)(ξ;0). The following result

gives sufficient conditions ong (∆ξ; θ) that preserves stability.

Theorem 3 Suppose that the system (19-22) has an asymptotically stable equilibriumpoint ξ given by (23), and consider a regionDr =

{∆ξ ∈ <3n| ‖∆ξ‖2 < r

}. Suppose

also that the perturbationg (∆ξ; θ) existing in (39) satisfies

‖g (∆ξ; θ)‖2 = |ge (∆ξ; θ)| ≤ γ ‖∆ξ‖2 + ρ, (40)

for some positive numbersγ, ρ, for all θ ∈K, and all∆ξ ∈ Dr. Therefore, for any initialcondition∆ξ (0) ∈ Dr the solution of the perturbed system (39) converges to a domainDr ⊂ Dr if

γ < γ =c

2 ‖P‖2, ρ < ρ = (γ − γ) r (41)

for some positive constantc < 1, whereP is the symmetric, definite positive matrix thatsolves the Lyapunov equationATP + PA = −I, and

r =c− 2 ‖P‖2 γ1− 2 ‖P‖2 γ

r.

Moreover, ifρ = 0 thenr = 0, and the origin of (39) will be asymptotically stable for anyperturbation verifying (40) ifγ satisfies (41) withc = 1.

Proof: This is a particular version of some results presented elsewhere [14], [16], so weonly present a sketch of the proof. MatrixA of system (39) is, by design, strictly Hurwitz.Therefore, there exists a (3n × 3n)-positive definite matrixP that solves the equationATP+ PA = −I. Now consider the positive definite functionV (∆ξ) = ∆ξTP∆ξ. Thederivative ofV along the trajectories of the perturbed system (39) satisfies

V (∆ξ) = −‖∆ξ‖22 + 2 ∆ξTPg (∆ξ; θ) ≤ −‖∆ξ‖22 + 2 ‖P‖2 ‖∆ξ‖2 ‖g (∆ξ; θ)‖2 .

From (40), and for some positive numberc < 1, we obtain

V (∆ξ) ≤ −‖∆ξ‖22 + 2 ‖P‖2 (γ ‖∆ξ‖2 + ρ) ‖∆ξ‖2

< − (1− c) ‖∆ξ‖22 + 2 ‖P‖2(

c

2 ‖P‖2− γ)

(r − ‖∆ξ‖2) ‖∆ξ‖2def= v (‖∆ξ‖2) .


The smooth, quadratic functionv satisfiesv (0) = v (r) = 0, it is positive in the inter-val (0, r), and negative for‖∆ξ‖2 > r, in particular in the interval(r, r], with a pos-

itive maximum atr/2. BecauseDr ⊂ Dr and V < v then V < 0 in Dr − Drdef=

{x ∈ Dr &x /∈ Dr}. Therefore, if∆ξ (0) ∈ Dr − Dr then∆ξ (t) approachesDr, andit is not difficult to show thatDr is invariant [14]. If ρ = 0 then the derivative ofValong the trajectories of (39) satisfiesV (∆ξ) ≤ − (1− 2 ‖P‖2 γ) ‖∆ξ‖22. Therefore, ifγ < 1/ (2 ‖P‖2) thenV < 0.

5. Examples

In this section we show the application of the IMC to three simulated plants: a pendulum,a chemical reactor, and a distillation column.

Example: [A simple pendulum] This system is represented by the equations

x1 = x2,

x2 = −a sinx1 − bx2 + cu,

wherex1 andx2 are the angular position and velocity, respectively;u is the torque in thejoint, anda 6= 0, b, andc 6= 0 are constant parameters. For a constant torqueu the systemhas the equilibrium pointx = (x1, x2) = (arcsin (cu/a) , 0), that exists if|cu/a| ≤ 1and it is stable ifb anda cos

(sin−1 (cu/a)

