Equilibrium controllability analysis and input design of linear systems

ABSTRACT In setpoint control, an equilibrium of the system to be controlled is typically employed as the setpoint. Thus, it is important to know the “possible equilibria” of the system, i.e. the value at which the state continues to stay by a suitable control input. Here, a possible equilibrium is called a controllable equilibrium. In this paper, we study two problems on controllable equilibria. First, we consider the problem of determining the set of controllable equilibria associated with constant inputs of magnitude one or less. Second, we address the problem of finding a constant input minimizing the distance between the resulting equilibrium and the desired state value. For each problem, we provide solutions in model-based and data-driven manners.


Introduction
Setpoint control is a widely used method to regulate the state or output around a desired value, called the setpoint.In the control scheme, an equilibrium of the system to be controlled is typically employed as the setpoint.Thus, it is important to know the "possible equilibria" of the system, i.e. the value at which the state continues to stay by a suitable control input.Here, a possible equilibrium is called a controllable equilibrium.
So far, several problems on controllable equilibria have been studied.For example, a design method of a constant input to generate a desired equilibrium has been developed in Refs.[1,2].Feedback controllers which provide the desired equilibrium under an unknown disturbance have been discussed in Ref. [3].Moreover, a solution has been presented for the problem of designing a controller which shifts the equilibrium to the desired equilibrium using only the network structure in Ref. [4].However, to the best of our knowledge, there is no comprehensive framework for the analysis and synthesis of controllable equilibria, though specific results have been sporadically obtained as mentioned above.
In order to establish a framework of analysis and design of controllable equilibrium, this paper addresses the following two problems on controllable equilibria for linear systems.
(i) Analysis problem: Determine the set of controllable equilibria associated with constant inputs of magnitude one or less.
(ii) Input design problem: For a given desired state value, find a constant input minimizing the distance between the resulting equilibrium and the desired state value.
We present model-based and data-driven solutions to the above problems.The model-based solution is applicable when we have a mathematical model of the system.The solution clarifies the geometric property of the solutions: the solution to (i) is the intersection of an ellipsoid and a subspace of the state space and the solution to (ii) is the projection of a vector onto a subspace of the state space.On the other hand, the data-driven solution is useful for the case where we have neither a model nor a dataset from which a model is constructed [5][6][7][8][9][10][11][12][13][14][15].In fact, the data-driven solution provides at least candidates of the solutions of (i) and (ii) even if a given dataset is insufficient to construct a model, and, if additional knowledge, such as domain knowledge, is available, we may be able to derive a practical approximate solution from the candidates by incorporating the additional knowledge.Such a situation often arises in biological systems because the state space is high-dimensional and the amount of data is limited due to the labour of experiment.Therefore, our result is demonstrated for a biological example.
The rest of this paper is organized as follows.In Section 2, we formulate the problems.Section 3 gives model-based solutions.In Section 4, we derive the solutions in a data-driven framework.In Section 5, the proposed approach is applied to a biological network.Section 6 concludes this paper.

Notation
We denote the set of real numbers by R and the set of nonnegative real numbers by R 0+ .The set of n × n real symmetric matrices is denoted by S(n).
We use 0 n and 0 n×m to represent an n-dimensional zero vector and an n × m zero matrix, respectively.
Let I n denote an n × n identity matrix.The transpose of matrix A is represented by A .If there exists a right inverse for the matrix A, it is denoted by A + .Let A † denote the pseudo-inverse matrix for the matrix A. For the vector x ∈ R n , x denotes its Euclidean norm.Finally, let e i be the i-th natural basis of R n .

Problem formulation
Consider the linear system where x(t) ∈ R n is the state, u(t) ∈ R m is the input, A ∈ R n×n and B ∈ R n×m are matrices.We do not impose the assumption for the matrix A but assume that m ≤ n and If a constant input u e ∈ R m is applied to the system (1), an equilibrium x e ∈ R n satisfies We call x e ∈ R n a controllable equilibrium if there exists a constant input u e ∈ R m satisfying u e ≤ 1 and (3).Then the set of controllable equilibria is expressed as The set X e is called the controllable equilibrium set.For example, consider the system In this paper, we address the following two problems.
Problem 1: For the system (1), determine X e .
Problem 2: For the system (1), suppose that a vector x ∈ R n is given.Then, find a pair (x e , u e ) minimizing x − x e under the condition in (3).
Problem 1 is the problem of finding the controllable equilibrium set, which is called the equilibrium controllability analysis problem.
On the other hand, Problem 2 corresponds to placing the equilibrium as close as possible to the given desired location x.
This problem is useful in terms of not only providing the best control input but also clarifying the limitations of generating an equilibrium in the system (1).If the equilibrium resulting from Problem 2 is not acceptable, the control designer can make the decision to shift to other options such as adding an actuator.

