Further Results on the Control Law via the Convex Hull of Ellipsoids

Recently, a new Lyapunov function based on the convex hull of ellipsoids was introduced for the study of uncertain and/or time-varying linear discrete-time systems with/without constraints. The new Lyapunov function has many attractive features such as: 1) the associated ellipsoids are not required to be robustly invariant; 2) the design conditions are formulated as linear matrix inequality constraints. The control law is obtained by solving a convex optimization problem online. This optimization problem generally does not have a closed-form solution, and hence it is solved by numerical methods. In this article, we intend to complement the results by analyzing the geometric structures of the solution to the optimization problem, and of the control law. In particular, we show that the control law is a piecewise linear and Lipschitz continuous function of the state.


I. INTRODUCTION
Lyapunov functions play a central role in the study of dynamical systems, and the construction of Lyapunov functions is one of the most fundamental problems in systems theory.The most direct application is stability analysis, but similar problems appear in performance analysis, and controller design.Consequently, methods for constructing Lyapunov functions are of great theoretical and practical interest.
For uncertain and/or time-varying linear discrete-time systems, the most popular types of Lyapunov functions are piecewise linear functions [3] and quadratic functions [4].Piecewise linear functions have been involved mostly in control problems with state and input constraints [2], [3], [6], [9].Their main strengths are: 1) the arbitrary approximation of the domain of attraction; 2) various analysis and design problems can be transformed into algebraic problems.The main weakness of piecewise linear functions lies in the construction of the corresponding polyhedral sets.In general, this is a difficult problem, especially for high-dimensional systems.In contrast with piecewise linear functions, quadratic functions are tractable because of the existence of the linear matrix inequality (LMI) technique.Combining the quadratic functions and the LMIs, several analysis and design problems can be converted into a convex optimization problem.However, the results obtained by the quadratic functions can be conservative.It is well known [11] that there are stable systems that are not quadratically stable, and stabilizable systems that are not quadratically stabilizable.
Nguyen [10], in a recent publication, proposed a new Lyapunov function that is based on the convex hull of ellipsoids.The new Lyapunov function has several advantages over the standard quadratic and piecewise linear Lyapunov functions.Compared to the quadratic one, the new Lyapunov function reduces the conservativeness as it does not require the quadratically stable or stabilizable assumptions.Compared to the piecewise linear Lyapunov function, the design conditions are formulated as LMI constraints.Hence, the new Lyapunov function overcomes the main challenge of the piecewise linear Lyapunov function.It is legitimate to say that the new Lyapunov function goes for the best of both quadratic and piecewise linear functions worlds.
Using the new Lyapunov function, the control law is obtained by solving a convex optimization problem online.This optimization problem does generally have a closed-form solution.Hence, the inherent structure that the new control strategy imposes on the controller is unclear.The objective of this article is to complement the results in [10].We will analyze the geometric structures of the solution of the optimization problem, and of the control law.
Note that the problem of studying the geometric structures of a solution to an optimization problem is known in the literature as multiparametric programming [1], [5].However, the efforts focus mostly on linear and quadratic optimization problems.
The rest of this article is organized as follows.Section II covers notation and preliminaries.Section III is dedicated to the problem formulation.Section IV is concerned with the question of the uniqueness of the solution.Then, in Section V, geometric structures of control law are presented.One simulated example is evaluated in Section VI.Finally, Section VII concludes this article.

II. NOTATION AND PRELIMINARIES
Notation: For a given set C, its boundary is denoted as Fr(C).We denote by 0 n /I n n × n zero/identity matrices, and by 0 the zero matrix of the appropriate dimension.A positive definite matrix P is denoted by P 0. We denote by R the set of real numbers, by R n×m the set of real n × m matrices, and by S n the set of n × n positive definite matrices.For a given P ∈ S n , E(P ) represents the following ellipsoid: We use 1, s to denote the set {1, 2, . . ., s}.The convex hull of ellipsoids E(P 1 ), E(P 2 ), . . ., E(P s ) is denoted as P is the smallest convex set containing E(P i ), ∀i = 1, s.For any x ∈ P, there exist v i and λ i , i = 1, s such that where v i ∈ E(P i ), s i=1 λ i = 1, and λ i ≥ 0 ∀i = 1, s. Definition 1 (Redundant Ellipsoid): For a given set P = Co(E(P i )), i = 1, s, the set P −j is defined by removing the jth ellipsoid E(P j ) from Co(E(P i )), i.e., The ellipsoid E(P j ) is redundant if and only if

