An Immersion and Invariance Algorithm for a DifferentialAlgebraic Systemg
Narayan S. Manjarekar1,∗, Ravi N. Banavar1,∗∗, Romeo Ortega2,∗∗∗1 Interdisciplinary Group in Systems and Control Engineering, IIT Bombay, Mumbai, India;2 Laboratoire des Signaux et Systèmes, CNRS-Supelec, Gif-sur-Yvette, France
A stability result based on the Immersion and Invariancecontrol strategy is derived for a class of systems describedby differential equations with algebraic constraints. Theresult is then applied to synthesize a stabilizing controllaw for a two machine electrical power system with acontrollable series capacitor (CSC). The dynamics of thetwo-machine system is given by differential equations andthe active power balance at the load buses is given byalgebraic equations. The CSC dynamics is described by afirst order differential equation.
Keywords: Immersion and Invariance, differential alge-braic systems, transient stability, electrical power systems.
1. Introduction
Electrical power systems are naturally described bynonlinear differential algebraic equations (DAEs). Thedynamics of the dynamical components in the systemsuch as the synchronous generators are given by a set ofdifferential equations, while various network constraintssuch as power balance equations at different nodes areexpressed by a set of algebraic equations. A nonlinearstrategy introduced in the recent past by [3, 2, 17, 1] istermed Immersion and Invariance (I&I). The philosophyof I&I is based on selecting a desired behaviour of the
closed loop dynamics and then ensuring that the sys-tem reaches this behaviour asymptotically. The desiredbehaviour – represented by a smaller number of variablesthan the original dynamical system – is “immersed” inthe larger (original) system configuration and this desiredbehaviour is rendered “invariant” – trajectories that enterthis smaller subsystem continue to remain in the subsys-tem. While this procedure has been shown for systemsdescribed by smooth differential equations (and affine inthe control), we extend the methodology to include alge-braic constraints as well. This class of systems is definedby the condition that, the Jacobian of the algebraic con-straints with respect to the algebraic variables is full rankin a region of interest. We demonstrate the methodologyon an application from power systems – to a system of twomachines with two load buses.
Herewe address transient stabilization of a twomachinesystem at a known operating equilibrium condition. Theexcitation control scheme has been widely used for tran-sient stabilization of power systems. Conventionally, lin-ear controllers are used with excitation control to improvethe transient performance. However, limited stability mar-gin and unpredictable load demand make the systemnonlinearities more dominant and call for better con-trol techniques. In [21], nonlinear control using turbinecontrol, and excitation control has been proposed. Theexcitation control law has been investigated to replacethe traditional Automatic Voltage Regulator (AVR) andthe Power System Stabilizer (PSS) control structure. In
Received 21 March 2010; Accepted 16 December 2010Recommended by A. Astolfi, A.J. van der Schaft
146 N.S. Manjarekar et al.
[8, 34, 18, 27, 20] feedback linearization is applied to thecontrol of single machine as well as multi-machine sys-tems, using output feedback and state observers. However,this method is fragile as it relies on nonlinearity cancel-lation, and the issue of robustness remains unanswered.This motivated the investigation of the energy-based con-trol technique for this problem. The use of energy functionfor control application has been given in [31]. The workbased ondamping injection controllers, also knownasLgVcontrollers, is found in [11, 28, 32, 33]. In [4, 5], a dynamicdamping injection controller is presented. It is shown thatthe domain of attraction becomes larger with this tech-nique. In [9, 30] a passivation technique is proposed forpower system stabilization. An observer-based controlleris given in [19]. Further, in [10] a passivity-based controllaw is proposed for the excitation control of a synchronousgenerator by shaping the total energy function via mod-ification of the energy transfer between the mechanicaland electrical components of the system. This control lawenlarges the domain of attraction, thus increasing the crit-ical clearing time. An observer-based (adaptive) control isgiven in [16]. In [26], an output feedback excitation controlof synchronous generators is proposed using a nonlinearobserver. [29] addresses the transient stabilization of amultimachine power system with nontrivial transfer con-ductances. Recently, in [14], energy shaping approach isapplied to a power system using direct mechanical damp-ing assignment. In [20] nonlinear controller design ofthyristor controlled series compensation is presented fordamping inter-area power oscillations. Transient stabiliza-tion of structure preserving power systems with excitationcontrol using an energy-shaping technique is given in [13].In [12] power systems with nonlinear loads are consideredand nonlinear decentralized disturbance attenuation exci-tation control is proposed based on theHamiltonian theory.In [15] direct mechanical damping injection is proposedto control power systems using energy-shaping approach.Recently in [7] a robust adaptive modulation controller(RAMC) for thyristor controlled series capacitor (TCSC)in interconnected power systems to damp low frequencyoscillation is proposed. The main idea of the controller isto drive the area centers of inertia (COI) to a stable equi-librium point to keep system synchronization and to damplow frequency oscillation. In [22] the IDA-PBC strategy isused for transient stabilization of a synchronous generatorusing a CSC. Further in [25] the CSC is modeled by a firstorder system and the IDA-PBC methodology is used forcontrol synthesis. The controller performance is found tobe effective in the sense that it gives an improved transientresponse for the closed-loop system. In [23], the I&I strat-egy is used to stabilize the nonlinear swing equationmodelof the single machine infinite bus (SMIB) using a CSC,where the control law is synthesized using the I&I strat-egy. In [24] the I&I methodology is extended to a class of
differential algebraic systems. This methodology has beendemonstrated on the SMIB system with a load bus. Herewe consider a two machine system described by a set ofdifferential algebraic equations, with aCSC.Wemodel theCSC as a first order system to take into account the actu-ator dynamics. The control objective is to asymptoticallystabilize the power system at a given equilibrium.
The paper is organized as follows: In Section 2 we pro-pose an I&I-based stability result for a class of dynamicalsystems with algebraic constraints. In Section 3 we con-sider a two machine system with a CSC and synthesizea stabilizing control law. We provide a few simulationresults in Section 4. Finally Section 5 concludes the paper.