)are positive. For a zero torque the Jacobian

matrix at the origin does not have eigenvalues at zero; therefore, the origin is always anisolated equilibrium point. We use a model with the same form, a second-order filter, andthe control law

u =−a · y + aM sinxM1 + bMxM2 + aF·xF − aF1 e1 + d2y∗/dt2

cM,

wherea, aF ∈ <2, y =(xM1 + xF1 − y∗, xM2 + xF2 − dy∗/dt

). In theξ-coordinates the

closed-loop system has the form given by (19-21) withn = 2,

φe = −aP sinxP1 − bPxP2 + a · y − aF · xF + aF1 e1 − d2y∗/dt2 + cPu,

φxF = −aF · xF + aF1 e1,

φy = −a · y.The equilibriaµ (y∗) of the closed-loop system is given by (23), where

e1 = µ (y∗) = y∗ − arcsin(aP cM

aMcPsin y∗

), (42)

which is defined if∣∣(aP cM/aMcP ) sin y∗

∣∣ ≤ 1.Suppose that the filter parameters areaF1 = 1/ε2 andaF2 = 2/ε, with ε > 0. Then the

coefficientsqi of (32) are

q1 =aMcP

cMε2cos (y∗ − e1) , q2 =

2aP

εcos y∗ −

bP − bMcP

cM

ε2,

q3 =2bP

ε+ aP cos y∗ +

cP

cMε2, q4 = bP +

2ε,


Figure 2. Response of the pendulum controlled with the IMC. The model parameters areaM = cM = 1,bM = 0.1. (a) Stable pendulum,M = P . (b) Unstable pendulum:bP = −0.1. (c) Perturbed, stable pendulum:aP = 0.1aM , bP = 5bM , cP = 10cM . (d) Perturbed, unstable pendulum:aP = 1.5aM , bP = −bM ,cP = 0.8cM .

and a direct application of theorem 2 leads to the following result.

Proposition 1 Suppose that the reference and the parameter values of the pendulumare such that the errore1 given by (42) is defined, and the filter is chosen in the form givenby (14). Then there exists a small enough, positive value ofaF such that the equilibriumpoint of the closed-loop system, given by (23) and (42), is asymptotically stable ifcM andcP have the same sign,aM cos (y∗ − e1) > 0, andbP < bMcP /cM .

Let us consider some numerical examples. Suppose thataP = cP = 1, bP = 0.1. Figure2a displays the response of the closed-loop system when the model is identical to the plant,showing the perfect tracking property of this structure. Now suppose thatbP is negative;this corresponds to an unstable system. In figure 2b we show the corresponding response.In this case it was necessary to keep positive the parameterbM to have stability of theclosed-loop system.

Application of theorem 3 permits to have a first idea of the robustness properties of thisexample. To simplify the analysis, consider thaty∗ = 0, aF1 = a1 = 0.25, aF2 = a2 = 1;then γ = 1/ (2 ‖P‖) ∼= 0.045. Moreover, for the plant parameters given before, andassuming the same uncertainty for all of them, i.e.,θ = θ0 + ∆θ, it is easy to show that‖g (∆ξ; ∆θ)‖2 = |ge (∆ξ; ∆θ)| ≤ γ ‖∆ξ‖2 ∼= 1.9∆θ ‖∆ξ‖2. Therefore, a conservativeestimation gives an uncertainty margin of approximately0.025. This means that stabilityis assured in the parameter rangeθ = θ0 ± 0.025 .


Figure 3. Response of a perturbed pendulum controlled with an internal-model controller. The model parametersareaM = cM = 1, bM = 0.1. (a) Stable pendulum:aP = 0.1aM , bP = 5bM , cP = 10cM . (b) Unstablependulum:aP = 1.5aM , bP = −bM , cP = 0.8cM .

A less conservative estimation of the uncertainty margin can be obtained from the lastproposition. In fact, the most important constraint is given by (42). This constraint canbe tested for the parameter values in the range of uncertainty, and the model parameterscan be adjusted accordingly to guarantee the existence of an equilibrium point. The otherimportant constraint is that the coefficient of the control input (cP ) must conserve its sign.In figure 3 we show the response of the controlled pendulum when there are plant-modelmismatches, for a stable and an unstable pendulum.

