**2.1 AUV kinematics and dynamics**

The mathematical models of marine vehicles consist of kinematic and dynamic part, where the kinematic model gives the relationship between speeds in a body-fixed frame and derivatives of positions and angles in an Earth-fixed frame, see Fig.1. The vector of positions and angles of an underwater vehicle *<sup>T</sup>* η = [*x*, *y*, *z*,ϕ,θ ,ψ ] is defined in the Earth-fixed coordinate system(E) and vector of linear and angular v *v uvw* = [,, , ,,] *p q r T* elocities is defined in a body-fixed(B) coordinate system, representing surge, sway, heave, roll, pitch and yaw velocity, respectively.

Fig. 1. Earth-fixed and body-fixed reference frames

According to the Newton-Euler formulation, the 6 DOF rigid-body equations of motion in the body-fixed coordinate frame can be expressed as:

$$\begin{cases} m\left[\left(\dot{u}\_r - v, r + w, q\right) - \mathbf{x}\_G \left(q^2 + r^2\right) + y\_G \left(pq - \dot{r}\right) + \mathbf{z}\_G \left(pr + \dot{q}\right)\right] = \mathbf{X} \\ m\left[\left(\dot{v}\_r - w, p + u\_r r\right) - y\_G \left(r^2 + p^2\right) + \mathbf{z}\_G \left(qr - \dot{p}\right) + \mathbf{x}\_G \left(qp + \dot{r}\right)\right] = \mathbf{Y} \\ m\left[\left(\dot{w}\_r - u, q + v, p\right) - \mathbf{z}\_G \left(p^2 + q^2\right) + \mathbf{x}\_G \left(rp - \dot{q}\right) + y\_G \left(rq + \dot{p}\right)\right] = \mathbf{Z} \\ I\_x \dot{p} + \left(I\_z - I\_y\right)qr + m\left[y\_G \left(\dot{w}\_r + pv\_r - qw\_r\right) - \mathbf{z}\_G \left(\dot{v}\_r + ru\_r - pw\_r\right)\right] = \mathbf{K} \\ I\_y \dot{q} + \left(I\_x - I\_z\right)rp + m\left[\mathbf{z}\_G \left(\dot{u}\_r + w\_r q - v\_r r\right) - \mathbf{x}\_G \left(\dot{w}\_r + pv\_r - u\_r q\right)\right] = \mathbf{M} \\ I\_z \dot{r} + \left(I\_y - I\_z\right)pq + m\left[\mathbf{x}\_G \left(\dot{v}\_r + u\_r r - pw\_r\right) - y\_G \left(\dot{u}\_r + qw\_r - v\_r r\right)\right] = \mathbf{N} \end{cases} \tag{1}$$

where *m* is the mass of the vehicle, *I I*, *x y* and *Iz* are the moments of inertia about the *<sup>x</sup>* ,*<sup>y</sup> b b* and *<sup>z</sup> <sup>b</sup>* -axes, *<sup>x</sup>* ,*<sup>y</sup> g g* and *zg* are the location of center of gravity, *u ,v ,w rr r* are relative 134 Autonomous Underwater Vehicles

Six degree of freedom vehicle simulations are quite important and useful in the development of undersea vehicle control systems. There are several processes to be modeled in the simulation including the vehicle hydrodynamics, rigid body dynamics, and actuator

The mathematical models of marine vehicles consist of kinematic and dynamic part, where the kinematic model gives the relationship between speeds in a body-fixed frame and derivatives of positions and angles in an Earth-fixed frame, see Fig.1. The vector of positions

= [*x*, *y*, *z*,

coordinate system(E) and vector of linear and angular v *v uvw* = [,, , ,,] *p q r T* elocities is defined in a body-fixed(B) coordinate system, representing surge, sway, heave, roll, pitch

According to the Newton-Euler formulation, the 6 DOF rigid-body equations of motion in

( ) 2 2 2 2

*rr r G G G*

*rr r G G G*

*rr r G G G*

 

( ) ( )( )

⎡ ⎤ ⎣ ⎦ ⎡ ⎤ ⎣ ⎦ ⎡ ⎤ ⎣ ⎦

*m u v r w q x q r y pq r z pr q X*

− + − + + −+ + =

− + − + + −+ + =

− + − + + −+ + = + − + +− − +− =

( )( )( )( )

