197

A modified model reference adaptive control approach for systems with noise or unmodelled dynamics S A Neild*, L Yang, and D J Wagg Department of Mechanical Engineering, University of Bristol, Bristol, UK The manuscript was received on 27 November 2007 and was accepted after revision for publication on 24 January 2008. DOI: 10.1243/09596518JSCE527

Abstract: In this paper, a modified model reference adaptive control (MRAC) strategy is developed for use on plants with noise or unmodelled high-frequency dynamics. MRAC consists of two parts, an adaptive control part and a fixed gain control part. The adaptive algorithm uses a combination of low- and high-pass filters such that the frequency range for the adaptive part of the strategy is limited. The mechanism for noise-induced gain wind-up is demonstrated analytically, and it is shown how MRAC can be modified to eliminate this windup. Further to this, an additional filter is proposed to improve MRAC robustness to highfrequency unmodelled dynamics. Two test plants, one with added noise and the other with unmodelled high-frequency dynamics, are considered. Both plants exhibit unstable behaviour when controlled using standard MRAC, but, with the modified strategy, robustness is significantly improved. Keywords:

1

model reference adaptive control, robustness, unmodelled dynamics

INTRODUCTION

Adaptive controllers, such as model reference adaptive control (MRAC), incorporate a mechanism by which the control gains are adjusted in real time on the basis of the system response [1]. Theoretically, they are attractive control solutions as they have the ability to respond to plant parameter variation and plant non-linearity and have been shown to be globally asymptotically stable [2, 3]. However, in practical implementation, MRAC strategies lack robustness in the presence of external disturbances or plant undermodelling [1–5]. It is noted that there are many robust linear control strategies that are capable of accurate control in the presence of noise and undermodelling, but they can be difficult to apply to plants with a high degree of parameter uncertainty such as parameter variation due to environmental conditions. It has been observed that adaptive control gains can ‘wind up’ (or ‘drift’) rather than settle to or oscillate about a steady value, and that this wind-up *Corresponding author: Department of Mechanical Engineering, University of Bristol, Queen’s Building, University Walk, Bristol BS8 1TR, UK. email: [email protected] JSCE527 F IMechE 2008

leads to control instability [6, 7]. Recently, Yang et al. [8, 9] studied an MRAC strategy applied to a firstorder plant subject to square-wave input. By performing a localized stability analysis (see, for example, references [10] to [12]), it was shown that a localized zero eigenvalue exists, leading to gain wind-up in the presence of noise. A modified control scheme, r-modified MRAC, proposed by Yang et al. [8], incorporates a leakage term that replaces the zero eigenvalue with a negative real eigenvalue. When applied to an experimental non-linear plant, r-modified MRAC could almost entirely eliminate wind-up in the presence of both noise and unmodelled dynamics. Plants with unmodelled high-frequency dynamics are one important class of plant uncertainty. Rohrs et al. [13] demonstrated that applying a first-order MRAC controller to a plant that had higher-order dynamics (first order with an additional higherfrequency critically damped resonance) can lead to system instability. Further studies have shown how this type of plant uncertainty can cause system instability in many real applications [14, 15]. This paper concentrates on the application of the MRAC strategy designed for first-order plants. In section 2, the mechanisms by which instability Proc. IMechE Vol. 222 Part I: J. Systems and Control Engineering

198

S A Neild, L Yang, and D J Wagg

occurs owing to noise and to higher-order unmodelled dynamics are investigated, and it is shown that they are fundamentally different phenomena. By considering the gain adaptation in the frequency domain, an extension to the r-modified MRAC strategy, rw-modified MRAC, is then proposed in section 3. In section 4, the way in which the r and w components significantly reduce the effects of external disturbance and higher-order unmodelled dynamics is demonstrated via experiments on a plant consisting of a reconfigurable electrical circuit.

feedback and feedforward gains respectively. The gain adaptation mechanism aims at minimizing ye, a measure of how well the plant output x matches that of a user-defined reference model output xm. In general, ye 5 Cexe, where Ce is a weighting vector applied to the various error states contained in vector xe 5 xm 2 x. However, in the case of first-order MRAC implementation, xe, and hence Ce, is scalar. The adaptive gains are defined via hyperstability [3], in a proportional plus integral formulation K ðt Þ~a

ðt 0

2

ROBUSTNESS OF MRAC Kr ðt Þ~a

2.1

ðt 0

MRAC strategy

MRAC has been described by numerous previous authors (see, for example, references [1] and [2]). A schematic of the strategy is shown in Fig. 1, in which r is the demand (or reference), u is the control signal, and K and Kr are the adaptive gains. For first-order MRAC, the plant is nominally assumed to have first-order dynamics x_ ðt Þ~{axðt Þzbuðt Þ

