Achievable dynamics for a class of nonlinear systems

8th IFAC Symposium on Nonlinear Control Systems University of Bologna, Italy, September 1-3, 2010

Achievable dynamics for a class of nonlinear systems Harsh Vinjamoor∗ and Arjan van der Schaft∗

Abstract— For a certain class of nonlinear systems we obtain necessary and sufficient conditions for the existence of a controller which, when ‘interconnected’ to the plant, yields a system which has the desired dynamics. The necessary and sufficient condition is obtained in terms of the equations of the plant and the desired system. Algorithms to verify whether the condition holds true or not depend on the class of systems under consideration. For linear systems this condition can always be verified. Owing to the geometric nature of the condition we believe it has potential to be computable for some special cases of nonlinear systems. Keywords: Nonlinear systems, bisimulations, achievability, canonical controller.

I. I NTRODUCTION AND MOTIVATION A basic question in systems and control theory is the following: given a plant system, by constructing another dynamical system called a controller and ‘interconnecting/attaching’ this to the plant, what are the possibilities of modifying the input-output behavior of the plant? Now, to decide whether the interconnected system does indeed have the desired dynamics, we need some notion of equivalence between systems. Also, we must specify what exactly we mean by ‘interconnection’. In this section we specify the class of systems we consider. In the following section, Section II, we explain precisely what we mean by ‘interconnection’ of the plant and the controller. Section III elaborates on the notational issues and also states the problem statement precisely. The main result of the paper is stated in Section IV together with an illustrative example. Due to lack of space we have skipped the proof of the main result. We conclude with some remarks and future directions in Section V. Throughout this paper we will have to deal with three systems viz. the plant P , the desired system

S and the controller system C. The goal is to find necessary and sufficient conditions under which there exists a controller such that when attached to P , the resulting systems behaves exactly like S; we will say that C achieves S. We will answer this question in terms of these three systems and their interconnections. We first describe the class of systems that we wish to consider. Let P denote the plant. It is a system which has an input vector uP , two output vectors yP and zP and a state vector xP in some state space X (P ). Given the state at time zero, xP (0), and the input, the outputs yP and zP are assumed to exist and are unique. For example, a time invariant state space description of a system with an input vector and two output vectors satisfies these requirements (under standard technical assumptions about smoothness of the vector fields involved). Next we have the desired system S. This is a system with state space X (S) and an output vector zS ; given the state xS (0) at time zero the output zS is assumed to be uniquely defined. Thus S is ‘autonomous’. The controller too is assumed to be a system with a state xC in state space X (C), an input uC and an output yC ; given the state at time zero xC (0) and the input uC the output yC is uniquely defined. To make things concrete, we shall explain and interpret our results for the following special case: let the plant be described by x˙ P = fP (xP ) + g(xP )uP where uP (t) ∈ R, xP (t) ∈ RnP yP = h(xP ); zP = cP (xP ) where yP (t), zP (t) ∈ R and fP and g are smooth vector fields while h and cP are smooth maps. Assume the desired system is described by

∗ The authors are with the Institute of Mathematics and Computing Science, University of Groningen. The corresponding author for this paper is Harsh Vinjamoor. Emails: [email protected], [email protected]

978-3-902661-80-7/10/$20.00 © 2010 IFAC

(1)

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x˙ S = fS (xS ); where xS (t) ∈ RnS zS = cS (xS ) where zS (t) ∈ R

(2)

10.3182/20100901-3-IT-2016.00008

NOLCOS 2010 Bologna, Italy, September 1-3, 2010

and fS is a smooth vector field while cS is a smooth map. Note that the theory that follows holds true for general systems; systems described in equations (1) and (2) are meant to illustrate the main ideas of the paper in the smooth vector fields case. The aim is to make sure that the variable zP behaves as desired i.e. as it behaves in S. We assume that only the variables (uP , yP ) of the plant are available for interconnection with the controller. Practically this is quite a common situation; for instance, if the variable zP is difficult to measure or if the sensor required is too expensive compared to the sensors for measuring yP . The input uP which is applied to the actuator is always available for the controller. Another situation is when zP has to be controlled but is not a physical variable i.e. it is just some function of the state which is required to be controlled. Hence the need to partition the variables of the plant into two sets namely zP and (uP , yP ). We shall call the variable zP ‘manifest’ (denoted by m) as it is the variable whose behaviour we are interested in. For the variables (uP , yP ) we shall use the term ‘control’ (denoted by c) variables since they are available for control. II. M ORE GENERAL INTERCONNECTIONS Now, the class of controller interconnections that we consider are more general than the ones usually seen in controller design techniques. We first motivate the need for our more general controller interconnection and then state precisely what type of interconnections we allow. Classical control theory deals with input/output controllers, i.e. controllers which accept the output of the plant as their input and produce an output which acts as an input to the plant. Thus a controller is looked at as a signal processing unit. These controllers have certain advantages. For instance, in the case of linear time invariant state space systems without feed-through terms, an input/output interconnection is guaranteed to be well-posed, in the sense that after attaching the controller, the space of initial conditions of the plant does not become a proper subspace of the plant state space; a property that is often desirable. However, there are desired systems S which can be achieved but not by this class of inter-

