Optimal Decoupling of Dynamical Systems
A Convex Approach With Aerospace Applications
Theses of Ph.D. Dissertation by Tam´ as Ba´ ar
In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
Department of Control for Transportation and Vehicle Systems Budapest University of Technology and Economics
Hungary 2023 Supervisor:
Tam´ as Luspay Ph.D.
Introduction
A large-scale, complex dynamic system is generally composed of several subsystems. These serve as building blocks, describing various parts of the underlying dynamics. Control of the system as a whole might be a challenging task. However, splitting the control problem into distinct subproblems may ease the control design process. A straightforward idea is the control of the individual building blocks (subsystems) where on this local level, the complexity of the design is significantly reduced. The thesis explores how to interact with a specific subsystem without interfering with the others. More specifically, static input and output transformation vectors are introduced, ensuring that a given controller will only interact with the targeted subsystem. This decoupling objective is in line with the recent trends of systems- and control engineering aiming for the design ofstructured controllers [1], where each block of the controller may correspond to a specific subsystem.
A conventional decoupling methodology in the control system literature is the so called input-output decoupling, which is a frequently used approach to simplify the control system design by enforcing a diagonal controller structure. This is a well established research field [20], and most approaches trace back to the application of a suitable method which converts the system into a diagonally dominant one (such as: decoupling by static and dynamic pre- and post-compensators [29, 23], decoupling by state feedback [33], etc.). A common feature is that the outputs are defined to be the controlled variables. These methods limit the interaction between certain loops, but due to the additive nature of the subsystems, they may contribute to several controlled outputs.
Another direction in decoupled control design is the approach of controlling selected subsys- tems by specific control laws. Here, the frequency-wise separation of the different subsystems is one well-known tool. Traditionally notchor roll-off filters are introduced to suppress certain frequency ranges, limiting the controller’s interaction with various parts of the dynamics [15, 26]. In this framework, an H∞ closed-loop shaping approach has been presented in [31], in- volving band-stop weighting functions to achieve decoupled interaction between the plant and the controller. The concept of dynamic filters often leads to satisfactory results however, the dimension of the controller is increased by each applied weighting filter.
The thesis proposes an approach which fits more closely into a recently developed trend in the decoupled control design; which issubsystem decoupling by suitable input and output trans- formations. The advantage of these approaches is the unaltered dimension of the underlying control problem: the static transformations convert the design into a SISO one without intro- ducing further dynamics. The method of ’Modal Isolation and Damping for Adaptive Aeroelastic Suppression’ (MIDAAS) [16] is a constrained least-squares optimization based algorithm, which designs controller for specifically damping the undesired dynamical components in the system, without affecting the remaining dynamics. A special combination of the available input and output signals is used for this purpose. Another static decoupling approach is presented in [27]
and [28], which relies on a jointH2 norm based input- and output-blend calculation method to maximize the controllability and observability of the selected modes, while minimize the transfer through the decoupled dynamics.
Despite their successful applications to aerospace systems, these methods do not have any extensions for uncertain or parameter varying systems.
The thesis discusses the synthesis of static input and output transformations, which isolate targeted subsystems in the system dynamics. For their computation, the subsystems are ren- dered into two groups. One consists of those to be controlled, while others belong to the ones to be decoupled. The transformation vectors are designed such that they maximize the transfer through the subsystems to be controlled while they minimize it through the other dynamics (to be decoupled). These transformation vectors are squaring down the problem for each sub- system to be a Single Input Single Output (SISO) one. On the input side, the transformation assures that the excitation of the controlled subsystem is maximized while interactions with the
decoupled one are minimized. Similarly, the output transformation maximizes the information regarding the controlled subsystem and at the same time, suppresses the effects of the remaining dynamics.
The thesis covers the decoupling of various system classes involving Linear Time Invariant (LTI), uncertain, and Linear Parameter Varying (LPV) systems as well. The proposed synthesis methods are heavily building on the minimum and maximum sensitivity characterizations of these systems. The maximum or peak gain of a system is a widely discussed topic in the control systems literature. However, the question of minimum sensitivity has received less attention. When the minimum sensitivity analysis conditions are not yet developed for a certain system class, the dissertation presents its extension in detail. The synthesis methods for the transformations are derived from the analysis conditions corresponding to the minimum and maximum sensitivity characterizations.
