2.4. Assumption for the conventional control

Taking into account that the mathematical model of the dc motor is of the second order, the two state variables are deducted (armature current, iA, and the speed, ωm). The mathematical model is characterized by two different time constants: Tm>>T<sup>A</sup> (the electromechanical time constant, Tm, is greater with one order than the electromagnetic ones, TA). This conclusion leads to the cascaded control with two loops: the armature current loop (the inner loop), and the angular velocity (the outer loop). By considering that the angular velocity should be controlled and the armature current should be limited at the maximum value, the angular velocity is the dc motor output. Taking into account that the dc motor is supplied by the six pulses ac-dc full bridge power converter, the modulus criterion is applied for tuning inner loop, and the symmetrical optimum for the outer loop (Kessler variant). The adequate operational block diagram is deducted according to Figure 2.

The dc drive is constituted by two parts: the power and the control. A unified [0, 10]V voltage system is taken into account with respect to the control part. Therefore, for the maximum value of 10 V the maximum allowable speed at the rated flux is obtained (i.e., n\* = 1,2 nr, by taken into consideration the 20% speed overshoot introduced by the Kessler criterion).

It is well-known that the symmetrical criterion supposes the ramp reference for the speed loop. In order to transform the step reference analogue signal from the potentiometer into ramp signal, an adequate filter has been designed with the following transfer function:

Figure 2. The conventional control of the dc drive.

Matlab-Simulink-Based Compound Model Reference Adaptive Control for DC Motor http://dx.doi.org/10.5772/intechopen.71758 121

$$H\_{fn}(\mathbf{s}) = \frac{1}{1 + sT\_{Fn}} \,\mathrm{\,\,\,}\tag{2}$$

where TFn is the time constant of the speed transducer.

The corresponding Simulink block diagram is shown below in Figure 3.

The control design procedure starts with the armature current loop design (inner loop). The imposed performances for the inner closed loop are:

I. the steady state regime:

2.2. Assumptions for the ac-dc/dc-dc power converters

switching time are negligible.

120 Adaptive Robust Control Systems

2.3. Assumptions for the load

2.4. Assumption for the conventional control

tional block diagram is deducted according to Figure 2.

Figure 2. The conventional control of the dc drive.

The uninterrupted conduction is taken into account (a high value for the additional armature inductance is designed), the conduction droop voltage on the power semiconductors and the

The load torque is considered as mathematical model of the process, the load is reduced appropriately to the dc motor shaft (by using both equivalent inertia moment and reduced

Taking into account that the mathematical model of the dc motor is of the second order, the two state variables are deducted (armature current, iA, and the speed, ωm). The mathematical model is characterized by two different time constants: Tm>>T<sup>A</sup> (the electromechanical time constant, Tm, is greater with one order than the electromagnetic ones, TA). This conclusion leads to the cascaded control with two loops: the armature current loop (the inner loop), and the angular velocity (the outer loop). By considering that the angular velocity should be controlled and the armature current should be limited at the maximum value, the angular velocity is the dc motor output. Taking into account that the dc motor is supplied by the six pulses ac-dc full bridge power converter, the modulus criterion is applied for tuning inner loop, and the symmetrical optimum for the outer loop (Kessler variant). The adequate opera-

The dc drive is constituted by two parts: the power and the control. A unified [0, 10]V voltage system is taken into account with respect to the control part. Therefore, for the maximum value of 10 V the maximum allowable speed at the rated flux is obtained (i.e., n\* = 1,2 nr, by taken

It is well-known that the symmetrical criterion supposes the ramp reference for the speed loop. In order to transform the step reference analogue signal from the potentiometer into ramp

into consideration the 20% speed overshoot introduced by the Kessler criterion).

signal, an adequate filter has been designed with the following transfer function:

equivalent speed by taking into account the specific transmission ratio).

	- a. current overshoot, σ = 4.3%;
	- b. response time depends on the cutting frequency of the loop: tr = 2.35/ωcI;
	- c. phase margin of the current loop: γ = 63�26<sup>0</sup> .