2. Immersion and Invariance for aDifferential Algebraic System (IIDAS)
Consider a smooth dynamical system of the form
x = f (x, y) + g(x, y)u (1a)
h(x, y) = 0 (1b)
on a smooth manifold X of dimension n + q. Locally,f : IRn × IRq → IRn, g : IRn × IRq → IRn×m and h :IRn × IRq → IRq are smooth functions in U ⊂ X andinput u ∈ IRm, x ∈ IRn are state variables, and y ∈ IRq arealgebraic link variables. Further, the function h satisfies
Rank
⎡⎢⎣
grad h1...
grad hq
⎤⎥⎦ = q, ∀(x, y) ∈ U.
We wish to propose a constructive procedure for asymp-totically stabilizing the system to an equilibrium (x�, y�)
in a region U ⊂ X . �.The rank condition on the gradient ensures that the sys-
tem evolves on an immersed submanifold S of dimensionn. S is described by the coordinate slice
S = {(x, y) ∈ IRn× IRq | h(x, y) = 0}.
Philosophy of immersion and invariance: The objec-tive of the control philosophy based on the Immersionand Invariance technique is to asymptotically stabilize thesystem to (x�, y�) in a region U ⊂ X in the followingway.
1. Construct an immersed submanifold M of S ⊂ X ofdimension p < n and containing (x�, y�). Synthesizea feedback control law ui to make M invariant and tomake the closed-loop system restricted to M asymp-totically stable at (x�, y�). This implies all trajectories
Immersion and Invariance for Differential Algebraic Systems 147
originating in M stay in M for all time (backward andforward) and asymptotically converge to (x�, y�).
2. Make the manifold M attractive in U⋂
S . Thisimplies that trajectories of the closed-loop system (1)with a feedback control law uoff (where the subscriptoff denotes being “off” the manifold M ) originatinganywhere in U
⋂S would be attracted towards M .
Further, we make this attraction “asymptotic.”
�.
Remark 2.1: Since M is an immersed submanifold ofX sitting in S (M ⊂ S ⊂ X ), M could be describedonce again as a coordinate slice as
M = {(x, y) ∈ U : φ1(x, y) = 0, . . . ,φn−p(x, y) = 0,
h1(x, y) = 0, . . . , hq(x, y) = 0} (2)
where φ1, . . . ,φn−p are smooth functions from U to IR,with the assumption
Assumption 2.1:
Rank
⎡⎢⎣
grad φ1...
grad φn−p
⎤⎥⎦ = n − p ∀(x, y) ∈ U ⊂ X
�.An alternate characterization of the manifold M in termsof p + r parameters (ξ , η): = (ξ1, . . . , ξp, η1, . . . , ηr) ∈IRp × IRr and a smooth mapping
The rank condition on the mapping � ensures that themanifold M is an immersed submanifold of dimension p.
For objective 1, we select the dynamics of the system inM to satisfy the invariance condition aswell as asymptoticconvergence. Through the parametrization (4), given
(x, y) = (�x(ξ , η),�y(ξ , η)) ∈ M
the dynamics on the manifold M is described by
x|(x,y)∈M = f (x, y) + g(x, y)ui
= f (�x(ξ , η),�y(ξ , η))
+ g(�x(ξ , η),�y(ξ , η))ui
= ∂�x(ξ , η)
∂ξξ + ∂�x(ξ , η)
∂ηη (5)
with the equilibrium defined as
(x�, y�): = (�x(ξ�, η�),�y(ξ�, η�)).
Objective 1 involves
1. Choosing a vector field α(ξ , η) that renders the evolu-tion of ξ given by
in a small neighbourhood of the equilibrium (ξ�, η�).2. The selection of the functions �x(ξ , η),�y(ξ , η) and a
feedback control ui(�x(ξ , η),�y(ξ , η)) to satisfy
f (�x(ξ , η),�y(ξ , η))
+ g(�x(ξ , η),�y(ξ , η))ui(�x(ξ , η),�y(ξ , η))
= ∂�x(ξ , η)
∂ξα(ξ) + ∂�x(ξ , η)
∂ηη (7)
which ensures
(x(0), y(0)) ∈ M ⇒ (x(s), y(s)) ∈ M
∀ s ∈ (−∞,∞), Invariance.
For objective 2, any initial condition x0 ∈ U \ M mustmove towards M . This involves choosing a feedback con-trol uoff (x, y,φ(x, y)) that renders the dynamics of the φissuch that
limt→∞ φi(x(t), y(t)) = 0 ∀ i = 1, . . . , n − p (8)
148 N.S. Manjarekar et al.
The dynamics of φ: = (φ1, . . . ,φn−p)T could be
written as
dφ(x, y)
dt= ∂φ(x, y)
∂xx + ∂φ(x, y)
∂yy = β(φ(x, y)) (9)
where β(·) ensures the previous condition (8). However,to synthesize the control law, an explicit expression for ywould be required.
Remark 2.2: Please note that so far the assumption isonly on the rank condition on the gradient of h. In manyapplications, y could be expressed as an implicit functionof x and hence the functions φis could be modified as φisthat are functions of x alone. In this case, we have analternate description for M in terms of a smooth functionφ : IRn → IRn−p as
M : ={(x, y) ∈ IRn× IRq | φ(x) = 0
}. (10)
So we wish to make the dynamics of φ(x) along thetrajectories of (1) with input uoff such that
limt→∞ φ(x(t)) = 0. (11)
Then, (9) reduces to
∂φ(x)
∂xx = β(φ(x))
�.
Substituting for the dynamics of x, we have
∂φ(x)
∂x(f (x, y) + g(x, y)uoff (x, y, φ(x))) = β(φ(x))
In otherwords, if there exists a control uoff (x, y, φ(x)) suchthat the trajectories of the system
˙φ(x) = ∂φ(x)
∂x
[f (x, y) + g(x, y)uoff (x, y, φ(x))
](12)
x = f (x, y) + g(x, y)uoff (x, y, φ(x)) (13)
h(x, y) = 0; (14)
are bounded and satisfy (11) then, the manifold Mis attractive for (1) with input uoff (x, y, φ(x)). Further,(x�, y�) is an asymptotically stable equilibrium of (1) withinput uoff (x, y, φ(x)).