ð1Þ

where a and b are the nominal plant parameters. The control signal, u(t), is generated with the aim that the plant response follows a first-order reference model x_ m ðt Þ~{am xm ðt Þzbm rðt Þ

ð3Þ

where K and Kr are the time-dependent adaptive

Fig. 1

ye rðt Þ dtzbye rðt ÞzKr0 ð4Þ

where a and b are adaptive control weightings representing the adaptive effort, and K0 and Kr0 are the initial gain values. In the case of a first-order implementation, since Ce is a scalar, it may be incorporated into the a and b adaptive control weightings, and hence Ce 5 1 is set in this paper without loss of generality. If the real plant dynamics exactly match the nominal plant dynamics (equation (1)), then ideal time-invariant gains exist, known as Erzberger gains (see, for example, reference [1]). Using equations (1) to (3) the error xe 5 xm 2 x may be written as x_ e ~{am xe zða{am {bK Þxzðbm {bKr Þr

ð5Þ

ð2Þ

where am and bm are user-defined reference model parameters, typically am 5 bm such that there is unity steady state response. The control signal input to the plant is given by uðt Þ~K ðt Þxðt ÞzKr ðt Þrðt Þ

ye xðt Þ dtzbye xðt ÞzK0

Schematic block diagram of MRAC

Proc. IMechE Vol. 222 Part I: J. Systems and Control Engineering

Therefore, for am . 0 the error system will give an ideal response if K ~K E ~

a{am b

and

Kr ~KrE ~

bm b

ð6Þ

where ( )E denotes the Erzberger gain values.

2.2

Robustness to external disturbance

It is well known that gain wind-up can cause controller instability [6, 7]. To understand how noise can cause gain wind-up, in this section the dynamics of the MRAC strategy and plant will be examined analytically in a similar way to that adopted by Hillis et al. [16] when studying an error-based modified MRAC controller. Using some simplifying assumptions, it will be possible to see the underlying mechanism causing the gain wind-up owing to noise by deriving an analytical expression for the gain wind-up trajectory. JSCE527 F IMechE 2008

A modified model reference adaptive control approach

Consider the case where the demand signal r is zero and the plant output is subject to high-frequency sinusoidal noise d 5 sin(vt) as shown in Fig. 2. Simulation results, for the case where there is unit amplitude 50 Hz noise, the plant dynamics Gp(s) are 8/(s + 8), the initial gain K0 5 4, and the adaptive weightings are a 5 10 and b 5 1, are shown in Fig. 3. Figure 3(a) shows the error between the reference model output and the plant output xe, and Fig. 3(b) shows the adaptive gain which is growing more negative with time. To analyse this system, first note that, if the controller is working perfectly, the plant output x will be zero. For this to be the case, the plant output prior to the addition of transducer noise, xp, must have the same frequency as the noise, v. The control will not be perfect, but it is assumed that xp is still dominated by frequency v. As the noise (and hence xp) is high frequency and so beyond the plant break frequency, the plant dynamics (equation (1)) can be reduced to the purely integral term, such that ð xp ðt Þ~ buðt Þ dt, x_ ~buðt Þzd_

ð7Þ

As the demand r is zero, the equation for the control signal may be written as ð

uðt Þ~K ðt Þxðt Þ~xðt Þa xe ðt Þxðt Þ dt zxðt Þbxe ðt Þxðt Þzxðt ÞK0

As x(t) is expected to be oscillatory in nature with frequency v, with time the right-hand side of equation (9) will be dominated by the term for the integral of the error squared, # x2(t). Ignoring the other terms on the right-hand side and substituting in the expression for the noise gives ð

v cosðvt Þ~baxðt Þ x2 ðt Þ dt

xðt Þ~At n cosðvt Þ

ð11Þ

where A and n are constants, gives ð v~baAt n A2 t 2n cos2 ðvt Þ dt

ð12Þ

However, from inspecting Fig. 3, it can be seen that, as time becomes large, the change in the amplitude of the sinusoidal x(t), Atn, is slow, i.e. n , 1. This is because x(t) is tending towards matching 2d(t) through gain adaptation. This observation allows an approximation to solve the integral in equation (12). It is assumed that over each cycle the amplitude of oscillation is constant, i.e. over any full cycle (centred at ¯t ) it is possible to write ð
t zp=v

ð8Þ

ð9Þ

ð10Þ

Substituting in a trial solution for x(t) of the form

A2 t 2n cos2 ðvt Þ dt&


t {p=v

Combining equations (7) and (8) and noting that xe 5 2x gives ð _d ~x_ zbxðt Þa x2 ðt Þ dtzbbx3 ðt Þ{bK0 xðt Þ

