On Designing Optimal Controllers with Limited Model Information
FARHAD FAROKHI, CEDRIC LANGBORT, KARL HENRIK JOHANSSON
Stockholm 2010
KTH - Royal Institute of Technology School of Electrical Engineering
Automatic Control
SE-100 44 Stockholm, Sweden
TRITA-EE 2010:039
On Designing Optimal Controllers with Limited Model Information
Farhad Farokhi
∗, Cedric Langbort
†, and Karl H. Johansson
∗October 1, 2010
Abstract
We introduce the family of limited model information designs, which construct controllers by accessing the plant’s model in a constrained manner. We investigate the closed loop performance of the best controller that they can produce. For a class of linear discrete-time, time invariant plants, we show that there exists a limited model information control design which results in a controller whose performance is in a bounded neighborhood of the optimal control design and we show that this controller is the best controller that one can design with limited information about the plant model. We investigate the plant model structure and the model information on this neighborhood.
1 Introduction
In past half a century, a large number of mathematicians and engineers tried to solve the problem of designing distributed and/or decentralized controllers for large and/or networked plants. In most of these researches, they tried to design the controller with entire model information, something that is nearly impossible for practical large and/or networked plants. Designing controllers with limited model information has received no attention for large and/or networked plants, in comparison to designing distributed and/or decentralized controllers. The goal of this paper is to solve the problem of designing optimal controller for large-scale interconnected plants and provide some bounds on the closed-loop performance of controllers designed with limited model information.
Consider a large or networked dynamical system composed of some sub-systems connected to each other. In control synthesis process, sometimes, it happens that the designer do not have access to full model of the plant. Either because the full model is simply not available, or because the subsystems may not wish to provide a complete description of themselves to the designer. The former case occurs naturally in the presence of uncertainties, while the latter should be expected in large-scale systems shared by private individuals. But with these restrictions on our information from the plant model, it is likely to know about the structure of the sub-systems interconnection. For instance, a complicated system consist of four interconnected sub-system to each other is shown in figure 1. Subsystem number three can only affect subsystem number one and it is affected by subsystems number one and two. This structure behind the the subsystem interconnection is called he plant structure later in this paper. In modeling stage of the problem, we know, in advance, that some of the subsystems cannot affect some of the other subsystems (maybe because these subsystems are spatially far away from each other one in real system and it is unlikely that they have something is common). For designing controller Cifor subsystem i, our knowledge of the model is limited to the dynamic of that particular subsystem, the way that the other subsystems can affect it, and the structure behind the plant. We do not know, how that particular subsystem affect the other subsystems and what is the other sub-systems dynamics. Every controller may or may not have access to full state vector for feedback purposes. For example, controller C1 only have access to state measurements of sub-system number one and three. If subsystem i, in process of stabilizing itself, do something strange, it may unstablize the whole system or may cause a great effort for the other subsystems to be able to stabilize themselves. Therefore, a natural question, then, is: “can such a limited information design process result in acceptable controllers, with guaranteed level of closed-loop performance and/or guaranteed stability of the closed loop?”.
∗F. Farokhi and K. H. Johansson are with KTH - Royal Institute of Technology, School of Electrical Engineering, Automatic Control, SE-100 44 Stockholm, Sweden. E-mails:{farokhi,kallej}@kth.se
†C. Langbort is with the Department of Aerospace Engineering, University of Illinois at Urbana-Champaign. E-mail:
langbort@uiuc.edu
P1
P2
P3 C1
C3 C2
P4 C4
Figure 1: Interpretation of class of limited-model-information design methods. In process of designing controller Ci for subsystem i, our knowledge of the model is limited to the dynamic of that particular subsystem, the way that the other subsystems can affect it, and the structure behind the plant.
For answering this question, we develop some new tools based on Langbort and Delvenne definitions in [1].
They formulated this question for the scalar sub-system case when input control dimension was equal to state vector dimension. They introduced two new performance criteria named “domination” and “competitive ratio” with inspiration from the fields of Economics and Distributed and On-Line Computing. They found the relationship between the controller design strategy, the closed loop system and these performance criteria and they tried to minimize these performances measures. Their controller design strategy had access to full state vector for feedback purposes. In this paper, we try to solve this problem in the most general way;
.i.e., for the non-scalar subsystems with arbitrary dimension and arbitrary control inputs, with full state feedback.