The analysis and synthesis methods are expressed as optimization problems with LMI con- straints because this formulation guarantees an efficient way to solve the underlying problems.
System uncertainties are handled by the most widely accepted methods, including polytopic modelling and Integral Quadratic Constraint (static and dynamic as well) based uncertainty descriptions. The thesis handles plant nonlinearities in two different ways. If the nonlinearity is induced by a continuously measurable parameter, then a Linear Parameter Varying framework is applied. Otherwise the previously mentioned uncertainty handling methodologies are followed.
Special attention is paid to providing aerospace-related application examples where the use of the proposed decoupling approach offers significant advantages.
Contributions of the thesis
This section collects the contributions of the dissertation, along with their brief discussion, to provide a high-level overview of the developed results.
Thesis 1
I have formalized a subsystem decoupling algorithm for Linear Time Invariant systems.
The approach synthesizes static input and output transformation vectors, which isolate the targeted subsystem(s). When applied to the system, they maximize the transfer through the targeted subsystem while rendering uncontrollable and unobservable the remaining dynamics.
An attached controller will interact with the desired dynamics through the transformed system interfaces. The proposed solution for the computation of the decoupling transformations is an efficient Linear Matrix Inequality based technique, where an arising rank constraint is satisfied by a sequence of alternating projections.
Related publications: [6, 12, 3, 7].
As a starting point, consider a continuous time Linear Time Invariant (LTI) dynamics given in the following generic state-space form
Pny×nu :
(x(t) =˙ Ax(t) +Bu(t),
y(t) =Cx(t) +Du(t), (1)
with the standard notations: x ∈ Rnx is the state vector, u ∈ Rnu is the input vector and y∈Rny is the output vector of the system. The system matrices are of appropriate dimensions.
In addition, assume that the system is given in the following subsystem form:
A=
Ac 0 0 Ad
, B = Bc
Bd
, C =
Cc Cd
, D= D
.
(2)
G(s) ku
Gc(s) Gd(s)
kyT
−λc(s)
u +
+
y
¯y u¯
Figure 1: Closed loop control scheme with input and output blending
Under the assumption of diagonalizable A, such a representation is always achievable with the respective similarity transformation, which is generally referred to as modal form [17]. In modal form theAmatrix has a block diagonal structure, where each block corresponds to a dynamical mode of the system. These dynamical modes can be represented by either real (R) or complex (with imaginary partI) eigenvaluesλ, which determine the structure of the corresponding block of matrix A= diag (A1, ..., An) as
Ai =
λi if I(λi) = 0
R(λi) I(λi)
−I(λi) R(λi)
if I(λi)̸= 0. (3)
Representation (2) can be considered a special modal form, without loss of generality, where the modes are grouped into two subsystems: one that we wish to control and another one that we wish to decouple (leave unaffected). The subsystems are denoted by indexes {·}c and {·}d
respectively. The corresponding transfer function matrix representation is given by G(s) = X
i∈{c,d}
Ci(sI−Ai)−1Bi+D/2 =Gc(s) +Gd(s), (4)
where Gc(s) and Gd(s) are the transfer functions of the subsystems to be controlled and decou- pled, respectively, with the standard notation of sbeing the Laplace variable and I being the identity matrix.
An immediate observation of the structure (2) shows that the two subsystems are coupled through only their input-output channels. In order to isolate the two subsystems from each other, it is natural to seek input and output transformations that weaken or resolve this coupling. For this purpose, input and output blending vectors are introduced such thatku ∈Rnuandky ∈Rny and ||ku||2 = ||ky||2 = 1. These blending vectors transform the signal vectors u and y onto a single dimension, consequently reducing the control problem into a SISO one. In Figure 1 the control input
¯u∈Ris distributed between the plant’s inputs (u=ku
¯u) in a way that they only excite the subsystem which one wishes to control. Similarly, the controller’s input
¯y=kTyy∈R is calculated such that the information content from the subsystem which has to be decoupled, is minimized. Furthermore, if the loop is closed by a λc(s) controller, then the ku and ky
transformations assure that it will mainly interact with the targeted Gc(s) dynamics.
