Complementarity Systems inConstrained Steady-state Optimal Control

A. Jokic, M. Lazar, and P.P.J. van den Bosch

Dept. of Electrical Eng., Eindhoven Univ. of Technology,P.O. Box 513, 5600 MB Eindhoven, The Netherlands{a.jokic, m.lazar, p.p.j.v.d.bosch}@tue.nl

Abstract. This paper presents a solution to the problem of regulating ageneral nonlinear dynamical system to a time-varying economically op-timal operating point. The system is characterized by a set of exogenousinputs as an abstraction of time-varying loads and disturbances. Theeconomically optimal operating point is implicitly defined as a solutionto a given constrained convex optimization problem, which is related tosteady-state operation. The system outputs and the exogenous inputsrepresent respectively the decision variables and the parameters in theoptimization problem. Complementarity systems are employed as build-ing blocks to construct a dynamic controller that solves the consideredregulation problem. The complementarity solution arises naturally via adynamic extension of the Karush-Kuhn-Tucker optimality conditions forthe steady-state related optimization problem.

1 Introduction

Although there has been earlier substantial work in specific research areas wherecombinations of differential equations are coupled with complementarity condi-tions, it is since their formal introduction in 1996 by Van der Scahft and Schu-macher [1], see also [2], that complementarity systems (CS) have become anextensive topic of research in the hybrid systems community. This particularclass of systems has certain structural properties that have already been suc-cessfully exploited in answering some of the fundamental theoretical questionslike well-posedness [2, 3], certain control synthesis problems [4], and recentlyalso Lyapunov stability analysis [5]. Complementarity systems naturally arise inmany application areas, such as constrained mechanical systems, electrical cir-cuit theory, dynamic optimization problems, oligopolistic markets and Leontieveconomy. For a more detailed presentation see the excellent overview given in [6].In this paper we present a new application, namely constrained steady-state op-timal control, where complementarity systems provide an attractive solution.In contrast to the vast majority of previously considered applications where CScome in as a natural modeling framework, here CS arise as a suitable control syn-thesis framework. More precisely, we propose a specific dynamic extension of theKarush-Kuhn-Tucker (KKT) optimality conditions to obtain a novel feedbackcontrol structure as a solution to the problem of regulating a general nonlinear

dynamical system to a time-varying economically optimal operating point. Theconsidered dynamical system is characterized with a set of exogenous inputs asan abstraction of time-varying loads and disturbances acting on the system. Eco-nomic optimality is defined through a convex constrained optimization problemwith system outputs as decision variables, and with the values of exogenous in-puts as parameters in the optimization problem. In the following paragraph wegive a flavor of possible regulation problems that motivate the results presentedin this paper.

In many production facilities, the optimization problem reflecting economicalbenefits of production is associated with a steady-state operation of the system.The control action is required to maintain the production in an optimal regimein spite of various disturbances, and to efficiently and rapidly respond to changesin demand. Furthermore, it is desirable that the system settles in a steady-statethat is optimal for novel operating conditions. The vast majority of control lit-erature is focused on regulation and tracking with respect to known setpoints ortrajectories, while coping with different types of uncertainties and disturbancesin both the plant and its environment. Typically, setpoints are determined off-line by solving an appropriate optimization problem and they are updated in anopen-loop manner. The optimization problem typically reflects variable costs ofproduction and economical benefits under the current market conditions, e.g. fuelor electricity prices, and accounts for physical and security limits of the plant.If a production system is required to follow a time-varying demand in real-time,e.g., if produced commodities cannot be efficiently stored in large amounts, itbecomes crucial to perform economic optimization on-line. A typical exampleof such a system are electrical power systems. Increase of the frequency withwhich the economically optimal setpoints are updated can result in a significantincrease of economic benefits accumulated in time. If the time-scale on whicheconomic optimization is performed approaches the time-scale of the underlyingphysical system, i.e. of the plant dynamics, dynamic interaction in between thetwo has to be considered. Economic optimization then becomes a challengingcontrol problem, especially since it has to cope with inequality constraints thatreflect the physical and security limits of the plant.

