Chapter 3 Nominal and robust performance This chapter presents approaches to formulate performance specifications for a control sy stem. d(8) rs e(s) K(8) s) G(s) Figure 3. 1: Standard feedback configuration Figure 3. 1 shows a st andard sy stem with feedback control. The controlled sy stem has an input reference r and a dist urb ance signal d. Since the two input s have the same transfer function to the error signal e(except for the sign difference), they are treated collectively, denoted by the signal u A compensator is traditionally designed for a specific input. This is true for classical design methods, where the design often aims at achieving cert ain characteristics for the closed loop resp onse to a step or ramp input as mentioned in Chapter 1. Likewise, linear quadratic control aims at minimizing the error for a given input signal In practice, it is often more relevant to design the compensator for a class of related inputs with the same characteristics. The exposition below aims at assessing the input error for ferent compensators exp osed to exogenous signals of the same 'size interpreted in a norr 3.1 Signal norms To measure the 'size' of a time domain signal, a norm will be applied. Predominantly, the 2 norm will be applied, which is defined by:
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3 buo hT Other ignals norm are possible, however. For a map dCs e.g. 3. 1 to be a norm of a time signal islit must possess following four properties dar pb 2. dia,t bu rs l de ddos trP 4. adc a It is left as an exercise to the reader to check that the 2 norm as well as the norms listed below actually possess these properties. The I norm dmd配 The oo norm E The l norm of a signal might represent a consumption of some ressource. The oo norm might be of relevance when checking boundedness of a signal, for instance for a system with physical limitations The 2 norm which will be used mostly in these notes, can be interpreted as the energy of a For each of the norms above we can define the linear sp ace of signals bulwhidh has a bounded value for either the 1 norm, the 2 norm or the oo norm. These function spaces are called Lr L and CT(Lebesgue spaces). In these notes, these spaces will not be treated further. The reader is refered to ZDG96 or [TC96 for a more thorough introduction In addition to the norms listed above, it is relevant to introduce a quantity, represent the 2 o hT laudi CCis not a norm does not possess property 2. It is usefull, however, because it power. If, howeve re)value of ru In the same way as in time domain, norms can be defined in frequency domain. In these notes,lower case letters will be used for time domain signals as well as for frequency domain the relevant will appear from the context. In frequency domain, the most relevant norms are the 2 norm and the oo norm
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The 2 norm w(w)ldw following import ant relationship(Parseval's The U(t)2 (3.6) where i is the fourier transform of u, This means that the 2 norm of a time domain signal equals the 2 norm of the associated frequen cy domain signal l gu) For the oo norm there is no equivalent relationship to Parseval's Theorem 3.6) 3.2 Norms for systems and Transfer functions Since signals at different places in a process are related by the dynamics of the sy stem, it is relevant to define norms related to the sy stem it self. Since a sy stem is charaterized by its impulse response g(t)or its transfer function G(s)=L(g(t)), the 2 norm for a system is defined by G(w) (3.8) 1g(t)12dt !|2 10) where Parseval's Theorem was applied for the second equal sign. For transfer functions, the Ho norm is defined in a similar fashion sup G(w) 3.3 Specification of inputs Some knowledge on the potential input s to a system is important in order to be able to specify The approach below involves describing a class of inputs, which are bounded by a norm. The next step is to specify performance by bounding the allowable norm of the output signal often the control error e or a filtered version of the error. For these specification the 2 norm or possibly the RMs value will be used The use of othe apensator specifications can be very relevant and is the subject of research presently. For example, the use of the 1-norm in time domain can be highly relevant, and is currently the subject of intensive research. For example, the use of the 1 norm in time domain can be relevant, if the objective is to reduce consumption of some ressource
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a class of signals can be defined as those signals v having a 2 norm bounded by 1 ju)*y(ju)du≤1 (3.12) By filtering u, a new class of signals v can be est ablished, characteristic as input to the process either as dist urb an ces or as reference signals V={u(s)=W()b(s):‖l1l2≤1} (3.13) By designing the control system such that it minimizes the possible error given a certain class of normbounded input signals, all potential input s are treated equally. This can be an advant age, as it is rarely known in the design phase for cert ain, which reference changes or disturber dition Hence, the reference r and the disturb an ce d can be d erized by prepending input filters to our st andard feedback loop, see Figure 3.2 d'(s Wa(s Wr(s) K(s G(s)Im( Figure 3.2: Control loop with unit norm bounded signals as inputs (a =le sup(In=12 =ecifica2ion) This example has been taken from/MZ8g. Assume that the reference is known only to change 7 step, and that only small disturbances are present such that d(s)0 and u(s=r(s). Then an inp ut weight W(s)must be chosen such that u(s)=s and u(s)EV. An obvious choice to be w(s=1/s and to introduce v(s)as an imp ulse (U()=1). Howev v(s)=l does not belong to the set V' since the integral in(3. 12) become infinite(an impulse does note have finite 2 norm). Hence, the weight W(s)=1/s does not provide the desired haracteristics for the reference r(s. Instead, the following weight can be used B>0 (3.14) With this filter, a step input is contained in V. For instance, assume that v(s) is given by 15) 8+a
