The Optimal Steady-State Control Problem

Transcription

1 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL. THIS VERSION: OCTOBER 31, The Optimal Steay-State Control Problem Liam S. P. Lawrence Stuent Member, IEEE, John W. Simpson-Porco, Member, IEEE, an Enrique Mallaa Member, IEEE arxiv: v1 [math.oc] 3 Oct 218 Abstract We consier the problem of esigning a feeback controller for a multivariable nonlinear system that regulates an arbitrary subset of the system states an inputs to the solution of a constraine optimization problem, espite parametric moelling uncertainty an time-varying exogenous isturbances; we term this the optimal steay-state (OSS) control problem. We erive necessary an sufficient conitions for the existence of an OSS controller by formulating the OSS control problem as an output regulation problem wherein the regulation error is unmeasurable. We introuce the notion of an optimality moel, an show that the existence of an optimality moel is sufficient to reuce the OSS control problem to an output regulation problem with measurable error. This yiels a esign framework for OSS control that unifies an extens many existing esigns in the literature. We present a complete an constructive solution of the OSS control problem for the case where the plant is linear time-invariant with structure parametric uncertainty, an isturbances are constant in time. We illustrate these results via an application to optimal frequency control of power networks, an show that our esign proceure recovers several frequency controllers from the recent literature. Inex Terms Reference tracking an isturbance rejection, output regulation, convex optimization, online optimization, robust control I. I N T R O D U C T I O N Many engineering systems are require to operate at an optimal steay-state, efine by the solution of a constraine optimization problem that seeks to minimize operational costs while satisfying equipment constraints. The current approach in inustrial practice for regulating a system to an optimal steay-state involves a time-scale separation between the tasks of computing the optimal setpoint an tracking this setpoint using feeback controllers. Consier, for example, optimizing the prouction setpoints of generators in an electric power system, while maintaining supply-eman balance an system stability. At present, optimal generation setpoints are compute offline using eman projections an a moel of the network, then the operating points are ispatche as reference commans to local controllers at each generation site [1]. This process is repeate with a fixe upate rate: a new optimizer is compute, ispatche, an tracke. If the supply an eman of power changes on a time scale that is slow compare to the upate rate, then this metho is perfectly acceptable. L. S. P. Lawrence an J. W. Simpson-Porco are with the Department of Electrical an Computer Engineering, University of Waterloo, Waterloo ON, N2L 3G1 Canaa. {lsplawre,jwsimpson}@uwaterloo.ca. E. Mallaa is with the Department of Electrical an Computer Engineering, The Johns Hopkins University, 34 N. Charles Street, Baltimore, MD 21218, mallaa@jhu.eu This work was supporte in part by the NSERC CGS-M Program, the NSERC Discovery Grant RGPIN , an by UW ECE start-up funing. If the optimizer changes rapily, however, as is the case for power networks with a high penetration of renewable energy sources, the conventional approach is inefficient [2]. Profit is reuce as a result of operating in a sub-optimal regime between optimizer upates. In the rapily-changing optimizer case, then, it woul be avantageous to eliminate the time-scale separation by combining the local generator controllers with an online optimization algorithm, so that the optimal operating conition coul be tracke in real time. Inee, this is the irection of much recent research in power system control [2] [22]. The same theme of real-time regulation of system variables to optimal values emerges in iverse areas, an much work on controller esign to implement online optimization exists in the literature. Fiels of application besies the power network control example mentione alreay inclue network congestion management [23] [27], chemical processing [28] [3], win turbine power capture [31], [32], active flow control an axial flow compressor control for aerospace applications [33], [34], temperature regulation in energy-efficient builings [35], an beam matching in particle accelerators [36]. The breath of applications motivates the nee for a general theory an esign proceure for controllers that regulate a system to a maximally efficient operating point efine by an optimization problem, even as the optimizer changes over time ue to changing market prices, isturbances to the system ynamics, an operating constraints that epen on external variables. We refer to the problem of esigning such a controller as the optimal steay-state (OSS) control problem. A number of recent publications have formulate problem statements an solutions for variants of the OSS control problem [37] [48]. Many of the currently-propose controllers, however, have limite applicability: some solutions only apply to systems of a special form [42], [43]; some attempt to optimize only the steay-state input [39] or output [38], [41], [44], [45] alone; some apply only to equality-constraine [47], [48] or unconstraine optimization problems [46]; an in all cases, the effects of parametric moelling uncertainty are omitte. As a result, a number of important questions are raise regaring the existence of solutions an the general architecture of OSS controllers we ientify three gaps in the OSS control literature. First, funamental controller existence theorems are lacking, leaving important questions unanswere: What conitions on the plant an optimization problem are necessary for the OSS control problem to be solvable? What properties must a controller satisfy to solve the OSS control problem? Secon, insufficient attention has been pai to unerstaning when real-time optimization can be performe robustly in the presence of parametric uncertainty. Thir, the literature

2 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL. THIS VERSION: OCTOBER 31, lacks a general, unifying architecture for OSS controllers that facilitates controller esign an connects to establishe esign methoologies. A. Contributions This paper contains two substantial contributions. First, in Section III, we present a formal statement of the OSS control problem as a generalization of the output regulation problem, in which a subset of the plant states an inputs must track the time-varying solution of an optimization problem parametrize by exogenous inputs. We leverage the relationship between OSS control an output regulation to state necessary an sufficient conitions for solvability of the OSS control problem. The feature that istinguishes OSS control from output regulation is the lack of knowlege of the reference comman, or, equivalently, the error signal. We introuce the concept of an optimality moel: a ynamic filter that is able to robustly prouce a proxy signal for the optimality error. The existence of an optimality moel converts the OSS control problem to a stanar output regulation problem, opening the oor to the application of esign techniques from output regulation theory. This moular controller esign framework unifies an generalizes the results present in the optimal steaystate control literature, an we provie examples of controllers from the literature which fall within our framework. Our secon major contribution (Section IV an V) is a complete constructive solution to the linear-convex OSS control problem, in which the plant is a linear time-invariant (LTI) system with (in general, structure) time-invariant parametric uncertainty, the optimization problem is convex, an the isturbances are constant in time. The LTI case provies a particularly clean setting for stuying when real-time optimization can be performe in the presence of uncertainty. To this en, we introuce two properties associate with the linearconvex OSS control problem the robust output subspace an robust feasible subspace properties. When either property hols, we ultimately show that the linear-convex OSS control problem can be reuce to a robust stabilization problem, an provie several strategies for stabilizer esign using techniques, e.g., robust control. We illustrate our results for the linearconvex OSS control problem on an example from power system control, an show that our esign framework recovers several well-stuie controllers from