Slide Mode Control (SMC) Techniques and Applications

Slide Mode Control
(SMC)
SOLO HERMELIN
Updated: 6.03.12
1

Table of Content
SOLO
Slide Mode Control (SMC)
2
Sliding Mode Control - Introduction
Control Statement of Sliding Mode
Existence of a Sliding Mode
Reachability: Attaining Sliding Manifold in Finite Time
Picard-Lindelöf Existence and Uniqueness of
a Differential Equations Solutions
Uniqueness of Sliding Mode Solutions
Sliding Motion Surface Keeping
Controller Design
Diagonalization Method
Other Methods – Relays with Constant Gains
Other Methods – Linear Feedback with Switched Gains
Other Methods – Linear Continuous Feedback
Other Methods – Univector Nonlinearity with Scale Factor
Chattering

Table of Content (continue - 1)
SOLO
3
Higher Order Sliding Mode Control
Sliding Order and Sliding Set
Second Order Sliding Modes
The Twisting Controller (Levantosky, 1985)
The Problem Statement
The Super-Twisting Controller
The Super-Twisting Controller (Shtessel version)
Sliding Mode Control for Linear Time Invariant (LTI) Systems
Regular Form of a LTI
Sliding Surface of a Regular Form of a LTI
Unit Vector Approach for a Controller of a Regular Form of a LTI
Output Feedback Variable Structure Controllers and State Estimators
for Uncertain Dynamic Systems
System Sliding Surface
Sliding Modes and System Zeros
Properties of the Sliding Modes
Design of a Sliding Surface (Hyperplane)

SOLO
Sliding Mode Control for Nonlinear Systems
Continuous Sliding Mode Control
Second Order Sliding Mode Control
Sliding Mode Observers
Sliding Mode Observers of Target Acceleration

SOLO
Slide Mode Control Examples
Control System of a Kill Vehicle
Equations of Motion of a KV (Attitude)
Fu, L-C, et all, Control System of a Kill Vehicle
Fu, L-C, et al, Solution for Attitude Control of KV
Fu, L-C, et al, Zero SM Guidance of a KV
Crassidis , et al -Attitude Control of the Kill Vehicle
Midcourse Intercept of a Ballistic in a Head On Scenario
Above a Minimal Altitude
HTK Guidance Using 2nd
Order Sliding Mode

Slide Mode Control (SMC) is a type of Variable Structure Control (VSC) that possess
Robust Characteristics to System Disturbances and Parameter Uncertainties.
A VSC System is a special type of Nonlinear System characterized by a discontinuous
control which change the System Structure when the States reach the Intersection of
Sets of Sliding Surfaces.
The System behaves independently of its general dynamical characteristics and system
disturbances once the controller has driven the System into a Sliding Mode.
SOLO
Such a High-Level of Performance requires High-Quality Actuators requiring a Very
Fast Responding and Fast Switching Action. This translates to a Very Wide Bandwidth
Actuators.
6
Sliding Mode Control - Introduction
A Few Examples are presented:
• Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude,
a solution using Sliding Mode given in 1975 (before the development of the Sliding
Mode Method)
• Control of a Kill Vehicle (because of the beauty of quaternion mathematics)
• Hit-to-Kill (HTK) Guidance Law using a Second Order Sliding Mode Control.

SOLO
In control theory, Sliding Mode Control, or SMC, is a form of Variable Structure
Control (VSC). It is a nonlinear control method that alters the dynamics of a
nonlinear system by application of a high-frequency switching control. The state-
feedback control law is not a continuous function of time. Instead, it switches from
one continuous structure to another based on the current position in the state space.
Hence, sliding mode control is a variable structure control method. The multiple
control structures are designed so that trajectories always move toward a switching
condition, and so the ultimate trajectory will not exist entirely within one control
structure. Instead, the ultimate trajectory will slide along the boundaries of the
control structures. The motion of the system as it slides along these boundaries is
called a Sliding Mode and the geometrical locus consisting of the boundaries is
called the sliding (hyper)surface. The sliding surface is described by σ = 0, and the
sliding mode along the surface commences after the finite time when system
trajectories have reached the surface. In the context of modern control theory, any
variable structure system, like a system under SMC, may be viewed as a special case
of a hybrid dynamical system.
Sliding Mode Control (SMC)
7

SOLO
Consider a Nonlinear Dynamical System Affine in Control:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) nxmnmn
xBtxftutx
tuxBtxftx
RRRR ∈∈∈∈
+=
,,,,
,
The components of the discontinuous feedback
are given by:
( )
( ) ( )
( ) ( )
mi
xiftxu
xiftxu
tu
ii
ii
i ,,2,1
0,
0,
=




<
>
= −
+
σ
σ
where σi (x) = 0 is the i-th component of the Sliding Surface, and
( ) ( ) ( ) ( )[ ] 0,,, 21 ==
T
m xxxx σσσσ  is the (n-m) dimensional Sliding Manifold
The sliding-mode control scheme involves:
1.Selection of a Hypersurface or a Manifold (i.e., the Sliding Surface) such that the
system trajectory exhibits Desirable Behavior when confined to this Manifold.
2.Finding discontinuous feedback gains so that the System Trajectory intersects and
stays on the Manifold.
8

SOLO
Consider a Nonlinear Dynamical System Affine in Control:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) nxmnmn
xBtxftutx
tuxBtxftx
RRRR ∈∈∈∈
+=
,,,,
,
( )
( ) ( )
( ) ( )
mi
xiftxu
xiftxu
tu
ii
ii
i ,,2,1
0,
0,
=




<
>
= −
+
σ
σ
where σi (x) = 0 is the i-th component of the Sliding Surface, and
( ) ( ) ( ) ( )[ ] 0,,, 21 ==
T
m xxxx σσσσ  is the (n-m) dimensional Sliding Manifold
A Sliding-Mode exists, if in the Vicinity of the
Switching Surface, σ (x) = 0, the Velocity Vector
of the State Trajectory, , is always
Directed Toward the Switching Surface.
( )tx
Because sliding mode control laws are not continuous, it has the ability to drive
trajectories to the sliding mode in finite time (i.e., stability of the sliding surface is better
than asymptotic). However, once the trajectories reach the sliding surface, the system
takes on the character of the sliding mode (e.g., the origin x=0 may only have
asymptotic stability on this surface).
9

SOLO
The Existence of the Sliding Mode requires Stability of the State Trajectory to the
Sliding Surface , σ (x) = 0, at least in a Neighborhood of the Sliding Surface, i.e., the
System State must approach the surface at least asymptotically.
From a Geometrical point of view, in the
Vicinity of the Switching Surface, σ (x) = 0, the
Velocity Vector of the State Trajectory, , is
always Directed Toward the Switching Surface.
( )tx
The Existence Problem can be seen as a Generalized Stability
Problem hence the Second Method of Lyapunov provides a
natural setting for Analysis.
Aleksandr Mikhailovich Lyapun
1857 - 1918
10

SOLO
1857 - 1918
Definition
A Domain D in a Manifold σ (x) = 0 is a Sliding Mode Domain if
for each ε > 0, there is a δ >0, such that any trajectory starting
within a n-dimensional δ-vicinity of D may leave the n-
dimensional δ-vicinity of D only through the n-dimensional ε-
vicinity of the boundary of D.
Second Method of Lyapunov
11

SOLO
1857 - 1918
For the (n-m) dimensional domain D to be the Domain of a Sliding
Mode, it is sufficient that in some n-dimensional domain Ω ϵ D,
there exists a function V (x,t,σ) continuously differentiable with
respect to all of its arguments, satisfying the following conditions:
1.V (x,t,σ) is positive definite with respect to σ, i.e., V (x,t,σ) > 0,
with σ ≠ 0 and arbitrary x,t, and V (x,t,σ=0) = 0; and on
the sphere ||σ|| = ρ, for all x ϵ Ω and any t the relations
holds, hρ and Hρ, depend on ρ (hρ ≠ 0 if ρ ≠0)
2.The Total Time Derivative of V (x,t,σ) for the System Dynamics
has a negative supremum for all x ϵ Ω except for x on the
Switching Surface where the control input are undefined, and hence
the derivative of V (x,t,σ) does not exist.
( )
( ) 0,,,sup
0,,,inf
>=
>=
=
=
ρρ
ρσ
ρρ
ρσ
σ
σ
HHtxV
hhtxV
12

SOLO
Unfortunately, there are no standard methods to find Lyapunov
Functions for Arbitrary Nonlinear Systems.
Existence of Sliding Mode
Consider a Lyapunov Function candidate:
( )[ ] ( ) ( ) ( ) ( )[ ] ( ) 000
2
1
2
1
2
=⇔=⇒≥== xxVxxxxV T
σσσσσσ
where ||*|| is the Euclidean norm (i.e. ||σ (x)||2 is the Distance away from the Sliding
Manifold where σ (x)=0 ). V (σ (x)) is Globally Positive Definite.
( ) ( ) ( ) ( )tuxBtxftx += ,
A Sufficient Condition for the Existence of the Sliding Mode is:
0<==
td
d
td
d
d
Vd
td
Vd T σ
σ
σ
σ
in a neighborhood of the surface σ (x)=0.
( ) ( )[ ]utxBtxf
xd
d
td
xd
xd
d
td
d
,, +==
σσσ
The feedback control law u (x) has a direct impact on .
td
d σ 13

SOLO
Existence of Sliding Mode (continue – 1)
( )[ ] ( ) ( ) ( ) ( )[ ] ( ) 000
2
1
2
1
2
=⇔=⇒≥== xxVxxxxV T
σσσσσσ
( ) ( ) ( ) ( )tuxBtxftx += ,
0<==
td
d
td
d
d
Vd
td
Vd T σ
σ
σ
σ
Roughly speaking (i.e., for the scalar control case when m = 1), to achieve ,
the feedback control law u (x) is picked so that σ and have opposite sign, that is
( ) ( )[ ]utxBtxf
xd
d
td
xd
xd
d
td
d
,, +==
σσσ
• u (x) makes negative when σ (x) is positive.xd
d σ
0<
td
dT σ
σ
td
d σ
• u (x) makes positive when σ (x) is negative.xd
d σ
14

SOLO
Reachability: Attaining Sliding Manifold in Finite Time
( ) ( ) ( ) ( )tuxBtxftx += ,
To ensure that the Sliding mode σ (x) = 0 in a Finite Time, dV/dt must be
Strongly Bounded Away From Zero. That is, if it vanished to quickly, the
Attraction to the Sliding Mode will only be Asymptotic. To ensure that the Sliding
Mode is entered in Finite Time
α
µV
td
Vd
−≤
where μ > 0 and 0 < α < 1 are constant
( )
( )
( )
( )[ ] ( )
( )0
1
0
00
1
111
0
ttVVd
Vtd
Vd
V
S
V
V
−−≤
−
−=⇒−≤ −
==
∫ µσ
α
µ α
σ
σ
αα
This shows that the time necessary to reach the Sliding Manifold σ [x(ts)] = 0 is
bounded by:
( )[ ] ( )
( )
100
1
1
0
0 <<>
−
≤−
−
α
αµ
σ α
V
ttS 15

SOLO
Note that for all single input functions a suitable Lyapunov
function is:
( ) ( )xtxV 2
2
1
,, σσ =
which is Globally Positive Definite.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]tutxBtxf
xd
xd
x
td
xd
xd
xd
x
td
xd
xtxV
td
d
,,,, +===
σ
σ
σ
σ
σ
σσ
Suppose that we can find u (t) such that in the neighborhood of V (x,t,σ=0)=0 we
have:
( ) ( ) ( ) ( ) 0,, <−≤= x
td
xd
xtxV
td
d
σµ
σ
σσ
( )
( )
( ) ( ) ( ) ( )
0sgn <−≤== µ
σσ
σ
σ
σ
σ
td
xd
td
xd
x
td
xd
x
x ( )[ ] ( )[ ] ( )00 tttxtx −−≤− µσσ
integration
This shows that the time necessary to reach the Sliding Manifold σ [x(ts)] = 0 is
bounded by: ( )[ ]
µ
σ 0
0
tx
ttS ≤−
Reachability: Attaining Sliding Manifold in Finite Time (continue – 1)
16

SOLO
Region of Attraction
Reachability: Attaining Sliding Manifold in Finite Time (continue – 2)
For the Dynamic System given by
and for the Sliding Surface σ (x) = 0, the subspace for which the
Sliding Surface is Reachable is given by ( ) ( ){ }0: <∈ xxRx Tn
σσ 
( ) ( ) ( ) ( )tuxBtxftx += ,
When Initial Conditions come from this Region, the Lyapunov Function Candidate
is a Lyapunov Function and the Space Trajectories are sure to
move toward Sliding Mode Surface σ (x) = 0. Moreover, if the Reachable Condition
is satisfied, the Sliding Mode will reach σ (x) = 0 in Finite
Time.
( )[ ] ( ) ( ) 2/xxxV T
σσσ =
( ) ( )10 <<−≤ αµ
α
VV
17

SOLO
Picard-Lindelöf Existence and Uniqueness of
a Differential Equations Solutions
Lipschitz Continuity Condition
Charles Émile
Picard
1856 - 1941
Ernst Leonard Lindelöf
1870 - 194618
Rudolf Otto Sigismund
Lipschitz
1832 – 7 1903
In mathematical analysis, Lipschitz continuity, named after Rudolf
Lipschitz, is a strong form of uniform continuity for functions.
A Function f (x) is called Lipschitz continuous if there exists a real
constant K ≥ 0 such that, for all x1 and x2 in X:
( ) ( ) 1212 xxKxfxf −≤−
Picard–Lindelöf Theorem
Consider the initial value problem
( ) ( ) [ ]εε +−∈== 0000 ,,,, tttxtxtxf
td
xd
Suppose f is Lipschitz continuous in x and continuous in t.
Then, for some value ε > 0, there exists a Unique Solution
x(t) to the initial value problem within the range [t0-ε,t0+ε].

