4.
ECI 660 LINEAR CONTROL SYSTEMS 3(3, 0) <ul><li>Course Contents </li></ul><ul><li>System Definition, Control System Archetechture, Input/Output System Models, Basic System Properties, Continuous and Discrete-Time Systems, System Modeling, State Space Representation, State Equations, System Analysis, Equivalence, Canonical Forms, Realizations, Stability, Sensitivity, Disturbance Rejection, Linearization, Controllability and Observability, Rank Tests, Linear Feedback Control, Fixed-Order Compensators (System Augmentation), Controller / Compensator Applications, Design Techniques: Root Locus, Frequency Response and Pole Placement, LQ Control, Luenberger Observers, Separation Principle (Estimated State Feedback). </li></ul>
5.
TEXT AND REFERENCE BOOKS <ul><li>Text Books </li></ul><ul><li>Chi-Tsong Chen : Linear System Theory and Design, 3 rd edition, Oxford University Press </li></ul><ul><li>Charlles L. Phillips, Royce D. Harbor : Feedback Control Systems, 4 th edition, New Jersey: Prentice Hall </li></ul><ul><li>Gene F. Franklin, J. David Powell: Feedback Control of Dynamic Systems, 5 th edition , New Jersey: Prentice Hall </li></ul><ul><li>Reference Books </li></ul><ul><li>Katsuhiko Ogata : Modern Control Engineering, 5 th edition, PHI Learning </li></ul><ul><li>Callier, F. M., Desoer, C. A: Linear System Theory , New York, Spronger-Verlag </li></ul><ul><li>Rugh, W : Linear System Theory , 2 nd edition, New Jersey: Prentice Hall </li></ul>
6.
CONTROL SYSTEM <ul><li>Definition </li></ul><ul><li>A control System is an interconnection of components to provide a desired function. The portion of the system to be controlled is called the plant (process, system), and the part doing the controlling is called the controller (compensator, filter). A control system designer has little or no design freedom with the plant; as it is fixed. The designer’s task is to develop a controller that will control the given plant acceptably. </li></ul>
7.
ARCHITECTURE OF CONTROL SYSTEMS <ul><li>Feed-forward or open-looped control system </li></ul><ul><ul><li>Input Output </li></ul></ul><ul><li>Feedback or closed-looped control system </li></ul><ul><li>Input Output </li></ul><ul><li>(unity-gain feedback) </li></ul>Controller Plant Controller Plant
8.
ARCHITECTURE OF CONTROL SYSTEMS <ul><li>Feedback or closed-looped control system </li></ul><ul><li>Input Output </li></ul><ul><li>(non-unity-gain feedback) </li></ul>Controller Plant
9.
DIGITAL CONTROL SYSTEMS <ul><li>Feedback control system using Digital Controller </li></ul><ul><li>Disturbances </li></ul><ul><li>(Desired response) (error Signal) (Response) </li></ul><ul><li>Input r(t) e(t) c(t) Output </li></ul><ul><li>(Sensor output) </li></ul><ul><li> </li></ul><ul><li> Disturbances </li></ul><ul><li> (non-unity-gain feedback) </li></ul>Digital Controller Plant Sensor A/D D/A
10.
CONTROL SYSTEM DEFINITIONS <ul><li>SISO System </li></ul><ul><li>A system with only one input and one output is called single-input single-output system. </li></ul><ul><li>MIMO System </li></ul><ul><li>A system with two or more input terminals and two or more output terminal is called multivariable system or multi-input multi-output system. </li></ul><ul><li>Continuous-time System </li></ul><ul><li>A system is called continuous-time system if it accepts continuous-time signals as input and generates continuous-time signals as its output. </li></ul><ul><li>Discrete-time System </li></ul><ul><li>A system is called discrete-time system if it accepts discrete-time signals as input and generates discrete-time signals as its output. </li></ul>
11.
CONTROL SYSTEM DEFINITIONS <ul><li>Memory less System </li></ul><ul><li>A system is called memory less system if its output y(t 0 ) depends only on the input applied at t 0 ; it is independent of the past and future input (input applied before or after t 0 ). </li></ul><ul><li>Casual System </li></ul><ul><li>A system is called casual or non-anticipatory if its current output depends on past and current inputs and not on future inputs. Every physical system is casual. </li></ul><ul><li>State of a System </li></ul><ul><li>The state x (t 0 ) of a system at time t 0 is the information at t 0 that, together with input u (t), for t ≥ t 0 , determines uniquely the output y (t 0 ) for all t ≥ t 0 . </li></ul><ul><li>Lumped System </li></ul><ul><li>A system is called lumped if its number of state variables is finite. </li></ul><ul><li>Disturbed System </li></ul><ul><li>A system is called disturbed if its number of state variables are infinite. </li></ul>
12.
LINEAR SYSTEM <ul><li>Linear System </li></ul><ul><li>A system is called a linear system if for every t 0 any state-input-output pairs </li></ul><ul><li>x i (t 0 ) </li></ul><ul><li>u i (t) </li></ul><ul><li> for i = 1, 2, we have </li></ul><ul><li>x 1 (t 0 ) + x 2 (t 0 ) </li></ul><ul><li>u 1 (t) + u 2 (t), t ≥ t 0 </li></ul><ul><li>and </li></ul><ul><li>α x i (t 0 ) </li></ul><ul><li>α u i (t), t ≥ t 0 </li></ul>y i (t), t ≥ t 0 y 1 (t) + y 2 (t), t ≥ t 0 (additivity) α y i (t), t ≥ t 0 (homogeneity)
13.
