Pros and Cons IIT Bombay In each case, we get highly nonlinear - PowerPoint PPT Presentation

Automation Lab Pros and Cons IIT Bombay � In each case, we get highly nonlinear optimization problem � Important to give good initial guess to obtain reliable model parameters: not an easy task � It is assumed that ‘structure’ of the linearized state space model / q-TF matrix is know a-priori, which may not be the case � Effect of unmeasured inputs (disturbances) on system dynamics is not captured by the model � Possible Remedy: Black box modeling 4/10/2012 System Identification 63

Automation Lab Summary IIT Bombay � If a reliable mechanistic / grey box model is available, then a linear perturbation model can be developed using the Taylor series expansion in the neighborhood of the operating point � If structure of linear perturbation model matrices is known, then the parameters can of the model can be estimated using input output data. 4/10/2012 System Identification 64

Automation Lab IIT Bombay Stability of Linear Dynamic Model Quick Review of Continuous Time Results Consider SISO Continuous Time Linear Perturbati on Model Transfer Function d x = + Ax B u ( ) ⎛ ⎞ n s [ ] ( ) dt − ⎜ ⎟ = = − y s u s C s I A 1 B u s ( ) ( ) ( ) ⎜ ⎟ d s ⎝ ⎠ = ( ) y Cx [ [ ] ] − n s C adj s I A B ( ) = x = ( 0 ) 0 [ ] − d s s I A ( ) det Roots of denominato r polynomial [ ] = I − d s s A ( ) det A i.e. eigen valu es of matrix determine the asymptotic behavior of the solution 4/10/2012 System Identification 65

Automation Lab IIT Bombay Stability of Linear Dynamic Model Case : All the roots of denominato r polynomial [ ] = − d s s I A ( ) det are in strictly Left Half Plane of complex s - plane ⇒ Dynamic System is Asymptotic ally Stable If some of the roots are on the imaginary axis and rest in LHP ⇒ Dynamic System is Marginaly Stable Case : Is nay root of denominato r polynomial [ ] = − d s s I A ( ) det are in Right Half Plane of complex s - plane ⇒ Dynamic System is Unstable 4/10/2012 System Identification 66

Automation Lab IIT Bombay Stability: Discrete Time Systems Consider SISO discrete time linear perturbation model ( ) n z + = + [ ] ( ) x k Φ x k Γ u k − ( 1 ) ( ) ( ) = = 1 y C I Φ Γ u z (z) z - ( ) d z ( ) = y k Cx k ( ) ( ) [ ] ( ) n z C adj I Φ Γ ( ) z - = x 0 = ( 0 ) [ ] d z I Φ ( ) det z - Analogous to the continuous time case Roots of denominato r polynomial [ ] = − Φ d z z I ( ) det Φ i.e. eigenvalue s of matrix determine the asymptotic behavior of the solution 4/10/2012 System Identification 67

Automation Lab IIT Bombay Stability: Discrete Time Systems = x Consider scenario ( 0 ) is non - zero and u(k) 0 = x Φ x ( 1 ) ( 0 ) = = x Φ x Φ x 2 ( 2 ) ( 1 ) ( 0 ) .......... .... k = k x Φ x ( ) ( 0 ) Consider special case ( ) ⎡ ⎤ k λ Φ 0 ... 0 where is diagonaliz able ⎢ ⎥ 1 ( ) k λ Φ = ΨΛΨ − ⎢ 0 ... 0 ⎥ 1 Λ = 2 ⎢ ⎥ − ... ... ... ... ⇒ Φ = ΨΛ Ψ 2 2 1 ⎢ ⎥ ( ) ⎥ k λ ⎢ ⎣ ⎦ 0 ... ... − ⇒ Φ = ΨΛ Ψ k k 1 n 4/10/2012 System Identification 68

Automation Lab IIT Bombay Stability of Linear Dynamic Model Case : All the roots of denominato r polynomial [ ] d ( z ) det z = − Φ I i.e. eigenvalue s of Φ are in strictly inside the unit circle in complex z - plane i.e. for i 1,2, ... n λ < = 1 i ( ) Then k as k λ → → ∞ 0 i [ ] [ ] k k k − ⇒ Λ → ⇒ Φ = ΨΛ Ψ → 1 0 0 ( k ) k ( ) ⇒ = Φ → x x 0 0 Dynamic System is Asymptotic ally Stable ⇒ 4/10/2012 System Identification 69

Automation Lab IIT Bombay Stability of Linear Dynamic Model Case : All the roots of denominato r polynomial [ ] d ( z ) det z = − Φ I i.e. eigenvalue s of Φ are in inside or on the unit circle in complex z - plane i.e. for i 1,2, ... n λ ≤ = 1 i ( ) Then k remains bounded as k λ → ∞ i k k remains bounded as k − ⇒ Φ = ΨΛ Ψ → ∞ 1 ( k ) k ( ) remains bounded as k ⇒ = Φ → ∞ x x 0 Dynamic System is Marginally Stable ⇒ 4/10/2012 System Identification 70