199

A2
t 2n

ð
t zp=v

1 cos2 ðvt Þ dt~ A2
t 2n 2
t {p=v

ð13Þ

The integral term from time 0 to time t may therefore be thought of as a sum of 12 A2
t 2n terms, where ¯t 5 p/v, 3p/v, …. Note that there will be a final incomplete cycle between ¯t 5 (2n 2 1)p/v and time t, where n is the largest integer that satisfies (2n 2 1)p/v ( t, which indicates that there will be a gain oscillation at frequency v. Using equation (13), and noting that at large t there are a large number of cycles, it is possible to write ð

A2 t 2n cos2 ðvt Þ dt&

ð 1 2 2n A t dt 2

ð14Þ

and equation (12) reduces to v~ Fig. 2

Schematic block diagram of the MRAC strategy applied to a plant with a noisy output and zero demand

JSCE527 F IMechE 2008

baA3 t 3nz1 2ð2nz1Þ

ð15Þ

Therefore, the constants A and n may be expressed as Proc. IMechE Vol. 222 Part I: J. Systems and Control Engineering

200

S A Neild, L Yang, and D J Wagg

Fig. 3

Simulation of the MRAC strategy applied to a plant with a noisy output and zero demand: (a) model-following error; (b) adaptive gains




1 n~{ , 3

2v 3ba

A~

13 ð16Þ

and hence the gain (extracted from equation (8) and using the assumption that the integral term is dominant) may be expressed as ð ð 2 K ~{a x2 dt~A2 t {3 cos2 ðvt Þ dt ~{aA

2

ð

2

t {3 cos2 ðvt Þ dt

ð17Þ

Applying the approximation that the frequency is large and the change in amplitude of the oscillation is small over an oscillation gives ð 1 {2 3 1 t 3 dt~{ aA2 t 3 K &{aA 2 2 2

2.3 ð18Þ

This approximate relationship is plotted as a dashed line in Fig. 3(b). It can be seen that the shape of the curve is very similar to that of the gain trajectory generated from the simulation. The difference is due to the plot starting from time t 5 0, whereas in the analysis it was assumed that time is large. To overcome this, it is therefore necessary to offset the time by some amount t* to compensate for the initial response period prior to the integral gain term becoming dominant, and hence also offset by some K * representing the gain generated up to the point when t 5 t*, giving the relationship 1 3 K ~{ aA2 ðt{t
Þ3 zK
2

Values of t* and K * can be found by fitting this relationship to the simulation data at two time points. Fitting to times t 5 50 s and t 5 55 s indicated by circles in Fig. 3(b) gives t* 5 9.05 and K * 5 7.87, which results in the analytical gain trajectory shown by a dotted line. It can be seen that the agreement with the simulation data is very good after approximately t 5 20 s. This analysis demonstrates how noise can cause gain wind-up and that this is due to the integral term in the relationship governing the gain adaptation. This wind-up leads to gain values tending to infinity rather than some large steady state value and that the gain is subject to oscillations at twice the frequency of the noise.

ð19Þ

Proc. IMechE Vol. 222 Part I: J. Systems and Control Engineering

Robustness to higher-order dynamics

Rohrs et al. [13] demonstrated that the MRAC strategy can become unstable when applied to a plant with unmodelled higher-order dynamics. They applied the MRAC strategy for first-order systems to a plant with dynamics X ðsÞ~

2 229 U ðs Þ 2 sz1 s z30sz229

ð20Þ

which was considered to have nominally first-order dynamics, 2/(s + 1), with higher-frequency unmodelled dynamics in the form of a resonance with approximately critical damping, 229/(s2 + 30s + 229). Using a reference model of 3/(s + 3), they showed that the system was unstable for a range of inputs. Figure 4 shows a simulation using these plant and reference model dynamics with the gains initially set to JSCE527 F IMechE 2008

A modified model reference adaptive control approach

Fig. 4

Simulation of the MRAC strategy applied to the Rohrs model with an impulse demand, a 5 50, and b 5 5: (a) model-following error; (b) adaptive gains

the nominal Erzberger values (KE 5 21 and KrE ~1:5), with a unity-amplitude impulse of width 0.1 s at zero time. The adaptive weightings were set as a 5 50 and b 5 a/10. It can be seen that the plant output is a stable decaying oscillation. However, if the adaptive weightings are increased to a 5 50.7 and b 5 a/10, then the plant output becomes unstable as shown in Fig. 5(a) owing to the gain wind-up shown in Fig. 5(b). If gain adaptation is slow, then over any short time period the gains can be approximated to constants, and therefore linear control analysis can be applied to the system. Assuming fixed gains, the closed-loop characteristic equation is given by   ðsz1Þ s2 z30sz229 {458K ~0

Fig. 5

201

ð21Þ

The natural frequency of the complex roots of the characteristic equation, owing to the presence of higher-order dynamics, may be tracked with varying gain K. This natural frequency is plotted as a dotted line in Fig. 5(c) against time for the instantaneous gains generated in the simulation (Fig. 5(b)). To demonstrate the importance of the instantaneous characteristic equation to the behaviour of the system, the frequency of oscillation of the output error xe can be considered. This frequency, calculated from zero crossing points in the time domain (Fig. 5(a)), is plotted as a dashed line in Fig. 5(c). It can be seen that the output error frequency at any time closely matches that calculated from the instantaneous closed-loop characteristic equation.

