CONTROL SYSTEM

Abstract

A control system (1) for controlling a plant (2) comprises a feedback loop including an integrator (7); a signal generator (32); and a scaling unit (10). The feedback loop comprises an input suitable for connection to an output (18) of the plant. The integrator integrates a signal received from the input to generate a state signal x. The signal generator generates a periodic base perturbation signal (34) with an initial amplitude. The scaling unit generates a scaling factor (30) having a first value if the variance of the state signal var(x) is zero, or a second value if the variance of the state signal is non-zero, wherein the second value is proportional to (formulae 1) The scaling unit is arranged to multiply (16) the initial amplitude of the periodic base perturbation signal by the scaling factor to produce a state dependent perturbation signal (35, 36), which is applied to an input of the plant.

Claims

1. A control system for controlling a plant, the control system comprising a feedback loop including an integrator; a signal generator; and a scaling unit, wherein the control system is arranged such that: the feedback loop comprises an input suitable for connection to an output of the plant; the integrator integrates a signal received from the input to generate a state signal {circumflex over (x)}; the signal generator generates a periodic base perturbation signal with an initial amplitude; the scaling unit generates a scaling factor having a first value if the variance of the state signal var({circumflex over (x)}) is zero, or a second value if the variance of the state signal is non-zero, wherein the second value is proportional to $e^{\frac{-1}{\mathrm{var}\left(\hat{x}\right)}};$ wherein the scaling unit is arranged to multiply the initial amplitude of the periodic base perturbation signal by the scaling factor to produce a state dependent perturbation signal, which is provided at an output of the control system suitable for connection to an input of the plant.

2. The control system of claim 1, wherein the signal generator is arranged to generate a periodic base perturbation signal that is sinusoidal.

3. The control system of claim 1, wherein a function .sub.n representing noise in the control system is bounded according to: $f_n-{\lim }_{T->}.Math.\frac{1}{T}.Math.{}_0^T.Math.n.Math..Math.\sin .Math..Math..Math..Math.t.Math..Math.\mathrm{dt};$ wherein sin t represents the periodic base perturbation signal having a perturbation frequency .

4. The control system of claim 1, wherein the second value has a lower limit substantially equal to the first value.

5. The control system of claim 1, wherein the second value is proportional to $e^{\frac{-}{\mathrm{var}\left(\hat{x}\right)}},$ wherein is a decay constant chosen to set the rate of exponential decay of the scaling factor.

6. The control system of claim 1, wherein the second value of the scaling factor is $.Math..Math.e^{\frac{-}{\mathrm{var}\left(\hat{x}\right)}},$ wherein is a decay constant chosen to set the rate of exponential decay of the scaling factor and is a perturbation gain.

7. The control system of claim 6, wherein the state dependent perturbation signal is represented as a function of the state signal {circumflex over (x)} according to: $f\left(\hat{x}\right)=\{\begin{array}{cc}.Math..Math.e^{\frac{-x}{\mathrm{var}\left(\hat{x}\right)}},& \mathrm{if}.Math..Math.\mathrm{var}\left(\hat{x}\right)<.Math..Math.\mathrm{and}.Math..Math.\mathrm{var}\left(\hat{x}\right)0\\ ,& \mathrm{otherwise}\end{array}.$

8. The control system of claim 5, wherein the constant is positive

9. The control system of claim 5, wherein the constant is selected such that var({circumflex over (x)})<.

10. The control system of claim 1, wherein the scaling unit is arranged to produce a state dependent perturbation signal having a gain that is large enough to excite the control system.

11. The control system of claim 1, wherein the feedback loop comprises a shifting unit arranged to provide an offset value.

12. The control system of claim 11, wherein the shifting unit is arranged such that the offset value is added to the state dependent perturbation signal.

13. The control system of claim 11, wherein the shifting unit is a compensator, for example a proportional-integral-derivative (PID) controller.

14. The control system of claim 1, wherein the feedback loop comprises a high pass filter having a cut-off frequency .sub.h that satisfies .sub.h<<, where is the frequency of the periodic base perturbation signal.

