Analysis and control of linear time–varying 2 systems
Before considering the actual subject matter, i.e., analysis and control design of nonlinear systems, at first linear time-varying systems or LTV systems of the form
x˙ A(t)x B(t)u, t t , x(t ) x (2.1a) = + > 0 0 = 0 y C(t)x D(t)u, t t (2.1b) = + ≥ 0 with x(t) Rn, u(t) Rm, y(t) Rp and ∈ ∈ ∈ n n n m p n p m A(t): R+ R × , B(t): R+ R × , C(t): R+ R × , D(t): R+ R × t 0 → t 0 → t 0 → t 0 → are considered. Here R+ : {t R t t 0} denotes the set of real numbers larger or equal to t0. t 0 = ∈ | ≥ The motivation behind this sequential approach is that LTV systems on the one hand show certain similarities to linear time–invariant systems but are distinguished by significant differences in their dynamic analysis. In this sense, they — at least to some extend — resemble nonlinear systems. On the other hand, the linearization of a nonlinear system around a solution trajectory t (x∗(t),u∗(t)) 7→ yields an LTV system. Hence the subsequently introduced tools can be used for control and observer design at least in a neighborhood of the trajectory (x∗(t),u∗(t)).
2.1 Transition matrix and solution of the state differential equations
At first free or autonomous systems
x˙ A(t)x, t t , x(t ) x (2.2) = > 0 0 = 0 are addressed. Under the assumption that any element of A(t) is bounded in the considered time interval t [t ,t ] the (local) existence and uniqueness theorem for autonomous nonlinear systems, ∈ 0 1 see, e.g., Khalil, 2002 or Meurer, 2019, Theorem 2.1, implies solution existence and uniqueness of (2.2). For this it is necessary to verify that A(t)x fulfills a Lipschitz condition. Let A(t) denote the induced k k matrix norm1, then
A(t)x A(t)x A(t)¡x x ¢ A(t) x x k 1 − 2k ≤ k 1 − 2 k ≤ k kk 1 − 2k implies the Lipschitz constant A(t) . Hence, the solution of (2.2) can be determined using successive k k approximation or Picard iteration, which starting at
x (t) x 0 = 0 1In finite–dimensional spaces such as Rn all norms are equivalent so that any norm can be bounded from below and from above by means of any other norm. There exist numerous matrix norms. Some commonly used norms are the Frobenius norm, the 1–norm (maximal column sum) or the –norm (maximal row sum). The notions 1–norm and ∞ –norm should not be confused with the L1–norm and the max– or sup–norms, respectively, that are sometimes also ∞ labeled by these terms.
23 implies the sequence
Z t ³ Z t ´ x1(t) x0 A(τ0)x0(τ0)dτ0 I A(τ0)dτ0 x0 = + t 0 = + t 0 Z t ³ Z τ0 ´ x2(t) x0 A(τ0) I A(τ1)dτ1 x0dτ0 = + t 0 + t 0 ³ Z t Z t ³ Z τ0 ´ ´ I A(τ0)dτ0 A(τ0) A(τ1)dτ1 dτ0 x0 = + t 0 + t 0 t 0 . . ³ Z t Z t Z τ0 xk (t) I A(τ0)dτ0 A(τ0) A(τ1)dτ1dτ0 = + t 0 + t 0 t 0 Z t Z τ0 Z τ1 A(τ0) A(τ1) A(τ2)dτ2dτ1dτ0 ... + t 0 t 0 t 0 + Z t Z τ0 Z τk 2 − ´ A(τ0) A(τ1) A(τk 1)dτk 1 dτ1dτ0 x0. + t 0 t 0 ··· t 0 − − ···
Here, I is the (n n) identity matrix. The solution of (2.2) is obtained as × x(t) Φ(t,t )x (2.3) = 0 0 in terms of the so–called Peano–Baker series
Z t Z t Z τ0 Φ(t,t 0) I A(τ0)dτ0 A(τ0) A(τ1)dτ1dτ0 = + t 0 + t 0 t 0 Z t Z τ0 Z τ1 A(τ0) A(τ1) A(τ2)dτ2dτ1dτ0 ... + t 0 t 0 t 0 + Z t Z τ0 Z τk 2 − A(τ0) A(τ1) A(τk 1)dτk 1 dτ1dτ0 . (2.4) + t 0 t 0 ··· t 0 − − ··· + ···
In direct analogy to linear time–invariant systems Φ(t,t 0) is called transition matrix.
Exercise 2.2. Prove (2.4) using induction.
It is an easy exercise to show that Φ(t ,t ) I. Thus, x(t) Φ(t,t )x fulfills both the differential 0 0 = = 0 0 equation (2.2) and the initial condition x(t ) Φ(t ,t )x x . Further properties of the transition 0 = 0 0 0 = 0 matrix are summarized below.
Lemma 2.1: Properties of the transition matrix
The transition matrix Φ(t,t 0) of a linear time–varying system fulfills the following identities
(i) Φ(t ,t ) I (Initial value) 0 0 = (ii) Φ(t ,t ) Φ(t ,t )Φ(t ,t ) (Product property) 2 0 = 2 1 1 0 1 (iii) Φ− (t,t ) Φ(t ,t) (Inversion) 0 = 0 d (2.5) (i v) Φ(t,t 0) A(t)Φ(t,t 0) (Differentiation) dt = µZ t ¶ ¡ ¢ 2 (i v) detΦ(t,t 0) exp tr A(τ) dτ (Determinant ). = t 0
2Herein tr(A) is the trace of the matrix A, i.e., the sum of the diagonal entries of A.
24 Chapter 2 Analysis and control of linear time–varying systems Exercise 2.3. Prove the properties (2.5).
Contrary to linear time–invariant systems there is no general solution approach for the determination of the transition matrix Φ(t,t 0) for linear time–varying systems. In particular the eigenvalues of the matrix A(t) in general do not provide information concerning the solution behavior.
Example 2.1. Let n 1 so that (2.1a) with A(t) a(t) reduces to the scalar differential equation = = x˙ a(t)x, t t , x(t ) x . (2.6) = > 0 0 = 0 Its solution can be determined straightforwardly using separation of variables, which yields
µZ t ¶ x exp a(τ)dτ x0. (2.7) = t 0
Let a(t) 1/t 2. In this case λ(t) 1/t 2 is the only eigenvalue of the matrix A(t). Moreover λ(t) 0 = = > for all finite t and λ(t) 0 as t . For t 0 the explicit solution of (2.6) follows as < ∞ = → ±∞ 0 6= µ 1 1¶ x exp x0. (2.8) = t 0 − t
Obviously x(t) is bounded for all t t and x(t) exp(1/t )x as t although the eigenvalue ≥ 0 → 0 0 → ∞ λ(t) has positive real part for all t t . This exemplarily confirms the statement that contrary to ≥ 0 linear time–invariant systems the eigenvalues of the system matrix in the time–varying case do not allow to deduce the solution dynamics.
Example 2.2. Contrary to linear time–invariant systems but in some analogy to nonlinear systems it is possible that the solution of a linear time–varying systems shows a finite escape time. To illustrate this consider 1 x˙ x, t t 0, x(t 0) x0 (2.9) = −(t t )2 > = − 1 for t t . Using separation of variables the solution is obtained as 1 > 0 µ 1 1 ¶ x exp x0. (2.10) = − t t + t t 0 − 1 − 1 Obviously, x(t) as t t , which implies the finite escape time. The solution to the differential → ∞ → 1 equation is hence only valid for the time interval t [t ,t ). ∈ 0 1
2.1.1 Systems admitting closed–form solutions
For the special case of a scalar linear time–varying differential equation Example 2.1 shows that
µZ t ¶ x exp a(τ)dτ x0 (2.11) = t 0 solves
x˙ a(t)x, t t , x(t ) x . (2.12) = > 0 0 = 0
2.1 Transition matrix and solution of the state differential equations 25 Coupled systems with dimx n 1 admit a closed–form solution, if the system matrix A(t) fulfills the = ≥ condition stated in Theorem 2.1 below.
Theorem 2.1 Let
A(t)A(τ) A(τ)A(t) (2.13) = for all t, τ. Then the transition matrix Φ(t,τ) of the linear time–varying system
x˙ A(t)x, t t , x(t ) x (2.14) = > 0 0 = 0 is given by3
µZ t ¶ Φ(t,τ) exp A(s)ds . (2.15) = τ
Proof. Taking into account the Taylor series expansion of the exponential function one obtains
Z t 1 Z t Z t Φ(t,τ) I A(s)ds A(s)ds A(p)dp . = + τ + 2! τ τ + ··· This implies
Z t 1 Z t Z t A(t)Φ(t,τ) A(t) A(t) A(s)ds A(t) A(s)ds A(p)dp . = + τ + 2! τ τ + ···
Differentiation of Φ(t,τ) with respect to t yields the series
d 1 µ Z t Z t ¶ Φ(t,τ) A(t) A(t) A(p)dp A(s)ds A(t) , dt = + 2! τ + τ + ···
which is identical to A(t)Φ(t,t) if
Z t Z t A(t) A(p)dp A(s)ds A(t). τ = τ
Differentiation with respect to τ and multiplication of both sides with 1 results in − A(t)A(τ) A(τ)A(t), = which corresponds to the preliminary assumption (2.13) in Theorem 2.1.