In order to satisfy the above mentioned performances the modulus criterion is used. The Kessler criterion guarantees of the above mentioned performances on condition that the open loop transfer function of the inner loop has the form:

$$H\_{cull}(\mathbf{s}) = \frac{1}{2\mathbf{s}T\_{\Sigma l}(1+\mathbf{s}T\_{\Sigma l})} \tag{3}$$

where, TΣ<sup>I</sup> – the parasitic time constant of the current loop is considerably lower than the armature time constant:

$$T\_{\Sigma I} = T\_{FI} + T\_{TI} << T\_A \tag{4}$$

Figure 3. Simulink block diagram of the dc drive system.

in which TFI is the current filter time constant, and TTI is the current transducer time constant. According to modulus criterion, the Proportional Integral (PI) controller is suitable to control the armature current:

$$H\_{RI}(s) = \frac{1 + s\tau\_1}{s\tau\_2},\tag{5}$$

Hcut,nð Þ¼ s

loop transfer function (11), the PI speed controller is obtained:

Moreover, the following speed controller parameters are obtained:

parameters are provided:

shown in Figure 5.

Figure 5. The numerical implementation of the dc drive system.

1 þ 4sTΣ<sup>n</sup>

1 þ sτ<sup>3</sup> sτ<sup>4</sup>

<sup>Σ</sup><sup>n</sup>ð Þ 1 þ sTΣ<sup>n</sup>

Matlab-Simulink-Based Compound Model Reference Adaptive Control for DC Motor

: (11)

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123

: (12)

: (14)

, (16)

, (17)

τ<sup>3</sup> ¼ 4TΣn, (13)

TΣ<sup>n</sup> ¼ 2TA þ TTN, (15)

8s<sup>2</sup>T<sup>2</sup>

By comparing the equivalent open loop transfer function (Figure 4) with the imposed open

HRnð Þ¼ s

<sup>τ</sup><sup>4</sup> <sup>¼</sup> <sup>8</sup>TΣ<sup>n</sup>

2 Cm � 30 KTIJπKTn

In order to well understand the tuning procedure of the speed controller, the additional

Cm <sup>¼</sup> Tlr IAr

> rot=min V

1:2 � nr

in which TΣ<sup>n</sup> is the parasitic time constant of the speed loop, Tlr is the rated load torque, Cm- the mechanical constant of the dc motor, and TTn is the time constant of the speed

The numerical implementation of the dc drive system is based on the Simulink block diagram

KTn <sup>¼</sup> <sup>10</sup>

transducer (it is obtained by imposing the time response of the speed loop).

with the appropriately controller parameters:

$$
\pi\_2 = T\_A \tag{6}
$$

$$
\pi\_2 = 2T\_{\Sigma I} K\_d K\_{T I} \frac{1}{R\_A} \,. \tag{7}
$$

Eq. (8) contains the mathematical model of the dc-dc power converter:

$$K\_d = \frac{\mathcal{U}\_{Ar}}{10} \,\text{\,\, \, ^\circ \text{\,\,}} \,\text{\,\, \,} \tag{8}$$

and the attenuation factor of the current transducer (obtained by imposing the time response of the current loop) has the form:

$$K\_{II} = \frac{10}{I\_{A\text{max}}} \left[ \frac{V}{A} \right]. \tag{9}$$

The closed loop armature current transfer function is as follows:

$$H\_{ol}(\mathbf{s}) = \frac{1}{k\_{Tl}(1 + \mathbf{2s}T\_{\Sigma l})} \tag{10}$$

The equivalent block diagram of the dc drive system is shown in Figure 4.

Taking into account the reduced block diagram of the dc drive system (Figure 4), the symmetrical optimum criterion could be applied.

According to symmetrical optimum criterion, by using the following open loop transfer function (Kessler) the required performances of the closed loop system are attained:

Figure 4. The operational model of the dc drive system.