15. The control system of claim 1, wherein the feedback loop comprises a low pass filter having a cut-off frequency .sub.l that satisfies .sub.l<<, where is the frequency of the periodic base perturbation signal.

16. The control system of claim 1, wherein the feedback loop comprises a bandpass filter comprising a high pass filter with a first corner frequency and a low pass filter with a second corner frequency, wherein the first corner frequency is greater than the second corner frequency.

17. The control system of claim 16, wherein the bandpass filter is arranged to filter the input before passing the filtered input signal to the integrator.

18. The control system of claim 16, wherein the output of the high pass filter is multiplied with the periodic base perturbation signal prior to passing through the low pass filter.

19. A method of controlling a plant, the method comprising: obtaining an output signal from the plant; integrating the output signal from the plant to generate a state signal {circumflex over (x)}; generating a periodic base perturbation signal with an initial amplitude; generating a scaling factor having a first value if the variance of the state signal var({circumflex over (x)}) is zero, or a second value if the variance of the state signal is non-zero, wherein the second value is proportional to $e^{\frac{-1}{\mathrm{var}\left(\hat{x}\right)}};$ multiplying the initial amplitude of the periodic base perturbation signal by the scaling factor to produce a state dependent perturbation signal; and inputting the state dependent perturbation signal to the plant so as to drive the plant to a desired operating point.

Description

[0037] Certain embodiments of the invention will now be described, by way of example only, with reference to the accompanying drawings in which:

[0038] FIG. 1 shows a graph that illustrates a typical extremum that might be located using the present invention;

[0039] FIG. 2 shows the simulated performance of two conventional control systems;

[0040] FIG. 3 shows a block diagram of an SDP-ESC control system in accordance with an embodiment of the invention;

[0041] FIG. 4 shows a comparison of typical input voltages for a conventional ESC control system and an SDP-ESC control system in accordance with an embodiment of the invention; and

[0042] FIGS. 5A and 5B show a comparison of typical input voltage duty cycles for a conventional ESC control system and an SDP-ESC control system in accordance with an embodiment of the invention.

[0043] FIG. 1 shows a graph that illustrates a typical extremum that might be located using the present invention. The curve 100 shows the relationship between the output P of a plant to be controlled and the input d to said plant. The input d can be controlled e.g. between two points 102, 104 on the curve 100, which varies the output P.

[0044] As can be seen from the curve 100, starting at point 102, the plant produces an initial output p.sub.0 for a given input d.sub.0. Moving towards the second point 104 by increasing the input to d.sub.1 causes the output to increase to p.sub.1. From this simple curve, it can be seen that eventually a maximum point 106 is reached. At the optimum point 106, the first derivative (i.e. the gradient) of the curve 100 is zero and it is therefore a stationary point. Furthermore, the second derivative of the curve 100 at the optimum point 106 is negative and therefore the stationary point is a maximum.

[0045] It is extrema such as this optimum point 106 which the present invention seeks to locate. While this curve 100 has a relatively simple relationship between the output P and the input d, it will be appreciated that the principles of the invention described herein apply to applications with far more complex input-output relationships.

[0046] FIG. 2 shows the simulated performance of two conventional control systems. More specifically, it is a simulation of the performance of a Bi.sub.2Te.sub.3 thermoelectric generator (TEG), simulating the output power P as a function of time t using different control methods.

[0047] The curve 200 depicted by the thicker dashed line shows the theoretical maximum performance of the TEG. The curve 206 depicted by the solid line with circular markers shows the performance of the TEG with no control mechanism used (i.e. the input is fixed).

[0048] A simulation of the results associated with controlling the TEG with the conventional perturb and observe (P&O) method as previously mentioned is shown by PO curve 204, while a simulation of a conventional Extremum Seeking Control (ESC) method ESC curve 202. While the ESC method begins further from the theoretical maximum than the PO method, as can be seen from the graph the ESC curve 202 on average is closer than the PO curve 204 to the theoretical curve 200 and converges faster. This indicates enhanced performance and greater convergence towards the optimal output power of the TEG.