It has to be noted that the preliminary (2.13) is not fulfilled in general, which restricts the applicability of this result. Also note that the verification of (2.13) only requires the evaluation of G(t,τ) A(t)A(τ) = since substituting τ for t and t for τ immediately shows, if G(t,τ) G(τ,t) holds true. =
Exercise 2.4. Determine the transition matrix for the linear time–varying system
· α exp( t)¸ x˙ − x A(t)x, t t , x(t ) x . = exp( t) α = > 0 0 = 0 − −
3It has to be noted that (2.15) denotes the matrix exponential function and not the application of the exponential function to each element of the matrix argument.
26 Chapter 2 Analysis and control of linear time–varying systems Solution 2.4. Since
" 2 (t τ) ¡ t τ¢# α e− + α e− e− A(t)A(τ) − + A(τ)A(t). ¡ t τ¢ 2 (t τ) = α e− e− α e− + = − + − based on (2.15) the transition matrix reads
µZ t ¶ µ· τ t ¸¶ α(t τ) e− e− Φ(t,τ) exp A(s)ds exp t − τ − . = = e− e− α(t τ) τ − − For the explicit evaluation of the matrix exponential function the matrix A(t) has to be transferred into diagonal form or Jordan canonical form, respectively (see, e.g., Meurer, 2019, Chapter 3). The eigenvalues of the matrix A(t) are given by λ (t) α iexp( t) and λ (t) α iexp( t) with the 1 = + − 2 = − − corresponding eigenvectors v [1 i]T and v [i 1]T . Taking into account the (time–invariant) 1 = 2 = transformation matrix V [v v ] one obtains = 1 2 · ¸ 1 λ1(t) 0 A˜(t) V − A(t)V = = 0 λ2(t)
and thus
· ¡ τ t ¢ ¡ τ t ¢¸ ¡ ˜ ¢ 1 α(t τ) cos e− e sin e− e Φ(t,τ) V exp A(t) V − e − ¡ τ− t ¢ ¡ τ − t ¢ . = = sin e− e cos e− e − − − Closed–form solutions of linear time–varying differential equations are available for special differential equations such as Bessel’s, Hermite’s, Laguerre’s or Mathieu’s differential equation, see, e.g., Bronstein et al., 1997.
2.1.2 Periodic matrices
If the system matrix contains only constant or periodic coefficients, then the so–called Floquet theory can be applied. Floquet theory does not provide the actual solution of the system of differential equations but allows to draw conclusions concerning the dynamic behavior of periodically varying linear systems.
Theorem 2.2 Consider the linear time–varying system
x˙ A(t)x, t t , x(t ) x (2.16) = > 0 0 = 0 with system matrix satisfying A(t) A(t ω). Then the transition matrix takes the form = + R(t τ) Φ(t,τ) P(t,τ)e − (2.17) = with P(t,τ) P(t ω,τ) and the constant matrix R. = +
The proof of this results can be found, e.g., in Wiberg, 1971. In particular (2.17) determines the structure of the solution of (2.16). The matrix P(t,τ) represents the periodic contribution to the solution while exp(R(t τ)) determines the envelope. In other words, exp(R(t τ)) is the transition − − matrix of the linear time–invariant system z˙(t) Rz(t) defining the envelope. If all eigenvalues of = R have strictly negative real part, then (2.16) is obviously exponentially stable. Analogously one can
2.1 Transition matrix and solution of the state differential equations 27 draw the conclusion that the envelope is periodic, if R has eigenvalues on the imaginary axis or grows exponentially, if R has eigenvalues with positive real part. In the latter case the linear time–varying system (2.16) is unstable.
2.1.3 Solution of the non–autonomous state differential equations
The previous remarks concerning the transition matrix of the autonomous systems can be immediately utilized to determine the solution of the non–autonomous state differential equations (2.1).
Theorem 2.3 The general solution of the linear time–varying system
x˙ A(t)x B(t)u, t t , x(t ) x (2.18a) = + > 0 0 = 0 y C(t)x D(t)u, t t (2.18b) = + ≥ 0 is given by
Z t x Φ(t,t 0)x0 Φ(t,τ)B(τ)u(τ)dτ (2.19a) = + t 0 Z t y C(t)Φ(t,t 0)x0 C(t)Φ(t,τ)B(τ)u(τ)dτ D(t)u (2.19b) = + t 0 +
The proof of this result is constructive by taking into account the variation of constants method. The homogeneous solution of (2.18a) for u(t) 0 is given by x(t) Φ(t,t )x . For u(t) 0 the ansatz = = 0 0 6= x(t) Φ(t,t )p(t) is used. Substitution into (2.18a) provides = 0 d d Φ(t,t 0)p Φ(t,t 0) p A(t)Φ(t,t 0)p B(t)u. dt + dt = +
Due to the differentiation property (iv) from (2.5), i.e., d Φ(t,t ) A(t)Φ(t,t ), the previous equation dt 0 = 0 reduces to d Φ(t,t 0) p B(t)u. dt =
1 Taking into account the inversion property (iii) from (2.5), i.e., Φ− (t,t ) Φ(t ,t), one obtains 0 = 0 d p Φ(t 0,t)B(t)u. dt = Integration with respect to t with p(t ) p yields 0 = 0 Z t p p0 Φ(t 0,τ)B(τ)udτ = + t 0
and hence Z t x Φ(t,t 0)p0 Φ(t,t 0)Φ(t 0,τ)B(τ)udτ. = + t 0 | {z } Φ(t,τ) = The introduced integration constant p can be easily determined from the initial condition x(t ) x 0 0 = 0 so that p x . This results in (2.19a) and the output equation (2.19b) is obtained by evaluation of 0 = 0 (2.18b).
28 Chapter 2 Analysis and control of linear time–varying systems 2.2 Similarity transformations
The choice of the state variable for the state space representation of a dynamic system is not unique so that given the linear time–varying system
x˙ A(t)x B(t)u, t t , x(t ) x (2.20a) = + > 0 0 = 0 y C(t)x D(t)u, t t (2.20b) = + ≥ 0 with state x(t) an equivalent representation in the new state z(t) can be obtained by a regular trans- formation
z V (t)x (2.21) = with the transformation matrix4 V (t). This requires that
• V (t) is regular in the considered time interval t [t ,t ] with t t and ∈ 0 1 1 > 0 • V (t) is at least once continuously differentiable for all t [t ,t ], i.e., V˙ (t) exists and is continu- ∈ 0 1 ous in t.
Application to (2.20) yields
£ ¤ 1 z˙ V˙ (t)x V (t)x˙ V˙ (t) V (t)A(t) V − (t)z V (t)B(t)u = + = + + 1 y C(t)V − (t)z(t) D(t)u. = + and hence
z˙ A˜(t)z B˜(t)u, t t , z(t ) z V (t )x (2.22a) = + > 0 0 = 0 = 0 0 y C˜(t)z D˜ (t)u, t t (2.22b) = + ≥ 0 with
£ ¤ 1 1 A˜(t) V˙ (t) V (t)A(t) V − (t), B˜(t) V (t)B(t), C˜(t) C(t)V − (t), D˜ (t) D(t). (2.22c) = + = = =
The two systems Σ(A(t),B(t),C(t),D(t)) and Σ(A˜(t),B˜(t),C˜(t),D˜ (t)) are called equivalent (systems) and (2.21) is called similarity transformation. The equivalence property is used later to construct a regular transformation according to (2.21) so that the system in the new states z(t) is best suited for the control and observer design.
2.3 Stability notions and Lyapunov transformations
As is shown in Example 2.1 given a linear time–varying system the location of the (time–dependent) eigenvalues of its system matrix in general does not allow to deduce stability properties. However, a characterization of the stability of autonomous linear time–varying systems
x˙ A(t)x, t t , x(t ) x = > 0 0 = 0
4Differing from Meurer, 2019 subsequently V (t) is used to denote the transformation from x(t) to z(t). This is done to 1 avoid the appearance of time–derivatives of V − (t) in the transformed representation (2.22).
2.2 Similarity transformations 29 can be performed by means of the transition matrix
x(t) Φ(t,t )x . = 0 0 The system is called stable (in the sense of Lyapunov), if there exists a positive constant M so that
Φ(t,t ) M, t,t . (2.23) k 0 k ≤ ∀ 0 The system is called asymptotically stable, if
lim x(t) lim Φ(t,t 0)x0 0 (2.24) t = t → →∞ →∞ holds true for all x(t ) x . The system is called exponentially stable, if there exist positive constants 0 = 0 M and ω so that
ω(t t ) Φ(t,t ) Me− − 0 for t t . (2.25) k 0 k ≤ ≥ 0 Since the actual verification of stability or instability for a linear time–varying system is difficult in general this topic is postponed to the subsequent sections addressing control and observer design. The approaches presented there make use of the proper transformation of the linear time–varying system into a linear time–invariant system, whose stability is much easier to assess.
