Robust Control and Stability Analysis of Computerized Numeric Controlled Machine Tool under Parametric Uncertainty

A computerized numeric controlled (CNC) machine is a highly integrated mechatronic system and it was developed to control the motion and operation of machine tools [1]. The CNCMT system is used mainly for drilling, cutting, milling, boring, finishing, welding, and aligning operations in automotive, aerospace, electrical, and instrumentation industries [2, 3]. In the CNCMT system, the information related to the structure of the workpiece is stored in the computer in the form of 'm-code'. The disturbances occurring during the machining process should be minimized to achieve high precision and operational efficiency [4-6]. The structured or parametric and unstructured uncertainties are presented in all practical control systems [7]. The source of parametric uncertainty in the CNCMT system is deficient in clear-cut knowledge of parameters, and few parameters like inertia and friction and varied under running conditions, therefore the system is called the uncertain CNCMT system. Hence the robustness analysis of the CNCMT system is essential and as proposed in this paper. To accomplish the most wanted position of machine table alongside the external disturbances and the parametric uncertainty three different robust controllers such as PI, IMC, and H_∞ controllers are presented. Due to the presence of the derivative 'D' term in the PID controller, a derivative kick occurs and as a result, the controller output becomes dangerously high which may damage the actuator, in the present case, a servomotor. An integral 'I' control, a steady-state error is nullified but overshoot (OS) reaches to very high magnitude. Therefore, a PI type controller is used for position control of uncertain CNCMT system.

Indeed, the design of a low order controller is often more difficult than the design of a high order complex controller [8]. A robust PI controller is designed using Kharitonov’s theorem with the worst gain margin (GM) and phase margin (PM) because it plays an important role concerning the robustness of the system [9, 10]. In 1978, Kharitonov published his work in form of a research paper in a journal which is well taken for robust control problem [11]. So far, Kharitonov's theorem has been applied to various engineering applications such as mechanical systems [12], vehicle system [13], and an electrical system like DC-DC converter [14], load frequency control [15], and permanent magnet synchronous motor [9]. IMC policy was developed by Garcia and Morari [16], the design strategy of IMC is broadly documented that can realize the robust performance for tracking and disturbance rejection from the system response [17, 18]. The industrial applications of IMC for an electro-mechanical system such as heavy-duty vehicles [17, 19, 20], power system [21], and milling CNCMT system [22] are discussed. The robust H_∞ controller using mixed sensitivity approach addresses the problem of system robustness by designing controllers in presence of noise and disturbance in the system [23, 24]. The method of selecting weighting functions for H_∞ control has been discussed by Tewari et al. [25-27]. The various applications of the H_∞ control theory is presented by Yadav et al. [13, 25, 26]. The modern and artificial intelligence-based control techniques such as fuzzy logic, adaptive, and artificial neural network-based controls of CNCMT is presented in the papers [1, 2, 22, 28, 29]. The benefits of artificial intelligence-based techniques such as fuzzy logic and artificial neural network are to reduce the mathematical complexity of controller design for the actual nonlinear, time-varying, and uncertain systems.

The effects of parametric uncertainty and disturbance on uncertain CNCMT systems with all three designed controllers are analyzed as proposed in this paper. The performance specifications such as OS, settling time (ST) and rise time (RT), and SM are considered for performance analysis of uncertain CNCMT system with designed controllers. A comparative performance analysis of controllers for the CNCMT system is presented to identify the superior controller over all the controllers designed in this paper. The robustness of the CNCMT system against parametric uncertainty using all designed controller is measured in terms of SM. Apart from OS, RT, ST, and SM performance parameters, the input disturbance and sensor noise rejection capability are also taken into consideration for performance analysis of different controllers.

The organization of the paper is as follows: In Section 2 the detailed modeling of uncertain CNCMT system is presented. The design of robust PI, IMC, and H_∞ controllers and stability analysis of the CNCMT system are presented in Section 3. In Section 4, simulation results and discussion of controllers as applied to the CNCMT system are presented, followed by the conclusion in Section 5.

3.1 Kharitonov’s theorem

The polynomial:

$P(s)=\sum_{i=0}^{n} a_{i} s^{n-i}=a_{0} s^{n}+a_{1} s^{n-1}+a_{2} s^{n-2}+\cdots+a_{n-1} s+a_{n}$    (3)

where, $\alpha_{i} \leq a_{i} \leq \beta_{i}, 0 \leq i \leq n$, and α_iand β_i are lower and upper limit values of polynomial respectively.

