Outline

The organization of the work is as follows. Chapter 1 includes the motivation and objective of this thesis, as well as the survey of relative works. Chapter 2 reviews the basic concept of SMC theories. The mathematical model of the PAC-3 missile is given in Chapter 3. The problem formulation and controller design via the CSMC, TSMC and NTSMC schemes are illustrated in Chapter 4. Then, in Chapter 5, the analytic results are applied to a simplified model to demonstrate the performances of the three SMC schemes.

Finally, the conclusions and suggestions for further research are made in Chapter 6.

CHAPTER TWO PRELIMINARIES

2.1 Conventional Sliding Mode Control (CSMC)

The history of CSMC up until the early 70’s has been described in [33]. By 1980, the main part of CSMC theory had been finished [34] and later reported by Russian Prof.

Utkin’s monograph in 1981 [35]. The main advantages of CSMC were the following [36]:

1) exact compensation (insensitivity) with respect to bounded matched uncertainties; 2) reduced order of sliding equations; 3) finite-time convergence to the sliding surface.

Consider a nth-order single-input system

x⁽ⁿ⁾ = f (x) + g(x)u + d(x) (2.1)

where x = [x ˙x · · · x⁽ⁿ⁻¹⁾]^T denotes the state vector and u is control input. In system (2.1), the functions f (x) and g(x) (in general, nonlinear) are not exactly known, but the extent of the imprecision on f (x) is upper bounded by a known continuous function of x, and control gain g(x) is of known sign and bounded by a known continuous function of x, respectively. And d(x) is set to combine the model uncertainties of f (x) and g(x) and external disturbances. The control problem is to get the state x to track a specific time-varying state x_d = [x_d ˙x_d · · · xd(n−1)]^T in the presence of d(x). In order to achieve the tracking task by using a finite control u, the initial desired state x_d(0) must be such that:

x_d(0) = x(0). (2.2)

Then, defining the tracking error

e = x− xd (2.3)

where the error vector composed of derivatives of error between output and desired output is denoted by

e = [e ˙e · · · e^(r−1)]^T (2.4)

In a second-order system, for example, position or velocity can not “jump”, so that any desire trajectory feasible from t=0 necessarily starts with the same position and velocity as those of the plant. Otherwise, tracking can only be achieved after a transient.

According to [37], the two-step procedure for sliding mode control design was clearly stated: 1) Sliding surface design. When the trajectory of closed-loop system is fixed in the sliding surface, it will be asymptotically stable. And, 2) Discontinuous controllers ensuring the sliding modes. The control law can let the trajectory of the closed-loop system reach the desired sliding surface in a finite time and stick on the the desired sliding surface. A typical phase portrait is illustrated in Fig. 2.1.

In first step, defining s(x) is a smooth scalar constraint function: IRⁿ→ IR, we select

Fig. 2.1. A typical phase portrait under sliding mode control

time-varying sliding surface to be s = 0 with

s = a^Te (2.5)

where the constant coefficient vector a^T = [a₁ · · · ar]. Here, a_i for i=1,· · · , r − 1 are selected constants and a_r = 1 is chosen such that

λ^r−1+ a_r−1λ^r−2+· · · + a2λ + a₁ (2.6) are Hurwitz polynomials. As the states trajectory remain on the sliding surface, i.e., s = 0, we can know Eq. (2.5) will be asymptotically stable, which means the error approaches zero as the time approaches infinity.

Second, designing the control law u, consisting of two parts

u = u^eq+ u^re (2.7)

where u^eq is continuous called a feedback control law and u^reis discontinuous or switched.