Matlab-Simulink-Based Compound Model Reference Adaptive Control for DC Motor http://dx.doi.org/10.5772/intechopen.71758 123

$$H\_{cut,n}(s) = \frac{1 + 4sT\_{\Sigma n}}{8s^2 T\_{\Sigma n}^2 (1 + sT\_{\Sigma n})} \,. \tag{11}$$

By comparing the equivalent open loop transfer function (Figure 4) with the imposed open loop transfer function (11), the PI speed controller is obtained:

$$H\_{Rn}(\mathbf{s}) = \frac{1 + s\tau\_3}{s\tau\_4}.\tag{12}$$

Moreover, the following speed controller parameters are obtained:

in which TFI is the current filter time constant, and TTI is the current transducer time constant. According to modulus criterion, the Proportional Integral (PI) controller is suitable to control

> 1 þ sτ<sup>1</sup> sτ<sup>2</sup>

> > 1 RA

, (5)

: (7)

τ<sup>2</sup> ¼ TA (6)

<sup>10</sup> , (8)

: (9)

(10)

HRIð Þ¼ s

τ<sup>2</sup> ¼ 2TΣIKdKTI

Kd <sup>¼</sup> UAr

and the attenuation factor of the current transducer (obtained by imposing the time response

Taking into account the reduced block diagram of the dc drive system (Figure 4), the symmet-

According to symmetrical optimum criterion, by using the following open loop transfer func-

V A 

1 kTIð Þ 1 þ 2sTΣ<sup>I</sup>

KTI <sup>¼</sup> <sup>10</sup> IAmax

HoIð Þ¼ s

tion (Kessler) the required performances of the closed loop system are attained:

The equivalent block diagram of the dc drive system is shown in Figure 4.

Eq. (8) contains the mathematical model of the dc-dc power converter:

The closed loop armature current transfer function is as follows:

the armature current:

122 Adaptive Robust Control Systems

with the appropriately controller parameters:

of the current loop) has the form:

rical optimum criterion could be applied.

Figure 4. The operational model of the dc drive system.

$$
\pi\_3 = 4T\_{\Sigma \mathfrak{n}\_{\prime}} \tag{13}
$$

$$\tau\_{44} = \frac{8T\_{\Sigma n}{}^2 \mathbb{C}\_m \cdot \mathbf{30}}{K\_{\text{TI}} \text{J} \pi K\_{\text{Tn}}}.\tag{14}$$

In order to well understand the tuning procedure of the speed controller, the additional parameters are provided:

$$T\_{\Sigma n} = 2T\_A + T\_{T\mathcal{N}\_{\mathcal{I}}} \tag{15}$$

$$\mathbf{C}\_{\text{m}} = \frac{T\_{lr}}{I\_{Ar}} \,\prime \tag{16}$$

$$K\_{\rm Tt} = \frac{10}{1.2 \cdot n\_{\rm r}} \left[ \frac{rot/\text{min}}{V} \right],\tag{17}$$

in which TΣ<sup>n</sup> is the parasitic time constant of the speed loop, Tlr is the rated load torque, Cm- the mechanical constant of the dc motor, and TTn is the time constant of the speed transducer (it is obtained by imposing the time response of the speed loop).

The numerical implementation of the dc drive system is based on the Simulink block diagram shown in Figure 5.

Figure 5. The numerical implementation of the dc drive system.

At the same time, the pole excess is known:

the reference model is chosen, n\*m = n\*p.

<sup>∂</sup> Dpð Þ<sup>s</sup> � � � <sup>∂</sup> Npð Þ<sup>s</sup> � � <sup>¼</sup> <sup>n</sup><sup>∗</sup>

Hmð Þ¼ s km

v<sup>1</sup> ¼

vu vy yp 3 7

2 6 4

The dynamic filters (Λ, h) are placed on the command v<sup>u</sup> and on the output of the process vyp:

<sup>u</sup>ðÞ¼ t Λvuð Þþ t huð Þt

<sup>y</sup>ðÞ¼ t Λvyð Þþ t hypð Þt

The solution of the dynamical filter is implemented in Matlab as in Figure 7 (applied only for

The (Λ, h) pair is chosen in controllable canonical form, Λ ∈ ℜ(np � 1) � (np � 1), h∈ ℜnp

cation error [2–6] implies, in this case, the use of the augmented error.

In order to obtain a stable system the following signals vector is inserted:

v o

(

v o

in which: λ1(s) is an arbitrary Hurwitz polynomial having the degree [7]:

The augmented error depends on the gain factor knowing.

3.1. The case of knowing only the sign of kp factor

the first equation).