Kharitonov’s theorem says that the P(s) is stable if four polynomials as given by (4) are stable.

$p_{1}(s)=\alpha_{0} s^{n}+\alpha_{1} s^{n-1}+\beta_{2} s^{n-2}+\beta_{3} s^{n-3}+\cdots$

$p_{2}(s)=\alpha_{0} s^{n}+\beta_{1} s^{n-1}+\beta_{2} s^{n-2}+\alpha_{3} s^{n-3}+\cdots$

$p_{3}(s)=\beta_{0} s^{n}+\alpha_{1} s^{n-1}+\alpha_{2} s^{n-2}+\beta_{3} s^{n-3}+\cdots$

$p_{4}(s)=\beta_{0} s^{n}+\beta_{1} s^{n-1}+\alpha_{2} s^{n-2}+\alpha_{3} s^{n-3}+\cdots$    (4)

The necessary and sufficient condition for robust stability of the third-order system and formula for calculation of SM is given by Hote et al. [14] which is as follows:

$\alpha_{1} \alpha_{2}>\beta_{0} \beta_{3}$ and $S M=\alpha_{1} \alpha_{2}-\beta_{0} \beta_{3}$    (5)

Hence, if the SM is high i.e. more positive, then the system will be more stable, if the SM is zero, then the system is marginally stable, otherwise, the system will be unstable.

3.2 Design of controller using the worst GM and PM

The closed-loop characteristic equation (CLCE) with controller C(s) is written as

$1+G(s) C(s)=0$    (6)

In order to determine the GM, a virtual gain k_v is introduced in series with G(s). Therefore, the CLCE in the form of a polynomial can be written as;

$P(s)=a_{0} s^{3}+a_{1} s^{2}+a_{2} s+a_{3} k_{v}$    (7)

The GM of the system is calculated simply from the s¹ row of the Routh array and the phase crossover frequency ѡ_cp is determined using an auxiliary equation which is derived from the s² row. Considering p₃(s) because it gives the worst value of GM and PM [9, 10] the equation for p₃(s) of (4) using (7) is as follows:

$p_{3}(s)=\beta_{0} s^{3}+\alpha_{1} s^{2}+\alpha_{2} s+\beta_{3} k_{v}$    (8)

Formulate the Routh table and taking higher-order coefficient is equal to 1.

From s¹ row of above Routh table,

$\left(\alpha_{1} \alpha_{2}-\beta_{3} k_{v}\right) / \alpha_{2}=0$    (9)

$k_{v}=\frac{\alpha_{1} \alpha_{2}}{\beta_{3}}=\frac{\text { coefficient of } s^{2} \times \text { coefficient of } s^{1}}{\text { coefficient of } s^{0}}$     (10)

$\mathrm{GM}$ in $\mathrm{dB}=20 \log \frac{\alpha_{1} \alpha_{2}}{\beta_{3}}$    (11)

The auxiliary equation, from s² row of Routh table, is written as:

$\alpha_{2} S^{2}+\beta_{3} k_{v}=0$    (12)

Put s=jw and $k_{v}=\frac{\alpha_{1} \alpha_{2}}{\beta_{3}}$ in (12), it gives w_cp

$\omega=\omega_{c p}=\sqrt{\alpha_{2}}=\sqrt{\text { coefficient of } s^{1}}$    (13)

In order to determine the PM, gain crossover frequency w_cg is determined first which is calculated using the empirical formula as given by [10]:

$\omega_{c g}=\omega_{c p}\left(\frac{1}{k_{v}}\right)^{0.5}$    (14)

From (10), (13), and (14); $\omega_{c g}=\sqrt{\beta_{3} / \alpha_{1}}$ and PM is as follows:

$P M=180^{0}-90^{0}-\tan ^{-1}\left(\sqrt{\alpha_{1}^{3} * \beta_{3}} /\left(\alpha_{1} \alpha_{2}-\beta_{3}\right)\right)$    (15)

In recent research, the typical range of GM and PM is 5-10dB and 50°-70° respectively. To design a robust PI controller the 10 dB of GM and 60° of PM is considered in this paper. From (1) and (6) the CLCE with PI controller can be written as:

$1+\left(\frac{\frac{K_{1} K_{2} n p}{\left(p^{2} m+n^{2} I_{m}\right)}}{s^{2}+\frac{K_{1} K_{2} n H_{2}}{\left(p^{2} m+n^{2} I_{m}\right)} s}\right)\left(K_{P}+\frac{K_{I}}{s}\right)=0$    (16)

where, K_P and K_I are parameters of the PI controller. Formulate a polynomial p(s) using p₃(s) of (4), (16) and values from Table 1, yields (17).