Finally, in figure 4 we compare the response of the IMC with a linear (pole-placement)controller using a Luenberger-type observer for the velocity and an additional integralterm for the position error, designed from a linear approximation of the pendulum. Theeigenvalues of the controller were placed at−10, this corresponds to the filter poles in theIMC.

Example: [A chemical reactor] The plant is a continuous-stirred-tank reactor where afirst-order, irreversible, exothermic kinetics takes place. We use the dimensionless modelproposed in [8] that consists on the equations

x1 = −x1 +D (1− x1) exp(

x2

1 + x2/γ

), (43)

x2 = − (1 + β)x2 +BD (1− x1) exp(

x2

1 + x2/γ

)+ βu,

wherex1 is the concentration,x2 the temperature,D the Damkohler number,γ the activationenergy,β the cooling rate,B the heat of reaction, andu the control input, related to the


Figure 4. Response of a linear controller (pole-placement plus integral action), compared with an IMC applied tothe pendulum. (a) Linear controller. (b) Linear and IMC with a perturbation of+100% in aP and+400% in bP .

coolant temperature. The control objective is to regulate the concentration. For a constantinput u the system has the equilibrium point(s)x = (x1, x2), wherex1 andx2 satisfy

x1 =[1 +D−1 exp

(− x2

1 + x2/γ

)]−1

, x2 =Bx1 + βu

1 + β. (44)

If all the coefficients and the nominal inputu are positive, then the concentration has valuesonly in the interval (1/ (1 + 1/D) , 1). Therefore, the region of interest is{

(x1, x2) ∈ <2|x1 ∈ (0, 1) , x2 ∈ <+}

. The system has a relative degree equal to2. Itsnormal form is given by (7), withn = 2 and

xP1 = x1 − x1, xP2 = f1 (x) def= −x1 +DP (1− x1) exp(

x2

1 + x2/γP

),

fP(xP)

= f1 (x)∂f1 (x)∂x1

+ f2 (x)∂f1 (x)∂x2

, gP(xP)

= βP f1 (x)∂f1 (x)∂x2

,

f2 (x) = −(1 + βP

)x2 +BPDP (1− x1) exp

(x2

1 + x2/γP

),

wheref1, f2 and their derivatives are evaluated atx1 = xP1 + x1 and

x2 =

(lnxP1 + xP2 + x1

DP(1− xP1 − x1

))−1

−(γP)−1

−1

.


Figure 5. Response of the CSTR controlled with an IMC and compared with a linear controller (pole-placementplus PI action). The model parameters areDM = 0.072,BM = 1, βM = 0.3, γM = 20. (a) IMC,P = M .(b) IMC, perturbed plant:DP = 1.5DM ,BP = 0.5BM , βP = 1.5βM , γP = 0.5γM . (c) Linear controller.(d) Linear and IMC, perturbation of+50% in DP andβP , and−50% in BP andγP .

Application of theorem 1 to (43) permits to conclude that the closed-loop system, for asmall referencey∗, will have an isolated equilibrium if

D

[B (1− x1)

(1 + β) (1 + x2/γ)2 − 1

]exp

(x2

1 + x2/γ

)6= 1.

Therefore, if it is possible to stabilize the system around the point (44), then the controllerwill be robust if this last inequality is satisfied.

If we use a model with the same form and a second-order filter, then the control law is

u =−a · y − L2

fMhM + aF · xF − aF1 e1 + d2y∗/dt2

LgMLfMhM,

wherea, aF ∈ <2, y =(xM1 + xF1 − y∗, xM2 + xF2 − dy∗/dt

), andfM, gM, hM have the

same structure thanfP, gP, hP , given by the original plant (5).Let us consider the parameter values proposed in [8]:DP = 0.072,BP = 1, βP = 0.3,

andγP = 20, with the filter poles and the eigenvalues ofAyy placed at−0.5. The resultsobtained for the ideal case (P = M ) are displayed in figure 5a. Figure 5b shows theresponse when there exists a plant-model mismatch. In this case we have a variation of+50% in parametersDP andβP ,−50% in BP andγP .

