Reaction-Jet Control System

This technology has been successfully implemented in PAC-3 since the Iraq War in 2003. This system, installed in front of the center of gravity of the missile or between the center of gravity and the top of the missile, yields lateral thrust changing the missile’s attitude immediately for additional auxiliary thrust mounted [51]. It is contented 180 impulse attitude control motors (IACMs), arraying in 10 circles (each one composed of 18 IACMs), staggered distributing along the O_xb-axis equably. Note that the IACM is disposable. Define ith circle (i = 1, 2,· · · , 10) for each circle from top to the center of gravity and jth IACM (j = 1, 2,· · · , 18) for the number in each circle, the odd and even number circles are shown in Fig. 2.12.

In Fig. 2.12, the layout of the odd number circles presents that the first IACM is opposite direction to the O_yb-axis, and the number of the others follows the direction of counterclockwise, respectively. In the similar manner, the layout of the even number circles presents that the first IACM is on the left 20 degrees of the opposite direction to the O_yb-axis. The angle of each IACM is described below:

Φ_ij =

Fig. 2.12. The layout scheme of IACMs, left side for odd number and right side for even number

where i^∗ = 2 when i is odd, and i^∗ = 1 when i is even. The force and moment of each (i, j) IACM is presented as





F_tby^ij = Kc_Φijs_ij F_tbz^ij =−KsΦijs_ij

(2.124)

and





M_tby^ij =−liF_tbz^ij M_tbz^ij = l_iF_tby^ij

(2.125)

where K is the force of each IACM; l_i is the moment arm of ith circle from the center of gravity of the missile to the location of the ith circle; and s_ij is defined as if (i, j) IAMC is untapped or used once, s_ij = 0 and if (i, j) IAMC is opened, s_ij = 1. The total components of the lateral force and moment are





F_tby =∑10 i=1

∑18 j=1F_tby^ij F_tbz =∑10

i=1

∑18 j=1F_tbz^ij

(2.126)

and





M_tby =∑10 i=1−li

∑18 j=1F_tbz^ij Mtbz =∑10

i=1li

∑18 j=1F_tby^ij

(2.127)

CHAPTER THREE

OUTPUT TRACKING CONTROL FOR A NONLIN-EAR SYSTEM

Similar to system stabilizability analysis and synthesis, the task of output tracking has received considerable attention in both theoretical and practical industry applications [52]-[54]. The objective of output tracking control is to design a feedback law such that the output of a controlled plant can track a desired reference signal. To solve the tracking-control problem effectively, many methods and techniques have been presented. Those include regulator-based approach [55], inversion-based approach [56]-[58], Lyapunov-based approach [59], Takagi-Sugeno (T-S) fuzzy model-based approach [60] and sliding mode control-based (SMC) approach [61]-[63]. In this thesis, we will study the output tracking problem from a blended control viewpoint via the following three techniques: CSMC, TSMC and NTSMC schemes.

3.1 Problem Formulation

Consider a nonlinear control system as described by [62]

˙x = f_o(x) + G_o(x)u (3.1)

and y = h(x) (3.2)

where x = [x₁,· · · , xn]^T ∈ IRⁿ, u = [u₁,· · · , um]^T ∈ IR^m , and y = [y₁,· · · , yv]^T ∈ IR^v denote the state variables, control inputs, and system outputs, respectively. The functions f_o(x) ∈ IRⁿ, G_o(x) = [g_o1(x),· · · , gom(x)]∈ IR^n×m and h(x) = [h₁(x),· · · , hv(x)]^T ∈ IR^v are smooth functions. Our interest is to construct a control input so that the output

approaches the sliding surface and achieves the desired value. For the decoupled input-output system, the new input-output form is obtained from differentiating several times until it is related to the input. That is, differentiating the output yj with respect to time, we obtain

