Novel Control Approaches Based on Projection Dynamics

In this letter, our objective is to explore how two well-known projection dynamics can be used as dynamic controllers for stabilization of nonlinear systems. Combining the properties of projection operators, Lyapunov stability theory and LaSalle’s theorem, we confirm that the projection dynamics on the feasible set and tangent cone are Krasovskii passive. To show the effectiveness of the proposed approach, we use the projection dynamics on the tangent cone for stabilizing boost converters in a DC microgrid while satisfying predefined input constraints.

system and interconnect it with the plant in a power-preserving way. Then, the stability of the closed-loop system can be analyzed relying on Lyapunov theory and invariance principle [4]. Several works on the topic focus on studying the passivity properties of continuous dynamical systems, whereas a limited number investigates the passivity properties of discontinuous dynamics [5]. Among the most typical discontinuous dynamics, projection dynamics are wildly used in mathematical programming, algorithm design [6], and controller design [7], [8]. Due to the deep relation between a projection operator and a variational inequality (VI) [9], projection dynamics are often used to solve mathematical programming problems, such as optimization and game problems [10]. In fact, in [11], the authors show that the equilibrium of the projection dynamics and the solution of a VI coincide. Furthermore, they also establish convergence by employing Lyapunov theory and the properties of the projection operator. With the advent of distributed systems, the focus shifted to combining the projection dynamics with distributed optimization [12] and control [13], [14], [15].
In summary, the research on projection dynamics in the mathematical programming field is exhaustive [16]. However, its potential to control nonlinear systems has not yet been fully explored, e.g., the property of Krasovskii passivity has not been studied in previous works, see [8], [17] for details on the topic. In this letter, we aim at investigating how two types of well-known projection dynamics can be used for controlling nonlinear systems. Particularly, we analyze their passivity properties. The main contributions are as follows.
• Compared to previous results [7], we analyze the interconnection of a nonlinear continuous-time dynamical system with projection dynamics. • In Section III, under some conditions on the Jacobian of the dynamics, we prove the asymptotic stability of the closed-loop system relying on monotonicity. • In Section IV, we consider passive nonlinear systems and show that the system dynamics projected on the tangent cone of a polyhedral convex and compact set are Krasovskii passive. Such a property is fundamental to establish convergence of the interconnected closed-loop system via standard passivity arguments. • We design a new controller for the boost converters of a direct current (DC) microgrid [18], ensuring that the control input remains within a predefined set.

II. PRELIMINARIES
To make this letter self-contained, we summarize in this section the main definitions and basic properties of the projection operator and monotone maps that we will use throughout this letter. After that, we state the problems studied in this letter.

A. Projection Operator
Hereafter, define as a nonempty, closed, and convex subset of R n , Let s ∈ R n and z ∈ R n , then the projection of z on is defined as P (z) = argmin s∈ z − s . According to [9,Th. 1.5.5] and [11,Th. 3.2], the following facts hold for any two vectors z, s ∈ R n .
Fact 1 (Projection properties): 3) The projection is non-expansive, i.e., and also co-coercive, i.e., Let T(z, ) be the tangent cone of at z ∈ , as defined in [9,Pg. 15]. Then, T (z, H(z)) represents the function projecting the vector-valued function H(z) onto T(z, ). Let N (z) denote a normal cone at z ∈ by Then, the inwards normals to at z ∈ can be defined as The following facts about the tangent cone hold.

B. Monotonicity and Variational Inequality
We next recall some definitions on monotone operators [9, Definition 2.3.1]. For all z, s ∈ , a map H : If ξ = 2, we say that it is strongly monotone. Now, we recall the definition of the variational inequality [9, Definition 1.1.1]. Given a feasible set ⊂ R n and a map H : → R n , VI(H, ) amounts to the problem of finding all z ∈ satisfying < s >≥ 0, ∀s ∈ . The solution set of VI(H, ) is denoted by Sol(H, ).
Monotonicity is often used to establish the existence and uniqueness of the solution set of the VI. We next introduce some classic results from [

