OPTIMAL CONTROL APPLICATIONS AND METHODS Optim. Control Appl. Meth., 2004; 25:235–262 Published online in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/oca.747
Robust continuous-time controller design via structural Youla–Kuc$ era parameterization with application to predictive control Z. Kowalczukn,y and P. Suchomski Department of Automatic Control, Faculty of Electronics, Telecommunication and Computer Science, Gdan!sk University of Technology, 80-952 Gdan!sk, Poland
SUMMARY This paper addresses the continuous-time control of uncertain linear SISO plants and its nominal and robust stability and nominal and robust performance objectives. A specific application of the Youla– Ku$cera (Q) parameterization concept leads to a new development of observer-like controller structures. This method is combined with a nominal design of continuous-time generalized predictive control suitable for both minimum-phase and non-minimum-phase plants. The subsequent design procedure consists of two steps. Firstly, the nominal stability and nominal performance of the control system are established by using an analytical design methodology, based on a collection of closed-loop prototype characteristics with definite time-domain specifications. And secondly, a generic structure of the controller is enhanced by suitable Q-parameters guaranteeing that the control system has the required robustness properties. The proposed structural (reduced-order) Q-parameterization relies on an observer structure of controllers, which can be easily enhanced with certain filters necessary for control robustification. To reduce the complexity of the resulting robust controllers, we suggest using a structural factorization, which allows for simple forms of robustifying (phase-lag) correctors of low order, easy for implementation, and convenient for optimization and tuning. Two numerical examples are given to illustrate the composed technique and its practical consequences. Copyright # 2004 John Wiley & Sons, Ltd. KEY WORDS:
continuous-time system design; optimal control; robust control; system parameterization; generalized predictive control; linear time-invariant systems
1. INTRODUCTION The continuous-time approach is a natural framework to represent models, parameters, and other characteristics of physical systems, which are basically continuous in time. In particular, the dynamics of such systems can be conveniently described by differential equations and other
n
Correspondence to: Professor Zdzis"aw Kowalczuk, Department of Automatic Control, Faculty of Electronics, Telecommunication and Computer Science, Gdan! sk University of Technology, Narutowicza 11/12, P.O. Box 612, 80-952 Gdan! sk, Poland y E-mail:
[email protected]
Copyright # 2004 John Wiley & Sons, Ltd.
Received 8 October 2002 Revised 6 September 2004
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related models, whose parameters have physical motivation. On the other hand the parameters of auxiliary discrete-time descriptions usually have no physical interpretation behind. Other drawbacks of the discrete-time modelling concern the technical characteristics and numerical conditioning of such parameterizations, which appear to be strongly influenced by the sampling frequency applied. Moreover, such models not properly describe the dynamics of an underlying continuous-time system whenever the sampling period chosen tends to zero (this is so, because the shift operator does not have a corresponding rational counterpart in the continuous-time framework). Therefore, in many cases, and especially with the advent of high-performance highfrequency computers the continuous-time alternative to the discrete-time approach (marked with such problems as pole clustering and aliasing, high-order models, numerical conditioning, arithmetic precision, nonminimum phase zeros, etc.) can be more suitable. For the digital implementation of continuous-time control algorithms one can use the wealth of discrete approximation methods [1] or resort to the delta approach (see References [2, 3], for instance). Recalling the interpretation argument, it is worth mentioning that also model uncertainty can be easily characterized with the use of the continuous-time system terms (gain, time constants, zero and poles, etc.). Having in mind the outlying background given above, the main purpose of this work is to present a straightforward robustification of a wide class of existing observer-based structure controllers resulting in a simple closed-form design. With a general applicability of this method its most efficient implementation is connected with an algebraic Diophantine design methodology. As an apt exemplary application of this methodology we shall use a special form of MPC, the continuous-time generalized predictive control (CGPC) problem solved with the use of an effective analytical design method of the pole placement type. In recent years the long-range model-based predictive control has been acknowledged as a useful approach to control design. The generalized predictive control (GPC) proposed by Clarke et al. [4, 5] as a discrete-time design methodology using a long-horizon quadratic cost function has attracted a great interest [6–12], which is due to its general applicability as compared to other control strategies. Similarly, the continuous-time GPC approach (CGPC) introduced by Demircioglu and Gawthrop [13, 14] has been found suitable for consideration both in robustand adaptive-control treatments [15–27]. Such control-system synthesis procedures are based on the so-called ‘emulator’ paradigm, in which physically unrealizable operations (predictions or derivatives) are replaced (emulated) by means of non-parametric or parametric system models [27]. In quest of stability bonds appropriate for continuous-time CGPC systems, two modified versions of the genuine approach of Demircioglu and Gawthrop [13] have been suggested in Reference [15]. These modifications exploiting the stability results of state–space recedinghorizon LQ control laws are based on two formulations of the so-called ‘end-point principle’, namely: end-point state constraints (EpC) and end-point state weighting (EpW). The EpC methodology uses the certainty adopted from the state–space receding-horizon LQ approach that if the state vector of the closed-loop control system at the end point is constrained to be zero, then the stability of the system is guaranteed. Since with the applied input–output model the system states are not available, they are emulated via a truncated Taylor-series expansion technique used for prediction of the system output. For a suitably chosen order of state prediction, when a corresponding truncated Taylor-series approximation of a system partial state is good, the resulting control system will be stable (this analytical result concerns a special selection of tuning parameters). Within the EpW approach to the CGPC control stability Copyright # 2004 John Wiley & Sons, Ltd.
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problem, a quadratic term of a weighted end-point state is included in a cost function. By the assumption that the truncated Taylor-series approximation of the system partial state is satisfactory, the corresponding EpW design is also able to result in stable closed-loop systems. A new pole-placement perspective and development of the CGPC design resulting in an explicit stable CGPC control design procedure for both minimum-phase and non-minimumphase SISO systems has been presented by Kowalczuk and Suchomski [21] (confer also References [2, 25]). Our completely analytical approach is founded on a collection of closed-loop prototype characteristics (different from the ones of [15]) with definite time-domain specifications. It assures both the nominal stability (NS) and the nominal performance (NP) closed-loop system requirements at the same time. This method represents a direct way of guaranteeing stability and an indirect way of assuring a limited control signal for a nominal tracking task. The idea of model-based prediction, concerning a future output (Case a) and a future partial state or a filtered output (Case a% ) of the plant, is based on suitable emulation of signal derivatives and is performed by solving a set of coupled Diophantine equations. The prediction of type a is suitable for the CGPC design of minimum-phase models of the controlled plant. If the model has non-minimum phase, we recommend emulating the derivatives of the output signal filtered by the numerator polynomial of the nominal transfer function of the controlled part of the plant (Case a% ). In consequence, on the basis of a generic CGPC scheme, two predictive control laws are told apart: the a-law based on the output emulation and restricted to minimum-phase plant models, and the a% -law utilizing the filtered output emulation and applicable both to minimum phase and non-minimum phase plant models. After a suitable analysis of the resulting closed-loop systems, explicit formulae for the system characteristic polynomials can be given [21, 25]. They serve as an efficient basis for analytical CGPC-design procedures (both ways, a and a% ), in which the principal CGPC ‘tuning knobs’, i.e. the output and control prediction orders as well as the horizon of observation, are directly related to typical time-domain design specifications. An analytical nature of the proposed methodology calls for analysis of the emulation-based design from the viewpoint of pole placement. Suchomski and Kowalczuk [2, 21, 25] have offered such a deliberation along with a catalogue of prototype characteristic polynomials. The considerations recollected above concern solely the nominal stability (NS) and the nominal performance (NP) issues of the CGPC design. Any real control system has to be designed so as to guarantee (to a certain extent) its robustness to differences between the actual and the nominal plant model. Clearly, a ‘natural’ approach to the robustness problem is to analyze the effect of the design parameters (the horizons and the weighting factor) on the closedloop performance [28]. Two general approaches to discrete-time GPC system robustification can be observed in the literature. The first one is based on the fact that the nominal tracking transfer function of the basic closed-loop GPC control system is not affected by the observer polynomial Cðq1 Þ used for improving the quality of output prediction in the presence of disturbances. Specifically, Clarke and Mohtadi have demonstrated [5] that the observer polynomial can be employed to shape the disturbance rejection properties of the GPC system with no influence on its tracking response. Consequently, the system tracking properties and the system robustness attributes (expressed in terms of the system sensitivity function) can be autonomously shaped when one considers the observer polynomial as a free design parameter. It is worth noticing here that there is a common opinion [5] that a successful identification of this element of the plant model is unlikely. Copyright # 2004 John Wiley & Sons, Ltd.