$p(s)=s^{3}+7.5 s^{2}+3 K_{P} s+6 K_{I}$    (17)

The lower and upper limit values of (17) are as follows: α₁=7.5, α₂=3 and β₀=1, β₃=6 respectively.

From (11) and (17),

$G M=20 \log \frac{7.5 * 3 * K_{P}}{6 K_{I}}=10 \Rightarrow K_{P}=0.84 * K_{I}$    (18)

From (15) and (17),

$P M=180^{\circ}-90^{\circ}-\tan ^{-1}\left(\frac{\sqrt{7.5^{3} * 6 * K_{I}}}{7.5 * 3 * K_{P}-6 K_{I}}\right)=60^{\circ}\Rightarrow\left(22.5 K_{P}-6 K_{I}\right)^{2}=7593.75 * K_{I}$    (19)

After solving (18) and (19); we obtain K_p=38.33 and K₁=45.65.

3.3 IMC design theory

IMC has many features such as noise and disturbance rejection capability, tracking of reference trajectory, and insensitive to parameter variations i.e. it makes a robust system [22]. In IMC, a single tunable parameter μ exists as given in (22). The procedure to obtain the suitable value of μ is discussed in appendix-B. The controller parameter μ represents a trade-off between tracking performance and robustness of the system and this is a unique feature of IMC. The generalized architecture of IMC is presented in Figure 3(a) and the proposed simplified IMC block diagram is shown in Figure 3(b), which gives similar performance as given by the IMC as shown in Figure 3(a). It consists of controller C(s), original system (G(s)), and internal model G₀(s). D(s) and N(s) are disturbance and noise in the system respectively.

3.png

Figure 3. (a) Generalized architecture (b) Proposed block diagram of IMC

From Figure 3(b); considering D(s) and N(s) are equal to zero.

$X_{0}(s)=\frac{G(s) C(s) X_{d}(s)}{1+C(s) G(s)}$    (20)

$C(s)=\frac{Q(s)}{1-G_{0}(s) Q(s)}$    (21)

The condition for perfect reference point tracking and effect of parameter variations are obtained, if G₀(s)=G(s) and Q(s)=1/G(s) [18, 22]. If G(s) is proper, then it's inverse i.e. 1/G(s) becomes improper. Hence to make a proper system a low pass filter is incorporated as a part of Q(s). Hence it is defined as:

$Q(s)=\frac{1}{(\mu s+1)^{\lambda}} G(s)^{-1}$    (22)

where, λ is the value which makes Q(s) proper. The low pass filter parameter μ influences the speed of response and robustness of the system. Here taking λ=2 and μ=1 for the design of IMC. The selection procedure of the suitable value of μ for different operations of the CNCMT system is discussed in appendix-B. Therefore, the transfer function of IMC is as follows:

$C(s)=\frac{0.21 s+2}{s+2}$    (23)

3.4 H_∞ control technique

The standard configuration of H_∞ control with weighting functions W₁, W_2, and W₃, and the proposed robust control block for uncertain CNCMT system is shown in Figure 4 (a) and (b) respectively. The motive of controller C(s) is to obtain both tracking and robustness performances under a specified range of system parameters.

In Figure 4(a), G(s) is the system with nominal values; z, y, SV, and u are the controlled output, the measured output, the set value, and the control input respectively, whereas of W₁(s), W₂(s) and W₃(s) are weights to the tracking error, controller transfer function, and system performance respectively. Figure 5 depicts the variation of singular values of the nominal system, plant uncertainty, and weighting function W₁ and W₃ with low to high frequency.

