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Multivariable Continuous-time Generalised Predictive Control: A State-space Approach to Linear and Nonlinear Systems CSC report: CSC{98001

Peter J. Gawthrop1, Huseyin Demircioglu2 & Irma I Siller-Alcala1 1. Centre for Systems and Control & Department of Mechanical Engineering, University of Glasgow, Glasgow G12 8QQ, Scotland . Email: P.Gawthrop@eng.gla.ac.uk

WWW: http://www.eng.gla.ac.uk/ peterg/

2. Hacettepe Universitesi, Elektrik ve Elektronik Muh. Bol. 06532 Beytepe, ANKARA, Turkey Email: demirci@eti.cc.hun.edu.tr 2 February 1998

trol, nonlinear control.

Keywords Predictive control, multivariable control, minimum-variance con-

1

The Multivariable Continuous-time Generalised Predictive Controller (CGPC) is recast in a state-space form and shown to include Generalised Minimum Variance (GMV) and an new algorithm, Predictive GMV (PGMV) as special cases. Comparisons are drawn with the exact linearisation methods of nonlinear control and it is noted that, unlike the transfer function approach, the state-space approach extends readily to the nonlinear case. The resulting state space design algorithms are conceptually and algorithmicly simpler than the corresponding transfer function based versions and have been realised as a freely available Matlab tool-box.

Abstract

2

MVCGPC: A State-space Approach Symbol SISO MIMO GMV IMC PGMV GPC CGPC

2nd February 1998

Meaning Single input { single output Multi input { multi output Generalised minimum variance (control) Internal model control Predictive generalised minimum variance (control) Generalised predictive control Continuous-time Generalised predictive control Table 1: Acronyms

1 Introduction

Emulator based control (EBC) provides a framework for control methods arising from Astrom's minimum-variance control; such methods include that of Clarke and Hastings-James (1971) the generalised minimum-variance controller of Clarke and Gawthrop(1975, 1979) and the Generalised Predictive control of Clarke, Mohtadi and Tu s (1987a, 1987b, 1989). EBC has been shown by Gawthrop, Jones, W., and Sbarbaro (1996) to have close relations with the Internal Model Control (IMC) of Morari and Za riou (1989); it follows that the results herein are also relevant to IMC. Generalised Predictive Control (GPC) is one of a wider set of methods called Model-based Predictive Control (Clarke 1994a; Morari 1994), hence results of this paper give a particular form of Model-based Predictive Control. Such methods have achieved success in a number of applications (see, for example the survey of Clarke (1994b) and the papers collected in Chapter 5 of Clarke (1994a)), and therefore we believe that CGPC will also nd industrial application. One dichotomy in the development of GPC is into the discrete-time approach of, for example Clarke, Mohtadi and Tu s (1987a, 1987b, 1989) and the continuous-time approach of Demircioglu (1989), Gawthrop and Demircioglu (1989, 1991), Demircioglu and Gawthrop (1991, 1992) and Demircioglu and Clarke (1992). We use a continuous-time setting here for the reasons given by Gawthrop (1986b, 1986a, 1987). In particular, we believe that a continuous-time approach exposes the fundamentals of the underlying control problem which are obscured by sampling: this is particularly the case for partially-known and nonlinear systems. Another dichotomy in GPC is into the transfer function approach of, for example Clarke, Mohtadi and Tu s (1987a, 1987b, 1989) and of Demircioglu

1

The emulator approach to control system design is that \where physically unrealisable operations such as prediction or taking derivatives can be emulated by making use of a parametric system model" (Gawthrop 1987)

1

1. Introduction

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MVCGPC: A State-space Approach

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Meaning System output, input and state v Input disturbance Auxiliary output w Setpoint YNy Output stacked derivatives UNu Input stacked derivatives Auxiliary output stacked derivatives N WNy Setpoint stacked derivatives A,B ,C System matrices c(s) Observer pole polynomial ONy Extended observability matrix HNy ;Nu Markov parameter matrix System relative degree ny ,nu,nx Output, input and state dimension Ny ,Nu Order of highest derivative of y and u N Order of highest derivative of Vector of time horizons diagonal matrix of T( ) Taylor expansion coe cient matrix T ( ; ) Integrated time matrix i ith derivative wrt time ^ Emulated variables Variables in receding time frame 0 Zero matrix of appropriate dimension 1 Unit matrix of appropriate dimension Table 2: Notation

1 2 ]

Symbol y,u,x

Equation 1, 2 1,17 18 26 6 7 36 44 1, 2 15 8 9 10 1, 2 6 36 35 38 38 55 6 16 42 { {

1. Introduction

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MVCGPC: A State-space Approach

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and Gawthrop (1991, 1992) on the one hand; and the state space approach of, for example, Gawthrop and Demircioglu (1991), Bitmead, Gevers, and Wertz (1990) Lee, Morari, and Garcia (1994) and Ordys and Clarke (1993) on the other. Ordys and Clarke (1993), suggest that, in the context of linear systems there is no signi cant di erence between the two approaches; however, Morari (1994) points out that in the multivariable case, the state-space approach has both conceptual and numerical advantages. As discussed by Ordys and Clarke (1993) in the context of nonlinear systems the state-space approach is essential. For both of these reasons, we choose the state-space approach in this paper. Following Gawthrop and Demircioglu (1991), the state-space approach gives a new look at the emulator, which is usually developed from the transfer function point of view. In their book, Bitmead, Gevers, and Wertz (1990) give a critical assessment of (discrete-time) GPC. To a large extent, these criticisms have been answered by the introduction of stable GPC methods by Clarke and Scattolini (1991), Demircioglu and Clarke (1992) and by Kouvaritakis, Rossiter, and Chang (1992), Rossiter and Kouvaritakis (1993) and Kouvaritakis and Rossiter (1993). Although we believe that their arguments can be answered in more general terms, su ce it to say here that their arguments are not directly relevant to nonlinear systems; the eventual goal of the research reported here. However, we do adopt the state-space observer/state feedback approach advocated by Bitmead, Gevers, and Wertz (1990) as also considered earlier by Gawthrop and Clarke (1980) and by Gawthrop and Demircioglu (1989). The continuous-time GPC of Demircioglu and Gawthrop (1991) provided one possible generalisation of the continuous-time GMV control in the same spirit as the generalisation of Clarke, Mohtadi and Tu s (1987a, 1987b, 1989) in the discrete-time context. As an intermediate step, a new algorithm, the Predictive Generalised Minimum Variance controller, is derived. Like GPC, but unlike GMV, this is a moving horizon (Mayne and Michalska 1990) controller. However, one particular feature of GMV (the so called P polynomial) was not used by Demircioglu and Gawthrop (1991) (though it did appear in a thesis Demircioglu (1989)). This paper provides this extension as a step towards the nonlinear GPC; but we believe it also has independent interest. The reasons why the P polynomial is important include: it provides an algorithm with a direct link to the exact linearisation methods of Isidori (1995), it provides a useful approach to disturbance response manipulation 1. Introduction Page 5.