*m v w p u r y r p z qr p x qp r Y*

( )( )( )( )

*m w u q v p z p q x rp q y rq p Z*

*x zy G r r r Gr r r*

*I p I I qr m y w pv qu z v ru pw*

*y x z Gr r r G r r r z yz Gr r r Gr r r*

⎪ ⎡ ⎤ ⎣ ⎦ ⎪ <sup>⎪</sup> ⎡ ⎤ <sup>⎩</sup> ⎣ ⎦

( ) ( )( )

*I q I I rp m z u w q v r x w pv u q M I r I I pq m x v u r pw y u qw v r N*

where *m* is the mass of the vehicle, *I I*, *x y* and *Iz* are the moments of inertia about the

+− + + − − + − = +− + + − − + − =

⎡ ⎤ ⎣ ⎦

*<sup>b</sup>* -axes, *<sup>x</sup>* ,*<sup>y</sup> g g* and *zg* are the location of center of gravity, *u ,v ,w rr r* are relative

( )( ) ( )( )

2 2

ϕ,θ ,ψ

] is defined in the Earth-fixed

*K*

(1)

η

**2. Mathematical modeling and simulation** 

and angles of an underwater vehicle *<sup>T</sup>*

Fig. 1. Earth-fixed and body-fixed reference frames

the body-fixed coordinate frame can be expressed as:

( ) ( )

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪

*<sup>x</sup>* ,*<sup>y</sup> b b* and *<sup>z</sup>*

**2.1 AUV kinematics and dynamics** 

and yaw velocity, respectively.

dynamics, etc.

translational velocities associated with surge, sway and heave to ocean current in the bodyfixed frame, here assuming the sea current to be constant with orientation in yaw only, which can be described by the vector <sup>T</sup> [ , , ,0,0, ] *Uc cc c c* = *uvw* α . The resultant forces *X* ,*Y* , *Z* , *K* , *M* , *N* includes positive buoyant *BW P* − = Δ ( since it is convenient to design underwater vehicles with positive buoyant such that the vehicle will surface automatically in the case of an emergency), hydrodynamic forces *XYZK MN HH H H H H* ,,, , , and thruster forces.

#### **2.2 Thrust hydrodynamics modeling**

The modeling of thruster is usually done in terms of advance ratio <sup>0</sup>*J* , thrust coefficients*KT* and torque coefficient *KQ* . By carrying out an open water test or a towing tank test, a unique curve where 0*J* is plotted against *KT* and *KQ* can be obtained for each propeller to depict its performance. And the relationship of the measured thrust force versus propeller revolutions for different speeds of advance is usually least-squares fitting to a quadratic model.

Here we introduce a second experimental method to modeling thruster dynamics. Fig.2 shows experimental results of thrusters from an open water test in the towing tank of the Key Lab of Autonomous Underwater Vehicles in Harbin Engineering University. The results are not presented in the conventional way with the thrust coefficient *KT* plotted versus the open water advance coefficient <sup>0</sup>*J* , for which the measured thrust is plotted as a function of different speeds of vehicle and voltages of the propellers.

The thrust force of the specified speed of vehicle under a certain voltage can be finally approximated by Atiken interpolation twice. In the first interpolation, for a certain voltage, the thrust forces with different speeds of the vehicle (e.g. 0m/s, 0.5m/s, 1.0m/s, 1.5m/s) can be interpolated from Fig.1, and plot it versus different speeds under a certain voltage. Then based on the results of the first interpolation, for the second Atiken interpolation we can find the thrust force for the specified speed of the vehicle.

Fig. 2. Measured thrust force as a function of propeller driving voltage for different speeds of vehicle

Modeling and Motion Control Strategy for AUV 137

in horizontal plane and the combined heave and pitch control for dive in vertical plane. And an improved S-surface control algorithm based on capacitor plate model is developed.