Finally, in figure 6 we compare the results given by the IMC and a linear controller plusa second-loop with PI action on the plant output to eliminate the steady-state error. Thiscontroller was designed from a linear approximation of the reactor about the equilibrium


Figure 6. Response of a linear controller compared with an IMC applied to the CSTR. (a) Linear, pole-placementcontroller. (b) Linear and IMC with a perturbation of+100% in DP andβP , and−50% in BP andγP .

point xP = (0.3, 1.9585). The eigenvalues of the linear controller were placed at−0.5,this corresponds to the poles of the filter in the IMC. The proportional and integral gainswere set at2 and1, respectively. Figure 5c shows the response for the ideal case (P = M ),and figure 5d shows the response of the linear controller and the IMC when there exists theplant-model mismatch given before (figure 5b).

Example: [A binary distillation column] This process consists of a set of separation stages(plates) where liquid-vapor equilibrium and heat exchange caused by vapor and liquid flowsthrough the plates occur. We consider a special class of column that distillates a binarymixture (water-methanol) with one feed flow and two outflowing products corresponding tothe distillate and the residual flows (figure 7). The column has two important manipulatedinputs: the reflux and the heating power. It also has some perturbations such as the feedflow and its concentration. The output of interest is anyone of the product concentrations:the distillate (top) or the residual (bottom).

Assuming some common hypothesis a physical, nonlinear model of the distillation columncan be obtained. We consider that the dynamics of the liquid concentration in thej-th plateof a column with a liquid feed flow in thejF -th plate, havingnp plates, a boiler, and acondenser, can be described by the following equation [6], [10],

cj xj = Lj−1 (xj−1 − xj) + Vj+1 (yj+1 − xj)− Vj (yj − xj) + δFF (z − xj) , (45)

for j = 0, 1, . . . , np + 1, wherecj is the molar hold-up,xj andyj are the molar concentra-tions of the liquid and vapor phases, respectively;Lj andVj are the liquid and vapor flows,respectively;F is the feed flow with a concentrationz, δF is 1 (0) if j = jF (j 6= jF ), and


Figure 7. Schematic diagram of a distillation column and its reduced model.

L−1 = V0 = Vnp+2 = 0. An approximate calculation of the flows is given byLj = L0

if j = 0, . . . , jF − 1; Lj = L0 + F if j = jF , . . . , np; Lnp+1 = L0 + F − V , andVj = V = kQ for j = 1, . . . , np + 1, whereQ is the heating power andk a propor-tional coefficient. The liquid and vapor concentrations in thej-th plate are related byyj = αjxj/ (1 + (αj − 1)xj), whereαj is the volatility of the first component, which wewill assume to be constant.

We present here the application of the IMC to a seven-plates distillation column fed at itsfifth plate, described by equation (45). However, because this model has a large dimensionwe will use a simplified model for the IMC design. This will show that even a very crudedescription of the process can be used with this control structure.

The simplified model is obtained as follows. Suppose that the control objective is toregulate the distillate concentration (x0) using the refluxL0 as the control input. Then wepropose to consider the column made up by only two compartments: the condenser and therest of the column (figure 7). Therefore, the reduced model is given by

cM1 xM1 = VM(yM2 − xM1

),

cM2 xM2 = −VMyM2 −(FM − VM

)xM2 + FMzM +

(xM1 − xM2

)L0,

yM2 =αM2 xM2

1 +(αM2 − 1

)xM2

, (46)

yM = xM1 .

In figure 8a we show the open-loop response of the model compared with the completeprocess given by (45), for some step changes in the reflux. The nominal values of the main


Figure 8. Response of the distillation column controlled with a reduced model: (a) Open-loop response to stepchanges of the reflux; (b) Closed-loop response. Response of the column with uncertainties,FP = 0.5FM ,V P = 0.58VM , zP = 1.4zM : (c) Plant output and reference, (d) Control input (reflux).

parameters and inputs areV = 1.2 mol/sec,F = 1.815 mol/sec,z = 0.5, α1 = 3.55,α2 = 3.9, α3 = 4.25, α4 = 4.6, α5 = 4.95, α6 = 5.3, α7 = 5.65, α8 = 6, c0 = 240mol, c1 = · · · = c7 = 60 mol, c8 = 900 mol,L0 = 0.415 mol/sec. The model parameterswere found according to some procedures developed elsewhere [6]; they have the valuesαM2 = 200, cM1 = 240 mol, cM2 = 900 mol. Note that, although the responses arequalitatively similar, there exists an important steady-state error.

