2.33

State Space Control I. VAJK

(2005)

Please note that in this section the language of control system is used, which differs somewhat from the terminology used by engineers working in the process industries. Therefore it might be of value to note that when the term system appears in this text, it refers to the process; when system output is used, that term refers to the controlled variable y(t); when system input is referred to, that term is equivalent to the manipulated variable u(t); and when the term reference signal is used, it, in the everyday PID terminology, means set point.

INTRODUCTION Using state space control, the poles of the system to be controlled can be placed to arbitrary positions. To achieve a satisfactory dynamic system behavior, the roots of the closedloop characteristic equations are to be assigned properly. The predefined system behavior can be achieved by applying proportional feedback for all the state variables of the system. The controller given by state variable feedback is realizable if and only if the system is controllable. The control strategy requires access to all the state variables of the system. In practice, however, this condition is rarely satisfied by most systems. This implies that the unmeasured state variables are supposed to be estimated. This is achievable if and only if the system is observable. Consequently, the complete control strategy will turn out to be a combination of a state observer and a state feedback, where the unmeasured state variables will be substituted by their estimated values. The state space control techniques using both state variable estimation and feedback can be converted to traditional feedback control schemes realized by transfer functions.

STATE SPACE DESCRIPTION Linear dynamic systems can be described in a number of mathematical forms. One of these is the impulse response description. Assuming that the impulse response is available, the system output [in process control, the controlled variable y(t)] at an arbitrary time can be determined, if up to that time the history of the system input [in process control, the manipulated

variable u(t)] is known. [The controlled variable y(t) — system output — can be calculated by evaluating a convolution integral.] Another way to describe a linear dynamic process is to introduce the use of state variables. In this case there is no need to know how the system developed to arrive to a certain state at a given time instant; the system output can be calculated by using the instantaneous value of the state variables and that of the system input. When selecting state variables in a system, it is reasonable to choose variables that are unable to exhibit abrupt changes. Consider the state space model (or in short, state model) of a system given by x (t ) = Ax(t ) + Bu(t ) y(t ) = Cx(t ) + Du(t )

2.33(1)

where x( t) is the state vector with n entries, while u(t ) and y(t ) are the system input and output signals, respectively. In the case of single-input single-output (SISO) systems both u(t ) and y(t) are scalar quantities. The A(n by n) and B (n by 1) matrices in Equations 2.33(1) will determine how the state variables develop from an initial state if a given input is applied to the system. The matrices C (1 by n) and D(1 by 1 ) tell how to find the system output once the state variables and the input signal are available for the calculations. The evaluation of Equations 2.33(1) suggests that the system dynamics is described by n first-order differential equations. Note that Equations 2.33(1) represent an appropriate way to uniformly describe linear systems with multiple inputs and multiple outputs (MIMO systems). Finally observe that the D matrix allows a direct contribution from the input u(t) to the output y(t). In all real processes it takes some time for a change in the manipulated variable to have an effect on the controlled variable. This delay between the input and the output results in D = 0. A state model is not unique in the sense that several sets of {A, B, C, D} exist to ensure input/output equivalence. Equivalent state models can be linked together by using a simple linear transformation, called similarity transformation. Assume that T (n by n) is an arbitrary nonsingular matrix, then x(t) can be transformed to a new state representation z(t) by z(t ) = Tx(t ).

2.33(2) 393

© 2006 by Béla Lipták

394

Control Theory

The transformed state model can simply be derived as z (t ) = TAT−1z(t ) + TBu(t )

2.33(3)

y(t ) = CT−1z(t ) + Du(t )

Among the state models several canonical forms exist (e.g., controllability, observability, diagonal forms). System controllability and observability are important features attached to state models: •

A system is controllable if there exists a finite control input to govern the system from an arbitrary initial state to an arbitrary final state within a finite time horizon. The ability to control (controllability) can be checked by considering the rank of the controllability matrix Q = [B AB A 2B ... A n−1B]

•

2.33(4)

The system is controllable if and only if the controllability matrix is of full rank (rank is n). In case of MIMO systems the controllability matrix is not a square matrix, so one way to check if the rank of Q is n is to T T consider QQ , which is an n × n matrix. If QQ is not singular it indicates that Q is of full rank and the system is controllable. A system is observable if from knowing the input and output records of a system for a finite time period the initial value of the state variables can be calculated. A system is observable if and only if the observability matrix

Equations 2.33(1) exhibit the fundamental description of SISO time-invariant linear dynamic systems. Using the Laplace transformation we obtain: sx(s) − x(t0 ) = Ax(s) + Bu(s) y(s) = Cx(s) + Du(s)

2.33(6)

where x(t0) is the initial value of the state vector and s stands for the Laplace operator. Rearranging Equations 2.33(6), the Laplace transform of the output signal can be expressed as follows: −1

y(s) = (C(sI − A) B + D)u(s) + C(sI − A) x(t0 )

2.33(7)

Observe the term H (s) = C(sI − A) B + D

© 2006 by Béla Lipták

C adj(sI − A)B B( s ) +D= det(sI − A) A(s)

2.33(9)

where B(s) and A(s) are both polynomials. It can be shown that for systems that are controllable and observable, B(s) and A(s) are relatively prime (often called coprime) polynomials, i.e., polynomials having no common roots. Note that if D = 0 holds, then the order of B(s) is less than the order of A(s). Systems with D = 0 are also called strictly proper systems. The rest of this section is devoted to strictly proper systems.

