[26] Willems, J.C., "Least-Squares Stationary Optimal Control and the Algebraic Riccati Equation, IEEE Trans. Aut Contr., AC-16(1971), pp 621-634 [27] Young, P. M, M. P. Newlin, and J C. Doyle, "Lets Get Real, "in Robust Control Theory, Springer Verlag, 1994, pp. 143-174. [28] Zames, G, "On the Input-Output Stability of Time-Varying Nonlinear Feedback Systems, Part I and Il, IEEE Trans. Aut Contr., AC-1l(1966), pp 228-238and465-476. 1-12
1 Introduction 1-12 [26] Willems, J.C., “Least-Squares Stationary Optimal Control and the Algebraic Riccati Equation,” IEEE Trans. Aut. Contr., AC–16 (1971), pp. 621-634. [27] Young, P. M., M. P. Newlin, and J. C. Doyle, “Let's Get Real,” in Robust Control Theory, Springer Verlag, 1994, pp. 143-174. [28] Zames, G., “On the Input-Output Stability of Time-Varying Nonlinear Feedback Systems, Part I and II,” IEEE Trans. Aut. Contr., AC–11 (1966), pp. 228-238 and 465-476
Uncertain Dynamical Systems Linear Time-Invariant Systems SYSTEM Matrix Time and Frequency Response Plots 2-6 Interconnections of Linear Systems Model Uncertainty 2-12 Uncertain State-Space Models 2-14 Models Affine Parameter-Dependent models 15 Quantification of Parameter Uncertainty 2-17 Simulation of Parameter-Dependent Systems 19 From Affine to Polytopic Models 2-20 Example Linear-Fractional Models of Uncertainty 2-23 How to derive such models Specification of the Uncertainty From affine to linear- Fractional models References
2 Uncertain Dynamical Systems Linear Time-Invariant Systems . . . . . . . . . . . 2-3 SYSTEM Matrix . . . . . . . . . . . . . . . . . . . 2-3 Time and Frequency Response Plots . . . . . . . . . 2-6 Interconnections of Linear Systems . . . . . . . . . 2-9 Model Uncertainty . . . . . . . . . . . . . . . . .2-12 Uncertain State-Space Models . . . . . . . . . . . .2-14 Polytopic Models . . . . . . . . . . . . . . . . . . .2-14 Affine Parameter-Dependent Models . . . . . . . . . . .2-15 Quantification of Parameter Uncertainty . . . . . . . . .2-17 Simulation of Parameter-Dependent Systems . . . . . . .2-19 From Affine to Polytopic Models . . . . . . . . . . . . .2-20 Example . . . . . . . . . . . . . . . . . . . . . .2-21 Linear-Fractional Models of Uncertainty . . . . . . .2-23 How to Derive Such Models . . . . . . . . . . . . . . .2-23 Specification of the Uncertainty . . . . . . . . . . . . .2-26 From Affine to Linear-Fractional Models . . . . . . . . .2-32 References . . . . . . . . . . . . . . . . . . . . .2-35
2 Uncertain Dynamical Syster The LMI Control Toolbox offers a variety of tools to facilitate the description and manipulation of uncertain dynamical systems. These include functions to: Manipulate the state-space realization of linear time-invariant (Lti) systems as a single SYSTEM matrix Form interconnections of linear systems Specify linear systems with uncertain state-space matrices(polytopic differential inclusions, systems with uncertain or time-varying physical parameters, etc. Describe linear-fractional models of uncertainty The next sections provide a tutorial introduction to uncertain dynamical systems and an overview of these facilities 2-2
2 Uncertain Dynamical Systems 2-2 The LMI Control Toolbox offers a variety of tools to facilitate the description and manipulation of uncertain dynamical systems. These include functions to: • Manipulate the state-space realization of linear time-invariant (LTI) systems as a single SYSTEM matrix • Form interconnections of linear systems • Specify linear systems with uncertain state-space matrices (polytopic differential inclusions, systems with uncertain or time-varying physical parameters, etc.) • Describe linear-fractional models of uncertainty The next sections provide a tutorial introduction to uncertain dynamical systems and an overview of these facilities