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Therefore, many authors [6, 28–34] have considered the possibilities of enhancing the robustness of GPC via tuning a suitably parameterized observer polynomial Cðq1 Þ: They incorporate a factor ð1 gq1 Þ with a free design knob g into this polynomial. For example, Robinson and Clarke [29] have discussed the influence of the specified-by-designer observer polynomial on the robustness of some GPC systems leading to either dead-beat or mean-level action. Simple guidelines for tuning such a one-parameter observer polynomial have also been given there}with an observation that there is a tradeoff between the closed-loop system robustness and the speed of rejection of disturbances. (In particular, slow observer poles are to be chosen for improving the system robustness, while fast poles are to be preferred for a suitable rejection of system disturbances.) Moreover, it has been reported by Soeterboek [6] that choosing Cðq1 Þ ¼ Aðq1 Þð1 gq1 Þ; for Aðq1 Þ being a stable denominator of the plant transfer function, gives an opportunity to improve the closed-loop robustness (by increasing the parameter g). Clearly, g ! 1 is a good choice when solely robustness is concerned, but this results in poor disturbance attenuation (see also References [32, 34]). A method for improving the robustness of the Smith predictor-based GPC system (SP-GPC) while maintaining its nominal closed-loop performance for stable plants has been proposed by Normey-Rico and Camacho [35]. Their approach consists of two steps: first the SP-GPC parameters are tuned to achieve nominal set-point specifications, and then a C-based pre-filter is selected for the disturbance rejection and robust performance purposes. A similar C-design technique has been considered within the continuous-time CGPC design by Kowalczuk and Suchomski [19, 20] in order to robustify control systems for delay plants. There is, in general, no systematic way of such optimization because of a complicated way in which the observer polynomial induces relevant sensitivity functions [7, 30, 31, 33]. Nevertheless, taking into account a specific quadratic criterion of robustness, a methodical approach to the C-design (within a two-degree-of-freedom principle) can also be shown [36] which aims at achieving robustly stabilizing receding-horizon predictive controllers. With respect to the generic H1 norm of a mixed-sensitivity function, the solutions obtained for the C-technique are sub-optimal (as it is not possible to globally minimize this norm by parameterizing all stable observer polynomials). This is in contrast with the general fact that the set of all stabilizing controllers minimizing the H1 norm of a mixed-sensitivity function can be effectively parameterized [30]. Another alternative approach to the robust control design is based on the Youla–Jabr– Bongiorno–Kuc$era parameterization paradigm [37, 38] known also as the Q-parameterization of all stabilizing controllers in terms of the set of all stable proper transfer functions [39, 40]. The Q-parameterization has been introduced to discrete-time GPC robustification by Kouvaritakis et al. [7], who presented a systematic approach for robust stabilization in the presence of plant uncertainties and solved the problem via the H1 norm minimization paradigm. A link between the Q parameter and the observer polynomial has also been established in this work. Since then many researchers have made similar observations and developments. The idea of constrained model-based predictive controllers based on the Q-parameterization of all stabilizing controllers has also been discussed by Fikar et al. [41]. The same Q-approach can be utilized in the GPC design for high-frequency disturbance-rejection purposes [42]. An interesting relationship between the C-design and the Q-parameterization has been analysed by Yoon and Clarke [30]. They have demonstrated that despite some structural limitations inherent to the application of observers (only to stable plants, for instance) in many cases the C-design provides robustness comparable with the effect obtained via the H1 method. Copyright # 2004 John Wiley & Sons, Ltd.
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A novel methodology for improving the original GPC design, in terms of robustification of the closed-loop system against plant uncertainty, has been presented by Ansay et al. [31, 33]. Their approach consists of two steps. In the first step, an optimal tracking GPC controller is obtained for Cðq1 Þ ¼ 1: In the second step, the controller is enhanced by a suitable form of the Q-parameter to achieve an acceptable compromise between the robustness and performance (disturbance attenuation) characteristics of the closed-loop system. An ultimate objective of this contribution is a new development of continuous-time controller designs resulting in a reduced-order Q-enhanced (robustly stable) design procedure suitable for both minimum-phase and non-minimum-phase uncertain SISO systems. With a broad-spectrum suitability of the intended method, its specific form will be presented and explained by means of a relevant CGPC design problem. Two key steps are distinguished in the proposed procedure. In the first step, the NS and NP attributes of the CGPC control system are fixed by the application of our analytical CGPC methodology [2, 21, 25]. In the second step, being an innovative part of this work, a basic structure of the CGPC controller is enhanced by the Q-parameters of a specific form. These parameters are designed in a way that the tracking properties of the resulting closed-loop control system are kept unaffected and, at the same time, both the robust stability (RS) and robust performance (RP) attributes of the system are achieved. It is worth emphasizing that our ‘structural’ reduced-order Youla–Ku$cera parameterization may be applied to any controller implemented in the observer-like structures (utilized in a wide class of pole-placement algorithms, for instance), which can be easily enhanced with certain filters necessary for its robustification. As the original affine parameterization of robust controllers has the disadvantage of enlarging their order, we shall take steps to reduce the controller complexity. Thus we will focus our attention on finding possibly simple forms of robustifying correctors, which should be of possibly low order, easy for implementation and tuning, as well as convenient for optimization. As a result, a consecutive method of ‘structural’ parameterization/factorization will be proposed which leads to effectively designing common phase-lag correctors. Our approach cannot be regarded as a simple continuous-time version of the method of Ansay et al. [31, 33], as there are several principal distinctions between them. Namely, within the CGPC framework the unity assignment CðsÞ ¼ 1 is generally ruled out because the degree of the observer polynomial CðsÞ is uniquely determined by the plant model order (a resulting degree of freedom in the design of this polynomial can be utilized to simplify or robustify the controller). Moreover, the presented method is completely analytical in terms of the nominal performance of the control system, what is achieved by employing a collection of closed-loop prototype characteristics with definite time-domain specifications. A short abstract of our previous results on nominal continuous-time linear and time-invariant generalized predictive control systems (referring to the origins of CGPC) is presented in Sections 2 and 3, which are used as an explicit background to the robustness material contained in Section 4 and illustrated in Section 5. The itemized content of this paper is as follows. A nominal minimal linear model of a controlled scalar continuous-time plant and the basics of the modelbased prediction of the future output and the partial state of the plant (Cases a and a% ; respectively) are introduced in Section 2. The prediction of type a (based on emulation of the output derivatives) is effective in the CGPC design for minimum-phase plant models, whereas the signal prediction of type a% (emulating the derivatives of the partial plant-state) is befitting for non-minimum-phase plant models. The corresponding two analytical design methods for predictive control are presented in Section 3. The question of how to achieve the robust stability Copyright # 2004 John Wiley & Sons, Ltd.
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and the robust performance of the CGPC systems at the same time is considered in Section 4. It is assumed that the plant model uncertainty is multiplicative. To counteract the abovementioned complexity problem [39] resulting from the standard Q-approach, several steps are considered to reduce the order of the controller. Primary, a ‘structural’ factorization is proposed in which procedures for stable and unstable plant models are differentiated, and further structural simplifications of the Q-enhanced CGPC controllers are achieved by considering the observer polynomial CðsÞ as a free design parameter. Two illustrative examples are given in Section 5.
2. CONTINUOUS-TIME MODEL-BASED PREDICTION Let a scalar linear continuous-time plant be described by the following minimal nominal model [13, 21]: YðsÞ ¼
BðsÞ CðsÞ UðsÞ þ VðsÞ AðsÞ AðsÞ
ð1Þ
where UðsÞ and YðsÞ are the plant’s input and output, VðsÞ is a disturbance transform, AðsÞ; BðsÞ; and CðsÞ are polynomials in the Laplace domain: AðsÞ is monic of deg AðsÞ ¼ NA 52; while deg BðsÞ ¼ NB and deg CðsÞ ¼ NC ¼ NA 1; and r ¼ NA NB ; r > 0; denotes the relative plantmodel order. Consider now two cases (a and a% ) of model-based emulation, which aims at predicting a future evolution of the following signals: the output (Case a) and the future partial state (Case a% ) of the plant. A rigorous description of the derivation concerning emulations of the output derivatives and the derivatives of the partial state has been given by Kowalczuk and Suchomski [21, 25]. Effectively, both the emulations can be constructed by solving a properly defined set of two coupled Diophantine equations presented below. 2.1. Emulation of output derivatives ðaÞ In order to emulate the kth ‘derivative’ of the plant output, Yk ðsÞ ¼ sk YðsÞ; k50; the following a-Diophantine basis, consisting of two pairs of coupled equations: primary (D1,D2) and secondary (D3,D4), are taken into account [21]: ðD1Þ: AðsÞ Ek ðsÞ þ Fk ðsÞ ¼ sk CðsÞ ðD2Þ: CðsÞ Hk ðsÞ þ Gk ðsÞ ¼ BðsÞ Ek ðsÞ ðD3Þ: AðsÞ Hk ðsÞ þ Lk ðsÞ ¼ sk BðsÞ ðD4Þ:
AðsÞ Gk ðsÞ þ BðsÞ Fk ðsÞ ¼ CðsÞ Lk ðsÞ
where the degrees of the polynomials are as follows: deg Ek ðsÞ ¼ k; deg Fk ðsÞ ¼ NA 1; deg Gk ðsÞ ¼ NA 2; deg Lk ðsÞ ¼ NA 1 (for k50Þ; and deg Hk ðsÞ ¼ k r (for k5r). Copyright # 2004 John Wiley & Sons, Ltd.