the literature. B. Notation The set of n m matrices with real entries is enote by R n m. For a continuously ifferentiable map f : R n R, f : R n R n enotes its graient. The symbol enotes a matrix of zeros whose imensions can be inferre from context, while n enotes the n-imensional column vector of zeros. For scalars or column vectors {v 1, v 2,..., v k }, col(v 1, v 2,..., v k ) is a column vector obtaine by vertical concatenation of v 1,..., v k. I I. B A C K G R O U N D O N T H E O U T P U T R E G U L AT I O N P R O B L E M This section recalls the basic problem setup an results for the global nonlinear output regulation problem from [49]; we will make extensive use of these ieas an the accompanying notation in our statement an solution of the OSS control problem in Section III. Our exposition is brief, an we refer the reaer to [49] [53] for etaile treatments. Output regulation is a generalization of integral control to cases where isturbances an/or reference signals are time-varying. Consier a nonlinear plant ẋ = f(x, u, w), x() X := R n (1) y m = h m (x, u, w), where x X is the state, u U := R m is the control input, y m R pm is the vector of available measurements, an w R nw is a set of exogenous inputs which might inclue isturbances to the plant ynamics, reference signals, or uncertain parameters. The function f is assume to be locally Lipschitz in x an continuous in u an w, while h m is assume to be continuous; note for later that y m may contain components of the input u. We efine an error signal e R p associate with the plant e = h e (x, u, w), (2) consisting of variables which shoul be protecte from the effects of the exogenous inputs an initial conitions. For example, e may be a vector of reference tracking errors, an shoul be riven to zero asymptotically using feeback control. The function h e is assume to be continuous. The class of exogenous inputs of interest is generate by the exosystem ẇ = s(w), w() W (3) where s is locally Lipschitz an W R nw is an open invariant set for the ynamics (3). Note that we can capture the effects of parametric uncertainty in the plant moel (1) by incluing such parameters as components of w with static ynamics ẇ i =. We enote the set of solutions of (3) by I s (W ), the corresponing ω-limit set by Ω(W ), an assume that solutions of the exosystem (3) are boune for all time t R. A general nonlinear feeback controller for (1) is given by ẋ c = f c (x c, y m ), u = h c (x c, y m ), x c () X c := R nc which processes measurements y m (t) an prouces the control signal u(t) in close-loop with the plant (1). The ynamics of the close-loop system are escribe by (1) an (4), with (3) generating the exogenous input to the close-loop system. The problem of output regulation is to esign the feeback controller such that the close-loop system satisfies a generalize stability criterion known as convergence with a uniformly boune steay-state (UBSS) (see [49] for etails) an such that the error signal e(t) is riven to zero. Problem 2.1 (Output Regulation): For the plant (1), esign, if possible, a ynamic feeback controller of the form (4) such that the close-loop system meets the following criteria: (i) well-poseness: the close-loop system is well-pose; (ii) global convergence: the close-loop system is globally uniformly convergent an satisfies the UBSS property for the class of inputs I s (W ); (iii) asymptotic error zeroing: for every initial conition (x(), x c ()) X X c of the close-loop system an (4)

3 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL. THIS VERSION: OCTOBER 31, initial conition w() W of the exosystem, the error signal (2) asymptotically tens to zero, i.e., lim e(t) = p. t The following theorem is a basic necessary conition for the output regulation problem to be solvable [49]. Theorem 2.2 (Regulator Equations): The output regulation problem is solvable only if there exist continuous mappings π : Ω(W ) X an ψ : Ω(W ) U which satisfy the regulator equations π(w) = f(π(w), ψ(w), w) t p = h e (π(w), ψ(w), w) for every solution of the exosystem w = w(t) satisfying w(t) Ω(W ) for all t R. The interpretation of Theorem 2.2 is that there must exist a steay-state feeforwar control input u(t) = ψ(w(t)) with corresponing steay-state state trajectory x(t) = π(w(t)) such that the error e = h e (x, u, w) is hel ientically equal to zero. The set of controllers that solve the output regulation problem is escribe in the next theorem [49]. Theorem 2.3 (Controller Conitions): The output regulation problem is solve by the controller (4) if an only if (i) there exists a mapping π c : Ω(W ) X c such that for some π an ψ satisfying the regulator equations (5) the mapping π c satisfies the generalize internal moel principle t π c(w) = f c (π c (w), h m (π(w), ψ(w), w)) ψ(w) = h c (π c (w), h m (π(w), ψ(w), w)) for every solution of the exosystem w = w(t) satisfying w(t) Ω(W ) for all t R; (ii) the close-loop system corresponing to this controller is globally uniformly convergent with the UBSS property for the class of inputs I s (W ). Theorem 2.3 (i) states that that there must exist a steaystate trajectory x c (t) = π c (w(t)) of the controller which can prouce the error-zeroing steay-state input u(t) = ψ(w(t)). The typical controller solving the output regulation problem consists of two sub-systems: an internal moel of the exosystem an a stabilizer; see [5, Equation (2.2)], [53, Equation (6.14)], [49, Equation (5.13)]. The internal moel ensures the controller satisfies the generalize internal moel property (12), an reuces the output regulation problem to the problem of (robustly) stabilizing the augmente plant, consisting of the internal moel an the plant in either a series or feeback configuration. The stabilizer ensures that the close-loop system satisfies the UBSS stability property. The output regulation problem is therefore solve using the following moular esign strategy: (i) esign an internal moel an construct the augmente plant; (ii) esign a stabilizer to ensure the close-loop system meets a stability criterion (e.g., global uniform convergence). (5) (6) Both of these steps are formiable in general, an methos for constructing internal moels are escribe in [53]. The specific case of a static exosystem (constant isturbances an uncertain parameters), however, has the well-known solution of integral control when the error e is measurable. Lemma 2.4 (Integral Control): Suppose the exosystem (3) is static so that ẇ =, an suppose the error e is available for measurement. Consier the controller η = e ẋ s = f s (x s, e, y m, η) u = h s (x s, e, y m, η). (7a) (7b) (7c) If the close-loop system with plant (1) an controller (7) is well-pose an has a globally asymptotically stable equilibrium point for every w() W, then the regulator equations (5) are solvable an the controller (7) solves the output regulation problem. The first component of the controller (7a) is the internal moel (in this case, a pure integrator), while the secon component (7b) an (7c) is the stabilizer. We now move on to our main focus: the optimal steay-state control problem. I I I. T H E O P T I M A L S T E A D Y- S TAT E C O N T R O L P R O B L E M A. Problem Statement Consier the general nonlinear plant (1) of the output regulation problem. Suppose that instea of trying to asymptotically zero an error signal, our objective is to esign a feeback controller of the form (4) so that an arbitrary subset of the control inputs an plant states are asymptotically riven to a cost-minimizing steay-state, etermine by the solution of a constraine optimization problem. This objective shoul be achieve espite parametric uncertainty in the plant, an in the presence of unknown exogenous isturbances. Formally, efine the regulate output y Y := R p as y = h r (x, u, w), (8) where h r is a continuous function. The cost-minimizing steaystate is etermine by an optimization problem with the regulate output y as the ecision variable. Specifically, consier the following nonlinear optimization problem parametrize by w W with ecision variable y Y, minimize y Y g(y, w) (9a) subject to l i (y, w) =, i {1,..., n ec } (9b) k j (y, w), j {1,..., n ic }. (9c) The cost function is g : Y W R. The constraints (9b) an (9c) represent n ec engineering equality constraints an n ic engineering inequality constraints which shoul be satisfie in the esire steay-state. The steay-state optimization problem (9) is flexible enough to encompasses many situations of interest. To wit, the components of w inclue in the cost function (9a) coul represent uncertain parameters, such as changing market prices. The engineering equality constraints (9b) might represent require setpoint tracking or