Theorem 2.3: Consider the nonlinear dynamical systemwith algebraic constraints (1) with an equilibrium (x�, y�)
to be stabilized. Let p < n and assume that we can findsmooth mapping � as defined in (3) such that the followinghold.
H1. The target dynamics given by (6) has an asymp-totically stable equilibrium (ξ�, η�) and (x�, y�) =�(ξ�, η�).
H2. The immersion condition (7) is satisfied for all(ξ , η) ∈ IRp × IRr.
H3. The set identity{(x, y) ∈ IRn× IRq | φ(x) = 0
}
={(�x(ξ , η),�y(ξ , η)) | (ξ , η) ∈ IRp × IRr ,
h(�x(ξ , η),�y(ξ , η)) = 0,
rank
(∂�x(ξ , η)
∂ξ
)= p, ∀(ξ , η) ∈ IRp
}
holds for all (ξ , η) ∈ IRp × IRr.H4. All trajectories of the system (12)–(14) are bounded
and satisfy (11)
Then, (x�, y�) is an asymptotically stable equilibrium ofthe closed-loop system (13)–(14).
Proof: We prove the result in two steps.
1. Attractivity of the equilibrium. Let (x0, y0) ∈ U bethe initial condition for (1). Then from (11) and (12)we have that all trajectories of the closed-loop sys-tem (1) with the input u = uoff (x, y, φ(x)) convergeto the manifold M . The manifold M is well-definedby (4), and (10). Further, from the target dynamics (6),the mapping (3) and the immersion condition (7), wehave that the manifold M is invariant and internallyasymptotically stable. And hence, all trajectories ofthe closed-loop system converge to x = x�. Note that,under the conditions of Remark 2.2 , we have y → y�
as x → x�. Thus, all trajectories of the closed-loopsystem finally converge to (x�, y�).
2. Lyapunov stability of the equilibrium. The system,(12) has a stable equilibrium at origin. Hence, for anyε1 > 0, there exists δ1 > 0 such that, |φ(x(0))| <
δ1 ⇒ |φ(x(t))| < ε1 for all t ≥ 0. We also havethat, y → y� as x → x�, (x�, y�) ∈ M and henceφ(x�) = 0. Let xp(t) denote a projection of x(t) on M
such that |x(t) − xp(t)| = γ (|φ(t)|) for some class-Kfunction γ (·) and such that xp(t) = �x(ξ(t)). Then,from the stability property of the target dynamics (6)we have that, for any ε2 > 0, there exists δ2 > 0 suchthat, |xp(0) − x�| < δ2 ⇒ |xp(t) − x�| < ε2 for allt ≥ 0. Selecting ε1 = ε2 = ε
Hence, for any ε > 0, there exists δ > 0, (δ dependenton δ1 and δ2) such that |x(0) − x�| < δ ⇒ |x(t) −x�| < ε for all t ≥ 0. Again note that, by continuous
Immersion and Invariance for Differential Algebraic Systems 149
Fig. 1. Immersion & Invariance for constrained systems
dependence of y on x we have that, for any εy > 0,there exists δy > 0 such that, |y(0) − y�| < δy ⇒|y(t)− y�| < εy for all t ≥ 0. Thus we have shown thatthe equilibrium (x�, y�) is Lyapunov stable.
�.
Remark 2.4: Please note that on the manifold M ,uoff = ui.�.
Remark 2.5: When the conditions of Remark 2.2 do notapply, we impose a stronger condition that[
∂h(x, y)
∂y
]q×q
is invertible in the region U. This would ensure an explicitform of the dynamics of y as
y = −[∂h(x, y)
∂y
]−1 [∂h(x, y)
∂x
](f (x, y)+g(x, y)u)
�.
The above theorem can be interpreted with the help ofFig. 1 as: Given the system (1) and the target dynamicalsystem (6), find, if possible, a submanifold M ⊂ S suchthat
1. restriction of the closed-loop system to M is the targetdynamics
2. M can be rendered invariant and attractive.
An implicit description of M is given by (2), while(4) gives a parametrized description. The control
law u = ui(�x(ξ , η),�y(ξ , η)) renders M invariant.A measure of the distance of the system trajectories toM is given by z = φ(x), called as off-the-manifold coor-dinate. Our aim is to design a control law u = uoff (x, y, z)that keeps the system trajectories bounded and drives thecoordinate z to zero.
An important class of systems to which the above resultproves very helpful in control synthesis is electrical powersystems with load buses, expressed in the structure pre-serving model (SPM) form. We now demonstrate theapplication of the IIDAS to a two machine power systemnetwork.
3. Two Machine Stabilization using a CSC
In this section we consider a two machine system with aCSC described by a set of differential algebraic equationsand synthesize a stabilizing control law based on theIIDAS strategy. The two machine system with a CSC isshown in Fig. 2. For generators G1 and G2, bus 1 and bus 3are the internal buses and, bus 2 and bus 4 are the terminalbuses, respectively. For the following description index iis used to denote the generators, for i = 1, 2. Let δi, θi andωi be the rotor angle, the terminal bus voltage angle androtor angular speed deviation, respectively, for the ith gen-erator with respect to a synchronously rotating reference.Let Di > 0, Mi > 0, Pi and PLi be the damping constant,moment of inertia constant, the mechanical power input,and the load power of the ith generator, respectively. LetXi be the transient reactance between the internal bus andthe terminal bus of the ith generator, and X12 be the reac-tance of the transmission line connecting the two terminalbuses. A CSC, denoted by a capacitive reactance of xc, is
150 N.S. Manjarekar et al.
Fig. 2. Two machine system with a CSC
connected in series with X12, and the effective reactancebetween the terminal buses is denoted by xl. We assumethat the voltages at all the buses are constant and equal to1 pu. Next, we assume that the rotor is round rotor type,and hence neglect the effect of the saliency of the rotor.