As designing the u^eq, there exists a condition which it must let the sliding surface s = 0 be invariant set relatively to the closed-loop system for unpresence of uncertainties or external disturbances of the matched type system [37]. That is s(x(t₀)) = 0 and s(x(t)) = 0,

∀ t ≥ t0. Moreover, the time derivative of s is given by

˙s = a^T˙e (2.8)

where

˙e = [ ˙e · · · e^(r)]^T (2.9)

Expanding Eq. (2.8), the sliding variable dynamics as follows

˙s = f (x) + g(x)u + d(x)− x^(r)d

+a_r−1e^(r−1)+· · · + a2e + a¨ ₁˙e (2.10) Herein, it is regardless of d and u^re such that u = u^eq can verify sliding condition. The equilibrium point of Eq. (2.10) will be s = 0. u^eq is designed as

u^eq =− 1 g(x)

[

f (x)− x^(r)_d + a_r−1e^(r−1)+· · · + a2e + a¨ ₁˙e ]

(2.11) The effect of u^eq is to eliminate the known form of Eq. (2.10). Substituting designed u^eq into Eq. (2.10), we get

˙s = g(x)u^re+ d(x) (2.12)

Obviously, if we do not consider the disturbance term d and just use the feedback controller u = u^eq, Eq. (2.12) exists a equilibrium point at s = 0. Next, we consider Eq. (2.12) and assume s(e(t0))̸= 0 to design u^re. The action of u^re is to make sliding variable s be zero in a finite time. That is the trajectory of the closed-loop system will achieve the sliding surface in limited time. To guarantee the reaching condition, we impose a assumption:

Assumption 2.1 There exists a nonnegative number ρ(x) such that

|d(x)| ≤ ρ(x) (2.13)

From Eq. (2.12), we obtain the other controller u^re designed as

u^re =− 1

g(x)[ρ + η] sgn(s) (2.14)

where

sgn(s) :=





1 if s > 0 0 if s = 0

−1 if s < 0

is the sign function (2.15)

and η > 0 is selected positive constant. Then, we substitute Eq. (2.14) into Eq. (2.12), the sliding variable dynamics becomes

˙s = − [ρ + η] sgn(s) + d(x) (2.16)

In order to prove the feasibility of Eq. (2.14), with V = 1/2s² as a Lyapunov function candidate for Eq. (2.16), we have

V = s ˙s =˙ − [ρ + η] s · sgn(s) + s · d(x) (2.17)

Using the relation of s· sgn(s) = |s| and Cauchy-Schwarz inequality to get s · d(x) ≤

|d(x)||s| and Assumption 2.1, the time derivative of Lyapunov function candidate becomes

V˙ = − [ρ + η] s · sgn(s) + s · d(x)

≤ − [ρ + η] |s| + |d(x)||s|

≤ − [ρ + η] |s| + ρ|s|

≤ −η|s| (2.18)

According to (2.18), we have known that the Lyapunov function converges. The result explains that the Eq. (2.16) is asymptotically stable, i.e., s → 0 as t → ∞. In other words, focusing on Eq. (2.12), u^re will make s approach the sliding surface in limited time when the sliding variable is not zero. Now, we discuss when the sliding variable reach the sliding surface. Actually, there is another form of the time derivative of Lyapunov function presented as

V =˙ d

dtV = 1 2

d

dt|s|² =|s| d

dt|s| (2.19)

As a result of Ieq. (2.18) and Eq. (2.19), we obtain

|s|d

dt|s| ≤ −η|s| (2.20)

That is

d

dt|s| ≤ −η (2.21)

It implies that |s| converges along with its slope less than or equal to −η. Integrating Ieq.

(2.21) with t on [0, t_r], we get

∫ tr

0

d|s(x(t))|

dt dt≤ −

∫ tr

0

ηdt (2.22)

According to the second fundamental theorem of calculus [38], Eq. (2.22) equals

|s(x(t))| − |s(x(0))| ≤ −ηt (2.23)

or

0≤ |s(x(t))| ≤ |s(x(0))| − ηt (2.24)

The above inequality shows that |s(x(t))| must converge before t = |s(x(0))|/η, which is illustrated in Fig. 2.2.

However, in order to account for the presence of modelling uncertainties and distur-bances, the control law has to be discontinuous across s(t). Since the implementation of the associate control switchings is necessarily imperfect (for example, in practice switching is not instantaneous, and the value s is not known with infinite precision), this leads to

Fig. 2.2. Sliding condition

chattering which is shown as Fig. 2.3. Now, chattering is undesirable in practice, since it involves high control activity and further may excite high-frequency dynamics neglected in the course of modelling such as unmodeled structure modes, neglected time-delays, and so on. Thus, in a second part, the discontinuous control law u^re is suitably smoothed to achieve an optimal trade-off between control bandwidth and tracking precision: while the first part accounts for parametric uncertainty, the second part achieves robustness to high-frequency unmodeled dynamics [20].