As a mathematical tool that is specifically suited for the stability analysis of linear time–varying systems Cesari’s theorem should be noted, which, however, imposes some limiting assumptions on the entries of the system matrix A(t), see, e.g., Cesari, 1971. In addition, Lyapunov’s stability theory provides different approaches for the stability analysis that in applications at least provide sufficient stability conditions (Wiberg, 1971; Khalil, 2002).
This section is closes with the introduction of so–called Lyapunov transformations that enable us to determine the stability of a system by analyzing an equivalent system formulation.
Definition 2.1: Lyapunov transformation 1 The transformation z(t) V (t)x(t) is called Lyapunov transformation, if V (t) and V − (t) are = bounded for all t t , i.e., if there exist positive constants κ and κ so that ≥ 0 1 2 1 V (t) κ , V − (t) κ , t t . k k ≤ 1 k k ≤ 2 ∀ ≥ 0
Given (2.20) and (2.22) Lyapunov transformations allow us to conclude about the stability relationship between the x–system and the z–system (Lyapunov, 1966).
Theorem 2.4: Stability of equivalent linear time–varying systems If the two systems Σ(A(t),B(t),C(t),D(t)) and Σ(A˜(t),B˜(t),C˜(t),D˜ (t)) are connected by a Lya- punov transformation according to Definition 2.1, then the exponential stability of the one system implies the exponential stability of the other system.
The proof of this (sufficient) claim is left to the reader as an exercise.
30 Chapter 2 Analysis and control of linear time–varying systems 2.4 Controllability and observability
The analysis of the controllability property of a system enables us to determine, whether the state of a dynamic system can be transferred from the initial state to an (arbitrary) final state in finite time by suitably adjusting the system input. This is typically a preliminary for control design.
Definition 2.2: Controllability of linear time–varying systems The system
x˙ A(t)x B(t)u, t t , x(t ) x = + > 0 0 = 0
is called completely controllable (or simply controllable) in the time interval [t 0,t 1], if for any initial state x(t ) x and any final state x there exists an input u(t), t [t ,t ] so that x(t ) x . 0 = 0 1 ∈ 0 1 1 = 1 The system is called totally controllable, if the transfer from x0 to x1 can be achieved for arbitrary small t t , i.e., the system is completely controllable in any subinterval of [t ,t ]. 1 > 0 0 1
In addition to complete and total controllability there are other controllability notions such as uniform or minimal controllability (Silverman and Meadows, 1967; Wiberg, 1971; Freund, 1971). Equivalent to the formulation in Theorem 2.2 is the analysis of the transition from the zero initial condition x 0 0 = to any state x at time t t or from any initial state x(t ) x to the zero final state x(t ) 0, which 1 = 1 0 = 0 1 = can be easily motivated by the solution (2.19a).
Observability is the dual concept to controllability. Loosely speaking this property enables us to deduce, if it is possible to determine the state x(t) from the knowledge of the output y(t).
Definition 2.3: Observability of linear time–varying systems The system
x˙ A(t)x B(t)u, t t , x(t ) x = + > 0 0 = 0 y C(t)x D(t)u, t t = + ≥ 0
is called completely observable (or simply observable) in the time interval [t 0,t 1], if the initial state x(t ) x can be uniquely determined from the knowledge of the input u(t), the output 0 = 0 y(t) and the system matrices in the interval t [t ,t ]. The system is called totally observable, ∈ 0 1 if x can be obtained from u(t), y(t) and the system matrices in t [t ,t ] for arbitrary small 0 ∈ 0 1 t t , i.e., the system is completely observable for any subinterval of [t ,t ]. 1 > 0 0 1
Similar to before there exist different observability notions such as uniform or minimal observability (Silverman and Meadows, 1967; Wiberg, 1971; Freund, 1971). For the verification of complete control- lability it is equivalent to prove that the initial state x(t ) x of the zero–input system (u(t) 0) can 0 = 0 = be uniquely determined based on the knowledge of the output y(t) in the interval t [t ,t ]. ∈ 0 1
Remark 2.1 For so–called analytic systems, where the elements of the system matrices A(t), B(t), C(t), and D(t) are analytic5 functions, complete controllability (observability) is identical to total controllability (observability) (Freund, 1971).
5Let D be an open subset of R. A function f (t): D R is called analytic, if f (t) C (D) and the Taylor series expansion → ∈ ∞ of f (t) converges in the neighborhood of any point t0 D. ∈
2.4 Controllability and observability 31 In some analogy to the Kalman controllability criterion for linear time–invariant systems it is possible to verify the following criterion for the time–varying case.
Theorem 2.5 The linear time–varying system
x˙ A(t)x B(t)u, t t , x(t ) x Rn = + > 0 0 = 0 ∈ is totally controllable in the interval [t ,t ], if and only if the (n nm)–matrix 0 1 × £ 0 1 n 1 ¤ S(A(t),B(t)) N B(t) N B(t) N − B(t) (2.26) = A A ··· A with the operator
k ¡ k 1 ¢ 1 0 N B(t) N N − B(t) , N B(t) N B(t) B˙(t) A(t)B(t), N B(t) B(t) (2.27) A = A A A = A = − + A =
has rank n for times t everywhere dense in [t 0,t 1].
Note that t being everywhere dense in [t 0,t 1] means that there can only exist isolated points t at which S(A(t),B(t)) can be of rank smaller n (Wiberg, 1971). The application of this theorem requires that A(t) and B(t) are (n 2)–times or respectively (n 1)–times continuously differentiable. − − Proof. The proof of Theorem 2.5 is sketched subsequently and is done in two steps.
a) rankS(A(t),B(t)) n in an interval containing η, t η t the system totally controllable: = 0 < < 1 ⇒ Without loss of generality x(t ) x 0 is assumed so that (2.19a) reduces to 0 = 0 =
Z t 1 x1 Φ(t 1,τ)B(τ)u(τ)dτ. = t 0
The explicit construction of the input u(t) so that this equality is identically fulfilled is based on the ansatz
n k 1 X d − ¯ u(τ) r δ(s)¯ . k k 1 ¯ = k 1 ds − s τ η = = − Taking into account the properties of the Dirac δ–function yields
Z t 1 n k 1 ¯ n k 1 ¯ X d − ¯ X k 1 d − £ ¤¯ x Φ(t ,τ)B(τ) r δ(s) dτ ( 1) − Φ(t ,τ)B(τ) r . (2.28) 1 1 k k 1 ¯ k 1 1 ¯ k = t 0 k 1 ds − ¯s τ η =k 1 − dτ − ¯τ η = = − = = In particular
d £ ¤ d d 1 Φ(t 1,τ)B(τ) Φ(t 1,τ) B(τ) Φ− (τ,t 1)B(τ), dτ = dτ + dτ holds true, which together with
d d £ 1 ¤ 1 d 1 0 I Φ(t,τ)Φ− (t,τ) A(t)Φ(t,τ)Φ− (t,τ) Φ(t,τ) Φ− (t,τ) = dt = dt = + dt and hence
d 1 1 Φ− (t,τ) Φ− (t,τ)A(t) dt = −
32 Chapter 2 Analysis and control of linear time–varying systems 1 as well as Φ− (τ,t) Φ(t,τ) implies = µ ¶ d £ ¤ d Φ(t 1,τ)B(τ) Φ(t 1,τ) B(τ) A(τ)B(τ) Φ(t 1,τ)N B(τ). dτ = dτ − = − A
Using an induction argument enables us to prove that
k 1 d − £ ¤ k 1 k Φ(t ,τ)B(τ) ( 1) − Φ(t ,τ)N B(τ). k 1 1 1 A dτ − = − Hence (2.28) simplifies to
r 1 . x Φ(t ,η)S(A(η),B(η)) . . 1 = 1 . r n
Since Φ(t ,η) is regular and rankS(A(η),B(η)) n holds true by assumption it is for any given x 1 = 1 possible to solve the latter equation for r 1, r 2,...,r n.
b) The system is totally controllable rankS(A(t),B(t)) n: The verification of this claim in princi- ⇒ = ple follows identically to the proof for the time–invariant case by taking into account the respective controllability Gramian (matrix) for linear time–varying systems. For details the reader is referred to Silverman and Meadows, 1967 or Wiberg, 1971, S. 137.
Remark 2.2 If rankS(A(t),B(t)) n for all t [t ,t ], then the system is called uniformly controllable. = ∈ 0 1
The duality between controllability and observability allows us to deduce the following criterion from Theorem 2.5 .