This paper is organized as follows. In section 2, we formulate the problem and define the performance metrics.
In section 3, we present the solution to the problem of finding the best controller design strategy when there is a known structure behind the system and the dimension of control input is equal to the dimension of the state vector for each subsystem when the designer’s knowledge for each subsystem is limited to that subsystem dynamical model. Section 4 generalize the last section results to arbitrary knowledge about the model. In section 5, we try to solve the problem for the case that the dimension of the control input is less than or equal to the dimension of the state vector for each subsystem. The robustness of the proposed controllers with respect to model uncertainty is discussed in section 6. Later in section 7 an illustrative example is mentioned to illustrate different aspects of the theoretical results on a practical control design problem. Finally, in section 8, we conclude the paper and discuss possible directions this research can take in future.
1.1 Notation
Sets will be denoted by calligraphic letters, such asP and A. If A is a subset of M then Acis the complement of A in M, i.e., the set of all elements of M which do not belong to A. When the set M is significant or unclear from context, we also writeM \ A.
Matrices are shown by capital roman letters such as A. Aj will denote the jth row of the A. Aij denotes the a sub-matrix i, j of matrix A. The dimension and the place of this sub-matrix will be defined in the text.
The entry in the ith row and the jth column of the matrix A is aij.
Let S++n (S+n) be the set of symmetric positive definite (semi-positive definite) matrices inRn×n. A > (≥)0 means symmetric matrix A ∈ Rn×n is positive definite (semi positive definite) and A > (≥)B means that A− B > (≥)0.
λ(Y ) and ¯λ(Y ) denote the smallest and the largest eigenvalues of the matrix Y , respectively. Similarly, σ(Y ) and ¯σ(Y ) will denote the smallest and the largest singular values of the matrix Y , respectively.
Finally, ei will denote the column-vector with all entries zero except the ith entry, which is equal to one.
2 Control Design With Limited Model Information
2.1 Plant Model
Consider the space of discrete time, linear time-invariant, plantsP described in state-space form by x(k + 1) = Ax(k) + Bu(k); x(0) = x0,
where x(k) ∈ Rn is the state vector and u(k) ∈ Rm is the control input, and A ∈ A ⊆ Rn×n and B ∈ B ⊆ Rn×m are model matrices. A plant P inP = A × B × Rn can be uniquely shown by the collection P = (A, B, x0). Now, suppose that both x(k) and u(k) are partitioned into to q sub-vectors as
x(k) =[
x1(k)T x2(k)T · · · xq(k)T ]T
, and
u(k) =[
u1(k)T u2(k)T · · · uq(k)T ]T
,
where each xi ∈ Rni represents the state vector of a sub-system for 1≤ i ≤ q and ∑q
i=1ni = n, and each ui∈ Rmi represents the control input of a sub-system for 1≤ i ≤ q and∑q
i=1mi = m. Now, each plant in P can be described by the interconnection of these sub-systems
xi(k + 1) = Aiixi(k) +∑
j̸=i
Aijxj(k) + Biiui(k) +∑
j̸=i
Bijuj(k),
where Aij ∈ Rni×nj and Bij ∈ Rni×mj are appropriate sub-matrices. The set of possible B matrices B is defined as
B = { ˜B∈ Rn×m| ˜Bij∈ Rni×mj is forced to be zero for all i, j such that i̸= j}.
Matrix A is assumed to have a specific structure shown by structure matrix S∈ {0, 1}q×q; i.e., A = { ˜A∈ Rn×n| ˜Aij ∈ Rni×nj is forced to be zero for all i, j such that sij = 0}.
This set of desired plants can be shown by a directed graph called the plant graph.
Definition 2.1 The plant graph GP ofP is an ordered pair GP = (VP, EP) comprising a set VP ={1, · · · , q}
of vertices together with a set EP ={(j, i)|sij ̸= 0} of edges. Note that, sij ̸= 0 means that sub-system number j is affecting sub-system number i, and thus, in the plant graph, there is an edge directed from node j to node i; i.e., (j, i)∈ EP. In the plant graph, paths like i1→ i2→ · · · → it→ i1 with distinct nodes i1,· · · , it
in a directed graph are called loops (of length t) and a sub-system is called sink, if there is no edge directed from this sub-system to any other sub-system.