˙

y_j =▽hj· ˙x = ▽hj· (fo+ G_ou) = L_foh_j(x) +

∑m i=1

L_goih_j(x)u_i (3.3)

where L_foh_j(x) and L_goih_j(x) are the Lie derivatives of h_j with respect to f_o and g_oi (for definition, please see e.g., [64]). If Lgihj(x) is equal to zero for i=1,· · · , m, then we have to differentiate the outputs y_j repeatedly until input appears. Assume that k_j is the smallest integer such that at least one of the inputs appears in y^(k_j ^j), then

y_j^(kj) = L^k_f^j

oh_j(x) +

∑m i=1

L_goiL^k_f^j−1

o h_j(x)u_i (3.4)

with L_goiL^k_f^j−1

o h_j(x)̸= 0 for at least one i in a neighborhood of the point x0. k_j is exactly the number of times one has to differentiate y_j in order to have the control u explicitly appearing, in which {k1,· · · , kv} is called the relative degree [64] of the system. We impose the following assumption:

Assumption 3.1 The System (3.1)-(3.2) has the following three properties:

I) The distribution △ := span {go1(x),· · · , gom(x)} is involutive.

II) It has relative degree {k1,· · · , kv}, that is, for all x ∈ IRⁿ, L_goiL^k_foh_j(x) = 0 for 1 ≤ i ≤ m, 1 ≤ j ≤ v and 0 ≤ k < kj − 1, while LgoiL^k_foh_j(x) ̸= 0 for 1 ≤ i ≤ m, 1≤ j ≤ v and k = kj− 1.

III) The control inputs u are divided into two parts u1 ∈ IR^m1 and u2 ∈ IR^m2 where m₁ ≥ v and m2 ≥ v.

Performing the above procedure for each output y_j yields



 y₁^(k1)

... yv^(kv)



 = f(x) + G(x)u (3.5)

where Equation (3.5) can also be rewritten as



Note that, we have introduced d in Eq. (3.10) to represent possible model uncertainties, measurement noise and external disturbances. In this study, we call u₁ the main inputs which are continuous and u₂ (with components u_2j for 1≤ j ≤ m2) the auxiliary inputs which are constant during a short time period once it was triggered with the following form:

u_2j :=

{ N_jK if t ∈ [toj, t_oj +△tp]

0 elsewhere (3.11)

where K denotes the minimum level of auxiliary control force; |Nj| is an integer which represents the number of actuators in u2j being activated; toj is the time instant that the actuator u_2j is triggered; and △tp denotes the time duration of the constant force.

Note that u₁ suffer from the output magnitude constraints, while u₂ only provide discrete values and the integer N_j, given by Eq. (3.11), satisfies |Nj| ≤ Nu, where N_u is a positive integer, i.e., N_j ∈ {0, ±1, ±2, · · · , ±Nu}. Besides, we assume that the output magnitude of the auxiliary inputs are much larger than those of the main inputs.

Before designing the control law, we have to check if the nonlinear system is minimum phase. The scalar k_r = k₁ +· · · + kv is called the total relative degree of the nonlinear

system [20]. The necessary and sufficient condition for the existence of a coordinate trans-formation and a feedback that can linearize the system completely from the Input/Output (I/O) point of view is the total relative degree kr being the same as the order of the sys-tem, i.e., k_r = n. If k_r < n, then, the nonlinear system can only be partially linearized.

In this case, the stability of the nonlinear system given by Eqs. (3.1) and (3.2) depends not only on the I/O linearized system, but also on the stability of the internal dynamics (or zero dynamics).

According to linear algebra theory, G₁(x) can be expressed as G₁(x) = G_1v1(x)G_1u1(x) where a diagonal matrix G_1v1(x)∈ IR^v×v and G_1u1(x)∈ IR^v×m1 satisfy rank (G_1v1(x)) = rank (G_1u1(x)) = v. Given a desired v, the minimum norm solution of u₁ that satisfies v1 = G1u1(x)u1 is u1 = G^T_1u1(x)(

G1u1(x)G^T_1u1(x))₋₁

v1, that is, u1 is easily constructed if v₁ has been designed. Note that, G₁(x)u₁ = G_1v1v₁. Therefore, we may assume without lose any generality that G₁(x) is a diagonal matrix and G₁(x) = diag[g₁₁(x),· · · , g1v(x)].