We choose the state variables for the system as x1 = δ1,x2 = δ2, x3 = ω1, x4 = ω2, and x5 = xl. Further,y1 = θ1 and y2 = θ2 denote the algebraic variables ofthe system. We denote the state vector of the system byx = (x1, x2, x3, x4, x5) ∈ S1× S1× IR3 and the vector ofalgebraic variables by y = (y1, y2) ∈ S1× S1. Then, thedynamics of the two machine system can be described bythe following set of DAEs:
x1 = x3 (15a)
x2 = x4 (15b)
x3 = 1
M1(P1 − D1x3 − b1 sin(x1 − y1)) (15c)
x4 = 1
M2(P2 − D2x4 − b2 sin(x2 − y2)) (15d)
x5 = 1
T(−x5 + x5� + u) (15e)
b1 sin(x1 − y1) − b12 sin(y1 − y2)
x5− PL1 = 0 (16a)
b2 sin(x2 − y2) + b12 sin(y1 − y2)
x5− PL2 = 0 (16b)
where b1 = 1X1
, b2 = 1X2
, b12 = 1, T is the time constantof the actuator dynamics and x5� is the effective open loopline reactance between bus 3 and bus 4, at the operatingequilibrium.
Equations (16) represent the power balance at bus 2and 3, and are the algebraic constraints of the form (1b)on the system dynamics. Thus, the system given by (15)is a dynamical system of the form (1a) and it evolves on amanifoldS ⊂ S1× S1× IR3× S1× S1 whereS is definedby the algebraic constraints (16) as
S : ={(x, y) ∈ S1× S1× IR3× S1× S1 |h(x, y) = 0
}.
(17)
Note that, the algebraic equations of the SPM model,given by (16), do not correctly describe the algebraic partfor the power system, since the impact of the reactivepower balances are not included. These constraints (reac-tive power balance) are indeed quite important and thevoltages at the load buses cannot be considered constant.However, in this attempt to use I&I for synthesizing stabi-lizing control laws, we have used this somewhat simplifiedmodel-(15) and (16). This modeling approach is similarto the one proposed by Bergen and Hill [6] where theyassume frequency dependent load and constant voltage atthe buses and thus, the reactive power constraints are notconsidered explicitly. For these control laws to be usedin practice, we need to include voltage variations in thecontrol synthesis procedure, and the future work has toconcentrate on the inclusion of such voltage variationsand reactive power balance.
3.1. Control objective
From practical considerations we assume,
Assumption 3.1: The region of operation is
D ={(x, y) ∈ S1× S1× IR3× S1× S1 | 0 < x1 − y1 <
π
2− d1,
0< x2 − y2 <π
2− d1, |y1 − y2| < π
2− d1, dl < x5 < dl
},
where d1 > 0, dl > 0 and dl > 0 are small numbers.
Remark 3.1: It is to be noted here that, rank( grad (h(x, y))) = 2 in D , and hence D
⋂S is a
5-dimensional immersed submanifold. The bounds 0 <
xi − yi < π2 − d1 ensure that the i-th machine is operat-
ing in generator mode for i = 1, 2. The bound dl < x5implies that the capacitive compensation provided by theCSC is always less than the inductive reactance of thetransmission line. That is, the net reactance between thebuses 2 and 3 is always inductive. On the other hand, thebound x5 < dl decides the limit on inductive compensa-tion of the line. Further, we note that in the region D
⋂S
Immersion and Invariance for Differential Algebraic Systems 151
we have rank (∂(h(x,y))
∂y ) = 2. This ensures that y is animplicit function of x, and the condition in Remark 2.5 issatisfied.
In general, it is difficult to determine the open loopequilibria of a nonlinear system of the form (1a)–(1b).To obtain the open loop equilibria of the system (15)–(16) we equate x = 0 on the manifold S . An openloop equilibrium of the system is of the form (x, y) =(x1, x2, 0, 0, x5�, y1, y2), where x1, x2, y1, and y2 are thesolutions to the simultaneous equations
P1 − b1 sin(x1 − y1) = 0
P2 − b2 sin(x2 − y2) = 0
b1 sin(x1 − y1) − b12 sin(y1 − y2)
x5�− PL1 = 0
b2 sin(x2 − y2) + b12 sin(y1 − y2)
x5�− PL2 = 0.
As mentioned earlier, we denote the operating stable equi-librium in D
⋂S by (x�, y�). We assume that (x�, y�) is
known to us and state the control objective as,
1. To synthesize a control law u(x, y) in order to makethe system given by (15)–(16) asymptotically stable at(x�, y�).
2. Synchronousgenerators generally exhibit poormechan-ical damping which results in sustained oscillations.The second control objective is to damp the oscillationseffectively, thus improving the transient response.
3. In this direction, the control objective is to choose asuitable lower order target dynamics (6) with a desiredenergy function on a lower dimensional submanifoldM ⊂ S which is defined using a mapping � asin (4), map the trajectories of the target dynamics tothe (original) higher order manifold using the map-ping �, and synthesize a control law to asymptoticallymatch the closed-loop system with the mapped targetdynamics.
3.2. Controller synthesis using IIDAS
In this section we synthesize an IIDAS based control lawto asymptotically stabilize the closed-loop system at thegiven equilibrium. The first step in the control synthesisis to choose an appropriate target dynamics. The openloop dynamics can be divided into two subsystems- onedescribing the two machine system, and the other describ-ing the CSC dynamics. We choose the target dynamicsdepending on the first subsystem as follows: Let ξ =[ξ1 ξ2 ξ3 ξ4]T ∈ S1× S1× IR× IR be the state vector andη ∈ S1 be an algebraic variable of the target dynamics.
