Uncertain State-Space Models Such models are also called polytopic linear differential inclusions in the literature 3] and arise in many practical situations, including Multimodel representation of a system, each model being derived around Nonlinear systems of the form x=A(x)x +b(x)u, y=C(x)x D(x)u State-space models depending affinely on time-varying parameters(see From Affine to Polytopic Models"on page 2-20) A simple example is the system whose state matrix A sin x ranges in the polytope A (E Co{-1,1}=[1,1] Polytopic systems are specified by the list of their vertex systems, i.e., by the SYSTEM matrices S1,..., Sk in(2-1). For instance, a polytopic model taking values in the convex envelope of the three LTI systems s1, s2, s3 is declared y polys psys(s1 s2 S3]) Affine Parameter-Dependent models The equations of physics often involve uncertain or time-varying coefficients When the system is linear, this naturally gives rise to parameter-dependent models(PDS)of the form e(p)x=A(p)x+ b(p) where A(),..., E( are known functions of some parameter vector p=(pl,., Pn). Such models commonly arise from the equations of motion, aerodynamics, circuits, ete The LMI Control Toolbox offers various tools to analyze the stability and formance of parameter-dependent systems with an affine dependence che parameter vector p=(pl,., Pn). That is, PDSs where nAn, b(p=Bo+PiB
Uncertain State-Space Models 2-15 Such models are also called polytopic linear differential inclusions in the literature [3] and arise in many practical situations, including: • Multimodel representation of a system, each model being derived around particular operating conditions • Nonlinear systems of the form = A(x) x + B(x) u, y = C(x) x + D(x) u • State-space models depending affinely on time-varying parameters (see “From Affine to Polytopic Models” on page 2-20) A simple example is the system = (sin x) x whose state matrix A = sin x ranges in the polytope A Œ Co{–1, 1} = [–1, 1]. Polytopic systems are specified by the list of their vertex systems, i.e., by the SYSTEM matrices S1, . . . , Sk in (2-1). For instance, a polytopic model taking values in the convex envelope of the three LTI systems s1, s2, s3 is declared by polsys = psys([s1 s2 s3]) Affine Parameter-Dependent Models The equations of physics often involve uncertain or time-varying coefficients. When the system is linear, this naturally gives rise to parameter-dependent models (PDS) of the form E(p) = A(p) x + B(p) u y = C(p) x + D(p) u where A(.), . . . , E(.) are known functions of some parameter vector p = (p1, . . . , pn). Such models commonly arise from the equations of motion, aerodynamics, circuits, etc. The LMI Control Toolbox offers various tools to analyze the stability and performance of parameter-dependent systems with an affine dependence on the parameter vector p = (p1, . . . , pn). That is, PDSs where A(p) = A0 + p1A1 + . . .+ pnAn, B(p) = B0 + p1B1 + . . .+ pnBn, x · x · x ·
Uncertain Dynamical Syster and so on Affine parameter-dependent models are well-suited for lyapunov- based analysis and synthesis and are easily converted to linear-fractional uncertainty models for small-gain-based design(see the nonlinear spring example in"Sector-Bounded Uncertainty"on page 2-30) With the notation S(p)=A(p)JE(P)B(p) A; +jEi b C(p) D(p) D the affine dependence on p is written more compactly in SYSTEM matrix terms P,s, The system"coefficients"So,., S, fully characterize the dependence on the uncertain parameters PI,..., Pn. Note that So,..., Sn need not represent meaningful dynamical systems. Only their combination S(p)is a relevant description of the problem Affine parameter-dependent systems are specified with psys by providing a description of the parameter vector p in terms of bounds on the parameter values and rates of variation The list of SYSTEM matrix coefficients So,..., S, For instance, the system P,S+p2s is defined by so= ltisys(a0, b0, co, do, e0) s1 ltisys(a1, b1, c1, d1, e1) s2= ltisys(a2, b2, c2, d2, e2) affsys= psys(pv, [so s1 s21) where pv is the parameter description returned by pvec( see next subsection for details). The output affsys is a structured matrix storing all relevant data. 2-16