CONTROL LAW DESIGN As it was indicated earlier, state variable feedback can ensure arbitrary pole locations for the closed-loop system. In other words, using state variable feedback, the poles of the original system (i.e., the poles of the system with no state variable feedback) can be moved to arbitrarily assigned positions in the complex plane. The question here is how to find the gains along the feedback path to achieve the required pole locations. Assume first that all the state variables are directly available (measured) for feedback. Later on, this condition will be relaxed.

Assume that the feedback is applied around the state model 2.33(5)

is of full rank (n). Again, for MIMO systems the sinT gularity of M M can alternatively be checked.

−1

H (s ) =

State Variable Feedback

 C    CA  M=  ...   n−1  CA 

−1

establishing the relation between the input and output signals. H(s) is the well-known transfer function of the system. Some algebra leads to

x (t ) = Ax(t ) + Bu(t ) y(t ) = Cx(t )

2.33(10)

by using the linear combination of the state variables according to u(t ) = −K x(t ) + u r (t )

2.33(11)

where ur (t) is the set point (scalar value in case of SISO systems) of the system and K is the gain matrix (essentially a row vector in case of SISO systems). The block diagram of the closed-loop system is shown in Figure 2.33a. Designing a controller is after all a procedure to find appropriate values for the gain matrix. Substituting Equation 2.33(11) back to Equation 2.33(10) results in the following differential equation for the closed-loop system: x (t ) = ( A − BK)x(t ) + B u r (t )

2.33(12)

Using the Laplace transforms, we obtain: y(s) = C(sIn − A + BK)−1 Bur (s) + C(sIn − A + BK)−1 x(t0 )

2.33(8)

2.33(13)

2.33 State Space Control

395

y(t)

C

Process

ur(t)

u(t)

1/s

B

x(t)

K

A

FIG. 2.33a Controlled process with state variable feedback.

and the matrix polynomial

or y( s ) =

C adj(sIn − A + BK) (Bur (s) + x(t0 )) 2.33(14) det(sIn − A + BK)

The above relation clearly shows that the closed-loop behavior is determined by the closed-loop characteristic polynomial:

α (s) = det(sIn − A + BK)

0

…

1]Q−1α ( A)

2.33(20)

2.33(15)

2.33(16)

Ignoring the effect of the nonzero initial value x(t0 ), Equation 2.33(16) shows that the state variable feedback just applied is equivalent to applying a series term A(s)/α (s) directly for the system given by the transfer function B(s)/ A(s). Next, the elements of the K gain matrix will be determined such that the roots of the closed-loop characteristic equation are {−s1, −s2, … −sn}. Using Equation 2.33(15) results in det(sIn − A + BK) = (s + s1 )(s + s2 )…(s + sn ) 2.33(17) Then the entries of K can be determined by matching the coefficients of the identical powers of s in the polynomials of the two sides in Equation 2.33(17). Ackermann derived a nice compact algorithm to find K directly. Introduce the polynomial

α (s) = (s + s1 )(s + s2 )…(s + sn ) = s n + α n−1s n−1 +  + α 0 2.33(18)

© 2006 by Béla Lipták

2.33(19)

The Ackermann formula presents the gain matrix by K = [0

Also, the overall relation between the set point and the output can be expressed by y( s ) B( s ) = C(sIn − A + BK)−1 B = ur (s) α (s )

α ( A) = A n + α n−1A n−1 +  + α 0 I

where Q denotes the controllability matrix. The Ackermann formula shows why the inverse of the controllability matrix should be nonsingular to get a gain matrix. If Q is nonsingular the gain matrix obtained by the Ackermann formula is unique. An even simpler way to determine the coefficients of the gain matrix can be obtained if the state model is assumed to be available in controllability form. Denoting the gain vector associated with the controllability form by Kc , consider a polynomial K(s) with coefficients delivered by Kc. Then, the characteristic polynomial of the closed-loop system simply becomes

α (s) = A(s) + K (s)

2.33(21)

Equation 2.33(21) offers a direct way to determine the feedback gain matrix; namely, K(s) can be calculated knowing the goal of the design, i.e., the location of the closed-loop poles. The presented state variable feedback can also be considered as a generalization of the traditional PD compensation. The generalization is accomplished in the sense that all the closed-loop poles are moved to desired positions. Consequently, all the components of the system response transients are under control. At the same time, according to the fundamental property of the PD compensation, the steady-state error will not go to zero if the system is disturbed by additive constant noise. This becomes only achievable by inserting integrator(s) into the loop.