LINEAR SYSTEM <ul><li> α 1 x 1 (t 0 ) + α 2 x 2 (t 0 ) </li></ul><ul><li> α 1 u 1 (t) + α 2 u 2 (t), t ≥ t 0 </li></ul><ul><li>for any real constants α 1 and α 1 , and is called the superposition </li></ul><ul><li>property. A system is called a nonlinear system if the superposition property does not hold. </li></ul><ul><li>If the input u(t) is identically zero for all t ≥ t 0 , the output will be exited exclusively by the initial state x(t 0 ). This output is called the zero-input response and is denoted by y zi </li></ul><ul><li>or </li></ul><ul><li>x(t 0 ) </li></ul><ul><li>u(t)= 0, t ≥ t 0 </li></ul>y zi (t), t ≥ t 0 α 1 y 1 (t) + α 2 y 2 (t), t ≥ t 0
14.
LINEAR SYSTEM <ul><li>If the initial state x(t 0 ) is zero, the output will be exited exclusively by the input u(t). This output is called the zero-state response and is denoted by y zs </li></ul><ul><li>or x(t 0 ) = 0 </li></ul><ul><li> u(t) , t ≥ t 0 </li></ul><ul><li>The additive property implies </li></ul><ul><li>x(t 0 ) x(t 0 ) </li></ul><ul><li>u(t) , t ≥ t 0 u(t) = 0 , t ≥ t 0 </li></ul><ul><li> x(t 0 ) = 0 u(t) , t ≥ t 0 </li></ul><ul><li>Response = zero-input response + zero-state response </li></ul><ul><li>The two responses can be studied separately and their sum yield the complete response. This is not true for nonlinear system, where the complete response can be different. </li></ul>y zs (t), t ≥ t 0 Output due to = output due to + output due to
15.
LINEAR SYSTEM <ul><li>Input-output Description </li></ul><ul><li>Zero-state response of a linear system: Consider a SISO linear system. Let δ Δ (t-t 1 ) be the pulse as shown: </li></ul><ul><li>It has a width Δ and height 1/ Δ and is located </li></ul><ul><li>at time t 1 . </li></ul><ul><li>Then every input u(t) can be approximated by a sequences of pulses as shown: </li></ul>t t 1 t 1 + Δ Δ 1/ Δ u(t i ) δ Δ (t-t 1 ) Δ u(t i ) t i t
16.
LINEAR SYSTEM <ul><li>If the pulse has a height of 1/ Δ then δ Δ (t-t 1 ) Δ has a height 1 and the left-most pulse with height u(t i ) can be expressed as u(t i ) δ Δ (t-t 1 ) Δ . Consequently, the input can be expressed as: </li></ul><ul><li>Let g Δ (t, t i ) be the output at time t excited by the pulse </li></ul><ul><li>u(t) = δ Δ (t-t 1 ) applied at time t i . Then we have </li></ul><ul><li>Thus the output y(t) excited by the input u(t) can be approximated by </li></ul>U(t) ≈ ∑ u(t i ) δ Δ (t-t i ) Δ y(t) ≈ ∑ g Δ (t,t i ) u(t i ) Δ δ Δ (t-t i ) g Δ (t, t i ) δ Δ (t-t i ) u(t i ) Δ g Δ (t, t i ) u(t i ) Δ (homogeneity) ∑ δ Δ (t-t i ) u(t i ) Δ ∑ g Δ (t, t i ) u(t i ) Δ (additivity)
17.
LINEAR SYSTEM <ul><li>now if Δ approaches zero, then the pulse δ Δ (t-t i ) becomes an impulse at t i , denoted by δ (t-t i ), and the corresponding output will be denoted by g(t, t i ). As Δ approaches zero, the approximation in </li></ul><ul><li>becomes an equality, the summation becomes an integration, the discrete t i becomes a continuum and can be replaced by זּ , and Δ can be written as d זּ . Thus the output can be expressed as: </li></ul><ul><li>Note that the impulse response g(t, זּ ) is a function of two variables. The second variable denotes the time at which the impulse input is applied; the first variable denotes the time at which the output is observed. </li></ul>y(t) ≈ ∑ g Δ (t,t i ) u(t i ) Δ y(t) = ʃ g(t, זּ ) u( זּ ) d זּ -∞ ∞
18.
LINEAR SYSTEM <ul><li>For a casual system the output will not appear before an input is applied. </li></ul><ul><li>Thus g(t, זּ ) = 0 for t < זּ </li></ul><ul><li>If the system is relaxed, its initial state at t 0 is 0. Therefore, for a linear system that is casual and relaxed at t 0 , the upper limit can be replaced by t and the lower limit by t 0 . The output can be expressed as: </li></ul><ul><li>If a linear system has r input and p output terminals then out is expressed as: </li></ul>y(t) = ʃ g(t, זּ ) u( זּ ) d זּ for t < זּ t 0 t y (t) = ʃ G (t, זּ ) u ( זּ ) d זּ for t < זּ t 0 t
19.
LINEAR SYSTEM <ul><li>where G (t, זּ ) is called the impulse response matrix of the system. </li></ul><ul><li>g 11 (t, זּ ) g 12 (t, זּ ) …. g 1r (t, זּ ) </li></ul><ul><li>g 21 (t, זּ ) g 22 (t, זּ ) …. g 2r (t, זּ ) </li></ul><ul><li> G (t, זּ ) = . . . </li></ul><ul><li> . . . </li></ul><ul><li>g p1 (t, זּ ) g p2 (t, זּ ) …. g pr (t, זּ ) </li></ul><ul><li> g ij (t, זּ ) is the response at time t at the ith output terminal due to an impulse applied at time זּ at the jth input terminal, the input at other terminals being identically zero. </li></ul>
20.