Figure 7. L-h (Λ, h) vu filter block.

Taking into account that the relative degree of the process is supraunitary, the second order of

Due to the supraunitary relative degree, the strictly real positive condition for the reference model cannot be accomplished. This condition that the tracking error differs from the identifi-

Nmð Þs

Matlab-Simulink-Based Compound Model Reference Adaptive Control for DC Motor

<sup>p</sup> > 1: (19)

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125

Dmð Þ<sup>s</sup> : (20)

<sup>5</sup>, (21)

detð Þ¼ sI � Λ Nmð Þ� s λ1ð Þs , (23)

(22)

, such that:

Figure 6. The dc drive numerical simulations results under load step variation.

By considering a dc motor with the following nameplate data: rated power P<sup>r</sup> = 5.1 kW, the maximum armature voltage U<sup>r</sup> = 440 V, the nominal armature current IAr = 17.8 A, the reduced moment of inertia J = 0.02 kgm<sup>2</sup> , the viscous force F<sup>v</sup> = 0.0006 Nms/rad, rated speed n<sup>r</sup> = 2700 rpm, the simulation results are obtained (Figure 6a–d). The motor data can be obtained based on the nameplate values by using the detailed Matlab software provided in [3].

Figure 6a–d shows the 0.7T<sup>r</sup> load starting simulation results of the dc conventional control based on the dc-dc full bridge power converter. Figure 6a contains the obtained armature voltage of the dc motor. The armature current varies according to Figure 6b, the speed varies as in Figure 6c under rated value of the load torque T<sup>l</sup> = 22.8 Nm (Figure 6d). The load torque is applied at t = 0.5 s.

#### 3. Adaptive control

There are three assumptions available [4, 5]: the mathematical model of the process is linear, strictly proper and of minimum phase, having the supraunitary relative degree n\* <sup>p</sup> = 2n<sup>∗</sup> <sup>p</sup> ¼ 2; the reference model has the relative degree greater than one (n\* <sup>m</sup> = 2), is stable and of minimum phase; the reference signal should be bounded limit, being a continuous function. The second order mathematical model of the dc motor is used in this chapter. This supposes the transfer function of the process has the form:

$$H\_p(s) = k\_p \frac{N\_p(s)}{D\_p(s)}.\tag{18}$$

At the same time, the pole excess is known:

$$
\left[\partial \left[D\_p(\mathbf{s})\right] - \partial \left[N\_p(\mathbf{s})\right] = n\_p^\* > 1. \tag{19}
$$

Taking into account that the relative degree of the process is supraunitary, the second order of the reference model is chosen, n\*m = n\*p.

$$H\_m(\mathbf{s}) = k\_m \frac{N\_m(\mathbf{s})}{D\_m(\mathbf{s})}.\tag{20}$$

Due to the supraunitary relative degree, the strictly real positive condition for the reference model cannot be accomplished. This condition that the tracking error differs from the identification error [2–6] implies, in this case, the use of the augmented error.

The augmented error depends on the gain factor knowing.

#### 3.1. The case of knowing only the sign of kp factor

By considering a dc motor with the following nameplate data: rated power P<sup>r</sup> = 5.1 kW, the maximum armature voltage U<sup>r</sup> = 440 V, the nominal armature current IAr = 17.8 A, the

speed n<sup>r</sup> = 2700 rpm, the simulation results are obtained (Figure 6a–d). The motor data can be obtained based on the nameplate values by using the detailed Matlab software provided

Figure 6a–d shows the 0.7T<sup>r</sup> load starting simulation results of the dc conventional control based on the dc-dc full bridge power converter. Figure 6a contains the obtained armature voltage of the dc motor. The armature current varies according to Figure 6b, the speed varies as in Figure 6c under rated value of the load torque T<sup>l</sup> = 22.8 Nm (Figure 6d). The load torque

There are three assumptions available [4, 5]: the mathematical model of the process is linear,

phase; the reference signal should be bounded limit, being a continuous function. The second order mathematical model of the dc motor is used in this chapter. This supposes the transfer

Npð Þs

Hpð Þ¼ s kp

strictly proper and of minimum phase, having the supraunitary relative degree n\*

the reference model has the relative degree greater than one (n\*

, the viscous force F<sup>v</sup> = 0.0006 Nms/rad, rated

<sup>p</sup> = 2n<sup>∗</sup>

<sup>m</sup> = 2), is stable and of minimum

Dpð Þ<sup>s</sup> : (18)

<sup>p</sup> ¼ 2;

reduced moment of inertia J = 0.02 kgm<sup>2</sup>

Figure 6. The dc drive numerical simulations results under load step variation.

in [3].

is applied at t = 0.5 s.