connections; these considerations are not new and have already been addressed; see for instance the example of the ‘door closing mechanism’ in [Kui95], [Wil97]. We now show an explicit mathematical example which illustrates the need for interconnections other than the standard feedback interconnection. Suppose we have a linear time invariant plant described by the differential equation     1 0 0 x˙ P = x + u 0 1 P 1   (3) zP = 1 0 xP   yP = 1 0 xP

Suppose the desired system S is just the zero system; hence the aim is to design a controller so that the output zP of the plant remains zero. The first step is to compute the largest controlled invariant subspace contained in the kernel of the output map corresponding to the output zP ; in this case it is 0 the subspace spanned by . If state feedback 1 were allowed the solution would be to set uP = F xP + Lw as is done in the theory of controlled invariance (see [BM92], [Won85]). However in our setting the state is not assumed to be available to the controller. Hence our only option is to influence uP and yP without any knowledge of the state. In this case, if we were to use the standard (well-posed) feedback configuration, the space of allowed initial conditions of the plant would continue to be the whole state space; thus the standard feedback configuration cannot keep the output zP zero. Now, if we set yP = 0 then the output zP will be identically zero. So yP = 0 is a controller which achieves the desired behaviour and is not in the standard feedback configuration. Thus it is clear that we can achieve more by using interconnections that are not necessarily in the standard feedback configurations. The following controller1 also does the job     1 0 ′ 0 ′ x˙ ′P = xP + u 0 1 1   (4) zP′ = 1 0 x′P = 0   yP′ = 1 0 x′P 1 This is the canonical controller (introduced in [vdS03] for behaviours) as adapted to state space systems in [VvdS09].

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with the following interconnection equations: u′ = u and yP = yP′ . This controller essentially does the same thing viz. sets the output yP to zero. The point is that this controller has been obtained from the equations of the plant and the desired system rather than from an adhoc design technique. This second controller indicates our approach towards control problems viz. to design controllers using the the ‘canonical controller’ as introduced in [vdS03]. For linear time invariant systems it can be shown that if there at all exists a controller then the ‘canonical controller’ also works; see [VvdS09]. Note that we are not claiming that the ‘canonical controller’ is the only controller. There may be several controllers that are not derived from the canonical controller. The importance of the canonical controller lies in the fact that given S and P it can immediately be constructed. The aim of this paper is to draw as much information as possible from the canonical controller to conclude about the existence of a controller which achieves the desired behaviour. We shall now explain what the dynamics of a system obtained by interconnection are. Towards this end we set up some notation and introduce a few definitions. III. D EFINITIONS AND NOTATION

that the number of variables in (uP , yP ) and in (uC , yC ) is the same. Then we define an interconnection of P and a controller C by the equations (uP , yP ) = Π(uC , yC ) where Π is a permutation matrix; i.e., a matrix obtained by permuting the columns of an identity matrix. We shall refer to Π as the interconnection matrix. More precisely, Π

k

P

2 Note that the first entry x (0) is a state vector while the P other entries are functions.

:=

{(xP (0), uP , yP , zP ) ×

(xC (0), uC ,yC ) | (xP (t), xC (t)) ∈ X (P ) × uP (t) u (t) X (C) and = Π C ∀t ≥ 0}. yP (t) yC (t) We denote the state space of such a system by X (P

Π

k C) ⊆ X (P ) × X (C). c

Note that this can be strictly smaller than X (P ) × X (C). Along the same lines, I