For ease of working with sub-system in the subsequent sections, the set of indices related to sub-system number j will be shown byIj ={∑j−1
i=1ni+ 1,· · · ,∑j i=1ni}.
2.2 Controller
The controller for these system are in the class of decentralized static feedback, thus u(k) = Kx(k),
where K∈ K ⊆ Rm×n is the controller gain. Partitioning K, using sub-systems dimensions results in
K =
K11 K12 · · · K1q
K21 K22 · · · K2q
... ... . .. ... Kq1 Kq2 · · · Kqq
,
where Kij ∈ Rmi×nj. The setK put some constraints on using other sub-systems state vector for feedback and this set can be visualized by a directed graph named as control graph.
Definition 2.2 The control graph GK= (VK, EK) is the directed graph with vertex set VK={1, · · · , q} and edge set defined as
EK={(j, i)| sub-matrix Kij ̸= 0 for at least one plant P = (A, B, x0)∈ P}.
Therefore, (j, i)∈ EK shows that the sub-system i controller can use the state measurements of sub-system j for feedback.
2.3 Controller Design
Controller design Γ∈ C ⊆ KA×Bis a mapping1 form the set of possible modelsA × B to the set of possible controller gainsK. Partitioning this mapping using sub-systems dimension results in
Γ =
Γ11 Γ12 · · · Γ1q
Γ21 Γ22 · · · Γ2q
... ... . .. ... Γq1 Γq2 · · · Γqq
,
where each sub-design is a mapping Γij :A × B → Rmi×nj. The set C of possible mappings from the set A × B to the set K is called the class of limited model information design strategies. The C can be shown by a directed graph named as design graph.
Definition 2.3 The design graph GC = (VC, EC) is the directed graph with vertex set VC ={1, · · · , q} and edge set defined as
EC ={(j, i)| if [
Γi1 · · · Γiq
] depends on [
Aj1 · · · Ajq
]and Bjj for at least one P ∈ P}.
Therefore, (j, i)∈ EC shows that the design mapping for the sub-system i is a function of the sub-system j dynamic model.
Putting q = n and B = I and using complete control graph, totally disconnected design graph with self loops only, and complete plant graph in this section definition results in the communication-less design strategy in [1].
In the rest of the paper, capital greek letters, such as Γ, will denote the controller design method, which we see as maps from a set of plants modelsA × B to a set of controllers K. Γishows the ith row of the controller design Γ, Γij shows the sub-design i, j of the design Γ (the place and the dimension will be mentioned in the text), and γij shows the design entry located in the ith row and the jth column of controller design Γ.
2.4 Performance Metrics
In this paper, we want to investigate the limitations of optimal controller design methods with limited model information. Therefore, we need to introduce two performance metrics for design methods. These performance metrics are the modified version of performance metrics that Langbort and Delvenne used in [1]. First, we need to introduce the cost function.
For all K∈ K, each plant P = (A, B, x0)∈ P is equipped with the performance criterion
JP(K) =
∑∞ k=1
x(k)TQx(k) +
∑∞ k=0
u(k)TRu(k), (1)
where Q∈ Q ⊆ S++n and R∈ R ⊆ S++n . It should be noted that Q and R are the set of block diagonal matrices with diagonal entries Qii ∈ S++ni and Rii ∈ S++mi, respectively. In the rest of the paper, we only consider Q = I and R = I, because for the general cost function in equation (1), we can use the change of variable ¯x(k) = Q12x(k) and ¯u(k) = R12u(k) which results in
JP(K) =
∑∞ k=1
¯
x(k)Tx(k) +¯
∑∞ k=0
¯
u(k)Tu(k),¯
and the state-space representation of the plant would become
¯
x(k + 1) = Q12AQ−12x(k) + Q¯ 12BR−12u(k) = ¯¯ A¯x(k) + ¯B ¯u(k).
This change of variable because of the special block diagonal form of the cost matrices Q and R would not affect the plant graph, the control graph, the design graph and the structure of matrix B.
Now we are ready to define the performance metrics which we are going to use in this paper.
1The set of all mappings from a setX to a set Y is denoted by YX.
Definition 2.4 (Competitive Ratio) There exists an optimal design K∗ :A × B → K such that, for every plant P = (A, B, x0)∈ P
JP(K∗(A, B))≤ JP(K), ∀K ∈ K.
