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//SYS21/D:/B&H3B2/ACE/REVISES(08-08-01)/ACEC08.3D ± 232 ± [232±271/40] 9.8.2001 2:34PM 8 State-space methods for control system design 8.1 The state-space-approach The classical control system design techniques discussed in Chapters 5±7 are gener- ally only applicable to (a) Single Input, Single Output (SISO) systems (b) Systems that are linear (or can be linearized) and are time invariant (have parameters that do not vary with time). The state-space approach is a generalized time-domain method for modelling, ana- lysing and designing a wide range of control systems and is particularly well suited to digital computational techniques. The approach can deal with (a) Multiple Input, Multiple Output (MIMO) systems, or multivariable systems (b) Non-linear and time-variant systems (c) Alternative controller design approaches. 8.1.1 The concept of state The state of a system may be defined as: `The set of variables (called the state variables) which at some initial time t 0 , together with the input variables completely determine the behaviour of the system for time t ! t 0 '. The state variables are the smallest number of states that are required to describe the dynamic nature of the system, and it is not a necessary constraint that they are measurable. The manner in which the state variables change as a function of time may be thought of as a trajectory in n dimensional space, called the state-space. Two-dimensional state-space is sometimes referred to as the phase-plane when one state is the derivative of the other. //SYS21/D:/B&H3B2/ACE/REVISES(08-08-01)/ACEC08.3D ± 233 ± [232±271/40] 9.8.2001 2:34PM 8.1.2 The state vector differential equation The state of a system is described by a set of first-order differential equations in terms of the state variables (x 1 , x 2 , FFF, x n ) and input variables (u 1 , u 2 , FFF, u n ) in the general form dx 1 dt a 11 x 1 a 12 x 2 ÁÁÁa 1n x n b 11 u 1 ÁÁÁb 1m u m dx 2 dt a 21 x 1 a 22 x 2 ÁÁÁa 2n x n b 21 u 1 ÁÁÁb 2m u m dx n dt a n1 x 1 a n2 x 2 ÁÁÁa nn x n b n1 u 1 ÁÁÁb nm u m (8:1) The equations set (8.1) may be combined in matrix format. This results in the state vector differential equation x Ax Bu (8:2) Equation (8.2) is generally called the state equation(s), where lower-case boldface represents vectors and upper-case boldface represents matrices. Thus x is the n dimensional state vector x 1 x 2 F F F x n P T T T R Q U U U S (8:3) u is the m dimensional input vector u 1 u 2 F F F u m P T T T R Q U U U S (8:4) A is the n  n system matrix a 11 a 12 FFF a 1n a 21 a 22 FFF a 2n F F F a n1 a n2 FFF a nn P T T T R Q U U U S (8:5) B is the n  m control matrix b 11 FFF b 1m b 21 FFF b 2m F F F b n1 FFF b nm P T T T R Q U U U S (8:6) State-space methods for control system design 233 //SYS21/D:/B&H3B2/ACE/REVISES(08-08-01)/ACEC08.3D ± 234 ± [232±271/40] 9.8.2001 2:34PM In general, the outputs ( y 1 , y 2 , FFF, y n ) of a linear system can be related to the state variables and the input variables y Cx Du (8:7) Equation (8.7) is called the output equation(s). Example 8.1 Write down the state equation and output equation for the spring±mass±damper system shown in Figure 8.1(a). Solution State variables x 1 y (8:8) x 2 dy dt x 1 (8:9) Input variable u P(t)(8:10) Now F y m y From Figure 8.1(b) P(t) À Ky À C y m y or d 2 y dt 2 À K m y À C m y 1 m P(t)(8:11) C K Pt () m m (a) (b) C y Ky Pt () y ,,y 3 y Fig. 8.1 Spring^mass ^damper system and free-body diagram. 