Fig. 2.3. Chattering as result of imperfect control switchings

2.2 Terminal Sliding Mode Control (TSMC)

Although the CSMC has received much attention as an efficient control technique for handling systems with large uncertainties, nonlinearities, and bounded external distur-bances and can guarantee finite-time convergence to the sliding surface, the closed-loop system states may only be guaranteed within infinite time. Thus, the terminal sliding mode control (TSMC) was evolved by Zak in the Jet Propulsion Laboratory (JPL) in 1988 [39]. The main idea of TSMC is the concept of terminal attractors which guarantee finite time convergence of the states. The TSMC was first introduced to the control of the dynamic systems based on second-order differential equations. After that, Yu and Man [40], [41] extended it to high-order system (2.1). The problem formulation is the same as Section 2.1. Defining s(x)_i for i=1,· · · , r − 1 is a smooth scalar constraint function:

IRⁿ → IR, the hierarchical terminal sliding mode structure is

s₁ = ˙s₀ + b₁s^q₀^1/p1 (2.25) s₂ = ˙s₁ + b₂s^q₁^2/p2 (2.26)

...

s_r−1 = ˙s_r−2+ b_r−1s^q_r^r−1₋₂^/pr−1 (2.27)

where s0 = e, bi > 0, pi > qi and pi, qi are positive odd integers This assumption allows us to achieve high-order continuous differentiation. For instance, the geometry plot for third-order system is shown in Fig. 2.4.

Fig. 2.4. The sliding mode of the third-order system

The control is divided into

u = u^eq+ u^re (2.28)

where u^eq is the equivalent control for system (2.1) without model uncertainties and external disturbances, such that s_r−1 = 0 and ˙s = 0 and u^re is to compensate the internal parameter variations and turbulence. Furthermore, the time derivative of s_r−1 is given by

d

dts_r−1 = d²

dt²s_r−2+ b_r−1 d

dts^q_r−2^r−1/pr−1 (2.29) Besides, it can be easily calculated that

d² Substituting Eqs. (2.52)-(2.34) into Eq. (2.29), we obtain

d Importing Eq. (3.4), the time derivative of s_r−1 will be

˙s_r−1 = f (x) + g(x)u + d(x)− x^r_d Thus, the controller u is designed as follows:

u^eq =− 1

Since |d| ≤ ρ, s · sgn(s) = |s|, Cauchy-Schwarz inequality s · d(x) ≤ |d(x)||s|, and selected positive constant η which have been accounted for in Section 2.1., the resulting expression is substituted into Eq. (2.37) and Eq. (2.38) and multiplied by sr−1 as

s_r−1˙s_r−1 = − [ρ + η] s · sgn(s) + s · d(x)

≤ − [ρ + η] |s| + |s||d(x)|

≤ − [ρ + η] |s| + ρ|s|

≤ −η|s| (2.39)

which means that the sliding mode s_r−1 = 0 will be reached in finite time along with its slope less than or equal to −η proved in Section 2.1. The finite time is directly proportional to the initial norm of s_r−1 and the selected positive constant η expressing as t_r ≤ |s(x(0))/η|. However, the magnitude of the designed controller will become infinity if s_i = 0 when ˙s_i ̸= 0. That is, it is the singularity problem. For example, the controller of second-order system is described

u =− 1 g(x)

[f (x)− x²d+ b₁(q₁/p₁) e^(q1/p1)−1˙e + (ρ + η) sgn(s)]

(2.40) The term b1(q1/p1) e^(q1/p1)−1˙e will occur singularity phenomenon since e^(q1/p1)−1 = 1/e^{((p1−q1)/p1} where (q1/p1)− 1 = (q1− p1)/p1 is negative constant causes 1/e^(p1−q1)/p1 → ∞ as e → 0.

