Nonlinear PIP Control System Design

Table of Contents CHAPTER 3................................................................................................................... 56 NONLINEAR CONTROL SYSTEM DESIGN .......................................................... 56 3.1 Dynamic Modelling and State Dependent Parameter Estimation......................... 57 3.2 Nonlinear Non-Minimum State Space form ......................................................... 63 3.3 Nonlinear State Dependent Parameter Control..................................................... 69 3.3.1 Nonlinear pole assignment............................................................................. 70 3.3.2 Linear Quadratic (LQ) optimization.............................................................. 76 3.4 Summary............................................................................................................... 80

List of Figures Figure 3.1 Comparison between estimated nonlinear SDP and linear TF model parameters for Example (3.1). Left hand side subplots show the time variation of the estimated SDP parameters a1 (top) and b1 (bottom). Right hand side subplots show the relationship between the parameters and the associated state variable. The right hand side graphs show the estimated SDP parameters (dots); the linear estimates (dashed); and the regression fit to the SDP estimates (solid)................................................................................. 62 Figure 3.2 Comparison between nonlinear SDP and linear TF model fit for Example (3.1). The upper subplot shows the actual (simulated) parameters (dots); the linear model response (dashed); and the nonlinear model response (solid). ................................................................................................... 63 Figure 3.3 Open-loop response of Example (3.3) showing zero response, at step input 1.25. ............................................................................................................ 68 Figure 3.4 The SDP-PIP control system......................................................................... 70 Figure 3.5 Unity step closed-loop response for the Hammerstein system in Example (3.4), with equal poles assigned to 0.5 on the complex z-plane. .......... 72 Figure 3.6 The closed-loop response for Example (3.5) at two different poles; poles at 0.5 (left plot) and 0.3 (right plot) on the complex z-plane both at reference input 0.3. The bottom plots shows the time-varying gains used in both cases.... 74 Figure 3.7 The Integral Absolute Error (IAE) for poles of 0.3, 0.4 and 0.5, indicating the Minimum Absolute Value (MAV) between the system input and the critical input, u critical = 0.125 . ................................................................ 76 Figure 3.8 The closed-loop response for Example (3.6) at reference input 0.3 with unity weighting matrix Q and unity scalar value R. ............................................ 79 Figure 3.9 The evaluated gains for Example (3.6). ........................................................ 79

i

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As discussed in Chapter 1, State-Dependent-Parameter, Proportional-Integral-Plus (SDP-PIP) control can be considered as a natural extension of linear PIP control methods. However, the nonlinear SDP-PIP approach allows the Transfer Function (TF) model parameters to vary as functions of their states, i.e. they are perhaps better termed SDP-TF models: see examples below. Such models provide a description of a wide class of nonlinear dynamic systems that even includes chaotic processes (Young, 1978). The main advantage of describing nonlinear dynamical systems by SDP models is the subsequent applicability of many aspects of linear systems control theory to the nonlinear system. In fact, this thesis shows that conventional SVF design procedures (Appendix A) can be exploited in the design of nonlinear controllers, in which the SDPPIP controller provides closed-loop feedback gains that are themselves state dependent.

The first stage of SDP-PIP control system design is to identify and estimate appropriate discrete-time SDP-TF models. Here, both recursive Kalman filtering and Fixed Interval Smoothing (FIS) algorithms are utilised, coupled with Maximum Likelihood (ML) estimation of the hyperparameters. Since the focus of the present thesis is nonlinear

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control, only a brief overview of the system identification stage will be discussed in the following section, before turning to control system design in subsequent sections.

3.1 Dynamic Modelling and State Dependent Parameter Estimation The idea of Time Variable Parameter (TVP) and State Dependent Parameter (SDP) modelling for nonstationary and nonlinear signal processing has been developed by Young (1978, 1983, 1993); Priestley (1988); and Young and Chotai (1996). These early publications do not take advantage of recursive FIS which was subsequently exploited by Young (1993, 2001).

When changes in the parameters are relatively slow in comparison to the rates of chance of the stochastic state variables of the system, the nonlinear system can often be approximated well by simple TVP models, for which the parameters can be estimated recursively (Appendix B). On the other hand, if the changes in the parameters are functions of the states or input variables, then the system is truly nonlinear and likely to exhibit sever nonlinear behaviour which cannot be approximated in a simple TVP manner. Therefore, recourse must be made to the alternative and more powerful SDP modelling approach, which again exploits recursive KF/FIS estimation but this time within an iterative ‘backfitting’ algorithm that involves special re-ordering of the time series data (Young, 2001). In this regard, the present research utilises two statistical stages for SDP estimation: 1- Identification of the state dependency using recursive methods and ‘backfitting’. 2- Parameterization of the identified non-parametric relationships, followed by statistically efficient estimation of the normally constant parameters that characterize these nonlinearities.