234 Advanced Control Engineering //SYS21/D:/B&H3B2/ACE/REVISES(08-08-01)/ACEC08.3D ± 235 ± [232±271/40] 9.8.2001 2:34PM From equations (8.9), (8.10) and (8.11) the set of first-order differential equations are x 1 x 2 x 2 À K m x 1 À C m x 2 1 m u (8:12) and the state equations become x 1 x 2 ! 01 À K m À C m P R Q S x 1 x 2 ! 0 1 m P R Q S u (8:13) From equation (8.8) the output equation is y [1 0] x 1 x 2 ! (8:14) State variables are not unique, and may be selected to suit the problem being studied. Example 8.2 For the RCL network shown in Figure 8.2, write down the state equations when (a) the state variables are v 2 (t) and v 2 (b) the state variables are v 2 (t) and i(t). Solution (a) x 1 v 2 (t) x 2 v 2 x 1 (8:15) From equation (2.37) LC d 2 v 2 dt 2 RC dv 2 dt v 2 v 1 (t)(8:16) From equations (8.15) and (8.16) the set of first-order differential equations are x 1 x 2 x 2 À 1 LC x 1 À RC LC x 2 1 LC u (8:17) R C it ( ) vt 1 ( ) L vt 2 ( ) Fig. 8.2 RCL network. State-space methods for control system design 235 //SYS21/D:/B&H3B2/ACE/REVISES(08-08-01)/ACEC08.3D ± 236 ± [232±271/40] 9.8.2001 2:34PM and the state equations are x 1 x 2 ! 01 À 1 LC ÀR L P R Q S x 1 x 2 ! 0 1 LC P R Q S u (8:18) (b) x 1 v 2 (t) x 2 i(t) (8:19) From equations (2.34) and (2.35) L di dt Àv 2 (t) À Ri(t) v 1 (t)(8:20) C dv 2 dt i(t)(8:21) Equations (8.20) and (8.21) are both first-order differential equations, and can be written in the form x 1 1 C x 2 x 2 À 1 L x 1 À R L x 2 1 L u (8:22) giving the state equations x 1 x 2 ! 01 À 1 L À R L P R Q S x 1 x 2 ! 0 1 L P R Q S u (8:23) Example 8.3 For the 2 mass system shown in Figure 8.3, find the state and output equation when the state variables are the position and velocity of each mass. Solution State variables x 1 y 1 x 2 y 1 x 3 y 2 x 4 y 2 System outputs y 1 , y 2 System inputs u P(t)(8:24) For mass m 1 F y m 1 y 1 K 2 ( y 2 À y 1 ) À K 1 y 1 P(t) À C 1 y 1 m 1 y 1 (8:25) For mass m 2 F y m 2 y 2 À K 2 ( y 2 À y 1 ) m 2 y 2 (8:26) 236 Advanced Control Engineering //SYS21/D:/B&H3B2/ACE/REVISES(08-08-01)/ACEC08.3D ± 237 ± [232±271/40] 9.8.2001 2:34PM From (8.24), (8.25) and (8.26), the four first-order differential equations are x 1 x 2 x 2 À K 1 m 1 À K 2 m 1 x 1 À C 1 m 1 x 2 K 2 m 1 x 3 1 m 1 u x 3 x 4 x 4 K 2 m 2 x 1 À K 2 m 2 x 3 (8:27) Hence the state equations are x 1 x 2 x 3 x 4 P T T R Q U U S 0100 À K 1 K 2 m 1 À C m 1 K 2 m 1 0 0001 K 2 m 2 0 À K 2 m 2 0 P T T T T T T R Q U U U U U U S x 1 x 2 x 3 x 4 P T T R Q U U S 0 1 m 1 0 0 P T T T R Q U U U S u (8:28) and the output equations are y 1 y 2 ! 