In this situation, if ˙e = 0, the designed controller diverges. The problem is unexpected and will be solved in later Section.

Next, we will discuss whether or not the closed-loop system states can converge within finite time when sliding condition is exactly verified. First, importing the second-order differential equations [42], basically a nonlinear switch line,

s = ˙e + be^q/p (2.41)

where e = x − xd, b > 0, p, q are positive odd integers and p > q. Similar to the conventional sliding mode control technique, if the controller is designed such that s converges to zero, then we say that the switching variable s reaches the terminal sliding mode

˙e + be^q/p = 0 (2.42)

It has been shown in Zak [39] that e = 0 is the terminal attractor of dynamics (2.42). For a error e(t_r) at t = t_r when s = 0, then we integrate the time derivative of ˙e =−be^q/p to predict the convergence time ts at the sliding regime. That is

∫ ts+tr

According to the conditions: 1) p, q are positive odd integers; and 2) p > q, we multiply

−b(p − q)/p into Eq. (2.44) and move the second term of left to the right

0≤ |e(tr+ t_s)|¹⁻

The expression (2.46) means that in terminal sliding mode (2.42) the state error e con-verges to zero in finite time, the same for ˙e. The total time reaching e = 0 is t = t_r+ t_s. Then, expanding to high-order continuous differentiation. With the structure (2.26)-(2.27), if sr−1 = 0 is reached, the stability and finite-time reachability of system equilib-rium will be guaranteed because it is a concatenation of r dynamics of Eq. (2.41) type.

If s_r−1 = 0 is reached at t = t_r = t_s0, then s_r−2 will reach s_r−2 = 0 at

The total time reaching e = 0 is

t = t_s0+

∑r−1 i=1

p_r−i

b_r−i(p_r−i− qr−i)s_r−1−i(t_s(i−1))¹⁻qr−1

p_r−1 (2.51)

TSMC adds nonlinear functions into the design of the sliding upper plane. A ter-minal sliding surface is constructed and the tracking errors on the sliding surface converge to zero in a finite time. Thus, the TSMC can guarantee that the system will achieve the desired output in finite time if the controller is designed by Eqs. (2.37) and (2.38).

2.3 Nonsingular Terminal Sliding Mode Control (NTSMC)

The TSMC is characterized, like the CSMC, by strong robustness to uncertainties and disturbances and guaranteed to achieve the desired state in finite time, yet it exists a singular problem for control law, for instance in second-order system, if ˙e ̸= 0 when e = 0; that is, u → ±∞ if ˙e ̸= 0 as e = 0. In order to overcome the singularity problem in the conventional TSMC systems, several methods have been proposed. For instance, one approach is to switch the sliding mode between terminal sliding mode and linear hyperplane based sliding mode [43]. Another approach is to transfer the trajectory to a pre-specified open region where TSMC is not singular [41]. These methods are adopting indirect approaches to avoid the singularity. Thus, in 2001, Feng [44] proposed a novel TSMC for second-order system, called nonsingular terminal sliding mode control (NTSMC) to overcome the singularity problem. The time taken to reach the manifold from any initial state and the time taken to reach the equilibrium point in the sliding mode can be guaranteed to be finite time. However, the NTSMC is just adapted to the second-order system. In other words, selecting n = 2 for system (2.1). Choosing the sliding surface of the second-order NTSMC:

s = e +1

c˙e^p1/q1 (2.52)

where c = b^p₁^1/q1, and p₁, q₁ are positive odd integers under the constraint 1 < (p₁/q₁) < 2.

