The document provides an overview of the course ECI 660 Linear Control Systems. The course covers topics such as system modeling, state space representation, stability analysis, controllability and observability, and linear feedback control design techniques including root locus and pole placement. The course uses text books on linear system theory and feedback control systems. Key concepts covered include modeling physical systems using differential and state space equations, analyzing system properties, and designing compensators to improve system performance.

2. INTRODUCTION

3. LINEAR CONTROL SYSTEMS

25. EFFECT OF GAIN ON SYSTEM PERFORMANCE

26. EFFECT OF GAIN ON SYSTEM PERFORMANCE

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 - - -

51. EFFECT OF GAIN ON SYSTEM PERFORMANCE

52. EFFECT OF GAIN ON SYSTEM PERFORMANCE

53. 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

54. 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

55. 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 - - -

57. 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)

58. 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

59. 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

60. 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

61. 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

62. 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 )

64. 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

65. 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

66. 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) .

67. 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) =

69. 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) = .

70. 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)

71. 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)

72. 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) .

73. 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) .

74. 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.

75. 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!

76. 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

77. 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

78. 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) =

79. 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 .

80. 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

81. 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

82. ANY QUESTION

83. THANK YOU