MVCGPC: A State-space Approach

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as discussed by Demircioglu (1989), it is a more useful way of de ning controller properties than using the R polynomial alone. In this paper, we have chosen not to use the usual control weighting (called Q(s) by Clarke and Gawthrop (1975) and Clarke, Mohtadi, and Tu s (1987a)) as we believe that it is not a very useful form of control weighting. However, it is straightforward to include such a term if required. Stability issues as such are not treated in this paper. However, it is noted that substantial progress has been made on GPC stability by an number of authors including Clarke and Scattolini (1991), Demircioglu and Clarke (1992) and by Kouvaritakis, Rossiter, and Chang (1992), Rossiter and Kouvaritakis (1993) and Kouvaritakis and Rossiter (1993). Nonlinear systems are treated in Section 6, they provide an important motivation for this paper which paves the way for future developments in this area. This is essentially because of the close relationship between the exact linearisation approach of nonlinear control (Isidori 1995; Marino and Tomei 1995) and the output derivative approach used here. In particular, Section 7 give a geometric interpretation of GMV along the lines of the books of Isidori (1995) and Marino and Tomei (1995). The outline of the paper is as follows. Section 2 introduces the emulator in a state-space setting. This provides the foundation for the Generalised Minimum Variance control (GMV) of Section 3, the Predictive Generalised Minimum Variance control (PGMV) of Section 4 and the Generalised Predictive control (GPC) of Section 5; Section 5.1 considers the closed-loop system arising from GPC. Section 6 introduces GPC in a non-linear context and Section 7 gives a geometric interpretation of GMV. Section 8 gives an illustrative example. Section 9 concludes the paper. The acronyms and notation used in this paper appear in Tables and 2 respectively. A Matlab tool-box has been created to accompany this paper. This tool-box, together with a number of examples, is available on WWW at URL http://www.mech.gla.ac.uk/ peterg/software/matlab/emulator/. See the README le for details. One the one hand, this removes the need for an extensive examples section; on the other hand, the reader is free to experiment with the e ect of various parameters on the performance of the controllers in this paper.

2. Introduction

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MVCGPC: A State-space Approach

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2 Emulators in state-space form

Emulators Gawthrop(1987, 1986b, 1986a, 1996) provide a useful conceptual foundation for a number of control algorithms. This section gives a selfcontained development in a state-space setting. As discussed in Section 7, the procedure used is closely related to the geometric methods based on Lie algebra (Isidori 1995; Marino and Tomei 1995). This section considers linear proper dynamic systems with the state-space representation: x = Ax + B (u + v) _ (1) y = Cx (2) where the output y, input u, input disturbance v and state x dimensions are ny , nu, nv = nu and nx respectively. This particular form of disturbance is used for simplicity and is considered further in Section 2.1; but, for simplicity, will be omitted until that point. As discussed elsewhere Gawthrop(1987, 1986b, 1986a), emulators are dynamic systems giving realisable approximations to unrealisable dynamic systems. One such unrealisable dynamic system is the multiple derivative operator given, in Laplace Transform terms, by sN . Such emulators form the basis of (continuous-time) generalised minimum-variance control (GMV) Gawthrop(1987, 1986b, 1986a) and generalised predictive control (GPC) Gawthrop and Demircioglu(1991, 1992) and the corresponding self-tuning controllers.

Example 2.1

This example system, of the form of Equation 1, is used throughout this paper to illustrate the development of the theory. ? 0 1 A = ?1 1 ; B = ? ; C = 1 0 (3) 1 The corresponding transfer function is: b(s) = ? s + 1 + (4) a(s) s ? s + 1 This system is unstable and, if > 0, has unstable inverse.

2

Repeated di erentiation up to Ny times of the output y with respect to time, together with repeated substitution of the system of Equations 1 gives: (5) YNy (t) = ONy x(t) + HNy ;Nu UNu (t) 2. Emulators in state-space form Page 7.

MVCGPC: A State-space Approach

2nd February 1998

where (at the moment) Nu = Ny . YNy (t) is a column vector (n (Ny + 1) 1): of output derivatives:

0y 1 By C B C YN (t) = B y C B ::: C @ A

1] 2]

y

(6)

y Ny

]

where i indicates the ith derivative with respect to time. UNu (t) is a column vector (nu(Nu + 1) 1): of input derivatives:

]

u Nu ONy is the (extended) observability matrix (ny (Ny + 1) nx): 0 C 1 B CA C B C (8) ONy = B CA C B ::: C @ A CANy HNy ;Ny is a (Ny + 1) (Ny + 1) matrix containing the Markov parameter matrices hi (ny nu ) as elements: 0h 0 0 ::: 0 1 Bh h 0 ::: 0 C C Bh h B h ::: 0 C (9) HNy ;Ny = B A @ : : : : : : : : : : : : : : :C hNy hNy ? hNy ? : : : h where hi is the ith Markov parameter (Kwakernaak and Sivan 1972; Kailath 1980) of the system of Equation 1; h = 0 and hi = CAi? B when i > 0. 0 is the ny nu zero matrix. The (Ny + 1) (Nu + 1) ( Nu < Ny ) matrix HNy ;Nu which comprises the rst Nu + 1 columns of the matrix displayed in Equation 9. In the SISO (ny = nu = 1) case, the relative degree of the system of Equations 1 is de ned as the minimum value of i for which hi 6= 0 (10)

] 2 0 1 0 2 1 0 1 2 0 0 1

0 u 1 Bu C B C UN (t) = B u C B ::: C @ A

1] 2]

u

(7)

2. Emulators in state-space form

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MVCGPC: A State-space Approach

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The relative degree of a system is an important piece of information in the design of controllers such as GMV; it is a strength of GPC that such information is not of crucial importance. We therefore do not go into the issues involved in multivariable extension of the concept of relative degree (Isidori 1995; Marino and Tomei 1995).

