Equivalence Analysis of Cascade Control for a Class of Cascade Integral Systems

A class of generalized proportional-integral-derivative (PID) control with feedforward compensation (FFC) is obtained for a class of cascade integral systems by equivalence analysis of the cascade control and such the analysis is obtained by using the information of the model and outermost loop feedback. Firstly, a new type of error related to the traditional error such as proportional (P) or proportional integral (PI) is given. Secondly, by analyzing cascade control for a class of ideal cascade integral systems, the generalized PID control with FFC is presented based on the proposed new type of error. Then, the generalized PID control with FFC is extended to a class of non-ideal cascade integral systems to reduce the number of feedback loop and sensors. Finally, the simulation results of the speed/position servo system of direct current motor are given to verify the theoretical analysis results.


I. INTRODUCTION
Electrical machine drive system used to develop electrification has been widely applied to aerospace, aviation, navigation, land transportation and other scenarios. Especially, developing high reliable and efficient electrification technology is of critical importance for the aviation industry due to the following advantages. Firstly, it can sharply decrease the workload of maintenance and enhance the flexibility of operability [1]. Secondly, it can significantly reduce manmade CO 2 emissions [2], which will make civil aircraft more environmentally friendly. Furthermore, the revolutionary of full electrification eliminates mechanical, hydraulic and pneumatic power systems for auxiliary power unit or starter/generator in aircraft, which can lighten the weight of aircraft and then reduces the specific fuel consumption of aircraft engine [1]. Furthermore, electrical machines are the significant part of the more electric aircraft [1].
The associate editor coordinating the review of this manuscript and approving it for publication was Qi Zhou.
Control structure is one of the most significant parts for electrical machine drive system, which affect the operation performance. The most common used control structure of electrical machine drive system is two-stage or three-stage cascade control structures (see [3], [4], [5]). To be specific, the starter/generator of gas turbine engine always adopts twostage proportional-integral (PI) cascade control to realize the speed tracking control or torque tracking control in starter mode and set-point voltage control in generator mode [6]. The same cascade control structure is also applied in electric fuel pumps scenarios. Furthermore, electro-hydrostatic actuators (EHA) and electro-mechanical actuators (EMA) adopt three-stage proportional-integral cascade control in more electric aircraft (MEA) and more electric engine (MEE) [7]. Besides, electric flap surface actuator, one of the most significant pieces of equipment on the aircraft, adopts three-stage PI cascade control to achieve a desired control effect [8], [9].
In addition, cascade control structures are introduced in [10], [11], [12], [13], and [14]. Cascade control structure is realized by sensors feedback. For example, the three-stage VOLUME 11, 2023 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ cascade control of electrical machines includes the outermost loop sensor named position loop, the middle loop named speed loop and the inner loop named current loop. The corresponding feedback is realized by position sensors, speed sensors and current sensors, respectively. However, not all states are measurable in the engineering scenarios, such as the solar panel of space station. Therefore, in such cases, the state feedback method may not be applied, which may be replaced by observers based on output feedback (see [15]). Or some other elaborated robust algorithms are thoroughly studied in [16], [17], [18], [19], and [20] to address the similar problems.
In the cascade control structures, while many sensors used for feedback control may increase hardware costs of control system and the probability of equipment failure (see [21], [22]). In addition, the existence of uncertainties or delays in the sensors is inevitable [23], [24]. Decreasing the number of sensors in cascade control is encouraged to be investigated. Thus, it is necessary to propose a new control structure to reduce the dependence on sensors feedback without sacrificing control performance. In the other hand, some models have also been transformed into a cascade form such that some analysis methods can be applied, i.e., the ''analytic'' cascade system. For example, the reference transforms linear systems into a cascade form to address the mismatching disturbances [25]. In [26], the method is used to analyze the existence of solution to the distributed optimal coordination. Motivated by the above observations, this paper proposes a class of new control structures relative to the generalized PID control by making the equivalent analysis of cascade control based on model information.
The main contributions of this paper can be summarized as follows. Firstly, for any cascade integral system, we propose a new generalized PID control structure with FFC based on generalized error, which can be used to make the equivalence analysis for different kinds of cascade control structures. This point is proved mathematically in this paper. Secondly, compared with the traditional methods [10], [11], [12], [13], [14], the proposed method uses only the outermost loop feedback which can reduce the number of feedback loops and sensors and without sacrificing control performance. In addition, direct current motor speed/position systems are given to verify the theoretical analysis and also show that the control performance of the generalized PID control with FFC is satisfactory.
The rest of this paper is organized as follows. Section 2 formulates the problem. The generalized PID control with FFC derived from cascade control structures is presented for ideal cascade integral systems in Section 3. Section 4 gives the generalized PID control with FFC for non-ideal cascade integral systems. Section 5 gives simulation results and discussions. Conclusions and future work appear in Section 6.