We choose the target dynamics as
ξ1 = ξ3 (18a)
ξ2 = ξ4 (18b)
ξ3 = − D1
M1ξ3 − β1 sin(ξ1 − η) (18c)
ξ4 = − D2
M3ξ4 − β2 sin(ξ2 − η) (18d)
0 = β1M1 sin(ξ1 − η) + β2M2 sin(ξ2 − η) (18e)
where βi is a positive constant, ξi: = ξi − ξi� and ξi� isthe operating equilibrium of the i-th generator. We assumethat the region of operation for the target dynamics is,
DT : ={(ξ , η) ∈ S1× S1× IR× IR× S1 | ξi − η
∈(−π
2,π
2
), i = 1, 2
}. (19)
Here, note that (18e) defines a smooth constraint manifoldST ⊂ DT for the target dynamics (18). For (18) the set ofequilibria is given by
ET : = {(a + ξ1�, a + ξ2�, 0, 0, a)
∈ S1× S1× IR× IR× S1| a ∈ S1}. (20)
Let the operating equilibrium of (18) be denoted by(ξ�, η�). We next show that the target dynamics (18) isasymptotically stable at (ξ�, η�) using the energy function
H(ξ , η) = −β1 cos(ξ1 − η) − β2 cos(ξ2 − η)
+ 1
2(ξ2
3 + ξ24 ). (21)
Clearly, the energy function is minimum on ET . The timerate of change of the energy function along the trajectoriesof the target system is given by
As H(ξ , η) is negative semidefinite, we can infer that thetarget dynamics is stable in the sense of Lyapunov. Using
152 N.S. Manjarekar et al.
LaSalle’s invariance principle we can show that (ξ�, η�) isasymptotically stable. Consider the situationwhere ξ3 ≡ 0and ξ4 ≡ 0. In that case, from (18) we have ξ1 = ξ2 =ξ3 = ξ4 ≡ 0 and hence ξ1 ≡ η ≡ ξ2. Thus, by LaSalle’sinvariance principle we conclude that the trajectories ofthe target dynamics asymptotically approach ET .
where �5(ξ , η), �6(ξ , η) and �7(ξ , η) are to be chosensuch that,
b1 sin(ξ1 − �6(ξ , η))
− b12 sin(�6(ξ , η) − �7(ξ , η))
�5(ξ , η)− PL1 = 0 (23a)
b2 sin(ξ2 − �7(ξ , η))
+ b12 sin(�6(ξ , η) − �7(ξ , η))
�5(ξ , η)− PL2 = 0.
(23b)
Further, consider the manifold M ⊂ S defined as
M : = {(x, y) ∈ S | ∃(ξ , η) ∈ ST
such that (x, y) = �(ξ , η)}. (24)
We have to choose the mapping �(ξ , η) for the targetdynamics (18) such that the immersion condition (7) issatisfied on M , that is,
⎡⎢⎢⎢⎢⎣
ξ3ξ4
1M1
[P1 − D1ξ3 − b1 sin(ξ1 − �6(ξ , η))]1
M2[P2 − D2ξ4 − b2 sin(ξ2 − �7(ξ , η))]
1T
[−�5(ξ , η) + x5�
]
⎤⎥⎥⎥⎥⎦+
⎡⎢⎢⎢⎢⎣00001T
⎤⎥⎥⎥⎥⎦ ui(�(ξ , η))
=
⎡⎢⎢⎢⎢⎣
1 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 0
∂�5(ξ ,η)∂ξ1
∂�5(ξ ,η)∂ξ2
∂�5(ξ ,η)∂ξ3
∂�5(ξ ,η)∂ξ4
∂�5(ξ ,η)∂η
⎤⎥⎥⎥⎥⎦
×
⎡⎢⎢⎢⎢⎣
ξ3ξ4
− D1M1
ξ3 − β1 sin(ξ1 − η)
− D2M3
ξ4 − β2 sin(ξ2 − η)
η
⎤⎥⎥⎥⎥⎦ (25)
with the constraint (23).
The first two rows of (25) are trivially satisfied for themapping (22) and the target dynamics (18). From the thirdrow we get,
1
M1(P1 − b1 sin(ξ1 − �6(ξ , η))) = −β1 sin(ξ1 − η)
or,
�6(ξ1, η) = ξ1 − arcsin
(P1 + β1M1 sin(ξ1 − η)
b1
).
(26)
In a similar way, from the fourth row we get,
�7(ξ2, η) = ξ2 − arcsin
(P2 + β2M2 sin(ξ2 − η)
b2
).
(27)
Note that �6(ξ1, η) is a functions of ξ1 and η, and�7(ξ2, η) is a function of ξ2 and η. Here, we make thefollowing assumptions:
Assumption 3.2: β1 < b1−P1M1
and β2 < b2−P2M2
to ensure existence of �6(ξ1, η) and �7(ξ2, η), respec-tively.
Next, we substitute for �6(ξ1, η) and �7(ξ2, η) in (23).Then, it is clear that we can solve (23) for �5(ξ , η) if
P1 + P2 + β1M1 sin(ξ1 − η)
+ β2M2 sin(ξ2 − η) − PL1 − PL2 = 0, (28)
that is, if
β1M1 sin(ξ1 − η) + β2M2 sin(ξ2 − η) = 0 (29)
as, from (15) and (16) at (x, y) = (x�, y�) it can be shownthat P1+P2−PL1−PL2 = 0. Notice that, (29) is the sameas (18e), the constraint defined on the target dynamics.Thus, we can compute �5(ξ , η) as
Immersion and Invariance for Differential Algebraic Systems 153
where we denote ζi: = arcsin(
Pi+βiMi sin(ξi−η)bi
)for i =
1, 2. Here we make the following assumption:
Assumption 3.3: β1 <|P1−PL1|
M1and β2 <
|P2−PL2|M2
.
This assumption makes �5(ξ1, ξ2, η) bounded for all(ξ1, ξ2, η) ∈ S1× S1× S1.
Thuswe have selected themapping�(ξ , η)whichmapsthe trajectories of the target dynamics from ST into themanifold M ⊂ S . Here, note that �5 is a function ofξ1, ξ2 and η.