S k P := {(xS (0), zS ) × (xP (0), uP , yP , zP ) | m

(xS (t), xP (t)) ∈ X (S) × X (P ) and zS (t) = zP (t)∀t ≥ 0}. As earlier we denote the state I

space by X (S k P ) ⊆ X (S) × X (P ); here m

the interconnection matrix is the identity matrix so that the interconnection equations become zP = zS . Interconnections of more than two systems are defined in a similar way. For example I

We shall use the notation (xP (0), uP , yP , zP ) ∈ P to indicate that starting with an initial condition xP (0), if we apply the input function uP to the system P , then (yP , zP ) will be the resulting output functions2 . Similarly, (xS (0), zS ) ∈ S if for initial condition xS (0) the output is zS and (xC (0), uC , yC ) ∈ C if for initial condition xC (0) and input uC the output is yC . Note that the states evolve in their respective state spaces X (P ), X (S) and X (C). The interconnection of two systems will always be either with respect to the manifest variables or the control variables; as mentioned earlier we shall indicate this by subscripts m and c respectively. Also, an interconnection means that we set some variables of one system equal to some variables of the other system. We assume that zP (t) and zS (t) are vectors of the same size. Similarly, we also assume

C

c

I

I

I

S k P k P is defined as (S k P ) k P . m

c

m

c

Before proceeding, we explain what the state space of an interconnected system is in a special case. Consider the systems described in equations (1) and (2). The explicit equations defining the I

interconnected system S k P are m

x˙ S = fS (xS ) x˙ P = fP (xP ) + g(xP )uP l := zP − zS = cP (xP ) − cS (xS ) = 0

(5)

yP = h(xP ) The state space of this system is the largest controlled invariant output nulling submanifold with l as the output (see [NvdS90], [Isi95] for a treatment of smooth systems). Also, computing the state space of the above interconnection is in this case equivalent to finding the largest simulation relation as explained for smooth systems in [vdS04, Section 7]. Thus the state spaces of

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interconnected systems are in general analogues of controlled invariant subsets. Note that for S we consider autonomous systems only; if the desired system S has an input then simulation relations and controlled invariant subsets are not the same objects. The relation between simulation relations and controlled invariance has been studied in several places like [vdS04], [HTP05], [Gra05] and also in the context of abstractions in [Pap02]. The special case of control affine nonlinear systems has been addressed in [TP04]. We need to address one more question before we can state the problem precisely viz. the notion of equivalence. Given a controller C, when do we Π

say that P k C behaves like S? One intuitive idea c

is that for every initial condition in S there should Π

exist an initial condition in P k C such that the c

outputs zP and zS are identical. The definition of bisimulation as introduced in [vdS04] (inspired by [Pap03] and followed up in [Gra07]) makes this idea precise. For this we use the notion of bisimulation.

For smooth nonlinear systems the above property means that R is a controlled invariant submanifold (see equation (5) and the discussion there after) while for linear time invariant system it means that R is a controlled invariant subspace. We say that R is a full simulation relation if R projects onto the whole of X (S). In fact, we say that S is simulated by P , written as S 4 P , if there exists a full simulation relation of S by P ; geometrically this is equivalent to X (S) ⊆ I

πS (X (S k P )). m

Finally, one more note about notational convenience. Often we will need to talk about I

states of the system S k P k P ; to make c

systems as S1

Π

of S and P k C. For bisimilar systems the state Π

space of S k P k C is called a bisimulation c

relation3 . Problem statement: Give P and S, find necessary and sufficient conditions for the existence of a controller C and an interconnection matrix Π

Π such that P k C ≈ S. c

IV. M AIN RESULT The main result that we shall soon state is in I

I

terms of the interconnected system S k P k m

c

P with the states of the first two components I

restricted to a certain subset of X (S k P ). This m 3 Our

m

(xS (t), xP (t)) ∈ R ∀ t ≥ 0.

c

Π

m

I

that (xS (0), z) × (xP (0), u, y, z) ∈ S k P and

things easier to write we shall often write these

S k P k C projects fully on to the state spaces

c I

m

Π

to S, denoted by P k C ≈ S if the state space of

c

k P ) is called

a simulation relation of S by P if for all (xS (0), xP (0)) ∈ R, there exists an input u such

m

c

m

I

Definition 2: R ⊆ X (S

I

Π

Definition 1: We say that P k C is bisimilar

I

subset has a desirable property which we now define.

definition is an adaptation of the definition in [vdS04].