SOLO
Aleksandr Mikhailovich Lyapu
1857 - 1918
Uniqueness of Sliding Mode Solutions
The Nonlinear Dynamical System Affine in Control:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) nxmnmn
xBtxftutx
tuxBtxftx
RRRR ∈∈∈∈
+=
,,,,
,
( )
( ) ( )
( ) ( )
mi
xiftxu
xiftxu
tu
ii
ii
i ,,2,1
0,
0,
=




<
>
= −
+
σ
σ
with the switching control, do not formally satisfy the classical
Picard-Lindelöf Existence and Uniqueness Solutions, since they
have discontinuou right-hand sides. Moreover the right-hand
sides usually are not defined on the discontinuous surfaces.
Charles Émile Picard
1856 - 1941
Ernst Leonard Lindelöf
1870 - 1946
Existence and Uniqueness of Differential Equations with Discontinuous
Right-hand Sides is was addressed by different researchers. One of the
straightforward approaches is the Method of Filippov (Filippov Aleksei
Fedorovich, “Differential Equations with Discontinuous Right Hand
Sides”, Kluwer, Dordrecht, the Nederlands)
19

SOLO
Uniqueness of Sliding Mode Solutions (continue – 1)
( ) ( )utxftx ,,=
( )
( ) ( )
( ) ( )



<
>
= −
+
0,
0,
xiftxu
xiftxu
tu
σ
σ
Method of Filippov (Filippov Aleksei Fedorovich)
Consider the n-order Single Input System:
with the following Control Strategy:
The System Dynamics is not defined on σ (x) = 0. Filippov has shown that the
solution on the Surface σ (x) = 0 is given by the equation:
( ) ( ) ( ) ( ) 10,,1,, 0
111 ≤≤=−+= −+
ααα nxnxnx futxfutxftx
The term α is a function of the System States and
can be specified in such a way that the “average”
dynamics f0 is tangent to the Surface σ (x) = 0 .
20

SOLO
Sliding Motion Surface Keeping
( ) ( ) ( ) ( ) 0, =
∂
∂
+
∂
∂
=
∂
∂
= equxB
x
txf
x
tx
x
x
σσσ
σ 
( ) 0=xσ
The Dynamic System will stay on the Sliding Surface σ (x) = 0,
if the equivalent control ueq will keep
( )xB
x
nxm
mxn






∂
∂ σ
If is nonsingular, i.e., the System has a kind of Controllability
that assures that we can find a controller to move a trajectory closer to σ (x) = 0,
then ( ) ( )txf
x
xB
x
ueq ,
1
∂
∂






∂
∂
−=
−
σσ
( ) ( ) ( )txf
x
xB
x
Itx ,
1








∂
∂






∂
∂
−=
−
σσ
The Dynamic System Equation is
Note that using ueq any trajectory
that starts at σ (x) = 0, remains on it,
Since . As a consequence
the Sliding Manifold σ (x) = 0 is an
Invariant Set.
( ) 0=xσ
Return to Chattering
21

SOLO
Controller Design
( ) ( ) ( ) ( )tutxB
x
txQtu ,,* 1






∂
∂
= − σ
We must choose Switched Feedback capable of forcing the Plant
State Trajectories to the Switching Surface and maintaining a
Sliding Mode Condition. We assume that the Sliding Surface has
already been designed.
Diagonalization Method
The Diagonalization Method converts the multi – input design problem into
m single-input design problems.
The Method is based on the construction of a new control vector u* through
a nonsingular transformation of ueq:
where Q (x,t) is an arbitrary mxm Diagonal Matrix with elements qi (x,t), i=1,…,m,
such that inf |qi (x,t)| > 0 for all t ≥ 0 and all x.
22

SOLO
Controller Design (continue – 1)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )tutxQtutxB
x
tutxB
x
txQtu *,,,,* 1
=





∂
∂
⇒





∂
∂
= − σσ
Diagonalization Method (continue – 1)
where Q (x,t) is an arbitrary mxm Diagonal Matrix with elements qi (x,t), i=1,…,m,
such that inf |qi (x,t)| > 0 for all t ≥ 0 and all x.
For existence and reachability of a Sliding Mode is enough to satisfy .0<
td
dT σ
σ
( ) ( ) ( ) ( ) ( ) ( ) mitutxqtxf
xd
d
td
d
ortutxQtxf
xd
d
td
xd
xd
d
td
d
ii
i
i
,,1,,,,
**
=+





=+==
σσσσσ
To satisfy the existence and reachability we choose each control u*i as
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )tutxQtxB
x
tu
xwhentxf
x
utxq
xwhentxf
x
utxq
i
i
n
j
j
j
ii
i
i
n
j
j
j
ii
*,,
0,,
0,, 1
1
*
1
*
−
=
+
=
+












∂
∂
=⇒







<





∂
∂
−>
>





∂
∂
−<
∑
∑
σ
σ
σ
σ
σ
23

SOLO
Other Methods
A possible structure for the control is: miuuu iNii eq
,,2,1 =+=
Where is continuous and
uiN is the discontinuous part
( ) ( )
i
i txf
x
xB
x
u eq








∂
∂






∂
∂
−=
−
,
1
σσ
( ) ( )( )
( ) ( ) ( ) ( ) NNeq
Neq
uxB
x
uxB
x
uxB
x
txf
x
uuxB
x
txf
xtd
d






∂
∂
=
∂
∂
+
∂
∂
+
∂
∂
=
+
∂
∂
+
∂
∂
=
σσσσ
σσσ
  
0
,
,
Several Design Methods are applicable
24

SOLO
Other Methods – Relays with Constant Gains
,,2,1 =+=
where is continuous and
( ) ( )
i
i txf
x
xB
x
u eq








∂
∂






∂
∂
−=
−
,
1
σσ
( ) NuxB
xtd
d






∂
∂
=
σσ
( ) ( )[ ] ( )





=
<−
>
==>





∂
∂
−=
−
00
01
01
sgn0sgn
1
xif
xif
xif
x
x
xxxB
x
u ii
i
iiN ασ
σ
α
This controller satisfies the reaching condition since:
( ) ( ) ( ) ( )[ ] ( ) ( ) 00sgn ≠<−=−= xifxxx
td
xd
x iiiiii
i
i σσασσα
σ
σ 25

SOLO
Other Methods – Linear Feedback with Switched Gains
,,2,1 =+=
( ) ( )
i
i txf
x
xB
x
u eq








∂
∂






∂
∂
−=
−
,
1
σσ
( ) NuxB
xtd
d






∂
∂
=
σσ
( ) [ ]




<<−
>>
=ΨΨ=Ψ








Ψ





∂
∂
=
−
00
001
jjij
iiij
ijij
i
iN
xif
xif
xxB
x
u
σβ
σασ
( ) ( ) ( )( ) 011 <Ψ++Ψ= ninii
i
i xxx
td
xd
x σ
σ
σ 26

SOLO
Control Methods (continue – 5)
Other Methods – Linear Continuous Feedback
,,2,1 =+=
( ) ( )
i
i txf
x
xB
x
u eq








∂
∂






∂
∂
−=
−
,
1
σσ
( ) NuxB
xtd
d






∂
∂
=
σσ
( )xLu nxnN σ−=
( ) ( ) ( ) ( ) ( ) 00 ≠<−= xifxLx
td
xd
x TT
σσσ
σ
σ
where Lnxn is a Positive Definite Constant Matrix
27

SOLO
Other Methods – Univector Nonlinearity with Scale Factor
,,2,1 =+=
( ) ( )
i
i txf
x
xB
x
u eq








∂
∂






∂
∂
−=
−
,
1
σσ
( ) NuxB
xtd
d






∂
∂
=
σσ
( ) ( )
( )
( ) ( ) ( )xxx
x
x
xu T
N σσσρ
σ
σ
ρ =>−= 2
2
&0
( ) ( ) ( ) ( ) 002
≠<−= xifx
td
xd
xT
σσρ
σ
σ
28

SOLO
Chattering
Due to the presence of external disturbance, noise
and inertia of the sensors and actuators the
switching around the Sliding Surface occurs at a
very high (but finite) frequency.
The main consequence is that the Sliding Mode
take place in a small neighbor of the Sliding
Manifold , which is called Boundary Layer, and
whose dimension is inversely proportional with the
Control Switching Frequency.
The effect of High Frequency Switching is known as Chattering.
The High Frequency Switching propagate through the System exciting the fast
dynamics and undesired oscillations that affect the System Output To prevent the
Chattering Effect different techniques are used. One of the techniques is the use of
continuous approximations of sign (.) (such as sat (.) function, the tanh (.) function,..)
in the implementation of the Control Law. A consequence of this method is that the
Invariance Property is Lost.
Invariance Definition
29

SOLO
Sliding Order and Sliding Set
The Sliding Order r is the number of continuous total derivative, including the zero one,
of the function σ = σ (t,x) whose vanishing defines the equations of the Sliding Manifold.
The Sliding Set of r – th order associated in the Manifold σ (t,x) = 0 is defined by the
equalities ( )
01
===== −r
σσσσ 
which forms an r – dimensional condition on the State of the Dynamic System.
The corresponding motion satisfying the equalities is
called an r – order Sliding Mode with respect to the
Manifold σ (t,x) = 0 .
30

SOLO
The Problem Statement
Consider a Dynamic Single Output System of
the form:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) 11
,,,,
,
nxnn
xbtxftutx
tuxbtxftx
RRRR ∈∈∈∈
+=
Let σ (t,x) = 0 be the chosen Sliding Manifold, then the Control Objective is to
enforce a Second Order Sliding Mode on the Sliding Manifold σ (t,x) = 0 , i.e.,
in Finite Time.
Let analyze the following two cases:
Case A: relative
degree
01 ≠
∂
∂
⇒= σ
u
r
Case B: relative degree 0,02 ≠
∂
∂
=
∂
∂
⇒= σσ 
uu
r 31
( ) ( ) 0,, == xtxt σσ 

SOLO
The Problem Statement (continue – 1)
( ) ( ) ( ) ( )tuxbtxftx += ,
Case A: relative degree 01 ≠
∂
∂
⇒= σ
u
r
( ) ( ) ( ) ( ) ( )[ ]
( ) ( ) ( )tuuxtuxt
tuxtbxtfxt
x
xt
t
AA


,,,,
,,,,
γϕσ
σσσ
+=
+
∂
∂
+
∂
∂
=
( ) ( ) ( ) ( ) ( ) ( )[ ]
( ) ( ) ( )







∂
∂
=
+
∂
∂
+
∂
∂
=
xtbxt
x
xt
tuxtbxtfxt
x
xt
t
uxt
A
A
,,:,
,,,,:,,
σγ
σσϕ


The control u is understand as an internal disturbance affecting the drift term φA.
The control derivative is used as an auxiliary control used to steer σ and to 0.
Note that affect the dynamics.
σu
u σ
32
( ) ( ) 0,,: == xtxtSurfaceSliding σσ 

SOLO
( ) ( ) ( ) ( )tuxbtxftx += ,
∂
∂
=
∂
∂
⇒= σσ 
uu
r
( ) ( ) ( ) ( )

( )
( ) ( ) ( )tuuxtuxt
tuxbxtfxt
x
xt
t
BB ,,,,
,,,
0
γϕσ
σσσ
+=








+
∂
∂
+
∂
∂
=


( ) ( ) ( ) ( )
( ) ( ) ( )