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In order to illustrate the approach, consider the following nth order Auto-Regressive eXogenous variables (ARX) model with state dependent parameters, sometimes called the SDARX model (e.g. Young, 2000), y k = − a1 ( χ k ) y k −1 − L − a n ( χ k ) y k −n + b1 ( χ k )u k −1 + b2 ( χ k )u k −2 + L + bm ( χ k )u k −m + ek

(3.1)

for which m is the highest lag for the input regression variables (or the order of the numerator polynomial in TF model terms: see later). This equation can be rewritten in the following form,

y k = w Tk pk + ek

(3.2)

where ek is a zero mean, white noise input with Gaussian normal amplitude distribution and variance σ 2 , and w Tk = [− y k −1 L − y k −n u k −1 L u k −m ] p k = [a1 ( χ k ) L a n ( χ k ) b1 ( χ k ) L bm ( χ k )]

T

(3.3)

Here, ai ( χ k ) ∀ i = 1, 2,L, n and b j ( χ k ) ∀ j = 1, 2, K , m are the state dependent parameters, which are assumed to be functions of the vector w k or any other vector S k of other variables that may affect the relationship between the two primary variables y k and u k . In general, therefore, ai ( χ k ) and b j ( χ k ) are functions of the following nonminimal state vector,

[

χ k = w Tk S kT

[

]

T

where S k = S1, k S 2, k L S r , k

(3.4)

]T .

Since the parameter vector p k is potentially state dependent, it may vary at a rate proportional to the temporal variation in the regression variables. For this reason, it

cannot immediately be assumed to take the form of a simple Generalised Random Walk

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(GRW), as is usually the case for TVP models: see equations (B.4) and (B.5) in Appendix B. In order to obviate these difficulties, it is necessary to perform the TVP estimation in a different manner, in which the data are ‘re-ordered’ prior to estimation and the recursive FIS algorithm is applied using a ‘back-fitting’ procedure.

The data re-ordering is a simple but very effective device for transforming the rapid TVP estimation into a much simpler and solvable, slow TVP estimation problem. It works on the basis that if, at any sample time k in an off-line (non-real-time) situation,

all the variables in an equation such as (3.2) are available for the purposes of estimation, then it is not necessary to consider each equation in the normal temporal order, k = 1,2,...N (Pedregal et al., 2003).

For instance, each equation and the variables appearing in this equation, can be reordered in some manner and the model parameters in the equation can then be recursively updated in this new, transformed data space. Furthermore, if the re-ordering is chosen such that, in this transformed data space, the variables and associated parameters are changing quite slowly, then recursive FIS estimation, based on the GRW class of models for the parameter variations, will provide sensible estimates of the parameter variations in the transformed data space. Transformation of these estimated SDPs back into the original data space then reveals their true rapid variation in natural temporal terms.

In order to clarify the backfitting procedure used in estimating the SDPs in equation (3.2), consider the following simple first order SDARX model, y k = − a1 ( x1, k ) y k −1 + bo ( x2, k ) u k + ek

(3.5)

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for which the time delay has been removed for simplicity (Pedregal et al., 2003). Here,

[

]

the vector of model parameters is pk = a1 ( x1, k ) bo ( x2, k ) T , where x1,k is an arbitrary state variable. The backfitting algorithm for the SDP model described in equation (3.5) takes the following form: 1- Assume that, without any sorting, FIS estimation has yielded prior TVP estimates

[

pˆ 1k | N = aˆ11, k | N bˆo1, k | N

]

T

for which N defines the number of samples in the data

series. Then the SDP estimation equation for a1, k can then be formulated as,

[y

k

− bˆo, k | N u k

]

sorted

= − a1sorted y ksorted ,k −1

(3.6)

for which all the terms in both sides have been sorted in terms of ascending order of

y k −1 . Subsequently, the application of the standard TVP algorithm to this single SDP sub-model (Appendix B) yields the FIS estimate of aˆ1sorted ,k | N . 2- Then, the aˆ1sorted , k | N is unsorted so that an SDP estimation equation for bo , k can be formulated as, sorted [yk − aˆ1, k | N yk −1 ]sorted = bosorted ,k uk

(3.7)

Here all the variables on both sides of the equation are sorted in ascending order of

u k . Once again, the application of the standard TVP algorithm to this single SDP sub-model reveals the FIS estimate of bˆosorted , k | N and hence the first iteration of the backfitting algorithm is complete. 3- This process is continued in an iterative manner. Each time unsorting, forming the modified SDP estimation equation, and sorting according to the current right hand side variable prior to TVP estimation using FIS algorithm, until the FIS estimates of

[

the SDPs pˆ 1k | N = aˆ11, k | N bˆo1, k | N

]

T

(for which each element is time series of length N)

do not change significantly between iterations.