The synthesis of these blend vectors relies on the minimum and maximum sensitivities of a transfer function, which are defined as
||G(s)||[−¯ω,¯ω]:= inf
ω∈[ω,¯¯ω]¯σ G(jω)
, (5)
||G(s)||∞:= sup
ω
¯ σ
G(jω)
, (6)
A Bw Bu
Cv Dvw Dvu
Cy Dyw Dyu
∆1
...
∆M
b, w=
w1 ...
wM
v=
v1
...
vM
u y
Figure 2: LFT modeling respectively, where [
¯ω,ω] denotes the investigated frequency range. Then the normalized vectors¯ ku and ky are such that, they maximize
||kTyGc(s)ku||[−¯ω,¯ω]> β, β ≥0, (7) while minimize
||kyTGd(s)ku||∞< γ, γ ≥0, (8) over the selected frequency range. Here β and γ are two positive constants referring to the minimum and maximum sensitivities, respectively. These blend vectors are computed in an iterative way, where first, an optimalkuinput transformation is found, and then a corresponding output blend is calculated. The synthesis algorithm is based on the Linear Matrix Inequality conditions for the minimum and maximum sensitivities of a system.
The proposed decoupling method is evaluated based on an academic example, highlighting the important properties of the approach and a batch test involving 1452 randomly generated state-space systems.Careful evaluation of the results revealed that the algorithm achieved suc- cessful decoupling in 86 % of the investigated test samples.
Thesis 2
I defined the robust minimum sensitivity for uncertain LTI systems as the system’s minimum gain over all possible uncertainties in the uncertainty set (∆ ∈∆). The uncertain system is modeled as Fu(M,∆), i.e. the upper LFT interconnection of a ∆ uncertainty block and a nominal system M. Then the robust minimum gain is defined as
||Fu(M,∆)||[Ω]∆−> β,
over the Ω frequency range. Relying on Integral Quadratic Constraints, Linear Matrix In- equality based analysis conditions are provided for the computation of the β lower bound.
Related publications: [9, 8].
Upper Linear Fractional Representation denotes the standard interconnection of a nominal, known dynamics (M) and an uncertain block (∆) as it is given in Figure 2. Its transfer function form is given as follows. Consider a ∆ ∈ Cnv×nw uncertainty block, and a transfer function matrix M ∈C(nv+ny)×(nw+nu) partitioned as
M=
M11 M12
M21 M22
. (9)
In order to simplify the notations in Figure 2, only dynamic ∆kuncertainty blocks are represented. However, the block may contain uncertainδkparameters as well, as it is shown in (12).
∆ M Ψ
w v
u y
z
Figure 3: Analysis Interconnection Structure Then the corresponding Upper LFT form is given by
Fu(M,∆) =M22+M21∆(I− M11∆)−1M12. (10) Throughout the thesis it is assumed thatFu(M,∆) is well-posed, i.e., (I− M11∆)−1 is nonsin- gular. By state-space equations, the system can be modelled as
˙
x(t) =Ax(t) +Bww(t) +Buu(t), v(t) =Cvx(t) +Dvww(t) +Dvuu(t), y(t) =Cyx(t) +Dyww(t) +Dyuu(t), w(t) = ∆v(t),
(11)
where ∆∈∆the set is defined by
∆= diag
δ1I, ..., δMI,∆1, ...,∆Mc ⊂Cny∆×nu∆, δk∈R,|δk| ≤1,∆k∈Crk×ck,σ(∆¯ k)≤1 . (12) Thev∈Rnv,w∈Rnw are the inputs and outputs of the uncertain ∆ block. The signalsu∈Rnu and y ∈Rny are the nominal inputs and outputs of the system. The input and output sizes of each complex ∆k uncertainty block are denoted by ck and rk, respectively.