[0049] However, as described previously, the ESC method must utilise a perturbation signal with a high amplitude in order to achieve convergence to the desired value. This high amplitude introduces oscillations and losses to the output, as will be described below with reference to FIGS. 4 to 6.

[0050] FIG. 3 shows a block diagram of a state dependent parameter extremum seeking control (SDP-ESC) system 1 in accordance with an embodiment of the invention. The control system 1 is arranged as a loop containing: a plant 2; a high pass filter 4; a low pass filter 6; an integrator 7; a compensator 8; a scaling unit 10; multipliers 12, 16; an adder 14; and a signal generator 32.

[0051] The plant 2 is the entity to be controlled by the control system 1, which as previously described might, for example, be: a thermoelectric generator; a photovoltaic cell; a wind turbine; a fuel cell; an anti-lock braking system; or a bio-reactor.

[0052] In order to control the plant 2, it is perturbed using a periodic perturbation signalin this particular embodiment, a sinusoidal signal 34 generated by the signal generator 32. However, rather than simply using the perturb and observe method outlined previously, this sinusoidal signal 34 is modulated by the control loop to achieve rapid convergence to a desired operating point within the plant 2. The sinusoidal signal 34, representing a periodic base perturbation signal having a perturbation frequency , may take the general form sin t.

[0053] The plant 2 produces an output 18, which is monitored by the control system 1. The output 18 is fed to the high pass filter 4 which filters any low frequency components from the output below an upper corner frequency, .sub.h, which generates a high pass filtered signal 20. The transfer function of the high pass filter 4 in the Laplace domain is given by Eq. 1 below:

[00012]$\begin{array}{cc}\mathrm{Transfer}.Math..Math.\mathrm{function}.Math..Math.\mathrm{of}.Math..Math.\mathrm{high}.Math..Math.\mathrm{pass}.Math..Math.\mathrm{filter}.Math..Math.4.Math..Math.F_{h.Math.p.Math.f}\left(s\right)=\frac{s}{s+{}_h}& \mathrm{Eq}..Math.1\end{array}$

wherein: .sub.hpf(s) is the transfer function of high pass filter 4; s is the complex frequency; and .sub.h is the upper corner frequency.

[0054] The high pass filtered signal 20 is then input to the first multiplier 12, which is also fed with the periodic perturbation signal 34. The multiplier 12 then generates a mixed signal 22.

[0055] The resulting mixed signal 22 is then input to the low pass filter 6, which filters any low frequency components from the output below a lower corner frequency, .sub.l, which generates a low pass filtered signal 24. The transfer function of the low pass filter 7 in the Laplace domain is given by Eq. 2 below:

[00013]$\begin{array}{cc}\mathrm{Transfer}.Math..Math.\mathrm{function}.Math..Math.\mathrm{of}.Math..Math.\mathrm{low}.Math..Math.\mathrm{pass}.Math..Math.\mathrm{filter}.Math..Math.6.Math..Math.F_{\mathrm{lpf}}\left(s\right)=\frac{{}_l}{s+{}_l}& \mathrm{Eq}..Math.2\end{array}$

wherein: .sub.lpf(s) is the transfer function of low pass filter 6; s is the complex frequency; and .sub.l is the lower corner frequency.

[0056] The low pass filtered signal 24 is then input to the integrator 7. The integrator 7 is arranged to sum the low pass filtered signal 24 over time, generating a state signal 26. This state signal 26 is then input to both the compensator 8 and the scaling unit 10. The transfer function of the integrator 7 in the Laplace domain is given by Eq. 3 below:

[00014]$\begin{array}{cc}\mathrm{Transfer}.Math..Math.\mathrm{function}.Math..Math.\mathrm{of}.Math..Math.\mathrm{the}.Math..Math.\mathrm{integrator}.Math..Math.7.Math..Math.F_{i.Math.n.Math.t}\left(s\right)=\frac{k}{s}& \mathrm{Eq}..Math.3\end{array}$

wherein: F.sub.int(s) is the transfer function of low pass filter 7; s is the complex frequency; and k is a constant.