C. Problem Formulation
Consider the nonlinear plant dynamicṡ where x ∈ R n , u ∈ R m , and y ∈ R p denote the state, control input, and output, respectively. The maps f : R n × R m → R n and h : R n → R p are continuously differentiable. Now, we formulate two control problems, which are solved in Sections III and IV, respectively. Pr. 1: Given the dynamical system (5), consider a projection-based controller associated with a natural map C [9, Sec. 1 where F : R p × R m → R m is a vector-valued map. Derive a condition such that a desired equilibrium (x * , u * ) of the closed-loop system is asymptotically stable. Pr. 2: Given the dynamical system (5) with m = p, establish Krasovskii passivity properties for the projection-based dynamic controller [19, eq. (2.2)], [5, eq. (5)] where G : In Problems 1 and 2, we investigate different projectionbased controllers, and their relations are explained below. In Problem 1, we provide a control design method that is applicable to a general class of plants (5). On the other hand, sometimes control design is simplified by utilizing the fundamental properties of plants. A representative method is passivity-based control, which illustrates that passive plants can be stabilized by passive feedback controllers. Motivated by this, we mention in Problem 2 that the controller (7) has kinds of passivity properties, which can be beneficial for the stabilization of a plant possessing the corresponding passivity property. The following remark introduces the relationship between the above two projection-based controllers.
Remark 1: According to the definition of the convex set, we can deduce that C(y, u) ∈ T(u, ) holds for all u ∈ . If is a polyhedron, then the feasible cone of at arbitrary u ∈ coincides with T(u, ) (refer to [9, Lemma 3.3.6]). Hence, if u − F(y, u) / ∈ T(u, ), then P (u − F(y, u)) must be on the boundary of T(u, ), which further implies that Moreover, if u − F(y, u) ∈ , then the following equality holds. The above equalities (8) and (9) show that the two controllers (6) and (7) are similar in some specific cases.

III. PROJECTION DYNAMICS ON THE FEASIBLE SET
In this section, we solve Problem 1. To this end, we rely on the following blanket assumptions, where A 0 (A 0) implies that the symmetric part of the matrix A is positive definite (semi definite).
Assumption 1 (Initial Value and Continuity): The initial value satisfies u(t 0 ) ∈ , and the vector-valued function

Assumption 2 (Positive Semi Definiteness of the Jacobian):
The Jacobian matrix of col(f (x, u), F(y, u)), denoted by Before proceeding with the closed-loop analysis, we explain that the set of equilibrium points (x * , u * ) of the closedloop system (5)-(6) is non-empty, and it coincides with the following set According to Fact 3.1 and [9, Proposition 2.3.2], Assumption 2 ensures that Φ is non-empty, bounded, and convex. Combining the fact that is convex with [9, Proposition 1.5.8], we conclude that Φ coincides with the set of equilibria of the closed-loop system. Next, we introduce the convergence results of the closed-loop system (5)- (6).
Proof: We first show that K is a positively invariant set for the system (5)- (6). We consider the distance of (x, u) ∈ R n × R m to K = R n × . This distance is equivalent to a distance of u ∈ R m to , which is nothing but D(u) introduced in Fact 1.4. By virtue of Fact 1.4 and taking the time derivative of D(u) along (6), it follows thaṫ where the last inequality holds by virtue of Fact 1.2 and the fact that P (u − F(y, u)) ∈ . Inequality (10) implies that D(u(t)) is non-increasing for all t ≥ t 0 . Combining this with Assumption 1, we can conclude that D(u(t 0 )) = 0 and u(t) = P (u(t)) holds for all t ≥ t 0 , which implies that K is a positively invariant set for the closed-loop system (5)-(6). Next, construct the following storage function where (x * , u * ) is an arbitrary point in Φ. The above analysis shows that if u(t 0 ) ∈ , then u(t) ∈ holds for all t ≥ t 0 . Based on such a fact, we can deduce that holds for all t ≥ t 0 , where the last inequality is based on Fact 1.2. By substituting (12) into (11), it follows that Combining this result with the fact that S I (x, u) is continuous, we can further deduce that, for arbitrary bounded initial value (x(t 0 ), u(t 0 )) ∈ K , there always exists an r > 0 such that S I (x, u) > S I (x(t 0 ), u(t 0 )) whenever col(x, u) > r. As a consequence, we can deduce that Next, we show thatṠ I (x, u) ≤ 0 holds for all t ≥ t 0 . By employing Fact 1.4 and the chain rule, it follows thaṫ Note that, by combining the fact that u(t) ∈ for all t ≥ t 0 , the inequality (2), and Assumption 1, it follows that the sum of the second, third, and fourth elements in (13) satisfies C(y, u) + F(y, u), C(y, u) + u − u * ≤ 0.
Moreover, Assumption 2 guarantees that the first item on the right-hand side of (13) is negative semi-definite. Combining this fact with (13)- (14), we can conclude thaṫ Recall that is nonempty, closed, and convex, and Assumption 2 guarantees that the inequality <F(y * , u * ), u − u * >≥ 0 holds for all u ∈ and (x * , u * ) ∈ Φ, where y * = h(x * ). Therefore, we havė By virtue of Assumption 2 and using [9, Proposition 2.3.2], we deduceṠ I (x, u) ≤ 0 for all t ≥ t 0 . Finally, we show the convergence. Combing (14)- (16) with Assumption 2, we can deduce that the first item and the sum of the later four items in (13) are non-positive, which further implies thatṠ I (x, u) = 0 if and only ifẋ = 0 andu = 0. Thus, for a given bounded (x(t 0 ), u(t 0 )) ∈ K , we can always find a bounded c such that the trajectory starting at it converges to Φ ∈ c . Then, we complete the proof by combining the LaSalle's theorem introduced in [4, Th. 4.4]. Now, by virtue of Assumption 3, the next theorem shows that the equilibrium of the system (5)-(6) is unique and asymptotically stable.
Proof: Combining (11)- (12) and the analysis in the proof of Theorem 1, it follows that S I (x, u) > 0 holds for all (x, u) ∈ K \Φ and S I (x, u) = 0 for (x, u) ∈ Φ. Since J(x, u) 0 holds for all (x, u) ∈ K , then it follows that Φ is a singleton by using Fact 3.2 and [9, Proposition 2.3.2]. Following the analysis in (13)

