整合SDRE和ISMC設計之非線性系統可靠度控制理論與應用

國

立

交

通

大

學

電控工程研究所

碩士論文

整合 SDRE 和 ISMC 設計

之非線性系統可靠度控制理論與應用

Reliable Nonlinear Control

via Combining SDRE and ISMC Approaches

研 究 生：林立岡

指導教授：梁耀文 博士

整合 SDRE 和 ISMC 設計

之非線性系統可靠度控制理論與應用

Reliable Nonlinear Control

via Combining SDRE and ISMC Approaches

研 究 生：林立岡

Student: Li-Gang Lin

指導教授：梁耀文 博士 Advisor: Dr. Yew-Wen Liang

國立交通大學電控工程研究所

碩士論文

A Thesis

Submitted to institute of Electrical and Control Engineering

College of Electrical Engineering and Computer Science

National Chiao Tung University

In Partial Fulfillment of the Requirements

For the Degree of Master

In

Electrical and Control Engineering

June 2010

Hsinchu, Taiwan, Republic of China

i 

整合SDRE和ISMC設計

之非線性系統可靠度控制理論與應用

研究生：林立岡

指導教授：梁耀文 博士

國立交通大學電控工程研究所

摘要

本論文探討非線性系統使用

SDRE)理論的

可靠度控制設計，並應用於衛星之姿態控制。由於引進了

(ISMC)理論合併使用，研究發現可以大大增進系統的穩健性和可靠度。

然而，傳統的SDRE設計必須先拆解漂流項成為

的形式，然後再利

用

及線性理論來判斷系統在該狀態的可穩定性和可觀測性以確保對應

之SDRE存在正定解。但當系統動態足夠複雜時，這些判斷條件不容易被檢

驗，此外，目前文獻也沒有提供不同拆解方式的分析與比較。因此，本論

文提出另一種SDRE的拆解方式，並探討能保證對應的Riccati方程式存在正

定唯一解的充分且必要條件，此充要條件只需要系統動態在該狀態之資訊。

本論文中也發現如果採用固定拆解方式的傳統SDRE設計方法能夠正常工

作，那麼本論文探討的SDRE拆解方式一樣可以正常工作。透過例子，我們

說明了本論文提出拆解方法的好處。

ii   

Reliable Nonlinear Control

via Combining SDRE and ISMC Approaches

Student: Li-Gang Lin

Advisor: Dr. Yew-Wen Liang

Department of Electrical and Control Engineering

National Chiao Tung University

ABSTRACT

In this thesis, we investigate the nonlinear reliable control issues via the state-dependent Riccati equation (SDRE) scheme with application to the attitude control of a satellite. Owing to incorporating with the integral sliding mode control (ISMC) design, both the robustness and the reliability performances are greatly improved. However, it is known that the conventional SDRE scheme has to symbolically factorize the drift term in the form of

f(x)=A(x)x, and then using this A(x) to check system's stabilizability and observability

symbolically at every nonzero state for ensuring the solvability of an associated SDRE. These checking conditions are in general not easy to implement when the system dynamics is complicated, and there is no guideline provided for performing the factorization. As a result, this study also presents an alternative approach of factorization, which only requires the information of the system dynamics at every state and guarantees the existence of a unique positive definite solution of the associated Riccati equation when a mild condition is satisfied. It is shown that the alternative approach always works if the conventional SDRE approach adopting any specific factorization for f(x) is successfully operated. An illustrative example is also given to demonstrate the benefits of the alternative approach of factorization.

iii 

誌 謝

由衷感激幫助我完成本篇論文的所有人

首先要先感謝學生我(以下簡稱學生)的指導教授 – 梁耀文博士。紮

實豐富的專業知識和認真嚴謹的研究態度，令學生獲益良多，於研究過程

中許多難題也因教授從旁協助而迎刃而解。接著要感謝口試委員鄧清政博

士、廖德程博士和徐勝均博士給予寶貴的建議與指導使本論文更加完整。

接著要感謝同窗研究的實驗室學長學弟們：徐勝均學長、丁立偉學長、

王士昕學長和吳家榮學長總是分享過來人的研究經驗，使學生一路走來更

加順遂，在我遇到困難時也總是提供專業意見和實質幫助。而學弟們旭志、

智強、榮仁、君豪和偉庭也都會適時的給予協助和可靠意見，並且為實驗

室營造和諧融洽的氣氛，讓整個研究生活變得更輕鬆愉快。要特別感謝的

一位，魏源廷同學，他跟我預計要同時完成碩士學業，於研究學習過程中

互相勉勵一起成長，因為有他，使我研究生活更加豐富。

最後要感謝我的家人和好友們，全力支持我拿到各階段的學業，沒有他

們，沒有現在的我。他們總是給我最大的鼓勵，讓我可以毫無後顧之憂努

力在學業上勇往直前，進而有機會完成研究所的學習。謝謝你們。

- 謹將本篇論文獻給我的女友怡樺 -

iv   

TABLE OF CONTENTS

ABSTRACT (Chinese) ...

ABSTRACT (English) ...

TABLE OF CONTENTS...

LIST OF TABLES...

LIST OF FIGURES...

1. INTRODUCTION

2. PRELIMINARIES

2.1. State Dependent Riccati Equation (SDRE)... 4

2.2. Integral Sliding Mode Control (ISMC)... 5

3. STUDY OF ROBUSTNESS PERFORMANCE OF SDRE+ISMC

SCHEME………..

3.1. Problem Statement ... 7

3.2. Design of Control Law ... 9

3.2.1. SDRE ………….………... 9

3.2.2. ISMC ……….………... 11

3.3. Application to Satellite Attitude Control ……….………. 13

3.3.1. Satellite Dynamics ... 13

3.3.2. Simulation Results ... 14

v 

4. STUDY OF RELIABILITY PERFORMANCE OF SDRE+ISMC

SCHEME………..

4.1. Problem Statement ... 24

4.2. Design of Active Reliable Control Law …... 26

4.2.1. SDRE ………... 26

4.2.2. ISMC ……….………... 26

4.3. Application to Satellite Attitude Control ……….………. 28

4.3.1. Design of Fault Diagnosis and Detection (FDD)…………... 28

4.3.2. Simulation Results ... 29

5. ON FACTORIZATION OF THE DRIFT TERM IN SDRE SCHEME…….

5.1. Problem Statement………... 36

5.2. Solvability Condition ……….…... 37

5.3. Implementation ……….…... 42

5.4. Algorithm ………... 47

5.5. Illustrative Example - 2-Dim Single-Input Affine System ………...………... 49

6. CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH

6.1. Conclusions ………... 52

6.2. Suggestions for Further Research …... 53

vi   

LIST OF TABLES

Table 3.1. Simulation parameters……... 14

Table 3.2. Comparison of performance... 18

Table 4.1. Simulation parameters……... 30

Table 4.2. Comparison of performance... 35

vii 

LIST OF FIGURES

Figure 3.1. Time history of the six state variables... 16

Figure 3.2. Time history of the four control inputs... 17

Figure 3.3. Time history of the sliding variables (SDRE+ISMCd)... 17

Figure 4.1. FDD diagram [38]…... 29

Figure 4.2. Time history of the six state variables... 32

Figure 4.3. Time history of the four control inputs... 33

Figure 4.4. Time history of the three residual signals... 33

Figure 4.5. Time history of the three alarm signals... 34

Figure 4.6. Time history of the three sliding variables (SDRE+ISMCr)... 34

CHAPTER

ONE

INTRODUCTION

1.1

Motivation

Numerous design methodologies are known to exist for the control design of highly nonlinear systems [12]. These include a huge number of linear design techniques used in conjunction with gain scheduling [38]; nonlinear design methodologies such as dynamic inversion [17] and sliding mode control [33]; and adaptive techniques which encompass both linear adaptive and nonlinear adaptive control. Lesser known but promising nonlin-ear design procedures are those that involve state-dependent Riccati equations (SDRE) [13]-[14].

Recently, the study of SDRE approach among the variety of control schemes for non-linear systems has attracted considerable attention (see e.g., [10]-[13], and [36]) due to its remarkable benefits. These include: 1) concept of SDRE approach is intuitive which directly adopts the LQR design at every nonzero state; 2) SDRE approach can directly address system performance through the specification of the performance index by adjust-ing the state and the control weightadjust-ings with predictable results, for instance, the engineer may tune up the weightings on system state to speed up the response at the expense of more control effort; 3) SDRE approach possesses an extra design degree of freedom arose from the non-uniqueness of the SDC representation of the nonlinear drift term, which can be utilized to enhance controller performance; 4) SDRE approach preserves the essential system nonlinearities, since it does not truncate any system’s nonlinear term. Many prac-tical and meaningful applications which are successfully performed by the SDRE design include advanced guidance law development, autopilot design, integrated guidance and control design, satellite and spacecraft control and estimation, process control, magnetic

levitation, control of systems with parasitic effects, control of artificial human pancreas, robotics, simultaneous state and parameter estimation, fan control, and various bench-mark problems (see [10], [15], [36] and the references therein).

SDRE scheme to the stabilization of nonlinear control systems is known to need to symbolically factorize the drift term in the form of f(x, t) = A(x, t)x, and then using this A(x, t) to check system’s stabilizability and observability symbolically at every state for ensuring the solvability of an associated state-dependent Riccati equations. In doing so, the SDRE algorithm fully captures the nonlinearities of the system, bringing the nonlin-ear system to a (non-unique) linnonlin-ear structure having state-dependent coefficient (SDC) matrices, and minimizing a nonlinear performance index having a quadratic-like struc-ture. Moreover, the nonuniqueness of the factorization creates extra degrees of freedom, which can be used to enhance controller performance, such as robustness. But, there is no guideline provided for the factorization f(x, t) = A(x, t)x, specifically to improve robustness. However, with the help of Integral-type Sliding Mode Control (ISMC), we can still improve robustness using SDC factorization. The ISMC approach does not have reaching phase and possesses the advantages of robustness and ease of implementation. When the uncertainty and disturbance are matched regarding the nominal healthy sub-system, the state trajectories of the nominal healthy subsystem and the uncertain system are identical. Thus, in this study, we adopt the SDRE strategy for the nominal system, and the ISMC strategy to completely nullify the matched uncertainty and disturbance. In addition to robustness, we are also interested in the reliability issue related to SDRE only and SDRE-ISMC combined designs.