∂
∂
=
∂
∂
+
∂
∂
=
xtbuxt
x
xt
xtfuxt
x
uxt
t
uxt
B
B
,,,:,
,,,,,:,,
σγ
σσϕ


It is assumed that , which means that the sliding variable has
relative degree two. In this case the actual actuator u is discontinuous.
( ) 0, ≠xtBγ 33

SOLO
( ) ( ) ( ) ( )tuxbtxftx += ,
∂
∂
=
∂
∂
⇒= σσ 
uu
r
Case A: relative degree 01 ≠
∂
∂
⇒= σ
u
r
Both Cases A and B can be dealt with an uniform treatment, because the structure of
the System to be stabilized is the same, i.e. a 2nd
Order System with Affine relevant
control signal (the control derivative in Case A, the actual control u in Case B).u
( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )




=+⋅=
==
=
xttvxtty
xttyty
xtty
,,
,
,
2
21
1
σγϕ
σ
σ


Case A: relative degree r = 1
( ) ( )
( ) ( )


=
=⋅
tutv
uxtA

,,ϕϕ
Case B: relative degree r = 2 ( ) ( )
( ) ( )


=
=⋅
tutv
uxtB ,,ϕϕ
Assume
( )
( ) 21 ,0 GxtG ≤≤<
Φ≤⋅
γ
ϕ
34
( ) ( )
( ) ( )


=
=⋅
tutv
uxtA

,,ϕϕ

SOLO
This Algorithm provides twisting around the origin of
the phase plane . This means that trajectories
perform rotations around the origin while converging
in finite time to the origin of the phase plane.
σσ O
( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )




=+⋅=
==
=
xttvxtty
xttyty
xtty
,,
,
,
2
21
1
σγϕ
σ
σ


( ) ( )
( )




≤>=−
≤≤=−
>−
=
1;0
1;0
1
211
211
uyyifysignV
uyyifysignV
uifu
tu
M
m
σσ
σσ


( )
( ) 21 ,0 GxtG ≤≤<
Φ≤⋅
γ
ϕ
35
( ) ( )
( ) ( )


=
=⋅
tutv
uxtA

,,ϕϕ
Levant, Arie ( formerly
Levantosky, Lev )
( ) ( ) ( ) ( )tuxbtxftx += ,

SOLO
This Algorithm provides twisting around the origin of
the phase plane . This means that trajectories
perform rotations around the origin while converging
in finite time to the origin of the phase plane.
σσ O
( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )




=+⋅=
==
=
xttvxtty
xttyty
xtty
,,
,
,
2
21
1
σγϕ
σ
σ


( ) ( )
( )
( )


>=−
≤=−
==
0
0
211
211
σσ
σσ


yyifysignV
yyifysignV
tutv
M
m
( )
( ) 21 ,0 GxtG ≤≤<
Φ≤⋅
γ
ϕ
The following conditions must be fulfilled for the
finite time convergence
( )
Φ+>Φ−
Φ
>
>
>
mM
m
m
mM
VGVG
G
V
G
V
VV
21
1
2
0
4
σ
36
Case B: relative degree r = 2
( ) ( )
( ) ( )


=
=⋅
tutv
uxtB ,,ϕϕ
( ) ( ) ( ) ( )tuxbtxftx += ,

SOLO
The Super-Twisting Controller
( ) ( ) ( ) ( )
( ) ( )



−=
+−=
121
1111
ysigntu
tuysigntytu
α
α
ρ

The control is given by:
( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )




=+⋅=
==
=
xttvxtty
xttyty
xtty
,,
,
,
2
21
1
σγϕ
σ
σ


( )
( ) 21 ,0 GxtG ≤≤<
Φ≤⋅
γ
ϕ
( )
( )
5.00
4
1
2
21
22
2
1
2
1
≤<
Φ
>
Φ−
Φ+Φ
≤
ρ
α
α
α
α
G
G
G
G
( ) ( ) ( ) ( )∫−−=
t
dtysignysigntytu
0
12111 αα
ρ
37
The following conditions must be fulfilled
for the finite time convergence
( ) ( ) ( ) ( )tuxbtxftx += ,
( ) ( )
( ) ( )


=
=⋅
tutv
uxtB ,,ϕϕ

SOLO
The Super-Twisting Controller (continue)
( ) ( ) ( )
( )
( )
( )
( )
( )



≤−
>−
=




≤−
>−
=
+=
01111
01101
2
12
1
21
1
1
σα
σσα
α
ρ
ρ
yifysigny
yifysign
tu
uifysign
uifu
tu
tututu

The control is given by:
( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )




=+⋅=
==
=
xttvxtty
xttyty
xtty
,,
,
,
2
21
1
σγϕ
σ
σ


( )
( ) 21 ,0 GxtG ≤≤<
Φ≤⋅
γ
ϕ
( )
( )
5.00
4
1
2
21
22
2
1
2
1
≤<
Φ
>
Φ−
Φ+Φ
≤
ρ
α
α
α
α
G
G
G
G
38
The following conditions must be fulfilled
for the finite time convergence
( ) ( )utxftx ,,=
General Nonlinear System
( ) ( ) ( ) ( )tuxbtxftx += ,

SOLO
Sliding Order Sliding Modes
( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )





>−=
>=+−=
=
0
0,
,
21
3/1
122
121
2/1
111
1
αα
ασα
σ
ysigntyty
xttyysigntyty
xtty


The 2nd
Order Sliding Mode is Given by
is Finite Time Stable, i.e., is Asymptotically Stable with a Finite Settling Time for any
solution and any initial conditions.
Proof Let choose the following Lyapunov Function candidate:
( )





==
≠≠>+
=
000
000
4
3
2,
21
21
3
4
12
2
2
21
yandyifonly
yandyify
y
yyV
α
( ) ( )
( ) ( )[ ] ( ) ( )[ ] 6/5
1211
3/1
12221
2/1
111
3/1
12
2211
3/1
122
2
1
1
21
21
,
yysigntyyyysignyysigny
yyyysignyy
y
V
y
y
V
yyV
td
d
yy
ααααα
α
−=−++−=
+=
∂
∂
+
∂
∂
=
    


( ) 00, 1
6/5
12121 ≠<−= yifyyyV
td
d
αα 39

SOLO
alpha1=3, alpha2 =3
alpha1=1, alpha2 =9
-5000 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 5000
-2000
-1000
0
1000
X1
X1dot
alpha1=1, alpha2 =1
-4000 -3000 -2000 -1000 0 1000 2000 3000 4000 5000
-400
-200
0
200
400
X1
X1dot
alpha1=3, alpha2 =1
-2000 -1000 0 1000 2000 3000 4000 5000
-400
-200
0
200
X1
X1dot
-3000 -2000 -1000 0 1000 2000 3000 4000 5000
-1000
-500
0
500
X1
X1dot
alpha1=3, alpha2 =9
-4000 -3000 -2000 -1000 0 1000 2000 3000 4000 5000
-1000
-500
0
500
1000
X1
X1dot
alpha1=9, alpha2 =1
-1000 0 1000 2000 3000 4000 5000
-1000
-500
0
500
X1
X1dot
-1000 0 1000 2000 3000 4000 5000
-1000
-500
0
500
X1
X1dot
alpha1=9, alpha2 =3
alpha1=9, alpha2 =9
-2000 -1000 0 1000 2000 3000 4000 5000
-1000
-500
0
500
X1
X1dot
Time = 100 sec
-4000 -3000 -2000 -1000 0 1000 2000 3000 4000 5000
-1000
-500
0
500
X1
X1dot
alpha1=1, alpha2 =3
We can see that to speed-up the
Convergence to Origin we must
Increase alpha1 and keep alpha2
Small relative to alpha1.
40

SOLO
( )
( )





==
≠≠>+
= ∫
000
000
2,
21
21
0
3/1
2
2
2
21
1
yandyifonly
yandyifdzzsignz
y
yyV
y
α
>> [X,Y]=meshgrid(-0.5:0.05:0.5);
>> alpha2=1;
>> Z=0.5*Y.^2+0.75*alpha2*abs(X).^1.3334+eps;
>> mesh(X,Y,Z,'EdgeColor','black')
>> contour(X,Y,Z)
-0.5
0
0.5
-0.5
0
0.5
0
0.1
0.2
0.3
0.4
0.5
1y2y
( )21, yyV
MATLAB:
1y
2y
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
41

SOLO
Theorem: Assume a Lyapunov Function that satisfies
( ) ( ) 00,&
000
000
, 121
21
21
21 ≠<



==
≠≠>
= yifyyV
td
d
yandyifonly
yandyif
yyV
Assume that in addition exists a Domain , that includes the origin
(y1 = 0, y2 = 0), and in this domain
( ) Dyy ∈21,
( ) ( ) ( )10&00,, 2121 <<>≤+ αα
kyyVkyyV
td
d
then V (y1, y2) → 0 in a Finite Time ts.
Proof:
( ) ( ) 0,, 2121 >−≤ yyV
td
d
yyVk
α ( )
( )
0
,
,1
21
21
>−≤ α
yyV
yyVd
k
td
0>td
q.e.d.
( )
( ) ( )
( ) ( ) ( ) ( )
( )
( )100
1
,
,,
1
1
1
211
21
1
0
210
0
0
<<←>
−
=








−
−
−≤−
−
−−
α
αα
α
αα
k
yyV
yyVyyV
k
tt
t
ttS S

We can see that ts at which V (y1, y2) → 0 is Finite.
Let integrate between an initial time t0 to a time ts at which V (y1, y2) → 0.
42

SOLO
Let find, if in our case, exists a Domain D, that includes the origin, and satisfies:
( ) ( ) ( )10&00,, 2121 <<>≤+ αα
kyyVkyyV
td
d
We have: ( )





==
≠≠>+
=
000
000
4
3
2,
21
21
3
4
12
2
2
21
yandyifonly
yandyify
y
yyV
α
( ) 00, 1
6/5
12121 ≠<−= yifyyyV
td
d
αα
( ) ( ) 6
5
1
21213
4
12
2
2
21
,1
4
3
2
, y
ktd
yyVd
k
y
y
yyV
αα
α
α
α
=−≤







+=
( ) 6
5
1
213
4
12
10
2
3
3
4
12
2
2
21
2
3
4
3
2
,
2
1
12
2
2
y
k
yy
y
yyV
yy
αα
αα
α
α
α
α
α
≤





≤







+=
<<
≤
Define the Domain D that includes the origin by: 1
2
3
1
3
4
12
2
2 <≤ yandyy α
α
ααα
α
1
6
5
1
1
213
4
12
2
3
y
k
y 





≤or:
If since1
6
5
1
1
6
5
<<⇒< α
α
11
3
4
,1
1
6
5
1 <<>< yand
α
from the Figure, and choosing some k>0, we can see that exists some small |y1s| such that for
we have3
4
102
2
2101
2
3
yyandyy α≤≤ ( ) ( ) ( )100,, 2121 <<≤+ αα
yyVkyyV
td
d
α
α
αα
α
1
21
2
1
6
5
3
4
10
3
2






=






−
k
y
equality
1y
2y
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
43

SOLO
Therefore for:
Return to Table of Content
( ) 3
4
10221
2
1
3
4
102202101
2
3
,
2
3
, 0
yyyVandyyyyy t
αα =





=≤≤
V (y1, y2) → 0 in Finite Time ts .
( ) ( )
( )
( )100
1
,
1
21
0
0
<<←>
−
≤−
−
α
α
α
k
yyV
tt
t
S
-0.5
0
0.5
-0.5
0
0.5
0
0.1
0.2
0.3
0.4
0.5
1y2y
( )21, yyV
1y
2y
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
44

SOLO
Regular Form of a LTI
Consider the following LTI:
mnRBRARuRxuBxAx nxmnxnmn
>∈∈∈∈+=
Assume rank (B) = m (i.e., matrix B is full rank) and the pair (A,B) is controllable.
Perform the Singular Value Decomposition (SVD) of B:
( )
( )H
B
xmmn
B
Bnxm mxm
mxm
nxn
VUB 1
1
1
0 






Σ
=
−
where H means Transpose of a matrix and complex conjugate of it’s elements, and:
mB
H
B
H
BBnB
H
B
H
BB IVVVVIUUUU ==== 11111111 ;
( )
( ) ( )mxmmnxnn
mmB
diagIdiagI
diagmxm
1,,1,1,1,,1,1
0,,,, 21211


==
>≥≥≥=Σ σσσσσσ
45

SOLO
Regular Form of a LTI (continue - 1)
nxmnxnmn
RBRARuRxuBxAx ∈∈∈∈+=
Define the Orthogonal Transformation Matrix:
( ) ( )
( )
( ) ( )
( ) ( )
( )H
nxn
xmmnmn
mmnmx
Bnxn
H
B
mnmxm
mnxmmn
nxn T
I
I
UTU
I
I
T nxnnxn
=