1.1 Nomenclature

For a matrix A ∈ Rm×n, [A]ij denotes the element in the i-th row and j-thcolumn of A. For a vector x ∈ Rn, [x]i denotes the i-th element of x. A vectorx ∈ Rn is said to be nonnegative (nonpositive) if [x]i ≥ 0 ([x]i ≤ 0) for alli ∈ {1, . . . n}, and in that case we write x ≥ 0 (x ≤ 0). The nonnegative orthantof Rn is defined by Rn

+ := {x ∈ Rn | x ≥ 0 }. The operator col(·, . . . , ·) stacksits operands into a column vector. For u, v ∈ Rk we write u ⊥ v if u>v = 0. Weuse the compact notational form 0 ≤ u ⊥ v ≥ 0 to denote the complementarityconditions u ≥ 0, v ≥ 0, u ⊥ v. The matrix inequalities A Â B and A º B meanA and B are Hermitian and A−B is positive definite and positive semi-definite,respectively. For a scalar-valued differentiable function f : Rn → R, ∇f(x)denotes its gradient at x = col(x1, . . . , xn) and is defined as a column vector,

i.e. ∇f(x) ∈ Rn, [∇f(x)]i = ∂f∂xi

. For a vector-valued differentiable functionf : Rn → Rm, f(x) = col(f1(x), . . . , fm(x)), the Jacobian at x = col(x1, . . . , xn)is the matrix Df(x) ∈ Rm×n and is defined by [Df(x)]ij = ∂fi(x)

∂xj. For a vector

valued function f : Rn → Rm, we will use ∇f(x) to denote the transpose of theJacobian, i.e. ∇f(x) ∈ Rn×m, ∇f(x) , Df(x)>, which is consistent with thegradient notation ∇f when f is a scalar-valued function. With a slight abuse ofnotation we will often use the same symbol to denote a signal, i.e. a function oftime, as well as possible values that the signal may take at any time instant.

2 Problem formulation

In this section we formally present the constrained steady-state optimal controlproblem considered in this paper. Furthermore, we list several standing assump-tions, which will be instrumental in the subsequent sections.

Consider a dynamical system

x = f(x,w, u), (1a)y = g(x,w), (1b)

where x(t) ∈ Rn is the state, u(t) ∈ Rm is the control input, w(t) ∈ Rnw is anexogenous input, y(t) ∈ Rm is the measured output, f : Rn × Rnw × Rm → Rn

and g : Rn × Rnw → Rm are arbitrary nonlinear functions.For a constant w ∈ W , with W ⊂ Rnw denoting a known bounded set,

consider the following convex optimization problem associated with the outputy of the dynamical system (1):

miny

J(y) (2a)

subject toLy = h(w), (2b)qi(y) ≤ ri(w), i = 1, . . . , k, (2c)

where J : Rm → R is a strictly convex and continuously differentiable function,L ∈ Rl×m is a constant matrix, h : Rnw → Rl and ri : Rnw → R, i = 1, . . . , k arecontinuous functions, while qi : Rm → R, i = 1, . . . , k are convex, continuouslydifferentiable functions. For the matrix L we require rank L = l < m.

For a constant exogenous signal w(t) = w ∈ W , the optimization problem (2)implicitly defines the optimal operating point in terms of the steady-state valueof the output vector y in (1). The constraints in (2) represent the security-type“soft” constraints for which some degree of transient violation may be accepted,but whose feasibility is required in steady-state. Note that in general not all ofthe elements of y that appear in the constraints (2b), (2c) need to appear in theobjective function J(·), and vice versa. The objective of the control input u isto drive the output y to the optimal steady-state operating point given by (2).

We continue by listing several assumptions concerning the dynamics (1) andthe optimization problem (2). Let Il denote the set of indices i for which the

function qi in (2c) is a linear function, and let In denote the set of indices cor-responding to nonlinear qi.