One can easily see that when s = 0, Eq. (2.52) is equivalent to Eq. (2.26) for n = 2 so that the time of convergence is the same as TSMC for n = 2 when s = 0. For convenience, we simplify p₁, q₁ as p, q, respectively. The finite time is taken to to travel from e(t_r)̸= 0

at t = t_r to e(t_r+ t_s) is given by

ts = p

b(p− q)|e(tr)|¹⁻ q

p (2.53)

Note that in using Eq. (2.52) the derivative of s along the system dynamics does not result in terms with negative powers, but the parameters p and q must satisfy the constraint 1 < p/q < 2 in addition. Next, we will account for the derivation process of the NTSMC controller. The controller is chosen

u = u^eq+ u^re (2.54)

where u^eq is the feedback control for system (2.1) for n = 2 without model uncertainties and external disturbances, such that s = 0 and ˙s = 0 and u^reis to compensate the internal parameter variations and turbulence. Furthermore, the time derivative of s is given by

˙s = ˙e + 1

Hence, the controller u can be designed as follows:

u^eq=− 1 selected positive constant η which have been accounted for in Section 2.1, the resulting expression of Eq. (2.55) is substituted into Eq. (2.56) and Eq. (2.57) and multiplied by s as

Because p and q are positive odd integers and 1 < p/q < 2, there is ˙e^(p/q)−1 > 0 for ˙e̸= 0. will the NTSMC s = 0 be reached within finite time? The answer is yes [45]. The condition for Lyapunov stability is satisfied for the case ˙e ̸= 0. According to Eq. (2.59), it implies that the slope of sliding variable is always negative expect for ˙e = 0. In addition, for the case ˙e = 0, substituting the control (2.56) and (2.57) into the second equation of (2.1) yields

It means ˙e(t) is monotonous decreasing and at least at the speed of η

2 cross the vicinity rate ˙e(0), we can obtain

¨

It means ˙e(t) is monotonous increasing and at least at the speed of η

2 cross the vicinity δe˙ within the finite time

t_δe˙ ≤ 2 ( ˙e(t_δe˙)− ˙e(0)) η = 4δ_e˙

η (2.66)

Therefore the crossing of trajectory from one boundary of the vicinity ˙e = δ_e˙ to the other boundary ˙e = −δe˙ for s > 0 and from ˙e = −δe˙ to ˙e = δ_e˙ for s < 0 is finite time. For the region outside the | ˙e| < δe˙, the time to reach the boundaries of the vicinity is finite.

Indeed, we can easily show that

s ˙s ≤ −δR( ˙e)|s| (2.67)

meaning the finite time reachability of the boundaries. The phase plane plot of the second-order system is presented in Fig. 2.5 as below:

Fig. 2.5. The phase plot of the second-order system

Therefore we can conclude that the switching line can be reached within finite time.

Furthermore, the designed controller does not contain the singularity term to occur singu-larity phenomenon compared with TSMC because the term c (q/p) ˙e^2−(p/q) of Eq. (2.56) does not yield singularity under the constraint 1 < (p/q) < 2. Thus, a NTSMC use the other nonlinear functions into the design of the sliding upper plane to not only over-come the singularity problem of TSMC but also verify the convergence of tracking desired output in finite time.

2.4 Mathematical Models of the Missile

2.4.1 Coordinate Systems

I) Definitions of Coordinate Frames

Before proceeding with the derivation, it is necessary to assume that the earth is an inertial reference, and unless otherwise stated the atmosphere is fixed with respect to the earth [46]. In addition, the coordinate systems adopted in the present discussion are right-handed axis systems.

i) Earth (Inertial) coordinate frame (Ox_gy_gz_g)

The origin O_g is at the ground tracker. Ox_g-axis is taken as north. The positive Oy_g-axis points upward in the vertical plane including Ox_g-axis. The positive Oz_g-axis is the right direction or completes the right-handed coordinate system.

ii) Body coordinate frame (Ox_by_bz_b)

The origin Ob is at the center of gravity of the missile. The positive Oxb-axis coincides with the center line (or longitudinal axis) of the missile or forward direction. The positive Oy_b-axis points upward in the vertical plane including Ox_b-axis. The Oz_b-axis completes the right-handed coordinate system.

iii) Ballistic coordinate frame (Ox_ty_tz_t)

The origin O_t is at the center of gravity of the missile. The positive Ox_t-axis coincides with the velocity of the missile. The positive Oy_t-axis points upward in the vertical plane including Ox_t-axis and perpendicular to the horizontal plane of the earth. The Ozt-axis completes the right-handed coordinate system.

iv) Wind coordinate frame (Ox_vy_vz_v)

The origin O_v is at the center of gravity of the missile. The positive Ox_v -axis coincides with the velocity of the missile. The positive Oy_v-axis points upward in the vertical plane including Ox_v-axis. The Oz_v-axis completes the right-handed coordinate system. Note that if the wind coordinate frame is nonrotating with respect to Ox_v-axis and the initial definition of the Oy_v-axis is including the vertical plane of the inertial coordinate frame (or the direction

of Oy_v-axis is always the same as Oy_t-axis), it is equal to ballistic coordinate frame.