Definition 2 (Minimal Representation):
The set P = Co(E (P i )) ∀i = 1, s has the minimal representation if and only if the removal of any ellipsoid would change P, i.e., there are no redundant ellipsoids.
Clearly, the minimal representation of P can be achieved by removing all redundant ellipsoids.Definition 3 (Supporting Hyperplane): For a given vector β ∈ R n , and a given convex set C, the hyperplane β T x = 1 is a supporting hyperplane of C if and only if β T x ≤ 1 ∀x ∈ C, and there exists at least one point x 0 ∈ Fr(C) such that β T x 0 = 1.
If C is an ellipsoid, then x 0 is unique [8].If C is the convex hull of ellipsoids, i.e., C = P, then there are several x 0 ∈ Fr(P) such that β T x 0 = 1.To characterize the set of x 0 , we recall the following two definitions [14].
Definition 4 (Face): A face of P is the intersection of P with a supporting hyperplane of P.
Definition 5 (Extreme Point): A point v ∈ Fr(P) is an extreme point of P if it cannot be represented as a convex combination of other points in P.
For a given x ∈ R n , its Euclidean norm is denoted as ||x||, i.e., ||x|| = √ x T x.To analyze the control law, we recall the following definition [15].
Definition 6 (Liptchitz Continuous): A function f : R n → R m is called Liptchitz continuous if there exists a positive scalar M such that, Any such M is referred to as a Liptchitz constant.

III. PROBLEM FORMULATION
In this section, we first summarize the results in [10].We then formulate the problems that need to be solved.
Consider the following uncertain and/or time-varying linear discretetime system: where where The vector of unknown and/or time-varying parameters One of many simple ways to control the system (6) is to employ a linear state feedback control law u(k) = Kx(k) and an associated quadratic Lyapunov function V (x) = x(k) T P −1 x(k).In this case, it is well known [4] that the problem of finding K and P can be converted into a convex semidefinite program (SDP).However, requiring the existence of a linear control law and a quadratic function can be quite restrictive.This is because the same control gain and the same Lyapunov matrix must verify for all vertices of the uncertain domain (7).
In [10], to overcome the conservative weakness of the quadratic Lyapunov function and of the linear control law, the convex hull of quadratic functions P = Co(E(P i )), P i ∈ S n and the associated matrix gains K i ∈ R m×n , i = 1, s are employed.It was shown that if the following LMI conditions, are satisfied, then P is robustly invariant for (6).Using (9), it should be noted that the associated ellipsoids E(P i ) are generally not robustly invariant.The gains K i = Y i P −1 i are also not robustly stabilizing.From this point on, using the conditions (9), it is assumed that P i , as well as At time instant k, for a given state x(k) ∈ P, the control action is computed as where λ * i (k) and v * i (k) are a solution of the following optimization problem: Using ( 2) and ( 3), problem (11) is feasible ∀x ∈ P. It was proved [10] that the control law (10), (11) guarantees recursive feasibility, i.e., is a Lyapunov function for the closed-loop system ( 6), (10), and (11).Hence, the control law ( 10), (11) robustly asymptotically stabilizes the system (6).
The optimization problem (11) is nonlinear and nonconvex due to the multiplication of λ i , v i .Using a change of variables, we can reformulate (11) as a convex optimization problem, for which there exists an efficient solver [10].
Problem (11) might have multiple solutions since the cost function ( 11) is linear.Multiple solutions are undesirable.This is because the control law might be a discontinuous function of the state due to a fast switching between the different control actions.It is well known [13] that in the presence of a small measurement noise, robustness cannot be guaranteed generally for a closed-loop system with a discontinuous control law.
In this article, we aim to answer the following three questions.Q1: What conditions need to be hold for (11) to have a unique solution?Q2: The implementation of the control law ( 10) is based on solving online the optimization problem (11).This problem generally does not have an analytical solution.Therefore, What is the form of the function u(x)?Q3: Is the control law u(k) = u(x(k)) a Liptschitz continuous function of the state?If yes, what is the Lipschitz constant?Remark 1: The results in [10] were obtained with/without state and input constraints.Because the aim of this article is to answer the questions Q1, Q2, Q3, these constraints are not considered here for simplicity.