=⇒








=
−−
−−
−
−−
0
0
0
0
: 1
1
1
( )
( ) ( ) midiagVUB immxmmxm
mxm
nxn BBBB
H
B
xmmn
B
Bnxm ,,10,,
0 111
1
1  =≠=Σ







Σ
=
−
λλλS.V.D. of B:
( )
( ) ( )
( )
( )
0det
000
2
211
1
1
≠⇒








=








Σ
=








Σ
=
−−−
mxm
mxmmxmmxm
mxm
mxm
B
BV
VBT
xmmn
H
BB
xmmnH
B
B
xmmn
nxmnxn
Perform the following Transformation of Variables: xTx nxnr =: 46

SOLO
Regular Form of a LTI (continue - 2)
nxmnxnmn
RBRARuRxuBxAx ∈∈∈∈+=
Perform the following Transformation of Variables: xTx nxnr =:
We obtain:
( ) ( ) ( ) ( )
( )
( ) ( )
0det
0
21
22
1
2221
1211
2
1
1
1
1
1
≠








+
















=








=
−−
−
−−−−
mxm
mxmmx
xmn
mxmmnmx
xmmnmnxmn
mx
xmn
Bu
Bx
x
AA
AA
x
x
x mx
xmmn
r
r
r
r
r



This is called the Regular Form of the LTI
47

SOLO
Sliding Surface of a Regular Form of a LTI
Consider the following Regular Form of a LTI:
( ) ( ) ( ) ( )
( )
( ) ( )
1
22
1
2221
1211
2
1 0
1
1
1
1
mx
xmmn
r
r
r
r
r u
Bx
x
AA
AA
x
x
x
mxmmx
xmn
mxmmnmx
xmmnmnxmn
mx
xmn








+
















=








=
−−
−
−−−−



Define a Switching Function s (t)
( ) ( )
[ ] ( )
[ ] 0det0 22211
2
1
211
1
1
≠=+=








=
−
− mxm
mx
xmn
mxmmnmx
SxSxS
x
x
SSts rr
r
r
S
mx
  

Therefore, on the Sliding Surface: ( ) ( ) ( )txMtxSStx rr
M
r 111
1
22 −=−=
−

( ) 112112121111 rrrr xMAAxAxAx −=+=Then we have:
By analogy to the “Classical” State-Feedback Theory, it can be seen that this is
the same problem of finding the State-Feedback matrix M for the Regulator Form
where xr2(t) plays the role of the “control” signal.
48

SOLO
Sliding Surface of a Regular Form of a LTI (continue - 1)
11
1
212112121111 r
M
rrr xSSAAxAxAx








−=+=
−

Then we have:
The Stability and Performance of the System depends on the Controllability of the
Regular Form Pair (A11, A12). It can be shown that (A11, A12) is Controllable, if
and only if the pair (A , B) is Controllable.
Controllability of the Regular Form is
[ ]
( )














+
+











 −
−


2
22221221
22121211
2
22
12
2
1
0
B
AAAA
AAAA
B
A
A
B
I
nrankfullBABAB
m
xmmn
r
n
rrrr
49

SOLO
The design of the Sliding Mode Controller must achieve:
• The design of the Matrix S=[S1,S2] to obtain the required performance and Stable
Dynamics for the Closed-Loop Sliding Mode System.
• The design of the Control Law to ensure that the Sliding Surface is Reached and
Maintained.
For the Sliding Surface Reachability Condition let define the Lyapunov Function:
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )



===
≠=≠
=
00
00
2
1
:
tststsif
tststsif
tststV
T
T
T
The Reachability Condition is:
( ) ( ) ( ) ( ) 0& >∀−≤= µµ someforttststs
td
tVd T 
50

SOLO
• The design of the Matrix S = [S1,S2] to obtain the required performance and Stable
Maintained.
( ) ( ) 0== tsts 
( )[ ] ( ) nxmnxnmn
RBRARxtDuRxxtDBuBxAx ∈∈∈∈++= ,,, ξξ
( )xtD ,ξ are the uncertainties in the input
On the Sliding Surface
( ) ( ) ( )[ ]{ } 0, =++== xtDBuBxAStxSts eq ξ
The Equivalent Control that Maintains the System on the Sliding Surface is
( ) ( ) ( )[ ]xtDBSxASBStueq ,
1
ξ+−=
−
51

SOLO
• The design of the Matrix S = [S1,S2] to obtain the required performance and Stable
Maintained.
( )[ ] ( ) nxmnxnmn
RBRARxtDuRxxtDBuBxAx ∈∈∈∈++= ,,, ξξ
( ) ( ) ( )[ ] ( ) 0,
1
=⇐+−=
−
tsxtDBSxASBStueq
ξ
( )[ ] ( )[ ] ( )[ ]xtDBSBSBIxASBSBIx n
P
n
S
,
0
11
ξ
    
 −−
−+−=
xAPx S=
52

SOLO
Unit Vector Approach for a Controller of a Regular Form of a LTI
Consider the System:
where is an unknown but bounded (matched) uncertainty that satisfies:
( )[ ] mn
m
nxmnxnmn
m RRRfRBRARuRxuxtfuBxAx xx,, 1
∈∈∈∈∈++=
( )uxtfm ,,
( ) ( ) ( )xttukuxtfm ,,, α+≤
By using the Orthogonal Transformation T we obtain:
( ) ( ) ( ) ( )
( )
( ) ( ) ( )
( )uxtf
B
u
Bx
x
AA
AA
x
x
x rm
xmmn
mx
xmmn
r
r
r
r
r
mxmmxmmx
xmn
mxmmnmx
xmmnmnxmn
mx
xmn
,,
00
2
1
22
1
2221
1211
2
1
1
1
1
1








+








+
















=








=
−−−
−
−−−−



The Switching Function can be written:
( ) [ ] 0det 2
2
1
1
1
22
2
1
21 ≠













=





=
−
S
x
x
ISSS
x
x
SSts
r
r
m
Mr
r



53

SOLO
Unit Vector Approach for a Controller of a Regular Form of a LTI (continue – 1)
Define the Sub-System:
Let Differentiate this:
( )
( )uxtf
BS
u
BSs
x
SASSASMASASMASAS
SAMAA
uxtf
B
u
Bs
x
SSS
I
AA
AA
SS
I
s
x
rm
r
rm
rr
,,
00
,,
0000
2222
1
1
2222
1
2121222212121111
1
2121211
22
1
1
21
1
22221
1211
21
1






+





+













+−+−
−
=














+





+











−












=





−−
−
−−


or:
( )
( )
( )
( )
( )
( )
( )
( ) 











−
=

















=





−−
ts
tx
SSS
I
tx
tx
tx
tx
SS
I
ts
tx r
r
r
r
r
T
r
S
1
1
21
1
22
1
2
1
21
1
0
&
0

( )uxtf
B
u
BsS
x
AAMMAMMAAMA
AMAA
sS
x
rm
rr
,,
00
22
1
2
1
221212221121
121211
1
2
1






+





+











+−−+
−
=








−− 

54

SOLO
The Sub-System:
In order to force s to zero, Φ must satisfy a Lyapunov Equation of the type:
( )uxtf
B
u
BsS
x
AA
AA
sS
x
rm
rr
,,
00
22
1
2
1
2221
1211
1
2
1






+





+













=








−− 

Choose: NonlinearLinear uuu +=
( ) suBSsSASxASs Linearr Φ=++=
−
22
1
22221212

( ) ( )[ ]sSASxASBSu rLinear Φ−+−=
−− 1
22221212
1
22
m
T
IPP −=Φ+Φ 22
( ) ( )
sP
sP
BSxtu rNonlinear
2
21
22,
−
−= ρChoose:
and:
The Linear Part must keep the System on the Sliding Manifold ( )0== ss
The Nonlinear Part must force the System to Reach the Sliding Manifold.
55

SOLO
The Sub-System:
( )uxtf
B
u
BsS
x
AA
AA
sS
x
rm
rr
,,
00
22
1
2
1
2221
1211
1
2
1






+





+













=








−− 

Choose: NonlinearLinear uuu +=
( ) ( )[ ]sSASxASBSu rLinear Φ−+−=
−− 1
22221212
1
22
( ) ( )
sP
sP
BSxtu rNonlinear
2
21
22,
−
−= ρ
The Linear Part must keep the System on the Sliding Manifold ( )0== ss
The Nonlinear Part must force the System to Reach the Sliding Manifold.
( ) ( ) ( )[ ] ( ) ( ) ( )uxtfB
sP
sP
BSxtBsSASxASBSBsSAxAsS rmrrr ,,, 2
2
21
222
1
22221212
1
222
1
222121
1
2 +−Φ−+−+=
−−−−−
ρ
( ) ( )uxtfBS
sP
sP
xtss rmr ,,, 22
2
2
+−Φ= ρ 56

SOLO
For the Sub-System:
002 ≠>= sifsPsV T
Choose a potential Lyapunov Function that shows the Reachability of the
Sliding Surface:
( ) ( )uxtfBS
sP
sP
xtss rmr ,,, 22
2
2
+−Φ= ρ








+−Φ+







+−Φ=+== m
T
T
m
TT
fBS
sP
sP
sPssPfBS
sP
sP
ssPssPsV
td
Vd
22
2
2
2222
2
2
22 ρρ
( ) ( ) m
TT
I
TT
fBSPssPPs
sP
sPPs 22222
2
22 2
1
2 +−Φ+Φ=
−
ρ

m
TT
fBSPssPss 2222 22 +−−= ρ
57

SOLO
For the Sub-System:
002 ≠>= sifsPsV T
Choose a potential Lyapunov Function that shows the Reachability of the
Sliding Surface:
( ) ( )uxtfBS
sP
sP
xtss rmr ,,, 22
2
2
+−Φ= ρ
m
TT
fBSPssPssV 2222 22 +−−= ρ
mm
T
fBSsPfBSPs 222222 ≤Using Cauchy-Schwarz Inequality:
( )mfBSsPsV 2222 −−−≤ ρ
We want to choose ρ (t,x) such that and( )( )xtukBSfBS m ,2222 αρ +≥>
( ) 02 222 <−−−≤ mfBSsPsV ρ 58

SOLO
( )mfBSsPsV 2222 −−−≤ ρ
We want to choose ρ (t,x) such that and( )( )xtukBSfBS m ,2222 αρ +≥>
( ) 02 222 <−−−≤ mfBSsPsV ρ
002 ≠>= sifsPsV T
( ) 1
22
1
22
−−
+≤+=+≤+= BSuBSuuuuuu LinearLinearNonlinearLinearNonlinearLinear ρρ
( )( )
k
xtukBS Linear
−
+
≥
1
,22 α
ρ
( ) ( ) ( ) 1
22
1
2222
1
2222
1
22 1
−−−−
≥⇒≤=⇒= BSBSBSBSIBSBSI
automatically fulfilled for k > 1
Define η (t,x) > 0 such that
1
22
−
+≥+ BSuu Linear ρη
( ) ( ) ( )( ) ( )( )xtBSkukBSxtukBSkfBS Linearm ,,
1
22222222 αραηηρ ++≥++≥+>
−
59

SOLO
where u1 and u2 are the known and unknown inputs, respectively.
System Description and Notation
212211
21212211
21
2121
mmppCrankmBrankmBrank
RCRyxCy
RBRBRARuRuRxuBuBxAx
pxn
pxnp
nxmnxmnxnmmn
nxmnxm
+≥===
∈∈=
∈∈∈∈∈∈++=
Assumptions:
• There exists a known nonnegative scalar function such that( )yt,ρ
( ) tytu ∀≤ ,2 ρ
• The pairs (A,B1), (A,B2) are controllable and (A,C) is observable with the matrices
B1, B2 and C being of full rank
• p ≥ m1+m2, that means that bthe number of output channels is greater or equal
then the number of inputs, and rank (CB1) =m1, rank (CB2) – m2
60
Output Feedback Variable Structure Controllers and State Estimators for Uncertain
Dynamic Systems

SOLO
where u1 and u2 are the known and unknown
inputs, respectively.
System Sliding Surface
xCy
uBuBxAx
=
++= 2211

Assume that the Sliding Surface is of the type: ( ) xSx 1=σ
On the Sliding Surface we must have: ( ) ( ) 0&0 == xx σσ 
( ) ( ) 0221111 =++== uBuBxASxSx σ
If (S B1) is nonsingular:
( ) ( )221
1
111 uBxASBSu eq
+−=
−
On the Sliding Surface the dynamics of the System is given by:
( )[ ]
01
1
1
111
=
−=
−
xS
xASBSBIx
Note that in the Sliding Mode the System is governed by a reduced order of
differential equations with the eigenvalues of , and are
not affected by the unknown inputs/disturbances.
( )[ ]ASBSBI 1
1
111
−
−
61
Output Feedback Variable Structure Controllers and State Estimators for Uncertain
Dynamic Systems