Assumption 1. For each w ∈ W the set

{y | Ly = h(w), qi(y) < ri(w) for i ∈ In, qi(y) ≤ ri(w) for i ∈ Il}

is nonempty. 2

Assumption 1 states that the convex optimization problem (2) satisfies Slater’sconstraint qualification [7] for each w ∈ W , implying that strong duality holdsfor the considered problem. Note also that due to strict convexity of the objec-tive function in (2), the optimization problem has an unique minimizer y(w) foreach w ∈ W .

Assumption 2. For each w ∈ W , in the optimization problem (2) the mini-mum is attained. 2

Assumption 3. For each w ∈ W , there is a unique pair (x(w), u(w)) such that

0 = f(x(w), w, u(w)), (3a)y(w) = g(x(w), w), (3b)

where y(w) denotes the corresponding minimizer in (2). 2

Assumption 3 guarantees that for all constant w(t) = w ∈ W , the outputvector y can be driven to the corresponding optimal steady-state point, whichis then characterized by a unique, constant value of the input signal u. In otherwords, Assumption 3 implies that the steady-state relations from (1) do not poseany additional constraints to the optimization problem (2), i.e. any constrainton the steady-state imposed by (1) is already included in (2).

Assumption 4. All components of w that appear in (2) are available at all timeinstants. 2

Assumption 4 implies that violations of the constraints (2b) and (2c) can bemeasured. Hence, they can be used for control purposes.

With the definitions and assumptions made so far, we are now ready toformally state the control problem considered in this paper.

Problem 1. Constrained steady-state optimal control.For a dynamical system given by (1), design a feedback controller that has yas input signal and u as output signal, such that the following objective is metfor any constant-valued exogenous signal w(t) = w ∈ W : the closed-loop systemglobally converges to an equilibrium point with y = y(w), where y(w) denotesthe corresponding minimizer of the optimization problem (2). 2

Note that Problem 1 includes the standard regulation problem, see e.g. Chap-ter 12 in [8], as a special case. More precisely, if (2) is modified to include onlyequality constraints and J(y) = 0, then Problem 1 reduces to the problem ofregulating the output y := Ly to the constant reference signal r := h(w). Fur-thermore, note that for a dynamical system (1) we have assumed the samedimension of the output signal y and the control input signal u. However, thisassumption can be relaxed. A typical example of such a relaxation is present inthe illustrative example in Section 5.

3 Dynamic KKT controllers

In this section we present a controller that guarantees the existence of an equi-librium point with y = y(w) as described in Problem 1. This is done using anappropriate dynamic extension of the Karush-Kuhn-Tucker optimality condi-tions for the optimization problem (2).

Assumption 1 implies that for each w ∈ W , the first order Karush-Kuhn-Tucker (KKT) conditions are necessary and sufficient conditions for optimality.For the optimization problem (2) these conditions are given by the following setof equalities and inequalities:

∇J(y) + L>λ +∇q(y)µ = 0, (4a)Ly − h(w) = 0, (4b)

0 ≤ −q(y) + r(w) ⊥ µ ≥ 0, (4c)

where q(y) := col(q1(y), . . . , qk(y)), r(w) := col(r1(w), . . . , rk(w)) and λ ∈ Rl,µ ∈ Rk are Lagrange multipliers. Since the above conditions are necessary andsufficient conditions for optimality, it is apparent that the existence of an equilib-rium point with y = y(w) is implied if for each w ∈ W the controller guaranteesthe existence of the vectors λ and µ, such that that the conditions (4) are fulfilledin a steady-state of the closed-loop system.

In what follows, we present two controllers that achieve this goal. Later inthis section it will be shown that there are certain insightful differences as wellas similarities between these two control structures.