Example 2.2

Choosing Ny = 5, the system of Equation 3 gives: 1 0 0 01 01 0 0 0 0 0 B? B0 1C 0 0 0 0 0C C B?1 1 C B C;H = B 1 ? B 0 0 0C B?1 0 C ; B1 + 1 ?0 0 0 0C (11) O =B C B C B C B 0 ?1C @ A @ 1+ 1 ? 0 0A ?1 1+ 1 ? 0 1 ?1 In this case, h = 0, h = ? and h = 1. It follows that the relative degree = 1 if 6= 0 and = 2 if = 0. This paper is concerned with systems where the state is not measurable and thus Equation 5 (containing x) cannot be implemented. For this reason, a standard state observer (Kwakernaak and Sivan 1972; Kailath 1980) is used given by: _ x = Ax + Bu + Le ^ ^ (12) y = Cx ^ ^ (13) e=y?y ^ (14) where L is the nx ny observer gain matrix is chosen to give observer poles (eigenvalues of A ? LC ) at appropriate places; see, for example, Kwakernaak and Sivan (1972). In particular we choose the observer poles to correspond to the roots of the N th order polynomial c(s); in other words c(s) = detA ? LC (15) It is, of course, possible to impose a stochastic setup on the system and derive an optimal observer: the Kalman lter (Kwakernaak and Sivan 1972). This idea has been used in a discrete-time context by Krauss, Dass, and Rake (1994). ^ The corresponding emulated Y (t), Y (t), is de ned as: ^ (16) ^ YNy (t) = ONy x(t) + HNy ;Ny UNy (t) Equation 16 is the same as Equation 5, except that x replaces x. ^ The emulator concept provides the basis for the three control algorithms considered here:

5 5 5 0 1 2

2. Emulators in state-space form

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MVCGPC: A State-space Approach

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1. Generalised minimum-variance control (GMV) 2. Predictive generalised minimum-variance control (PGMV) 3. Generalised predictive control (GPC) GMV is used as a stepping stone leading to GPC, and its disadvantages motivate the development of GPC.

2.1 Disturbance modelling

It is always important to model disturbances as well as the system itself to achieve satisfactory control. In particular, a practically useful assumption is that each control input is associated with a constant additive disturbance v as in Equation 1. Each such disturbance may be modelled by adding an additional state to the system equations which is not driven by the control inputs but adds on to the control inputs. Thus it is wise in practice to augment the system matrices A, B and C to give Aa , Ba and Ca as follows:

? Aa = 0 A 0 B ; Ba = 0 B ; Ca = C 0 nu n u nu nu nu n x

(17)

As discussed elsewhere (Gawthrop 1986b), we believe that this approach of explicitly including the disturbance in the system model (and hence giving controller integral action as a consequence) is more natural than arbitrarily forcing such integral action onto the controller.

3 Generalised minimum-variance control (GMV)

Generalised minimum-variance control (GMV) was originally derived in discretetime form by Clarke and Gawthrop (1975, 1979) (based on the Self-tuning Regulator of Astrom and Wittenmark (1973)) and more recently in continuoustime form Gawthrop(1986b, 1986a, 1987). In each case, GMV had a transferfunction formulation and was SISO. This section focuses on one particular version of GMV, the model-reference version Gawthrop(1987). The contribution is to supply a multivariable statespace version. and relate it, albeit in the linear case, to the exact linearisation approach presented by , for example, Isidori (1995) and Marino and Tomei (1995). In the state-space formulation, this is equivalent to de ning the unrealisable vector (t) (dimension n ) given by (18) (t) = PYNp (t) 3. Generalised minimum-variance control (GMV) Page 10.

MVCGPC: A State-space Approach and its realisable emulated version ^ ^(t) = P YNp (t) ^ where and the pi are n de ned as:

2nd February 1998

(19) (20)

P = p p : : : pNp

0 1

?

ny matrices. A corresponding polynomial matrix can be p(s) =

N X

p

i=0

pisi

(21)

As far as this paper is concerned, GMV is used with the following assumptions:

Assumptions 3.1 (GMV)

1. The system is square: nu = ny . 2. The dimension of and y are the same: n = ny . 3. Ny = Np 4. the system structure and the model structure are such that:

PH = ( ; 0nu

where and 0nu 5. det 6= 0. 6. The system inverse is stable.

nu

nu

; : : : ; 0 nu

nu

)

(22)

are nu nu matrices

Assumptions 4 and 5 are very restrictive. For example, in the single-input single-output (nu = ny = 1) case it implies that the system relative degree is known. The multivariable case is discussed by, for example, Kailath (1980). Using equation 16, ^ can be rewritten as ^(t) = PO x(t) + PH ; U (t) ^ (23) 3. Generalised minimum-variance control (GMV) Page 11.

MVCGPC: A State-space Approach Using Assumption 4, it follows that ^(t) = PO x(t) + u(t) ^ where =p h by

2nd February 1998

(24) (25)

and, from Assumption 5, det 6= 0. Following Gawthrop (1987), the corresponding GMV control is de ned ^(t) = w(t) (26)

where w is the set-point, or reference, signal of dimension n . This gives a form of model matching control where, using Equations 19 and 26

p y+p y + ^ ^

1] 0 1

^ + pNy y Ny = w(t)

]

(27)

and the error e = y ? y evolves via the observer dynamics of Equation 12. ^ Substituting from Equation 24 gives the implementation

u = kw w ? k x x ^

where

(28) (29) (30)

kw =

and

?