The IQC framework assumes that, ∆ imposes constraints on thev ∈ Ln2v, w∈ Ln2w signals [25]. The description of these constraints by a quadratic integral formula replaces the unknown
∆ block in the Fu(M,∆) interconnection by a known Ψ filter (see Figure 3). Then the input and output signals of the uncertainty block satisfy the IQC defined by the (Ψ, M) pair, if
ZT 0
z(t)TM z(t)dt≥0, ∀T ≥0, (13)
holds, whereM ∈Snz is a symmetric matrix and Ψ∈RHn∞z×(nv+nw)is a stable, invertible, linear system with the following frequency domain realization:
Ψ :=CΨ(jωI−AΨ)−1
BΨv BΨw +
DΨv DΨw
(14) and state-space representation:
˙
xΨ(t) =AΨxΨ(t) +BΨvv(t) +BΨww(t),
z(t) =CΨxΨ(t) +DΨvv(t) +DΨww(t). (15) If T = ∞, then (Ψ, M) is called a soft factorization, otherwise, it is referred to as hard fac- torization. Then through the proper selection of the Ψ filter, various sources of uncertainties and nonlinearities can be characterized. A robust, IQC-based maximum sensitivity analysis condition is developed in [25] in terms of Linear Matrix Inequalities.
The novelty of Thesis 2 is that it provides robust analysis conditions, relying on LMIs, for characterizing an uncertain system’s minimum sensitivity over finite and infinite frequency
0.001 0.01 0.1 1 10 100
−40
−20 0 20 40
frequency [rad/s]
magnitude[dB]
G(s)
Existing in [25]
Thesis 2
Figure 4: Robust sensitivity analysis example
Mc+Md
ku ky
∆
−λc(s)
u y
¯y
¯u
v w
Figure 5: Closed loop control scheme with input and output blending
ranges. These contributions are shown in Figure 4, based on an elevator toaznormal acceleration transfer function, corresponding to a fixed-wing aircraft. The model is taken from [13], and it describes the Aerosonde UAV in a trimmed straight and level flight at 33 m/s, and a ±5%
inaccuracy in the Cmα parameter leads to the set of LTI systems in the shaded area. The already existing robust maximum sensitivity analysis condition is plotted by the golden dotted line, while the newly developed results are shown by the red dashed lines.
Thesis 3
Following the results in Thesis 1, I have developed a robust subsystem decoupling algorithm. I derived Integral Quadratic Constraints based LMI synthesis conditions for the computation of the robust input and output transformations. For this purpose, the subsystems in an Fu(M,∆)uncertain interconnection, with∆∈∆, ||∆||∞≤1are combined into two groups, according to which one(s) should be controlled (amplified) or decoupled (suppressed). These are denoted by Fu(Mc,∆c) and Fu(Md,∆d), respectively. The designed ku ∈ Rnu and ky ∈ Rny blend vectors are such that they maximize the transfer through the controlled subsystem while minimizing it through Fu(Md,∆d).
Related publication: [8].
This thesis strongly builds on the results of Thesis 2. Similarly for Thesis 1, it is supposed that system Mis brought to a subsystem form withM=Mc+Md. The indices are denoting that the subsystems should be controlled or decoupled, respectively. The decoupling of these two subsystems is characterized by the minimum and maximum sensitivities, defined as
||Mc||∆−= inf
∆∈∆||Fu(Mc,∆)||−, (16)
||Md||∆∞= sup
∆∈∆||Fu(Md,∆)||∞. (17)
The ||M||∆∞ term is usually labeled in the literature as the worst-case gain of the Fu(M,∆) interconnection [29]. As it can be depicted from its definition, we are using this metric for the subsystem, which has to be decoupled, while the minimum gain is used for the targeted subsystem (see eq. (16)). Note also the subscript ∆ in the above definitions, explicitly indicating that these metrics are computed over the uncertainty set.
Based on Figure 5, the decoupling problem can be stated as follows. Design an environ- ment (denoted by the dashed frame) which allows the control of the subsystem Mc(s) by a corresponding controller λc(s) and minimizes the interaction with Md(s). This is achieved by finding linear combinations of the input- and output- signals of the system, respectively. The normalized vectors ku ∈Rnu and ky ∈Rny are such that, they maximize
||kyTMcku||[∆−¯ω,¯ω]> β, β≥0, (18) while minimize
||kyTMdku||∆∞< γ, γ ≥0, (19) over the selected frequency range. Here β and γ are two positive constants referring to the minimum and maximum sensitivities, respectively. These blend vectors are computed in an iterative way as well, where first an optimalku input transformation is found, and then a corre- sponding output blend is calculated. By relying on polytopic modeling, and static and dynamic IQC-based uncertainty descriptions, three different decoupling algorithms were developed for the synthesis of the robust blend vectors. Similarly to Thesis 1, all approaches are based on the Linear Matrix Inequality conditions for the minimum and maximum sensitivities of a system.
