On Finite-Dimensional Transformations of Anisochronic Controllers Designed by Algebraic Means: A User Interface

2.1. Input-output quasipolynomial model

Since the authors’ interest lies in single-input single-output (SISO) TDS and their input-output models, they are concerned here. Namely, transfer functions in the form of quasipolynomial fractions giving rise to the meromorphic representation are taken as initial models to be approximated.

For both lumped (point-wise) and distributed input-output and internal delays in the system or model, the Laplace transform of a state space model (considering zero initial conditions) can be formulated as follows

whereb(s), a(s)are quasipolynomials of the general form

where n is the order of a quasipolynomial which usually agrees with the system degree (of a state-space model), mijare real numbers and ϑij≥0represent delays. If  ∑j=1hnmnjexp(−ϑnjs)does not equal a real constant, the system is called neutral; otherwise, the system is retarded.

2.2. TDS stability

A retarded system (1) is said to be asymptotically (exponentially) stable if all poles are located in the open left half plane, ℂ0−, i.e. there is no s satisfying

Condition b(s)≠0in (3) is taken into accont if the system contains distributed delays since there hence exist common roots of the transfer function numerator and denominator which are not system poles.

A rather more complicated TDS stability conditions are given regarding to neutral systems, since there may exist vertical strips of system poles tending to the imaginary axis of infinite amplitude.Moreover, these strips can be sensitive to even infinitesimally small deviations in delays, i.e. the position of system poles in the real axis is not continuous with respect to delays. A neutral system (1) is exponentially stable if there is no s such that

for arbitrarily ε>0.

A system is strongly stable if so-called associated difference equation (in state-space formulation) remains asymptotically stable when subjected to small variations in delays (Hale &Verduyn Lunel, 1993), i.e. a TDS remains formally stable, see details in (Byrnes et al., 1984; Loiseau et al., 2002). Formal stability, roughly speaking, means that the rightmost vertical strip of poles does not cross the imaginary axis. If this holds also under small delay changes, the system is strongly stable. A necessary and sufficient strong stability (and thus also formal stability) condition in the Laplace transform can be formulated as

according to e.g. (Zítek & Vyhlídal, 2008), where mnjare real coefficients for the highest s-power in the denominator a(s)of (1).

2.3. TDS model over RMS ring

Algebraic approaches for analysis and control of TDS can be performed either in the state space or in the realm of input-output models where fields, modules and rings as principal algebraic notions and tools are utilized. Usually, commensurate delays, i.e. those which can be expressed as integer multiples of the smallest one, are assumed; however, delays are naturally real-valued and thus this assumption is rather restrictive for real applications.Non-commensurate or rationally unapproximated delays results in a fraction of quasipolynomials as presented above. However, these transfer function representations are not suitable in order to satisfy some basic control requirements, e.g. controller feasibility, closed-loop (Hurwitz) and formal (strong) stability.

Rather more general approaches (Vidyasagar, 1985; Kučera, 1993) utilize a field of fractions where a transfer function is expressed as a ratio of two coprime elements of a suitable ring. A ring is a set closed for addition and multiplication, with a unit element for addition and multiplication and an inverse element for addition, i.e. division is not generally allowed. A powerful algebraic tool ensuring some basic control requirements, such as internal closed-loop stability and controller properness, is a ring of stable and proper RQ-meromorphic functions (R_MS). Since the original definition of R_MS in (Zítek & Kučera, 2003) does not constitute a ring, some minor changes in the definition were made e.g. in (Pekař & Prokop, 2009). Namely, although the retarded structure of TDS is considered only, the minimal ring conditions require the use of neutral quasipolynomials at least in the numerator as well. In this chapter, the ring definition is reformulated once more to comprise models of neutral type, distributed delays and formal stability.

A term T(s)from R_MSring is represented by a proper ratio of two quasipolynomials y(s)/x(s)where a denominator is a quasipolynomial of degree n and a numerator can be factorized as

where y˜(s)is a quasipolynomial of degree l andτ0. Note that the degree of a quasipolynomial means its highest s-power. The element is analytic and bounded in ℂ+, particularly, there is no pole s₀ such thatRes0≥0(or Res0≥−ε, ε>0,for neutral terms, more precisely, which can be taken as a generalization) - in other words, all possible roots of x(s)in ℂ+are those of y(s).Thus, it lies in the space H∞(ℂ+) providing the finite norm defined as

It is said that T(s)is H∞stable (Partington & Bonnet, 2004). That is, the system has finite L2(0,∞)to L2(0,∞)gain where L2(0,∞)norm of an input or output signal h(t)is defined as

Notice, for instance, that T(s)having no pole in the right-half complex plane but with a sequence of poles with real part converging to zero can be H_∞ unstable due to an unbounded gain at the imaginary axis.