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4- Note that ML optimization is carried out prior of each iteration for smoothing the hyperparameters required for FIS estimation.

Example (3.1) Consider the following 1st order Hammerstein system with unity samples time delay,

y k = 0.7 y k −1 + u k −1u k −1 + ek

(3.8)

where e k is a white noise component, introduced here to represent measurement noise in the simulation experiments. This model is based on a straightforward difference equation but includes a nonlinear term in the input, i.e. u k2 −1 . By exciting (3.8) with a PRBS input and analysing the simulation data, the following SDP model is obtained1,

y k = − a1 ( χ k ) y k −1 + b1 ( χ k ) u k −1

(3.9)

As expected, the estimated a1 ( χ k ) = −0.7238 is found to be time invariant, while the estimated b1 ( χ k ) ≈ u k −1 is changing over time, as shown in Figure (3.1). Note that, in this example, there is a small off-set in the estimate of a1 ( χ k ) because of the measurement noise and particular stochastic realisation. Here, the SDP analysis is based only on the assumption that a1 ( χ k ) is an unknown function of the lagged output variable y k −1 and that b1 ( χ k ) is a similarly unknown function of the lagged input variable u k −1 . Figure (3.1) shows that both functions have been correctly identified for this simulation example. By contrast, if the same PRBS input is used for conventional linear modelling, then the following linear equation is obtained,

y k = −a1 y k −1 + b1 u k

(3.10)

For one particular operating condition (i.e. u k ≈ 0.5 ), the time invariant model parameters are a1 = −0.7 and b1 = 0.697 . Clearly equation (3.10) represents a linearised

1

All the identification results in this thesis are obtained using the CAPTAIN Toolbox for MATLAB®: Taylor et al. (2006). The toolbox can be downloaded from: www.es.lancs.ac.uk/cres/captain

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approximation of the nonlinear system (3.8) for a given operating condition. By contrast, the SDP estimates stated in equation (3.9) approximate the actual nonlinear system.

Figure (3.1) compares the estimated linear and nonlinear model parameters. Here, the variations of the model parameters plotted against the sample number are shown on the left-hand side subplots, while the state dependencies are shown on the right-hand side subplots. Finally, Figure (3.2) illustrates the linear and nonlinear model fit, where it is clear that the nonlinear model yields a significantly improved coefficient of determination RT2 = 0.9955 , in comparison to the linear model where RT2 = 0.8193 . Further details and examples of the SDP identification and estimation approach may be found in Pedregal et al. (2003) or Taylor et al. (2006).

Figure 3.1 Comparison between estimated nonlinear SDP and linear TF model parameters for Example (3.1). Left hand side subplots show the time variation of the estimated SDP parameters a1 (top) and b1 (bottom). Right hand side subplots show the relationship between the parameters and the associated state variable. The right hand side graphs show the estimated SDP parameters (dots); the linear estimates (dashed); and the regression fit to the SDP estimates (solid).

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Figure 3.2 Comparison between nonlinear SDP and linear TF model fit for Example (3.1). The upper subplot shows the actual (simulated) parameters (dots); the linear model response (dashed); and the nonlinear model response (solid).

3.2 Nonlinear Non-Minimum State Space form One of the advantages of the SDP model (3.1) is its similarity to the structural form of a conventional linear TF: see equations (1.1) and (1.2). This becomes apparent when equation (3.1) is rewritten in the following SDP-TF form, yk =

B( χ k , z −1 ) A( χ k , z −1 )

uk

(3.11)

where, A( χ k , z −1 ) = 1 + a1 ( χ k +1 ) z −1 + a 2 ( χ k + 2 ) z −2 + L + a n ( χ k + n ) z − n B( χ k , z −1 ) = b1 ( χ k +1 ) z −1 + b2 ( χ k + 2 ) z −2 + L + bm ( χ k + m ) z −m

(3.12)

Here, the polynomial parameters, ai ( χ k +i ) for i = 1, L n and b j ( χ k + j ) for j = 1, L , m , may vary between samples k to k + 1 but are constant during the sampling period itself.