Thesis 4
I proposed two algorithms for synthesizing static parameter dependent input and output transformations for the decoupling of linear parameter varying subsystems. It is assumed that the LPV system is given in a G(ϱ) = Gc(ϱ) +Gd(ϱ) subsystem form, describing the dynamics to be controlled and decoupled, respectively. When applied to the system, theku(ϱ) : Rnϱ →Rnu and ky(ϱ) :Rnϱ →Rny normalized decoupling vector functions, are maximizing the transfer through Gc(ϱ), while suppressing Gd(ϱ). The developed synthesis algorithms are building on the polytopic and grid-based LPV system representations, and the corresponding design conditions are formalized in terms of coupling Linear Matrix Inequalities.
Related publication: [11].
Thesis 2 and Thesis 3 developed techniques to handle uncertain parameters in the analysis and decoupling synthesis phase. This thesis investigates the case when parameters in the system vary along a measurable trajectory. The focus is on plant models where the changes in certain parameters are responsible for the system’s nonlinear behavior. This time varying system class is often refered as Linear Parameter Varying (LPV) systems. The varying parameters together are labeled as the ϱ scheduling parameter vector. The correspondingG(ϱ) system has state-space form
˙
x(t) =A(ϱ)x(t) +B(ϱ)u(t),
y(t) =C(ϱ)x(t) +D(ϱ)u(t), (20)
with the standard notation ofx(t)∈Rnx,u(t)∈Rnu and y(t)∈Rny being the state, input and output vector, respectively, depending on the continuous time variablet. The trajectories of the
Time dependence ofϱis omitted to ease the notation.
time-varying scheduling vector ϱ ∈ Rnϱ are unknown apriori but measurable online, and they are assumed to be constrained in the parameter variation set
FPV ={ϱ∈ Cl(R+, Rnϱ) :ϱ∈ P, ϱ˙ ∈ V,∀t≥0}, (21) where Cl is the class of piece-wise continuously differentiable functions, with
P :={ϱ∈ Rnϱ :ϱi∈[
¯ϱi,¯ϱi]}, and V :={ϱ˙ ∈ Rnϱ :|ϱ˙i| ≤νi, ∀i= 1, ..., nϱ}. A trajectory is said to be rate unbounded if ¯νi =∞ and
¯νi =−∞. Hence, LPV systems are a special class of Linear Time Varying systems, where system matrices are continuous functions of the time-varying scheduling vector.
The developed decoupling algorithms use the subsystem form of an LPV system. This, in structure, is similar to the LTI case (2), but the subsystems are parameter dependent and given in the form
A(ϱ) =
Ac(ϱ) 0 0 Ad(ϱ)
, B(ϱ) =
Bc(ϱ) Bd(ϱ)
, C(ϱ) =
Cc(ϱ) Cd(ϱ)
, D(ϱ) = D(ϱ)
,
(22)
where the subsystems were grouped according to which one should be controlled or decou- pled. This special form may originate naturally from the properties of the underlying dynamics or achievable by suitable parameter-varying transformations. Such transformations yielding a parameter-dependent modal form are discussed in [21] and [32]. Both approaches result in a block-diagonal and continuous A(ϱ) function, where each block represents a dynamical mode of the dynamics.
The minimum and maximum sensitivities of a parameter varying system are defined as
||G(ϱ)||[Ω]− := inf
ϱ∈FPV inf
u∈L2,Ω, u̸=0
||y||2
||u||2
, (23)
||G(ϱ)||i2 := sup
ϱ∈FPV
sup
u∈L2, u̸=0
||y||2
||u||2
. (24)
Note that the system’s minimum sensitivity in (23) is defined over an Ω finite frequency interval.
This means that, the u input signal is constrained to have a finite Ω frequency spectra [30].
Furthermore it is important to emphasize that the analysis conditions corresponding to (23) are based on the Minimum Gain Lemma, which can be proved based on time domain techniques [14]. Further infomration about the minimum sensitivity of an LPV system is given in [18] and [19]. In the robust control literature the maximum sensitivity in (24) is labeled as the inducedL2
norm of the system. For the computation of the (23) and (24) sensitivity bounds LMI techniques are used.