396

Control Theory

Process

ur(t)

u(t)

1/s

B

yr (t) C x(t)

ki /s

y(t) K

A

FIG. 2.33b State feedback with PI action.

State Feedback with PI Control The state variable feedback concept outlined in the previous section represents a certain generalization of the PD compensation technique. To ensure zero steady-state error, an additional integrator should be inserted into the loop. This can easily be done by augmenting the size of the state vector by one more entry. Denoting the new state variable by z(t) we have z(t ) = y(t ) − yr (t )

2.33(22)

where yr (t) denotes the reference signal. The description of the augmented system becomes: x (t ) = Ax (t ) + Bu(t ) + Br yr (t ) y(t ) = Cx (t )

2.33(23)

 x(t )  A 0  B x (t ) =   A=  B=   z (t )  C 0  0 0 Br =   C = [C 0 ]  −1

2.33(24)

where

The state variable feedback will be realized by using the augmented state vector for feedback, according to u(t ) = −Kx (t ) = −Kx(t ) − ki z (t ) = − K

 x(t )  ki     z (t )  2.33(25)

Applying the above feedback, the characteristic polynomial of the closed-loop system can be written as  sI − A + BK Bki  α (s) = det(sIn+1 − A + BK) = det  n  C s   = s det(sIn − A + BK) + Cadj(sI n − A + BK)Bki 2.33(26)

© 2006 by Béla Lipták

Using the transfer functions introduced earlier, Equation 2.33(26) leads to

α (s) = s( A(s) + K (s)) + B(s)ki

2.33(27)

Relation to Equation 2.23(21) is clearly seen from the above form. Given α (s), A(s), and B(s), the feedback polynomial K(s) and the ki coefficient can both be determined. The block scheme of the state variable feedback extended by the integrator is shown in Figure 2.33b. The augmented system is capable of rejecting nonzero steady-state error. Note that the closed-loop system exhibits identical operational performance for following the reference signal (servo property), as well as for rejecting additive disturbances acting on the system output. A final remark here relates to the size of the feedback gain matrix. For SISO systems K is a vector with n entries, while for MIMO systems K is an m × n matrix, where m denotes the number of inputs. The entries of the gain matrix are adjusted to locate the closed-loop poles; however, there are only n poles to be located, resulting in n equations to determine m × n coefficients in the gain matrix. Consequently, there exist several gain matrices resulting in identical closedloop characteristic equations.

OBSERVER DESIGN So far a state variable feedback technique has been studied assuming that all the state variables are available for the feedback. This condition is rarely met in practice. In the following, how to modify the control strategy derived so far to be able to incorporate the unavailable state variables into the feedback structure will be discussed. The procedure developed to estimate the state variables based on the input/output records of the system is called state estimation or state reconstruction, and the related functional unit is called an observer.

2.33 State Space Control

Full Order Observer The state variables are to be estimated by the following linear model xˆ (t ) = Qxˆ (t ) + Ru(t ) + Sy(t )

xˆ (t ) = Axˆ (t ) + Bu(t ) + L(y(t ) − yˆ (t )) yˆ (t ) = Cxˆ (t )

2.33(29)

x (t ) = xˆ (t ) − x(t ),

x (t ) = ( A − LC)x (t ) + Ly(t )

2.33(31)

The error in the state estimation develops according to the characteristic polynomial

β (s) = det(sIn − A + LC)

2.33(32)

Various choices for the gain matrix L allow locating the poles of the above characteristic polynomial.

Process x(t)

C

2.33(33)

should be solved. Again, this can be done by matching the coefficients of the left-side and right-side polynomials in a very similar way as the feedback gain vector was determined by Equation 2.33(17) earlier. Introducing the polynomial

β (s) = (s + s1 )(s + s2 )…(s + sn ) = s n + βn−1s n−1 +  + β0 and the matrix polynomial

β ( A) = A n + βn−1A n−1 +  + β0 I

2.33(35)

the related Ackermann formula takes the following form:

2.33(30)

whose development is described by the following differential equation:

1/s

det(sIn − A + LC) = (s + s1 )(s + s2 )…(s + sn )

2.33(34)

The above expression clearly shows how the estimated system output is derived from the estimated state vector. The related block scheme, also called the asymptotic state observer, is shown in Figure 2.33c. The error in the state estimation can be expressed by

B

Assume working with single output systems. To design the way as the error in the state estimation decays, the poles of the characteristic Equation 2.33(32) should be located to predefined −s1, −s2, … −sn values. To determine the entries of the vector L, polynomial equation

2.33(28)

where xˆ (t ) denotes the estimated state vector. Formally, designing an observer is equivalent to determining the Q, R, and S matrices. One classical solution of this problem is to use a feedback in the observer itself driven by the deviation between the measured and estimated system outputs:

u(t)

