Pole Placement and Reduced-Order Modelling for Time-Delayed Systems Using Galerkin Approximations

If the eigenvalues of A lie in the left half of the complex plane, the system is stable. If the rank of the controllability matrix CM , [B AB A2B · · · An−1B] is n, then the system represented by Eqs.

Pole placement for ODEs

Example of a TDS

We can see that a small change in τ significantly affects the stability and dynamics of the system. Note that only k1 and k2 can be varied to control an infinite-dimensional system, and that the rightmost root must be in the left half of the complex plane for the DDE to be stable.

Literature review

Spectrum of DDEs
Pole placement for TDS
Algebraic pole placement frameworks for TDS
Reduced-order modelling of TDS

However, none of these studies investigated the experimental validation of the pole placement problem using real-time experiments. Thomson [100] performed some of the first work in the field of reduced-order modeling for TDS.

Motivation and outline of the thesis

Second-order Galerkin method

The boundary conditions for the original DDE (Eq. 2.1) are given as follows:. where m, c and k are calculated as follows: 2.13). The boundary conditions can be incorporated into Eq. 2.9 using the spectral-tau method [62], ultimately resulting in a system of first-order ODEs: . 2.14) approximates the original DDE (Eq.

First-order Galerkin method

We use transformjoy(s, t) =w(t+s) and differentiate with respect to t and separately with respect to obtain the following relation (analogous to equation 2.18). Thus, we transformed the original first-order DDE (Eq. 2.1)) into an equivalent first-order PDE system (Eq. 2.18)) with the boundary conditions given by Eq. The matrix of boundary conditions is obtained by approximating the series solution (equation 2.28) to that of the original DDE (equation

These basis functions allow matricesMandC in the second-order formulation (Eq. 2.9)) and matrices P and R in the first-order formulation (Eq.

Results and discussion

Example 1

For the second and first order formulations, taking into account the N terms in the solution of the series (eqs.

Example 2

Source of spurious roots

Chapter summary

Problem definition

Given A, Bq and delays τq, the objective is to determine the feedback gains Kq needed to stabilize the system (ie, to move all poles into the left half of the complex plane). J =Re (λmax(Kq)) +α2, (3.2) where Re (λmax) is the real part of the rightmost eigenvalue, which is a function of feedback gain Kq, and α >0 is a parameter specifying the desired distance between λmax and the imaginary axis.

Mathematical modeling

Convergence can be checked by substituting the calculated eigenvalues of G into Eq.3.4 (the characteristic equation of Eq.3.3) and calculating the absolute error (E). Also note that we use shifted Legendre polynomials as basis functions: 3.26c) Shifted Legendre polynomials have shown good convergence properties [62] and facilitate expressing the inputs of matrices C(p) and D(p), as defined in Eq.

Verification of pseudoinverse method

First-order DDE with two delays

The characteristic equation is obtained by substituting x(t) =est: s−a− 1. 3.29) To test the robustness of the proposed approach, we find the rightmost roots of the equation. The proposed pseudo-inverse Galerkin approximation method was further compared with the PSD method using Monte Carlo. In Galerkin's method, N corresponds to the number of terms in the set solution (equation 3.11); in the PSD method, N is the number of collocation points.

On average, more roots converged with the proposed Galerkin method than with the PSD method, as shown in Table 3.1.

Second-order DDE with three delays

3.3, the results obtained using the proposed Galerkin method are consistent with the results obtained using the spectral tau method [49, 62].

Results and discussion

Example from Niu et al

We choose an initial guess of k = 0.8, which leads to the rightmost eigenvalues shown in Fig. To stabilize this system, we set α = 1 in the objective function (Eq. 3.2) and solve the minimization problem using the Nelder–Mead algorithm. in MATLAB via the fminsearch function. The optimal gain was found to be k, which leads to the most correct eigenvalues shown in Fig.