Max-based KKT controller . Let Kλ ∈ Rl×l, Kµ ∈ Rk×k, Kc ∈ Rm×m andKo ∈ Rk×k be diagonal matrices with non-zero elements on the main diagonaland Kµ  0, Ko  0. Consider a dynamic controller with the following structure:

xλ = Kλ(Ly − h(w)), (5a)xµ = Kµ(q(y)− r(w)) + v, (5b)

xc = Kc(L>xλ +∇q(y)xµ +∇J(y)), (5c)0 ≤ v ⊥ Ko xµ + Kµ(q(y)− r(w)) + v ≥ 0, (5d)

u = xc, (5e)

where xλ, xµ and xc denote the controller states and the matrices Kλ, Kµ, Kc

and Ko represent the controller gains. Note that the input vector v(t) ∈ Rk in(5b) is at any time instant required to be a solution to a finite-dimensional linearcomplementarity problem (5d). 2

Saturation-based KKT controller . Let Kλ ∈ Rl×l, Kµ ∈ Rk×k and Kc ∈Rm×m be diagonal matrices with non-zero elements on the main diagonal andKµ Â 0. Consider a dynamic controller with the following structure:

xλ = Kλ(Ly − h(w)), (6a)xµ = Kµ(q(y)− r(w)) + v, (6b)

xc = Kc(L>xλ +∇q(y)xµ +∇J(y)), (6c)0 ≤ v ⊥ xµ ≥ 0, (6d)

u = xc, (6e)xµ(0) ≥ 0, (6f)

where xλ, xµ and xc denote the controller states and the matrices Kλ, Kµ andKc represent the controller gains. Note that the input vector v(t) ∈ Rk in (6b)is at any time instant required to be a solution to a finite-dimensional linearcomplementarity problem (6d). The initialization constraint (6f) is required as anecessary condition for well-posedness, conform with the inequality in the com-plementarity condition (6d). 2

The choice of names max-based KKT controller and saturation-based KKTcontroller will become clear later in this section. Notice that both controllersbelong to the class of complementarity systems [1, 2].

Theorem 1. Let w(t) = w ∈ W be a constant-valued signal, and suppose thatAssumption 1 and Assumption 3 hold. Then the closed-loop system, i.e. thesystem obtained from the system (1) connected with controller (5) or (6) ina feedback loop, has an equilibrium point with y = y(w), where y(w) denotes thecorresponding minimizer of the optimization problem (2).

Proof. We first consider the closed-loop system with the max-based KKT con-troller, i.e. controller (5). By setting the time derivatives of the closed-loop sys-tem states to zero and by exploiting the non-singularity of the matrices Kλ, andKc, we obtain the following complementarity problem:

0 = f(x, w, xc), (7a)y = g(x, w), (7b)0 = Ly − h(w), (7c)0 = Kµ(q(y)− r(w)) + v, (7d)

0 = L>xλ +∇q(y)xµ +∇J(y), (7e)0 ≤ v ⊥ Ko xµ + Kµ(q(y)− r(w)) + v ≥ 0, (7f)

with the closed-loop system state vector xcl := col(x, xλ, xµ, xc) and the vectorv as variables. Any solution xcl to (7) is an equilibrium point of the closed-loopsystem. By substituting v = −Kµ(q(y) − r(w)) from (7d) and utilizing Kµ  0and Ko  0, the complementarity condition (7f) reads as 0 ≤ −q(y) + r(w) ⊥xµ ≥ 0. With λ := xλ and µ := xµ, the conditions (7c),(7d),(7e),(7f) thereforecorrespond to the KKT conditions (4) and, under Assumption 1, they necessarilyhave a solution. Furthermore, for any solution (y, xλ, xµ, v) to (7c),(7d),(7e),(7f),it necessarily holds that y = y(w). It remains to show that (7a), (7b) admits asolution in (x, xc) for y = y(w). This is, however, the hypothesis of Assumption 3.Moreover, Assumption 3 implies uniqueness of x and xc in an equilibrium.