397

L = β ( A)M −1[0

…

0

1]T

2.33(36)

where M still denotes the observability matrix. Equation 2.33(36) shows that the observer feedback gain vector L exists if and only if M is invertible. In other words, to be able to apply the Ackermann formula and to have a unique solution for the observer gain vector L, the system should be observable. The structure of the observer shown in Figure 2.33c can be transferred to an equivalent form shown in Figure 2.33d. According to this structure, the observer itself can be interpreted as a servo controller with a reference signal represented by the system output and a controlled signal represented by the estimated output signal. Further on the system parameters are stored in A, B, and C, while the system state variables are the estimated states. Comparing Figures 2.33a and 2.33d, it is seen that the structure of the state variable control and that of the full order

y(t) u(t)

A

B

L B

1/s

y(t) ˆ x(t)

C

L

1/s

yˆ(t) A

A

FIG. 2.33c Block scheme of the full order observer.

© 2006 by Béla Lipták

FIG. 2.33d Full order observer as a servo problem.

ˆ x(t)

C

ˆ y(t)

398

Control Theory

observer resemble each other to a large extent. The output signal, as well as the L and C matrices, play an identical role in the observer as the control signal, as well as the B and K matrices do in the state variable control. Parameters in matrix L and parameters in matrix K are to be freely adjusted for the observer and for the state variable control, respectively. In a sense, calculating the controller and observer feedback gain matrices represents dual problems. In this case, duality means that any one of the structures shown in Figures 2.33a and 2.33d can be turned to its dual form by reversing the direction of the signal propagation, interchanging the input and output signals (u ↔ y), and transforming the summation points to signal nodes and vice versa. The function realized by the observer can also be expressed in the form of a transfer function. Assuming zero initial state variable conditions, the solution of Equation 2.33(29) results in yˆ (s) = C(sIn − A + LC)−1 (Bu(s) + Ly(s))

2.33(37)

Elementary matrix operations used for Equation 2.33(32) lead to a polynomial separation:

β (s) = det(sIn − A + LC) = A(s) + L (s)

2.33(38)

where A(s) = det(sIn − A)

2.33(39)

and

observer intends to follow contains a constant, nondecaying, nonzero component (typically derived from constant disturbance acting on the input), the state estimation will be biased. The concept of duality suggests that this bias can be eliminated by adding an integrator to the observer circuit. Consider a system together with an unknown constant additive noise (augmented system): x (t ) = Ax(t ) + B(u(t ) + w(t )) w (t ) = 0 y(t ) = Cx(t )

Following the technique applied earlier, the state variables of the augmented system can be estimated along the following equations: xˆ (t ) = Axˆ (t ) + B(u(t ) + wˆ (t )) + L( y(t ) − yˆ (t )) wˆ (t ) = li ( y(t ) − yˆ (t ))

2.33(40)

Using the observability form for the state model studied, coefficients of the L(s) polynomial are identical to the entries of matrix L. Then using the above polynomials, Equation 2.33(37) turns to

β (s) yˆ (s) = ( A(s) + L (s)) yˆ (s) = B(s)u(s) + L (s) y(s) 2.33(41) For MIMO systems the L matrix consists of n × p entries, where p denotes the number of the system outputs. However, the required location of the observer poles defines only n equations, thus the problem — just as when designing a MIMO state variable feedback control— is underdetermined. Full Order Observer for Constant Error It is a well-known fact from classical control theory that the application of proportional controllers ensures finite, nonzero control error when constant set points or constant disturbances are applied. Not surprisingly at all, if the output signal the

© 2006 by Béla Lipták

2.33(43)

yˆ (t ) = Cxˆ (t ) The characteristic polynomial of the augmented system is  sI − A + LC β (s) = det  n liC 

−B  s 

2.33(44)

The roots of the characteristic equation can be affected via the feedback elements in L and li . Using equivalent transformations for the above determinant, Equation 2.33(44) can be separated by

β (s) = s( A(s) + L (s)) + B(s)li L (s) = Cadj(sIn − A)L

2.33(42)

2.33(45)

Then in the knowledge of β (s), A(s), and B(s), the coefficients in L(s), and li itself, can be calculated. Reduced Order Observer If no significant disturbance is acting on the system output, there is no need to construct an observer providing an error in the estimated output signal converging to zero only asymptotically, rather than having an error in the estimated output signal being constantly zero. It is seen in Figure 2.33d that the feedback is utilizing the error in the estimated output signal and the state estimation is influenced by this error via the linear gain L. The error in the estimated output signal is forced to be constantly zero if the state estimation provides y(t ) = Cxˆ (t )

2.33(46)

If the output signals of the system are linearly independent from each other, they can be considered as state variables. Introduce the state vector such that all the output variables

2.33 State Space Control

show up in the state vector. Then the state vector can be separated according to  y(t )  z(t ) =    zr (t ) 