[0057] The scaling unit 10 is arranged to produce a gain factor 30 depending on the variance of the state signal 26. If the variance of the state signal 26 is zero, the gain factor 30 is set to a first value , where this first value is sufficiently large to be able to excite the plant 2. However, if the variance of the state signal 26 is non-zero, the gain factor 30 is set to a second value proportional to

[00015]$e^{\frac{-1}{\mathrm{var}\left(\hat{x}\right)}}.$

The relationship is shown mathematically below in Eq. 4:

[00016]$\begin{array}{cc}\mathrm{State}.Math..Math.\mathrm{signal}.Math..Math.\mathrm{dependent}.Math..Math.\mathrm{function}.Math..Math.f\left(\hat{x}\right)=\{\begin{array}{cc}.Math.e^{\frac{-}{v.Math.a.Math.r\left(\hat{x}\right)}},& \mathrm{if}.Math..Math.\mathrm{var}.Math..Math.\left(\hat{x}\right)<.Math..Math.\mathrm{and}.Math..Math.\mathrm{var}.Math..Math.\left(\hat{x}\right)0\\ ,& \mathrm{otherwise}\end{array}& \mathrm{Eq}..Math.4\end{array}$

[0058] wherein: {circumflex over (x)} is the state signal 26; ({circumflex over (x)}) is the gain factor 30; is the first value as described above; and is a decay constant which can be chosen to set the rate of exponential decay with respect to the state signal 26.

[0059] Because, at least in time-based systems, the plant 2 has a faster response to changes than the feedback loop, the decay constant is chosen such that the filters 4, 6 are faster than the exponential function that sets the gain factor 30. The decay constant is therefore dependent on the dynamics of the plant 2 to be controlled, and a is then chosen after the upper and lower filter corner frequencies .sub.h and .sub.l are set.

[0060] The resulting gain factor 30 is applied as an input to the multiplier 16, which is also provided with the sinusoidal signal 34. This multiplier 16 then generates a scaled sinusoidal signal 35.

[0061] The compensator 8 is arranged to produce an offset value 28 that will be applied to the sinusoidal signal 34. While the scaling unit 10 scales the amplitude of the perturbation signal to be applied to the plant 2, the compensator 8 is arranged to shift the average value of the perturbation signal up or down as appropriate for optimal convergence to the desired operating point. Accordingly, the compensator 8 uses proportional-integral control based on the state signal 26 to generate the offset value 28. This offset value 28 is then input to the adder 14 along with the scaled sinusoidal signal 35 to generate a state-dependent perturbation signal 36, which is then input to the plant 2. The transfer function of the compensator 8 in the Laplace domain is given below in Eq. 5:

[00017]$\begin{array}{cc}\mathrm{Transfer}.Math..Math.\mathrm{function}.Math..Math.\mathrm{of}.Math..Math.\mathrm{the}.Math..Math.\mathrm{compensator}.Math..Math.8.Math..Math.F_{\mathrm{comp}}\left(s\right)=\frac{s.Math.k_c+1}{s}& \mathrm{Eq}..Math.5\end{array}$

wherein: F.sub.comp(s) is the transfer function of the compensator 8; s is the complex number frequency; and k.sub.c is a constant.

[0062] In relation to the integrator 7 and compensator 8, the constants k.sub.c and k should be chosen to be sufficiently large so as to adapt to deliberate perturbations while avoiding the detection of small variations caused by noise. In other words, these parameters should be selected such that, noise is not detected within the SDP-ESC feedback loop. However, if k.sub.c and k are too large, this may increase oscillations due to continuous detection of every small variation within the feedback loop.