Now, consider the closed-loop system with the saturation-based KKT con-troller, i.e. controller (6). The difference in this case comes only through (6b)and (6d). It is therefore sufficient to show that (6b) and (6d) imply 0 ≤ −q(y)+r(w) ⊥ xµ ≥ 0. This implication is obvious since Kµ Â 0. 2

In the following subsection we concentrate on those parts of the KKT con-trollers that are directly affected by the algebraic complementarity conditions(5d) and (6d). We present a systematic procedure for implementing these con-ditions, which involves the definition of complementary integrators as the basicbuilding blocks for imposing steady-state complementarity conditions. In turn,complementarity integrators form the basic building blocks of the developeddynamic KKT controllers.

3.1 Complementarity integrators

The main distinguishing feature between the max-based KKT controller (5) andthe saturation-based KKT controller (6) is in the way the steady-state comple-mentarity slackness condition (4c) is enforced. Although characterized by thesame steady-state relations, the two controllers, and therefore the correspondingclosed-loop systems, have some significantly different dynamical features whichwill be discussed further in this section. In the following two paragraphs ourattention is on the equations (5b),(5d) and (6b),(6d), and the goal is to showthe following:• The max-based KKT controller, i.e. controller (5), can be represented as adynamical system in which certain variables are coupled by means of static,continuous, piecewise linear characteristics;• The saturation-based KKT controller, i.e. controller (6), can be represented asa dynamical system with state saturations.

Max-based complementarity integrator. Let η = [q(y) − r(w)]i, ξ = [xµ]i,ν = [v]i, ko = [Ko]ii and kµ = [Kµ]ii, for some i ∈ {1, . . . , k}. Then the i-th rowin (5b) and (5d) is respectively given by

ξ = kµη + ν, (8a)0 ≤ ν ⊥ koξ + kµη + ν ≥ 0, (8b)

Fig. 1: Complementarity integrators. (a) Max-based complementarity integrator.(b) Saturation-based complementarity integrator.

where ko > 0 and kµ > 0. Let a, b and c be real scalars related through thecomplementarity condition 0 ≤ c ⊥ a + b + c ≥ 0. It is easily verified, e.g.by checking all possible combinations, that this complementarity condition isequivalent to b + c = max(a + b, 0) − a. Now, by taking c = ν, a = koξ andb = kµη, it follows that (8) can be equivalently described by

ξ = max(ko ξ + kµη, 0)− ko ξ. (9)

Figure 1a presents a block diagram representation of (9). The block labeled“Max” in the figure, represents the scalar max relation as a static piecewiselinear characteristic.

With ko > 0 and kµ > 0, it is easy to verify that, if the system in Figure 1ais in steady-state, then the value of its input signal η and the value of its outputsignal ξ necessarily satisfy the complementarity condition 0 ≤ ξ ⊥ −η ≥ 0. 2

Saturation-based complementarity integrator. Let η = [q(y) − r(w)]i,ξ = [xµ]i, ν = [v]i and kµ = [Kµ]ii, for some i ∈ {1, . . . , k}. Then the i-throw in (6b),(6d) and (6f) is respectively given by

ξ = kµη + ν, (10a)0 ≤ ν ⊥ ξ ≥ 0, (10b)ξ(0) ≥ 0, (10c)

where kµ > 0. The dynamical system (10) can equivalently be described by

ξ = φSCI(ξ, η) :=

0 if ξ = 0 and kµη < 0,

kµη if ξ = 0 and kµη ≥ 0,

kµη if ξ > 0.

(11)

Figure 1b presents a block diagram representation of (11), which is a saturatedintegrator with the lower saturation point equal to zero. The equivalence of thedynamics (10) and the saturated integrator defined by (11) directly follows fromthe equivalence of gradient-type complementarity systems (GTCS) ((10) belongsto the GTCS class) and projected dynamical systems (PDS) ((11) belongs to thePDS class). For the precise definitions of GTCS and PDS system classes and forthe equivalence results see [9] and [10].

