1. INTRODUCTION
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
hxTQ(x)x + uTR(x)uidt (2.2)
where QT(x) = Q(x) ≥ 0, RT(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.
And u1 is a discontinuous control input designed to compensate disturbances such that
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)]TTss|| 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, t0) = 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 = 12sTs, differentiate V and from (2.4)-(2.8), we know
V˙ = sT˙s = sTDB(x, t) · (u + dm− u0)
= sTDB(x, t) ·
(
−ρ(x, t) · [DB(x, t)]Ts
||[DB(x, t)]Ts|| + dm
)
≤ −ρ(x, t) · ||[DB(x, t)]Ts|| + ||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) guarantees the system remain on the sliding manifold, i.e., s = 0, ∀t ∈ [t0, ∞).
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 = (d1, · · · , dn)T ∈ IRn denote possible model uncertainties and/or external disturbances and
(·)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) = [xT2 ... ˜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
hxTQ(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: contribute in the (1, 2), (1, 3), and (1, 6) entries of the corresponding SDC matrix, respec-tively. contribute in the (1, 1), and (1, 3) entries of the corresponding SDC matrix, respectively.
x5sin(x3) sin(x2) 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) = 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. (3.7)
• Solve the SDRE for P (x) to produce the SDRE controller u = −R−1(x)BT(x, t)P (x)x.
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 u˜ 0 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
hf(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 ) · u0(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)u0]. 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)]Ts
||[D ˜B(x, t)]Ts|| (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
= sTD · [ ˜B(x, t)u + d − ˜B(x, t)u0]
= sTD ˜B(x, t) · [u0+ u1+ ˜B+(x, t)d − u0]
≤ −ρ(x, t) · ||[D ˜B(x, t)]Ts|| + || ˜B+(x, t)d|| · ||[D ˜B(x, t)]Ts||
≤ [−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 ˜[D ˜B(x,t)]B(x,t)]TTss|| 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) = 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 ˜[D ˜B(x,t)]B(x,t)]TTss|| if ||[D ˜B(x, t)]Ts|| ≥ ǫ
u0− ρ(x, t) · [D ˜B(x,t)]ǫ Tǫ if ||[D ˜B(x, t)]Ts|| < 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
• SDREd : the disturbed 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 uTu, quadratic performance index R(xTx + uTu), 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 uTu, state regulationR xTx, quadratic performance indexR(xTx + uTu) 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 uTu, state regulationR xTx, quadratic per-formance index R(xTx + uTu) 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
Fig. 3.1. Time history of the six state variables.
0 5 10 15 20
Fig. 3.2. Time history of the four control inputs.
0 2 4 6 8 10 12 14 16 18 20
Fig. 3.3. Time history of the sliding variables (SDRE+ISMCd).
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
• 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 :
+1
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 :
a52 = 1
−1
4I3ω0sx3sx2sx1. 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, uH ∈ IRk and uF ∈ IRm−k are the control inputs. f(x, t) ∈ IRn and ˜B(x, t) = [ ˜BH(x, t) ∈ IRn×k ... ˜BF(x, t) ∈ IRn×(m−k)] ∈ IRn×m, where m ≥ k ≥ n.
d = (d˜ 1, · · · , 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) = [xT2 ... ˜fT(x, t)]T, B(x, t) = [0Tn×m ... ˜BT(x, t)]T, u = (uTH ... uTF)T, and d = (0Tn×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
hxTQ(x)x + uTR(x)uidt (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
uF = ˆuF + ∆uF (4.4)
where ˆuF and ∆uF 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 uH ∈ IRk and ˆuF, ∆uF ∈ IRm−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 BH(x, t) = [0Tn×(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)BHT(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 = ˜BH+(x, t) · [ ˜BF(x, t)∆uF + ˜d] and ˜BH+(x, t) is the pseudo-inverse matrix of B˜H(x,t).
Assumption 4.3 : There exist ρm(x, t) > 0 such that
||∆dm|| ≤ ρm(x, t) (4.7)
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)u0i≤ −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 ) · u0(τ )i· dτ
= 0 (4.11)
where D ∈ IRn×n and D ˜BH(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 · [ ˜BH(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 uH 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 uH =
( 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 ˜BH(x, t)]Ts
||[D ˜BH(x, t)]Ts|| (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 = 12sTs, then differentiate V and substitute into (4.11), (4.13) and (4.14),
V˙ = sT˙s
= sTD · [ ˜BH(x, t) · (uH+ ∆dm) + ˜BF(x, t)ˆuF − ˜B(x, t)u0]
= sTD · [ ˜B(x, t)u0− ˜BF(x, t)ˆuF + ˜BH(x, t)u1 + ˜BH(x, t)∆dm+ ˜BF(x, t)ˆuF − ˜B(x, t)u0]
= sTD ˜BH(x, t) ·
(
−ρ(x, t) [D ˜BH(x, t)]Ts
||[D ˜BH(x, t)]Ts|| + ∆dm
)
≤ −||[D ˜BH(x, t)]Ts|| · ρ(x, t) + ||[D ˜BH(x, t)]Ts|| · ||∆dm||
≤ ||[D ˜BH(x, t)]Ts|| · [−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. (4.15)
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 ˜[D ˜BBHH(x,t)](x,t)]TTss|| if ||[D ˜BH(x, t)]Ts|| ≥ ǫ B˜H+(x, t)[ ˜B(x, t)k0− ˜BF(x, t)ˆuF] − ρ(x, t)[D ˜BH(x,t)]ǫ Ts if ||[D ˜BH(x, t)]Ts|| < 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
In addition, in Table 4.2, we also compare the performances under the Sliding Mode