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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