2.33(47)

y(t) T –1

T

u(t)

B

C

yˆ(t)

T

1/s

where zr (t) contains the state variables not measured directly. The transformation leading to the state vector by Equation 2.33(47) can be found by  y(t )   y(t )  C  z(t ) =   = Tx(t ) =   x(t ) =   Tx(t )  T   zr (t ) 

2.33(48)

The goal now is to find an estimation for the inaccessible zr (t) component of the state vector as the state vector of the following linear state model: zˆ r (t ) = Qzˆ r (t ) + Ru(t ) + Sy(t )

z r (t ) = zˆ r (t ) − zr (t )

2.33(50)

asymptotically converges to zero. The state model using the new (transformed) state vector takes the following form:  y (t )  C    =   A  H  z r (t )   T 

 y(t )  C  F    +   Bu(t )  zr (t )   T   y(t )  F     zr (t ) 

y(t ) = C  H

2.33(52)

has been applied. Performing the above operations, Equation 2.33(51) becomes z r (t ) = TAFzr (t ) + TAHy(t ) + TBu(t ) y (t ) = CAFzr (t ) + CAHy(t ) + CBu(t )

C

y(t)

FIG. 2.33e Block scheme of the reduced order observer.

observer; see Figure 2.33e for the block diagram. The error related to the estimation of the inaccessible state variables can be expressed by z r (t ) = TAFz r (t ),

2.33(55)

while the error related to the original state vector can be written as x (t ) = FTAx (t ) = (I n − HC)x (t )

2.33(56)

The initial condition of the original state vector can be calculated by x (t0 ) = Fz r (t0 ). Selecting appropriate values in H — keeping in mind that CH = Ip holds — the error by Equation 2.33(56) can arbitrarily be reduced. Note that here p denotes the number of the system outputs. Using the constraint by HC + FT = I n

−1

F 

xˆ (t)

F

2.33(51)

where the notation by C    =  H T 

H

A

2.33(49)

Find the matrices Q, R, and S in Equation 2.33(49) such that the estimation error

399

2.33(57)

(see Equation 2.33[52]), the reduced order observer shown in Figure 2.33e can equivalently be transformed to the form shown in Figure 2.33f. Note that the observer is using one single matrix (H) to reconstruct the state vector. Selecting the elements in matrix H, the constraint by CH = Ip should be taken into account. The number of the state variables involved in the observer shown in Figure 2.33f is not minimal; however, it is operating as a reduced order observer. Assuming one single output signal and a state model where this single output is one of the state variables, we have C = 1

2.33(53)

y(t ) = CFzr (t ) + CHy(t ) = y(t )

0

…

0  ,

2.33(58)

as well as a matrix H Then, using Equation 2.33(53), the following relation is derived zˆ r (t ) = TAFzˆ r (t ) + TAHy(t ) + TBu(t ) xˆ (t ) = Fzˆ r (t ) + Hy(t )

2.33(54)

to estimate the state variables. The functional unit realizing the above state estimation algorithm is called reduced order

© 2006 by Béla Lipták

H = [1

h1

…

hn−1 ]T = [1

 T ]T H

2.33(59)

of special structure. Separating the states as before leads to A A =  11  A 21

A12   A 22 

and

B  B =  1  B2 

2.33(60)

400

Control Theory

y(t) constraint: CH = Ip u(t)

B

C

H ˆ x(t)

1/s

C

y(t)

A

FIG. 2.33f Reduced order observer with extra state variables.

and the form of the reduced order observer is    ˆ zˆ r (t ) = ( A 22 − HA 12 )(z r (t ) + Hy(t )) + ( A 21 − HA11 ) y(t )  )u(t ) + (B − HB 2

ur(t)

u(t)

Process B

2.33(61)

Further on, the error in the state estimation is expressed by   z (t ) = (A 22 − HA 12 )z(t )

B

1/s A

Based on Equation 2.33(62), the disturbance rejection property of the closed-loop system equipped with the reduced order observer will be determined by the roots of the

K

ˆ x(t)

C

ˆ y(t)

2.33(63) can be considered as a complete controller (see Figure 2.33g). State reconstruction is realized using the measured variables. In the above part of this chapter, two state variable feedback strategies (a proportional and an extended one by an integrator), as well as four state reconstruction strategies (full order and reduced order observers with or without handling constant input disturbance) were presented. Any pair combined by a selected state variable feedback strategy and a selected observer strategy is applicable.