With kµ > 0, it is easy to verify that if the system in Figure 1b is in steady-state, the value of the input signal η and the value of its output signal ξ neces-sarily satisfy the complementarity condition 0 ≤ ξ ⊥ −η ≥ 0. 2

The above presented complementarity integrators present us with the basicbuilding blocks for imposing steady-state complementarity conditions. We willuse the term max-based complementarity integrator (MCI) to refer to the system(8), i.e. the system with the structure as depicted in Figure 1a, and we will usethe term saturation-based complementarity integrator (SCI) for the system (10),i.e. the system in Figure 1b. Together with a pure integrator, complementarityintegrators form the basic building block of a KKT controller.

Remark 1. For the MCI given by (8) the following holds:(i) If ξ(0) < 0 then ξ(t) → 0 as t → ∞. Indeed for ξ(t) < 0, from (8) it followsthat ξ(t) > 0, irrespective of the value of the input signal η(t).(ii) If ξ(0) ≥ 0, then ξ(t) ≥ 0 for all t ∈ R+. Indeed for ξ(t) = 0, from (8) itfollows that ξ(t) ≥ 0, irrespective of the value of the input signal η(t). Therefore,similarly to the behavior of the saturation-based KKT controller, if xµ(0) ≥ 0in the max-based KKT controller (5), then xµ(t) ≥ 0 for all t ∈ R+.

In what follows, we point out an interesting relation between the dynamicalbehavior of the two types of complementarity integrators. Consider the MCI (8)and let ξ(0) ≥ 0. Note that according to Remark 1 it follows that ξ(t) ≥ 0 forall t ∈ R+. For ξ(t) ≥ 0, the dynamics (8) can be equivalently represented in apiecewise-linear form as follows:

ξ = φMCI(ξ, η) :=

{kµη if ξ ≥ −kµ

koη,

−koξ if ξ < −kµ

koη.

(12)

Now, suppose that the gain kµ has the same value in (11) and (12). For agiven η(t) < ∞, we define the set D := {ξ | ξ ≥ 0, φSCI(ξ, η) 6= φMCI(ξ, η)}.By inspection it can easily be observed that for any η(t) < ∞, the Lebesguemeasure of the set D tends to zero as ko tends to ∞. This implies that the SCIcan be considered as a special case of the MCI when the gain ko is set to infinity.In the same sense, the saturation-based KKT controller can be considered as aspecial case of the max-based KKT controller.

4 Well-posedness and stability of the closed-loop system

In this subsection we shortly present some results concerning the well-posednessand stability analysis problems of the closed-loop system, i.e. of the system (1)interconnected with a dynamic KKT controller in a feedback loop. We referto [11] for a more detailed treatment of these topics.

4.1 Well-posedness

Since the function max(·, 0) is globally Lipschitz continuous, for checking well-posedness of the closed-loop system with max-based KKT controller one canresort on standard Lipschitz continuity conditions.

Notice that the system (1) in closed loop with a saturation-based KKT con-troller belongs to a specific class of gradient-type complementarity systems forwhich sufficient conditions for well-posedness have been presented in [9] and [10].More precisely, it was shown that the hypermonotonicity property plays a crucialrole in establishing well-posedness, see [9] and [10] for details.

It can be easily verified, see [11] for details, that Lipschitz continuity implieshypermonotonicity, and therefore we can state the following unified condition forwell-posedness of the system (1) in closed loop with a dynamic KKT controller(irrespective of the KKT controller type):

Proposition 1. Suppose that the functions q, ∇J and all entries in ∇q areglobally Lipschitz. Then the system (1) in closed loop with a dynamic KKTcontroller (5) or (6) is globally well-posed.

4.2 Stability analysis

Stability analysis for a fixed w ∈ W . Theorem 1 states that for anyconstant-valued exogenous signal w(t) ∈ W , the closed-loop system necessarilyhas an equilibrium. Furthermore, from the proof of this theorem it follows thatfor all corresponding equilibrium points the values of the state vectors (x, xc)are unique. For a given w(t) = w ∈ W , the necessary and sufficient conditionfor uniqueness of the remaining closed-loop system state vectors (xλ, xµ), andtherefore a necessary and sufficient condition for uniqueness of the closed-loopsystem equilibrium, corresponds to the condition for uniqueness of the Lagrangemultipliers in (4). This condition is known as the strict Mangasarian-Fromovitzconstraint qualification (SMFCQ) and is presented in [12].

