Robust IMC–PID tuning for cascade control systems with gain and phase margin specifications

Abstract

In this article, an internal model control plus proportional-integral-derivative (IMC–PID) tuning procedure for cascade control systems is proposed based on the gain and phase margin specifications of the inner and outer loop. The internal model control parameters are adjusted according to the desired frequency response of each loop with a minimum interaction between the inner and outer PID controllers, obtaining a fine tuning and the desired gain and phase margins specifications due to an appropriate selection of the PID controller gains and constants. Given the design specifications for the inner and outer loop, this tuning procedure adjusts the IMC parameter of each controller independently, with no interference between the inner and outer loop obtaining a robust method for cascade controllers with better performance than sequential tuning or other frequency domain-based methods. This technique is accurate and simple, providing a convenient technique for the PID tuning of cascade control systems in different applications such as mechanical, electrical or chemical systems. The proposed tuning method explained in this article provides a flexible tuning procedure in comparison with other tuning procedures because each loop is tuned simultaneously without modifying the robustness characteristics of the inner and outer loop. Several experiments are shown to compare and validate the effectiveness of the proposed tuning procedure over other sequential or cascade tuning methods; some experiments under different conditions are done to test the performance of the proposed tuning technique. For these reasons, a robustness analysis based on sensitivity is shown in this article to analyze the disturbance rejection properties and the relations of the IMC parameters.

Appendices

Appendix 1

The derivations of the inner loop IMC parameters are obtained by substituting (19) and (20) in (30)–(33) obtaining

$$\frac{{\tau_{1} \omega_{p1} }}{{K_{c1} K_{1} }}\sqrt {\frac{{\left( {\tau_{1}^{2} \omega_{p1}^{2} + 1} \right)(\tau_{2}^{2} \omega_{p1}^{2} + 1)}}{{\left( {\omega_{p1}^{2} \tau_{i1}^{2} + 1} \right)(\tau_{d1}^{2} \omega_{p1}^{2} + 1)}}} = A_{m1}$$

(55)

$$- \frac{\pi }{2} + \arctan (\tau_{i1} \omega_{p1} ) + \arctan (\tau_{d1} \omega_{p1} ) + - \arctan (\tau_{1} \omega_{p1} ) - \arctan (\tau_{2} \omega_{p1} ) - \omega_{p1} \theta_{1} + \pi = 0$$

(56)

$$\frac{\pi }{2} + \arctan (\tau_{i1} \omega_{g1} ) + \arctan (\tau_{d1} \omega_{g1} ) + - \arctan (\tau_{1} \omega_{g1} ) - \arctan (\tau_{2} \omega_{g1} ) - \omega_{g1} \theta_{1} = \phi_{m1}$$

(57)

$$\frac{{K_{c1} K_{1} }}{{\tau_{1} \omega_{g1} }}\sqrt {\frac{{\left( {\omega_{g1}^{2} \tau_{i1}^{2} + 1} \right)(\tau_{d1}^{2} \omega_{g1}^{2} + 1)}}{{\left( {\tau_{1}^{2} \omega_{g1}^{2} + 1} \right)(\tau_{2}^{2} \omega_{g1}^{2} + 1)}}} = 1$$

(58)

Therefore, the relations of the gain and phase margin with the controller parameters are obtained. These equations provide all the relations between the gain and phase margin and their respective crossover frequencies, necessary for the design of the IMC–PID tuning method for the inner loop feedback controller.

Substituting the IMC–PID gains and constants for the feedback controller C ₁(s) defined in (21)–(23) yields

$$A_{m1} = \frac{{\omega_{p1} }}{{\omega_{g1} }}$$

(59)

With the crossover frequency ω _p1

$$\omega_{p1} = \frac{{\frac{\pi }{2}}}{{\theta_{1} }}$$

(60)

And

$$\omega_{g1} = \frac{{\frac{\pi }{2} - \phi_{m1} }}{{\theta_{1} }}$$

(61)

The IMC parameter in terms of the gain margin A _m1 is given by

$$\lambda_{1} = \frac{{A_{m1} }}{{\omega_{p1} }} - \theta_{1}$$

(62)

The IMC parameter λ ₁ is now adjusted according to the desired gain margin and phase margin specifications. The resulting inner loop feedback system is tuned independently of the outer loop, and the robustness requirements can be analyzed with the sensitivity peak of the inner given in (63)

$$M_{s1} = \mathop {\hbox{max} }\limits_{\omega } \left| {S_{1} (j\omega )} \right| = \mathop {\hbox{max} }\limits_{\omega } \frac{1}{{\left| {1 + C_{1} \left( {j\omega } \right)G_{1} (j\omega )} \right|}}$$

(63)

The gain margin of the inner loop process can be defined in terms of the maximum sensitivity peak as

$$A_{m1} = \frac{1}{{\left| {C_{1} \left( {j\omega_{p1} } \right)G_{1} (j\omega_{p1} )} \right|}} = \frac{{M_{s1} - 1}}{{M_{s1} }}$$

(64)