Finally, from (30) and the last row of (25) we get,
ui(�(ξ , η)) = − x5�+ �5(ξ1, ξ2, η)
+ T
(∂�5(ξ1, ξ2, η)
∂ξ1ξ3
+ ∂�5(ξ1, ξ2, η)
∂ξ2ξ4
+ ∂�5(ξ1, ξ2, η)
∂ηη
),
or
ui(�(ξ , η))
= − x5�+
b12 sin(ξ1 − ξ2 − ζ1 + ζ2
)P1 − PL1 + β1M1 sin(ξ1 − η)
+Tb12 cos
(ξ1 − ξ2 − ζ1 + ζ2
)P1 − PL1 + β1M1 sin(ξ1 − η)
×
⎛⎜⎜⎜⎜⎝ξ3 − ξ4 +
⎛⎜⎜⎜⎜⎝
β1M1 cos(ξ1 − η) (−ξ3 + η)
b1
√1 −
(P1+β1M1 sin(ξ1−η)
b1
)2
⎞⎟⎟⎟⎟⎠
+
⎛⎜⎜⎜⎜⎝
β2M2 cos(ξ2 − η) (ξ4 − η)
b2
√1 −
(P2+β2M2 sin(ξ2−η)
b2
)2
⎞⎟⎟⎟⎟⎠
⎞⎟⎟⎟⎟⎠
+Tb12β1M1 cos(ξ1 − η) sin
(ξ1 − ξ2 − ζ1 + ζ2
)(−ξ3 + η)(
P1 − PL1 + β1M1 sin(ξ1 − η))2
(31)
This input u = ui(�(ξ , η)) given by (31) makes themanifold M invariant.
Next, we design a control law u = uoff (·)which ensuresthat the trajectories of the closed-loop system are boundedand converge to the manifold M . Under the mapping �
we have that, ξi → xi for i = 1, . . . , 4. Then, it is clear
that η(ξ1, ξ2) → ηx := η(x1, x2) and η(ξ) → ηx :=η(x1, x2, x3, x4). From (18e) we can obtain an expressionfor ηx as
ηx = tan−1(
β1M1 sin x1 + β2M2 sin x2β1M1 cos x1 + β2M2 cos x2
). (32)
Then, an implicit description of M can be given as
M : ={(x, y) ∈ S | φ(x) = 0
}. (33)
where
φ(x, ηx) = x5 − �5(x1, x2, ηx)
= x5 − sin(x1 − x2 − y1 + y2
)P1 − PL1 + β1M1 sin(x1 − ηx)
. (34)
with (16) to be satisfied, where xi: = xi − xi� and yi: =arcsin
(Pi+βiMi sin(xi−ηx)
bi
)for i = 1, 2.
Let z = φ(x, ηx) denote the off-the-manifold coordi-nate. Then, we have
z = x5 − �5(x1, x2, ηx)
= x5 − ∂�5(x1, x2, ηx)
∂x1x1 − ∂�5(x1, x2, ηx)
∂x2x2
− ∂�5(x1, x2, ηx)
∂ηxηx
= 1
T
[−x5 + x5�+ uoff (x, y, z)
]− ∂�5(x1, x2, ηx)
∂x1x3
− ∂�5(x1, x2, ηx)
∂x2x4 − ∂�5(x1, x2, ηx)
∂ηxηx (35)
where we have substituted for x1, x2 and x5 from (15).To ensure the boundedness of the trajectories of the off-
the-manifold coordinate z and also that limt→∞ z(t) = 0we take
z = − γ z, γ > 0. (36)
Then, from (35) and (36) we can write
uoff (x, ηx , z)
= x5 − x5�− γ T (x5 − �5(x1, x2, ηx))
+ T
(∂�5(x1, x2, ηx)
∂x1x3 + ∂�5(x1, x2, ηx)
∂x2x4
+ ∂�5(x1, x2, ηx)
∂ηxηx
). (37)
154 N.S. Manjarekar et al.
0 2 4 6 8 100
0.2
0.4
Time (s)
x 1 (r
ad)
0 2 4 6 8 10
0
0.2
0.4
Time (s)
x 2 (r
ad)
0 2 4 6 8 10
0
2
Time (s)
x 3 (r
ad/s
)
0 2 4 6 8 10
0
2
Time (s)
x 4 (r
ad/s
)
0 2 4 6 8 10
5
0
Time (s)
y 1 (r
ad)
0 2 4 6 8 10
5
0
Time (s)
y 2 (r
ad)
0 2 4 6 8 10
0.2
0.4
0.6
Time (s)
x 5 (p
.u.)
Fig. 3. Response of the twomachine system (15)–(16) with the I&I control law (38): Dotted line (open loop response), solid line (closed-loop responsewith β1 = 5, β2 = 5 and γ = 5).
Thus a stabilizing IIDAS control law is synthesized as
u(x) = uoff (x, ηx , φ(x))
= x5 − x5�− γ T
×(
x5 − b12 sin(x1 − x2 − y1 + y2
)P1 − PL1 + β1M1 sin(x1 − ηx)
)
+ Tb12 cos(x1 − x2 − y1 + y2
)P1 − PL1 + β1M1 sin(x1 − ηx)
×
⎛⎜⎜⎝x3 − x4 + β1M1 cos(x1 − ηx) (−x3 + ηx)
b1
√1 −
(P1+β1M1 sin(x1−ηx)
b1
)2
+ β2M2 cos(x2 − η) (x4 − ηx)
b2
√1 −
(P2+β2M2 sin(x2−η)
b2
)2⎞⎟⎟⎠
+ Tb12 sin(x1 − x2 − y1 + y2
)(P1 − PL1 + β1M1 sin(x1 − ηx))
2
× (β1M1 cos(x1 − ηx)) (−x3 + ηx) . (38)
Finally, we establish boundedness of the trajectoriesof the closed-loop system (15) with the control law (38)and the off-the-manifold coordinate z. The closed-loopsystem is
x1 = x3 (39a)
x2 = x4 (39b)
x3 = 1
M1
[P1 − D1x3 − b1 sin(x1 − y1)
](39c)
x4 = 1
M2
[P2 − D2x4 − b2 sin(x2 − y2)
](39d)
x5 = 1
T
[−x5 + x5�+ u]
(39e)
0 = b1 sin(x1 − y1) − b12 sin(y1 − y2)
x5− PL1 (39f)
0 = b2 sin(x2 − y2) + b12 sin(y1 − y2)
x5− PL2 (39g)
z = −γ z. (39h)
Here x1, x2, y1, y2 ∈ S1, and hence we have x1, x2, y1, y2 ∈L∞ where L∞ denotes the space of bounded functions.