1

kx =

?

1

PO

Equations 28 and 12 together de ne the linear dynamic system with inputs w and y and output u forming the GMV control. This exact model-matching is the fundamental problem associated with GMV; it is well known that exact model-matching implies severe restriction on the process and the Assumptions given above re ect this. It is therefore natural to relax the assumptions by relaxing the exact model matching requirement. This is the motivation for Generalised Predictive Control; but rst, Predictive GMV is considered as an intermediate step. A geometric interpretation of GMV is given in Section 7. 3. Generalised minimum-variance control (GMV) Page 12.

MVCGPC: A State-space Approach

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Example 3.1

Assuming that 6= 0, the system of Equation 3 has relative degree = 1. Choosing Ny = = 1 and P = (1 p) gives:

^(t) = PO x(t) + PH ; U (t) ^ ? ? 0 0 ^ = 1 p 1 0 x(t) + 1 p ? 0 0 1 = x + px ? p u ^ ^

1 1 1 1 1 2

(31) (32) (33) (34)

The corresponding controller is of the form of Equation 28 but with: ? kw = ? 1 and kx = ? p ?

p

1

1

This gives an unstable closed-loop system unless < 0; this is a consequence of zero cancellation. The controller has in nite gain if = 0; this is a consequence of the system relative order changing from 1 to 2 at = 0.

The fundamental problems with GMV are : the need to know the system relative order (and its multivariable equivalent) precisely and the fact that it cancels system zeros. To some extent, this problem can be overcome using control weighting (Clarke and Gawthrop 1975; Clarke and Gawthrop 1979; Gawthrop 1987); however, this requires detailed design in its own right and is not considered further here. Instead, predictive moving horizon control is used to provide a more satisfactory solution to the problem. Section 5 provides the GPC solution, but a new algorithm, predictive GMV (PGMV) provides an intermediate step.

Assumptions 4.1 (PGMV)

4 Predictive generalised minimum-variance control (PGMV)

1. The system is square: nu = ny . 2. The dimension of and y are the same: n = ny .

4. Predictive generalised minimum-variance control (PGMV)

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MVCGPC: A State-space Approach

3. Ny = N + Np

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Notice that Assumptions 3.1 numbers 4, 5 and 6 have been dropped. The approach used here is to combine the twin concepts of recedinghorizon control and predictive control in, for example the GPC of Clarke, Mohtadi and Tu s (1987a, 1987b, 1989) and the corresponding continuous-time version of Demircioglu and Gawthrop (1991, 1992). This section presents a new algorithm based on these ideas which is developed in a state-space context. The next section derives the corresponding GPC. There are four distinct, but related concepts associated with the predictive control considered here: 1. prediction via Taylor series expansion, 2. moving-horizon control, 3. control constraints (within the moving horizon time-frame), 4. and optimisation. In a continuous-time formulation (Demircioglu and Gawthrop 1991; Demircioglu and Gawthrop 1992), prediction is accomplished via a Taylor series expansion of the system output y; here this concept is extended slightly to use an corresponding expansion of . A di erent prediction horizon i for each is possible; the n time horizons are collected together in the n 1 vector given by:

0 1 B C = B: : :C @ A

1 2

(35)

n

De ning N to be a column vector (dimension n (Ny + 1))containing and its derivatives:

N

0 1 B C B C =B C B ::: C @ A

1] 2]

(36)

N

]

Then the Taylor series expansion (with N terms) giving at times in the future based on N at time t is: ( ; t) = T ( )

N

(t)

(37) Page 14.

4. Predictive generalised minimum-variance control (PGMV)

MVCGPC: A State-space Approach where T ( ) is a row vector with n

2nd February 1998

n matrix elements and given by :::

i!

i

T ( ) = In

where

n

:::

N

N

!

(38)

is the n n diagonal matrix with ii = i and In n is the n n unit matrix. Following Demircioglu and Gawthrop (1991), the unrealisable derivatives in Equation 36 and 37 are replaced by the emulated versions to give: ^( ; t) = T ( ) ^ (t) (39) where ^ (t) is given in terms of the state estimate of Equation 12 by: ^ (t) = YNy (t) ^ (40)

where is the (N +1) (Ny +1) matrix with n 20) or 0n ny

ny elements pi (Equation

0 p B0n n B = B0n n B ::: @

0

y y

0n 0n

p p

1

0

0n

:::

ny ny

ny

0n

p p p :::

2 1 0

ny

: : : 0 n ny : : : 0 n ny C C : : : 0 n ny C C ::: ::: A : : : pn

1

(41)

The concept of moving-horizon control has been discussed in detail elsewhere (Mayne and Michalska 1990; Demircioglu and Gawthrop 1991). The basic idea is to design with a moving time frame located at time t regarding x(t) as the initial condition of a state trajectory x ( ; t) driven by an input ^ u ( ; t) together with associated predicted outputs y ( ; t) and ( ; t). None of the starred variables have a direct relationship with the actual variables, in particular u(t + ) 6= u (t; ) except when = 0 Within this moving time frame, the predicted value of at time is given by an equation of the same form as Equation 39. ^ ( ; t) = T ( ) (t) = T ( ) ONy x(t) + HNy ;Nu UNu (t; 0) (42) where UNu (t; 0) is a vector containing the derivatives of u ( ; t) of the same form as Equation 7. Also, within this time frame it is assumed that the reference signal w (t; ) has a Taylor series expansion of the form:

w (t; ) = T ( )W (t)

4. Predictive generalised minimum-variance control (PGMV)

(43) Page 15.