Using the model (46) the control law has the formu = n (x) /d (x), where

n (·) = −a · y +(VM

cM1

)2 (yM2 − xM1

)+ aF · xF − aF1

(yP − yM

)+ d2y∗/dt2

+(VMαM2cM1 cM2

) (VM − FM

)xM2 + FMz − VMyM2

1 +(αM2 − 1

)xM2

,

d (·) =(VMαM2cM1 cM2

)xM1 − xM2

1 +(αM2 − 1

)xM2

.

The application of this control law to the complete model of the distillation column isshown in figure 8b. In this case we assumed that the perturbation inputs (feed flow, feedconcentration, and vapor flow) are perfectly known. In spite of the great difference betweenthe plant and the model an almost perfect tracking is observed. Finally, in figures 8c and8d we show the response of the same controller when uncertainties in the perturbationinputs exist. A good tracking is observed, and a zero steady-state error is obtained.


Figure 9. Response of the IMC for a distillation column with uncertainties in the perturbation inputs.FP =0.5FM , V P = 0.58VM , zP = 1.4zM . (a) Plant output and reference. (b) Control input (reflux).

6. Conclusions

In this paper we have proposed a controller with the so-called internal-model control struc-ture, designed for a class of nonlinear systems, which has shown a good performance in thelinear framework. This controller has been designed for nonlinear systems that are fullylinearizable by coordinates transformation and state feedback. The proposed procedure canbe applied to minimum phase nonlinear systems in essentially the same way; we have notincluded the details of calculation for not to complicate excessively the notation.

Two important features of this scheme are that it does not need an explicit design of anonlinear observer, and that if the closed-loop system has an asymptotically stable equi-librium point then the plant output converges to the reference. This is independent of theplacement of the equilibrium, so a plant-model mismatch can be tolerated.

We have obtained sufficient conditions for the existence and stability of equilibria of theclosed-loop system in such a way that the plant output converges to the desired, steady-state constant reference. Also, we gave some conditions to have a robust closed-loopsystem with respect to parameter uncertainties, estimating a convergence region in thestate-parameter space. The application of the procedures to three nonlinear models ofphysical processes have shown a good performance of the controller, even for unstableopen-loop plants (pendulum) or for very crude approximations of the process (distillationcolumn).


Appendix

Notation

A bold character likev denotes a vector, withvi its i-th component.The superscriptsP , M , F , andO denote variables associated to the plant, the model, thefilter, and an observer, respectively.A bar over a variable, likev, denotes the steady-state ofv, or an equilibrium point.B : heat of reaction (CSTR).cj : molar hold-up of thej-th plate (distillation column).D : Damkohler number (CSTR).e = xP−xM : plant-model error.eP = yP − y∗ : plant output error.ePM = yP − yM : plant-model output error.F , z : feed flow and concentration (distillation column).Lj , Vj : liquid and vapor flows in thej-th plate (distillation column).Q : heating power (distillation column).v ·w : scalar product of vectorsv andw.x, u, y : state vector, control input, and output.yg = y1 = yM + yF − y∗ : output of the global system.y =

(yg, dyg/dt, . . . , d

n−1yg/dtn−1)

: vector formed by the global output and its deriva-tives.y∗, y∗ : reference signal for the plant outputyP and its steady-state value.y∗ =

(y∗,dy∗/dt, . . . ,dn−1y∗/dtn−1

), y∗ = (y∗,0) ∈ <n : the reference vector and

its steady-state value.αj : volatility of the first component in thej-th plate (distillation column).β : cooling rate (CSTR).γ : activation energy (CSTR).ξ =

(e,xF,y

), ξ : transformed state vector of the overall, closed-loop system, and its

steady-state value.

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