Note that the mapping K∗ is not necessarily in the set C and it may depend on all entries of the model matrices A and B. The competitive ratio of a controller design method Γ∈ C is defined as
rP(Γ) = sup
P∈P
JP(Γ(A, B)) JP(K∗(A, B)), with the convention that “00” equals one.
Definition 2.5 (Domination) A control design method Γ1 ∈ C is said to dominate another control design method Γ2∈ C if
JP(Γ1(A, B))≤ JP(Γ2(A, B)), ∀ P = (A, B, x0)∈ P, with strict inequality holding for at least one plant inP.
2.5 Problem Formulation
With the definitions in previous subsections, we can reformulate the main high-level question of this paper regarding the connection between closed-loop performance and limited model information control design as follows. For a given plant graph, control graph, and design graph, we would like to solve the following optimization problem
Γ∗= arg min
Γ∈CrP(Γ). (2)
Considering that this problem may have more than one solution, we would like to find the undominated solution of the problem. Because, it is possible that other limited information design strategies exist which, while having the same worst-case performance, perform better on some instances and never worse. Such strategies would clearly be desirable because they would provide a better trade-off between average and worst-case performance.
First, we try to solve this problem for arbitrarily plant graph, complete control graph and totally disconnected design graph with self-loops only. Then, we generalize the results to different design graphs.
3 Control Input Dimension Equal to State Dimension for Each Subsystem
In this section, we assume that the dimension of the control input is equal to the dimension of the state vector. It means that the dynamical system is fully actuated. And we try to solve the optimization problem in equation (2) for totally disconnected design graph with self-loops only, complete control graph, and an arbitrary plant graph. Therefore, our model information is only limited to each subsystem model and the interconnection pattern of the subsystems which is a natural assumption because in large and/or networked systems, the spatial map of the sub-systems usually enforces the interconnection pattern.
Before getting involved with mathematical results, it should be noted that, the plant graph may have some isolated nodes. This isolated nodes are not important because we can design the controller for these nodes locally without full model knowledge (since these sub-systems do not affect any other sub-system and they are not affected by other ones). If the plant graph has some isolated nodes, we remove isolated nodes from the graph (we design the controller for these isolated nodes locally) and then we deal with the rest of the plant graph. Thus, without loss of generality, we assume that the plant graph does not have any isolated node.
Consider a plant P = (A, B, x0)∈ P. Without loss of generality, we can assume that the matrix B is full rank, because otherwise everything which can be achieved with u can equally be achieved with an input vector of smaller dimension. Furthermore, we assume that
ϵ = inf
B∈Bσ(B) > 0. (3)
This assumption is stronger than B being subset of the set invertible matrices. Thus, every sub-system Bii matrix has a singular value decomposition Bii = UiiΣiiViiT with Σii ≥ ϵIni×ni. Combining these singular value decompositions together results in a singular value decomposition for matrix B = U ΣVT
where U = diag(U11, U22,· · · , Uqq), Σ = diag(Σ11, Σ22,· · · , Σqq), and V = diag(V11, V22,· · · , Vqq). Defining
¯
x(k) = UTx(k) and ¯u(k) = VTu(k) results in
¯
x(k + 1) = UTAU ¯x(k) + UTBV ¯u(k),
because of the block triangular form of the matrix U , this change of variable is an internal change of variable for each sub-system and it will not affect the plant graph, the control graph and the design graph. With this change of variable, the cost function would become
JP(K) =
∑∞ k=1
x(k)Tx(k) +
∑∞ k=0
u(k)Tu(k)
=
∑∞ k=1
¯
x(k)TUTU ¯x(k) +
∑∞ k=0
¯
u(k)TVTV ¯u(k)
=
∑∞ k=1
¯
x(k)Tx(k) +¯
∑∞ k=0
¯
u(k)Tu(k),¯
because both U and V are unitary matrices and ¯B = UTBV = Σ would become a diagonal matrix. Therefore, without loss of generality, we assume that the matrix B is always diagonal with singular values as its diagonal entries and the setB would be the set of invertible diagonal matrices with positive diagonal entries.
Before stating the important mathematical results of this section, we need to define the dead-beat controller design strategy and to find the competitive ratio of the dead-beat controller design strategy.
