... sds,1.8where P and Q are positive matrices and γ is a positive scalar.Delay independent criteria ofstability for some classes of delay neutral systems aredeveloped in 10. The stabilityofsystems ... and P. Yu, Stabilityof Dynamical Systems, vol. 5 of Monograph Series on NonlinearScience and Complexity, Elsevier, Amsterdam, the Netherlands, 2007.12 Ju H. Park and S. Won, “A note on stability ... exponential stabilityof system 1.1 as well ason estimates of the norms of its solutions and their derivatives in the case of exponential stability and in the case of exponential stability being...
... Discrete systemsof the form (1.1) containing only one delay are oftencalled systems with pure delay. The main goal of the present paper is to extend the notion of the delayed exponential of a matrix ... type technique and retract principle), some of theseproblems have been investigated, for example, in the recent papers [2–9, 11–13]. REPRESENTATION OF SOLUTIONS OF LINEAR DISCRETE SYSTEMS WITH ... CONSTANTCOEFFICIENTS AND PURE DELAYJ. DIBL´IK AND D. YA. KHUSAINOVReceived 16 January 2006; Accepted 22 January 2006The purpose of this contribution is to develop a method for construction of solutions of linear...
... theory of ODE we show that each solution of (3.1) is asolution of a linear equation of the form The Central Exponent and Asymptotic Stability 13Now we consider again the case of nonlinear ... consisting of a system of ordinary differential equations(ODEs) and a system of algebraic equations so that we can use methods and results of the theory of ODEs. Many results on stability properties of ... exponents oflinear DAEs of index 1. In Sec. 3 we investigate exponential asymptotic stabilityof linear DAEs with respect to small linear as well as nonlinear perturbation.2. The Central Exponent of...
... determinist systems: the concept of state 44 2.1.2. Equations of state and equations of measurement for continuous systems 46 2.1.3. Case oflinearsystems 47 2.1.4. Case of continuous and invariant ... oflinearand invariant systems 66 2.4.3. Canonic representation of partially controllable systems 69 2.4.4. Scalar representation of partially controllable systems 73 2.5. Observability of ... ofsystems 74 2.5.1. General definitions 74 2.5.2. Observability oflinearand invariant systems 74 2.5.3. Case of partially observable systems 77 2.5.4. Case of partially controllable and...
... by equations of state. Asymptotic stability Stability in the broad senseUnstable Figure 1.7. Concepts ofstability Another point of view can be adopted where the stabilityof a system can ... ofLinearSystems Figure 1.19. Unit-step response of the second order system ξ ≥ 1 1=ξ: critical state. The roots of the denominator of the transfer function are real and merged, and ... Analysis and Control ofLinearSystems In general, the Dirac impulse is a very simplified model of any impulse phenomenon centered in ott = , with a shorter period than the time range of the systems...
... 2002. 94 Analysis and Control ofLinearSystems NOTE 3.3.– the class of rational systems that can be described by [3.16] or [3.18] is a sub-class of DLTI systems. To be certain of this, let us ... Analysis and Control ofLinearSystems Figure 3.5. Bode diagram of the continuous system and its discretization 3.4.5. The problem of sub-sampling Let us consider the case of a “standardized” ... in is the multiplicity order of iλ and ith the eigenvalue of A. 5 A canonical form called controllable companion. 88 Analysis and Control ofLinearSystems Hence, if we suppose...
... the use of rollouts and in particular the fixed poles and the remaining degrees of freedom, we can mention [MAL 93, MAL 97] and [MAR 94, MAR 99]. 126 Analysis and Control ofLinearSystems ... 0): 110 Analysis and Control ofLinearSystems The object of this chapter is to describe certain structural properties oflinear systems that condition the resolution of numerous control ... illustrates the invariance properties of the various structures of these canonical forms (indices of controllability, of observability, finite and infinite zeros) and of the associated transformation...
... of ν0. 142 Analysis and Control ofLinear Systems A discrete-time deterministic signal y[k],k ∈Zis, by definition, a sequence of complex numbers:y =y[k]k∈ZIn short, we often speak of ... y2)=y1y2[5.9] 152 Analysis and Control ofLinear Systems Figure 5.1. Typical power spectrum of an MA (left)or AR (right) modelIn the case of a single denominator (nc=0), we talk of an AR (autoregressive)model, ... Signals: Deterministic and Statistical Models 1535.4. Modeling of LTI systemsand ARMAX modelingLet us take a linear time-invariant (LTI) system, of impulse response g. Theresponse of this system...
... the condition of detectability by a careful choice of matrix Q. 162 Analysis and Control ofLinearSystems case of multi-control systems. As indicated in Chapter 2, the equations of state can ... at the expense of a stronger control. 176 Analysis and Control ofLinearSystems Figure 6.6. Stabilization by quadratic optimization 166 Analysis and Control ofLinearSystems Figure ... 174 Analysis and Control ofLinearSystems there is a unique matrix P, symmetric andpositive semi-defined, solution of the following equation (called discrete...
... from the point of view of calculation, in terms oflinear regression (or linearized), i.e. it leads to a quadratic 196 Analysis and Control ofLinear Systems in performing the roles of the procedure, ... K1N101+12n1N10[7.21] 222 Analysis and Control ofLinear Systems iteration, unstable), interrupt the simulation and the calculation of sensitivity func-tions, hence of the gradient, on a preset upper ... calculation of a criterion whose theoretical value for the current set of parameters diverges (for example, a transfer output error whose poles are, during an 218 Analysis and Control ofLinear Systems following...
... Analysis and Control ofLinear Systems Consequently, the field of absolute stabilityof multi-interval methods (implicit ornot depending on the value of 0) is the set of à C so that the roots of ... requirements of speed and accuracyaccessible in simulation.8.2. Standard linear equations8.2.1. Definition of the problemWe will adopt the notations usually used to describe the state formsand linear dynamic ... desired and to perform the corre-sponding substitutions. Let us take the example of the classic PID regulator. Let e(t) 230 Analysis and Control ofLinear Systems The sizes of blocks 0 and I...
... study the case of open loop stable systems, that of integrator systemsand finally the case of open loop unstable systems. 270 Analysis and Control ofLinearSystems We note that when ξ tends ... and the phase is of π− . The image of the ray [0,]`−∞− is the symmetric curve, with respect to the axis of real numbers. The image of the semicircle of radius ε is an arc of circle of ... other hand. The quality of the feedback control is mainly translated through its stabilityand the follow-up precision of the output on the input and consequently of the dynamics of this...