Note that the optimal value of the objective function J∗ = 0 is obtained when α = 1 in the objective function (Equation 3.2), indicating that α could be increased to achieve an even larger stability limit.

Figure 3.4: Rightmost characteristic roots of Eq. 3.31 using initial feedback gain k = 0.8.

Example from Michiels et al

Note that the rightmost eigenvalue has a real component of -1, indicating that stability has been achieved. As shown in Fig.3.5(b), the system will remain stable for delays substantially larger than τ = 1 when the feedback gain k∗ is used. Unlike the example of Section 3.3.1, the optimal value of the objective function is greater than zero (J in this case, indicating that the margin of stability cannot be increased by increasing α.

Figure 3.6: Rightmost characteristic roots of Eq. 3.32 with delay τ = 5 (a) and the variation of the rightmost roots with respect to delay τ (b) using initial feedback

Experimental validation

We first assess the stability of the system using the proposed pseudoinverse-based Galerkin method and then verify our predictions. We repeat the optimization procedure, increasing α by δα = 1 at each iteration, until the real component of the rightmost pole cannot reach −α. Thus, adding to the inherent delay of 2 ms, the physical system exhibited a critical delay between 17 and 17.5 ms, which is within 1–4% of the predicted critical delay of τ = 17.7 ms.

The pseudoinverse-based Galerkin method provided a simple and reliable tool to predict and optimize the stability of the rotating system of the inverted pendulum.

Table 3.2: Parameter values for the rotary inverted pendulum apparatus [2].

Chapter summary

This chapter presents a pole-placement technique for TDS that combines the strengths of the receptance method and an optimization-based strategy. In this chapter, an optimization-based strategy is proposed to address the limitations of the receptance method. In Section 4.2, a detailed mathematical derivation of the Galerkin approximations to find the characteristic roots of quadratic TDS with a single delay is given.

The proposed optimization strategy uses the characteristic root information provided by the Galerkin approximations.

Mathematical modelling

The characteristic equation of Eq. 4.8) can be obtained by substituting ¯x(t) = ¯x0est and setting the determinant equal to zero:. We first convert the system of DDEs (Eq. 4.8)) to a system of PDEs with time-dependent boundary conditions. As N (the number of terms retained in the series solution, Eq. 4.19)) increases, the eigenvalues of Z converge to the characteristic roots of Eq.

The absolute error is defined as the value of the characteristic equation (Eq. 4.10)) after replacing the eigenvalues of Z.

Problem definition

Equation (4.28) is a system of ODEs, the response of which approaches the response of the original system of DDEs (Eq. In this work, we define the convergence criterion to <10−4 and thus obtain the spectrum of the original DDE system ( The equation solves the problem of constrained optimization, whose constraints ensure that the gains are within the physically feasible region.In this work, the objective function given by Equation 4.29) is minimized using the PSO technique.

Results and discussion

Example 1

We now solve the pole placement problem for the same system using the proposed optimization-based strategy. 4.2(a)): Although spillover still occurs over a similar range of τ, the deviation from the desired polar location is significantly reduced. Similar results are observed when using different optimization parameters, such as the relative tolerance and the number of generations.

Because Galerkin approximations are used in the proposed optimization-based strategy, the robustness of the system can be evaluated in propagation cases without requiring any additional analysis.

Figure 4.1: Locations of the four roots of Eq. (4.1) as delay τ varies, using MoR with location sets (a) S 1 and (b) S 2

Example 2

Experimental validation

To validate these results, we deliberately introduced a delay into the experimental system and calculated the feedback gain using the proposed optimization-based strategy for four values of τ: 131 ms, 140 ms, 150 ms, and 160 ms. Table 4.1. Figures 4.6(b), 4.7(a) and 4.7(b) illustrate the system response for the same reference signal when the delay exceeds 131 ms.