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The operator form of the predictable part Ykn ðsÞ of Yk ðsÞ becomes Ykn ðsÞ ¼ Yk ðsÞ þ Ykþ ðsÞ; k50; where Yk ðsÞ denotes an ‘observer’ part Yk ðsÞ ¼
Gk ðsÞ Fk ðsÞ UðsÞ þ YðsÞ CðsÞ CðsÞ
having the control signal filtered by a strictly proper transfer function Gk ðsÞ=CðsÞ and the plant output filtered by a proper transfer function Fk ðsÞ=CðsÞ; while Ykþ ðsÞ ¼ Hk ðsÞ UðsÞ is a ‘predictor’ part that is completely determined by the input UðsÞ and the secondary quotient polynomial Hk ðsÞ: 2.2. Emulation of partial state derivatives ð%aÞ First, note that only the non-trivial case NB 51 is considered beneath, and that there are certain problems arising from potential non-minimality of plant models (with non-coprime pairs ðAðsÞ; BðsÞÞ) as discussed by Kowalczuk and Suchomski [21]. Let thus ðAðsÞ; BðsÞÞ be coprime. In that case the following four Diophantine equations make a suitable a% -Diophantine basis (with k50) for a common CGPC design development [21]: % 1Þ: AðsÞ E% k ðsÞ þ BðsÞ F% k ðsÞ ¼ sk CðsÞ ðD % 2Þ: CðsÞ H% k ðsÞ þ G% k ðsÞ ¼ E% k ðsÞ ðD % 3Þ: AðsÞ H% k ðsÞ þ L% k ðsÞ ¼ sk ðD % 4Þ: ðD
AðsÞ G% k ðsÞ þ BðsÞ F% k ðsÞ ¼ CðsÞ L% k ðsÞ
where the degrees of the polynomials are: deg E% k ðsÞ ¼ maxfNB 1; k 1g; deg F% k ðsÞ ¼ NA 1; deg G% k ðsÞ ¼ NA 2; deg L% k ðsÞ ¼ NA 1 (for k50), and deg H% k ðsÞ ¼ k NA (for k5NA ). Let Y% nk ðsÞ denote the operator form of the predictable part of the kth ‘derivative’ Y% k ðsÞ %þ ¼ sk Y% ðsÞ of Y% ðsÞ: The emulator equation for Y% nk ðsÞ then becomes Y% nk ðsÞ ¼ Y% k ðsÞ þ Y k ðsÞ; k50; in which Y% ðsÞ stands for a corresponding ‘observer’ part k F% k ðsÞ G% k ðsÞ UðsÞ þ YðsÞ Y% k ðsÞ ¼ CðsÞ CðsÞ with the control signal filtered via a strictly proper transfer function G% k ðsÞ=CðsÞ and the plant % output filtered by a proper transfer function F% k ðsÞ=CðsÞ; and Y% þ k ðsÞ ¼ H k ðsÞ UðsÞ is a ‘predictor’ part. 2.3. Estimation of the future output and partial state of the plant (a & a% ) Let t# be a variable of future time and t 2 ½0; T; T50; stand for a relative variable of future time: t ¼ t# t: From the time-domain forms of the emulated derivatives Ykn ðsÞ and Y% nk ðsÞ we conclude that the future output yðt#Þjt#¼tþt and the future partial state y%ðt#Þjt#¼tþt can be approximated by suitably truncated series y#ðt#Þ and y#%ðt#Þ; respectively. These series can be Copyright # 2004 John Wiley & Sons, Ltd.
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described with the use of two basic ‘horizons’: the plant output prediction order Ny 50 and the control prediction order Nu 50 [21].
3. CGPC DESIGN Two analytical design methods for predictive control will be briefly described: the a-design that utilizes the emulated future output of the plant (recommended for minimum-phase plant models), and the a% -design that applies emulation of the future partial state (appropriate for handling both minimum-phase and non-minimum-phase plant models). 3.1. Basic CGPC designs Supposing that wðtÞ represents the reference signal, we denote the future output reference by ðaÞ: w# ðt#Þjt#¼tþt ¼ wðtÞ and the future partial state reference by ð%a Þ: w#% ðt# Þjt#¼tþt ¼ wðtÞ tT0;Ny ðtÞb% Ny ffi L1 ½WðsÞ=BðsÞ where b% i ¼ ½b%0 b%i T is composed of P the MarkovP parameters b%i associated with the all-pole 1 % i T B i m n inverse-plant-allied system 1=BðsÞ ¼ 1= N b s ¼ i¼0 bi s ; while tm;n ðtÞ ¼ ½t =m! t =n! : i¼0 i Let us define two future control error signals: ðaÞ:
eðt# Þ ¼ rw# ðt# Þ y#ðt# Þ
ð%a Þ: e%ðt# Þ ¼ r%w#% ðt# Þ y#%ðt# Þ where r 2 R and r% 2 R are pre-scaling coefficients. Minimization of the following quadratic indices: Z T Z e2 ðt þ tÞ dt and 0
T
e%2 ðt þ tÞ dt
0
defined for an observation horizon T > 0; yields (according to the generic receding-horizon control paradigm) the optimal control input uðtÞ for Cases a and a% ; respectively. Let kNy 2 RNy and k% Ny 2 RNy stand for the resulting optimal gains of the CGPC controllers. Moreover, let us introduce the following notational conventions: DNy ðsÞ ¼ kTNy DNy sND DNy ¼ ½d 0 d Ny T ;
d k ¼ ½dk;0 dk;ND T
P D i where si ¼ ½s0 s1 si T ; i50; and D0k ðsÞ ¼ d Tk sND ¼ N i¼0 dk;i s is a design polynomial. The corresponding closed-loop control law can be described as UðsÞ ¼ grWðsÞ Mu ðsÞ UðsÞ My ðsÞ YðsÞ Copyright # 2004 John Wiley & Sons, Ltd.
