Angular displacement model is given by

(1)

where ^{_{Tm (t}} ) is the torque generated by the electrical element given by

(2)

and ^{_{Te (t}} ) the sum of viscous friction and load torque ^{_{T (t}} )^,

(3)

Derivatives of angular position ^{_{θm (t}} ) are represented in terms of angular frequency ^{_{ωm (t}} ) interchangeably. The armature circuit model is given by

(4)

where, ^{_{Im (t}} ) ^{_{, Vn (t}} ) and ^{_{ωm (t}} ) are the inductance current, voltage input and angular frequency respectively. The back EMF (electro motive force) is defined by

(5)

Under a constant flux and by assuming maximum efficiency the model can be written as

(6)

Angular position is limited to

(7)

Process parameters are summarized in Table 1.

Table 1

Source: Adapted from [15].

Control requirements are summarized next. Given a reference angular position signal ^{_{yr (t}} ), the shaft angular position ^{_{θm (t}} ) is delayed and modified as follows

(8)

where ^{_{ηθ (t}} ) is the residual tracking error defined as the difference between the total tracking error and the nominal tracking error imposed by the target delay and the performance rate ^{_{yr (t}} ), i.e.,

(9)

Continuous time notation is used since it is expected these requirements must be fulfilled by true angular position. However, errors are calculated and used by the controller from discrete time signals and measurements, as it will be shown in section 0. The disturbance rejection requirements are expressed in frequency terms, using the sensitivity function

(10)

where ^_ωrd correspond to the common called 0dB output disturbance rejection (Sensitivity) bandwidth.

The design model can be represented as a pair of elements as in Fig. 2 the design dynamics and the error dynamics.

For the design dynamics, the state vector ^x can be partitioned in a controllable part state ^xc and a disturbance state vector ^xd , whose dynamics is represented by * and ^D in Fig. 2. Design dynamics is driven by the command signal ^u and the noise ^w .The overall noise can be w

subdivided into, ^u, the noise entering the controllable states, and ^w _d, the signal driving the disturbance state.

Parametric uncertainty is denoted as a bounded function m (•) defined as

(12)

where, m (•) stands for the known information term whereas Δm (•) describes the unknown part, dependent on a bounded parameters vector p _m . The last uncertainty in the design model is the neglected dynamics and interconnections denoted with E in Fig. 2. Its output e _m together with the model output y _m , provides what is expected to be the process output y. According to this the design model can be written as

(13)

where,

(14)

with and . Assume n = nc + nd and ny = nu. No stability assumptions are imposed, the pairs (Ac, Bc) and (A, G) are controllable and pair (A, C) is observable.

The unknown disturbance dynamics is given by

(15)

The term d_m is the response to w and belongs to a signal class D _m , which has a forced response that at time i + k, k > 0 is independent of x _c (i - h) and u (i - h), h ≥ 0. Term d _m may affect any controllable state (it is referred to as non-collocated), whereas the interaction m is collocated.

The model error e _m is the output of the fractional error dynamics E accounting for neglected dynamics or interconnections. Model error can be written as

(16)

3.1.1. The case study

According to the design model in section 0, at least a second order integrator chain must be considered, that is,

(17)

The design model block diagram is in Fig. 3. A first order disturbance dynamics has been accounted. The neglected dynamics is due to the electrical part (armature circuit) and the transfer function ^{_{Ve (z}} ) in (16) is

(18)

where ^_τe = L / RT.

The embedded model makes use of the controllable part and the disturbance dynamics imposed in the design model. Different notations are used for distinguishing between design model and EM. For the EM 'hat' and 'bar' notation is included. 'hat' applies to one-step predicted variables ,'bar' applies to signals which cannot be predicted, e.g., noises. For the EM the parametric uncertainty is set to be only the known component, i.e and ^em = ⁰ is assumed this is,

(19)

Here the EM is considering that the disturbance is absorbing the unknown term associated to Δm, this pose a stability problem and must be taken into account during the design process by the definition of a proper transfer function H (z) defined in [12]. For the case study m (•) is related to the neglected back EMF interconnection thus a discrete time integrator

(20)

The Stability of the entire control loop is guaranteed by the stability conditions introduced in [12], this is,

(21)

Fig. 3 Source: The Authors.

In addition to (17)-(19), the implementation of the EM must be completed with the uncertainty estimator. The general form of the dynamic uncertainty estimator is [18]:

(22)

this is, the model e-ror is filtered by the dynamical system .

Asymptotic stability conditions are imposed to (22). This transfer function closes the loop with (19), and provides updated estimations of the noises feeding the disturbance dynamics and the controllable states. Then, the overall state prediction equation in compact form is given by

(23)

where,

(24)

It can be proven that prediction error dynamics reads

(25)

with,

(26)

This fact allows the prediction error to be written as a transfer function of uncertain input signals, i.e.,

(27)

Term Λ (F_m) represents the set of eigenvalues of matrix ^_Fm . Equation (27) explains the roles of V _m and S _m . Here, the larger is the S _m bandwidth, the better the estimation of d . On the other hand the shorter is the V _m bandwidth, the better the rejection capacity of uncertainty effect on e _m . An important result can be established.