4 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL. THIS VERSION: OCTOBER 31, balance conitions. The engineering inequality constraints (9c) can be use to ensure that states an inputs o not excee their maximum continuous operation ratings. For each w W, let y (w) := argmin {g(y, w) (9b) an (9c) hol} (1) y Y enote the optimal solution function of the problem (9). In general, the optimal solution function is set-value, an its value for a particular w may be empty. Going forwar we will assume that the optimization problem (9) has a unique minimizer for each w W, an hence y is a single-value map. We further assume that y : W Y is continuous this assumption is essential for the application of output regulation results to the optimal steay-state control problem. Assumption 3.1 (Properties of y ): The optimal solution function y : W Y efine in (1) is single-value an continuous on W. The goal in optimal steay-state control is to asymptotically guie the regulate output y(t) to the time-varying optimizer y (w(t)), where w(t) is generate by the exosystem (3). We efine the optimal steay-state control problem in the language of the output regulation problem, Problem 2.1, with error signal e := y y (w). Problem 3.2 (Optimal Steay-State Control): Design, if possible, a ynamic feeback controller of the form (4) for the nonlinear plant (1) such that the close-loop system meets the following criteria: (i) well-poseness: the close-loop system is well-pose; (ii) global convergence: the close-loop system is globally uniformly convergent an satisfies the UBSS property for the class of inputs I s (W ); (iii) asymptotic optimality: For every initial conition (x(), x c ()) X X c of the close-loop system an initial conition w() W of the exosystem, the regulate output y is asymptotically brought into agreement with the optimizer lim (h r(x(t), u(t), w(t)) y (w(t)) = p. t The OSS control problem is a generalization of the output regulation problem; inee, one may recover the stanar output regulation problem by setting g = in (9a), omitting the inequality constraints (9c), an efining the error signal e component-wise via (9b) as e i := l i (h r (x, u, w), w). An important feature that istinguishes the general OSS control problem from the output regulation problem is the absence of knowlege of the error signal: the optimal solution function y is generally unknown, an the time-varying optimizer y (w(t)) epens on the unknown exogenous isturbance w. Technically, the statement of the output regulation problem oes not require the error signal e to be measurable, but stanar controller esigns make such an assumption again, see [5, Equation (2.2)], [53, Equation (6.14)], [49, Equation (5.13)]. This creates a new set of challenges for optimal steay-state control beyon the substantial challenges alreay present in the output regulation problem. B. Solvability Conitions for the OSS Control Problem Before outlining a general esign strategy for the OSS control problem, we present solvability theorems that follow immeiately from output regulation results. The optimal steaystate control problem is efine as the output regulation problem with continuous error signal e = h e (x, u, w) := h r (x, u, w) y (w). The results of Section II therefore apply; in particular, we have the following necessary conition for solvability which follows from Theorem 2.2. Theorem 3.3 (OSS Regulator Equations): The OSS control problem is solvable only if there exist continuous mappings π an ψ that satisfy the OSS regulator equations π(w) = f(π(w), ψ(w), w) t y (w) = h r (π(w), ψ(w), w) (11) for every solution of the exosystem w = w(t) satisfying w(t) Ω(W ) for all t R. Theorem 3.3 for the optimal steay-state control problem amits a similar interpretation to Theorem 2.2 for the output regulation problem: for every exogenous input signal w(t) Ω(W ), there must exist a control input u(t) = ψ(w(t)) that prouces the state trajectory x(t) = π(w(t)) in the plant such that the regulate output is optimal, i.e., y(t) = y (w(t)). An alternative interpretation is that (11) expresses compatibility between the set of all time-varying optimizers an the set of possible steay-state behaviours of the plant. The necessary an sufficient conitions for a controller to solve the optimal steay-state control problem follow from Theorem 2.3. Theorem 3.4 (OSS Controller Conitions): The OSS control problem is solve by the controller (4) if an only if (i) there exists a mapping π c : Ω(W ) X c such that for some π an ψ satisfying the OSS regulator equations (11) the mapping π c satisfies the generalize internal moel principle t π c(w) = f c (π c (w), h m (π(w), ψ(w), w)) ψ(w) = h c (π c (w), h m (π(w), ψ(w), w)) (12) for every solution of the exosystem w = w(t) satisfying w(t) Ω(W ) for all t R; (ii) the close-loop system corresponing to this controller is globally uniformly convergent with the UBSS property for the class of inputs I s (W ). C. Optimal Steay-State Controller Design We will now escribe a framework for the esign of optimal steay-state controllers that encompasses an generalizes many esigns present in the literature. As we have alreay state, the optimal steay-state control problem is more ifficult to solve than the output regulation problem because the optimizer set is unknown an therefore the regulation error is not measurable. However, if we coul prouce a measurable proxy for the optimality error, then we coul mirror the esign techniques from the output regulation literature.

5 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL. THIS VERSION: OCTOBER 31, Recall that in the esign of a controller to solve the output regulation problem, employing an internal moel of the exosystem reuces the problem of output regulation to a problem of robust stabilization. Inspire by this approach, we will introuce the iea of an optimality moel to reuce the problem of optimal steay-state control to an output regulation problem with a measurable error signal. An optimality moel is a filter applie to the measure output of the plant that, when in steay-state, prouces an output ɛ which is a proxy for the optimality error e = y y (w). To make this iea precise, consier a filter with state ξ Ξ := R n ξ, inputs y m (plant measurements) an v V := R nv (auxiliary stabilizing input), an output ɛ escribe by ξ = ϕ(ξ, y m, v) ɛ = h ɛ (ξ, y m, v), (13) an efine the filtere plant as the original plant (1) in cascae with the filter (13): ẋ = f(x, u, w) y m = h m (x, u, w) ξ = ϕ(ξ, y m, v) ɛ = h ɛ (ξ, y m, v) y f = col(ξ, y m, ɛ). (14) The state of the filtere plant is col(x, ξ), the control inputs are u an v, the error output is ɛ, an the measure output is y f = col(ξ, y m, ɛ) consisting of the filter state, original plant measurements, an error output. The purpose of the aitional input v will become clear shortly. Consier now the steaystate behaviours of the filtere plant that woul lea to our error proxy signal ɛ being ientically zero. In other wors, consier the solutions (π, π ξ, ψ u, ψ v ) : Ω(W ) X Ξ U V of the regulator equations for the filtere plant: t π(w) = f(π(w), ψ u(w), w), t π ξ(w) = ϕ(π ξ (w), h m (π(w), ψ u (w), w), ψ v (w)), = h ɛ (π ξ (w), h m (π(w), ψ u (w), w), ψ v (w)). This leas us to the efinition of an optimality moel. (15) Definition 3.5 (Optimality Moel): The filter (13) is sai to be an optimality moel (for the OSS control problem, Problem 3.2) if the following implication hols: if the quaruple (π, π ξ, ψ u, ψ v ) is a solution of the regulator equations (15) for the filtere plant (14), then the pair (π, ψ u ) satisfies the OSS regulator equations (11). An optimality moel encoes sufficient conitions for optimality uring steay-state operation with the plant. For instance, given a convex optimization problem where strong uality hols, the optimality moel might encoe the KKT conitions when it is in ynamic steay-state with the plant; we explore this case further in Section V. Just as knowlege of an internal moel can be use to reuce the output regulation problem to a robust stabilization problem, an optimality moel can be use to reuce the optimal steay-state control problem to an output regulation problem with measurable error, as the following theorem shows. A proof is containe in the appenix. Theorem 3.6 (Reuction of OSS to Output Regulation): Suppose that the filter (13) is an optimality moel for the OSS control problem (Problem 3.2), an consier the filtere plant (14). If the controller ẋ c = f c (x c, ξ, y