LINEAR SYSTEM <ul><li>State-space description: Every linear lumped system can be defined by a set of first order coupled equations of the form: </li></ul><ul><li>(set of n differential equations) </li></ul><ul><li>(set of p algebraic equations) </li></ul><ul><li>For r input and q output system, u is a r x 1 vector and y is a q x 1 vector. In the system has n state variables then x is an n x 1 vector. A , B , C and D must be n x n, n x r, p x n and p x r matrices. </li></ul><ul><li> </li></ul>y (t) = C (t) u (t) + D (t) u (t) x (t) = A (t) u (t) + B (t) u (t) .
21.
LINEAR SYSTEM <ul><li>Linear Time-Invariant (LTI) Systems: A system is said to be time invariant if for every state-input-output pair </li></ul><ul><li>and for any T, we have </li></ul><ul><li>It means that if the initial state is shifted to time t 0 + T and the same waveform is applied from t 0 + T instead of t 0 , then the output waveform will be the same except that it starts to appear from time t 0 + T. In other words, if the initial state and the input are same, no matter at what time they are applied, the output waveform will always be the same. </li></ul><ul><li> </li></ul>x(t 0 ) u(t), t ≥ t 0 y(t), t ≥ t 0 x(t 0 +T) u(t -T), t ≥ t 0 + T y(t - T), t ≥ t 0 + T (time shifting)
22.
LINEAR SYSTEM <ul><li>Input-output description for LTI Systems: For time invariant system </li></ul><ul><li> g(t, זּ ) = g(t + T, זּ + T) = g(t - זּ , 0) = g(t - זּ ) </li></ul><ul><li>for any T </li></ul><ul><li>where we have replaced t 0 by 0 and the above integration is called convolution integral. Unlike the time varying case, where g is a function of two variables, g is a function of single variable in time invariant case. g(t) = g(t - 0) is the output at time t due to impulse in put at time 0. </li></ul><ul><li>State-space description for LTI Systems: </li></ul>y (t) = C u (t) + D u (t) y(t) = ʃ g(t - זּ ) u( זּ ) d זּ = ʃ g( זּ ) u(t - זּ ) d זּ 0 t 0 t x (t) = A u (t) + B u (t) .
23.
CONTROL PROBLEM <ul><li>A physical system or process is to be controlled, so that the output (response) is adjusted as required by the error signal. The error signal is a measure of the difference between the system response as determined by the sensor and the desired response (input). </li></ul><ul><li>The controller is required to process the error signal such that: </li></ul><ul><ul><li>To track (or follow) the reference input r(t) </li></ul></ul><ul><ul><li>To reject (or not respond to) the disturbances </li></ul></ul><ul><ul><li>To reduce steady state errors </li></ul></ul><ul><ul><li>To obtain desired transient response. </li></ul></ul><ul><ul><li>We hope that the sensor has a transfer function close to 1, so often we ignore sensor dynamics </li></ul></ul><ul><ul><li>If the sensor dynamics are significant, they should be included in the analysis </li></ul></ul><ul><ul><li>If r(t) is always 0, the special case is called a regulator . </li></ul></ul><ul><ul><li>The block labeled digital controller can be implemented with any sort of hardware: </li></ul></ul><ul><ul><ul><li>Computer running real-time operating system </li></ul></ul></ul><ul><ul><ul><li>Digital signal processor (DSP) or other special-purpose digital hardware </li></ul></ul></ul>
24.
ANALYSIS OF PHYSICAL SYSTEMS <ul><li>Empirical Methods </li></ul><ul><li>Applying various inputs to a physical system and measuring its response. If performance is not satisfactory, some of its parameters are adjusted or connected to a compensator to improve its performance. </li></ul><ul><li>Analytical Methods </li></ul><ul><li>The analytical study of a physical system consists of four parts: </li></ul><ul><ul><li>Modeling </li></ul></ul><ul><ul><li>Development of mathematical description </li></ul></ul><ul><ul><li>Analysis (quantitative & qualitative) </li></ul></ul><ul><ul><li>Design (controller or compensator) </li></ul></ul>
27.
SYSTEM MODELING or Electrical Circuit PI Controller R f R i C f R i V i V f R f sC f R i V f V i 1 0 V f V i R f R i sR i C f 1 V f V i K p s K i
28.
SYSTEM MODELING Electrical Circuit PID Controller V(s) = R I(s) + sL I(s) - LI(0) + V c (s) I(t) = sCV c (s) - CV c (0) i(t) = C dv c dt v(t) = R i(t) + L + v c (t) dt di v v c L R C i sC 1 = R + sL + I(s) V(s) V(s) I(s) H(s) = = sC 1 R + sL + 1
29.
STATE SPACE MODEL Electrical Circuit PID Controller i(t) Set of first order differential equations i(t) = C dv c dt v(t) = R i(t) + L + v c (t) dt di = - - v c (t) - v(t) di dt d dt R L 1 L 1 L = i(t) dv c dt 1 C 1 L dv c dt 1 C R L di dt 1 L 0 i(t) v c (t) + = v(t) 0 - - -
30.
ECI 660 LINEAR CONTROL SYSTEMS 3(3, 0) <ul><li>Course Contents </li></ul><ul><li>Input/Output System Models, Basic System Properties, Continuous and Discrete-Time Systems, System Modeling, State Space Representation, State Equations, System Analysis, Equivalence, Canonical Forms, Realizations, Stability, Sensitivity, Disturbance Rejection, Linearization, Controllability and Observability, Rank Tests, Linear Feedback Control, Fixed-Order Compensators (System Augmentation), Controller / Compensator Applications, Design Techniques: Root Locus, Frequency Response and Pole Placement, LQ Control, Luenberger Observers, Separation Principle (Estimated State Feedback). </li></ul>
31.