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where g ¼ k0 is an effective scalar controller gain, Mu ðsÞ ¼ GNy ðsÞ=CðsÞ is a strictly proper transfer function and My ðsÞ ¼ FNy ðsÞ=CðsÞ is a proper transfer function. In Case a the characteristic polynomial of the resulting closed-loop system takes the form PðsÞ ¼ P0 ðsÞ CðsÞ; with P0 ðsÞ ¼ AðsÞ þ LNy ðsÞ: Analogous formulae can be given for the ‘bar’ case ð%aÞ: 3.2. Analytical CGPC designs The CGPC a-design. With the output prediction order being settled at its minimal value Ny ¼ r þ Nu ; the polynomial P0 ðsÞ; being a factor of the resulting closed-loop characteristic polynomial takes the form P0 ðsÞ ¼ BðsÞ Kr;Nu ðsÞ; where Kr;Nu ðsÞ ¼ kTrþNu srþNu : The CGPC a% -design. When the output prediction order is established as Ny ¼ NA þ Nu ; the factor P0 ðsÞ becomes P0 ðsÞ ¼ K% NA ;Nu ðsÞ with K% NA ;Nu ðsÞ ¼ k% TNA þNu sNA þNu : By defining a normalized variable p ¼ Ts; the followingP set of ðN; Nu Þ-parameter prototype i *i *N closed-loop polynomials can be acquired: K* N;Nu ðpÞ ¼ N i¼0 kN;Nu p with kN;Nu ¼ 1: The coefficients of these monic polynomials K* N;Nu ðpÞ have been derived explicitly and their Hurwitz property has been examined in References [21, 25]. For given N ¼ r (Case a) and N ¼ NA (Case a% ) we can stabilize the closed-loop system taking the ‘coupled’ pair of the CGPC parameters ðNy ¼ N þ Nu ; Nu Þ what allows for a suitable control system design with a properly chosen r control prediction order Nu : Consequently, we have Kr;Nu ðsÞ ¼ h1 K* r;Nu ðpÞjp¼Ts for Case a; r T 1 N and K% NA ;Nu ðsÞ ¼ h%NA T A K* NA ;Nu ðpÞjp¼Ts for Case a% : Explicit formulas for the closed-loop polynomials and the prototype design characteristic polynomials are presented and catalogued by Kowalczuk and Suchomski [21, 25]. It is important that the choice of the observation horizon T does not affect the closed-loop stability. On the other hand, in a simplest approach it can be treated as an additional time-scaling factor (tuner). Precisely speaking, the parameter T can be established by considering common timedomain design specifications as overshoots, peak times and settling times. Also the shape of the control signal (and its maximal value) as well as the steady-state tracking error for the reference signal of unit velocity can be tuned by using T [21, 25]. It follows from the above that the a-design rule is restricted to minimum-phase models of the plant. The tracking transfer function Tyw ðsÞ ¼ YðsÞ=WðsÞ for the nominal (generic) CGPC ¼ ¼ system acquires the following forms: Tyw ðsÞ ¼ rk0 =Kr;Nu ðsÞ for Case a; and Tyw ðsÞ ¼ r%k%0 BðsÞ =K% NA ;Nu ðsÞ for Case a% : In order to assure the unity DC gain of Tyw ðsÞ; we should assume r% ¼ 1 for Case a and r% ¼ 1=b0 for Case a% : The prototype closed-loop transfer functions Tyw ðsÞ can easily be scanned [25] to yield their basic time-domain specifications for different pairs ðr; Nu Þ: It is thus clear that the two distinct cases of the nominal design (taking into account the phase characteristics of the plant, and differentiated by the ‘bar’ sign) of the CGPC controller result directly from the two goals of prediction (emulation) discussed in Section 2. For the following discussion it is important that the above introduced prototype characteristics will serve as a basis for a fully analytical design procedure assuring both the nominal stability and nominal performance specifications.
4. ROBUST CGPC DESIGN In this section the problem of achieving the RS and RP of the CGPC systems is considered. A controller provides RS if the internal stability of the control system is guaranteed for Copyright # 2004 John Wiley & Sons, Ltd.
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every model belonging to a set of all admissible models of the controlled plant. RP with respect of a certain characteristic means that this characteristic holds for every allowable plant model. 4.1. Nominal performance of CGPC closed-loop system NS and NP attributes of the CGPC control system can be established for both the analytical a- and a% -design methodologies. A block diagram of the closed-loop CGPC system is given in Figure 1, where DðsÞ denotes an equivalent output disturbance and NðsÞ is a measurement noise. The zero steady-state error property for positional references can be robustified by employing an integrator in the control loop, namely in the forward control path [18–23, 25]. Thus, in the sequel, a pair ðsAðsÞ; BðsÞÞ with explicitly indicated integral action will solely be considered, which means that the parameter N of the prototype polynomials should be taken as follows: N ¼ r ¼ NA NB þ 1 in Case a; and N ¼ NA þ 1 for Case a% : 4.2. Uncertainty modeling and robust stability Let the controlled channel of a ‘real’ plant be described by the following multiplicative model of unstructured uncertainty: ð1 þ DðsÞ Ws ðsÞÞ BðsÞ=ðs AðsÞÞ; where jjDðsÞjj1 51 for DðsÞ 2 RH1 ; and ws ðsÞ 2 RH1 is a weighting function depending on a particular type of model uncertainty [9, 40]. By using the small gain theorem [40] we conclude that the closed-loop system, characterized by the noise sensitivity function Tyn ðsÞ ¼ YðsÞ=NðsÞ; is robustly internally stable if and only if jjTyn ðsÞ Ws ðsÞjj1 41
ð2Þ
It follows from the above that the system’s robustness to model uncertainties may be obtained by shaping its noise sensitivity function Tyn ðsÞ: Since Tyd ðsÞ Tyn ðsÞ ¼ 1; where Tyd ðsÞ ¼ YðsÞ=DðsÞ denotes the output sensitivity function, this shaping usually means a trade-off between disturbance attenuation and robust stability. To facilitate the following discussion (see also Section 3.2) we represent a nominal form ¼ of the noise sensitivity function by Tyn ðsÞ; and a nominal output sensitivity function by ¼ Tyd ðsÞ:
D ( s) W ( s)
U ( s) gr
Plant
_
M u (s)
Y( s)
+
+ N ( s) + M y (s)
+ +
Figure 1. Generic CGPC control system. Copyright # 2004 John Wiley & Sons, Ltd.
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ROBUST CONTINUOUS-TIME CONTROLLER DESIGN
4.3. Q-enhanced CGPC controller: primary structural factorization Let us assume that the generic CGPC system is nominally stable. A key element of the proposed enhancement follows from the observation that the nominal tracking transfer function Tyw ðsÞ does not change when in place of the original pair ðMu ðsÞ; My ðsÞÞ the pair ðMu ðsÞ þ Xu ðsÞ; My ðsÞ þ Xy ðsÞÞ is used, where Xu ðsÞ ¼ BðsÞ QðsÞ and Xy ðsÞ ¼ s AðsÞ QðsÞ are correcting terms and QðsÞ 2 RH1 denotes a certain Q-parameter, i.e. the Youla–Ku$cera parameter being a stable transfer function for which My ðsÞ þ Xy ðsÞ remains proper. Such a Q-enhanced CGPC controller results in Tyn ðsÞ ¼
BðsÞ ðMy ðsÞ þ Xy ðsÞÞ ; P0 ðsÞ
Tyd ðsÞ ¼
sAðsÞ ð1 þ Mu ðsÞ þ Xu ðsÞÞ P0 ðsÞ
In the sequel, it will be shown that the simple choice QðsÞ ¼ My ðsÞ Q0 ðsÞ with a Q-corrector function Q0 ðsÞ 2 RH1 ; having the relative order greater than NA ; can be a good candidate for the Q-parameter. The robust QCGPC controller, resulting from such a Q-enhancement, can be implemented in the structure shown in Figure 2.
4.4. Secondary structural factorization: RS-specifications of the Q-parameterization As Tyn ðsÞ Ws ðsÞ is affine in QðsÞ; the problem described by (2) can be recognized as a modelmatching problem [43]. It is a common fact that the standard Q-parameterization methodology has a disadvantage of increasing the controller order [39]. Therefore some additional steps have to be undertaken in order to reduce the complexity of the resulting controller. It is worth noticing that with the parameterization of the Q-enhanced system proposed above the factor My ðsÞ already exists in the other of the robustifying branches of the applied parallel controller sub-structure. Thus the order of the controller is kept non-increased in this setting. What is more, a further ‘structural’ development of the Q-parameter shown below leads to a simple model of an effectual tuner. Q-enhanced CGPC design for stable plants (I). Let AðsÞ be a Hurwitz polynomial. In such a case we shall use Q0 ðsÞ ¼ Q00 ðsÞ=AðsÞ with a Q-subcorrector Q00 ðsÞ 2 RH1 of a relative order
D(s) Y(s) W(s)
U(s) gr
N(s)
+
+
Plant
_ B(s).Q0(s)
sA(s).Q0(s) _
M u(s))
+
M y (s)
+ +
Figure 2. Robust Q-enhanced CGPC control system. Copyright # 2004 John Wiley & Sons, Ltd.
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Z. KOWALCZUK AND P. SUCHOMSKI
greater than one. This assumption reflects in QðsÞ ¼
My ðsÞ Q00 ðsÞ AðsÞ
ð3Þ
¼ and Tyn ðsÞ ¼ Tyn ðsÞ ZðsÞ; where ZðsÞ ¼ 1 þ s Q00 ðsÞ is a free design factor, which directly models the noise sensitivity function. The simplest choice for the Q-subcorrector
k sþg
Q00 ðsÞ ¼
ð4Þ
with two design parameters k and g > 0; leads to ¼ Tyn ðsÞ ¼ Tyn ðsÞ Zgk ðsÞ and
Zgk ðsÞ ¼ Z
sþb sþg
ð5Þ
where Z ¼ k þ 1 and b ¼ g=ðk þ 1Þ: ¼ Let us denote Tw¼ ðsÞ ¼ Tyn ðsÞ Ws ðsÞ and assume that the generic CGPC system requires ¼ robustification, i.e. jjTw ðsÞjj1 ¼ Tm > 1; and jTw¼ ðjom Þj ¼ Tm for some finite pick-frequency om : From (5) it follows that the necessary condition for RS takes the form of inequality Z51: On the basis of the Bode plot of Figure 3, in order to assure RS, equivalent to jjTyn ðsÞ Ws ðsÞjj1 ¼ jjTw¼ ðsÞ Zgk ðsÞjj1 41; it is sufficient that the following ðg; kÞ-tuning rule is observed: g¼
Zo o m ; ZT T m
k¼
1 ZT Tm ZT T m
where for safety reasons ZT > 1 and 05Zo 51 (as b ¼ Zo om ). Note that, by virtue of (4) with Z ¼ 1=ðZT Tm Þ51; it is appropriate to use a negative gain ðk50Þ of the Q-subcorrector. Clearly, the most simple (though not necessarily optimal) is the decade-distance rule of ‘0.1’ resulting in Zo ¼ 0:1: To establish a proper value of Zo ; any direct searching procedure can be employed. As the promptitude of disturbance elimination improves while increasing g; there is no interest in excessively large ZT : Practically, the choice ZT ¼ 1:1–1.5 can be recommended. On the other hand, we have to bear in mind that increasing the ‘safety factor’ ZT
Tw= ( j ω )
Tm
1
γ
β
ω ωm γ
jω +β
β
jω +γ
η
Figure 3. Tuning the Q-subcorrector of the enhanced CGPC controller. Copyright # 2004 John Wiley & Sons, Ltd.