II. PROBLEM FORMULATIONS
The current and speed cascade system of a direct current (DC) motor shown in (1) and the current, speed and position cascade system of a DC motor presented in (2) are prevalent.
where i a is the armature current, θ is the angle of the DC motor rotor, R a is the armature resistance, L a is the armature inductance, ω m is the rotor angle speed, K b is the back electromotive force coefficient, υ a is the input voltage, J m is the rotor inertia, B m is the friction coefficient, T L is the load torque disturbances and K t is the torque constant.
The traditional error is defined as the error between the reference value and the feedback value, i.e., e 0 = x ref 11 − x 11 , where x ref 11 and x 11 are given in Figure 1. In this paper, we employ the definition of generalized error in control systems, namely P, integral (I), proportional-derivative (PD), PID or other types of the traditional error, to obtain the corresponding generalized PID control.
Note that (1) illustrates that the outermost loop controlled variable is the speed. (2) shows that the outermost loop controlled variable is the position. The three-stage and twostage cascade PI control structures are prevalently adopted to regulate the position and the speed in electrical machine systems, respectively. By analyzing the cascade control, we aim to reveal the relationship between the cascade control and its corresponding generalized PID control. Firstly, we analyze the ideal cascade integral systems. Then the corresponding results are extended to the non-ideal cascade integral systems. Furthermore, the generalized PID control will be presented as the PI m 1 I m 2 D n 1 D n 2 control, where n i ∈ N , m i ∈ N and i ∈ {1, 2}. N is a set of nonnegative integers.k stands for dk dt .

III. GENERALIZED PID CONTROL FOR IDEAL CASCADE INTEGRAL SYSTEMS
In this section, for a class of cascade integral systems, the analysis of cascade control structures, namely PI control, P control and hybrid control of PI and P, is presented in details.
(1) and (2) can be recast as where . In order to simplify the analysis of cascade control, when f 1 = 0 and f 2 = 0, (3) can be generalized to be described by (5), (6). (4) can be generalized to be described by (7), (8) and (9). Thus, an ideal second-order cascade system and an ideal third-order cascade system, illustrated in Figure 1 and Figure 2, are ideal cascade integral systems.
The prerequisite is that x ref 11 is constant reference value. x 11 , x 21 and x 31 are the feedback values which can be measured. u 1 is control input. Y is a measurable output vector. a 121 , a 122 , b 2 , b 3 are positive constants. kp 11 , kp 21 , kp 31 , ki 11 , ki 21 and ki 31 are gain coefficients of controllers.  To be more specific, the plant in the second-order cascade integral system of the closed-loop in Figure 1 can be given as follows:ẋ The plant in the third-order cascade integral system of the closed-loop in Figure 2 can be described as follows:

A. EQUIVALENCE ANALYSIS OF PI CASCADE CONTROL
In this subsection, one lemma is presented to show such generalized PID control.

Lemma 1: For the systems which can be changed into the ideal cascade integral systems, the corresponding cascade control structures based on PI control can be interpreted as a class of the generalized PID control with FFC based on the generalized error.
Proof: In this subsection, two-stage cascade control system will be first presented. The C11 and C21 in Figure 1 can be shown as follows: The equations (10) and (11) can be illustrated in Figure 3.
Substituting (5) and (10) into (11), we obtain Remark 1: The generalized error can be defined as the PI of the traditional error, which are shown in equations (13), (18) and (39). Because x ref 1 is constant, then (12) can be recast as Define the generalized error then we have The equation (14) can be shown in Figure 4, which reveals that (14) can be interpreted as a generalized PID controller with FFC. Next, we show the analysis of three-stage cascade control system. In Figure 2, C11, C21 and C31 adopt PI controllers yielding The equations (15), (16) and (17) can be described as in Figure 5. Substituting (7), (8), (15) and (16) into (17), we have    Define the generalized error Then u 1 is rewritten as The (19) can be seen as a generalized PII 2 DD 2 controller with FFC, which can be illustrated in Figure 6. The proof of Lemma 1 is finished. ■