Since both types of complementarity integrators can be presented in a piece-wise affine framework [13], for a given w(t) = w ∈ W characterized by a uniqueequilibrium, one can perform a global asymptotic stability analysis based on: i)the analysis procedures from [14, 15] in case when (2) is a quadratic programand (1) is a linear system; ii) the analysis procedure from [16] in case when (2)is given by a (higher order) polynomial objective function and (higher order)polynomial inequality constraints, while (1) is a general polynomial system.

In the case when w(t) = w ∈ W is such that the SMFCQ does not hold, theclosed-loop system is characterized by a set of equilibria (not a singleton), whichis then an invariant set for the closed-loop system. Each equilibrium in this set ischaracterized by different values of the state vectors (xλ, xµ), but unique valuesof the remaining states. Under additional generalized Slater constraint qualifi-cation, see [17] for details, the set of equilibria is guaranteed to be bounded. Forstability analysis with respect to this set, one could invoke a suitable extensionof LaSalle’s invariance theorem [18].

Stability analysis for all w ∈ W . A possibility to perform stability analysisfor all possible constant values of the exogenous signal w(t), i.e. for all w(t) = wwhere w is any constant in W , is to formulate a corresponding robust stabilityanalysis problem. For instance, consider the max-based KKT control structure,which is particulary suitable for this approach. Let M denote the set of au-tonomous systems that contains all the closed-loop systems that correspond toone fixed w ∈ W . Furthermore, suppose that each system in M has the ori-gin the equilibrium, after an appropriate state transformation. Then, it can beshown that any for any closed-loop system in M the static nonlinearity of theMCI, see Figure 1a, fulfills certain sector bound conditions. Therefore, stabilityof all the closed-loop systems in the set M can be established using the integralquadratic constraint approach [19]. See [11] for a complete description that alsodeals with non-unique equilibria.

5 Illustrative example

To illustrate the theory, in this section we present the following example thatincludes nonlinear constraints on the steady-state operating point. Consider athird-order system of the form (1) given by

x1

x2

x3

=

−2.5 0 −5

0 −5 −150.1 0.1 −0.2

x1

x2

x3

+

00

−0.1

w +

2.5 00 50 0

(u1

u2

), (13a)

y = col(x1, x2, x3), (13b)

where u = col(u1, u2) collects the control inputs.With xp := col(x1, x2), the associated steady-state related optimization prob-

lem is defined as follows:

minxp

12x>p Hxp + a>xp (14a)

subject tox1 + x2 = w, (14b)

(x1 − 4.7)2 + (x2 − 4)2 ≤ 3.52, (14c)

where H = diag(6, 2), a = col(−4,−4), and the value of the exogenous signalw is limited in the interval W = [4, 11.5]. It can be verified that for this Wand the constraints (14b) and (14c), Assumption 1 holds. Furthermore, it caneasily be verified that Assumption 3 holds. From the dynamics of the state x3,it follows that in steady-state the equality x1 + x2 − 2x3 = w holds. Therefore,in steady-state, x3 = 0 implies fulfilment of the constraint (14b). This impliesthat for control we can directly use the value of the state x3 as a measure ofviolation of this constraint. Hence, explicit knowledge of w is not required.

Simulations of the closed-loop system response to stepwise changes in theexogenous input w(t), which is presented in Figure 2a, have been performed.