124 Adaptive Robust Control Systems

3. Adaptive control

function of the process has the form:

In order to obtain a stable system the following signals vector is inserted:

$$\mathbf{v}\_1 = \begin{bmatrix} \mathbf{v}\_{\mathbf{u}} \\ \mathbf{v}\_{\mathbf{y}} \\ \mathbf{y}\_{\mathbf{p}} \end{bmatrix}, \tag{21}$$

The dynamic filters (Λ, h) are placed on the command v<sup>u</sup> and on the output of the process vyp:

$$\begin{cases} \stackrel{\bullet}{\mathbf{v}}\_{\mathbf{u}}(t) = \mathbf{A}\mathbf{v}\_{\mathbf{u}}(t) + \mathbf{h}\mathbf{u}(t) \\ \stackrel{\bullet}{\mathbf{v}}\_{\mathbf{y}}(t) = \mathbf{A}\mathbf{v}\_{\mathbf{y}}(t) + \mathbf{h}\mathbf{y}\_{\mathbf{p}}(t) \end{cases} \tag{22}$$

The solution of the dynamical filter is implemented in Matlab as in Figure 7 (applied only for the first equation).

The (Λ, h) pair is chosen in controllable canonical form, Λ ∈ ℜ(np � 1) � (np � 1), h∈ ℜnp , such that:

$$\det(\mathbf{sI} - \mathbf{A}) = N\_m(\mathbf{s}) \cdot \lambda\_1(\mathbf{s}),\tag{23}$$

in which: λ1(s) is an arbitrary Hurwitz polynomial having the degree [7]:

Figure 7. L-h (Λ, h) vu filter block.

$$\partial[\mathbf{N}\_{\rm m}(\mathbf{s})\lambda\_1(\mathbf{s})] = \mathbf{n}\_{\rm p} - \mathbf{1}.\tag{24}$$

3.2.2. Deduction of the variable structure parameter θ<sup>v</sup>

vector of the parameter estimation errors

Therefore, the compound adaptive law is used:

unmodelled dynamics.

stability [4].

3.3. The stability of the solution

v1

ξ

L -1(s)

θ1 T

ΦT

θT

L -1(s)

Figure 8. The block diagram of the augmented error determination ec.

4. Numerical simulation results

u tðÞ¼ θ<sup>g</sup> þ θ<sup>v</sup> <sup>T</sup>

The parameter θ<sup>v</sup> can be deducted by using the following [2]:

θ<sup>v</sup> ¼ θ<sup>v</sup> e

θ o

Kec<sup>ξ</sup> � <sup>1</sup> <sup>=</sup> <sup>e</sup>

Kec<sup>ξ</sup> <sup>þ</sup> <sup>1</sup> sign Kp

The block diagram of obtaining the augmented error is depicted in Figure 8, in which Φ is the

In the adaptive control, there is a commutation function; usually the signum function conducts toward a sliding mode regime such that the evolution to the equilibrium point is very fast.

The vector of the parameter θ is obtained by using the compound structure: θ = θ<sup>g</sup> + θv.

v þ �γpsign Kp

The adaptive control provides robust characteristics to external disturbances and to

The perturbation of the dc drive system can leads to the instability of the system. The signals in the variable structure law are bounded. Therefore, the adaptive control assures a global

Hm(s)


e0


ea

+

ξ1 T ξ1

Hm(s)L(s)

+ +

ec

Taken into account the dc machine from the conventional control, operating at the constant flux, a speed cycle is applied in order to test the compound adaptive control. The speed cycle contains the dynamic regimes (starting, braking, reversing) and the steady state regime (Figures 9–22).