B. EQUIVALENCE ANALYSIS OF P CASCADE CONTROL
In this subsection, we use one lemma to show the analysis of P cascade control. Lemma 2: For the systems which can be changed into the ideal cascade integral systems, the corresponding cascade control structures based on P control can be interpreted as a class of the generalized PID control based on the generalized error.
Proof: We first consider two-stage cascade control system. In Figure 1, both C11 and C21 use P control, which can be described as follows: The equations (20) and (21) are shown in Figure 7. Substituting (5) and (20) into (21), we can obtain Remark 2: The generalized error can be defined as the P of the traditional error, which are illustrated in the equations (23), (29) and (34).
Define the generalized error  Then (22) can be rewritten as Therefore, (24) shown in Figure 8 can be seen as a PD controller. Next, three-stage cascade control system is analyzed. All C11, C21 and C31 employ P control, which can be written as follows: The equations (25), (26) and (27) are shown in Figure 9. Substituting (7), (8), (25) and (26) So (30) can be regarded as a generalized proportional derivative derivative-derivative (PDD 2 ) controller, which is shown in Figure 10. The proof of Lemma 2 is finished. ■

C. EQUIVALENCE ANALYSIS OF HYBRID CASCADE CONTROL
In some industrial scenarios, the outer loop adopts PI control structure and the inner loop uses P control structure so as to increase the response speed of the manipulated variables. We use the following lemma to show this point.

Lemma 3: For the systems which can be changed into the ideal cascade integral systems, the corresponding cascade control structures based on hybrid control of P and PI can be interpreted as a class of the generalized PID control with FFC based on the generalized error.
Proof: We first consider two-stage cascade control system. C11 and C21 in Figure 1 are designed as PI and P controllers, respectively, which are shown in Figure 11. The mathematical expressions can be described as follows: Substituting (5) and (31) Thus, (35) can be regarded as a generalized PID controller, which is illustrated in Figure 12. Now we consider three-stage cascade control system. Controllers C11, C21 and C31 in Figure 2 employ P, PI and PI, respectively, which can be illustrated as follows: The equations (36), (37) and (38) are shown in Figure 13. Substitute (7), (8), (36) and (37) into (38).

Remark 5:
The generalized error can be defined as the PI of the traditional error. and define the generalized error Then we obtain So (40) can be treated as a generalized PIDD 2 controller with FFC, which is illustrated in Figure 14. The proof of Lemma 3 is finished. ■

D. A GENERALIZED PID CONTROL
In subsections A, B, C derivation process is another way to analyze cascade control for the second-order and third-order cascade integral systems. To show the analysis of cascade control, we present the following theorem based on the conclusions of Lemma 1, Lemma 2 and Lemma 3. Theorem 1: For the systems which can be changed into the cascade integral systems, the corresponding cascade control structures based on P control, PI control or hybrid control of P and PI can be interpreted as a class of the generalized PID control with FFC based on the generalized error.
Proof: The conclusion of Theorem 1 can be easily proved by referring to Lemmas 1, 2 and 3. To be more specific, Lemma 1 illustrates that the cascade control based on PI control can be interpreted as a class of the generalized PID control with FFC; Lemma 2 presents that the cascade control based on P control is regarded as a class of the generalized PID control; Lemma 3 shows that the cascade control based on hybrid control of P and PI can be interpreted as a class of the generalized PID control with FFC. So the proof of Theorem 1 is finished. ■

IV. GENERALIZED PID CONTROL FOR NON-IDEAL CASCADE INTEGRAL SYSTEMS
Section III illustrates the generalized PID control for ideal cascade integral systems. However, it is prevalent that the controlled plants are non-ideal cascade integral systems. Thus, it is meaningful to extend the results of Section III to non-ideal cascade integral systems. Thus, we propose the generalized PID control with FFC for speed/position servo systems of direct current motor, which is originated from the cascade control.

Lemma 4: For the systems (1), the corresponding cascade control structures based on PI control with FFC can be interpreted as a class of the generalized PID control with FFC.
Proof: As for (1), there exists a PI-PI cascade structure, which can be illustrated as follows: where ω * m and i * a are reference values. Substituting (42) into (41), we can obtain According to i a = J mωm K t + ω m B m K t + T L K t , (43) can be recast as Defining the error e 7 = ω c (ω * m −ω m ), (44) can be rewritten as In (45), the mathematical terms −

Lemma 5: For the systems (2), the corresponding cascade control structures based on hybrid control of P and PI control with FFC can be interpreted as a class of the generalized PID control with FFC.
Proof: For (2), it is prevalent that the position loop adopts P controller, and the speed loop and current loop both use PI controllers. That is to say, ω * m =k p (θ * m − θ m ), where θ * m is a reference value. Then we have Fromθ m = ω m , (46) can be recast as Defining e 8 = k p (θ * m − θ m ), (47) can be rewritten as In (48), the mathematical terms −ω c L a dt can be seen as FFC. Thus, (48) can be regarded as a PII 2 controller with FFC. The proof of Lemma 5 is finished. ■