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ROBUST CONTINUOUS-TIME CONTROLLER DESIGN
D(s) Y(s) N(s) W(s)
U(s) gr
+
Plant
_
_
B(s) sA(s)
+
+
s Q00 (s)
M u(s) +
+ +
M y (s)
+
Figure 4. Robust IMC-QCGPC system for Hurwitz AðsÞ:
constitutes an effective instrument for improving the performance robustness of the control system, at least in the sense of the tracking properties. The Q-branch of the resulting controller can be implemented in the internal model control (IMC) structure shown in Figure 4. The assumed simple form of Zgk ðsÞ; as a first-order phase-lag transfer function, follows from ¼ the fact that only the RS condition is taken into account. As Tyn ðsÞ ¼ Tyn ðsÞ Zgk ðsÞ; in the case k of spectral requirements one can consider employing Zg ðsÞ of a more complicated construction. ¼ Shaping Tud ðsÞ ¼ Tud ðsÞ Zgk ðsÞ may require a further complication of the corrector. Certainly, all such design developments have to be checked against basic nominal-performance (NP) control-system requirements (like noise attenuation, disturbance attenuation, and control signals restrictions, for instance). In both cases (RS and NP), common frequency-domain techniques for ‘serial’ correction can be used. Q-enhanced C-design for stable plants (II). With the assumption that the observer polynomial CðsÞ is a free design parameter, the designed robust QCGPC system can be additionally simplified. Let CðsÞ ¼ AðsÞ; deg CðsÞ ¼ NA : Consequently, we have ðaÞ: FNy ðsÞ ¼ gAðsÞ;
GNy ðsÞ ¼ BðsÞENy ðsÞ AðsÞ
ð%aÞ: F% Ny ðsÞ ¼ gAðsÞ=b0 ;
G% Ny ðsÞ ¼ E% Ny ðsÞ AðsÞ
The above leads to: My ðsÞ ¼ my ¼ g in Case a; and My ðsÞ ¼ m% y ¼ g=b0 for Case a% : Consider Case a along with the Q-subcorrector (4) in the following form: Q00 ðsÞ ¼
k my ðs þ gÞ
ð6Þ
The input and output robustness-correcting terms are then Xu ðsÞ ¼
kBðsÞ sk and Xy ðsÞ ¼ ðs þ gÞ AðsÞ sþg
Thus the stability robustness condition based on the corrected noise-sensitivity function is jjTw¼ ðsÞ Zgk ðsÞjj1 41; where Zgk ðsÞ is defined as in (5) for Z ¼ g=b ¼ 1 þ k=my : Also the whole Copyright # 2004 John Wiley & Sons, Ltd.
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Z. KOWALCZUK AND P. SUCHOMSKI
Y(s) W(s) _
U(s) my
_
Plant
_
D(s)
N(s)
+
+
B(s) sA(s)
M u (s) κs s+ γ
_
+
Figure 5. Three-loop structure of IMC-QCGPC system for Hurwitz CðsÞ ¼ AðsÞ:
previously-given ðg; kÞ-tuning judgment holds: g ¼ ðZo om Þ=ðZT Tm Þ and k ¼ my ð1 ZT Tm Þ= ðZT Tm Þ: It can easily be shown that in Case a% only the quantity m% y should be used in place of my : As demonstrated in Figure 5, a ‘direct’ IMC implementation of the resulting QCGPC controller has three control loops: the positional loop, the robustness loop with the Q-corrector in the IMC (model-error) structure, and the dynamical input-feedback part of the generic CGPC controller. Q-enhanced CGPC design for unstable plants (I). If AðsÞ is not Hurwitz, QðsÞ cannot follow the prescription of (3). At the same time, there are no particular reasons for considering the two separate tracks (a and a% ) of the analytical CGPC design methodologies. In order to find a suitable QCGPC controller of a possibly simple structure, let AðsÞ be factored as AðsÞ ¼ A ðsÞ Aþ ðsÞ with A ðsÞ of deg A ðsÞ ¼ NA having all zeros in the open left half-plane and Aþ ðsÞ of degAþ ðsÞ ¼ NAþ with all zeros in the right-half plane (RHP). By assuming that QðsÞ ¼ My ðsÞ Q0 ðsÞ with the Q-corrector Q0 ðsÞ ¼ Q 0 ðsÞ=A ðsÞ composed of a free-design Q-subcorrector Q0 ðsÞ 2 RH1 of the relative order greater than NAþ ; we obtain the following Q-parameter: QðsÞ ¼
My ðsÞ Q 0 ðsÞ A ðsÞ
ð7Þ
and the noise sensitivity function of the corresponding QCGPC control system Tyn ðsÞ ¼ ¼ Tyn ðsÞ ZðsÞ with ZðsÞ ¼ 1 þ s Aþ ðsÞ Q 0 ðsÞ: It is clear that, this time, the factor ZðsÞ cannot be chosen arbitrarily as the achievable robustness is constrained by right half-plane poles of the plant model. By using the following simple form of the Q-subcorrector: Q 0 ðsÞ ¼
k þ
ðs þ gÞNA þ1
;
g>0
ð8Þ
¼ we get the sensitivity function Tyn ðsÞ ¼ Tyn ðsÞ Zgk ðsÞ with the noise-sensitivity shaping function, which again plays the role of an effective robustness corrector þ
Zgk ðsÞ ¼
Copyright # 2004 John Wiley & Sons, Ltd.