SOLO
The Transmission Zeros of the System are defined as the solutions for λ of:
Sliding Modes and System Zeros
( ) 1
1
1
1
1
0
0
0
det mn
S
BAI
rankwhichforor
S
BAI
z
nn
+<




 −−
=




 −−
=
λ
λ
λ
λ
The solutions are not affected if we multiply the Square Matrix by Non-
Singular Square Matrices that are not functions of λ:
( )





 −−−
=

























 −−





 −−−
00
00
00
0
1
1
1
1
1
1
1
1
111
TSM
NBTTFBAIT
rank
N
T
IF
I
S
BAI
M
T
rank
xmm
nxn
m
nn
mxm
nxn λλ
( )





 −−−
=



















 −−





 −





 −−−
00
0
000
0
1
1
1
1
1
1
1
1
111
TSM
NBTTSHAIT
rank
N
T
S
BAI
I
HI
M
T
rank
xmm
nxnn
m
n
mxm
nxn λλ
xnmm
nxmnxnmn
RSRxS
RBRARuRxuBxAx
11
11
11
1111
∈∈=
∈∈∈∈+=
σσ

62

SOLO
Sliding Modes and System Zeros (continue – 1)
( ) 0
0
det
1
1
=




 −−
=
S
BAI
z
nλ
λ
The System Zeros are not affected under the following set of transformations:
• Nonsingular State Transformation ( )xTx =~
• State Feedback ( A+B F)
• Output Injection (A + H S)
• Nonsingular Input Control Transformations ( )uNu =~
• Nonsingular Output Signal Transformations ( )σσ M=~
xnmm
nxmnxnmn
RSRxS
RBRARuRxuBxAx
11
11
11
1111
∈∈=
∈∈∈∈+=
σσ

63

SOLO
( )
( )
( )
( ) ( )[ ]1
1
1
1
1
1
1
1
11
1
detdet
0
0
det
0
det
BAISAI
BAIS
BAII
IS
AI
S
BAI
z
nn
n
nn
m
nn
−
−
−
−−=
















−
−−





 −
=




 −−
=
λλ
λ
λλλ
λ
Hence:
( )[ ] ( )
( )AI
z
BAIS
n
n
−
=−
−
λ
λ
λ
det
det 1
1
1
xnmm
nxmnxnmn
RSRxS
RBRARuRxuBxAx
11
11
11
1111
∈∈=
∈∈∈∈+=
σσ

64

SOLO
xnmm
nxmnxnmn
RSRxS
RBRARuRxuBxAx
11
11
11
1111
∈∈=
∈∈∈∈+=
σσ

( )
( )
( )( )
( )( )[ ] ( )( )[ ]{ }1
11
1111
1
111
1
1
1
11
1
1
111
11
detdet
0
det
0
0
det
0
det
BASBSBIISASBSBII
S
BASBSBII
IASBS
I
S
BAI
S
BAI
z
nn
n
m
nnn
−−−
−
−
−−−−=





 −−−
=
















−





 −−
=




 −−
=
λλ
λ
λλ
λ
65

SOLO
using:
Let compute:
( ) ( )( )[ ] ( )( )[ ]{ }
( )( )[ ] ( ) ( )( )[ ] ( )11
1
11111
11
111
1
11
1111
1
111
detdetdetdet
detdet
1
BSASBSBIIBSASBSBII
BASBSBIISASBSBIIz
m
nn
nn
−−−−
−−−
−−=−−=
−−−−=
λλλλ
λλλ
( )( )[ ] 1
11
1111 BASBSBIIS n
−−
−−λ
( ) ( ) 111111 −
××
−
×
−
××××
−
×
−
×
−
××× +−=+ mmmnnmmmmnnnnmmmmmmnnmmm CBDCBIDCCBDC
( )( )[ ] ( )( )[ ]
( )( ) 0since
1
111111
1
1
1
111
11
11
11
1111
=−=
−+=−−
−−
−−−−−
SBSBISBS
BSBSBIIISBASBSBIIS nnn
λ
λλλ 
Therefore:
( )( )[ ] ( ) ( )[ ] 1
11
1
111 detdet 1 −−
=−− BSzASBSBII m
n λλλ
We found that the Poles of the System on the Sliding Surface S1x = 0 are defined
by the Zeros of the triple (A, B1, S1).
66

SOLO
Properties of the Sliding Modes
The following are a Summary of the Properties of the Sliding Modes:
• The System behaves as a Reduced Order motion which (apparently) does not
depend on the control signal u (t).
• There are (n-m) States that determines the dynamics of the Closed Loop System.
• The Closed-Loop Sliding Motion depends only on the choise of the Sliding Surface.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) 1111 112112121111 xmnxmmnmnxmnmxxmmnxmnmnxmnxmn rmnmxrrr xMAAxAxAx −−−−−−−−− −−=+=
• The Poles of the Sliding Motion are given by the Invariant Zeros of the System
Triple (A, B, S)
67

SOLO
Design of a Sliding Surface (Hyperplane)
The following “Classical” Methods can be used to obtain the Matrix 1
1
2 SSM
−
=
• Quadratic Minimization
• Robust Eigen-structure Assignment
• Direct Eigen-structure Assignment
68

SOLO
Design of a Sliding Surface (Hyperplane) by Quadratic Minimization
( ) ( ) ( ) ( )
( )
( ) ( )
1
22
1
2221
1211
2
1 0
1
1
1
1
mx
xmmn
r
r
r
r
r u
Bx
x
AA
AA
x
x
x
mxmmx
xmn
mxmmnmx
xmmnmnxmn
mx
xmn








+
















=








=
−−
−
−−−−



And the following Optimization (Minimization) Problem:
( ) ( )∫∫
∞∞
++==
SS t
r
T
rr
T
rr
T
r
t
r
T
r tdxQxxQxxQxtdxQxJ 222221211111 2
2
1
2
1
Let rewrite:
( ) ( )
( ) ( ) 112
1
22121112
1
22222112
1
222
112
1
22121112
1
2222112
1
222222112
1
222
112
1
2222
1
22121112
1
2222
1
22121112
1
22222222
1
221212222
222211222121
r
TT
rr
T
r
T
r
T
r
r
TT
rr
TT
r
T
rr
T
r
T
r
r
T
I
T
rr
T
I
T
rr
T
I
T
rr
I
T
rr
T
r
r
T
rr
T
rr
TT
r
xQQQxxQQxQxQQx
xQQQxxQQQxQQxxQxQQx
xQQQQQxxQQQQQxxQQQxxQQQxxQx
xQxxQxxQx
−−−
−−−−
−−−−−−
−++=
−+++=
−+++=
++

69

SOLO
Design of a Sliding Surface (Hyperplane) by Quadratic Minimization (continue -1)
( ) ( ) ( ) ( )
( )
( ) ( )
1
22
1
2221
1211
2
1 0
1
1
1
1
mx
xmmn
r
r
r
r
r u
Bx
x
AA
AA
x
x
x
mxmmx
xmn
mxmmnmx
xmmnmnxmn
mx
xmn








+
















=








=
−−
−
−−−−



And the following Optimization (Minimization) Problem:
( ) ( ) ( )[ ]∫
∞
−−−
+++−=
St
r
T
r
T
r
T
rr
TT
r tdxQQxQxQQxxQQQQxJ 112
1
22222112
1
222112
1
2212111
2
1
Define: 112
1
22212
1
221211 :&:ˆ
r
T
r
T
xQQxvQQQQQ
−−
+=−=
Therefore: ( )∫
∞
+=
St
T
r
T
r tdvQvxQxJ 221111
ˆ
2
1
( ) vAxQQAAxAxAx r
T
rrr 12112
1
2212112121111 +−=+=
−
70

SOLO
Design of a Sliding Surface (Hyperplane) by Quadratic Minimization (continue - 2)
We ended up with the following Optimization Problem:
The “Optimal Control” is:
112
1
22212
1
2212111112
1
221211 :&:ˆ,:ˆ
r
T
r
TT
xQQxvQQAAAQQQQQ
−−−
+=−=−=
where P1 is given by the Riccati Equation:
( )∫
∞
+=
St
T
r
T
r tdvQvxQxJ 221111
ˆ
2
1
vAxAx rr 121111
ˆ +=
( ) 1112
1
22* rxPAQv
−
−=
0ˆˆˆ
112
1
2212111 =+−+
−
QPAQAPAPPA
TT
We obtained: ( ) 1112112
1
22112
1
222 * rr
TT
r
T
r xMxQPAQxQQvx =+−=−=
−−
Therefore: ( )TT
QPAQSSM 12112
1
221
1
2 +−==
−−
where S2 is arbitrary.
71

SOLO
Consider a Sliding Manifold σ (x,t). We want to design the Control ueq that keeps
the trajectory on the manifold:
( ) ( ) ( )tutftx
td
d
eq+= ,, σσ
72
where f (σ,t) is a known or unknown but bounded function.

SOLO
( ) ( ) ( )tutxftx
td
d
eq+= ,,σ
73
When f (σ,t) is a known function a solution for ueq is:
( ) ( ) ( ) 0,
2/1
>−−= ρσσρ signtxftueq
Continuous Sliding Mode Control
We obtain: ( ) ( ) 0,
2/1
=+ σσρσ signtx
td
d
Let choose the following Lyapunov Function: ( )



==
≠>
=
00
00
2
1 2
σ
σ
σσV
( ) 0
2/32/1
≤−=⋅−=⋅= σρσσρσ
σ
σ sign
td
d
td
Vd
( ) ρσρσσρ
α
αα
<≤+−=+−=+
<=<
kforkkVk
td
Vd
0
2/3
1
4
3
0
22/3
Therefore σ→0 in a Finite Time
When f (σ,t) is a unknown we can use an Sliding Mode Observer to estimate it.
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20

SOLO
( ) ( ) ( )tutxftx
td
d
eq+= ,,σ
74
Continuous Sliding Mode Control (continue – 1)
Lyapunov Function: ( )



==
≠>
=
00
00
2
1 2
σ
σ
σσV
>> [X,Y]=meshgrid(-0.5:0.05:0.5);
>> Z=0.5*Y.^2+0.5*X.^2+eps;
>> contour(X,Y,Z)
MATLAB:
-0.5
0
0.5
-0.5
0
0.5
0
0.1
0.2
0.3
0.4
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5

SOLO
( ) ( ) ( )tutxftx
td
d
eq+= ,,σ
75
When f (σ,t) is a known function a Second Order SM Control for ueq is:
( ) ( ) ( ) ( ) 0,, 100
2/1
1
0
>−−−= ∫ ρρσρσσρ
t
t
eq dtsignsigntxftu
Second Order Sliding Mode Control
We obtain: ( ) ( ) ( ) 0,
0
0
2/1
1 =++ ∫
t
t
dtsignsigntx
td
d
σρσσρσ
Let choose the following Lyapunov Function:
( ) ( )



===
≠≠>
=+=+= ∫ 0,00
0,00
22
, 0
2
0
0
2
z
zz
dsign
z
zV
σ
σ
σρσσρσ
σ
Rewrite:
( )
( )



=
−−=
σρ
σσρσ
signz
zsign
0
2/1
1



SOLO
( ) ( ) ( )tutxftx
td
d
eq+= ,,σ
76
Second Order Sliding Mode Control (continue -1)
( ) ( )



===
≠≠>
=+=+= ∫ 0,00
0,00
22
, 0
2
0
0
2
z
zz
dsign
z
zV
σ
σ
σρσσρσ
σ
Let check:
( ) ( ) ( ) ( ) ( )[ ] 2/1
10
2/1
1000, σρρσσρσρσρσσρσ
σ
−=−−⋅+⋅=⋅+⋅=
  


zsignsignsignzsignzzzV
td
d
z
For: ( ) 0
2
,
0
0
2
==≤+= zat
C
C
z
zV Max
ρ
σσρσ
( ) kforkCk
Cz
kVk
td
Vd
>≤−−=+−≤





++−=+
<=<
1
2/1
01
2/1
0
12/10
2/1
0
100
2
2/1
10 0
2
ρρρρ
ρ
ρρσρσρρ
α
α
α
α
Therefore σ→0 in a Finite Time
When f (σ,t) is a unknown we can use an Sliding Mode Observer to estimate it.