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It should also be noted that the use of the backward shift operator z − q (for which q represents i or j) is to ensure that the state dependent parameters ai ( χ k +i ) and b j ( χ k + j ) differ from those defined in equation (3.1) by q sampling intervals. In

particular, equation (3.12) represent the q-step ahead values of ai ( χ k ) and b j ( χ k ) .

The definition of the nonlinear SDP model as described in equations (3.11) and (3.12) allows for the exploitation of many aspects of linear systems theory, including the representation of the system using a non-minimal state space (NMSS) form as follows, x k = Fk x k −1 + g k u k −1 + d rk

yk = h xk

(3.13)

Here, the SDP-NMSS representation gives a full description of the nonlinear dynamical behaviour of the system, since it is determined directly from the nonlinear SDP-TF model. The n + m dimensional non-minimal state vector takes the following form, x k = [ y k y k −1 L y k −n+ 2 y k −n+1 u k −1 u k −2 L u k −m+ 2 u k −m+1 z k ]T

(3.14)

where z k is the integral of error between the between the reference command input rk , and the sampled output y k as follows, zk =

1 1 − z −1

(rk − y k )

(3.15)

In a similar manner to the linear NMSS representation (Appendix A), the state transition matrix Fk , the input vector g k , the command input vector d and the output vector h are suitably defined (in this case variable) matrices with appropriate dimensions, as follows,

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65

bm ( χ k ) − a1 ( χ k ) L − a n−1 ( χ k ) − a n ( χ k ) b2 ( χ k ) L bm−1 ( χ k )  O 0 O 0   I M 0 M  O 0 O 0  Fk =  0 L 0 0 0 L 0 0  0 O 0  O  0 M I M  O 0 O 0   a (χ ) L a (χ ) a n ( χ k ) − b2 ( χ k ) L − bm−1 ( χ k ) − bm ( χ k ) n−1 k  1 k

0 0 M  0 0  0 M  0 1

g k = [b1 ( χ k ) 0 L 0 0 1 0 L 0 0 − b1 ( χ k )]T d = [0 0 L 0 0 0 0 L 0 0 1]T h = [1 0 L 0 0 0 0 L 0 0 0]

(3.16)

Example (3.2)

By considering the 1st order Hammerstein system (3.8), it is found that the estimated nonlinear equation has the following form, y k = 0.7238 y k −1 + u k −1u k −1

(3.17)

Therefore the SDP-TF model can be written according to equations (3.11) and (3.12) as follows, yk = =

b1 ( χ k +1 ) z −1 1 + a1 ( χ k +1 ) z −1 u k z −1 1 − 0.7238 z −1

uk

uk

Also, from equation (3.17), the SDP-NMSS model (3.16) is given by,

(3.18)

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NONLINEAR CONTROL SYSTEM DESIGN

 y k   0.7238 0  y k −1   u k −1  0   z  = − 0.7238 1  z  + − u u k −1 + 1 rk   k −1   k −1     k  y  y k = [1 0] k   zk 

66

(3.19)

By contrast, the linear (time invariant) TF can be constructed from equation (3.10) as follows, yk =

0.697 z −1 1 - 0.7 z -1

uk

(3.20)

The associated NMSS model (A.8) is defined by,  y k   0.7 0  y k −1   0.697  0   z  = − 0.7 1  z  + − 0.697 u k −1 + 1 rk   k −1       k  y  y k = [1 0] k   zk 

(3.21)

Controllability

If the NMSS model is to be used as the basis for the design of SVF control systems, it is important to evaluate the conditions for controllability of the system model. These conditions are described in a theorem for linear systems in Appendix A. This theorem is now modified to give the minimum requirements for the controllability of the nonlinear SDP-NMSS model, as follows: Given a Single-Input Single-Output (SISO) discrete-time system described by the SDP-TF model, equations (3.11) and (3.12), the NMSS representation, equations (3.13) and (3.16), as defined by the pair [ Fk , g k ] , should satisfy the following two conditions at each sample: 1. The polynomials A( χ k , z −1 ) and B( χ k , z −1 ) are coprime, and m

2.

∑ b j (χ k + j ) ≠ 0 . j =1

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67

The variation of the parameters A( χ k , z −1 ) and B( χ k , z −1 ) in the nonlinear SDP-TF model ensures that, in some circumstances, these conditions may be satisfied at one moment in time but not at others. In this case, difficulties may arise during the design of SDP-PIP control systems. The next example shows how the variations of SDP-TF parameters could cause a particular model to become quasi-uncontrollable.