Simulation of the MRAC strategy applied to the Rohrs model with an impulse demand, a 5 50.7, and b 5 5.07: (a) model-following error; (b) adaptive gains; (c) oscillation frequency of xe compared with that of the complex roots of the instantaneous closedloop characteristic equation

JSCE527 F IMechE 2008

Proc. IMechE Vol. 222 Part I: J. Systems and Control Engineering

202

S A Neild, L Yang, and D J Wagg

Further to this, using the instantaneous closed-loop characteristic equation, the gain corresponding to neutral stability is K 5 217.03. From the simulation data, the point at which the amplitude of the oscillations in xe switch from decaying to growing is at 29.4 s, which corresponds to a gain of around K 5 217. This indicates that the simulation instability is caused by instability of the instantaneous closed-loop characteristic equation. Note that, in the case where a 5 50, the initial gain adaptation due to the impulse input is not sufficient for the feedback gain K to grow sufficiently negative for instability to occur, K . 215 for all time. This analysis shows that for plants with higherorder dynamics the instability mechanism is due to closed-loop instability, whereas the instability mechanism due to noise is caused by integral gain wind-up. 3

A MODIFIED MRAC APPROACH

The r-modified MRAC strategy proposed by Yang et al. [8] will firstly be discussed, and then, by considering the frequency domain, a second modification to the strategy – the rw-modified MRAC strategy – will be proposed. 3.1

adaptive control gains (from equation (4)) in the Laplace domain, and K *(s) and Kr
ðsÞ are constant gains. It was shown for best results that K *(s) and Kr
ðsÞ should be set to the Erzberger values based on the nominal plant dynamics, K *(s) 5 KE and Kr
ðsÞ~KrE . This modification may be thought of as a complementary filter pair in the frequency domain. A high-pass filter is used to reduce the frequency content of the adaptive gains at low frequency and is replaced with fixed gain control, as indicated by the Bode plot in Fig. 6(a). The effect of this modification on the wind-up mechanism can be seen by reconsidering the analysis discussed in section 2.2. As before (equation (7)), the plant dynamics may be approximated as x_ {d_ ~buðt Þ~bKmr x

ð24Þ

Rewriting the expression for the Kmr gain (equation (23)) in the time domain and substituting in the expression for K (which can be extracted from equation (8)) gives ð Kmr ðt Þzr2 Kmr {K E dt~ ð

{a x2 ðt Þ dt{bx2 ðt ÞzK0

ð25Þ

MRAC with r modification

Yang et al. [8, 9] performed a localized stability analysis on the first-order MRAC system. They considered the case where there was a square-wave demand and showed that a localized time-invariant zero eigenvalue exists. This zero eigenvalue results in an infinite number of gain solutions that can allow wind-up. They proposed a modified scheme, rmodified MRAC, incorporating a leakage-type term that was designed to replace the zero eigenvalue with an eigenvalue at 2r2. The control signal for the modified algorithm is given by uðt Þ~Krmr ðt Þrðt ÞzKmr ðt Þxðt Þ

ð22Þ

where Kmr and Krmr are modified adaptive gains. These modified gains are given by Kmr ðsÞ~

s r2 K ð s Þz K
ðsÞ szr2 szr2

Krmr ðsÞ~

s r2 K ð s Þz K
ðs Þ r szr2 szr2 r ð23Þ

where r is a constant, K(s) and Kr(s) are the standard Proc. IMechE Vol. 222 Part I: J. Systems and Control Engineering

Fig. 6

(a) r-modified MRAC adaptive gain structure. The solid line represents the fixed gain control part, K * or Kr
, and the dash-dot line represents the adaptive gain part, K or Kr. The vertical dash line shows the value of r2 corresponding to the complementary filter break point. (b) rand w-modified MRAC structure. The solid line represents the fixed gain control part, K * or Kr
, and the dash-dot line represents the adaptive gain part, K or Kr. The vertical dash line shows the value of r2 and w2 JSCE527 F IMechE 2008