SOLO
( ) ( ) ( )tutxftx
td
d
eq+= ,,σ
77
Second Order Sliding Mode Control (continue -2)
Lyapunov Function:
>> [X,Y]=meshgrid(-0.5:0.05:0.5);
>> rho0=2;
>> Z=0.5*Y.^2+rho0*abs(X).^1+eps;
>> contour(X,Y,Z)
MATLAB:
( )



===
≠≠>
=+=
0,00
0,00
2
, 0
2
z
zz
zV
σ
σ
σρσ
-0.5
0
0.5
-0.5
0
0.5
0
0.5
1
1.5
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5

SOLO
Sliding Mode Observers
In most of the Linear and Nonlinear Unknown Input Observers proposed so far,
the necessary and sufficient conditions for the construction of Observers is that the
Invariant Zeros of the System must lie in the open Left Half Complex Plane, and the
transfer Function Matrix between Unknown Inputs and Measurable Outputs satisfies
the Observer Matching Condition.
Observers are Dynamical Systems that are used to Estimate the State of a Plant using
its Input-Output Measurements; they were first proposed by Luenberger.
David G. Luenberger
Professor
Management Science
and Engineering
Stanford University
In some cases, the inputs of the System are unknown or partially
unknown, which led to the development of the so-called Unknown
Input Observers (UIO), first for Linear Systems. Motivated by the
development of Sliding-Mode Controllers, Sliding Mode UIOs
have been developed.
The main advantage of using Sliding-Mode Observer over their
Linear counterparts is that while in Sliding, they are Insensitive to
the Unknown Inputs and, moreover, they can be used to
Reconstruct Unknown Inputs which can be a combination of
System Disturbances, Faults or Nonlinearities.
78

79
Guidance of Intercept
Sliding Mode Observers of Target Acceleration
Kinematics: ( )
→→
⋅−⋅+Λ−=Λ tataRR
td
d
MT 11

We want to Observe (Estimate) the Unknown Target Acceleration Component :
→
⋅ taT 1

Define: 0:1_ vtaestAt
Est
T =





⋅=
→
( ) mEstEst AvRz
td
d
−+Λ−= 00

The Differential Equation of the Observer will be a copy of the kinematics:
mM Ata =





⋅
→
:1

Define the Observer Error: EstEstO Rz Λ−= 
0:σ
Define the Sliding Mode Observers that must drive σO→0:
( )
( )
( )
22
11
1232
201
2/1
0121
1
3/2
10
1.1
5.1
2
vz
vz
vzsigntv
zvzsignvztv
zsigntv OO
=
=
−⋅⋅=
+−−⋅⋅=
+⋅⋅⋅=


σσ
Missile Command Acceleration
( ) ( )
( )
estAtdRsignRRsignRRNa
EstEstRSM
EstEstEstEstEstEstEstEstEstEstC _'
2
3/1
2
2/1
1 +ΛΛ+ΛΛ+Λ−=
Λ
∫   


µαα
t1, t2, t3 are Design Parameters
Observer 4: Variation of 1
( )
( )
( )
22
11
122
201
2/1
01
2/1
1
1
3/23/1
0
1.1
5.1
2
vz
vz
vzsignLv
zvzsignvzLv
zsignLv OO
=
=
−⋅⋅=
+−−⋅⋅=
+⋅⋅⋅=


σσ
Observer 1:
( )
( )
02
11
3/1
21
1
2/1
10
=
=
⋅⋅−=
+⋅⋅−=
z
vz
signv
zsignv
OOO
OOO


σσα
σσα ( )
( )
02
11
21
1
2/1
10
=
=
⋅−=
+⋅⋅−=
z
vz
signv
zsignv
OO
OOO


σρ
σσρ
L is a Design Parameter are Design Parameters21, OO αα
are Design Parameters21, OO ρρ
Observer 2: Observer 3:

80
Sliding Mode Observer of Target Acceleration: MATLAB Listing
% Nonlinear Sliding Mode Target Acceleration Observers
At_est=0;
v0=0;
z0=x1;
z1=0;
z2=0;
Observer=1;
%First Observer Parameter
L=10;
%Second Observer Parameters
alphaO1=30;
alphaO2=1;
%Third Observer Parameters
rho1=20;
rho2=3;
%Fourth Observer Parameters
t1=10;
t2=3;
t3=1;
%Second Order Sliding Mode
SigmaSM=Range_est*Lamdadot_est;
y2 = alpha1*sign(SigmaSM)*abs(SigmaSM)^0.5+x2;
x2_dot =alpha2*sign(SigmaSM)*abs(SigmaSM)^(1/3);
%Nonlinear Sliding Mode Target Acceleration Observers
z0_dot=v0-Rdot_est*Lamdadot_est-Am;
SigmaO=z0-SigmaSM;
if(Observer==1)
v0=-2*L^(1/3)*abs(SigmaO)^(2/3)*sign(SigmaO)+z1;
v1=-1.5*L^(1/2)*abs(z1-v0)^(1/2)*sign(z1-v0)+z2;
v2=1.1*L*sign(z2-v1);
z1_dot=v1;
z2_dot=v2;
v0_dot=0;
At_est=v0;
end
if(Observer==2)
v0=-alphaO1*abs(SigmaO)^(1/2)*sign(SigmaO)+z1;
v1=-alphaO2*abs(SigmaO)^(1/3)*sign(SigmaO);
z1_dot=v1;
z2_dot=0;
v0_dot=0;
At_est=v0;
end
if (Observer==3)
v0=-rho1*abs(SigmaO)^(1/2)*sign(SigmaO)+z1;
v1=-rho2*sign(SigmaO);
z1_dot=v1;
z2_dot=0;
v0_dot=0;
At_est=v0;
end
if(Observer==4)
v0=-2*t1*abs(SigmaO)^(2/3)*sign(SigmaO)+z1;
v1=-1.5*t2*abs(z1-v0)^0.5*sign(z1-v0)+z2;
v2=1.1*t3*sign(z2-v1);
z1_dot=v1;
z2_dot=v2;
v0_dot=0;
At_est=v0;
end
%Missile Acceleration Command and Autopilot
Ac=-N*Rdot_est*Lamdadot_est+y2+At_est;
N = 3;
alpha1 =10;
alpha2 = 1;
%Nonlinear Sliding Mode Target Acceleration
% Observer State Integration
z0=z0+z0_dot* delta_time;
v0=v0+v0_dot* delta_time;
( )
( )
( )
22
11
1232
201
2/1
0121
1
3/2
10
1.1
5.1
2
vz
vz
vzsigntv
zvzsignvztv
zsigntv OO
=
=
−⋅⋅=
+−−⋅⋅=
+⋅⋅⋅=


σσ
( )
( )
( )
22
11
122
201
2/1
01
2/1
1
1
3/23/1
0
1.1
5.1
2
vz
vz
vzsignLv
zvzsignvzLv
zsignLv OO
=
=
−⋅⋅=
+−−⋅⋅=
+⋅⋅⋅=


σσ
( )
( )
02
11
3/1
21
1
2/1
10
=
=
⋅⋅−=
+⋅⋅−=
z
vz
signv
zsignv
OOO
OOO


σσα
σσα
( )
( )
02
11
21
1
2/1
10
=
=
⋅−=
+⋅⋅−=
z
vz
signv
zsignv
OO
OOO


σρ
σσρ

81
( )
( )
( )
22
11
1232
201
2/1
0121
1
3/2
10
1.1
5.1
2
vz
vz
vzsigntv
zvzsignvztv
zsigntv OO
=
=
−⋅⋅=
+−−⋅⋅=
+⋅⋅⋅=


σσ
( )
( )
( )
22
11
122
201
2/1
01
2/1
1
1
3/23/1
0
1.1
5.1
2
vz
vz
vzsignLv
zvzsignvzLv
zsignLv OO
=
=
−⋅⋅=
+−−⋅⋅=
+⋅⋅⋅=


σσ
Observer 1:
( )
( )
02
11
3/1
21
1
2/1
10
=
=
⋅⋅−=
+⋅⋅−=
z
vz
signv
zsignv
OOO
OOO


σσα
σσα ( )
( )
02
11
21
1
2/1
10
=
=
⋅−=
+⋅⋅−=
z
vz
signv
zsignv
OO
OOO


σρ
σσρ
Scenario: R0=10000 m, Rdot=-1000 m/s, alpha1=10, alpha2=1, N=3, Ldot0=0.05 rad/s
A step pulse Target acceleration At=100 m/s2
starting at t=3 s and finishing at t=7s
With Not Noise
10=L 1,30 21 == OO αα 3,20 21 == OO ρρ 1,3,10 321 === ttt
0 1 2 3 4 5 6 7 8 9 10
-200
0
200
Atest
0 1 2 3 4 5 6 7 8 9 10
-1000
0
1000
z0
0 1 2 3 4 5 6 7 8 9 10
-100
0
100
z1
0 1 2 3 4 5 6 7 8 9 10
-10
0
10
SigmaO
0 1 2 3 4 5 6 7 8 9 10
-200
0
200
Atest
0 1 2 3 4 5 6 7 8 9 10
-1000
0
1000
z0
0 1 2 3 4 5 6 7 8 9 10
-20
0
20
z1
0 1 2 3 4 5 6 7 8 9 10
-50
0
50
SigmaO
0 1 2 3 4 5 6 7 8 9 10
-200
0
200
Atest
0 1 2 3 4 5 6 7 8 9 10
-1000
0
1000
z0
0 1 2 3 4 5 6 7 8 9 10
-10
0
10
z1
0 1 2 3 4 5 6 7 8 9 10
-20
0
20
SigmaO
0 1 2 3 4 5 6 7 8 9 10
-100
0
100
Atest
0 1 2 3 4 5 6 7 8 9 10
-1000
0
1000
z0
0 1 2 3 4 5 6 7 8 9 10
-200
0
200
z1
0 1 2 3 4 5 6 7 8 9 10
-200
0
200
SigmaO
Sliding Mode Observer of Target acceleration - MATLAB Results

82
( )
( )
( )
22
11
1232
201
2/1
0121
1
3/2
10
1.1
5.1
2
vz
vz
vzsigntv
zvzsignvztv
zsigntv OO
=
=
−⋅⋅=
+−−⋅⋅=
+⋅⋅⋅=


σσ
( )
( )
( )
22
11
122
201
2/1
01
2/1
1
1
3/23/1
0
1.1
5.1
2
vz
vz
vzsignLv
zvzsignvzLv
zsignLv OO
=
=
−⋅⋅=
+−−⋅⋅=
+⋅⋅⋅=


σσ
Observer 1:
( )
( )
02
11
3/1
21
1
2/1
10
=
=
⋅⋅−=
+⋅⋅−=
z
vz
signv
zsignv
OOO
OOO


σσα
σσα ( )
( )
02
11
21
1
2/1
10
=
=
⋅−=
+⋅⋅−=
z
vz
signv
zsignv
OO
OOO


σρ
σσρ
0 1 2 3 4 5 6 7 8 9 10
-200
0
200
Atest
0 1 2 3 4 5 6 7 8 9 10
-1000
0
1000
z0
0 1 2 3 4 5 6 7 8 9 10
-100
0
100
z1
0 1 2 3 4 5 6 7 8 9 10
-100
0
100
SigmaO
0 1 2 3 4 5 6 7 8 9 10
-200
0
200
Atest
0 1 2 3 4 5 6 7 8 9 10
-1000
0
1000
z0
0 1 2 3 4 5 6 7 8 9 10
-10
0
10
z1
0 1 2 3 4 5 6 7 8 9 10
-20
0
20
SigmaO
0 1 2 3 4 5 6 7 8 9 10
-200
0
200
Atest
0 1 2 3 4 5 6 7 8 9 10
-1000
0
1000
z0
0 1 2 3 4 5 6 7 8 9 10
-20
0
20
z1
0 1 2 3 4 5 6 7 8 9 10
-50
0
50
SigmaO
0 1 2 3 4 5 6 7 8 9 10
-200
0
200
Atest
0 1 2 3 4 5 6 7 8 9 10
-1000
0
1000
z0
0 1 2 3 4 5 6 7 8 9 10
-100
0
100
z1
0 1 2 3 4 5 6 7 8 9 10
-20
0
20
SigmaO
starting at t=3 s and finishing at t=7s
With Lamda_dot Noise Filtered with Time Constant of 200msec
10=L 1,30 21 == OO αα 3,20 21 == OO ρρ 1,3,10 321 === ttt

83
No Target acceleration , No Measurement Noises
Observer Output
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-5
0
5
Atest
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-100
0
100
z0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-1
0
1
z1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.1
0
0.1
SigmaO
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-50
0
50
100
X1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-400
-200
0
200
X1
d
ot
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.5
1
X2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-500
0
500
Am
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.1
0
0.1
Lamda
d
ot
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.5
0
0.5
Lamda
d
ot2

84
starting at t=0.3 s and finishing at t=0.6s
Without Noise
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-50
0
50
100
X1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-400
-200
0
200
X1
d
ot
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.5
1
X2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-500
0
500
Am
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.1
0
0.1
Lamdad
ot
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-5
0
5
Lamdad
ot2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-100
0
100
Atest
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-100
0
100
z0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-20
0
20
z1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-10
0
10
SigmaO
Observer Output