[0063] This control system 1 is arranged such that the feedback loop then allows for a dynamically variable state-dependent perturbation signal 36 based on the output 18 of the plant 2, so as to drive the output 18 to the desired value. The plant 2 itself has a transfer function expressed below in Eq. 6:

[00018]$\begin{array}{cc}\mathrm{Transfer}.Math..Math.\mathrm{function}.Math..Math.\mathrm{of}.Math..Math.\mathrm{the}.Math..Math.\mathrm{plant}.Math..Math.2.Math..Math.f\left(d\right)=P_{m.Math..Math.\mathrm{ax}}+\frac{P^{}}{2}.Math.{\left(d-d_{\mathrm{opt}}\right)}^2& \mathrm{Eq}..Math.6\end{array}$

wherein: (d) is the transfer function of the plant 2; d is the state-dependent perturbation signal 36; P.sub.max is the maximum theoretical output of the plant 2; d.sub.opt is the theoretical optimum perturbation signal and P is the second derivative of the output 18 of the plant 2.

[0064] For example, if the control system of the present invention were to be used to control the output of a thermoelectric generator (TEG) such that it provides maximum power transfer to a load, the desired output voltage from the plant 2 (i.e. the photovoltaic cell) P.sub.max would be half of the open circuit voltage of the TEG.

[0065] Using the TEG as a worked example, first consider a model of the TEG (i.e. the plant 2) approximated using a Taylor series expansion as per Eq. 7:

[00019]$\begin{array}{cc}\mathrm{Taylor}.Math..Math.\mathrm{series}.Math..Math.\mathrm{expansion}.Math..Math.\mathrm{of}.Math..Math.a.Math..Math.\mathrm{TEG}.Math..Math.y\left(u\right)=y^{*}+\frac{y^{}}{2}.Math.{\left(u-u^{*}\right)}^2& \mathrm{Eq}..Math.7\end{array}$

where in u is the duty cycle, u* is the duty cycle at the extremum point, y is the measured output and y* is the output at the extremum point.

[0066] Two assumptions can be made. Firstly, it is assumed that the low pass filter 6 is not essential for convergence analysis since the integrator 7 will typically attenuate high frequencies. However, it should be noted that for practical implementations the low pass filter 6 will be included. The second assumption made here is that the while the TEG model will include other dynamics, the state-dependent perturbation signal 36 used in the control system 1 is considered to be sufficiently slow so as to treat the plant 2 as a static nonlinear map.

[0067] The objective of SDP-ESC is to minimise the quantity (u*) such that the measured output y approaches y* i.e. y(u)y*. The difference between u* and is called estimation error u.sub.e and is given as per Eq. 8 below:

u.sub.e=u*Eq. 8: Estimation error

[0068] This quantity is modulated such by ({circumflex over (x)}) sin t to obtain u. The difference between u and u* is given as per Eq. 9:

uu*=({circumflex over (x)})sin tu.sub.eEq. 9: Difference between u and u*

[0069] Substituting Eq. 9 into Eq. 8 reformulates the objective of SDP-ESC as per Eq. 10:

[00020]$\begin{array}{cc}\mathrm{Reformulated}.Math..Math.\mathrm{objective}.Math..Math.\mathrm{of}.Math..Math.\mathrm{SDP}.Math.\text{-}.Math.\mathrm{ESC}.Math..Math.y\left(u\right)=y^{*}+\frac{y^{}}{2}.Math.{\left(u_e-f\left(\hat{x}\right).Math.\sin .Math..Math..Math..Math.t\right)}^2& \mathrm{Eq}..Math.10\end{array}$

[0070] Furthermore, if var({circumflex over (x)})> and var({circumflex over (x)})0, substituting Eq. 4 into Eq. 10 provides Eq. 11:

[00021]$\begin{array}{cc}\mathrm{Reformulated}.Math..Math.\mathrm{objective}.Math..Math.\mathrm{of}.Math..Math.\mathrm{SDP}.Math.\text{-}.Math.\mathrm{ESC}.Math..Math.\mathrm{using}.Math..Math.\mathrm{state}.Math..Math.\mathrm{dependent}.Math..Math.\mathrm{parameter}.Math..Math.\mathrm{function}.Math..Math..Math.y\left(u\right)=y^{*}+\frac{y^{}}{2}.Math.{\left(u_e-.Math.e^{-}.Math.\sin .Math..Math..Math..Math.t\right)}^2.Math..Math..Math.\mathrm{where}.Math..Math.=\frac{}{\mathrm{var}.Math..Math.\left(\hat{x}\right)}.& \mathrm{Eq}..Math.11\end{array}$