Theorem 2.6 The linear time–varying system
x˙ A(t)x B(t)u, t t , x(t ) x Rn = + > 0 0 = 0 ∈ y C(t)x D(t)u, t t = + ≥ 0 is called totally observable in the interval [t ,t ], if and only if the (np n)–matrix 0 1 × 0 MAC(t) M 1 C(t) A O(C(t), A(t)) . (2.29) = . . n 1 MA− C(t)
with the operator
k ¡ k 1 ¢ 1 0 M C(t) M M − C(t) , M C(t) M C(t) C˙(t) C(t)A(t), M C(t) C(t). (2.30) A = A A A = A = + A =
has rank n for times t everywhere dense in [t 0,t 1].
As before A(t) and C(t) are supposed to be (n 2)–times or respectively (n 1)–times continuously − − differentiable. The proof of Theorem 2.6 is subsequently omitted. Nevertheless the necessity of the
2.4 Controllability and observability 33 claim can be easily motivated by considering the output y(t) and its successive time–derivatives. For the sake of simplicity let u(t) 0 (the general case can be analyzed similarly). This yields the = sequence
y C(t)x = d d y £C(t)x¤ £C˙(t) C(t)A(t)¤x M C(t)x dt = dt = + = A . . n 1 d − d £ n 2 ¤ n 1 n 1 y MA− C(t)x MA− C(t)x dt − = dt = ··· = and hence
y . . O(C(t), A(t))x. n 1 = d − n 1 y dt −
If rankO(C(t), A(t)) n for all t [t ,t ], then the resulting linear system of equations can be uniquely = ∈ 0 1 solved for x(t) for all t [t ,t ]. This formal procedure is also applied for the observability analysis of ∈ 0 1 nonlinear systems.
Remark 2.3 If rankO(C(t), A(t)) n for all t [t ,t ], then the system is called uniformly observable. = ∈ 0 1
Exercise 2.5. Let Σ(A(t),B(t),C(t),D(t)) and Σ(A˜(t),B˜(t),C˜(t),D˜ (t)) be equivalent systems un- der the transformation z(t) V (t)x(t). Determine the matrices S˜(A˜(t),B˜(t)) and O˜(C˜(t), A˜(t)) = of the transformed system (2.22). How are S(A(t),B(t)) and S˜(A˜(t),B˜(t)) and O(C(t), A(t)) and O˜(C˜(t), A˜(t)) related? What can be said about the controllability or observability, respectively, of equivalent systems?
Solution 2.5. The following relationships hold true
S˜(A˜(t),B˜(t)) V (t)S(A(t),B(t)) (2.31) = 1 O˜(C˜(t), A˜(t)) O(C(t), A(t))V − (t). (2.32) =
Exercise 2.6. Analyze the total controllability and observability of the system
·1 0¸ ·t ¸ x˙ x u, t t , x(t ) x = 0 2 + 1 > 0 0 = 0 £ t 2t ¤ y e− e− x, t t . = ≥ 0
Solution 2.6. The system is totally controllable but not observable.
34 Chapter 2 Analysis and control of linear time–varying systems 2.5 State feedback control design
For the state feedback control design for linear time–varying systems it is in following distinguished between the single–input single–output (SISO) case and the multiple-input multiple–output (MIMO) case. The presented approaches are mainly devoted to extensions of the well–known Ackermann formula, which is based on the determination of the respective canonical form of the linear system.
2.5.1 SISO systems
Our starting point is the linear time–varying SISO system
x˙ A(t)x b(t)u, t t , x(t ) x (2.33a) = + > 0 0 = 0 y cT (t)x d(t)u, t t . (2.33b) = + ≥ 0 In direct analogy to the time–invariant case, see, e.g., Meurer, 2019, a similarity transformation according to Section 2.2 in the form
z V (t)x (2.34) = n n is sought, with the regular, continuously differentiable transformation matrix V (t) R × that trans- ∈ forms (2.33) into the so–called controller canonical form for linear time–varying systems
z 0 1 0 0 z 0 1 ··· 1 z 0 0 1 0 z 0 2 2 d . . ··· . . . . . . . . u, t t 0 (2.35a) dt . = . . . + . > zn 1 0 0 0 1 zn 1 0 − ··· − zn a˜0(t) a˜1(t) a˜2(t) a˜n 1(t) zn b˜n 1(t) | {z } | − − −{z ··· − − }| {z } | −{z } z z A˜(t) = b˜(t) = = = z(t ) z (2.35b) 0 = 0 z1 z 2 £ ¤ . ˜ y c˜0(t) c˜1(t) c˜n 1(t) . d(t)u, t t 0. (2.35c) = ··· − + ≥ | {z } T zn 1 c˜ (t) − = zn | {z } z =
The coefficient b˜n 1(t) provide a design degree–of–freedom (DOF) and is supposed to satisfy b˜n 1(t) 0 − − 6= for all t in the considered time interval.
2.5.1.1 Transformation into SISO controller canonical form
The transformation matrix V (t) into controller canonical form is determined constructively. According to (2.22c) the matrix V (t) has to fulfill the following conditions
£ ¤ 1 A˜(t) V˙ (t) V (t)A(t) V − (t) (2.36a) = + b˜(t) V (t)b(t). (2.36b) =
2.5 State feedback control design 35 Using the ansatz
z w T (t)x (2.37) 1 = the successive differentiation with respect to t and comparison with the respective row in (2.35a) yields the sequence of equations
z˙ (t) w˙ T (t)x w T (t)A(t)x w T (t)b(t)u ¡M 0 w T (t)¢b(t) 0 1 = + + ⇒ A = ¡M 1 w T (t)¢x z (t) = A = 2 d ¡ 1 T ¢ ¡ 1 T ¢ ¡ 1 T ¢ ¡ 1 T ¢ z˙2(t) M w (t) x M w (t) Ax M w (t) b(t)u M w (t) b(t) 0 = dt A + A + A ⇒ A = ¡M 2 w T (t)¢x z (t) = A = 3 . .
d ¡ n 2 T ¢ ¡ n 2 T ¢ ¡ n 2 T ¢ ¡ n 2 T ¢ z˙n 1(t) MA− w (t) x MA− w (t) Ax MA− w (t) b(t)u MA− w (t) b(t) 0 − = dt + + ⇒ = ¡ n 1 T ¢ M − w (t) x z (t) = A = n d ¡ n 1 T ¢ ¡ n 1 T ¢ ¡ n 1 T ¢ ¡ n 1 T ¢ z˙n(t) MA− w (t) x MA− w (t) Ax MA− w (t) b(t)u MA− w (t) b(t) 0 = dt + + ⇒ | {z } 6= b˜n 1(t) = − ¡ n T ¢ 1 ˜ MA w (t) V − (t)z bn 1(t)u. = + −
The arising equality constraints follow from the fact that in the controller canonical form the input only enters the differential equation for zn(t). In summary the relationship
0 T MA w (t) M 1 w T (t) A z . x V (t)x (2.38) = . = . n 1 T MA− w (t)
is obtained together with the equations
0 T MA w (t) 0 M 1 w T (t) 0 A . b(t) . (2.39) . = . . . n 1 T M − w (t) b˜n 1(t) A − determining the vector w T (t). For the explicit computation of w T (t) the following Lemma proves useful.
Lemma 2.2 Let w T (t) and v(t) be sufficiently often continuously differentiable. Then the two statements are equivalent for k 0: ≥ a) ¡M 0 w T (t)¢v(t) 0, ¡M 1 w T (t)¢v(t) 0, ..., ¡M k w T (t)¢v(t) 0 A = A = A = b) w T (t)¡N 0 v(t)¢ 0, w T (t)¡N 1 v(t)¢ 0, ..., w T (t)¡N k v(t)¢ 0. A = A = A =
36 Chapter 2 Analysis and control of linear time–varying systems The proof of Lemma 2.2 is left to the reader as an exercise but is based on the relationship that
d w T (t)b(t) 0 0 ¡w T (t)b(t)¢ w˙ T (t)b(t) w T (t)b˙(t) w˙ T (t)b(t) w T (t)b˙(t). = ⇒ = dt = + ⇒ = −
Exercise 2.7. Prove Lemma 2.2.
With Lemma 2.2 the equations (2.39) imply
T £ 0 1 n 1 ¤ £ ˜ ¤ w (t) NAb(t) NAb(t) NA− b(t) 0 0 bn 1(t) (2.40) | {z ··· } = ··· − S(A(t),b(t)) = and hence
T £ ¤ 1 w (t) 0 0 b˜n 1(t) S− (A(t),b(t)). (2.41) = ··· − The row vector w T (t) is the scaled last row of the inverse time–varying controllability matrix. This yields the following result.
Theorem 2.7: Transformation into SISO controller canonical form The linear time–varying system
x˙ A(t)x b(t)u, t t , x(t ) x = + > 0 0 = 0 y cT (t)x d(t)u, t t = + ≥ 0 can be brought into controller canonical form (2.35) by means of the transformation z(t) = V (t)x(t) with
0 T MA w (t) M 1 w T (t) A T £ ˜ ¤ 1 V (t) . , w (t) 0 0 bn 1(t) S− (A(t),b(t)), (2.42) = . = ··· − . n 1 T MA− w (t)
if and only if the SISO controllability matrix S(A(t),b(t)) is regular for t t . ≥ 0
The design DOF b˜n 1(t) can be herein used to simplify the necessary computations. −
2.5.1.2 Eigenvalue assignment using Ackermann’s formula
The controller canonical form (2.35) is particularly suited for control design using eigenvalue assign- ment since
• all time–varying functions of the transformed system are concentrated in the last line of the system matrix A˜(t) and
• the control input only acts in the last line of the normal form.