A modified model reference adaptive control approach

noting that xe 5 2x. As before, on the right-hand side the integral of the plant output squared will be the dominant term with increasing time. However, now on the left-hand side there is also an integral term that allows the adaptive gain Kmr to settle to a constant-amplitude oscillation about an average value. For simplicity, time is defined such that the oscillations of the plant output are purely sinusoidal, which has the effect of phase shifting the disturbance signal, d 5 sin(vt + w). Now, if it is assumed that the plant output settles to a constant amplitude, such that x 5 Asin(vt), and that with time the gain ¯ , then the integral oscillates about an average value K terms in equation (25) become dominant with time, allowing the approximation ð ð
{K E dt~{aA2 sin2 ðvt Þ dt r2 K

¯ 5 87.6 and A 5 0.41, equations (27) and (28) gives K ¯ 5 84.6 and A 5 0.395. whereas in simulation, K Therefore, for this example, it can be seen that the r modification allows the gain to settle to a constant mean value.

3.2

MRAC with rw modification

Through consideration of the r modification in the frequency domain, a second modification, a second complementary filter at a high frequency, is proposed. The rw-modified MRAC control gains are described in the Laplace domain as Km ðsÞ~

ð26Þ

w2 s r2   K ðsÞz K
ðs Þ 2 2 szr 2 ðszr Þ szw z

where any oscillations in Kmr have been ignored, as ¯ term will become the integral of the constant K dominant with time. Therefore, it is possible to write 2

aA K
~K E { 2 2r

ð28Þ

Figure 7 shows simulation results for the plant considered in section 2.2 using the modified controller with a 5 1000, b 5 100, and r2 5 1. Solving

s2   K
ðs Þ ðszr2 Þ szw2

w2 s r2   Kr ðsÞz K
ðs Þ szr2 r ðszr2 Þ szw2 z

 v2  K
2 ~ 2 2 1{A2 b A

Fig. 7

Krm ðsÞ~

ð27Þ

Substituting the expressions for d and x into equ¯ ation (24), a second expression in terms of A and K may be derived

203

s2   Kr
ðsÞ 2 2 ðszr Þ szw ð29Þ

where r and w are constants that need to be selected by the control designer, and K * and Kr
are steady state gains. Ideally, these gains are set to the values of the Erzberger gains, as before. This modification will limit adaptation to a frequency window as shown in Fig. 6(b). The motivation to limit high-frequency adaptation is to reduce the effect of unmodelled

Simulation of the r-modified MRAC strategy applied to a plant with a noisy output and zero demand

JSCE527 F IMechE 2008

Proc. IMechE Vol. 222 Part I: J. Systems and Control Engineering

204

S A Neild, L Yang, and D J Wagg

higher-frequency dynamics. It will also reduce the effect of high-frequency noise (although the w modification on its own will not eliminate wind-up due to noise as the gain adaptation is just limited, not eliminated, at higher frequencies). An example using this controller is given in the next section.

4

EXPERIMENTAL VALIDATION

In this section the performance of the modified MRAC strategy is assessed experimentally. The plant is a reconfigurable electronic circuit, a Quansar ‘analogue plant simulator’, and the controller is implemented in dSPACE, a real-time DSP programmable via Matlab-Simulink. Two set-ups are considered: a first-order plant with noise and a plant with higher-order dynamics.

4.1

Fig. 8

First-order plant with high-frequency noise

This part gives an example of a plant with highfrequency noise and a unit step input as the demand signal. This example is used to demonstrate the different function between r and w. The plant used is a first-order system created using the reconfigurable electronic circuit with transfer function 2.101/(s + 1.1001). This circuit has a certain amount of inherent noise, but, for demonstration purposes, additional high-frequency noise was added to the plant output. The added noise is a sinusoid wave with an amplitude of 0.05 (about 5 per cent of the input signal amplitude) and a frequency of 50 Hz. The demand signal is a unit step at time t 5 1. The reference model parameters are chosen to be am 5 bm 5 3, and hence (using equation (6)) the Erzberger gains are KE 5 20.951 and KrE ~1:428. Figure 8 shows the performance of standard MRAC with adaptive weights set to a 5 b 5 10. It can be seen from the figure that both control gains K and Kr are winding up, and the rate of this wind-up increases with increasing adaptive weights (not shown). The large oscillation of the gains is caused by the high-frequency noise. Figure 9 shows the control result by using MRAC with w modification, and w2 5 62.8 (corresponding to a break frequency of 10 Hz). Comparison with Fig. 8 shows that the amplitude of the gain oscillations is much smaller. This is because the w-modified part works as a low-pass filter, which cuts down the adaptive power at frequencies above 10 Hz. Hence, the w modification decreases MRAC adaptation to Proc. IMechE Vol. 222 Part I: J. Systems and Control Engineering

Plant with 50 Hz noise controlled by standard MRAC. The input signal is a unit step after 1, and the adaptive power is a 5 b 5 5: (a) modelfollowing error; (b) adaptive feedforward gain Kr; (c) adaptive feedback gain K

50 Hz noise. However, wind-up still exists in both gains. Figure 10 shows the control result by using MRAC with r modification. Comparison with the standard MRAC control result (Fig. 8) shows that the control gains settle at around 10 s. This is because the use of r modification increases the influence of the steady state gains (Erzberger gains) at low frequency. Hence, this decreases the system settling time. However, the control gains still have large oscillation.