[0071] Expanding Eq. 11 and replacing sin t with (1cos 2t) yields Eq. 12:

[00022]$\begin{array}{cc}\mathrm{Expanded}.Math..Math.\mathrm{objective}.Math..Math.\mathrm{of}.Math..Math.\mathrm{SDP}.Math.\text{-}.Math.\mathrm{ESC}.Math..Math.\mathrm{using}.Math..Math.\mathrm{state}.Math..Math.\mathrm{dependent}.Math..Math.\mathrm{parameter}.Math..Math.\mathrm{function}.Math..Math.y\left(u\right)=y^{*}+\frac{y^{}.Math.u_e^2}{2}+\frac{y^{}}{4}.Math.{}^2.Math.e^{-2.Math.}-y^{}.Math.u_e.Math..Math.e^{-}.Math.\sin .Math..Math.t-\frac{y^{}}{4}.Math.{}^2.Math.e^{-2.Math.}.Math.\cos .Math.2.Math..Math.t& \mathrm{Eq}..Math.12\end{array}$

[0072] The high-pass filter 4 will remove any slow DC component of and thus Eq. 12 can be approximated as:

[00023]$\begin{array}{cc}.Math.\mathrm{Approximation}.Math..Math.\mathrm{after}.Math..Math.\mathrm{high}.Math.\text{-}.Math.\mathrm{pass}.Math..Math.\mathrm{filter}.Math..Math.4.Math..Math.\frac{s}{s+{}_h}\left[y\right]y^{*}+\frac{y^{}.Math.u_e^2}{2}+\frac{y^{}}{4}.Math.{}^2.Math.e^{-2.Math.}-y^{}.Math.u_e.Math..Math.e^{-}.Math.\sin .Math..Math..Math..Math.t-\frac{y^{}}{4}.Math.{}^2.Math.e^{-2.Math.}.Math.\cos .Math.2.Math..Math.t& \mathrm{Eq}..Math.13\end{array}$

[0073] The signal of Eq. 13 is then demodulated by multiplying with the dither signal sin t to obtain:

[00024]$\begin{array}{cc}.Math.\mathrm{Demodulated}.Math..Math.\mathrm{signal}.Math..Math.\mathrm{post}.Math.\text{-}.Math.\mathrm{dithering}.Math..Math.\frac{y^{}.Math.u_e^2}{2}.Math.\sin .Math..Math.t+\frac{y^{}}{4}.Math.{}^2.Math.e^{-2.Math.}.Math.\sin .Math..Math..Math..Math.t-y^{}.Math.u_e.Math..Math.e^{-}.Math.{\sin }^2.Math..Math.t-\frac{y^{}}{4}.Math.{}^2.Math.e^{-2.Math.}.Math.\cos .Math.2.Math..Math.t.Math.\sin .Math..Math.t& \mathrm{Eq}..Math.14\end{array}$

[0074] By replacing the cos 2t sin t term with the identity (sin 3tsin t) demodulated signal in Eq. 14 yields Eq. 15 below:

[00025]$\begin{array}{cc}.Math.\mathrm{Demodulated}.Math..Math.\mathrm{signal}.Math..Math.\mathrm{post}.Math.\text{-}.Math.\mathrm{dithering}.Math..Math.\frac{y^{}.Math.u_e^2}{2}.Math.\sin .Math..Math.t+\frac{y^{}}{4}.Math.{}^2.Math.e^{-2.Math.}.Math.\sin .Math..Math..Math..Math.t-y^{}.Math.u_e.Math..Math.e^{-}.Math.{\sin }^2.Math..Math.t-\frac{y^{}}{8}.Math.{}^2.Math.e^{-2.Math.}\left(\sin .Math..Math.3.Math..Math..Math.t-\sin .Math..Math..Math..Math.t\right)& \mathrm{Eq}..Math.15\end{array}$