In the same manner, we also may assume that G₂(x) is a diagonal matrix and G₂(x) = diag[g₂₁(x),· · · , g2v(x)]. Under the setting, System (3.5) can be rewritten in more simpler form as follows:

y^(k_j ^j)= fj(x) + g1j(x)u1j + g2j(x)u2j + dj (3.12)

where f_j, g_1j, g_2j : IRⁿ → IR, and u1j and u_2j are the jth component of u₁ and u₂, respectively, for j = 1,· · · , v. In addition, we define the output errors to be

e_j(t) = y_j(t)− yjd(t), j = 1,· · · , v (3.13)

where y_jd(t) is the desired output trajectory.

The main goal of this thesis is then to design suitable control laws which integrate the main inputs u1j and auxiliary inputs u2j such that the output tracking performance can be achieved, i.e., ej(t)→ 0 as quickly as possible.

3.2 Design of Blended Controller

In this section, we will incorporate the main inputs (i.e., tail controllers) with auxiliary inputs (i.e., lateral thrusters), called blended control, through the CSMC, TSMC and

NTSMC schemes. The idea behind the design is as follows. First, a boundary layer (BL) of the sliding surface is determined from the region where the system states will be forced out or on this BL using the minimum level of auxiliary control inputs in one time duration or period. Inside or on the BL, only the main inputs are used to keep the system states close to the sliding surface as better as possible. When the system states are outside the BL, both main and auxiliary inputs will be activated for better convergence rate of system states to the sliding surface, compared with only considering the main inputs. The level of the auxiliary inputs will be determined from the distance between the system states and the sliding surface. Moreover, because the magnitude of the auxiliary inputs are much larger than those of the main inputs, the main inputs are used to compensate for only the deterministic dynamics, while the auxiliary inputs are responsible for disturbance rejection and reaching the sliding surface. For better understanding of the design, a block diagram for the blended control is shown in Fig. 3.1.

Under the CSMC, TSMC and NTSMC schemes, the presented blended control design

Fig. 3.1. Block diagram of blended control

include the following two features:

I) Outside the BL, the auxiliary inputs are used to account for the convergence speed of the system states to the sliding surface because they only provide constant control force, which is much larger than the main inputs, while the main inputs are employed

to compensate for deterministic dynamics and the drastic change of states produced by the activation of the auxiliary inputs for maintaining the rate of sliding variable being zero.

II) Inside or on the BL, because the auxiliary inputs are not activated so that the states variations are smooth. Therefore, only the main inputs are used for output the tracking purpose.

3.2.1 Control Design via CSMC scheme

The CSMC design consists of the following two steps: I) choose an appropriate sliding surface in terms of error states and II) construct a control law in form of

u = u^re+ u^eq (3.14)

to realize the tracking performance, where u^re plays the role of making the error states reach the sliding surface in finite time, while u^eq keeps the sliding surface an invariant set and directs the output tracking errors to the origin. For the first step, we choose the sliding surface to be s_j(t) = 0 with

s_j = e^(k_j ^j−1)(t) + a_j(kj−1)e^(k_j ^j−2)(t) +· · · + aj2˙e_j(t) + a_j1e_j(t) (3.15)

for j = 1,· · · , v. Here, ajk for k = 1,· · · , (kj−1) are selected constants and the polynomial

λ^k_j^j−1+ a_j(kj−1)λ^k_j^j−2+· · · + aj2λ_j + a_j1 (3.16)

for j = 1,· · · , v are Hurwitz. Obviously, the output tracking performance can be achieved if the system states keep lying on the sliding surface, that is, e_j → 0 as t → ∞. Further-more,

˙s_j = f_j(x) + g_1j(x)u_1j + g_2j(x)u_2j + d_j− yjd^(kj)

+a_j(kj−1)e^(k_j ^j−1)(t) +· · · + aj2e¨_j(t) + a_j1˙e_j(t) (3.17)

For second step, the controller design is divided into two parts: I) main inputs and II) auxiliary inputs, as described below:

I) Design of Main Inputs

The control law of each main input is designed to be the form of u^eq_1j =− 1

g_1j(x) ·[

f_j(x)− y^(k_jd^j)(t) + a_j(kj−1)e^(k_j ^j−1)(t) +· · · + aj2e¨_j(t) + a_j1˙e_j(t) ] (3.18) to accomplish the demand of making the sliding surface an invariant set. To guar-antee the reaching condition, we assume that d_j is bounded as follows

Assumption 3.3 There exists nonnegative functions ρj(x, t), j = 1,· · · , v, such that

|dj| ≤ ρj(x, t) (3.19)

Let ϵ_j be the BL width associated with s_j. Choose

u^re_1j =









− 1

g_1j(x) · (ρj + η_mj)· sgn(sj) if |sj| ≤ ϵj and u_2j = 0

0 otherwise

(3.20)

where η_mj for j = 1,· · · , v are selected positive constants.