Fig. 9

Contribution of w-modified MRAC to a plant with high-frequency noise. The input signal is a unit step, a 5 b 5 5, and w2 5 62.8: (a) modelfollowing error; (b) adaptive feedforward gain Kr; (c) adaptive feedback gain K JSCE527 F IMechE 2008

A modified model reference adaptive control approach

Fig. 10 Contribution of r-modified MRAC to a plant with high-frequency noise. The input signal is a unit step, a 5 b 5 5, and r2 5 0.5: (a) modelfollowing error; (b) adaptive feedforward gain Kr; (c) adaptive feedback gain K

As a result, it is observed that the r and w modifications have fundamentally different functions. The r modification cuts down the adaptive control power by increasing the fixed gain control weighting at low frequency. Hence, the modification will make the system settle faster and resolves the gain wind-up problem. On the other hand, the w modification reduces the adaptive control power above a certain frequency. This modification is an effective way of removing the deleterious effect of noise and unmodelled high-frequency dynamics on the adaptive control system. Figure 11 shows the control result by using MRAC with both r and w modifications. As expected, the control gains are stable (settling within 10 s), and the gain oscillations are decreased as well. This is a combination control result from both r and w. Note that the error amplitude remains at about 5 per cent of the step amplitude for all the controllers. 4.2

Plant with higher-order unmodelled dynamics

The application of the modified MRAC strategy to a plant with higher-order unmodelled dynamics will now be considered. A practical motivation of this work is the control of hydraulic shaking tables. Hydraulic shaking tables are widely used in the earthquake engineering community for dynamic testing of structures subjected to extreme loading. Adaptive control is desirable owing to the unknown and changing dynamics of the test specimen attached to the table when exposed to extreme loading [17]. JSCE527 F IMechE 2008

Fig. 11

205

Plant with high-frequency noise controlled by MRAC with both r and w modification. The input signal is a unit step, a 5 b 5 5, r2 5 0.5, and w2 5 62.8: (a) model-following error; (b) adaptive feedforward gain Kr; (c) adaptive feedback gain K

Under low loading, hydraulic actuators may approximately be modelled as nominally first-order systems [18]. However, owing to large inertial loading, such as the table and payload, shaking-table actuators can exhibit significant lightly damped (around 10 per cent of critical damping) higher-frequency dynamics owing to oil column resonance [14, 15, 19]. In the tests presented here, the reconfigurable electric circuit is used to create a nominal first-order plant with an additional lightly damped resonance representing the higher-frequency dynamics. A system identification of the circuitry shows that a third-order representation of the dynamics may be written as G ðsÞ~

2:108 229:521 ðsz1:147Þ ðs2 z2:895sz231:516Þ

ð30Þ

where the nominal first-order plant is 2.108/(s + 1.147), and the unmodelled higher-frequency resonance is represented by 229.521/(s2 + 2.895s + 231.516). A Bode plot of the plant is given in Fig. 12(a). The reference model parameters were set to am 5 bm 5 3, as in the previous tests. The sinusoidal input demand signal is given as rðt Þ~0:3z1:85 sinð0:8t Þ

ð31Þ

The analogue circuitry is subject to external disturbance in the form of electronic noise. The standard MRAC strategy exhibits gain wind-up, resulting in system instability when applied to the plant with higher-frequency dynamics. Proc. IMechE Vol. 222 Part I: J. Systems and Control Engineering

206

S A Neild, L Yang, and D J Wagg

Figure 13 shows the control performance using rmodified MRAC (with r2 5 0.5 and a 5 b 5 0.5). It can be seen from this figure that the system is unstable in spite of the r modification. This is because the r modification is designed to remove wind-up rather than the gain oscillations that occur when the unmodelled higher-order dynamics have low damping. Figure 14 shows the system response using the w-modified MRAC algorithm (with w2 5 5 and a 5 b 5 0.5). The value of w has been selected to reduce gain adaptation at the oil column resonance frequency at 11 rad/s. The system is stable, with the error and both gains settling within around 150 s. In contrast to the r modification, the w modification results in filtering of the unmodelled high-frequency dynamics directly to avoid the system adapting to these undesirable dynamics. Finally, Fig. 15 shows the control result using the combined rw-modified MRAC algorithm (with a 5 b 5 0.5, r2 5 0.5, and w2 5 5). The system has a stable response, with the error and gains settling within around 10 s – faster than when w-modified MRAC was used. The reason is that, by increasing r, the fixed gain contribution to the controller, which requires no time to settle, becomes more dominant.