TEXT AND REFERENCE BOOKS <ul><li>Text Books </li></ul><ul><li>Chi-Tsong Chen : Linear System Theory and Design, 3 rd edition, Oxford University Press </li></ul><ul><li>Charlles L. Phillips, Royce D. Harbor : Feedback Control Systems, 4 th edition, New Jersey: Prentice Hall </li></ul><ul><li>Gene F. Franklin, J. David Powell: Feedback Control of Dynamic Systems, 5 th edition , New Jersey: Prentice Hall </li></ul><ul><li>Reference Books </li></ul><ul><li>Katsuhiko Ogata : Modern Control Engineering, 5 th edition, PHI Learning </li></ul><ul><li>Callier, F. M., Desoer, C. A: Linear System Theory , New York, Spronger-Verlag </li></ul><ul><li>Rugh, W : Linear System Theory , 2 nd edition, New Jersey: Prentice Hall </li></ul>
32.
CONTROL SYSTEM <ul><li>Definition </li></ul><ul><li>A control System is an interconnection of components to provide a desired function. The portion of the system to be controlled is called the plant (process, system), and the part doing the controlling is called the controller (compensator, filter). A control system designer has little or no design freedom with the plant; as it is fixed. The designer’s task is to develop a controller that will control the given plant acceptably. </li></ul>
33.
ARCHITECTURE OF CONTROL SYSTEMS <ul><li>Feed-forward or open-looped control system </li></ul><ul><ul><li>Input Output </li></ul></ul><ul><li>Feedback or closed-looped control system </li></ul><ul><li>Input Output </li></ul><ul><li>(unity-gain feedback) </li></ul>Controller Plant Controller Plant
34.
ARCHITECTURE OF CONTROL SYSTEMS <ul><li>Feedback or closed-looped control system </li></ul><ul><li>Input Output </li></ul><ul><li>(non-unity-gain feedback) </li></ul>Controller Plant
35.
DIGITAL CONTROL SYSTEMS <ul><li>Feedback control system using Digital Controller </li></ul><ul><li>Disturbances </li></ul><ul><li>(Desired response) (error Signal) (Response) </li></ul><ul><li>Input r(t) e(t) c(t) Output </li></ul><ul><li>(Sensor output) </li></ul><ul><li> </li></ul><ul><li> Disturbances </li></ul><ul><li> (non-unity-gain feedback) </li></ul>Digital Controller Plant Sensor A/D D/A
36.
CONTROL SYSTEM DEFINITIONS <ul><li>SISO System </li></ul><ul><li>A system with only one input and one output is called single-input single-output system. </li></ul><ul><li>MIMO System </li></ul><ul><li>A system with two or more input terminals and two or more output terminal is called multivariable system or multi-input multi-output system. </li></ul><ul><li>Continuous-time System </li></ul><ul><li>A system is called continuous-time system if it accepts continuous-time signals as input and generates continuous-time signals as its output. </li></ul><ul><li>Discrete-time System </li></ul><ul><li>A system is called discrete-time system if it accepts discrete-time signals as input and generates discrete-time signals as its output. </li></ul>
37.
CONTROL SYSTEM DEFINITIONS <ul><li>Memory less System </li></ul><ul><li>A system is called memory less system if its output y(t 0 ) depends only on the input applied at t 0 ; it is independent of the past and future input (input applied before or after t 0 ). </li></ul><ul><li>Casual System </li></ul><ul><li>A system is called casual or non-anticipatory if its current output depends on past and current inputs and not on future inputs. Every physical system is casual. </li></ul><ul><li>State of a System </li></ul><ul><li>The state x (t 0 ) of a system at time t 0 is the information at t 0 that, together with input u (t), for t ≥ t 0 , determines uniquely the output y (t 0 ) for all t ≥ t 0 . </li></ul><ul><li>Lumped System </li></ul><ul><li>A system is called lumped if its number of state variables is finite. </li></ul><ul><li>Disturbed System </li></ul><ul><li>A system is called disturbed if its number of state variables are infinite. </li></ul>
38.
LINEAR SYSTEM <ul><li>Linear System </li></ul><ul><li>A system is called a linear system if for every t 0 any state-input-output pairs </li></ul><ul><li>x i (t 0 ) </li></ul><ul><li>u i (t) </li></ul><ul><li> for i = 1, 2, we have </li></ul><ul><li>x 1 (t 0 ) + x 2 (t 0 ) </li></ul><ul><li>u 1 (t) + u 2 (t), t ≥ t 0 </li></ul><ul><li>and </li></ul><ul><li>α x i (t 0 ) </li></ul><ul><li>α u i (t), t ≥ t 0 </li></ul>y i (t), t ≥ t 0 y 1 (t) + y 2 (t), t ≥ t 0 (additivity) α y i (t), t ≥ t 0 (homogeneity)
39.
LINEAR SYSTEM <ul><li> α 1 x 1 (t 0 ) + α 2 x 2 (t 0 ) </li></ul><ul><li> α 1 u 1 (t) + α 2 u 2 (t), t ≥ t 0 </li></ul><ul><li>for any real constants α 1 and α 1 , and is called the superposition </li></ul><ul><li>property. A system is called a nonlinear system if the superposition property does not hold. </li></ul><ul><li>If the input u(t) is identically zero for all t ≥ t 0 , the output will be exited exclusively by the initial state x(t 0 ). This output is called the zero-input response and is denoted by y zi </li></ul><ul><li>or </li></ul><ul><li>x(t 0 ) </li></ul><ul><li>u(t)= 0, t ≥ t 0 </li></ul>y zi (t), t ≥ t 0 α 1 y 1 (t) + α 2 y 2 (t), t ≥ t 0
40.