4.png

Figure 4. (a) Standard configuration of H_∞ control with W₁, W_2, and W₃ weighting functions; (b) Proposed robust control block diagram for uncertain CNCMT system

5.png

Figure 5. Plot of singular values of the nominal system, plant uncertainty, and W₁ and W₃

The C(s) is designed such that the H_∞ norm of the sensitivity matrix ||S(jω)||_∞ and the mixed-sensitivity matrix, ||M(jω)||_∞ are minimized, where M(jω) is given by the following equation [13, 25].

$M(j \omega)=\left[\begin{array}{c}W_{1}(j \omega) S(j \omega) \\ W_{2}(j \omega) C(j \omega) S(j \omega) \\ W_{3}(j \omega) T(j \omega)\end{array}\right]$    (24)

The weighting matrices for tracking and robustness performance of H_∞ control is represented as follows [25]:

$\sigma_{\max }(S(j \omega)) \leq \sigma_{\max }\left(W_{1}^{-1}(j \omega)\right)$

$\sigma_{\max }(C(j \omega) S(j \omega)) \leq \sigma_{\max }\left(W_{2}^{-1}(j \omega)\right)$

$\sigma_{\max }(T(j \omega)) \leq \sigma_{\max }\left(W_{3}^{-1}(j \omega)\right)$    (25)

where, σ_max is the maximum singular value. The H_∞ control algorithm used in this work is as follows:

$\|\vartheta M(j \omega)\|_{\infty} \leq 1$    (26)

where, ϑ is a scaling factor. Using a robust control toolbox of MATLAB command ‘hinfopt’ which iterates ϑ until a stabilizing solution which satisfies (26) is obtained. The selection of W₁(s), W₂(s), and W₃(s) are determined with the help of Yang et al. [26]. The weighting transfer functions used in the simulation are given as below:

$W_{1}(s)=\frac{20}{s+0.01}, W_{2}(s)=1$ and $W_{3}(s)=\frac{1}{s+0.1}$    (27)

Iteration number 14 gives an answer under the tolerance of 0.0100, which has ϑ=1.3794*10^-2. The order of the resulting controller is fourth, which is the same as the order of the augmented plant. The stabilizing C(s) is obtained as follows:

$C(s)=\frac{20.12 s^{3}+197.2 s^{2}+19.52 s+2.72 * 10^{-13}}{s^{4}+14.23 s^{3}+54.16 s^{2}+5.879 s+0.05339}$    (28)

Using order reduction command ‘reduce' of MATLAB, the original controller given by (28) is reduced to the first order for easy calculation of robust stability. The reduced-order controller gives almost similar performance as the original higher-order controller. The original and reduced order controller gives almost the same singular value from low to high-frequency range as shown in Figure 6.

6.png

Figure 6. Plot of singular values of original forth and reduced first-order controller C(s)

The transfer function of reduced first order stabilized controller C(s) is as follows:

$C(s)=\frac{20.97}{s+5.5}$    (29)

3.5 Robustness analysis of uncertain CNCMT system in terms of SM

The robust stability analysis of uncertain CNCMT systems with robust PI, IMC, and H_∞ controllers, and using the Kharitonov theorem is presented in this section. For the robust stability analysis and calculation of SM, it is required to calculate the α₁, α_2, β₀ and β₃ as represented in (5). The CLCE of CNCMT system with a robust PI controller is as follows:

$s^{3}+\frac{K_{1} K_{2} n H_{2}}{\left(p^{2} m+n^{2} I_{m}\right)} s^{2}+\frac{K_{1} K_{2} n p}{\left(p^{2} m+n^{2} I_{m}\right)} K_{P} s+\frac{K_{1} K_{2} n p}{\left(p^{2} m+n^{2} I_{m}\right)} K_{I}=0$    (30)

where, K_p=38.33, K₁=45.65, and β₀=1. The uncertainty of CNCMT system parameters as shown in Table 1 is used for calculation of α₁, α_2, and β₃ as shown below:

$\alpha_{1}=\min \left\{\frac{K_{1} K_{2} n H_{2}}{\left(p^{2} m+n^{2} I_{m}\right)}\right\}=7.5$

$\alpha_{2}=\min \left\{\frac{K_{1} K_{2} n p}{\left(p^{2} m+n^{2} I_{m}\right)} K_{P}\right\}=115$ and

$\beta_{3}=\max \left\{\frac{K_{1} K_{2} n p}{\left(p^{2} m+n^{2} I_{m}\right)} K_{I}\right\}=273.9$    (31)

With these values satisfying the condition (5); gives 7.5*15>273.89, hence the system is found stable. From (5) SM is calculated as 588.6.