Definition 3.1 (The Dead-Beat Controller Design Strategy) A controller design strategy Γ :A × B → K is called the deadbeat controller design strategy if
Γ(A, B) =−B−1A, for all P = (A, B, x0)∈ P.
Using this controller design strategy the plant will reach the origin just in one time-step. This controller is in the class of limited information control design methods C because the [
Γi1(A, B) · · · Γiq(A, B) ] equals to Bii−1[
Ai1 · · · Aiq
].
We use the notation Γ∆ to indicate the dead-beat controller design strategy.
Theorem 3.2 The competitive ratio for dead-beat controller design strategy is 1 + 1/ϵ2, where ϵ is the lower bound for singular values of the matrices B∈ B as it is defined in equation (3).
Proof: The algebraic Riccati equation for finding the optimal controller design K∗:A × B → K is X = ATXA− ATXB(I + BTXB)−1BTXA + I, (4) where X∈ S++n is the solution of the Riccati equation and the optimal cost is JP(K∗(A, B)) = xT0(X−I)x0. Putting BB−1 before every matrix A and B−TBT after every matrix AT in equation (4), results in
X−I = ATB−TBTXBB−1A
− ATB−TBTXB(I + BTXB)−1BTXBB−1A. (5) Naming BTXB as Y simplify equation (5) as
X− I = ATB−T[Y − Y (I + Y )−1Y ]B−1A. (6) Note that BTXB is a positive definite matrix because X is positive definite and B is full rank. Let us denote the right-hand side of (6) by ATB−Tg(Y )B−1A. Then we can make the following two claims regarding the rational function g(.).
Claim 1: For any scalar y > 0, g(y) = y/(1 + y) is a monotone increasing function.
Claim 2: Let Y ∈ S++n and D, T be diagonal and unitary matrices, respectively, such that Y = TTDT . Then g(Y ) = TTdiag(g(dii))T , where the dii are the diagonal elements of D (and the eigenvalues of Y ).
Claim 1 is proved by computing the derivative of g(y) over y > 0, while Claim 2 follows from the fact that all matrices involved in the computation of g(Y ) can be diagonalized in the same basis. Using these two claims, we find that, for all Y , with eigenvalues denoted by λ1(Y ),· · · , λn(Y )
X− I = ATB−Tg(Y )B−1A
= ATB−TTTdiag(g(λi(Y )))T B−1A
≥ (g(λ(Y )))ATB−TB−1A,
(7)
where λ(Y ) is a positive number because matrix Y is a positive definite matrix. It should be noted that according to C.-H. Lee in [2], we have
λ(X)≥ λ(AT(I + BBT)−1A + I)≥ σ2(A)
1 + ¯σ2(B)+ 1, (8)
Using equation (8) in inequality λ(Y )≥ σ2(B)λ(X) gives λ(Y )≥σ2(B)σ2(A)
1 + ¯σ2(B) + σ2(B), and
g(λ(Y ))≥ σ2(B)[σ2(A) + ¯σ2(B) + 1]
1 + ¯σ2(B) + σ2(B)[σ2(A) + ¯σ2(B) + 1]
≥ σ2(B) σ2(B) + 1.
(9)
Combining equations (7) and (9) results in
X− I ≥ σ2(B)
σ2(B) + 1ATB−TB−1A, and therefore
JP(Γ∆(A, B))
JP(K∗(A, B)) = xT0(ATB−TB−1A)x0
xT0(X− I)x0
≤ σ2(B) + 1 σ2(B)
≤ 1 + 1 ϵ2.
Now we have to find an example to show that this not only an upper bound. Assume that i1 ∈ Ii and j1 ∈ Ij where 1 ≤ i ̸= j ≤ q and sij ̸= 0 (you can find these indices i and j because we assumed that there is no isolated node in the plant graph). Consider matrix A defined as A = rei1eTj
1 and matrix B defined as B = ϵI. The unique solution of the Riccati equation is X = I + [r2/(1 + ϵ2)]ej1eTj
1 and J(A,B,e
j1)(K∗(A, B)) = r2/(1 + ϵ2). On the other hand Γ∆(A, B) =−[r/ϵ]ei1eTj
1 and J(A,B,e
j1)(Γ∆(A, B)) = r2/ϵ2. Therefore, rP = 1 + 1/ϵ2.