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Following Demircioglu and Gawthrop (1991), one such possibility is to choose W (t) = R y(t) + R(w(t) ? y(t)) (44) where R is a column vector containing the Markov parameters of a reference dynamic system and R has rst element the unit matrix and the rest zero matrices of appropriate dimensions. Both R and R have the same dimension ny (N + 1) ny . A particularly simple case arises when the reference is just a unit gain and so R = R : W (t) = R w(t) (45) In fact, as discussed by Demircioglu (1989), the use of the P polynomial weighting is more e ective that R in most cases and so the use of Equation 45 is recommended. Following Demircioglu and Gawthrop (1991), the control u (t; ) (within the moving time frame) is constrained to be a polynomial of order Nu function of time. This is achieved by replacing the vector UNy by UNu . The optimisation problem can now be formulated as the minimisation with respect to UNu (t; 0) of the non-dynamic cost function: JP GMV (UNu ) = (t) ? w (t)]T (t) ? w (t)] 1 = 2 ONy x + HNy ;Nu UNu ? W ]T T T ( ) (46) T ( ) ONy x + HNy ;Nu UNu ? W ] The minimising vector UNu is: T T UNu (t; 0) = HNy ;Nu T T T ( )T ( ) HNy ;Nu ]? HNy ;Nu T T T ( ) ^ (47) T ( )(W ? ONy x(t)) The control is calculated by setting u(t) = u (t; 0) , the rst element of UNu (t; 0) This gives the linear, time invariant, controller: u = kw w ? k x x ? k y y ^ (48) where kw , kx and ky are given by the rst nu rows of Kw , Kx and Ky respectively where: Kw = ?R (49) Ky = ?(R ? R) (50) (51) Kx = ? ONy T ? HT T T T T ? = HNy ;Nu T ( )T ( ) HNy ;Nu ] Ny ;Nu T ( )T ( ) (52)

0 0 0 0 0 1 0 1

5. Predictive generalised minimum-variance control (PGMV)

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5 Generalised predictive control (GPC)

The CGPC of Demircioglu and Gawthrop (1991, 1992) was not a direct generalisation of the PGMV of the previous section - the P polynomial was not used. This section provides the derivation of the more general case based on PGMV. The PGMV cost function focuses on a single time instant . In contrast, the GPC cost averages the error over the time intervals speci ed by and . To allow for the di erent time intervals corresponding to each in the context of a single integral, is parameterised by the scalar 2 0 1]: = + ( ? ) (53) The cost function is:

1 2 1 2 1

JGP C (UNu ) =

=

Z

1

ONy x + HNy ;Nu UNu ? W ]T T T ( )T ( ) ONy x + HNy ;Nu UNu ? W ]d = ONy x + HNy ;Nu UNu ? W ]T T ( ; ) (54) ONy x + HNy ;Nu UNu ? W ]

0 1 2

Z

0 1

( ; t) ? w ( ; t)]T

( ; t) ? w ( ; t)]d

Where

T( ; ) =

1 2

Z

0

1

T T ( )T ( )d

( 1)!( 1)! 2

(55)

j Using Equation 38. the ij th element of T T ( )T ( ) is i?i?1 j??1 It follows that the ijth (matrix) element of T ( ; ) is:

1

Once again, this is a non-dynamic optimisation. Taking the rst derivative of J with respect to UNu and setting the result to zero gives the same result as in Equation 47 except that T ( )T T ( ) is replaced by T ( ; ). T T ^ UNu (t; 0) = HNy ;Nu T T ( ; ) HNy ;Nu ]? HNy ;Nu T T ( ; )(W ? ONy x(t)) (57) It follows that the GPC is also described by the controller equation u = kw w ? k x x ? k y y ^ (58)

1 2 1 1 2 1 2

Tij ( ; ) = (i ? 1)!(j ? ? i + j ? 1) 1)!(

2 1 1 2

i+j ?1

i+j ?1

(56)

5. Generalised predictive control (GPC)

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MVCGPC: A State-space Approach

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1 2

This is identical to Equations 48 { 52, except that T T T is replaced by T ( ; ) in 52 to give T T ? = HNy ;Nu T T ( ; ) HNy ;Nu ]? HNy ;Nu T T ( ; ) (59)

1 1 2 1 2

Following Kwakernaak and Sivan (1972), collecting together the equations for the system (1), the observer 12 and the controller (28), (48) or (58) gives the closed-loop system of the form xc = Acxc + Bcuc _ (60) yc = Ccxc + Dcuc (61) where: and

5.1 Closed-loop system

y xc = x ; uc = w ; yc = u x ^ v

(62) (63) (64)

As shown by Kwakernaak and Sivan (1972) the poles of the closed loop system (eigenvalues of Ac) comprise the separately designed controller poles and observer poles; this is a manifestation of the separation principle (Kwakernaak and Sivan 1972).

A Ac = LC? Bky C A ? ?Bkx Bk ; Bc = Bkw B Bkw 0 ? Bky C LC ? x C Cc = ?k C ?0 ; Dc = k0 0 kx y w 0

6 Nonlinear Emulator-based Control (NEBC)

The PGMV and GPC controllers for linear systems derived Sections 4 and 5 are based on taking multiple derivatives of the system output with respect to time. In principle, this procedure can be equally well applied to the outputs of non-linear systems: this is the fundamental idea behind this paper. It is also the basis of much of the geometric theory of nonlinear systems (Isidori 1995). This section considers nonlinear dynamic systems with the state-space representation: x = F (x; u) _ (65) y = H (x) (66) 6. Nonlinear Emulator-based Control (NEBC) Page 18.

MVCGPC: A State-space Approach

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where the function F and H are smooth (to be precise di erentiable Ny times with respect to each argument). and, as in the linear case, the outputy, input u and state x dimensions are ny , nu and nx respectively. A special case of Equation 65 is one where the control enters in a linear fashion : x = f (x) + g(x)u _ (67) y = h(x) (68) where f (x) + g(x)u = F (x; u) and h(x) = H (x). Much of geometric control theory (see, for example, Isidori (1995) for an exposition) is built on systems of the form of Equation 67 rather than that of Equation 65. However, Equation 67 has no particular advantage for our purposes so the more general case of Equation 65 is considered here.