The CLCE with IMC is given by (32).

$s^{3}+\left\{2+\frac{K_{1} K_{2} n H_{2}}{\left(p^{2} m+n^{2} I_{m}\right)}\right\} s^{2}$

$+\left\{2 * \frac{K_{1} K_{2} n H_{2}}{\left(p^{2} m+n^{2} I_{m}\right)}+0.21\right.$

$\left.* \frac{K_{1} K_{2} n p}{\left(p^{2} m+n^{2} I_{m}\right)}\right\} s+2$

$* \frac{K_{1} K_{2} n p}{\left(p^{2} m+n^{2} I_{m}\right)}=0$    (32)

The values of α₁, α_2, and β₃ are calculated and as given below:

$\alpha_{1}=\min \left\{2+\frac{K_{1} K_{2} n H_{2}}{\left(p^{2} m+n^{2} I_{m}\right)}\right\}=9.5$,

$\alpha_{2}=\min \left\{2 * \frac{K_{1} K_{2} n H_{2}}{\left(p^{2} m+n^{2} I_{m}\right)}+0.21 * \frac{K_{1} K_{2} n p}{\left(p^{2} m+n^{2} I_{m}\right)}\right\}=16.575$

and $\beta_{3}=\max \left\{2 * \frac{K_{1} K_{2} n p}{\left(p^{2} m+n^{2} I_{m}\right)}\right\}=12$    (33)

The stability condition as given by (5) is satisfied; therefore, the uncertain system is found as stable. The SM from (5) is calculated as 9.5*16.575-12=145.4.

The CLCE with H_∞ controller is as follows:

$s^{3}+\left\{5.5+\frac{K_{1} K_{2} n H_{2}}{\left(p^{2} m+n^{2} I_{m}\right)}\right\} s^{2}+\left\{5.5 * \frac{K_{1} K_{2} n H_{2}}{\left(p^{2} m+n^{2} I_{m}\right)}\right\} s+20.97 * \frac{K_{1} K_{2} n p}{\left(p^{2} m+n^{2} I_{m}\right)}=0$    (34)

The values of α₁, α_2, and β₃ are calculated and as given below:

$\alpha_{1}=\min \left\{5.5+\frac{K_{1} K_{2} n H_{2}}{\left(p^{2} m+n^{2} I_{m}\right)}\right\}=13$

$\alpha_{2}=\min \left\{5.5 * \frac{K_{1} K_{2} n H_{2}}{\left(p^{2} m+n^{2} I_{m}\right)}\right\}=41.25$ and

$\beta_{3}=\max \left\{20.97 * \frac{K_{1} K_{2} n p}{\left(p^{2} m+n^{2} I_{m}\right)}\right\}=125.82$    (35)

The stability condition is given by (5) is satisfied; hence the uncertain system is stable. The SM is calculated as follows: 13*41.25-125.82=410.4, which is slightly less than as obtained using a robust PI controller but it is much more than that obtained using IMC. With high SM controller can handle a wider range of parametric uncertainty in the CNCMT system.

In this paper the robust PI, IMC, and H_∞ controllers are designed successfully for uncertain CNCMT system that is subjected to input disturbance and sensor noise. Comparative performance analysis of the controllers is performed and shows that H_∞ controller gives a better and desirable performance index and reasonable SM among all other reported controllers. H_∞ controller and IMC have eliminated measurement noise and disturbance completely as compared to the robust PI controller. Hence H_∞ controller is considered best out of other designed controllers in this paper. This robust control technique may also be applied to aerospace and other electromechanical systems.

This section presents the modeling of the CNCMT system and derivation of the transfer function of the same as represented by (1). Consider Figures 1 and 2 for the complete modeling of the CNCMT system [4].

1. Power Amplifier: The output of the power amplifier is a controlled input E(s) to the DC servomotor as follows:

$E(s)=K_{1}(U(s)-B(s))$    (36)

where, K₁ is the gain of the power amplifier.