A lower bound for the phase margin of the inner loop tuning method can be established using the relation between the phase margin and the maximum sensitivity peak given in (64)

$$\phi_{m1} \ge 2\sin^{ - 1} \left( {\frac{1}{{2M_{s1} }}} \right)$$

(65)

Using (64) and the crossover frequency ω _p1 (60), the lower bound for the phase margin ϕ _m1 is given by

$$\phi_{m1} \ge 2\sin^{ - 1} \left( {0.5\left( {1 - \frac{{K_{c1} K_{1} }}{{\omega_{p1} \tau_{I1} }}} \right)} \right)$$

(66)

Appendix 2

The outer loop IMC parameters are obtained by substituting (25) and (36) in the properties of the gain and phase margin (30)–(33) with i = 2 yields

$$\frac{{\tau_{i2} \omega_{p2} }}{{K_{c2} K_{2} }}\sqrt {\frac{{\left( {\lambda_{1}^{2} \omega_{p2}^{2} + 1} \right)(\tau_{3}^{2} \omega_{p2}^{2} + 1)}}{{\left( {\tau_{i2}^{2} \omega_{p2}^{2} + 1} \right)(\tau_{d2}^{2} \omega_{p2}^{2} + 1)}}} = A_{m2}$$

(67)

$$- \frac{\pi }{2} + \arctan (\tau_{i2} \omega_{p2} ) + \arctan (\tau_{d2} \omega_{p2} ) + - \arctan (\lambda_{1} \omega_{p2} ) - \arctan (\tau_{3} \omega_{p2} ) - \omega_{p2} (\theta_{1} + \theta_{2} ) + \pi = 0$$

(68)

$$\frac{\pi }{2} + \arctan (\tau_{i2} \omega_{g2} ) + \arctan (\tau_{d2} \omega_{g2} ) + - \arctan (\lambda_{1} \omega_{g2} ) - \arctan (\tau_{3} \omega_{g2} ) - \omega_{g2} (\theta_{1} + \theta_{2} ) = \phi_{m2}$$

(69)

$$\frac{{K_{c2} K_{2} }}{{\tau_{i2} \omega_{g2} }}\sqrt {\frac{{\left( {\tau_{i2}^{2} \omega_{g2}^{2} + 1} \right)(\tau_{d2}^{2} \omega_{g2}^{2} + 1)}}{{\left( {\lambda_{1}^{2} \omega_{g2}^{2} + 1} \right)(\tau_{3}^{2} \omega_{g2}^{2} + 1)}}} = 1$$

(70)

Substituting the IMC–PID gains and constants for the feedback controller C ₂(s) described in (27)–(29) yields the following relations

$$A_{m2} = \frac{{\omega_{p2} }}{{\omega_{g2} }}$$

(71)

With the crossover frequency ω _p2

$$\omega_{p2} = \frac{{\frac{\pi }{2}}}{{(\theta_{1} + \theta_{2} )}}$$

(72)

And

$$\omega_{g2} = \frac{{\frac{\pi }{2} - \phi_{m2} }}{{\theta_{1} + \theta_{2} }}$$

(73)

The tuning parameter of the IMC–PID controller for the outer loop λ ₂ is adjusted in terms of the specified gain margin for the outer loop, and as it can be seen, this tuning parameter is independent of the tuning parameters of the inner loop, so as it can be seen on the following section, the calculated values for the tuning parameters for λ ₁ and λ ₂ yield an independent frequency domain controller design methodology for the inner and outer loop taking into account the robustness considerations.

The controller parameter λ₂ is adjusted by the formulae (74)

$$\lambda_{2} = \frac{{A_{m2} }}{{\omega_{p2} }} - (\theta_{1} + \theta_{2} )$$

(74)

The bounds for the gain and phase margin for the tuning method of the outer loop can be obtained from the maximum sensitivity peak of the equivalent cascade process. For the outer loop, the gain and phase margin relations of the tuning method obtained from the maximum sensitivity peak of the equivalent cascade process are derived from the equivalent system C ₂(s)G _p2(s) as it is shown in (75)

$$M_{s2} = \mathop {\hbox{max} }\limits_{\omega } \left| {S_{2} (j\omega )} \right| = \mathop {\hbox{max} }\limits_{\omega } \frac{1}{{\left| {1 + C_{2} \left( {j\omega } \right)G_{p2} (j\omega )} \right|}}$$

(75)

Then, the gain margin in terms of the maximum sensitivity peak is given by

$$A_{m2} = \frac{1}{{\left| {C_{2} \left( {j\omega_{p2} } \right)G_{p2} (j\omega_{p2} )} \right|}} = \frac{{M_{s2} - 1}}{{M_{s2} }}$$

(76)

The phase margin lower bound for the tuning method of the outer loop in terms of the maximum sensitivity peak is given by

$$\phi_{m2} \ge 2\sin^{ - 1} \left( {\frac{1}{{2M_{s2} }}} \right)$$

(77)

Using (76) and ω _p2 described in (72), the lower bound for the phase margin of the inner loop feedback controller is as follows:

$$\phi_{m2} \ge 2\sin^{ - 1} \left( {0.5\left( {1 - \frac{{K_{c2} K_{2} }}{{\omega_{p2} \tau_{I2} }}} \right)} \right).$$

(78)