Fig. 12

Fig. 13

Plant with unmodelled high-frequency dynamics and a damping ratio of 0.1, controlled by r-modified MRAC. The input signal r(t) 5 0.3 + 1.85sin(1t), a 5 b 5 0.5, r2 5 0.5: (a) model-following error; (b) adaptive feedforward gain Kr; (c) adaptive feedback gain K. The system is unstable

(a) Bode diagram of the hydraulic shaking-table model. The dashed line shows the firstorder nominal plant. The solid line shows the hydraulic shaking table model described in equation (30). The solid vertical line shows an input signal frequency of 0.8 rad/s, and the dashed vertical line is the frequency of the high-frequency dynamics peak response. (b) Bode diagram of rw-modified MRAC. The dash-dot line illustrates the adaptive gain control, and the solid lines show the steady state gain control. The two vertical dash lines show the r2 and w2 positions separately. By using r and w modification, an adaptive window is created. The purpose is to let the nominal plant break frequency and demand frequency sit inside the window. This avoids adaptation of MRAC to an undesired frequency

Proc. IMechE Vol. 222 Part I: J. Systems and Control Engineering

JSCE527 F IMechE 2008

A modified model reference adaptive control approach

Fig. 14 Plant with unmodelled high-frequency dynamics and a damping ratio of 0.1, controlled by w-modified MRAC. The input signal r(t) 5 0.3 + 1.85sin(1t), a 5 b 5 0.5, and w2 5 5: (a) model-following error; (b) adaptive feedforward gain Kr; (c) adaptive feedback gain K. The system is stable, with the error and gains settling within around 150 s

5

CONCLUSION

This paper has considered MRAC applied to nominally first-order systems. The mechanisms that result in noise-induced gain wind-up and instability owing to higher-order unmodelled dynamics have been

207

identified. By considering a system with zero demand, it has been shown analytically that gain wind-up is caused by the integral term in the adaptive algorithm. In contrast, instability because of higher-order unmodelled dynamics was shown to be due to instability of the instantaneous closedloop characteristic equation. Based on these insights, a rw-modified MRAC strategy has been proposed and tested. In the frequency domain, the gain adaptation mechanism within the modified MRAC is made up of three distinct regions: an adaptive control region and two fixed gain control regions. First-order complementary filters are used to define these regions, with r used to define the limit of the low-frequency fixed gain region and w used to define the start of the high-frequency fixed gain region. The r filter limits the gain adaptation at low frequency and hence eliminates gain wind-up due to noise, which is demonstrated both analytically and in simulation. The w filter is used to reduce adaptation at high frequencies and hence minimize the influence of both higher-frequency unmodelled dynamics and high-frequency noise. Tests were conducted on a reconfigurable electrical circuit with controllers implemented using dSPACE, a real-time programmable DSP. First, a noisy system was considered, and it was shown that the r modification successfully removed the gain wind-up present in standard MRAC. The w modification reduced gain oscillations due to noise, as expected, in view of the fact that the adaptive power at higher frequencies is limited by the modification; however, wind-up remained. Second, a plant with a higher-order lightly damped resonance was considered. System instability occurred using both standard MRAC and r-modified MRAC for this plant. However, when w modification was used, the plant was controlled successfully. Future work on developing this technique will focus on its application to mechanical systems such as a shaking table.

ACKNOWLEDGEMENTS

Fig. 15 Plant with unmodelled high-frequency dynamics and a damping ratio of 0.1, controlled by rw-modified MRAC. The input signal r(t) 5 0.3 + 1.85sin(1t), a 5 b 5 0.5, r2 5 0.5, and w2 5 5: (a) model-following error; (b) adaptive feedforward gain Kr; (c) adaptive feedback gain K. The system is stable, with the error and gains settling within around 10 s JSCE527 F IMechE 2008

The authors would like to acknowledge the support of the EPSRC. Lin Yang is supported by the Dorothy Hodgkin Postgraduate Award scheme (EPSRC-BP) and David Wagg by an Advanced Research Fellowship.