Moreover, T(s)is also formally stable which is guaranteed by condition (5) forx(s). Unfortunately, strong stability can not be included in the ring definition since the product of two strongly stable terms can be strongly unstable. Although neutral TDS that are strongly stable can be H∞stable, see (Partington & Bonnet, 2004), yet they are not H∞(nor BIBO) stabilizable, (Loiseau et al., 2002). It can be shown that neglecting formal stability can bring the problem with ring axioms in controller design, i.e. H∞stability is not sufficient.

In addition, the ratio is proper, i.e. ln. More precisely, there exists a real number R> 0 for which holds that

2.4. Coprime factorization in RMS

Let the plant be initially described by the transfer function

where a(s), b(s)are quasipolynomials. Hence, using a coprime factorization, a plant model has the form

where A(s), B(s)∈RMSare coprime, i.e. there does not exist a non-trivial (non-unit) common factor of both elements. Details about divisibility can be analogously deduced from notes presented in (Pekař & Prokop, 2009). Note that a system of neutral type can induce problem since there can exist a coprime pair A(s), B(s)which is not, however, Bézout coprime – which implies that the system can not be stabilized by any feedback controller admitting the Laplace transform (Loiseau et al., 2002); for instance when the system is not formally stable. More precisely, two coprime elements A(s), B(s)∈RMSform a Bézout factorization if and only if

3.1. Closed loop stabilization

Given a Bézout coprime pair A(s), B(s)∈R_MS the closed-loop system is stable if and only if there exists a pair P(s),Q(s)∈RMSsatisfying the Bézout identity

A particular stabilizing solution of (17), sayP0(s),Q0(s), can be then parameterized as

where Z(s)∈RMS. Parameterization (18) is used to satisfy remaining control and performance requirements, such as reference tracking, disturbance rejection etc.

The proof of the statement above can be done analogously as in (Zítek & Kučera, 2003) where a three-step proof for a similar ring was presented. Condition (12) ensures i.a. that there can exist the ring inversion of M(s)since it proves that there is no common zero of A(s),B(s)in ℂ+(including infinity).

3.2. Reference tracking

The task of this subsection is to find Z(s)∈RMSin (18) so that the reference signal is being tracked. The solution idea results from the form of GWE(s)defined in (14). Consider the limit

where ⋅Wexpresses that the signal is a response to the reference not influenced by other external inputs. Limit (19) reaches zero if lims→0EW(s)<∞and EW(s)is analytic and bounded in the right half-plane, i.e. EW(s)∈H∞(ℂ⁺) – this requirement also satisfies that  limt→∞∫0teW(τ)dτ<∞. Moreover, it must hold that GWE(s)is proper (or, equivalently, EW(s)is strictly proper) because of the feasibility (impulse free modes) of eW(t). If one wants to prevent the closed loop stability from the sensitivity to small delays, or to preserve formal stability at least, the denominator of EW(s)must be a (quasi)polynomial satisfying (5). This implies, in other words, that the reference tracking is fulfilled if EW(s)∈RMS.

Alternatively, FW(s)must divide the product A(s)P(s)in RMS. Hence, all unstable zeros (including infinity) of FW(s)must be those of A(s)P(s)and, moreover, the quasipolynomial numerator of FW(s)is formally (strongly) stable. It means that one has to set all unstable zeros of FW(s)(with corresponding multiplicities) as zeros of P(s)- if there is no one already contained inB(s). Recall that zeros mean zero points of a whole term in RMS, not only those of a quasipolynomial numerator.

Note that the controller approach fails for formally unstable controlled processes since then the feedback loop remains formally unstable (neutral term can not be affected by a controller).

3.3. Load disturbance rejection

The attenuation of the load disturbance signal entering a plant model can be done analogously as for reference tracking. Thus, Z(s) is chosen so that YD(s)∈RMSwhich is clear from

where ⋅Dmeans that the output is influenced only by the disturbance. Or, FD(s)must divide the product B(s)P(s)in RMS.

One has to be careful when deciding about the form of Z(s)since both divisibility conditions must be fulfilled simultaneously. A detailed procedure of reference tracking and disturbance rejection briefly described above was presented e.g. in (Pekař & Prokop, 2011).

4.1. Padé approximation

In the second half of the 19^th century, a French mathematician Henry Padé devised a simple and, nowadays, one of the most used and favorite rational approximations which is based on the comparison of derivatives of the approximating and approximated functions in zero. More precisely, let F(s)with F(0)≠0be analytic in the neighborhood of zero. Then, the n-n Padé approximation is the function Φ(s)=Nn(s)/Dn(s)where Nn(s), Dn(s)are polynomials of the nth order with Dn(0)=1and it holds that F(i)(0)=Φ(i)(0),i=1,...,2n.