IV. PROJECTION DYNAMICS ON THE TANGENT CONE
In this section, we solve Problem 2. Namely, we mention that the projection-based dynamic controller (7) on the tangent cone of can be either shifted or Krasovskii passive under suitable assumptions. Consider the projection-based dynamic controller (7) on the tangent cone of with, where g : R m → R m is differentiable. We assume that the following two assumptions hold for the controller dynamics.
In general, the equilibrium setsΦ for this setup is different from Φ defined in Section III.
Assumption 5 (Monotonicity): The function g(u) is monotone for all u ∈ .

A. Shifted Passivity
To make this letter self-consistent, we present the following proposition, proven in [5, Sec. "Projected-gradient play and passivity"], that ensures that the projection-based dynamic controller (7) has the following shifted passivity property [1,Sec. 4.7].
Proposition 1 (Shifted Passivity): If Assumptions 1, 4, and 5 hold for the controller dynamics (7), then S s (u) = for any (y, u) ∈ R m × and any (y * , u * ) ∈Φ. By Proposition 1, the controller dynamics (7) is shifted passive with respect to the storage function S s (z), input −y, and output u. Now, we suppose that the system (5) is shifted passive with respect to the input u and output y, i.e., satisfiesV s (x) ≤ y − y * , u − u * for some storage function V s (x). Then, the closed-loop system satisfiesV s (x) +Ṡ s (u) ≤ 0. Based on this inequality, one can proceed with the closedloop stability analysis by invoking the results of the passive interconnection analysis, e.g., Lyapunov and LaSalle's theorems.

B. Krasovskii Passivity
When is a polyhedral set, we can also show that the controller (7) is Krasovskii passive [8,Definition 2.8].
Theorem 3 (Krasovskii Passivity): If is a polyhedron, and Assumptions 1, 4, and 5 hold for the controller dynamics (7), then the following scalar-valued function in the sense of Carathéodory for almost all t ≥ t 0 , all u(t 0 ) ∈ , and all continuously differentiable y : R → R m . Proof: According to the analysis in [19,Pg. 27], if u(t 0 ) ∈ holds, then we have u(t) ∈ holds for all t ≥ t 0 and for all continuously differentiable y(t) ∈ R m , t ≥ t 0 . Next, we analyze all the possible scenarios for the trajectory of (7). First, if u(t) ∈ int( ), then it follows thaṫ Combining Assumption 5 with (20), we obtain (19). Second, we consider the case that u(t) located on bnd( ) is moving to int( ). In such a case, the right-hand side of (7) is continuous, and thus (19) follows from the analysis in (20). Third, if u(t) is switching from int( ) to bnd( ), the storage function S k (y, u) is not differentiable. However, following the analysis in [21,Lemma 4.2] and combining (1), we can deduce that S k (y, u) is non-increasing during the switching. Since is polyhedral, and F(y, u) is continuous, the number of switching from int( ) to bnd( ) is finite. Finally, according to Fact. 2.2, when u(t) is moving on bnd( ), the dynamic (7) reduces tȯ where A p (u, ) represents the projection matrix projecting −F(y, u) on bnd( ). Using this representation (21), we havė Combining the fact that A p (u, ) is idempotent with (22) and using Assumption 5, we complete the proof. Theorem 3 implies that the controller dynamics (7) is Krasovskii passive with respect to the storage function S k (y, u), input −ẏ, and outputu. A similar discussion as shifted passivity holds for the closed-loop stability analysis. That is, suppose the system (5) is Krasovskii passive with respect to the inputu and outputẏ, i.e., satisfiesV k (x, u) ≤<ẏ,u> for some storage function V k (x, u). Then, the closed-loop system satisfiesV k (x, u) +Ṡ k (y, u) ≤ 0, and we can proceed with the stability analysis based on this inequality and invoke Lyapunov and LaSalle's theorems. Note that Krasovskii passivity is a property ofu andẏ. Thus, instead of the original output h(x) of the system (5), we can feed its shifted signal h(x) +ȳ by arbitrary constantȳ into the input y of the controller dynamics (7); the closed-loop system can still be interpreted as the feedback interconnection of a Krasovskii passive plant and controller. A similar discussion holds for u.