Example 6.1

This nonlinear system is used as an example within this section. x = x ? tanh(u) _ x = ?x + x + u _ y=x The zero dynamics are speci ed by setting y = x = 0 (Isidori 1995). If 6= 0, this gives the rst order zero dynamics

1 2 2 2 2 1 1 1

x = x + tanh? x _ u = tanh? x

2 2 1 2 1 2

2

If > 0, these dynamics are unstable. If, on the other hand, = 0, there are no zero dynamics; moreover, the system is of the form of Equation 67 with f (x) = ?x x+ x ; g(x) = 0 ; h(x) = x (71) 1

2 1 2 2 1

6.1 Emulators in state-space form

Following Section 2, repeated di erentiation Ny times of the output y with respect to time, together with repeated substitution of the system equation 65 gives a nonlinear equation relating the output derivative vector Y to the state x and the input derivative vector U : Y (t) = O(x(t); U (t)) (72) 6. Nonlinear Emulator-based Control (NEBC) Page 19.

MVCGPC: A State-space Approach

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where Y (t) and U (t) are given by Equations 6 and 7 respectively and O(x; U ) is a (nonlinear) function of x(t) and U (t). In the SISO case , O is an (Ny + 1) 1 vector of nonlinear functions giving y : : : y Ny . Following Isidori (1995), the relative degree of this nonlinear system is the least value of i for which y i 6= 0.

] ]

Taking the particular example of a constant input u = u and repeatedly di erentiating the output y of system of Example 6.1, the elements of the Y vector (Yi = y i ) are given by y =x y = ?(tanh (u ) ? x ) y =u ?x +x y = tanh (u ) + 2u x ? 2x x + 2x ? x y = ?((4x ? 8x + 1)u + (8x ? 1)x ? 2 tanh (u ) x ? 2u ? 2x ? 6x + 3x ) y = 6 tanh (u ) u ? 6 tanh (u ) x + 10 tanh (u ) x ? tanh (u ) + 16u x ? 32u x x + 40u x ? 10u x + 16x x ? 40x x + 10x x + 24x ? 14x + x (73)

0 ] 0] 1 1] 0 2 2] 2 2 0 1 3] 3 2 0 0 2 1 2 2 4] 2 2 2 2 2 0 2 1 4 2 2 2 1 0 1 0 2 5] 2 2 0 0 0 1 0 2 0 0 2 0 1 2 3 2 2 1 3 2 0 0 2 2 1 5 2 3 2 1 2 2

Example 6.2

In this case, the expression for y explicitly contains u if 6= 0; the expression for y explicitly contains u . It follows that the relative degree = 1 if 6= 0 and = 2 if = 0.

1] 0 2] 0

In a similar fashion to the linear case (Equation 16) the emulated value ^ of Y (t), Y (t) is de ned as: ^ Y (t) = O(^(t); U (t)) x (74) where x(t) is an estimate of the state x(t). ^ Unlike the linear case, however, there is no general theory of state estimation for non-linear systems. For the purposes of this paper, observers are taken to be of the form: _ x = f (^; u) + L(^)e ^ x x (75) y = g(^) ^ x (76) e=y?y ^ (77) Unlike the linear case, the stability of such an observer is not guaranteed in general and its design is non trivial (Walcott, Corless, and Zak 1987; 6. Nonlinear Emulator-based Control (NEBC) Page 20.

MVCGPC: A State-space Approach

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Hunt and Verma 1994). Although observer design is an important issue for nonlinear GPC it is beyond the scope of this paper. It is an area of continuing research by the authors; in particular we are looking at physical model-based observers (Gawthrop, Jones, and MacKenzie 1992; Gawthrop and Smith 1996). In exactly the same way as in Section 3, the quantity ^ is de ned in term of ^ the emulated output derivative vector Y as ^ (t) = P Y (t) = P O(^(t); U (t)) x (78) Because, in the context of GMV, Ny = (the relative order of the the system) then only u(t) (not its derivatives) will appear in Equation 78. The NGMV control is thus implicitly de ned (at each time t) by: ^(t) = P O(^(t); U (t)) = w x (79) This equation may have none, one or many solutions depending on the form of P O(^(t); U (t)) and the current value of x(t). In general, the equation x ^ must be solved numerically online. The precise conditions for solution are not of concern; we merely note that, for the reasons given in Section 3, GMV is not a practically useful controller. However, if the system of Equation 65 has the special form of Equation 67 then (and noting that Ny = ) the implicit control of Equation 79 becomes

6.2 Generalised minimum-variance control (NGMV)

PO(^(t)) + PH (^(t))u(t) = w x x

This has the obvious solution

(80) (81)

u = PH ? (^(t)) w(t) ? PO(^(t))] x x

1 1

if PH ? (^(t)) 6= 0. x A geometric interpretation of this special case appears in Section 7.2.

Example 6.3

1

Considering rst the case where 6= 0, ? as P = 1 p This gives the controller:

1 1 2 1

= 1 and so P can be chosen (82) Page 21.

^ ^ u = tanh? x + x ? w p 6. Nonlinear Emulator-based Control (NEBC)

MVCGPC: A State-space Approach

2nd February 1998

Give the unstable zero dynamics, this is not a useful controller. Turning to the case where = 0, = 2 and so P can be chosen as ? P = 1 p p This gives the controller:

1 2

x ^ ^ u = w ? (1 ? p )^p ? p x ? p x

2 1 1 2 2 2

2 2

(83)

As, in this case, there are no inverse dynamics, this is a sensible controller. In each case, the controller dynamics arises from the observer, Equations 75.