Padé approximation of F(s)=exp(−sT),T≥0is given by the following relation (Partington, 2004)

where n is the order of the approximation. Obviously, one can approximate another function, e.g. the whole transfer function.

A method called diagonal Padé approximation,which is distinguished by some authors, see (Battle & Miralles, 2000;Richard, 2003) can be expressed as

However, it is easy to verify, that (21) and (22) represent the same approximations. In fact,

4.2. Shift operator approximations

Approaches based on operator shifting yields from the fact that a delay term exp(−sT)can be perceived as a shift operator and can be subjected to Maclaurin series expansion. Moreover, the variable s can be vied as a derivative operator.

Hence

A concise overview of some important shift operator approaches follows.

5.1. H₂ norm

The H₂,sometimes called quadratic, norm of a stable strictly proper transfer function is defined as

The norm is finite for strictly proper stable systems having no pole on the imaginary axis, and the meaning of H₂ is energy of G. A generalized strict properness can be expresses as follows

Note that for TDS with distributed delays, there can exist a denominator root (or roots) of Gwhich is not the system pole.

In computer practice, i.e. working with discrete samples, the integral in (36) is calculated as a sum within a finite range of nonnegative frequencies, ω∈[0,ωmax]. Due to the symmetry, the resulting value is doubled finally. The value of ωmaxcan be chosen so that the frequency gain is “small enough”.

Residual expansion can be used when analytic (and continuous) calculation of‖G‖2, (Štecha & Havlena, 2000). It holds that

where siare (stable) system poles and resuduum of a complex function F(s)reads

where m is the multiplicity of a pole.

5.2. H_∞ norm

The H-infinity (H∞) norm is defined as

i.e. it expresses the supreme of the amplitude (gain) frequency characteristics of G, see Fig. 2. If the system is asymptotically (exponentially) stable and provides a finite H_∞norm, it is said that it is H_∞ stable and lies in the space H∞(+) of functions analytic and bounded in the right-half complex plane.

The norm is also called L₂ gain, That is, the H_∞ stable system has finite L2(0,∞)to L2(0,∞)gain where L2(0,∞)norm of an input or output signal h(t)is defined as

The frequency characteristics supreme can be easily found by standard analytic means, or by mapping the values of G(jω)when using digital computers.

Note, for instance, that a transfer function having no pole on the imaginary axis but a sequence of poles with real part converging to zero can have an infinite H_∞ norm due to an unbounded gain, see (Partington & Bonnet, 2004).

7.1. Stable system with stepwise reference

Let the plant be described by the transfer function of a stable first order TDS model with both internal and input-output delays as

Condition aϑ∈(0,2π)ensures the asymptotic stability of the model.

Coprime (Bézout) factorization is the first step of controller design in R_MSring as follows

where m(s)is an appropriate stable (quasi)polynomial of the first order (due to the coprimeness). A suitable form of m(s)is contentious and depends on user’s requirements, let m(s)=s+m0, m0>0.

Consider the simplest practical case that both external inputs are from the class of step functions, hence

where mw(s)and md(s)are arbitrary stable (quasi)polynomials of degree one, say, for the simplicity, s+m0again, HW(s),FW(s),HD(s),FD(s)∈RMSand w0and d0are real constants.

Find a stabilizing particular solution by (17). Set e.g. Q0=1which yields

Now parameterize the solution according to (18) to obtain controllers asymptotically rejecting the load disturbance and tracking the stepwise reference

The numerator of P(s) has to have at least one zero root. Moreover, it is appropriate to have P(s) in a simple form, which is fulfilled e.g. when

providing

Thus, final controller’s structure is the following

The obtained control structure can be easily compared with the well-known Smith predictor structure and note that the controller is of the anisochronic type because of delay in the transfer function denominator. It is naturally possible to take m(s)as a quasipolynomial instead of polynomial; however, this option would make a controller more complicated. The importance of m(s)reveals from the closed loop transfer function

i.e. m(s)appears as a characteristic (quasi)polynomial of the closed loop.

Model (41) can to fit the dynamics of a high order undelayed system; for instance, a tenth order system governed by the transfer function

can be estimated by model (41) witha=b=6.5⋅10−2,τ=15.3,ϑ=6.7, see (Zítek & Vyhlídal, 2003; Pekař &Prokop, 2008) for details. Hence, let the system has these parameters and m0=0.5.