As already remarked in Section 2.1 linear time–varying systems do not allow to deduce stability properties by means of the location of the eigenvalues of the system matrix. However, the special structure of the controller canonical form enables us
2.5 State feedback control design 37 (i) to compensate the time–varying elements of A˜(t) and
(ii) to assign desired constant eigenvalues to time–invariant matrix obtained in the previous step.
Introducing the vector
T £ ¤ ¡ n T ¢ 1 a˜ (t) a˜0(t) a˜1(t) a˜n 1(t) MA w (t) V − (t) = ··· − = − the choice of the control input
1 u ¡a˜ T (t)z v¢ (2.43) = b˜n 1(t) + − with the new input v(t) results in the integrator chain
z˙ z (2.44) 1 = 2 z˙ z (2.45) 2 = 3 . . (2.46) z˙ v (2.47) n = of length n. Note that the system (2.44) is also called Brunovsky normal form. Let
n n 1 n Y p∗(λ) p0 p1λ ... pn 1λ − λ (λ λ∗j ) (2.48) = + + + − + = j 1 − = denote the desired characteristic polynomial. By assigning the new input as
v p0z1 p1z2 ... pn 1zn (2.49) = − − − − − and substituting this expression into (2.43), i.e.,
1 u ¡a˜ T (t) pT ¢z k˜ T (t)z, (2.50) = b˜n 1(t) − = − − where
T £ ¤ p p0 p1 pn 1 , (2.51) = ··· − the dynamics of the closed–loop control system is obtained in the form
0 1 0 0 ··· 0 0 1 0 ··· ¡ T ¢ . . z˙ A˜(t) b˜(t)k˜ (t) z . . z. (2.52) = − = . . 0 0 0 1 ··· p0 p1 p2 pn 1 − − − ··· − − T In particular, A˜g A˜(t) b˜(t)k˜ (t) is a time–invariant matrix, whose characteristic polynomial p(λ) = − n 1 n = p∗(λ) p0 p1λ ... pn 1λ − λ is given by (2.48) (see, e.g., Meurer, 2019, Lemma 6.2). As a = + + + − + result, (2.50) and (2.38) directly enables us to assign eigenvalues for the closed–loop control system and hence to design its dynamical properties. This construction is subsumed in the so–called Ackermann formula.
38 Chapter 2 Analysis and control of linear time–varying systems Theorem 2.8: Ackermann’s formula for linear time–varying SISO system Let the linear time–varying system
x˙ A(t)x b(t)u, t t , x(t ) x = + > 0 0 = 0 y cT (t)x d(t)u, t t = + ≥ 0 be uniformly controllable for t t , i.e., the controllability matrix ≥ 0 £ 0 1 n 1 ¤ S(A(t),b(t)) N b(t) N b(t) N − b(t) = A A ··· A with operator
k ¡ k 1 ¢ 1 0 N b(t) N N − b(t) , N b(t) b˙(t) A(t)b(t), N b(t) b(t) A = A A A = − + A = has rank n for all t t . Then the state feedback ≥ 0 u kT (t)x (2.53a) = − with the time–varying feedback gain
T 1 h 0 1 n 1 ni T k (t) p0MA p1MA ... pn 1MA− MA w (t) (2.53b) = b˜n 1(t) + + + − + ◦ − for
T £ ¤ 1 w (t) 0 0 b˜n 1(t) S− (A(t),b(t)) (2.53c) = ··· − T T T 1 yields the time–invariant system matrix A˜ A˜(t) b˜(t)k˜ (t), k˜ (t) k (t)V − (t) of the closed– g = − = loop control system in the transformed state z(t) V (t)x(t). The characteristic polynomial n 1 n = 6 p0 p1λ ... pn 1λ − λ of A˜g can be assigned as an arbitrary Hurwitz polynomial by the + + + − + proper choice of the coefficients p , j 0,1,...,n 1. In this case A˜ is a Hurwitz matrix. j = − g If the transformation z(t) V (t)x(t) with = 0 T MA w (t) M 1 w T (t) A V (t) . = . . n 1 T MA− w (t)
is a Lyapunov transformation into controller canonical form according to Definition 2.1, then Theorem 2.4 implies the exponential stability of the closed–loop control system
x˙ ¡A(t) b(t)kT (t)¢x = | − {z } A (t) = g
in the original state x(t) provided that A˜g is a Hurwitz matrix.
6A polynomial is called a Hurwitz polynomial, if all roots of the polynomial have negative real part.
2.5 State feedback control design 39 Example 2.3. To illustrate the previous developments consider the state feedback control design for the linear time–varying SISO system
·t sin(t)¸ · 2 ¸ x˙ x u, t t 0, x(t ) x . = 1 1 + 1 > 0 = 0 = 0 − − i
40 Chapter 2 Analysis and control of linear time–varying systems 2.5 State feedback control design 41 The extension of Ackermann’s formula to trajectory tracking control for ζ(t) w T (t)x(t) is left to the = reader as an exercise.
Exercise 2.8 (Trajectory tracking control for linear time–varying SISO systems using Ackermann’s formula). Let the preliminaries of Theorem 2.8 as well as the two assumptions
• the coefficients p , j 0,1,...,n 1 are coefficients of a Hurwitz polynomial and j = − •V (t) is a Lyapunov transformation
hold true. Show that the application of the time–varying, inhomogeneous state feedback
T ¡ ¢ u k (t) x x∗ u∗ (2.54a) = − − + with kT (t) from (2.53b) and the feedforward contribution
n 1 ³ d ¡ n T ¢ ´ u∗ n ζ∗ MA w (t) x∗ (2.54b) = b˜n 1(t) dt − − yields exponentially stable tracking behavior
ω(t t ) x x∗ Me− − 0 x x∗ , M, ω 0 (2.55) k − k ≤ k 0 − 0 k > for
ζ∗ d ζ∗ 1 dt x∗ V − (t)z∗, z∗ . . (2.56) = = . n 1 d − n 1 ζ∗ dt −
The constant ω is thereby determined by the coefficients p j used in the eigenvalue assignment.
It has to be noted that the reference (desired) trajectory ζ∗(t) corresponds to the desired evolution of ζ(t) w T (t)x(t) with w T (t) defined in (2.53c). In other words trajectory tracking is achieved so that = n ζ(t) converges to ζ∗(t) exponentially. In particular, ζ∗(t) C for t t or t [t ,t ], respectively, is ∈ ≥ 0 ∈ 0 1 required for the evaluation of the feedforward part u∗(t).
2.5.2 MIMO systems
The extension to the MIMO case in principle follows the lines of the SISO case with the major difference that so–called controllability indices %j have to be introduced. Under the assumption that the linear time–varying MIMO system
m X x˙ A(t)x B(t)u A(t)x b j (t)u j , t t 0, x(t 0) x0 (2.57a) = + = + j 1 > = = y C(t)x D(t)u, t t , (2.57b) = + ≥ 0 is (completely, totally or uniformly, respectively) controllable, the controllability indices enable us to systematically extract n linearly independent column vectors from the (n nm) controllability matrix × S(A(t),B(t)).
42 Chapter 2 Analysis and control of linear time–varying systems Remark 2.4 If not stated otherwise in the following controllability is associated with uniform controllability. In addition it is assumed that rankB(t) m holds true for all t t . = ≥ 0
Given the representation (2.57a) the controllability matrix (2.26) can be written as
h ¯ 1 1 ¯ ¯ n 1 n 1 i S(A(t),B(t)) b (t) b (t) ¯ N b (t) N b (t) ¯ ¯N − b (t) N − b (t) . (2.58) = 1 ··· m ¯ A 1 ··· A m ¯ ··· ¯ A 1 ··· A m If the system is controllable, then rankS(A(t),B(t)) n. The rank is identical to the rank of the reduced = controllability matrix
m h %1 1 ¯ ¯ %m 1 i X ¯ − ¯ ¯ − S(A(t),B(t)) b1(t) NA b1(t) ¯ ¯ bm(t) NA bm(t) , %j n, (2.59) = ··· ··· ··· j 1 = = which is composed of n linear independent column vectors of S(A(t),B(t)). This motivates the following definition.
Definition 2.4
The jth controllability index %j of the controllable linear time–varying systems (2.57) is the % 1 j − smallest integer for which the column vector NA b j (t) is linear independent from any column vector located to its left in (2.58).
2.5.2.1 Transformation into MIMO controller canonical form
The transformation into the controller canonical form for the MIMO case is in principle identical to the approach in the SISO case but requires to consider the individual controllability indices.