LINEAR SYSTEM <ul><li>If the initial state x(t 0 ) is zero, the output will be exited exclusively by the input u(t). This output is called the zero-state response and is denoted by y zs </li></ul><ul><li>or x(t 0 ) = 0 </li></ul><ul><li> u(t) , t ≥ t 0 </li></ul><ul><li>The additive property implies </li></ul><ul><li>x(t 0 ) x(t 0 ) </li></ul><ul><li>u(t) , t ≥ t 0 u(t) = 0 , t ≥ t 0 </li></ul><ul><li> x(t 0 ) = 0 u(t) , t ≥ t 0 </li></ul><ul><li>Response = zero-input response + zero-state response </li></ul><ul><li>The two responses can be studied separately and their sum yield the complete response. This is not true for nonlinear system, where the complete response can be different. </li></ul>y zs (t), t ≥ t 0 Output due to = output due to + output due to
41.
LINEAR SYSTEM <ul><li>Input-output Description </li></ul><ul><li>Zero-state response of a linear system: Consider a SISO linear system. Let δ Δ (t-t 1 ) be the pulse as shown: </li></ul><ul><li>It has a width Δ and height 1/ Δ and is located </li></ul><ul><li>at time t 1 . </li></ul><ul><li>Then every input u(t) can be approximated by a sequences of pulses as shown: </li></ul>t t 1 t 1 + Δ Δ 1/ Δ u(t i ) δ Δ (t-t 1 ) Δ u(t i ) t i t
42.
LINEAR SYSTEM <ul><li>If the pulse has a height of 1/ Δ then δ Δ (t-t 1 ) Δ has a height 1 and the left-most pulse with height u(t i ) can be expressed as u(t i ) δ Δ (t-t 1 ) Δ . Consequently, the input can be expressed as: </li></ul><ul><li>Let g Δ (t, t i ) be the output at time t excited by the pulse </li></ul><ul><li>u(t) = δ Δ (t-t 1 ) applied at time t i . Then we have </li></ul><ul><li>Thus the output y(t) excited by the input u(t) can be approximated by </li></ul>U(t) ≈ ∑ u(t i ) δ Δ (t-t i ) Δ y(t) ≈ ∑ g Δ (t,t i ) u(t i ) Δ δ Δ (t-t i ) g Δ (t, t i ) δ Δ (t-t i ) u(t i ) Δ g Δ (t, t i ) u(t i ) Δ (homogeneity) ∑ δ Δ (t-t i ) u(t i ) Δ ∑ g Δ (t, t i ) u(t i ) Δ (additivity)
43.
LINEAR SYSTEM <ul><li>now if Δ approaches zero, then the pulse δ Δ (t-t i ) becomes an impulse at t i , denoted by δ (t-t i ), and the corresponding output will be denoted by g(t, t i ). As Δ approaches zero, the approximation in </li></ul><ul><li>becomes an equality, the summation becomes an integration, the discrete t i becomes a continuum and can be replaced by זּ , and Δ can be written as d זּ . Thus the output can be expressed as: </li></ul><ul><li>Note that the impulse response g(t, זּ ) is a function of two variables. The second variable denotes the time at which the impulse input is applied; the first variable denotes the time at which the output is observed. </li></ul>y(t) ≈ ∑ g Δ (t,t i ) u(t i ) Δ y(t) = ʃ g(t, זּ ) u( זּ ) d זּ -∞ ∞
44.
LINEAR SYSTEM <ul><li>For a casual system the output will not appear before an input is applied. </li></ul><ul><li>Thus g(t, זּ ) = 0 for t < זּ </li></ul><ul><li>If the system is relaxed, its initial state at t 0 is 0. Therefore, for a linear system that is casual and relaxed at t 0 , the upper limit can be replaced by t and the lower limit by t 0 . The output can be expressed as: </li></ul><ul><li>If a linear system has r input and p output terminals then out is expressed as: </li></ul>y(t) = ʃ g(t, זּ ) u( זּ ) d זּ for t < זּ t 0 t y (t) = ʃ G (t, זּ ) u ( זּ ) d זּ for t < זּ t 0 t
45.
LINEAR SYSTEM <ul><li>where G (t, זּ ) is called the impulse response matrix of the system. </li></ul><ul><li>g 11 (t, זּ ) g 12 (t, זּ ) …. g 1r (t, זּ ) </li></ul><ul><li>g 21 (t, זּ ) g 22 (t, זּ ) …. g 2r (t, זּ ) </li></ul><ul><li> G (t, זּ ) = . . . </li></ul><ul><li> . . . </li></ul><ul><li>g p1 (t, זּ ) g p2 (t, זּ ) …. g pr (t, זּ ) </li></ul><ul><li> g ij (t, זּ ) is the response at time t at the ith output terminal due to an impulse applied at time זּ at the jth input terminal, the input at other terminals being identically zero. </li></ul>
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LINEAR SYSTEM <ul><li>State-space description: Every linear lumped system can be defined by a set of first order coupled equations of the form: </li></ul><ul><li>(set of n differential equations) </li></ul><ul><li>(set of p algebraic equations) </li></ul><ul><li>For r input and q output system, u is a r x 1 vector and y is a q x 1 vector. In the system has n state variables then x is an n x 1 vector. A , B , C and D must be n x n, n x r, p x n and p x r matrices. </li></ul><ul><li> </li></ul>y (t) = C (t) u (t) + D (t) u (t) x (t) = A (t) u (t) + B (t) u (t) .