However, we encounter some difficulties during the SDRE design. If the system dynam-ics is sufficiently complicated, the checking conditions of stabilizability and observability are generally not easy to implement, and there is no guideline provided for performing the factorization fulfilling some predetermined control objectives. Moreover, if SDRE fails some checking conditions at a system state, then the system may just stuck in the state since SDRE can not guarantee a feasible control related to the unique positive definite solution of the associated Riccati equation. As a result, this study also presents an alter-native approach for the factorization, which only requires the information of the system

dynamics at every state and guarantees the existence of a unique positive definite solution of the associated Riccati equation when a mild condition is satisfied. To be more detailed, we give a necessary and sufficient condition for that solution as well as the implementing algorithm on how to factorize f(x, t). Moreover, it is shown that the alternative approach always works if the conventional SDRE approach adopting any specific factorization for f(x,t) is successfully operated. An illustrative example is also given to demonstrate that we adopt conventional approach at almost all system states, but at some states (which fails to operate under conventional approach), instead we resort to the alternative approach for a different factorization of A which works.

1.2

Outline

The rest of this thesis is organized as follows. Chapter 2 sketches the SDRE and ISMC designs. Then we investigate the robustness and reliability issues related to both the SDRE and SDRE+ISMC combined designs with analytical simulation results. In Chapter 5, we formulate an alternative SDRE problem and describes our solution. Finally, we provide a short conclusion and give some suggestions of future research related afterwards.

CHAPTER

TWO

PRELIMINARIES

2.1

State Dependent Riccati Equation (SDRE)

Consider the following class of time-variant nonlinear control systems

˙x = f(x, t) + B(x, t)u (2.1)

where x ∈ IRn and u ∈ IRm denote the system states and control inputs, respectively,

f(x, t) ∈ IRn, B(x, t) ∈ IRn×m and f(0) = 0. In addition, we consider the following

performance index J = Z ∞ 0 h xT_{Q(x)x + u}T_R(x)ui dt (2.2)

where QT_{(x) = Q(x) ≥ 0, R}T_{(x) = R(x) > 0 and (·)}T _{denotes the transpose of a vector}

or a matrix. In this study, we assume that B(x, t) 6= 0 and Q(x) 6= 0 for any nonzero state x.

SDRE techniques are increasingly being used in nonlinear control applications [15] and entails factorization of the nonlinear dynamics into the state vector and the product of a matrix-valued function that depends on the state itself [10]. In doing so, the SDRE algorithm fully captures the nonlinearities of the system, bringing the nonlinear system to a (non-unique) linear structure having state-dependent coefficient (SDC) matrices, and minimizing a nonlinear performance index having a quadratic-like structure.

To solve the SDRE problem, almost all the existing studies adopted the following procedure:

• Symbolically factorize f(x, t) into the form of f(x, t) = A(x, t)x, where A(x, t) ∈ IRn×n_.

• Check the stabilizability of [A(x, t), B(x, t)] and the observability of [A(x, t), C(x)] symbolically, where C(x) ∈ IRp×n has full rank and satisfies Q(x) = CT_{(x)C(x), to}

ensure the solvability of the following SDRE [24]: AT

(x, t)P (x) + P (x)A(x, t) − P (x)B(x, t)R−1(x)BT_{(x, t)P (x) + Q(x) = 0. (2.3)}

• Solve the SDRE for P (x) to produce the SDRE controller u = −R−1_(x)BT_{(x, t)P (x)x.}

2.2

Integral Sliding Mode Control (ISMC)

The design concept of Integral Sliding Mode Control (ISMC) is quite similar to Sliding Mode Control (SMC, see e,g, [18], [26], [44], and [45]), and the main difference is that ISMC adopts the integral-type sliding surface and results no reaching phase, i.e., the system trajectories will start on the sliding manifold from the first time instant. Moreover, when the system is on the sliding manifold, the system trajectories is determined by the control law applied to the related nominal subsystem, and this control law can be any control laws fulfilling design objectives. In the following, we describe the design of ISMC([6], [7], and [28]).

Consider the following class of time-variant nonlinear control systems

˙x = f(x, t) + B(x, t)u + d (2.4)

where x ∈ IRn and u ∈ IRm denote the system states and control inputs, respectively.

f(x, t) ∈ IRn and B(x, t) ∈ IRn×m are both smooth functions. d denotes possible system uncertainties and disturbances. Here we assume that d has only matched part with regard to B, thus we write (2.4) as:

˙x = f(x, t) + B(x, t)(u + dm) (2.5)

where dm = B+(x, t) · d, B+(x, t) is the pseudo-inverse matrix of B(x, t), and ||dm|| ≤

ρm(x, t), ρm(x, t) > 0. Then we design the control law composed of two parts:

u = u0+ u1 (2.6)

where u0 is the control input applied to the nominal subsystem, ˙x = f(x, t) + B(x, t) · u.

the system trajectories can remain on the sliding manifold, as follows: u1 = ( 0 if s = 0 −ρ(x, t) · _||[DB(x,t)][DB(x,t)]TTs_s|| if s 6= 0 (2.7) where ρ(x, t) > ρm(x, t), and the sliding surface is designed to be

s(x, t) = D ·  x(t) − x(t0) − Z t t0 [f(x(τ ), τ ) + B(x(τ ), τ ) · u0(τ )] · dτ  = 0 (2.8)

with D ∈ IRm×n _{and DB(x, t) having full rank. From (2.8), we observe that s(x, t}

0) = 0,

which implies the system trajectories start on the manifold from the first time instant (t0). On the other hand, when system is on the sliding manifold, i.e., s = ˙s = 0, from

(2.4) and (2.8), we obtain

˙s = D · { ˙x − [f + B(x, t)u0]}

= D · {[f + B(x, t)u + B(x, t)dm] − [f + B(x, t)u0]}

= DB(x, t) · (u + dm− u0)

thus u = u0− dm, substitute into (2.4) and obtain

˙x = f(x, t) + B(x, t) · u0

which explains that the system trajectories staying on the sliding manifold is identical to that of the nominal system.

On the other hand, to see that u1 keeps the system stay on the sliding manifold. When

s 6= 0, we choose the Lyapunov function V = 12s

T_{s, differentiate V and from (2.4)-(2.8),}

we know ˙ V = sT_{˙s = s}T DB(x, t) · (u + dm− u0) = sT_{DB(x, t) ·} ( −ρ(x, t) · [DB(x, t)] T_s ||[DB(x, t)]T_s|| + dm ) ≤ −ρ(x, t) · ||[DB(x, t)]T s|| + ||dm|| · ||[DB(x, t)]Ts|| ≤ [−ρ(x, t) + ρm(x, t)] · ||[DB(x, t)]Ts|| < 0.

Since DB(x,t) is assumed full rank and s(x, t0) = 0, the control law (2.6) and (2.7)

CHAPTER

THREE

STUDY OF ROBUSTNESS PERFORMANCE

OF SDRE+ISMC SCHEME

SDRE can be used to enhance the performance of robustness through the extra de-sign degree of freedom arose from the non-uniqueness of the SDC representation of the nonlinear drift term (see e.g., [10] and [12]). But, there is no guideline provided for the factorization f(x, t) = A(x, t)x to improve robustness. However, with the help of ISMC, we can still improve robustness using SDC factorization. The ISMC approach does not have reaching phase and possesses the advantages of robustness and ease of implementa-tion. When the uncertainty and disturbance are matched regarding the nominal healthy subsystem, the state trajectories of the nominal healthy subsystem and the uncertain system are identical. Thus, in this chapter, we adopt the SDRE strategy for the nom-inal system, and the ISMC strategy to completely nullify the matched uncertainty and disturbance.

In Section 3.1, we define the system type, cost function, and control objective. Then we detailed the design of control law of SDRE and SDRE+ISMC in Section 3.2. Finally, we apply the control law to the satellite attitude control and analyze the simulating results.

3.1

Problem Statement

Consider a set of n 2nd-order time-variant nonlinear control systems as described by

(

˙x1 = x2

˙x2 = ˜f(x, t) + ˜B(x, t)u + ˜d.

(3.1) Here, x1 = (x1, · · · , xn)T ∈ IRn, x2 = (xn+1, · · · , x2n)T ∈ IRn and x = (xT1, xT2)T are

the system states, u = (u1, · · · , um)T ∈ IRm are the control inputs and m ≥ n, ˜d =

(·)T _{denotes the transpose of a vector or a matrix. Note that System(3.1) is equivalent to}

the following system dynamic:

˙x = f(x, t) + B(x, t)u + d (3.2)

where f(x, t) = [xT

2 ... ˜fT(x, t)]T, B(x, t) = [0Tn×m ... ˜BT(x, t)]T, and d = (0Tn×1 ... ˜dT)T.

Assumption 3.1 : f(x, t) ∈ IRn and B(x, t) ∈ IRn×m are smooth functions with f(0) = 0.

Assumption 3.2 : For all states, B(x, t) is full rank.

Moreover, we define the quadratic performance index J = Z ∞ 0 h xTQ(x)x + uTR(x)uidt (3.3) where Q(x) = Q(x)T ≥ 0 and R(x) = R(x)T _{> 0.}

The control objective is to compare the performances of the two control strategies (SDRE and SDRE+ISMC) when there are possible model uncertainties and/or external disturbances. To be more precisely, we study whether the system can be stabilized and use the cost function (and others mentioned later) defined in (3.3) as an index to compare the performance.