85
Return to Table of Content
starting at t=0.3 s and finishing at t=0.6s
With Lamda_dot Noise Filtered with Time Constant of 20msec
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-50
0
50
100
X1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-400
-200
0
200
X1
d
ot
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.5
1
1.5
X2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-500
0
500
Am
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.1
0
0.1
Lamda
d
ot
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-5
0
5
Lamda
d
ot2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-200
0
200
Atest
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-100
0
100
z0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-20
0
20
z1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-10
0
10
SigmaO
Observer
Output 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.1
0
0.1
Lamda
d
ot
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-5
0
5
Lamdad
ot2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.05
0
0.05
Noise
L
amdadot
Lamda_dot
Noise

SOLO
86
Slide Mode Control Examples
Fu, L-C, et all, Control System of a Kill Vehicle
Fu, L-C, et al, Solution for Attitude Control of KV
Fu, L-C, et al, Zero SM Guidance of a KV
Crassidis , et al -Attitude Control of the Kill Vehicle
Midcourse Intercept of a Ballistic in a Head On Scenario
Above a Minimal Altitude
HTK Guidance Using 2nd
Order Sliding Mode

87
Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude
SOLO
Problem: Develop the Midcourse Guidance Law to intercept in a Head On Scenario a
Ballistic Missile, by an Vertical Launched Interceptor at altitude above a given value hmin.
Solution Method:
• Start with a Planar Approximation where the Ballistic Trajectory is a
known Straight Line and the Launch Point define the Plane.
• Assume a Constant Velocity Bounded Maneuver Interceptor and don’t
consider gravitation.
• Define the Straight Line Ballistic Trajectory as the Sliding Curve and
define a Linear Guidance Law and a Lyapunov Volume around the
Sliding Curve such that the Entering Interceptor Trajectory will not
escape and will Converge to the Ballistic Trajectory in a Head On situation.
• Using Optimal Control Theory define the Interceptor Trajectory from the
Launching Point that will enter the Sliding Lyapunov Volume, above the minimal height hmin, in
Minimum Time.
• Define the Interceptor Midcourse Guidance Law as the combination of the Optimal Trajectory that
reaches the Sliding Lyapunov Surface followed by the Linear Guidance Law to reach the Ballistic
Trajectory in Head On.
• Transfer the Planar Guidance Law to a Three Dimensional Law with the same structure.
• Simulate the result using a real 6 DOF Interceptor Model.

88
SOLO
• Start with a Planar Approximation where the Ballistic Trajectory is a
known Straight Line and the Launch Point define the Plane.
In this plane choose a Coordinate System xOy, having the x axis on the
Ballistic Straight Line approximation pointing toward the Ballistic Target,
the origin at the intersection of the straight line with the ground and y axis
pointing above the ground.
• Assume a Constant Velocity Bounded Maneuver Interceptor and don’t
consider gravitation.
Under those conditions the Interceptor equation of motion are given by:
MAXM
M
M
M
M
aa
V
a
td
xd
V
td
yd
V
td
xd
≤=
=
=
γ
γ
sin
cos
where γ is the angle between Interceptor Velocity Vector and the Ballistic
Trajectory. To drive the Interceptor to fly on the Ballistic Trajectory we
want to bring γ, y and dy/dt to zero, therefore let choose a Interceptor
Guidance Law as:
MAXMMMM aaVk
td
yd
kyk
td
d
Va ≤−−−== γ
γ
321

89
SOLO
• Define the Straight Line Ballistic Trajectory as the Sliding Curve and
define a Linear Guidance Law and a Lyapunov Volume around the
Sliding Curve such the Entering Interceptor Trajectory will not escape
and will Converge to the Ballistic Trajectory in a Head On situation.
Define:
( ) ( )γγγγ
γ
cos1sin
0
−==Ψ ∫ MM VdV
Choose a Lyapunov Function candidate:
( ) ( ) ( ) 0cos1
2
1
2
1
:, 1
1
2
2
1
2
>−+=Ψ+= k
k
V
y
k
V
yyV MM
γγγ
We can see that:
( )



±±====
≠≠>
,...2,1,0200
000
,
jjandy
andy
yV
πγ
γ
γ
MAXM
M
M
M
M
aa
V
a
td
xd
V
td
yd
V
td
xd
≤=
=
=
γ
γ
sin
cos
MAXMMMM aaVk
td
yd
kykVa ≤−−−== γγ 321
Guidance Law:

90
SOLO
• Define the Straight Line Ballistic Trajectory as the Sliding Curve and define a Linear Guidance
Law and a Lyapunov Volume around the Sliding Curve such the Entering Interceptor Trajectory
will not escape and will Converge to the Ballistic Trajectory in a Head On situation.
Choose a Lyapunov Function candidate:
( ) ( ) 0
,...2,1,0200
00
cos1
2
1
:, 1
1
2
2
>



±±====
≠>
−+= k
jjandy
y
k
V
yyV M
πγ
γγ
( )
πγγγγ
γγγγγ
γγ
<>≤−−=






−−−−=+=
&0,,0sinsin
sinsinsinsin
,
321
2
1
322
1
2
32
1
1
2
1
2
kkkforV
k
k
V
k
k
kky
V
k
k
V
Vy
td
d
k
V
td
yd
y
td
yVd
MM
M
M
M
M
We can see that the Lyapunov Function confirms the convergence of the system
to y=o, γ =0.
MAXM
MMMM
M
M
aakky
V
k
k
td
yd
V
k
y
V
k
V
a
td
d
V
td
yd
≤−−−=−−−==
=
γγγ
γ
γ
32
1
3
21
sin
sin
Guidance Law

SOLO
The contour of the Lyapunov Function in the y, γ plane is defined by
( ) ( ) const
k
V
yyV M
==−+= 2
1
2
2
2
1
cos1
2
1
:, εγγ
( ) ( ) ( )γγγ −−=−= ,,, yVyVyV








±==
±==
−
MV
k
y
y
2
sin2,0
0
11
maxmin,
maxmin,
ε
γ
εγ
The Maximum Contour Ω where the trajectories
Converge to y=0, γ = 0 is given for
1
maxmin,
1
maxmin,
22
k
V
y
k
V MM
±=±=→=→±= εεπγ
Note: when
the Guidance Law will assure
convergence
( ) ( )
1
2
1
2
2 2
cos1
2
1
:,
k
V
k
V
yyV MM
≤−+= γγ

SOLO
To define the values k1, k2, k3 we use Isoclines method to define the Trajectories behavior:
MAXMM
M
M
aaVkky
V
k
td
d
V
td
yd
≤−−−=
=
γγ
γ
γ
32
1
sin
sin
The Trajectories
in y, γ plane are:
The slope (inclination) of the Trajectories in y, γ plane is given by: N
Vkky
V
k
V
d
yd
M
M
M
=
++
−=
γγ
γ
γ
32
1
sin
sin
The curves in y, γ plane for which the slope N is constant are called Isoclines and are given by:
( ) γγγ 3
1
2
1
sin k
k
V
N
V
k
k
V
y MMM
N −





+= ( )
( )










−=→−=
−=→∞→
±±==→=
−=
∞→
=
γγ
γγγ
πγ
3
12
3
1
2
1
0
2
sin
,2,1,0,0
k
k
V
y
k
V
N
k
k
V
k
k
V
yN
kkN
M
k
V
N
M
MM
N
N
M


SOLO
O Singular Stable Points
* Singular Non-Stable Points
( ) γγγ 3
1
2
1
sin k
k
V
N
V
k
k
V
y MMM
N −





+= ( )
( )










−=→−=
−=→∞→
±±==→=
−=
∞→
=
γγ
γγγ
πγ
3
12
3
1
2
1
0
2
sin
,2,1,0,0
k
k
V
y
k
V
N
k
k
V
k
k
V
yN
kkN
M
k
V
N
M
MM
N
N
M

Isoclines

SOLO
To define the values k1, k2, k3 we use Isoclines. We found that if
econvergencyoscillatornonkkk
econvergencyoscillatorkkk
−⋅≤
⋅>
321
321
MAXMMMM aaVk
td
yd
kykVa ≤−−−== γγ 321
Guidance Law:

SOLO
Un-Saturated Acceleration
Region M
MAX
MM
MAX
V
a
kky
V
k
V
a
<−−−=<− γγγ 32
1
sin
From which: ( ) ( ) 2
1
32
11
32
1
1 :sinsin: y
k
a
kk
k
V
y
k
a
kk
k
V
y MAXMMAXM
=−+−>>++−= γγγγ

SOLO
In can be shown that all the trajectories that enter the non-saturated region for
will stay in the un-saturated region, and will finally reach the origin (y=0,γ=0).
11 γγγ <<−
where: ( ) ( )
( )
( )






+≥
+<








+
+








+=
−−
2
1
2
23
2
1
2
232
1
2
2
31
2
1
2
2
21
1
/
/
/
sin
/
sin
:
MAXM
MAXM
MAXMMAXM
aVkkk
aVkkk
aVkk
k
aVkk
k
π
γ
The unsaturated region around the
origin bounded by –γ1<γ<γ1 ,
defined as Ω1 , is a Capture Zone
for the trajectories.

97
SOLO
Block Diagram of the Linear Guidance Law that assure convergence of the
Interceptor Trajectory to the Ballistic Target Trajectory in a Head On
situation.

98
SOLO
Minimum Time.
Choose the Coordinate System xOy, having the x axis on the
Ballistic Straight Line approximation pointing toward the
Ballistic Target, the origin at the intersection of the straight
line with the ground and y axis pointing above the ground.
• Assume a Constant Velocity Bounded Maneuver Interceptor
and don’t consider gravitation.
( ) ( ) ( )
( ) ( )
( ) ( ) freeTTttt
Tzuuz
TygivenyyzVy
MAX
MAXM
====
==≤=
===⋅=
,001
0,/01
0,0sin
0
0



γγγ
γ
We want to reach the Ox line, in minimum time T, ( y (T)=0 ), and with angle γ (T)=0.
Start without considering the constraint of minimal height hmin. The system equations are:
The Hamiltonian of the Optimal Problem is : ( ) 321 sin λλγλ ++⋅= uzVH MAXM

( )
0
cos
0
3
12
1
=
∂
∂
−=
⋅−=
∂
∂
−=
=
∂
∂
−=
t
H
zV
y
H
y
H
MAXMMAX
λ
γγλλ
λ



( ) ( )
( ) ( ) 133
11
−=
∂
∂
−==
==
=Tt
t
J
Tt
constTt
λλ
λλ
( )TtJ
uu
minmin =

99
SOLO
Minimum Time.
Since H is not an explicit function of time, we have
H =constant,
Therefore:
( ) 1sin 21 −+⋅= uzVH MAXM λγλ 
( )  ( ) 0coscos 211
2
=+⋅−⋅=
−
uuzVuzV
td
Hd
MAXMMAX
z
MAXMAXM

  


λγγλγγλ
λ
( )
( )








=→≤≤=
==→==→=→≤≤=
⇒==
constutttu
or
uzconstzttt
u
td
Hd
MAX
21
2212
2
0
0
1
2
00
0





γ
π
λλ
λ
The optimum is obtained using: ( )[ ]1sinminargminargminarg 21
*
−+⋅=== uzVHJu MAXM
uuu
λγλ 
( )



<<=
≠−
=
212
22*
00
0
ttt
sign
u
λ
λλ

100
SOLO
Minimum Time.
The trajectories that ends at the origin are given by
integrating the equations of state assuming optimal control,
and are given by:
( )



<<=
≠−
=
212
22*
00
0
ttt
sign
u
λ
λλ
( ) ( ) ( )
( ) ( )
( ) ( ) freeTTttt
Tzuuz
TygivenyyzVy
MAX
MAXM
====
==≤=
===⋅=
,001
0,/01
0,0sin
0
0



γγγ
γ
( ) ( ) ( )
( ) ( ) ( ) 1&00cos1
1&00cos1
*
*
−==≤−−=
==≥−=
uT
V
ty
uT
V
ty
MAX
M
II
MAX
M
I
γγ
γ
γγ
γ


( ) ( ) ( ) ( )γγγ
γ
signusign
V
ty
MAX
M
III −=−−= *
, cos1

or
Extremal (non necessarlly
Optimal) Trajectories

101
SOLO
Minimum Time.
Singular Trajectories:
( ) ( )
( ) ( )
( )
( )
( )
( )
21
2
2
2
0&0
0&0
0&0
ttt
ttI
ttI
ttI
nn
<<









==
==
==



λ
λ
λ
( ) ( )
( )
uzVtH
tI
MAXM 21 sin1 λγλ +⋅+−=
  

For Singular Trajectories to occur for t1<t<t2, the following conditions must be satisfied:
( ) ( )
( )
( ) ( ) 
( ) ( ) 0cos
0cos
0
0sin1
12
1
2
1
=⋅−=
=⋅=
=
=⋅+−=
MAXMMAX
z
MAXMAXM
MAXM
zVt
uzVtI
t
zVtI
γγλλ
γγλ
λ
γλ




2
π
γ ±=⋅ MAXz 
MV/11 ±=λ
0=u

102
SOLO
The Optimal Trajectories are function of the Initial Conditions y0, γ0.
If the constraint of minimal convergence altitude hmin is disregarded,
the Optimal Trajectories are given by:
• Turn toward the x axis in Ox direction with
maximum turn rate.
• If γ = +π/2 or - π/2 and the trajectory is toward Ox
and the distance is y > R (Turning Radius), than the
trajectory will be the straight line normal to Ox. When
y = R turn, with maximum turn rate toward Ox
direction, on yI,II (t), to reach y (T) = 0 and γ (T) = 0.
• If during the first turn we get close to Ox line before
reaching γ = +π/2 or - π/2 , we reverse the maximum
turn, on yI,II (t), to reach y (T) = 0 and γ (T) = 0.
( )[ ] ( ) ( )[ ]




<<−≠
−+−=−−=
2/2/2/
cos1,
*
πγππγ
γγγ signRysignyysignu III
I
( )
( ) ( ) ( )


−−=
−=
γγγ
γ
cos1,
*
signRy
signu
III
Converging to
y=0. γ=0
II
( ) ( )




−−≠
=
=
γγ
πγ
cos1
2/
0*
signRy
u
On Singular Arcs
III
There are three classes of optimal paths defined by:

103
SOLO
Minimum Time.
Block Diagram of the Optimal Law that assure convergence in minimum
time of the Interceptor Trajectory to the Ballistic Target Trajectory in a Head
On situation.

