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Extra info for Analysis and control of linear systems
Hence, vector x(t ) = ⎡⎢ ⎣v(t ) ⎥⎦ makes it possible to describe this evolution. Thus, knowing the state of a system at instant t ' < t and the controls u(t ) applied to the system between instants t ' and t , the system output is written as: y(t ) = ht ,t ' [ x(t '), u(τ )] for t ' ≤ τ ≤ t Similarly, the evolution of the state will be expressed by the relation: x(t ) = ϕt ,t ' [ x(t '), u(τ )] for t ' ≤ τ ≤ t We note that x(t ' ) can be expressed from x(t" ) (t" < t ' ) and the controls u(τ ' ) applied to the system between instants t " and t ' : x(t ') = ϕt ',t "[ x(t "), u(τ ')] for t " ≤ τ ' ≤ t ' which leads to: x(t ) = ϕt ,t ' [ϕt ',t "[ x(t "), u(τ ')], u(τ )] for t " ≤ τ ' ≤ t ' ≤ τ ≤ t Between t " and t , we have: x(t ) = ϕt ,t " [ x(t "), u(τ ")] for t " ≤ τ " ≤ t 46 Analysis and Control of Linear Systems The comparison between these two results leads to the property of transition, which is essential for the systems that we are analyzing here: ϕ t , t " [x(t "), u(τ ")] = ϕ t , t ' [ϕ t ', t " [x(t "), u(τ ')], u(τ )] t" ≤ τ " ≤ t t" ≤ τ ' ≤ t ' ≤ τ ≤ t and which characterizes the transition of a state x(t" ) to a state x(t ) by going through x(t ' ) .
Cours de mathématiques, vol. III, Masson, Paris, 1971. , Automatique, Hermès, Paris, 1996 (2nd edition). , Distributions et Transformation de Fourier, Ediscience, 1971. , Distributions et signal, Eyrolles, Paris, 1990. Chapter 2 State Space Representation Control techniques based on spectral representation demonstrated their performances though numerous industrial implementations, but they also revealed their limitations for certain applications. The objective of this chapter is to provide the basis for a more general representation than the one adopted for the frequency approach and to offer the necessary elements to comprehend time control through the state approach.
3. Properties As we have already seen, the Fourier and Laplace transforms reveal the same concept adapted to the type of signal considered. Thus, these transforms have similar properties and we will sum up the main ones in the following table. We recall that U (t ) designates the unit-step function. Fourier transform Linearity TF (λx + µy ) = λTF (x ) + µTF ( y ) Laplace transform TL(λx + µy ) = λTL(x ) + µTL( y ) The convergence domain is the intersection of each domain of basic transforms. x(t ) ⎯⎯⎯→ X ( f ) TF X ( p) ⎧ TL x(t ) ⎯⎯⎯→⎨ ⎩σ 1 < Re( p ) < σ 2 Delay x(t − τ ) ⎯⎯⎯→ X ( f ) e −2πjfτ TF Time reverse x(− t ) ⎯⎯⎯→ X (− f ) = X * ( f ) TF ⎧⎪ X ( p ) e −τp TL x(t − τ ) ⎯⎯⎯→⎨ ⎪⎩σ 1 < Re( p ) < σ 2 X (− p ) ⎧ TL x(− t ) ⎯⎯⎯→⎨ ⎩− σ 2 < Re( p ) < −σ 1 Signal derivation The signal is modeled by a continuous function: dx TF ⎯⎯⎯→(2πjf )X ( f ) dt pX ( p ) ⎧ dx TL ⎯⎯⎯→⎨ dt ⎩σ 1 < Re( p ) < σ 2 This property verifies that the signal is The signal is modeled by a function that has a modeled by a function or a distribution.