The prediction error in (27) is bounded, if and only if is asymptotically stable, i.e., and, Δm and e _m are bounded. Proof. See [12].

3.3.1. Case study

For the system presented in (17), the uncertainty estimator must be selected to be a dynamic one. This is,

(28)

This selection has a physical meaning. The rigid body displacement involves a second order chain of integrators, as said before. However, the input to velocity-position integrator should not include a noise signal as can be seen in Fig. 3. The reason for this is that the derivative relationship between position/velocity is a mathematical abstraction, but physically there is no signal noise affecting this link. The lack of this noise fedback imposes the necessity of considering the dynamic filter X _p (t) in (28). Two eigenvalues sets were designed. The first set, Λ₁ ( ^_Fm ) is such to guarantee a flat frequency response of the transfer function from reference to measurement. The second set Λ₂ ( ^_Fm ) is such to perform better disturbance rejection. Both sets of eigenvalues are in Table 2.

The Fig. 4 show the sennsitivity function ^_Sq and complement function ^_Vq =1- ^_Sq of the state predictor. A BW wider than 15 Hz is necessary to achieve the disturbance rejection performance. For the case study, stability conditions (21) are guar_banteed by a proper selection of the embedded model gain b_a. Proof of stability conditions by means of a polar plot is in Fig. 5, where it can be observed how the overall sensitivity ^S is capable to force ^H (dashed line) to be inside the unit circle as demanded by (21) that for this study case becomes the more critical stability condition.

Fig. 4 Source: The authors.

Fig. 5 Source: The authors.

The reference generator or reference dynamics (RD) has the same expression than the EM (19), but free of noise and disturbance dynamics, leaving only the controllable dynamics. The RD only accounts for known terms, i.e.,

(29)

Roughly speaking, reference dynamics implements the state trajectories that fulfill input. Nominal trajectory generation may follow diverse methodologies, linear or nonlinear, closed or open loop, according to what is required for reference tracking, here two methodologies are presented

3.4.1. Closed loop reference generator

For the case study a linear closed loop system is selected, by defining a nominal tracking error . A simple approach has been followed for trajectories generation, by considering the same structure of the embedded model, without the disturbance dynamics. In order to keep the derivative relationship between position/velocity free of noise a dynamic filter is introduced in the same way as in the disturbance estimator

(30)

where the state is included for closed loop stability. The block scheme of the reference generator is presented in Fig. 6.

The frequency response of the reference generator is designed to be as flat as possible within the desired BW. The BW is set at 10 Hz in section 0, which can be achieved by a proper selection of the eigenvalue set Λ(A_R) of the closed loop dynamics formed by the combination of (29)-(30). The signal to be tracked by the closed loop is the one provided by this system, i.e., the pair given nominal command .

Fig. 6 Source: The authors.

3.4.2. A stepwise guidance as a reference generator.

The reference generator is in charge of computing the EM reference trajectories, these trajectories may be subject to constrains in command or states and must respond to real-time operation requests. It can cope with common control problems like not uniform overshoot in controllers subjected to stepwise user operation requests and persistent excitations typical in industrial applications [19]. By the inclusion of command profile defined as

(31)

where the coefficients are selected to guarantee the desired settling time N (command saturation dependent) and desired EM states values by using the following closed form solution

(32)

The control law aims to provide the performance to the control loop. Define the 'true' tracking error as the reference minus the design variables, this is

(33)

The ideal stabilizing control law is given by

(34)

The control law (34) includes disturbance cancelation terms and it is called ideal since it is able to reject disturbance d in (15) with the exception of the noise. Disturbance x _d has been included in error (33) with the objective of non-collocated disturbance cancelation.

Assume w is bounded and zero mean. The tracking error (33) is bounded and the mean value tends to zero with control law (34) if and only if the Davison-Francis relationship

(35)

Has a solution and the matrix is asymptotically stable, with eigenvalues set . Proof. See [12].

3.5.1. Case study

From (17) and (35) disturbance compensation matrices can be easily found by solving

(36)

Then, Q = [0 0]^T and m = 1. After replacing these parameters and by considering gain matrix K = [ ^{_{kθ kω}} ], the control law can be written as

(37)

There is no room for a complete development of an EMC design that guarantees stability and disturbance rejection performance based on (27) and (21). Eigenvalues obtained from that process for control parameters in former case study sections 0, 0 and 0 are summarized in Table 2.