With the mentioned properties of the deadbeat control design strategy, now, we are ready to solve the optimization problem in equation (2).
For finding the optimal design strategy for different plant graphs, two different scenarios in the plant graph may happen. In the first scenario, the plant graph does not have any sink, but in the second scenario, the plant graph has at least one sink. We deal with these two scenarios in two different subsections.
3.1 Plant Graphs with No Sinks
Theorem 3.3 The dead-beat controller design method is undominated if and only if there is no sink in the plant graph.
Proof: For ease of notation in this proof and the other proofs, we use [Γ]i=[
Γi1 Γi2 · · · Γiq
] and [A]i=[
Ai1 Ai2 · · · Aiq
].
First, we prove that if∀j∃k, k ̸= j, such that skj̸= 0; i.e., there is no sink in the plant graph [3], then the dead-beat controller is undominated. For proving this claim, we are going to prove that for any controller design Γ ∈ C\{ Γ∆}, there exits a plant P = (A, B, x0) ∈ P such that JP(Γ(A, B)) > JP(Γ∆(A, B)) = xT0[ATB−TB−1A]x0. We will proceed in several steps, which require us to partition the set of limited model information control design methodsC as follows
C = Lc∪ W1∪ W2∪ {Γ∆}, where
L := {Γ ∈ C|∃Λj:Rnj×n× Rnj×nj → Rnj×nj, [Γ(A, B)]j= Λj([A]j, Bjj)[A]j, for all j = 1,· · · , q},
W1:={Γ ∈ L|∃j, i ̸= j and Aij ∈ Rni×nj nonzero s.t. I + BiiΛi([
0 · · · 0 Aij 0 · · · 0 ]
, Bii)̸= 0}, and
W2:={Γ ∈ L \ W1|∃i ∈ {1, · · · , q}, [A]i∈ Rni×n, with appropriate structure s.t. I + BiiΛi([A]i, Bii)̸= 0}.
First, we prove that the dead-beat control design method is undominated by control design strategies inLc. Let Γ∈ Lc and let j be such that∃j1∈ Ij which Γj1( ¯A, B)T cannot be written as a linear combination of vectors in the set { ¯ATi,∀i ∈ Ij} for some matrix ¯A and matrix B. Let aTi = ¯Ai for all i∈ Ij and consider matrix A such that the row Ai = aTi for all i ∈ Ij and Ai = 0 for all i ∈ Ijc. If Γ(0, B) ̸= 0, then Γ cannot dominate Γ∆(since Γ∆(0, B) = 0 for all x0) and, thus, there is no loss of generality in assuming that Γ(0, B) = 0 for all x0, and, in turn that Γi(A, B) = 0 for all i∈ Ijc. Let us also denote Γ(A, B) by K and Γi(A, B) = Γi( ¯A, B) by KiT for all i∈ Ij. For all x0,
J(A,B,x0)(Γ(A, B))≥ xT0[KTK + (A + BK)T(A + BK)]x0, and
J(A,B,x0)(Γ(A, B))− J(A,B,x0)(Γ∆(A, B))≥
xT0[AT(I− B−TB−1)A + ATBK + KTBTA + KT(I + BTB)K]x0. (10) We know that null(A) = span{ATi ,∀i ∈ Ij}⊥ ̸= {0}, because n1 < n. On the other hand, we know that there exists an j1∈ Ij such that Kj1∈ span{A/ Ti,∀i ∈ Ij} which shows that
span{ATi,∀i ∈ Ij} span{ATi ,∀i ∈ Ij} + span{KiT,∀i ∈ Ij},
Thus, we can choose an initial condition x0 ∈ null(A) such that Kx0 ̸= 0. Using this x0 in equation (10) results in
J(A,B,x0)(Γ(A, B))− J(A,B,x0)(Γ∆(A, B))≥ xT0[KT(I + BTB)K]x0> 0. (11) Therefore, the controller design strategies in Lc cannot dominate the dead-beat controller design strategy Γ∆.
Second, we prove that the dead-beat control design strategy is undominated by control design methods in W1. Let Γ∈ W1and let j be such that (I + BiiΛi([
0 · · · 0 ¯Aij 0 · · · 0 ]
, Bii))̸= 0 for some i ̸= j.