2.33(64)

Augmenting the state equation by the above relation, the number of the state variables will be increased, and an additional integrator will be part of the observer. The augmented reduced order observer is capable of rejecting disturbances of unknown constant values. COMBINED OBSERVER–CONTROLLER The state variable feedback and observer strategies discussed in the previous sections can be applied simultaneously, as well. The state variable feedback driven by the reconstructed states

© 2006 by Béla Lipták

y(t)

FIG. 2.33g Block scheme of the combined observer-controller.

 repcharacteristic equation. Note that the coefficients in H resent the freedom to locate the roots of the closed-loop characteristic equation. To be able to complete the design in that way, the subsystem belonging to the states in the reduced observer must be observable. In case of constant disturbances, even reduced order observers need special treatment. Assume a constant additive disturbance signal acting on the input: w = 0

C

L

2.33(62)

 β (s) = det(sIn−1 − A 22 + HA 12 )

x(t)

A

1

  y(t ) xˆ (t ) =   y(t )  ˆ z ( t ) + H   r

1/s

Combined Observer-Controller Behavior First consider a closed-loop system with state variable feedback using a full order observer to reconstruct the unmeasured state variables. Again, the system is described by x (t ) = Ax(t ) + Bu(t ) y(t ) = Cx(t )

2.33(65)

while the full order observer is given by xˆ (t ) = Axˆ (t ) + Bu(t ) + L(y(t ) − Cxˆ (t ))

2.33(66)

2.33 State Space Control

Applying the control strategy of u(t ) = −K xˆ (t ) + u r (t )

2.33(67)

Equations 2.33(65) through 2.33(67) allow deriving the following state equation for the closed-loop system:  x (t )   A   =   xˆ (t )   HC

  x(t )   B  − BK  +   u r (t )  A − BK − HC   xˆ (t )   B 

2.33(68)

where the system state vector and the observer state vector together form the complete state vector of the closed-loop system.  Selecting the state estimation error x(t ) = xˆ (t ) − x(t ) to replace the estimated state vector in the complete state vector, the following form can be obtained:  x (t )   A − BK    =   x(t )   0

− BK   x(t )   B      +   u r (t ) A − HC   x(t )   0 

2.33(69)

The above form allows the derivation of the characteristic polynomial of the closed-loop system in an easy way:   sI − A + BK  BK det   n  0 sIn − A + LC   = det(sIn − A + BK) det(sIn − A + LC)

Equation 2.33(70) shows that the roots of the closed-loop characteristic equation contain roots determined by the state variable feedback law and roots introduced by the observer. It is also seen that the state variable feedback and the observer can be designed independently from each other. In other words, the setting of the coefficients in the feedback gain vector does not depend on whether the feedback is using the state vector itself or its reconstructed value. In the literature, this property is referred to as the separation property or separation theory. As far as the concept of setting the dynamics of the state variable feedback and that of the observer, it is reasonable to design fast enough transient dynamics for the observer to force the error in the state vector reconstruction to converge quickly to zero. “Fast enough transient” means here that the observer dynamics are to be designed faster than the dynamics of the system being observed. Assigning too-short time constants for the observer may lead to unwanted closed-loop behavior; specifically, the state estimation may become extremely sensitive, especially in cases of noisy measurements. Considering SISO systems, find the transfer functions between the reference signal and the control input, as well as between the system output and the control input, respectively. Based on Equations 2.33(66) and 2.33(67), the control input can be written by u(s) = −K(sI n − A + LC + BK)−1 L( y(s) − yr (s)) 2.33(71)

© 2006 by Béla Lipták

For the sake of simplicity, assume zero initial conditions for the state vectors involved. Also, selecting a special reference signal, a closed-loop system exhibiting identical servo and disturbance rejection properties will be designed. Equation 2.33(71) can be expressed in the form of a rational transfer function: u(s ) = −

F (s ) ( y(s) − yr (s)) G (s )

2.33(72)

where the nth order denominator polynomial above can be expressed by G (s) = det(sIn − A + LC + BK),

2.33(73)

while the numerator polynomial F(s) is given by F (s) = K adj(sIn − A + LC + BK)L.

2.33(74)

Using the above polynomials the characteristic equation of the closed-loop system is P(s) = G (s) A(s) + F (s) B(s)

2.33(70)

401

2.33(75)

In harmony with Equation 2.33(70), it is seen that the closed-loop characteristic equation can be derived by multiplying the characteristic equations represented by the state variable feedback and the observer, respectively: P(s) = α (s)β (s) = G (s) A(s) + F (s) B(s)

2.33(76)

Studying the behavior of the closed-loop system, no distinction can be made between the poles of the state variable feedback and the observer. Theoretically, all the poles of the closed-loop system can be assigned for the design procedure. Given a system with A(s) and B(s) and a design goal specified by P(s), the polynomials F(s) and G(s) are to be determined to complete the closed-loop control. Assuming that controllability and observability both hold, as well as ord ( P(s)) = 2n, solving the Diophantine Equation 2.33(76), a unique solution is obtained for F(s) and G(s). The controller derived is strictly proper. For stable systems the presented control structure can also be transformed into an IMC (internal model control) form (see Figure 2.33h). In this structure a parallel process model is applied, preceded by a serial compensator. The overall transfer function becomes y ( s ) F ( s ) B( s ) = yr (s) α (s)β (s)

2.33(77)

Note that the number of state variables realizing the IMC structure is 3n, while the characteristic equation of the system remains as α (s)β (s) after eliminating the common terms.