[0075] The magnitude of u.sub.e.sup.2 is considered to be small and can be neglected accordingly. Eq. 15 is then reduced to:

[00026]$\begin{array}{cc}.Math.\mathrm{Demodulated}.Math..Math.\mathrm{signal}.Math..Math.\mathrm{post}.Math.\text{-}.Math.\mathrm{dithering}.Math..Math.\mathrm{after}.Math..Math.\mathrm{neglecting}.Math..Math.u_e^2.Math..Math.-\frac{y^{}}{2}.Math.u_e.Math..Math.e^{-}+\frac{y^{}}{4}.Math.{}^2.Math.e^{-2.Math.}.Math.\sin .Math..Math.t-\frac{y^{}}{2}.Math.u_e.Math..Math.e^{-}.Math.\cos .Math.2.Math..Math.t-\frac{y^{}}{8}.Math.{}^2.Math.e^{-2.Math.}\left(\sin .Math.3.Math..Math.t-\sin .Math..Math..Math..Math.t\right)& \mathrm{Eq}..Math.15\end{array}$

[0076] Equation 15 comprises a number of high frequency signals which when passed through the integrator 7, yields Eqs. 16 and 17 below:

[00027]$\begin{array}{cc}\mathrm{Approximation}.Math..Math.\mathrm{of}.Math..Math.\mathrm{the}.Math..Math.\mathrm{state}.Math..Math.\mathrm{signal}.Math..Math.\hat{x}-\frac{k}{s}.Math.\frac{\left(e^{-}.Math..Math..Math.y^{}\right)}{2}.Math.u_e& \mathrm{Eq}..Math.16\end{array}$

[0077] Similarly:

[00028]$\begin{array}{cc}\mathrm{Approximation}.Math..Math.\mathrm{of}.Math..Math.\mathrm{the}.Math..Math.\mathrm{optimal}.Math..Math.\mathrm{perturbation}.Math..Math.\mathrm{signal}.Math..Math.\hat{u}\frac{s.Math.k_c+1}{s}.Math.\hat{x}& \mathrm{Eq}..Math.17\end{array}$

[0078] Substituting Eq. 16 into Eq. 17 yields:

{circumflex over ({dot over (u)})}zk.sub.cu.sub.ez{dot over (u)}

[0079] Eq. 18: Approximation of the rate of change of the optimal perturbation signal where

[00029]$=\frac{\left(k.Math..Math.e^{-}.Math.y^{}\right)}{2}.$

[0080] As u* is constant as shown in Eq. 8, its derivative can be written as:

{dot over (u)}.sub.e={circumflex over ({dot over (u)})}Eq. 19: Derivative of u* with respect to

[0081] Substituting Eq. 18 into Eq. 19 yields:

[00030]$\begin{array}{cc}\mathrm{Estimation}.Math..Math.\mathrm{error}.Math..Math.\mathrm{converging}.Math..Math.\mathrm{to}.Math..Math.\mathrm{extremum}.Math..Math.\mathrm{point}.Math..Math.\stackrel{.}{u_e}=\frac{z}{1-z}.Math.k_c.Math.u_e& \mathrm{Eq}..Math.20\end{array}$

[0082] As

[00031]$u_e.Math.\stackrel{\mathrm{yields}}{}.Math.0,$

and e.sup. converges to a small region such that 0<e.sup.<1, then converges within a smaller region of u* with minimised oscillations. For the scheme described hereinabove wherein the low pass filter 6 is removed from the SDP-ESC loop, the output error yy* achieves local exponential convergence to an O(.sup.2e.sup.2) of the operating point with minimum oscillations, provided that the exponential decay is bounded such that 0<e.sup.<1.

