Finally, the functions sloop and sift form basic feedback interconnections The function sloop computes the closed-loop mapping between r and y in the loop of Figure 2-3. The result is a state-space realization of the transfer G,Ga)c here e=±1 G1(s) G2(s) Figure 2-3: sloop This function is useful to specify simple feedback loops. For instance, the closed-loop system clsys corresponding to the two-degree-of-freedom tracking K G is derived by setting G,(s)=G(s)K(s)and Go(s)=1: clsys smult(C, sloop(smult(k, g), 1)) The function sift forms the more general feedback interconnection of figure 2-4 and returns the closed-loop mapping from/o (aa)to (2). To form this interconnection when u e R2 andy e R, the command is sIft(P1
2 Uncertain Dynamical Systems 2-10 Finally, the functions sloop and slft form basic feedback interconnections. The function sloop computes the closed-loop mapping between r and y in the loop of Figure 2-3. The result is a state-space realization of the transfer function (I – εG1G2) –1G1 where ε = ±1. Figure 2-3: sloop This function is useful to specify simple feedback loops. For instance, the closed-loop system clsys corresponding to the two-degree-of-freedom tracking loop is derived by setting G1(s) = G(s)K(s) and G2(s) = 1: clsys = smult(c, sloop(smult(k,g),1)) The function slft forms the more general feedback interconnection of Figure 2-4 and returns the closed-loop mapping from to . To form this interconnection when u ∈ R2 and y ∈ R3, the command is slft(P1,P2,2,3) r + y −ε G1(s) G2(s) r + y − C K G ω1 ω 2 z1 z 2
Interconnections of Linear Systems The last two arguments dimension u and y. This function is useful to compute linear-fractional interconnections such as those arising in H. theory and its extensions(see"How to Derive Such Models"on page 2-23) P2(s) Figure 2-4: sift 2-11
Interconnections of Linear Systems 2-11 The last two arguments dimension u and y. This function is useful to compute linear-fractional interconnections such as those arising in H∞ theory and its extensions (see “How to Derive Such Models” on page 2-23). Figure 2-4: slft w1 z1 u y w2 z2 P1(s) P2(s)
2 Uncertain D Model Uncertainty For control design purposes, the possibly complex behavior of dynamical ory The notion of uncertain dynamical system is central to robust control the systems must be approximated by models of relatively low complexity. The gap between such models and the true physical system is called the model uncertainty. another cause of uncertainty is the imperfect knowledge of some components of the system, or the alteration of their behavior due to changes in perating conditions, aging, etc. Finally, uncertainty also stems from physical parameters whose value is only approximately known or varies in time. Note that model uncertainty should be distinguished from exogenous actions such as disturbances or measurement noise The LMI Control Toolbox focuses on the class of dynamical systems that can be approximated by linear models up to some possibly nonlinear and/or time-varying model uncertainty. When deriving the nominal model and estimating the uncertainty, two fundamental principles must be remembered Uncertainty should be small where high performance is desired (tradeoff between performance and robustness). In other words, the linear model should be sufficiently accurate in the control bandwidth The more information you have about the uncertainty (phase, structure, time invariance, etc. ) the higher the achievable performance will There are two major classes of uncertainty Dynamical uncertainty, which consists of dynamical components neglected in the linear model as well as of variations in the dynamical behavior during operation. For instance, high-frequency flexible modes, nonlinearities for large inputs, slow time variations, etc Parameter uncertainty, which stems from imperfect know ledge of the physical parameter values, or from variations of these parameters during operation. Examples of physical parameters include stiffness and damping coefficients in mechanical systems, aerodynamical coefficients in flying devices, capacitors and inductors in electric circuits, ete Other important characteristics of uncertainty include whether it is linear or nonlinear, and whether it is time invariant or time varying Model uncertainty is generally a combination of dynamical and parametric uncertainty, and may arise at several different points in the control loop For instance, there may be 2-12
2 Uncertain Dynamical Systems 2-12 Model Uncertainty The notion of uncertain dynamical system is central to robust control theory. For control design purposes, the possibly complex behavior of dynamical systems must be approximated by models of relatively low complexity. The gap between such models and the true physical system is called the model uncertainty. Another cause of uncertainty is the imperfect knowledge of some components of the system, or the alteration of their behavior due to changes in operating conditions, aging, etc. Finally, uncertainty also stems from physical parameters whose value is only approximately known or varies in time. Note that model uncertainty should be distinguished from exogenous actions such as disturbances or measurement noise. The LMI Control Toolbox focuses on the class of dynamical systems that can be approximated by linear models up to some possibly nonlinear and/or time-varying model uncertainty. When deriving the nominal model and estimating the uncertainty, two fundamental principles must be remembered: • Uncertainty should be small where high performance is desired (tradeoff between performance and robustness). In other words, the linear model should be sufficiently accurate in the control bandwidth. • The more information you have about the