Solution to Problem 1
We provide the solution to Problem 1.
Therefore, we obtain the following result.is the solution to Problem 1.
From this result, we see that X e is the intersection of the ellipsoid {x e ∈ R n | x e A 1 A 1 x e ≤ 1} and the subspace {x e ∈ R n | 0 n−m = A 2 x e } on R n .

Example 3.2:
Consider Problem 1 for the system From Theorem 3.1, the solution to Problem 1 is given by because n = m, B = I 2 , and A = A 1 .The set (10) is illustrated in Figure 1.
Example 3.3: Consider the system  In this case, A 1 = [ −2 1 ] and A 2 = [ 2 −3 ] from (5).It follows from Theorem 3.1 that the solution to Problem 1 is given by which is illustrated in Figure 2.

Solution to Problem 2
Next, a solution to Problem 2 is obtained as follows.
Then, the pair (x This theorem is the straightforward consequence of the following facts. (a) The vector x * e is a solution to (P2) and 0 n = Ax * e + Bu * e holds.(b) Suppose that a vector x ∈ R n is given.If x is a solution to (P2), then x is a solution to (P1).(c) Suppose that a vector x ∈ R n is given.If x is a solution to (P1) and 0 n = Ax + Bu * e holds, then (x, u * e ) is a solution to Problem 2.
Proof of (a).First, we prove that x * e is a solution to (P2).It is clear that (P2) is equivalent to the following optimization problem: where y := x − x e .Since A 2 x ∈ Im(A 2 ), there exists a solution to the equation A 2 y = A 2 x for the unknown y.By noting this, it follows from Lemma A.1 in the appendix section that we have ȳ = A † 2 A 2 x as a solution to (15).This, together with the definition of y, implies that x − A † 2 A 2 x is a solution to (P2).Therefore, x * e is a solution to (P2).
Next, we show that 0 n = Ax * e + Bu * e holds.Lemma A.2 in the appendix section, (2), ( 5), (13), and ( 14) provide This completes the proof.Proof of (b).It is straightforwardly given by Lemma A.3 in the appendix section and the relation (16) The relation ( 16) is obtained as follows.Suppose that an element x e of the set on the left-hand side of ( 16) is arbitrarily given.Because (3) is represented as ( 6) and ( 7), we have ⇒ ∃u e ∈ R m satisfying (6) and ( 7) This proves (16).
Proof of (c).Since Problem 2 is equivalently expressed as min it is clear that the pair of a solution x to (P1) and u * e is a solution to Problem 2 if 0 n = Ax + Bu * e .
Example 3.5: Consider Problem 2 for the system and x = [1 2 3] .For this system, we obtain because A 2 = 0 1 −2 from (5).It follows from Theorem 3.4 that is a solution to Problem 2.

Data-driven solutions
In this section, we provide a data-driven solution to the two problems.

Problem formulation
Let us formulate the corresponding problems with the data of state trajectories.
Consider the system (1).For the time t 1 ∈ R 0+ and the state x 1 ∈ R n , let x(t, t 1 , x 1 ) denote the solution of the system (1) under x(t 1 ) = x 1 and u(t) ≡ 0 m .Let X([t 1 , t 2 ], x 1 ) be the state trajectory on the time interval [t 1 , t 2 ] under the same condition, i.e.X([t 1 , t 2 ], x 1 ) = t∈[t 1 ,t 2 ] {(t, x(t, t 1 , x 1 ))}, where t 1 < t 2 .Then, suppose that the matrix A is unknown but the dataset In this situation, our problems are formulated as follows.
Problem 1 : For the system (1), suppose that a dataset D is given.Assume that the matrix B is known but the matrix A is unknown.Then, determine X e .
Problem 2 : For the system (1), suppose that a vector x ∈ R n and a dataset D are given.Assume that the matrix B is known but the matrix A is unknown.Then, find a pair (x e , u e ) minimizing x − x e under the condition in (3).