2 Uncertain Dynamical Systems 2-16 and so on. Affine parameter-dependent models are well-suited for Lyapunovbased analysis and synthesis and are easily converted to linear-fractional uncertainty models for small-gain-based design (see the nonlinear spring example in “Sector-Bounded Uncertainty” on page 2-30). With the notation the affine dependence on p is written more compactly in SYSTEM matrix terms as S(p) = S0 + p1S1 + . . . + pnSn. The system “coefficients” S0, . . . , Sn fully characterize the dependence on the uncertain parameters p1, . . . , pn. Note that S0, . . . , Sn need not represent meaningful dynamical systems. Only their combination S(p) is a relevant description of the problem. Affine parameter-dependent systems are specified with psys by providing • A description of the parameter vector p in terms of bounds on the parameter values and rates of variation • The list of SYSTEM matrix coefficients S0, . . . , Sn For instance, the system S(p) = S0 + p1S1 + p2S2 is defined by s0 = ltisys(a0,b0,c0,d0,e0) s1 = ltisys(a1,b1,c1,d1,e1) s2 = ltisys(a2,b2,c2,d2,e2) affsys = psys(pv, [s0 s1 s2]) where pv is the parameter description returned by pvec (see next subsection for details). The output affsys is a structured matrix storing all relevant data. S p( ) A p( ) + jE p( ) B p( ) C p( ) D p( ) Si Ai jEi + Bi Ci Di = , =
Uncertain State-Space Models Important: By default, ltisys sets the e matrix to the identity. omitting the arguments e0, e1, e2 altogether results in setting E(p)=I+(p1+ p2L. To specify an affine PdS with a parameter-independent E matrix(e.g E(p)=D, you must explicitly set El=E2=0 by typing s1 ltisys(al, b1, c1, d1, 0) 2=1 tisys(a2,b2,c2,d2,0) Quantification of parameter Uncertaint Parameter uncertainty is quantified by the range of parameter values and possibly the rates of parameter variation. This is done with the function pvec The characteristics of parameter vectors defined with pvec are retrieved with The parameter uncertainty range can be described as a box in the parameter space. This corresponds to cases where each uncertain or time-varying parameter p; ranges between two empirically determined extremal values p and pi P:∈[2F (2-1) If p=(pl,., pn) is the vector of all uncertain parameters, (2-2)delimits a hyperrectangle of the parameter space R" called the parameter box. Consider the example of an electrical circuit with uncertain resistor p and capacitor c ranging in p∈600,10001c∈[1,5] The corresponding parameter vector p=(p, c)takes values in the box drawn Figure 2-5. This uncertainty range is specified by the commands range=[6001000,15] p= pvec(box, range The i-th row of the matrix range lists the lower and upper bounds on p 2-17
Uncertain State-Space Models 2-17 Important: By default, ltisys sets the E matrix to the identity. Omitting the arguments e0, e1, e2 altogether results in setting E(p) = I + (p1 + p2)I. To specify an affine PDS with a parameter-independent E matrix (e.g., E(p) = I), you must explicitly set E1 = E2 = 0 by typing s0 = ltisys(a0,b0,c0,d0) s1 = ltisys(a1,b1,c1,d1,0) s2 = ltisys(a2,b2,c2,d2,0) Quantification of Parameter Uncertainty Parameter uncertainty is quantified by the range of parameter values and possibly the rates of parameter variation. This is done with the function pvec. The characteristics of parameter vectors defined with pvec are retrieved with pvinfo. The parameter uncertainty range can be described as a box in the parameter space. This corresponds to cases where each uncertain or time-varying parameter pi ranges between two empirically determined extremal values and : (2-1) If p = (p1, . . . , pn) is the vector of all uncertain parameters, (2-2) delimits a hyperrectangle of the parameter space Rn called the parameter box. Consider the example of an electrical circuit with uncertain resistor ρ and capacitor c ranging in ρ ∈ [600, 1000] c ∈ [1, 5] The corresponding parameter vector p = (ρ, c) takes values in the box drawn in Figure 2-5. This uncertainty range is specified by the commands: range = [600 1000,1 5] p = pvec('box',range) The i-th row of the matrix range lists the lower and upper bounds on pi. pi pi pi pi pi ∈ [ ] ,