m, ɛ) u = h u c (x c, ξ, y m, ɛ) v = h v c(x c, ξ, y m, ɛ) (16) solves the output regulation problem for the filtere plant (14) with error signal ɛ, then the controller ξ = ϕ(ξ, y m, v) ẋ c = f c (x c, ξ, y m, h ɛ (ξ, y m, v)) u = h u c (x c, ξ, y m, h ɛ (ξ, y m, v)) v = h v c(x c, ξ, y m, h ɛ (ξ, y m, v)) solves the optimal steay-state control problem. (17) Base on Theorem 3.6, we obtain the following moular esign strategy for solving the OSS control problem: (i) esign an optimality moel an construct the filtere plant; (ii) esign a controller that solves the output regulation problem for the filtere plant. A controller solving the OSS control problem will therefore typically consist of three cascae subsystems: an optimality moel, an internal moel of the exosystem, an a stabilizer. See Figure 1 for a iagram of this propose scheme. The purpose of the auxiliary input v to the optimality moel can now be mae clear: it provies aitional inputs for stabilization of the close-loop system. u Plant Exosystem ẇ = s(w) ẋ = f(x, u, w) y m = h m (x, u, w) v w ẋ s = f s (x s, η, ξ, y m, ɛ) u = h u s (x s, η, ξ, y m, ɛ) v = h v s (x s, η, ξ, y m, ɛ) Stabilizer y m η ξ Optimality Moel ξ = ϕ(ξ, y m, v) ɛ = h ɛ (ξ, y m, v) ɛ η = γ(η, ɛ) Internal Moel Fig. 1: A general architecture for OSS controllers. The plant an optimality moel are place in cascae to form the filtere plant. The internal moel an stabilizer solve the output regulation problem for the filtere plant with error signal ɛ. The overall controller is containe in the shae blue region. D. Previous OSS Controller Designs as Special Cases Several controllers in the literature for solving particular instances of the OSS control problem have been implicitly using the esign strategy of Figure 1 to enforce the necessary an sufficient solvability conitions of Theorem 3.4. For three such examples, we now ientify the respective optimality moels an output-regulating controllers (we use the notation

6 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL. THIS VERSION: OCTOBER 31, of the cite papers, apart from our explicit introuction of the error variable ɛ). One can verify that, for the OSS control problems efine in each paper, the ientifie optimality moel satisfies the requirements of Definition 3.5 (none of the following examples, however, uses the auxiliary stabilizing input v). First, consier the controller [38, Equation (4)]. The optimality moel is given by ẋ λ = K λ (Ly h(w)) ẋ µ = K µ (q(y) r(w)) + v v K x µ + K µ (q(y) r(w)) + v ɛ = K c (L T x λ + q(y)x µ + J(y)), while the output-regulating controller is pure integral control: ẋ c = ɛ, u = x c. Secon, consier the controller [48, Equation (6)]. The optimality moel is λ = µ ( M gt µ(y 2 + µλ) λ) ɛ = f t (u) Π T 1u h t (y 1 ) Π T 2u M gt µ(y 2 + µλ), an the output-regulating controller is pure integral control: u = ɛ. Thir an finally, consier the controller [4, Equation (13)]. The optimality moel is given by ˆx = (A LC)ˆx + (B LD)u + Ly ɛ = ϕ(ˆx), an the output-regulating controller is a PI controller: ẋ c = ɛ, u = K I x c + K p ɛ. The concept of an optimality moel an the OSS control structure in Figure 1 generalizes an extens these previous esigns from the literature. We conclue by noting that the architecture of Figure 1 is not a priori necessary for the solution of the OSS control problem, nor is a given esign solving the OSS control problem base on Figure 1 necessarily minimal in terms of orer; jugment shoul be use to reuce the orer, if possible, in particular applications. Nonetheless, the existence of an optimality moel as efine in Definition 3.5 appears to be the key to obtaining tractable problem instances which can then be aresse using stanar tools. I V. L I N E A R - C O N V E X O P T I M A L S T E A D Y- S TAT E C O N T R O L For an arbitrary nonlinear plant, exosystem, an optimization problem, the OSS control problem is likely intractable. The remainer of this paper focuses in etail on the important case of a LTI plant with time-invariant parametric uncertainty, a static exosystem, an a convex steay-state optimization problem. We call this case linear-convex optimal steay-state control, an leverage the results of Section III to provie a complete solution to this problem. Here in Section IV, we consier the funamentals of linear-convex OSS control, before presenting a constructive controller esign proceure in Section V. A. Uncertain LTI Plant We specialize the nonlinear plant (1) an the output to be optimize (8) to the case of linear time-invariant ynamics with structure parametric uncertainty in the system matrices: ẋ = A(δ)x + B(δ)u + B w (δ)w, y = C(δ)x + D(δ)u + Q(δ)w, y m = C m (δ)x + D m (δ)u + Q m (δ)w. x() X (18) Note the ifference in notation from Section III, as we have split the exogenous input into two components, w an δ. The vector w R nw moels exogenous isturbances an setpoints, while δ δ R n δ is a vector that characterizes structure parametric moel uncertainty an belongs to a set δ containing the origin (δ = is the nominal moel). All matrices are assume to be continuous functions of δ δ. We assume the corresponing exosystems are static: w nw =, w() R nw, δ() δ, (19) t δ nδ which yiels constant signals w an δ. B. Robust Steay-State Optimization Problem Recall from Theorem 3.3 that a necessary conition for solvability of the OSS control problem is that the OSS regulator equations (11) have a solution; that is, there must exist a steaystate operation of the plant that yiels the optimal output. We can therefore either assume that the OSS regulator equations have a solution, or we can constrain our optimization problem to guarantee that solutions exist. We opt for the latter strategy by embeing the steay-state operation constraint into the steay-state optimization problem. Consier the equilibrium outputs ȳ that can be generate from (18) by an equilibrium state x an input ū: n = A(δ) x + B(δ)ū + B w (δ)w ȳ = C(δ) x + D(δ)ū + Q(δ)w. (2) We efine the set-value mapping Y : W δ Y so that Y (w, δ) is the set of all such achievable equilibrium regulate outputs ȳ for fixe values of w an δ: Y (w, δ) := {ȳ Y there exists an ( x, ū) such that ( x, ū, ȳ) satisfy (2)}. (21) For each (w, δ), the set Y (w, δ) is an affine subset of Y, which we assume is nonempty. 1 We shall inclue ȳ Y (w, δ) as a constraint of the steay-state optimization problem to ensure compatibility between the optimizers an steay-state operation of the plant, thereby ensuring solvability of the OSS regulator equations (Theorem 3.3). The cost-minimizing equilibrium point is etermine by the convex optimization problem minimize y Y g(y, w) (22a) subject to y Y (w, δ) (22b) Hy = Lw (22c) k i (y, w), i {1,..., n ic } (22) 1 Equivalently, we assume that range B w(δ) range [ A(δ) B(δ) ] for all δ δ.

7 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL. THIS VERSION: OCTOBER 31, in which g : Y W R is assume to be a continuous function of all of its arguments, an ifferentiable an convex in y for each w. The constraint (22b) is the steay-state constraint just iscusse. The constraints (22c) an (22) represent n ec engineering equality constraints an n ic engineering inequality constraints which shoul be satisfie in the esire steay-state. To ensure the optimization problem is convex, the engineering equality constraints must be linear an the functions k i : Y W R of the engineering inequality constraints must be convex in y for each w. The matrices H, L an functions k i are part of the esign specifications, an are therefore not subject to parametric uncertainty. Proceeing from (1), as before y : W δ Y is the optimal solution function of (22), an we suppose y satisfies Assumption 3.1. We further assume that a strictly feasible point exists for the optimization problem (22); that is, we assume there exists a point ỹ Y that satisfies ỹ Y (w, δ), Hỹ = Lw, an k i (ỹ, w) < for all i {1,..., n ic }. The existence of a strictly feasible point ensures Slater s constraint qualification hols, an therefore guarantees that the Karush-Kuhn-Tucker (KKT) conitions are necessary an sufficient for optimality [54, Sections an 5.5.3]. The optimality conition associate with the constraint y Y (w, δ) involves the unique subspace associate with the affine set Y (w, δ). Recall that any affine set in Eucliean space can be written as the sum of a (unique) subspace an a (non-unique) offset vector. It follows from the efinition of Y that for each (w, δ), there exists an offset vector y(w, δ) an a unique subspace V (δ) such that Y (w, δ) = y(w, δ) + V (δ). (23) For each (w, δ), the optimal solution y