Example (3.3)

Consider the following nonlinear system: y k = 0.9 y k −1 − 0.08 y k − 2 + 0.5 u k −1 − 0.4 u k2 − 2

(3.22)

where y k is the output variable and u k is the input variable. This nonlinear model can be rewritten in the following discrete TF form, yk =

0.5 z −1 − (0.4 u k ) z −2 1 − 0.9 z −1 + 0.08 z − 2

(3.23)

uk

It can be easily seen that for u k = 1.25 , the numerator becomes zero and the input has no effect on the system, as illustrated in Figure (3.3). In this case, the system is quasiuncontrollable due to the invalidation of the second condition of controllability. Furthermore, at u k = 0.125 and u k = 1 , the system shows again quasi-uncontrollability because of pole-zero cancellations. In these cases, the numerator and denominator polynomials are not coprime since they share the same roots of 0.1 and 0.8 respectively.

Note that the first controllability condition is equivalent to the requirement that the controllability matrix S1 associated with the ‘regulator’ NMSS model (i.e. no integral of error),

[

S1 = g F g F 2 g L F n + m − 2 g

]

(3.24)

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68

has full rank n + m − 1 , i.e. that it is non-singular (Young et al., 1987). The SDP-NMSS representation, without the integrated control error state z k , of the nonlinear system described in equation (3.13) is,

Figure 3.3 Open-loop response of Example (3.3) showing zero response, at step input 1.25. ~ x~k = Fk ~ x k −1 + g~k u k −1 ~ y =h~ x k

(3.25)

k

for which the reference input rk is omitted and the state vector has the following T ~ = [y y regulator form, x k k k −1 u k −1 ] . For the system (3.23), the nonlinear ‘regulator’ ~ ~ transition matrix Fk , input vector g~k and output vector h are defined as,

− a1 ( χ k ) − a 2 ( χ k ) b2 ( χ k ) ~ ~ Fk =  1 0 0  , g~k = [b1 ( χ k ) 0 1]T , h = [1 0 0]   0 0 0  

Then the controllability matrix (3.24) will be,

S1(u =0.125)

0.5 0.4 0.32 =  0 0 .5 0 .4    0  1 0

S1(u =1)

0.5 0.05 0.005 =  0 0.5 0.05    0 0  1

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69

It is obvious that both matrices are singular which supports the idea that the system is uncontrollable at these specific values of the input signal, i.e. 0.125, 1 and 1.25. In other words, the system is quasi-uncontrollable. In this thesis, the term ‘quasi-uncontrollable’ is used to mean that the system is uncontrollable for certain state values but that if these particular cases can be avoided, the system is otherwise controllable.

3.3 Nonlinear State Dependent Parameter Control Assuming that the SDP-NMSS system, [Fk , g k , d , h] , is globally defined and fully controllable2 for all k, within a certain range of input u k and output y k , then there exists a valid vector of feedback gains that forms the SVF control law, given by, u k = −k k x k

(3.26)

where k k is the n + m dimensional SVF control gain vector,

[

k k = f o, k f1, k L f n −1, k g1, k g 2, k L g m −1, k − k I , k

]T

(3.27)

in which the feedback control gains are themselves functions of the states: for brevity, the subscript k is utilised above, to indicate that the control gains change over time. In practice, these changes are defined by the SDP model that has previously been estimated off-line – in other words, the controller is a scheduled rather than adaptive design, as discussed further below.

Here, at each sampling interval, the SVF control problem is solved such that either the closed-loop poles “eigenvalues” are at pre-assigned positions in the complex z-plane; or alternatively, that the system is optimised in some manner, such as in the sense of a

2

As shown in Example (3.3), the variation of the parameters possibly causes the SDP-NMSS to become uncontrollable. However, if the variation of these parameters is limited to within a certain range, perhaps because of the physical characteristic of the system itself, then the SDP-NMSS will be controllable for all k within this range. This is the case for all the practical examples considered in this thesis.

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70

Linear Quadratic (LQ) cost function. The former approach is utilised by Stables (2006), whilst the present thesis focuses on the latter, which has proven to work very well in the context of the construction robot examples. Both approaches are briefly reviewed below.

In either case, the SDP-PIP control system is illustrated by Figure (3.4), which can be compared with the equivalent linear PIP controller shown in Figure (A.1), Appendix A.

rk

ek

uk

k I ,k

B( χ k , z −1 )

yk

A( χ k , z −1 )

1 − z −1 G1, k (z −1 ) Fk (z −1 )

Figure 3.4 The SDP-PIP control system.