3.2

Design of Control Law

3.2.1 SDRE

Under Assumption 3.1, we can factorize the drift term in the form of f(x, t) = A(x, t)x and let every element of system (3.1) state appearing in f(x, t) contributes as an element in A(x, t), i.e. capture their state dependency in the proper entry of SDC matrix. To achieve this goal, we adopt some factorizing techniques given by [10]. The following are some examples to illustrate:

x6cos(x3) cos(x2) = x6 cos(x2) − 1 x2 x2 + x6 cos(x3) − 1 x3 x3+ [1 + (cos(x3) − 1)(cos(x2) − 1)]x6 = h 0 x6cos(x 2)−1 x2 x6 cos(x3)−1 x3 0 0 [1 + (cos(x3) − 1)(cos(x2) − 1)] i           x1 x2 x3 x4 x5 x6           . (3.4) The drift term, x6cos(x3) cos(x2), has three state components, x2, x3, and x6, thus they

contribute in the (1, 2), (1, 3), and (1, 6) entries of the corresponding SDC matrix, respec-tively. 1 2cos2(x3) sin(2x1) = 1 4 sin(2x1) x1 x1+ 1 4cos 2_(x 3) sin(2x1) x1 x1+ 1 4 cos2_(x 3) − 1 x3 sin(2x1)x3 = h 1 4 sin(2x1) x1 + 1 4cos2(x3) sin(2x1) x1 0 1 4 cos2 (x3)−1 x3 sin(2x1) 0 0 0 i           x1 x2 x3 x4 x5 x6           . (3.5) The drift term, 1₂cos2_(x

3) sin(2x1), has three state components, x1, and x3, thus they

x5sin(x3) sin(x2) = 1 3x5sin(x3) sin(x2) x2 x2+ 1 3x5 sin(x3) x3 sin(x2)x3+ 1 3sin(x3) sin(x2)x5 = h 0 1 3x5sin(x3) sin(x2) x2 1 3x5 sin(x3) x3 sin(x2) 0 1 3sin(x3) sin(x2) 0 i           x1 x2 x3 x4 x5 x6           . (3.6) The drift term, x5sin(x3) sin(x2), has three state components, x2, x3, and x5, thus they

contribute in the (1, 2), (1, 3), and (1, 5) entries of the corresponding SDC matrix, respec-tively.

After symbolically factorize f(x, t) into the form of f(x, t) = A(x, t)x, where A(x, t) ∈ IRn×n_{, we adopt the following procedures to solve the SDRE problem:}

• Check the stabilizability of [A(x, t), B(x, t)] and the observability of [A(x, t), C(x)] symbolically, where C(x) ∈ IRp×n _{has full rank and satisfies Q(x) = C}T_{(x)C(x), to}

ensure the solvability of the following SDRE [24]:

AT_{(x, t)P (x) + P (x)A(x, t) − P (x)B(x, t)R}−1(x)BT(x, t)P (x) + Q(x) = 0. (3.7)

3.2.2 ISMC

Consider System (3.1), first we need following assumptions.

Assumption 3.3 : There exist ρm(x, t) > 0 such that

||˜dm|| ≤ ρm(x, t) (3.8)

where ˜dm = ˜B+(x, t) · ˜d, and ˜B+(x, t) is the pseudo-inverse matrix of ˜B(x, t).

Assumption 3.4 : The origin of the nominal subsystem ˙x1 = x2 and ˙x2 = ˜f(x, t) +

˜

B(x, t)u is uniformly asymptotically stabilizable, that is, there exists a control u0 and a

continuously differentiable function V (x, t) such that

γ1(||x||) ≤ V (x, t) ≤ γ2(||x||) (3.9) and ∂V (x, t) ∂t + ∂V (x, t) ∂x !T ·h˜f(x, t) + ˜B(x, t)u0 i ≤ −γ3(||x||) (3.10)

where γ1, γ2 : IR+ → IR+ are class K∞ functions and γ3 is a class K function.

Under Assumptions 3.3 and 3.4, the control law is designed into two parts:

u = u0+ u1 (3.11)

where u0 can be any control law which satisfies Assumption 3.4 and creates a desired

system trajectory for the state of the uncertain system to follow. In this chapter, u0

adopts the SDRE strategy. On the other hand, u1 is designed to compensate for the

disturbances such that the system state can remain on the sliding manifold.

Along the ISMC design procedure, the sliding manifold is introduced as (3.12) below: s = s(x, t) := D ·  x(t) − x(t0) − Z t t0 h f(x(τ ), τ ) + B(x(τ ), τ ) · u0(τ ) i · dτ  (3.12) where D = (D1, D) and D1 ∈ IRm×n, D ∈ IRm×n. Note that D · B(x, t) = D · ˜B(x, t).

Differentiate (3.12), the sliding manifold (3.12) is simplified to be s(x, t) = D ·  x2(t) − x2(t0) − Z t t0 h ˜f(x(τ), τ) + ˜_{B(x(τ ), τ ) · u}0(τ ) i · dτ  = 0. (3.13)

Note that ∀x, D ˜B(x, t) is full rank.

When the system is on the sliding manifold, x = 0 and ˙s = 0. From (3.1) and (3.12), it is obtained that ˙s = D ·n˙x2− [˜f + ˜B(x, t)u0] o = D ·n˜f + ˜B(x, t)u + ˜_{d − [˜f + ˜}B(x, t)u0] o = D · [ ˜B(x, t)u + ˜_{d − ˜}B(x, t)u0] = 0. Hence u = − ˜B+_{(x, t) · [˜d + ˜B(x, t)u}

0]. By substituting this u into (3.1), the system

resembles the nominal system.

The other part of control law, u1, the discussion separate into two cases: one is when

s = 0, u1 = 0; The other is when s 6= 0, u1 is designed to keep s = 0, let

u1 = −ρ(x, t)

[D ˜B(x, t)]T_s

||[D ˜B(x, t)]T_s|| (3.14)

where ρ(x, t) > ρm(x, t). By choosing the Lyapunov function as V = 1₂sTs, then

differen-tiate V and substitute into (3.11) and (3.12), ˙ V = sT˙s = sT D · [ ˜B(x, t)u + d − ˜B(x, t)u0] = sT_{D ˜} B(x, t) · [u0+ u1+ ˜B+(x, t)d − u0] ≤ −ρ(x, t) · ||[D ˜B(x, t)]T_{s|| + || ˜}B+_{(x, t)d|| · ||[D ˜}B(x, t)]T_s|| ≤ [−ρ(x, t) + ρm(x, t)] · ||[D ˜B(x, t)]Ts|| < 0.

To conclude, the following theorem is presented.

Theorem 3.1 For the nonlinear 2nd-order system (3.1) under Assumptions 3.2-3.4, if

adopting the following control law: u =    u0 if s = 0 u0− ρ(x, t) · [D ˜B(x,t)] T_s ||[D ˜B(x,t)]T_s|| if s 6= 0 (3.15) then the origin of this system is globally asymptotically stable (GAS).

3.3

Application to Satellite Attitude Control

3.3.1 Satellite Dynamics

An attitude model for a spacecraft along a circular orbit can be described in the same form as (3.1) with n = 3 [32]. The three Euler’s angles (φ, θ, ψ) and their derivatives are adopted as the six state variables. For simplicity, we assume in this study that the thruster is the only applied control force. Let x = (φ, θ, ψ, ˙φ, ˙θ, ˙ψ)T _{and ˜f(x, t) =}

( ˜f1(x, t), ˜f2(x, t), ˜f3(x, t))T. The overall system dynamics has parameters described as

below: ˜ f1(x, t) = ω0x6cx3cx2 − ω0x5sx3sx2+ Iy− Iz Ix  x5x6 + ω0x5cx1sx3sx2+ ω0x5cx3sx1 +ω0x6cx3cx1+ 1 2ω 2 0s(2x3)c2x1sx2+ 1 2w 2 0c2x3s(2x1) − ω0x6sx3sx2sx1 −1 2ω 2 0s2x2s2x3s(2x1) − 1 2ω 2 0s(2x3)sx2s2x1− 3 2ω 2 0c2x2s(2x1)  , (3.16) ˜ f2(x, t) = ω0x6sx3cx1+ ω0x4cx3sx1+ ω0x6cx3sx2sx1+ ω0x5sx3cx2sx1+ ω0x4sx3sx2cx1 +Iz − Ix Iy  x4x6+ ω0x4cx1sx3sx2 + ω0x4cx3sx1− ω0x6sx3cx2 −1₂ω₀2s(2x2)s2x3cx1− 1 2w 2 0cx2sx1s(2x3) + 3 2ω 2 0s(2x2)cx1  , (3.17) ˜ f3(x, t) = ω0x4sx1sx3sx2− ω0x6cx1cx3sx2− ω0x5cx1sx3cx2+ ω0x6sx3sx1− ω0x4cx3cx1 +Ix− Iy Iz  x4x5+ ω0x4cx3cx1− ω0x4sx3sx2sx1− ω0x5sx3cx2 −1 2ω 2 0s(2x3)cx2cx1+ 1 2w 2 0s2x3sx1s(2x2) − 3 2ω 2 0s(2x2)sx1  , (3.18) ˜ B(x, t) = B =˜    0.67 0.67 0.67 0.67 0.69 −0.69 −0.69 0.69 0.28 0.28 _{−0.28 −0.28}   . (3.19)

Here, Ix, Iy, and Iz are the inertia with respect to the three body coordinate axes, ω0

de-notes the constant orbital rate, and c and s denote the cos and sin functions, respectively. Note that, Assumptions 3.1 and 3.2 are obviously satisfied, since B(x, t) is a constant matrix and any three columns taking from B is invertible. Therefore, the system is found to be controllable for any control inputs and Assumption 3.4 is also satisfied.

The control objective is to compare the performances of the two control strategies (SDRE and SDRE+ISMC) when there are possible model uncertainties and/or external disturbances. To be more precisely, we study whether the system can be stabilized and

use the cost function (and others mentioned later) defined in (3.3) as an index to compare the performance.

3.3.2 Simulation Results

In this section, we use MATLAB software to simulate the satellite attitude control un-der SDRE and ISMC approach. For both control approaches, we check whether the sys-tem with disturbances can be stabilized and compare their performances (e.g. quadratic performance index and convergence time).

The Table 3.1 shows the simulating parameters in this chapter: (Note that for SDRE approach, the procedure of factorizing f(x, t) = A(x, t)x is described in Appendix)

Table 3.1. Simulation parameters.