REFERENCES 1 Astro¨m, K. J. and Wittenmark, B. Adaptive control, 2nd edition, 1995 (Addison-Wesley). Proc. IMechE Vol. 222 Part I: J. Systems and Control Engineering

208

S A Neild, L Yang, and D J Wagg

2 Landau, Y. D. Adaptive control: the model reference approach, 1979 (Marcel Dekker). 3 Popov, V. M. Hyperstability of control systems, 1973 (Springer). 4 Anderson, B. D. O., Bitmead, R. R., Johnson Jr, C. R., Kokotovic, P. V., Kosut, R. L., Mareels, I. M. Y., Praly, L., and Riedle, B. D. Stability of adaptive systems: passivity and averaging analysis, 1987 (MIT Press). 5 Sastry, S. and Bodson, M. Adaptive control: stability, convergence and robustness, 1989 (Prentice-Hall). 6 Ioannou, P. A. and Kokotovic, P. V. Instability analysis and improvement of robustness of adaptive control. Automatica, 1984, 20(5), 583–594. 7 Virden, D. and Wagg, D. J. System identification of a mechanical system with impacts using model reference adaptive control. Proc. IMechE, Part I: J. Systems and Control Engineering, 2005, 219(I2), 121–132. 8 Yang, L., Neild, S. A., Wagg, D. J., and Virden, D. W. Model reference adaptive control of a nonsmooth dynamical system. Nonlinear Dynamics, 2006, 46(3), 323–335. 9 Yang, L. A modified model reference adaptive control algorithm to improve system robustness. MSc Thesis, University of Bristol, UK, 2004. 10 Gu, E. Y. L. Dynamic systems analysis and control based on a configuration manifold model. Nonlinear Dynamics, 2000, 19, 113–134. 11 Townley, S. Topological aspects of universal adaptive stabilization. SIAM J. Control and Optimization, 1996, 34(3), 1044–1070. 12 Ronki Lamooki, G. R., Townley, S., and Osinga, H. M. Normal forms, bifurcations and limit dynamics in adaptive control systems. Int. J. Bifurcation and Chaos, 2005, 15(5), 1641–1664. 13 Rohrs, C., Valavani, L., Athans, M., and Stein, G. Robustness of continuous-time adaptive control algorithms in the presence of unmodeled dynamics. IEEE Trans. Autom. Control, 1985, AC-30, 881–889. 14 Nikzad, K., Ghaboussi, J., and Paul, S. L. Actuator dynamics and delay compensation using neurocontrollers. J. Engng Mechanics, 1996, 122(10), 966–975. 15 Neild, S. A., Stoten, D. P., Drury, D., and Wagg, D. J. Control issues relating to real-time substructuring experiments using a shaking table. Earthquake Engng and Struct. Dynamics, 2005, 34(9), 1171–1192. 16 Hillis, A. J., Neild, S. A., Stoten, D. P., and Harrison, A. J. L. A minimal control synthesis algorithm for narrow-band applications. Proc. IMechE, Part I: J. Systems and Control Engineering, 2005, 219(I8), 591–607. ´ mez, E. Real-time adaptive 17 Stoten, D. P. and Go control of shaking tables using the minimal control synthesis algorithm. Philosophical Trans. R. Soc., Lond. A, 2001, 359, 1697–1723. 18 Neild, S. A., Drury, D., and Stoten, D. P. An improved substructuring control strategy based on Proc. IMechE Vol. 222 Part I: J. Systems and Control Engineering

the MCS control algorithm. Proc. IMechE, Part I: J. Systems and Control Engineering, 2005, 219(I5), 305–317. 19 Crewe, A. J. The characterisation and optimisation of earthquake shaking table performance. PhD Thesis, University of Bristol, UK, 1998.

APPENDIX Notation a am

X ye

plant denominator coefficient reference model denominator coefficient trial solution scaling parameter plant numerator coefficient reference model numerator coefficient error state weighting vector plant disturbance adaptive feedback gain Erzberger feedback gain rw-modified adaptive feedback gain r-modified adaptive feedback gain adaptive feedforward gain rw-modified adaptive feedforward gain r-modified adaptive feedforward gain initial feedforward gain fixed feedback gain Erzberger feedforward gain fixed feedforward gain initial feedback gain averaged feedback gain feedback gain offset trial solution exponent parameter demand (or reference) signal Laplace-transform variable time time (used in averaging over an oscillation) plant input (or control) signal plant input in Laplace domain plant output model-following error 5 xm 2 x reference model output plant output prior to added disturbance plant output in Laplace domain weighted model-following error

a b r w v

weighting on integral adaptation weighting on proportional adaptation r-modified MRAC control parameter w-modified MRAC control parameter angular frequency

A b bm Ce d K KE Km Kmr Kr Krm Krmr Kr0 K* KrE Kr
K0 ¯ K , K n r s t ¯t u U x xe xm xp

JSCE527 F IMechE 2008