As a nonlinear function method to construct the controller, S-surface control has been proven quite effective in sea trial for motion control of AUV in Harbin Engineering

where *e* ,*e* are control inputs, and they represent the normalized error and change rate of the error, respectively; *u* is the normalized output in each degree of freedom; 1 *k* , <sup>2</sup> *k* are control parameters corresponding to control inputs *e* and *e* respectively, and we only need

Based on the experiences of sea trials, the control parameters 1 *k* , <sup>2</sup> *k* can be manually adjusted to meet the fundamental control requirements, however, whichever combination of <sup>1</sup> *k* , <sup>2</sup> *k* we can adjust, it merely functions a global tuning which dose not change control structure. Here the improved S-surface control algorithm is developed based on the capacitor with each couple of plates putting restrictions on the control variables *e* , *e* respectively, which can provide flexible gain selection with proper physical meaning.

The capacitor plate model as shown in Fig.3 demonstrates the motion of a charged particle driven by electrical field in capacitor is coincident with the motion of a controlled vehicle from current point(,) *e e* to the desired point, for which the capacitor plate with voltage

*u* = 2.0 1.0 exp( ) 1.0 ( + −− − *ke ke* 1 2 ) (5)

University (Li et al., 2002). The nonlinear function of S-surface is given as:

to adjust them to meet different control requirements.

**3.1 Control algorithm** 

Fig. 3. Capacitor plate model

Compared with conventional procedure to obtain thrust that is usually done firstly by linear approximating or least-squares fitting to *KT 0 - J* plot (open water results), then using formulation *2 4 Ft t =Kn D* to compute the thrust *Ft* . The experimental results of open water can be directly used to calculate thrust force without using the formulation, which also can be applied to control surface of rudders or wings, *etc*.

### **2.3 General dynamic model**

To provide a form that will be suitable for simulation and control purposes, some rearrangements of terms in Eq.(1) are required. First, all the non-inertial terms which have velocity components were combined with the fluid motion forces and moments into a fluid vector denoted by the subscript vis (viscous). Next, the mass matrix consisting all the coefficient of rigid body's inertial and added inertial terms with vehicle acceleration components *uvw* ,, ,,, *p q r* was defined by matrix *E* , and all the remaining terms were combined into a vector denoted by the subscript else, to produce the final form of the model:

$$
\dot{EX} = F\_{\text{vis}} + F\_{\text{else}} + F\_t \tag{2}
$$

where <sup>T</sup> *X* =[,, ,,,] *uvwpqr* is the velocity vector of vehicle with respect to the body-fixed frame.

Hence, the 6 DOF equations of motion for underwater vehicles yield the following general representation:

$$\begin{cases} \dot{X} = E^{-1}(F\_{\text{vis}} + F\_{\text{else}} + F\_t) \\ \dot{\eta} = J(\eta)X \end{cases} \tag{3}$$

with

$$\mathbf{E} = \begin{bmatrix} m - X\_{\dot{u}} & 0 & 0 & 0 & mz\_{\mathcal{G}} & -my\_{\mathcal{G}} \\ 0 & m - Y\_{\dot{v}} & 0 & -mz\_{\mathcal{G}} - Y\_{\dot{p}} & 0 & mx\_{\mathcal{G}} - Y\_{\dot{r}} \\ 0 & 0 & m - Z\_{\dot{w}} & my\_{\mathcal{G}} & -mx\_{\mathcal{G}} - Z\_{\dot{q}} & 0 \\ 0 & -mz\_{\mathcal{G}} - K\_{\dot{v}} & 0 & I\_{x} - K\_{\dot{p}} & 0 & -K\_{\dot{r}} \\ mz\_{\mathcal{G}} & 0 & -mx\_{\mathcal{G}} - M\_{\dot{w}} & 0 & I\_{y} - M\_{\dot{q}} & 0 \\ 0 & mx\_{\mathcal{G}} - N\_{\dot{v}} & 0 & -N\_{\dot{p}} & 0 & I\_{z} - N\_{\dot{r}} \end{bmatrix} \tag{4}$$

where *J*( ) η is the transform matrix from body-fixed frame to earth-fixed frame, η is the vector of positions and attitudes of the vehicle in earth-fixed frame.

The general dynamic model is powerful enough to apply it to different kinds of underwater vehicles according to its own physical properties, such as planes of symmetry of body, available degrees of freedom to control, and actuator configuration, which can provide an effective test tool for the control design of vehicles.