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LINEAR SYSTEM <ul><li>Linear Time-Invariant (LTI) Systems: A system is said to be time invariant if for every state-input-output pair </li></ul><ul><li>and for any T, we have </li></ul><ul><li>It means that if the initial state is shifted to time t 0 + T and the same waveform is applied from t 0 + T instead of t 0 , then the output waveform will be the same except that it starts to appear from time t 0 + T. In other words, if the initial state and the input are same, no matter at what time they are applied, the output waveform will always be the same. </li></ul><ul><li> </li></ul>x(t 0 ) u(t), t ≥ t 0 y(t), t ≥ t 0 x(t 0 +T) u(t -T), t ≥ t 0 + T y(t - T), t ≥ t 0 + T (time shifting)
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LINEAR SYSTEM <ul><li>Input-output description for LTI Systems: For time invariant system </li></ul><ul><li> g(t, זּ ) = g(t + T, זּ + T) = g(t - זּ , 0) = g(t - זּ ) </li></ul><ul><li>for any T </li></ul><ul><li>where we have replaced t 0 by 0 and the above integration is called convolution integral. Unlike the time varying case, where g is a function of two variables, g is a function of single variable in time invariant case. g(t) = g(t - 0) is the output at time t due to impulse in put at time 0. </li></ul><ul><li>State-space description for LTI Systems: </li></ul>y (t) = C u (t) + D u (t) y(t) = ʃ g(t - זּ ) u( זּ ) d זּ = ʃ g( זּ ) u(t - זּ ) d זּ 0 t 0 t x (t) = A u (t) + B u (t) .
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CONTROL PROBLEM <ul><li>A physical system or process is to be controlled, so that the output (response) is adjusted as required by the error signal. The error signal is a measure of the difference between the system response as determined by the sensor and the desired response (input). </li></ul><ul><li>The controller is required to process the error signal such that: </li></ul><ul><ul><li>To track (or follow) the reference input r(t) </li></ul></ul><ul><ul><li>To reject (or not respond to) the disturbances </li></ul></ul><ul><ul><li>To reduce steady state errors </li></ul></ul><ul><ul><li>To obtain desired transient response. </li></ul></ul><ul><ul><li>We hope that the sensor has a transfer function close to 1, so often we ignore sensor dynamics </li></ul></ul><ul><ul><li>If the sensor dynamics are significant, they should be included in the analysis </li></ul></ul><ul><ul><li>If r(t) is always 0, the special case is called a regulator . </li></ul></ul><ul><ul><li>The block labeled digital controller can be implemented with any sort of hardware: </li></ul></ul><ul><ul><ul><li>Computer running real-time operating system </li></ul></ul></ul><ul><ul><ul><li>Digital signal processor (DSP) or other special-purpose digital hardware </li></ul></ul></ul>
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ANALYSIS OF PHYSICAL SYSTEMS <ul><li>Empirical Methods </li></ul><ul><li>Applying various inputs to a physical system and measuring its response. If performance is not satisfactory, some of its parameters are adjusted or connected to a compensator to improve its performance. </li></ul><ul><li>Analytical Methods </li></ul><ul><li>The analytical study of a physical system consists of four parts: </li></ul><ul><ul><li>Modeling </li></ul></ul><ul><ul><li>Development of mathematical description </li></ul></ul><ul><ul><li>Analysis (quantitative & qualitative) </li></ul></ul><ul><ul><li>Design (controller or compensator) </li></ul></ul>
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SYSTEM MODELING or Electrical Circuit PI Controller R f R i C f R i V i V f R f sC f R i V f V i 1 0 V f V i R f R i sR i C f 1 V f V i K p s K i
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SYSTEM MODELING Electrical Circuit PID Controller V(s) = R I(s) + sL I(s) - LI(0) + V c (s) I(t) = sCV c (s) - CV c (0) i(t) = C dv c dt v(t) = R i(t) + L + v c (t) dt di v v c L R C i sC 1 = R + sL + I(s) V(s) V(s) I(s) H(s) = = sC 1 R + sL + 1
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STATE SPACE MODEL Electrical Circuit PID Controller i(t) Set of first order differential equations i(t) = C dv c dt v(t) = R i(t) + L + v c (t) dt di = - - v c (t) - v(t) di dt d dt R L 1 L 1 L = i(t) dv c dt 1 C 1 L dv c dt 1 C R L di dt 1 L 0 i(t) v c (t) + = v(t) 0 - - -
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MODELING MECHANICAL SYSTEMS <ul><li>Mechanical elements (linear) </li></ul><ul><li>Mass: </li></ul><ul><li>Spring: </li></ul><ul><li>Friction: </li></ul>f = K x f x dx dt f = B v = B f = M a = M 2 dx dt 2 M f x K B x f
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LINEAR MECHANICAL SYSTEM output equation y(t) = x(t) State equations State Space Model B x f K M M + B + K x = f 2 dx dt 2 dx dt = - v(t) - x(t) + f(t) dv dt K M 1 M B M dx dt = v(t) v(t) B M K M 1 M dx dt = dv dt - + 0 1 - x(t) 0 f(t) y(t) = 0 1 x(t)
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TRANSFER FUNCTION Taking Laplace Transform s M X(s) - s M x(0) – M x(0) + s B X(s)– B x(0) + K X(s) = F (s) 2 With zero initial conditions s 2 M X(s) + s B X(s) + K X(s) = F (s) The system transfer function: M + B + K x = f 2 dx dt 2 dx dt X(s) F(s) = H(s) = 1 M s + B s + K 2
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ROTARY MECHANICAL SYSTEM System Dynamics: The system transfer function: K Ƭ , ɵ J J + B + K ɵ = Ƭ 2 dɵ dt 2 dɵ dt θ (s) Ƭ(s) = H(s) = 1 J s + B s + K 2