0 100 200 300 400 500 600 700 800 9003

4

5

6

7

8

9

10

11

12

13

t

x 1(t

)+x 2

(t)

w(t)

w1

w2

w3

w4

x1(t)+x

2(t)

saturation−based

max−based

,

(a)

0 100 200 300 400 500 600 700 800 900−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

t

(x1(t

)−4

.7)2

+(x

2(t

)−4

)2−

3.5

2

w1 w

2 w

3 w

4

(x1(t)−4.7)2+(x

2(t)−4)2−3.52

saturation−based

max−based

Ko=0.5

Ko=1

,

(b)

Fig. 2: (a) The values of w and x1 + x2, i.e. the right hand and the left hand side ofthe equality constraint (14b), as a function of time. (b) Violation of the inequalityconstraint (14c) as a function of time. When the curves are above zero (horizontaldashed line), the constraint is violated.

Figure 2 and Figure 3 present the results of the simulation when the systemis controlled with a saturation-based and when it is controlled with a max-based KKT controller with different values of the gain Ko. Both controllers wereimplemented with the gains Kλ = 0.15, Kµ = 0.1, Kc = −0.7I2, and the gainKo in the max-based controller was set to 0.5 and 1. In each figure, a legend isincluded to indicate which trajectory belongs to each controller.

Figure 2a and Figure 2b clearly illustrate that the controllers continuouslydrive the closed-loop system towards the steady-state where the constraints(14b), (14c) are satisfied. Figures 2b and 3a illustrate fulfilment of the com-plementarity slackness condition (4c) in steady-state.

Finally, Figure 3b illustrates that the controllers drive the system towardsthe correspondent optimal operating point as defined by (14). In this figure thestraight dashed lines labeled wi, i = 1, . . . , 4, represent the equality constraintx1 +x2 = wi where the values of wi, i = 1, . . . , 4 are the ones given in Figure 2a.The dashed circle represents the inequality constraint (14c), i.e. the steady-statefeasible region for xp is within this circle. Thin dotted lines represent the contourlines of the objective function (14a), while the dash-dot line represents the locusof the optimal point xp(w) for the whole range of values w in the case when theinequality constraint (14c) would be left out from the optimization problem.

From the simulations we can observe that by increase of the gain Ko in themax-based controller, the trajectory of the closed-loop system with the max-based KKT controller approaches the trajectory of the closed-loop system withsaturation-based KKT controller.

0 100 200 300 400 500 600 700 800 900−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

t

x µ(t)

w1 w

2 w

3 w

4

sat.−based

max−based

xµ(t) K

o=0.5

Ko=1

,

(a)

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

x1

x 2 w1

w2

w3

w4

xp(0)

sat.−based

max−based

Ko=0.5

Ko=1

,

(b)

Fig. 3: (a) Simulated trajectory of the controller state xµ. (b) Simulated trajectory ofthe state vector xp for the close-loop system.

Remark 2. The presented KKT control methodology has a great potential forapplication in real-time, price-based power balance and network congestion con-trol of electrical energy transmission systems. This is considered to be one of thetoughest problems in operation and control of market-based power systems [20].This topic has recently gained a significant attention in power systems commu-nity, following the large restructuring processes occurring in this sector. Specifi-cally, the KKT control structure is suitable for this particular application since itexplicitly manipulates with the Lagrange multipliers, which, in power systems,have an interpretation of nodal prices for electricity. The interested reader isrefereed to [11] for more details on this topic.

6 Conclusions

In this paper we have considered the problem of regulating a general nonlineardynamical system to a time-varying economically optimal operating point. Theeconomically optimal operating point was implicitly defined as a solution to agiven constrained convex optimization problem, which is related to steady-stateoperation. We have shown that complementarity systems arise naturally as asolution to this problem. More precisely, the solution that we proposed in thispaper is based on the specific dynamic extension of the Karush-Kuhn-Tuckeroptimality conditions for the steady-state related optimization problem. An ad-vantageous feature of the proposed solution is that it offers an explicit controllaw, i.e. the implementation of the controller does not require solving on-line thecorresponding optimization problem. Some results and tools that can be usedfor well-posedness and stability analysis of the resulting closed-loop system havealso been discussed.

Acknowledgments. The authors would like to thank Dr. Maurice Heemels forvaluable discussions.

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