Theorem 2.9: Transformation into MIMO controller canonical form The linear time–varying system
m X x˙ A(t)x b j (t)u j , t t 0, x(t 0) x0 = + j 1 > = = y C(t)x D(t)u, t t = + ≥ 0 can be brought into controller canonical form by means of the transformation z(t) V (t)x(t) = with
0 T MA w 1 (t) . . % 1 1− T MA w 1 (t) . V (t) . (2.60a) = 0 T MA w m(t) . . % 1 m − T MA w m(t)
2.5 State feedback control design 43 and
length %1 length %m z }|= { z }|= { T ˜ ˜ w 1 (t) 0 0 b1,% 1(t) ¯ ... ¯ 0 0 b1,% 1(t) ··· 1− ¯ ¯ ··· m − . . . ¯ ¯ . . ¯ 1 . . . ¯ ¯ . . S− (A(t),B(t)), = . . ¯ ¯ . . T ˜ ¯ ¯ ˜ w m(t) 0 0 bm,% 1(t) ... 0 0 bm,% 1(t) ··· 1− ··· m − | {zm n } R × ∈ (2.60b)
if and only if the reduced controllability matrix S¯(A(t),B(t)) is regular for all t t . ≥ 0 ˜ The functions b j,% 1(t) are again design DOFs that can be chosen so that the evaluation effort is j − significantly reduced.
As an alternative to the matrix representation used in the SISO case in the following a more compact formulation is considered that directly exploits the properties of the canonical form. In particular any subsystem has an upper adjacent diagonal filled with ones. Hence, let
ζ w T (t)x, (2.61) j = j then
%1 d %1 T %1 1 T %1 1 T ζ1 M w (t) (M − w (t))b1(t) (M − w (t))bm(t) dt %1 A 1 A 1 ··· A 1 . . . . . . x . . u. (2.62) . = . + . . %m d %m T %m 1 T %m 1 T ζm M w m(t) (M − w m(t))b1(t) (M − w m(t))bm(t) dt %m A A ··· A | {z m }n | m m{z } A (t) R × K (t) R × (coupling matrix) = ∈ = ∈
The MIMO controller canonical form is composed of m subsystems of length % , j 1,...,m. Their j = explicit representation is not necessary to determine the MIMO version of Ackermann’s formula and is left to the reader as an exercise.
2.5.2.2 Decoupling and eigenvalue assignment using Ackermann’s formula
Based on the previous remarks the following result can be deduced.
Theorem 2.10: Ackermann’s formula for linear time–varying MIMO systems Let the linear time–varying system
m X x˙ A(t)x b j (t)u j , t t 0, x(t 0) x0 = + j 1 > = = y C(t)x D(t)u, t t = + ≥ 0 be uniformly controllable for t t , i.e., the reduced controllability matrix ≥ 0 m h %1 1 ¯ ¯ %m 1 i X ¯ − ¯ ¯ − S(A(t),B(t)) b1(t) NA b1(t) ¯ ¯ bm(t) NA bm(t) , %j n, = ··· ··· ··· j 1 = =
44 Chapter 2 Analysis and control of linear time–varying systems with the controllability indices % , j 1,...,m, and the operator j = k ¡ k 1 ¢ 1 0 N b(t) N N − b(t) , N b(t) b˙(t) A(t)b(t), N b(t) b(t) A = A A A = − + A = has rank n for all t t . Let in addition the coupling matrix ≥ 0 %1 1 T %1 1 T (M − w (t))b1(t) (M − w (t))bm(t) A 1 ··· A 1 . . K (t) . . = . . % 1 % 1 (M m − w T (t))b (t) (M m − w T (t))b (t) A m 1 ··· A m m be regular for all t t . Then the state feedback ≥ 0 u K (t)x (2.63a) = − with the time–varying feedback gain matrix
¡ 0 1 %1 1 %1 ¢ T p1,0M p1,1M ... p1,% 1M − M w (t) A + A + + 1− A + A ◦ 1 1 . K (t) K − (t) . (2.63b) = . ¡ 0 1 %m 1 %m ¢ T pm,0M pm,1M ... pm,%m 1M − M w m(t) A + A + + − A + A ◦ for w T (t), j 1,...,m from (2.60b) yields the time–invariant system matrix A˜ A˜(t) B˜(t)K˜(t), j = g = − 1 K˜(t) K (t)V − (t) of the closed–loop control system in the transformed state z(t) V (t)x(t). = = The characteristic polynomial
m Y¡ %j 1 %j ¢ p j,0 p j,1λ ... p j,%j 1λ − λ j 1 + + + − + =
of A˜g can be assigned as an arbitrary Hurwitz polynomial by the proper choice of the coefficients p j,i , i 0,...,%j 1, j 1,...,m. In this case A˜g is a Hurwitz matrix. = − = If the transformation z(t) V (t)x(t) with V (t) defined in (2.60a) is a Lyapunov transforma- = tion into controller canonical form according to Definition 2.1, then Theorem 2.4 implies the exponential stability of the closed–loop control system
x˙ ¡A(t) B(t)K (t)¢x = | − {z } A (t) = g
in the original state x(t) if A˜g is a Hurwitz matrix.
Theorem 2.10 is based on the introduction of the new input v(t) Rm in (2.62) by making use of the ∈ feedback transformation
1 ¡ ¢ u K − (t) v A (t)x . = −
This implies the decoupling of the individual states ζj (t) such that m integrator chains of length %j , j 1,...,m, i.e., = d%j ζj v j dt %j =
2.5 State feedback control design 45 are obtained. Similar to the SISO case given this time–varying Brunovsky normal form the new input v j (t) is chosen to assign desired eigenvalues ensuring exponential convergence.
Example 2.4. For the linear time–varying MIMO system
1 0 2 0 1 t −t x˙ e 1 1x 1u1 e u2, t t 0, x(t 0) x0 = + + > = 2 0 2 0 1
we seek for the system representation in MIMO controller canonical form and want to determine a suitable state feedback control. With
0 0 1 N b1(t) N b1(t) 1 A = A = 0
and
1 1 0 −t 1 t N b2(t) e , N b2(t) 1 e , A = A = − 1 0
the reduced controllability matrix (2.59) is obtained as
0 1 1 £ 1 ¤ −t t S¯(A(t),B(t)) b1(t) b2(t) N b2(t) 1 e 1 e . = A = − 0 1 0
Hence, % 1, % 2 and S¯(A(t),B(t)) is invertible due to detS¯(A(t),B(t)) 1. With this, the 1 = 2 = = preliminaries of Theorem 2.9 are fulfilled. Evaluation of (2.60b) yields
· T ¸ · ˜ ˜ ¸ w 1 (t) b1,0(t) 0 b1,1(t) ¯ 1 T ˜ ˜ S− (A(t),B(t)) w 2 (t) = b2,0(t) 0 b2,1(t) · ¡ t ¢ ¸ b˜1,1(t) e 1 b˜1,0(t) b˜1,0(t) b˜1,1(t) b˜1,0(t) + ¡ − ¢ − . = b˜ (t) et 1 b˜ (t) b˜ (t) b˜ (t) b˜ (t) 2,1 + − 2,0 2,0 2,1 − 2,0 A proper choice of the DOFs b˜ (t) 1, b˜ (t) 1, b˜ (t) 0, b˜ (t) 1 provides 1,0 = 1,1 = 2,0 = 2,1 = w T (t) £et 1 0¤, w T (t) £1 0 1¤. 1 = 2 = Substitution into the equation determining the transformation matrix (2.60a) yields
T t w 1 (t) e 1 0 T V (t) w 2 (t) 1 0 1. = 1 T = MA w 2 (t) 3 0 4
By evaluating the transformation z(t) V (t)x(t) the MIMO controller canonical form of the linear = time–varying system
¡ ¢ 1 z˙ V˙ (t) V (t)A(t) V − (t)z V (t)b (t)u V (t)b (t)u = + + 1 1 + 2 2 1 2et 3 1 1 0 − 0 0 1z 0u1 0u2 = + + 0 2 3 0 1
46 Chapter 2 Analysis and control of linear time–varying systems is obtained, which confirms that the coupling matrix
·1 0¸ K (t) I = = 0 1
is regular.