Note that Figure (3.4) takes a similar structural form to conventional ProportionalIntegral (PI) control, albeit with time varying gains and with the following higher order feedback compensators, G1,k (z −1 ) and Fk (z −1 ) , G1,k (z −1 ) = g1,k z −1 + L + g m−1,k z − m+1 Fk (z −1 ) = f o,k + f1,k z −1 + L + f n−1,k z − n+1

(3.28)

3.3.1 Nonlinear pole assignment One approach to obtain the nonlinear SDP gain vector k k , is to determine the closedloop control system in transfer function form and to use this for pole assignment. The closed-loop TF is obtained directly by reducing the block diagram shown in Figure (3.4) as follows,

CHAPTER 3

yk =

NONLINEAR CONTROL SYSTEM DESIGN k I ,k B( χ k , z −1 )

[

]

∆ Gk (z -1 ) A( χ k , z -1 ) + Fk (z -1 ) B( χ k , z -1 ) + k I ,k B( χ k , z -1 )

71

(3.29)

rk

for which ∆ = 1 − z -1 is the difference operator and Gk (z −1 ) = 1 + G1, k (z −1 ) . Straightforward polynomial algebra manipulation of the characteristic equation,

[

]

∆ Gk (z -1 ) A( χ k , z -1 ) + Fk (z -1 ) B( χ k , z -1 ) + k I ,k B ( χ k , z -1 ) = (1 − p1z −1 ) L (1 − p m+ n z −1 ) (3.30)

is utilized to find the SVF gains that yield pre-determined positions of the poles pi , ∀ i = 1, 2, L , n + m , in the complex z-plane.

Example (3.4)

Consider the same Hammerstein system introduced in Example (3.1). The NMSS model is described by equation (3.19), while the control algorithm (3.26) will be,

[

u k = − f o,k − k I ,k

Given that

]

xk

A( χ k , z −1 ) = 1 + a1 z −1 ,

(3.31) B( χ k , z −1 ) = b1 ( χ k +1 ) z −1 , Gk (z −1 ) = 1 and

Fk (z −1 ) = f o, k , and substituting in equation (3.30) then

[

]

[

]

1 + a1 + f o, k b1 ( χ k +1 ) − 1 + k I , k b1 ( χ k +1 ) z −1 − a1 + f o, k b1 ( χ k +1 ) z −2 = 1 − ( p1 + p2 ) z −1 + p1 p2 z − 2 By comparing the left hand and right hand sides, and solving the two simultaneous equations in f o, k and k I , k , then f o,k =

− a1 − p1 p 2 b1 ( χ k +1 )

1 − a1 + ( p1 p 2 + a1 ) − ( p1 + p 2 ) k I ,k = b1 ( χ k +1 )

Given that b1 ( χ k +1 ) = u k , this leads to the following control equation,

(3.32)

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NONLINEAR CONTROL SYSTEM DESIGN

u k2 = (a1 + p1 p 2 ) y k + [1 − a1 + ( p1 p 2 + a1 ) − ( p1 + p 2 )]z k

72

(3.33)

= −0.45 y k + 0.25 z k given that a1 = −0.7 and p1 = p 2 = 0.5 .

Figure (3.5) shows the closed-loop response of Example (3.2) at 0.5 poles on the complex z-plane.

Figure 3.5 Unity step closed-loop response for the Hammerstein system in Example (3.4), with equal poles assigned to 0.5 on the complex z-plane.

This example illustrates a singularity in the value of the integral gain k I ,k when u k = 0 . This directly relates to the 2nd controllability condition stated in Section 3.2, i.e. the system is uncontrollable for b1 ( χ k +1 ) = 0 where, in the present example, b1 ( χ k +1 ) = u k . However, it is straightforward to avoid this problem by the algebraic manipulation shown in equation (3.33).

The next example illustrates another type of singularity, this time due to the variation of the SDP-TF parameters. In this case, the problem relates to the 1st controllability

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73

condition, i.e., the nonlinear dynamic system is potentially uncontrollable (or ‘quasiuncontrollable’) because the numerator and denominator polynomials are not necessarily coprime at each sampling instant.