I

k m

I

P1

k P2 and refer to c

states and trajectories in various components by using different subscripts/superscripts/primes whenever necessary. For example (xS (0), z) × I

I

m

c

(xP1 (0), u, y, z) × (xP2 (0), u, y, z ′ ) ∈ S1 k P1 k P2 has three states xS (0) ∈ X (S), xP1 ∈ X (P ) and xP2 ∈ X (P ). Similarly it has output z = zS = zP1 , input u = uP1 = uP2 , output y = yP1 = yP2 and output z ′ = zP2 . Note that S1 , P1 and P2 are merely copies of the systems S and P i.e., the equations defining P1 , P2 and P are the same and likewise, the equations for S1 and S are the same. However, the initial conditions need not be the same i.e., there exist trajectories (xS (0), z)×(xP1 (0), u, y, z)×(xP2 (0), u, y, z ′ ) ∈ I

I

S1 k P1 k P2 where xP1 (0) 6= xP2 (0). m

c

We now introduce a system which will turn out to be the controlled system i.e. the system bisimilar to the desired system S. Let R be a full simulation relation of S by P . We define the

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I

I

system denoted by CR as S1 k P1 k P2 with the m

c

states of the first two components restricted to a full simulation relation R. Note that since R is a full simulation relation and is hence controlled invariant, CR is well defined. Hence the system I

I

I

S1 k P1 k P2 k S2 with the states of the first m

c

m

two components restricted to R is denoted by I

I

CR k S2 . As we shall see, S1 k P1 with state m

m

space as the full simulation relation R will be the I

controller we use. Note that the system S1 k P1 m

is nothing but the canonical controller introduced in [vdS03] in a behavioural context. Restricting to a full simulation relation R is thus restricting the canonical controller to an invariant subset. Some simulation relations of S by P have a desirable property which we use crucially in proving the main result. We now state this property. Definition 3: A simulation relation R of S by I

P is said to be symmetric if R = πP2 S2 {X (CR k m

S)}. The systems that we are considering are quite general; since we only demand that given the state and input the output is unique. Consequently, as expected, the main result does not hold in complete generality. We need a certain assumption that appears rather technical at first sight. We later illustrate some common situations where the assumption holds true. Assumption 4: We assume that the systems S and P satisfy the following property: if (xS (0), z)×(xP1 (0), u, y, z)×(xP2 (0), u, y, z ′ ) ∈ I

I

k

S1

P1

m

k

P2

and

c

(xS (0), z) ×

(xP1 , (0)u′ , y ′ , z) × (xP2 , (0)u′ , y ′ , z ′′ ) I

I

m

c

∈

S1 k P1 k P2 then z ′ = z ′′ . Note that all linear time invariant systems always satisfy assumption 4. Remark 5: There is another class of systems which satisfy assumption 4. Consider the system

ically satisfied. Consider for example a plant as described in equation (1). Suppose the desired systems is just the zero system, i.e., the output zP should be identically zero. Further assume that the plant has a well defined relative degree r ≤ nP with respect to the input uP and output zP ; we assume that r is the same at every point of the state space and that Lg Lr−1 cP (xP ) 6= 0 for all xP ∈ f {xP | Lg Lkf cP (xP ) = 0; 0 ≤ k < r} =: V . We assume that V is a well defined smooth manifold. Then we know that starting from an initial condition on V we can keep the output zero by choosing uP := −Lrf cP (xP (t))/Lg Lr−1 cP (xP (t)); f see [Isi95, Section 4.3, page 169]. Thus, given the state xP , uP is uniquely defined. Consequently the problem of zeroing the output (or equivalently that of keeping the output constant) for a SISO control affine nonlinear system satisfies assumption 4. We now state the main result of this paper. Theorem 6: Given S and P satisfying assumption 4 the following are equivalent: 1) There exists a controller C and an interconΠ

nection matrix Π such that P k C ≈ S c

2) ∃ a full simulation relation R such that CR 4 S. 3) ∃ a full symmetric simulation relation R′ such that CR′ 4 S. According to the above theorem, given S and P , it is enough to look at the simulation relations of S by P to conclude about the existence of a controller. Moreover, the last statement in the above theorem tells us that it is enough to consider only symmetric simulation relations. A. Illustration of the main result We now state a mathematical example where the canonical controller does not achieve the desired system but when restricted to an invariant subset it does indeed achieve the desired system. Consider a plant given by the equations x˙ P = xP (1 + uP ) yP = x2P