Ix 2000 N · m · s2

Iy 400 N · m · s2

Iz 2000 N · m · s2

ω0 1.0312 × 10−3 rad/s

˜

d (0.05 sin(t), 0.05 cos(2t), 0.05 sin(3t))T

A(x, t) see Appendix 3A

D I3 Q I6 R I4 u0 SDRE approach ρ(x, t), ρm(x, t) ||B+(x, t)d||∞+ 1 x0 (−0.7, −0.07, 1.5, 0.3, 1.3, −0.2)T

Furthermore, to alleviate chattering, we modify the control law (3.15) into: u =    u0− ρ(x, t) · [D ˜B(x,t)] T_s ||[D ˜B(x,t)]T_s|| if ||[D ˜B(x, t)] T s|| ≥ ǫ u0− ρ(x, t) · [D ˜B(x,t)] T_ǫ ǫ if ||[D ˜B(x, t)] T s|| < ǫ (3.20) where we choose ǫ = 0.02.

The simulation results are shown in Figs. 3.1-3.3, and the summary of comparison of performance are shown in Table 3.2.

We denote the results:

• SDRE : the system without disturbance (nominal system) under SDRE approach only

• SDRE+ISMCd : the disturbed system using SDRE-ISMC combined approaches In addition, in Table 3.2, we also compare the performances under the Sliding Mode Control (SMC, see Section 3.2.3 in [42]), LQR (see Section 3.2.2 in [42]), and LQR-ISMC combined approach (see Section 3.2.1 in [42]), respectively.

• SMC : the disturbed system under nonlinear SMC approach only

• LQR : the system without disturbance (nominal system) under nonlinear LQR ap-proach only

• LQRd : the disturbed system under nonlinear LQR approach only

• LQR+ISMCd : the disturbed system using LQR-ISMC combined approaches From Fig. 3.1, we observe that SDRE approach stabilizes the nominal system but fails to stabilize when there exists disturbances. However, resorting to ISMC, the system with disturbances can still be stabilized. In addition, it is interesting to find that the trajectory of SDRE+ISMCd and SDRE for nominal design are almost identical (this is why we seem to see only two trajectories in this figure), this agrees with the theoretical conclusion. Moreover, the persistent oscillation of the state trajectory of SDREd comes from the effect of the disturbance ˜d, which also contributes to the oscillating control inputs of SDREd and SDRE+ISMCd in Fig. 3.2. From Fig. 3.2, we see that the control inputs of SDRE+ISMCd experiences larger oscillating amplitude than SDREd, this is

because the additional part of control inputs in SDRE+ISMCd than SDREd, u1 in (3.11),

which contributes to compensate disturbances while SDRE control scheme has no such mechanism. Finally, in Fig. 3.3, it is obvious that sliding variables of SDRE+ISMCd start on the sliding manifold and remain on it afterwards, which again agrees with the theoretical results that ISMC has no reaching phase.

Table 3.2 shows the comparison of performance, including energy consumptionR

uT_u,

quadratic performance index R

(xT_{x + u}T_{u), required maximum control magnitude ||u||} ∞,

and convergence time (when the magnitude of state is less than 0.01 at first time). For nominal system, LQR [42] approach seems to have better performance than SDRE in

energy consumptionR

uT_{u, state regulation}R

xT_{x, quadratic performance index}R

(xT_{x +}

uT_{u) and convergence time. But SDRE scheme has smaller maximum control magnitude}

||u||∞ since LORd uses Taylor’s series approximation up to 3rd-order for the real LQR

solution associated to a Hamiltonian-Jacobian equation of the the nonlinear system [49]. For the system with disturbances, LQR+ISMCd approach also have better performance

than SDRE+ISMCd in energy consumption R

uT_{u, state regulation}R

xT_{x, quadratic}

per-formance index R

(xT_{x + u}T_{u) and convergence time. Moreover, both SDRE+ISMCd and}

LQR+ISMCd consumes more control energy than the corresponding nominal control law

SDRE and LQR [42], respectively. This is because the additional part, u1 in (3.11), is

required in the ISMC design. Last but not least, we see that SMCd has the least con-vergence time among all approaches, and can be explained by the fact that Sliding Mode Control (SMC) inherently possesses robustness to model uncertainties and/or external disturbances [6]-[7], [16], [18], and [45].

To sum up, we conclude that SDRE (so as LQR) is not a robust control law.

0 5 10 15 20 −0.5 0 0.5 1 t x₁ 0 5 10 15 20 −0.5 0 0.5 t x₂ 0 5 10 15 20 −2 −1 0 1 t x₃ 0 5 10 15 20 −1 −0.5 0 0.5 t x₄ 0 5 10 15 20 −2 −1 0 1 t x₅ 0 5 10 15 20 −0.5 0 0.5 1 t x₆ SDRE SDRE SDRE SDRE SDREd SDREd SDRE+ISMCd SDREd SDRE+ISMCd SDRE+ISMCd SDRE+ISMCd SDREd SDREd SDREd SDRE SDRE+ISMCd SDRE SDRE+ISMCd

0 5 10 15 20 −0.5 0 0.5 1 1.5 2 t u₁ 0 5 10 15 20 −1.5 −1 −0.5 0 0.5 t u₂ 0 5 10 15 20 −2.5 −2 −1.5 −1 −0.5 0 0.5 t u₃ 0 5 10 15 20 −0.2 0 0.2 0.4 0.6 t u₄ SDRE SDRE SDRE+ISMCd SDRE+ISMCd SDREd SDREd SDREd SDRE SDRE+ISMCd SDREd SDRE _SDRE+ISMCd

Fig. 3.2. Time history of the four control inputs.

0 2 4 6 8 10 12 14 16 18 20 −0.1 −0.05 0 0.05 0.1 t s₁ 0 2 4 6 8 10 12 14 16 18 20 −0.5 0 0.5 t s₂ 0 2 4 6 8 10 12 14 16 18 20 −0.5 0 0.5 t s₃

Table 3.2. Comparison of performance. Performance Index Controller ||x(t)||t_→∞ < 10−3 R uTu R xTx R(xTx + uTu) ||u||_∞ Convergence time LQR+ISMCd 2.1259 4.6294 6.7553 2.5099 5.173 SDRE+ISMCd Yes 2.7979 4.7618 7.5597 2.3917 11.981 SMCd Yes 2.4605 4.8981 7.3586 2.6305 4.633 LQRd SDREd No X X X X X LQR 1.9517 4.6277 6.5794 2.5099 5.157 SDRE Yes 2.7756 4.7638 7.5395 2.3917 11.598

Appendix 3A

We denote

• I1, I2, and I3 denote Iy_I−I_xz, Iz_I−I_yx, and Ix_I−I_z y, respectively.

• c and s denote the cos and sin functions, respectively.

As Section 3.2.1 illustrates, first we reformulate the drift terms (3.16)-(3.18) into:

Reformed Drift Terms : ˜ f1 is reformed as: ω0x6 cx2− 1 x2 x2+ ω0x6 cx3− 1 x3 x3+ ω0[1 + (cx3 − 1)(cx2− 1)]x6− 1 3ω0x5sx3 sx2 x2 x2 −1 3ω0x5 sx3 x3 sx2x3− 1 3ω0sx3sx2x5+ 1 2I1x6x5+ 1 2I1x5x6+ 1 4I1ω0x5 cx1− 1 x1 sx3sx2x1 +1 4I1ω0x5cx1sx3 sx2 x2 x2+ 1 4I1ω0x5cx1 sx3 x3 sx2x3+ 1 4I1ω0(cx1sx3sx2+ sx3sx2)x5 +1 3I1ω0x5cx3 sx1 x1 x1+ I1ω0 1 3x5 cx3− 1 x3 sx1x3+ 1 3I1ω0(cx3sx1+ sx1)x5+ I1ω0x6 cx1 − 1 x1 x1 +I1ω0x6 cx3− 1 x3 x3+ I1ω0[(cx3 − 1)(cx1− 1) + 1]x6 + 1 6I1ω 2 0s(2x3) sx2 x2 x2 +1 6I1ω 2 0s(2x3) c2_x 1− 1 x1 sx2x1+ 1 6I1ω 2 0sx3 sx2 x2 x2+ 1 6I1ω 2 0 s(2x3) x3 c2x1sx2x3+ 1 4I1ω 2 0 sx1 x1 x1 +1 4I1ω 2 0c2x3 s(2x1) x1 x1+ 1 4I1ω 2 0 c2_x 3− 1 x3 s(2x1)x3− 1 4I1ω0x6sx3sx2 sx1 x1 x1 −1₄I1ω0x6sx3 sx2 x2 sx1x2− 1 4I1ω0x6 sx3 x3 sx2sx1x3− 1 4I1ω0sx3sx2sx1x6 +1 6I1ω 2 0s2x2s2x3 s(2x1) x1 x1− 1 6I1ω 2 0 s2_x 2 x2 s2x3s(2x1)x2− 1 6I1ω 2 0s2x2 s2_x 3 x3 s(2x1)x3 −1₆I1ω02s(2x3)sx2 s2_x 1 x1 x1− 1 6I1ω 2 0s(2x3) sx2 x2 s2x1x2− 1 6I1ω 2 0 s(2x3) x3 sx2s2x1x3 −3 4I1ω 2 0c2x2 s(2x1) x1 x1− 3 4I1ω 2 0 s(2x1) x1 x1 − 3 4I1ω 2 0 c2_x 2− 1 x2 s(2x1)x2. (3A.1) ˜ f2 is reformed as: 1 3ω0x6sx3 cx1− 1 x1 x1+ 1 3ω0x6 sx3 x3 cx1x3+ 1 3ω0(sx3cx1+ sx3)x6 + 1 3ω0x4cx3 sx1 x1 x1 +1 3ω0x4 cx3− 1 x3 sx1x3+ 1 3ω0(cx3sx1+ sx1)x4+ 1 4ω0x6cx3sx2 sx1 x1 x1+ 1 4ω0x6cx3 sx2 x2 sx1x2 +1 4ω0x5 cx3− 1 x3 sx2sx1x3+ 1 4ω0(cx3sx2sx1+ sx2sx1)x6 + 1 4ω0x5sx3cx2 sx1 x1 x1