The transformed system representation in controller canonical form can be directly exploited for the control design. With u (t) k˜ T (t)z and u (t) k˜ T (t)z the equations of the closed–loop control 1 = − 1 2 = − 2 system in transformed states is given as
1 k˜ (t) 2et 3 k˜ (t) 1 k˜ (t) − 1,1 − − 1,2 − 1,3 z˙ 0 0 1 z. = k˜2,1(t) 2 k˜2,2(t) 3 k˜2,3(t) | − −{z − } A˜ (t) = g The coefficients of the state feedback are chosen so that (i) A˜ (t) is time–invariant (A˜ (t) A˜ ) and g g = g (ii) the eigenvalues of A˜g are placed at desired locations in the complex left–half plane. For the considered example this can be achieved by assigning
k˜ (t) 1, k˜ (t) 2et 3 1, k˜ (t) 1 1,1 = 1,2 = − − 1,3 = k˜ (t) p , k˜ (t) 2 p , k˜ (t) 3 p . 2,1 = 0 2,2 = + 1 2,3 = + 2 The characteristic polynomial of A˜ is reduced to λ3 p λ2 p λ p . It is important to note that g + 2 + 1 + 0 the choice of the coefficients k˜i,j (t) is not unique. Indeed there are many different possibilities to solve this design problem. The corresponding state feedback in original states follows from K (t) [k˜ T (t) k˜ T (t)]V (t). = 1 | 2 For comparison purposes consider the direct evaluation of Ackermann’s formula (2.63b), which yields the gain matrix
" ¡p M 0 M 1 ¢ w T (t) # 1 1,0 A A 1 K (t) K − (t) + ◦ = ¡p M 0 p M 1 M 2 ¢ w T (t) 2,0 A + 2,1 A + A ◦ m ·1 0¸· (p 3)et p 1 2et 1 ¸ 1,0 + 1,0 + + = 0 1 3p p 11 0 4p p 14 2,1 + 2,0 + 2,1 + 2,0 + · (p 3)et p 1 2et 1 ¸ 1,0 + 1,0 + + = 3p p 11 0 4p p 14 2,1 + 2,0 + 2,1 + 2,0 + and hence the time–varying state feedback u(t) K (t)x(t). = − 1 As presented in Theorem 2.10 the stability argument requires that V (t) and V − (t), respectively, define a Lyapunov transformation. To verify this property it is necessary to shown that V (t) and 1 V − (t) remain bounded for t t 0. Due to the term V1,1(t) exp(t) in V (t) this is obviously only ≥ 1 = fulfilled for t (related terms arise in V − (t). As a result, stability can be only ensured for finite < ∞ time intervals t [t ,t ] with t . ∈ 0 1 1 < ∞ The extension of this design concept to trajectory tracking control for the m variables ζ (t) w T (t)x(t), j = j j 1,...,m is left as an exercise to the reader. =
2.5 State feedback control design 47 Exercise 2.9 (Trajectory tracking control for linear time–varying MIMO systems using Acker- mann’s formula). Let the preliminaries of Theorem 2.10 as well as the two assumptions
• the coefficients i 0,...,%j 1, j 1,...,m are coefficients of a Hurwitz polynomial and = − = •V (t) is a Lyapunov transformation
hold true. Show that the application of the time–varying, inhomogeneous state feedback ¡ ¢ u K (t) x x∗ u∗ (2.64a) = − − + with K (t) from (2.63b) and the feedforward contribution
d%1 %1 T dt %1 ζ1 MA w 1 (t) 1 . . u∗ K − (t) . . x∗ (2.64b) = . − . d%m %m T dt %m ζm MA w m(t)
yields exponentially stable tracking behavior
ω(t t ) x x∗ Me− − 0 x x∗ , M, ω 0 (2.65) k − k ≤ k 0 − 0 k > for
· % 1 ¯ ¯ % 1 ¸T 1 d 1− ¯ ¯ d m − x∗ V − (t)z∗, z∗ . (2.66) ζ1∗ % 1 ζ1 ¯ ¯ ζm∗ % 1 ζm = = ··· dt 1− ¯ ··· ¯ ··· dt m −
The constant ω is thereby determined by the coefficients p j used in the eigenvalue assignment.
Similar to the SISO case analyzed in Exercise 2.8 tracking is achieved so that each ζ (t) w T (t)x(t) j = j converges to ζ∗j (t) exponentially. Sufficient regularity (differentiability) of the desired trajectory T ζ∗(t) [ζ∗(t),...,ζ∗ (t)] has to be guaranteed by proper trajectory planning. = 1 m
2.6 State observer design
The state feedback control design methods introduced before rely on the availability (knowledge) of the complete state vector x(t) for all t t . Since this assumption is not fulfilled in general it is ≥ 0 necessary to amend the control loop by a state observer to reconstruct x(t) from the knowledge of the output y(t), the input u(t) and the system matrices. In what follows the design of a Luenberger observer, see, e.g., Meurer, 2019, Chapter 6.5 is considered for the linear time–varying system
x˙ A(t)x B(t)u, t t , x(t ) x (2.67a) = + > 0 0 = 0 y C(t)x D(t)u, t t . (2.67b) = + ≥ 0 The state observer is composed of a copy of the system (simulation term) and a correction term. Let xˆ(t) denote the observer state, then
xˆ˙ A(t)xˆ B(t)u L(t)(y yˆ) , t t 0, xˆ(t 0) xˆ 0 (2.68a) = | {z+ }+ | {z− } > = simulation term correction term yˆ C(t)xˆ D(t)u, t t . (2.68b) = + ≥ 0
48 Chapter 2 Analysis and control of linear time–varying systems The observer error dynamics in the (observer) error state x˜(t) x(t) xˆ(t) with x(t) from (2.67a) and = − xˆ(t) from (2.68a) hence reads
x˜˙ ¡A(t) L(t)C(t)¢x˜ A (t)x˜, x˜(t ) x˜ x xˆ . (2.69) = − = b 0 = 0 = 0 − 0 For the design of the observer correction matrix L(t) weighting the output injection both the SISO and the MIMO case are subsequently analyzed.
2.6.1 SISO systems
n 1 T 1 n For the SISO case, i.e., (2.67) with dimu dim y 1, B(t) b(t) R × , C(t) c (t) R × , and = = = ∈ = ∈ D(t) d(t) R, the observer design is based on the compensation of the time–variance by exploiting = ∈ the SISO observer canonical form
z 0 0 0 a˜ (t) z b˜ (t) 1 ··· − 0 1 0 z 1 0 0 a˜ (t) z b˜ (t) 2 1 2 1 d . . ··· − . . . . . . . . u, t t 0, z(t 0) z0 (2.70a) dt . = . . . + . > = ˜ zn 1 0 0 0 a˜n 2(t)zn 1 bn 2(t) − ··· − − − − zn 0 0 1 a˜n 1(t) zn b˜n 1(t) | {z } | ··· {z − − }| {z } | −{z } z ˜ z ˜ = A(t) = b(t) = = z1 z 2 £ ¤ . ˜ y 0 0 0 c˜n 1(t) . d(t)u, t t 0 (2.70b) = ··· − + ≥ | {z } T zn 1 c˜ (t) − = zn | {z } z = followed by the eigenvalue assignment for the resulting linear time–invariant system.
2.6.1.1 Transformation into SISO observer canonical form
Transferring the procedure introduced in Section 2.5.1.1 enables us to prove the following result.
Theorem 2.11: Transformation into SISO observer canonical form The linear time–varying system
x˙ A(t)x b(t)u, t t , x(t ) x = + > 0 0 = 0 y cT (t)x d(t)u, t t = + ≥ 0 can be brought into observer canonical form (2.70) by means of the transformation z(t) = V (t)x(t) with 0
1 £ 0 1 n 1 ¤ 1 T 0 V − (t) N v(t) N v(t) N − v(t) , v(t) O− (c (t), A(t)) (2.71) = A A ··· A = ··· c˜n 1(t), − if and only if the SISO observability matrix O(cT (t), A(t)) is regular for t t . ≥ 0
2.6 State observer design 49 The function c˜n 1(t) provides a design DOF, which can be used to simplify the computations. −
Exercise 2.10. Prove Theorem 2.11.
2.6.1.2 Eigenvalue assignment using Ackermann’s formula
Dual to the exposition for the determination of the Ackermann formula in course of state feedback control design the individual steps for the design of the observer gain l(t) can be specified.
Theorem 2.12: Ackermann’s formula for linear time–varying SIS systems Let the linear time–varying system
x˙ A(t)x b(t)u, t t , x(t ) x = + > 0 0 = 0 y cT (t)x d(t)u, t t = + ≥ 0 be uniformly observable for t t , i.e., the observability matrix ≥ 0 0 T MAc (t) M 1 cT (t) T A O(c (t), A(t)) . = . . n 1 T MA− c (t)
with operator
k T ¡ k 1 T ¢ 1 T T T 0 T T M c (t) M M − c (t) , M c (t) c˙ (t) c (t)A(t), M c (t) c (t) A = A A A = + A = has rank n for all t t . Then the time–varying observer gain vector ≥ 0
1 h 0 1 n 1 ni l(t) p0NA p1NA ... pn 1NA− NA v(t) (2.72a) = c˜n 1(t) + + + − + ◦ − for 0 0 1 T v(t) O− (c (t), A(t)) . (2.72b) = . . c˜n 1(t) − yields the time–invariant system matrix A˜ A˜(t) l˜(t)c˜T (t), l˜(t) V (t)l(t) of the observer error b = − = dynamics in the transformed state z(t) V (t)x(t). The characteristic polynomial p0 p1λ ... n 1 n = + + + pn 1λ − λ of A˜b can be assigned as an arbitrary Hurwitz polynomial by the proper choice of − + the coefficients p , j 0,1,...,n 1. In this case A˜ is a Hurwitz matrix. j = − b If the transformation z(t) V (t)x(t) with = 1 £ 0 1 n 1 ¤ V − (t) N v(t) N v(t) N − v(t) = A A ··· A
50 Chapter 2 Analysis and control of linear time–varying systems is a Lyapunov transformation into observer canonical form according to Definition 2.1, then Theorem 2.4 implies the exponential stability of the observer error dynamics
x˜˙ ¡A(t) l(t)cT (t)¢x˜ = | − {z } A (t) = b in the original state x˜(t) provided that A˜b is a Hurwitz matrix.