Obviously, the best result for H_∞ is given alongside by the Padé approximation and Laguerre shift of the first order, whereas, amazingly, higher orders make results worse. The Fourier analysis based methods yields almost the same score for all studied orders.The benchmark results for the H₂ norm with ωmax=15introduced in Table 1 are almost identical with those for H_∞, i.e. the Padé approximation and Laguerre shift of the first order are the best and the Fourier analysis based methods gives almost the same results for all orders.The corresponding gain frequency responses for the approximations of orders as in the last column in Table 1 are displayed in Fig. 4.

The approximating transfer function by Padé approximation and Laguerre shift with n = 1 is given by (51), which agrees with a conventional PID controller

7.2. Control of the roller skater on a swaying bow

Consider an unstable system describing roller skater on a swaying bow, (Zítek et al., 2008), governed by the transfer function

see Fig. 5, where y(t)is the skater’s deviation from the desired position, u(t)expresses the slope angle of a bow caused by force P, delays τ, ϑmeans the skater’s and servo latencies, respectively, and b, a are real parameters. Skater controls the servo driving by remote signals into servo electronics.

Let the model parameters be b = 0.2, a = 1, τ=0.3s, ϑ=0.1s, as in the literature, and design the controller structure analogously as in the previous subsection 7.1. Consider that the reference and load disturbance are in the form of step-wise functions. Then the final controller has the structure given by the following transfer function

where p₂, p₁, p₀, q₃, q₂, q₁, q₀∈are free parameters, see details in (Pekař & Prokop, 2011b). Using a quasi-optimal tuning algorithm, the parameters were set as

and e.g. m0=5.

The comparison of the best controller rational approximations can be found in Table 2. Again, the method based on the Fourier series expansion very slowly approaches the limit value of the H₂norm (≈7.811) with the increasing n. The only method evincing the better asymptotical results with the higher order approximation is the Kautz shift. Again, the Padé and Laguerre approximations of the first order give very good results with the approximating controller transfer function

Fig. 6 displays Bode magnitude plots for the best orders for H₂ (the last column in Table 2), which verifies a very good performance of all the approaches.

7.3. Control of a circuit heating plant

The laboratory heating plant, a photo and a sketch of which, respectively, are displayed in Fig. 7, was assembled at the Faculty of Applied Informatics of Tomas Bata University in Zlín in order to test control algorithms for systems with dead time. The original description of the apparatus and its electronic circuits can be found in (Dostálek et al., 2008).

The heat transferring fluid (namely distilled water) is transported using a continuously controllable DC pump {6} into a flow heater {1} with maximum power P(t)of 750 W. The temperature of a fluid at the heater output is measured by a platinum thermometer giving value ofϑHO(t). Warmed liquid then goes through a 15 meters long insulated coiled pipeline {2} which causes the significant delay in the system. The air-water heat exchanger (cooler) {3} with two cooling fans {4, 5} represents a heat-consuming appliance. The speed of the first fan can be continuously adjusted, whereas the second one is of on/off type. Input and output temperatures of the cooler are measured again by platinum thermometers giving ϑCI(t), resp. ϑCO(t). The expansion tank {7} compensates for the expansion effect of the water.

Originally, it was intended to control input delays only; however, it was shown that the plant contains internal delays as well. A detailed mathematical model was presented in (Pekař et al., 2009). A linearized model of the relation between the power to the heater P(t)and ϑCO(t)can be expressed by the transfer function

where all real parameters in the model are complex algebraic functions of physical quantities in the circuit and input and output steady states, see details in the literature. It was determined that for a certain working point, the parameters are

The controller structure obtained by controller design in RMSyields

Table 3 displays the best values for approximations at their ordersn={1,2,3,4,5}. H₂ norm is measured within the frequency range ω∈[0.001;0.5]with the discretization step Δω=0.001.

Immense values of H_∞ norm are caused by a pair controller poles which is very close to the imaginary axis (si=0.0002±0.094j) and the controller is obviously unstable. Padé approximation of the 4^th order provides the best value for H_∞, whereas Laguerre shift of the 2^nd order gives the best approximation measured by H₂ norm. Again, this example indicates that the higher order of the approximation does not mean the more accurate result automatically.

The comparison of Bode magnitude plots for the best orders (which are placed in the last column in Table 3) for H₂ can be seen in Fig. 8.

To conclude study cases above, it is startling that the best approximations measured by H₂ and H_∞ norms are mostly given by the well known and widely used Padé approximation of the first (or a low) order. By simulations, the higher order of an approximation does not generally yields the more accuracy finite dimensional model, which is in the contradiction with a general expectation and analytic results for rational approximations for TDS with input-outputs delays. Comparative Bode plots above indicate the usability and a very good efficiency of selected methods.