ðs þ gÞNA þ1 þ ksAþ ðsÞ þ
ðs þ gÞNA þ1 Optim. Control Appl. Meth. 2004; 25:235–262
ROBUST CONTINUOUS-TIME CONTROLLER DESIGN
249
¼ Let Tw¼ ðsÞ ¼ Tyn ðsÞ Ws ðsÞ be such that jjTw¼ ðsÞjj1 ¼ jTw¼ ðjom Þj ¼ Tm > 1: For the following discussion it is important that typical model uncertainty (and om ) is placed at a high frequency. It can easily be shown that the robustness shaping function Zgk ðsÞ has similar asymptotic properties to the previously given function (5), i.e. Zgk ð0Þ ¼ 1 and Z ¼ lims!1 Zgk ðsÞ ¼ 1 þ k: By applying the classical root-locus method to a virtual open-loop system described by YðsÞ ¼ þ sAþ ðsÞ=ðs þ gÞNA þ1 with a negative gain k; we easily conclude that for 8k 2 ð1; 0Þ the þ corresponding (virtual) characteristic polynomial ððs þ gÞNA þ1 þ ksAþ ðsÞÞ has always one (adequate) negative zero lower than g on the real axis of the (s-) root plane [44]. Moreover, if Aþ ðsÞ has an odd degree ðNAþ Þ then the above characteristic polynomial has, for 8k 2 ð1; 0Þ; a negative zero greater than g: This implies that the correcting function expressed on a Bode diagram has its multiple pole g above this zero, what results in jjZgk ðsÞjj1 > 1: It is also clear that ¼ with jTyn ð0Þj ¼ 1 the RS condition jjTw¼ ðsÞ Zgk ðsÞjj1 41 can be satisfied only for Ws ðsÞ of a highfrequency nature. On the other hand, in the case of an even degree of Aþ ðsÞ; it may as well happen (depending on unstable poles of the plant embodied in Aþ ðsÞ and, especially, for greater absolute values of k) that the robustness conditioning (in terms of jjZgk ðsÞjj1 41) can improve. What is more, simple judgment proves that for a sufficiently low g; such a small k (tending to 1þ ; in order to minimize Z) can be chosen that it does not move the ‘principal’ real zero b behind om on the frequency axis of the Bode plot. In the case of going with b above om ; we have a chance of obtaining other zeros greater than g and smaller than om (which come from the loci directed to the unstable poles of the plant model) that will assure enough attenuation. Thus it is possible to state the following (conditional) RS tuning rule:
g ¼ Zgo om ;
k¼
1 ZT Tm ZT T m
ð9Þ
with the similar-as-before safety coefficient ZT and a suitably chosen 05ngo 51: Note that the effort we have made for enlarging the parameter g results from a vital interest in improving the closed-loop system ability to eliminate disturbances. Q-enhanced C-design for unstable plants (II). By considering CðsÞ as a free design parameter, we simplify the Q-enhanced CGPC controller by assuming CðsÞ ¼ A ðsÞ C ðsÞ; where C ðsÞ of deg C ðsÞ ¼ NAþ is an arbitrary Hurwitz polynomial with roots placed sufficiently far (to the left) from the origin. From D1 it follows that FNy ðsÞ ¼ FNy ðsÞA ðsÞ; where a factor FNy ðsÞ has a reduced degree NAþ ; what results in a suitably reduced-in-order output observer My ðsÞ ¼ FNy ðsÞ=C ðsÞ in Case a: Analogous formulae can be readily shown for the ‘bar’ case ð%a Þ: 4.5. RP-specifications of the structural Q-parameterization Recall that the analytically designed (ACGPC) systems, obtained by the ða=%aÞ methodology, are characterized by appropriate NS and NP properties. Assume that the modulus margin D¼ m ¼ ¼ 1=jjTyd ðsÞjj1 can perform as a sufficient tool to express a simple ‘synthetic’ frequency domain characteristic [9, 45]. By defining Wp ¼ D¼ m =ZD as a primitive frequency-independent weighting function with ZD > 1; standing for a factor allowing for an ‘admissible deterioration’ of the designed system with respect to the nominal one, we can express the NP requirements in terms of the ‘weighted’ output sensitivity function jjTyd ðsÞ Wp jj1 41 Copyright # 2004 John Wiley & Sons, Ltd.
ð10Þ Optim. Control Appl. Meth. 2004; 25:235–262
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Z. KOWALCZUK AND P. SUCHOMSKI
With the above settings, it is clear that 05Wp 51 always. Putting together the conditions given in (2) and (10) leads to the following combined ‘two-disk’ condition for RP [43]: max ðjTyn ð joÞ Ws ð joÞj þ jTyd ð joÞ Wp jÞ41 o
ð11Þ
This condition can be rewritten in a conservative form max ½jTyn ð joÞj ðjWs ð joÞj þ Wp Þ41 Wp o
ð12Þ
which allows us to compute the following worst-point robustness indices: ¼ Tmr ¼ maxo ½jTyn ð joÞj ðjWs ð joÞj þ Wp Þ ¼ orm ¼ arg maxo ½jTyn ð joÞj ðjWs ð joÞj þ Wp Þ
ð13Þ
Let us confine our considerations to the case Tmr > 1 Wp ; when the additional tuning of the generic CGPC system discussed above is obligatory from the stability and performance robustness viewpoints. In general, the RP requirements can be represented in terms of the weighted output sensitivity function Tyd ðsÞ: jjTyd ðsÞ Wp ðsÞjj1 41; where Wp ðsÞ 2 RH1 stands for a frequency dependent weighting function [45] selected by the designer. Nominal performance specifications should thus be captured by an upper bound, 1=jWp ðsÞj; on the magnitude of Tyd ðsÞ: A simple rule for tuning Wp ðsÞ can be derived by considering observations given by Suchomski [3]. Q-enhanced CGPC design for stable plants (I) and (II). The Q-corrector (I) has the form given by (3) and (4) with g¼
Zo orm ð1 Wp Þ ; ZT Tmr
k¼
1 ZT Tmr Wp ZT Tmr
In Case a we may also consider the C-design (II) with CðsÞ ¼ AðsÞ; which results accordingly in the Q-subcorrector of the form (6) with g¼
Zo orm ð1 Wp Þ ; ZT Tmr
k¼
my ð1 ZT Tmr Wp Þ ZT Tmr
while m% y should be used in place of my if we apply the ACGPC design of Case a% : Q-enhanced CGPC design for unstable plants (I) and (II). Taking into considerations the Qsubcorrector (8) and rewriting (9) immediately leads to the following correcting parameters: g ¼ Zgo orm ;
k¼
1 ZT Tmr Wp ZT Tmr
and the partial compensation used within the Q-enhanced C-design for unstable plants (II) of Section 4.4 can also be put into practice. Copyright # 2004 John Wiley & Sons, Ltd.
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ROBUST CONTINUOUS-TIME CONTROLLER DESIGN
251
5. ILLUSTRATIVE EXAMPLES 5.1. Robust QCGPC for a plant with uncertain gain and delay Consider the Q-enhanced CGPC (QCGPC) design ensuring RS and RP for an unstable nominal model of the controlled part of the plant kp esT0 =½sðs þ 1:2Þðs 0:2Þ with an uncertain gain kp 2 ½kp min ; kp max and an uncertain transportation delay T0 2 ½0; T0 max : Let kp min ¼ 3; kp max ¼ 5 and T0 max ¼ 0:3 s: Such a system can be represented by a delay-free nominal model k#p BðsÞ ¼ sAðsÞ sðs þ 1:2Þðs 0:2Þ
ð14Þ
with the mean gain k#p ¼ ðkp min þ kp max Þ=2 ¼ 4 and multiplicative uncertainty characterized by the rational weight Ws ðsÞ 2 RH1 derived in Reference [45]: Ws ðsÞ ¼
rk þ sT0 max ð1 þ rk =2Þ 1 þ s 2 0:838T0 max þ s2 ðT0 max =2:363Þ2 1 þ sT0 max =2 1 þ s 2 0:685T0 max þ s2 ðT0 max =2:363Þ2
where rk ¼ ðkmax kmin Þ=ðkmax þ kmin Þ ¼ 0:25 denotes a relative uncertainty in the gain. Thus AðsÞ ¼ 0:24 þ s þ s2 ; A ðsÞ ¼ s þ 1:2; Aþ ðsÞ ¼ s 0:2; BðsÞ ¼ 4; NA ¼ 2; NA ¼ 1; NAþ ¼ 1 and NB ¼ 0: ACGPC design. Let us additionally assume that CðsÞ ¼ A ðsÞ C ðsÞ ¼ ðs þ 1:2Þ ðs þ 6Þ ¼ 7:2 þ 7:2s þ s2 : By accepting the following specifications concerning the step response of the nominal model: overshoot k0 40:1; settling time Ts2% 41:5 s and uð0Þ450; we obtain the subsequent exemplary settings for the a-design of ACGPC [21, 25]: Nu ¼ 1; Ny ¼ 4; K* r;Nu ðpÞ ¼ K* 3;1 ðpÞ ¼ 67:2 þ 33:6p þ 8p2 þ p3 ; T ¼ 0:75 s and g ¼ 39:822: This yields GNy ðsÞ ¼ 119:9067 þ9:6667s and FNy ðsÞ ¼ 286:72 þ 119:9067s þ 135:7776s2 : The reduced degree numerator of the output observer takes the form of FNy ðsÞ ¼ f0 þ f1 s ¼ 238:9333 þ 135:7776 s: By examining ¼ ¼ Tw¼ ðsÞ ¼ Tyn ðsÞ Ws ðsÞ with Tyn ðsÞ ¼ FNy ðsÞ=ðC ðsÞKr;Nu ðsÞÞ we learn that Tm ¼ 2:8595 and om ¼ 5:237 rad=s: The generic CGPC system is thus not robustly stable. Q-enhanced C-design. The robustness property can be achieved by employing the following Q-parameter: QðsÞ ¼
FNy ðsÞ
k CðsÞ ðs þ gÞ2
ð15Þ
Letting ZT ¼ 1:2 results in k ¼ 0:7086 and Z ¼ 0:2914: The parameter Zgo is chosen such that the RS condition jjTw¼ ðsÞ Zgk ðsÞjj1 41 is satisfied with a possibly large g > 0: Taking into account that now Zgk ðsÞ ¼
ðs þ gÞ2 þ ksAþ ðsÞ ðs þ gÞ2
we accept g ¼ 0:3938 corresponding to Zgo ¼ 0:075: The matching exploration for a suitable RS term of QCGPC is illustrated in Figure 6. The resulting control system is characterized by the following indices: gain margin Dg ¼ 5:24 dB; phase margin Dp ¼ 82:858; modulus margin Dm ¼ 3:0 dB; delay margin Dd ¼ 0:878 s; gain crossover frequency ocg ¼ 1:65 rad=s and phase crossover frequency ocp ¼ 8:355 rad=s: Copyright # 2004 John Wiley & Sons, Ltd.