Example (3.5)

Consider the same nonlinear dynamical system described in Example (3.3). It is shown that the non-minimal state space vector is defined as x k = [ y k y k −1 u k −1 z k ]T while the SDP-NMSS of the system is straightforward to generate using equations (3.13) and (3.16) as follows,  y k   0.9 − 0.08 − 0.4 u k − 2 y   1 0 0  k −1  =  u k −1   0 0 0     z k  − 0.9 0.08 0.4 u k −2  yk  y  k −1  y k = [1 0 0 0] u k −1     zk 

0  y k −1   0.5  0       0  0  y k − 2  0  + u k −1 +   rk 0  0  u k − 2   1        1  z k −1  − 0.5 1 

(3.34)

The characteristic equation describing the closed-loop TF is obtained from equation (3.29) using the following polynomials, A( χ k , z −1 ) = 1 + (−0.9)z −1 + (0.08)z −2 B( χ k , z −1 ) = (0.5)z −1 + (−0.4 u k )z −2 F ( χ k , z −1 ) = f o,k + f1,k z −1

(3.35)

G ( χ k , z −1 ) = 1 + g1,k z −1

Since the polynomials A( χ k , z −1 ) , B( χ k , z −1 ) , F ( χ k , z −1 ) and G ( χ k , z −1 ) are of a higher order than for Example (3.4), it is difficult to obtain the closed-form of the gains using direct algebra. In particular, as shown in equation (3.35), the polynomial B( χ k , z −1 ) is a function of the input signal required at the current sample, i.e. u k .

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However, a straightforward iterative procedure is performed to overcome this difficulty: see Appendix C for details.

Note that the selected poles could cause the input to reach one of the critical values considered in Example (3.3) above, i.e. 0.125, 1 or 1.25. Therefore, a trial and error procedure for selecting the appropriate poles is utilized to obtain a suitable closed-loop response: see Figure (3.6).

(a) poles = 0.5

(b) poles = 0.3

Figure 3.6 The closed-loop response for Example (3.5) at two different poles; poles at 0.5 (left plot) and 0.3 (right plot) on the complex z-plane both at reference input 0.3. The bottom plots shows the time-varying gains used in both cases.

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Figure (3.6a) shows an oscillatory closed-loop response for poles at 0.5 on the z-plane. This happens because the control input is close to one of the critical values, i.e. ucritical = 0.125 , which causes the SDP-TF to become uncontrollable. However, if faster poles of 0.3 are selected, the nonlinear system yields a good tracking response because the control input is away from this critical value, as shown in Figure (3.6b). Also as shown in Figure (3.7), the Integral of Absolute Error (IAE) never reaches a constant value when the poles are placed at 0.5 on complex z-plane.

For this example, it is found that the minimum absolute value (MAV) between the control input u k and the critical input, ucritical = 0.125 , is 2.5665 × 10 −4 and 0.0014 for poles of 0.5 and 0.3, respectively. This supports the idea that at poles of 0.5 the control input is closer to the critical input than for the case with poles at 0.3.

Finally, the response of the closed-loop system shown in Figure (3.6) is compared with the ‘designed-for’ linear response using the same pole positions (dashed trace). Such a response is obtained by simulating a transfer function with these poles in open-loop, together with a scalar numerator chosen to ensure a unity gain. When the poles are placed at 0.5 on the complex z-plane, then this theoretical response diverges from the simulated SDP-PIP response shown in Figure (3.6); by contrast, with poles at 0.3 on the complex z-plane, then the theoretical and simulated responses are almost identical as would be expected from the discussion above.

Note that such controllability problems are all overcome using the partial and exact linearization approaches developed later in the thesis (Chapters 5 and 6). Indeed, the present examples provide motivation for the development of these new approaches.

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2.5

76

poles = 0.3 poles = 0.4 poles = 0.5

Integral of Absolute Error [IAE]

2

MAV=2.5665x10

-4

MAV=7.2329x10

-4

1.5

1

MAV=0.0014 0.5

0

0

10

20

30

40

50 60 Samples

70

80

90

100

Figure 3.7 The Integral Absolute Error (IAE) for poles of 0.3, 0.4 and 0.5, indicating the Minimum Absolute Value (MAV) between the system input and the critical input, ucritical = 0.125 .

3.3.2 Linear Quadratic (LQ) optimization As an alternative to pole assignment, the SVF gain vector may determined by the optimization of the following LQ cost function, ∞

J=

∑ {xiT Q xi + R ui2 }

(3.36)

i =0

Here, the optimum SVF gain vector is obtained such that it satisfies predetermined conditions or weighting criteria. The problem of optimization for a linear SISO discrete NMSS form is briefly reviewed in Appendix A, whilst the present chapter considers the modifications required in the nonlinear case.