104
SOLO
• Define the Interceptor Midcourse Guidance Law as the combination of the Optimal Trajectory that
reaches the Sliding Lyapunov Surface followed by the Linear Guidance Law to reach the Ballistic
Trajectory in Head On.
• Start with the Optimal Law toward the Sliding Surface until
reaching the Captive Volume Ω1.
• Switch to Linear Guidance Law that keeps the trajectory
inside the Captive Zone Ω1 and converges to the origin
y=0 and γ = 0, to the Head On with the Ballistic Target.

105
SOLO
Block Diagram of the Guidance Law that starts with the Optimal Law that assure
convergence in minimum time to the Capture Zone, followed by the Linear Law that
assre convergence to the Ballistic Target Trajectory in a Head On situation.

106
SOLO
Minimal Altitude hmin of the Trajectory
From the Figure we can see that for
y0>0 and γ0>0
and for
y0<0 and γ0<0
There may be situation when the
minimum time trajectory is not
feasible and we must change it.

107
SOLO
( ) ( )αγγα −−+= 00 coscos RRhh
( ) ( )00min cos1 γγα −−=== Rhhh
From the Figure
Therefore
We want to find hmin as function of
y0, γ0
( ) RRyRRdd
dRl
lh
−−=−−=
−=
=
0001
01
0
cos/cos/
sin
cos
γγ
γ
γ
( )000000 sincossincossin γγγγγ +++−= Ryh
( )1sincossincossin 000000min −+++−= γγγγγ Ryh

108
SOLO
A solution to the altitude problem is to
choose on the Ballistic Trajectory the point
of the minimum altitude hmin as the point
below which convergence of the Intercept
Trajectory is not acceptable. By doing this if
exists a Singular Arc it will be at |γ| < π/2

109
SOLO
Define:
M – the Interception Position
P - the point at the Ballistic Missile Trajectory
at altitude hmin.
Ix1 – unit vector in the Straight Line Ballistic
Trajectory Approximation, pointing up.
h

– the vector pointing from P to M.
d

– the vector distance from Straight Line
Ballistic Trajectory to M, pointing to M.
dhxf I

−=:1
MV

– Interceptor Velocity at M.
– angle between to .γ
MV

Ix1
– angle between to .hγ MV

h








 ×
= −
hV
hV
M
M
h

1
sinγ







 ×
= −
M
IM
V
xV 1
sin 1

γ

110
SOLO
( ) ( )
( )
( )
( ) MAXC
MIM
MIM
M
MM
MM
C aa
VxV
VxV
Vk
dVV
dVV
dkdka ≤
××
××
+
××
××
+= 




1
1
321 γ
( )
( )
( ) MAXCMM
h
h
MMM
MM
hMC aahVV
hV
k
hVV
hVV
Vka ≤××





=
××
××
=



γ
γ
γ
sin
( ) ( )
( )
321
32 /sin
cos11:
kkk
dVkk
Rd
SWSWSWM
SWSW
=
+==
−∆+=
γγ
γ
When the denominators are close to zero, they
will be bounded above zero, to prevent
numerical problems.
Guidance Law B: Converges to the Ballistic
Trajectory in H.O. Scenario
Guidance Law A: Reaches the Ballistic
Trajectory above Altitude hmin
When the Interceptor distance to the Ballistic
trajectory is less than dSW (defined bellow) we
switch to Guidance Law B.

SOLO
Assume a Divert Attitude Control System (DACS).
The Divert Control Thrusts are located near the Mass Center of the Vehicle, and are
aligned with the two axes perpendicular to the longitudinal axis of the Kill Vehicle,
so as to generate the Pure but Arbitrary Divert Motion (axB, ayB, azB), where Attitude
Control Thrusts are located and aligned such that only Three Pure Rotational
Moments about the principal axes are produced (TxB,TyB,TzB). All those Thrusts are
Pulse Type, i.e., they only have ON/OFF states with fixed amplitude.
Thrust Command
Thrust Output
See “Kill Vehicle Guidance
& Control Using Sliding
Mode” Presentation for
more details
The goal of the Control System is to track a design quaternion and corresponding
angular velocity .
dq
dω


112
SOLO
Desired Attitude and Angular Rate of the Kill Vehicle (KV)
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( )
( ) 









−
+
−−+
=






















+−−+−
−−+−+
+−−−+
=
===










dddd
dddd
dddd
dddddddddddd
dddddddddddd
dddddddddddd
Bd
Bd
TBd
I
Bd
Bd
I
Bd
I
zI
yI
xI
qqqq
qqqq
qqqq
qqqqqqqqqqqq
qqqqqqqqqqqq
qqqqqqqqqqqq
xCxCd
d
d
d
3120
3021
2
3
2
2
2
1
2
0
2
3
2
2
2
1
2
010323120
3210
2
3
2
2
2
1
2
03021
20312130
2
3
2
2
2
1
2
0
2
2
0
0
1
22
22
22
111
1
1
1
Kill Vehicle (KV)Control
Since the KV roll is free, let choose q1d = 0
We obtain:
0
3
0
20
2
1
,
2
1
,
2
11
q
d
q
q
d
q
d
q
yI
d
zI
d
xI
d ==
+
=
The Main KV Engine is aligned to xB direction of the KV.
The KV Divert Thrusters act normal to to the xB direction.
Suppose that we want that the KV xB direction shall follow a
given direction and it’s inertial derivative .
The rotation position of the KV is free.
The Desired KV Attitude is defined as Bd. The following
relation must be satisfied:
d1
I
d
td
d
1

113
SOLO
Desired Attitude and Angular Rate of the Kill Vehicle (KV)
We found the quaternion from inertia (I) to Desired Body (Bd) Attitude:
0
3
0
210
2
1
,
2
1
,0,
2
11
q
d
q
q
d
qq
d
q
yI
d
zI
dd
xI
d ===
+
=
Taking the derivatives:
( ) td
qd
q
d
d
td
d
qtd
qd
q
td
qd
q
d
d
td
d
qtd
qd
q
td
qd
qd
td
d
dtd
qd
q d
d
yI
yI
d
d
d
d
d
zI
zI
d
d
d
d
dxI
xI
d
d
0
2
00
3
3
0
2
00
2
2
1
1
0
0
2
1
1
2
1
,
2
1
1
2
1
,0,1
112
1
−





==−





====





+
== 












=
d
d
d
d
Bd
I
q
q
q
q
q
3
2
1
0
where:
Using those results
we can find:
( )






















−−
−−
−−
=←
d
d
d
d
dddd
dddd
dddd
I
IBd
q
q
q
q
qqqq
qqqq
qqqq
3
2
1
0
0123
1032
2301
2








ω
( )
( ) [ ][ ]












×+−=Θ=←
td
d
td
qd
Iqq
dt
d
q
d
d
dxdd
Bd
I
Bd
I
TI
IBd
ρ
ρρω 



0
33022

SOLO
114

SOLO
115

SOLO
1
2
0
2
3
2
2
2
1
2
0
0
4210
=+=+++






=+++==←
ρρ
ρρ

   
T
B
IIB
qqqqq
q
qkqjqiqqq
Coordinate Systems
Inertial Coordinates (x,y,z), and
Body Coordinates at the Body Center of Gravity (xB, yB, zB)
The Rotation Matrix from Inertial Coordinates (x,y,z),
to Body Coordinates at the Body Center of Gravity
(xB, yB, zB), CI
B
, is defined via the Quaternions:
1−=⋅⋅=⋅=⋅=⋅ kjikkjjii

kijji

=⋅−=⋅
ijkkj

=⋅−=⋅
jkiik

=⋅−=⋅
( ) ( ) ( )
( ) ( ) ( ) I
B
B
I
B
I
B
I
B
I
B
IIB
qqqqqqqqq
qqkqjqiqqq
==⇒=+++=
−=−−−+==
−
−
←
*12
3
2
2
2
1
2
0
*
04210
**
1
ρ
ρ

   
Quaternion Complex Conjugate: kkjjii

−=−=−= ***
( ) ( ) 1ˆˆ2/sinˆ2/cos0 === nnnq T
θρθ

116

117
SOLO
Kill Vehicle (KV) Control
Product of Quaternions
1−=⋅⋅=⋅=⋅=⋅ kjikkjjii

kijji

=⋅−=⋅ ijkkj

=⋅−=⋅ jkiik

=⋅−=⋅
⊗
( )
( )





×++
⋅−
=





⊗





=⊗
BAABBA
BABA
B
B
A
A
BA
qq
qqqq
qq
ρρρρ
ρρ
ρρ



00
0000
[ ] [ ] 













×−
−
=













×+
−
=





⊗





=⊗
A
A
BxBB
T
BB
B
B
AxAA
T
AA
B
B
A
A
BA
q
Iq
qq
Iq
qqq
qq
ρρρ
ρ
ρρρ
ρ
ρρ





0
330
00
330
000
( ) ( )32103210
00
BBBBAAAA
B
B
A
A
BA qkqjqiqqkqjqiq
qq
qq

 +++⋅+++=





⊗





=⊗
ρρ
Using this definition the Product of two Quaternions is given by:BA qq ⊗
Let define:
or in Matrix Product Form:

118
SOLO
Product of Quaternions ⊗
Let compute:
( )
1
0
1
0
00
1
2
0
001
=





=






















×−−
+
=





−
⊗





=⊗
−



  


AAAAAA
A
T
AA
A
A
A
A
AA
qq
q
qq
qq
ρρρρ
ρρ
ρρ

SOLO
Coordinate Transformations
( )
( )
( )
( )
[ ] [ ][ ]{ } ( ) 





××+×−
=





⊗





⊗





−
=⊗⊗=





= AA
B
A
AB
AB
B
vqI
q
v
q
qvq
v
v 
ρρρρρ 22
000
0
00*
Given the vector described in (A) coordinates by , and in (B) coordinates byv
 ( )A
v
 ( )B
v

[ ] [ ][ ]{ }
( ) ( )
( ) ( )
( ) ( ) 











+−−−+
+−+−−
−+−−+
=××+×−=
2
3
2
2
2
1
2
010323120
3210
2
3
2
2
2
1
2
03021
20312130
2
3
2
2
2
1
2
0
0
22
22
22
22
qqqqqqqqqqqq
qqqqqqqqqqqq
qqqqqqqqqqqq
qIC
B
A ρρρ

[ ][ ]
( ) [ ]
( )
  



  







q
x
T
q
x
B
A
Iq
Iq
qqq
qqq
qqq
qqq
qqqq
qqqq
qqqq
C
T
Θ
Ξ 









×−
−
×−−=
















−
−
−
−−−










−−
−−
−−
=
ρ
ρ
ρρ
330
330
012
103
230
321
0123
1032
2301
119
The Rotation Matrix from A to B, CA
B
, is defined as:

Slide Mode Control (SMC) Techniques and Applications

Slide Mode Control (SMC) Techniques and Applications

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