For discussing the parameter varying subsystem decoupling, introduce the notationG(ϱ) = Gc(ϱ) +Gd(ϱ), with indices denoting the controlled and decoupled subsystems respectively. The aim of the decoupling problem is to find ku(ϱ) : Rnϱ → Rnu and ky(ϱ) : Rnϱ → Rny vector functions, such that,||ku(ϱ)||2 =||ky(ϱ)||2= 1, ∀ϱ∈ FPV and they maximize
||ky(ϱ)TGc(ϱ)ku(ϱ)||[Ω]− > β, β≥0, (25) while minimize
||ky(ϱ)TGd(ϱ)ku(ϱ)||∞< γ, γ≥0, (26) over the selected Ω frequency range. Here β and γ are two positive constants referring to the minimum and maximum sensitivities, respectively.
These blend vector functions are computed in an iterative way as well, where first, an optimal ku(ϱ) input transformation is found, and then a corresponding output blend is calculated. By re- lying on polytopic and grid-based LPV system descriptions, two different decoupling algorithms were developed for the synthesis of the parameter varying blend vector functions. A detailed evaluation and comparison of the two approaches is provided in Chapter 5 of the dissertation. In Chapter 6, the grid-based decoupling method is applied to a real aerospace system for isolating a flexible mode responsible for wing oscillations from the rigid body dynamics. The importance of this decoupling is that, it may facilitate the synthesis of a controller targeting the flexible dynamics solely, without interactions with the baseline flight control system.
Thesis 5
In this thesis I investigate the optimal subsystem decoupling for flutter suppression control law synthesis based on the model of a flexible wing aircraft. I show that by designing input and output transformations according to the computation methods in Thesis 1, Thesis 3 and Thesis 4, the aircraft’s flutter modes can be isolated from the rigid body dynamics. I evaluate the decoupling performance for LTI, uncertain and LPV system models through frequency and time domain methods as well. Related publications: [6, 12, 8, 11].
Flutter is a coupled aeroelastic phenomenon which originates from the interaction of the aerodynamic and structural forces, and it leads to a lightly damped or unstable oscillating motion of the structure. The results in Thesis 1, Thesis 3 and Thesis 4 assured that the flutter modes can be decoupled from the rigid body dynamics in the LTI, uncertain and LPV sense as well. These findings were supported by frequency domain transfer characteristics (singular value curves) of the various subsystems, and by time domain simulations.
Furthermore the dissertation presents possible flutter suppression techniques, which are based on isolating the targeted flutter modes by static transformations. The novelty of the approaches is that they employ Ku and Ky blend matrices, which do not necessarily convert the dynamics to SISO ones. This allows to synthesize control laws for the targeted dynamics by many conventional control design techniques, such as state-feedback, static and dynamic output-feedback. The decoupling transformations assure that they will not interact with the re- maining dynamics. From the aerospace control perspective, the benefit of the proposed control law synthesis approach is straightforward: facilitating the separate design of a flutter suppression controller without changing the rigid-body autopilot.
Next, the static output-feedback technique is discussed briefly. The objective of the control is the stabilization of the unstable flutter modes, and so drive any nonzero initial conditions to zero. For this a partial pole placement technique is proposed, which only affects the targeted poles and leaves the remaining ones untouched. A necessary condition for static output feedback design is that, the blended system satisfiesnu,b×ny,b≥nx[34]. In order to satisfy this constraint, Ku ∈Rnu×nu,b and Ky ∈Rny×ny,b blend matrices were synthesized, where nu,b and ny,b denote the input and output dimensions after the blend matrices are applied to the system.
Suppose that the system is partitioned according to (2) and that the F output feedback matrix is synthesized based on the controlled dynamics. The closed-loop is then given as
Ac 0 0 Ad
− Bc
Bd
KuF KyT
Cc Cd
=
Ac−BcKuF KyTCc −BcKuF z }| {≈0
KyTCd
−BdKu
| {z }
≈0
F KyTCc Ad−BdKu
| {z }
≈0
F KyTCd
| {z }
≈0
≈
Ac−BcKuF KyTCc 0
0 Ad
.