Solution to Problem 1
Now, we provide the solution to Problem 1 .The idea is to construct the matrices A 1 A 1 and A 2 in (8) Since the left-hand side of ( 24) is expressed as it follows from (24) that and, furthermore, from (5).
Next, we can calculate the square of the norm for the left-hand side of (27) as follows: Note that the relation (28) holds for every i ∈ {1, 2, . . ., N}.Therefore, we have the following lemma.
Lemma 4.1: Suppose that a dataset D is given.Consider the linear equation where S ∈ S(n) is the unknown.If there exists a unique solution S on S(n) to (29), then the solution is equal to A 1 A 1 .
Proof: Since the relation (28) holds for any datum in D, the matrix A 1 A 1 is a solution to (29).Moreover, Next, we show that A 2 ∈ R (n−m)×n is obtained from D. Equations ( 5) and ( 26) imply which holds for any datum in D. Thus, we have the following lemma.
Lemma 4.2: Suppose that a dataset D is given.Consider the linear equation x(t, t 1i , x 1i ) dt where M ∈ R (n−m)×n is the unknown.If there exists a unique solution M on R (n−m)×n to (31), then the solution is equal to A 2 .
Proof: By using (30), we can prove this lemma in the same manner as Lemma 4.1.
From Lemmas 4.1 and 4.2, we obtain the solution to Problem 1 .
which provide the solution to Problem 1 as This is consistent with (12). Figure 5 shows the resulting X e .
In practice, the data may be corrupted by noise.In this case, it is reasonable to use the least-squares technique to (29) and (31), i.e. min and min Example 4.5: Consider Problem 1 for the system (11).Suppose that the data is given as in Figure 4, which corresponds to the data in Figure 3 that is distorted by the Gaussian noise with mean of 0 and variance of 0.05.The right-hand side of Figure 5 illustrates the set in (36).This result is sufficiently close to the result by noise-free data, from which we can conclude that the proposed method is useful for noisy data.

Solution to Problem 2
In this section, we provide closed-form and optimization-based solutions to Problem 2 .

Closed-form solution
We first prepare the following lemma.
Lemma 4.6: Suppose that a dataset D is given.Consider the linear equation x(t, t 1i , x 1i ) dt where L ∈ R m×n is the unknown.If there exists a unique solution L on R m×n to (37), then the solution is equal to A 1 .
Proof: By using (27), we can prove this lemma in the same manner as Lemma 4.1.
Then, we have the following result.

Optimization-based solution
For the solution of Theorem 4.7, there is no flexibility for incorporating prior knowledge into the process of solving Problem 2 .Thus, a solution which allows us to incorporate prior knowledge is useful.In this subsection, we propose an optimization-based solution.
For the dataset D, let Moreover, we define For the optimization Problem (42), we have the following theorem.The proof is shown at the end of this subsection.The condition in Theorem 4.9 implies that the number N of data is more than or equal to n and the data are rich.(43) as a solution to Problem 2 .This is consistent with (39).
If the data are not rich, we cannot obtain the solution to Problem 2 from (42).However, we may derive the solution by incorporating prior knowledge about the solution into (42).

Example 4.11:
Consider Problem 2 for the same system and vector x as in Example 3.5.Suppose that the dataset D is composed of the left and middle figures in Figure 6, where N = 2, but we know that the resulting x e is composed of nonnegative elements and u e satisfying u e1 ≤ −0.5 and u e2 ≥ 3, where u ei is the ith element of u e .Then, we can incorporate this prior knowledge into (42), which results in min By solving this, we obtain [0.3972 2.8851 1.7944] , [−0.5000 3.6700] .
(44) We see that (44) is an approximation of (43), i.e. of a solution to Problem 2 .Now, we prove Theorem 4.9.First, we show the relation between the system (1) and the system defined by the transpose of A, i.e.
where z(t) ∈ R n is the state.In a similar manner to x(t, t 1 , x 1 ) for system (1), let z(t, t 1 , z 1 ) denote the solution of the system (45) for z(t 1 ) = z 1 .
The following lemma is useful for understanding the relation between ( 1) and (45).
Lemma 4.12 implies that the state trajectory of ( 45) is obtained from the data generated by (1).
Based on Lemma 4.12, we obtain the following result.
Lemma 4.13: For the system (1), suppose that a dataset D is given.If E 0 E † 0 = I n holds and there exists a right inverse D + int for D int , then Proof: For the system in (45), we have By integrating the both sides of (47) on [0, h], we obtain By calculating the integral of the left-hand side of (48) and applying Lemma 4.12 to it, we have Meanwhile, the right-hand side of (48) is expressed as from Lemma 4.12.Equations ( 49) and (50) imply By considering (40), (41), and the relation (51) holds for every i ∈ {1, 2, . . ., N}, we obtain If there exists a right inverse D + int for D int , we see which implies (46).
Proof of Theorem 4.9: By comparing (42) with ( 17), the theorem is proven by showing that the condition (3) is equivalent to This is proven as follows.If D int has full row rank, then there exists a right inverse D + int for D int .It follows from Lemma 4.13 that the condition (3) is equivalently rewritten as By applying Lemma A.4 in the appendix section to (55), this is transformed into Moreover, by considering that D int (D + int ) D diff = D diff holds from ( 52) and (53), we see that the condition (56) is equivalent to (54).This completes the proof.