6.3 Predictive generalised minimum-variance control (NPGMV)

As in the linear case, it is useful to de ne a predicted ^: ^( ; t) = T ( ) ^ (t) = T ( ) Y (t) ^ (84) In the non-linear case, however: ( ; t) = T ( ) (t) = T ( ) O(^(t); U (t; 0)) x (85) Using the same cost function as Equation 46: JP GMV (U ) = (t) ? W (t)]T (t) ? W (t)] 1 x = 2 O(^(t); U (t; 0)) ? W ]T T T ( ) T ( ) O(^(t); U (t; 0) ? W ] x This non-dynamic optimisation problem equation must be solved numerically for U (t; 0) at each time t. This situation can be contrasted with the linear case where the control law (Equation 48) is an algebraic equation. The corresponding closed loop system is a mixture of an ODE (arising from the system and observer) and an algebraic equation (the state feedback). In the linear case, the fact that the resulting DAE can be immediately recast as an ODE is so obvious as to hardly need mention. However the nonlinear case gives rise to a mathematical structure which is akin to a DAE but with the algebraic equation replaced by a non-dynamic optimisation. The precise mathematical description and investigation of such di erential-optimisation equations remains an open question. From a practical point of view, however, the corresponding di erential equations may be discretised in time and the optimisation performed at each time step { see Section 8 for an example. 6. Nonlinear Emulator-based Control (NEBC) Page 22.

(86)

MVCGPC: A State-space Approach Using the same cost function as Equation 54:

2nd February 1998

6.4 Generalised predictive control (NGPC)

JGP C (U ) =

=

Z

1

O(^(t); U (t; 0)) ? W ]T T T ( )T ( ) O(^(t); U (t; 0)) ? W ]d x x = O(^(t); U (t; 0)) ? W ]T T ( ; ) O(^(t); U (t; 0)) ? W ] (87) x x

1 0 1 2

Z

0

( ; t) ? w (t)]T

( ; t) ? w (t)]d

This non-dynamic optimisation problem equation must be solved numerically for U (t; 0) at each time t. As discussed by Clarke (1994b) and Kuznetsov and Clarke (1994), the notion of constraints leads to practically useful control. Exploration of this idea is beyond the scope of this paper; but we note that NCGPC has the necessary ingredients, in particular: a built in optimisation and the ability to de ne as many components of as required to specify constraints.

This section gives a geometric interpretation of linear GMV and draws parallels with a linear version of the (nonlinear) exact linearisation strategy laid out in Section 4.2 of Isidori (1995). For simplicity ( and following Section 4.2 of Isidori (1995)) the discussion is restricted to the SISO (ny = nu = n = 1) case and it is assumed that the state x is available for measurement or, equivalently that x = x. ^ At this point, it is useful to de ne a new state vector X (t) as a linear transformation of x(t)

7.1 Linear systems

7 Geometric interpretation of GMV

X = Onx nx x

(88)

where Onx nx is the nx nx observability matrix of Equation 8. Assuming that the system is observable Onx nx is a (nonsingular) linear transformation matrix that puts the system into observability canonical form (Kailath 1980); that is _ X = Ao X + B o u (89)

7. Geometric interpretation of GMV

Page 23.

MVCGPC: A State-space Approach where

2nd February 1998

00 1 0 B0 0 1 B ? Ao = On n AOn n = B : : : : : : : : : B0 0 0 @ 0 h 1? ? ? Bh C B C Bo = On n B = B : : : C Bh C @ n?A

x x x

1

x

1

2

3

::: 0 ::: 0 C C ::: ::: C C ::: 1 A : : : ? nx

1

(90)

1 2

x

x

(91)

hnx

x

1

Recalling that the ith Markov parameter hi = 08i < and comparing Equation 5 with 88 it follows that the rst elements of X are identical to the corresponding elements of Y and thus X may be written as

X= Y Z Z is a vector of dimension nx ? .

(92)

Substituting the GMV controller of Equation 28 (in the new coordinates) into Equation 89 gives _ X = AcoX + Bcow (93) where the closed-loop system matrices Aco and Bco are A = A ? 1 B PO O? ; B = 1 B

co o o

1

co

o

(94)

From the closed-loop Equation 27 it follows that Aco and Bco are of the form A = Pco 0 (95)

co

Azy Azz

where Pco is the

companion matrix 00 1 0 B0 0 1 B Pco = B : : : : : : : : : B0 0 0 @

p p p ? p1 ? p2 ? p3

::: 0 1 ::: 0 C C ::: ::: C C : : : p1 A : : : ? p?1

(96)

7. Geometric interpretation of GMV

Page 24.

MVCGPC: A State-space Approach

2nd February 1998

Azy is a

nx ? matrix and Azz is a nx ?

001 B0C B ::: C B C B C Bco = B h 1+1 C B C Bh C B C B ::: C @h A

h

nx

nx ? matrix.

(97)

Example 7.1

The closed-loop system therefore consists of two subsystems: 1. The system of order with state Y with dynamics decoupled from the state Z and with dynamics determined by Equation 27. 2. The system of order nx ? and with state Z with dynamics determined by Azz and partially coupled to the rst system via Ayz . Following Isidori (1995), this is called the zero dynamics of the closed-loop system. These dynamics are unobservable from the system output and correspond to zero cancellation. In the special case that pi = 0 8i < and p = 1, The GMV feedback of Equation 28, together with the state transformation of Equation 88 gives the closed-loop system y =w (98) This is the linear version of the exact linearisation by feedback strategy of Isidori (1995). If, on the other hand, p s is a desired closed-loop system, this corresponds to exact linearisation by feedback followed by pole placement.

] 1 ( )

The system of Equation 3 is already in normal form. Using the GMV controller of Section 3 (with state feedback) gives the closed-loop system 1 1 x = ?px + pw _ (99) x = 1 ?1 x + 1+ x + 1 w _ (100)

1 1 2

The rst state x is uncoupled from the second state x and (noting that y = x ) gives the desired closed loop output: py + y = w _ (101) The second state is unobservable. Its dynamics correspond to the system zero at s = and is unstable if > 0. If = 0 the equation becomes singular.

1 2 1 1+

p

1

2

p

7. Geometric interpretation of GMV

Page 25.