The procedure to solve this minimization problem is by means of the Algebraic Riccati Equation (ARE), derived from the standard LQ cost function in equation (3.36). In this case, the control gain vector k k is obtained recursively at every sample k based on the SDP-NMSS system matrices defined at that sampling instant as follows,

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[

k k = g kT P (i +1) g k + R

]

−1 T gk

P (i +1) Fk

77

(3.37)

P (i ) = FkT P (i +1) [Fk − g k k k ] + Q

where Q is a square positive definite matrix with dimension n + m , which for simplicity is often defined as a diagonal matrix (A.17) and R is an additional scalar weight on input. Also P is symmetrical-positive definite matrix and its initial value, P (i +1) , equals the weighting matrix Q; finally, k k is the control gain vector (3.27).

Earlier research has either used a ‘frozen-parameter’ system defined as a sample member of the family of NMSS models or has solved the discrete-time algebraic Riccatti equation at each sampling instant: see e.g. McCabe et al. (2000).

In the former case, the frozen parameter system for state transition matrix, Fk′ , and input vector, g k′ , can be used for which the frozen system is defined as a single sample member of the family of Fk and g k (Young et al., 2002; Kontoroupis et al., 2003a)3. In this case, the matrix P will be a time invariant symmetrical positive-definite matrix solution for the ARE described in equation (3.37). By merging the two equations in (3.37), the discrete-time solution of the P matrix will be P (i ) = Fk′T P (i +1)  Fk′ − g ′k g ′kT P (i +1) g k′ + R 

(

)

−1

g k′T P (i +1) Fk′  + Q 

(3.38)

Once the value of the positive-definite matrix P is obtained off-line from equation (3.38), the value of state dependent gain vector, k k , can be obtained at each sample using the first equation in equation (3.37), while keeping the system matrix Fk and input vector g k unfrozen, as follows:

[

k k = g kT P g k + R 3

]

−1 T gk

P Fk

(3.39)

In all the examples existed in this thesis, the frozen parameter system has been chosen at u k −i = 0 and y k −i = 0 for simplicity unless stated otherwise. This is because, for the examples considered, no significant improvement has been achieved by using other values.

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Example (3.6)

Consider the same nonlinear system as given in Example (3.3), for which the SDP-TF is described in equation (3.23) and its NMSS representation is equation (3.34). By setting u k − 2 = 0 , then the fixed state transition matrix of the NMSS form will be  0.9 − 0.08  1 0 Fk′ =  0  0 − 0.9 0.08

0 0 0 0

0 0  0 1

(3.40)

However, since the input vector g k , the command input vector d and the output vector h are time-invariant, they have the same values as in equation (3.34). The second step is the selection of the weighting matrix Q and weighting scalar R to verify an acceptable response. Here, Q is set to the unity matrix and R = 1 . The solution to the ARE is obtained and an evaluation of the P matrix is found as follows  5.4943 − 0.3243 − 0.3243 1.0302 P=  0 0   − 1.5917 0.1449

0 − 1.5917 0 0.1449  1 0   0 2.8244 

(3.41)

By substituting into equation (3.39), this yields,

k k = (4.8755) −1 [4.9413 − 0.4601 − 2.3004 u k −2 − 2.2081]

(3.42)

As illustrated in equation (3.38), the frozen parameter approach for determining the P matrix is straightforward and does not need a priori knowledge about the nonlinear TF parameters. Also it helps in linearizing the gains that correspond to the constant parameters, if any, in the SDP-TF.

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Figure (3.8) shows the closed-loop response of the nonlinear dynamic system by using an updated P matrix at every sample, compared with the fixed P matrix approach discussed above. There is no significant difference between the two techniques, whilst clearly the fixed P matrix is simpler to implement in practice. Moreover, the frozen P matrix approach helps to linearizing the gains f o , f1 and k I as shown in Figure (3.9).

Figure 3.8 The closed-loop response for Example (3.6) at reference input 0.3 with unity weighting matrix Q and unity scalar value R.

Figure 3.9 The evaluated gains for Example (3.6).

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3.4 Summary The present chapter has shown how a nonlinear dynamical system can be represented and estimated as a state dependent parameter model, using either a Transfer Function (SDP-TF) or Non-Minimal State Space (SDP-NMSS) form. Two approaches to find the associated State Variable Feedback (SVF) control law have been introduced, namely closed-loop pole assignment and optimization of a Linear Quadratic (LQ) cost function. Both approaches yield nonlinear Proportional-Integral-Plus (SDP-PIP) control algorithms that are relatively straightforward to implement in practice. Illustrative examples have been included throughout the discussion to clarify each step of the proposed SDP-PIP control system design process.

Of particular importance to this thesis, the term ‘quasi-uncontrollability’ has been introduced, i.e. the nonlinear SDP model can lose controllability because of changes in the parameters over time. Novel solutions to this problem are considered in the following two Chapters 4 and 5.