402

Control Theory

Process yr(t)

F(s)A(s) b(s)a(s)

u(t)

y(t)

B(s) A(s)

B(s) A(s)

FIG. 2.33h Observer-controller as internal model controller.

If the measured control input is available for realizing the control strategy, it is reasonable to derive another form of the controller utilizing the control signal. Assume that u (t) is the control input directly driving the process and u0 (t) is the control signal released by the controller (see Figure 2.33i). The actual feedback and observer relations can be written as xˆ (t ) = ( A − LC)xˆ (t ) + Bu(t ) + L(y(t ) − y r (t )) u 0 (t ) = −Kxˆ (t )

2.33(78)

are strictly proper transfer functions. The transfer function form of the combined observer–controller is shown in Figure 2.33i. The presented versions require various amounts of computing power but result in identical control actions for a stable, linear, disturbance-free process. At the same time, for practical applications with slightly nonlinear behavior or input saturation, the exact calculations may lead to different closed-loop behavior. Note that the polynomial approach shown above can accomplish even more complex control strategies. The state variable feedback technique followed so far required the polynomial P(s) to be separable according to P(s) = α (s)β (s). A possible generalization may assume P(s) to be freely selectable. The control problem then is set up as follows: given A(s), B(s), P(s), and β (s) find the polynomials F(s) and G(s) such that F (s ) β (s ) and G (s ) − β (s ) β (s )

or for SISO systems in the form of transfer functions: u0 (s) = −K(sIn − A + LC)−1 (Bu(s) + L( y(s) − yr (s))) 2.33(79) With some straightforward manipulations:

β (s)u0 (s) = − F (s)( y(s) − yr (s)) − (G (s) − β (s))u(s). 2.33(80) In the above expression F (s ) β (s )

are both strictly proper. The control strategy derived in that way is more general than the state variable feedback control supported by an observer. Due to the limited available space, not all the eight versions (two feedback solutions and four observers) will be discussed here. In general, each additional integrator or constant input disturbance estimation will increase the order of the controller by one. Applying a reduced order observer will reduce the order of the controller. The resulting F(s)/β (s) and F(s)/G(s) blocks will be just proper, not strictly proper transfer functions. Transfer-Function Interpretation

and

A general controller is driven by the reference signal, the output signal, and the control signal. Assuming that the controller is linear and the number of the states used by the

G (s ) − β (s ) β (s )

Process yr(t)

F(s)

uo(t)

G(s) – b(s) b(s)

FIG. 2.33i Transfer function form of the combined observer-controller.

© 2006 by Béla Lipták

u(t)

B(s) A(s)

b(s)

y(t)

2.33 State Space Control

403

TABLE 2.33j Orders of the Transfer-Function Polynomials Equivalent to the Combined Observer–Controller Feedback

Observer

ord α (s)

ord β (s)

No. States

No. Integr.

ordG(s)

ordF(s)

P

O

n

n

n

0

n

n–1

PI

O

n+1

n

n+1

1

n

n

P

EO

n

n+1

n+1

1

n

n

PI

EO

n+1

n+1

n+2

2

n

n+1

P

R

n

n–1

n–1

0

n–1

n–1

PI

R

n+1

n–1

n

1

n–1

n

P

ER

n

n

n

1

n–1

n

PI

ER

n+1

n

n+1

2

n–1

n+1

controller is q, the most complicated possible controller can be described by p f (s)u(s) = pyr (s) yr (s) − py (s) y(s) − pu (s)u(s)

2.33(81)

The above controller is realizable if and only if ord( pf (s)) ≥ ord(pyr (s)), ord( p f (s)) ≥ ord( py (s)) and ord(pf (s)) > ord(pu (s)) are held for the polynomials involved. Designing identical servo and disturbance rejection properties we have pyr (s) = py(s). Then the control law by Equation 2.33(81) simplifies to p f (s)u(s) = py (s)( yr (s) − y(s)) − pu (s)u(s)

2.33(82)

resulting in u(s ) =

py (s ) p f (s) + pu (s)

( yr (s) − y(s))

2.33(84)

Solution of the above Diophantine equation results in the controller polynomials. The control law itself turns out to be u(s ) =

F (s ) ( y (s) − y(s)). s iG ( s ) r

2.33(85)

Comparing Equations 2.33(83) and 2.33(85), it is seen that the state variable feedback control is equivalent to a polynomialbased controller with py (s) = F (s) and p f (s) + pu (s) = s iG (s). If the control input is available and used for the control law calculations, the integrators included in the controller should be separated. In the case of state variable feedback control, the control input can be calculated by