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MODELING ELECTROMECHANICAL SYSTEMS DC Generator is driven mechanically by a prime mover. The shaft excite the field winding The equation for the field circuit is: E f (s) = (s L f + R f ) I f (s) or e f i L e g e a L a R a Z L i f R f L f Field Circuit Load Armature Circuit e f = R f i f + L f dt di f A
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MODELING ELECTROMECHANICAL SYSTEMS DC Generator The equation for the armature circuit is: The armature voltage v g is generated through field flux as shown by the equation: The flux ɸ is directly proportional to the field current, as shown by the equation: e g = K g i f E g = [s L a + R a + Z L (s)] I a (s) or e a = i a Z L E a = I a (s) Z L (s) E g (s) = K g i f (s) K is a parameter determined by physical structure of the generator & angular velocity of the armature is assumed to be constant e g = R a ia + L a + e a dt di a e g = K ɸ dt dɵ C D B
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MODELING ELECTROMECHANICAL SYSTEMS DC Generator From equations A, B, C, and D The system transfer function: The system block diagram is: G(s) = E f (s) E a (s) (sL f + R f ) [s L a + R a + Z L (s)] K g Z L (s) = E a (s) I a (s) E f (s ) 1 [s L a + R a + Z L (s)] 1 (s L f + R f ) K g Z L (s) I f (s ) E g (s )
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MODELING ELECTROMECHANICAL SYSTEMS Servomotor (DC Motor) • Apply a dc source to the armature • Excite the field (sets up air-gap flux) Stationary field winding, or Permanent magnets <ul><li>Commutator works as an inverter, </li></ul><ul><li>converts dc terminal voltage to ac </li></ul><ul><li>voltage on rotating armature winding </li></ul>The voltage generated in the armature coil because of the motion of the coil in the motor’s magnetic field is called the back emf e m (t) = K Φ d θ dt , θ e s B L a R a J e a e m R s
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MODELING ELECTROMECHANICAL SYSTEMS Servomotor The equation for the armature circuit is: Where K is a motor parameter, Φ is filed flux and θ is the angle of motor shaft. If we assume that the flux Φ is constant , then E s (s) = [s L a + R s + R a ] I a (s) + E m (s) E m (s) = K m s Θ (s) e s (t) = (R s + R a ) i a (t) + L a + e m (t) dt di a 2 1 e m (t) = K m d θ dt I a (s) = E s (s) - E m (s) s L a + R s + R a
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MODELING ELECTROMECHANICAL SYSTEMS or The torque is proportional to the flux and the armature current. Servomotor For the mechanical load the torque equation is Ƭ(s) = [s 2 J(s) +s B] Θ (s) Equations 1,2,3 and 4 will give us the system block diagram 3 = K i Φ i a (t) (t) = K i a (t) (t) Ƭ(s) = K I a(s ) J + B = (t) d 2 θ dt 2 d θ dt 4
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MODELING ELECTROMECHANICAL SYSTEMS Block Diagram of Servomotor I a (s) E s (s) H(s)=s K m G 1 (s)= 1 s L a + R s + R a G 2 (s)= S 2 J + s B 1 K E m (s) Θ (s) Ƭ(s) E s (s) - E m (s) I a (s) E s (s) H(s) = K m 1 s L a + R s + R a S J + B 1 K E m (s) Θ (s) Ƭ(s) E s (s) - E m (s) 1 s Θ (s) .
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MODELING ELECTROMECHANICAL SYSTEMS Transfer function of Servomotor Approximation can be made by ignoring the armature inductance G(s) = s 3 J L a + s 2 (J R s +J R a + B L a ) + s ( B R s + B R a + K m K ) K G(s) = s 3 K 1 + s 2 K 2 + s K 3 K G(s) = s(s 2 K 1 + s K 2 + K 3 ) K G(s) = s 2 (J R s + J R a ) + s ( B R s + B R a + K m K ) K G(s) = s 2 J R + s ( B R+ K m K ) K G(s) = E s (s) Θ (s) G 1 (s) K G 2 (s) 1 + K G 1 (s) G 2 (s) H(S) =
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STATE SPACE MODEL <ul><li>Definition: </li></ul><ul><li>State Variable or State Space Model is a set of first order coupled differential equations, usually written in vector matrix form. It is a mathematical model of a physical system as a set of input, output and state variables related by first-order coupled differential equations, which preserve the input- output relationship. </li></ul><ul><li>Advantages </li></ul><ul><li>In addition to the input- output characteristics, the internal characteristics of the system are represented. </li></ul><ul><li>Provides a convenient and compact way to model and analyze systems with multiple inputs and outputs (MIMO). </li></ul><ul><li>Computer aided analysis and design of state models are performed more easily on digital computer for higher order systems. </li></ul><ul><li>We can feedback more information(internal variables) about the plant to perform more or complete control of the system. </li></ul><ul><li>This model is required for simulation of complex systems. </li></ul>
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EXAMPLE OF STATE SPACE MODEL Linear Mechanical translational system: The differential equation model is The transfer function model is This model gives a description of position y(t) as a function of force f(t). If we also want information of velocity, the state variable model give the solution by defining two state variables as X 1 (t) = y(t) f(t) M K B y(t) M + B + K y = f 2 dy dt 2 dy dt = - - y + f(t) 2 dy dt 2 dy dt B M K M 1 M Y(s) F(s) = G(s) = 1 M s + B s + K 2 dy dt X 2 (t) = 2 dy dt 2 X 2 (t) = .