+1 4ω0x5sx3 cx2− 1 x2 sx1x2+ 1 4ω0x5 sx3 x3 cx2sx1x3+ 1 4ω0(sx3cx2sx1+ sx3sx1)x5 +1 4ω0x4sx3sx2 cx1− 1 x1 x1+ 1 4ω0x4sx3 sx2 x2 cx1x2+ 1 4ω0x4 sx3 x3 sx2cx1x3 +1 4ω0(sx3sx2cx1+ sx3sx2)x4+ 1 2I2x6x4+ 1 2I2x4x6+ 1 4I2ω0x4 cx1− 1 x1 sx3sx2x1 +1 4I2ω0x4cx1sx3 sx2 x2 x2+ 1 4I2ω0x4cx1 sx3 x3 sx2x3+ 1 4I2ω0(cx1sx3sx2+ sx3sx2)x4 +1 3I2ω0x4cx3 sx1 x1 x1+ 1 3I2ω0x4 cx3− 1 x3 sx1x3+ 1 3I2ω0(cx3sx1+ sx1)x4 −1₃I2ω0x6sx3 cx2− 1 x2 x2− 1 3I2ω0x6 sx3 x3 cx2x3− 1 4I2ω0(sx3cx2+ sx3)x6 −1₆I2ω02sx2s2x3 cx1− 1 x1 x1− 1 6I2ω 2 0 s(2x2) x2 (sx 3cx1 + sx3)x2− 1 6I2ω 2 0s(2x2) s2_x 3 x3 cx1x3 −1₆I2ω02cx2 sx1 x1 s(2x3)x1− 1 6I2ω 2 0 cx2− 1 x2 sx1s(2x3)x2 − 1 6I2ω 2 0 sx1 x1 s(2x3)x1 −1 6I2ω 2 0cx2sx1 s(2x3) x3 x3+ 3 4I2ω 2 0s(2x2) cx1− 1 x1 x1+ 3 4I2ω 2 0 s(2x2) x2 (cx1+ 1)x2. (3A.2) ˜ f3 is reformed as: 1 4ω0x4 sx1 x1 sx2sx3x1+ 1 4ω0x4sx1 sx2 x2 sx3x2+ 1 4ω0sx1sx2 sx3 x3 x3+ 1 4ω0sx1sx2sx3x4 −ω0x6 cx1 − 1 x1 (cx3− 1)sx2x1 − ω0x6(cx1− 1) sx2 x2 x2− ω0x6 cx3− 1 x3 sx2x3− ω0sx2x6 −ω0x5sx3 cx1 x1 x1− ω0x5sx3(cx1− 1) cx2− 1 x2 x2− ω0x5 sx3 x3 (cx2− 1)x3− ω0sx3x5 +1 3ω0x6sx3 sx1 x1 x1+ 1 3ω0x6 sx3 x3 sx1x3+ 1 3ω0sx3sx1x6− ω0x4 cx1 − 1 x1 x1− ω0x4 cx3 − 1 x3 x3 −ω0[(cx3 − 1)(cx1 − 1) + 1]x4+ 1 2I3x5x4+ 1 2I3x4x5+ I3ω0x4 cx1− 1 x1 x1+ I3ω0x4 cx3− 1 x3 x3 +I3ω0[(cx3− 1)(cx1− 1) + 1]x4− 1 4I3ω0x4sx3sx2 sx1 x1 x1− 1 4I3ω0x4sx3 sx2 x2 sx1x2 −1₄I3ω0x3 sx3 x3 sx2sx1x3− 1 4I3ω0sx3sx2sx1x4− 1 3I3ω0x5sx3 cx2− 1 x2 x2 −1₃I3ω0 sx3 x3 cx2x3 1 3I3ω0(sx3cx2+ sx3)x5− 1 2I3ω 2 0s(2x3) cx1− 1 x1 x1 − 1 2I3ω 2 0s(2x3) cx2− 1 x2 x2 −1₂I3ω02 s(2x3) x3 [(cx2− 1)(cx1− 1) + 1]x3+ 1 6I3ω 2 0s2x3 sx1 x1 s(2x2)x1+ 1 6I3ω 2 0s2x3sx1 sx2 x2 x2 +1 6I3ω 2 0 s2_x 3 x3 sx1s(2x2)x3− 3 4I3ω 2 0s(2x2) sx1 x1 x1 − 3 4I3ω 2 0 s(2x2) x2 sx1x2. (3A.3)

Then we can factorize the drift term of System (3.1) into f = A(x, t) · x and the elements of A(x, t) is described as below:

Factorization of the Drift Terms : A(x, t) = [aij(x, t)] a1j = 0, j = 1, 2, 3, 5, 6; and a14 = 1. a2j = 0, j = 1, 2, 3, 4, 6; and a15 = 1. a3j = 0, j = 1, 2, 3, 4, 5; and a16 = 1. a41 = 1 4I1ω0x5 cx1− 1 x1 sx3sx2+ 1 3I1ω0x5cx3 sx1 x1 + I1ω0x6 cx1− 1 x1 +1 6I1ω 2 0s(2x3) c2_x 1− 1 x1 sx2+ 1 4I1ω 2 0 sx1 x1 +1 4I1ω 2 0c2x3 s(2x1) x1 −1₄I1ω0x6sx3sx2 sx1 x1 + 1 6I1ω 2 0s2x2s2x3 s(2x1) x1 − 1 6I1ω 2 0s(2x3)sx2 s2_x 1 x1 −3₄I1ω20c2x2 s(2x1) x1 − 3 4I1ω 2 0 s(2x1) x1 . a42 = ω0x6 cx2− 1 x2 − 1 3ω0x5sx3 sx2 x2 + 1 4I1ω0x5cx1sx3 sx2 x2 +1 6I1ω 2 0s(2x3) sx2 x2 +1 6I1ω 2 0sx3 sx2 x2 − 1 4I1ω0x6sx3 sx2 x2 sx1 −1 6I1ω 2 0 s2_x 2 x2 s2x3s(2x1) − 1 6I1ω 2 0s(2x3) sx2 x2 s2x1 − 3 4I1ω 2 0 c2_x 2− 1 x2 s(2x1). a43 = ω0x6 cx3− 1 x3 − 1 3ω0x5 sx3 x3 sx2+ 1 4I1ω0x5cx1 sx3 x3 sx2 +I1ω0 1 3x5 cx3− 1 x3 sx1+ I1ω0x6 cx3− 1 x3 + 1 6I1ω 2 0 s(2x3) x3 c2_x 1sx2 +1 4I1ω 2 0 c2_x 3 − 1 x3 s(2x1) − 1 4I1ω0x6 sx3 x3 sx2sx1− 1 6I1ω 2 0s2x2 s2_x 3 x3 s(2x1) −1 6I1ω 2 0 s(2x3) x3 sx2s2x1. a44 = 0. a45 = − 1 3ω0sx3sx2 + 1 2I1x6 + 1 4I1ω0(cx1sx3sx2 + sx3sx2) + 1 3I1ω0(cx3sx1+ sx1). a46 = ω0[1 + (cx3− 1)(cx2− 1)] + 1 2I1x5+ I1ω0[(cx3 − 1)(cx1 − 1) + 1] −1₄I1ω0sx3sx2sx1. a51 = 1 3ω0x6sx3 cx1− 1 x1 + 1 3ω0x4cx3 sx1 x1 +1 4ω0x6cx3sx2 sx1 x1 +1 4ω0x5sx3cx2 sx1 x1 +1 4ω0x4sx3sx2 cx1− 1 x1 + 1 4I2ω0x4 cx1− 1 x1 sx3sx2 +1 3I2ω0x4cx3 sx1 x1 − 1 6I2ω 2 0sx2s2x3 cx1− 1 x1 − 1 6I2ω 2 0cx2 sx1 x1 s(2x3) −1₆I2ω20 sx1 x1 s(2x3) + 3 4I2ω 2 0s(2x2) cx1− 1 x1 .

a52 = 1 4ω0x6cx3 sx2 x2 sx1+ 1 4ω0x5sx3 cx2− 1 x2 sx1+ 1 4ω0x4sx3 sx2 x2 cx1 +1 4I2ω0x4cx1sx3 sx2 x2 − 1 3I2ω0x6sx3 cx2− 1 x2 − 1 6I2ω 2 0 s(2x2) x2 (sx 3cx1+ sx3) −1₆I2ω20 cx2− 1 x2 sx1s(2x3) + 3 4I2ω 2 0 s(2x2) x2 (cx1+ 1). a53 = 1 3ω0x6 sx3 x3 cx1 + 1 3ω0x4 cx3− 1 x3 sx1+ 1 4ω0x5 cx3 − 1 x3 sx2sx1 +1 4ω0x5 sx3 x3 cx2sx1 + 1 4ω0x4 sx3 x3 sx2cx1 + 1 4I2ω0x4cx1 sx3 x3 sx2 +1 3I2ω0x4 cx3− 1 x3 sx1− 1 3I2ω0x6 sx3 x3 cx2− 1 6I2ω 2 0s(2x2) s2_x 3 x3 cx1 −1₆I2ω20cx2sx1 s(2x3) x3 . a54 = 1 3ω0(cx3sx1+ sx1) + 1 4ω0(sx3sx2cx1+ sx3sx2) + 1 4I2ω0(cx1sx3sx2+ sx3sx2) +1 3I2ω0(cx3sx1+ sx1). a55 = 1 4ω0(sx3cx2sx1+ sx3sx1). a56 = 1 3ω0(sx3cx1+ sx3) + 1 4ω0(cx3sx2sx1+ sx2sx1) + 1 2I2x6x4+ 1 2I2x4 −1 4I2ω0(sx3cx2 + sx3). a61 = 1 4ω0x4 sx1 x1 sx2sx3− ω0x6 cx1− 1 x1 (cx3− 1)sx2− ω0x5sx3 cx1 x1 +1 3ω0x6sx3 sx1 x1 − ω 0x4 cx1− 1 x1 + I3ω0x4 cx1− 1 x1 −1₄I3ω0x4sx3sx2 sx1 x1 − 1 2I3ω 2 0s(2x3) cx1− 1 x1 +1 6I3ω 2 0s2x3 sx1 x1 s(2x2) −3 4I3ω 2 0s(2x2) sx1 x1 . a62 = 1 4ω0x4sx1 sx2 x2 sx3− ω0x6(cx1− 1) sx2 x2 − ω0 x5sx3(cx1− 1) cx2− 1 x2 −1₄I3ω0x4sx3 sx2 x2 sx1− 1 3I3ω0x5sx3 cx2 − 1 x2 − 1 2I3ω 2 0s(2x3) cx2− 1 x2 +1 6I3ω 2 0s2x3sx1 sx2 x2 − 3 4I3ω 2 0 s(2x2) x2 sx1. a63 = 1 4ω0sx1sx2 sx3 x3 − ω0 x6 cx3− 1 x3 sx2− ω0x5 sx3 x3 (cx2− 1) +1 3ω0x6 sx3 x3 sx1− ω0x4 cx3− 1 x3 + I3ω0x4 cx3− 1 x3 −1₄I3ω0x3 sx3 x3 sx2sx1− 1 2I3ω 2 0 s(2x3) x3 [(cx2− 1)(cx1− 1) + 1] +1 6I3ω 2 0 s2_x 3 x3 sx1s(2x2). a64 = 1 4ω0sx1sx2sx3− ω0[(cx3− 1)(cx1− 1) + 1] + 1 2I3x5+ I3ω0[(cx3− 1)(cx1− 1) + 1]