II) Definitions of Angles

Herein, in order to clearly understand the definitions of the angles, the plus or minus sign of each angle is according to a rule which the rotation axis directs to the reader.

i) Angles between wind frame and body frame

α: It is between O_xv-axis and the plane composed of O_xb-axis and O_zb-axis and defined the sign is positive when O_xv-axis is under that plane.

β: It is between Oxv-axis and the plane composed of Ox_b-axis and Oy_b-axis and defined the sign is positive when O_xv-axis is on the right of that plane.

Fig. 2.6. Definitions of α and β

ii) Angles between ballistic frame and wind frame

γ_v: It is between O_zv-axis and the plane composed of O_xt-axis and O_yt-axis and defined the sign is positive when Ozv-axis is on the left of that plane.

iii) Angles between inertial frame and ballistic frame

θ: It is between Oxt-axis and the plane composed of Oxg-axis and Ozg-axis and defined the sign is positive when O_xt-axis is on the above of that plane.

ψ_v: It is between O_xt-axis and the plane composed of O_xg-axis and O_yg-axis

Fig. 2.7. Definition of γv

and defined the sign is positive when O_xt-axis is on the left of that plane.

Fig. 2.8. Definitions of θ and ψ_v

iv) Angles between inertial frame and body frame

ϑ: It is between O_xb-axis and the plane composed of O_xg-axis and O_yg-axis and defined the sign is positive when O_xb-axis is on the above of that plane.

ψ: It is between O_xb-axis and the plane composed of O_xg-axis and O_zg-axis and defined the sign is positive when O_xb-axis is on the left of that plane.

γ: It is between O_yb-axis and the plane composed of O_x′

g-axis and O_yg-axis and defined the sign is positive when O_yb-axis is on the left of that plane.

III) Coordinate Transformation

Define M_k(ϕ) to be the rotation by k axis with angle ϕ which any one frame is rotated counterclockwise away from other one with. and it is called direct cosine

Fig. 2.9. Definitions of ϑ, ψ, and γ

matrix (DCM). So far as the surface-to-air missile is concerned, each one coordinate follows three rotated steps to other one: 1) yaw, 2) pitch, and 3) roll, and the derivation of the transformation only is established in the direction presented in Fig. 2.10. The DCM rotated by the axes y, z, and x will be:

M_y(ϕ) =



 cϕ 0 −sϕ

0 1 0

s_ϕ 0 c_ϕ





M_z(ϕ) =



 cϕ sϕ 0

−sϕ c_ϕ 0

0 0 1



 (2.68)

M_x(ϕ) =



 1 0 0

0 c_ϕ s_ϕ 0 −sϕ c_ϕ





where c_ϕ and s_ϕ denote cos ϕ and sin ϕ, respectively.

Fig. 2.10. The relationships among coordinates

And the each coordinate transformation is described separately as follows:

i) Wind frame transforms to body frame

ii) Ballistic frame transforms to wind frame

T_t^v(γ_v) =

iii) Inertial frame transforms to ballistic frame

T_g^t(θ, ψ_v) =

iv) Inertial frame transforms to body frame

T_g^b(γ, ϑ, ψ) = 2.4.2 Rigid-Body Equations of Motion

In this section we will consider a typical missile and derive the equations of motion according to Newton’s laws. In deriving the rigid-body equations of motion, the following assumptions will be made [46], [47], [48]:

I) The missile is a rigid body, that is, the missile does not undergo any change in size and shape.

II) The missile is approximate a cylinder, that is, it is an axisymmetric or rotational

II) The missile is approximate a cylinder, that is, it is an axisymmetric or rotational

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