Application to a biological network model
In this section, our approach is applied to a biological network.The scenario is as follows.Consider a biological network whose state stays at an equilibrium state, corresponding to the undesirable state.We want to shift the state to a new equilibrium, corresponding to a desirable state, generated by applying a constant input.To this end, we have to determine the pair of the new equilibrium state and the constant input.This is exactly the case of Problem 2. Here, we provide a solution for the case where the model is not available but the data of state trajectories are available.
Note here that the problem of shifting equilibrium has been originally addressed in Ref. [4], where a solution with a partial information of the system, which is the network structure, has been proposed.Also, a similar problem can be found in economics [18].However, the result differs from the following result based on a data-driven approach.Now, let us demonstrate this scenario for the model of a biological network with 100 nodes, developed in Ref. [15].Consider the network system with the five inputs, nodes 1-5, as shown in Figure 8.The dynamics of node k is given by  57) and (58) for u i (t) ≡ 0 m , [t 1i , t 2i ] = [0, 1], and x 1i that is drawn from the uniformly distribution from the interval (0, 1).Moreover, x is the vector whose 9-th, 12-th, 17-th, 29-th, 77-th, and 87-th elements are one, and the others are all zero.
In this case, there exists a unique solution L on R m×n to (37) and a unique solution M on R (n−m) The distance between the given desirable state x and the resulting new equilibrium state x e is obtained as follows: x − x e = 1.9942.

Conclusion
In this paper, we have addressed two problems on controllable equilibria for linear systems.First, we formulate the problems of determining the region where an equilibrium is placed by a constant input and of finding a constant input such that the resulting equilibrium is close to the desirable state value.To the problems, model-based and data-driven solutions have been developed.Finally, we have introduced and demonstrated a scenario in biology that the proposed framework will be a useful tool.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
This work was partly supported by JST FOREST Program #JPMJFR2123 and JST Moonshot RD Grant Number JPMJMS2023.

Notes on contributors
Takumi Iwata received the B.E. and M.E.degrees from Nanzan University, Nagoya, Japan, in 2017 and 2019, respectively.He is currently a doctoral student at the Graduate School of Engineering, Nagoya University, Nagoya, Japan.His research interests include data-driven analysis and design.Jun-ichi Imura received the M.E.degree in applied systems science and the Ph.D. degree in mechanical engineering from Kyoto University, Kyoto, Japan, in 1990 and 1995, respectively.He served as a Research Associate with the Department of Mechanical Engineering, Kyoto University, from 1992 to 1996, and as an Associate Professor with the Division of Machine Design Engineering, Faculty of Engineering, Hiroshima University, Hiroshima, Japan, from 1996 to 2001.From May 1998 to April 1999, he was a Visiting Researcher with the Faculty of Mathematical Sciences, University of Twente, Enschede, The Netherlands.Since 2001, he has been with the Tokyo Institute of Technology, Tokyo, Japan, where he is currently a Professor with the Department of Systems and Control Engineering.His research interests include modeling, analysis, and synthesis of nonlinear systems, hybrid systems, and large-scale network systems with applications to power systems, intelligent transportation systems, biological systems, and industrial process systems.Dr. Imura is a Member of Institute of Electrical and Electronics Engineers (IEEE), The Institute of Systems, Control and Information Engineers (ISCIE), and The Robotics Society of Japan.

Theorem 3 . 1 :
Consider Problem 1.Then, the set {x e ∈ R n | x e A 1 A 1 x e ≤ 1 and 0 n−m = A 2 x e } (8)

Figure 1 .
Figure 1.The set X e for the system (9).

Figure 2 .
Figure 2. The set X e for the system (11).