−1₄I3ω0sx3sx2sx1. a65 = −ω0sx3+ 1 2I3x4− 1 3I3ω0 sx3 x3 cx2x3 1 3I3ω0(sx3cx2 + sx3). a66 = −ω0sx2+ 1 3ω0sx3sx1. (3A.4)

CHAPTER

FOUR

STUDY OF RELIABILITY PERFORMANCE

OF SDRE+ISMC SCHEME

In Chapter 3, we found that SDRE is not a robust scheme via numerical simulation, but when combined with Integral-type Sliding Mode Control (ISMC), the closed-loop system is less sensitive to disturbances. In this chapter, we investigate the reliability issue of SDRE. Since there is no guideline provided for the factorization f(x, t) = A(x, t)x, specifically, to improve reliability, we resort to ISMC approach to improve robustness performance for a specific factorization of the nonlinear drift term.

From the approach viewpoint, reliable control can be classified as active [3]-[5], [19], [31], [35], [39], [50], [51] or passive [23], [27], [29], [46]-[49]. In a passive reliable design, we need to separate the healthy actuators from those actuators that might malfunction before it applies on the system. Nevertheless, it is difficult to retrieve such information in advance. On the contrary, in the active reliable control design, faults are detected and identified by a fault detection and diagnosis (FDD) mechanism, and then the controllers are reconfigured in real time in accordance with the online detection results. Therefore, we only consider the active reliable design in this chapter.

In Section 4.1, we define the system type, cost function, and control objective. Then we detailed the design of FDD and control law of SDRE and ISMC in Section 4.2. Finally, we apply the control law to the satellite attitude control and analyze the simulating results.

4.1

Problem Statement

In this study, we assume that the actuators’ fault has been successfully detected and diagnosed by a Fault Detection and Diagnosis (FDD) mechanism. The fault may be time varying and include degradation, amplification and outage [30], [41]. Before the

occurrence of faults, the engineers may take any kind of control strategy to fulfill their desired system performance. When the fault is detected and diagnosed, the control law is guided to switch to an active reliable law for ensuring system performance. Thus, after the fault is detected, we may divide the actuators into two groups H and F, within which we assume that all of the actuators in H are healthy, while those in F experience faults. Therefore, System (3.1) can be rewritten as

(

˙x1 = x2

˙x2 = ˜f(x, t) + ˜BH(x, t)uH+ ˜BF(x, t)uF+ ˜d (4.1)

where x1 = (x1, · · · , xn)T ∈ IRn, x2 = (xn+1, · · · , x2n)T ∈ IRn and x = (xT1, xT2)T are

the system states, u_{H ∈ IR}k and u_{F ∈ IR}m−k _{are the control inputs. f(x, t) ∈ IR}n

and ˜B(x, t) = [ ˜B_{H(x, t) ∈ IR}n×k ... ˜B

F(x, t) ∈ IRn×(m−k)] ∈ IRn×m, where m ≥ k ≥ n.

˜

d = (d1, · · · , dn)T ∈ IRn denote possible model uncertainties and/or external disturbances.

Note that System(4.1) is equivalent to the following system dynamic:

˙x = f(x, t) + B(x, t)u + d (4.2)

where f(x, t) = [xT

2 ... ˜fT(x, t)]T, B(x, t) = [0Tn_×m ... ˜BT(x, t)]T, u = (uT_H ... uT_F)T, and

d = (0T

n_×1 ... ˜dT)T.

Assumption 4.1 : f(x, t) and B(x, t) are smooth functions with f(0) = 0.

Assumption 4.2 : For all states, B(x, t) is full rank.

Moreover, we define the quadratic performance index J = Z ∞ 0 h xT_{Q(x)x + u}T_R(x)ui dt (4.3) where Q(x) = Q(x)T _{≥ 0 and R(x) = R(x)}T _{> 0.}

The control objective is to compare the performances of the two control strategies (SDRE and SDRE+ISMC) when there are possible model uncertainties and/or external disturbances, especially when some actuators malfunction. To be more precisely, We study whether the system can be stabilized and use the cost function (and others mentioned later) defined in (4.3) as an index to compare the performance.

4.2

Design of Active Reliable Control Law

We assume that the output values of the faulty actuators are successfully diagnosed by an FDD mechanism as

u_F = ˆu_F + ∆u_F (4.4)

where ˆu_F and ∆u_F denote the estimated value and the estimated error, respectively.

Then System (4.1) can be written as

˙x1 = x2 and ˙x2 = ˜f(x, t) + ˜BH(x, t)uH+ ˜BF(x, t)(ˆuF + ∆uF) + ˜d (4.5)

where u_{H ∈ IR}k and ˆu_F, ∆u_{F ∈ IR}m−k.

4.2.1 SDRE

Similar to Section 3.2.1 but with slight modification, we symbolically factorize f(x, t) into the form of f(x, t) = A(x, t)x (see Appendix 3A in Chapter 3), where A(x, t) ∈ IRn×n, and then adopt the following procedures to solve the SDRE problem:

• Check the stabilizability of [A(x, t), BH(x, t)] and the observability of [A(x, t), C(x)]

symbolically, where B_H(x, t) = [0T

n_×(m−k) ... ˜BHT(x, t)]T, C(x) ∈ IRp×n has full rank

and satisfies Q(x) = CT_{(x)C(x), to ensure the solvability of the corresponding}

SDRE [24].

• Solve the SDRE for P (x) to produce the SDRE controller u = −R−1_(x)BT

H(x, t)P (x)x.

4.2.2 ISMC

Under Assumption 4.2, System (4.5) is rewritten into:

˙x1 = x2 and ˙x2 = ˜f(x, t) + ˜BH(x, t) · (uH+ ∆dm) + ˜BF(x, t) · ˆuF (4.6)

where ∆dm = ˜B_H+(x, t) · [ ˜B_F(x, t)∆u_F + ˜d] and ˜B_H+(x, t) is the pseudo-inverse matrix of

˜

B_H(x,t).

Assumption 4.3 : There exist ρm(x, t) > 0 such that

Assumption 4.4 : The origin of the nominal subsystem

(

˙x1 = x2

˙x2 = ˜f(x, t) + ˜B(x, t)u (4.8)

is uniformly asymptotically stabilizable, that is, there exists a control u0 and a

continu-ously differentiable function V (x, t) such that

γ1(||x||) ≤ V (x, t) ≤ γ2(||x||) (4.9) and ∂V (x, t) ∂t + ∂V (x, t) ∂x !T ·h˜f(x, t) + ˜B(x, t)u0 i ≤ −γ3(||x||) (4.10)

where γ1, γ2 : IR+ → IR+ are class K∞ functions and γ3 is a class K function.

Along the ISMC design procedure, the sliding manifold is introduced as (4.11) below: s(x, t) = D ·  x2(t) − x2(t0) − Z t t0 h ˜f(x(τ), τ) + ˜_{B(x(τ ), τ ) · u}0(τ ) i · dτ  = 0 (4.11) where D ∈ IRn×n _{and D ˜B} H(x, t) is full rank ∀x.

When the system’s trajectory is on the sliding manifold, s = 0, ˙s = 0, from (4.6) and (4.11), it is obtained that

˙s = D · [x2− ˜f − ˜B(x, t)u0]

= D · [ ˜B_{H(x, t) · (uH}+ ∆um) + ˜BF(x, t)ˆuF − ˜B(x, t)u0] = 0

⇒ uH= − ˜BH+(x, t) · [ ˜BF(x, t)ˆuF − ˜B(x, t)u0] − ∆dm.

Substitute u_H into (4.6), the equivalent system dynamics is obtained

(

˙x1 = x2

˙x2 = ˜f(x, t) + ˜B(x, t)u0

(4.12) which agrees with the nominal system defined in Assumption (4.4).

In order to keep the system state on the sliding manifold, it is chosen that

u_H = ( _˜ B+ H(x, t) · [ ˜B(x, t)u0− ˜BF(x, t)ˆuF] if s = 0; ˜ B_H+_{(x, t) · [ ˜}B(x, t)u0− ˜BF(x, t)ˆuF] + u1 if s 6= 0 (4.13) and u1 = −ρ(x, t) [D ˜B_H(x, t)]T_s ||[D ˜B_H(x, t)]T_s|| (4.14)

where ρ(x, t) > ρm(x, t). Note that u1is designed to keep the state on the sliding manifold.

By choosing the Lyapunov function as V = 1₂sT_{s, then differentiate V and substitute into}

(4.11), (4.13) and (4.14), ˙

V = sT_˙s

= sT_{D · [ ˜}B_{H(x, t) · (uH}+ ∆dm) + ˜B_F(x, t)ˆu_F − ˜B(x, t)u0]

= sT

D · [ ˜B(x, t)u0− ˜BF(x, t)ˆuF + ˜BH(x, t)u1 + ˜BH(x, t)∆dm+ ˜BF(x, t)ˆuF − ˜B(x, t)u0]

= sT_{D ˜B} H(x, t) · ( −ρ(x, t) [D ˜BH(x, t)] T_s ||[D ˜B_H(x, t)]T_s|| + ∆dm ) ≤ −||[D ˜B_H(x, t)]T_{s|| · ρ(x, t) + ||[D ˜}B_H(x, t)]T_{s|| · ||∆d}m|| ≤ ||[D ˜B_H(x, t)]T s|| · [−ρ(x, t) + ρm(x, t)] < 0.