I

S k P . If it satisfies the property that given

zP =

m I

(xS (0), xP (0)) ∈ X (S k P ) the input u is m

uniquely defined, then assumption 4 is automat-

(x2P

(6) − 4)(xP − 1)

Suppose the desired system is the zero system, i.e., we require zP to be identically zero. It is clear that one must have an initial condition in

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the set {2, −2, 1}. By choosing uP = −1 these points become equilibria. Further note that this is the only choice of uP which ensures that zP is identically zero. Thus assumption 4 is satisfied in this case; see Remark 5. Consider R := {(0, x) | x ∈ {2, −2, 1}} which is a full simulation relation (where 0 is assumed to be the state space of S). Consider the system CR . On computing the state space one finds that (0, 0) × (xP1 = 1, uP1 = −1, yP1 = 1, zP1 = 0) × (xP2 = −1, uP2 = −1, yP2 = 1, zP2 = 6) ∈ CR . Thus zP2 6= 0. Hence CR is not bisimilar to S. However consider R′ := {(0, x) | x ∈ {2, −2}}. Then on carrying out the computation one finds that CR′ ≈ S. V. C ONCLUSIONS Given a plant, we have obtained a necessary and sufficient condition for the achievability of a given desired system. From the proof of Theorem 6 we see that if there at all exists a controller Π

such that P k C ≈ S, then there exists a full c

simulation R′ of S by P (not necessarily the I

maximal simulation relation) such that, S k P m

with state space restricted to R′ is a controller i.e. CR′ ≈ S. Thus, provided assumption 4 is I

satisfied, the system S k P contains all the m

information needed to draw conclusions about the existence of a controller. However, although crisp and general, Theorem 6 may not always be easy to verify. For linear time invariant systems the condition can always be checked, see [VvdS09]. Also, an important problem is to provide methods to ‘list/compute’ simulation relations, other than the maximal simulation relation, of S by P . Finally, we still have to estimate how reasonable assumption 4 is. It does not appear to be overly restrictive (see text after Assumption 4 and Remark 5 ); however more classes of systems and specifications which satisfy assumption 4 need to be studied. Also, computationally testing the validity of assumption 4 is a topic for future study.

[Gra05]

Kevin Grasse. Lifting of trajectories of control systems related by smooth mappings. Systems & Control Letters, 54:195–205, 2005. [Gra07] K.A. Grasse. Simulation and bisimulation on nonlinear control systems with admissible classes of inputs and disturbances. SIAM Journal of Control and Optimization, 46(2):562–584, 2007. [HTP05] Esfandiar Haghverdi, Paulo Tabuada, and George Pappas. Bisimulatin relations for dynamical, control, and hybrid systems. Theoretical Computer Science, 342:229–261, 2005. [Isi95] Alberto Isidori. Nonlinear control systems. Communications and control engineering. Springer Verlag, London Limited, 1995. [Kui95] Margreet Kuijper. Why do stabilizing controllers stabilize? Automatica, 31(4):621–625, 1995. [NvdS90] Henk Nijmeijer and Arjan van der Schaft. Nonlinear dynamical systems. Springer Verlag, New York, 1990. [Pap02] George Pappas. Consistent abstractions of affine control systems. Systems & Control Letters, 47(5):745–756, May 2002. [Pap03] G.J. Pappas. Bisimilar linear systems. Automatica, 39(12):2035–2047, 2003. [TP04] P. Tabuada and G.J. Pappas. Bismilar control affine systems. Systems & Control letters., 52:49–58, 2004. [vdS03] A.J. van der Schaft. Achievable behavior of general systems. Systems & Control Letters, 49:141–149, 2003. [vdS04] A.J. van der Schaft. Equivalence of dynamical systems by bisimulation. IEEE Transaction on Automatic Control, 49(12):2160–2172, December 2004. [VvdS09] Harsh Vinjamoor and Arjan van der Schaft. On achievable bisimulations for linear time invariant systems. in Proc. European Control Conference, Budapest, Hungary, August 23-26 2009. [Wil97] J.C. Willems. On interconnections, control and feedback. IEEE Trasactions on Automatic Control, 42:326–339, 1997. [Won85] W.M. Wonham. Linear multivariable control: a geometric approach. Springer, New York, Third Edition, 1985.

R EFERENCES [BM92]

G. Basile and G. Marro. Controlled and conditioned invariants in linear system theory. Prentice Hall, Englewood Cliffs, 1992.

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