1000 0010 ! x 1 x 2 x 3 x 4 P T T R Q U U S (8:29) K 1 y t 1 () Pt () m 1 C 1 yt 2 () m 2 m 2 Ky 11 Pt () C 11 y y 222 , y , 3 y 111 , y 3 , Ky 2 ()– 21 y y 2 > y 1 (a) (b) K 2 m 1 Fig. 8.3 Two-mass system and free-body diagrams. State-space methods for control system design 237 //SYS21/D:/B&H3B2/ACE/REVISES(08-08-01)/ACEC08.3D ± 238 ± [232±271/40] 9.8.2001 2:34PM 8.1.3 State equations from transfer functions Consider the general differential equation d n y dt n a nÀ1 d nÀ1 y dt nÀ1 ÁÁÁa 1 dy dt a 0 y b nÀ1 d nÀ1 u dt nÀ1 ÁÁÁb 1 du dt b 0 u (8:30) Equation (8.30) can be represented by the transfer function shown in Figure 8.4. Define a set of state variables such that x 1 x 2 x 2 x 3 F F F F F F x n Àa 0 x 1 À a 1 x 2 ÀÁÁÁÀa nÀ1 x n u (8:31) and an output equation y b 0 x 1 b 1 x 2 ÁÁÁb nÀ1 x n (8:32) Then the state equation is x 1 x 2 F F F x nÀ1 x n P T T T T T R Q U U U U U S 010FFF 0 001FFF 0 F F F 000FFF 1 Àa 0 Àa 1 Àa 2 FFF Àa nÀ1 P T T T T R Q U U U U S x 1 x 2 F F F x nÀ1 x n P T T T T T R Q U U U U U S 0 0 F F F 0 1 P T T T T R Q U U U U S u (8:33) The state-space representation in equation (8.33) is called the controllable canonical form and the output equation is y [ b 0 b 1 b 2 FFF b nÀ1 ] x 1 x 2 x 3 F F F x n P T T T T T R Q U U U U U S (8:34) Example 8.4 (See also Appendix 1, examp84.m) Find the state and output equations for Y U (s) 4 s 3 3s 2 6s 2 Us () Ys () bs bsb n –110 n –1 + ++ sas asa n n ++ ++ n –1 10 –1 Fig. 8.4 Generalized transfer function. 238 Advanced Control Engineering //SYS21/D:/B&H3B2/ACE/REVISES(08-08-01)/ACEC08.3D ± 239 ± [232±271/40] 9.8.2001 2:34PM Solution State equation x 1 x 2 x 3 P R Q S 010 001 À2 À6 À3 P R Q S x 1 x 2 x 3 P R Q S 0 0 1 P R Q S u (8:35) Output equation y [400] x 1 x 2 x 3 P R Q S (8:36) Example 8.5 Find the state and output equations for Y U (s) 5s 2 7s 4 s 3 3s 2 6s 2 Solution The state equation is the same as (8.35). The output equation is y [475] x 1 x 2 x 3 P R Q S (8:37) 8.2 Solution of the state vector differential equation Consider the first-order differential equation dx dt ax(t) bu(t)(8:38) where x(t) and u(t) are scalar functions of time. Take Laplace transforms sX(s) À x(0) aX(s) bU(s)(8:39) where x(0) is the initial condition. From equation (8.39) X(s) x(0) (s À a) b (s À a) U(s)(8:40) Inverse transform x(t) e at x(0) t 0 e a(tÀ) bu()d (8:41) where the integral term in equation (8.41) is the convolution integral and is a dummy time variable. Note that e at 1 at a 2 t 2 23 ÁÁÁ a k t k k3 (8:42) State-space methods for control system design 239 //SYS21/D:/B&H3B2/ACE/REVISES(08-08-01)/ACEC08.3D ± 240 ± [232±271/40] 9.8.2001 2:34PM Consider now the state vector differential equation x Ax Bu (8:43) Taking Laplace transforms sX(s) À x(0) AX(s) BU(s)(8:44) (sI À A)X(s) x(0) BU(s) Pre-multiplying by (sI À A) À1 X(s) (sI À A) À1 x(0) (sI À A) À1 BU(s)(8:45) Inverse transform x(t) e At x(0) t 0 e A(tÀ) BU()d (8:46) if the initial time is t 0 , then x(t) e A(tÀt 0 ) x(0) t t 0 e A(tÀ) Bu()d (8:47) The exponential matrix e At in equation (8.46) is called the state-transition matrix F(t) and represents the natural response of the system. Hence F(s) (sI À A) À1 (8:48) F(t) l À1 (sI À A) À1 e At (8:49) Alternatively F(t) I At A 2 t 2 23 ÁÁÁ A k t k k3 (8:50) Hence equation (8.46) can be written x(t) F(t)x(0) t 0 F(t À )Bu()d (8:51) In equation (8.51) the first term represents the response to a set of initial conditions, whilst the integral term