IV. GEOMETRICAL PROPERTIES OF THE SOLUTION
In this section, we aim to answer the question Q1 by revealing the geometrical properties of the solution of (11).For this purpose, we will first propose a procedure to eliminate redundant ellipsoids from the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.convex hull of ellipsoids.We will then study the geometrical properties of the solution.

A. Removing Redundant Ellipsoid
The objective of this section is to propose a procedure to remove redundant ellipsoids in the convex hull of ellipsoids.This redundancy elimination has the following two purposes.1) To reduce the online computational burden of the optimization problem (11).Obviously, the online computational burden is lower if the number of constraints in ( 11) is smaller.
2) It will be shown that if there is no redundant ellipsoid in the convex hull of ellipsoids, then the solution is unique.For a given set of matrices P i ∈ S n , i = 1, s, consider the following optimization problem: Denote γ * i ∀i = 1, s; i = j as an optimal solution of ( 12).The following theorem holds.
Theorem 1: The ellipsoid E(P j ) is redundant in Co(E(P i )) if and only if s i=1,i =j γ * i ≤ 1. Proof: Consider the set P −j in (4).Using the proof of [10, Th. 1], it follows that P −j can be parameterized as In other words, x belongs to P −j if and only if there exist γ i satisfying (13) such that Using ( 5) and ( 14), it follows that E(P j ) is redundant if and only if ∃γ i satisfying (13), i = 1, s, i = j, such that: or equivalently P j s i=1,i =j γ i P i .This completes the proof.Remark 2: Theorem 1 is the first one that provides a convex condition to verify if a given ellipsoid is redundant in the convex hull of ellipsoids.To the best of the author's knowledge, there does not exist any condition in the literature.
Using Theorem 1, Algorithm 1 can be used to remove redundant ellipsoids in P = Co(E(P i )), i = 1, s.

B. Uniqueness of Solution
Using Algorithm 1, it is assumed that P = Co(E(P i )), i = 1, s has the minimal representation.The main aim of this section is to show that with this assumption, the solution to (11) is unique.
The following result concerns a geometrical property of the optimal solution.
Lemma 1: For a given state x(k), (λ * i , v * i ) is an optimal solution of (11) if and only if

1)
i is a solution of (11) if and only if either Consider now the case when x(k) is strictly inside P. If x(k) = 0, then g * = 0, and λ * i = 0, v * i = 0 ∀i = 1, s. Otherwise there exists 0 < g * < 1 such that x f (k) ∈ Fr(P), where x f (k) = 1 g * x(k) (see Fig. 1).Define λ f,i = 1 g * λ i ∀i = 1, s. Rewrite the problem (11) as Since x f ∈ Fr(P), one has s i=1 λ * f,i = 1.It follows that: One also has either The proof is complete.
Remark 3: Using the proof of Lemma 1, three observations can be made as follows.1) One has x(k) = g * x f (k), 0 ≤ g ≤ 1.Hence, x(k) lies on the line segment joining x f (k) and the origin.
2) The level sets of the optimal value function are given by scaling the boundary of P.