Immersion and Invariance for Differential Algebraic Systems 155
0 0.1 0.2 0.3 0.4 0.5 0.6
5
5
0
0.5
1
1.5
2
x1 (rad)
x 3 (r
ad/s
)
1 0 0.1 0.2 0.3 0.4 0.5
5
5
0
0.5
1
1.5
2
x2 (rad)
x 4 (r
ad/s
)
Fig. 4. Phase plots of the two machine system (15)–(16) with the I&I control law (38): Dotted line (open loop response), solid line (closed-loopresponse with β1 = 5, β2 = 5 and γ = 5).
Next, we can rewrite (39c) and (39d) as
x3 = − D1
M1x3 + �1(x1, y1) (40a)
x4 = − D2
M2x4 + �2(x2, y2) (40b)
where �1(x1, y1) = 1M1
[P1 − b1 sin(x1 − y1)
]and
�2(x2, y2) = 1M2
[P2 − b2 sin(x2 − y2)
]. Clearly, both
�1(x1, y1) ∈ L∞ and �2(x2, y2) ∈ L∞. As we haveD1 > 0 and M1 > 0, (40a) is an asymptotically sta-ble linear system in x3 with a bounded driving function�1(x1, y1). This implies x3 ∈ L∞. In a similar way wecan show that x4 ∈ L∞.
Next, we have x5 = z + �5(x1, x2). We have, from(36), that z is bounded and limt→∞ z(t) = 0. Also,from Assumption 3.3 we have that �5(x1, x2) is boundedfor all (x1, x2) ∈ S1× S1, and hence we can concludeboundedness of x5.
Thus, we have shown that the trajectories of (39) arebounded and limt→∞ z(t) = 0. The above discussion onthe control synthesis can be summarized in the followingproposition which is an important result in this paper:
Proposition 3.2: The closed-loop system (15)–(16) withthe control law (38) is locally asymptotically stable at(x�, y�).
Proof:Based on the arguments given above.
4. Simulation
The simulation parameters for the two machine systemshown in Fig. 2 are assumed as follows: M1 = M2 = 8
We assume that a transient condition (a sudden dip involtage) occurs at bus 2 for a duration of 0.1 s at t = 1s.The rotors of both the machines start swinging in responseto the transient. Due to poor mechanical damping theoscillations sustain for a long period as shown by dottedplots in Fig. 3. The closed-loop response for the tuningparameters β1 = β2 = 5 and γ = 5 is denoted bysolid lines. For the closed-loop system the oscillations inboth generators die out in about 5 s. The phase portraitsfor the open loop response and the closed-loop responseare shown in Fig. 4. The power variations are shown inFig. 5.
156 N.S. Manjarekar et al.
0 1 2 3 4 5 6 7 8 9 100.8
1
1.2
1.4
1.6
1.8
Time (s)
p G1
(p.u
.)
0 1 2 3 4 5 6 7 8 9 100.8
1
1.2
1.4
1.6
1.8
2
Time (s)
p G2
(p.u
.)
Fig. 5. Power variations of the two machine system (15)–(16) with the I&I control law (38): Dotted line (open loop response), solid line (closed-loopresponse with β1 = 5, β2 = 5 and γ = 5).
5. Conclusion
The I&Imethodology proposed for unconstrained dynam-ical systems was extended to a class of constraineddynamical systems. The result was then used to synthe-size an asymptotically stabilizing control law for a twomachine systemwith a set of algebraic constraints, using aCSC as an actuator. The power system was modeled usingthe SPMmodel, where the rotor dynamics of eachmachinewas described by the swing equation model and the powerbalance equations at the generator terminal buses weregiven by the algebraic constraints. The CSC was modeledas a first order system. The simulation results show signif-icant improvement in terms of the magnitude and settlingtime of the oscillations.
Acknowledgment
This project was partially supported by an Indo-Frenchresearch project (Project No. 3602-1) under the aegis ofIFCPAR.
The authors thank Prof. AnilM. Kulkarni of the Depart-ment of Electrical Engineering, IIT Bombay for sharingmany insights into the engineering aspects of the problem.
References
1. Acosta JÁ, Ortega R, Astolfi A, Sarras I. A constructivesolution for stabilization via immersion and invariance: The
cart and pendulum system. Automatica, 2008; 44(9):2352–2357.
2. Astolfi A, Ortega R. Immersion and Invariance: A New Toolfor Stabilization andAdaptiveControl ofNonlinear Systems.IEEE Trans Automatic Contr, 2003; 48(4): 590–606.
3. Astolfi A, Ortega R. Invariant manifolds, asymptotic immer-sion and the (adaptive) stabilization of nonlinear systems.In Zinober A and Owens D, editors, Nonlinear and Adap-tive Control, volume 281 of Lecture Notes in Control andInformation Sciences, pages 1–20. Springer, 2003.
4. Bazanella AS, e Silva AS, Kokotovic PV. Lyapunov designof excitation control of synchronousmachines. In IEEE Con-ference on Decision and Control, pages 211–216, SanDiego,1997.
5. Bazanella AS, Kokotovic PV, e Silva AS. A dynamic exten-sion for LgV controllers. IEEE Trans Automatic Contr, 1999;44(3): 588–592.
6. Bergen AR, Hill DJ. A structure preserving model for powersystem stability analysis. IEEE Trans. Power Apparatus Syst,1981; PAS-100(1): 25–35.
7. Cai Z, Zhu L, Lan Z, Gan D, Ni Y, Shi L, Bi T. A studyon robust adaptive modulation controller for TCSC based onCOI signal in interconnected power systems. Electr PowerSyst Research, 2008; 78(1): 147–157.
8. Chapman JW, Ilic MD, King CA, Eng L, Kaufman H.Stabilizing a multimachine power system via decentralizedfeedback linearizing excitation control. IEEE Trans. PowerSyst, 1993; 8(3): 830–839.