#### **3. Motion control strategy**

In this section, the design of motion control system of AUV-XX is described. The control system can be cast as two separate designs, which include both position and speed control in horizontal plane and the combined heave and pitch control for dive in vertical plane. And an improved S-surface control algorithm based on capacitor plate model is developed.

#### **3.1 Control algorithm**

136 Autonomous Underwater Vehicles

Compared with conventional procedure to obtain thrust that is usually done firstly by linear approximating or least-squares fitting to *KT 0 - J* plot (open water results), then using formulation *2 4 Ft t =Kn D* to compute the thrust *Ft* . The experimental results of open water can be directly used to calculate thrust force without using the formulation, which also can

To provide a form that will be suitable for simulation and control purposes, some rearrangements of terms in Eq.(1) are required. First, all the non-inertial terms which have velocity components were combined with the fluid motion forces and moments into a fluid vector denoted by the subscript vis (viscous). Next, the mass matrix consisting all the coefficient of rigid body's inertial and added inertial terms with vehicle acceleration components *uvw* ,, ,,, *p q r* was defined by matrix *E* , and all the remaining terms were combined into a vector denoted by the subscript else, to produce the final form of the model:

where <sup>T</sup> *X* =[,, ,,,] *uvwpqr* is the velocity vector of vehicle with respect to the body-fixed

Hence, the 6 DOF equations of motion for underwater vehicles yield the following general

= ++

*m X mz my*

− −

0 0 0

*G G w y q*

*mz mx M I M*

*X EF F F*

vis else ( )

*u G G*

*v G p G r w GG*

 *z r I N*

*m Y mz Y mx Y m Z my mx*

*G v x p r*

*mz K I K K*

− − − − − − −

is the transform matrix from body-fixed frame to earth-fixed frame,

The general dynamic model is powerful enough to apply it to different kinds of underwater vehicles according to its own physical properties, such as planes of symmetry of body, available degrees of freedom to control, and actuator configuration, which can provide an

In this section, the design of motion control system of AUV-XX is described. The control system can be cast as two separate designs, which include both position and speed control

<sup>⎡</sup> <sup>⎤</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎢</sup> <sup>⎥</sup> ⎢⎣ ⎥⎦ <sup>−</sup>

− −− −

*t*

1

( )

η

 0 0 0 0 0 0

<sup>−</sup> − − <sup>=</sup>

0 0 0

0 0 0

− −

*mx N N*

vector of positions and attitudes of the vehicle in earth-fixed frame.

*G v p*

<sup>−</sup> ⎧⎪ ⎨ ⎪⎩

= 

η*J X*

0 0

effective test tool for the control design of vehicles.

*E*

**3. Motion control strategy** 

= vis else + + *<sup>t</sup> EXF F F* (2)

(3)

(4)

η

is the

0

*q*

*Z*

be applied to control surface of rudders or wings, *etc*.

**2.3 General dynamic model** 

frame.

with

where *J*( ) η

representation:

As a nonlinear function method to construct the controller, S-surface control has been proven quite effective in sea trial for motion control of AUV in Harbin Engineering University (Li et al., 2002). The nonlinear function of S-surface is given as:

$$
\mu = 2.0/(1.0 + \exp(-k\_1 e - k\_2 \dot{e})) - 1.0\tag{5}
$$

where *e* ,*e* are control inputs, and they represent the normalized error and change rate of the error, respectively; *u* is the normalized output in each degree of freedom; 1 *k* , <sup>2</sup> *k* are control parameters corresponding to control inputs *e* and *e* respectively, and we only need to adjust them to meet different control requirements.

Based on the experiences of sea trials, the control parameters 1 *k* , <sup>2</sup> *k* can be manually adjusted to meet the fundamental control requirements, however, whichever combination of <sup>1</sup> *k* , <sup>2</sup> *k* we can adjust, it merely functions a global tuning which dose not change control structure. Here the improved S-surface control algorithm is developed based on the capacitor with each couple of plates putting restrictions on the control variables *e* , *e* respectively, which can provide flexible gain selection with proper physical meaning.