V. THE STABILITY ANALYSIS OF CLOSED-LOOP UNDER EQUIVALENCE CONTROL
In this part, the stability analysis of closed-loop for PI cascade, P cascade control and hybrid cascade control under equivalence control is presented. As for two-stage PI cascade control, its transfer function can be described as: where X 11 (s) and X As for three-stage PI cascade control, its transfer function can be depicted by As for two-stage P cascade control, its transfer function can be presented as: where N 3 = kp 11 kp 21 a 121 b 2 , D3 = s 2 + kp 21 b 2 s + kp 11 kp 21 a 121 b 2 . The transfer function of its equivalence control is the same as equation (51). According to theory of frequency domain analysis, there exists kp 11 , kp 21 such that the poles of the closed-loop transfer function (51) locate in the left half plane of the complex frequency domain.
As for three-stage P cascade control, its transfer function can be illustrated as: where N 4 = kp 11  As for the P-PI-PI cascade control, its transfer function can be shown as: where As for the PI-P cascade control, its transfer function can be depicted as: where N 6 = a 121 b 2 kp 11 kp 21 s + a 121 b 2 ki 11 kp 21 , D6 = s 3 + (b 2 kp 21 )s 2 + (a 121 b 2 kp 11 kp 21 )s + a 121 b 2 ki 11 kp 21 . The transfer function of its equivalence control is the same as equation (54). According to theory of frequency domain analysis, there exists kp 11 , ki 11 , kp 21 such that the poles of the closed-loop transfer function (54) locate in the left half plane of the complex frequency domain.

VI. SIMULATION AND DISCUSSIONS
In order to verify the control effect of the proposed generalized PID control, some related simulation results are illustrated as follows: R a = 0.605 , L a = 0.210e −3 H , J m = 86.57e −7 kg · m 2 , B m = 4.2167e −5 Nm/(rad/s), K b = 0.0233V /(rad/s) and K t = 0.0234Nm/A. The parameters of controllers are shown as follows: ω c = 2000, ω s = 100 and k p = 5. We adopt integral time multiplied absolute error (ITAE), integral of squared error(ISE),integral of time multiplied squared error(ITSE) and integral of absolute error(IAE) to evaluate the performance of controllers.

A. PARAMETER TUNING RULES
In order to ensure the fairness of simulation comparison, we give the corresponding parameter tuning rules. The corresponding parameters of the PII 2 D is from the equation (45). The corresponding parameters of the PII 2 D with FFC is from the equation (45). The corresponding parameters of the PII 2 is from the equation (48). The corresponding parameters of the PII 2 with FFC is from the equation (48). The parameters of the above controllers can be calculated by using equation (45), equation (48), the parameters of (1) and (2), ω c = 2000, ω s = 100 and k p = 5.

B. SIMULATION RESULTS AND DISCUSSIONS
For the constant speed tracking and constant position tracking conditions, the plant faces step disturbance (T L step change) at t=5s. Figure 15 illustrates that the closed-loop response of PII 2 D control with FFC is the same as that of PI-PI control with FFC structure, which verifies that the PII 2 D control with FFC is equivalence control of the PI-PI control with FFC structure. It also shows that the PII 2 control has slight overshoot without consideration of FFC. The worst one is the PI-PI control structure. From the ITAE value perspective, the PII 2 D control with FFC and the PI-PI control with FFC are slightly better than the PII 2 D control and PI-PI control structure in Figure 16.

1) CONSTANT SPEED TRACKING
However, from Figures 17-19, the PII 2 D control is superior to the the PII 2 D control with FFC and the PI-PI control with FFC in consideration of IAE, ISE and ITSE. Thus, the evaluation index of controller performance also affects the    selection of controllers. Furthermore, other factors should be considered in the selection of controllers, which is beyond the scope of this article.      faces the disturbances, which is illustrated in Figure 20. Figures 20-24 validate that the PII 2 control with FFC is the equivalence control of the P-PI-PI with FFC. Figures 20-24 also illustrate that the PII 2 control is inferior to the P-PI-PI with FFC and PII 2 control with FFC.

VII. CONCLUSION
This paper has obtained the generalized PID control by equivalence analysis of cascade control for a class of cascade integral systems, which is based on the renewal type of the error. To be more specific, it has proved that the PI-PI cascade control, PI-PI-PI cascade control, P-PI-PI cascade control, PI-P cascade control, P-P cascade control and P-P-P cascade control can be interpreted as the proposed generalized PID control with FFC for ideal cascade integral systems. Moreover, the generalized PID control with FFC is extended to the non-ideal cascade integral systems, which has been verified in the MATLAB/Simulink environment. The possible future work will consider input delays based on the proposed analysis method.