9. Espinosa-PérezG,Godoy-AlcantarM,Guerrero-RamirezG.Passivity-based control of synchronous generators. In Pro-ceedings of the IEEE International Symposium on IndustrialElectronics, pages SS101–SS106, Portugal, 1997.
Immersion and Invariance for Differential Algebraic Systems 157
10. Galaz M, Ortega R, Bazanella AS, Stankovic AM. Anenergy-shaping approach to the design of excitation con-trol of synchronous generators. Automatica, 2003; 39(1):111–119.
11. Ghandhari M, Andersson G, Pavella M, Ernst D. A controlstrategy for controllable series capacitor in electric powersystems. Automatica, 2001; 37(10): 1575–1583.
12. Jin Hao, Chen Chen, Libao Shi, Jie Wang. Nonlinear decen-tralized disturbance attenuation excitation control for powersystems with nonlinear loads based on the Hamiltoniantheory. IEEE Trans. Ener Conver, 2007; 22(2): 316–324.
13. Bin He, Xiubin Zhang, Xingyong Zhao. Transient stabiliza-tion of structure preserving power systems with excitationcontrol via energy-shaping. Int J Electr Power Ener Syst,2007; 29(10): 822–830.
14. Jiao X, Sun Y, Shen T. Energy-shaping control of powersystems with direct mechanical damping assignment. InProceedings of IFAC, pages 281–286, Nogoya, 2006.
15. Xiaohong Jiao, Yuanzhang Sun, Tielong Shen. An energy-shaping approach with direct mechanical damping injectionto design of control for power systems. In Francesco Bulloand Kenji Fujimoto, editors, Lagrangian and HamiltonianMethods for Nonlinear Control 2006, volume 366 of LectureNotes in Control and Information Sciences, pages 353–364.Springer-Verlag, 2007.
16. Karagiannis D, Astolfi A, Ortega R. Adaptive output feed-back stabilization of a class of nonlinear systems. In IEEEConference on Decision and Control, pages 1491–1496,Las Vegas, 2002.
17. Karagiannis D, Ortega R, Astolfi A. Nonlinear adaptive sta-bilization via system immersion: Control design and applica-tions. In F. Lamnabhi-Lagarrigue, A. Loria, and E. Panteley,editors, Advanced Topics in Control Systems Theory, volume311 of Lecture Notes in Control and Information Sciences,pages 1–21. Springer, 2005.
18. King CA, Chapman JW, Ilic MD. Feedback linearizing exci-tation control on a full-scale power system model. IEEETrans Power Syst, 1994; 9(2): 1102–1109.
19. De Leon-Morales J, Espinosa-Pérez G, Macias-Cardoso I.Observer-based control of a synchronous generator: aHamil-tonian approach. Int J Electr Power Ener Syst, 2002; 24(8):655–663.
20. Li XY. Nonlinear controller design of thyristor controlledseries compensation for damping inter-area power oscilla-tions. Electr Power Syst Research, 2006; 76(12): 1040–1046.
21. LuQ, SunYZ.Nonlinear stabilizing control ofmultimachinesystems. IEEE Trans Power Syst, 1989; 4(1): 236–241.
22. Manjarekar NS, Banavar RN, Ortega R. Application ofPassivity-based Control to Stabilization of the SMIB Systemwith Controllable Series Devices. In Proceedings of IFAC,pages 5581–5586, Seoul, 2008.
23. Manjarekar NS, Banavar RN, Ortega R. Nonlinear Con-trol Synthesis for Asymptotic Stabilization of the SwingEquation using a Controllable Series Capacitor via Immer-sion and Invariance. In IEEE Conference on Decision andControl, pages 2493–2498, Cancun, Mexico, 2008.
24. Manjarekar NS, Banavar RN, Ortega R. An Application ofImmersion and Invariance to a Differential Algebraic Sys-tem: A Power System. In European Control Conference2009, pages 838–843, Budapest, Hungary, 2009.
25. Manjarekar NS, Banavar RN, Ortega R. Application of Inter-connection andDampingAssignment to the Stabilization of aSynchronousGeneratorwith aControllable SeriesCapacitor.Int J Electr Power Ener Syst, 2010; 32(1): 63–70.
26. Maya-Ortiz P, Espinosa-Pérez G. Output feedback excitationcontrol of synchronous generators. Int J Robust NonlinearContr, 2004; 14: 879–890.
27. Mielczarski W, Zajaczkowski AM. Nonlinear field volt-age control of a synchronous generator using feedbacklinearization. Automatica, 1994; 30(10): 1625–1630.
28. Moon Y-H, Choi B-K, Roh T-H. Estimating the domainattraction for power systems via a group of damping-reflectedenergy functions. Automatica, 2000; 36(3): 419–425.
29. Ortega R, Galaz M, Astolfi A, Sun Y, Shen T. Transientstabilization of multimachine power systems with nontrivialtransfer conductances. IEEE Trans Automatic Contr, 2005;50(1): 60–75.
30. Ortega R, Stankovic A, Stefanov P. A passivation approachto power systems stabilization. In IFAC Symp. Nonlin-ear Control Systems Design, pages 1–3, Enschede, NL,1998.
31. Pai MA. Energy Function Analysis for Power System Stabil-ity. Kluwer Academic Publishers, Boston, MA, 1989.
32. Shen T, Ortega R, Lu Q, Mei S, Tamura K. Adaptive L2 dis-turbance attenuation ofHamiltonian systemswith parametricperturbation and application to power systems. In IEEE Con-ference on Decision and Control, pages 4939–4944, Sydney,Australia, December, 2000.
33. SunYZ, SongYH,LiX.Novel energy-basedLyapunov func-tion for controlled power systems. IEEE Power Eng Rev,2000; 20(5): 55–57.
34. Wang Y, Hill DJ, Middleton RH, Gao L. Transient stabilityenhancement and voltage regulation of power systems. IEEETrans Power Syst, 1993; 8(2): 620–627.