II) Design of Auxiliary Inputs

Because the auxiliary inputs involve the following two characteristics: I) being zero or nonzero constant during a time duration depending on whether or not they are triggered; II) with output magnitudes being much larger than the main inputs if they are triggered. Consequently, the control law of each auxiliary input is designed to be

u^eq_2j = 0 (3.21)

and u^re_2j = N_jK. (3.22)

Now, we will discuss how N_j is selected. The method is based on the sliding condition d

dt|sj| ≤ −ηrj^′ (3.23)

where η_rj^′ is a fictitious positive constant. The geometry of Condition (3.23) is shown in Fig. 3.2 below:

According to Ineq. (3.23), the system state will reach s_j(x) = 0 within the time of

Fig. 3.2. The time response of |sj(x(t))|

|sj(x(0))| /η_rj^′ . When△tpis given, the minimum η_rj^′ that makes the system state reaching the manifold s_j(x) = 0 within △tp is

η^′_rj = |0 − sj(x(t))|

△tp

. (3.24)

We choose

u^re_2j^′ =− 1 g2j ·(

ρ_ju+ η_rj^′

)· sgn(sj) (3.25)

where u^re_2j^′ is the fictitious control input of u^re_2j and ρ_ju is the upper bound of ρ_j. Then,

s_j(t) ˙s_j(t) = −sj(t) (

ρ_ju+ η_rj^′

)· sgn(sj) + s_j(t)· dj

≤ −(

ρju+ η^′_rj

)|sj| + ρju|sj|

≤ −η_rj^′ |sj| (3.26)

Inequality (3.26) accounts for the sliding variable s_j will converge in finite time. However, in fact, u^re_2j^′ must accord with the form of NjK. One method to determine Nj is such that the value NjK as close u^re_2j^′ as possible, that is, we round off

(

u^re_2j^′/K )

to determine the

constant integer N_j. In more detail, Eq. (3.25) is replaced by Eq. (3.22) described below

Note that the function round(·) is defined to round off a scalar. Then, the virtual con-vergence speed is approximate to be:

η_rj = N_j^′Kg_2j − ρju. (3.30)

Although the actual reaching time |sj(x(0))| /ηrj can not exactly coincide with the ex-pected reaching time |sj(x(0))| /η^′rj, that is, this will cause inaccuracy of the reaching time which is at most in △tp, s_j can still converge according to the next paragraph.

Herein, we verify whether the sliding variable will converge. When s_j is outside the BL, from (3.17), we have

s_j(t) ˙s_j(t) = s_j(t)N Kg_2j + s_j(t)· dj

≤ −Nj^′Kg_2j|sj| + ρju|sj|

≤ −(ρju+ ηrj)|sj| + ρju|sj|

≤ −ηrj|sj| (3.31)

Clearly, the system states will approach the sliding surface with a convergence speed at

Similarly, the system states will reach the sliding surface in a finite time t_rj =|sj(x(t_aj))|/ηmj

for j=1, · · · , v [20] where taj is the time when any auxiliary input is not activated. Ac-cording to above analysis, it implies that we can select bigger ηrj advisably outside the BL to make s_j approach s_j = 0 faster and it still satisfies the sliding condition after s_j is inside the BL.

In addition, according to Eq. (3.28), the minimum nonzero integer of N_j^′ is 1, that is, it implies that

From Eqs. (3.24) and (3.33), the BL can be derived as follows 

Hence, we have the next result:

Theorem 3.1 Suppose that System (3.1)-(3.2) is minimum phase and satisfies Assump-tion 3.1 and 3.2 having input-output relaAssump-tion (3.12) with relative degree (k₁,· · · , kv) and

k_j ≥ 1. Then, the output tracking performance yj → yjd for j=1,· · · , v can be accom-plished by the CSMC blended controller (3.18), (3.20)-(3.22), and (3.29) if the designed forces fulfill the physical constraints for each control channel and ρj(x, t) satisfies As-sumption 3.3.