<sup>ξ</sup>e<sup>0</sup> <sup>þ</sup> <sup>θ</sup>

<sup>T</sup>

o v

(34)

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127

ξ: (37)

<sup>v</sup> ¼ �λθ<sup>v</sup> � γ<sup>v</sup> ξec j j: (35)

Matlab-Simulink-Based Compound Model Reference Adaptive Control for DC Motor

<sup>Φ</sup> <sup>¼</sup> <sup>θ</sup> � <sup>θ</sup><sup>0</sup> (36)

<sup>e</sup><sup>K</sup>ξe<sup>0</sup> � <sup>1</sup> eKξe<sup>0</sup> þ 1

Therefore, the λ1(s) polynomial is a design component.

The Hurwitz polynomial L(s) is chosen such that the transfer function Hm(s)L(s) becomes SPR. The degree of the L(s) is <sup>∂</sup>[L(s)] = n\*m�1. If the L(s) is Hurwitz polynomial, then L�<sup>1</sup> is stable.

The parameters vectors v, ξ∈ ℜ2np consist of

$$\mathbf{v} = \begin{bmatrix} v\_u \\ v\_{yp} \\ y\_p \\ r \end{bmatrix}, \xi = L^{-1}(\mathbf{s})[\mathbf{v}(t)], \tag{25}$$

The v1, ξ1∈ ℜ2np � <sup>1</sup> are defined as follows:

$$\mathbf{v}\_1 = \begin{bmatrix} v\_u \\ v\_{yp} \\ y\_p \end{bmatrix}, \quad \xi\_1 = \begin{bmatrix} \xi\_u \\ \xi\_y \\ \xi\_p \end{bmatrix}. \tag{26}$$

The auxiliary error is computed on-line:

$$\mathbf{e}\_{a} = \boldsymbol{\Theta}^{T}\boldsymbol{\xi} - \boldsymbol{L}^{-1}(\mathbf{s}) \left[\boldsymbol{\Theta}^{T}(\mathbf{t})\mathbf{v}(t)\right],\tag{27}$$

where:

$$\mathfrak{E}(t) = L^{-1}(s)[\mathbf{v}(t)].\tag{28}$$

The augmented error is defined as:

$$\mathbf{e}\_{\mathbf{c}} = \mathbf{e}\_{0} + H\_{m}(\mathbf{s})L(\mathbf{s})\left[\mathbf{K}\_{1}\mathbf{e}\_{\mathbf{a}} - \boldsymbol{\xi}\_{1}^{T}\boldsymbol{\xi}\_{1}\mathbf{e}\_{\mathbf{c}}\right],\tag{29}$$

and the on-line gradient adjustable parameter K1 depends only by the augmented error:

$$\stackrel{0}{K}\_{1} = -\mathfrak{e}\_{\mathfrak{e}} \mathfrak{e}\_{\mathfrak{e}} \tag{30}$$

#### 3.2. The parametric adjustment laws for the compound adaptive control

#### 3.2.1. Gradient

The gradient law [2–4] is expressed as:

$$\stackrel{\circ}{\partial}\_{\mathcal{S}} = -\gamma\_{\mathcal{g}} \text{sign}(\mathcal{K}\_{\mathcal{p}}) e\_{\mathfrak{e}} \xi(t) \frac{1}{1 + \underline{\xi}^{T} \underline{\xi}} \tag{31}$$

$$\mathbf{e}\_{a} = \left(\boldsymbol{\theta}\_{\mathcal{S}} + \boldsymbol{\theta}\_{v}\right)^{T} \boldsymbol{\xi} - \boldsymbol{L}^{-1}(\mathbf{s}) \left[\left(\boldsymbol{\theta}\_{\mathcal{S}} + \boldsymbol{\theta}\_{v}\right)^{T} \mathbf{v}(t)\right] \tag{32}$$

$$\boldsymbol{\xi}(t) = \boldsymbol{L}^{-1}(\boldsymbol{s})[\mathbf{v}(t)]; \text{where } \boldsymbol{\xi} = \begin{bmatrix} \boldsymbol{\xi}\_{u} \\ \boldsymbol{\xi}\_{yp} \\ \boldsymbol{\xi}\_{p} \end{bmatrix}. \tag{33}$$