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low frequences is consistent with anticipated low frequent reference changes or disturbances whereas the control system is not expected to be able to attenuate disturbances above a certain frequency The configuration shown in Figure 3. 2 can be used also if the controller should be designed or a specific input. v can also be chosen as an impulse, v(s)=l v(s) then becomes a signal with the same Laplace transform as W(s) or a sinusoidal input v, the 2 norm does no longer apply, since the integral in(3. 1)becomes infinite. In this case it is more appropriate to use the RMs value If a filter W(s is applied in this case, the frequen cy response for wGw) specifies the amplitudes of v that are anticipated in (or sp ecified for) the control sy stem at each frequen cy 3.4 Requirements for Performance The ultimate requirement to the compensator is, that it works well for the real system. Thi requirements can be sub divided into the following four categories 1. Nominal st ability: the compensator must ensure internal stability in the controlled system, provided the model is correct 2. Nominal performance: the compensat or must minimize the error e.(E. g. expressed as the 2 norm of the error for given input s 3. Robust st ability: for all mo dels in g(see(2. 2) the compensator must ensure internal stability 4. Robust performance: for all models in g the compensator must ensure that the error is within a specified bound The requirements(1) and(3) has been described above. The performance specifications will be further described below The main objective of the compensator is to minimize the error e occuring in the face of reference r and / or a dist urb ance d. The most important transfer functions, see Figure 3. 1 are e(s e(s y(s) (3.19) r(8 ()a)-1+(s)k(s=5() y(s G(sK(s) (S)K(s) T(s (3.20) The quantity s(s)is called the sensitivity function. Usually, it is desired to make S(s)small, due to the wish to minimize the error e. For physical systems G(s)K(s is proper(the order of the ator is not larger than the order of the denominator). In practice, it will even be strictly proper(the order of the numerator is smaller than the order of the denominator). This corresponds to the fact that usually very high frequencies will not pass through the sy stem G(s)K(s)=0 (3.21) Thi (3.22)
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In order to find the optimal compensator K, for a given K it must first be established for which input the largest value of the 2 norm is achieved sup‖el S(w)w(u )y(w)l-dwl (3.31) 00 It can be shown, that this integral assumes its largest value if v is a sinusoid, modified in such a way that the 2 norm exists. This could for example be a signal equal to a const ant times a sinusoid for a long time interval and zero otherwise. The const ant must be chosen such that the signal has unit norm. The angular frequency of the sinusoid has to be the value for which S(u)w(w)l assumes its maximal value. Then sup‖ell=sup|s(ju)W(j)=‖sW‖ (3.32) Hence. the value of the error assumed for any v E v simply equals the Hoo norm of ompensator is the solution to the following minimization problem P2=SWHx=是s|s()W( (3.33) In other words, the Hoo optimal control minimizes the Hoo norm of the sensitivity function s weighted by w In comp arison, the H2 optimal compensator minimizes the mean value over all frequencies of the square of S(ju)w(u )12, whereas the Hoo optimal compensator minimizes the maximal S(u)w(w) By scaling wow) it is possible to formulate the Hoo nominal performance condition in the form Sgu)w gw)< 1 (3.35) The advant ages by applying the Hoo norm for specification of the performance requirements above the先2 norm are: The designer can bound the peak value of S(u) directly by chosing the input weight w Gw) appropriately Also using the input weight www), the designer can specify the desired bandwidth of the sensitivity function S(w), defined as the value of w where S(a) st ays beyond (3dB) Moreover, the Hoo norm facilit ates a tool for sp ecifying robust performance as we shall see t section 3. 7 Robust performance If a compensator is designed only based on requirement s for nominal performance and robust st ability, g might cont ain a model which is close to instability for the closed loop sy stem This is likely to give a very poor performance. To ensure the compensator to work well for all models in g, robust performance should is required for the models in g
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Hereby, conditions have been formulated both for robust performance and for robust st ability for Siso sy stems An example of such a specification is the control of a water supply pump with a maximal performance of w lv at 2( xf, The requirements for the specific sy stem in mind was originally formulated in time domain as conditions on thethe error of the pressure by a step in water consumptions at u(6>l t maksimal transient error: 0.4 bar t settling time for error of 0.1 bar.2 sec t maksimal st ationary error: 0.1 bar The water consumption infuen ces the pressure in a way which resembles a first order linear transfer function. Thus, the disturb ance can be mo deled as T TO (343) where TOe 1(wxf vs and T e u(y), for the sensitivity function, the corre sponding response to f disturbance can now be computed. A first order sensitivity bound kp with a high frequency kain slightly ab ove 1 was chosen Knurls (3.44) .2 Figure3.4:Tm≤dm△ny≈≤ te disturbanc≤ And carr<spanding er quency rx≈机≤ s<ns itivity sp<ci sAtian kpis
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