Chapter 2 Nominal and robust stability In order to be able to design a robust compensator to control a given pro cess, it is necessary not only to specify a nominal mo del of the process, but also the mo del uncert ainty to whid the control sy stem has to be robust. The compensator is required to make the output follow variations in the reference signal and to attenuate disturbances. Hence, to design the com- ensator, it is also necessary to know the anticipated character of the reference signal and of the disturb ances. Given these, performance specifications can be expressed as requirements of a cert ain level of error reduction in the presence of these input Design of a robust compensator is based on A model of the process a description of the model uncert ainty Knowledge of the character of the inputs(for reference and disturb ances Performance specifications A compensator is said to be robust, if the performance specifications are satisfied both for the nominal model of the process as for any other model cont ained in the admissible set of models as specified by the mo del uncertainty. systems are formulated. For multi variable(MIMO)systems, see Chapter 5 on page A o. In the following sections, a framework for specification of the model, of the mo del uncertain of the character of the inputs and of performance specifications for single variable (S 2.1 A Model of the proces The concepts and met hods used in the sequel are based on linear time invariant models 2.2 Model Uncertainty Model uncert ainty is often specified in the frequency domain. This can lead to a region for each frequency w in the Nyquist plane in which the real'model of the process GaGw) is
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known to be cont ained in. The shape of this region g(w) is determined by the way in whid the uncertainty is specified. Thus, specification of uncert aunties in amplitude and phase will lead to a sector bound Figure 2. la. Several of the results and met ho ds described in the following are b ased on norm bounded mo del deviations. For SIso systems this means that the uncert ainty at each frequency is bounded by a circle of radius la(w)in the Nyquist plane, see Figure 2. 1b G(w) radius la(w) auncertaintyin amplitude b. norm bounded uncertainty and phas Figure 2.1: Uncertainty regions by spec fication of(a) amplitude and phase uncertainty (6) norm bounded uncertainty Based on this, a family of models can be defined c={G△:|G△()-G(j)≤Ca(u)} (21) {G△:|G△(元)-G(ju)≤Cm(u)G(j) (22) where G(u denote the nominal model, and Ga(u) denote possilbe models of the process ea(w) is the maximal additive model uncert ainty, and em(w) is the maximal multiplicative (relative)uncertainty. An arbitrary member of g can be described as G△(ju)=G(ju)(1+△m(u)=G(u)+△a(ju) where the actual multiplicative or additive model deviation(Am or Aa) is bounded by 1△m(ju)≤nm(u) (24) △a(ju)≤a(a) (25) A multiplicative model uncert ainty description often increase with increasing frequences the models applied for compensator design are derived with emphasis on the description of the dominating dynamics If em(w)> 1, the norm bounded model deviation allow the mo dels to have a different number of zeros in the right half plane. A zero on the jw axis is faciliated e.g. by Am gw)=-1
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This uncert ainty description, however, is not very suited for describing model deviations, where the number of poles in the right half plane might vary, since this means that em(w) must assume the value oo Note, that la and lm are frequency dependent scalars and, hence, that they are functions of w(rather than ju ) Often, however, ea()and em(w) will be represented by normal trans functions, where mainly the amplitude will be of significance 2.3 Nominal Stability A control sy stem is internally st able, if an excit ation by a bounded signal anywhere in the system can not stimulate an unbounded signal somewhere. In Figure 2.2 a controlled process is shown with three inputs(o, u, d) and three outputs(e, u, y to the overall system r)()()x()as G(s)m(s) u s Figure 2.2: Controlled s ystem with inp uts to the analysis of internal stability Analyzing internal st ability does not necessarily imply testing a 3 x 3 matrix, since several of the signals contain the same information from a st ability point of view. The signals d and r, for inst ance, have the same influence on the output u with respect to stability. Choosing r and u as inputs and y and u as output s, the transfer matrix below can be determined y(s r(s (2.6) From(2.6)it can be seen, that the controlled system is internally st able, only if none of the four elements in the matrix have poles in the right half plane. In a similar fashion, if both G(s and K(s)are stable, it suffices to analyze the characteristic equation 1+GK(S=0 Note that if G (s) has an unstable pole, it does not suffice to apply a compensator which exactly cancels the unstable pole in order to achieve internal st ability. This can be seen fromg(2.6), since a bounded input ul imply an unbounded output 2. 4 Robust stability. In the sequel, conditions for the controlled sy stem to be robustly stable will be studied. This means that the system is stable for all mo dels of the process cont ained in g. It is assumed that all mo dels in g has the same number of poles n in the right half plane Under these conditions, the controlled sy stem is stable, only if the Nyquist curve for GKgw) encomp asses the Ny quist point (-1, 0) exactly n times counter-clockwise
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Imag 1+G△R 1+GKGw) igure 2.3: Distance to the Nyquist point (1, 0)for the nominal system G(s) and for an arbitrary model in g If K(s stabilizes the nominal system G(s), the condition for K(s) to st abilize all mo dels in g, is that the number of encirclements of (-1, 0) does not change. This is equivalent to the region in the Ny quist plane, covered by all possible Gak(w)not to include(1,0), or that the distance from G(u)to(1, 0)is larger than Gk)e(w) 1+GK(u)>|GK(j)(a), This expression can be rewritten as Gou) 1+GK(w) (u)<1 (28) ju)(a)<1 29 Here T (s)is the closed loop transfer function from reference to output, which is also called the complementary sensitivity function. Hence, if the model uncert ainty is specified by bounding the norm of the mo del deviation, an analytical experession is obtained for robust stability, which offer the possibility to apply the condition directly in design methods that, e. g, involve mI he model uncert ainty is given in other ways, for le by specifying the uncert aunties for amplitude and phase, it is more obvious to apply design methods, where hical il that the uncertainty region for the open loop transfer function does not contain the Nyquist point(1, 0)
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