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0.3
Tyn( j ω) .Ws ( j ω )
100 0.15
ACGPC
0.075
ηγ =
QCGPC
ω
10-2
10-3 10-1
100
101
102
ω [rad/s]
Figure 6. Bode plots of the RS term Tyn ðsÞ Ws ðsÞ:
c0 = 100 = ( jω ) . W ( j ω ) Tyn s
0.5 0.02
0.1
10-2
10-3 10-1
100
101
102
ω [rad/s] ¼ Figure 7. Bode plots of the RS term Tyn ðsÞ Ws ðsÞ: ineffective C-technique.
Exclusive C-technique. In order to illustrate some general limitations in robustness that are produced by plant instability, let us examine the possibility of robustification by using solely the C-design technique (i.e. QðsÞ ¼ 0). Let CðsÞ ¼ A ðsÞ C ðsÞ; where the factor C ðsÞ ¼ c0 þ s is ¼ equipped with a free parameter c0 > 0: Bode plots of the nominal RS terms Tyn ðsÞ Ws ðsÞ obtained for a widely distributed c0 > 0 are given in Figure 7. Clearly, such a one-degree-offreedom C-technique enhancement of the CGPC design practically does not supply any relevant robustification. Some improvement can be achieved by assuming that the whole polynomial CðsÞ is subject to tuning. Note that this usually excludes the possibility of simplification in My ðsÞ: Moreover, each Copyright # 2004 John Wiley & Sons, Ltd.
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ROBUST CONTINUOUS-TIME CONTROLLER DESIGN
0.5
c0 =
= ( ω) . Tyn Ws ( j ω) j
100 0.3 0.1
10-2
10-4 10-1
100
101
102
ω [rad/s] ¼ Figure 8. Bode plots of the RS term Tyn ðsÞ Ws ðsÞ: successful C-technique.
time CðsÞ is changed the polynomials FNy ðsÞ and GNy ðsÞ have to be redesigned. Letting CðsÞ ¼ ðc0 þ sÞ2 with a free coefficient c0 > 0 provides a simple opportunity to system robusti¼ fication. Bode plots of the nominal RS term Tyn ðsÞ Ws ðsÞ for c0 ¼ 0:1; 0:3 and 0.5 are given in Figure 8. In this case the C-technique happens to be partially successful. Namely, the choice c0 ¼ 0:1 satisfies the RS condition. Nevertheless, by comparing CðsÞ with the denominator of the previously used parameter QðsÞ of (15), which has much smaller time constants, we conclude that the system reaction to the output disturbances is disadvantageously degraded. The plots of the nominal plant outputs given in Figure 9, where the QCGPC system reaction is given for comparative purposes, do confirm this claim. RP design. Let Dm 7 dB (or equivalently jjTyd ðsÞjj1 ¼ 2:25) be an acceptable value of the modulus margin of the designed closed-loop control system. This value is slightly smaller (ZD 1:03) as compared to the modulus margin of the generic ACGPC system (since for ¼ r QðsÞ ¼ 0 we have D¼ m ¼ 6:72 dB and jjTyd ðsÞjj1 ¼ 2:167). Thus Wp ¼ 0:45 and then Tm ¼ 3:658 r and om ¼ 5:057 rad=s: A suitable searching procedure results in the Q-parameter (15) with g ¼ 0:2023 and k ¼ 0:8997; which, as shown in Figure 10, guarantees that the RP condition holds. The resulting control system is described by the following margins and crossover frequencies: Dg ¼ 7:997 dB; Dp ¼ 70:08; Dm ¼ 1:697 dB; Dd ¼ 1:89 s; ocg ¼ 0:65 rad=s; and ocp ¼ 7:097 rad=s: Step responses of 50 QCGPC control systems with plants of randomly distributed gain and T0 ¼ T0max (the worst case) are illustrated in Figures 11 (while taking into account only the RS condition) and 12 (when the ‘full’ RP condition is satisfied). 5.2. Robust QCGPC for a plant with uncertain zero Let now 100ð1 þ sTz Þ=½sðs þ 10Þ2 describe the controlled part of a real (and potentially nonminimum-phase) plant, where Tz }determining an uncertain zero of the plant}is bounded by Copyright # 2004 John Wiley & Sons, Ltd.
Optim. Control Appl. Meth. 2004; 25:235–262
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Z. KOWALCZUK AND P. SUCHOMSKI
0.8
y(t)
0.4
QCGPC 0 ACGPC (c0 = 0.1) -0.4
0
5
10
15
20
25
t [s]
Figure 9. CGPC system reactions to step output disturbance.
Tyn( jω ) . Ws ( j ω ) + Tyd ( j ω) . Wp
101
ACGPC
100
QCGPC
10-1 10-2
10-1
100 ω [rad/s]
101
102
Figure 10. Plots of the RP term jTyn ð joÞ Ws ð joÞj þ jTyd ð joÞ Wp j:
Tz min 4Tz 4Tz max with Tz min ¼ 0:15 s and Tz max ¼ 0:3 s: This leads to the following nominal model of the plant: BðsÞ 100ð1 þ T# z sÞ ¼ ; sAðsÞ sð10 þ sÞ2 Copyright # 2004 John Wiley & Sons, Ltd.
Tz min þ Tz max ¼ 0:075 s T# z ¼ 2
ð16Þ
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y(t)
2
1
0
0
5
10 t [s]
15
20
Figure 11. Step responses of 50 QCGPC systems satisfying solely the RS condition.
and the weighting function of the multiplicative characteristic of model uncertainty Ws ðsÞ ¼ rz
T# z s ; 1 þ T# z s
rz ¼
Tz max Tz min ¼3 Tz max þ Tz min
where T# z represents the mean value of Tz ; and rz is a relative uncertainty in this parameter. The nominal model is then represented by AðsÞ ¼ 100 þ 20s þ s2 and BðsÞ ¼ 100 þ 75s; with NA ¼ 2: ACGPC design. Let the nominal performance, defined for the closed-loop step response, be described by: ko ffi 0:05; Ts2% 40:2 s and uð0Þ4100: As the nominal model is minimum phase, Case a can be applied with the simplest C-design choice CðsÞ ¼ AðsÞ: The polynomial P0 ðsÞ; being a factor of the closed-loop characteristic polynomial PðsÞ ¼ CðsÞ P0 ðsÞ; has the form P0 ðsÞ ¼ BðsÞ KðsÞ with KðsÞ ¼ K* r;Nu ðpÞjp¼sT ; where K* r;Nu ðpÞ is a two-parameter ðr; Nu Þ prototype (closed-loop) polynomial in a normalized complex variable p. The first parameter r equals to the relative order of the controlled channel of the nominal plant (r ¼ 2), and the second parameter stands for the control prediction order. The simplest choice Nu ¼ 1 is acceptable [21, 25]. This implies that K* r;Nu ðpÞ ¼ K* 2;1 ðpÞ ¼ k*02;1 þ k*12;1 p þ p2 ¼ 15 þ 6p þ p2 : As uð0Þ ¼ my ¼ g ¼ k*02;1 =ðbNB T 2 Þ; the time scaling factor T is constrained by T5½k*02;1 =ðbNB uð0ÞÞ1=2 ¼ 0:1414 s: Letting T ¼ 0:1414 s yields the nominal settling time Ts2% ¼ T T* s2% ¼ 0:196 s50:2 s: The resulting input observer takes the form Mu ðsÞ ¼ ð465:685 þ 35:760sÞ=AðsÞ and the output observer is static: My ðsÞ ¼ g ¼ 100: The following indices: Dp ¼ 68:778; Dm ¼ 1:833 dB; Dd ¼ 0:073 s and ocg ¼ 16:48 rad=s; mark the system properties. As Tm ¼ 2:069 appears at frequency om ¼ 17:423 rad=s; the generic CGPC system is not robustly stable. Copyright # 2004 John Wiley & Sons, Ltd.
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y (t)
2
1
0 0
5
10 t [s]
15
20
Figure 12. Step responses of 50 QCGPC systems satisfying the RP condition.