3) For any
In other words, for any i = 1, s if the ellipsoidal constraint (v i ) T P −1 i v i ≤ 1 is active, then the constraint λ i ≥ 0 is inactive, and vice versa.For a given x(k), if there is only one active ellipsoidal constraint, i.e., there exists only one index 1 ≤ j ≤ s such that v * j P −1 j v * j = 1 and v * i = 0, ∀i = 1, s, i = j.Then one obtains In addition, if x(k) ∈ Fr(P), then λ * j = g * = 1 and v * j = x(k).In this case, x(k) is an extreme point of P, as it cannot be represented as the convex combination of other points in P.
Consider now the case where we have more than one active ellipsoidal constraint.Without loss of generality, it is assumed that for a given x(k), the first s a ellipsoidal constraints are active, 2 ≤ s a ≤ s, i.e., The following result holds.Lemma 2: The optimal solution v * i ∀i = 1, s a and x(k) g * belong to the same supporting hyperplane of P, i.e., where 0 < g * ≤ 1 is a scalar such that x(k) g * ∈ Fr(P).The normal vector β * ∈ R n satisfies the following set of equations: Proof: Following the proof of Lemma 1, if v T i P −1 i v i = 1 ∀i = 1, s a , then one gets: Rewrite the problem (11) as Consider the Lagrange function ( The factor 1/2 introduced in the Lagrange function is for the scaling purpose.An optimal solution (g * , β * , η * i , e * i , ρ * , μ * i ) satisfies the following conditions: Using (21), one gets It follows that: Substituting ( 22) to (21), one obtains Because (e * i ) Using (20) and sa i=1 η * i = 1, it follows that: Using (25), one has ρ * = g * .As a consequence, using (21) and ( 23) Hence, v * i ∀i = 1, s a and x g * belong to the same supporting hyperplane.
Using (22) and since η * i = g * μ * i , one gets e * i = g * P i β * .Using the fact that (e * i ) T P −1 i e * i = (g * ) 2 , one obtains The proof is complete.Remark 4: For β * given in (17), on has The hyperplane (β * ) T x = 1 touches the ellipsoid E(P i ) at the extreme point v * i = P i β * ∀i = 1, s a .Hence (β * ) T x = 1 is a supporting hyperplane of E(P i ) ∀i = 1, s a .Because there is no redundant ellipsoid in P, it follows that v * i ∀i = 1, s a are also extreme points of P.
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We are now ready to state the main theorem of this section.Theorem 2: If P = Co(E(P i )), i = 1, s has the minimal representation, then the solution of ( 11) is unique.
Proof: For a given x(k), using Theorem 3, the normal vector β * of the supporting hyperplane can be found by solving the set of (17).It is clear that one needs at most n equations in (17) to obtain β * , since β * ∈ R n .It follows that the number of points v * i = P i β * is at most equal to n for the given supporting hyperplane (β * ) T x = 1.Combining with the fact that v * i ∈ R n are extreme points of P, i.e., they cannot be represented as the convex combination of other points in P, one concludes that v * i are linearly independent.Now, suppose, on the contrary that x(k) can be decomposed as where q ≤ n, and q i=1 λ i = q i=1 ζ i .Using (27), one obtains Because v * i are linearly independent, (28) holds if and only if λ i = ζ i ∀i = 1, q.In other words, the solution of ( 11) is unique.
Remark 5: The number of active extreme points q in (27) can be different from the number of active ellipsoidal constraints s a in (18).Indeed, one always has q ≤ s a .This is because two or many nonredundant ellipsoids can share the same extreme points.For example, consider the following matrices P 1 , P 2 : It is clear that the vector [1 0] T is an extreme point of both P 1 and P 2 .Hence, if x(k) = [1 0] T , then both ellipsoidal constraints are active, i.e., p a = 2.However, there is only one active extreme point [1 0] T , i.e., q = 1.

V. GEOMETRICAL PROPERTIES OF THE CONTROL LAW
In this section, we aim to answer the questions Q2 and Q3 by studying the geometrical properties of the control law.