Fig. 3. Capacitor plate model

The capacitor plate model as shown in Fig.3 demonstrates the motion of a charged particle driven by electrical field in capacitor is coincident with the motion of a controlled vehicle from current point(,) *e e* to the desired point, for which the capacitor plate with voltage

Modeling and Motion Control Strategy for AUV 139

Since AUV-XX is equipped with two transverse tunnel thrusters in the vehicle fore and aft respectively and two main thrusters (starboard and port) aft in horizontal plane, which can produce a force in the *x*-direction needed for transit and a force in the *y*-direction for maneuvering, respectively. So both speed and position controllers are designed in

Speed control is to track the desired surge velocity with fixed yaw angle and depth, which is usually used in long distance transfer of underwater vehicles. Before completing certain kind of undersea tasks, the vehicle needs to experience long traveling to achieve the destination. In this chapter, speed control is referred to a forward speed controller in surge based on the control algorithm we introduced in above section, its objective is to make the vehicle transmit at a desired velocity with good and stable attitudes such as fixed yaw and

Position control enables the vehicle to perform various position-keeping functions, such as maintaining a steady position to perform a particular task, following a prescribed trajectory to search for missing or seek after objects. Accurate position control is highly desirable when the vehicle is performing underwater tasks such as cable laying, dam security inspection and mine clearing. To ensure AUV-XX to complete work assignments of obstacles avoiding, target recognition, and mine countermeasures, we design position controllers for surge, sway, yaw and depth respectively for equipping the vehicle with abilities of diving at fixed deepness, navigating at desired direction, sailing to given points and following the given

As for the desired or target position or speed in the control system, it is the path planning system who decides when to adopt and switch control scheme between position and speed, the desired position that the vehicle is supposed to reach, and the velocity at which the

**3.2 Speed and position control in horizontal plane** 

Fig. 4. Position and speed control loop

horizontal plane.

depth.

track, *etc*.

serves as the controller, and the equilibrium point of electrical field is the desired position that the vehicle is supposed to reach.

Due to the restriction of two couples of capacitor plates put on control variables *e* and *e* , the output of model can be obtained as

$$y = u^{+\mathcal{U}\_0} + u^{-\mathcal{U}\_0} = F(L\_1, L\_2)(+\mathcal{U}\_0) + F(L\_2, L\_1)(-\mathcal{U}\_0) \tag{6}$$

where 1 2 *L L*, are horizontal distances from the current position of the vehicle to each capacitor plate, respectively, and the restriction function *F*(\*,\*) is defined to be hyperbolic function of 1 2 *L L*, by Ren and Li (2005):

$$\begin{cases} F(L\_1, L\_2) = \frac{L\_1^{-k}}{L\_1^{-k} + L\_2^{-k}} \\ F(L\_2, L\_1) = \frac{L\_2^{-k}}{L\_1^{-k} + L\_2^{-k}} \end{cases} \tag{7}$$

The restriction function *F*(\*,\*) reflects the closer the current position(,) *e e* of vehicle moving to capacitor plate, the stronger the electrical field is. Choosing *U*<sup>0</sup> = 1 , the output of capacitor plate model yields:

$$\mu = \frac{L\_1^{-k} - L\_2^{-k}}{L\_1^{-k} + L\_2^{-k}} \\ \mathcal{U}\_0 = \frac{\left(e\_0 + e\right)^k - \left(e\_0 - e\right)^k}{\left(e\_0 + e\right)^k + \left(e\_0 - e\right)^k} \tag{8}$$

where 0*e* is the distance between the plate and field equilibrium point of capacitor. An improved S-surface controller based on the capacitor plate model is proposed, that is

$$\begin{cases} u\_{ei} = \text{[2.0]} \left( \mathbf{1.0} + \left( \frac{e\_0 - e\_i}{e\_0 + e\_i} \right)^{ki\cdot} \right) - \mathbf{1.0} \end{cases}$$

$$\begin{cases} u\_{ei} = \text{[2.0]} \left( \mathbf{1.0} + \left( \frac{e\_0 - \dot{e}\_i}{e\_0 + \dot{e}\_i} \right)^{ki\cdot} \right) - \mathbf{1.0} \end{cases} \tag{9}$$

$$\begin{cases} f\_i = \mathbf{K}\_{ei} \cdot \mathbf{u}\_{ei} + \mathbf{K}\_{ei} \cdot \mathbf{u}\_{ei} \end{cases} \tag{9}$$