(27)
The blend matrices are designed such that they minimize the transfer through the decoupled
−40
−20 0 20 40
xc(t)
Stabilize without blend
−40
−20 0 20 40
Stabilize with blend
0 0.5 1 1.5 2
−5 0 5
time [s]
xd(t)
0 0.5 1 1.5 2
0.1 0.05 0
−0.05
time [s]
Figure 6: LTI example: numerical simulation with static output-feedback (with for controlled and for decoupled subsystems)
subsystem. By the assumptionsBdKu ≈0 andKyTCd≈0, (27) is approximately blockdiagonal, where the Ad block is not affected by the feedback. However, note that the cross terms on the off-diagonal are such that the BcKu and KyTCc are maximizing the transfer through the targeted subsystem. This shows that, a successful decoupling necessitates ||BdKu||2 ≪ 1 and
||KyTCd||2 ≪ 1, and in the extreme case of Ku ∈ ker(Bd), Ky ∈ ker(CdT), the approach may lead to perfect decoupling. In light of the before mentioned arguments, partial pole placement is not guaranteed by the proposed method, but it is often achieved depending on the actual parameters of the underlying system.
The dissertation evaluates the proposed technique based on the numerical model of a flexible- wing aircraft. Figure 6 compares two scenarios. In the first case, the input and output blend matrices are not used, i.e. Ku =Inu and Ky =Iny in (27), and F has been synthesized for sta- bilizing the closed-loop. Note the strong interactions between the subsystems, as the decoupled rigid body dynamics are excited as well. On the other hand, if the decoupling transforma- tions are applied, and F is such that it stabilizes the G(s) subsystem, then the interactions are significantly limited.
Publications of the Author
[2] Tam´as Ba´ar and P´eter Bauer. “A rep¨ul´esi biztons´ag n¨ovel´ese leveg˝oh¨oz k´epesti sebess´eg m´as szenzorokra t´amaszkod´o becsl´es´evel”. In:Rep¨ul´estudom´anyi k¨ozlem´enyek 30.1 (2018), pp. 161–183.
[3] Tam´as Ba´ar, P´eter Bauer, and Tam´as Luspay. “Decoupling of Discrete-time Dynamical Systems Through Input-Output Blending”. In:IFAC-PapersOnLine 53.2 (2020), pp. 921–
926.
[4] Tam´as Ba´ar, P´eter Bauer, Zolt´an Szab´o, B´alint Vanek, and J´ozsef Bokor. “Evaluation of multiple model adaptive estimation of aircraft airspeed in close to real conditions”. In:
25th Mediterranean Conference on Control and Automation. IEEE. 2017, pp. 271–276.
[5] Tam´as Ba´ar, Bence Beke, P´eter Bauer, B´alint Vanek, and J´ozsef Bokor. “Smoothed multi- ple model adaptive estimation”. In:European Control Conference. IEEE. 2016, pp. 1135–
1140.
[6] Tam´as Ba´ar and Tam´as Luspay. “AnH−/H∞blending for mode decoupling”. In:Ameri- can Control Conference. IEEE. 2019, pp. 175–180.
[7] Tam´as Ba´ar and Tam´as Luspay. “Dinamikus rendszerek sz´etcsatol´asa be- ´es kimeneti transzform´aci´okkal”. In: Alkalmazott Matematikai Lapok 39.1 (2022), pp. 1–19.
[8] Tam´as Ba´ar and Tam´as Luspay. “Robust decoupling of uncertain subsystems”. In:Inter- national Journal of Robust and Nonlinear Control 32.10 (2022), pp. 6086–6109.
[9] Tam´as Ba´ar and Tam´as Luspay. “Robust minimum gain lemma”. In:60th Conference on Decision and Control. IEEE. 2021.
[10] P´eter Bauer, Tam´as Ba´ar, Tam´as P´eni, B´alint Vanek, and J´ozsef Bokor. “Application of input and state multiple model adaptive estimator for aircraft airspeed approximation”.
In:IFAC-PapersOnLine 49.17 (2016), pp. 76–81.
[11] Tam´as Ba´ar, P´eter Bauer, and Tam´as Luspay. “Parameter Varying Mode Decoupling for LPV Systems”. In:IFAC-PapersOnLine 53.2 (2020). 21st IFAC World Congress, pp. 1219–
1224.
[12] Tam´as Ba´ar and Tam´as Luspay. “Decoupling through input–output blending”. In:Inter- national Journal of Control 94.12 (2021), pp. 3491–3505.
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