In this design idea, the features of the blended controller include:

I) Outside the BL, in order to achieve the output tracking performance as soon as possible, the auxiliary inputs provide large constant force to let the sliding variable approach the sliding surface as quickly as possible, while the main inputs compen-sate the deterministic dynamics and drastic change of states produced by auxiliary inputs.

II) Inside or on the BL, since the auxiliary inputs are not activated and the states variation will be smaller than those outside the BL, u_1j has more chance to avoid saturation. Thus, we only use u_1j to keep the system states close the sliding surface as better as possible.

3.2.2 Control Design via TSMC scheme

This scheme can deal with that each output of System (3.1)-(3.2) has relative degree more or equal to 2. The TSMC design consists of the following two steps: I) choosing an appropriate sliding surface in terms of error states and II) constructing a control law in form of Eq. (3.14). First, we choose sliding surface presented as

sj1 = ˙sj0+ bj1s^q_j0^j1/pj1 (3.35) s_j2 = ˙s_j1+ b_j2s^q_j1^j2/pj2 (3.36)

...

sjk = ˙s_j(k−1)+ bjks^q_j(k^jk/p₋₁₎^jk (3.37)

for j=1,· · · , v and k=1,· · · , (kj − 1), where sj0 = e_j, s_jk = s_j, b_jk > 0, p_jk > q_jk and p_jk, q_jk are positive odd integers. Taking time derivative on s_jk, we have

˙s_j(kj−1) = d^(kj) For second step, the controller design is divided into two parts: I) main inputs and II) auxiliary inputs.

I) Design of Main Inputs We design

where η_mj is selected positive constants.

II) Design of Auxiliary Inputs

Because the auxiliary inputs involve the following two characteristics: I) being zero or nonzero constant during a time duration depending on whether or not they are triggered; II) with output magnitudes being much larger than the main inputs if they are triggered. Consequently, the control law of each auxiliary input is designed to be

u^eq_2j = 0 (3.41)

and u^re_2j = N_jK. (3.42)

Now, we will discuss how N_j is selected. The method is based on the sliding condition d

dt|sj| ≤ −ηrj^′ (3.43)

where η^′_rj is a fictitious positive constant. According to Condition (3.43), the system state

Inequality (3.46) accounts for the sliding variable s_j will converge in finite time. However, in fact, u^re_2j^′ must accord with the form of N_jK. One method to determine N_j is such that constant integer Nj. In more detail, Eq. (3.45) is replaced by Eq. (3.42) described below

u^re_2j = round

Then, the virtual convergence speed is approximate to be:

η_rj = N_j^′Kg_2j − ρju. (3.50)

Although the actual reaching time |sj(x(0))| /ηrj can not exactly coincide with the ex-pected reaching time |sj(x(0))| /η^′rj, that is, this will cause inaccuracy of the reaching time which is at most in △tp, s_j can still converge according to the next paragraph.

Herein, we verify whether the sliding variable will converge. When s_j is outside the BL, from (3.38), we have

s_j(t) ˙s_j(t) = s_j(t)N Kg_2j + s_j(t)· dj

≤ −N_j^′Kg_2j|sj| + ρju|sj|

≤ −(ρju+ η_rj)|sj| + ρju|sj|

≤ −ηrj|sj| (3.51)

Clearly, the system states will approach the sliding surface with a convergence speed at least η_rj for j=1, · · · , v in a finite time [20] whenever the system states are outside the BL. In contrast, when s_j is inside or on the BL, from (3.17), we get

s_j(t) ˙s_j(t) = −(ρj+ η_mj)s_j(t)sgn(s_j) + s_j(t)· dj

≤ −(ρj+ η_mj)|sj| + ρj|sj|

≤ −ηmj|sj| (3.52)

Similarly, the system states will reach the sliding surface in a finite time trj =|sj(x(taj))|/ηmj

for j=1, · · · , v [20] where taj is the time when any auxiliary input is not activated. Ac-cording to above analysis, it implies that we can select bigger η_rj advisably outside the selected boundary to make s_j approach s_j = 0 faster and it still satisfies the sliding con-dition after s_j is inside or on the BL.