is characterize as the unique vector such that there exists µ R nec an ν R nic such that (y, µ, ν ) satisfy the KKT conitions n ic g(y, w) + H T µ + νi k i (y, w) V (δ) i=1 (24a) = ν i k i (y, w), ν i, i {1,..., n ic } (24b) along with the primal feasibility conitions (22b) (22). The graient conition (24a) is written in a non-stanar manner in terms of the subspace V (δ), which can be interprete as the subspace of first-orer feasible variations for the affine constraint y Y (w, δ) see [55, Section 3.1] for etails. We raw the reaer s attention to a secon, equivalent, way to write the graient conition (24a). There exists a (y, µ, ν ) satisfying (24) if an only if there exists a (y, ν ) satisfying n ic g(y, w) + νi k i (y, w) (V (δ) null H) i=1 (25a) = ν i k i (y, w), ν i, i {1,..., n ic } (25b) Going forwar, we will make use of both formulations (24) an (25) when appropriate. Remark 4.1 (Comments on Linear-Convex OSS Formulation): The assumption that H, L in (22c) are free of parametric uncertainty can be relaxe without much ifficulty. One coul relax the assumption of ifferentiability of g by using subgraients or proximal operator methos, as in [48]; we o not pursue this here. C. Linear-Convex OSS Regulator Equations Recall that solvability of the OSS regulator equations (11) is necessary for the solvability of the OSS control problem. The inclusion of the equilibrium constraint (22b) in the optimization problem ensures this necessary conition is met. Proposition 4.2 (Solvability of OSS Regulator Equations for Linear-Convex OSS Control): For the linear-convex OSS control problem with plant (18), exosystem (19), an convex optimization problem (22), there exist functions π(w, δ) an ψ(w, δ) satisfying the OSS regulator equations (11). Proof: We consier whether there exist functions π an ψ satisfying (11) for the LTI ynamics (18) with exosystem (19). That is, we consier solutions to n = A(δ)π(w, δ) + B(δ)ψ(w, δ) + B w (δ)w y (w, δ) = C(δ)π(w, δ) + D(δ)ψ(w, δ) + Q(δ)w. (26) Since, by the constraints of the optimization problem, y (w, δ) Y (w, δ), it follows from (2) an (21) that the mappings π an ψ exist. Remark 4.3 (Necessity of Steay-State Constraints): Failing to inclue the steay-state constraints (22b) in the optimization problem (22) can result in an instance of the OSS control problem in which y (w, δ) / Y (w, δ) for some (w, δ). That is, the optimizer of (22) might be inconsistent with steay-state operation of the plant (18) for some (w, δ). In this case, the OSS regulator equations (11) will fail to have globally efine solutions, an the OSS control problem will be insolvable. V. C O N S T R U C T I V E S O L U T I O N S F O R L I N E A R - C O N V E X O P T I M A L S T E A D Y- S TAT E C O N T R O L This section presents constructive solutions to the linearconvex OSS control problem outline in Section IV. A. Robust Subspaces Following the esign strategy outline in Section III-C, we must construct an optimality moel for the linear-convex OSS control problem, an then esign a controller solving the output regulation problem for the series interconnection of the LTI plant an the optimality moel (Theorem 3.6). A major roablock to constructing optimality moels is the presence of parametric uncertainty, an we therefore evote significant effort in this subsection to stuying it. Inee, consier the KKT conitions (24) or (25), an notice that both involve the subspace V (δ), which epens on the uncertain parameters δ. It is therefore impossible for our controller to incorporate the graient conition within an optimality moel without knowlege of δ unless V (δ) or V (δ) null H is, in fact, inepenent of δ. While we cannot expect such a result to hol for an arbitrary uncertain LTI plant, this uncertaintyinepenence always hols when V (δ) = R p for all δ. What s more, this property may hol even when V (δ) R p for all δ,

8 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL. THIS VERSION: OCTOBER 31, V (δ 1) V (δ2) null H span the subspace V (δ), an hence V (δ) = range R(δ). Proof: We can view the affine space Y from (21) as being constructe by a two-step process. In the first step, we examine the steay-state solutions ( x, ū) to A(δ) x + B(δ)ū + B w (δ)w = n. (28) Fig. 2: An illustration of the robust feasible subspace property. For two istinct values of uncertainty δ 1 an δ 2, the subspaces V (δ 1) an V (δ 2) are ifferent. However, the intersecte subspaces V (δ 1) null H an V (δ 2) null H, shown by the ashe line, are equal. provie that the manner in which the uncertainty enters our moel possesses some structure. 2 We now explore the means by which one can verify the robustness (i.e., uncertainty inepenence) of these subspaces to parametric uncertainty, an some of the consequences. Our first efinition makes precise the notion of V (δ) null H being inepenent of δ. Definition 5.1 (Robust Feasible Subspace (RFS)): Let V (δ) be the unique subspace associate with Y (w, δ) as in (23). The optimization problem (22) is sai to satisfy the robust feasible subspace (RFS) property if there exists a fixe l N an a fixe set of vectors {v 1, v 2,..., v l } R p such that V (δ) null H = span(v 1, v 2,..., v l ) for all δ δ. The robust feasible subspace property is illustrate in Figure 2. In this example, one can visualize the subspace V (δ) rotating along the ashe-line axis as δ changes in value. So long as the subspaces V (δ) an null H are never equal, there exists a fixe basis, inepenent of δ, for the subspace V (δ) null H. In this case, the basis consists of one vector in the irection of the ashe line. A sufficient conition for V (δ) null H to be inepenent of δ is that V (δ) is itself inepenent of δ; this leas to our secon efinition. Definition 5.2 (Robust Output Subspace (ROS)): The uncertain LTI plant (18) is sai to satisfy the robust output subspace (ROS) property if there exists a fixe l N an a fixe set of vectors {v 1, v 2,..., v l } R p such that V (δ) = span(v 1, v 2,..., v l ) for all δ δ. We next iscuss how to verify the RFS an ROS properties algebraically through the construction of certain matrices whose column vectors form a basis of the subspace V (δ) or V (δ) null H. These matrices will play a key role in efining optimality moels for the linear-convex OSS control problem. We first present the construction of a matrix whose range is V (δ), which will be useful for assessing both the RFS an ROS properties. Lemma 5.3 (Construction of V (δ)): Let N AB (δ) be any matrix such that range N AB (δ) = null [ A(δ) B(δ) ]. Then the columns of the matrix R(δ) := [ C(δ) D(δ) ] N AB (δ) (27) 2 We show that this special structure exists in, for example, power system moels in Section VI. In the secon step, we compute each corresponing output as ȳ = C(δ) x+d(δ)ū+q(δ)w an place ȳ into Y. Let the set of solutions to the linear equations (28) be enote L R n R m. The set L is affine, an therefore can be written as the sum of a subspace an an offset vector. Fix a particular solution ( x, ũ) to (28), an note that L = ( x, ũ) + null [ A(δ) B(δ) ]. The set Y can then be written as Y (w, δ) = {C(δ) x + D(δ)ū + Q(δ)w ( x, ū) L} = C(δ) x + D(δ)ũ + Q(δ)w + }{{} y(w,δ) ( ) C(δ) D(δ) null A(δ) B(δ). }{{} V (δ) It follows that the construction escribe in the statement of the lemma inee yiels the subspace V (δ). We now consier how to verify when the robust feasible subspace property hols. For δ δ, let R (δ) be a matrix satisfying null R (δ) = V (δ). 3 We then have V (δ) null H = null R (δ) null H R (δ) = null. H (29) If the null space of the last line of (29) has a basis inepenent of δ, then the robust feasible subspace property hols. These observations lea us to the following result. Proposition 5.4 (Algebraic Characterization of Robust Feasible Subspace Property): Let R (δ) be any matrix satisfying null R (δ) = V (δ) for all δ. The optimization problem (22) satisfies the robust feasible subspace property if an only if there exists a fixe matrix T such that R (δ) range T = null (3) H for all δ δ. Proof: First suppose the optimization problem (22) satisfies the RFS property. Then there exists an l N an a set of vectors {v 1, v 2,..., v l } R p such that V (δ) 3 One can either construct R (δ) from R(δ) by requiring that R (δ)r(δ) = an [ R(δ) R T (δ)] is full rank, or one can use a more irect proceure. First, construct a matrix Γ(δ) such that T range Γ(δ) T A(δ) B(δ) = null, for all δ δ. C(δ) D(δ) Then, partition Γ(δ) as Γ(δ) = [ X(δ) Z(δ) ] where X(δ) has n columns an Z(δ) has p columns. One can show that V (δ) = null Z(δ) an therefore one may use R (δ) := Z(δ) in Proposition 5.4.