represents the response to a forcing function. Characteristic equation Using a state variable representation of a system, the characteristic equation is given by j(sI À A)j0(8:52) 240 Advanced Control Engineering //SYS21/D:/B&H3B2/ACE/REVISES(08-08-01)/ACEC08.3D ± 241 ± [232±271/40] 9.8.2001 2:34PM 8.2.1 Transient solution from a set of initial conditions Example 8.6 For the spring±mass±damper system given in Example 8.1, Figure 8.1, the state equations are shown in equation (8.13) x 1 x 2 ! 01 À K m À C m P R Q S x 1 x 2 ! 0 1 m P R Q S u (8:53) Given: m 1kg, C 3 Ns/m, K 2 N/m, u(t) 0. Evaluate, (a) the characteristic equation, its roots, ! n and (b) the transition matrices f(s) and f(t) (c) the transient response of the state variables from the set of initial conditions y(0) 1:0, y(0) 0 Solution Since x 1 y and x 2 y, then x 1 (0) 1:0, x 2 (0) 0. Inserting values of system parameters into equation (8.53) gives x 1 x 2 ! 01 À2 À3 ! x 1 x 2 ! 0 1 ! u (a) (sI À A) s 0 0 s ! À 01 À2 À3 ! s À1 2(s 3) ! (8:54) From equation (8.52), the characteristic equation is j(sI À A)js(s 3) À(À2) s 2 3s 2 0(8:55) Roots of characteristic equation s À1, À2(8:56) Compare equation (8.55) with the denominator of the standard form in equation (3.43) ! 2 n 2 i.e ! n 1:414 rad/s 2! n 3 i.e 1:061 (8:57) (b) The inverse of any matrix A (see equation A2.17) is A À1 Adjoint A det A (8:58) From equation (8.48) F(s) (sI À A) À1 State-space methods for control system design 241 [...]... //SYS21/D:/B&H3B2/ACE/REVISES(0 8- 0 8- 0 1)/ACEC 08. 3D ± 2 48 ± [232±271/40] 9 .8. 2001 2:34PM 2 48 Advanced Control Engineering 8. 4 8. 4.1 Control of multivariable systems Controllability and observability The concepts of controllability and observability were introduced by Kalman (1960) and play an important role in the control of multivariable systems A system is said to be controllable if a control vector u(t) exists... 0 0 ! 0 I 1 (8: 1 18) Thus proving that equation (8. 1 08) is already in the controllable canonical form Since TÀ1 is also I, substitute (8. 1 18) into (8. 116) K [4 0 ]I [ 4 0] (8: 119) (c) Ackermann's formula: From (8. 103) K [0 1 ]MÀ1 (A) (8: 120) From (8. 117) MÀ1 1 À4 À1 À1 ! À1 4 0 1 From (8. 104) (A) A2 1 A 0 I 1 0 ! (8: 121) //SYS21/D:/B&H3B2/ACE/REVISES(0 8- 0 8- 0 1)/ACEC 08. 3D ± 254 ± [232±271/40]... ! 0:0045 % (8: 85) 0: 086 17 (b) Using the values of A(T ) and B(T ) given in equations (8. 81) and (8. 84), together with the matrix vector difference equation (8. 76), the first few recursive steps of the discrete solution to a step input to the system is given in equation (8. 86) //SYS21/D:/B&H3B2/ACE/REVISES(0 8- 0 8- 0 1)/ACEC 08. 3D ± 247 ± [232±271/40] 9 .8. 2001 2:34PM State-space methods for control system... //SYS21/D:/B&H3B2/ACE/REVISES(0 8- 0 8- 0 1)/ACEC 08. 3D ± 260 ± [232±271/40] 9 .8. 2001 2:34PM 260 Advanced Control Engineering Using the definition of (A) in equation (8. 104) Ke (A2 1 A 0 I) Ke 8. 4.4 À2 À3 ! 6 7 ! 98 7 1 14 98 14 0 10 À20 À30 ! ! 0 0 ! 1 0 0 1 !À1 100 0 0 0 1 !! 100 ! (8: 149) 1 0 0 ! 1 77 0 1 1 ! ! ! 7 0 7 77 1 77 0 ! 1 (8: 150) Effect of a full-order state observer on a closed-loop system Figure 8. 10... (8. 