Similarly, bounds on the rate of variation p; of p (t)are specified by adding a third argument rate to the calling list of pvec. For instance the constraints 01≤p(t)≤1,|e(t)≤0.001 are incorporated by p= pvec(box,range, rate) All parameters are assumed to be time-invariant when rate is omitted. Slowly varying parameters can be specified in this manner. In general, robustness gainst fast parameter variations is more stringent than robustness against constant but uncertain parameters 5 range of parameter 1000 Figure 2-5: Parameter box Iternatively, uncertain parameter vectors can be specified as ranging in a polytopic of the parameter space r" like the one drawn in Figure 2-6 for n= 2 This polytope is characterized by the three vertices I1=(1,4),∏2=(3,8),∏3=(10,1) An uncertain parameter vector p with this range of values is defined by pi1=[1,4],pi2=[3,8],pi3=[10,1] p= pvec(pol,[pi1, pi2, pi3])
2 Uncertain Dynamical Systems 2-18 Similarly, bounds on the rate of variation of pi(t) are specified by adding a third argument rate to the calling list of pvec. For instance, the constraints are incorporated by rate = [0.1 1, 0.001 0.001] p = pvec('box',range,rate) All parameters are assumed to be time-invariant when rate is omitted. Slowly varying parameters can be specified in this manner. In general, robustness against fast parameter variations is more stringent than robustness against constant but uncertain parameters. Figure 2-5: Parameter box Alternatively, uncertain parameter vectors can be specified as ranging in a polytopic of the parameter space Rn like the one drawn in Figure 2-6 for n = 2. This polytope is characterized by the three vertices: Π1 = (1, 4), Π2 = (3, 8), Π3 = (10, 1). An uncertain parameter vector p with this range of values is defined by pi1=[1,4], pi2=[3,8], pi3=[10,1] p = pvec('pol',[pi1,pi2,pi3]) p· i 0.1 ρ · ( )t 1 c · ≤ ≤ , ( )t ≤ 0.001 c p 600 1000 1 5 range of parameter values
Uncertain State-Space Models The string 'pol indicates that the parameter range is defined as a polytope p2 range of values of p I11 10 igure 2-6: Polytopic parameter range Simulation of Parameter-Dependent Systems The function pdsimul simulates the time response of affine parameter dependent systems along given parameter trajectories. The parameter vector p(t)is required to range in a box(type box' of pvec). The parameter trajectory is defined by a function p= fun(t) returning the value of p at time t For simul(pds, ' traj') plots the step response of a single-input parameter-dependent system pds long the parameter trajectory defined in traj. m To obtain other time responses, specify an input signal as in pdsimul(pds, traj, 1, sin) This command plots the response to a sine wave between t=0 and t= l s. Multi-input multi-output (MIMO) systems are simulated similarly by specifying an appropriate input function
Uncertain State-Space Models 2-19 The string 'pol' indicates that the parameter range is defined as a polytope. Figure 2-6: Polytopic parameter range Simulation of Parameter-Dependent Systems The function pdsimul simulates the time response of affine parameterdependent systems along given parameter trajectories. The parameter vector p(t) is required to range in a box (type 'box' of pvec). The parameter trajectory is defined by a function p = fun(t) returning the value of p at time t. For instance, pdsimul(pds,'traj') plots the step response of a single-input parameter-dependent system pds along the parameter trajectory defined in traj.m. To obtain other time responses, specify an input signal as in pdsimul(pds,'traj',1,'sin') This command plots the response to a sine wave between t = 0 and t = 1 s. Multi-input multi-output (MIMO) systems are simulated similarly by specifying an appropriate input function. p2 p1 3 10 1 8 4 1 Π2 Π3 Π1 range of values of p