Moreover, according to Eq. (3.48), the minimum nonzero integer of N_j^′ is 1, that is, it implies that

 

(ρ_ju+ η^′_rj) g_2j



 = K

2 . (3.53)

From Eqs. (3.44) and (3.53), the BL can be derived as follows 



(ρ_ju+ η_rj^′ ) g_2j

 = K

2

⇒

(ρ_ju+ η_rj^′ )

|g2j| = K 2

⇒ (ρ_ju+ ϵ_j/△tp)

|g2j| = K 2

⇒ ϵj =△tp

(K|g2j| 2 − ρju

)

. (3.54)

Thus, we have the next result:

Theorem 3.2 Suppose that System (3.1)-(3.2) is minimum phase and satisfies Assump-tion 3.1 and 3.2 having input-output relaAssump-tion (3.12) with relative degree (k₁,· · · , kv) with k_j ≥ 2. Then, the output tracking performance yj → yjd for j=1,· · · , v can be accom-plished by the TSMC blended controller (3.39), (3.40)-(3.42), and (3.49) if the control forces fulfill the physical constraints for each control channel and ρ_j(x, t) satisfies As-sumption 3.3.

In this design idea, the features of the blended controller include:

I) Outside the BL, in order to achieve the output tracking performance as soon as possible, the auxiliary inputs provide large constant force to let the sliding variable approach the sliding surface as quickly as possible, while the main inputs compen-sate the deterministic dynamics and drastic change of states produced by auxiliary inputs.

II) Inside or on the BL, since the auxiliary inputs are not activated and the states variation will be smaller than those outside the BL, u_1j has more chance to avoid saturation. Thus, we only use u_1j to keep the system states close the sliding surface as better as possible.

Nevertheless, this method of TSMC scheme confronts the singularity problem for the controller. In other words, this problem occurs in Eq. (3.39) when ˙s_jk ̸= 0 but sjk = 0.

3.2.3 Control Design via NTSMC scheme

Finally, we introduce another controller design via NTSMC scheme to avoid the sin-gularity phenomenon yielding from TSMC scheme. However, this scheme only can deal with that each output of System (3.1)-(3.2) has relative degree 2. Choose sliding surface presented as

The controller design is divided into two parts: I) main inputs and II) auxiliary inputs.

I) Design of Main Inputs We choose

where η_mj is selected positive constants.

II) Design of Auxiliary Inputs

Because the auxiliary inputs involve the following two characteristics: I) being zero or nonzero constant during a time duration depending on whether or not they are triggered; II) with output magnitudes being much larger than the main inputs if they are triggered. Consequently, the control law of each auxiliary input is designed to be

u^eq_2j = 0 (3.59)

and u^re_2j = N_jK. (3.60)

Now, we will discuss how N_j is selected. The method is based on the sliding condition

where η_rj^′ is a fictitious positive constant. Making s_j(x(t)) achieve the sliding surface at least within △tp and ˙s_j0 is constant in this time duration as the time duration is in a short time, then we obtain

η^′_rj = c_j1

where toj is the time instant when we decide to activate the jth auxiliary input. Although Eq. (3.62) is undefined as ˙s_j0 = 0, in this case we will adopt allowable maximum value of u_2j under such situation. In addition, at ˙s_j0 = 0 there is an advantage for choosing maximum u_2j since ¨s_j0 ≤ −ηrj and ¨s_j0 ≥ ηrj for both s_j1 > 0 and s_j1 < 0, respectively (discussed in Eq. (2.61) of Section 2.3). That is to say that if ˙s_j0 = 0, choosing maximum η_rj^′ can make ˙s_j0 leave ˙s_j0 = 0 fastest. Then, we choose

Inequality (3.64) accounts for the sliding variable s_j will converge in finite time. However,

Inequality (3.64) accounts for the sliding variable s_j will converge in finite time. However,

在文檔中 應用可變結構控制技術於反戰術彈道飛彈具有側向力與氣動力之複合控制技術之研究 (頁 47-0)