To sum up with an important theorem,

Theorem 4.1 : Suppose that System (4.1) experiences actuator faults at the control

channels in F with estimated value ˆuF and error ∆uF given by FDD mechanism (4.5).

Then the origin of System (4.1) under Assumptions 4.1 - 4.4 and the control law given by (4.13)-(4.14) is globally asymptotically stable (GAS).

4.3

Application to Satellite Attitude Control

In this section, we use the same satellite attitude control model as in Section 3.3.1. In the following, we first detail the design of fault detection and diagnosis (FDD) and compare the simulating results using different control methods.

4.3.1 Design of Fault Diagnosis and Detection (FDD)

In this section, we investigate the deign of FDD observer mentioned in Section 3.3.1 for the satellite attitude control. The main idea of this design is to decouple the control input so that the fault associated with each channel can be diagnosed and distinguished from the healthy ones. And the following system dynamics, same as (3.1), is considered.

(

˙x1 = x2

˙x2 = ˜f(x, t) + ˜B(x, t)u + ˜d.

Fig. 4.1. FDD diagram

Fig. 4.1 shows the relation between FDD and system. Since the three Euler rates can be expressed in terms of angular velocity vector, which is available through accelerometer and gyroscope [34], in this section, we assume that all of the state variables are available for measurement and that ˜B(x, t) in (4.15) is a constant matrix. We adopt the observer and residual signals ri from [32] as (4.16) and (4.17) below:

ξi = finew(z) + ui+ liu4+ ki· (zi+3− ξi), i = 1, 2, 3 (4.16)

and

ri = zi+3− ξi, i = 1, 2, 3 (4.17)

where ki > 0. It was shown in [32] that any single actuator fault can be detected and

diagnosed at an exponential rate depending on ki.

When the residual signals are larger than a selected threshold, the alarm will be set to be on.

4.3.2 Simulation Results

In this section, we still use MATLAB software to simulate the satellite attitude control under SDRE and ISMC approach. For both control approaches, we check whether the sys-tem with disturbances can be stabilized and compare their performances (e.g. quadratic performance index and convergence time).

The following Table 4.1 shows the simulating parameters in this chapter: (Note that for SDRE approach, the procedure of factorizing f = A(x, t)x is described in Appendix)

Table 4.1. Simulation parameters. Ix 2000 N · m · s2 Iy 400 N · m · s2 Iz 2000 N · m · s2 ω0 1.0312 × 10−3 rad/s ˜

d (0.01 sin(t), 0.01 cos(2t), 0.01 sin(3t))T

A(x, t) see Appendix 3A in Chapter 3

D I3 Q I6 R I4 u0 SDRE approach ρ(x, t), ρm(x, t) ||∆dm||∞+ 0.5 x0 (0.7, 0.07, −1.5, −0.3, −1.3, 0.2)T k1 10 k2 10 k3 10

Furthermore, to alleviate chattering, we modify the control law (4.13) into: u =    ˜ B_H+(x, t)[ ˜B(x, t)k0− ˜BF(x, t)ˆuF] − ρ(x, t) [D ˜BH(x,t)] T_s ||[D ˜BH(x,t)]Ts|| if ||[D ˜BH(x, t)] T s|| ≥ ǫ ˜ B_H+(x, t)[ ˜B(x, t)k0− ˜BF(x, t)ˆuF] − ρ(x, t)[D ˜BH(x,t)] T_s ǫ if ||[D ˜BH(x, t)] T s|| < ǫ(4.18) where we choose ǫ = 0.02. We simulate the faulty situation by that u2 fails at time 1 and

alarm signals as soon as |ri| ≥ 0.01.

The simulation results are shown in Figs. 4.2-4.6, and the summary of comparison of performance are shown in Table 4.2.

We denote the results:

• SDRE : the system without disturbance (nominal system) under SDRE approach only

• SDREr : the disturbed and actuator-failed system under SDRE approach only • SDRE+ISMCr : the disturbed and actuator-failed system using SDRE-ISMC

com-bined approaches

In addition, in Table 4.2, we also compare the performances under the Sliding Mode Control (SMC, see Section 4.2.2 in [42]), LQR (see Section 4.2.3 in [42]), and LQR-ISMC combined approach (see Section 4.2.1 in [42]), respectively.

• SMCr : the disturbed and actuator-failed system under nonlinear SMC approach only

• LQR : the system without disturbance (nominal system) under nonlinear LQR ap-proach only

• LQRr : the disturbed system and actuator-failed under nonlinear LQR approach only

• LQR+ISMCr : the disturbed and actuator-failed system using LQR-ISMC combined approaches

It is observed from Fig. 4.2 that the stabilization performance is, as expected, achieved for the SDRE and the SDRE+ISMC designs. Besides, the state trajectories of the ISMC and those for nominal design (SDRE) are found almost identical, which agrees with the theoretical conclusion. From Fig. 4.6, the sliding variables of the SDRE+ISMC design are seen to keep at zero all the time. It implies that the system states remain on the sliding manifold for all t, which also agrees with the main results. In Fig. 4.4, the actuator fault is successfully detected by both designs, since the magnitude of the second residual signal exceeds the threshold near tSDRE+ISMC ≈ 1.04 and tSDRE ≈ 1.067, respectively. This can

also be seen from the alarm signals given in Fig. 4.5 where alarm2 denotes the fault of

the second actuator. After the fault is successfully detected, the associated active reliable controllers are activated and the magnitude of the residual signals soon decreases, as shown in Fig. 4.4. The persistent oscillation of the residual signal comes from the effect of the disturbance ˜d, which also contributes to the oscillating control inputs (u1, u3, u4) of

SDREr and SDRE+ISMCr in Fig. 4.3. It is also noted from Figs. 4.3 that SDRE+ISMC design is observed to require larger control efforts than SDRE design due to the additional

control u1 in (4.13) and (4.14). Finally, since the SDRE+ISMC design of this example

adopts the SDRE scheme for the nominal healthy subsystem, its performances are close to those of SDRE except for the requirement of extra control component to compensate for the uncertainties.

Table 4.2 shows the comparison of performance, including energy consumptionR

quadratic performance index R

(xT_{x + u}T

u), required maximum control magnitude ||u||∞,

and convergence time (when the magnitude of state is less than 0.01 at first time). For nominal system, LQR [42] approach has better performance than SDRE for all consid-ered performance indexes. For the system with disturbances, LQR+ISMCr approach also has better performance than SDRE+ISMCr. Moreover, both SDRE+ISMCr and LQR+ISMCr consumes more control energy than the corresponding nominal control law

SDRE and LQR [42], respectively. This is because the additional part, u1 in (4.13) and

(4.14), is required in the ISMC design. Last but not least, we see that SMCr [42] and SDREr succeeds to stabilize, but LQRr [42] fails. To sum up, we conclude that in this study SDRE control law possesses certain robustness but not reliable.

0 5 10 15 20 −0.5 0 0.5 1 t x₁ 0 5 10 15 20 −0.5 0 0.5 t x₂ 0 5 10 15 20 −2 −1 0 1 t x₃ 0 5 10 15 20 −1 −0.5 0 0.5 t x₄ 0 5 10 15 20 −2 −1 0 1 t x₅ 0 5 10 15 20 −0.5 0 0.5 1 t x₆ SDRE SDRE SDRE SDRE SDRE SDRE SDRE+ISMCr SDRE+ISMCr SDREr SDRE+ISMCr SDRE+ISMCr SDREr SDREr SDRE+ISMCr SDRE+ISMCr SDREr SDREr SDREr

0 5 10 15 20 −1 −0.5 0 0.5 1 1.5 2 t u₁ 0 5 10 15 20 −1.5 −1 −0.5 0 0.5 t u₂ 0 5 10 15 20 −2.5 −2 −1.5 −1 −0.5 0 0.5 t u₃ 0 5 10 15 20 −0.2 0 0.2 0.4 0.6 t u₄ SDRE SDRE SDRE SDREr SDRE+ISMCr SDRE+ISMCr SDRE+ISMCr SDREr SDREr SDREr SDRE+ISMCr SDRE

Fig. 4.3. Time history of the four control inputs.

0 2 4 6 8 10 12 14 16 18 20 −0.01 0 0.01 0.02 t r₁ 0 2 4 6 8 10 12 14 16 18 20 −0.02 0 0.02 0.04 t r₂ 0 2 4 6 8 10 12 14 16 18 20 −0.01 0 0.01 0.02 t r₃ SDREr SDREr SDRE+ISMCr SDRE+ISMCr SDRE+ISMCr SDREr

0 2 4 6 8 10 12 14 16 18 20 −1 −0.5 0 0.5 1 t alarm₁ 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 t alarm₂ 0 2 4 6 8 10 12 14 16 18 20 −1 −0.5 0 0.5 1 t alarm₃ SDREr SDREr SDREr SDRE+ISMCr SDRE+ISMCr SDRE+ISMCr

Fig. 4.5. Time history of the three alarm signals.

0 2 4 6 8 10 12 14 16 18 20 −0.2 0 0.2 0.4 t s₁ 0 2 4 6 8 10 12 14 16 18 20 −0.5 0 0.5 t s₂ 0 2 4 6 8 10 12 14 16 18 20 −0.5 0 0.5 t s₃

Table 4.2. Comparison of performance. Performance Index Controller ||x(t)||t→∞ <10−3 R uTu R xTx R(xTx + uTu) ||u||∞ Convergence time LQR+ISMCr 1.6149 4.4142 6.0291 2.1232 8.844 SDRE+ISMCr Yes 4.0208 4.7619 8.7827 2.3917 11.61 SMCr Yes 1.8763 6.0829 7.9592 2.2829 7.094 LQRr No X X X X X SDREr Yes 3.5434 5.066 8.6094 2.3917 16.982 LQR 1.5576 4.4156 5.9732 2.1232 8.799 SDRE Yes 2.7756 4.7638 7.5395 2.3917 11.598

CHAPTER

FIVE

ON FACTORIZATION OF THE DRIFT TERM IN

SDRE SCHEME

5.1

Problem Statement

Although the SDRE algorithm fully captures the nonlinearities of the system, bringing the nonlinear system to a (non-unique) linear structure having state-dependent coefficient (SDC) matrices, and minimizing a nonlinear performance index having a quadratic-like structure, it has some drawbacks. First, it is known that the conditions“[A(x, t), B(x, t)] is stabilizable” and“[A(x, t), C(x)] is observable” are required for the existence of a unique positive definite solution P (x) in Eq. (2.3) [24]; however, these symbolic checking con-ditions are in general not easy to implement, especially when the system dynamics is complicated. Next, there is no guideline provided for the factorization f(x, t) = A(x, t)x. To avoid these difficulty, in this study, we consider the following approach instead.