It means that there exists at least i1 ∈ Ii and j1 ∈ Ij such that ¯ai1j1 ̸= 0 and ¯ai1j1 + bi1i1γi1j1(A, B)̸= 0.
Using the structure matrix, we know that there exits a l ̸= i such that sli ̸= 0. Choose an index l1 ∈ Il. Consider the matrix A defined by [A]i= [ ¯A]i, al1i1 = r and all other entries equal to zero. Then, [Γ(A, B)]i= Λi([A]i, Bii)[A]i, [Γ(A, B)]l = Λl([A]l, Bll)[A]l (because Γ∈ L), and [Γ(A, B)]z = 0 for all z ̸= i, l. Denote Γ(A, B) by K. We have
J(A,B,x0)(Γ(A, B))≥ xT0[(A + BK)TKTK(A + BK) + ((A + BK)2)T(A + BK)2]x0. Using x0= ej1 results in
J(A,B,e
j1)(Γ(A, B))− J(A,B,ej1)(Γ∆(A, B))≥ [kl21i1+ (r + bl1l1kl1i1)2](ai1j1+ bi1i1ki1j1)2−∑
z∈Ii
a2zj
1
b2zz . (12) Note that, irrespective of the choice of the controller gain kl1i1,
kl21i1+ (r + bl1l1kl1i1)2≥ r2 1 + b2l
1l1
, and as a result,
r→+∞lim [k2l1i1+ (r + bl1l1kl1i1)2](ai1j1+ bi1i1ki1j1)2=∞,
because ai1j1+ bi1i1ki1j1 ̸= 0. Hence, we can always construct A with appropriate choice of index l and a scalar r large enough to make the right hand side of the expression (12) positive. As a result, Γ∈ W1cannot dominate Γ∆.
Third, we prove that the dead-beat control design strategy is undominated by control design methods inW2. Let Γ∈ W2and index i and vector [ ¯A]i be such that I + Λi([ ¯A]i, Bii)̸= 0. Thus we know that there exists
at least i1∈ Iisuch that ¯Ai1 ̸= 0 and ¯Ai1+ bi1i1Γi1( ¯A, B)̸= 0. Based on the structure matrix we know that there exits l̸= i such that sli ̸= 0. Choose an index l1 ∈ Il. Consider the matrix A defined by [A]i = [ ¯A]i
and al1i1 = r and all other entries of A equal to zero. Then [A]i+ [Γ(A, B)]i= (I + Λi([A]i, Bii))[A]i and [A]j+ [Γ(A, B)]j= 0 for all j ̸= i (and, in particular, j = l since Γ does not belong to W1). Again, K will stand for Γ(A, B). We have
KTK + (A + BK)TKTK(A + BK)− ATB−TB−1A
≥ (Ai1+ bi1i1Γi1(A, B))T(Ai1+ bi1i1Γi1(A, B))× r2/(bl1l1)2−∑
z∈Ii
(ATzAz)/(b2zz),
and hence, since Ai1+ bi1i1Γi1(A, B)̸= 0, we can choose r large enough to ensure that this matrix has a strictly positive eigenvalue. Thus, the control design strategy Γ∈ W2cannot dominate Γ∆.
Now, we have to prove the only if part of the theorem. Proving this part is equivalent to proving the fact that if there exists j such that for every i̸= j, sij = 0, then there exists a controller Γ which can dominate the dead-beat controller. Without loss of generality we can assume that siq= 0 for all i̸= q. In this situation, we can partition the matrix A as
A =
A11 A12 · · · A1,q−1 0 A21 A22 · · · A2,q−1 0 ... ... . .. ... ... Aq−1,1 Aq−1,2 · · · Aq−1,q−1 0 Aq1 Aq2 · · · Aq,q−1 Aqq
,
Define ¯x0= [ x1(0) · · · xq−1(0) ]T. Let K = Γ(A, B) be
K =
−B−111A11 · · · −B11−1A1,q−1 0
−B−122A21 · · · −B22−1A2,q−1 0 ... . .. ... ... Kq1 · · · Kq,q−1 Kqq
,
and let ¯K =[
Kq1 · · · Kq,q−1 Kqq
]be the sub-controller gain for the last sub-system. The cost of this controller would be
J(A,B,x0)(Γ(A, B)) = J(A,B,x(1)
0)+ J(A,B,x(2)
0)( ¯K), where
J(A,B,x(1)
0)=¯xT0
A11 · · · A1,q−1 ... . .. ... Aq−1,1 · · · Aq−1,q−1
T
A11 · · · A1,q−1 ... . .. ... Aq−1,1 · · · Aq−1,q−1
¯x0,
and J(A,B,x(2)
0)( ¯K) is the cost associated with plant ˆ
x(k + 1) = ˆAˆx(k) + ˆB ˆu(k), with cost function
J(A,B,x(2)
0)( ¯K) =
∑∞ k=1
ˆ
x(k)TQˆx(k) +
∑∞ k=0
ˆ
u(k)Tu(k),ˆ where
Q =
0 0 · · · 0 0 0 0 · · · 0 0 ... ... . .. ... ... 0 0 · · · 0 0 0 0 · · · 0 I
.