Automation Lab IIT Bombay Stability of Linear Dynamic Model Case : If any one or more roots of denominato r polynomial [ ] d ( z ) det z = − Φ I i.e. eigenvalue s of Φ are outside inside the unit circle of complex z - plane i.e. for some j λ > 1 j ( ) Then k as k λ → ∞ → ∞ j [ ] [ ] k k k − ⇒ Λ → ∞ ⇒ Φ = ΨΛ Ψ → ∞ 1 ( k ) k ( ) ⇒ = Φ → ∞ x x 0 Linear Dynamic System is Unstable ⇒ 4/10/2012 System Identification 71

Automation Lab Poles in s-plan and z-plane IIT Bombay [ ] e e -1 T T -1 Λ = ΨΛΨ ⇒ Φ = = Ψ Ψ A A This implies that Matrix and Matrix have identical eigenvecto rs Φ A Also, if is eigenvalue of A then e is eigenvalue of T λ λ Φ Let j represent an eigenvalue of λ = α + β A [ ] e T e T j T e T cos( T ) j sin( T ) λ α + β α = = β + β [ ] cos( T ) j sin( T ) β + β = 1 Location of e T in z - plane is determined by e T λ α ⇒ 4/10/2012 System Identification 72

Automation Lab Poles in s-plan and z-plane IIT Bombay If 0 then e T α α < < 1 All poles in left half of s - plane map inside a unit circle ⇒ around the origin in z - plane If 0 then e T α α = = 1 All poles on the imaginary axis in s - plane ⇒ map on the boundary of the unit circle in z - plane If 0 then e T α α > > 1 All poles in left half of s - plane ⇒ map outsidesid e the unit circle in z - plane 4/10/2012 System Identification 73

Automation Lab Poles in s-plan and z-plane IIT Bombay Conformal map z = exp(sT) 4/10/2012 System Identification 74

Automation Lab Poles in s-plan and z-plane IIT Bombay Note : For every continuous time linear system we can find an equivalent discrete time system once the samplingin terval is specified. However, the reverse may not be always possible. Example + = + y(k ) - . y(k) u(k) 1 0 5 − = z domain pole : z - 0.5 There is no continuous time system of the form dy τ + = κ y t u t ( ) ( ) dt τ τ = − with real value of such that exp(-T/ ) 0 . 5 4/10/2012 System Identification 75

Reference Textbooks Automation Lab IIT Bombay � Astrom, K. J., and B. Wittenmark, Computer Controlled Systems, Prentice Hall India (1994). � Franklin, G. F., Powell, J. D., and M. L. Workman, Digital Control Systems, Addison Wesley, 1990. � Ljung, L., Glad, T., Modeling of Dynamic Systems, Prentice Hall, N. J., 1994. 4/10/2012 System Identification 76

Development of Control Relevant Linear Perturbation Models Part II: Development of Black Box Models Sachin C. Patwardhan Dept. of Chemical Engineering, IIT Bombay Email: sachinp@iitb.ac.in

Automation Lab IIT Bombay Data Driven Models Development of linear state space/transfer models starting from first principles/gray box models is impractical proposition in many situations. Practical Approach • Conduct experiments by perturbing process around chosen operating point • Collect input-output data • Fit a differential equation or difference equation model using optimization Difficulties • Measurements are inaccurate • Process is influenced by unknown disturbances • Models are approximate 4/10/2012 System Identification System Identification 2 2

Automation Lab Discrete Model Development IIT Bombay Excite plant around the desired operating point by injecting input perturbations Measurement Noise 3.2 2.9 3 2.8 2.7 2.8 Measured Output 2.6 Manipulated Input 2.6 Process 2.5 2.4 2.4 2.3 2.2 2.2 2 2.1 1.8 2 0 5 10 15 20 0 2 4 6 8 10 12 14 16 18 20 Sampling Instant Sampling Instant Unmeasured Input excitation for Measured output Disturbances model identification response 4/10/2012 System Identification System Identification 3 3

Automation Lab Black Box Models IIT Bombay � Black Box Models: Static maps (correlations) and or dynamic models (difference equations) developed directly from historical input-output data � Valid over limited operating range covered by the training data set � Provide NO insight into internal working of system under consideration � Development process: much less time consuming and comparatively easy. Does not need services of a domain expert 4/10/2012 System Identification System Identification 4 4

Automation Lab Two Non-Interacting Tanks Setup IIT Bombay Tank 2 Control Tank 1 Valve 2 Control Valve 1 4/10/2012 System Identification System Identification 5 5