β (s)u(s) =

F (s ) ( yr (s) − y(s)) − (G (s)s c − β (s))u(s) 2.33(86) si

© 2006 by Béla Lipták

P

State variable feedback (proportional feedback from the state variables)

PI

State variable feedback plus extra integral feedback from the output

O

Basic full order asymptotic observer

EO

Extended asymptotic observer (observer with constant disturbance)

2.33(83)

On the other hand, the general form of the controller design using the state variable feedback technique is based on the following form (i is the number of pure integrators in the controller to be designed): P(s) = α (s)β (s) = G (s) A(s)s i + F (s) B(s)

In other words, the control input calculations are always filtered according to the characteristic polynomial of the observer. Table 2.33j summarizes the state variable feedback and observer methods discussed in this section. It contains the order of the polynomials in the Diophantine equations and the number of the state variables involved for various control strategies. The following notations are used:

R

Reduced order observer

ER

Extended reduced order observer (reduced order observer with constant disturbance)

No. states

Number of the state variables in the controller

No. integr.

Number of the integrators in the controller

The table shows that the state variable feedback extended by an integrator and the proportional state variable feedback with an observer extended by constant disturbance estimation result in identical control actions. Although these control configurations are associated with identical F(s) and G(s) polynomials, under practical conditions (e.g., control signal saturation, handling slight nonlinearities) they will exhibit different closed-loop behavior because the detailed realizations of the algorithms are different. CONCLUSIONS This section overviewed the state variable feedback control of continuous-time linear dynamic processes. Using static gain coefficients for the state variables, the location of the closed-loop poles can be assigned in an arbitrary way. In order to improve both the servo and disturbance rejection properties

404

Control Theory

of the closed-loop system, it is reasonable to extend this basic feedback structure by a term integrating the output signal. State variables that are not directly accessible but are needed to obtain the feedback signal can be replaced by their estimates. The state estimate can be obtained by full order or reduced order observers. It has been shown how the state variable control together with an observer can be interpreted in the form of transfer functions. Though only continuous-time problems have been discussed, all the control methods and control algorithms that were discussed can also be transformed to handle discretetime systems. Instead of describing the design criterion by the desired location of the roots of the closed-loop characteristic equation, the feedback gain matrix can be found as a result of optimizing a loss function (LQ problem). In that case the feedback gain matrix will be calculated as the solution of a Riccati equation. The estimated states can be found not only by assigning the poles of the observer characteristic polynomial, but also by a well-elaborated solution based on stochastic considerations and assumptions related to the disturbances (Kalman filter technique). The two approaches are based on different assumptions; however, they lead to identical control actions for SISO systems.

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© 2006 by Béla Lipták

Bertsekas, D. P., Dynamic Programming and Optimal Control, Vols. I and II, Belmont, MA: Athena Scientific, 2001. Bryson, A. E., Dynamic Optimization, Reading, MA: Addison-Wesley, 1999. Bryson, A. E., and Ho, Y., Applied Optimal Control, New York: Hemisphere/Wiley, 1975. Burl, J. B., Linear Optimal Control: H2 and H-Infinity Methods, Reading, MA: Addison-Wesley, 1999. Control system toolbox for use with MATLAB. User’s Guide. The Math Works, 1998. Decarlo, R. A., Linear Systems. A State Variable Approach with Numerical Implementation, Englewood Cliffs, NJ: Prentice Hall, 1989. D’Souza, A. F., Design of Control Systems, Englewood Cliffs, NJ: Prentice Hall, 1988. Goodwin, G. C., Graebe, S. F., and Salgado, M. E., Control System Design, Englewood Cliffs, NJ: Prentice Hall, 2001. Iserman, R., Digital Control Systems. Volume I. Fundamentals, Deterministic Control, Heidelberg: Springer-Verlag, 1989. Iserman, R., Digital Control Systems. Volume II. Stochastic Control, Multivariable Control, Adaptive Control, Applications, Heidelberg: Springer-Verlag, 1991. Kailath, T., Linear Systems, Englewood Cliffs, NJ: Prentice Hall, 1980. Kwakernaak, H., and Sivan, R., Linear Optimal Control Systems, New York: John Wiley & Sons, 1972. Levine, W. S. (Ed.), The Control Handbook, Boca Raton, FL: CRC Press, 1996. Lewis, F. L., and Syrmos, V. L., Optimal Control, New York: John Wiley & Sons, 1995. Lyashko, S. I., Generalized Optimal Control of Linear Systems with Distributed Parameters, Dordrecht: Kluwer, 2002. Maciejowski, J. M., Multivariable Feedback Design, Reading, MA: AddisonWesley, 1989. Middleton, R. H., and Goodwin, G. C., Digital Control and Estimation. A Unified Approach, Englewood Cliffs, NJ: Prentice Hall, 1990. Naidu, D. S., Optimal Control Systems, Boca Raton, FL: CRC Press, 2003. Sinha, P. K., Multivariable Control. An Introduction, New York: Marcel Dekker, 1984.