9 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL. THIS VERSION: OCTOBER 31, null H = span(v 1, v 2,..., v l ) for all δ. It follows that T := v 1 v 2... v l satisfies R (δ) range T = V (δ) null H = null H for all δ. Conversely, suppose there exists a matrix T satisfying (3) for all δ. Then V (δ) null H = range T for all δ, hence the column vectors of T span V (δ) null H for all δ, an therefore the problem (22) satisfies the ROS property. We now have an algebraic characterization of the ROS property the existence of a matrix T satisfying (3). We can make an analogous statement for the robust output subspace property. The proof of the following proposition is essentially ientical to the proof of Proposition 5.4. Proposition 5.5 (Algebraic Characterization of Robust Output Subspace Property): Let R(δ) be the matrix efine in Lemma 5.3. The LTI plant (18) satisfies the robust output subspace property if an only if there exists a fixe matrix R such that range R = range R(δ) for all δ δ. The next result gives a strong sufficient conition for the ROS property. Proposition 5.6 (Robust Full Rank Implies ROS): The LTI system (18) satisfies the ROS property if A(δ) B(δ) rank = n + p, for all δ δ. (31) C(δ) D(δ) Moreover, (i) the requirements of Proposition 5.5 are satisfie by the matrix R := I p ; (ii) the requirements of Proposition 5.4 are satisfie by any matrix T such that range T = null H. Proof: If (31) hols, then Y (w, δ) = R p for all (w, δ), an therefore V (δ) = R p for all δ. The stanar basis vectors {e 1, e 2,..., e p } R p therefore satisfy V (δ) = span(e 1, e 2,..., e p ) for all δ, an hence the plant satisfies the ROS property. Since range R(δ) = V (δ) = R p for all δ δ, one may inee take R = I p in Proposition 5.5. Finally, consiering Proposition 5.4, the only vali choice of the matrix R (δ) such that null R (δ) = V (δ) for all δ δ is R (δ) :=. Hence the matrix T[ can be selecte as any matrix satisfying range T = null = null H. H] Note that (31) can hol only when the number of outputs to be optimize is less than or equal to the number of control inputs. The rank conition (31) of Proposition 5.6 sometimes referre to as the non-resonance conition [52, Lemma 4.1] is a stanar assumption of the linear output regulation problem with constant isturbances. We emphasize that (31) is only a sufficient conition for the ROS property, which itself is merely sufficient for the RFS property. The relationships between these conitions, an the optimality moels of the following section, are summarize in Figure 3. Remark 5.7 (Enforcing the RFS Property): Two potential remeies exist if the RFS property fails to hol for an instance Prop. 5.5 Prop. 5.4 Prop. 5.6 ROS Property RFS Property OM (34) OM (32) Fig. 3: Relationships between robust subspace results an optimality moels. of OSS control. First, the esigner can consier aing aitional engineering[ equality] constraints (rows of H) to ensure R (δ) the subspace null of Proposition 5.4 is inepenent H of δ. This can be accomplishe by observing the manner in which the uncertain parameters δ enter the matrix R (δ) an aing rows of H accoringly. Secon, the esigner can consier changing the selection of system variables to be optimize, i.e. changing the efinition of y, to moify the matrices C(δ) an D(δ) an hence the matrix R(δ) of Lemma 5.3. This moification might consist of reucing the number of regulate outputs p to a value less than or equal to the number of control inputs m, to make it possible for the full-rank conition of Proposition 5.6, an therefore the ROS property, to hol. B. Optimality Moels for Linear-Convex OSS Control We now consier the construction of optimality moels for linear-convex OSS control. The options available to us epen on which of the two previously-efine subspace robustness properties hol. For simplicity, we omit the auxiliary stabilizing input v from consieration in the remainer of our iscussion. We assume the constraint violations Hy Lw an k i (y, w), an objective function graient, g(y, w), are available for feeback, in that they are either irectly measurable or can be calculate using measurements. Incorporating the inequality constraints an associate ual variable conitions relies on the following lemma, which is straightforwar to prove by checking every sign combination. Lemma 5.8: For real numbers a an b, the pair (a, b) satisfies a = max(a+b, ) if an only if a, b, an ab =. It follows that, for each i {1,..., n ic }, the conitions ν i, k i (y, w), an ν i k i (y, w) = are equivalent to ν i = max(ν i +k i (y, w), ). In compact notation, we write ν = max(ν + k(y, w), ), where max evaluates max elementwise an k(y, w) := col(k 1 (y, w),..., k nic (y, w)). Proposition 5.9 (Robust Feasible Subspace Optimality Moel (RFS-OM)): Suppose the optimization problem (22) satisfies the robust feasible subspace property, an let T be any matrix satisfying the statement of Proposition 5.4. Then ν = max(ν + k(y, w), ) ν [ ] Hy Lw ɛ = T T ( g(y, w) + n ic i=1 ν i k i (y, w)) (32)

10 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL. THIS VERSION: OCTOBER 31, is an optimality moel for the linear-convex OSS control problem. Proof: For each (w, δ), consier the solutions ( x, µ, ν, ū) of the regulator equations for the LTI plant (18) an optimality moel (32) in series: = A(δ) x + B(δ)ū + B w (δ)w (33a) ȳ = C(δ) x + D(δ)ū + Q(δ)w (33b) = max( ν + k(ȳ, w), ) ν (33c) = Hȳ Lw (33) ( = T T g(ȳ, w) + n ic ) ν i k i (ȳ, w). (33e) i=1 All time erivatives on the left-han sie of the regulator equations (33) are zero since the exosystem is static. We show that the regulator equations are equivalent to the KKT conitions (25). The first two equations (33a) an (33b) imply ȳ Y (w, δ). The equation (33) is the engineering equality constraint. The engineering inequality constraints an remaining KKT conitions (25) are encoe by (33c) an (33e). Since the KKT conitions are sufficient for optimality, the following implication hols for all (w, δ): if ( x, µ, ν, ū) satisfy the regulator equations (33), then ( x, ū) satisfy the linear-convex OSS regulator equations (26). The filter (32) satisfies the criterion of Definition 3.5, an is therefore an optimality moel. The above optimality moel may be employe whenever the RFS property hols. If, furthermore, the ROS property hols, then we have a secon option. Proposition 5.1 (Robust Output Subspace Optimality Moel (ROS-OM)): Suppose the plant (18) satisfies the robust output subspace property, an let R be any matrix satisfying the statement of Proposition 5.4. Then µ = Hy Lw ν = max(ν + k(y, w), ) ν ( ɛ = R T g(y, w) + H T µ + n ic ) (34) ν i k i (y, w) i=1 is an optimality moel for the linear-convex OSS control problem. Proof: The proof is almost ientical to the proof of Proposition 5.9, except that we compare the graient conition to (24a) instea of (25a). If the robust output subspace property hols, then we are free to employ either (32) or (34) as our optimality moel. 4 In each optimality moel, the choice of the matrix T or R provies a great eal of esign flexibility. When combine with ifferent controller esign options solving the output regulation problem for the filtere plant, this gives a huge variety of esign options for synthesizing OSS controllers. 4 In fact, even more variations are possible by consiering other equivalent formulations of the KKT conitions an eveloping appropriate robust subspace notions; for brevity we omit the etails here. C. Linear-Convex OSS Controller Suppose that we have constructe an optimality moel for our linear-convex OSS control problem, perhaps using one of the two optimality moels of the previous section. We represent the optimality moel generically by ξ = ϕ(ξ, y m ) ɛ = h ɛ (ξ, y m ). We nee to construct a controller that solves the output regulation problem for the filtere plant comprising the LTI plant an optimality moel in series: ẋ = A(δ)x + B(δ)u + B w (δ)w y = C(δ)x + D(δ)u + Q(δ)w y m = C m (δ)x + D m (δ)u + Q m (δ)w ξ = ϕ(ξ, y m ) ɛ = h ɛ (ξ, y m ) (35) Observe that the filtere plant is a nonlinear system subject to constant exogenous inputs with a measurable error ɛ Lemma 2.4 immeiately yiels the solution. We must employ an integrator an stabilizer to solve the output regulation problem for the filtere plant (35): η = ɛ, ẋ s = f s (x s, y m, ξ, η, ɛ) u = h s (x s, y m, ξ, η, ɛ). (36) If the close-loop system is well-pose an stable, the linearconvex OSS control problem is solve. Proposition 5.11 (Constructive Solution of Linear-Convex OSS Control Problem): Let (ϕ, h ɛ ) be an optimality moel for the linear-convex OSS control problem. If the stabilizer (f s, h s ) is esigne such that the close-loop system of the filtere plant (35) an controller (36) in feeback is well-pose an has a globally asymptotically stable equilibrium point for all (w, δ), then the controller ξ = ϕ(ξ, y m ) η = h ɛ (ξ, y m ) ẋ s = f s (x s, y m, ξ, η, h ɛ (ξ, y m )) u = h s (x s, y m, ξ, η, h ɛ (ξ, y m )) (37) solves the linear-convex optimal steay-state control problem. D. Stabilizer Design Using an optimality moel an a bank of integrators, Proposition 5.11 tells us that we have reuce the linear-convex OSS control problem to a stabilization problem. In general, the stabilizer is a nonlinear system, an may be ifficult to optimally esign. Nonetheless, linear static stabilizer esigns such as PI controllers [37], [4] u = K P ɛ K I η (38) can be effective for some applications. Inee, a classic result of Davison [56, Lemma 3] is that if an LTI system is internally stable, then one may always achieve robust tracking an closeloop stability via an integral controller with sufficiently low gain; this yiels a simple an straightforwar stabilizer esign