Problem A: At any nonzero state x and time t, f := f(x, t) is a constant vector, while

B := B(x, t) and C := C(x) are constant matrices. Find a matrix A := A(x, t) ∈ IRn×n

pointwise such that Ax = f, (A, B) is stabilizable and (A, C) is observable.

To demonstrate the benefits of the alternative approach, we give an example below which shows the traditional SDRE scheme does not work when a specific factorization of f = Ax is adopted, but the alternative approach do work.

Example: Let f = (x1+ x21x32, x12x22)T, B = (0, 1)T and C = I2. Suppose that a specific

factorization for f = Ax is given as A := 1 x

2 1x22

0 x2 1x2

!

. Clearly, (A, C) is observable, but (A, B) is not stabilizable when x1 = 0 or x2 = 0. Thus, the SDRE, given by (2.3), might

fail to have a positive definite solution P (x) when x1 = 0 or x2 = 0, which will result in

the SDRE scheme failing to operate. However, since Q(x) = CT_{(x)C(x) = I}

A is solvable for this case (see Corollary 5.2).

It is also worth noting that Problem A is always solvable if the SDRE problem for some specific factorization can be continuously operated. We first consider the Problem A at a specific nonzero state, as described in Problem B below:

Problem B: Given two constant vectors x, f ∈ IRn

, and two constant matrices B ∈

IRn×m _{and C ∈ IR}p×n _{with x 6= 0, rank(B) ≥ 1 and rank(C) ≥ 1, when does there exist}

a matrix A ∈ IRn×n _{pointwise such that Ax = f, (A, B) is stabilizable and (A, C) is}

observable?

Note that, Problem A (and B) are always solvable for the case of n = 1. Therefore, in the following we only consider the case of n > 1. To answer Problem B, we denote (IRn_)∗

= {xT_{|x ∈ IR}n

}, which is known to be the dual space of IRn _{[22]. Suppose}

that p1, · · · , pk ∈ IRn and qT1, · · · , qTl ∈ (IR n_)∗ . We denote {p1, · · · , pk}⊥ = {qT ∈ (IRn_)∗ | qT_p i = 0 for 1 ≤ i ≤ k} and {qT1, · · · , qTl }⊥ = {p ∈ IR n |qT i p = 0 for 1 ≤ i ≤ l}.

In addition, we denote B⊥ _{:= {q}T _{∈ (IR}n_)∗

|qT_{B = 0} and C}⊥_{:= {p ∈ IR}n

|Cp = 0}.

5.2

Solvability Condition

We assume that the matrix A is diagonalizable in the form of

A = MDM−1 (5.1)

where D =diag[λ1, · · · , λn] ∈ IRn×n, M = [p1, · · · , pn] ∈ IRn×n, M−1 = [q1, · · · , qn]T ∈

IRn×n_{, and λ}

1, · · · , λn are distinct. Clearly, λ1, · · · , λn are the eigenvalues of A, pi and qTi

are the right and the left eigenvectors of A associated with eigenvalues λi, respectively.

We have the following lemma:

Lemma 5.1 Let A be factorized in the form of (5.1). Then

(i) Ax = f ⇐⇒ λiqTi x = qTi f for all i ⇐⇒ qTi (λix − f) = 0 for all i.

(ii) (A, C) is observable if and only if pi 6∈ C⊥ for all i = 1, · · · , n.

(iii) (A, B) is controllable if and only if qT

i 6∈ B⊥ for all i = 1, · · · , n.

(iv) (A, B) is stabilizable if and only if qT

i 6∈ B⊥ whenever λi ≥ 0.

Proof: (i) The result follows from writing Ax = f in the form of DM−1_{x = M}−1_{f and}

(ii) It is known from the PBH test [8] that the pair (A, C) is observable if and only if rank C λiI − A !!

= n for all i = 1, · · · , n, i.e., _λ C

iI − A

!

p 6= 0 for any p 6= 0. It is clear that (λiI − A)p = 0 if and only if (λi, p) is an eigenpair of A or p = 0. It

follows that (A, C) is observable if and only if Cpi 6= 0 for all i, that is, pi 6∈ C⊥ for all

i = 1, · · · , n.

(iii) It is known that (A, B) is controllable if and only if (AT_{, B}T_{) is observable [8].}

Since (λi, qi), i = 1, · · · , n, are eigenpairs of AT, we have from the proof of (ii) that

(AT_{, B}T_{) is observable if and only if B}T_q

i 6= 0, i.e., qTi B 6= 0, for all i. Thus, (A, B) is

controllable if and only if qT

i 6∈ B⊥ for all i.

(iv) (A, B) is stabilizable if and only if rank([λiI − A ... B]) = n for those i in which

λi ≥ 0 [8]. This is equivalent to qTi B 6= 0 whenever λi ≥ 0, that is, qTi 6∈ B⊥ whenever

λi ≥ 0.

We also need the following three results:

Lemma 5.2 Let V be a k dimensional vector subspace of (IRn_)∗

, k < n, and {qT

1, · · · , qTk}

are linearly independent (LI) vectors with qT

1 6∈ V. Then there exists qTk+1 ∈ V such that

{qT

1, · · · , qTk+1} are LI.

Proof: Suppose that such qT

k+1 does not exist. Then V ⊂ span{qT1, · · · , qTk}. Since

both V and span{qT

1, · · · , qTk} have dimension k, we have V =span{qT1, · · · , qTk}, and thus

qT

1 ∈ V, a contradiction. This completes the proof.

Lemma 5.3 Let V be a k−1 dimensional vector subspace of (IRk)∗ _{and {v}T

1, · · · , vkT} be

a basis of (IRk)∗_{with v}T

i 6∈ V for all i. Besides, let Wi := span{vT₁, · · · , viT₋₁, vTi+1, · · · , vTk}.

Then V 6⊂ ∪k

i=1Wi. As a result, there exists a nonzero vT ∈ V such that vT =Pki=1αivTi

and αi 6= 0 for all i = 1, · · · , k.

Proof: Note that, for all i = 1, · · · , k, Wi is a vector space of dimension k − 1 and

V 6= Wi; Otherwise, vTj ∈ V for all j 6= i, which contradicts to the assumption vjT 6∈ V

for all j. Since ∪k

i=1Wi is not a vector space, we thus have V 6⊂ ∪ki=1Wi. This fact

together with {vT

1, · · · , vTk} being a basis implies there exists a nonzero vT ∈ V such that

vT ₌Pk

i=1αiviT with αi 6= 0 for all i; Otherwise, each v ∈ V will belong Wi for some i,

which contradicts V 6⊂ ∪k i=1Wi.

Lemma 5.4 Let {qT

1, · · · , qTn₋₁, cT} are LI and qTn := αccT +Pnj=1−1αjqTj, αc 6= 0 and

αj 6= 0 for all j = 1, · · · , n − 1. Then

(i) {qT

1, · · · , qTn} are LI.

(ii) For any i ∈ {1, · · · , n}, the n vectors {qT

1, · · · , qTi−1, qTi+1, · · · , qTn, cT} are LI.

(iii) For any i ∈ {1, · · · , n}, {qT

1, · · · , qTi−1, qTi+1, · · · , qTn}⊥6⊂ (cT)⊥.

Proof: (i) Suppose that Pn

i=1kiqTi = 0T. Inserting the expression of qTn into the

equation yields Pn−1

i=1(ki+ knαi)qTi + knαccT = 0T. Since {qT1, · · · , qTn−1, cT} are LI, we

have knαc = 0 and ki+ knαi = 0 for all i = 1, · · · , n − 1. Since αc 6= 0, we have kn = 0

and ki = 0 for i = 1, · · · , n − 1. This proves the linear independency of {qT1, · · · , qTn}.

(ii) Suppose that P

j6=ikjqTj + kccT = 0T. Inserting qTn into the equation, we have

Pn−1

j_6=i(kj+ knαj)qTj + knαiqTi + (knαc+ kc)cT = 0T. Since {qT1, · · · , qTn₋₁, cT} are LI and

αi 6= 0, we have from the coefficient of qTi that kn = 0, and thus kc = 0 and kj = 0 for all

j 6= i and j ≤ n − 1. Thus, {qT

1, · · · , qTi−1, qTi+1, · · · , qTn, cT} are LI.

(iii) Suppose, on the contrary, that {qT

1, · · · , qTi−1, qTi+1, · · · , qTn}⊥ ⊂ (cT)⊥. Then any

nonzero vector p ∈ {qT

1, · · · , qTi−1, qTi+1, · · · , qTn}⊥ has the property cTp = 0 and qTjp = 0

for all j 6= i. Since {qT

1, · · · , qTi−1, qTi+1, · · · , qTn, cT} is a basis for (IR

n_{)∗, it follows that p}

must be a zero vector, which contradicts the fact that {qT

1, · · · , qTi−1, qTi+1, · · · , qTn}⊥ is a

vector space of dimension 1. This proves that {qT

1, · · · , qTi−1, qTi+1, · · · , qTn}⊥ 6⊂ (cT)⊥.

In the following, we denote IR− the set of negative real numbers. A necessary and

sufficient condition for Problem B is now stated as Theorem 5.1 below:

Theorem 5.1 Problem B is unsolvable if and only if {x, f} are linearly dependent (LD)

and Cx = 0.

Proof: We divide the proof into the following four cases: Case 1: ({x, f} are LI and C[x, f] 6= 0)

Note that, C[x, f] 6= 0 implies that there exists a nonzero row vector cT _{of C with}

cT

6∈ {x, f}⊥_{. Choose λ}

1, · · · , λn ∈ IR−such that the n vectors {λix − f | i = 1, · · · , n} are

distinct and cT_(λ

ix−f) 6= 0 for all i = 1, · · · , n. If n > 2, since dim((λix−f)⊥) = n−1 for

all i, we may easily choose qT