Linear Time-Invariant Systems Linear Time-Invariant Systems The LMI Control Toolbox provides streamlined tools to manipulate state-space representations of linear time-invariant (LTI) systems. These tools handle general LTI models of the form E=Ax(t)+Bu(t) y(t)= Cx(t)+Du(t) where A.B. C.d. E are real matrices and e is invertible. as well as their discrete-time counterpart Exk+1=Axk+ Bu Cx, +Du Recall that the vectors x(t), u(t), y(t) denote the state, input, and output trajectories. Similarly, xk, uk, yk denote the values of the state, input, and output vectors at the sample time k The“ descriptor” formulation(E≠D) proves useful when specifying parameter-dependent systems(see "Affine Parameter-Dependent Models" on page 2-15)and also avoids inverting E when this inversion is poorly conditioned. Moreover, many dynamical systems are naturally written in descriptor form. For instance, the second-order system admits the state-space representation 01 1,0)xX 0 where(t):x(t) x(t) SYSTEM Matrix For convenience, the state-space realization of LTI systems is stored as a single MATLAB matrix called a SYSTEM matrix. Specifically, a continuous-or
Linear Time-Invariant Systems 2-3 Linear Time-Invariant Systems The LMI Control Toolbox provides streamlined tools to manipulate state-space representations of linear time-invariant (LTI) systems. These tools handle general LTI models of the form where A, B, C, D, E are real matrices and E is invertible, as well as their discrete-time counterpart Recall that the vectors x(t), u(t), y(t) denote the state, input, and output trajectories. Similarly, xk, uk, yk denote the values of the state, input, and output vectors at the sample time k. The “descriptor” formulation ( ) proves useful when specifying parameter-dependent systems (see “Affine Parameter-Dependent Models” on page 2-15) and also avoids inverting E when this inversion is poorly conditioned. Moreover, many dynamical systems are naturally written in descriptor form. For instance, the second-order system . admits the state-space representation where . SYSTEM Matrix For convenience, the state-space realization of LTI systems is stored as a single MATLAB matrix called a SYSTEM matrix. Specifically, a continuous- or Edx dt ------ = Ax t( ) + Bu t( ) y t( ) = Cx t( ) + Du t( ) Exk + 1 Axk Buk = + yk Cxk Duk = + E I ≠ mx·· fx · + + kx = u yx , = 1 0 0 m dX dt ------- 0 1 –k –f X 0 1 = + u y , = ( ) 1 0, X X t( ) := x t( ) x · ( )t
discrete-time LTI system with state-space matrices A, B, C, D, Eis represented A+j(E-I)B C 0 w-1. The upper right entry n corresponds to the number of states (i.e AE RX)while the entry Inf is used to differentiate SYSTEM matrices from regular matrices. This data structure is similar to that used in the u The functions ltisys and ltiss create SYSTEM matrices and extract state- space data from them. For instance specifies the Lti system x To retrieve the values of A, B, C, D from the SYSTEM matrix sys, type [a, b, C, d]= liss (sys) For single-input/single-output (SISO) systems, the function ltitf returns the numerator/denominator representation of the transfer function G(s)=D+C(SE-A-B=n(s) Conversely, the command ltisys( tf, n, d) returns a state-space realization of the SiSo transfer function n(s)d(s)in SYSTEM format. Here n and d are the vector representation of n(s)and d(s)(type help poly for details) The number of states, inputs, and outputs of a linear system are retrieved from the system matrix with sinfo
2 Uncertain Dynamical Systems 2-4 discrete-time LTI system with state-space matrices A, B, C, D, E is represented by the structured matrix where . The upper right entry n corresponds to the number of states (i.e., A ∈ Rn×n) while the entry Inf is used to differentiate SYSTEM matrices from regular matrices. This data structure is similar to that used in the µ Analysis and Synthesis Toolbox. The functions ltisys and ltiss create SYSTEM matrices and extract statespace data from them. For instance, sys = ltisys( 1,1,1,0) specifies the LTI system To retrieve the values of A, B, C, D from the SYSTEM matrix sys, type [a,b,c,d] = ltiss(sys) For single-input/single-output (SISO) systems, the function ltitf returns the numerator/denominator representation of the transfer function Conversely, the command sys = ltisys('tf',n,d) returns a state-space realization of the SISO transfer function n(s)/d(s) in SYSTEM format. Here n and d are the vector representation of n(s) and d(s) (type help poly for details). The number of states, inputs, and outputs of a linear system are retrieved from the SYSTEM matrix with sinfo: … A jE I + ( ) – B n 0 C D 0 0 –Inf j = –1 x · = – x + u yx , = G s( ) D C sE A ( ) – –1 B n s( ) d s( ) = + = -----------