11 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL. THIS VERSION: OCTOBER 31, G K (a) G K Fig. 4: A robust control framework for stabilizer esign an analysis. (a) Robust controller synthesis, where system uncertainty δ an optimality moel nonlinearities g an max(, ) are extracte into, all certain LTI ynamics are retaine in G, an a stabilizer K is esigne for stability/performance. (b) Robust stability/performance analysis, which can be use to assess the performance of a specifie stabilizer esign K in the presence of nonlinearities an uncertainties. for internally stable systems. Analogous results exist for lowgain integral control applie to LTI systems with input an output nonlinearities [57], [58] such as the graient term g(y, w). Other times, a particular stabilizer may suggest itself base on problem-specific jugment. 5 Regarless, given any chosen LTI stabilizer, one can apply integral quaratic constraint (IQC) theory to assess close-loop stability/performance (Figure 4b). See our previous work [37] for a result in this irection, an [59] for a complete an etaile exposition on IQC theory. If one s attention is restricte to linear time-invariant stabilizers, then a constructive stabilizer synthesis proceure can be formulate via robust control theory; we iscuss this approach in brief. If the inequality constraints (22) of the optimization problem are linear, an if we employ the optimality moel of Proposition 5.9 or Proposition 5.1, the only nonlinearities in the close-loop system are the static, slope-restricte nonlinearities max(, ) an g(, w). The function max(, ) satisfies a [, 1] slope restriction, while y g(y, w) satisfies a [κ, L] slope restriction if y g(y, w) is κ-strongly convex an y g(y, w) is L-Lipschitz continuous. As a result, we can manipulate the close-loop system into the stanar configuration of the robust control synthesis problem epicte in Figure 4a (see [6, Chapter 9] for backgroun). Stabilizer esign can then be accomplishe by escribing the block with integral quaratic constraints an applying stanar linear matrix inequality tools. In general, this proceure will result in a high-orer ynamic LTI stabilizer, the orer of which can then be reuce using stanar tools. For an example of linear-convex OSS controller esign using this approach, see our previous work [37]. 5 Interestingly, a special case may occur when the optimality error ɛ contains the control input u. If one can solve the equation ɛ = for u, then one obtains the control input which instantaneously zeros the optimality error; see (5) for an instance of this. (b) V I. A P P L I C AT I O N T O O P T I M A L F R E Q U E N C Y R E G U L AT I O N I N P O W E R S Y S T E M S This final section illustrates the application of our theory to a power system control problem. Our main objective is to work through the constructions presente in Section IV, an to simultaneously illustrate the many sources of esign flexibility within our propose framework. In particular, we will show that several centralize an istribute frequency controllers propose in the literature are recoverable as special cases of our framework. The ynamics of synchronous generators in a connecte AC power network with n buses an n t transmission lines is moelle in a reuce-network framework by the swing equations. The vectors of angular frequency (eviations from nominal) ω R n an real power flows p R nt along the transmission lines obey the ynamic equations M(δ) ω = P D(δ)ω Ap + u ṗ = B(δ)A T ω, (39) in which M(δ) is the (iagonal) inertia matrix, D(δ) is the (iagonal) amping matrix, A {, 1, 1} n nt is the signe noe-ege incience matrix of the network, B(δ) is the iagonal matrix of transmission line susceptances, P R n is the vector of uncontrolle power injections (generation minus eman) at the buses, an u R n is the controllable reserve power prouce by the generator. The iagonal elements of the inertia, amping, an branch susceptance matrices are uncertain but positive; for example, they coul be known within some bouns. See [61, Section VII] for a first-principles erivation of this moel. The incience matrix satisfies null A T = span(1 n ), an strictly for simplicity we assume that the network is acyclic, in which case n t = n 1 an null A = {}. We consier the optimal frequency regulation problem (OFRP), wherein we minimize the total cost i J i(u i ) of reserve power prouction in the system subject to networkwie balancing of supply an eman. We will consier two equivalent formulations, yieling two ifferent OSS control problems. In both formulations, we take steay-state operation of the plant (39) as an implicit constraint, as in the efinition of the optimization problem (22) for linear-convex OSS control. A. Economic Dispatch Formulation of OFRP The first formulation of the optimization problem requires balance between power supply an eman minimize J(u) := n J i(u i ) u i=1 subject to 1 T nu = 1 T np. (4) With state vector x := col(ω, p), the ynamics (39) can be put into the stanar LTI form (18) with matrices M(δ) A(δ) := 1 D(δ) M(δ) 1 A B(δ)A T M(δ) 1 M(δ) 1 B(δ) := B w (δ) :=.

12 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL. THIS VERSION: OCTOBER 31, Base on (4), we efine the output to be optimize as y := u. Therefore C = [ ], D = I n. We assume the measure output y m consists of the inputs u an the exogenous isturbance term 1 T np, so that y m = col(u, 1 T np ). As a consequence, the constraint violation 1 T nu+ 1 T np is measurable. We begin by etermining whether this OSS control problem satisfies the robust feasible subspace property of Definition 5.1. We first check whether the robust output subspace property (Definition 5.2) hols by constructing the matrix R(δ) as outline in Lemma 5.3. We construct a matrix N AB (δ) satisfying range N AB (δ) = null [ A(δ) B(δ) ] by examining M(δ) null 1 D(δ) M(δ) 1 A M(δ) 1 B(δ)A T One may verify that choosing N AB (δ) := 1 n I n (41) D(δ)1 n A yiels the require property. We fin that the matrix R(δ) = [ C D ] NAB (δ) is given by R(δ) = [ D(δ)1 n A ]. Because range A = (null A T ) = (span(1 n )) an 1 T nd(δ)1 n > for all δ, the matrix R(δ) is full column rank for all δ. Therefore, we may choose R := I n as a matrix satisfying range R = range R(δ) for all δ. The robust output subspace property therefore hols by Proposition Because the plant satisfies the robust output subspace property, we have access to the optimality moels for both the ROS property (Proposition 5.1) an the RFS property (Proposition 5.9). The ROS optimality moel (34) reuces to µ = 1 T nu + 1 T np, ɛ = J(u) + µ1 n, an applying Proposition 5.11, two possible OSS controllers corresponing to ifferent stabilizer choices are µ = 1 T nu + 1 T np η = J(u) + 1 n µ an, if each J i is strictly convex u = η (42) µ = 1 T nu + 1 T np, u = ( J) 1 ( µ1 n ). (43) In (42) the optimality error ɛ is integrate to zero by the internal moel, while in (43) we instea instantaneously zero ɛ through selection of u. The former can be consiere as a primalual algorithm (see [7]), while the latter woul be calle ual ascent. Both esigns are feeforwar OSS controllers, in that neither uses feeback from the system ynamics; in this application one is free to a aitional negative frequency feeback to the control. We omit the calculations for the RFS optimality moel (32) for this formulation of the optimization problem, as the RFS- OM will be illustrate for the secon formulation. We note, 6 One may also reach the same conclusion by observing that the full-rank conition of Proposition 5.6 hols for this instance of OSS control. however, that application of the RFS-OM recovers another control scheme from the literature. Inspire by approaches in multi-agent control, we introuce a connecte, weighte an irecte communication graph G c = ({1,..., n}, E c ) between the buses, with associate Laplacian matrix L c R n n. If the irecte graph G c contains a globally reachable noe 7, then 1 T ɛ = n u + 1 T np (44) L c J(u) is one option for an optimality moel. If, furthermore, G c is weight-balance, then from the optimality moel (44) we can recover the controller η = L c J(u) + 1 ( 1 T n n u + 1 T np ) 1 n, u = η. (45) Equation (45) is the controller [63, Equation (7)] specialize to an economic ispatch problem with ifferentiable cost function an without box constraints. B. Frequency Constraint Formulation of OFRP The secon formulation of the optimization problem explicitly requires zero steay-state frequency eviations: minimize J(u) := n J i(u i ) u,ω i=1 subject to F ω = r. (46) The matrix F R r n is assume to satisfy 1 n / null F. Options for the matrix F inclue F := I n, to enforce ω = n, or F := e T 1, to enforce ω 1 =, or F := c T, where c R n is a vector of convex combination coefficients satisfying c i an n i=1 c i = 1. We ientify the regulate output as y := col(u, ω). Therefore C := [ ] I n D := In. (47) We assume the measure output y m consists of the inputs u an the term F ω, so that y m = col(u, F ω). As a consequence, the constraint violation F ω is measurable. We ientify the matrix H of the engineering equality constraints in (22) as H := [ F ]. Using (41) an (47), we may calculate R(δ) = [ C R(δ) = D ] N AB (δ) to be ]. [ D(δ)1n A 1 n The subspace range R(δ) varies with δ, an therefore there cannot exist a fixe matrix R such that range R(δ) = range R for all δ. The robust output subspace property fails by Proposition 5.5. However, it is still possible that the robust feasible subspace property hols. To check whether this is the case, we first construct a matrix R (δ) R n 2n satisfying null R (δ) = range R(δ). We fin that selecting R (δ) := [ 1 n 1 T n (1 T nd(δ)1 n )I n ] yiels the require property. Following (3), we now ask whether there exists a fixe matrix T such that 1n 1 range T = null T n (1 T nd(δ)1 n )I n (48) F 7 See [62, Chapter 6] for etails.