2. DC servo motor: The applied voltage e(t) is described as follows:

$e(t)=L_{f} \frac{d i_{f}}{d t}+R_{f} i_{f}$    (37)

where, L_f and R_f are the motor parameters, and i_f is the field current. The motor torque T_m $\propto i_{a} i_{f}$, since $i_{a}$ is constant for field controlled DC motor, therefore T_m=K_f i_f, where K_f is the constant. Assuming field time constant L_f/R_f is small as compared to the mechanical time constant of the motor. Therefore, T_m is as follows:

$T_{M}(s)=K_{2} E(s)$    (38)

The equivalent block diagram of field controlled DC servo motor is shown in Figure 14.

14.png

Figure 14. Equivalent field controlled DC servo motor

3. Gearbox, Lead Screw, and Machine Table: The free body diagram of the gearbox with motor and lead screw shafts are shown in Figure 15.

15.png

Figure 15. Free body diagram of the gearbox with motor and lead screw shafts

For the motor shaft:

$\sum M=I_{m} \frac{d^{2} \theta_{m}}{d t^{2}}$    (39)

$T_{m}(t)-C_{m} \frac{d \theta_{m}}{d t}-a X(t)=I_{m} \frac{d^{2} \theta_{m}}{d t^{2}}$    (40)

where, X(t) is gear tooth reaction force, θ_m is the angular position of the motor, a is the radius of gear connected to the motor and I_m is the equivalent inertia of the motor. Rearranging (40),

$X(t)=\frac{1}{a}\left(T_{m}(t)-C_{m} \frac{d \theta_{m}}{d t}-I_{m} \frac{d^{2} \theta_{m}}{d t^{2}}\right)$    (41)

Taking Laplace transform of (41) with C_m=0.

$X(s)=\frac{1}{a}\left(T_{m}(s)-I_{m} s^{2} \theta_{m}(s)\right)$    (42)

The Free body diagram of the machine table and lead screw shaft is shown in Figure16.

For lead screw shaft assuming: work in = workout.

$b X(t) \theta_{0}(t)=F(t) x_{0}(t)$    (43)

where, b is the radius of gear connected to the lead screw shaft, θ₀ is the angular position of lead screw, F(t) is the force acting on the machine table and x₀ is the linear position of the machine table.

16.png

Figure 16. Free body diagram of the machine table and lead screw shaft

The pitch p of the lead screw is defined as:

$p=\frac{x_{0}(t)}{\theta_{0}(t)}$    (44)

From (43) and (44),

$F(t)=\frac{b X(T)}{p}$    (45)

The equation of motion for the machine table is written as follows:

$F(t)=m \ddot{x}_{0}$    (46)

Equating (45) and (46) and taking Laplace transform.

$X(s)=\frac{1}{b}\left(p m s^{2} X_{0}(s)\right)$    (47)

Equating (42) and (47) gives (48).

$p m s^{2} X_{0}(s)=\frac{b}{a}\left(T_{m}(s)-I_{m} s^{2} \theta_{m}(s)\right)$    (48)

where, $\frac{b}{a}=n$ i.e. gear ratio and

$\theta_{m}(s)=n \theta_{0}(s)$    (49)

Hence,

$s^{2} \theta_{m}(s)=n s^{2} \theta_{0}(s)$    (50)

From (44),

$G_{0}(s)=\frac{X_{0}(s)}{p}$    (51)

From (48) to (51),

$p m s^{2} X_{0}(s)=n T_{m}(s)-n I_{m} \frac{n}{p} s^{2} X_{0}(s)$    (52)

Hence the open-loop transfer function of the gearbox, lead screw, and machine table is obtained after rearranging (52), which is as follows:

$\frac{s X_{0}(s)}{T_{m}(s)}=\frac{n}{\left(p m+\frac{n^{2} I_{m}}{p}\right) s}$    (53)

4. Tachogenerator: The feedback signal B(s) is given as:

$B(s)=H_{2} s \theta_{0}(s)$ or $B(s)=H_{2} s X_{0}(s) / p$    (54)

where, H₂ is the gain of tachogenerator.

5. Position transducer: Feedback signal X_m(s) is given as follows:

$X_{d}(s)=H_{1} X_{0}(s)$    (55)

where, H₁ is the gain of shaft encoder. The overall system is represented in Figure 2 is a block diagram of the CNCMT system. The complete open-loop transfer function of the CNCMT system is given by (1).