[0083] With respect to the choice of the parameters themselves, there are a number of considerations that must be made. Firstly, the frequency of the perturbation signal 34 must be sufficiently large but not equal to the frequency of any significant noise components else the tracking error will increase. The bounded noise is assumed to be uncorrected with perturbation signal, therefore noise should be bounded as per Eq. 21:

[00032]$\begin{array}{cc}\mathrm{Bounded}.Math..Math.\mathrm{noise}.Math..Math.\mathrm{function}.Math..Math.f_n=\underset{T.fwdarw.}{\lim }.Math.\frac{1}{T}.Math.{}_0^T.Math.n.Math..Math.\sin .Math..Math..Math..Math.\mathrm{tdt}& \mathrm{Eq}..Math.21\end{array}$

[0084] Secondly, the perturbation gain must be large enough to excite the plant 2 as well as to achieve a desired convergence speed. A large perturbation gain will increase the convergence speed with minimum oscillations due to the exponentially decaying effect cause by the SDP-ESC function. If is too small it may fail to excite the plant 2, especially when the SDP-ESC control system 1 is applied to low voltage applications. As the SDP-ESC function will decay exponentially close to zero, can be selected to increase the rate of convergence to the extremum, as well as providing sufficient excitation to the plant 2.

[0085] The upper corner frequency .sub.h and the lower corner frequency .sub.l should be chosen after determining the frequency of the perturbation signal 34 such that (.sub.h, .sub.1<<). The upper corner frequency .sub.h and the lower corner frequency .sub.l should be bounded such that the high pass filter 4 removes any unwanted DC components. On the other hand, the low pass filter 6 should attenuate any unwanted high frequency components. The dynamics of these filters 4, 6 should be sufficiently fast enough to respond to perturbations.

[0086] Finally, it is important to select the exponential decaying constant such that, var({circumflex over (x)})<. If =0, there is no will be no effect on limit cycle minimisation, and the resulting SDP-ESC control system 1 will merely provide the same performance as a conventional ESC-based control system. Conversely, if <0 is selected, the SDP-ESC function ({circumflex over (x)}) will increase exponentially, causing the control system 1 to become unstable. Since the state signal dependent function ({circumflex over (x)}) of Eq. 4 causes the perturbation gain to decay exponentially to a small value when >0 is selected, it is therefore apparent that selecting a large value of introduces to the control system 1 neither unwanted oscillations nor sensitivity to noise. In order to successfully minimise limit cycles as well as ensure the control system 1 responds correctly, should be bounded such that var({circumflex over (x)})<.

[0087] FIG. 4 shows a comparison of typical input voltages for a conventional ESC control system and an SDP-ESC control system in accordance with an embodiment of the invention. The upper plot 402 shown in FIG. 4 shows the input voltage V.sub.in applied to the TEG system as described previously with reference to FIG. 2 using a conventional ESC control system. The lower plot 404 shows the input voltage V.sub.in applied to the TEG system using an SDP-ESC control system in accordance with an embodiment of the invention.

[0088] As can be seen by comparing the two plots, there is significantly less noise on the lower plot 404 than is present on the upper plot 402. This noise is due to the oscillations introduced by the ESC control system when large initial amplitudes are used. These oscillations hinder the performance of the system and thus the SDP-ESC control system is clearly an improvement over the conventional ESC method.

[0089] FIGS. 5A and 5B show a comparison of typical input voltage duty cycles for a conventional ESC control system and a control system in accordance with an embodiment of the invention. Both the ESC curve 502 and the SDP-ESC curve 504 both suffer from large oscillations in the input duty cycle initially. However, the oscillations on the SDP-ESC curve 504 are reduced in a relatively short period of time while the oscillations on the ESC curve 502 remain relatively constant over time.

[0090] FIG. 5B shows a further comparison of typical input voltage duty cycles at a later time. As can be seen, even once the system has been given time to settle, the oscillations are still present on the ESC curve 502 whereas the SDP-ESC curve 504 is relatively smooth.

[0091] Thus it will be seen that a control system and method have been described which achieves fast convergence while minimising oscillations and losses on the output of a controlled plant. Although particular embodiments have been described in detail, it will be appreciated by those skilled in the art that many variations and modifications are possible using the principles of the invention set out herein. The scope of the present invention is defined by the following claims.