13 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL. THIS VERSION: OCTOBER 31, for all δ. This null space is spanne by vectors of the form col(v, n ) where 1 T nv =. If L c is, as before, the Laplacian matrix of a communication igraph G c with a globally reachable noe, then L T T := c, is an eligible choice for T. Therefore, the optimization problem satisfies the robust feasible subspace property by Proposition 5.4. Noting that T T g(y, w) = [ L c ] J(u) = L c J(u), we apply Proposition 5.9 to obtain the optimality moel F ω ɛ =. (49) L c J(u) Therefore, one option for the linear-convex OSS controller of Proposition 5.11 is η 1 = F ω η 2 = L c J(u) u = K 1 η 1 K 2 η 2 K 3 ω, where K 1, K 2, an K 3 are gain matrices that shoul be selecte for close-loop stability/performance. With F = I n, K 1 = K 2 = 1 k I n for k >, an K 3 =, this esign reuces to the istribute-averaging proportional-integral (DAPI) frequency control scheme; see [3], [5], [16], [64], [65]. Other choices of F with this same stabilizer esign lea to various centralize/ecentralize controller esigns. The so-calle gather-an-broacast scheme of [8] can be recovere as follows. Assume that each J i is strictly convex, set F = c T as iscusse previously, an retain the integral controller η = c T ω, which integrates a weighte average of the frequency eviations. Next, select the input u in (49) to zero the secon component of ɛ: L c J(u) = n α R s.t. J(u) = α1 n α R s.t. u = ( J) 1 (α1 n ). Selecting α = η leas to the hierarchical gather-an-broacast controller η = n i=1 c iω i, u i (t) = ( J i ) 1 (η(t)). (5) In summary, many recent frequency control schemes can be recovere as special cases of our general control framework. The full potential of our methoology for the esign of improve power system control will be an area for future stuy. V I I. C O N C L U S I O N S We have efine an presente a etaile iscussion of the optimal steay-state control problem, wherein the goal is to regulate a combination of states an inputs of a nonlinear ynamical system to an optimal steay-state, in the presence of exogenous time-varying isturbances an moel uncertainty. Necessary an sufficient conitions for solvability of the problem were presente, along with a constructive esign framework that revolves aroun the introuction of an optimality moel, the purpose of which is to robustly prouce a proxy for the error between the optimize variables an their esire optimal values. This optimality moel converts the OSS control problem into an output regulation problem; one esigns an output-regulating controller for the plant an optimality moel in cascae to solve the OSS control problem. We then stuie in etail the special case of the linearconvex OSS control problem, wherein the plant is an uncertain LTI system, the exogenous isturbances are constant, an the optimization problem is convex with linear constraints. A complete controller esign proceure was presente, an two properties the robust feasible subspace an robust output subspace properties were ientifie as important for unerstaning cases where optimizing robustly with respect to parametric moelling uncertainty is achievable. Applying our linear-convex OSS proceures to a frequency regulation problem from power systems, we recovere a number of existing controller esigns from the recent literature. Immeiate future work will present the analogous iscretetime an sample-ata OSS control problems, along with a more etaile stuy of applications in power system control. 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15 SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL. THIS VERSION: OCTOBER 31, [63] A. Cherukuri an J. Cortés, Initialization-free istribute coorination for economic ispatch uner varying loas an generator commitment, Automatica, vol. 74, pp , 216. [64] M. Anreasson, D. V. Dimarogonas, H. Sanberg, an K. H. Johansson, Distribute control of networke ynamical systems: Static feeback, integral action an consensus, IEEE Transactions on Automatic Control, vol. 59, no. 7, pp , 214. [65] C. Zhao, E. Mallaa, an F. Dörfler, Distribute frequency control for stability an economic ispatch in power networks, in American Control Conference, Chicago, IL, USA, Jul. 215, pp Liam Lawrence (S 17) receive the B. Eng. egree in engineering physics from McMaster University, Hamilton, ON, Canaa in 217. He is currently a MASc stuent in the Electrical & Computer Engineering epartment at the University of Waterloo, Waterloo, ON, Canaa. He is a recipient of the NSERC Canaa Grauate Scholarship. His research interests inclue control, optimization, applie mathematics, an physics. A P P E N D I X Proof of Theorem 3.6: Suppose the controller (16) solves the output regulation problem for the filtere plant (14). One consequence of Theorem 2.3 is that there must exist a solution (π, π ξ, ψ u, ψ v ) to the regulator equations for the filtere plant (15). Define y m (w) := h m (π(w), ψ u (w), w) to be the steaystate plant measurements an y f (w) := col(π ξ (w), y m (w), ) to be the steay-state output of the filtere plant note that ɛ = since (π, π ξ, ψ u, ψ v ) solve the regulator equations for the filtere plant. Also by Theorem 2.3, we can conclue there exists a mapping π c such that t π c(w) = f c (π c (w), y f (w)) ψ u (w) = h u c (π c (w), y f (w)) ψ v (w) = h v c(π c (w), y f (w)) an the close-loop system corresponing to this controller is globally uniformly convergent with the UBSS property for the class of inputs I s (W ). Define the OSS controller with state col(ξ, x c ) an ynamics given by (17). Since the filter is an optimality moel an (π, π ξ, ψ u, ψ v ) solve the regulator equations for the filtere plant, it follows from Definition 3.5 that the pair (π, ψ u ) solve the OSS regulator equations (11). Therefore, by Theorem 3.4, there exist mappings π ξ, π c such that for the solution (π, ψ u ) of the OSS regulator equations, the OSS controller satisfies t π ξ(w) = ϕ(π ξ (w), y m (w), h v c(π c (w), y f (w))) t π c(w) = f c (π c (w), y f (w)) ψ u (w) = h u c (π c (w), y f (w)) an the close-loop system corresponing to this controller is globally uniformly convergent with the UBSS property for the class of inputs I s (W ). Employing Theorem 2.3 in the other irection, we conclue the OSS controller solves the optimal steay-state control problem. John W. Simpson-Porco (S 11 M 18) receive the B.Sc. egree in engineering physics from Queen s University, Kingston, ON, Canaa in 21, an the Ph.D. egree in mechanical engineering from the University of California at Santa Barbara, Santa Barbara, CA, USA in 215. He is currently an Assistant Professor of Electrical an Computer Engineering at the University of Waterloo, Waterloo, ON, Canaa. He was previously a visiting scientist with the Automatic Control Laboratory at ETH Zürich, Zürich, Switzerlan. His research focuses on feeback control theory an applications of control in moernize power gris. Prof. Simpson-Porco is a recipient of the IFAC Automatica Prize an the Center for Control, Dynamical Systems an Computation Best Thesis Awar an Outstaning Scholar Fellowship. Enrique Mallaa (S 9-M 18) is an Assistant Professor of Electrical an Computer Engineering at Johns Hopkins University. Prior to joining Hopkins in 216, he was a Post-Doctoral Fellow in the Center for the Mathematics of Information at Caltech from 214 to 216. He receive his Ingeniero en Telecomunicaciones egree from Universia ORT, Uruguay, in 25 an his Ph.D. egree in Electrical an Computer Engineering with a minor in Applie Mathematics from Cornell University in 214. Dr. Mallaa was aware the NSF CAREER awar in 218, the ECE Director s PhD Thesis Research Awar for his issertation in 214, the Center for the Mathematics of Information (CMI) Fellowship from Caltech in 214, an the Cornell University Jacobs Fellowship in 211. His research interests lie in the areas of control, ynamical systems an optimization, with applications to engineering networks such as power systems an the Internet.