V. SIMULATION
In this section, we use the projection dynamics (7) and Krasovskii passivity to design a controller for a DC microgrid with 4 nodes in a ring topology, where each node includes a boost converter supplying a constant impedance and a constant current load. Let z ∈ R n , then we define [z] diag(z 1 , . . . , z n ). The dynamics of the considered microgrid can be expressed as in [18], i.e., where the used symbols are explained in Table I. The network is described by an undirected ring graph with incidence matrix denoted by B ∈ R 4×4 . For convenience, let x col(I, V, I l ) and rewrite (23) as the following compact form where Q 0 is a diagonal matrix that can be easily attained by inspection of (23). Next, we analyze the passivity property of (24). Consider the storage function H k (x) = 1 2 ẋ 2 Q , which satisfiesḢ Our goal is to design a controller such that (24) converges to a desired equilibrium and the control input solves the following optimization problem where u r ∈ denotes the reference control input, E 0, and ε > 0. Note that (25) has a unique solution u = u r . Now, we show the closed-loop system consisting of (24) and a controller based on (7), i.e., where λ ∈ R n is the Lagrange multiplier, andu andλ are interconnected in a passive way. Next, we construct the following storage function where the inequality follows from A − Y(u) 0. Let σ min (·) and σ max (·) denote the smallest and largest eigenvalue of a matrix, respectively. If for all t ≥ t 0 , thenV s (x, u, λ) ≤ 0 holds for almost all t ≥ t 0 , since E 0 and Z −1 L 0. Note that, since (26b) has a unique equilibrium u * = u r , then x * and λ * = −ε[I * ]V * are also unique. By combining the above analysis and the result in [4, Lemma 4.1], we then prove that (26) converge to the unique equilibrium in the sense of Carathéodory if ε satisfies (28) for all t ≥ t 0 . Next, we introduce the detailed parameter settings. The parameters of all the distributed generator units (DGUs) are reported in Table II Although such a selection may cause a deviation from the desired value, it avoids to require information on I * . Then, we obtain u r = col(0.2923, 2932, 0.2941, 0.2950). Moreover, we set = [u r − 10 −3 1, u r + 10 −3 1]. From Figure 1(a), we observe that the control input u is always within the feasible set . Figure 1(b) shows thatλ converges to 0 after a short transient, which also confirms that the deviation between the control input and the reference value converges to 0. Figure 2 shows that the currents and voltages of all the DGUs converge towards the desired values within a short time. Figure 3(a) shows the transmission line currents. Since we neglect R in the computation of u r , we observe from    It is worth noting that shifted passivity can also be used for control design. However, such a controller requires information on the equilibrium x * including the load parameters I L and Z L that are usually unknown. In contrast, the proposed Krasovskii passivity-based controller can be implemented only by knowing the desired voltage. Compared with the controller in [18], the proposed controller does not require information onẋ, which makes it easier to be implemented and less sensitive to measurement noises.

VI. CONCLUSION
Projection dynamics can be connected to a differentiable dynamical system, ensuring asymptotic stability of the closedloop system under certain assumptions on the associated Jacobian matrix. The projection dynamics on the tangent cone of a convex set are passive and, in the case of a polyhedral set, Krasovskii passive. These findings are valuable for designing controllers for nonlinear systems, such as power converters, to meet operational constraints. However, further investigation is needed to understand the impact of relaxing these assumptions on the convergence of the closed-loop system