Exercise 2.11. Prove Theorem 2.12.
Example 2.5. Given the linear time–varying system
· 0 1¸ x˙ x, t t , x(t ) x = α(t) 0 > 0 0 = 0 y £0 1¤x, t t , = ≥ 0 design a state observer. i
2.6 State observer design 51 52 Chapter 2 Analysis and control of linear time–varying systems 2.6.2 MIMO systems
The generalization of the previous concept to the MIMO case follows basically the lines of the SISO case with the difference that so–called observability indices ρ j have to be introduced. Under the assumption that the linear time–varying MIMO system
x˙ A(t)x B(t)u, t t , x(t ) x (2.73a) = + > 0 0 = 0 y cT (t)x d T (t)u, j 1,...,p, t t (2.73b) j = j + j = ≥ 0 is (completely, totally or uniformly, respectively) observable, the observability indices enable us to systematically extract n linearly independent row vectors from the (np n) observability matrix × O(C(t), A(t)).
Remark 2.5 If not stated otherwise in the following observability is associated with uniform observability. In addition it is assumed that rankC(t) p holds true for all t t . = ≥ 0
With (2.73) the observability matrix (2.29) can be written as
T c1 (t) . . T c p (t) M 1 cT (t) A 1 . . 1 T O(C(t), A(t)) MAc p (t) . (2.74) = . . n 1 T MA− c1 (t) . . n 1 T MA− c p (t)
If the system is observable, then rankO(C(t), A(t)) n. Obviously the rank of O(C(t), A(t)) is identical = to the rank of the reduced observability matrix
T c1 (t) . . ρ 1 1− T MA c1 (t) p ¯ . X O(C(t), A(t)) . , ρ j n, (2.75) = T j 1 = c (t) = p . . ρ 1 p − T MA c p (t)
2.6 State observer design 53 which is composed of the n linear independent row vectors of O(C(t), A(t)). This motivates the following definition.
Definition 2.5
The j th observability index ρ j of the observable linear time–varying system (2.73) is the smallest ρ j T integer for which the row vector MA c j (t) is linearly independent from any row vector located above it in (2.74).
2.6.2.1 Transformation into MIMO observer canonical form
The determination of the transformation matrix into observer canonical form for the MIMO case follows the lines of the SISO cases but requires to take into account the observability indices.
Theorem 2.13: Transformation into MIMO observer canonical form The linear time–varying system
x˙ A(t)x B(t)u, t t , x(t ) x = + > 0 0 = 0 y cT (t)x d T (t)u, j 1,...,p, t t j = j + j = ≥ 0 can be brought into observer canonical form by means of the transformation z(t) V (t)x(t) = with h i 1 0 ρ1 1 0 ρp 1 V − (t) N v (t) N − v (t) N v (t) N − v (t) (2.76a) = A 1 ··· A 1 ··· A p ··· A p and
£ ¤ 1 T v 1(t) v p (t) O¯ − (C(t), A(t))C¯ (t) (2.76b) ··· = for
0 0 ··· . . . . length ρ 0 0 1 1 ··· 1 c˜1,ρ 1(t) c˜p,ρ 1(t) 1− ··· 1− . . C¯T (t) . . , (2.76c) = . . 0 0 ··· . . . . length ρp 0 0 p ··· p c˜1,ρ 1(t) c˜p,ρ 1(t) p − ··· p − | {zn p } R × ∈ if and only if the reduced observability matrix O¯(C(t), A(t)) is regular for all t t . ≥ 0
54 Chapter 2 Analysis and control of linear time–varying systems j The functions c˜ (t) herein provide design DOFs that can be chosen to simplify the computations. j,ρ j 1 However, it has to ensured− that matrix
1 1 c˜1,ρ 1(t) c˜p,ρ 1(t) 1− ··· 1− . . p p C (t) . . R × (2.77) = p p ∈ c˜1,ρ 1(t) c˜p,ρ 1(t) p − ··· p − is regular.
Exercise 2.12. Prove Theorem 2.13.
Since the determination of the observer gain matrix L(t) is independent of the input u(t) subsequently without loss of generality u(t) 0 is assumed (see, also the equations governing the observer error = dynamics in the x˜–state). The MIMO observer canonical form is composed of p coupled subsystems of length ρ , j 1,...,p, that can be expressed in the form j = 0 0 0 0 ··· 1 0 0 0 . ··· . 0 0 . .. . ··· . . . 0 0 1 0 ··· .. z˙ 0 . 0 z = 0 0 0 0 ··· 1 0 0 0 0 0 . ··· . ··· . .. . . . . 0 0 1 0 ··· a˜1 (t) a˜1 (t) 0,1 ··· 0,p . . . . z a˜1 (t) a˜1 (t) ρ1 ρ1 1,1 ρ1 1,p − ··· − zρ ρ . . 1+ 2 . . . − p p . a˜ (t) a˜ (t) 0,1 ··· 0,p z . . ρ1 ρ2 ρp . . + +···+ . . | {z p } p p z¯ R a˜ρ 1,1(t) a˜ρ 1,p (t) = ∈ p − ··· p − y C¯(t)z. =
2.6.2.2 Decoupling and eigenvalue assignment using Ackermann’s formula
The proof of the Ackermann formula for the MIMO observer design is left to the reader but can be found, e.g., in meurer_studienarbeit:99; Freund, 1971; Rothfuß, 1997.
Theorem 2.14: Ackermann’s formula for linear time–varying MIMO systems Let the linear time–varying system
x˙ A(t)x B(t)u, t t , x(t ) x = + > 0 0 = 0 y cT (t)x d T (t)u, j 1,...,p, t t j = j + j = ≥ 0
2.6 State observer design 55 be uniformly observable for t t , i.e., the reduced observability matrix ≥ 0 T c1 (t) . . ρ 1 1− T MA c1 (t) p ¯ . X O(C(t), A(t)) . , ρ j n = T j 1 = c (t) = p . . ρ 1 p − T MA c p (t)
with the observability indices ρ , j 1,...,p, and the operator j = k T ¡ k 1 T ¢ 1 T T T 0 T T M c (t) M M − c (t) , M c (t) c˙ (t) c (t)A(t), M c (t) c (t) A = A A A = + A = has rank n for all t t . Let in addition the matrix ≥ 0 p c˜1 (t) c˜ (t) 1,ρ1 1 ··· p,ρ1 1 −. .− C (t) . . = p c˜1 (t) c˜ (t) 1,ρp 1 p,ρ 1 − ··· p − to be regular for all t t . Then the time–varying observer gain matrix ≥ 0 1 £ ¤ 1 L(t) V − (t) A¯(t) P C − (t) (2.78a) = + 1 for V − (t) from (2.76a) and matrices
h ρ i A¯(t) V (t) N ρ1 v (t) N p v (t) (2.78b) = − A 1 ··· A p as well as
p p p 1,0 1,0 ··· 1,0 . . . . . . p p p 1,ρ1 1 1,ρ1 1 1,ρ1 1 − − ··· − 0 p2,0 p2,0 ··· . . . . . . P (2.78c) = 0 p2,ρ 1 p2,ρ 1 2− ··· 2− . . . . . . 0 0 pp,0 ··· . . . . . . 0 0 pp,ρ 1 ··· p − yields the time–invariant system matrix A˜ A˜(t) L˜(t)C˜(t), L˜(t) V (t)L(t) of the observer error b = − = dynamics in the transformed state z(t) V (t)x(t). The characteristic polynomial = p Y¡ ρ j 1 ρ j ¢ p j,0 p j,1λ ... p j,ρ j 1λ − λ j 1 + + + − + =
56 Chapter 2 Analysis and control of linear time–varying systems of A˜b can be assigned as an arbitrary Hurwitz polynomial by the proper choice of the coefficients p j,i , i 0,...,ρ j 1, j 1,...,p. In this case A˜b is a Hurwitz matrix. = − = 1 If the transformation z(t) V (t)x(t) with V − (t) defined in (2.76a) is a Lyapunov transforma- = tion into observer canonical form according to Definition 2.1, then Theorem 2.4 implies the exponential stability of the observer error dynamics
x˜˙ ¡A(t) L(t)C(t)¢x˜ = | −{z } A (t) = b in the original state x˜(t) if A˜b is a Hurwitz matrix.
Exercise 2.13. Prove Theorem 2.14.
2.6 State observer design 57 References
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58 Chapter 2 Analysis and control of linear time–varying systems