uncertainty (phase, structure, time invariance, etc.), the higher the achievable performance will be. There are two major classes of uncertainty: • Dynamical uncertainty, which consists of dynamical components neglected in the linear model as well as of variations in the dynamical behavior during operation. For instance, high-frequency flexible modes, nonlinearities for large inputs, slow time variations, etc. • Parameter uncertainty, which stems from imperfect knowledge of the physical parameter values, or from variations of these parameters during operation. Examples of physical parameters include stiffness and damping coefficients in mechanical systems, aerodynamical coefficients in flying devices, capacitors and inductors in electric circuits, etc. Other important characteristics of uncertainty include whether it is linear or nonlinear, and whether it is time invariant or time varying. Model uncertainty is generally a combination of dynamical and parametric uncertainty, and may arise at several different points in the control loop. For instance, there may be
Model Uncertainty dynamical uncertainty on the system actuators, and parametric uncertainty on some sensor coefficients. Two representations of model uncertainty are used in the lmi control toolbox Uncertain state-space models. This representation is relevant for systems described by dynamical equations with uncertain andor time-varying Linear-fractional representation of uncertainty. Here the uncertain system is described as an interconnection of known Lti systems with uncertain omponents called"uncertainty blocks. Each uncertainty block A, ( represents a family of systems of which only a few characteristics are known For instance, the only available information about A may be that it is a time-invariant nonlinearity with gain less than 0.01 Determinant factors in the choice of representation include the available model (state-space equations, frequency-domain model, etc )and the analysis or synthesis tool to be used
Model Uncertainty 2-13 dynamical uncertainty on the system actuators, and parametric uncertainty on some sensor coefficients. Two representations of model uncertainty are used in the LMI Control Toolbox: • Uncertain state-space models. This representation is relevant for systems described by dynamical equations with uncertain and/or time-varying coefficients. • Linear-fractional representation of uncertainty. Here the uncertain system is described as an interconnection of known LTI systems with uncertain components called “uncertainty blocks.” Each uncertainty block ∆i(.) represents a family of systems of which only a few characteristics are known. For instance, the only available information about ∆i may be that it is a time-invariant nonlinearity with gain less than 0.01. Determinant factors in the choice of representation include the available model (state-space equations, frequency-domain model, etc.) and the analysis or synthesis tool to be used
2 Uncertain Dynamical Systems Uncertain State-Space Models Physical models of a system often lead to a state-space description of its dynamical behavior. The resulting state-space equations typically involve physical parameters whose value is only approximately known, as well approximations of complex and possibly nonlinear phenomena. In other words, the system is described by an uncertain state-space model Ex =Ax+ Bu, y= Cx Du where the state-space matrices A, B, C, D, E depend on uncertain and/or time-varying parameters or vary in some bounded sets of the space of matrices Of particular relevance to this toolbox are the classes of polytopic or parameter-dependent models discussed next. We collectively refer to such models as P-systems,“P-” standing for“ polytopic or parameter-dependent.” P-systems are specified with the function psys and manipulated like ordinary LTI systems except for a few specific restrictions olyt。 pic Models We call polytopic system a linear time-varying system E(t)x=A(t)x+B(t) whose SYSTEM matrix S(t)=A()+JE(t)B(t C(t) Du varies within a fixed polytope where S1,..., Sk are given vertex systems In other words, S(t)is a convex combination of the SYSTEM matrices The nonnegative numbers al,..., ak are called the polytopic coordinates ofs 2-14
2 Uncertain Dynamical Systems 2-14 Uncertain State-Space Models Physical models of a system often lead to a state-space description of its dynamical behavior. The resulting state-space equations typically involve physical parameters whose value is only approximately known, as well as approximations of complex and possibly nonlinear phenomena. In other words, the system is described by an uncertain state-space model E = Ax + Bu, y = Cx + Du where the state-space matrices A, B, C, D, E depend on uncertain and/or time-varying parameters or vary in some bounded sets of the space of matrices. Of particular relevance to this toolbox are the classes of polytopic or parameter-dependent models discussed next. We collectively refer to such models as P-systems, “P–” standing for “polytopic or parameter-dependent.” P-systems are specified with the function psys and manipulated like ordinary LTI systems except for a few specific restrictions. Polytopic Models We call polytopic system a linear time-varying system E(t) = A(t) x + B(t) u y = C(t) x + D(t) u whose SYSTEM matrix varies within a fixed polytope of matrices, i.e., where S1, . . . , Sk are given vertex systems: (2-1) In other words, S(t) is a convex combination of the SYSTEM matrices S1, . . . , Sk. The nonnegative numbers α1, . . . , αk are called the polytopic coordinates of S. x · x · S t( ) A t( ) + jE t( ) B t( ) C t( ) D t( ) = S t( ) Co S1 … Sk ∈ { } , , := αi Si i = 1 k ∑ : αi 0 αi i = 1 k ≥ , ∑ = 1 S1 A1 jE1 + B1 C1 D1 ,…Sk Ak jEk + Bk Ck Dk = =