Q-enhanced C-design. RS can be achieved by employing the Q-subcorrector (6) with g ¼ 3:5085; k ¼ 59:7259; and the design parameters Zo ¼ 0:5; ZT ¼ 1:2; and Z ¼ 0:4027: The solution provides: Dp ¼ 97:918; Dm ¼ 0:841 dB; Dd ¼ 0:348 s: At the same time, however, the open-loop gain crossover frequency ocg ¼ 4:907 rad=s has been decreased. RP design. Assume that Dm ¼ 6 dB stands for a typical ‘hard’ acceptable value of the modulus margin of the designed robust closed-loop control system. Hence the NP weight is ¼ Wp ¼ Dm ¼ 0:5: By examining the plot of jTyn ðjoÞj ½jWs ðjoÞj þ Wp one discovers that Tmr ¼ r 2:507 > 0:5; which occurs at om ¼ 16:57 rad=s: Let us assume that Zo ¼ 0:5 and ZT ¼ 1:2 what results in g ¼ 1:3768 and k ¼ 83:3824: Figure 13 shows the corresponding plot of jTyn ð joÞ Ws ð joÞj þ jTyd ð joÞ Wp j; from which we conclude that, contrary to the generic ACGPC system, the QCGPC system satisfies the RP condition of (11). In particular, it holds: Dp ¼ 94:868; Dm ¼ 0:358 dB; Dd ¼ 1:08 s; and ocg ¼ 1:534 rad=s: Step responses of 50 QCGPC systems satisfying the full RP condition and designed for the plant with the zero 1=Tz ; having its time constant randomly distributed within Tz min 4Tz 4 Tz max ; are given in Figure 14. Three characteristic QCGPC system’s responses to the step output disturbance are given in Figure 15. For comparative purposes, the QCGPC systems satisfying only the RS condition have also been considered, and their corresponding responses are shown in Figures 16 and 17. It is clear that robustness always relates to nominal (stability or performance) characteristics of the control system. Consequently, all systems are accepted until they fulfill all assumed criteria. Specifically, the oscillations observed in Figures 11, 12 and 14 result from the (typical) Copyright # 2004 John Wiley & Sons, Ltd.
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Tyn( j ω) . Ws ( j ω) + Tyd ( j ω) . Wp
101
ACGPC 100
QCGPC 10-1
10-2 10-1
101
100
102
ω [rad/s]
Figure 13. Structural singular value jTyn ð joÞ Ws ð joÞj þ jTyd ð joÞ Wp j:
3
y (t)
2
1
0
-1
0
1
2
3
t [s]
Figure 14. Step responses of 50 QCGPC systems satisfying the RP condition.
design objectives applied and, in particular, from the necessity of balancing between the speed of reaction and the overshoot of the control system. Note that in the considered ACGPC design approach, for instance, via decreasing the design parameter T one accelerates the closed-loop system’s step response and, at the same time, undesirably increases the system overshoot/ Copyright # 2004 John Wiley & Sons, Ltd.
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2 Tz = Tz min ^ Tz 1 y (t)
Tz max
0
-1
0
1
2
3
t [s]
Figure 15. Step output disturbance attenuation in QCGPC systems satisfying the RP condition.
3
y (t)
2
1
0
-1
0
1
2
3
t [s]
Figure 16. Step responses of 50 QCGPC systems satisfying solely the RS condition. Copyright # 2004 John Wiley & Sons, Ltd.
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2
Tz = ^ Tz
Tz min
y (t)
1
T z max
0
-1
0
1
2
3
t [s]
Figure 17. Step output disturbance attenuation in QCGPC systems satisfying only the RS condition.
undershoot. The validating part of the above design examples in the form of the presented characteristics shows that the design objectives are retained even in the case of quite strong model discrepancies. Clearly, the oscillations illustrated in Figures 16 and 17 are obvious consequences of caring about the robust stability and forgetting about robustifying the system performance.
6. CONCLUSIONS In this paper the issue of robust design of closed-loop continuous-time observer-based controlsystem properties in the presence of unstructured model uncertainties has been considered. Additionally, this work has been motivated by a general acceptance gained by long-range model-based predictive control in industry and a recently observed interest in the continuoustime approach to design of digital control systems. Certainly, the pertinent task of robust synthesis of linear controllers can be solved without making use of the predictive approach: One may consider, for instance, any of the wellestablished methods of H2 - and H1 -techniques and their extensions to the m-design [40, 43, 45]. The alternative approach used in this paper consists of two design steps. Firstly, the problem of nominal performance and nominal stability is solved with the application of any design methodology of a general use (we follow here the track of the CGPC design). Next, the issue of robust performance RP and robust stability RS is resolved based on the Youla–Ku$cera (Q) parameterization technique. As has been shown in this paper, the result of the Q-approach is equivalent to the solution of the standard problem of model matching [43]. It is thus clear that the deliberated Q-enhanced design (based on the multiplicative model of unstructured uncertainty) can easily be extended to manage also other types of model uncertainties. (In the Copyright # 2004 John Wiley & Sons, Ltd.
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case of additive uncertainty we shape the input sensitivity function, while the feedback uncertainty requires a suitable arrangement of the output sensitivity function.) Such a ‘direct’ approach to the issue of robustification of the control systems is burden with the basic disadvantage (which concerns most ‘direct’ applications of the affine Youla parameterization of robust controllers) that lies in essentially enlarging the order of controllers [39]. Having this in mind, we took several steps to reduce the complexity of the resulting controller in this paper. Our structural Youla–Ku$cera proposition basically relies on the observer structure of a controller, which is enhanced with the Q elements responsible for control robustification. Yet, with the system complexity in view, we have focused our attention on searching for possibly simple forms of robustifying correctors, which should be of a possibly low order, easy for implementation and parameterization (with a small number of parameters to be tuned), and convenient for optimization (in the considered case any scalar direct searching procedure can be employed; e.g. a low-cost bisection method over a closed interval). In particular, the resulting simplicity of our approach originates both in the Youla–Ku$cera parameterization concept and in our proposition of its structural factorization (starting from the primary one with QðsÞ ¼ My ðsÞ Q0 ðsÞ). Effectively, the proposed design steps consist in a specific tuning of a corrector, which enhance the controller structure, and are based on implementing an elementary mechanism of serial phase-lag correction in the frequency domain (as explained in Figure 3). It is worth emphasizing that our corrector is purposely simply and appears as one element of the observer control structure applied. Several important features, possible catches and consequences are evidenced by means of few examples. It can thus be considered that, in effect, we have simply solved the issue of the robust stability and performance of continuous-time systems, governed by any controller implemented in the observer-like structure. Our analytical approach to CGPC design, applied here, can be treated as a representative of any nominal design methodologies. It results in a desired set of nominal system attributes, such as: stability margins, step-response parameters (static accuracy, overshoot, settling time), and acceptable control signals for nominal excitations. Within this context we can also take a hard-constrains viewpoint, because some practical limitations can easily be included in the proposed design (with respect to the analytically-designed controlledplant output and the initial control-signal action). Other constrained predictive control algorithms equipped with time-varying Q-elements can be found in References [46, 47], for instance. The proposed solution for the Q-enhanced CGPC generally confirms the conviction expressed by Skogestad and Postlethwaite [45] that RP is not a ‘big issue’ for SISO systems. This also approves the fact that the RP problem can easily be solved by ‘de-tuning’ the Q-parameter, derived with respect to the system’s RS, and enlightens the supposition that by safely deepening the RS characteristics (via increasing the RS-safety coefficient ZT ; for instance) we improve the system’s performance robustness as well. Another practical hint is that the performance-safety coefficient ZD can also be interpreted as a measure of an allowable degradation of the system modulus margin that may result from the assumed plant-model uncertainties. It is also worth noticing that within our nominal analytical ACGPC control design approach the choice of the observation horizon T does not exert any influence on the closed-loop stability. Instead, this parameter solely plays the role of a user-defined time-scaling factor (though it allows extra balancing between the speed of reaction and the overshoot of the control system). This simply distinguishes our ACGPC design of guaranteed stability and the discrete-time Copyright # 2004 John Wiley & Sons, Ltd.
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predictive control approach of Scokaert and Rawlings [10], where the use of the infinite horizon is essential for ensuring the closed-loop stability. Other related detailed problems of model nonminimality, design-solution existence, and numerical issues related to the CGPC design have been discussed by Kowalczuk and Suchomski in [21]. Though we have submitted only the CGPC control problem and simple rational controllers to the presented sensitivity analysis, the developed methodology for the Q-enhancement of observer-based control algorithms can be easily modified so as to robustify other more involved control systems. It can be, for instance, the Smith-predictor paradigm applied for plants with an arbitrary large nominal delay [19, 20]. The only question is whether a robust solution for the uncertain delay exists. In the case of a wider range of parameter variations we would rather recommend using an adaptive approach, within which one can also exercise the continuous-time approach for system identification [48–50].
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