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EXAMPLE OF STATE SPACE MODEL Linear Mechanical translational system: is position x 1 (t) = y(t) is velocity y(t) = x 1 (t) 1 2 3 1 and 2 are first order state equations and 3 is the output equation, represent the second order system. These equations are usually written in vector matrix form (standard form), are called state equations of the system, which can be manipulated easily. dy dt x 2 (t) = x 1 (t) = x 2 (t) . 2 dy dt 2 X 2 (t) = . = - - + f(t) B M K M 1 M x 2 (t) x 1 (t)
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EXAMPLE OF STATE SPACE MODEL State Space Model The internal state variables are the smallest possible subset of system variables that can represent the entire state of the system at any given time. The minimum number of state variables required to represent a given system, n , is usually equal to the order of the system's defining differential equation, or is equal to the order of the transfer function's denominator after it has been reduced to a proper fraction. - - B M K M 1 M = + 0 1 x 1 (t) 0 f(t) x 2 (t) x 1 (t) . x 2 (t) . y(t) = 0 1 x 1 (t) x 2 (t)
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STANDARD FORM OF STATE SPACE MODEL y (t) = C x(t) + D u(t) Where x(t) = state vector (n × 1) vector of the states of an nth-order system u(t) = input vector (r × 1) vector composed of the system input functions y(t) = output vector (p × 1) vector composed of the defined outputs of the system A = (n × n) system matrix B = (n × r) input matrix C = (p × n) output matrix D = (p × r) feed-forward matrix (usually it is zero) x (t) = A x(t) + B u(t) .
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SOLUTION OF STATE EQUATIONS The standard form of state equation is given by The Laplace transform in matrix form can be written as: Where x(0) = [ x 1 (0) x 2 (0) . . . x n (0) ] T ---------------- 1 The inverse Laplace transform will give the solution of state equation, the state vector x(t). sX (s) - x(0)= A X(s) + B U(s) sX (s) - A X(s) = x(0) + B U(s) (sI – A) X(s) = x(0) + B U(s) X(s) = (sI – A) -1 [ x(0) + B U(s) ] x (t) = A x (t) + B u(t) .
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SOLUTION OF STATE EQUATIONS The matrix (s I – A ) -1 is called the resolvant of A and is written as: Φ (s) = (s I – A ) -1 The inverse Laplace transform of this term is defined as the state transition matrix: φ (t) = £ -1 [(s I – A ) -1 ] This matrix is also called the fundamental matrix and is (n×n) for nth order system. the state matrix can be written as: X(s) = Φ (s) x(0) + Φ (s) B U(s) ] The inverse Laplace transform of the 2 nd term in this equation can be expressed as a convolution integral. x(t) = φ (t) x(0) + φ (t) B u(t - ) d ---------- 2 Both equations 1& 2 can be used for the solution of state equations.
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SOLUTION OF STATE EQUATIONS Properties of state transition matrix φ (t) : φ (0) = I (identity matrix) φ (t) is nonsingular for finite elements in A φ -1 (t) = φ (-t) φ (t 1 – t 2 ) φ (t 2 – t 3 ) = φ (t 1 – t 3 ) φ (T) φ (T) = φ (2T) The state transition matrix φ (t) satisfies the homogenous state equation, Thus Let e At is the solution then Therefore, the state transition matrix φ (t) is also defined as: dx(t) dt = A x(t) d φ (t) dt = A φ (t) de At dt = A e At φ (t) = e At = I + A t + A 2 t 2 + A 3 t 3 + . . . 1 3! 1 2!
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SIMULATION DIAGRAMS A simulation diagram is a type of either block diagram or signal flow diagram that is constructed to have a specified transfer function or to model specified set of differential equations. It is useful for construction computer simulation of a system. It is very easy to get a state model from the simulation diagram. The basic element of the simulation diagram is the integrator. If y(t) = x(t) dt The Laplace Transform of this equation is Y(s) = X(s) y(t) x(t) Y(s) X(s) 1 s x(t) x(t) . 1 s 1 s
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SIMULATION DIAGRAMS From system differential equations The transfer function of the device that integrate is , if output of the integrator is y(t) then the input is . Similarly, if input is then out put of the integrator will be . Lets take the differential equation of mechanical translational system. The simulation diagram can be constructed from the differential equation by combination of integrators, gain and summing junction as: y(t) . y(t) .. . 2 dy dt 2 = - - + f(t) B M K M 1 M y(t) y(t) y(t) . 1 s y(t) . y(t) f(t) B M K M 1 M y(t) .. 1 s 1 s
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SIMULATION DIAGRAMS If simulation diagram is constructed from the differential equations then it will be unique, but if it is constructed from system transfer function then it not unique. The general form of system transfer function is: Two different type of simulation diagrams can be constructed from the general form of transfer function, for example if n = 3 (a) Control canonical form (b) Observer canonical form From system transfer functions b n-1 s n-1 +b n-2 s n-2 + ……. b 0 s n + a n-1 s n-1 +a n-2 s n-2 + ……. a 0 G(s) = b 2 s 2 + b 1 s + ……. b 0 s 3 + a 2 s 2 +a 1 s + ……. a 0 G(s) =
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SIMULATION DIAGRAMS Control Canonical Form x 2 . a 0 y(t) f(t) 1 s 1 s 1 s a 1 a 2 b 1 b 0 b 2 x 1 x 1 . x 3 x 2 x 3 .
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SIMULATION DIAGRAMS Observer Canonical Form Once simulation diagram is constructed, the state model of the system can easily be obtained by assigning a state variable to the out put of each integrator and write equation for each state and system output. x 2 . x 1 . x 3 . y(t) a 0 u(t) 1 s 1 s 1 s a 1 a 2 b 1 b 0 b 2 x 1 x 3 x 2
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STATE MODEL FROM SIMULATION DIAGRAMS State model of the control canonical fo rm State model of the observer canonical form x = . -a 0 u 1 -a 1 -a 2 x + 0 0 0 0 0 0 1 1 y = x b 1 b 0 b 2 . x = -a 0 1 -a 1 -a 2 0 0 0 0 1 u x + b 1 b 0 b 2 0 0 1 y = x
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