MVCGPC: A State-space Approach

2nd February 1998

There has been a recent resurgence of interest in non-linear control driven by Geometrical Control Theory; to avoid proliferation of references the book of Isidori (1995) is used as a summary of such results. The purpose of this section is to recast our results in a geometric setting. In particular, the construction of the nonlinear GMV is shown to follow the same steps as the development of the Exact Linearisation via Feedback given in Section 4.2 of Isidori (1995) . For simplicity, the SISO case is considered; and, following Isidori (1995), state feedback is considered so that x = x. The linear case ^ appears in Section 7. Following Isidori (1995), the special (linear in the control) system of Equation 67 is considered in this section. Di erentiating the output y with respect to time gives:

7.2 Nonlinear systems

@h y = @h (x)x = @h (x)f (x) + @x (x)g(x)u(t) = o (x) + h (x)u (102) _ @x @x If h (x) = @h (x)g(x) 6= 0 the procedure terminates; otherwise the second @x derivative of the output y with respect to time is written as: y = @o (x)x = @o (x)f (x) + @o (x)g(x)u(t) = o (x) + h (x)u(t) (103) @x _ @x @x This procedure is repeated until h 6= 0. This is the de nition of relative

1] 1 1 1 2] 1 1 1 2 2

order in the nonlinear context (Isidori 1995). At this point Equation 72 can be written (with Ny = ) as

Y = O (x) + H ; (x)U

where:

(104)

0 0 0o (x)1 B O (x) = @ : : : A ; H ; (x) = B : 0: @ :

1

o (x)

h (x)

1 C C A

(105)

This can be written in the Lie notation of Geometrical Control Theory Isidori (1995) as

ok = Lk h(x) f

and

(106) (107) Page 26.

h = Lg Lf? h(x)

1

7. Geometric interpretation of GMV

MVCGPC: A State-space Approach The GMV controller of Equation 81 then becomes: P w ? i pioi(x) u= p Ph (x)Li h(x) w ? i pi f = p Lg Lf? h(x)

=1 =1 1

2nd February 1998

(108) (109)

The externally, the closed-loop system is de ned by:

X

i=1

pi y i = w

]

(110)

As in Equation 98, if

pi = 0 8i < ; p = 1

then

(111) (112) (113) (114)

?o u = w h (x(x) ) w ? Lf h(x) = Lg Lf? h(x)

1

and

y =w

]

This is precisely the situation described in Proposition 4.2.1 of Isidori (1995). It follows that the special case of NGMV using state feedback and Equation 111 is equivalent to the exact linearisation by feedback described by Isidori (1995). Following the same procedure as in Section 7, de ne the (nonlinear) function Onx? in the same way as Equation 105 but with replaced by nx ? 1. The new state X is then de ned as:

1

X = Y = Onx? (x) Z

1

(115)

Exactly as discussed in Section 7 this represents a decomposition into two subsystems.

8. Geometric interpretation of GMV

Page 27.

MVCGPC: A State-space Approach

1.5

2nd February 1998

1

y/w0

0.5

0

?0.5 0

0.5

1

1.5

2

2.5 t

3

3.5

4

4.5

5

Figure 1: Simulation: Non-linear system with NCGPC

8 Examples

The MATLAB code to generate the examples in this section is available at URL

http://www.mech.gla.ac.uk/~peterg/software/matlab/emulator/

See the README le for details. For reasons of space, illustrative linear examples have been omitted from this paper. The purpose of this section is to illustrate the advantages of using a nonlinear algorithm to control a nonlinear system. In particular, although it is not possible to perform exact linearisation (due to the unstable inverse dynamics) the closed loop system is approximately linearised. To this end, the nonlinear system of Example 6.1 is simulated within the Matlab/Simulink environment using Euler integration with a step size of 0:01 and using two separate controllers: the (non-linear) NCGPC and a (linear) GPC based on the linearisation of the non-linear system about y = u = 0. In each case, the design parameters were chosen as p(s) = 1 + s, Nu = 0, and = 0:5. Each closed-loop system was simulated for step inputs of magnitude 0:1; 0:5; 1 and 2. The corresponding step responses were normalised by dividing by the step magnitude and plotted in Figures 1 and

2

8. Examples

Page 28.

MVCGPC: A State-space Approach

1.5

2nd February 1998

1

y/w0

0.5

0

?0.5 0

0.5

1

1.5

2

2.5 t

3

3.5

4

4.5

5

Figure 2: Simulation: Non-linear system with (linear) CGPC 2 respectively. In each gure, the rm line represents the model response de ne by P and the dashed lines the actual system outputs. Figure 1 shows that the normalised step responses for the non-linear controller are relatively una ected by step magnitude as compared for the corresponding results for the linear controller 2. In this sense, the closed-loop system (with the non-linear controller) is approximately linear. The cost minimisation was performed on-line using the Matlab function \fmin"; in the simulation, this function was called at each evaluation step of the integration process.

9 Conclusion

A state-space formulation of multivariable continuous-time GPC has been presented. Although beyond the scope of this paper, the results can be extended to give stability results using constraints as in Demircioglu and Clarke (1992). This state-space approach: provides an alternative (to the transfer function) approach to CGPC; gives a simple set of algorithms for both the SISO and MIMO cases appropriate for Matlab implementation; 9. Conclusion Page 29.

MVCGPC: A State-space Approach

2nd February 1998

illuminates a hitherto unrecognised relationships with the geometric approach; readily extends to non-linear systems. An example has been presented to illustrate the approach and more examples, and a corresponding tool-box, are available from the rst author's home page. Future work will include: extension of the current stability results of Clarke and Scattolini (1991), Demircioglu and Clarke (1992),Kouvaritakis, Rossiter, and Chang (1992), Rossiter and Kouvaritakis (1993) and Kouvaritakis and Rossiter (1993) to the methods of this paper; investigation of the properties of the di erential-optimisation equations appearing in Section 6; experimental evaluation on industrial processes.

10 Acknowledgements

Irma Siller-Alcala was supported by CONACYT and Peter Gawthrop by EPSRC grant GR/H41942. The authors gratefully acknowledge the contribution of other members of the Centre for Systems and Control.

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