Theorem 4 . 3 :Example 4 . 4 :
Consider Problem 1 .If there exists a unique solution S on S(n) to (29) and a unique solution M on R (n−m)×n to (31), then the set {x e ∈ R n | x e Sx e ≤ 1 and 0 n−m = Mx e } is the solution to Problem 1 .In Theorem 4.3, the solution to Problem 1 is presented subject to the existence and uniqueness of the solutions to (29) and (31).If the data are insufficient in number and variety, then (29) and (31) may have multiple solutions.On the other hand, the condition on existence and uniqueness can be easily checked because (29) and (31) are linear equations that can be represented in the form of Cx = d.Here, it is well known that Cx = d has a unique solution if and only if rank(C) = rank([C d]) and C is of full column rank[16].Consider Problem 1 for the system(11).Suppose that the dataset D is given as shown in Figure3, where N = 3.Then, (29) and (31) are obtained as follows: ⎧ ⎨ ⎩ 1.9065 = [1.13530.8899] S [1.1353 0.8899] , 0.5605 = [0.50320.2578] S [0.5032 0.2578] , 0.2658 = [0.76101.0064] S [0.7610 1.0064] , 3991 = M [1.1353 0.8899] , 0.2330 = M [0.5032 0.2578] , −1.4973 = M [0.7610 1.0064] .By solving these equations, we have S = 4.0000 −2.0000 −2.0000 1.0000 , M = 2.0000 −3.0000 ,

Theorem 4 . 9 :
Consider Problem 2 .If E 0 E † 0 = I n holds and D int is of full row rank, then a solution to the optimization Problem (42) is a solution to Problem 2 .

Example 4 . 10 :
Consider Problem 2 for the same system and vector x as in Example 3.5.Suppose that the dataset D is given as Figure6, where N = 3.In this case, E 0 E † 0 = I n holds and D int is of full row rank.By solving the optimization Problem (42), we obtain [1.0000 2.8000 1.4000] , [−0.8000 4.0000]

Figure 8 .
Figure 8. Network structure of the system considered in Section 5.The diamond-shaped node represents the input node.
Azuma received the Ph.D. degree in mechanical and environmental informatics from the Tokyo Institute of Technology, Tokyo, Japan, in 2004.He was an Assistant Professor and an Associate Professor with the Graduate School of Informatics, Kyoto University, Kyoto, Japan, from 2005 to 2011 and 2011 to 2017, respectively, and a Professor with the Graduate School of Engineering, Nagoya University, Nagoya, Japan.He is currently a Professor with the Graduate School of Informatics, Kyoto University.His research interests include the analysis and control of network systems and hybrid systems.RyoAriizumi received the B.E., M.E., and Ph.D. degrees from Kyoto University, Kyoto, Japan, in 2010, 2012, and 2015, respectively.He was a Research Fellow with the Japan Society for the Promotion of Science working at Kyoto University from 2014 to 2015 and an Assistant Professor at Nagoya University, Nagoya, Japan, from 2015 to 2023.He is currently an Associate Professor with the Tokyo University of Agriculture and Technology, Tokyo, Japan.His research interests include the control of redundant robots and the optimization of robotic systems.Dr. Ariizumi has received awards, including the IEEE Robotics and Automation Society Japan Chapter Young Award (IROS 2014) in 2014 and the Best Paper Award from the Robotics Society of Japan (RSJ) in 2018.Toru Asai received the B.E., M.E., and Ph.D. degrees in engineering from the Tokyo Institute of Technology, Tokyo, Japan, in 1991, 1993, and 1996, respectively.He was a Research Fellow with JSPS from 1996 to 1998.In 1999, he joined the faculty of Osaka University, Suita, Japan.Since 2015, he has been an Associate Professor with the Subdepartment of Mechatronics, Nagoya University, Nagoya, Japan.His research interests include robust control and switching control.Dr. Asai is a member of the Society of Instrument and Control Engineers and the Institute of Systems Control and Information Engineers.
(t) ∈ R and u k (t) ∈ R are the state and input of node k, a kj is a real number, and N k is the index set of the neighbours that affect the state of node k.The system has an equilibrium at the origin.Here, suppose that the equilibrium is an undesirable state and e 9 + e 12 + e 17 + e 29 + e 77 + e 87 is a desirable state.Assuming that the information of the model is unavailable but the data of state trajectories are available, we address Problem 2 .The dataset D is of N = 100 and artificially generated by the model defined by (