A. Explicit Control Law
Definition 7 (Dimension of Face): Suppose that a given supporting hyperplane intersects with P at q extreme points, q ≥ 1.A face of dimension q − 1 is the convex hull of all the q extreme points.
Following the proof of Theorem 2, one has q ≤ n.The boundary of P is the union of faces of dimension 0, 1, . . ., n − 1.For example, in R 2 , the boundary of P is composed of elliptical arcs and line segments.The elliptical arcs have dimension 0, and the line segments have dimension 1.In R 3 , the boundary of P consists of the following three different kinds of faces. 1) Elliptical faces, which are parts of the ellipsoids.The dimension of the elliptical faces is 0. 2) Conical faces, which are the convex hull of two extreme points.
The dimension of the conical faces is 1. 3) Planar faces, which are the convex hull of three extreme points, i.e., triangles.The dimension of the planar faces is 2. The following definition is borrowed from [12].Definition 8 (Critical Region): A critical region (CR) is the set of all states x that have the same set of active extreme points for the optimization problem (11).
For example, the origin is a CR, because, in this case, the set of active extreme points is empty.Otherwise, for any x = 0, there is always at least one active extreme point.Consider an extreme point v, such that a corresponding supporting hyperplane contains only this point.Using Remark 3 -point 1, it follows that v is active for any x = 0 belonging to the line segment connecting v and the origin.Hence, this line segment is a CR.Note that the origin is excluded from this CR.
Consider the case x(k) ∈ Co −0 (0, v i ) where v i ∈ E(P i ) is an extreme point of P, i = 1, p.In this case, x(k) is rewritten as The control action is computed as If x(k) = 0, then using the control law (30), one has u(k) = 0. Hence, ∀x ∈ Co(0, v i ), the control law is (30).Without loss of generality, consider now the case where the first q extreme points v i ∈ E(P i ), ∀i = 1, q are active, 2 ≤ q ≤ n, and One has where λ i ≥ 0, i = 1, p. Rewrite (31) in a compact vector form as where Because v 1 , v 2 , . . ., v q are linearly independent, one gets rank(V ) = q.Using the singular decomposition (SVD), rewrite the matrix V ∈ R n×q as follows that the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
diagonal elements of S v are positive, and the matrix S v is invertible.Using (32) and (33), one obtains The control action is computed as Thus, with Combining with (34), one obtains where The proof is complete by noting that if x(k) = 0 then using (35), one gets u(k) = 0. Hence, for any x ∈ Co(0, v 1 , v 2 , . . ., v q ), the control law is (35).Remark 6: We separate two cases with one, and with more than one active extreme point only for clarity.The SVD technique (33) also works with one active extreme point.
Remark 7: The problem of constructing offline the control law (35) in R n is complex.This is because one needs to list all the faces of dimension q − 1, 1 ≤ q ≤ n of the convex hull of ellipsoids.Note that it is enough to consider q ≥ 2, since the faces of dimension 0 are elliptical faces, which are parts of the ellipsoids.The faces are computed by finding the associated extreme points, or equivalently, the normal vectors.They are in turn computed by solving a set of (17).It is clear that this is a combinatorial problem, which is NP hard.A possible way to reduce the computational burden is to construct an approximate solution, for example, as in [7].However, this is beyond the scope of the article.
Remark 8: With a slight abuse of notation, a partition of dimension q is the convex hull of the origin and of a face of dimension q − 1, 1 ≤ q ≤ n.Note that except the partitions of dimension n in R n , the other partitions are degenerate.Consider now the partitions of dimension 1. Recall that these partitions are the convex hull of the origin and of faces of dimension 0. If these faces belong to the same elliptical arc, then the control gains for these partitions are the same.Hence, the partitions with faces of the same elliptical arc can be merged to create a new full-dimensional partition.Since the boundary of P is smooth, one can have the same behavior for the partitions of dimension 2, . . ., n − 1.As a result, the partitions have a full dimension after merging.

B. Particular Case: n = 2
The aim of this section is to illustrate graphically the discussions in Section V-A for the case n = 2.
Consider the convex hull of ellipsoids P = Co(E(P 1 ), . . ., E(P s )), P i ∈ S 2 , and the associated control gains It is clear that in R 2 to compute the state space partitions and the control gains, our main problem is to construct the partitions of dimension 2 as well as the control gains in these partitions.Our first step is to calculate all possible normal vectors β of all faces of dimension 1.This is done by solving the following set of equations, (37) Fig. 2. Graphical illustration for the proof of Theorem 6.
Once the normal vector β is computed, the extreme points v i 1 , v i 2 are given as Since v i 1 , v i 2 are linearly independent, and In this case, one does not need to perform the SVD technique to factorize where the control gain F i 1 i 2 is computed as