5.1.1. Slew rate

The slew rate was tested through a square-wave reference position. An angular displacement of n4 (45°) was used. The Fig. 7 shows the position reference, a 0.2 Hz square wave, denoted with a continuous line; the measured angular position response is depicted with a dashed line for the EMC, a dash-dotted for EMC with the stepwise guidance (SWG), and a dotted line for the PID. The Fig. 8 shows the response to a decreasing position jump, which requires a negative voltage variation. It can be noticed from Fig. 8 the considerable overshoot exhibited by the PID control of around 15%, whereas for the case of EMC the overshoot was reduced to 3% with the inclusion of the SWG. The settling time for the EMC is around 80ms while the PID exhibit a settling time around 420ms.

The performance of the EMC controller by considering the reference generator for stepwise operation request is presented in Fig. 9. The measured angular position is depicted with a dash-dotted line for the EMC and dotted line for the PID, while the operation request ^{_{yr (i}} ) and reference trajectory are presented in dashed and solid line respectively? It can be noticed that the non-uniform overshot exhibit by the PID controller, is not present in the EMC as expected due to the inclusion of the reference command (31) In Fig. 10 an enlargement of the voltage control law around 1s is presented for both controllers. It can be observed that the PID controller shows a command saturation. This behavior explains the non- uniform overshoot presented in the measured angular positions in Fig. 7-10. It is also presented in solid line the reference command for the EMC control law in (34) for disturbance/uncertainties compensation, the closed loop term can also be appreciated.

Fig. 7 Source: The authors.

Fig. 8 Source: The authors.

5.1.2. Time delay and tracking

The time response to a 2 Hz triangular external input is shown in Fig. 11. Here an angular displacement of π/4 (45°) was used, it can be seen that time delay of the measured angle (denoted with dash-dotted line) with respect to external reference (continuous line) is smaller than 20 ms, as required in section 0. Same can be said for PID response (dotted line) but an overshoot when there is a change of sign in the derivative is observed.

The Fig. 13 shows the tracking errors for the case of triangular input, in this figure can be appreciated the effect of the overshoot already mentioned. Theoretically, this tracking error must be close to a square-shaped signal. However, the error reported for the case of PID controller shows the spikes produced by the time trajectory overshoot. The residual tracking error is shown as a dotted line, presenting the distance between the nominal error and the measurement.

Fig. 9 Source: The authors

Fig. 10 Source: The authors

Fig. 11 Source: The authors

5.1.3. Disturbance rejection

Here a constant reference of 45° was introduced, but in this case disturbance rejection properties are analyzed. An external torque disturbance was electrically added by means of an equivalent voltage. The Fig. 13 presents the tirae trajectories of the equivalent disturbance torque ^_Tl (coii)inuous line), and the EM disturbance state prediction . The external disturbance is modelled as a 40 mN step. plus a first-order colored noise introduced at 10 s. The same tests have been applied to a PID controller.

From the sets of eigenvalues defined in section 0 (see Fig. 4 and Table 2), the first one, presents a narrower BW whose disturbance tracking is clearly slower than the perturbation as can be seen in Fig. 13 where the disturbance dynamics X ^{_{d (i)}} could not track properly the load disturbance which certainly produced a performance degradation in the tracking errors as reported in Fig. 14. Dashed line of Fig. 14 shows the tracking error for this setup; after the introduction of the disturbance, the system tried to bring the error to zero, but it takes a longer time in comparison with the same situation for the PID case. However, when reaches zero mean its standard deviation remains lower than that of the PID as expected by the difference in the low frequency sensitivity slope.

Fig. 12 Source: The authors

Fig. 13 Source: The authors.

The second set performs better in term of disturbance rejection that the others controllers according to time trajectories in Fig. 13 this set is able to track the external disturbance properly by producing an almost instantaneous zero mean tracking error.

Fig. 14 Source: The authors

Fig. 15 Source: The authors.

As a complement to performance analysis and in order to verify the initial requirements, additional frequency tests were carried out. The Fig. 15 presents the harmonic responses of the closed loop system. Here the EMC system responses for both sets of eigenvalues are presented and compared with that of the PID controller.

First of all, the PID transfer function (plus-marker line in Fig. 15) presents a considerable overshoot along a wide range of frequencies. This frequency overshoot explains the spike observed in the tracking error in Fig. 14. On the other hand, there are the two sets of eigenvalues of the EMC. The narrower set transfer function (circle-marker line) presents a flat frequency response presented in Fig. 15. For the second case, i.e., a wider BW produced a transfer function with an overshoot. This poses a performance trade-off to be considered, because a flat reference-to-measurement frequency response is always desirable, but it can be obtained at expenses of a poor disturbance rejection capability.

Fig. 16 Source: The authors.

The PSD of the resulting tracking errors is presented in Fig. 16. It can be noticed that the difference between the rejection factors within the BW is proportional to the difference between the low frequency slopes of sensitivity functions for both controllers as expected.