Chapter summary

At the same time, the reconstructed solution of the reduced-order TDS described by Eq. 5.23). Taking the Laplace transform of Eq. 5.14) and assuming that the initial conditions are zero, we get:. 5.25) Equation (5.25) is the Galerkin approximated Nth order LTI transfer function for the original TDS given by Eq. 5.1) and here called the full order transfer function (FOTF), and the obtained frequency response is represented by ˆY(s).

It is only the Nc (< N) poles of the transfer function given by Eq. 5.25) that converge to the poles of the TDS within a tolerance of

Numerical results

For the examples covered in this chapter, it was noted that the time it took for MATLAB to compute the eigenvalues and eigenvectors of took less than 1 second. Now, as the dimension ofLin increases, the computation time required to compute the eigenvalues and eigenvectors also increases. For systems of large dimension (N >103), depending on the desired order of the ROM, only the right-hand eigenvalues Dr and eigenvectors Pr of L need to be calculated.

For different test cases, using the procedure described in Section 5.1, the characteristic roots are obtained for the DDE given by Eq. The average Nc is obtained by running 10,000 Monte Carlo simulations with parameters from uniform distributions for Eq. rounded to the nearest integer) for each value of N and.

Table 5.1: Average N c from Monte Carlo simulations of Eq. (5.29)

Numerical Examples

5.3(a) that, for a forcing function of the form f(t) = sin(t), the time responses of the TDS and the 6th-order GEVD system are identical. Next, we compare the frequency responses of the ROMs with the TDS given by Eq. Next, the frequency responses of the ROMs are compared with those of the TDS given by Eq.

5.10[(a) – (c)] that more natural frequencies of the system can be included in the GETF depending on the requirement.

Figure 5.3: Time response and error plot of Eq. (5.31) for f(t) = sin(t).

Experimental validation

It can be observed that for a sinusoidal input in the form f(t) = 0.2 sin(2t), the predicted trajectories θy(t), θp(t) and θr(t) using the 12th-order GEVD system closely match the experimentally obtained ones θy(t), θp(t) and θr(t). Then they were for the same delay, ie. τ = 20 ms, performed experiments with a combination of sine waveforms with incommensurate frequencies of the form f(t) = 0.2 sin(2t) + 0.1 sin(πt). It can be observed that for the combination of sinusoidal waveforms with disproportionate frequencies in the form f(t) = 0.2 sin(2t) + 0.1 sin(πt) the trajectories θy(t), θp(t) and θr(t ) using the 12th-order GEVD system are in close agreement with the experimentally obtained θy(t), θp(t), and θr(t).

At τ = 25 ms, two sets of experiments were performed with first a sinusoidal input of the form f(t) = 0.2 sin(2t) and then a combination of sinusoidal waveforms with uncompensated frequencies of the form f(t) = 0.2 sin (2t) + 0.1 sin(πt).

Figure 5.12: Roots of Eq. (5.38) using QPmR and spectral methods.

Chapter summary

Ulsoy, “Eigenvalue assignment via the Lambert W function for control of time-delay systems,” Journal of Vibration and Control, vol. Z'ıtek, "Control design for time-delay systems based on quasi-direct pole placement," Journal of Process Control, vol. Ding, “Eigenvalue assignment for control of time-delay systems via the generalized Runge-Kutta method,” Journal of Dynamic Systems, Measurement, and Control , vol.

Meerbergen, »Krylov-based model order reduction of time-delay systems«, SIAM Journal on Matrix Analysis and Applications, vol.

Voltage of (a) motors A and C, and (b) motors B and D in the 3D

Roots of Eq. (5.31) using QPmR and spectral-tau methods

Frequency response and error plot of Eq. (5.31)

Frequency response and error plot of Eq. (5.31) for different degrees of

Roots of Eq. (5.34) using QPmR and spectral-tau methods

Frequency response and error plots of GETFs given in Eqs. (5.36a –

Frequency response plots and the associated error plots for different

Roots of Eq. (5.38) using QPmR and spectral methods