C. Lipschitz Continuous Piecewise Linear Control Law
In this section, we assume that the partitions have a full dimension as they are obtained after merging.Define M as the maximum of the Euclidean norm of F l , where F l is the control gain in (36), and l ranges over the set of indices of partitions.The following results hold.
Theorem 3: Consider the control laws (10) and (11). 1) This control law can be represented as a piecewise linear function of the state.2) This control law is a Lipschitz continuous function of the state with the Lipschitz constant M .Piecewise linear control law proof: The proof comes directly by using (30) and (35), and by the fact that the set P is the union of the partitions.
Lipschitz continuous control law proof: First of all, we show that the control law (10) and ( 11) is a continuous function of the state.The proof comes from the following two facts.1) The partitions are closed sets.
2) The control law is continuous in any partition.Now we show that the control law (10), ( 11) is Lipschitz continuous with the Lipschitz constant M .For any two points x S ∈ P and x E ∈ P, there exists p + 1 points z l , l = 0, p that lie on the interval connecting x S and x E such that: 1) z 1 = x S , z p = x E ; 2) z l , z l+1 is the intersection between x S , x E and the boundary of some partition P l , i.e., z l , z l+1 = x S , x E Fr(P l ), see Fig. 2 .
Since the control laws (10) and ( 11) are a continuous function of the state, one has Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
It follows that: The last equality holds since z 1 = x S , z p+1 = x E and the points z l ∀l = 1, p + 1 belong to the same interval z 1 , z p+1 .Remark 9: Denote l * as an index of partition such that ||F l * || = M .For any two points x S , x E belonging to the same partition P l * , one has It follows that the Lipschitz constant M is tight.

VI. EXAMPLE
In this section, we will demonstrate the obtained results via an example taken from [10].Consider the system (6) with In [10], input constraints were also considered: The goal is to design a robust stabilizing controller.As written in [10], it can be verified that (42) is not quadratically stabilizable.LMI conditions for designing a linear feedback gain and an associated quadratic Lyapunov function are not feasible.Also, we were not able to construct a robustly controlled invariant polyhedral set using procedures in [3].
Using [10], one obtains the matrices P 1 , P 2 , K 1 , K 2 as ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩    ( Using the control law (46), for the initial condition x(0) = [−4.754.2] T , Fig. 5 presents the state trajectories of the closed-loop system as functions of time.Fig. 6 shows the input trajectory and the realization of α as a function of time.Finally, Fig. 7 presents the optimal value of the cost λ * 1 + λ * 2 in (11) as a function of time.As expected, this function is strictly decreasing.

VII. CONCLUSION
In this article, we complement the recent results in [10] by studying the geometric structures of the solution of the optimization problem, and of the control law.We proposed a procedure to remove redundant ellipsoids from the convex hull of ellipsoids.We proved that if the convex hull of ellipsoid has the minimal representation, then the solution of the optimization problem is unique.We also showed that the control law is a piecewise linear and Lipschitz continuous function of the state.We used an unstable uncertain time-varying second-order system example to validate the obtained theoretical results.

Further
Results on the Control Law via the Convex Hull of Ellipsoids Hoai-Nam Nguyen Abstract-Recently, a new Lyapunov function based on the convex hull of ellipsoids was introduced for the study of uncertain and/or time-varying linear discrete-time systems with/without constraints.The new Lyapunov function has many attractive features such as: 1) the associated ellipsoids are not required to be robustly invariant; 2) the design conditions are formulated as linear matrix inequality constraints.The control law is obtained by solving a convex optimization problem online.This optimization problem generally does not have a closed-form solution, and hence it is solved by numerical methods.In this article, we intend to complement the results by analyzing the geometric structures of the solution to the optimization problem, and of the control law.In particular, we show that the control law is a piecewise linear and Lipschitz continuous function of the state.Index Terms-Convex hull of ellipsoids, invariant set, linear matrix inequality (LMI), Lyapunov function, uncertain and/or timevarying linear discrete-time system.