104) Example 8. 12 (See also Appendix 1, examp812.m) A system is described by ! ! ! ! x1 0 1 x1 0 u À2 À3 x2 1 x2 ! x1 y [1 0] x2 (8: 134) //SYS21/D:/B&H3B2/ACE/REVISES(0 8- 0 8- 0 1)/ACEC 08. 3D ± 2 58 ± [232±271/40] 9 .8. 2001 2:34PM 2 58 Advanced Control Engineering Design a full-order observer that has an undamped natural frequency of 10 rad/s and a damping ratio of 0.5 Solution From equation (8. 89),... method (b) The controllable canonical form method (c) Ackermann's formula such that the closed-loop poles have the values s À2, s À2 Solution From the open-loop transfer function y 4y u (8: 105) x1 y (8: 106) x1 x2 x2 À4x2 u (8: 107) Let Then //SYS21/D:/B&H3B2/ACE/REVISES(0 8- 0 8- 0 1)/ACEC 08. 3D ± 252 ± [232±271/40] 9 .8. 2001 2:34PM 252 Advanced Control Engineering Equation (8. 106) provides...//SYS21/D:/B&H3B2/ACE/REVISES(0 8- 0 8- 0 1)/ACEC 08. 3D ± 242 ± [232±271/40] 9 .8. 2001 2:34PM 242 Advanced Control Engineering Using the standard matrix operations given in Appendix 2, equation (A2.12) ! (s 3) 2 Minors of F(s) À1 s ! (s 3) À2 Co-factors of F(s) 1 s The Adjoint matrix is the transpose of the Co-factor matrix ! (s 3) 1 Adjoint of F(s) À2 s (8: 59) Hence, from equations (8. 58) and (8. 48) P Q (s 3)... ! x1 y [ 1 À1 ] x2 Solution From equation (8. 88) the controllability matrix is M [B X AB] where AB À2 0 3 À5 ! ! ! 1 À2 0 3 hence 0 M [B X AB] 1 À2 3 ! (8: 90) //SYS21/D:/B&H3B2/ACE/REVISES(0 8- 0 8- 0 1)/ACEC 08. 3D ± 249 ± [232±271/40] 9 .8. 2001 2:34PM State-space methods for control system design 249 Equation (8. 90) is non-singular since it has a non-zero determinant Also the two row and column... (8: 140) //SYS21/D:/B&H3B2/ACE/REVISES(0 8- 0 8- 0 1)/ACEC 08. 3D ± 259 ± [232±271/40] 9 .8. 2001 2:34PM State-space methods for control system design 259 From equation (8. 140) (3 ke1 ) 10, ke1 7 (8: 141) (3ke1 2 ke2 ) 100 ke2 100 À 2 À 21 77 (8: 142) (b) Observable canonical form method: From equation (8. 132) 4 5 0 À a0 Ke Q 1 À a1 4 5 100 À 2 Q 10 À 3 4 5 98 Q 7 (8: 143) From equation (8. 133)... ]TÀ1 (8: 100) where T is a transformation matrix that transforms the system state equation into the controllable canonical form (see equation (8. 33)) T MW (8: 101) //SYS21/D:/B&H3B2/ACE/REVISES(0 8- 0 8- 0 1)/ACEC 08. 3D ± 251 ± [232±271/40] 9 .8. 2001 2:34PM State-space methods for control system design 251 where M is the controllability matrix, P a1 a2 T a2 a3 T T F WT F T F R anÀ1 1 1 0 equation (8. 88) Q . À2 13 ! (8: 90) 2 48 Advanced Control Engineering //SYS21/D:/B&H3B2/ACE/REVISES(0 8- 0 8- 0 1)/ACEC 08. 3D ± 249 ± [232±271/40] 9 .8. 2001 2:34PM Equation (8. 90) is non-singular since it has a non-zero. y 3 , Ky 2 ()– 21 y y 2 > y 1 (a) (b) K 2 m 1 Fig. 8. 3 Two-mass system and free-body diagrams. State-space methods for control system design 237 //SYS21/D:/B&H3B2/ACE/REVISES(0 8- 0 8- 0 1)/ACEC 08. 3D ± 2 38 ± [232±271/40] 9 .8. 2001 2:34PM 8. 1.3. ! State-space methods for control system design 247 //SYS21/D:/B&H3B2/ACE/REVISES(0 8- 0 8- 0 1)/ACEC 08. 3D ± 2 48 ± [232±271/40] 9 .8. 2001 2:34PM 8. 4 Control of multivariable systems 8. 4.1 Controllability
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