A =ˆ
0 0 · · · 0 0
0 0 · · · 0 0
... ... . .. ... ...
0 0 · · · 0 0
Aq1 Aq2 · · · Aq,q−1 Aqq
,
and
B =ˆ
0 0 · · · 0 0 0 0 · · · 0 0 ... ... . .. ... ... 0 0 · · · 0 0 0 0 · · · 0 Bqq
.
Using dead-beat controller design strategy for the upper part, decouples the cost of the lower part of the system J(A,B,x(2)
0)( ¯K) from the cost of the upper part of the system J(A,B,x(1)
0). Thus one can design the controller for the lower part without the model-information of the upper part. The Riccati equation for finding the optimal controller for this discrete time system is
AˆTX ˆˆA− ˆATX ˆˆB(I + ˆBTX ˆˆB)−1BˆTX ˆˆA− ˆX + Q = 0, The unique solution to this Riccati equation is
X = Q + ˆˆ ATW ˆA,
where W is
0 · · · 0
... . .. ...
0 · · · Xqq− XqqBqq(I + BqqTXqqBqq)−1BqqTXqq
,
and
ATqqXqqAqq− ATqqXqqBqq(I + BqqTXqqBqq)−1BqqTXqqAqq− Xqq+ I = 0.
which is solvable with limited model information because it is a function of last subsystem model only. Using the unique solution of the mentioned Riccati equation, the optimal controller gain would be
K =ˆ −(I + BqqTXqqBqq)−1BqqTXqq
[ Aq1 Aq2 · · · Aq,q−1 Aqq
].
Using this controller for any matrix A with structure matrix S, the cost is always less than or equal to dead-beat controller and for the case which ˆA does not equals to zero, the cost of Γ is strictly less than the dead-beat controller, because
Xqq− XqqBqq(I + BqqTXqqBqq)−1BqqTXqq = B−Tqq g(Yqq)Bqq−1,
where Yqq = BqqTXqqBqq and g(Yqq) is defined in the proof of Theorem 3.2. Using claim 2 in the proof of Theorem 3.2 results in
Bqq−Tg(Y )B−1qq ≤ g(¯λ(Y ))Bqq−TBqq−1< Bqq−TBqq−1, which is equivalent to
W < ˆB−TBˆ−1. Therefore, there exists an initial condition x0 such that
J(A,B,x0)(Γ(A, B)) = J(A,B,x(1)
0)+ J(A,B,x(2)
0)( ¯K) < J(A,B,x0)(Γ∆(A, B)).
Thus, the controller design method Γ dominates the dead-beat controller design method Γ∆.
Lemma 3.4 Γ∈ C has bounded competitive ratio only if aij = 0 for all i ∈ Il where 1≤ l ≤ q results in γij(A, B) = 0 for all i∈ Il.
Proof: Assume that this claim is not correct. Consider matrix ¯A defined such that ¯Ai= Ai for all i∈ Il
and ¯Ak = 0 for all k /∈ Il. Based on the definition of limited-model-information controller design methods, we know Γi(A, B) = Γi( ¯A, B) for all i∈ Iland Γi( ¯A, B) = 0 for all i /∈ Il. For x = ej, we have
J( ¯A,B,e
j)(Γ( ¯A, B))≥∑
i∈Il
γij( ¯A, B)2=∑
i∈Il
γij(A, B)2> 0.