where *<sup>i</sup> f* is the outputted thrust force of controller for each DOF, and *KKK ei ei i* = = is the maximal thrust force in *i* th DOF, therefore the control output can be reduced to

$$\begin{cases} u\_i = u\_{ei} + u\_{ii} = \text{[2.0]} \left( 1.0 + (\frac{e\_0 - e\_i}{e\_0 + e\_i})^{\text{k}\text{i}1} \right) - \text{1.0]} + \text{[2.0]} \left( 1.0 + (\frac{e\_0 - \dot{e}\_i}{e\_0 + \dot{e}\_i})^{\text{k}\text{i}2} \right) - \text{1.0]}\\ f\_i = \text{K}\_i \times u\_i \end{cases} \tag{10}$$

The capacitor model's S-surface control can provide flexible gain selections with different forms of restriction function to 1 2 *L L*, to meet different control requirements for different phases of control procedure.

138 Autonomous Underwater Vehicles

serves as the controller, and the equilibrium point of electrical field is the desired position

Due to the restriction of two couples of capacitor plates put on control variables *e* and *e* , the

where 1 2 *L L*, are horizontal distances from the current position of the vehicle to each capacitor plate, respectively, and the restriction function *F*(\*,\*) is defined to be hyperbolic

1 2

*<sup>L</sup> FL L*

*<sup>L</sup> FL L*

(,)

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

− − − −

*ei*

*u*

*ei*

*u*

⎪

⎪ ⎪ ⎪⎩ 2 1

where 0*e* is the distance between the plate and field equilibrium point of capacitor. An improved S-surface controller based on the capacitor plate model is proposed, that is

<sup>⎧</sup> ⎛ ⎞ <sup>⎪</sup> ⎜ ⎟ <sup>⎪</sup> ⎝ ⎠

<sup>⎪</sup> ⎛ ⎞ <sup>⎨</sup> ⎜ ⎟ <sup>⎪</sup> ⎝ ⎠

*i ei ei ei ei*

maximal thrust force in *i* th DOF, therefore the control output can be reduced to

*e e ki e e ki uu u*

<sup>⎧</sup> ⎛ ⎞⎛ ⎞ <sup>⎪</sup> <sup>=</sup> ⎜ ⎟⎜ ⎟ <sup>⎨</sup> ⎝ ⎠⎝ ⎠ <sup>⎪</sup>

*f Ku Ku*

= ⋅+ ⋅

(,)

12 0 21 0 ( , )( ) ( , )( ) *U U*

1

− − − − − −

*k k k k k k*

1 2 2

*L L*

<sup>=</sup> <sup>+</sup>

<sup>=</sup> <sup>+</sup>

The restriction function *F*(\*,\*) reflects the closer the current position(,) *e e* of vehicle moving to capacitor plate, the stronger the electrical field is. Choosing *U*<sup>0</sup> = 1 , the output of

> 1 2 0 0 0 12 0 0

*k k k k k k k k L L ee ee u U L L ee ee*

1 2

( )( ) ( )( )

> 0 1 0

*i i ki*

*i ki*

[2.0 1.0 ( ) 1.0]

<sup>−</sup> =+ −

<sup>−</sup> =+ −

where *<sup>i</sup> f* is the outputted thrust force of controller for each DOF, and *KKK ei ei i* = = is the

− − += + − + + −

The capacitor model's S-surface control can provide flexible gain selections with different forms of restriction function to 1 2 *L L*, to meet different control requirements for different

*e e*

*e e e e*

+

[2.0 1.0 ( ) 1.0]

*e e*

+

0 2 0

*i*

0 0 0 0 1 2 [2.0 1.0 ( ) 1.0] [2.0 1.0 ( ) 1.0] *i i*

*e e e e*

+ +

*i i*

<sup>−</sup> +−− <sup>=</sup> <sup>=</sup> + ++− (8)

*L L*

*y u u FL L U FL L U* + − = + = ++ − (6)

(7)

(9)

(10)

0 0

that the vehicle is supposed to reach.

output of model can be obtained as

function of 1 2 *L L*, by Ren and Li (2005):

capacitor plate model yields:

*i ei ei*

*i ii*

phases of control procedure.

*f Ku*

= ×

⎩
